**A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems**

Sheng Zeng and Emmanuel Fernandez

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50488

#### **1. Introduction**

Adaptive control has attracted a lot of research attention in control theory for many decades. In the certainty equivalence based adaptive controller design [4, 5], the unknown parameters of the uncertainty system are substituted by their online estimates, which are generated through a variety of identifiers, as long as the estimates satisfy certain properties independent of the controller. This approach leads to structurally simple adaptive controllers and has been demonstrated its effectiveness for linear systems with or without stochastic disturbance inputs [10] when long term asymptotic performance is considered. Yet, the certainty equivalence approach is unsuccessful to generalize to systems with severe nonlinearities. Also, early designs based on this approach were shown to be nonrobust [13] when the system is subject to exogenous disturbance inputs and unmodeled dynamics. Then, the stability and the performance of the closed-loop system becomes an important issue. This has motivated the study of robust adaptive control in the 1980s and 1990s, and the study of nonlinear adaptive control in the 1990s.

The topic of adaptive control design for nonlinear systems was studied intensely in the last decade after the celebrated characterization of feedback linearizable or partially feedback linearizable systems [7]. A breakthrough is achieved when the integrator backstepping methodology [8] was introduced to design adaptive controllers for parametric strict-feedback and parametric pure-feedback nonlinear systems systematically. Since then, a lot of important contributions were motivated by this approach, and a complete list of references can be found in the book [9]. Moreover, this nonlinear design approach has been applied to linear systems to compare performance with the certainty equivalence approach. However, simple designs using this approach without taking into consideration the effect of exogenous disturbance inputs have also been shown to be nonrobust when the system is subject to exogenous disturbance inputs.

©2013 Zeng and Fernandez, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ©2013 Zeng and Fernandez, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The robustness of closed-loop adaptive systems has been an important research topic in late 1980s and early 1990s. Various adaptive controllers were modified to render the closed-loop systems robust [6]. Despite their successes, they still fell short of directly addressing the disturbance attenuation property of the closed-loop system.

proposed controller achieves disturbance attenuation level *zero* with respect to the measured disturbances, which further leads to a stronger asymptotic tracking property, namely, the tracking error converges to zero when the unmeasured disturbances are L<sup>2</sup> ∩ L∞, and the

The balance of this Chapter is organized as follows. In Section 2, we list the notations used in the Chapter. In Section 3, we present the formulation of the adaptive control problem and discuss the general solution methodology. In Section 4, we first obtain parameter identifier and state estimator using the *cost-to-come function* analysis in Subsection 4.1, then we derive the adaptive control law in Subsection 4.2. We present the main results on the robustness of the system in Section 5, and the example in Section 6. The Chapter ends with some concluding

We denote IR to be the real line; IR*e* to be the extended real line; IN to be the set of natural numbers; C to be the set of complex numbers. For a function *f* , we say that it belongs to C if it is continuous; we say that it belongs to C*<sup>k</sup>* if it is *k*-times continuously (partial) differentiable.

the vector −→*M* is formed by stacking up its column vectors. For any symmetric matrix *M*,

denotes the vector formed by stacking up the column vector of the lower triangular part of *M*. For *n* × *n*-dimensional symmetric matrices *M*<sup>1</sup> and *M*2, where *n* ∈ IN, we write *M*<sup>1</sup> > *M*<sup>2</sup> if *M*<sup>1</sup> − *M*<sup>2</sup> is positive definite; we write *M*<sup>1</sup> ≥ *M*<sup>2</sup> if *M*<sup>1</sup> − *M*<sup>2</sup> is positive semi-definite. For *n* ∈ IN, the set of *n* × *n*-dimensional positive definite matrices is denoted by S+*n*. For *n* ∈ IN ∪ {0}, *In* denotes the *n* × *n*-dimensional identity matrix. For any matrix *M*, �*M*�*<sup>p</sup>* denotes its *p*-induced norm, 1 ≤ *p* ≤ ∞. L<sup>2</sup> denotes the set of square integrable functions and L<sup>∞</sup> denotes the set of bounded functions. For any *n*, *m* ∈ IN ∪ {0}, **0***n*×*<sup>m</sup>* denotes the *n* × *m*-dimensional matrix whose elements are zeros. For any *n* ∈ IN and *k* ∈ {1, ··· , *n*}, *en*,*<sup>k</sup>*

We consider the robust adaptive control problem for the system which is described by the

y

1

w

<sup>1</sup> <sup>2</sup>

S

1

y

w1

⎧ ⎨ ⎩

*<sup>z</sup>*)1/2. For any vector *<sup>z</sup>* <sup>∈</sup> IR*n*, and any

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

> 2 *<sup>M</sup>* = *z*�

−1 *b* < 0 0 *b* = 0 1 *b* > 0

*Mz*. For any matrix *M*,

. For

109

←−*M*

For any matrix *A*, *A*� denotes its transpose. For any *b* ∈ IR, sgn(*b*) =

any vector *<sup>z</sup>* <sup>∈</sup> IR*n*, where *<sup>n</sup>* <sup>∈</sup> IN, <sup>|</sup>*z*<sup>|</sup> denotes (*z*�

01×(*k*−1) 1 01×(*n*−*k*)

w w

u 2 2

y

2

S2

**Figure 1.** Diagram of two sequentially interconnected SISO linear systems.

**3. Problem Formulation**

block diagram in Figure 1.

*n* × *n*-dimensional symmetric matrix *M*, where *n* ∈ IN, |*z*|

�� .

measured disturbances are L<sup>∞</sup> only.

remarks in Section 7.

**2. Notations**

denotes �

The objectives of robust adaptive control are to improve transient response, to accommodate unmodeled dynamics, and to reject exogenous disturbance inputs, which are the same as the objectives to motivate the study of the *H*∞- optimal control problem. *H*∞-optimal control was proposed as a solution to the robust control problem, where these objectives are achieved by studying only the disturbance attenuation property for the closed-loop system. The game-theoretic approach to *H*∞-optimal control developed for the linear quadratic problems, offers the most promising tool to generalize the results to nonlinear systems [3]. Worst-case analysis based adaptive control design was proposed in late 1990s to address the disturbance attenuation property directly, and it is motivated by the success of the game-theoretic approach to *H*∞-optimal control problems [2]. In this approach, the robust adaptive control problem is formulated as a nonlinear *H*<sup>∞</sup> control problem under imperfect state measurements. By *cost-to-come function* analysis, it is converted into an *H*<sup>∞</sup> control problem with full information measurements. This full information measurements problem is then solved using nonlinear design tools for a suboptimal solution. This design scheme has been applied to worst-case parameter identification problems [11], which has led to new classes of parametrized identifiers for linear and nonlinear systems. It has also been applied to adaptive control problems [1, 12, 14, 15, 18, 19], and the convergence properties is studied in [20]. In [14], adaptive control for a strict-feedback nonlinear systems was considered with noiseless output measurements, and more general class of nonlinear systems was studied in [1]. In [12], single-input and single output (SISO) linear systems were considered with noisy output measurements. SISO linear systems with partly measured disturbance was studied in [18], which leads to a disturbance feed-forward structure in the adaptive controller. [19] generalizes the results of [12] to the adaptive control design for SISO linear systems with zero relative degree under noisy output measurements. In [17], adaptive control for a sequentially interconnected SISO linear system was considered, and a special class of unobservable systems was also studied using the proposed approach. More recently, [16] generalized the result of [17] to adaptive control design for a linear system under simultaneous driver, plant and actuation uncertainties.

In this Chapter, we study the adaptive control design for sequentially interconnected SISO linear systems, **S**<sup>1</sup> and **S**2(see Figure 1), under noisy output measurements and partly measured disturbance using the similar approaches as [12] and [17]. We assume that the linear systems satisfy the same assumption as [17], and the adaptive control design follows the same design method discussed above. The robust adaptive controller achieves asymptotic tracking of the reference trajectories when disturbance inputs are of finite energy. The closed-loop system is totally stable with respect to the disturbance inputs and the initial conditions. Furthermore, the closed-loop system admits a guaranteed disturbance attenuation level with respect to the exogenous disturbance inputs, where ultimate lower bound for the achievable attenuation performance level is equal to the noise intensity in the measurement channel of **S**1. The results are as same as those in [17]. In addition, the controller achieves arbitrary positive distance attenuation level with respect to the measured disturbances by proper scaling. Moreover, if the measured disturbances satisfy the assumption 2 for *w*ˇ 1,*<sup>b</sup>* and *w*ˇ 2,*b*, the proposed controller achieves disturbance attenuation level *zero* with respect to the measured disturbances, which further leads to a stronger asymptotic tracking property, namely, the tracking error converges to zero when the unmeasured disturbances are L<sup>2</sup> ∩ L∞, and the measured disturbances are L<sup>∞</sup> only.

The balance of this Chapter is organized as follows. In Section 2, we list the notations used in the Chapter. In Section 3, we present the formulation of the adaptive control problem and discuss the general solution methodology. In Section 4, we first obtain parameter identifier and state estimator using the *cost-to-come function* analysis in Subsection 4.1, then we derive the adaptive control law in Subsection 4.2. We present the main results on the robustness of the system in Section 5, and the example in Section 6. The Chapter ends with some concluding remarks in Section 7.

#### **2. Notations**

2 Will-be-set-by-IN-TECH

The robustness of closed-loop adaptive systems has been an important research topic in late 1980s and early 1990s. Various adaptive controllers were modified to render the closed-loop systems robust [6]. Despite their successes, they still fell short of directly addressing the

The objectives of robust adaptive control are to improve transient response, to accommodate unmodeled dynamics, and to reject exogenous disturbance inputs, which are the same as the objectives to motivate the study of the *H*∞- optimal control problem. *H*∞-optimal control was proposed as a solution to the robust control problem, where these objectives are achieved by studying only the disturbance attenuation property for the closed-loop system. The game-theoretic approach to *H*∞-optimal control developed for the linear quadratic problems, offers the most promising tool to generalize the results to nonlinear systems [3]. Worst-case analysis based adaptive control design was proposed in late 1990s to address the disturbance attenuation property directly, and it is motivated by the success of the game-theoretic approach to *H*∞-optimal control problems [2]. In this approach, the robust adaptive control problem is formulated as a nonlinear *H*<sup>∞</sup> control problem under imperfect state measurements. By *cost-to-come function* analysis, it is converted into an *H*<sup>∞</sup> control problem with full information measurements. This full information measurements problem is then solved using nonlinear design tools for a suboptimal solution. This design scheme has been applied to worst-case parameter identification problems [11], which has led to new classes of parametrized identifiers for linear and nonlinear systems. It has also been applied to adaptive control problems [1, 12, 14, 15, 18, 19], and the convergence properties is studied in [20]. In [14], adaptive control for a strict-feedback nonlinear systems was considered with noiseless output measurements, and more general class of nonlinear systems was studied in [1]. In [12], single-input and single output (SISO) linear systems were considered with noisy output measurements. SISO linear systems with partly measured disturbance was studied in [18], which leads to a disturbance feed-forward structure in the adaptive controller. [19] generalizes the results of [12] to the adaptive control design for SISO linear systems with zero relative degree under noisy output measurements. In [17], adaptive control for a sequentially interconnected SISO linear system was considered, and a special class of unobservable systems was also studied using the proposed approach. More recently, [16] generalized the result of [17] to adaptive control design for a linear system under simultaneous

In this Chapter, we study the adaptive control design for sequentially interconnected SISO linear systems, **S**<sup>1</sup> and **S**2(see Figure 1), under noisy output measurements and partly measured disturbance using the similar approaches as [12] and [17]. We assume that the linear systems satisfy the same assumption as [17], and the adaptive control design follows the same design method discussed above. The robust adaptive controller achieves asymptotic tracking of the reference trajectories when disturbance inputs are of finite energy. The closed-loop system is totally stable with respect to the disturbance inputs and the initial conditions. Furthermore, the closed-loop system admits a guaranteed disturbance attenuation level with respect to the exogenous disturbance inputs, where ultimate lower bound for the achievable attenuation performance level is equal to the noise intensity in the measurement channel of **S**1. The results are as same as those in [17]. In addition, the controller achieves arbitrary positive distance attenuation level with respect to the measured disturbances by proper scaling. Moreover, if the measured disturbances satisfy the assumption 2 for *w*ˇ 1,*<sup>b</sup>* and *w*ˇ 2,*b*, the

disturbance attenuation property of the closed-loop system.

driver, plant and actuation uncertainties.

We denote IR to be the real line; IR*e* to be the extended real line; IN to be the set of natural numbers; C to be the set of complex numbers. For a function *f* , we say that it belongs to C if it is continuous; we say that it belongs to C*<sup>k</sup>* if it is *k*-times continuously (partial) differentiable.

For any matrix *A*, *A*� denotes its transpose. For any *b* ∈ IR, sgn(*b*) = ⎧ ⎨ ⎩ −1 *b* < 0 0 *b* = 0 1 *b* > 0 . For

any vector *<sup>z</sup>* <sup>∈</sup> IR*n*, where *<sup>n</sup>* <sup>∈</sup> IN, <sup>|</sup>*z*<sup>|</sup> denotes (*z*� *<sup>z</sup>*)1/2. For any vector *<sup>z</sup>* <sup>∈</sup> IR*n*, and any *n* × *n*-dimensional symmetric matrix *M*, where *n* ∈ IN, |*z*| 2 *<sup>M</sup>* = *z*� *Mz*. For any matrix *M*, the vector −→*M* is formed by stacking up its column vectors. For any symmetric matrix *M*, ←−*M* denotes the vector formed by stacking up the column vector of the lower triangular part of *M*. For *n* × *n*-dimensional symmetric matrices *M*<sup>1</sup> and *M*2, where *n* ∈ IN, we write *M*<sup>1</sup> > *M*<sup>2</sup> if *M*<sup>1</sup> − *M*<sup>2</sup> is positive definite; we write *M*<sup>1</sup> ≥ *M*<sup>2</sup> if *M*<sup>1</sup> − *M*<sup>2</sup> is positive semi-definite. For *n* ∈ IN, the set of *n* × *n*-dimensional positive definite matrices is denoted by S+*n*. For *n* ∈ IN ∪ {0}, *In* denotes the *n* × *n*-dimensional identity matrix. For any matrix *M*, �*M*�*<sup>p</sup>* denotes its *p*-induced norm, 1 ≤ *p* ≤ ∞. L<sup>2</sup> denotes the set of square integrable functions and L<sup>∞</sup> denotes the set of bounded functions. For any *n*, *m* ∈ IN ∪ {0}, **0***n*×*<sup>m</sup>* denotes the *n* × *m*-dimensional matrix whose elements are zeros. For any *n* ∈ IN and *k* ∈ {1, ··· , *n*}, *en*,*<sup>k</sup>* denotes � 01×(*k*−1) 1 01×(*n*−*k*) �� .

#### **3. Problem Formulation**

We consider the robust adaptive control problem for the system which is described by the block diagram in Figure 1.

**Figure 1.** Diagram of two sequentially interconnected SISO linear systems.

We assume that the system dynamics for **S1** and **S2** are given by,

$$
\dot{\mathfrak{x}}\_1 = \dot{A}\_1 \mathfrak{x}\_1 + \mathcal{B}\_1 \mathfrak{y}\_2 + \mathcal{D}\_1 \mathfrak{w}\_1 + \dot{\mathcal{D}}\_1 \mathfrak{w}\_1;\tag{1a}
$$

**Assumption 2.** *The measured disturbance w*ˇ <sup>1</sup> *can be partitioned as: w*ˇ <sup>1</sup> =

Based on Assumption 2, the matrix *<sup>D</sup>*<sup>ˇ</sup> *<sup>i</sup>* can be partitioned into �

**<sup>0</sup>**(*ri*−1)×*q*ˇ*i*,*<sup>b</sup> D*ˇ *<sup>i</sup>*,*b*<sup>0</sup> *<sup>D</sup>*<sup>ˇ</sup> *<sup>i</sup>*,*bri*

model, and make the following two assumptions.

*<sup>θ</sup>i*) <sup>≤</sup> <sup>1</sup>}*; moreover,* <sup>∀</sup> ¯

We make the following assumption about the reference signal, *yd*.

*, where y*(0)

The uncertainty of subsystem **S1** is *ω*` <sup>1</sup> := (*x*1,0, *θ*1, *w*` <sup>1</sup>[0,∞), *w*ˇ <sup>1</sup>[0,∞),*Yd*0, *y*

*<sup>d</sup>* (0), ··· , *y*

(0)

(*r*1+*r*2) *<sup>d</sup>* ] �

boundedness of the estimate of *θ*<sup>1</sup> and *θ*2.

(0) *<sup>d</sup>* , ··· , *y*

*j* = 1, ··· ,*r*<sup>1</sup> + *r*2*; define Yd*<sup>0</sup> := [*y*

**Assumption 3.** *For i* = 1, 2*, the matrices Ei are such that EiE*�

<sup>2</sup> and *Li* := *DiE*�

the following structure

Define *ζ<sup>i</sup>* := (*EiE*�

*<sup>θ</sup><sup>i</sup>* <sup>∈</sup> IR*σ<sup>i</sup>* <sup>|</sup> *Pi*(¯

*vector Yd* := [*y*

{ ¯

*D*ˇ *<sup>i</sup>*,*<sup>b</sup>* =

*i* )<sup>−</sup> <sup>1</sup>

*Hi*(*s*), *bi*,0, is equal to *bi*,*p*<sup>0</sup> + *A*¯

parameter vectors *θ*<sup>1</sup> and *θ*2.

⎡ ⎣

have *ni* <sup>×</sup> *<sup>q</sup>*ˇ*i*,*a*- and *ni* <sup>×</sup> *<sup>q</sup>*ˇ*i*,*b*-dimensional, respectively; and *<sup>D</sup>*<sup>ˇ</sup> *<sup>i</sup>*,*b*, *<sup>A</sup>*¯

⎤ <sup>⎦</sup> ; *<sup>A</sup>*¯

where *<sup>D</sup>*<sup>ˇ</sup> *<sup>i</sup>*,*b*<sup>0</sup> and *<sup>A</sup>*¯*i*,213 *<sup>j</sup>*0, *<sup>j</sup>* <sup>=</sup> *<sup>q</sup>*ˇ*i*,*<sup>a</sup>* <sup>+</sup> 1, ··· , *<sup>q</sup>*ˇ*i*,*<sup>a</sup>* <sup>+</sup> *<sup>q</sup>*ˇ*i*,*b*, are row vectors, *<sup>i</sup>* <sup>=</sup> 1, 2.

*i*

*<sup>i</sup>*,212 0*θi*, *i* = 1, 2.

, *i* = 1, 2.

*degree less than r*<sup>1</sup> + *r*2*; the measured disturbance w*ˇ <sup>2</sup> *can be partitioned as: w*ˇ <sup>2</sup> =

*q*ˇ1,*<sup>a</sup> dimensional, q*ˇ1,*<sup>a</sup>* ∈ IN ∪ {0}*, and the transfer function from each element of w*ˇ 1,*<sup>a</sup> to y*<sup>1</sup> *has relative*

*w*ˇ 2,*<sup>a</sup> is q*ˇ2,*<sup>a</sup> dimensional, q*ˇ2,*<sup>a</sup>* ∈ IN ∪ {0}*, and the transfer function from each element of w*ˇ 2,*<sup>a</sup> to y*<sup>2</sup> *has relative degree less than r*2*.* �

*<sup>i</sup>*,213 *<sup>j</sup>* =

Since we will base our design of adaptive controllers using the model (2), we call (2) the design

Due to the structures of *Ai*, *A*¯*i*,212 and *Bi*, the high frequency gain of the transfer function

To guarantee the stability of the identified system, we make the following assumption on the

**Assumption 4.** *The sign of bi*,0 *is known; there exists a known smooth nonnegative radially-unbounded strictly convex function Pi* : IR*σ<sup>i</sup>* <sup>→</sup> IR*, such that the true value <sup>θ</sup><sup>i</sup>* <sup>∈</sup> <sup>Θ</sup>*<sup>i</sup>* :<sup>=</sup>

Assumption 4 delineates *a priori* convex compact sets where the parameter vectors *θ*<sup>1</sup> and *θ*<sup>2</sup> lie in, respectively. This will guarantee the stability of the closed-loop system and the

**Assumption 5.** *The reference trajectory, yd, is r*<sup>1</sup> + *r*<sup>2</sup> *times continuously differentiable. Define*

*for feedback.* �

IR*n*<sup>1</sup> <sup>×</sup> <sup>Θ</sup><sup>1</sup> ×C×C× IR*r*1+*r*<sup>2</sup> × C, which comprises the initial state *<sup>x</sup>*1,0, the true value of the parameters *θ*1, the unmeasured disturbance waveform *w*` <sup>1</sup>[0,∞), the measured disturbance waveform *w*ˇ <sup>1</sup>[0,∞), the initial conditions of the reference trajectory *Yd*0, and the waveform

*<sup>d</sup>* <sup>=</sup> *yd, and y*(*j*)

(*r*1+*r*2−1)

*<sup>θ</sup><sup>i</sup>* <sup>∈</sup> <sup>Θ</sup>*i,* sgn(*bi*,0)(*bi*,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*¯*i*,212 0 ¯

⎡ ⎣

**<sup>0</sup>**(*ri*−1)×*σ<sup>i</sup> A*¯*i*,213 *<sup>j</sup>*<sup>0</sup> *A*¯ *i*,213 *jri*

⎤

� *w*ˇ� 1,*<sup>a</sup> w*ˇ� 1,*b* ��

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

*D*ˇ *<sup>i</sup>*,*<sup>a</sup> D*ˇ *<sup>i</sup>*,*<sup>b</sup>*

�

⎦ , *j* = *q*ˇ*i*,*<sup>a</sup>* + 1, ··· , *q*ˇ*<sup>i</sup>*

*<sup>i</sup>* > 0*.* �

*θi*) > 0*, i* = 1, 2*.* �

*<sup>d</sup> is the jth order time derivative of yd,*

(*r*1+*r*2)

*<sup>d</sup>*[0,∞) ) <sup>∈</sup> <sup>W</sup>` <sup>1</sup> :<sup>=</sup>

*<sup>d</sup>* (0)]� <sup>∈</sup> IR*r*1+*r*<sup>2</sup> *. The signal Yd is available*

� *w*ˇ� 2,*<sup>a</sup> w*ˇ� 2,*b* �� *where*

*where w*ˇ 1,*<sup>a</sup> is*

111

, where *D*ˇ *<sup>i</sup>*,*<sup>a</sup>* and *D*ˇ *<sup>i</sup>*,*<sup>b</sup>*

*<sup>i</sup>*,213 (*q*ˇ*i*,*<sup>a</sup>*+1), ··· , *<sup>A</sup>*¯*i*,213 *<sup>q</sup>*ˇ*<sup>i</sup>* have

$$
\Delta y\_1 = \triangle\_1 \pounds\_1 + \triangle\_1 \w\_1 \tag{1b}
$$

$$
\dot{\mathfrak{x}}\_2 = \dot{A}\_2 \mathfrak{x}\_2 + \mathfrak{B}\_2 u + \dot{A}\_{2,y} \mathfrak{y}\_2 + \dot{D}\_2 \mathfrak{w}\_2 + \dot{D}\_2 \mathfrak{w}\_2;\tag{1c}
$$

$$
\hat{y}\_2 = \triangle\_2 \hat{x}\_2 + \triangle\_2 \psi\_2 \tag{1d}
$$

where *x*`*<sup>i</sup>* is the *ni*-dimensional state vectors with initial condition *x*`*i*(0) = *x*`*i*,0, *ni* ∈ IN; *u* is the scalar control input; *yi* is the scalar measurement output; *w*`*<sup>i</sup>* is *q*`*i*-dimensional unmeasured disturbance input vector, *q*`*<sup>i</sup>* ∈ IN; *w*ˇ*<sup>i</sup>* is *q*ˇ*i*-dimensional measured disturbance input vector, *<sup>q</sup>*ˇ*<sup>i</sup>* <sup>∈</sup> IN; the elements of *<sup>w</sup>*ˇ*<sup>i</sup>* are *w*ˇ*i*,1 ··· *w*ˇ*i*,*q*ˇ*<sup>i</sup>* � ; *y*´2 = *y*1; the matrices *A*`*i*, *A*` *<sup>i</sup>*,*y*, *B*` *<sup>i</sup>*, *C*` *<sup>i</sup>*, *<sup>D</sup>*` *<sup>i</sup>*, ` *D*ˇ *<sup>i</sup>*, and *E*` *<sup>i</sup>* are of the appropriate dimensions, generally unknown or partially unknown, *i* = 1, 2. For subsystem **S1**, the transfer function from *y*<sup>2</sup> to *y*<sup>1</sup> is *H*1(*s*) = *C*` <sup>1</sup>(*sIn*<sup>1</sup> <sup>−</sup> *<sup>A</sup>*` <sup>1</sup>)−1*B*` 1, for subsystem **S2**, the transfer function from *u* to *y*<sup>2</sup> is *H*2(*s*) = *C*` <sup>2</sup>(*sIn*<sup>2</sup> <sup>−</sup> *<sup>A</sup>*`2)−1*B*` 2. All signals in the system are assumed to be continuous.

The subsystems **S1** and **S2** satisfy the following assumptions,

**Assumption 1.** *For i* = 1, 2*, the pair* (*A*`*i*, *C*` *<sup>i</sup>*) *is observable; the transfer function Hi*(*s*) *is known to have relative degree ri* ∈ IN*, and is strictly minimum phase. The uncontrollable part of* **S1** *(with respect to y*2*) is stable in the sense of Lyapunov; any uncontrollable mode corresponding to an eigenvalue of the matrix A*` <sup>1</sup> *on the jω-axis is uncontrollable from w*`� <sup>1</sup> *w*ˇ� 1 � *. The uncontrollable part of* **S2** *(with respect to u) is stable in the sense of Lyapunov; any uncontrollable mode corresponding to an eigenvalue of the matrix <sup>A</sup>*` <sup>2</sup> *on the jω-axis is uncontrollable from w*`� <sup>2</sup> *y*´2 *w*ˇ� 2 � *.* �

Based on Assumption 1, for *i* = 1, 2, there exists a state diffeomorphism: *xi* = *T*` *ix*`*i*, and a disturbance transformation: *wi* = *M*` *iw*`*i*, such that **Si** can be transformed into the following state space representation,

$$\begin{aligned} \dot{x}\_1 &= A\_1 x\_1 + (y\_1 \bar{A}\_{1,211} + y\_2 \bar{A}\_{1,212} + \sum\_{j=1}^{\tilde{q}\_1} \tilde{w}\_{1,j} \bar{A}\_{1,213j}) \theta\_1 + B\_1 y\_2 + D\_1 w\_1 + \tilde{D}\_1 \ddot{w}\_1; \\ y\_1 &= C\_1 x\_1 + E\_1 w\_1 \\ \dot{x}\_2 &= A\_2 x\_2 + (y\_2 \bar{A}\_{2,211} + u \bar{A}\_{2,212} + \sum\_{j=1}^{\tilde{q}\_2} \tilde{w}\_{2,j} \bar{A}\_{2,213j} + \dot{y}\_2 \bar{A}\_{2,214}) \theta\_2 + B\_2 u + A\_{2,\dot{y}} \dot{y}\_2 + D\_2 w\_2 + \dot{D}\_2 \ddot{w}\_2; \\ y\_2 &= C\_2 x\_2 + E\_2 w\_2 \end{aligned}$$

where *θ<sup>i</sup>* is the *σi*-dimensional vector of unknown parameters for the subsystem **S***i*, *σ<sup>i</sup>* ∈ IN; the matrices *Ai*, *A*¯ *<sup>i</sup>*,211, *A*¯ *<sup>i</sup>*,212, *A*¯ *<sup>i</sup>*,2131, ··· , *<sup>A</sup>*¯ *i*,213*q*ˇ*<sup>i</sup>* , *A*¯ 2,214, *A*2,*y*, *Bi*, *Di*, *D*ˇ *<sup>i</sup>*, *Ci*, and *Ei* are known and have the following structures, *Ai* = (*ai*,*jk*)*ni*×*ni* ; *ai*,*j*(*j*+1) = 1, *ai*,*jk* = 0, for 1 ≤ *j* ≤ *ri* − 1 and *<sup>j</sup>* <sup>+</sup> <sup>2</sup> <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *ni*; *<sup>A</sup>*¯*i*,212 <sup>=</sup> **<sup>0</sup>***σi*×(*ri*−1) *<sup>A</sup>*¯� *<sup>i</sup>*,2120 *<sup>A</sup>*¯� *i*,212*ri* � , *Ci* = [ <sup>1</sup> **<sup>0</sup>**1×(*ni*−1) ], *Ai*,2120 is a row vector, *Bi* = **<sup>0</sup>**1×(*ri*−1) *bi*,*p*<sup>0</sup> ··· *bi*,*p*(*ni*−*ri*) � , *bi*,*pj j* = 0, 1, ··· , *ni* − *ri* are constants. We denote the elements of *<sup>x</sup>*<sup>1</sup> and *<sup>x</sup>*<sup>2</sup> by *x*1,1 ··· *x*1,*n*<sup>1</sup> � and *x*2,1 ··· *x*2,*n*<sup>2</sup> � , with initial conditions *x*1,0 and *x*2,0, respectively.

**Assumption 2.** *The measured disturbance w*ˇ <sup>1</sup> *can be partitioned as: w*ˇ <sup>1</sup> = � *w*ˇ� 1,*<sup>a</sup> w*ˇ� 1,*b* �� *where w*ˇ 1,*<sup>a</sup> is q*ˇ1,*<sup>a</sup> dimensional, q*ˇ1,*<sup>a</sup>* ∈ IN ∪ {0}*, and the transfer function from each element of w*ˇ 1,*<sup>a</sup> to y*<sup>1</sup> *has relative degree less than r*<sup>1</sup> + *r*2*; the measured disturbance w*ˇ <sup>2</sup> *can be partitioned as: w*ˇ <sup>2</sup> = � *w*ˇ� 2,*<sup>a</sup> w*ˇ� 2,*b* �� *where w*ˇ 2,*<sup>a</sup> is q*ˇ2,*<sup>a</sup> dimensional, q*ˇ2,*<sup>a</sup>* ∈ IN ∪ {0}*, and the transfer function from each element of w*ˇ 2,*<sup>a</sup> to y*<sup>2</sup> *has relative degree less than r*2*.* �

Based on Assumption 2, the matrix *<sup>D</sup>*<sup>ˇ</sup> *<sup>i</sup>* can be partitioned into � *D*ˇ *<sup>i</sup>*,*<sup>a</sup> D*ˇ *<sup>i</sup>*,*<sup>b</sup>* � , where *D*ˇ *<sup>i</sup>*,*<sup>a</sup>* and *D*ˇ *<sup>i</sup>*,*<sup>b</sup>* have *ni* <sup>×</sup> *<sup>q</sup>*ˇ*i*,*a*- and *ni* <sup>×</sup> *<sup>q</sup>*ˇ*i*,*b*-dimensional, respectively; and *<sup>D</sup>*<sup>ˇ</sup> *<sup>i</sup>*,*b*, *<sup>A</sup>*¯ *<sup>i</sup>*,213 (*q*ˇ*i*,*<sup>a</sup>*+1), ··· , *<sup>A</sup>*¯*i*,213 *<sup>q</sup>*ˇ*<sup>i</sup>* have the following structure

$$\mathbf{^j\breve{D}}\_{i,b} = \begin{bmatrix} \mathbf{0}\_{(r\_l-1)\times\breve{q}\_{l\dot{l}}} \\ \breve{D}\_{i,b0} \\ \breve{D}\_{i,br\_l} \end{bmatrix}; \quad \breve{A}\_{i,213} = \begin{bmatrix} \mathbf{0}\_{(r\_l-1)\times\sigma\_l} \\ \breve{A}\_{i,213} \boldsymbol{j} \\ \breve{A}\_{i,213} \boldsymbol{j} \boldsymbol{r}\_l \end{bmatrix}, \boldsymbol{j} = \breve{q}\_{i,a} + 1, \cdots, \breve{q}\_{i\dot{l}}$$

where *<sup>D</sup>*<sup>ˇ</sup> *<sup>i</sup>*,*b*<sup>0</sup> and *<sup>A</sup>*¯*i*,213 *<sup>j</sup>*0, *<sup>j</sup>* <sup>=</sup> *<sup>q</sup>*ˇ*i*,*<sup>a</sup>* <sup>+</sup> 1, ··· , *<sup>q</sup>*ˇ*i*,*<sup>a</sup>* <sup>+</sup> *<sup>q</sup>*ˇ*i*,*b*, are row vectors, *<sup>i</sup>* <sup>=</sup> 1, 2.

Since we will base our design of adaptive controllers using the model (2), we call (2) the design model, and make the following two assumptions.

**Assumption 3.** *For i* = 1, 2*, the matrices Ei are such that EiE*� *<sup>i</sup>* > 0*.* �

Define *ζ<sup>i</sup>* := (*EiE*� *i* )<sup>−</sup> <sup>1</sup> <sup>2</sup> and *Li* := *DiE*� *i* , *i* = 1, 2.

4 Will-be-set-by-IN-TECH

<sup>1</sup>*y*<sup>2</sup> <sup>+</sup> *<sup>D</sup>*` <sup>1</sup>*w*` <sup>1</sup> <sup>+</sup> `

where *x*`*<sup>i</sup>* is the *ni*-dimensional state vectors with initial condition *x*`*i*(0) = *x*`*i*,0, *ni* ∈ IN; *u* is the scalar control input; *yi* is the scalar measurement output; *w*`*<sup>i</sup>* is *q*`*i*-dimensional unmeasured disturbance input vector, *q*`*<sup>i</sup>* ∈ IN; *w*ˇ*<sup>i</sup>* is *q*ˇ*i*-dimensional measured disturbance input vector,

*<sup>i</sup>* are of the appropriate dimensions, generally unknown or partially unknown, *i* = 1, 2. For

*have relative degree ri* ∈ IN*, and is strictly minimum phase. The uncontrollable part of* **S1** *(with respect to y*2*) is stable in the sense of Lyapunov; any uncontrollable mode corresponding to an eigenvalue of the*

> *w*`� <sup>1</sup> *w*ˇ� 1 �

*to u) is stable in the sense of Lyapunov; any uncontrollable mode corresponding to an eigenvalue of the*

*w*`� <sup>2</sup> *y*´2 *w*ˇ� 2 �

disturbance transformation: *wi* = *M*` *iw*`*i*, such that **Si** can be transformed into the following

where *θ<sup>i</sup>* is the *σi*-dimensional vector of unknown parameters for the subsystem **S***i*, *σ<sup>i</sup>* ∈ IN;

*<sup>i</sup>*,2120 *<sup>A</sup>*¯�

�

*x*1,1 ··· *x*1,*n*<sup>1</sup>

*i*,212*ri*

� and

�

*i*,213*q*ˇ*<sup>i</sup>* , *A*¯

Based on Assumption 1, for *i* = 1, 2, there exists a state diffeomorphism: *xi* = *T*`

*q*ˇ1 ∑ *j*=1

*q*ˇ2 ∑ *j*=1

*<sup>i</sup>*,2131, ··· , *<sup>A</sup>*¯

**<sup>0</sup>**1×(*ri*−1) *bi*,*p*<sup>0</sup> ··· *bi*,*p*(*ni*−*ri*)

*w*ˇ 2,*jA*¯

**<sup>0</sup>***σi*×(*ri*−1) *<sup>A</sup>*¯�

�

*<sup>x</sup>*`2 <sup>=</sup> *<sup>A</sup>*` <sup>2</sup>*x*`2 <sup>+</sup> *<sup>B</sup>*`2*<sup>u</sup>* <sup>+</sup> *<sup>A</sup>*`2,*yy*´2 <sup>+</sup> *<sup>D</sup>*` <sup>2</sup>*w*` <sup>2</sup> <sup>+</sup> `

*D*ˇ <sup>1</sup>*w*ˇ 1; (1a)

*D*ˇ <sup>2</sup>*w*ˇ 2; (1c)

*<sup>i</sup>*,*y*, *B*` *<sup>i</sup>*, *C*`

<sup>1</sup>(*sIn*<sup>1</sup> <sup>−</sup> *<sup>A</sup>*` <sup>1</sup>)−1*B*`

*<sup>i</sup>*) *is observable; the transfer function Hi*(*s*) *is known to*

*. The uncontrollable part of* **S2** *(with respect*

2,213*<sup>j</sup>* + *y*´2*A*¯2,214)*θ*<sup>2</sup> + *B*2*u* + *A*2,*yy*´2 + *D*2*w*<sup>2</sup> + *D*ˇ <sup>2</sup>*w*ˇ 2;

2,214, *A*2,*y*, *Bi*, *Di*, *D*ˇ *<sup>i</sup>*, *Ci*, and *Ei* are known

; *ai*,*j*(*j*+1) = 1, *ai*,*jk* = 0, for 1 ≤ *j* ≤ *ri* − 1

, *bi*,*pj j* = 0, 1, ··· , *ni* − *ri* are constants.

*x*2,1 ··· *x*2,*n*<sup>2</sup>

, *Ci* = [ <sup>1</sup> **<sup>0</sup>**1×(*ni*−1) ], *Ai*,2120

� , with initial

*.* �

*<sup>i</sup>*, *<sup>D</sup>*` *<sup>i</sup>*, `

1, for subsystem

2. All signals in the system

*D*ˇ *<sup>i</sup>*, and

*ix*`*i*, and a

<sup>1</sup>*w*` <sup>1</sup> (1b)

<sup>2</sup>*x*`2 + *E*`2*w*` <sup>2</sup> (1d)

; *y*´2 = *y*1; the matrices *A*`*i*, *A*`

<sup>2</sup>(*sIn*<sup>2</sup> <sup>−</sup> *<sup>A</sup>*`2)−1*B*`

*w*ˇ 1,*jA*¯ 1,213*j*)*θ*<sup>1</sup> + *B*1*y*<sup>2</sup> + *D*1*w*<sup>1</sup> + *D*ˇ <sup>1</sup>*w*ˇ 1;

We assume that the system dynamics for **S1** and **S2** are given by,

*x*`1 = *A*` <sup>1</sup>*x*`1 + *B*`

<sup>1</sup>*x*`1 + *E*`

*w*ˇ*i*,1 ··· *w*ˇ*i*,*q*ˇ*<sup>i</sup>*

˙

˙

**S2**, the transfer function from *u* to *y*<sup>2</sup> is *H*2(*s*) = *C*`

<sup>1</sup> *on the jω-axis is uncontrollable from*

**Assumption 1.** *For i* = 1, 2*, the pair* (*A*`*i*, *C*`

*matrix <sup>A</sup>*` <sup>2</sup> *on the jω-axis is uncontrollable from*

*<sup>q</sup>*ˇ*<sup>i</sup>* <sup>∈</sup> IN; the elements of *<sup>w</sup>*ˇ*<sup>i</sup>* are

are assumed to be continuous.

state space representation,

*y*<sup>1</sup> = *C*1*x*<sup>1</sup> + *E*1*w*<sup>1</sup>

*y*<sup>2</sup> = *C*2*x*<sup>2</sup> + *E*2*w*<sup>2</sup>

the matrices *Ai*, *A*¯

is a row vector, *Bi* =

*x*˙1 = *A*1*x*<sup>1</sup> + (*y*1*A*¯ 1,211 + *y*2*A*¯1,212 +

*x*˙2 = *A*2*x*<sup>2</sup> + (*y*2*A*¯ 2,211 + *uA*¯ 2,212 +

and *<sup>j</sup>* <sup>+</sup> <sup>2</sup> <sup>≤</sup> *<sup>k</sup>* <sup>≤</sup> *ni*; *<sup>A</sup>*¯*i*,212 <sup>=</sup>

*<sup>i</sup>*,211, *A*¯

We denote the elements of *<sup>x</sup>*<sup>1</sup> and *<sup>x</sup>*<sup>2</sup> by

conditions *x*1,0 and *x*2,0, respectively.

*<sup>i</sup>*,212, *A*¯

and have the following structures, *Ai* = (*ai*,*jk*)*ni*×*ni*

*E*`

*matrix A*`

*y*<sup>1</sup> = *C*`

*y*<sup>2</sup> = *C*`

subsystem **S1**, the transfer function from *y*<sup>2</sup> to *y*<sup>1</sup> is *H*1(*s*) = *C*`

The subsystems **S1** and **S2** satisfy the following assumptions,

Due to the structures of *Ai*, *A*¯*i*,212 and *Bi*, the high frequency gain of the transfer function *Hi*(*s*), *bi*,0, is equal to *bi*,*p*<sup>0</sup> + *A*¯ *<sup>i</sup>*,212 0*θi*, *i* = 1, 2.

To guarantee the stability of the identified system, we make the following assumption on the parameter vectors *θ*<sup>1</sup> and *θ*2.

**Assumption 4.** *The sign of bi*,0 *is known; there exists a known smooth nonnegative radially-unbounded strictly convex function Pi* : IR*σ<sup>i</sup>* <sup>→</sup> IR*, such that the true value <sup>θ</sup><sup>i</sup>* <sup>∈</sup> <sup>Θ</sup>*<sup>i</sup>* :<sup>=</sup> { ¯ *<sup>θ</sup><sup>i</sup>* <sup>∈</sup> IR*σ<sup>i</sup>* <sup>|</sup> *Pi*(¯ *<sup>θ</sup>i*) <sup>≤</sup> <sup>1</sup>}*; moreover,* <sup>∀</sup> ¯ *<sup>θ</sup><sup>i</sup>* <sup>∈</sup> <sup>Θ</sup>*i,* sgn(*bi*,0)(*bi*,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*¯*i*,212 0 ¯ *θi*) > 0*, i* = 1, 2*.* �

Assumption 4 delineates *a priori* convex compact sets where the parameter vectors *θ*<sup>1</sup> and *θ*<sup>2</sup> lie in, respectively. This will guarantee the stability of the closed-loop system and the boundedness of the estimate of *θ*<sup>1</sup> and *θ*2.

We make the following assumption about the reference signal, *yd*.

**Assumption 5.** *The reference trajectory, yd, is r*<sup>1</sup> + *r*<sup>2</sup> *times continuously differentiable. Define vector Yd* := [*y* (0) *<sup>d</sup>* , ··· , *y* (*r*1+*r*2) *<sup>d</sup>* ] � *, where y*(0) *<sup>d</sup>* <sup>=</sup> *yd, and y*(*j*) *<sup>d</sup> is the jth order time derivative of yd, j* = 1, ··· ,*r*<sup>1</sup> + *r*2*; define Yd*<sup>0</sup> := [*y* (0) *<sup>d</sup>* (0), ··· , *y* (*r*1+*r*2−1) *<sup>d</sup>* (0)]� <sup>∈</sup> IR*r*1+*r*<sup>2</sup> *. The signal Yd is available for feedback.* �

The uncertainty of subsystem **S1** is *ω*` <sup>1</sup> := (*x*1,0, *θ*1, *w*` <sup>1</sup>[0,∞), *w*ˇ <sup>1</sup>[0,∞),*Yd*0, *y* (*r*1+*r*2) *<sup>d</sup>*[0,∞) ) <sup>∈</sup> <sup>W</sup>` <sup>1</sup> :<sup>=</sup> IR*n*<sup>1</sup> <sup>×</sup> <sup>Θ</sup><sup>1</sup> ×C×C× IR*r*1+*r*<sup>2</sup> × C, which comprises the initial state *<sup>x</sup>*1,0, the true value of the parameters *θ*1, the unmeasured disturbance waveform *w*` <sup>1</sup>[0,∞), the measured disturbance waveform *w*ˇ <sup>1</sup>[0,∞), the initial conditions of the reference trajectory *Yd*0, and the waveform of the (*r*<sup>1</sup> + *r*2) th order derivative of the reference trajectory, *y* (*r*1+*r*2) *<sup>d</sup>*[0,∞) . The uncertainty for subsystem **S2** is *<sup>ω</sup>*` <sup>2</sup> := (*x*2,0, *<sup>θ</sup>*2, *<sup>w</sup>*` <sup>2</sup>[0,∞), *<sup>w</sup>*<sup>ˇ</sup> <sup>2</sup>[0,<sup>∞</sup>)) <sup>∈</sup> <sup>W</sup>` <sup>2</sup> :<sup>=</sup> IR*n*<sup>2</sup> <sup>×</sup> <sup>Θ</sup><sup>2</sup> ×C×C, which comprises the initial state *x*2,0, the true value of the parameters *θ*2, the unmeasured disturbance waveform *w*` <sup>2</sup>[0,∞), and the measured disturbance waveform *w*ˇ <sup>2</sup>[0,∞).

Our objective is to derive a control law, which is generated by the following mapping,

$$
\mu(t) = \mu(y\_{2[0,t]^\prime} \circ\_{2[0,t]^\prime} Y\_{d[0,t]^\prime} \,\ntoarrow \nu\_1, \nu\_2) \tag{3}
$$

Clearly, when the inequality (4) is achieved, the squared L<sup>2</sup> norm of the output tracking error *<sup>C</sup>*1*x*<sup>1</sup> <sup>−</sup> *yd* is bounded by *<sup>γ</sup>*<sup>2</sup> times the squared <sup>L</sup><sup>2</sup> norm of the transformed disturbance input

> *i* , *x*� *i* ] �

> > *ξ*<sup>1</sup> +

*<sup>j</sup>*=<sup>1</sup> *<sup>w</sup>*<sup>ˇ</sup> 1,*jA*¯ 1,213*<sup>j</sup> <sup>A</sup>*<sup>1</sup>

*D*ˇ <sup>1</sup>*w*ˇ <sup>1</sup>

2,213*<sup>j</sup>* + *y*´2*A*¯

<sup>2</sup>*<sup>u</sup>* <sup>+</sup> *<sup>A</sup>*¯2,*yy*´2 <sup>+</sup> *<sup>D</sup>*¯ <sup>2</sup>*w*<sup>2</sup> <sup>+</sup> ¯

The worst-case optimization of the cost function (4) can be carried out in two steps as depicted

 2 ∑ *i*=1

(*r*1+*r*2)

The inner supremum operators will be carried out first. We maximize over *ω<sup>i</sup>* given that the measurement *ω<sup>m</sup>* is available for estimator design, *i* = 1, 2. In this step, the control input, *u*, is a function only depended on *ωm*, then *u* is an open-loop time function and available for the optimization. Using *cost-to-come* function analysis, we derive the dynamics of the estimators

The outer supremum operator will be carried out second. In this step, we use a backstepping

2,214

*D*ˇ <sup>2</sup>*w*ˇ <sup>2</sup>

sup *<sup>ω</sup>*` <sup>1</sup>∈W` 1, *<sup>ω</sup>*` <sup>2</sup>∈W` <sup>2</sup>|*ωm*∈W*<sup>m</sup>*

sup *ω*1∈W1,*ω*2∈W2|*ωm*∈W*<sup>m</sup>*

> sup *ωi*∈W*i*|*ωm*∈W*<sup>m</sup>*

the squared <sup>L</sup><sup>2</sup> norm of *<sup>C</sup>*1*x*<sup>1</sup> <sup>−</sup> *yd* is also finite, which implies lim*t*→<sup>∞</sup>

<sup>1</sup>*y*<sup>2</sup> <sup>+</sup> *<sup>D</sup>*¯ <sup>1</sup>*w*<sup>1</sup> <sup>+</sup> ¯

*<sup>j</sup>*=<sup>1</sup> *<sup>w</sup>*<sup>ˇ</sup> 2,*jA*¯

*Jγtf*

= sup *ωm*∈W*<sup>m</sup>*

≤ sup *ωm*∈W*<sup>m</sup>*

= sup *ωm*∈W*<sup>m</sup>*

, plus some constant. When the L<sup>2</sup> norm of *w*` 1, *w*` 2, *w*ˇ 1, and *w*ˇ <sup>2</sup> are finite,

, *i* = 1, 2, and note that ˙

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

> **0** *D*<sup>1</sup> *w*<sup>1</sup> +

 **0** *B*2 *u* + **0** *A*2,*<sup>y</sup> y*´2

*Jγtf*

*Jγtf*

*<sup>d</sup>*[0,∞) ) ∈ W*<sup>m</sup>* :<sup>=</sup> C×C×C×C× IR*r*1+*r*<sup>2</sup> × C.

*Ji*,*γtf* 

 **0** *B*1 *y*<sup>2</sup> +

**0** *A*2  *ξ*<sup>2</sup> + (*C*1*x*1(*t*) − *yd*(*t*)) = 0,

*θ<sup>i</sup>* = 0, we have

113

(6)

 **0** *D*ˇ 1 *w*ˇ 1

 *w*� <sup>1</sup> *w*ˇ� 1,*<sup>a</sup> w*� <sup>2</sup> *w*ˇ� 2,*a* �

> ˙ *ξ*<sup>1</sup> =

> ˙ *ξ*<sup>2</sup> =

=: *A*¯

=: *C*¯

+ **0** *D*<sup>2</sup> *w*<sup>2</sup> +

> **0** *C*<sup>2</sup>

in the following equations.

<sup>2</sup>*ξ*<sup>2</sup> + *E*2*w*<sup>2</sup>

**0** *C*<sup>1</sup> 

<sup>1</sup>*ξ*<sup>1</sup> + *E*1*w*<sup>1</sup>

*y*2*A*¯ 2,211 + *uA*¯

=: *A*¯2(*y*1, *y*2, , *w*ˇ 2, *u*)*ξ*<sup>2</sup> + *B*¯

*ξ*<sup>2</sup> + *E*2*w*<sup>2</sup>

*<sup>y</sup>*<sup>1</sup> =

*<sup>y</sup>*<sup>2</sup> =

=: *C*¯

under additional assumptions.

Let *ξ<sup>i</sup>* denote the expanded state vector *ξ<sup>i</sup>* = [*θ*�

the following expanded dynamics for system (2),

*<sup>y</sup>*1*A*¯ 1,211 <sup>+</sup> *<sup>y</sup>*2*A*¯ 1,212 <sup>+</sup> <sup>∑</sup>*q*ˇ1

<sup>1</sup>(*y*1, *y*2, *w*ˇ <sup>1</sup>)*ξ*<sup>1</sup> + *B*¯

*ξ*<sup>1</sup> + *E*1*w*<sup>1</sup>

**0**

 **0** *D*ˇ 2 *w*ˇ 2

sup *<sup>ω</sup>*` <sup>1</sup>∈W` 1, *<sup>ω</sup>*` <sup>2</sup>∈W` <sup>2</sup>

where *ωm* is the measured signals of the system, and defined as

This completes the formulation of the robust adaptive control problem.

*ω<sup>m</sup>* := (*y*1[0,∞), *y*2[0,∞), *w*ˇ <sup>1</sup>[0,∞), *w*ˇ <sup>2</sup>[0,∞),*Yd*0, *y*

for subsystem **S**<sup>1</sup> and **S**<sup>2</sup> independently.

procedure to design the controller *μ*.

**0 0**

2,212 <sup>+</sup> <sup>∑</sup>*q*ˇ2

where *μ* : C×C×C×C×C → IR, such that *x*1,1 can asymptotically track the reference trajectory *yd*, while rejecting the uncertainty (*ω*` 1, *<sup>ω</sup>*` <sup>2</sup>) <sup>∈</sup> <sup>W</sup>` <sup>1</sup> <sup>×</sup> <sup>W</sup>` 2, and keeping the closed-loop signals bounded. The control law *<sup>μ</sup>* must also satisfy that, <sup>∀</sup>(*ω*` 1, *<sup>ω</sup>*` <sup>2</sup>) <sup>∈</sup> <sup>W</sup>` <sup>1</sup> <sup>×</sup> <sup>W</sup>` 2, there exists a solution *x*`1[0,∞) and *x*`2[0,∞) to the system (1), which yields a continuous control signal *u*[0,∞). We denote the class of these admissible controllers by M*μ*.

For design purposes, instead of attenuating the effect of *w*`� <sup>1</sup> *w*ˇ� <sup>1</sup> *w*`� <sup>2</sup> *w*ˇ <sup>2</sup> � we design the adaptive controller to attenuate the effect of *w*� <sup>1</sup> *w*ˇ� <sup>1</sup> *w*� <sup>2</sup> *w*ˇ <sup>2</sup> � . This is done to allow our design paradigm to be carried out. This will result in a guaranteed attenuation level with respect to *ω*` <sup>1</sup> and *ω*` 2. To simplify the notation, we take the uncertainty *ω*<sup>1</sup> := (*x*1,0, *θ*1, *w*1[0,∞), *w*ˇ <sup>1</sup>[0,∞),*Yd*0, *y* (*r*1+*r*2) *<sup>d</sup>*[0,∞) ) ∈ W<sup>1</sup> :<sup>=</sup> IR*n*<sup>1</sup> <sup>×</sup> <sup>Θ</sup><sup>1</sup> ×C×C× IR*r*1+*r*<sup>2</sup> × C, and *<sup>ω</sup>*<sup>2</sup> :<sup>=</sup> (*x*2,0, *<sup>θ</sup>*2, *<sup>w</sup>*2[0,∞), *<sup>w</sup>*<sup>ˇ</sup> <sup>2</sup>[0,<sup>∞</sup>)) ∈ W<sup>2</sup> :<sup>=</sup> IR*n*<sup>2</sup> <sup>×</sup> <sup>Θ</sup><sup>2</sup> ×C×C.

We state the control objective precisely as follows,

**Definition 1.** *A controller μ* ∈ M*<sup>μ</sup> is said to achieve* disturbance attenuation level *γ with respect to w*� <sup>1</sup> *w*ˇ� 1,*<sup>a</sup> w*� <sup>2</sup> *w*ˇ� 2,*a* � *, and* disturbance attenuation level zero *with respect to w*ˇ� 1,*<sup>b</sup> w*ˇ� 2,*b* � *, if there exists functions l*1(*t*, *θ*1, *x*1, *y*1[0,*t*], *y*2[0,*t*] , *w*ˇ <sup>1</sup>[0,*t*] , *w*ˇ <sup>2</sup>[0,*t*] , *Yd*[0,*<sup>t</sup>*])*, l*2(*t*, *θ*2, *x*2, *y*1[0,*t*], *y*2[0,*t*], *w*ˇ <sup>1</sup>[0,*t*], *w*ˇ <sup>2</sup>[0,*t*] ,*Yd*[0,*t*] )*, and a known nonnegative constant l*0(*x*ˇ1,0, *x*ˇ2,0, ˇ *θ*1,0, ˇ *θ*2,0)*, such that*

$$\sup\_{\forall t\_1 \in \mathring{\mathcal{W}}\_1, \nexists t\_2 \in \mathring{\mathcal{W}}\_2} J\_{\gamma t\_f} \le 0; \quad \forall t\_f \ge 0$$

*and l*<sup>1</sup> ≥ 0 *and l*<sup>2</sup> ≥ 0 *along the closed-loop trajectory, where*

$$\mathbf{l}\_{\uparrow t}\mathbf{i}\_{f} \ddot{=}\mathbf{l}\_{1,\uparrow t\_{f}} + \mathbf{l}\_{2,\uparrow t\_{f}} \tag{5a}$$

$$J\_{1\gamma t\not\!\vdash} = \int\_0^{t\_f} \left( (\mathbb{C}\_1 \mathbf{x}\_1 - y\_d)^2 + l\_1 - \gamma^2 |w\_1|^2 - \gamma^2 |\mathbb{W}\_{1,d}|^2 \right) \mathrm{d}\tau - \gamma^2 \left| \left[ \theta\_1' - \theta\_{1,0}' \,\mathbf{x}\_{1,0}' - \mathbf{x}\_{1,0}' \right]' \right|\_{Q\_{1,0}}^2 \text{(5b)}$$

$$J\_{2,\gamma t\uparrow} \coloneqq \int\_0^{t\_f} \left(l\_2 - \gamma^2 |w\_2|^2 - \gamma^2 |\psi\_{2,a}|^2\right) \mathrm{d}\tau - l\_0 - \gamma^2 \left|\left[\theta\_2' - \check{\theta}\_{2,0}' \, \mathfrak{x}\_{2,0}' - \check{\mathfrak{x}}\_{2,0}'\right]'\right|\_{\check{Q}\_{2,0}}^2 \tag{5c}$$

ˇ *<sup>θ</sup>i*,0 <sup>∈</sup> <sup>Θ</sup>*<sup>i</sup> is the initial guess of <sup>θ</sup>i; <sup>x</sup>*ˇ*i*,0 <sup>∈</sup> IR*ni is the initial guess of xi*,0*; <sup>Q</sup>*¯*i*,0 <sup>&</sup>gt; <sup>0</sup> *is a* (*ni* + *σi*) × (*ni* + *σi*)*-dimensional weighting matrix, quantifying the level of confidence in the estimate* ˇ *θ*� *<sup>i</sup>*,0 *x*ˇ� *i*,0 � *; Q*¯ <sup>−</sup><sup>1</sup> *<sup>i</sup>*,0 *admits the structure <sup>Q</sup>*−<sup>1</sup> *<sup>i</sup>*,0 *<sup>Q</sup>*−<sup>1</sup> *<sup>i</sup>*,0 Φ� *i*,0 Φ*i*,0*Q*−<sup>1</sup> *<sup>i</sup>*,0 <sup>Π</sup>*i*,0 <sup>+</sup> <sup>Φ</sup>*i*,0*Q*−<sup>1</sup> *<sup>i</sup>*,0 Φ� *i*,0 *, Qi*,0 *and* Π*i*,0 *are σ<sup>i</sup>* × *σiand ni* × *ni-dimensional positive definite matrices, respectively, i* = 1, 2*.*

Clearly, when the inequality (4) is achieved, the squared L<sup>2</sup> norm of the output tracking error *<sup>C</sup>*1*x*<sup>1</sup> <sup>−</sup> *yd* is bounded by *<sup>γ</sup>*<sup>2</sup> times the squared <sup>L</sup><sup>2</sup> norm of the transformed disturbance input *w*� <sup>1</sup> *w*ˇ� 1,*<sup>a</sup> w*� <sup>2</sup> *w*ˇ� 2,*a* � , plus some constant. When the L<sup>2</sup> norm of *w*` 1, *w*` 2, *w*ˇ 1, and *w*ˇ <sup>2</sup> are finite, the squared <sup>L</sup><sup>2</sup> norm of *<sup>C</sup>*1*x*<sup>1</sup> <sup>−</sup> *yd* is also finite, which implies lim*t*→<sup>∞</sup> (*C*1*x*1(*t*) − *yd*(*t*)) = 0, under additional assumptions.

6 Will-be-set-by-IN-TECH

subsystem **S2** is *<sup>ω</sup>*` <sup>2</sup> := (*x*2,0, *<sup>θ</sup>*2, *<sup>w</sup>*` <sup>2</sup>[0,∞), *<sup>w</sup>*<sup>ˇ</sup> <sup>2</sup>[0,<sup>∞</sup>)) <sup>∈</sup> <sup>W</sup>` <sup>2</sup> :<sup>=</sup> IR*n*<sup>2</sup> <sup>×</sup> <sup>Θ</sup><sup>2</sup> ×C×C, which comprises the initial state *x*2,0, the true value of the parameters *θ*2, the unmeasured disturbance

, *y*´2[0,*t*],*Yd*[0,*t*]

*w*� <sup>1</sup> *w*ˇ� <sup>1</sup> *w*� <sup>2</sup> *w*ˇ <sup>2</sup> �

design paradigm to be carried out. This will result in a guaranteed attenuation level with respect to *ω*` <sup>1</sup> and *ω*` 2. To simplify the notation, we take the uncertainty *ω*<sup>1</sup> :=

**Definition 1.** *A controller μ* ∈ M*<sup>μ</sup> is said to achieve* disturbance attenuation level *γ with respect*

*, and* disturbance attenuation level zero *with respect to*

:=*J*1,*γtf* + *J*2,*γtf* (5a)

2 

> *θ*� <sup>2</sup> <sup>−</sup> <sup>ˇ</sup> *θ*� 2,0 *x*�

<sup>d</sup>*<sup>τ</sup>* <sup>−</sup> *<sup>γ</sup>*<sup>2</sup>

*<sup>i</sup>*,0 Φ� *i*,0

> *<sup>i</sup>*,0 Φ� *i*,0

 *θ*� <sup>1</sup> <sup>−</sup> <sup>ˇ</sup> *θ*� 1,0 *x*�

<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2|*w*<sup>ˇ</sup> 1,*a*<sup>|</sup>

<sup>d</sup>*<sup>τ</sup>* <sup>−</sup> *<sup>l</sup>*<sup>0</sup> <sup>−</sup> *<sup>γ</sup>*<sup>2</sup>

*<sup>θ</sup>i*,0 <sup>∈</sup> <sup>Θ</sup>*<sup>i</sup> is the initial guess of <sup>θ</sup>i; <sup>x</sup>*ˇ*i*,0 <sup>∈</sup> IR*ni is the initial guess of xi*,0*; <sup>Q</sup>*¯*i*,0 <sup>&</sup>gt; <sup>0</sup> *is a* (*ni* + *σi*) × (*ni* + *σi*)*-dimensional weighting matrix, quantifying the level of confidence in the estimate*

*<sup>i</sup>*,0 *<sup>Q</sup>*−<sup>1</sup>

*<sup>i</sup>*,0 <sup>Π</sup>*i*,0 <sup>+</sup> <sup>Φ</sup>*i*,0*Q*−<sup>1</sup>

, *w*ˇ <sup>2</sup>[0,*t*]

where *μ* : C×C×C×C×C → IR, such that *x*1,1 can asymptotically track the reference trajectory *yd*, while rejecting the uncertainty (*ω*` 1, *<sup>ω</sup>*` <sup>2</sup>) <sup>∈</sup> <sup>W</sup>` <sup>1</sup> <sup>×</sup> <sup>W</sup>` 2, and keeping the closed-loop signals bounded. The control law *<sup>μ</sup>* must also satisfy that, <sup>∀</sup>(*ω*` 1, *<sup>ω</sup>*` <sup>2</sup>) <sup>∈</sup> <sup>W</sup>` <sup>1</sup> <sup>×</sup> <sup>W</sup>` 2, there exists a solution *x*`1[0,∞) and *x*`2[0,∞) to the system (1), which yields a continuous control

Our objective is to derive a control law, which is generated by the following mapping,

(*r*1+*r*2)

*w*`� <sup>1</sup> *w*ˇ� <sup>1</sup> *w*`� <sup>2</sup> *w*ˇ <sup>2</sup>

*<sup>d</sup>*[0,∞) ) ∈ W<sup>1</sup> :<sup>=</sup> IR*n*<sup>1</sup> <sup>×</sup> <sup>Θ</sup><sup>1</sup> ×C×C× IR*r*1+*r*<sup>2</sup> × C, and *<sup>ω</sup>*<sup>2</sup> :<sup>=</sup>

*θ*1,0, ˇ

*<sup>d</sup>*[0,∞) . The uncertainty for

� we design the

, *w*ˇ <sup>1</sup>[0,*t*],

2

*Q*¯ 1,0 (5b)

(5c)

. This is done to allow our

 *w*ˇ� 1,*<sup>b</sup> w*ˇ� 2,*b* � *, if*

1,0 − *x*ˇ� 1,0 � 

2

*Q*¯ 2,0

*, Qi*,0 *and* Π*i*,0 *are σ<sup>i</sup>* × *σi-*

, *Yd*[0,*<sup>t</sup>*])*, l*2(*t*, *θ*2, *x*2, *y*1[0,*t*], *y*2[0,*t*]

*θ*2,0)*, such that*

*Jγtf* ≤ 0; ∀*tf* ≥ 0 (4)

2,0 − *x*ˇ� 2,0 � 

, *w*ˇ 1, *w*ˇ <sup>2</sup>) (3)

of the (*r*<sup>1</sup> + *r*2) th order derivative of the reference trajectory, *y*

waveform *w*` <sup>2</sup>[0,∞), and the measured disturbance waveform *w*ˇ <sup>2</sup>[0,∞).

*u*(*t*) = *μ*(*y*2[0,*t*]

signal *u*[0,∞). We denote the class of these admissible controllers by M*μ*.

For design purposes, instead of attenuating the effect of

(*r*1+*r*2)

)*, and a known nonnegative constant l*0(*x*ˇ1,0, *x*ˇ2,0, ˇ

sup *<sup>w</sup>*` <sup>1</sup>∈W` 1,*w*` <sup>2</sup>∈W` <sup>2</sup>

> 2

> > *<sup>Q</sup>*−<sup>1</sup>

Φ*i*,0*Q*−<sup>1</sup>

adaptive controller to attenuate the effect of

(*x*2,0, *<sup>θ</sup>*2, *<sup>w</sup>*2[0,∞), *<sup>w</sup>*<sup>ˇ</sup> <sup>2</sup>[0,<sup>∞</sup>)) ∈ W<sup>2</sup> :<sup>=</sup> IR*n*<sup>2</sup> <sup>×</sup> <sup>Θ</sup><sup>2</sup> ×C×C. We state the control objective precisely as follows,

*there exists functions l*1(*t*, *θ*1, *x*1, *y*1[0,*t*], *y*2[0,*t*], *w*ˇ <sup>1</sup>[0,*t*]

*and l*<sup>1</sup> ≥ 0 *and l*<sup>2</sup> ≥ 0 *along the closed-loop trajectory, where*

(*C*1*x*<sup>1</sup> <sup>−</sup> *yd*)<sup>2</sup> <sup>+</sup> *<sup>l</sup>*<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*2|*w*1<sup>|</sup>

*<sup>i</sup>*,0 *admits the structure*

<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2|*w*<sup>ˇ</sup> 2,*a*<sup>|</sup>

*and ni* × *ni-dimensional positive definite matrices, respectively, i* = 1, 2*.*

*<sup>l</sup>*<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2|*w*2<sup>|</sup>

(*x*1,0, *θ*1, *w*1[0,∞), *w*ˇ <sup>1</sup>[0,∞),*Yd*0, *y*

*to w*� <sup>1</sup> *w*ˇ� 1,*<sup>a</sup> w*� <sup>2</sup> *w*ˇ� 2,*a* �

*w*ˇ <sup>2</sup>[0,*t*]

*Jγtf*

*J*1*γtf* := *tf* 0 

*J*2,*γtf* := *tf* 0 

ˇ

 ˇ *θ*� *<sup>i</sup>*,0 *x*ˇ� *i*,0 � *; Q*¯ <sup>−</sup><sup>1</sup>

,*Yd*[0,*t*]

Let *ξ<sup>i</sup>* denote the expanded state vector *ξ<sup>i</sup>* = [*θ*� *i* , *x*� *i* ] � , *i* = 1, 2, and note that ˙*θ<sup>i</sup>* = 0, we have the following expanded dynamics for system (2),

˙ *ξ*<sup>1</sup> = **0 0** *<sup>y</sup>*1*A*¯ 1,211 <sup>+</sup> *<sup>y</sup>*2*A*¯ 1,212 <sup>+</sup> <sup>∑</sup>*q*ˇ1 *<sup>j</sup>*=<sup>1</sup> *<sup>w</sup>*<sup>ˇ</sup> 1,*jA*¯ 1,213*<sup>j</sup> <sup>A</sup>*<sup>1</sup> *ξ*<sup>1</sup> + **0** *B*1 *y*<sup>2</sup> + **0** *D*<sup>1</sup> *w*<sup>1</sup> + **0** *D*ˇ 1 *w*ˇ 1 =: *A*¯ <sup>1</sup>(*y*1, *y*2, *w*ˇ <sup>1</sup>)*ξ*<sup>1</sup> + *B*¯ <sup>1</sup>*y*<sup>2</sup> <sup>+</sup> *<sup>D</sup>*¯ <sup>1</sup>*w*<sup>1</sup> <sup>+</sup> ¯ *D*ˇ <sup>1</sup>*w*ˇ <sup>1</sup> *<sup>y</sup>*<sup>1</sup> = **0** *C*<sup>1</sup> *ξ*<sup>1</sup> + *E*1*w*<sup>1</sup> =: *C*¯ <sup>1</sup>*ξ*<sup>1</sup> + *E*1*w*<sup>1</sup> ˙ *ξ*<sup>2</sup> = **0** *y*2*A*¯ 2,211 + *uA*¯ 2,212 <sup>+</sup> <sup>∑</sup>*q*ˇ2 *<sup>j</sup>*=<sup>1</sup> *<sup>w</sup>*<sup>ˇ</sup> 2,*jA*¯ 2,213*<sup>j</sup>* + *y*´2*A*¯ 2,214 **0** *A*2 *ξ*<sup>2</sup> + **0** *B*2 *u* + **0** *A*2,*<sup>y</sup> y*´2 + **0** *D*<sup>2</sup> *w*<sup>2</sup> + **0** *D*ˇ 2 *w*ˇ 2 =: *A*¯2(*y*1, *y*2, , *w*ˇ 2, *u*)*ξ*<sup>2</sup> + *B*¯ <sup>2</sup>*<sup>u</sup>* <sup>+</sup> *<sup>A</sup>*¯2,*yy*´2 <sup>+</sup> *<sup>D</sup>*¯ <sup>2</sup>*w*<sup>2</sup> <sup>+</sup> ¯ *D*ˇ <sup>2</sup>*w*ˇ <sup>2</sup> *<sup>y</sup>*<sup>2</sup> = **0** *C*<sup>2</sup> *ξ*<sup>2</sup> + *E*2*w*<sup>2</sup> =: *C*¯ <sup>2</sup>*ξ*<sup>2</sup> + *E*2*w*<sup>2</sup>

The worst-case optimization of the cost function (4) can be carried out in two steps as depicted in the following equations.

$$\begin{aligned} \sup\_{\omega\_1 \in \mathcal{W}\_1, \,\,\omega\_2 \in \mathcal{W}\_2} & J\_{\gamma t\_f} = \sup\_{\omega\_m \in \mathcal{W}\_m} \sup\_{\substack{\omega\_1 \in \mathcal{W}\_1, \,\,\omega\_2 \in \mathcal{W}\_2 \,\vert \,\omega\_m \in \mathcal{W}\_m}} J\_{\gamma t\_f} \\ & \le \sup\_{\omega\_m \in \mathcal{W}\_m} \sup\_{\substack{\omega\_1 \in \mathcal{W}\_1, \,\,\omega\_2 \in \mathcal{W}\_2 \,\vert \,\omega\_m \in \mathcal{W}\_m}} J\_{\gamma t\_f} \\ & = \sup\_{\omega\_m \in \mathcal{W}\_m} \left( \sum\_{i=1}^2 \sup\_{\omega\_i \in \mathcal{W}\_i \,\vert \,\omega\_m \in \mathcal{W}\_m} J\_{i, \gamma t\_f} \right) \end{aligned} \tag{6}$$

where *ωm* is the measured signals of the system, and defined as

$$\omega\_{\mathfrak{m}} := (y\_{1[0,\infty)}, y\_{2[0,\infty)}, \psi\_{1[0,\infty)}, \psi\_{2[0,\infty)}, Y\_{d0^c} y\_{d[0,\infty)}^{(r\_1+r\_2)}) \in \mathcal{W}\_{\mathfrak{m}} := \mathcal{C} \times \mathcal{C} \times \mathcal{C} \times \mathcal{C} \times \mathbb{R}^{r\_1+r\_2} \times \mathcal{C} \dots$$

The inner supremum operators will be carried out first. We maximize over *ω<sup>i</sup>* given that the measurement *ω<sup>m</sup>* is available for estimator design, *i* = 1, 2. In this step, the control input, *u*, is a function only depended on *ωm*, then *u* is an open-loop time function and available for the optimization. Using *cost-to-come* function analysis, we derive the dynamics of the estimators for subsystem **S**<sup>1</sup> and **S**<sup>2</sup> independently.

The outer supremum operator will be carried out second. In this step, we use a backstepping procedure to design the controller *μ*.

This completes the formulation of the robust adaptive control problem.

#### **4. Adaptive control design**

In this section, we present the adaptive control design, which involves estimation design and control design. First, we discuss estimation design.

Then the dynamics of Σ1, Φ1, Π<sup>1</sup> are given as follows with initial conditions *γ*−2*Q*−<sup>1</sup>

*q*ˇ1 ∑ *j*=1 *A*¯

the covariance matrix Σ<sup>1</sup> upper and lower bounded as summarized in the following Lemma

**Lemma 1.** *Consider the dynamic equation (11a) for the covariance matrix* <sup>Σ</sup>1*. Let K*1,*<sup>c</sup>* <sup>≥</sup> *<sup>γ</sup>*2Tr(*Q*1,0)*,*

*<sup>γ</sup>*2Tr(*Q*1,0) <sup>≤</sup> Tr(Σ1(*t*))−<sup>1</sup> <sup>≤</sup> *<sup>K</sup>*1,*c*; <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *tf* ]

<sup>1</sup> online, we define *<sup>s</sup>*1,<sup>Σ</sup> <sup>=</sup> Tr(Σ−<sup>1</sup>

<sup>1</sup>*L*1*C*<sup>1</sup> *is not Hurwitz, then the desired disturbance attenuation level <sup>γ</sup>* <sup>&</sup>gt; *<sup>ζ</sup>*−<sup>1</sup>

This assumption implies that the achievable disturbance attenuation level *γ* is no smaller than

<sup>1</sup> . Under this assumption, we initialize Π<sup>1</sup> as the unique positive definite solution of its

**Assumption 7.** *The initial weighting matrix* Π1,0 *is chosen as the unique positive definite solutions*

To guarantee the estimates parameter to be bounded and the estimate of high frequency gain to be bounded away from zero, projection function scheme is applied to modify the dynamics

*<sup>θ</sup>*<sup>1</sup> <sup>∈</sup> IR*σ*<sup>1</sup> , *<sup>b</sup>*1,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*¯

By Assumption 4 and Lemma 2 in [19] we have 1 < *ρ*<sup>1</sup> ≤ ∞. Fix any *ρ*1,*<sup>o</sup>* ∈ (1, *ρ*1), and define

1*C*1Π1+*D*1*D*�

Riccati Differential Equation (11b), which is summarized as the following assumption.

−Π1*C*� 1*ζ*2

1*L*1*C*1)�

*<sup>θ</sup>*1) <sup>|</sup> ¯

1,*<sup>c</sup> <sup>I</sup>*1,*σ*<sup>1</sup> <sup>≤</sup> <sup>Σ</sup>1(*t*) <sup>≤</sup> <sup>Σ</sup>1(0) = *<sup>γ</sup>*−2*Q*−<sup>1</sup>

*Q*¯ 1, we have the following assumption to guarantee the boundedness of Σ<sup>1</sup> and *s*1,Σ,

<sup>1</sup> *, choose <sup>β</sup>*1,<sup>Δ</sup> <sup>≥</sup> <sup>0</sup> *such that A*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

<sup>1</sup>*L*1*C*1)� <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

<sup>1</sup> − 1)*C*1Φ1Σ<sup>1</sup> (11a)

<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*−2) is Hurwitz. By picking *<sup>γ</sup>* <sup>≥</sup> *<sup>ζ</sup>*−<sup>1</sup>

<sup>1</sup> − Π1*C*�

<sup>1</sup> (*ζ*<sup>2</sup>

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

1,213 *jw*ˇ 1,*<sup>j</sup>* (11c)

<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*−2)*C*1Π<sup>1</sup>

<sup>1</sup> ). Based on the structure of

<sup>1</sup>*L*1*C*<sup>1</sup> + *β*1,Δ/2*In*<sup>1</sup> *is Hurwitz. If the*

1*L*1*L*�

<sup>1</sup> <sup>+</sup> *<sup>γ</sup>*2Δ<sup>1</sup> (11b)

<sup>1</sup> *, and �*<sup>1</sup> *be given by either (10b) or (10b). Then, the matrix* Σ<sup>1</sup> *is upper and lower*

1,0 ;

<sup>1</sup>*L*1*C*<sup>1</sup> *is Hurwitz, then the desired disturbance attenuation level*

1−*ζ*<sup>2</sup> 1*L*1*L*�

1,212 0 ¯

*θ*) < *ρ*1,*o*}. Our control design will guarantee that the

and Φ1,0 respectively,

where *<sup>A</sup>*1, *<sup>f</sup>* :<sup>=</sup> *<sup>A</sup>*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

*<sup>Q</sup>*1,0 <sup>&</sup>gt; <sup>0</sup>*, <sup>γ</sup>* <sup>≥</sup> *<sup>ζ</sup>*−<sup>1</sup>

Π˙

[12].

*<sup>γ</sup>* <sup>≥</sup> *<sup>ζ</sup>*−<sup>1</sup>

*ζ*−<sup>1</sup>

of ˇ *ξ*1. Define

*matrix A*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

(*A*1−*ζ*<sup>2</sup>

the open set <sup>Θ</sup>1,*<sup>o</sup>* :<sup>=</sup> { ¯

<sup>Σ</sup>˙ <sup>1</sup> = (*�*<sup>1</sup> <sup>−</sup> <sup>1</sup>)Σ1Φ�

+*D*1*D*�

<sup>Φ</sup>˙ <sup>1</sup> = *<sup>A</sup>*1, *<sup>f</sup>* <sup>Φ</sup><sup>1</sup> + *<sup>y</sup>*1*A*¯

To avoid the calculation of Σ−<sup>1</sup>

<sup>1</sup> *. In case <sup>γ</sup>* <sup>=</sup> *<sup>ζ</sup>*−<sup>1</sup>

**Assumption 6.** *If the matrix A*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

*to the following algebraic Riccati equations:*

<sup>1</sup>*L*1*C*1)Π1+Π<sup>1</sup> (*A*1−*ζ*<sup>2</sup>

*<sup>ρ</sup>*<sup>1</sup> :<sup>=</sup> inf{*P*1(¯

*<sup>θ</sup>*<sup>1</sup> <sup>∈</sup> IR*σ*<sup>1</sup> <sup>|</sup> *<sup>P</sup>*1(¯

<sup>1</sup> = (*A*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

1*C*� <sup>1</sup> (*γ*2*ζ*<sup>2</sup>

<sup>1</sup>*L*1*C*<sup>1</sup> − Π1*C*�

*bounded as follows, whenever* Φ<sup>1</sup> *is continuous on* [0, *tf* ]*,*

*K*−<sup>1</sup>

<sup>1</sup>*L*1*C*1)Π<sup>1</sup> <sup>+</sup> <sup>Π</sup><sup>1</sup> (*A*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

1,211 + *y*2*A*¯ 1,212 +

<sup>1</sup>*C*<sup>1</sup> (*ζ*<sup>2</sup>

1,0 , Π1,0,

115

<sup>1</sup> , we have

<sup>1</sup> *.* �

<sup>1</sup>+*γ*2Δ<sup>1</sup> <sup>=</sup>**0***n*1×*n*<sup>1</sup> (12)

*θ*<sup>1</sup> = 0} (13)

#### **4.1. Estimation design**

In this subsection, we present the estimation design for the adaptive control problem formulated. First, we will derive the identifier of subsystem **S**1. In this step, the measurement waveform *y*1, *y*<sup>2</sup> and measured disturbance *w*ˇ <sup>1</sup> are assumed to be known. Then we can obtain the identifier of subsystem **S1** from a *game-theoretic* solution methodology – *cost-to-come* function analysis.

We first set function *<sup>l</sup>*<sup>1</sup> in the definition to be <sup>|</sup>*ξ*<sup>1</sup> <sup>−</sup> <sup>ˆ</sup> *ξ*1| 2 *Q*¯ 1 + 2(*ξ*<sup>1</sup> − *l*1,1)� *l*1,2 +ˇ *l*1, where ˆ *ξ*<sup>1</sup> = [ˆ *θ*� <sup>1</sup>, *x*ˆ� 1] � is the worst-case estimate for the expanded state *<sup>ξ</sup>*1, *<sup>Q</sup>*¯ <sup>1</sup>(*y*1[0,*τ*],*Yd*[0,*τ*], *<sup>w</sup>*<sup>ˇ</sup> <sup>1</sup>[0,*τ*]) is a matrix-valued weighting function, *l*1,1(*y*1[0,*τ*],*Yd*[0,*τ*], *w*ˇ <sup>1</sup>[0,*τ*]), *l*1,2(*y*1[0,*τ*],*Yd*[0,*τ*], *w*ˇ <sup>1</sup>[0,*τ*]) and ˇ *l*1(*y*1[0,*τ*],*Yd*[0,*τ*], *w*ˇ <sup>1</sup>[0,*τ*]) are three design functions to be introduced later, the cost function of subsystem **S1** is then of the a linear quadratic structure.

The robust adaptive problem for **S1** becomes an *H*<sup>∞</sup> control of affine quadratic problem, and admits a finite dimensional solution. By *cost-to-come* function analysis, we obtain the dynamics of worst-case covariance matrix Σ¯ 1, and state estimator ˇ *ξ*1, which are given by

$$\begin{aligned} \dot{\Sigma}\_{1} &= (\bar{A}\_{1} - \zeta\_{1}^{2} \boldsymbol{L}\_{1} \boldsymbol{\zeta}\_{1}) \boldsymbol{\Sigma}\_{1} + \bar{\boldsymbol{\Sigma}}\_{1} (\bar{A}\_{1} - \zeta\_{1}^{2} \boldsymbol{L}\_{1} \boldsymbol{\zeta}\_{1})^{\prime} - \bar{\boldsymbol{\Sigma}}\_{1} (\gamma^{2} \zeta\_{1}^{2} \boldsymbol{\zeta}\_{1}^{\prime} \boldsymbol{\zeta}\_{1} - \boldsymbol{\zeta}\_{1}^{\prime} \boldsymbol{\zeta}\_{1} - \boldsymbol{\zeta}\_{1}) \boldsymbol{\Sigma}\_{1} \\ &+ \gamma^{-2} \boldsymbol{D}\_{1} \boldsymbol{D}\_{1}^{\prime} - \gamma^{-2} \zeta\_{1}^{2} \boldsymbol{L}\_{1} \boldsymbol{L}\_{1}^{\prime}; \quad \boldsymbol{\Sigma}\_{1}(0) = \gamma^{-2} \boldsymbol{Q}\_{1,0}^{-1} \\ \dot{\xi}\_{1} &= (\bar{A}\_{1} + \boldsymbol{\Sigma}\_{1} (\bar{C}\_{1}^{\prime} \boldsymbol{\zeta}\_{1} + \bar{Q}\_{1})) \dot{\xi}\_{1} + \bar{\boldsymbol{B}}\_{1} y\_{2} + \zeta\_{1}^{2} (\gamma^{2} \boldsymbol{\Sigma}\_{1} \boldsymbol{\zeta}\_{1}^{\prime} + \boldsymbol{L}\_{1}) (y\_{1} - \boldsymbol{\zeta}\_{1} \boldsymbol{\xi}\_{1}) \end{aligned} \tag{7a}$$

$$-\Sigma\_1 \left( \tilde{\mathcal{L}}\_1' y\_d + \tilde{Q}\_1 \xi\_1 - l\_{1,2} \right) + \tilde{D}\_1 w\_1; \quad \tilde{\xi}\_1(0) = \left[ \tilde{\theta}\_{1,0}' \, \tilde{\mathcal{x}}\_{1,0}' \right]' \tag{7b}$$

where *<sup>L</sup>*¯ <sup>1</sup> is defined as *<sup>L</sup>*¯ <sup>1</sup> = **<sup>0</sup>**1×*σ*<sup>1</sup> *<sup>L</sup>*� 1 �

We partition Σ¯ <sup>1</sup> as the same structure as

$$
\Sigma\_1 = \begin{bmatrix}
\Sigma\_1 & \Sigma\_{1,12} \\
\Sigma\_{1,21} & \Sigma\_{1,22}
\end{bmatrix} = \begin{bmatrix}
\Sigma\_1 & \Sigma\_1 \Phi\_1' \\
\Phi\_1 \Sigma\_1 & \frac{1}{\gamma^2} \Pi\_1 + \Phi\_1 \Sigma\_1 \Phi\_1'
\end{bmatrix} \tag{8}
$$

where <sup>Φ</sup>1(*t*) :<sup>=</sup> <sup>Σ</sup>¯ 1,21(*t*)(Σ1(*t*))−<sup>1</sup> and <sup>Π</sup>1(*t*) :<sup>=</sup> *<sup>γ</sup>*<sup>2</sup> (Σ¯ 1,22(*t*) <sup>−</sup> <sup>Σ</sup>¯ 1,21(*t*)(Σ1(*t*))−1Σ¯ 1,12(*t*)), <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *tf* ]. Then the weighting matrix Σ¯ <sup>1</sup> is positive definite if and only if Σ<sup>1</sup> and Π<sup>1</sup> are positive definite. To guarantee the boundedness of Σ1, we choose weighing matrix *Q*¯ <sup>1</sup> as follows,

.

$$\boldsymbol{Q}\_{1} = \boldsymbol{\Sigma}\_{1}^{-1} \begin{bmatrix} \mathbf{0}\_{\boldsymbol{\sigma}\_{1} \times \boldsymbol{\sigma}\_{1}} & \mathbf{0}\_{\boldsymbol{\sigma}\_{1} \times \boldsymbol{\eta}\_{1}} \\ \mathbf{0}\_{\boldsymbol{\eta}\_{1} \times \boldsymbol{\sigma}\_{1}} & \boldsymbol{\Delta}\_{1}(t) \end{bmatrix} \boldsymbol{\Sigma}\_{1}^{-1} + \begin{bmatrix} \boldsymbol{\varepsilon}\_{1} \boldsymbol{\Phi}\_{1}^{\prime} \mathbf{C}\_{1}^{\prime} (\boldsymbol{\gamma}^{2} \boldsymbol{\zeta}\_{1}^{2} - 1) \mathbf{C}\_{1} \boldsymbol{\Phi}\_{1} \ \mathbf{0}\_{\boldsymbol{\sigma}\_{1} \times \boldsymbol{\eta}\_{1}} \\ \mathbf{0}\_{\boldsymbol{\eta}\_{1} \times \boldsymbol{\sigma}\_{1}} & \mathbf{0}\_{\boldsymbol{\eta}\_{1} \times \boldsymbol{\eta}\_{1}} \end{bmatrix} \tag{9}$$

where <sup>Δ</sup>1(*t*) = *<sup>γ</sup>*−2*β*1,ΔΠ1(*t*) + <sup>Δ</sup>1,1, with *<sup>β</sup>*1,<sup>Δ</sup> <sup>≥</sup> 0 being a constant and <sup>Δ</sup>1,1 being an *<sup>n</sup>*<sup>1</sup> <sup>×</sup> *<sup>n</sup>*1 dimensional positive-definite matrix, and *�*<sup>1</sup> is a scalar function defined by

$$\varepsilon\_1(t) := \text{Tr}(\Sigma\_1(t))^{-1} / \mathcal{K}\_{1,\varepsilon} \quad \forall t \in [0, t\_f] \tag{10a}$$

or

$$\epsilon\_1(t) := 1 \quad \forall t \in [0, t\_f] \tag{10b}$$

Then the dynamics of Σ1, Φ1, Π<sup>1</sup> are given as follows with initial conditions *γ*−2*Q*−<sup>1</sup> 1,0 , Π1,0, and Φ1,0 respectively,

$$\begin{split} \dot{\Sigma}\_{1} &= (\epsilon\_{1} - 1)\Sigma\_{1} \Phi\_{1}^{\prime} \mathbb{C}\_{1}^{\prime} (\gamma^{2} \zeta\_{1}^{2} - 1) \mathbb{C}\_{1} \Phi\_{1} \Sigma\_{1} \\ \dot{\Pi}\_{1} &= (A\_{1} - \zeta\_{1}^{2} L\_{1} \mathbb{C}\_{1}) \Pi\_{1} + \Pi\_{1} \left(A\_{1} - \zeta\_{1}^{2} L\_{1} \mathbb{C}\_{1} \right)^{\prime} - \zeta\_{1}^{2} L\_{1} L\_{1}^{\prime} - \Pi\_{1} \mathbb{C}\_{1}^{\prime} (\zeta\_{1}^{2} - \gamma^{-2}) \mathbb{C}\_{1} \Pi\_{1} \end{split} \tag{11a}$$

$$+D\_1D\_1' + \gamma^2 \Delta\_1 \tag{11b}$$

$$
\dot{\Phi}\_1 = A\_{1,f}\Phi\_1 + y\_1\bar{A}\_{1,211} + y\_2\bar{A}\_{1,212} + \sum\_{j=1}^{\dot{q}\_1} \bar{A}\_{1,213j} \check{w}\_{1,j} \tag{11c}
$$

where *<sup>A</sup>*1, *<sup>f</sup>* :<sup>=</sup> *<sup>A</sup>*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup> <sup>1</sup>*L*1*C*<sup>1</sup> − Π1*C*� <sup>1</sup>*C*<sup>1</sup> (*ζ*<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>γ</sup>*−2) is Hurwitz. By picking *<sup>γ</sup>* <sup>≥</sup> *<sup>ζ</sup>*−<sup>1</sup> <sup>1</sup> , we have the covariance matrix Σ<sup>1</sup> upper and lower bounded as summarized in the following Lemma [12].

**Lemma 1.** *Consider the dynamic equation (11a) for the covariance matrix* <sup>Σ</sup>1*. Let K*1,*<sup>c</sup>* <sup>≥</sup> *<sup>γ</sup>*2Tr(*Q*1,0)*, <sup>Q</sup>*1,0 <sup>&</sup>gt; <sup>0</sup>*, <sup>γ</sup>* <sup>≥</sup> *<sup>ζ</sup>*−<sup>1</sup> <sup>1</sup> *, and �*<sup>1</sup> *be given by either (10b) or (10b). Then, the matrix* Σ<sup>1</sup> *is upper and lower bounded as follows, whenever* Φ<sup>1</sup> *is continuous on* [0, *tf* ]*,*

$$\begin{aligned} \mathsf{K}\_{1,\boldsymbol{\varepsilon}}^{-1} \boldsymbol{I}\_{1,\sigma\_1} &\leq \Sigma\_1(t) \leq \Sigma\_1(0) = \gamma^{-2} \mathsf{Q}\_{1,0}^{-1}; \\ \gamma^2 \mathrm{Tr}(\boldsymbol{Q}\_{1,0}) &\leq \mathrm{Tr}(\Sigma\_1(t))^{-1} \leq \mathsf{K}\_{1,\boldsymbol{\varepsilon}}; \qquad \forall t \in [0, t\_f] \end{aligned}$$

To avoid the calculation of Σ−<sup>1</sup> <sup>1</sup> online, we define *<sup>s</sup>*1,<sup>Σ</sup> <sup>=</sup> Tr(Σ−<sup>1</sup> <sup>1</sup> ). Based on the structure of *Q*¯ 1, we have the following assumption to guarantee the boundedness of Σ<sup>1</sup> and *s*1,Σ,

**Assumption 6.** *If the matrix A*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup> <sup>1</sup>*L*1*C*<sup>1</sup> *is Hurwitz, then the desired disturbance attenuation level <sup>γ</sup>* <sup>≥</sup> *<sup>ζ</sup>*−<sup>1</sup> <sup>1</sup> *. In case <sup>γ</sup>* <sup>=</sup> *<sup>ζ</sup>*−<sup>1</sup> <sup>1</sup> *, choose <sup>β</sup>*1,<sup>Δ</sup> <sup>≥</sup> <sup>0</sup> *such that A*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup> <sup>1</sup>*L*1*C*<sup>1</sup> + *β*1,Δ/2*In*<sup>1</sup> *is Hurwitz. If the matrix A*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup> <sup>1</sup>*L*1*C*<sup>1</sup> *is not Hurwitz, then the desired disturbance attenuation level <sup>γ</sup>* <sup>&</sup>gt; *<sup>ζ</sup>*−<sup>1</sup> <sup>1</sup> *.* �

This assumption implies that the achievable disturbance attenuation level *γ* is no smaller than *ζ*−<sup>1</sup> <sup>1</sup> . Under this assumption, we initialize Π<sup>1</sup> as the unique positive definite solution of its Riccati Differential Equation (11b), which is summarized as the following assumption.

**Assumption 7.** *The initial weighting matrix* Π1,0 *is chosen as the unique positive definite solutions to the following algebraic Riccati equations:*

$$(A\_1 - \zeta\_1^2 L\_1 \mathbb{C}\_1) \Pi\_1 + \Pi\_1 \left( A\_1 - \zeta\_1^2 L\_1 \mathbb{C}\_1 \right)' - \Pi\_1 \mathbb{C}\_1' \zeta\_1^2 \mathbb{C}\_1 \Pi\_1 + D\_1 D\_1' - \zeta\_1^2 L\_1 L\_1' + \gamma^2 \Delta\_1 = \mathbf{0}\_{\mathbb{N} \times \mathbb{N}} \tag{12}$$

To guarantee the estimates parameter to be bounded and the estimate of high frequency gain to be bounded away from zero, projection function scheme is applied to modify the dynamics of ˇ *ξ*1.

Define

8 Will-be-set-by-IN-TECH

In this section, we present the adaptive control design, which involves estimation design and

In this subsection, we present the estimation design for the adaptive control problem formulated. First, we will derive the identifier of subsystem **S**1. In this step, the measurement waveform *y*1, *y*<sup>2</sup> and measured disturbance *w*ˇ <sup>1</sup> are assumed to be known. Then we can obtain the identifier of subsystem **S1** from a *game-theoretic* solution methodology – *cost-to-come*

� is the worst-case estimate for the expanded state *<sup>ξ</sup>*1, *<sup>Q</sup>*¯ <sup>1</sup>(*y*1[0,*τ*],*Yd*[0,*τ*]

*l*1(*y*1[0,*τ*],*Yd*[0,*τ*], *w*ˇ <sup>1</sup>[0,*τ*]) are three design functions to be introduced later, the cost function of

The robust adaptive problem for **S1** becomes an *H*<sup>∞</sup> control of affine quadratic problem, and admits a finite dimensional solution. By *cost-to-come* function analysis, we obtain the dynamics

<sup>1</sup>*L*¯ <sup>1</sup>*C*¯

<sup>1</sup>*y*<sup>2</sup> + *ζ*<sup>2</sup>

*D*ˇ <sup>1</sup>*w*ˇ 1; ˇ

Φ1Σ<sup>1</sup>

where <sup>Φ</sup>1(*t*) :<sup>=</sup> <sup>Σ</sup>¯ 1,21(*t*)(Σ1(*t*))−<sup>1</sup> and <sup>Π</sup>1(*t*) :<sup>=</sup> *<sup>γ</sup>*<sup>2</sup> (Σ¯ 1,22(*t*) <sup>−</sup> <sup>Σ</sup>¯ 1,21(*t*)(Σ1(*t*))−1Σ¯ 1,12(*t*)), <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *tf* ]. Then the weighting matrix Σ¯ <sup>1</sup> is positive definite if and only if Σ<sup>1</sup> and Π<sup>1</sup> are positive definite. To guarantee the boundedness of Σ1, we choose weighing matrix *Q*¯ <sup>1</sup> as follows,

where <sup>Δ</sup>1(*t*) = *<sup>γ</sup>*−2*β*1,ΔΠ1(*t*) + <sup>Δ</sup>1,1, with *<sup>β</sup>*1,<sup>Δ</sup> <sup>≥</sup> 0 being a constant and <sup>Δ</sup>1,1 being an *<sup>n</sup>*<sup>1</sup> <sup>×</sup> *<sup>n</sup>*1-

<sup>1</sup>; <sup>Σ</sup>¯ <sup>1</sup>(0) = *<sup>γ</sup>*−2*Q*¯ <sup>−</sup><sup>1</sup>

*ξ*1| 2 *Q*¯ 1

<sup>1</sup>)� <sup>−</sup> <sup>Σ</sup>¯ <sup>1</sup> (*γ*2*ζ*<sup>2</sup>

<sup>1</sup> (*γ*2Σ¯ <sup>1</sup>*C*¯�

 ˇ *θ*� 1,0 *x*ˇ� 1,0 �

1

*<sup>γ</sup>*<sup>2</sup> <sup>Π</sup><sup>1</sup> + <sup>Φ</sup>1Σ1Φ�

*�*1(*t*) :<sup>=</sup> Tr(Σ1(*t*))<sup>−</sup>1/*K*1,*<sup>c</sup>* <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *tf* ] (10a)

*�*1(*t*) := 1 ∀*t* ∈ [0, *tf* ] (10b)

*ξ*1(0) =

<sup>Σ</sup><sup>1</sup> <sup>Σ</sup>1Φ�

1

+ 2(*ξ*<sup>1</sup> − *l*1,1)�

, *w*ˇ <sup>1</sup>[0,*τ*]), *l*1,2(*y*1[0,*τ*],*Yd*[0,*τ*]

*ξ*1, which are given by

<sup>1</sup> <sup>+</sup> *<sup>L</sup>*¯ <sup>1</sup>)(*y*<sup>1</sup> <sup>−</sup> *<sup>C</sup>*¯

1

<sup>1</sup> − 1)*C*1Φ<sup>1</sup> **0***σ*1×*n*<sup>1</sup> **0***n*1×*σ*<sup>1</sup> **0***n*1×*n*<sup>1</sup>

1*C*¯� 1*C*¯ <sup>1</sup> <sup>−</sup> *<sup>C</sup>*¯� 1*C*¯ *l*1,2 +ˇ

<sup>1</sup> <sup>−</sup> *<sup>Q</sup>*¯ <sup>1</sup>)Σ¯ <sup>1</sup>

1,0 (7a)

1 ˇ *ξ*1) *l*1, where

, *w*ˇ <sup>1</sup>[0,*τ*]) and

, *w*ˇ <sup>1</sup>[0,*τ*])

(7b)

(8)

(9)

**4. Adaptive control design**

**4.1. Estimation design**

function analysis.

¯˙ Σ<sup>1</sup> = (*A*¯

˙ ˇ *ξ*<sup>1</sup> = (*A*¯

ˆ *ξ*<sup>1</sup> = [ˆ *θ*� <sup>1</sup>, *x*ˆ� 1]

ˇ

control design. First, we discuss estimation design.

We first set function *<sup>l</sup>*<sup>1</sup> in the definition to be <sup>|</sup>*ξ*<sup>1</sup> <sup>−</sup> <sup>ˆ</sup>

is a matrix-valued weighting function, *l*1,1(*y*1[0,*τ*],*Yd*[0,*τ*]

subsystem **S1** is then of the a linear quadratic structure.

of worst-case covariance matrix Σ¯ 1, and state estimator ˇ

<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*−2*ζ*<sup>2</sup>

1*C*¯

<sup>1</sup>*yd* <sup>+</sup> *<sup>Q</sup>*¯ <sup>1</sup> <sup>ˆ</sup>

<sup>1</sup>)Σ¯ <sup>1</sup> <sup>+</sup> <sup>Σ</sup>¯ <sup>1</sup> (*A*¯ <sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

<sup>1</sup>*L*¯ <sup>1</sup>*L*¯ �

*<sup>ξ</sup>*<sup>1</sup> <sup>−</sup> *<sup>l</sup>*1,2) + ¯

*ξ*<sup>1</sup> + *B*¯

 =

 Σ¯ <sup>−</sup><sup>1</sup> <sup>1</sup> + *�*1Φ� 1*C*� 1(*γ*2*ζ*<sup>2</sup>

dimensional positive-definite matrix, and *�*<sup>1</sup> is a scalar function defined by

<sup>1</sup> + *Q*¯ <sup>1</sup>))ˇ

**<sup>0</sup>**1×*σ*<sup>1</sup> *<sup>L</sup>*� 1 � .

 <sup>Σ</sup><sup>1</sup> <sup>Σ</sup>¯ 1,12 Σ¯ 1,21 Σ¯ 1,22

**0***σ*1×*σ*<sup>1</sup> **0***σ*1×*n*<sup>1</sup> **0***n*1×*σ*<sup>1</sup> Δ1(*t*)

or

<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup> <sup>1</sup>*L*¯ <sup>1</sup>*C*¯

+*γ*−2*D*¯ <sup>1</sup>*D*¯ �

<sup>−</sup>Σ¯ <sup>1</sup> (*C*¯�

*Q*¯ <sup>1</sup> = Σ¯ <sup>−</sup><sup>1</sup> 1 

We partition Σ¯ <sup>1</sup> as the same structure as

Σ¯ <sup>1</sup> =

where *<sup>L</sup>*¯ <sup>1</sup> is defined as *<sup>L</sup>*¯ <sup>1</sup> =

<sup>1</sup> + Σ¯ <sup>1</sup> (*C*¯�

$$\rho\_1 := \inf \{ P\_1(\bar{\theta}\_1) \mid \bar{\theta}\_1 \in \mathbb{R}^{\sigma\_1}, b\_{1,p0} + \bar{A}\_{1,212} \theta\_1 = 0 \} \tag{13}$$

By Assumption 4 and Lemma 2 in [19] we have 1 < *ρ*<sup>1</sup> ≤ ∞. Fix any *ρ*1,*<sup>o</sup>* ∈ (1, *ρ*1), and define the open set <sup>Θ</sup>1,*<sup>o</sup>* :<sup>=</sup> { ¯ *<sup>θ</sup>*<sup>1</sup> <sup>∈</sup> IR*σ*<sup>1</sup> <sup>|</sup> *<sup>P</sup>*1(¯ *θ*) < *ρ*1,*o*}. Our control design will guarantee that the estimate ˇ *<sup>θ</sup>*<sup>1</sup> lies in <sup>Θ</sup>1,*o*, which immediately implies <sup>|</sup>*b*1,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*¯ 1,212 0 ˇ *θ*1| > *c*1,0 > 0, for some *c*1,0 > 0. Moreover, the convexity of *P*<sup>1</sup> implies the following inequality

$$\frac{\partial P\_1}{\partial \theta\_1}(\check{\theta}\_1) \left(\theta\_1 - \check{\theta}\_1\right) < 0 \quad \forall \check{\theta}\_1 \in \mathbb{R}^{\sigma\_1} \backslash \Theta\_1$$

We set *l*1,1 = ˇ *<sup>ξ</sup>*1, and *<sup>l</sup>*1,2 = � <sup>−</sup>(*P*1,*r*(<sup>ˇ</sup> *<sup>θ</sup>*1))� **<sup>0</sup>**1×*n*<sup>1</sup> �� , where

$$P\_{1,r}(\check{\theta}\_1) := \begin{cases} \frac{\frac{1}{t^{1-P\_1(\theta\_1)}} \left(\frac{\partial P\_1}{\partial \theta\_1}(\check{\theta}\_1)\right)'}{\left(\rho\_{1,o} - P\_1(\check{\theta}\_1)\right)^3} \, \forall \theta\_1 \in \Theta\_{1,o} \,\backslash \Theta\_1\\ \mathbf{0}\_{\sigma\_1 \times 1} \qquad \forall \theta\_1 \in \Theta\_1 \end{cases}$$

$$\coloneqq p\_{1,r}(\check{\theta}\_1) \left(\frac{\partial P\_1}{\partial \theta\_1}(\check{\theta}\_1)\right)' \tag{14}$$

Next, we will derive the estimator for subsystem **S2**. In this step, the measurements waveform *ωm* is assumed to be known. Since the control input, *u*, is a causal function of *ωm*, then it is known. Again, we will apply the *cost-to-come function* methodology to derive the estimator.

of subsystem **S2** is then of a linear quadratic structure. By *cost-to-come* function analysis, we

then the weighting matrix Σ¯ <sup>2</sup> is positive definite if and only if Σ<sup>2</sup> and Π<sup>2</sup> are positive definite.

� + *�*2Φ� 2*C*� 2*γ*2*ζ*<sup>2</sup>

where <sup>Δ</sup>2(*t*) = *<sup>γ</sup>*−2*β*2,ΔΠ2(*t*) + <sup>Δ</sup>2,1, with *<sup>β</sup>*2,<sup>Δ</sup> <sup>≥</sup> 0 being a constant and <sup>Δ</sup>2,1 being an *n*<sup>2</sup> × *n*2- dimensional positive-definite matrix, and *�*<sup>2</sup> is a scalar function defined by *�*<sup>2</sup> =

<sup>2</sup> ) or *�*<sup>2</sup> <sup>=</sup> 1. *<sup>K</sup>*2,*<sup>c</sup>* <sup>≥</sup> *<sup>γ</sup>*2Tr(*Q*2,0) is a design constant, *<sup>Q</sup>*2,0 is an *<sup>σ</sup>*<sup>2</sup> <sup>×</sup> *<sup>σ</sup>*2-dimensional

*Q*2,0

*<sup>γ</sup>*2Tr(*Q*2,0) <sup>≤</sup> Tr(Σ2(*t*))−<sup>1</sup> <sup>≤</sup> *<sup>K</sup>*2,*c*, whenever it exists on [0, *tf* ] and <sup>Φ</sup><sup>2</sup> is continuous on [0, *tf* ].

To guarantee the estimates parameter to be bounded and the estimate of high frequency gain to be bounded away from zero without persistently exciting signals, we introduce the

*c*2,0 > 0, for some *c*2,0 > 0. Moreover, the convexity of *P*<sup>2</sup> implies the following inequality:

2,212 0 ¯

*<sup>θ</sup>*<sup>2</sup> <sup>|</sup> *<sup>P</sup>*2(¯

*<sup>θ</sup>*<sup>2</sup> lies in <sup>Θ</sup>2,*o*, which immediately implies <sup>|</sup>*b*2,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*¯

*<sup>θ</sup>*<sup>2</sup> <sup>∈</sup> IR*σ*<sup>2</sup> \Θ2. To incorporate the modifier to the estimates dynamics,

<sup>2</sup> online, we define *<sup>s</sup>*2,<sup>Σ</sup> <sup>=</sup> Tr(Σ−<sup>1</sup>

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2Δ2,1; <sup>Π</sup>2(0) = <sup>Π</sup>2,0 (17b)

<sup>+</sup> <sup>2</sup>(*ξ*<sup>2</sup> <sup>−</sup> <sup>ˇ</sup>

*ξ*2)�

*l*2,2 + ˇ

*l*<sup>2</sup> are two design functions to be introduced later, the cost function

*l*2, where ˆ

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

*ξ*<sup>2</sup> is the estimate of *ξ*2, *Q*¯ <sup>2</sup> is a matrix-valued

<sup>2</sup> and <sup>Π</sup><sup>2</sup> :<sup>=</sup> *<sup>γ</sup>*2(Σ¯ 2,22 <sup>−</sup> <sup>Σ</sup>¯ 2,21Σ−<sup>1</sup>

<sup>2</sup>*C*2Φ<sup>2</sup> **0***σ*2×*n*<sup>2</sup> **0***n*2×*σ*<sup>2</sup> **0***n*2×*n*<sup>2</sup>

<sup>2</sup>*L*2*C*<sup>2</sup> + *β*2,Δ/2*In*<sup>2</sup> )� − Π2*C*�

*A*¯ 2,213*jw*ˇ 2,*<sup>j</sup>* + *y*´2*A*¯ 2,214; Φ2(0) = Φ2,0 (17c)

<sup>2</sup> is Hurwitz. By Lemma [12], we have the covariance

<sup>2</sup> ).

2,*<sup>c</sup> <sup>I</sup>σ*<sup>2</sup> <sup>≤</sup> <sup>Σ</sup>2(*t*) <sup>≤</sup> <sup>Σ</sup>2(0) = *<sup>γ</sup>*−2*Q*−<sup>1</sup>

*θ*<sup>2</sup> = 0}, we have 1 < *ρ*<sup>2</sup> ≤ ∞. Fix any

*θ*) < *ρ*2,*o*}. Our control design will

*ξ*<sup>2</sup> = [ ˆ *θ*� <sup>2</sup>, *x*ˆ� 2] � is the

2*ζ*2

*ξ*2. We partition

<sup>2</sup> <sup>Σ</sup>¯ 2,12),

117

(16)

(17a)

2,0 ,

2

<sup>2</sup>*C*2Π<sup>2</sup> + *D*2*D*�

2,212 0 ˇ

*θ*2| >

*ξ*2| 2 *Q*¯ 2

obtain the dynamics of worst-case covariance matrix Σ¯ 2, and state estimator ˇ

and introduce Φ<sup>2</sup> := Σ¯ 2,21Σ−<sup>1</sup>

To guarantee the boundedness of Σ2, we choose weighing matrix *Q*¯ <sup>2</sup> as follows,

positive-definite matrix. Then the dynamics of Σ2, Φ2, Π<sup>2</sup> are given as follows,

<sup>2</sup>*C*2Φ2Σ2; <sup>Σ</sup>2(0) = *<sup>γ</sup>*−<sup>2</sup>

*q*ˇ2 ∑ *j*=1

2*C*2*ζ*<sup>2</sup>

*<sup>θ</sup>*<sup>2</sup> <sup>∈</sup> IR*σ*<sup>2</sup> , *<sup>b</sup>*2,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*¯

<sup>2</sup>*L*2*C*<sup>2</sup> <sup>+</sup> *<sup>β</sup>*2,Δ/2*In*<sup>2</sup> )Π<sup>2</sup> <sup>+</sup> <sup>Π</sup><sup>2</sup> (*A*<sup>2</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

<sup>2</sup>*L*2*C*<sup>2</sup> − Π2*C*�

matrix Σ<sup>2</sup> upper and lower bounded as follows, *K*−<sup>1</sup>

following soft projection design on the parameter estimate.

*<sup>θ</sup>*2) <sup>|</sup> ¯

*<sup>ρ</sup>*2,*<sup>o</sup>* <sup>∈</sup> (1, *<sup>ρ</sup>*2), we define the open set <sup>Θ</sup>2,*<sup>o</sup>* :<sup>=</sup> { ¯

We briefly summarize the estimation design for **S**<sup>2</sup> as follows.

Set function *<sup>l</sup>*<sup>2</sup> in definition to be <sup>|</sup>*ξ*<sup>2</sup> <sup>−</sup> <sup>ˆ</sup>

weighting function, *l*2,2 and ˇ

*Q*¯ <sup>2</sup> = −Φ� 2 *In*2

<sup>Σ</sup>˙ <sup>2</sup> = (*�*<sup>2</sup> <sup>−</sup> <sup>1</sup>)Σ2Φ�

<sup>2</sup> = (*A*<sup>2</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

<sup>−</sup>*ζ*<sup>2</sup> 2*L*2*L*�

where *<sup>A</sup>*2, *<sup>f</sup>* :<sup>=</sup> *<sup>A</sup>*<sup>2</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

Define *<sup>ρ</sup>*<sup>2</sup> :<sup>=</sup> inf{*P*2(¯

*<sup>θ</sup>*2) (*θ*<sup>2</sup> <sup>−</sup> <sup>ˇ</sup>

guarantee that the estimate ˇ

*<sup>θ</sup>*2) <sup>&</sup>lt; <sup>0</sup> <sup>∀</sup> <sup>ˇ</sup>

To avoid the calculation of Σ−<sup>1</sup>

 <sup>Σ</sup><sup>2</sup> <sup>Σ</sup>¯ 2,12 <sup>Σ</sup>¯ 2,21 <sup>Σ</sup>¯ 2,22

Σ¯ <sup>2</sup> as Σ¯ <sup>2</sup> =

*K*−<sup>1</sup> 2,*<sup>c</sup>* Tr(Σ−<sup>1</sup>

Π˙

*∂P*<sup>2</sup> *∂θ*<sup>2</sup> (ˇ

worst-case estimate for the expanded state *ξ*2, ˇ

2*C*� 2*γ*2*ζ*<sup>2</sup>

Φ˙ <sup>2</sup> = *A*2, *<sup>f</sup>* Φ<sup>2</sup> + *y*2*A*¯ 2,211 + *uA*¯ 2,212 +

*γ*4Π−<sup>1</sup>

<sup>2</sup> <sup>Δ</sup>2Π−<sup>1</sup> 2 −Φ� 2 *In*2

then, we obtain

$$\dot{\tilde{\xi}}\_{1}^{\tilde{\xi}} = -\boldsymbol{\Sigma}\_{1} \left[ (P\_{1,r}(\check{\theta}\_{1}))' \, \mathbf{0}\_{1 \times n\_{1}} \right]' + \bar{A}\_{1}\check{\xi}\_{1} + \bar{\Sigma}\_{1}\mathsf{C}\_{1}'(y\_{d} - \mathsf{C}\_{1}\check{\xi}\_{1}) - \bar{\Sigma}\_{1}\bar{Q}\_{1}(\mathsf{P}\_{1}, s\_{1,\Sigma})\xi\_{1,\varepsilon} + \bar{B}\_{1}y\_{2}$$
 
$$+ \zeta\_{1}^{2} (\gamma^{2} \mathsf{E}\_{1} \mathsf{C}\_{1}' + \bar{L}\_{1})(y\_{1} - \mathsf{C}\_{1}\check{\xi}\_{1}) + \bar{D}\_{1}\vartheta\_{1}; \quad \xi\_{1}(0) = \begin{bmatrix} \theta\_{1,0}' \ \mathsf{x}\_{1,0}' \end{bmatrix}' \tag{15}$$

where *ξ*1,*<sup>c</sup>* = ˆ *<sup>ξ</sup>*<sup>1</sup> <sup>−</sup> <sup>ˇ</sup> *ξ*1.

We summarize the equations for subsystem **S**<sup>1</sup> as follows,

**<sup>0</sup>** = (*A*1−*ζ*<sup>2</sup> <sup>1</sup>*L*1*C*1)Π1+Π<sup>1</sup> (*A*1−*ζ*<sup>2</sup> 1*L*1*C*1)� −Π1*C*� 1(*ζ*<sup>2</sup> 1−*γ*−2)*C*1Π1+*D*1*D*� 1−*ζ*<sup>2</sup> 1*L*1*L*� <sup>1</sup>+*γ*2Δ<sup>1</sup> <sup>Σ</sup>˙ <sup>1</sup> <sup>=</sup> <sup>−</sup>(<sup>1</sup> <sup>−</sup> *�*1)Σ1Φ� 1*C*� <sup>1</sup> (*γ*2*ζ*<sup>2</sup> <sup>1</sup> − 1)*C*1Φ1Σ<sup>1</sup> *s*˙1,<sup>Σ</sup> = (*γ*2*ζ*<sup>2</sup> <sup>1</sup> − 1) (1 − *�*1)*C*1Φ1Φ� 1*C*� 1 *�*<sup>1</sup> = *K*−<sup>1</sup> 1,*<sup>c</sup> s*1,<sup>Σ</sup> or 1 *<sup>A</sup>*1, *<sup>f</sup>* <sup>=</sup> *<sup>A</sup>*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup> <sup>1</sup>*L*1*C*<sup>1</sup> − Π1*C*� <sup>1</sup>*C*<sup>1</sup> (*ζ*<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>γ</sup>*−2) Φ˙ <sup>1</sup> = *A*1, *<sup>f</sup>* Φ<sup>1</sup> + *y*1*A*¯ 1,211 + *y*2*A*¯ 1,212 + *q*ˇ1 ∑ *j*=1 *A*¯ 1,213 *jw*ˇ 1,*<sup>j</sup>* ˙ ˇ *<sup>θ</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>Σ1*P*1,*r*(<sup>ˇ</sup> *θ*1) − Σ1Φ� 1*C*� <sup>1</sup> (*yd* <sup>−</sup> *<sup>C</sup>*1*x*ˇ1) <sup>−</sup>� Σ<sup>1</sup> Σ1Φ� 1 � *Q*¯ <sup>1</sup>*ξ*1,*<sup>c</sup>* + *γ*2*ζ*<sup>2</sup> 1Σ1Φ� 1*C*� <sup>1</sup> (*y*<sup>1</sup> − *C*1*x*ˇ1) ˙ *<sup>x</sup>*ˇ1 <sup>=</sup> <sup>−</sup>Φ1Σ1*P*1,*r*(<sup>ˇ</sup> *<sup>θ</sup>*1) + *<sup>A</sup>*1*x*ˇ1 <sup>−</sup> (*γ*−2Π<sup>1</sup> <sup>+</sup> <sup>Φ</sup>1Σ1Φ� 1)*C*� <sup>1</sup> (*yd* <sup>−</sup> *<sup>C</sup>*1*x*ˇ1) + *<sup>B</sup>*1*y*<sup>2</sup> <sup>+</sup> *<sup>D</sup>*<sup>ˇ</sup> <sup>1</sup>*w*<sup>ˇ</sup> <sup>1</sup> − � Φ1Σ<sup>1</sup> *γ*−2Π<sup>1</sup> + Φ1Σ1Φ� 1 � *Q*¯ <sup>1</sup>*ξ*1,*<sup>c</sup>* + (*y*1*A*¯ 1,211 + *y*2*A*¯ 1,212 + *q*ˇ1 ∑ *j*=1 *w*ˇ 1,*jA*¯ 1,213*j*)ˇ *θ*1 +*ζ*<sup>2</sup> <sup>1</sup> (Π1*C*� <sup>1</sup> <sup>+</sup> *<sup>γ</sup>*2Φ1Σ1Φ� 1*C*� <sup>1</sup> + *L*1)(*y*<sup>1</sup> − *C*1*x*ˇ1)

This completes the estimation design of **S1**.

Next, we will derive the estimator for subsystem **S2**. In this step, the measurements waveform *ωm* is assumed to be known. Since the control input, *u*, is a causal function of *ωm*, then it is known. Again, we will apply the *cost-to-come function* methodology to derive the estimator. We briefly summarize the estimation design for **S**<sup>2</sup> as follows.

10 Will-be-set-by-IN-TECH

*<sup>θ</sup>*1) <sup>&</sup>lt; <sup>0</sup> <sup>∀</sup> <sup>ˇ</sup>

�� , where

*θ*1))

**<sup>0</sup>***σ*1×<sup>1</sup> ∀*θ*<sup>1</sup> ∈ <sup>Θ</sup><sup>1</sup>

<sup>1</sup>(*yd*−*C*¯

*D*ˇ <sup>1</sup>*w*ˇ 1; ˇ

−Π1*C*�

1,213 *jw*ˇ 1,*<sup>j</sup>*

Σ<sup>1</sup> Σ1Φ� 1 �

1)*C*�

*Q*¯ <sup>1</sup>*ξ*1,*<sup>c</sup>* + (*y*1*A*¯ 1,211 + *y*2*A*¯ 1,212 +

1(*ζ*<sup>2</sup>

1 ˇ

*ξ*1(0) =

� ˇ *θ*� 1,0 *x*ˇ� 1,0 ��

1−*γ*−2)*C*1Π1+*D*1*D*�

*Q*¯ <sup>1</sup>*ξ*1,*<sup>c</sup>* + *γ*2*ζ*<sup>2</sup>

1Σ1Φ� 1*C*�

<sup>1</sup> (*yd* <sup>−</sup> *<sup>C</sup>*1*x*ˇ1) + *<sup>B</sup>*1*y*<sup>2</sup> <sup>+</sup> *<sup>D</sup>*<sup>ˇ</sup> <sup>1</sup>*w*<sup>ˇ</sup> <sup>1</sup>

*q*ˇ1 ∑ *j*=1

*w*ˇ 1,*jA*¯

*<sup>ξ</sup>*1)−Σ¯ <sup>1</sup>*Q*¯ <sup>1</sup>(Φ1,*s*1,Σ)*ξ*1,*c*+*B*¯1*y*<sup>2</sup>

1−*ζ*<sup>2</sup> 1*L*1*L*�

1,212 0 ˇ

*<sup>θ</sup>*<sup>1</sup> <sup>∈</sup> IR*σ*<sup>1</sup> \Θ<sup>1</sup>

<sup>3</sup> ∀*θ*<sup>1</sup> ∈ Θ1,*o*\Θ<sup>1</sup>

*θ*1| > *c*1,0 > 0, for some

(14)

(15)

<sup>1</sup>+*γ*2Δ<sup>1</sup>

<sup>1</sup> (*y*<sup>1</sup> − *C*1*x*ˇ1)

1,213*j*)ˇ *θ*1

*<sup>θ</sup>*<sup>1</sup> lies in <sup>Θ</sup>1,*o*, which immediately implies <sup>|</sup>*b*1,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*¯

*c*1,0 > 0. Moreover, the convexity of *P*<sup>1</sup> implies the following inequality

*<sup>θ</sup>*1) (*θ*<sup>1</sup> <sup>−</sup> <sup>ˇ</sup>

⎧ ⎪⎨ *e* 1 <sup>1</sup>−*P*1(<sup>ˇ</sup> *θ*1) � *<sup>∂</sup>P*<sup>1</sup> *∂θ*<sup>1</sup> (<sup>ˇ</sup> *θ*1) ��

⎪⎩

:= *p*1,*r*(ˇ

�� +*A*¯ <sup>1</sup> ˇ

*<sup>θ</sup>*1))� **<sup>0</sup>**1×*n*<sup>1</sup>

*θ*1)

1 ˇ *<sup>ξ</sup>*1) + ¯

1*L*1*C*1)�

<sup>1</sup> − 1)*C*1Φ1Σ<sup>1</sup>

<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*−2)

*q*ˇ1 ∑ *j*=1 *A*¯

1,212 +

<sup>1</sup> (*yd* <sup>−</sup> *<sup>C</sup>*1*x*ˇ1) <sup>−</sup>�

*<sup>θ</sup>*1) + *<sup>A</sup>*1*x*ˇ1 <sup>−</sup> (*γ*−2Π<sup>1</sup> <sup>+</sup> <sup>Φ</sup>1Σ1Φ�

1 �

<sup>1</sup> + *L*1)(*y*<sup>1</sup> − *C*1*x*ˇ1)

1*C*�

1*C*� 1

<sup>1</sup>*C*<sup>1</sup> (*ζ*<sup>2</sup>

1*C*�

(*ρ*1,*<sup>o</sup>*−*P*1(<sup>ˇ</sup>

� *∂P*<sup>1</sup> *∂θ*<sup>1</sup> (ˇ *θ*1) ��

*ξ*1+Σ¯ <sup>1</sup>*C*¯�

*∂P*<sup>1</sup> *∂θ*<sup>1</sup> (ˇ

*P*1,*r*(ˇ

*<sup>θ</sup>*1))� **<sup>0</sup>**1×*n*<sup>1</sup>

<sup>1</sup> <sup>+</sup> *<sup>L</sup>*¯ <sup>1</sup>)(*y*<sup>1</sup> <sup>−</sup> *<sup>C</sup>*¯

We summarize the equations for subsystem **S**<sup>1</sup> as follows,

<sup>1</sup>*L*1*C*1)Π1+Π<sup>1</sup> (*A*1−*ζ*<sup>2</sup>

1*C*� <sup>1</sup> (*γ*2*ζ*<sup>2</sup>

<sup>1</sup> − 1) (1 − *�*1)*C*1Φ1Φ�

<sup>1</sup>*L*1*C*<sup>1</sup> − Π1*C*�

*θ*1) − Σ1Φ�

Φ1Σ<sup>1</sup> *γ*−2Π<sup>1</sup> + Φ1Σ1Φ�

<sup>1</sup> <sup>+</sup> *<sup>γ</sup>*2Φ1Σ1Φ�

<sup>−</sup>(*P*1,*r*(<sup>ˇ</sup>

*θ*1) :=

*<sup>ξ</sup>*1, and *<sup>l</sup>*1,2 = �

estimate ˇ

We set *l*1,1 = ˇ

then, we obtain ˙ ˇ *<sup>ξ</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>Σ¯ <sup>1</sup>

where *ξ*1,*<sup>c</sup>* = ˆ

**<sup>0</sup>** = (*A*1−*ζ*<sup>2</sup>

*s*˙1,<sup>Σ</sup> = (*γ*2*ζ*<sup>2</sup>

*�*<sup>1</sup> = *K*−<sup>1</sup>

*<sup>A</sup>*1, *<sup>f</sup>* <sup>=</sup> *<sup>A</sup>*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

*<sup>θ</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>Σ1*P*1,*r*(<sup>ˇ</sup>

*<sup>x</sup>*ˇ1 <sup>=</sup> <sup>−</sup>Φ1Σ1*P*1,*r*(<sup>ˇ</sup>

− �

+*ζ*<sup>2</sup>

<sup>1</sup> (Π1*C*�

This completes the estimation design of **S1**.

˙ ˇ

˙

<sup>Σ</sup>˙ <sup>1</sup> <sup>=</sup> <sup>−</sup>(<sup>1</sup> <sup>−</sup> *�*1)Σ1Φ�

1,*<sup>c</sup> s*1,<sup>Σ</sup> or 1

Φ˙ <sup>1</sup> = *A*1, *<sup>f</sup>* Φ<sup>1</sup> + *y*1*A*¯ 1,211 + *y*2*A*¯

� (*P*1,*r*(ˇ

*<sup>ξ</sup>*<sup>1</sup> <sup>−</sup> <sup>ˇ</sup> *ξ*1.

<sup>1</sup> (*γ*2Σ¯ <sup>1</sup>*C*¯�

+*ζ*<sup>2</sup>

Set function *<sup>l</sup>*<sup>2</sup> in definition to be <sup>|</sup>*ξ*<sup>2</sup> <sup>−</sup> <sup>ˆ</sup> *ξ*2| 2 *Q*¯ 2 <sup>+</sup> <sup>2</sup>(*ξ*<sup>2</sup> <sup>−</sup> <sup>ˇ</sup> *ξ*2)� *l*2,2 + ˇ *l*2, where ˆ *ξ*<sup>2</sup> = [ ˆ *θ*� <sup>2</sup>, *x*ˆ� 2] � is the worst-case estimate for the expanded state *ξ*2, ˇ *ξ*<sup>2</sup> is the estimate of *ξ*2, *Q*¯ <sup>2</sup> is a matrix-valued weighting function, *l*2,2 and ˇ *l*<sup>2</sup> are two design functions to be introduced later, the cost function of subsystem **S2** is then of a linear quadratic structure. By *cost-to-come* function analysis, we obtain the dynamics of worst-case covariance matrix Σ¯ 2, and state estimator ˇ *ξ*2. We partition Σ¯ <sup>2</sup> as Σ¯ <sup>2</sup> = <sup>Σ</sup><sup>2</sup> <sup>Σ</sup>¯ 2,12 <sup>Σ</sup>¯ 2,21 <sup>Σ</sup>¯ 2,22 and introduce Φ<sup>2</sup> := Σ¯ 2,21Σ−<sup>1</sup> <sup>2</sup> and <sup>Π</sup><sup>2</sup> :<sup>=</sup> *<sup>γ</sup>*2(Σ¯ 2,22 <sup>−</sup> <sup>Σ</sup>¯ 2,21Σ−<sup>1</sup> <sup>2</sup> <sup>Σ</sup>¯ 2,12), then the weighting matrix Σ¯ <sup>2</sup> is positive definite if and only if Σ<sup>2</sup> and Π<sup>2</sup> are positive definite. To guarantee the boundedness of Σ2, we choose weighing matrix *Q*¯ <sup>2</sup> as follows,

$$\mathbf{Q}\_2 = \begin{bmatrix} -\Phi\_2'\\ I\_{\eta\_2} \end{bmatrix} \gamma^4 \Pi\_2^{-1} \Lambda\_2 \Pi\_2^{-1} \begin{bmatrix} -\Phi\_2'\\ I\_{\eta\_2} \end{bmatrix}' + \begin{bmatrix} \varepsilon\_2 \Phi\_2' \mathbf{C}\_2' \gamma^2 \mathbf{C}\_2^2 \mathbf{C}\_2 \Phi\_2 \ \mathbf{0}\_{\sigma\_2 \times \eta\_2} \\ \mathbf{0}\_{\eta\_2 \times \sigma\_2} & \mathbf{0}\_{\eta\_2 \times \eta\_2} \end{bmatrix} \tag{16}$$

where <sup>Δ</sup>2(*t*) = *<sup>γ</sup>*−2*β*2,ΔΠ2(*t*) + <sup>Δ</sup>2,1, with *<sup>β</sup>*2,<sup>Δ</sup> <sup>≥</sup> 0 being a constant and <sup>Δ</sup>2,1 being an *n*<sup>2</sup> × *n*2- dimensional positive-definite matrix, and *�*<sup>2</sup> is a scalar function defined by *�*<sup>2</sup> = *K*−<sup>1</sup> 2,*<sup>c</sup>* Tr(Σ−<sup>1</sup> <sup>2</sup> ) or *�*<sup>2</sup> <sup>=</sup> 1. *<sup>K</sup>*2,*<sup>c</sup>* <sup>≥</sup> *<sup>γ</sup>*2Tr(*Q*2,0) is a design constant, *<sup>Q</sup>*2,0 is an *<sup>σ</sup>*<sup>2</sup> <sup>×</sup> *<sup>σ</sup>*2-dimensional positive-definite matrix. Then the dynamics of Σ2, Φ2, Π<sup>2</sup> are given as follows,

$$
\dot{\Sigma}\_2 = (\varepsilon\_2 - 1)\Sigma\_2 \Phi\_2' \mathcal{C}\_2' \gamma^2 \mathcal{C}\_2 \Phi\_2 \Sigma\_2; \quad \Sigma\_2(0) = \frac{\gamma^{-2}}{\mathcal{Q}\_{2,0}} \tag{17a}
$$

$$\begin{aligned} \Pi\_{2} &= (A\_{2} - \zeta\_{2}^{2} \mathcal{L}\_{2} \mathbb{C}\_{2} + \beta\_{2,\Lambda} / 2 \mathcal{I}\_{\mathbb{H}2}) \Pi\_{2} + \Pi\_{2} \left( A\_{2} - \zeta\_{2}^{2} \mathcal{L}\_{2} \mathbb{C}\_{2} + \beta\_{2,\Lambda} / 2 \mathcal{I}\_{\mathbb{H}2} \right)' - \Pi\_{2} \zeta\_{2}' \zeta\_{2}^{2} \mathcal{C}\_{2} \Pi\_{2} + D\_{2} \mathcal{O}\_{2}' \\ &- \zeta\_{2}^{2} \mathcal{L}\_{2} \mathcal{L}\_{2}' + \gamma^{2} \Delta\_{2,\mathbb{H}} ; \quad \Pi\_{2}(0) = \Pi\_{2,0} \end{aligned} \tag{17b}$$

$$\dot{\Phi}\_2 = A\_{2,f}\Phi\_2 + y\_2\bar{A}\_{2,211} + \mu\bar{A}\_{2,212} + \sum\_{j=1}^{\delta\_2} \bar{A}\_{2,213j}\psi\_{2,j} + \dot{y}\_2\bar{A}\_{2,214}; \qquad \Phi\_2(0) = \Phi\_{2,0} \tag{17c}$$

where *<sup>A</sup>*2, *<sup>f</sup>* :<sup>=</sup> *<sup>A</sup>*<sup>2</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup> <sup>2</sup>*L*2*C*<sup>2</sup> − Π2*C*� 2*C*2*ζ*<sup>2</sup> <sup>2</sup> is Hurwitz. By Lemma [12], we have the covariance matrix Σ<sup>2</sup> upper and lower bounded as follows, *K*−<sup>1</sup> 2,*<sup>c</sup> <sup>I</sup>σ*<sup>2</sup> <sup>≤</sup> <sup>Σ</sup>2(*t*) <sup>≤</sup> <sup>Σ</sup>2(0) = *<sup>γ</sup>*−2*Q*−<sup>1</sup> 2,0 , *<sup>γ</sup>*2Tr(*Q*2,0) <sup>≤</sup> Tr(Σ2(*t*))−<sup>1</sup> <sup>≤</sup> *<sup>K</sup>*2,*c*, whenever it exists on [0, *tf* ] and <sup>Φ</sup><sup>2</sup> is continuous on [0, *tf* ]. To avoid the calculation of Σ−<sup>1</sup> <sup>2</sup> online, we define *<sup>s</sup>*2,<sup>Σ</sup> <sup>=</sup> Tr(Σ−<sup>1</sup> <sup>2</sup> ).

To guarantee the estimates parameter to be bounded and the estimate of high frequency gain to be bounded away from zero without persistently exciting signals, we introduce the following soft projection design on the parameter estimate.

Define *<sup>ρ</sup>*<sup>2</sup> :<sup>=</sup> inf{*P*2(¯ *<sup>θ</sup>*2) <sup>|</sup> ¯ *<sup>θ</sup>*<sup>2</sup> <sup>∈</sup> IR*σ*<sup>2</sup> , *<sup>b</sup>*2,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*¯ 2,212 0 ¯ *θ*<sup>2</sup> = 0}, we have 1 < *ρ*<sup>2</sup> ≤ ∞. Fix any *<sup>ρ</sup>*2,*<sup>o</sup>* <sup>∈</sup> (1, *<sup>ρ</sup>*2), we define the open set <sup>Θ</sup>2,*<sup>o</sup>* :<sup>=</sup> { ¯ *<sup>θ</sup>*<sup>2</sup> <sup>|</sup> *<sup>P</sup>*2(¯ *θ*) < *ρ*2,*o*}. Our control design will guarantee that the estimate ˇ *<sup>θ</sup>*<sup>2</sup> lies in <sup>Θ</sup>2,*o*, which immediately implies <sup>|</sup>*b*2,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*¯ 2,212 0 ˇ *θ*2| > *c*2,0 > 0, for some *c*2,0 > 0. Moreover, the convexity of *P*<sup>2</sup> implies the following inequality: *∂P*<sup>2</sup> *∂θ*<sup>2</sup> (ˇ *<sup>θ</sup>*2) (*θ*<sup>2</sup> <sup>−</sup> <sup>ˇ</sup> *<sup>θ</sup>*2) <sup>&</sup>lt; <sup>0</sup> <sup>∀</sup> <sup>ˇ</sup> *<sup>θ</sup>*<sup>2</sup> <sup>∈</sup> IR*σ*<sup>2</sup> \Θ2. To incorporate the modifier to the estimates dynamics,

#### 12 Will-be-set-by-IN-TECH 118 Game Theory Relaunched A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems <sup>13</sup>

we introduce *<sup>l</sup>*2,2 = [−(*P*2,*r*(<sup>ˇ</sup> *<sup>θ</sup>*2)� **<sup>0</sup>**1×*n*<sup>2</sup> ] � , where

$$\begin{split} P\_{2,r}(\boldsymbol{\delta}\_{2}) &:= \begin{cases} \frac{\exp\left(\frac{1}{1-P\_{2}(\boldsymbol{\delta}\_{2})}\right)}{\left(\rho\_{2,\boldsymbol{\rho}}-P\_{2}(\boldsymbol{\delta}\_{2})\right)^{\mathsf{T}}} \left(\frac{\partial P\_{2}}{\partial \boldsymbol{\theta}\_{2}}(\boldsymbol{\delta}\_{2})\right)' \,\forall \boldsymbol{\theta}\_{2} \in \Theta\_{2,o} \,\backslash \Theta\_{2} \\\ 0\_{\sigma\_{2} \times 1} & \forall \theta\_{2} \in \Theta\_{2} \end{cases} \\ &:= p\_{2,r}(\boldsymbol{\theta}\_{2}) \left(\frac{\partial P\_{2}}{\partial \boldsymbol{\theta}\_{2}}(\boldsymbol{\theta}\_{2})\right)' \end{split}$$

**4.2. Control design**

sup *<sup>w</sup>*` <sup>1</sup>∈W` 1,*w*` <sup>2</sup>∈W` <sup>2</sup>

≤sup *ωm*∈W*<sup>m</sup>*

≤sup *ωm*∈W*<sup>m</sup>*

state vector ˆ

where *vi* =

view of *y*<sup>2</sup> = *ζ*−<sup>1</sup>

� *v*� 1,*<sup>a</sup> v*� 2,*a* �� �

<sup>2</sup> *e*�

**S1**, where *q*ˇ*<sup>a</sup>* = *q*ˇ1,*<sup>a</sup>* + *q*ˇ2,*<sup>a</sup>* .

*ζ<sup>i</sup>* (*yi* − *Cix*ˇ*i*) *w*ˇ�

For *i* = 1, 2, we introduce the matrix *Mi*, *<sup>f</sup>* :=

*ni*-dimensional vector such that the pair (*Ai*, *<sup>f</sup>* , *pi*,*ni*

where function ˇ

and are therefore neglected.

*Jγtf*

sup *ω*1∈W1|*ωm*∈W*<sup>m</sup>*


�

�

� � *tf* 0

*<sup>l</sup>*1(*τ*, *<sup>θ</sup>*1, *<sup>x</sup>*1, *<sup>y</sup>*1[0,*τ*],*Yd*[0,*τ*], *<sup>w</sup>*<sup>ˇ</sup> <sup>1</sup>), and <sup>ˇ</sup>

*ξ*<sup>1</sup> and ˆ

In this section, we describe the controller design for the uncertain system under consideration. Note that, we ignored some terms in the cost function (5) in the identification step, since they are constant when *y*1, *y*2, *w*ˇ 1, *w*ˇ <sup>2</sup> and *y*´2 are given. In the control design step, we will include such terms. Then, based on the cost function (5), the controller design is to guarantee that the

following supremum is less than or equal to zero for all measurement waveforms,

*J*1,*γtf* + sup

� |*ξi*,*c*| 2 *Q*¯ *i* +ˇ *li*−*γ*2*ζ*<sup>2</sup>

2+ 2 ∑ *i*=1

*v* =

*<sup>i</sup>*,*<sup>a</sup> w*ˇ� *i*,*b* ��

By the special structure of the system, we define *vi*,*<sup>a</sup>* = �

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ *ω*2∈W2|*ωm*∈W*<sup>m</sup>*

*l*2(*τ*, *θ*2, *x*2, *y*2[0,*τ*],*Yd*[0,*τ*], *w*ˇ <sup>2</sup>) to be designed, which are constants in the identifier design step

By equation (21), we observe that the cost function is expressed in term of the states of the estimator we derived, whose dynamics are driven by the measurement *y*1, *y*2, *w*ˇ 1, *w*ˇ 2, *y*´2, the reference trajectory *yd*, the input *u*, and the worst-case estimate for the expanded

nonlinear *H*∞-optimal control problem under full information measurements. Since *y*´2 = *y*<sup>1</sup> in the adaptive system under consideration, we can equivalently deal with the following transformed variables instead of considering *y*1, *y*2, *w*ˇ 1, *w*ˇ 2, and *y*´2 as the maximizing variable,

> *ζ*<sup>1</sup> (*y*<sup>1</sup> − *C*1*x*ˇ1) *w*ˇ 1,*<sup>a</sup> w*ˇ 1,*<sup>b</sup> ζ*<sup>2</sup> (*y*<sup>2</sup> − *C*2*x*ˇ2) *w*ˇ 2,*<sup>a</sup> w*ˇ 2,*<sup>b</sup>*

, *i* = 1, 2.

*J*2,*γtf*

�

*<sup>i</sup>* |*yi*−*Cix*ˇ*i*|

*l*2(*τ*, *y*2[0,*τ*],*Yd*[0,*τ*], *w*ˇ <sup>2</sup>) is part of the weighting function

*l*1(*τ*, *y*1[0,*τ*],*Yd*[0,*τ*], *w*ˇ <sup>1</sup>) is part of the weighting function

*ξ*2, which are signals we either measure or can construct. This is then a

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

= � *v*1 *v*2

, and we will attenuate disturbance *va*, and cancel the disturbance *w*ˇ 1,*<sup>b</sup>* and *w*ˇ 2,*b*. In

� *Ani*−<sup>1</sup>

*<sup>q</sup>*ˇ*a*+2,*q*ˇ1,*a*+2*va* + *x*ˇ2,1, we will treat *x*ˇ2,1 as the virtual control input of subsystem

�

*ζ<sup>i</sup>* (*yi* − *Cix*ˇ*i*) *w*ˇ�

*<sup>i</sup>*, *<sup>f</sup> pi*,*ni* ··· *Ai*, *<sup>f</sup> pi*,*ni pi*,*ni*

*i*,*a* ��

�

) is controllable. We note that *y*´2 = *y*1, then

, *i* = 1, 2, *va* =

, where *pi*,*ni* is a

<sup>2</sup>−*γ*2|*w*ˇ*i*,*a*<sup>|</sup>

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

> 2 � � d*τ* �

(21)

119

and the dynamics of ˇ *ξ*<sup>2</sup> is then given as follows,

$$\begin{split} \dot{\tilde{\xi}}\_{2}^{\tilde{\mathsf{x}}} &= -\mathsf{\dot{\Sigma}}\_{2} \left[ (P\_{2,r}(\dot{\theta}\_{2}))' \, \mathbf{0}\_{1 \times n\_{2}} \right]' + \bar{A}\_{2} \dot{\xi}\_{2} + \bar{B}\_{2} u + \bar{\zeta}\_{2}^{2} \left( \gamma^{2} \Sigma\_{2} \mathsf{C}\_{2}' + \bar{A}\_{2,y} \dot{y}\_{2} + \bar{L}\_{2} \right) (y\_{2} - \mathsf{C}\_{2} \dot{\xi}\_{2}) \\ &+ \bar{D}\_{2} \vartheta \underline{v}\_{2} - \Sigma\_{2} \bar{Q}\_{2} (\dot{\xi}\_{2} - \dot{\xi}\_{2}) \end{split}$$

where ˇ *ξ*<sup>2</sup> = [ˇ *θ*� <sup>2</sup> *x*ˇ2] � with initial condition [ˇ *θ*� 2,0 *x*ˇ� 2,0] � , and *<sup>L</sup>*¯ <sup>2</sup> is defined as *<sup>L</sup>*¯ <sup>2</sup> = [**0**1×*σ*<sup>2</sup> *<sup>L</sup>*� 2] � . This completes the estimation design of **S2**.

Associated with the above identifier and estimator of subsystem **S***i*, *i* = 1, 2, we introduce the value function *Wi* : IR*ni*+*σ<sup>i</sup>* <sup>×</sup> IR*ni*+*σ<sup>i</sup>* × S+(*ni*+*σi*) <sup>→</sup> IR and the time derivative are as follows

*Wi*(*ξi*, ˇ *<sup>ξ</sup>i*, <sup>Σ</sup>¯ *<sup>i</sup>*) = <sup>|</sup>*θi*<sup>−</sup> <sup>ˇ</sup> *θi*| 2 Σ−<sup>1</sup> *i* <sup>+</sup> *<sup>γ</sup>*2|*xi* <sup>−</sup> *<sup>x</sup>*ˇ*<sup>i</sup>* <sup>−</sup> <sup>Φ</sup>*<sup>i</sup>* (*θ<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup> *θi*)| 2 Π−<sup>1</sup> *i* (18) *<sup>W</sup>*˙ <sup>1</sup> <sup>=</sup> −|*x*1,1 <sup>−</sup> *yd*<sup>|</sup> <sup>2</sup> <sup>−</sup> *<sup>γ</sup>*4|*x*<sup>1</sup> <sup>−</sup> *<sup>x</sup>*ˆ1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> (*θ*<sup>1</sup> <sup>−</sup> <sup>ˆ</sup> *θ*1)| 2 Π−<sup>1</sup> <sup>1</sup> <sup>Δ</sup>1Π−<sup>1</sup> 1 + |*C*1*x*ˇ1 − *yd*| 2 <sup>−</sup>*�*1( *<sup>γ</sup>*2*ζ*<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup>)|*θ*<sup>1</sup> <sup>−</sup> <sup>ˆ</sup> *θ*1| 2 Φ� 1*C*� <sup>1</sup>*C*1Φ<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*2*ζ*<sup>2</sup> <sup>1</sup>|*y*<sup>1</sup> − *C*1*x*ˇ1| <sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*w*1<sup>|</sup> <sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2|*w*<sup>1</sup> <sup>−</sup> *<sup>w</sup>*1,∗| 2 <sup>+</sup><sup>2</sup> (*θ*<sup>1</sup> <sup>−</sup> <sup>ˇ</sup> *θ*1)� *P*1,*r*(ˇ *θi*) + |*ξ*1,*c*| 2 *<sup>Q</sup>*¯ <sup>1</sup> (19) *<sup>W</sup>*˙ <sup>2</sup> <sup>=</sup> <sup>−</sup>*γ*4|*x*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*ˆ2 <sup>−</sup> <sup>Φ</sup><sup>2</sup> (*θ*<sup>2</sup> <sup>−</sup> <sup>ˆ</sup> *θ*2)| 2 Π−<sup>1</sup> <sup>2</sup> <sup>Δ</sup>2Π−<sup>1</sup> 2 <sup>−</sup> *�*<sup>2</sup> *<sup>γ</sup>*2*ζ*<sup>2</sup> <sup>2</sup>|*θ*<sup>2</sup> <sup>−</sup> <sup>ˆ</sup> *θ*2| 2 Φ� 2*C*� <sup>2</sup>*C*2Φ<sup>2</sup> <sup>+</sup> <sup>|</sup>*ξ*2,*c*<sup>|</sup> 2 *Q*¯ 2 *P*2,*r*(ˇ

<sup>−</sup>*γ*2*ζ*<sup>2</sup> <sup>2</sup>|*y*<sup>2</sup> − *C*2*x*ˇ2| <sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*w*2<sup>|</sup> <sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2|*w*<sup>2</sup> <sup>−</sup> *<sup>w</sup>*2,∗| <sup>2</sup> <sup>+</sup> <sup>2</sup> (*θ*<sup>2</sup> <sup>−</sup> <sup>ˇ</sup> *θ*2)� *θ*2) (20) where *wi*,<sup>∗</sup> is the worst-case disturbance, given by *wi*,<sup>∗</sup> : IR<sup>×</sup> IR*ni*+*σ<sup>i</sup>* <sup>×</sup> IR*ni*+*σ<sup>i</sup>* × S+(*ni*+*σi*) −→ IR

$$w\_{i,\*} (\mathfrak{xi}\_i, \mathfrak{F}\_i, \Sigma\_i, w\_i) = \zeta\_i^2 E\_i' \left( y\_i - \mathfrak{C}\_i \mathfrak{z}\_i \right) + \gamma^{-2} \left( \mathbf{I}\_{q\_i} - \mathfrak{z}\_i^2 E\_i' \mathbf{E}\_i \right) \mathcal{D}\_i' \Sigma\_i^{-1} \left( \mathfrak{z}\_i - \mathfrak{f}\_i \right); \quad i = 1, 2$$

We note that (18) holds when <sup>Σ</sup>*<sup>i</sup>* <sup>&</sup>gt; 0 and *<sup>θ</sup><sup>i</sup>* <sup>∈</sup> <sup>Θ</sup>*i*,0, and the last term in *<sup>W</sup>*˙ *<sup>i</sup>* is nonpositive, zero on the set <sup>Θ</sup>*<sup>i</sup>* and approaches <sup>−</sup><sup>∞</sup> as <sup>ˇ</sup> *θ<sup>i</sup>* approaches the boundary of the set Θ*i*,*o*, which guarantees the boundedness of ˇ *θi*, *i* = 1, 2.

Then (5) can be equivalently written as, *i* = 1, 2:

$$\begin{split} J\_{1,\gamma t\_f} &= \int\_0^{t\_f} \left( |\mathbb{C}\_1 \mathbb{1}\_f - y\_d|^2 + |\xi\_{1,\boldsymbol{\epsilon}}|\_{\tilde{Q}\_1}^2 + I\_1 - \gamma^2 \xi\_1^2 |y\_1 - \mathbb{C}\_1 \mathbb{1}\_1|^2 - \gamma^2 |w\_1 - w\_{1,\*}|^2 - \gamma^2 |\mathbb{i}b\_{1,\boldsymbol{\epsilon}}|^2 \right) d\tau \\ &- l\_{1,0} - |\xi\_1(t\_f) - \xi\_1(t\_f)|\_{\left(\tilde{\Sigma}\_1(t\_f)\right)^{-1}}^2 \\ J\_{2,\gamma t\_f} &= \int\_0^{t\_f} \left( |\mathbb{S}\_{2,\boldsymbol{\epsilon}}|\_{\tilde{Q}\_2}^2 + \check{l}\_2 - \gamma^2 \xi\_2^2 |y\_2 - \mathbb{C}\_2 \mathbb{A}\_2|^2 - \gamma^2 |w\_2 - w\_{2,\*}|^2 - \gamma^2 |\mathbb{i}\_{2,\boldsymbol{\epsilon}}|^2 \right) d\tau \\ &- l\_{2,0} - |\xi\_2(t\_f) - \check{\xi}\_2(t\_f)|\_{\left(\tilde{\Sigma}\_2(t\_f)\right)^{-1}}^2 \end{split}$$

This completes the identification design step.

#### **4.2. Control design**

12 Will-be-set-by-IN-TECH

∀*θ*<sup>2</sup> ∈ Θ2,*o*\Θ<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>A</sup>*¯ 2,*yy*´2 <sup>+</sup> *<sup>L</sup>*¯ <sup>2</sup>) (*y*<sup>2</sup> <sup>−</sup> *<sup>C</sup>*¯

, and *<sup>L</sup>*¯ <sup>2</sup> is defined as *<sup>L</sup>*¯ <sup>2</sup> = [**0**1×*σ*<sup>2</sup> *<sup>L</sup>*�

+ |*C*1*x*ˇ1 − *yd*|

*θ*2)� *P*2,*r*(ˇ

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*w*1<sup>|</sup>

*<sup>Q</sup>*¯ <sup>1</sup> (19)

<sup>2</sup>|*θ*<sup>2</sup> <sup>−</sup> <sup>ˆ</sup> *θ*2| 2 Φ� 2*C*�

*<sup>i</sup>*Σ¯ <sup>−</sup><sup>1</sup>

<sup>2</sup>−*γ*2|*w*<sup>1</sup> <sup>−</sup> *<sup>w</sup>*1,∗|

<sup>2</sup><sup>−</sup> *<sup>γ</sup>*2|*w*<sup>ˇ</sup> 2,*a*<sup>|</sup>

2 � d*τ*

<sup>2</sup> <sup>+</sup> <sup>2</sup> (*θ*<sup>2</sup> <sup>−</sup> <sup>ˇ</sup>

*<sup>i</sup>* (*ξ<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup>

*θ<sup>i</sup>* approaches the boundary of the set Θ*i*,*o*, which

2

<sup>2</sup>*C*2Φ<sup>2</sup> <sup>+</sup> <sup>|</sup>*ξ*2,*c*<sup>|</sup>

*ξi*); *i* = 1, 2

<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2|*w*<sup>ˇ</sup> 1,*a*<sup>|</sup>

2 � d*τ*

<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2|*w*<sup>1</sup> <sup>−</sup> *<sup>w</sup>*1,∗|

2 *Q*¯ 2

*θ*2) (20)

2 ˇ *ξ*2)

> 2] � .

> > (18)

2

<sup>0</sup>*σ*2×<sup>1</sup> ∀*θ*<sup>2</sup> ∈ <sup>Θ</sup><sup>2</sup>

<sup>2</sup> (*γ*2Σ¯ <sup>2</sup>*C*¯�

*<sup>θ</sup>*2)� **<sup>0</sup>**1×*n*<sup>2</sup> ]

exp � <sup>1</sup> <sup>1</sup>−*P*2(<sup>ˇ</sup> *θ*2) �

⎧ ⎨ ⎩

:= *p*2,*r*(ˇ

*ξ*<sup>2</sup> is then given as follows,

�� + *A*¯2 ˇ

� with initial condition [ˇ

<sup>1</sup> <sup>−</sup> <sup>1</sup>)|*θ*<sup>1</sup> <sup>−</sup> <sup>ˆ</sup>

*θ*1)� *P*1,*r*(ˇ

<sup>2</sup>|*y*<sup>2</sup> − *C*2*x*ˇ2|

*<sup>i</sup> E*�

*<sup>i</sup>* (*yi* <sup>−</sup> *<sup>C</sup>*¯

<sup>2</sup>+|*ξ*1,*c*<sup>|</sup> 2 *Q*¯ 1 +ˇ *<sup>l</sup>*1−*γ*2*ζ*<sup>2</sup>

*ξ*1(*tf*)| 2 (Σ¯ <sup>1</sup>(*tf*))−<sup>1</sup>

*<sup>l</sup>*<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2*ζ*<sup>2</sup>

*ξ*2(*tf*)| 2 (Σ¯ <sup>2</sup>(*tf*))−<sup>1</sup>

*<sup>W</sup>*˙ <sup>2</sup> <sup>=</sup> <sup>−</sup>*γ*4|*x*<sup>2</sup> <sup>−</sup> *<sup>x</sup>*ˆ2 <sup>−</sup> <sup>Φ</sup><sup>2</sup> (*θ*<sup>2</sup> <sup>−</sup> <sup>ˆ</sup>

� , where

(*ρ*2,*<sup>o</sup>*−*P*2(<sup>ˇ</sup>

*θ*2)

<sup>+</sup> *<sup>γ</sup>*2|*xi* <sup>−</sup> *<sup>x</sup>*ˇ*<sup>i</sup>* <sup>−</sup> <sup>Φ</sup>*<sup>i</sup>* (*θ<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup>

*θ*1| 2 Φ� 1*C*�

*θi*) + |*ξ*1,*c*|

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*w*2<sup>|</sup>

*θi*, *i* = 1, 2.

<sup>2</sup>|*y*<sup>2</sup> − *C*2*x*ˇ2|

<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*4|*x*<sup>1</sup> <sup>−</sup> *<sup>x</sup>*ˆ1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> (*θ*<sup>1</sup> <sup>−</sup> <sup>ˆ</sup>

2

where *wi*,<sup>∗</sup> is the worst-case disturbance, given by *wi*,<sup>∗</sup> : IR<sup>×</sup> IR*ni*+*σ<sup>i</sup>* <sup>×</sup> IR*ni*+*σ<sup>i</sup>* × S+(*ni*+*σi*) −→ IR

*<sup>i</sup>ξi*) + *<sup>γ</sup>*−<sup>2</sup> (*Iqi* <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

We note that (18) holds when <sup>Σ</sup>*<sup>i</sup>* <sup>&</sup>gt; 0 and *<sup>θ</sup><sup>i</sup>* <sup>∈</sup> <sup>Θ</sup>*i*,0, and the last term in *<sup>W</sup>*˙ *<sup>i</sup>* is nonpositive,

*θ*2)| 2 Π−<sup>1</sup> <sup>2</sup> <sup>Δ</sup>2Π−<sup>1</sup> 2

*θ*2)) 3 � *<sup>∂</sup>P*<sup>2</sup> *∂θ*<sup>2</sup> (ˇ *θ*2) ��

� *∂P*<sup>2</sup> *∂θ*2 (ˇ *θ*2) ��

*ξ*<sup>2</sup> + *B*¯

*θ*� 2,0 *x*ˇ� 2,0] �

Associated with the above identifier and estimator of subsystem **S***i*, *i* = 1, 2, we introduce the value function *Wi* : IR*ni*+*σ<sup>i</sup>* <sup>×</sup> IR*ni*+*σ<sup>i</sup>* × S+(*ni*+*σi*) <sup>→</sup> IR and the time derivative are as follows

> *θi*)| 2 Π−<sup>1</sup> *i*

<sup>1</sup>*C*1Φ<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*2*ζ*<sup>2</sup>

*θ*1)| 2 Π−<sup>1</sup> <sup>1</sup> <sup>Δ</sup>1Π−<sup>1</sup> 1

<sup>1</sup>|*y*<sup>1</sup> − *C*1*x*ˇ1|

<sup>−</sup> *�*<sup>2</sup> *<sup>γ</sup>*2*ζ*<sup>2</sup>

*<sup>i</sup> E*� *<sup>i</sup>Ei*)*D*¯ �

<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2|*w*<sup>2</sup> <sup>−</sup> *<sup>w</sup>*2,∗|

<sup>1</sup>|*y*1−*C*1*x*ˇ1|

<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2|*w*<sup>2</sup> <sup>−</sup> *<sup>w</sup>*2,∗|

<sup>2</sup>*u* + *ζ*<sup>2</sup>

we introduce *<sup>l</sup>*2,2 = [−(*P*2,*r*(<sup>ˇ</sup>

and the dynamics of ˇ

+ ¯

*ξ*<sup>2</sup> = [ˇ *θ*� <sup>2</sup> *x*ˇ2]

*<sup>ξ</sup>i*, <sup>Σ</sup>¯ *<sup>i</sup>*) = <sup>|</sup>*θi*<sup>−</sup> <sup>ˇ</sup>

� (*P*2,*r*(ˇ

*<sup>D</sup>*<sup>ˇ</sup> <sup>2</sup>*w*<sup>ˇ</sup> <sup>2</sup> <sup>−</sup> <sup>Σ</sup>¯ <sup>2</sup>*Q*¯ <sup>2</sup>( <sup>ˆ</sup>

This completes the estimation design of **S2**.

*θi*| 2 Σ−<sup>1</sup> *i*

<sup>−</sup>*�*1( *<sup>γ</sup>*2*ζ*<sup>2</sup>

<sup>+</sup><sup>2</sup> (*θ*<sup>1</sup> <sup>−</sup> <sup>ˇ</sup>

<sup>−</sup>*γ*2*ζ*<sup>2</sup>

*<sup>ξ</sup>i*, <sup>Σ</sup>¯ *<sup>i</sup>*, *wi*) = *<sup>ζ</sup>*<sup>2</sup>

guarantees the boundedness of ˇ

� *tf* 0 �

� *tf* 0 � |*ξ*2,*c*| 2 *<sup>Q</sup>*¯ <sup>2</sup> <sup>+</sup> <sup>ˇ</sup>

zero on the set <sup>Θ</sup>*<sup>i</sup>* and approaches <sup>−</sup><sup>∞</sup> as <sup>ˇ</sup>


<sup>−</sup>*l*1,0 − |*ξ*1(*tf*) <sup>−</sup> <sup>ˇ</sup>

<sup>−</sup>*l*2,0 − |*ξ*2(*tf*) <sup>−</sup> <sup>ˇ</sup>

This completes the identification design step.

Then (5) can be equivalently written as, *i* = 1, 2:

*wi*,∗(*ξi*, <sup>ˇ</sup>

*J*1,*γtf* =

*J*2,*γtf* =

*<sup>W</sup>*˙ <sup>1</sup> <sup>=</sup> −|*x*1,1 <sup>−</sup> *yd*<sup>|</sup>

˙ ˇ *<sup>ξ</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>Σ¯ <sup>2</sup>

where ˇ

*Wi*(*ξi*, ˇ

*P*2,*r*(ˇ

*θ*2) :=

*<sup>θ</sup>*2))� **<sup>0</sup>**1×*n*<sup>2</sup>

*<sup>ξ</sup>*<sup>2</sup> <sup>−</sup> <sup>ˇ</sup> *ξ*2) In this section, we describe the controller design for the uncertain system under consideration. Note that, we ignored some terms in the cost function (5) in the identification step, since they are constant when *y*1, *y*2, *w*ˇ 1, *w*ˇ <sup>2</sup> and *y*´2 are given. In the control design step, we will include such terms. Then, based on the cost function (5), the controller design is to guarantee that the following supremum is less than or equal to zero for all measurement waveforms,

$$\begin{aligned} &\sup\_{\boldsymbol{\vartheta}\boldsymbol{\omega}\in\mathcal{W}\_{1},\boldsymbol{\vartheta}\_{2}\in\mathcal{W}\_{2}} \; \mathcal{I}\_{\mathcal{I}\_{f}} \\ &\leq \sup\_{\boldsymbol{\omega}\_{m}\in\mathcal{W}\_{m}} \left( \sup\_{\boldsymbol{\omega}\_{1}\in\mathcal{W}\_{1}|\boldsymbol{\omega}\_{m}\in\mathcal{W}\_{m}} \boldsymbol{I}\_{1,\gamma t\_{f}} + \sup\_{\boldsymbol{\omega}\_{2}\in\mathcal{W}\_{2}|\boldsymbol{\omega}\_{m}\in\mathcal{W}\_{m}} \boldsymbol{I}\_{2,\gamma t\_{f}} \right) \\ &\leq \sup\_{\boldsymbol{\omega}\_{m}\in\mathcal{W}\_{m}} \left\{ \int\_{0}^{t\_{f}} \left( |\mathcal{C}\_{i}\boldsymbol{\tilde{x}}\_{1} - \boldsymbol{y}\_{d}|^{2} + \sum\_{i=1}^{2} \left( |\boldsymbol{\xi}\_{i,c}|\_{\boldsymbol{\xi}\_{i}}^{2} + \bar{I}\_{i} - \gamma^{2}\bar{\xi}\_{i}^{2}|\boldsymbol{y}\_{i} - \mathsf{C}\_{i}\boldsymbol{\tilde{x}}\_{i}|^{2} - \gamma^{2}|\boldsymbol{\tilde{w}}\_{i,d}|^{2} \right) \right) \, \mathrm{d}\tau \right\} \end{aligned} (21)$$

where function ˇ *l*1(*τ*, *y*1[0,*τ*],*Yd*[0,*τ*], *w*ˇ <sup>1</sup>) is part of the weighting function *<sup>l</sup>*1(*τ*, *<sup>θ</sup>*1, *<sup>x</sup>*1, *<sup>y</sup>*1[0,*τ*],*Yd*[0,*τ*], *<sup>w</sup>*<sup>ˇ</sup> <sup>1</sup>), and <sup>ˇ</sup> *l*2(*τ*, *y*2[0,*τ*],*Yd*[0,*τ*], *w*ˇ <sup>2</sup>) is part of the weighting function *l*2(*τ*, *θ*2, *x*2, *y*2[0,*τ*],*Yd*[0,*τ*], *w*ˇ <sup>2</sup>) to be designed, which are constants in the identifier design step and are therefore neglected.

By equation (21), we observe that the cost function is expressed in term of the states of the estimator we derived, whose dynamics are driven by the measurement *y*1, *y*2, *w*ˇ 1, *w*ˇ 2, *y*´2, the reference trajectory *yd*, the input *u*, and the worst-case estimate for the expanded state vector ˆ *ξ*<sup>1</sup> and ˆ *ξ*2, which are signals we either measure or can construct. This is then a nonlinear *H*∞-optimal control problem under full information measurements. Since *y*´2 = *y*<sup>1</sup> in the adaptive system under consideration, we can equivalently deal with the following transformed variables instead of considering *y*1, *y*2, *w*ˇ 1, *w*ˇ 2, and *y*´2 as the maximizing variable,

$$v = \begin{bmatrix} \zeta\_1 \left( y\_1 - \mathcal{C}\_1 \mathbb{1}\_1 \right) \\\\ \psi\_{1,a} \\\\ \frac{\mathcal{U}\_{1,b}}{\mathcal{L}\_2 \left( y\_2 - \mathcal{C}\_2 \mathbb{1}\_2 \right)} \\\\ \psi\_{2,a} \\\\ \psi\_{2,b} \end{bmatrix} = \left[ \frac{v\_1}{v\_2} \right]$$

where *vi* = � *ζ<sup>i</sup>* (*yi* − *Cix*ˇ*i*) *w*ˇ� *<sup>i</sup>*,*<sup>a</sup> w*ˇ� *i*,*b* �� , *i* = 1, 2.

By the special structure of the system, we define *vi*,*<sup>a</sup>* = � *ζ<sup>i</sup>* (*yi* − *Cix*ˇ*i*) *w*ˇ� *i*,*a* �� , *i* = 1, 2, *va* = � *v*� 1,*<sup>a</sup> v*� 2,*a* �� , and we will attenuate disturbance *va*, and cancel the disturbance *w*ˇ 1,*<sup>b</sup>* and *w*ˇ 2,*b*. In view of *y*<sup>2</sup> = *ζ*−<sup>1</sup> <sup>2</sup> *e*� *<sup>q</sup>*ˇ*a*+2,*q*ˇ1,*a*+2*va* + *x*ˇ2,1, we will treat *x*ˇ2,1 as the virtual control input of subsystem **S1**, where *q*ˇ*<sup>a</sup>* = *q*ˇ1,*<sup>a</sup>* + *q*ˇ2,*<sup>a</sup>* .

For *i* = 1, 2, we introduce the matrix *Mi*, *<sup>f</sup>* := � *Ani*−<sup>1</sup> *<sup>i</sup>*, *<sup>f</sup> pi*,*ni* ··· *Ai*, *<sup>f</sup> pi*,*ni pi*,*ni* � , where *pi*,*ni* is a *ni*-dimensional vector such that the pair (*Ai*, *<sup>f</sup>* , *pi*,*ni* ) is controllable. We note that *y*´2 = *y*1, then the following 3*n*<sup>1</sup> + 4*n*<sup>2</sup> + *q*ˇ1 + *q*ˇ2-dimensional prefiltering system for *y*1, *y*2, *u*, *w*ˇ 1, *w*ˇ 2, and *y*´2 generates the Φ<sup>1</sup> and Φ<sup>2</sup> online:

step 1 with the virtual control law *α*1,1, which guarantees *V*˙

*x*ˇ2,1 := *α*1,*r*<sup>1</sup> . This completes the control design for subsystem **S1**.

*η*˙2,*<sup>d</sup>* = *A*2, *<sup>f</sup> η*2,*<sup>d</sup>* + *p*2,*n*2*α*1,*r*<sup>1</sup> + *p*2,*n*<sup>2</sup> *e*

To stabilize *η*2, we introduce variable *η*2,*<sup>d</sup>* as below,

*W*<sup>1</sup> + *W*<sup>2</sup> + *V*2,*r*<sup>2</sup> , and its time derivative is given by

*θj*) + |*η*˜*j*|

2 

> 

2

*Q*¯ 1 + *ξ*2,*<sup>c</sup>* + 1 2 *ς*2,*r*<sup>2</sup> 

2,1*ν*1,*r*<sup>1</sup> <sup>+</sup> *<sup>γ</sup>*−<sup>2</sup> (*Iq*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

2,2*ν*2,*r*<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*−<sup>2</sup> (*Iq*<sup>2</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

*ς*1,(*r*1+*r*2)

2 − 2 ∑ *j*=1 

*θj*)� *Pj*,*r*(ˇ

<sup>+</sup>*γ*2|*w*<sup>ˇ</sup> *<sup>j</sup>*,*<sup>a</sup>* <sup>−</sup> *<sup>w</sup>*<sup>ˇ</sup> *<sup>j</sup>*,*opt*<sup>|</sup>

1*e* �

2*e* �

Choosing value function *V*2,0 := |*η*<sup>2</sup> − *η*2,*d*|

will guarantee the *V*˙

*<sup>j</sup>*=1(|*η*˜*j*| 2 *Zj* <sup>+</sup> <sup>∑</sup>*rj k*=1 1 <sup>2</sup> *<sup>z</sup>*<sup>2</sup>

*<sup>U</sup>*˙ <sup>=</sup> −|*x*1,1 <sup>−</sup> *yd*<sup>|</sup>

<sup>−</sup><sup>2</sup> (*θ<sup>j</sup>* <sup>−</sup> <sup>ˇ</sup>

+ *ξ*1,*<sup>c</sup>* + 1 2

*w*1,*opt* = *ζ*1*E*�

*w*2,*opt* = *ζ*2*E*�

 **0**�

 **0**�

*w*ˇ 1,*opt* =

*w*ˇ 2,*opt* =

*<sup>V</sup>*2,*r*<sup>2</sup> = <sup>∑</sup><sup>2</sup>

that *V*˙

error.

the variable *z*1,2 = *x*ˇ1,2 − *α*1,1 for step 2. Repeating the backstepping procedure until step *r*1, the virtual control input *x*ˇ2,1 will appear in the dynamic of *z*˙1,*<sup>r</sup>*<sup>1</sup> . Using the similar procedure

�

and is the reference trajectory for *η*<sup>2</sup> to track, where *ν*1,*r*<sup>1</sup> is a function obtained after step *r*1.

Riccati equation. We complete the step *r*<sup>1</sup> + 1 with the virtual control law *α*2,0 = *α*1,*r*<sup>1</sup> , which

step *r*<sup>1</sup> + *r*<sup>2</sup> + 1, the virtual control input *u* will appear in the dynamic of *z*˙2,*<sup>r</sup>*<sup>2</sup> . Introduce

For the closed-loop adaptive nonlinear system, we have the following value function, *U* =

*θj*)| 2 Π−<sup>1</sup> *<sup>j</sup>* <sup>Δ</sup>*j*Π−<sup>1</sup> *j*

*<sup>j</sup>*,*<sup>k</sup>* <sup>−</sup> *<sup>γ</sup>*2|*wj*<sup>|</sup>

<sup>2</sup>*C*2Φ<sup>2</sup> <sup>−</sup> <sup>1</sup> 4 *<sup>ς</sup>*1,(*r*1+*r*2)

2

*Q*¯ 2

<sup>1</sup>Σ¯ <sup>−</sup><sup>1</sup>

<sup>2</sup>Σ¯ <sup>−</sup><sup>1</sup>

<sup>1</sup> (*ξ*<sup>1</sup> <sup>−</sup> <sup>ˇ</sup>

<sup>2</sup> (*ξ*<sup>2</sup> <sup>−</sup> <sup>ˇ</sup>

where *ς*1,*r*1+*r*<sup>2</sup> and *ς*2,*r*<sup>2</sup> are functions obtained after step *r*<sup>1</sup> + *r*<sup>2</sup> + 1, *w*1,*opt* and *w*2,*opt* are the

*<sup>β</sup>j*,*kz*<sup>2</sup>

*<sup>γ</sup>*4|*xj*−*x*ˆ*j*−Φ*j*(*θj*<sup>−</sup> <sup>ˆ</sup>

*rj* ∑ *k*=1

> *θ*2|Φ� 2*C*�

worst case disturbance with respect to the value function *U*, which are given by

1*E*� <sup>1</sup>*E*1)*D*¯ �

2*E*� <sup>2</sup>*E*2)*D*¯ �

<sup>1</sup>×(2+*q*ˇ1,*a*+*q*ˇ2,*a*) *<sup>e</sup>*(2+*q*ˇ1,*a*+*q*ˇ2,*a*),1 ··· *<sup>e</sup>*(2+*q*ˇ1,*a*+*q*ˇ2,*a*),*q*ˇ1,*<sup>a</sup>* **<sup>0</sup>**�

where *ν*1,*r*<sup>1</sup> and *ν*2,*r*<sup>2</sup> are functions obtained after backstepping design.

(2+*q*ˇ1,*a*)×(2+*q*ˇ1,*a*+*q*ˇ2,*a*) *<sup>e</sup>*(2+*q*ˇ1,*a*+*q*ˇ2,*a*),1 ··· *<sup>e</sup>*(2+*q*ˇ1,*a*+*q*ˇ2,*a*),*q*ˇ2,*<sup>a</sup>*

2 *Yj* +

<sup>−</sup>*�*2|*θ*<sup>2</sup> <sup>−</sup> <sup>ˆ</sup>

2,*r*<sup>2</sup> ≤ 0 under *u* := *μ*. Later, we will show that the control law *μ* will guarantee the boundedness of the closed-loop system states and the asymptotic convergence of tracking

2

*<sup>q</sup>*ˇ*a*+2,*q*ˇ1,*a*+2*ν*1,*r*<sup>1</sup> ; *η*2,*d*(0) = *η*2,*d*<sup>0</sup>

*<sup>j</sup>*,*k*), we then can derive the robust adaptive controller *μ* such

+ *�<sup>j</sup>* (*γ*2*ζ*<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*wj* <sup>−</sup> *wj*,*opt*<sup>|</sup>

 2 *Q*¯ 1 − 1 <sup>4</sup> <sup>|</sup>*ς*2,*r*<sup>2</sup> <sup>|</sup> 2 *Q*¯ 2

*ξ*1) + *ζ*<sup>2</sup> 1*E*�

*ξ*2) + *ζ*<sup>2</sup> 2*E*�

*<sup>j</sup>* <sup>−</sup> <sup>1</sup>)|*θ<sup>j</sup>* <sup>−</sup> <sup>ˆ</sup>

<sup>1</sup>*C*<sup>1</sup> (*x*ˇ1 − *x*1)

<sup>2</sup>*C*<sup>2</sup> (*x*ˇ2 − *x*2)

� *ν*1,*r*<sup>1</sup>

(1+*q*ˇ2,*a*)×(2+*q*ˇ1,*a*+*q*ˇ2,*a*)

� *ν*2,*r*<sup>2</sup> *θj*| 2 Φ� *j C*� *j Cj*Φ*<sup>j</sup>*

<sup>2</sup> <sup>−</sup> *<sup>γ</sup>*2|*w*<sup>ˇ</sup> *<sup>j</sup>*,*a*<sup>|</sup>

2

2,0 ≤ 0 under *x*ˇ2,1 = *α*2,0. Repeating the backstepping procedure until

*<sup>Z</sup>*<sup>2</sup> + *V*1,*r*<sup>1</sup> , where *Z*<sup>2</sup> is the solution to an algebraic

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

as previous steps, we can derive the robust adaptive controller *<sup>α</sup>*1,*r*<sup>1</sup> such that *<sup>V</sup>*˙

1,1 ≤ 0 under *x*ˇ1,2 = *α*1,1. Define

1,*r*<sup>1</sup> ≤ 0 under

121

*η*˙1 = *A*1, *<sup>f</sup> η*<sup>1</sup> + *p*1,*n*<sup>1</sup> *y*1; *η*˙*w*<sup>ˇ</sup> 1,*<sup>j</sup>* = *A*1, *<sup>f</sup> ηw*<sup>ˇ</sup> 1,*<sup>j</sup>* + *p*1,*n*1*w*ˇ 1,*j*; *ηw*ˇ*i*,*j*(0) = *ηw*ˇ*i*,*<sup>j</sup>*0, *i* = 1, ··· , *q*ˇ*<sup>i</sup> <sup>λ</sup>*˙ <sup>1</sup> = *<sup>A</sup>*1, *<sup>f</sup> <sup>λ</sup>*<sup>1</sup> + *<sup>p</sup>*1,*n*<sup>1</sup> *<sup>y</sup>*2; *<sup>λ</sup>*1(0) = *<sup>λ</sup>*1,0 Φ<sup>1</sup> = *An*1−<sup>1</sup> 1, *<sup>f</sup> η*<sup>1</sup> ··· *A*1, *<sup>f</sup> η*<sup>1</sup> *η*<sup>1</sup> *M*−<sup>1</sup> 1, *<sup>f</sup> <sup>A</sup>*¯ 1,211 + *An*1−<sup>1</sup> 1, *<sup>f</sup> λ*<sup>1</sup> ··· *A*1, *<sup>f</sup> λ*<sup>1</sup> *λ*<sup>1</sup> *M*−<sup>1</sup> 1, *<sup>f</sup> <sup>A</sup>*¯ 1,212 + *q*ˇ1 ∑ *j*=1 *An*1−<sup>1</sup> 1, *<sup>f</sup> ηw*<sup>ˇ</sup> 1,*<sup>j</sup>* ··· *A*1, *<sup>f</sup> ηw*<sup>ˇ</sup> 1,*<sup>j</sup> ηw*<sup>ˇ</sup> 1,*<sup>j</sup> M*−<sup>1</sup> 1, *<sup>f</sup> <sup>A</sup>*¯ *i*,213*j η*˙2 = *A*2, *<sup>f</sup> η*<sup>2</sup> + *p*2,*n*<sup>2</sup> *y*2; *η*˙*w*<sup>ˇ</sup> 2,*<sup>j</sup>* = *A*2, *<sup>f</sup> ηw*<sup>ˇ</sup> 2,*<sup>j</sup>* + *p*2,*n*2*w*ˇ 2,*j*; *ηw*<sup>ˇ</sup> 2,*j*(0) = *ηw*<sup>ˇ</sup> 2,*<sup>j</sup>*0, *j* = 1, ··· , *q*ˇ*<sup>i</sup> <sup>λ</sup>*˙ <sup>2</sup> = *<sup>A</sup>*2, *<sup>f</sup> <sup>λ</sup>*<sup>2</sup> + *<sup>p</sup>*2,*n*2*u*; *<sup>λ</sup>*1(0) = *<sup>λ</sup>*1,0 *η*˙2,*<sup>y</sup>* = *A*2, *<sup>f</sup> η*2,*<sup>y</sup>* + *p*2,*n*<sup>2</sup> *y*´2; *η*2,*y*(0) = *η*2,*y*<sup>0</sup> Φ<sup>2</sup> = *An*2−<sup>1</sup> 2, *<sup>f</sup> η*<sup>1</sup> ··· *A*2, *<sup>f</sup> η*<sup>2</sup> *η*<sup>2</sup> *M*−<sup>1</sup> 2, *<sup>f</sup> <sup>A</sup>*¯ 2,211 + *An*2−<sup>1</sup> 2, *<sup>f</sup> λ*<sup>2</sup> ··· *A*2, *<sup>f</sup> λ*<sup>2</sup> *λ*<sup>2</sup> *M*−<sup>1</sup> 2, *<sup>f</sup> <sup>A</sup>*¯ 2,212 + *q*ˇ2 ∑ *j*=1 *An*2−<sup>1</sup> 2, *<sup>f</sup> ηw*<sup>ˇ</sup> 2,*<sup>j</sup>* ··· *A*2, *<sup>f</sup> ηw*<sup>ˇ</sup> 2,*<sup>j</sup> ηw*<sup>ˇ</sup> 2,*<sup>j</sup> M*−<sup>1</sup> 2, *<sup>f</sup> <sup>A</sup>*¯ 2,213*j* + *An*2−<sup>1</sup> 2, *<sup>f</sup> η*2,*<sup>y</sup>* ··· *A*2, *<sup>f</sup> η*2,*<sup>y</sup> η*2,*<sup>y</sup> M*−<sup>1</sup> 2, *<sup>f</sup> <sup>A</sup>*¯ 2,214

The variables to be designed at this stage include *x*ˇ2,1, *u*, *ξ*1,*c*, and *ξ*2,*c*. Note that the structures of *A*<sup>1</sup> and *A*<sup>2</sup> in the dynamics is in strict-feedback form, we will use the backstepping methodology, see [9], to design the control input *u*, which will guarantee the global boundedness of the closed-loop system states and the asymptotic convergence of the tracking error. Since there are the nonnegative definite weighting on *ξ*1,*<sup>c</sup>* and *ξ*2,*<sup>c</sup>* in the cost function (21), we can not use integrator backstepping to design feedback law for *ξ*1,*<sup>c</sup>* and *ξ*2,*c*. Hence, we set *ξ*1,*<sup>c</sup>* = *ξ*2,*<sup>c</sup>* = 0 in the backstepping procedure. After the completion of the backstepping procedure, we will then optimize the choice of *ξ*1,*<sup>c</sup>* and *ξ*2,*<sup>c</sup>* based on the value function obtained. Note that Σ1, Π1, *s*1,Σ, ˇ *θ*1, Σ2, Π2, *s*2,Σ, and ˇ *θ*<sup>2</sup> are always bounded by the design in Section 4.1. Since Φ<sup>1</sup> is driven by control *y*2, and Φ<sup>2</sup> is explicitly driven by *u*, they can not be stabilized in conjunction with *x*ˇ1 and *x*ˇ2 in the backstepping design. We will assume they are bounded and prove later they are indeed so under the derived control law.

We carry out the backstepping design for subsystem **S1** first, and treat *x*ˇ2,1 as the virtual control input of subsystem **S1** in view of *y*<sup>2</sup> = *ζ*−<sup>1</sup> <sup>2</sup> *e*� *<sup>q</sup>*ˇ*a*+2,*q*ˇ1,*a*+2*va* + *x*ˇ2,1. To stabilize *η*1, we introduce variable *η*1,*d*, which satisfies *η*˙1,*<sup>d</sup>* = *A*1, *<sup>f</sup> η*1,*<sup>d</sup>* + *p*1,*n*<sup>1</sup> *yd* with initial condition *η*1,*d*(0) = *η*1,*<sup>d</sup>*0, and is the reference trajectory for *η*<sup>1</sup> to track. Choosing value function *V*1,0 := |*η*<sup>1</sup> − *η*1,*d*| 2 *Z*1 , where *Z*<sup>1</sup> is the solution to an algebraic Riccati equation. Treating *x*ˇ1,1 as the virtual control input, we complete the step 0 with the virtual control law *α*1,0 = *yd*, which will guarantee the *V*˙ 1,0 ≤ 0 under *x*ˇ1,1 = *α*1,0. At step 1, we introduce *z*1,1 := *x*ˇ1,1 − *yd*, and choose value function *V*1,1 = *V*1,0 + <sup>1</sup> 2 *z*2 1,1. Treating *x*ˇ1,2 as the virtual control input, we end the step 1 with the virtual control law *α*1,1, which guarantees *V*˙ 1,1 ≤ 0 under *x*ˇ1,2 = *α*1,1. Define the variable *z*1,2 = *x*ˇ1,2 − *α*1,1 for step 2. Repeating the backstepping procedure until step *r*1, the virtual control input *x*ˇ2,1 will appear in the dynamic of *z*˙1,*<sup>r</sup>*<sup>1</sup> . Using the similar procedure as previous steps, we can derive the robust adaptive controller *<sup>α</sup>*1,*r*<sup>1</sup> such that *<sup>V</sup>*˙ 1,*r*<sup>1</sup> ≤ 0 under *x*ˇ2,1 := *α*1,*r*<sup>1</sup> . This completes the control design for subsystem **S1**.

To stabilize *η*2, we introduce variable *η*2,*<sup>d</sup>* as below,

14 Will-be-set-by-IN-TECH

the following 3*n*<sup>1</sup> + 4*n*<sup>2</sup> + *q*ˇ1 + *q*ˇ2-dimensional prefiltering system for *y*1, *y*2, *u*, *w*ˇ 1, *w*ˇ 2, and *y*´2

1,211 + *An*1−<sup>1</sup>

2,211 + *An*2−<sup>1</sup>

 *M*−<sup>1</sup> 2, *<sup>f</sup> <sup>A</sup>*¯ 2,214

The variables to be designed at this stage include *x*ˇ2,1, *u*, *ξ*1,*c*, and *ξ*2,*c*. Note that the structures of *A*<sup>1</sup> and *A*<sup>2</sup> in the dynamics is in strict-feedback form, we will use the backstepping methodology, see [9], to design the control input *u*, which will guarantee the global boundedness of the closed-loop system states and the asymptotic convergence of the tracking error. Since there are the nonnegative definite weighting on *ξ*1,*<sup>c</sup>* and *ξ*2,*<sup>c</sup>* in the cost function (21), we can not use integrator backstepping to design feedback law for *ξ*1,*<sup>c</sup>* and *ξ*2,*c*. Hence, we set *ξ*1,*<sup>c</sup>* = *ξ*2,*<sup>c</sup>* = 0 in the backstepping procedure. After the completion of the backstepping procedure, we will then optimize the choice of *ξ*1,*<sup>c</sup>* and *ξ*2,*<sup>c</sup>* based on the value

the design in Section 4.1. Since Φ<sup>1</sup> is driven by control *y*2, and Φ<sup>2</sup> is explicitly driven by *u*, they can not be stabilized in conjunction with *x*ˇ1 and *x*ˇ2 in the backstepping design. We will assume they are bounded and prove later they are indeed so under the derived control law. We carry out the backstepping design for subsystem **S1** first, and treat *x*ˇ2,1 as the virtual

we introduce variable *η*1,*d*, which satisfies *η*˙1,*<sup>d</sup>* = *A*1, *<sup>f</sup> η*1,*<sup>d</sup>* + *p*1,*n*<sup>1</sup> *yd* with initial condition *η*1,*d*(0) = *η*1,*<sup>d</sup>*0, and is the reference trajectory for *η*<sup>1</sup> to track. Choosing value function

the virtual control input, we complete the step 0 with the virtual control law *α*1,0 = *yd*, which

2 *z*2

*θ*1, Σ2, Π2, *s*2,Σ, and ˇ

<sup>2</sup> *e*�

, where *Z*<sup>1</sup> is the solution to an algebraic Riccati equation. Treating *x*ˇ1,1 as

1,0 ≤ 0 under *x*ˇ1,1 = *α*1,0. At step 1, we introduce *z*1,1 := *x*ˇ1,1 − *yd*, and

1,1. Treating *x*ˇ1,2 as the virtual control input, we end the

 *M*−<sup>1</sup> 2, *<sup>f</sup> <sup>A</sup>*¯ 2,213*j*

 *M*−<sup>1</sup> 1, *<sup>f</sup> <sup>A</sup>*¯ *i*,213*j*

1, *<sup>f</sup> λ*<sup>1</sup> ··· *A*1, *<sup>f</sup> λ*<sup>1</sup> *λ*<sup>1</sup>

2, *<sup>f</sup> λ*<sup>2</sup> ··· *A*2, *<sup>f</sup> λ*<sup>2</sup> *λ*<sup>2</sup>

 *M*−<sup>1</sup> 1, *<sup>f</sup> <sup>A</sup>*¯ 1,212

 *M*−<sup>1</sup> 2, *<sup>f</sup> <sup>A</sup>*¯ 2,212

*θ*<sup>2</sup> are always bounded by

*<sup>q</sup>*ˇ*a*+2,*q*ˇ1,*a*+2*va* + *x*ˇ2,1. To stabilize *η*1,

*η*˙*w*<sup>ˇ</sup> 1,*<sup>j</sup>* = *A*1, *<sup>f</sup> ηw*<sup>ˇ</sup> 1,*<sup>j</sup>* + *p*1,*n*1*w*ˇ 1,*j*; *ηw*ˇ*i*,*j*(0) = *ηw*ˇ*i*,*<sup>j</sup>*0, *i* = 1, ··· , *q*ˇ*<sup>i</sup>*

1, *<sup>f</sup> ηw*<sup>ˇ</sup> 1,*<sup>j</sup>* ··· *A*1, *<sup>f</sup> ηw*<sup>ˇ</sup> 1,*<sup>j</sup> ηw*<sup>ˇ</sup> 1,*<sup>j</sup>*

*η*˙*w*<sup>ˇ</sup> 2,*<sup>j</sup>* = *A*2, *<sup>f</sup> ηw*<sup>ˇ</sup> 2,*<sup>j</sup>* + *p*2,*n*2*w*ˇ 2,*j*; *ηw*<sup>ˇ</sup> 2,*j*(0) = *ηw*<sup>ˇ</sup> 2,*<sup>j</sup>*0, *j* = 1, ··· , *q*ˇ*<sup>i</sup>*

 *M*−<sup>1</sup> 2, *<sup>f</sup> <sup>A</sup>*¯

2, *<sup>f</sup> ηw*<sup>ˇ</sup> 2,*<sup>j</sup>* ··· *A*2, *<sup>f</sup> ηw*<sup>ˇ</sup> 2,*<sup>j</sup> ηw*<sup>ˇ</sup> 2,*<sup>j</sup>*

2, *<sup>f</sup> η*2,*<sup>y</sup>* ··· *A*2, *<sup>f</sup> η*2,*<sup>y</sup> η*2,*<sup>y</sup>*

 *M*−<sup>1</sup> 1, *<sup>f</sup> <sup>A</sup>*¯

generates the Φ<sup>1</sup> and Φ<sup>2</sup> online:

Φ<sup>1</sup> = *An*1−<sup>1</sup>

Φ<sup>2</sup> = *An*2−<sup>1</sup>

*V*1,0 := |*η*<sup>1</sup> − *η*1,*d*|

will guarantee the *V*˙

2 *Z*1

choose value function *V*1,1 = *V*1,0 + <sup>1</sup>

+ *q*ˇ2 ∑ *j*=1 *An*2−<sup>1</sup>

+ *An*2−<sup>1</sup>

+ *q*ˇ1 ∑ *j*=1 *An*1−<sup>1</sup>

*η*˙1 = *A*1, *<sup>f</sup> η*<sup>1</sup> + *p*1,*n*<sup>1</sup> *y*1;

*η*˙2 = *A*2, *<sup>f</sup> η*<sup>2</sup> + *p*2,*n*<sup>2</sup> *y*2;

*<sup>λ</sup>*˙ <sup>1</sup> = *<sup>A</sup>*1, *<sup>f</sup> <sup>λ</sup>*<sup>1</sup> + *<sup>p</sup>*1,*n*<sup>1</sup> *<sup>y</sup>*2; *<sup>λ</sup>*1(0) = *<sup>λ</sup>*1,0

1, *<sup>f</sup> η*<sup>1</sup> ··· *A*1, *<sup>f</sup> η*<sup>1</sup> *η*<sup>1</sup>

*<sup>λ</sup>*˙ <sup>2</sup> = *<sup>A</sup>*2, *<sup>f</sup> <sup>λ</sup>*<sup>2</sup> + *<sup>p</sup>*2,*n*2*u*; *<sup>λ</sup>*1(0) = *<sup>λ</sup>*1,0 *η*˙2,*<sup>y</sup>* = *A*2, *<sup>f</sup> η*2,*<sup>y</sup>* + *p*2,*n*<sup>2</sup> *y*´2; *η*2,*y*(0) = *η*2,*y*<sup>0</sup>

2, *<sup>f</sup> η*<sup>1</sup> ··· *A*2, *<sup>f</sup> η*<sup>2</sup> *η*<sup>2</sup>

function obtained. Note that Σ1, Π1, *s*1,Σ, ˇ

control input of subsystem **S1** in view of *y*<sup>2</sup> = *ζ*−<sup>1</sup>

$$\dot{\eta}\_{2,d} = A\_{2,f}\eta\_{2,d} + p\_{2,\eta\_2}a\_{1,r\_1} + p\_{2,\eta\_2}e'\_{\tilde{\eta}\_d + 2,\tilde{\eta}\_{1,d} + 2}\upsilon\_{1,r\_1}; \eta\_{2,d}(0) = \eta\_{2,d0}$$

and is the reference trajectory for *η*<sup>2</sup> to track, where *ν*1,*r*<sup>1</sup> is a function obtained after step *r*1. Choosing value function *V*2,0 := |*η*<sup>2</sup> − *η*2,*d*| 2 *<sup>Z</sup>*<sup>2</sup> + *V*1,*r*<sup>1</sup> , where *Z*<sup>2</sup> is the solution to an algebraic Riccati equation. We complete the step *r*<sup>1</sup> + 1 with the virtual control law *α*2,0 = *α*1,*r*<sup>1</sup> , which will guarantee the *V*˙ 2,0 ≤ 0 under *x*ˇ2,1 = *α*2,0. Repeating the backstepping procedure until step *r*<sup>1</sup> + *r*<sup>2</sup> + 1, the virtual control input *u* will appear in the dynamic of *z*˙2,*<sup>r</sup>*<sup>2</sup> . Introduce *<sup>V</sup>*2,*r*<sup>2</sup> = <sup>∑</sup><sup>2</sup> *<sup>j</sup>*=1(|*η*˜*j*| 2 *Zj* <sup>+</sup> <sup>∑</sup>*rj k*=1 1 <sup>2</sup> *<sup>z</sup>*<sup>2</sup> *<sup>j</sup>*,*k*), we then can derive the robust adaptive controller *μ* such that *V*˙ 2,*r*<sup>2</sup> ≤ 0 under *u* := *μ*. Later, we will show that the control law *μ* will guarantee the boundedness of the closed-loop system states and the asymptotic convergence of tracking error.

For the closed-loop adaptive nonlinear system, we have the following value function, *U* = *W*<sup>1</sup> + *W*<sup>2</sup> + *V*2,*r*<sup>2</sup> , and its time derivative is given by

$$\begin{split} \dot{M} &= -|\mathbf{x}\_{1,1} - y\_d|^2 - \sum\_{j=1}^2 (\gamma^4 |\mathbf{x}\_j - \mathbf{x}\_j - \Phi\_j(\theta\_j - \theta\_j)|^2\_{\mathbf{II}\_j^{-1}\Delta\_1 \mathbf{II}\_j^{-1}} + \epsilon\_f (\gamma^2 \xi\_j^2 - 1) |\theta\_j - \dot{\theta}\_j|^2\_{\Phi\_j^\* C\_1 \mathbf{C}\_1 \Phi\_1} \\ &- 2 \left(\theta\_j - \dot{\theta}\_j \right)' \mathbf{P}\_{j,r}(\theta\_j) + |\dot{\eta}\_j|^2\_{\mathbf{V}\_j} + \sum\_{k=1}^{r\_l} \beta\_{j,k} \mathbf{z}\_{j,k}^2 - \gamma^2 |w\_j|^2 + \gamma^2 |w\_j - w\_{j, opt}|^2 - \gamma^2 |\dot{w}\_{j,a}|^2 \\ &+ \gamma^2 |\dot{w}\_{j,a} - \dot{w}\_{j,opt}|^2 \right) - \epsilon \mathbf{z} |\theta\_2 - \dot{\theta}\_2|\_{\Phi\_2^\* C\_2^\* \mathbf{C}\_2 \Phi\_2} - \frac{1}{4} \left| \zeta\_{1,(r\_1 + r\_2)} \right|\_{\dot{Q}\_1}^2 - \frac{1}{4} \left| \zeta\_{2, r\_2} \right|\_{\dot{Q}\_2}^2 \\ &+ \left| \widetilde{\xi}\_{1,c} + \frac{1}{2} \zeta\_{1,(r\_1 + r\_2)} \right|\_{Q\_1}^2 + \left| \widetilde{\xi}\_{2,c} + \frac{1}{2} \zeta\_{2,r\_2} \right|\_{Q\_2}^2 \end{split}$$

where *ς*1,*r*1+*r*<sup>2</sup> and *ς*2,*r*<sup>2</sup> are functions obtained after step *r*<sup>1</sup> + *r*<sup>2</sup> + 1, *w*1,*opt* and *w*2,*opt* are the worst case disturbance with respect to the value function *U*, which are given by

*w*1,*opt* = *ζ*1*E*� 1*e* � 2,1*ν*1,*r*<sup>1</sup> <sup>+</sup> *<sup>γ</sup>*−<sup>2</sup> (*Iq*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup> 1*E*� <sup>1</sup>*E*1)*D*¯ � <sup>1</sup>Σ¯ <sup>−</sup><sup>1</sup> <sup>1</sup> (*ξ*<sup>1</sup> <sup>−</sup> <sup>ˇ</sup> *ξ*1) + *ζ*<sup>2</sup> 1*E*� <sup>1</sup>*C*<sup>1</sup> (*x*ˇ1 − *x*1) *w*2,*opt* = *ζ*2*E*� 2*e* � 2,2*ν*2,*r*<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*−<sup>2</sup> (*Iq*<sup>2</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup> 2*E*� <sup>2</sup>*E*2)*D*¯ � <sup>2</sup>Σ¯ <sup>−</sup><sup>1</sup> <sup>2</sup> (*ξ*<sup>2</sup> <sup>−</sup> <sup>ˇ</sup> *ξ*2) + *ζ*<sup>2</sup> 2*E*� <sup>2</sup>*C*<sup>2</sup> (*x*ˇ2 − *x*2) *w*ˇ 1,*opt* = **0**� <sup>1</sup>×(2+*q*ˇ1,*a*+*q*ˇ2,*a*) *<sup>e</sup>*(2+*q*ˇ1,*a*+*q*ˇ2,*a*),1 ··· *<sup>e</sup>*(2+*q*ˇ1,*a*+*q*ˇ2,*a*),*q*ˇ1,*<sup>a</sup>* **<sup>0</sup>**� (1+*q*ˇ2,*a*)×(2+*q*ˇ1,*a*+*q*ˇ2,*a*) � *ν*1,*r*<sup>1</sup> *w*ˇ 2,*opt* = **0**� (2+*q*ˇ1,*a*)×(2+*q*ˇ1,*a*+*q*ˇ2,*a*) *<sup>e</sup>*(2+*q*ˇ1,*a*+*q*ˇ2,*a*),1 ··· *<sup>e</sup>*(2+*q*ˇ1,*a*+*q*ˇ2,*a*),*q*ˇ2,*<sup>a</sup>* � *ν*2,*r*<sup>2</sup>

where *ν*1,*r*<sup>1</sup> and *ν*2,*r*<sup>2</sup> are functions obtained after backstepping design.

Then the optimal choice for the variable *ξi*,*<sup>c</sup>* and ˆ *ξi*, *i* = 1, 2, are:

$$\begin{aligned} \xi\_{1,\mathcal{L}\*} &= -\frac{1}{2}\xi\_{1,r\_1+r\_2} \Longleftrightarrow \xi\_{1,\*} = \check{\xi}\_1 - \frac{1}{2}\xi\_{1,r\_1+r\_2}, \\ \xi\_{2,\mathcal{L}\*} &= -\frac{1}{2}\xi\_{2,r\_2} \Longleftrightarrow \xi\_{2,\*} = \check{\xi}\_2 - \frac{1}{2}\xi\_{2,r\_2} \end{aligned}$$

*1. Given cw* ≥ 0*, and cd* ≥ 0*, there exists a constant cc* ≥ 0 *and compact sets* Θ1,*<sup>c</sup>* ⊂ Θ1,*o,*

*and* (*x*2,0, *<sup>θ</sup>*2, *<sup>w</sup>*` 2,[0,∞), *<sup>w</sup>*<sup>ˇ</sup> 2,[0,<sup>∞</sup>)) <sup>∈</sup> <sup>W</sup>` <sup>2</sup> *with* <sup>|</sup>*x*1,0| ≤ *cw*; <sup>|</sup>*x*2,0| ≤ *cw*; <sup>|</sup>*w*` <sup>1</sup>(*t*)| ≤ *cw*; <sup>|</sup>*w*` <sup>2</sup>(*t*)| ≤ *cw*; <sup>|</sup>*w*<sup>ˇ</sup> <sup>1</sup>(*t*)| ≤ *cw*; <sup>|</sup>*w*<sup>ˇ</sup> <sup>2</sup>(*t*)| ≤ *cw*; <sup>|</sup>*Yd*(*t*)| ≤ *cd*; <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, <sup>∞</sup>) *all closed-loop state variables x*1*, <sup>x</sup>*ˇ1*,* <sup>ˇ</sup>

*<sup>ξ</sup>*<sup>1</sup> <sup>≤</sup> *cu,* <sup>ˆ</sup>

*cu* ≥ 0*. Furthermore, there exists constant c<sup>λ</sup>* ≥ 0 *such that* |*λi*,0(*t*)| ≤ *cλ,* |*λi*(*t*)| ≤ *cλ, i* = 1, 2*,*

*<sup>w</sup>*<sup>ˇ</sup> 2,[0,<sup>∞</sup>)) <sup>∈</sup> <sup>W</sup>` <sup>2</sup> *the controller <sup>μ</sup>* ∈ M *achieves disturbance attenuation level <sup>γ</sup> with respect to w*<sup>1</sup> *and w*2*, arbitrary disturbance attenuation level γ*ˇ *with respect to w*ˇ 1,*<sup>a</sup> and w*ˇ 2,*a, and disturbance*

*<sup>w</sup>*<sup>ˇ</sup> <sup>2</sup>[0,<sup>∞</sup>)) <sup>∈</sup> <sup>W</sup>` <sup>2</sup> *with <sup>w</sup>*` <sup>1</sup>[0,∞) ∈ L<sup>2</sup> ∩ L∞*, <sup>w</sup>*` <sup>2</sup>[0,∞) ∈ L<sup>2</sup> ∩ L∞*, <sup>w</sup>*<sup>ˇ</sup> 1,*a*[0,∞) ∈ L<sup>2</sup> ∩ L∞*, w*ˇ 2,*a*[0,∞) ∈ L<sup>2</sup> ∩ L<sup>∞</sup> *w*ˇ 1,*b*[0,∞) ∈ L∞*, w*ˇ 2,*b*[0,∞) ∈ L∞*, and Yd*[0,∞) ∈ L∞*, the noiseless output of*

*<sup>t</sup>*→<sup>∞</sup> (*x*1,1(*t*) <sup>−</sup> *yd*(*t*)) <sup>=</sup> <sup>0</sup> *4. The ultimate lower bound on the achievable performance level is only relevant to the Subsystem* **S1***,*

**Proof** For the first statement, fix *cw* ≥ 0, and *cd* ≥ 0 consider any uncertainty (*x*1,0, *x*2,0,


We define [0, *Tf*) to be the maximal length interval on which the closed system (22) has a solution that lies in D. Note that we have Σ1, Σ2, *s*1,<sup>Σ</sup> and *s*2,<sup>Σ</sup> are uniformly upper bounded

> *θ*2)� *η*˜� <sup>1</sup> *η*˜�

˜ *θi*| 2 Π−<sup>1</sup> *i* + 2 ∑ *i*=1 |*η*˜*i*| 2 *Zi* +

˜ *θi*| 2 Π−<sup>1</sup> *i* + 2 ∑ *i*=1 |*η*˜*i*| 2 *Zi* +

<sup>2</sup> *z*1,1 ··· *z*1,*r*<sup>1</sup> *z*2,1 ··· *z*2,*r*<sup>2</sup>

*r*1 ∑ *j*=1

> *r*1 ∑ *j*=1

*<sup>γ</sup>*1,*jz*<sup>2</sup> 1,*<sup>j</sup>* +

> *γ*1,*jz*<sup>2</sup> 1,*<sup>j</sup>* +

�

*<sup>γ</sup>*2,*jz*<sup>2</sup> 2,*j*

> *γ*2,*jz*<sup>2</sup> 2,*j*

*r*2 ∑ *j*=1

> *r*2 ∑ *j*=1

*<sup>d</sup>*[0,∞) ) that satisfies:

(*r*1+*r*2)

*θ*2*,* Σ2*, s*2,Σ*, η*2*, η*2,*d,* Φ2,*<sup>u</sup> are bounded as follows,* ∀*t* ∈ [0, ∞)*,*

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

*<sup>i</sup>*,0 ; *<sup>γ</sup>*2Tr(*Qi*,0)<sup>≤</sup> *si*,Σ(*t*) <sup>≤</sup> *Ki*,*c*; *<sup>i</sup>*=1, 2

*ξ*<sup>2</sup> ≤ *cu,* ∀*t* ∈ [0, ∞)*, for some constant*

*<sup>d</sup>*[0,∞) ) <sup>∈</sup> <sup>W</sup>` <sup>1</sup>*, and* (*x*2,0, *<sup>θ</sup>*2, *<sup>w</sup>*` 2,[0,∞),

*<sup>d</sup>*[0,∞) ) <sup>∈</sup> <sup>W</sup>` <sup>1</sup>*, and* (*x*2,0, *<sup>θ</sup>*2, *<sup>w</sup>*` <sup>2</sup>[0,∞),

*θi*(*t*) ∈ Θ*i*,*c*; |*ηi*(*t*)| ≤ *cc*; |*ηi*,*w*ˇ,1(*t*)| ≤ *cc*; ···|*ηi*,*w*ˇ,*q*ˇ*<sup>i</sup>*

(*r*1+*r*2)

(*r*1+*r*2)

(*r*1+*r*2) *<sup>d</sup>*[0,∞) ) <sup>∈</sup> <sup>W</sup>` <sup>1</sup> 123


*and* Θ2,*<sup>c</sup>* ⊂ Θ2,*<sup>o</sup> such that for any uncertainty* (*x*1,0, *θ*1, *w*` 1,[0,∞), *w*ˇ 1,[0,∞),*Yd*0, *y*

*<sup>i</sup>*,*<sup>c</sup> <sup>I</sup>* <sup>≤</sup> <sup>Σ</sup>*i*(*t*)<sup>≤</sup> *<sup>γ</sup>*−2*Q*−<sup>1</sup>

*<sup>θ</sup>*1*,* <sup>Σ</sup>1*, s*1,Σ*, <sup>η</sup>*1*, <sup>η</sup>*1,*d,* <sup>Φ</sup>1,*u, x*2*, <sup>x</sup>*ˇ2*,* <sup>ˇ</sup>

<sup>|</sup>*ηi*,*d*(*t*)| ≤ *cc*; <sup>|</sup>Φ*i*,*u*(*t*)| ≤ *cc*; *<sup>K</sup>*−<sup>1</sup>

*and* |*η*2,*y*(*t*)| ≤ *cλ,* ∀*t* ≥ 0*.*

*i.e., <sup>γ</sup>* <sup>≥</sup> *<sup>ζ</sup>*−<sup>1</sup>

<sup>1</sup> *or <sup>γ</sup>* <sup>&</sup>gt; *<sup>ζ</sup>*−<sup>1</sup>

*θ*1, *θ*2, *w*` 1,[0,∞), *w*` 2,[0,∞), *w*ˇ 1,[0,∞), *w*ˇ 2,[0,∞), *y*

Introduce the vector of variables *Xe* := ˇ *θ*� 1 ˇ *θ*�

> 2 ∑ *i*=1

2 ∑ *i*=1 *<sup>γ</sup>*2|˜ *θi*| 2 *Qi*,0 +

*Ki*,*c*|˜ *θi*| <sup>2</sup> + 2 ∑ *i*=1

*UM*(*Xe*) :=

*Um*(*Xe*) :=

<sup>1</sup> *.*

and uniformly bounded away from 0 as desired by Section 4.

*<sup>θ</sup>*1)� (*x*˜2 <sup>−</sup> <sup>Φ</sup><sup>2</sup> ˜

and two nonnegative and continuous functions defined on IR2*n*1+2*n*2+*σ*1+*σ*2+*r*1+*r*<sup>2</sup>

*<sup>γ</sup>*2|*x*˜*<sup>i</sup>* <sup>−</sup> <sup>Φ</sup>*<sup>i</sup>*

*<sup>γ</sup>*2|*x*˜*<sup>i</sup>* <sup>−</sup> <sup>Φ</sup>*<sup>i</sup>*

2 ∑ *i*=1

<sup>2</sup> (*x*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜

<sup>|</sup>*xi*(*t*)| ≤ *cc*; <sup>|</sup>*x*ˇ*i*(*t*)| ≤ *cc*; <sup>ˇ</sup>

*The inputs are also bounded* <sup>|</sup>*u*(*t*)| ≤ *cu, and* <sup>ˆ</sup>

*2. For any uncertainty* (*x*1,0, *θ*1, *w*` 1,[0,∞), *w*ˇ 1,[0,∞),*Yd*0, *y*

*attenuation level zero with respect to w*ˇ 1,*<sup>b</sup> and w*ˇ 2,*b, . 3. For any uncertainty* (*x*1,0, *θ*1, *w*` <sup>1</sup>[0,∞), *w*ˇ <sup>1</sup>[0,∞),*Yd*0, *y*

*the system, x*1,1*, asymptotically tracks the reference trajectory, yd, i*.*e*.*,* lim

which yields that the closed-loop system is dissipative with storage function *U* and supply rate with optimal choice for ˆ *ξi*, *i* = 1, 2:

$$-|x\_{1,1} - y\_d|^2 + \gamma^2 |w\_1|^2 + \gamma^2 |w\_2|^2 + \gamma^2 |\psi\_{1,a}|^2 + \gamma^2 |\psi\_{2,a}|^2$$

This completes the adaptive controller design step. We will discuss the robustness and tracking properties of the proposed adaptive control laws.

#### **5. Main result**

In this Section, we present the main result by stating two theorems.

For the adaptive control law, with the optimal choice of *<sup>ξ</sup>i*,*c*∗, the closed-loop system dynamics are

$$\dot{X} = F(X, y\_d^{(r\_1+r\_2)}) + G(X) \left[ w\_1' \; w\_2' \right] + G\_{\mathbb{H}}(X) \left[ \; \vartheta\_1' \; \vartheta\_2' \right]; \\ X(0) = X\_0 \tag{22}$$

where *F*, *G* and *GM* are smooth mapping of D × IR, D and D, respectively; and the initial condition *<sup>X</sup>*<sup>0</sup> ∈ D<sup>0</sup> :<sup>=</sup> {*X*<sup>0</sup> ∈ D| *<sup>θ</sup><sup>i</sup>* <sup>∈</sup> <sup>Θ</sup>*i*, <sup>ˇ</sup> *<sup>θ</sup>i*,0 <sup>∈</sup> <sup>Θ</sup>*i*, <sup>Σ</sup>*i*(0) = *<sup>γ</sup>*−2*Q*−<sup>1</sup> *<sup>i</sup>*,0 > 0, Tr (Σ*i*(0))−<sup>1</sup> <sup>≤</sup> *Ki*,*c*,*si*,Σ(0) = *<sup>γ</sup>*2Tr(*Qi*,0); *<sup>i</sup>* <sup>=</sup> 1, 2}. And the value function *<sup>U</sup>* satisfies an Hamilton-Jacobi-Isaacs equation, ∀*X* ∈ D, ∀*y* (*r*1+*r*2) *<sup>d</sup>* ∈ IR.

$$\begin{split} \frac{\partial \mathcal{U}}{\partial \mathbf{X}}(\mathbf{X}) F(\mathbf{X}, y\_d^{(r\_1+r\_2)}) + \frac{1}{4\gamma^2} \frac{\partial \mathcal{U}}{\partial \mathbf{X}}(\mathbf{X}) \left[ \mathbf{G}(\mathbf{X}) \ \mathbf{G}\_{\vec{\mathbf{w}}}(\mathbf{X}) \right] \left[ \mathbf{G}(\mathbf{X})' \ \mathbf{G}\_{\vec{\mathbf{w}}}(\mathbf{X})' \right]' \left( \frac{\partial \mathcal{U}}{\partial \mathbf{X}}(\mathbf{X}) \right)' \\ + Q(\mathbf{X}, y\_d^{(r\_1+r\_2)}) = 0; \end{split}$$

where *Q* : D × IR → IR is smooth and given by

$$\begin{split} Q(\mathbf{X}, \mathbf{y}\_d^{(r\_1+r\_2)}) &= |\mathbf{x}\_{1,1} - y\_d|^2 + \sum\_{j=1}^2 \left( \gamma^4 |\mathbf{x}\_j - \mathbf{x}\_j - \Phi\_j(\theta\_j - \hat{\theta})\_j|^2\_{\Pi\_j^{-1}\Delta\_j \Pi\_j^{-1}} \\ &+ \epsilon\_j \left( \gamma^2 \xi\_j^2 - 1 \right) |\theta\_j - \hat{\theta}\_j|^2\_{\Phi\_j'C\_j'\mathcal{C}\_l\Phi\_l} - 2\left(\theta\_j - \hat{\theta}\_j\right)' P\_{j,r}(\hat{\theta}\_j) + |\theta\_j|^2\_{Y\_j} + \sum\_{k=1}^{r\_j} \theta\_{j,k} z\_{j,k}^2 \right) \\ &+ \frac{1}{4} \left| \zeta\_{1,(r\_1+r\_2)} \right|\_{\hat{Q}\_1}^2 + \frac{1}{4} \left| \zeta\_{2,r\_2} \right|\_{Q\_2}^2 + \epsilon\_2 |\theta\_2 - \hat{\theta}\_2| \Phi\_2 \zeta\_2' \mathbf{c}\_2 \Phi\_2 \end{split}$$

The closed-loop adaptive system possesses a strong stability property, which will be stated precisely in the following theorem.

**Theorem 1.** *Consider the robust adaptive control problem formulated and assumptions in Section 3. The robust adaptive controller μ with the optimal choice of ξi*,*c, achieves the following strong robustness properties for the closed-loop system.*

*1. Given cw* ≥ 0*, and cd* ≥ 0*, there exists a constant cc* ≥ 0 *and compact sets* Θ1,*<sup>c</sup>* ⊂ Θ1,*o, and* Θ2,*<sup>c</sup>* ⊂ Θ2,*<sup>o</sup> such that for any uncertainty* (*x*1,0, *θ*1, *w*` 1,[0,∞), *w*ˇ 1,[0,∞),*Yd*0, *y* (*r*1+*r*2) *<sup>d</sup>*[0,∞) ) <sup>∈</sup> <sup>W</sup>` <sup>1</sup> *and* (*x*2,0, *<sup>θ</sup>*2, *<sup>w</sup>*` 2,[0,∞), *<sup>w</sup>*<sup>ˇ</sup> 2,[0,<sup>∞</sup>)) <sup>∈</sup> <sup>W</sup>` <sup>2</sup> *with* <sup>|</sup>*x*1,0| ≤ *cw*; <sup>|</sup>*x*2,0| ≤ *cw*; <sup>|</sup>*w*` <sup>1</sup>(*t*)| ≤ *cw*; <sup>|</sup>*w*` <sup>2</sup>(*t*)| ≤ *cw*; <sup>|</sup>*w*<sup>ˇ</sup> <sup>1</sup>(*t*)| ≤ *cw*; <sup>|</sup>*w*<sup>ˇ</sup> <sup>2</sup>(*t*)| ≤ *cw*; <sup>|</sup>*Yd*(*t*)| ≤ *cd*; <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, <sup>∞</sup>) *all closed-loop state variables x*1*, <sup>x</sup>*ˇ1*,* <sup>ˇ</sup> *<sup>θ</sup>*1*,* <sup>Σ</sup>1*, s*1,Σ*, <sup>η</sup>*1*, <sup>η</sup>*1,*d,* <sup>Φ</sup>1,*u, x*2*, <sup>x</sup>*ˇ2*,* <sup>ˇ</sup> *θ*2*,* Σ2*, s*2,Σ*, η*2*, η*2,*d,* Φ2,*<sup>u</sup> are bounded as follows,* ∀*t* ∈ [0, ∞)*,*

$$|\mathbf{x}\_{i}(t)| \le c\_{\varepsilon} \cdot |\mathfrak{x}\_{i}(t)| \le c\_{\varepsilon} \cdot \check{\theta}\_{i}(t) \in \Theta\_{i,\varepsilon'} \cdot |\eta\_{i}(t)| \le c\_{\varepsilon'} \cdot |\eta\_{i,\nexists \mathfrak{k},1}(t)| \le c\_{\varepsilon} \cdot \dots \cdot |\eta\_{i,\mathfrak{k},\tilde{\rho}\_{i}|}| \le c\_{\varepsilon'} \cdot \check{\theta}\_{i}(t)$$

$$|\eta\_{i\cdot \mathbf{i}}(t)| \le c\_{\mathbf{i}\cdot} |\Phi\_{\mathbf{i},\mathbf{i}}(t)| \le c\_{\mathbf{i}\cdot} \mathbf{K}\_{\mathbf{i},\mathbf{i}}^{-1} \mathbf{I} \le \Sigma\_{\mathbf{i}}(t) \le \gamma^{-2} \mathbf{Q}\_{\mathbf{i},0}^{-1}; \quad \gamma^{2} \text{Tr}(\mathbf{Q}\_{\mathbf{i},0}) \le s\_{\mathbf{i},\Sigma}(t) \le \mathbf{K}\_{\mathbf{i},\mathbf{i}}; \quad \mathbf{i} = \mathbf{1}, \mathbf{2}$$

*The inputs are also bounded* <sup>|</sup>*u*(*t*)| ≤ *cu, and* <sup>ˆ</sup> *<sup>ξ</sup>*<sup>1</sup> <sup>≤</sup> *cu,* <sup>ˆ</sup> *ξ*<sup>2</sup> ≤ *cu,* ∀*t* ∈ [0, ∞)*, for some constant cu* ≥ 0*. Furthermore, there exists constant c<sup>λ</sup>* ≥ 0 *such that* |*λi*,0(*t*)| ≤ *cλ,* |*λi*(*t*)| ≤ *cλ, i* = 1, 2*, and* |*η*2,*y*(*t*)| ≤ *cλ,* ∀*t* ≥ 0*.*


$$\lim\_{t \to \infty} \left( x\_{1,1}(t) - y\_d(t) \right) = 0$$

*4. The ultimate lower bound on the achievable performance level is only relevant to the Subsystem* **S1***, i.e., <sup>γ</sup>* <sup>≥</sup> *<sup>ζ</sup>*−<sup>1</sup> <sup>1</sup> *or <sup>γ</sup>* <sup>&</sup>gt; *<sup>ζ</sup>*−<sup>1</sup> <sup>1</sup> *.*

**Proof** For the first statement, fix *cw* ≥ 0, and *cd* ≥ 0 consider any uncertainty (*x*1,0, *x*2,0, *θ*1, *θ*2, *w*` 1,[0,∞), *w*` 2,[0,∞), *w*ˇ 1,[0,∞), *w*ˇ 2,[0,∞), *y* (*r*1+*r*2) *<sup>d</sup>*[0,∞) ) that satisfies:


We define [0, *Tf*) to be the maximal length interval on which the closed system (22) has a solution that lies in D. Note that we have Σ1, Σ2, *s*1,<sup>Σ</sup> and *s*2,<sup>Σ</sup> are uniformly upper bounded and uniformly bounded away from 0 as desired by Section 4.

Introduce the vector of variables

16 Will-be-set-by-IN-TECH

*<sup>ς</sup>*1,*r*1+*r*<sup>2</sup> ⇐⇒ <sup>ˆ</sup>

*<sup>ς</sup>*2,*r*<sup>2</sup> ⇐⇒ <sup>ˆ</sup>

*ξi*, *i* = 1, 2, are:

*<sup>ξ</sup>*<sup>2</sup> <sup>−</sup> <sup>1</sup> 2 *ς*2,*r*<sup>2</sup>

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*w*<sup>ˇ</sup> 1,*a*<sup>|</sup>

+ *Gw*ˇ(*X*)

 *w*ˇ� <sup>1</sup> *w*ˇ� 2 

*<sup>G</sup>*(*X*)� *Gw*ˇ(*X*)� �

*θ*)*j*| 2 Π−<sup>1</sup> *<sup>j</sup>* <sup>Δ</sup>*j*Π−<sup>1</sup> *j*

*θj*)� *Pj*,*r*(ˇ

*θ*2|Φ� 2*C*� <sup>2</sup>*C*2Φ<sup>2</sup>

*θj*) + |*η*˜*j*|

2 *Yj* +

*rj* ∑ *k*=1

*<sup>β</sup>j*,*kz*<sup>2</sup> *j*,*k* 

<sup>−</sup> <sup>2</sup> (*θ<sup>j</sup>* <sup>−</sup> <sup>ˇ</sup>

*<sup>Q</sup>*¯ <sup>2</sup> <sup>+</sup> *�*2|*θ*<sup>2</sup> <sup>−</sup> <sup>ˆ</sup>

*<sup>ξ</sup>*<sup>1</sup> <sup>−</sup> <sup>1</sup> 2

*ς*1,*r*1+*r*<sup>2</sup> ;

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*w*<sup>ˇ</sup> 2,*a*<sup>|</sup>

2

; *X*(0) = *X*<sup>0</sup> (22)

*<sup>i</sup>*,0 >

�

*<sup>θ</sup>i*,0 <sup>∈</sup> <sup>Θ</sup>*i*, <sup>Σ</sup>*i*(0) = *<sup>γ</sup>*−2*Q*−<sup>1</sup>

*∂U <sup>∂</sup><sup>X</sup>* (*X*)

*<sup>ξ</sup>*1,<sup>∗</sup> <sup>=</sup> <sup>ˇ</sup>

*<sup>ξ</sup>*2,<sup>∗</sup> <sup>=</sup> <sup>ˇ</sup>

which yields that the closed-loop system is dissipative with storage function *U* and supply

This completes the adaptive controller design step. We will discuss the robustness and

For the adaptive control law, with the optimal choice of *<sup>ξ</sup>i*,*c*∗, the closed-loop system dynamics

where *F*, *G* and *GM* are smooth mapping of D × IR, D and D, respectively; and the

(Σ*i*(0))−<sup>1</sup> <sup>≤</sup> *Ki*,*c*,*si*,Σ(0) = *<sup>γ</sup>*2Tr(*Qi*,0); *<sup>i</sup>* <sup>=</sup> 1, 2}. And the value function *<sup>U</sup>* satisfies

*G*(*X*) *Gw*ˇ(*X*)

*<sup>γ</sup>*4|*xj* <sup>−</sup> *<sup>x</sup>*ˆ*<sup>j</sup>* <sup>−</sup> <sup>Φ</sup>*j*(*θ<sup>j</sup>* <sup>−</sup> <sup>ˆ</sup>

The closed-loop adaptive system possesses a strong stability property, which will be stated

**Theorem 1.** *Consider the robust adaptive control problem formulated and assumptions in Section 3. The robust adaptive controller μ with the optimal choice of ξi*,*c, achieves the following strong robustness*

(*r*1+*r*2) *<sup>d</sup>* ∈ IR.

 *w*� <sup>1</sup> *w*� 2 

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*w*2<sup>|</sup>

Then the optimal choice for the variable *ξi*,*<sup>c</sup>* and ˆ

−|*x*1,1 − *yd*|

*X*˙ = *F*(*X*, *y*

rate with optimal choice for ˆ

**5. Main result**

are

0, Tr

*∂U*

*Q*(*X*, *y*

*<sup>∂</sup><sup>X</sup>* (*X*)*F*(*X*, *<sup>y</sup>*

+*Q*(*X*, *y*

(*r*1+*r*2)

*<sup>ξ</sup>*1,*c*<sup>∗</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

*<sup>ξ</sup>*2,*c*<sup>∗</sup> <sup>=</sup> <sup>−</sup><sup>1</sup>

tracking properties of the proposed adaptive control laws.

(*r*1+*r*2)

an Hamilton-Jacobi-Isaacs equation, ∀*X* ∈ D, ∀*y*

(*r*1+*r*2) *<sup>d</sup>* ) + <sup>1</sup>

where *Q* : D × IR → IR is smooth and given by

+*�<sup>j</sup>* (*γ*2*ζ*<sup>2</sup>

(*r*1+*r*2) *<sup>d</sup>* ) = 0;

*<sup>d</sup>* ) = |*x*1,1 − *yd*|

+ 1 4 *ς*1,(*r*1+*r*2)

precisely in the following theorem.

*properties for the closed-loop system.*

2

2

*ξi*, *i* = 1, 2:

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*w*1<sup>|</sup>

In this Section, we present the main result by stating two theorems.

*<sup>d</sup>* ) + *G*(*X*)

initial condition *<sup>X</sup>*<sup>0</sup> ∈ D<sup>0</sup> :<sup>=</sup> {*X*<sup>0</sup> ∈ D| *<sup>θ</sup><sup>i</sup>* <sup>∈</sup> <sup>Θ</sup>*i*, <sup>ˇ</sup>

4*γ*<sup>2</sup>

<sup>2</sup> + 2 ∑ *j*=1

*<sup>j</sup>* <sup>−</sup> <sup>1</sup>)|*θ<sup>j</sup>* <sup>−</sup> <sup>ˆ</sup>

 2 *Q*¯ 1 + 1 <sup>4</sup> <sup>|</sup>*ς*2,*r*<sup>2</sup> <sup>|</sup> 2

*∂U <sup>∂</sup><sup>X</sup>* (*X*) 

*θj*| 2 Φ� *j C*� *j Cj*Φ*<sup>j</sup>*

$$X\_{\mathcal{C}} := \left[\delta\_1' \,\delta\_2' \, (\mathfrak{x}\_1 - \Phi\_1 \mathfrak{f}\_1)' \, (\mathfrak{x}\_2 - \Phi\_2 \mathfrak{f}\_2)' \, \mathfrak{h}\_1' \, \mathfrak{h}\_2' \, z\_{1,1} \cdots z\_{1,r\_1} \, z\_{2,1} \cdots z\_{2,r\_2}\right]'$$

and two nonnegative and continuous functions defined on IR2*n*1+2*n*2+*σ*1+*σ*2+*r*1+*r*<sup>2</sup>

$$\begin{split} \mathcal{U}\_{M}(\boldsymbol{X}\_{\varepsilon}) &:= \sum\_{i=1}^{2} K\_{i,\varepsilon} |\boldsymbol{\theta}\_{i}|^{2} + \sum\_{i=1}^{2} \gamma^{2} |\mathfrak{x}\_{i} - \boldsymbol{\Phi}\_{i}\boldsymbol{\theta}\_{i}|\_{\Pi\_{i}^{-1}}^{2} + \sum\_{i=1}^{2} |\boldsymbol{\eta}\_{i}|\_{\boldsymbol{Z}\_{i}}^{2} + \sum\_{j=1}^{r\_{1}} \gamma\_{1,j} z\_{1,j}^{2} + \sum\_{j=1}^{r\_{2}} \gamma\_{2,j} z\_{2,j}^{2} \\ \mathcal{U}\_{\mathcal{U}}(\boldsymbol{X}\_{\varepsilon}) &:= \sum\_{i=1}^{2} \gamma^{2} |\boldsymbol{\theta}\_{i}|\_{\boldsymbol{Q}\_{i} \boldsymbol{0}}^{2} + \sum\_{i=1}^{2} \gamma^{2} |\mathfrak{x}\_{i} - \boldsymbol{\Phi}\_{i}\boldsymbol{\theta}\_{i}|\_{\Pi\_{i}^{-1}}^{2} + \sum\_{i=1}^{2} |\boldsymbol{\eta}\_{i}|\_{\boldsymbol{Z}\_{i}}^{2} + \sum\_{j=1}^{r\_{1}} \gamma\_{1,j} z\_{1,j}^{2} + \sum\_{j=1}^{r\_{2}} \gamma\_{2,j} z\_{2,j}^{2} \end{split}$$

then, we have

$$\mathcal{U}\_{\mathfrak{M}}(X\_{\mathfrak{e}}) \le \mathcal{U}(t, X\_{\mathfrak{e}}) \le \mathcal{U}\_{\mathcal{M}}(X\_{\mathfrak{e}}), \quad \forall (t, X\_{\mathfrak{e}}) \in [0, T\_f) \times \mathbb{R}^{2(n\_1 + n\_2) + \sigma\_1 + \sigma\_2 + r\_1 + r\_2}$$

the reference system, we conclude Φ1,*us* <sup>1</sup> is uniformly bounded by bounding Lemma. Because

1,212 0 ˜

*θ*1, Φ1,*us* 1, and *η*1. Since *z*1,1 = *x*ˇ1,1 − *yd*, and *z*1,1, *yd* are both uniformly bounded,

**<sup>0</sup>**1×*r*<sup>1</sup> *<sup>b</sup>*1,*p*<sup>1</sup> ··· *<sup>b</sup>*1,*p n*1−*r*<sup>1</sup>

*θ*<sup>1</sup> − *η*�

<sup>1</sup>*T*1,1 ˜ *θ*1

*θ*<sup>1</sup> is uniformly bounded in view of the boundedness

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

<sup>1</sup> <sup>−</sup> *<sup>γ</sup>*−2), we generated the signal *<sup>x</sup>*1,1 <sup>−</sup> *<sup>b</sup>*1,0*λ*1,*b*<sup>1</sup>

*y*<sup>2</sup> + *A*¯ 1,211*θ*1*y*<sup>1</sup> + *D*1*M*` <sup>1</sup>*w*` <sup>1</sup> + (*ζ*<sup>2</sup>

**<sup>0</sup>**1×*r*<sup>1</sup> *<sup>b</sup>*1,*p*<sup>1</sup> ··· *<sup>b</sup>*1,*p n*1−*r*<sup>1</sup>

� *y*2

*<sup>γ</sup>*<sup>2</sup> )) (*y*<sup>1</sup> <sup>−</sup> *<sup>E</sup>*1*M*` <sup>1</sup>*w*` <sup>1</sup>) + *<sup>A</sup>*¯ 1,211*θ*1*y*<sup>1</sup> <sup>+</sup> *<sup>D</sup>*1*M*` <sup>1</sup>*w*` <sup>1</sup>

*θ*1) is also uniformly bounded. Since *x*ˇ1,1 is uniformly bounded

*θ*1))−1*P*1(ˇ

*θ*1)

<sup>1</sup>*L*<sup>1</sup>

125

� *y*2

*θ*1(*t*) ∈ Θ1,*c*,

*<sup>θ</sup>*<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*1,*b*1*A*¯

<sup>1</sup>*C*<sup>1</sup> (*ζ*<sup>2</sup>

*A*¯ 1,212*r*<sup>1</sup> *θ*<sup>1</sup>

*<sup>γ</sup>*<sup>2</sup> )) (*y*<sup>1</sup> <sup>−</sup> *<sup>E</sup>*1*M*` <sup>1</sup>*w*` <sup>1</sup>) +

Now we will separate the above dynamics into *y*<sup>1</sup> dependent and *y*<sup>2</sup> dependent parts by the linearity of the system, *x*1,1 − *b*1,0*λ*1,*b*<sup>1</sup> := *x*1,*<sup>u</sup>* <sup>1</sup> + *x*1,*<sup>y</sup>* 1, which are respectively given by,

We observe that the signal *x*1,*<sup>u</sup>* <sup>1</sup> has relative degree at least *r*<sup>1</sup> + 1 with respect to *y*2, take *η*1,*<sup>L</sup>* and *y*<sup>2</sup> as output and input of the reference system, we conclude *x*1,*<sup>u</sup>* <sup>1</sup> is uniformly bounded by bounding Lemma . Since *x*1,*<sup>y</sup>* <sup>1</sup> has relative degree at least 1 with respect to *y*1, take *η*1,*<sup>L</sup>* and *y*<sup>1</sup> as output and input of the reference system, we conclude *x*1,*<sup>y</sup>* <sup>1</sup> is uniformly bounded by bounding Lemma. Then, *x*1,1 − *b*1,0*λ*1,*b*<sup>1</sup> is uniformly bounded. It follows that

*θ*<sup>1</sup> is uniformly bounded away from 0, we have *λ*1,*b*<sup>1</sup> is uniformly bounded. That further imply Φ1,1, i.e., *C*1Φ1, is uniformly bounded. Furthermore, since *x*1,1 − *b*1,0*λ*1,*b*<sup>1</sup> and *w*` <sup>1</sup> are

*θ*<sup>1</sup> is:

1,212 0 ˜

1,213,*jw*ˇ 1,*jθ*<sup>1</sup> + *D*ˇ <sup>1</sup>*w*ˇ <sup>1</sup>

 *<sup>y</sup>*<sup>2</sup> +

> <sup>1</sup> (*ζ*<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup>

bounded, we have that the signals of *x*1,1 and *y*<sup>1</sup> are uniformly bounded.

Next, we need to prove the existence of a compact set <sup>Θ</sup>1,*<sup>c</sup>* <sup>⊂</sup> <sup>Θ</sup>1,*<sup>o</sup>* such that <sup>ˇ</sup>

<sup>Υ</sup><sup>1</sup> :<sup>=</sup> *<sup>U</sup>* + (*ρ*1,*<sup>o</sup>* <sup>−</sup> *<sup>P</sup>*1(<sup>ˇ</sup>

*<sup>x</sup>*˜1,1 <sup>−</sup> <sup>Φ</sup>1,*us* <sup>1</sup> ˜

<sup>1</sup>*L*1*C*<sup>1</sup> − Π1*C*�

1,*<sup>b</sup>* <sup>=</sup> *<sup>A</sup>*1, *<sup>f</sup>* (*x*<sup>1</sup> <sup>−</sup> *<sup>b</sup>*1,0*λ*1,*b*) + **<sup>0</sup>***r*1×<sup>1</sup>

the first row element of *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜

we can conclude that *<sup>x</sup>*˜1,1 <sup>−</sup> *<sup>λ</sup>*1,*b*1*A*¯

we have that *x*ˇ1,1 is also uniformly bounded.

+Π1*C*�

+ *q*ˇ1 ∑ *j*=1 *A*¯

*x*1,1 − *b*1,0*λ*1,*b*<sup>1</sup> = *C*<sup>1</sup> (*x*<sup>1</sup> − *b*1,0*λ*1,*b*)

*x*˙1,*<sup>u</sup>* = *A*1, *<sup>f</sup> x*1,*<sup>u</sup>* +

*<sup>x</sup>*˙1,*<sup>y</sup>* = *<sup>A</sup>*1, *<sup>f</sup> <sup>x</sup>*1,*<sup>y</sup>* + (*ζ*<sup>2</sup>

+ *q*ˇ1 ∑ *j*=1

*<sup>x</sup>*ˇ1,1 <sup>−</sup> *<sup>λ</sup>*1,*b*<sup>1</sup> (*b*1,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*1,212 0 <sup>ˇ</sup>

∀*t* ∈ [0, *Tf*). First introduce the function

and ˇ

*x*1,*<sup>y</sup>* <sup>1</sup> = *C*1*x*1,*<sup>y</sup>*

*x*1,*<sup>u</sup>* <sup>1</sup> = *C*1*x*1,*<sup>u</sup>*

<sup>1</sup> (*ζ*<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup>

 **<sup>0</sup>***r*1×<sup>1</sup> *<sup>A</sup>*¯1,212*<sup>r</sup>*<sup>1</sup> *<sup>θ</sup>*

<sup>1</sup>*L*<sup>1</sup> + Π1*C*�

*A*¯ 1,213,*jw*ˇ 1,*jθ*<sup>1</sup> + *D*ˇ <sup>1</sup>*w*ˇ <sup>1</sup>

*θ*1, ˜

Notice that *<sup>A</sup>*1, *<sup>f</sup>* <sup>=</sup> *<sup>A</sup>*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup>

*<sup>x</sup>*˙1 <sup>−</sup> *<sup>b</sup>*1,0*λ*˙

of *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜

by:

Since *Um*(*Xe*) is continuous, nonnegative definite and radially unbounded, then ∀*α* ∈ IR, the set *<sup>S</sup>*1*<sup>α</sup>* :<sup>=</sup> {*Xe* <sup>∈</sup> IR2(*n*1+*n*2)+*σ*1+*σ*2+*r*1+*r*<sup>2</sup> <sup>|</sup> *Um*(*Xe*) <sup>≤</sup> *<sup>α</sup>*} is compact or empty. Since |*w*` <sup>1</sup>(*t*)| ≤ *cw*, and |*w*` <sup>2</sup>(*t*)| ≤ *cw*, ∀*t* ∈ [0, ∞), there exists a constant *c* > 0 such that we have the following inequality for the derivative of *U*:

$$\dot{M} \leq -\sum\_{i=1}^{2} \left( \frac{\gamma^4}{2} |\mathbf{x}\_i - \check{\mathbf{x}}\_i - \boldsymbol{\Phi}\_i(\boldsymbol{\theta}\_i - \check{\boldsymbol{\theta}}\_i)|\_{\Pi^{-1}\_l \Delta\_l \Pi^{-1}\_l}^2 - 2\left(\boldsymbol{\theta}\_i - \check{\boldsymbol{\theta}}\_i\right)' \boldsymbol{P}\_{l,r}(\check{\boldsymbol{\theta}}\_i) + |\tilde{\eta}\_i|\_{\mathbf{Y}\_l}^2 + \sum\_{j=1}^{r\_l} c\_{i,\boldsymbol{\theta}\_j} z\_{i,j}^2 \right) + c\tau$$

$$\text{Since } -\sum\_{i=1}^{2} (\frac{\gamma^4}{2} |\mathbf{x}\_i + \mathbf{\tilde{x}}\_i - \Phi\_i(\theta\_i - \tilde{\theta}\_i)|^2\_{\Pi\_l^{-1}\Lambda\_l\Pi\_l^{-1}} + |\boldsymbol{\eta}\_i|^2\_{\mathbf{Y}\_l} - 2\left(\theta\_i - \tilde{\theta}\_i\right)\boldsymbol{\prime}\boldsymbol{P}\_{i,\mathbf{y}}(\tilde{\theta}\_i) + \sum\_{j=1}^{r\_l} \mathbf{c}\_{\theta\_{l,j}} z^2\_{i,j}\text{) will tend}$$

to <sup>−</sup><sup>∞</sup> when *Xe* approaches the boundary of <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> , then there exists a compact set <sup>Ω</sup>1(*cw*) <sup>⊂</sup> <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> , such that *<sup>U</sup>*˙ <sup>&</sup>lt; 0 for <sup>∀</sup>*Xe* <sup>∈</sup> <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> \Ω1.

Then we have *<sup>U</sup>*(*t*, *Xe*(*t*)) <sup>≤</sup> *<sup>c</sup>*1, and *Xe*(*t*) is in the compact set *<sup>S</sup>*1*c*<sup>1</sup> <sup>⊆</sup> IR2(*n*1+*n*2)+*σ*1+*σ*2+*r*1+*r*<sup>2</sup> , <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *Tf*). It follows that the signal *Xe* is uniformly bounded, namely, ˜ *θ*1, ˜ *<sup>θ</sup>*2, *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜ *θ*1, *<sup>x</sup>*˜2 <sup>−</sup> <sup>Φ</sup><sup>2</sup> ˜ *θ*2, *η*˜1, *η*˜2, *z*1,1, ··· , *z*1,*r*<sup>1</sup> and *z*2,1, ··· , *z*2,*r*<sup>2</sup> are uniformly bounded.

Based on the dynamics of *η*1,*d*, we have *η*1,*<sup>d</sup>* is uniformly bounded. Since *η*˜1 = *η*<sup>1</sup> − *η*1,*<sup>d</sup>* is uniformly bounded, then *η*<sup>1</sup> is also uniformly bounded. Furthermore, there is a particular linear combination of the components of *η*1, denoted by *η*1,*L*,

$$\begin{aligned} \dot{\eta}\_1 &= A\_{1,f}\eta\_1 + p\_{1,n\_1}y\_1 \\\\ \eta\_{1,L} &= T\_{1,L}\eta\_1 \end{aligned}$$

which is strictly minimum phase and has relative degree 1 with respect to *y*1. Then the signal *η*1,*<sup>L</sup>* has relative degree *r*<sup>1</sup> + 1 with respect to the input *y*2, and is uniformly bounded. The composite system of *η*<sup>1</sup> and *x*`1 with input *w*` <sup>1</sup> and *y*<sup>2</sup> and output *η*1,*<sup>L</sup>* may serve as a reference system in the application of bounding Lemma [12].

Note Φ<sup>1</sup> = Φ1,*<sup>y</sup>* + Φ1,*<sup>u</sup>* and Φ1,*<sup>y</sup>* is uniformly bounded. To prove Φ<sup>1</sup> is bounded, we need to prove Φ1,*<sup>u</sup>* is uniformly bounded. Define the following equations to separate Φ1,*<sup>u</sup>* into two part:

$$
\Phi\_{1,u} = \Phi\_{1,u\_\*} + \lambda\_{1,b} \bar{A}\_{1,2120} \tag{23a}
$$

$$
\dot{\lambda}\_{1,b} = A\_{1,f}\lambda\_{1,b} + e\_{n\_1,r\_1}y\_2; \quad \lambda\_{1,b}(0) = \mathbf{0}\_{n\_1 \times 1} \tag{23b}
$$

$$\dot{\Phi}\_{1,u\_s} = A\_{1,f}\Phi\_{1,u\_s} + y\_2 \begin{bmatrix} \mathbf{0}\_{r\_1 \times \sigma\_1} \\ \vec{A}\_{1,212r\_1} \end{bmatrix}; \quad \Phi\_{1,u\_s}(0) = \Phi\_{1,u0} \tag{23c}$$

We observe that the relative degree for each element of Φ1,*us* <sup>1</sup> is at least *r*<sup>1</sup> + 1 with respect to the input *y*2, and is the output of a stable linear system. Take *η*1,*<sup>L</sup>* and *y*<sup>2</sup> as output and input of the reference system, we conclude Φ1,*us* <sup>1</sup> is uniformly bounded by bounding Lemma. Because the first row element of *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜ *θ*<sup>1</sup> is:

18 Will-be-set-by-IN-TECH

*Um*(*Xe*) <sup>≤</sup> *<sup>U</sup>*(*t*, *Xe*) <sup>≤</sup> *UM*(*Xe*), <sup>∀</sup>(*t*, *Xe*) <sup>∈</sup> [0, *Tf*) <sup>×</sup> IR2(*n*1+*n*<sup>2</sup> )+*σ*1+*σ*2+*r*1+*r*<sup>2</sup>

Since *Um*(*Xe*) is continuous, nonnegative definite and radially unbounded, then ∀*α* ∈ IR, the set *<sup>S</sup>*1*<sup>α</sup>* :<sup>=</sup> {*Xe* <sup>∈</sup> IR2(*n*1+*n*2)+*σ*1+*σ*2+*r*1+*r*<sup>2</sup> <sup>|</sup> *Um*(*Xe*) <sup>≤</sup> *<sup>α</sup>*} is compact or empty. Since |*w*` <sup>1</sup>(*t*)| ≤ *cw*, and |*w*` <sup>2</sup>(*t*)| ≤ *cw*, ∀*t* ∈ [0, ∞), there exists a constant *c* > 0 such that we have the

> + |*η*˜*i*| 2 *Yi*

to <sup>−</sup><sup>∞</sup> when *Xe* approaches the boundary of <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> , then there exists a compact set <sup>Ω</sup>1(*cw*) <sup>⊂</sup> <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> , such that *<sup>U</sup>*˙ <sup>&</sup>lt; 0 for <sup>∀</sup>*Xe* <sup>∈</sup> <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup>

Then we have *<sup>U</sup>*(*t*, *Xe*(*t*)) <sup>≤</sup> *<sup>c</sup>*1, and *Xe*(*t*) is in the compact set *<sup>S</sup>*1*c*<sup>1</sup> <sup>⊆</sup> IR2(*n*1+*n*2)+*σ*1+*σ*2+*r*1+*r*<sup>2</sup> ,

Based on the dynamics of *η*1,*d*, we have *η*1,*<sup>d</sup>* is uniformly bounded. Since *η*˜1 = *η*<sup>1</sup> − *η*1,*<sup>d</sup>* is uniformly bounded, then *η*<sup>1</sup> is also uniformly bounded. Furthermore, there is a particular

*η*˙1 = *A*1, *<sup>f</sup> η*<sup>1</sup> + *p*1,*n*<sup>1</sup> *y*<sup>1</sup>

which is strictly minimum phase and has relative degree 1 with respect to *y*1. Then the signal *η*1,*<sup>L</sup>* has relative degree *r*<sup>1</sup> + 1 with respect to the input *y*2, and is uniformly bounded. The composite system of *η*<sup>1</sup> and *x*`1 with input *w*` <sup>1</sup> and *y*<sup>2</sup> and output *η*1,*<sup>L</sup>* may serve as a reference

Note Φ<sup>1</sup> = Φ1,*<sup>y</sup>* + Φ1,*<sup>u</sup>* and Φ1,*<sup>y</sup>* is uniformly bounded. To prove Φ<sup>1</sup> is bounded, we need to prove Φ1,*<sup>u</sup>* is uniformly bounded. Define the following equations to separate Φ1,*<sup>u</sup>* into two

> **<sup>0</sup>***r*1×*σ*<sup>1</sup> *<sup>A</sup>*¯ 1,212*r*<sup>1</sup>

We observe that the relative degree for each element of Φ1,*us* <sup>1</sup> is at least *r*<sup>1</sup> + 1 with respect to the input *y*2, and is the output of a stable linear system. Take *η*1,*<sup>L</sup>* and *y*<sup>2</sup> as output and input of

<sup>−</sup> <sup>2</sup> (*θ<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup>

*θi*)� *Pi*,*r*(ˇ

<sup>−</sup> <sup>2</sup> (*θ<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup>

*θi*)� *Pi*,*r*(ˇ *θi*) +

*θi*) + |*η*˜*i*|

1,212 0 (23a)

(0) = Φ1,*<sup>u</sup>* <sup>0</sup> (23c)

*<sup>λ</sup>*˙ 1,*<sup>b</sup>* <sup>=</sup> *<sup>A</sup>*1, *<sup>f</sup> <sup>λ</sup>*1,*<sup>b</sup>* <sup>+</sup> *en*1,*r*<sup>1</sup> *<sup>y</sup>*2; *<sup>λ</sup>*1,*b*(0) = **<sup>0</sup>***n*1×<sup>1</sup> (23b)

; Φ1,*us*

2 *Yi* +

> *ri* ∑ *j*=1 *cβi*,*<sup>j</sup> z*2 *i*,*j*

*ri* ∑ *j*=1 *ci*,*β<sup>j</sup> z*2 *i*,*j* + *c*

*θ*1, ˜

) will tend

*<sup>θ</sup>*2, *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜

*θ*1,

then, we have

*<sup>U</sup>*˙ ≤ −

Since −

*<sup>x</sup>*˜2 <sup>−</sup> <sup>Φ</sup><sup>2</sup> ˜

part:

2 ∑ *i*=1

2 ∑ *i*=1 ( *γ*4

IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> \Ω1.

*γ*<sup>4</sup>

following inequality for the derivative of *U*:

<sup>2</sup> <sup>|</sup>*xi* <sup>−</sup> *<sup>x</sup>*ˇ*<sup>i</sup>* <sup>−</sup> <sup>Φ</sup>*<sup>i</sup>* (*θ<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup>

<sup>2</sup> <sup>|</sup>*xi* <sup>+</sup> *<sup>x</sup>*ˇ*<sup>i</sup>* <sup>−</sup> <sup>Φ</sup>*<sup>i</sup>* (*θ<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup>

*θi*)| 2 Π−<sup>1</sup> *<sup>i</sup>* <sup>Δ</sup>*i*Π−<sup>1</sup> *i*

<sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *Tf*). It follows that the signal *Xe* is uniformly bounded, namely, ˜

*θ*2, *η*˜1, *η*˜2, *z*1,1, ··· , *z*1,*r*<sup>1</sup> and *z*2,1, ··· , *z*2,*r*<sup>2</sup> are uniformly bounded.

*η*1,*<sup>L</sup>* = *T*1,*Lη*<sup>1</sup>

*θi*)| 2 Π−<sup>1</sup> *<sup>i</sup>* <sup>Δ</sup>*i*Π−<sup>1</sup> *i*

linear combination of the components of *η*1, denoted by *η*1,*L*,

system in the application of bounding Lemma [12].

<sup>Φ</sup>1,*<sup>u</sup>* = <sup>Φ</sup>1,*us* + *<sup>λ</sup>*1,*bA*¯

<sup>Φ</sup>˙ 1,*us* = *<sup>A</sup>*1, *<sup>f</sup>* <sup>Φ</sup>1,*us* + *<sup>y</sup>*<sup>2</sup>

$$\pounds\_{1,1} - \Phi\_{1,\mu\_s 1} \check{\theta}\_1 - \lambda\_{1,b1} \check{A}\_{1,212} \check{\theta}\_1 - \eta\_1' T\_{1,1} \check{\theta}\_1$$

we can conclude that *<sup>x</sup>*˜1,1 <sup>−</sup> *<sup>λ</sup>*1,*b*1*A*¯ 1,212 0 ˜ *θ*<sup>1</sup> is uniformly bounded in view of the boundedness of *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜ *θ*1, ˜ *θ*1, Φ1,*us* 1, and *η*1. Since *z*1,1 = *x*ˇ1,1 − *yd*, and *z*1,1, *yd* are both uniformly bounded, we have that *x*ˇ1,1 is also uniformly bounded.

Notice that *<sup>A</sup>*1, *<sup>f</sup>* <sup>=</sup> *<sup>A</sup>*<sup>1</sup> <sup>−</sup> *<sup>ζ</sup>*<sup>2</sup> <sup>1</sup>*L*1*C*<sup>1</sup> − Π1*C*� <sup>1</sup>*C*<sup>1</sup> (*ζ*<sup>2</sup> <sup>1</sup> <sup>−</sup> *<sup>γ</sup>*−2), we generated the signal *<sup>x</sup>*1,1 <sup>−</sup> *<sup>b</sup>*1,0*λ*1,*b*<sup>1</sup> by:

$$\begin{aligned} \dot{\mathbf{x}}\_{1} - b\_{1,0}\dot{\lambda}\_{1,b} &= A\_{1,f} \left( \mathbf{x}\_{1} - b\_{1,0}\lambda\_{1,b} \right) + \begin{bmatrix} \mathbf{0}\_{1 \times 1} \\ \bar{A}\_{1,212\gamma}\theta\_{1} \end{bmatrix} \mathbf{y}\_{2} + \bar{A}\_{1,211}\theta\_{1}\mathbf{y}\_{1} + D\_{1}\dot{M}\_{1}\dot{\omega}\_{1} + \begin{pmatrix} \mathbf{y}\_{1}^{2}\mathbf{I}\_{1} \\ \mathbf{y}\_{1} \end{pmatrix} \mathbf{z}\_{1} \\ &+ \Pi\_{1}\mathbf{C}\_{1}^{\prime} \left(\zeta\_{1}^{2} - \frac{1}{\gamma^{2}}\right) \left(\mathbf{y}\_{1} - E\_{1}\dot{M}\_{1}\dot{\omega}\_{1}\right) + \begin{bmatrix} \mathbf{0}\_{1 \times r\_{1}} \ b\_{1,p1} \cdot \cdots \ b\_{1,p\,n\_{1} - r\_{1}} \end{bmatrix} \mathbf{y}\_{2} \\ &+ \sum\_{j=1}^{\delta\_{1}} \bar{A}\_{1,213,j}\dot{\omega}\_{1,j}\theta\_{1} + \dot{D}\_{1}\dot{\omega}\_{1} \\ \mathbf{x}\_{1,1} - b\_{1,0}\lambda\_{1,b1} = \mathbf{C}\_{1} \left(\mathbf{x}\_{1} - b\_{1,0}\lambda\_{1,b}\right) \end{aligned}$$

Now we will separate the above dynamics into *y*<sup>1</sup> dependent and *y*<sup>2</sup> dependent parts by the linearity of the system, *x*1,1 − *b*1,0*λ*1,*b*<sup>1</sup> := *x*1,*<sup>u</sup>* <sup>1</sup> + *x*1,*<sup>y</sup>* 1, which are respectively given by,

$$\begin{aligned} \dot{\mathbf{x}}\_{1,u} &= A\_{1,f}\mathbf{x}\_{1,u} + \left[\frac{\mathbf{0}\_{r\_1 \times 1}}{\bar{A}\_{1,212r\_1}\theta}\right]y\_2 + \left[\mathbf{0}\_{1\times r\_1}\,b\_{1,p1}\,\cdots\,b\_{1,p\,n\_1\cdots n\_1}\right]'y\_2\\ \mathbf{x}\_{1,u} &= \mathbf{C}\_1\mathbf{x}\_{1,u} \\ \dot{\mathbf{x}}\_{1,y} &= A\_{1,f}\mathbf{x}\_{1,y} + \left(\zeta\_1^2L\_1 + \Pi\_1C\_1'\left(\zeta\_1^2 - \frac{1}{\gamma^2}\right)\right)(y\_1 - E\_1\dot{M}\_1\dot{\boldsymbol{w}}\_1) + \bar{A}\_{1,211}\theta\_1y\_1 + D\_1\dot{M}\_1\dot{\boldsymbol{w}}\_1\\ &+ \sum\_{j=1}^{\tilde{g}\_1} \bar{A}\_{1,213,j}\theta\_{1,j}\theta\_1 + \bar{D}\_1\dot{\boldsymbol{w}}\_1\\ \mathbf{x}\_{1,y} &= \mathbf{C}\_1\mathbf{x}\_{1,y} \end{aligned}$$

We observe that the signal *x*1,*<sup>u</sup>* <sup>1</sup> has relative degree at least *r*<sup>1</sup> + 1 with respect to *y*2, take *η*1,*<sup>L</sup>* and *y*<sup>2</sup> as output and input of the reference system, we conclude *x*1,*<sup>u</sup>* <sup>1</sup> is uniformly bounded by bounding Lemma . Since *x*1,*<sup>y</sup>* <sup>1</sup> has relative degree at least 1 with respect to *y*1, take *η*1,*<sup>L</sup>* and *y*<sup>1</sup> as output and input of the reference system, we conclude *x*1,*<sup>y</sup>* <sup>1</sup> is uniformly bounded by bounding Lemma. Then, *x*1,1 − *b*1,0*λ*1,*b*<sup>1</sup> is uniformly bounded. It follows that *<sup>x</sup>*ˇ1,1 <sup>−</sup> *<sup>λ</sup>*1,*b*<sup>1</sup> (*b*1,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*1,212 0 <sup>ˇ</sup> *θ*1) is also uniformly bounded. Since *x*ˇ1,1 is uniformly bounded and ˇ *θ*<sup>1</sup> is uniformly bounded away from 0, we have *λ*1,*b*<sup>1</sup> is uniformly bounded. That further imply Φ1,1, i.e., *C*1Φ1, is uniformly bounded. Furthermore, since *x*1,1 − *b*1,0*λ*1,*b*<sup>1</sup> and *w*` <sup>1</sup> are bounded, we have that the signals of *x*1,1 and *y*<sup>1</sup> are uniformly bounded.

Next, we need to prove the existence of a compact set <sup>Θ</sup>1,*<sup>c</sup>* <sup>⊂</sup> <sup>Θ</sup>1,*<sup>o</sup>* such that <sup>ˇ</sup> *θ*1(*t*) ∈ Θ1,*c*, ∀*t* ∈ [0, *Tf*). First introduce the function

$$\mathcal{Y}\_1 := \mathcal{U} + (\rho\_{1,\sigma} - P\_1(\check{\theta}\_1))^{-1} P\_1(\check{\theta}\_1)^{\frac{1}{2}}$$

We notice that, when ˇ *θ*<sup>1</sup> approaches the boundary of Θ1,*o*, *P*1(ˇ *θ*1) approaches *ρ*1,*o*. Then Υ<sup>1</sup> approaches <sup>∞</sup> as *Xe* approaches the boundary of <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*<sup>2</sup> )+*r*1+*r*<sup>2</sup> . There exist some constant *c* > 0 such that the following inequalities hold.

<sup>3</sup>◦: First, we need to show that <sup>Φ</sup>1,*us <sup>k</sup>*<sup>+</sup>1, *<sup>x</sup>*˜1,*k*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*1,*b k*+1*A*¯

*<sup>x</sup>*˜1,*k*+<sup>1</sup> <sup>−</sup> <sup>Φ</sup>1,*us <sup>k</sup>*+<sup>1</sup> ˜

(*k*)

*<sup>x</sup>*˙1 <sup>−</sup> *<sup>b</sup>*1,0*λ*˙ 1,*<sup>b</sup>* <sup>=</sup> *<sup>A</sup>*1, *<sup>f</sup>* (*x*<sup>1</sup> <sup>−</sup> *<sup>b</sup>*1,0*λ*1,*b*) + **<sup>0</sup>***r*1×<sup>1</sup>

<sup>1</sup> (*ζ*<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup>

*<sup>n</sup>*1,*k*+<sup>1</sup> (*x*<sup>1</sup> − *b*1,0*λ*1,*b*)

 *<sup>y</sup>*<sup>2</sup> +

> <sup>1</sup> (*ζ*<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup>

 **<sup>0</sup>***r*1×<sup>1</sup> *<sup>A</sup>*¯ 1,212 *<sup>r</sup>*<sup>1</sup> *<sup>θ</sup>*<sup>1</sup>

<sup>1</sup>*L*<sup>1</sup> + Π1*C*�

1,213,*jw*ˇ 1,*jθ*<sup>1</sup> + *D*ˇ <sup>1</sup>*w*ˇ <sup>1</sup>

bounded. It follows that *<sup>x</sup>*ˇ1,*k*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*1,*b k*+<sup>1</sup> (*b*1,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*¯1,212 0 <sup>ˇ</sup>

+Π1*C*�

+ *q*ˇ1 ∑ *j*=1 *<sup>d</sup>* , <sup>Φ</sup>1,*u k*, and <sup>ˇ</sup>

*A*¯ 1,213,*jw*ˇ 1,*jθ*<sup>1</sup> + *D*ˇ <sup>1</sup>*w*ˇ <sup>1</sup>

We can conclude that *<sup>x</sup>*˜1,*k*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*1,*b k*+1*A*¯1,212 0 ˜

The signal *x*1,*k*+<sup>1</sup> − *b*1,0*λ*1,*b k*+<sup>1</sup> is generated by:

*<sup>d</sup>* , Φ1,*<sup>u</sup>* 1, ··· *x*ˇ1,*k*, *y*

*θ*1, ˜

reference system has input *y*<sup>2</sup> and output *y*1. Note that *<sup>k</sup>* <sup>+</sup> 1st row element of *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜

*x*`1 *<sup>k</sup>*+<sup>1</sup> are bounded.

From equation (23c), we note that every element of Φ1,*us <sup>k</sup>*+<sup>1</sup> has relative degree of at least *r*<sup>1</sup> − *k* + 1 with respect to *y*2, and is the output of a stable linear system. Since the boundedness

*θ*<sup>1</sup> is

*<sup>θ</sup>*<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*1,*b k*+1*A*¯

*z*1,*k*+<sup>1</sup> = *x*ˇ1,*k*+<sup>1</sup> − *α*1,*k*, and *z*1,*k*+<sup>1</sup> is uniformly bounded, we have that *x*ˇ1,*k*+<sup>1</sup> is also uniformly

*<sup>A</sup>*¯ 1,212 *<sup>r</sup>*<sup>1</sup> *<sup>θ</sup>*<sup>1</sup>

*<sup>γ</sup>*<sup>2</sup> )) (*y*<sup>1</sup> <sup>−</sup> *<sup>E</sup>*1*M*` <sup>1</sup>*w*` <sup>1</sup>) +

Now we will separate the above dynamics into *y*<sup>1</sup> dependent and *y*<sup>2</sup> dependent parts by the linearity of the system, *x*1,*k*+<sup>1</sup> − *b*1,0*λ*1,*b k*+<sup>1</sup> := *x*1,*u k*+<sup>1</sup> + *x*1,*y k*<sup>+</sup>1, which are respectively given

We observe that the signal *x*1,*u k*+<sup>1</sup> has relative degree at least *r*<sup>1</sup> − *k* + 1 with respect to *y*2.

Lemma 11 in [12], where the reference system with input *y*<sup>2</sup> and output *y*1. We conclude *x*1,*y k*+<sup>1</sup> is uniformly bounded since *y*<sup>1</sup> is bounded. Then, *x*1,*k*+<sup>1</sup> − *b*1,0*λ*1,*b k*+<sup>1</sup> is uniformly

**<sup>0</sup>**1×*r*<sup>1</sup> *<sup>b</sup>*1,*p*<sup>1</sup> ··· *<sup>b</sup>*1,*p n*1−*r*<sup>1</sup>

*x*`1,1*<sup>k</sup>* are uniformly bounded, we conclude *x*1,*u k*+<sup>1</sup> is uniformly bounded by

*x*`1*k*, we conclude Φ1,*us <sup>k</sup>*+<sup>1</sup> is uniformly bounded by Lemma 11 in [12], where the

1,212 0 ˜

*θ*<sup>1</sup> − *η*�

*θ*1, Φ1,*us <sup>k</sup>*<sup>+</sup>1, and *η*1. Since the boundedness of *yd*, *s*1,Σ, *η*1, *η*1,*d*,

<sup>1</sup>*T*1,*k*+<sup>1</sup> ˜ *θ*1

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

*θ*<sup>1</sup> is uniformly bounded in view of the

*y*<sup>2</sup> + *A*¯ 1,211*θ*1*y*<sup>1</sup> + *D*1*M*` <sup>1</sup>*w*` <sup>1</sup> + (*ζ*<sup>2</sup>

**<sup>0</sup>**1×*r*<sup>1</sup> *<sup>b</sup>*1,*p*<sup>1</sup> ··· *<sup>b</sup>*1,*p n*1−*r*<sup>1</sup>

� *y*2

*<sup>γ</sup>*<sup>2</sup> )) (*y*<sup>1</sup> <sup>−</sup> *<sup>E</sup>*1*M*` <sup>1</sup>*w*` <sup>1</sup>) + *<sup>A</sup>*¯ 1,211*θ*1*y*<sup>1</sup> <sup>+</sup> *<sup>D</sup>*1*M*` <sup>1</sup>*w*` <sup>1</sup>

*θ*1) is also uniformly bounded. Since

<sup>1</sup>*L*<sup>1</sup>

127

� *y*2

*θ*1(*t*) ∈ Θ1,*c*, ∀*t* ∈ [0, *Tf*), *α*1,*<sup>k</sup>* is bounded. Since

*<sup>λ</sup>*1,*b k*<sup>+</sup>1, <sup>Φ</sup>1,*u k*<sup>+</sup>1, *<sup>x</sup>*1,*k*<sup>+</sup>1, and ¯

boundedness of *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜

*x*1,*k*+<sup>1</sup> − *b*1,0*λ*1,*b k*+<sup>1</sup> = *e*�

*x*˙1,*<sup>u</sup>* = *A*1, *<sup>f</sup> x*1,*<sup>u</sup>* +

� *<sup>n</sup>*1,*k*+1*x*1,*<sup>u</sup>*

+ *q*ˇ1 ∑ *j*=1 *A*¯

> � *<sup>n</sup>*1,*k*+1*x*1,*<sup>y</sup>*

*<sup>x</sup>*˙1,*<sup>y</sup>* = *<sup>A</sup>*1, *<sup>f</sup> <sup>x</sup>*1,*<sup>y</sup>* + (*ζ*<sup>2</sup>

*x*1,*u k*+<sup>1</sup> = *e*

*x*1,*y k*+<sup>1</sup> = *e*

*<sup>x</sup>*`1,11, ··· , ¯

Since ¯

(1)

of ¯

*<sup>x</sup>*`11, ··· , ¯

Σ1, *x*ˇ1,1, *y*

bounded.

by,

1,212 0 ˜

*θ*1, *x*ˇ1,*k*<sup>+</sup>1, *x*1,*k*+<sup>1</sup> − *b*1,0*λ*1,*b k*<sup>+</sup>1,

$$\begin{split} \dot{Y}\_{1} &= \dot{\mathcal{U}} + (\rho\_{1,o} - P\_{1}(\dot{\theta}\_{1}))^{-2} \rho\_{1,o} \frac{\partial P\_{1}}{\partial \theta\_{1}}(\dot{\theta}\_{1}) \dot{\theta}\_{1} \\ &\leq -\sum\_{i=1}^{2} (\frac{\gamma^{4}}{2} |\mathbf{x}\_{i,1} - \dot{\mathbf{x}}\_{i,1} - \Phi\_{i}(\theta\_{i} - \dot{\theta}\_{i})|\_{\Pi\_{i}^{-1}\Delta\_{i}\Pi\_{i}^{-1}}^{2} - 2\left(\theta\_{i} - \dot{\theta}\_{i}\right)' P\_{i,r}(\dot{\theta}\_{i}) + |\dot{\eta}\_{i}|\_{\tilde{\mathbf{Y}}\_{i}}^{2} + \sum\_{j=1}^{r\_{i}} c\_{i,\theta\_{j}} z\_{i,j}^{2} \\ &- \left| \left(\frac{\partial P\_{1}}{\partial \theta\_{1}}(\dot{\theta}\_{1})\right)' \right|^{2} (\rho\_{1,o} - P\_{1}(\dot{\theta}\_{1}))^{-4} \left(\mathcal{K}\_{1,\mathcal{I}}^{-1} \rho\_{1,o} p\_{1,r}(\dot{\theta}\_{1}) \left(\rho\_{1,o} - P\_{1}(\dot{\theta}\_{1})\right)^{2} - c \right) + c \end{split}$$

Since <sup>Υ</sup>˙ <sup>1</sup> will tend to <sup>−</sup><sup>∞</sup> when *Xe* approaches the boundary of <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> , then there exists a compact set <sup>Ω</sup>1,2(*cw*) <sup>⊂</sup> <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> , such that <sup>∀</sup>*Xe* <sup>∈</sup> <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> \Ω1,2, <sup>Υ</sup>˙ <sup>1</sup>(*Xe*) <sup>&</sup>lt; 0.

Then there exists a compact set <sup>Θ</sup>1,*<sup>c</sup>* <sup>⊂</sup> <sup>Θ</sup>1,*o*, such that <sup>ˇ</sup> *θ*1(*t*) ∈ Θ1,*c*, ∀*t* ∈ [0, *Tf*). Moreover, <sup>Υ</sup>1(*t*, *Xe*(*t*)) <sup>≤</sup> *<sup>c</sup>*2, and *Xe*(*t*) is in the compact set *<sup>S</sup>*1,2*c*<sup>2</sup> <sup>⊆</sup> <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> , <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup> [0, *Tf*).

To derive the uniformly boundedness of the closed-loop system states, we separate the relative degree, *r*1, into two cases: *r*<sup>1</sup> = 1, and *r*<sup>1</sup> ≥ 2. First, we consider the case 1: *r*<sup>1</sup> = 1.

Taking *x*1,1 and *y*<sup>2</sup> as the output and input of the reference system, we note that *x*1,1 is strictly minimum phase and has relative degree *r*<sup>1</sup> with respect to input *y*2. Since the state *x*<sup>1</sup> can be viewed as stably filtered output signals of *y*<sup>2</sup> and *y*1, it is uniformly bounded. Since *λ*<sup>1</sup> is also some stably filtered signals of *y*<sup>1</sup> and *y*2, it is uniformly bounded. It further implies Φ is uniformly bounded. Then we can conclude *x*ˇ1 is uniformly bounded from the boundedness of *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜ *θ*1. This further implies that the inputs *x*ˇ2,1 and ˆ *ξ*<sup>1</sup> are uniformly bounded.

Case 2: *r*<sup>1</sup> ≥ 2. Considering the canonical form (78) in [12] for the true system (1), we denote the elements of ¯ *x*` by ¯ *<sup>x</sup>*`11 ··· ¯ *x*`1*<sup>r</sup>*<sup>1</sup> � . We will use the mathematical induction to derive the boundedness of <sup>Φ</sup>1,*us <sup>i</sup>*, *<sup>x</sup>*˜1,*<sup>i</sup>* <sup>−</sup> *<sup>λ</sup>*1,*biA*¯ 1,212 0 ˜ *<sup>θ</sup>*1, *<sup>x</sup>*ˇ1,*i*, *<sup>x</sup>*1,*<sup>i</sup>* <sup>−</sup> *<sup>b</sup>*1,0*λ*1,*bi*, *<sup>λ</sup>*1,*bi*, <sup>Φ</sup>1,*u i*, *<sup>x</sup>*1,*i*, ¯ *x*`1,1*i*, ∀*i* = {1, ··· ,*r*1}. For the boundedness of ¯ *x*`1*i*, we will show that ¯ *x*`1*<sup>i</sup>* is a linear combination of *x*1,1, ··· , *<sup>x</sup>*1,*i*, ¯ *x*`3, and ¯ *x*`4, i.e.,

$$\bar{\mathfrak{X}}\_{1i} = \mathfrak{z}\_{1,1i}\mathfrak{x}\_{1,1} + \dots + \mathfrak{z}\_{1,i-1,i}\mathfrak{x}\_{1,i-1} + \mathfrak{x}\_{1,i} + \tilde{T}\_{1,3}\bar{\mathfrak{x}}\_3 + \tilde{T}\_{1,i4}\bar{\mathfrak{x}}\_4; \quad 1 \le i \le r\_1 \tag{24}$$

where *<sup>a</sup>*˜1,1*i*, ··· , *<sup>a</sup>*˜1,*<sup>i</sup>*−<sup>1</sup> *<sup>i</sup>* are constants, *<sup>T</sup>*˜ 1,*<sup>i</sup>*3, *T*˜ 1,*i*<sup>4</sup> are constant matrices, and ¯ *x*`3 and ¯ *x*`4 are defined at (78) in [12].

<sup>1</sup>◦: We have deduced that *<sup>η</sup>*1, *<sup>η</sup>*1,*d*, *<sup>η</sup>*1,*L*, <sup>Φ</sup>1,*us* 1, *<sup>x</sup>*˜1,1 <sup>−</sup> *<sup>λ</sup>*1,*b*1*A*¯ 1,212 0 ˜ *θ*1, *x*ˇ1,1, *x*1,1 − *b*1,0*λ*1,*<sup>b</sup>*1, *λ*1,*<sup>b</sup>*1, Φ1,*<sup>u</sup>* 1, and *x*1,1 are uniformly bounded in [0, *Tf*). ¯ *<sup>x</sup>*`11 is bounded in view of *<sup>x</sup>*1,1 <sup>−</sup> *<sup>C</sup>*` 3 ¯ *<sup>x</sup>*`3 <sup>−</sup> *<sup>C</sup>*` 4 ¯ *x*`4.

<sup>2</sup>◦: We assume that <sup>Φ</sup>1,*us <sup>i</sup>*, *<sup>x</sup>*˜1,*<sup>i</sup>* <sup>−</sup> *<sup>λ</sup>*1,*biA*¯ 1,212 0 ˜ *<sup>θ</sup>*1, *<sup>x</sup>*ˇ1,*i*, *<sup>x</sup>*1,*<sup>i</sup>* <sup>−</sup> *<sup>b</sup>*1,0*λ*1,*bi*, *<sup>λ</sup>*1,*bi*, <sup>Φ</sup>1,*u i*, *<sup>x</sup>*1,*i*, and ¯ *x*`1*<sup>i</sup>* are bounded, and

$$\bar{\mathfrak{X}}\_{1i} = \mathfrak{A}\_{1,1i}\mathfrak{x}\_{1,1} + \dots + \mathfrak{A}\_{1,i-1,i}\mathfrak{x}\_{1,i-1} + \mathfrak{x}\_{1,i} + \tilde{T}\_{1,i3}\bar{\mathfrak{X}}\_{3} + \tilde{T}\_{1,i4}\bar{\mathfrak{X}}\_{4}; \quad \forall i \in \{1, \dots, k\} \tag{25}$$

where 1 ≤ *k* < *r*1.

<sup>3</sup>◦: First, we need to show that <sup>Φ</sup>1,*us <sup>k</sup>*<sup>+</sup>1, *<sup>x</sup>*˜1,*k*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*1,*b k*+1*A*¯ 1,212 0 ˜ *θ*1, *x*ˇ1,*k*<sup>+</sup>1, *x*1,*k*+<sup>1</sup> − *b*1,0*λ*1,*b k*<sup>+</sup>1, *<sup>λ</sup>*1,*b k*<sup>+</sup>1, <sup>Φ</sup>1,*u k*<sup>+</sup>1, *<sup>x</sup>*1,*k*<sup>+</sup>1, and ¯ *x*`1 *<sup>k</sup>*+<sup>1</sup> are bounded.

From equation (23c), we note that every element of Φ1,*us <sup>k</sup>*+<sup>1</sup> has relative degree of at least *r*<sup>1</sup> − *k* + 1 with respect to *y*2, and is the output of a stable linear system. Since the boundedness of ¯ *<sup>x</sup>*`11, ··· , ¯ *x*`1*k*, we conclude Φ1,*us <sup>k</sup>*+<sup>1</sup> is uniformly bounded by Lemma 11 in [12], where the reference system has input *y*<sup>2</sup> and output *y*1.

Note that *<sup>k</sup>* <sup>+</sup> 1st row element of *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜ *θ*<sup>1</sup> is

20 Will-be-set-by-IN-TECH

approaches <sup>∞</sup> as *Xe* approaches the boundary of <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*<sup>2</sup> )+*r*1+*r*<sup>2</sup> . There exist

Since <sup>Υ</sup>˙ <sup>1</sup> will tend to <sup>−</sup><sup>∞</sup> when *Xe* approaches the boundary of <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> , then there exists a compact set <sup>Ω</sup>1,2(*cw*) <sup>⊂</sup> <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> , such that <sup>∀</sup>*Xe* <sup>∈</sup>

<sup>Υ</sup>1(*t*, *Xe*(*t*)) <sup>≤</sup> *<sup>c</sup>*2, and *Xe*(*t*) is in the compact set *<sup>S</sup>*1,2*c*<sup>2</sup> <sup>⊆</sup> <sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> , <sup>∀</sup>*<sup>t</sup>* <sup>∈</sup>

To derive the uniformly boundedness of the closed-loop system states, we separate the relative

Taking *x*1,1 and *y*<sup>2</sup> as the output and input of the reference system, we note that *x*1,1 is strictly minimum phase and has relative degree *r*<sup>1</sup> with respect to input *y*2. Since the state *x*<sup>1</sup> can be viewed as stably filtered output signals of *y*<sup>2</sup> and *y*1, it is uniformly bounded. Since *λ*<sup>1</sup> is also some stably filtered signals of *y*<sup>1</sup> and *y*2, it is uniformly bounded. It further implies Φ is uniformly bounded. Then we can conclude *x*ˇ1 is uniformly bounded from the boundedness

Case 2: *r*<sup>1</sup> ≥ 2. Considering the canonical form (78) in [12] for the true system (1), we denote

1,*<sup>i</sup>*3, *T*˜

*x*`1*i*, we will show that ¯

1,*i*<sup>3</sup> ¯ *x*`3 + *T*˜

1,*i*<sup>3</sup> ¯ *x*`3 + *T*˜

degree, *r*1, into two cases: *r*<sup>1</sup> = 1, and *r*<sup>1</sup> ≥ 2. First, we consider the case 1: *r*<sup>1</sup> = 1.

*θ*1. This further implies that the inputs *x*ˇ2,1 and ˆ

*x*`1*<sup>r</sup>*<sup>1</sup> �

*<sup>x</sup>*`1*<sup>i</sup>* <sup>=</sup> *<sup>a</sup>*˜1,1*ix*1,1 <sup>+</sup> ··· <sup>+</sup> *<sup>a</sup>*˜1,*<sup>i</sup>*−<sup>1</sup> *ix*1,*i*−<sup>1</sup> <sup>+</sup> *<sup>x</sup>*1,*<sup>i</sup>* <sup>+</sup> *<sup>T</sup>*˜

<sup>1</sup>◦: We have deduced that *<sup>η</sup>*1, *<sup>η</sup>*1,*d*, *<sup>η</sup>*1,*L*, <sup>Φ</sup>1,*us* 1, *<sup>x</sup>*˜1,1 <sup>−</sup> *<sup>λ</sup>*1,*b*1*A*¯

*<sup>x</sup>*`1*<sup>i</sup>* <sup>=</sup> *<sup>a</sup>*˜1,1*ix*1,1 <sup>+</sup> ··· <sup>+</sup> *<sup>a</sup>*˜1,*<sup>i</sup>*−<sup>1</sup> *ix*1,*i*−<sup>1</sup> <sup>+</sup> *<sup>x</sup>*1,*<sup>i</sup>* <sup>+</sup> *<sup>T</sup>*˜

Φ1,*<sup>u</sup>* 1, and *x*1,1 are uniformly bounded in [0, *Tf*). ¯

<sup>2</sup>◦: We assume that <sup>Φ</sup>1,*us <sup>i</sup>*, *<sup>x</sup>*˜1,*<sup>i</sup>* <sup>−</sup> *<sup>λ</sup>*1,*biA*¯ 1,212 0 ˜

<sup>−</sup> <sup>2</sup> (*θ<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup>

*θ*1) 

1,*<sup>c</sup> <sup>ρ</sup>*1,*<sup>o</sup> <sup>p</sup>*1,*r*(<sup>ˇ</sup>

*θi*)� *Pi*,*r*(ˇ

*<sup>ρ</sup>*1,*<sup>o</sup>* <sup>−</sup> *<sup>P</sup>*1(<sup>ˇ</sup>

*θi*) + |*η*˜*i*|

*θ*1) <sup>2</sup> <sup>−</sup> *<sup>c</sup>* + *c*

*θ*1(*t*) ∈ Θ1,*c*, ∀*t* ∈ [0, *Tf*). Moreover,

*ξ*<sup>1</sup> are uniformly bounded.

*x*`1*<sup>i</sup>* is a linear combination of *x*1,1,

*x*`4; 1 ≤ *i* ≤ *r*<sup>1</sup> (24)

*θ*1, *x*ˇ1,1, *x*1,1 − *b*1,0*λ*1,*<sup>b</sup>*1, *λ*1,*<sup>b</sup>*1,

*x*`4; ∀*i* ∈ {1, ··· *k*} (25)

*x*`3 and ¯

3 ¯ *<sup>x</sup>*`3 <sup>−</sup> *<sup>C</sup>*` 4 ¯ *x*`4.

*x*`1,1*i*, ∀*i* =

*x*`4 are

*x*`1*<sup>i</sup>* are

. We will use the mathematical induction to derive the

1,*i*<sup>4</sup> ¯

*<sup>x</sup>*`11 is bounded in view of *<sup>x</sup>*1,1 <sup>−</sup> *<sup>C</sup>*`

*<sup>θ</sup>*1, *<sup>x</sup>*ˇ1,*i*, *<sup>x</sup>*1,*<sup>i</sup>* <sup>−</sup> *<sup>b</sup>*1,0*λ*1,*bi*, *<sup>λ</sup>*1,*bi*, <sup>Φ</sup>1,*u i*, *<sup>x</sup>*1,*i*, and ¯

1,*i*<sup>4</sup> are constant matrices, and ¯

1,212 0 ˜

1,*i*<sup>4</sup> ¯

*<sup>θ</sup>*1, *<sup>x</sup>*ˇ1,*i*, *<sup>x</sup>*1,*<sup>i</sup>* <sup>−</sup> *<sup>b</sup>*1,0*λ*1,*bi*, *<sup>λ</sup>*1,*bi*, <sup>Φ</sup>1,*u i*, *<sup>x</sup>*1,*i*, ¯

2 *Yi* +

*ri* ∑ *j*=1 *ci*,*β<sup>j</sup> z*2 *i*,*j* )

*θ*1) approaches *ρ*1,*o*. Then Υ<sup>1</sup>

*θ*<sup>1</sup> approaches the boundary of Θ1,*o*, *P*1(ˇ

some constant *c* > 0 such that the following inequalities hold.

(*ρ*1,*<sup>o</sup>* <sup>−</sup> *<sup>P</sup>*1(<sup>ˇ</sup>

*∂P*<sup>1</sup> *∂θ*<sup>1</sup> (ˇ *<sup>θ</sup>*1) ˙ ˇ *θ*1

> *θi*)| 2 Π−<sup>1</sup> *<sup>i</sup>* <sup>Δ</sup>*i*Π−<sup>1</sup> *i*

*θ*1))−<sup>4</sup> *K*−<sup>1</sup>

*θ*1))−2*ρ*1,*<sup>o</sup>*

<sup>2</sup> <sup>|</sup>*xi*,1 <sup>−</sup> *<sup>x</sup>*ˇ*i*,1 <sup>−</sup> <sup>Φ</sup>*<sup>i</sup>* (*θ<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup>

2

<sup>Θ</sup>1,*<sup>o</sup>* <sup>×</sup> <sup>Θ</sup>2,*<sup>o</sup>* <sup>×</sup> IR2(*n*1+*n*2)+*r*1+*r*<sup>2</sup> \Ω1,2, <sup>Υ</sup>˙ <sup>1</sup>(*Xe*) <sup>&</sup>lt; 0.

Then there exists a compact set <sup>Θ</sup>1,*<sup>c</sup>* <sup>⊂</sup> <sup>Θ</sup>1,*o*, such that <sup>ˇ</sup>

We notice that, when ˇ

<sup>Υ</sup>˙ <sup>1</sup> <sup>=</sup> *<sup>U</sup>*˙ + (*ρ*1,*<sup>o</sup>* <sup>−</sup> *<sup>P</sup>*1(<sup>ˇ</sup>

 *∂P*<sup>1</sup> *∂θ*<sup>1</sup> (ˇ *θ*1) � 

2 ∑ *i*=1 ( *γ*4

≤ −

− 

[0, *Tf*).

of *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜

··· , *<sup>x</sup>*1,*i*, ¯

the elements of ¯

*x*`3, and ¯

¯

defined at (78) in [12].

bounded, and

¯

where 1 ≤ *k* < *r*1.

*x*` by ¯

{1, ··· ,*r*1}. For the boundedness of ¯

*x*`4, i.e.,

where *<sup>a</sup>*˜1,1*i*, ··· , *<sup>a</sup>*˜1,*<sup>i</sup>*−<sup>1</sup> *<sup>i</sup>* are constants, *<sup>T</sup>*˜

boundedness of <sup>Φ</sup>1,*us <sup>i</sup>*, *<sup>x</sup>*˜1,*<sup>i</sup>* <sup>−</sup> *<sup>λ</sup>*1,*biA*¯ 1,212 0 ˜

*<sup>x</sup>*`11 ··· ¯

$$\pounds\_{1,k+1} - \Phi\_{1,\mu\_s k+1}\pounds\_1 - \lambda\_{1,b:k+1}\pounds\_{1,212}\pounds\_1 - \eta'\_1T\_{1,k+1}\pounds\_1$$

We can conclude that *<sup>x</sup>*˜1,*k*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*1,*b k*+1*A*¯1,212 0 ˜ *θ*<sup>1</sup> is uniformly bounded in view of the boundedness of *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜ *θ*1, ˜ *θ*1, Φ1,*us <sup>k</sup>*<sup>+</sup>1, and *η*1. Since the boundedness of *yd*, *s*1,Σ, *η*1, *η*1,*d*, Σ1, *x*ˇ1,1, *y* (1) *<sup>d</sup>* , Φ1,*<sup>u</sup>* 1, ··· *x*ˇ1,*k*, *y* (*k*) *<sup>d</sup>* , <sup>Φ</sup>1,*u k*, and <sup>ˇ</sup> *θ*1(*t*) ∈ Θ1,*c*, ∀*t* ∈ [0, *Tf*), *α*1,*<sup>k</sup>* is bounded. Since *z*1,*k*+<sup>1</sup> = *x*ˇ1,*k*+<sup>1</sup> − *α*1,*k*, and *z*1,*k*+<sup>1</sup> is uniformly bounded, we have that *x*ˇ1,*k*+<sup>1</sup> is also uniformly bounded.

The signal *x*1,*k*+<sup>1</sup> − *b*1,0*λ*1,*b k*+<sup>1</sup> is generated by:

$$\begin{aligned} \dot{\mathbf{x}}\_{1} - b\_{1,0}\dot{\lambda}\_{1,b} &= A\_{1,f} \left( \mathbf{x}\_{1} - b\_{1,0}\lambda\_{1,b} \right) + \begin{bmatrix} \mathbf{0}\_{r\_{1}\times 1} \\ \bar{A}\_{1,212,r\_{1}}\theta\_{1} \end{bmatrix} y\_{2} + \bar{A}\_{1,211}\theta\_{1} y\_{1} + D\_{1}\dot{M}\_{1}\dot{w}\_{1} + \left( \zeta\_{1}^{2}L\_{1} \right) \dot{w}\_{2} \\ &+ \Pi\_{1}C\_{1}' \left( \zeta\_{1}^{2} - \frac{1}{\gamma^{2}} \right) \left( y\_{1} - E\_{1}\dot{M}\_{1}\vartheta\_{1} \right) + \begin{bmatrix} \mathbf{0}\_{1\times r\_{1}} \ b\_{1,p1} \cdot \cdots \ b\_{1,p\cdot n\_{1}-r\_{1}} \end{bmatrix}' y\_{2} \\ &+ \sum\_{j=1}^{\tilde{\theta}\_{1}} \bar{A}\_{1,213,j}\tilde{w}\_{1,j}\theta\_{1} + \tilde{D}\_{1}\ddot{w}\_{1} \\ \mathbf{x}\_{1,k+1} - b\_{1,0}\lambda\_{1,b\cdot k+1} &= e\_{n,k+1}' \left( \mathbf{x}\_{1} - b\_{1,0}\lambda\_{1,b} \right) \end{aligned}$$

Now we will separate the above dynamics into *y*<sup>1</sup> dependent and *y*<sup>2</sup> dependent parts by the linearity of the system, *x*1,*k*+<sup>1</sup> − *b*1,0*λ*1,*b k*+<sup>1</sup> := *x*1,*u k*+<sup>1</sup> + *x*1,*y k*<sup>+</sup>1, which are respectively given by,

$$\begin{aligned} \dot{\mathbf{x}}\_{1,u} &= A\_{1,f}\mathbf{x}\_{1,u} + \left[\frac{\mathbf{0}\_{r\_1 \times 1}}{\bar{A}\_{1,212,r}\theta\_1}\right]y\_2 + \left[\mathbf{0}\_{1 \times r\_1}\,\mathbf{b}\_{1,p1} \cdot \cdots \,\mathbf{b}\_{1,p\,u\_1 - r\_1}\right]'y\_2\\ \mathbf{x}\_{1,u,k+1} &= e\_{n,k+1}'\mathbf{x}\_{1,u} \end{aligned}$$
 
$$\begin{aligned} \dot{\mathbf{x}}\_{1,y} &= A\_{1,f}\mathbf{x}\_{1,y} + \left(\xi\_1^2 L\_1 + \Pi\_1 \mathbf{C}\_1' \left(\xi\_1^2 - \frac{1}{\gamma^2}\right)\right) \left(y\_1 - E\_1 \dot{M}\_1 \dot{w}\_1\right) + \bar{A}\_{1,211}\theta\_1 y\_1 + D\_1 \dot{M}\_1 \dot{w}\_1\\ &+ \sum\_{j=1}^{\frac{\theta\_1}{\beta\_1}} \bar{A}\_{1,213,j} \dot{w}\_{1,j}\theta\_1 + \bar{D}\_1 \dot{w}\_1\\ \mathbf{x}\_{1,y,k+1} &= e\_{n,k+1}' \mathbf{x}\_{1,y} \end{aligned}$$

We observe that the signal *x*1,*u k*+<sup>1</sup> has relative degree at least *r*<sup>1</sup> − *k* + 1 with respect to *y*2. Since ¯ *<sup>x</sup>*`1,11, ··· , ¯ *x*`1,1*<sup>k</sup>* are uniformly bounded, we conclude *x*1,*u k*+<sup>1</sup> is uniformly bounded by Lemma 11 in [12], where the reference system with input *y*<sup>2</sup> and output *y*1. We conclude *x*1,*y k*+<sup>1</sup> is uniformly bounded since *y*<sup>1</sup> is bounded. Then, *x*1,*k*+<sup>1</sup> − *b*1,0*λ*1,*b k*+<sup>1</sup> is uniformly bounded. It follows that *<sup>x</sup>*ˇ1,*k*+<sup>1</sup> <sup>−</sup> *<sup>λ</sup>*1,*b k*+<sup>1</sup> (*b*1,*p*<sup>0</sup> <sup>+</sup> *<sup>A</sup>*¯1,212 0 <sup>ˇ</sup> *θ*1) is also uniformly bounded. Since

*<sup>x</sup>*ˇ1,*k*+<sup>1</sup> is uniformly bounded and *<sup>b</sup>*1,*p*<sup>0</sup> + *<sup>A</sup>*¯ 1,212 0 <sup>ˇ</sup> *θ*<sup>1</sup> is uniformly bounded away from 0, we have *λ*1,*b k*+<sup>1</sup> is uniformly bounded. That further imply Φ1,*u k*+<sup>1</sup> is uniformly bounded. Furthermore, since *x*1,*k*+<sup>1</sup> − *b*1,0*λ*1,*b k*+<sup>1</sup> and *λ*1,*b k*+<sup>1</sup> are bounded, we have that the signals of *x*1,*k*+<sup>1</sup> is uniformly bounded.

then, we establish the second statement.

 ∞ 0

> ∞ 0

systems were simulated using SIMULINK.

2d*<sup>τ</sup>* <sup>≤</sup>

By the first statement, we notice that

 ∞ 0

it follows that


Then, we have

**6. Example**

 ∞ 0

*<sup>U</sup>*˙ <sup>d</sup>*<sup>τ</sup>* <sup>≤</sup>

For the third statement, we consider the following inequality,

*<sup>γ</sup>*2|*w*<sup>ˇ</sup> 1,*a*<sup>|</sup>

For the last statement, it's easy to establish by Section 4.

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*w*<sup>ˇ</sup> 1,*a*<sup>|</sup>

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*M*` <sup>1</sup>*w*` <sup>1</sup><sup>|</sup>

sup 0≤*t*<∞

lim

Consider the following linear systems with zeros initial conditions:

˙

˙

˙

˙

interconnected with additional output measurement,

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*M*` <sup>1</sup>*w*` <sup>1</sup><sup>|</sup>

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*w*<sup>ˇ</sup> 2,*a*<sup>|</sup>


*<sup>t</sup>*→<sup>∞</sup> <sup>|</sup>*x*1(*t*) <sup>−</sup> *yd*(*t*)<sup>|</sup> <sup>=</sup> <sup>0</sup>

This complete the proof of the theorem. �

In this section, we present one example to illustrate the main results of this Chapter. The designs were carried out using MATLAB symbolic computation tools, and the closed-loop

where *θ*1, *θ*<sup>2</sup> and *θ*<sup>3</sup> are three unknown parameters with true value 0s. The coefficient terms, 0.1 and 1, reflect the *a priori* knowledge that the disturbances *w*` <sup>1</sup> and *w*` <sup>2</sup> are weak in power relative to that of the disturbance *w*ˇ <sup>1</sup> and *w*ˇ 2. We note that (27) is an unobservable system. We can decompose (27) into the following two SISO linear systems, **S1** and **S2**, sequentially

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*w*<sup>ˇ</sup> 2,*a*<sup>|</sup>

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*M*` <sup>2</sup>*w*` <sup>2</sup><sup>|</sup>

*x*`1 = *x*`1 + *x*`2 + *x*`3 + 0.1*w*` 1; (27a)

*x*`2 = (1 + *θ*1)*x*`3 + (1 + *θ*2)*w*ˇ <sup>1</sup> (27b)

*x*`3 = −*x*`1 − *x*`3 + *x*`4 + *u* + *w*ˇ <sup>2</sup> (27c)

*x*`4 = *x*`1 + (2 + *θ*3)*u* + 0.1*w*` <sup>2</sup> + *w*ˇ <sup>2</sup> (27d)

*y* = *x*`1 + 0.1*w*` <sup>1</sup> (27e)

*x*˙11 = *x*<sup>11</sup> + *x*<sup>12</sup> + *y*<sup>2</sup> + *w*11; (28a)

*x*˙12 = (1 + *θ*1)*y*<sup>2</sup> + (1 + *θ*2)*w*ˇ <sup>1</sup> + *w*12; (28b)

<sup>2</sup> <sup>+</sup> *<sup>γ</sup>*2|*M*` <sup>2</sup>*w*` <sup>2</sup><sup>|</sup>

2  <sup>2</sup>)d*τ*

129

d*τ* + *U*(0) < +∞

(−|*x*<sup>1</sup> − *yd*|

Next, we need to show ¯ *x*`1,1 *<sup>k</sup>*+<sup>1</sup> is satisfied equation (24). Comparing the design model (2) and the canonical form (78) in [12], we have ¯ *C*` ¯ *x*` = *C*1*x*1. It further implies

$$
\bar{\mathcal{C}}\_1 \bar{A}\_1^k \bar{\mathfrak{x}}\_1 = \mathcal{C}\_1 (A\_1 + \bar{A}\_{1,211} \theta\_1 \mathcal{C}\_1)^k \mathfrak{x}\_1
$$

Hence, we have

$$\bar{\mathfrak{x}}\_{1k+1} = \mathfrak{z}\_{1,1k+1}\mathfrak{x}\_1 + \dots + \mathfrak{z}\_{1,kk+1}\mathfrak{x}\_{1,k} + \mathfrak{x}\_{1,k+1} + \tilde{T}\_{1,k+1}\bar{\mathfrak{x}}\_3 + \tilde{T}\_{1,k+14}\bar{\mathfrak{x}}\_4 \tag{26}$$

where *<sup>a</sup>*˜1,1 *<sup>k</sup>*<sup>+</sup>1, ··· , *<sup>a</sup>*˜1,*k k*+<sup>1</sup> are constants, and *<sup>T</sup>*˜ 1,*k*<sup>+</sup>1 3, *<sup>T</sup>*˜ 1,*k*+1 4 are constant matrices.

Then, we have the boundedness of ¯ *x*`1 *<sup>k</sup>*<sup>+</sup>1. Thus, we can conclude the boundedness of Φ1,*us <sup>i</sup>*, *<sup>x</sup>*˜1,*<sup>i</sup>* <sup>−</sup> *<sup>λ</sup>*1,*b iA*¯ 1,212 0 ˜ *<sup>θ</sup>*1, *<sup>x</sup>*ˇ1,*i*, *<sup>x</sup>*1,*<sup>i</sup>* <sup>−</sup> *<sup>b</sup>*1,0*λ*1,*b i*, *<sup>λ</sup>*1,*b i*, <sup>Φ</sup>1,*u i*, *<sup>x</sup>*1,*i*, and ¯ *x*`1 *<sup>i</sup>*, ∀*i* ∈ {1, ···*r*1}.

Since the state *x*<sup>1</sup> can be viewed as stably filtered output signals of *y*<sup>2</sup> and *y*1, it is uniformly bounded. Also, *η*1, *λ*<sup>1</sup> are some stably filtered signals of *y*<sup>2</sup> and *y*1, they are uniformly bounded. It further implies Φ<sup>1</sup> is uniformly bounded. Then we can conclude *x*ˇ1 is uniformly bounded from the boundedness of *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜ *θ*1. This further implies that the control input *x*ˇ2,1 is uniformly bounded. Therefore, it follows *Tf* = ∞ and the complete system states are uniformly bounded on [0, ∞).

The boundedness of closed-loop state variables of **S2** can be proven with the similar line of reasoning above. Thus, we have established statement 1 in all cases.

We define *l*<sup>0</sup> = *l*1,0 + *l*2,0 = *V*2,*r*<sup>2</sup> (*X*1(0), *X*2(0)), and

*l*<sup>1</sup> + *l*<sup>2</sup> := 2 ∑ *i*=1 *<sup>γ</sup>*4|*xi* <sup>−</sup> *<sup>x</sup>*ˆ*<sup>i</sup>* <sup>−</sup> <sup>Φ</sup>*<sup>i</sup>* (*θ<sup>i</sup>* <sup>−</sup> <sup>ˆ</sup> *θi*)| 2 Π−<sup>1</sup> *<sup>i</sup>* <sup>Δ</sup>*i*Π−<sup>1</sup> *i* + |*η*˜*i*| 2 *Yi* <sup>−</sup> <sup>2</sup> (*θ<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup> *θi*)� *Pi*,*r*(ˇ *θi*) + *ri* ∑ *j*=1 *βi*,*jz*<sup>2</sup> *i*,*j* + 1 4 |*ς*1,(*r*1+*r*2)| 2 *<sup>Q</sup>*¯ <sup>1</sup> + 1 4 |*ς*2,*r*<sup>2</sup> | 2 *<sup>Q</sup>*¯ <sup>2</sup> <sup>+</sup> *�*<sup>1</sup> (*γ*2*ζ*<sup>2</sup> <sup>1</sup> <sup>−</sup> <sup>1</sup>)|*θ*<sup>1</sup> <sup>−</sup> <sup>ˆ</sup> *θ*1| 2 Φ� 1*C*� <sup>1</sup>*C*1Φ<sup>1</sup> <sup>+</sup> *�*2|*θ*<sup>2</sup> <sup>−</sup> <sup>ˆ</sup> *θ*2| 2 Φ� 2*C*� <sup>2</sup>*C*2Φ<sup>2</sup> sup *<sup>w</sup>*` <sup>1</sup>∈W` 1,*w*` <sup>2</sup>∈W` <sup>2</sup> *tf* 0 (*x*1,1 <sup>−</sup> *yd*)<sup>2</sup> <sup>+</sup> *<sup>l</sup>*<sup>1</sup> <sup>+</sup> *<sup>l</sup>*<sup>2</sup> <sup>−</sup> 2 ∑ *i*=1 *<sup>γ</sup>*2|*wi*<sup>|</sup> 2 − 2 ∑ *i*=1 *<sup>γ</sup>*2|*wi*,*a*<sup>|</sup> 2 d*τ* − 2 ∑ *i*=1 *γ*2 *θ*� *<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup> *θ*� *<sup>i</sup>*,0 *x*� *<sup>i</sup>*,0 − *x*ˇ� *i*,0 2 *Q*¯*i*,0 − *l*<sup>0</sup> ≤ sup *<sup>w</sup>*` <sup>1</sup>∈W` 1,*w*` <sup>2</sup>∈W` <sup>2</sup> *tf* 0 (*x*1,1 <sup>−</sup> *yd*)<sup>2</sup> <sup>+</sup> *<sup>l</sup>*<sup>1</sup> <sup>+</sup> *<sup>l</sup>*<sup>2</sup> <sup>−</sup> 2 ∑ *i*=1 *<sup>γ</sup>*2|*wi*<sup>|</sup> 2 − 2 ∑ *i*=1 *γ*ˇ 2| *γ γ*ˇ *w*ˇ*i*,*a*| 2 d*τ* − *l*<sup>0</sup> <sup>−</sup>*γ*<sup>2</sup> 2 ∑ *i*=1 *θ*� *<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup> *θ*� *<sup>i</sup>*,0 *x*� *<sup>i</sup>*,0 − *x*ˇ� *i*,0 2 *Q*¯*i*,0 + *tf* 0 *<sup>U</sup>*˙ <sup>d</sup>*<sup>τ</sup>* <sup>−</sup> *<sup>U</sup>*(*t*) + *<sup>U</sup>*(0) ≤ −*U*(*t*) ≤ 0

then, we establish the second statement.

For the third statement, we consider the following inequality,

$$\int\_0^\infty \dot{\mathcal{U}} \mathrm{d}\tau \le \int\_0^\infty (-|\mathbf{x}\_1 - y\_d|^2 + \gamma^2 |\psi\_{1,d}|^2 + \gamma^2 |\dot{\mathcal{M}}\_1 \psi\_1|^2 + \gamma^2 |\psi\_{2,d}|^2 + \gamma^2 |\dot{\mathcal{M}}\_2 \psi\_2|^2) \mathrm{d}\tau$$

it follows that

22 Will-be-set-by-IN-TECH

have *λ*1,*b k*+<sup>1</sup> is uniformly bounded. That further imply Φ1,*u k*+<sup>1</sup> is uniformly bounded. Furthermore, since *x*1,*k*+<sup>1</sup> − *b*1,0*λ*1,*b k*+<sup>1</sup> and *λ*1,*b k*+<sup>1</sup> are bounded, we have that the signals

*x*`1 = *C*1(*A*<sup>1</sup> + *A*¯ 1,211*θ*1*C*1)*kx*<sup>1</sup>

1,*k*<sup>+</sup>1 3, *<sup>T</sup>*˜

Since the state *x*<sup>1</sup> can be viewed as stably filtered output signals of *y*<sup>2</sup> and *y*1, it is uniformly bounded. Also, *η*1, *λ*<sup>1</sup> are some stably filtered signals of *y*<sup>2</sup> and *y*1, they are uniformly bounded. It further implies Φ<sup>1</sup> is uniformly bounded. Then we can conclude *x*ˇ1 is uniformly

*x*ˇ2,1 is uniformly bounded. Therefore, it follows *Tf* = ∞ and the complete system states are

The boundedness of closed-loop state variables of **S2** can be proven with the similar line of

+ |*η*˜*i*| 2

> *θ*1| 2 Φ� 1*C*�

*<sup>γ</sup>*2|*wi*<sup>|</sup>

*<sup>i</sup>*,0 − *x*ˇ� *i*,0 2 *Q*¯*i*,0 − *l*<sup>0</sup> 

*<sup>γ</sup>*2|*wi*<sup>|</sup>

2 − 2 ∑ *i*=1 *γ*ˇ 2| *γ γ*ˇ *w*ˇ*i*,*a*| 2 

*<sup>U</sup>*˙ <sup>d</sup>*<sup>τ</sup>* <sup>−</sup> *<sup>U</sup>*(*t*) + *<sup>U</sup>*(0)

2 ∑ *i*=1 2 − 2 ∑ *i*=1

<sup>1</sup> <sup>−</sup> <sup>1</sup>)|*θ*<sup>1</sup> <sup>−</sup> <sup>ˆ</sup>

<sup>2</sup> <sup>+</sup> *<sup>l</sup>*<sup>1</sup> <sup>+</sup> *<sup>l</sup>*<sup>2</sup> <sup>−</sup>

2 ∑ *i*=1 *Yi* <sup>−</sup> <sup>2</sup> (*θ<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup>

*C*` ¯

*<sup>x</sup>*`1 *<sup>k</sup>*+<sup>1</sup> <sup>=</sup> *<sup>a</sup>*˜1,1 *<sup>k</sup>*+1*x*<sup>1</sup> <sup>+</sup> ··· <sup>+</sup> *<sup>a</sup>*˜1,*k k*+1*x*1,*<sup>k</sup>* <sup>+</sup> *<sup>x</sup>*1,*k*+<sup>1</sup> <sup>+</sup> *<sup>T</sup>*˜

reasoning above. Thus, we have established statement 1 in all cases.

*<sup>θ</sup>*1, *<sup>x</sup>*ˇ1,*i*, *<sup>x</sup>*1,*<sup>i</sup>* <sup>−</sup> *<sup>b</sup>*1,0*λ*1,*b i*, *<sup>λ</sup>*1,*b i*, <sup>Φ</sup>1,*u i*, *<sup>x</sup>*1,*i*, and ¯

*θi*)| 2 Π−<sup>1</sup> *<sup>i</sup>* <sup>Δ</sup>*i*Π−<sup>1</sup> *i*

*<sup>Q</sup>*¯ <sup>2</sup> <sup>+</sup> *�*<sup>1</sup> (*γ*2*ζ*<sup>2</sup>

(*x*1,1 <sup>−</sup> *yd*)<sup>2</sup> <sup>+</sup> *<sup>l</sup>*<sup>1</sup> <sup>+</sup> *<sup>l</sup>*<sup>2</sup> <sup>−</sup>

− 2 ∑ *i*=1 *γ*2 *θ*� *<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup> *θ*� *<sup>i</sup>*,0 *x*�

(*x*1,1 − *yd*)

*<sup>i</sup>*,0 − *x*ˇ� *i*,0 2 *Q*¯*i*,0 + *tf* 0

*x*`1,1 *<sup>k</sup>*+<sup>1</sup> is satisfied equation (24). Comparing the design model (2) and

1,*k*+1 3 ¯

*x*`1 *<sup>k</sup>*<sup>+</sup>1. Thus, we can conclude the boundedness of Φ1,*us <sup>i</sup>*,

*x*`3 + *T*˜

*x*`1 *<sup>i</sup>*, ∀*i* ∈ {1, ···*r*1}.

*θ*1. This further implies that the control input

*θi*)� *Pi*,*r*(ˇ *θi*) +

<sup>1</sup>*C*1Φ<sup>1</sup> <sup>+</sup> *�*2|*θ*<sup>2</sup> <sup>−</sup> <sup>ˆ</sup>

*<sup>γ</sup>*2|*wi*,*a*<sup>|</sup> 2 d*τ*

*ri* ∑ *j*=1

*θ*2| 2 Φ� 2*C*� <sup>2</sup>*C*2Φ<sup>2</sup>

*βi*,*jz*<sup>2</sup> *i*,*j* 

d*τ* − *l*<sup>0</sup>

1,*k*+1 4 are constant matrices.

1,*k*+1 4 ¯

*x*`4 (26)

*x*` = *C*1*x*1. It further implies

*θ*<sup>1</sup> is uniformly bounded away from 0, we

*<sup>x</sup>*ˇ1,*k*+<sup>1</sup> is uniformly bounded and *<sup>b</sup>*1,*p*<sup>0</sup> + *<sup>A</sup>*¯ 1,212 0 <sup>ˇ</sup>

¯ *C*` 1 ¯ *A*` *k* 1 ¯

of *x*1,*k*+<sup>1</sup> is uniformly bounded.

the canonical form (78) in [12], we have ¯

where *<sup>a</sup>*˜1,1 *<sup>k</sup>*<sup>+</sup>1, ··· , *<sup>a</sup>*˜1,*k k*+<sup>1</sup> are constants, and *<sup>T</sup>*˜

bounded from the boundedness of *<sup>x</sup>*˜1 <sup>−</sup> <sup>Φ</sup><sup>1</sup> ˜

We define *l*<sup>0</sup> = *l*1,0 + *l*2,0 = *V*2,*r*<sup>2</sup> (*X*1(0), *X*2(0)), and

*<sup>γ</sup>*4|*xi* <sup>−</sup> *<sup>x</sup>*ˆ*<sup>i</sup>* <sup>−</sup> <sup>Φ</sup>*<sup>i</sup>* (*θ<sup>i</sup>* <sup>−</sup> <sup>ˆ</sup>

Then, we have the boundedness of ¯

1,212 0 ˜

uniformly bounded on [0, ∞).

2 ∑ *i*=1


sup *<sup>w</sup>*` <sup>1</sup>∈W` 1,*w*` <sup>2</sup>∈W` <sup>2</sup>

≤ sup *<sup>w</sup>*` <sup>1</sup>∈W` 1,*w*` <sup>2</sup>∈W` <sup>2</sup>

> <sup>−</sup>*γ*<sup>2</sup> 2 ∑ *i*=1

≤ −*U*(*t*) ≤ 0

2 *<sup>Q</sup>*¯ <sup>1</sup> + 1 4 |*ς*2,*r*<sup>2</sup> | 2

 *tf* 0

 *tf* 0

 *θ*� *<sup>i</sup>* <sup>−</sup> <sup>ˇ</sup> *θ*� *<sup>i</sup>*,0 *x*�

Next, we need to show ¯

¯

Hence, we have

*<sup>x</sup>*˜1,*<sup>i</sup>* <sup>−</sup> *<sup>λ</sup>*1,*b iA*¯

*l*<sup>1</sup> + *l*<sup>2</sup> :=

+ 1 4

$$\int\_0^\infty |\mathbf{x}\_1 - y\_d|^2 \mathbf{d}\tau \le \int\_0^\infty \left(\gamma^2 |\vec{w}\_{1,a}|^2 + \gamma^2 |\dot{M}\_1 \vec{w}\_1|^2 + \gamma^2 |\vec{w}\_{2,a}|^2 + \gamma^2 |\dot{M}\_2 \vec{w}\_2|^2\right) \mathbf{d}\tau + \mathcal{U}(0) < +\infty$$

By the first statement, we notice that

$$\sup\_{0 \le t < \infty} |\dot{\mathfrak{x}}\_1 - \dot{\mathfrak{y}}\_d| < \infty.$$

Then, we have

$$\lim\_{t \to \infty} |\mathfrak{x}\_1(t) - y\_d(t)| = 0$$

For the last statement, it's easy to establish by Section 4.

This complete the proof of the theorem. �

#### **6. Example**

In this section, we present one example to illustrate the main results of this Chapter. The designs were carried out using MATLAB symbolic computation tools, and the closed-loop systems were simulated using SIMULINK.

Consider the following linear systems with zeros initial conditions:

$$
\dot{\mathfrak{x}}\_1 = \dot{\mathfrak{x}}\_1 + \dot{\mathfrak{x}}\_2 + \dot{\mathfrak{x}}\_3 + 0.1 \dot{w}\_1;\tag{27a}
$$

$$
\dot{\mathfrak{x}}\_2 = (1 + \theta\_1)\mathfrak{x}\_3 + (1 + \theta\_2)\mathfrak{w}\_1 \tag{27b}
$$

$$
\dot{\mathfrak{X}}\_3 = -\mathfrak{x}\_1 - \mathfrak{x}\_3 + \mathfrak{x}\_4 + \mathfrak{u} + \mathfrak{w}\_2 \tag{27c}
$$

$$
\dot{\mathfrak{X}}\_4 = \mathfrak{x}\_1 + (2 + \theta\_3)\mu + 0.1\vartheta\nu\_2 + \vartheta\nu\_2 \tag{27d}
$$

$$y = \pounds\_1 + 0.1\dot{w}\_1\tag{27e}$$

where *θ*1, *θ*<sup>2</sup> and *θ*<sup>3</sup> are three unknown parameters with true value 0s. The coefficient terms, 0.1 and 1, reflect the *a priori* knowledge that the disturbances *w*` <sup>1</sup> and *w*` <sup>2</sup> are weak in power relative to that of the disturbance *w*ˇ <sup>1</sup> and *w*ˇ 2. We note that (27) is an unobservable system. We can decompose (27) into the following two SISO linear systems, **S1** and **S2**, sequentially interconnected with additional output measurement,

$$
\dot{\mathbf{x}}\_{11} = \mathbf{x}\_{11} + \mathbf{x}\_{12} + \mathbf{y}\_2 + w\_{11};\tag{28a}
$$

$$
\dot{x}\_{12} = (1 + \theta\_1)y\_2 + (1 + \theta\_2)\vartheta\_1 + w\_{12};\tag{28b}
$$

$$y\_1 = x\_{11} + w\_{13} \,\text{\AA} \tag{28c}$$

$$\dot{\mathbf{x}}\_{21} = -\mathbf{x}\_{21} + \mathbf{x}\_{22} + \boldsymbol{\mu} - \mathbf{y}\_1 + \mathbf{\upbeta}\_2 + w\_{21};\tag{28d}$$

$$\dot{x}\_{22} = (2+\theta\_3)u + y\_1 + \vartheta\_2 + w\_{22};\tag{28e}$$

$$y\_2 = x\_{21} + w\_{23} \tag{28f}$$

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> −0.8

(a)

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> −1.5

(c)

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> −12

(e)

*w*` <sup>2</sup> = 0, *w*` <sup>3</sup> = 0, *w*ˇ <sup>1</sup>(*t*) = sin(12*t* + *<sup>π</sup>*

(f) State estimation error;

time[second]

time [sec]

control input

time[second]

parameter estimation

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> −0.01

(b)

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> −0.15

(d)

state estimation error of S1

and S2

<sup>3</sup> ). (a) Parameter estimate; (b)

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> −0.2

(f)

time[second]

time [sec]

time[second]

Tracking Error

Tracking error

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

131

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

−0.1 −0.05 0 0.05 0.1 0.15 0.2

> −0.1 0 0.1 0.2 0.3 0.4 0.5

state estimation error

**Figure 2.** System response for Example under command input *d*(*t*) = 0.4 sin(0.1*t*) + sin(0.6*t*), *w*` <sup>1</sup> = 0,

Tracking error; (c) Parameter estimate(based on [17]); (d) Tracking error(based on [17]); (e) control input;

<sup>9</sup> ) + 0.8 sin(3*t*), and *<sup>w</sup>*<sup>ˇ</sup> <sup>2</sup>(*t*) = 3 sin(3*<sup>t</sup>* <sup>+</sup> *<sup>π</sup>*

xgv1 − yd

tracking error

parameter estimation

−0.6 −0.4 −0.2 0 0.2 0.4 0.6

−1 −0.5 0 0.5 1 1.5

> −10 −8 −6 −4 −2 0 2 4 6

control input

thk

unknown parameter estimation

where

$$\begin{aligned} \mathbf{x}\_{1} &= \begin{bmatrix} \mathbf{x}\_{11} \\ \mathbf{x}\_{12} \end{bmatrix} = \begin{bmatrix} \dot{\mathbf{x}}\_{1} \\ \dot{\mathbf{x}}\_{2} \end{bmatrix}; \mathbf{x}\_{2} = \begin{bmatrix} \mathbf{x}\_{21} \\ \mathbf{x}\_{22} \end{bmatrix} = \begin{bmatrix} \dot{\mathbf{x}}\_{3} \\ \dot{\mathbf{x}}\_{4} \end{bmatrix}; \\\ w\_{1} &= \begin{bmatrix} w\_{11} \\ w\_{12} \\ w\_{13} \end{bmatrix} = \begin{bmatrix} 0.1\dot{\boldsymbol{\nu}}\_{1} - \dot{\boldsymbol{\nu}}\_{3} \\ -(1 - \theta\_{1})\dot{\boldsymbol{\nu}}\_{3} \\ 0.1\dot{\boldsymbol{\nu}}\_{1} \end{bmatrix}; w\_{2} = \begin{bmatrix} w\_{21} \\ w\_{22} \\ w\_{23} \end{bmatrix} = \begin{bmatrix} 0.1\dot{\boldsymbol{\nu}}\_{1} \\ -0.1\dot{\boldsymbol{\nu}}\_{1} + 0.1\dot{\boldsymbol{\nu}}\_{2} \\ \dot{\boldsymbol{\nu}}\_{3} \end{bmatrix}. \end{aligned}$$

Here *w*` <sup>3</sup> is the measurement disturbance of the state *x*`3. It is easy to check that **S1** and **S2** in (28) satisfied the assumptions 1–5.

For the adaptive control design, we set the desired disturbance attenuation level *γ* = 10. We select the true value of the parameters in subsystem **S1** and subsystem **S2** are zeros, and belong to the interval [−1, 1]. The projection function *P*1(*θ*1) and *P*2(*θ*2) are chosen as *P*1(*θ*1) = 0.5(*θ*<sup>2</sup> <sup>1</sup> <sup>+</sup> *<sup>θ</sup>*<sup>2</sup> <sup>2</sup> ), *<sup>P</sup>*2(*θ*2) = *<sup>θ</sup>*<sup>2</sup> <sup>3</sup>. The reference trajectory, *yd*, is generated by the following linear system *x*˙*d*,1 = −*xd*,2, *x*˙*d*,2 = *xd*,1 − *xd*,2 + *d*, *yd* = *xd*,1 with zeros initial condition, where *d* is the command input signal. The objective is to achieve asymptotic tracking of *x*`1 to the reference trajectory *yd*.

For design and simulation parameters of **S***i*, (*i* = 1, 2), we select

$$\begin{aligned} \pounds\_{1,0} &= \left[0.2 \ 0 \right]'; \; \pounds\_{2,0} = \left[0.1 \ 0 \right]'; \dot{\ $\!\_{1,0}} = \left[0.5 \ -0.5 \right]'; \; \dot{\$ \!\_{2,0}} = -1/2; \; \mathcal{Q}\_{i,0} = 0.001 \textit{i};\\ K\_{i,\mathcal{L}} &= 0.2; \; \Delta\_{i} = \textit{I}\_{2}; \; p\_{i,n\_{i}} = e\_{2,\mathcal{D}}; \; \Phi\_{i,0} = \mathbf{0}\_{2\times 1}; \; \rho\_{i,\rho} = 2; \; \dot{\rho}\_{i,\Delta} = 0; \; \varepsilon\_{i} = K\_{i,\mathcal{L}}^{-1} s\_{i\Sigma}; \; \lambda\_{i,0} = \mathbf{0}\_{2\times 1};\\ \pounds\_{i,1} &= 0.5; \quad \eta\_{i,0} = \mathbf{0}\_{2\times 1}; \; Z\_{1} = \begin{bmatrix} 0.0893 & -0.0081\\ -0.0081 & 0.0097 \end{bmatrix}; \; Z\_{2} = \begin{bmatrix} 0.1094 & -0.0099\\ -0.0099 & 0.0099 \end{bmatrix} \end{aligned}$$

We present one set of simulation results in this example to illustrate the regulatory behavior of the adaptive controller. We set *d*(*t*) = 0.4 sin(0.1*t*) + sin(0.6*t*), *w*` <sup>1</sup>(*t*) = 0, *w*` <sup>2</sup>(*t*) = 0, *w*` <sup>3</sup>(*t*) = 0, *w*ˇ <sup>1</sup>(*t*) = sin(12*t* + *<sup>π</sup>* <sup>9</sup> ) + 0.8 sin(3*t*), and *<sup>w</sup>*<sup>ˇ</sup> <sup>2</sup>(*t*) = 3 sin(3*<sup>t</sup>* <sup>+</sup> *<sup>π</sup>* <sup>3</sup> ). The results are shown in Figure 2(a)–(f). To illustrate that the proposed controller can improve the system performance by incorporating the measurements and/or the estimation of the significant external disturbances into the control design, the simulation results based on [17] are presented in Figure 2(c)(d), where the measured disturbances *w*ˇ <sup>1</sup> and *w*ˇ <sup>2</sup> are treated as arbitrary disturbances and *θ*<sup>3</sup> is treated as constant in control design. We observe that the output tracking error asymptotically converges to zero and the parameter estimates asymptotically converge to its true value 0 in (a) and (b) even if there is a non-zero measured disturbance in the system. But the parameter estimates doesn't converge to the true value,

24 Will-be-set-by-IN-TECH

where

0.5(*θ*<sup>2</sup>

<sup>1</sup> <sup>+</sup> *<sup>θ</sup>*<sup>2</sup>

*<sup>x</sup>*ˇ1,0 = �

reference trajectory *yd*.

0.2 0 ��

*x*<sup>1</sup> =

*w*<sup>1</sup> =

� *<sup>x</sup>*<sup>11</sup> *x*<sup>12</sup>

> *w*<sup>11</sup> *w*<sup>12</sup> *w*<sup>13</sup>

⎡ ⎢ ⎢ ⎣

(28) satisfied the assumptions 1–5.

<sup>2</sup> ), *<sup>P</sup>*2(*θ*2) = *<sup>θ</sup>*<sup>2</sup>

; *<sup>x</sup>*ˇ2,0 = �

*<sup>β</sup>i*,1 = 0.5; *<sup>η</sup>i*,0 = **<sup>0</sup>**2×1; *<sup>Z</sup>*<sup>1</sup> =

0, *w*` <sup>3</sup>(*t*) = 0, *w*ˇ <sup>1</sup>(*t*) = sin(12*t* + *<sup>π</sup>*

� = � *<sup>x</sup>*`1 *x*`2

⎤ ⎥ ⎥ ⎦ = ⎡ ⎢ ⎢ ⎣ �

For design and simulation parameters of **S***i*, (*i* = 1, 2), we select

; ˇ *<sup>θ</sup>*1,0 = �

*Ki*,*<sup>c</sup>* <sup>=</sup> 0.2; <sup>Δ</sup>*<sup>i</sup>* <sup>=</sup> *<sup>I</sup>*2; *pi*,*ni* <sup>=</sup> *<sup>e</sup>*2,2; <sup>Φ</sup>*i*,0 <sup>=</sup> **<sup>0</sup>**2×1; *<sup>ρ</sup>i*,*<sup>o</sup>* <sup>=</sup> 2; *<sup>β</sup>i*,<sup>Δ</sup> <sup>=</sup> 0; *�<sup>i</sup>* <sup>=</sup> *<sup>K</sup>*−<sup>1</sup>

0.1 0 ��

; *x*<sup>2</sup> =

0.1*w*` <sup>1</sup> − *w*` <sup>3</sup> −(1 − *θ*1)*w*` <sup>3</sup> 0.1*w*` <sup>1</sup>

� *x*<sup>21</sup> *x*<sup>22</sup> � = � *x*`3 *x*`4

Here *w*` <sup>3</sup> is the measurement disturbance of the state *x*`3. It is easy to check that **S1** and **S2** in

For the adaptive control design, we set the desired disturbance attenuation level *γ* = 10. We select the true value of the parameters in subsystem **S1** and subsystem **S2** are zeros, and belong to the interval [−1, 1]. The projection function *P*1(*θ*1) and *P*2(*θ*2) are chosen as *P*1(*θ*1) =

system *x*˙*d*,1 = −*xd*,2, *x*˙*d*,2 = *xd*,1 − *xd*,2 + *d*, *yd* = *xd*,1 with zeros initial condition, where *d* is the command input signal. The objective is to achieve asymptotic tracking of *x*`1 to the

0.5 <sup>−</sup>0.5 ��

� 0.0893 <sup>−</sup>0.0081 −0.0081 0.0097

We present one set of simulation results in this example to illustrate the regulatory behavior of the adaptive controller. We set *d*(*t*) = 0.4 sin(0.1*t*) + sin(0.6*t*), *w*` <sup>1</sup>(*t*) = 0, *w*` <sup>2</sup>(*t*) =

results are shown in Figure 2(a)–(f). To illustrate that the proposed controller can improve the system performance by incorporating the measurements and/or the estimation of the significant external disturbances into the control design, the simulation results based on [17] are presented in Figure 2(c)(d), where the measured disturbances *w*ˇ <sup>1</sup> and *w*ˇ <sup>2</sup> are treated as arbitrary disturbances and *θ*<sup>3</sup> is treated as constant in control design. We observe that the output tracking error asymptotically converges to zero and the parameter estimates asymptotically converge to its true value 0 in (a) and (b) even if there is a non-zero measured disturbance in the system. But the parameter estimates doesn't converge to the true value,

; ˇ

�

; *Z*<sup>2</sup> =

<sup>9</sup> ) + 0.8 sin(3*t*), and *<sup>w</sup>*<sup>ˇ</sup> <sup>2</sup>(*t*) = 3 sin(3*<sup>t</sup>* <sup>+</sup> *<sup>π</sup>*

⎤ ⎥ ⎥ ⎦ ; *w*<sup>2</sup> =

*y*<sup>1</sup> = *x*<sup>11</sup> + *w*13; (28c)

*x*˙21 = −*x*<sup>21</sup> + *x*<sup>22</sup> + *u* − *y*<sup>1</sup> + *w*ˇ <sup>2</sup> + *w*21; (28d)

*x*˙22 = (2 + *θ*3)*u* + *y*<sup>1</sup> + *w*ˇ <sup>2</sup> + *w*22; (28e)

*y*<sup>2</sup> = *x*<sup>21</sup> + *w*<sup>23</sup> (28f)

� ;

*w*<sup>21</sup> *w*<sup>22</sup> *w*<sup>23</sup> ⎤ ⎥ ⎥ ⎦ =

<sup>3</sup>. The reference trajectory, *yd*, is generated by the following linear

⎡ ⎢ ⎣

0.1*w*` <sup>1</sup> −0.1*w*` <sup>1</sup> + 0.1*w*` <sup>2</sup> *w*` 3

*θ*2,0 = −1/2; *Qi*,0 = 0.001*I*2;

� 0.1094 <sup>−</sup>0.0099 −0.0099 0.0099

⎤ ⎥ ⎦

*<sup>i</sup>*,*<sup>c</sup> si*,Σ; *<sup>λ</sup>i*,0 = **<sup>0</sup>**2×1;

<sup>3</sup> ). The

�

⎡ ⎢ ⎢ ⎣

**Figure 2.** System response for Example under command input *d*(*t*) = 0.4 sin(0.1*t*) + sin(0.6*t*), *w*` <sup>1</sup> = 0, *w*` <sup>2</sup> = 0, *w*` <sup>3</sup> = 0, *w*ˇ <sup>1</sup>(*t*) = sin(12*t* + *<sup>π</sup>* <sup>9</sup> ) + 0.8 sin(3*t*), and *<sup>w</sup>*<sup>ˇ</sup> <sup>2</sup>(*t*) = 3 sin(3*<sup>t</sup>* <sup>+</sup> *<sup>π</sup>* <sup>3</sup> ). (a) Parameter estimate; (b) Tracking error; (c) Parameter estimate(based on [17]); (d) Tracking error(based on [17]); (e) control input; (f) State estimation error;

#### 26 Will-be-set-by-IN-TECH 132 Game Theory Relaunched A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems <sup>27</sup>

and the tracking error doesn't converge to zero in (c) and (d). State estimation error, *x*<sup>1</sup> − *x*ˇ1 and *x*<sup>2</sup> − *x*ˇ2, converge to zero in (f), and the transient performance behaves well as in (e).

[2] Ba¸sar, T. & Bernhard, P. [1995]. *H*∞*-Optimal Control and Related Minimax Design Problems:*

A Game Theoretic Approach Based Adaptive Control Design for Sequentially Interconnected SISO Linear Systems

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[3] Didinsky, G. [1994]. *Design of minimax controllers for nonlinear systems using cost-to-come*

[4] Goodwin, G. C. & Mayne, D. Q. [1987]. A parameter estimation perspective of

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[6] Ioannou, P. A. & Sun, J. [1996]. *Robust Adaptive Control*, Prentice Hall, Upper Saddle

[8] Kanellakopoulos, I., Kokotovi´c, P. V. & Morse, A. S. [1991]. Systematic design of adaptive controllers for feedback linearizable systems, *IEEE Transactions on Automatic Control*

[9] Krsti´c, M., Kanellakopoulos, I. & Kokotovi´c, P. V. [1995]. *Nonlinear and Adaptive Control*

[10] Kumar, P. R. [1985]. A survey of some results in stochastic adaptive control, *SIAM Journal*

[11] Pan, Z. & Ba¸sar, T. [1996]. Parameter identification for uncertain linear systems with partial state measurements under an *H*<sup>∞</sup> criterion, *IEEE Transactions on Automatic Control*

[12] Pan, Z. & Ba¸sar, T. [2000]. Adaptive controller design and disturbance attenuation for SISO linear systems with noisy output measurements, *CSL report*, University of Illinois

[13] Rohrs, C. E., Valavani, L., Athans, M. & Stein, G. [1985]. Robustness of continuous-time adaptive control algorithms in the presence of unmodeled dynamics, *IEEE Transactions*

[14] Tezcan, I. E. & Ba¸sar, T. [1999]. Disturbance attenuating adaptive controllers for parametric strict feedback nonlinear systems with output measurements, *Journal of Dynamic Systems, Measurement and Control, Transactions of the ASME* 121(1): 48–57. [15] Zeng, S. [2010]. Adaptive controller design and disturbance attenuation for a general class of sequentially interconnected siso linear systems with noisy output measurements, *Proceedings of the 49th IEEE Conference on Decision and Control(CDC)*, Atlanta, GA,

[16] Zeng, S. [2011]. Worst-case analysis based adaptive control design for siso linear systems with plant and actuation uncertainties, *Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC)*, Orlando, FL, pp. 6349–6354. [17] Zeng, S. & Fernandez, E. [2010]. Adaptive controller design and disturbance attenuation for sequentially interconnected siso linear systems under noisy output measurements,

[18] Zeng, S. & Pan, Z. [2009]. Adaptive controller design and disturbance attenuation for SISO linear systems with noisy output measurements and partly measured disturbances,

[19] Zeng, S., Pan, Z. & Fernandez, E. [2010]. Adaptive controller design and disturbance attenuation for SISO linear systems with zero relative degree under noisy output measurements, *International Journal of Adaptive Control and Signal Processing* 24: 287–310.

[7] Isidori, A. [1995]. *Nonlinear Control Systems*, 3rd edn, Springer-Verlag, London.

*A Dynamic Game Approach*, 2nd edn, Birkhäuser, Boston, MA.

*methods*, PhD thesis, University of Illinois, Urbana, IL.

continuous time adaptive control, *Automatica* 23: 57–70.

Englewood Cliffs.

River, NJ.

36: 1241–1253.

41: 1295–1311.

pp. 2608–2613.

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*on Control and Optimization* 23(3): 329–380.

at Urbana-Champaign, Urbana, IL.

*on Automatic Control* 30: 881–889.

*IEEE Transactions on Automatic Control* 55: 2123–2129.

*International Journal of Control* 82(2): 310–334.

#### **7. Conclusions**

In this Chapter, we present the game-theoretical approach based adaptive control design for a special class of MIMO linear systems, which is composed of two sequentially interconnected SISO linear systems, **S**<sup>1</sup> and **S**2. We assume the subsystem under studied subject to noisy output measurements, unknown initial state conditions, linear unknown parametric uncertainties, measured and unmeasured additive exogenous disturbance input uncertainties. Our design objective is to address the asymptotical tracking, the transient response and robustness of the closed-loop system, which are the same as the objectives to motivate the study of the *H*∞- optimal control problem. In view of the similar solution between *H*<sup>∞</sup> optimal control design and *zero sum differential game*, we convert the original adaptive control design problem into a zero-sum game with soft constraints on the disturbance input uncertainties and the unknown initial state uncertainties, which incorporates the measures of transient response, disturbance attenuation, and asymptotic tracking into a single game-theoretic cost function and formulates the design problem as a nonlinear *H*<sup>∞</sup> control problem under imperfect state measurements. A game-theoretical approach, *cost-to-come function* analysis, is then applied to obtain the finite dimensional estimators of **S1** and **S2** independently, which is also converted the control design as an *H*<sup>∞</sup> control problem with full information measurements. The integrator backstepping methodology is finally applied on this full information measurements problem to obtain a suboptimal solution. The controller achieves the same result as [17], namely the total stability of the closed-loop system, the desired disturbance attenuation level, and asymptotic tracking of the reference trajectory when the disturbance is of finite energy and uniformly bounded. In addition, the proposed controller may achieve arbitrary positive disturbance attenuation level with respect to the measured disturbances by proper scaling. The contribution of the measurements of part of the disturbance inputs is that we can design an adaptive controller with disturbance feedforward structure with respect to *w*ˇ 1,*<sup>b</sup>* and *w*ˇ 2,*<sup>b</sup>* to eliminate their effect on the squared L<sup>2</sup> norm of the tracking error. Moreover, the asymptotic tracking is achieved even if the measured disturbances are only uniformly bounded without requiring them to be of finite energy.

#### **Author details**

Sheng Zeng *Critical Care R&D Engineering, Carefusion Corporation, CA, 92887, USA*

Emmanuel Fernandez *School of Electronics & Computing Systems, University of Cincinnati, OH, 45221-0030, USA*

#### **8. References**

[1] Arslan, G. & Ba¸sar, T. [2001]. Disturbance attenuating controller design for strict-feedback systems with structurally unknown dynamics, *Automatica* 37(8): 1175–1188.

[2] Ba¸sar, T. & Bernhard, P. [1995]. *H*∞*-Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach*, 2nd edn, Birkhäuser, Boston, MA.

26 Will-be-set-by-IN-TECH

and the tracking error doesn't converge to zero in (c) and (d). State estimation error, *x*<sup>1</sup> − *x*ˇ1 and *x*<sup>2</sup> − *x*ˇ2, converge to zero in (f), and the transient performance behaves well as in (e).

In this Chapter, we present the game-theoretical approach based adaptive control design for a special class of MIMO linear systems, which is composed of two sequentially interconnected SISO linear systems, **S**<sup>1</sup> and **S**2. We assume the subsystem under studied subject to noisy output measurements, unknown initial state conditions, linear unknown parametric uncertainties, measured and unmeasured additive exogenous disturbance input uncertainties. Our design objective is to address the asymptotical tracking, the transient response and robustness of the closed-loop system, which are the same as the objectives to motivate the study of the *H*∞- optimal control problem. In view of the similar solution between *H*<sup>∞</sup> optimal control design and *zero sum differential game*, we convert the original adaptive control design problem into a zero-sum game with soft constraints on the disturbance input uncertainties and the unknown initial state uncertainties, which incorporates the measures of transient response, disturbance attenuation, and asymptotic tracking into a single game-theoretic cost function and formulates the design problem as a nonlinear *H*<sup>∞</sup> control problem under imperfect state measurements. A game-theoretical approach, *cost-to-come function* analysis, is then applied to obtain the finite dimensional estimators of **S1** and **S2** independently, which is also converted the control design as an *H*<sup>∞</sup> control problem with full information measurements. The integrator backstepping methodology is finally applied on this full information measurements problem to obtain a suboptimal solution. The controller achieves the same result as [17], namely the total stability of the closed-loop system, the desired disturbance attenuation level, and asymptotic tracking of the reference trajectory when the disturbance is of finite energy and uniformly bounded. In addition, the proposed controller may achieve arbitrary positive disturbance attenuation level with respect to the measured disturbances by proper scaling. The contribution of the measurements of part of the disturbance inputs is that we can design an adaptive controller with disturbance feedforward structure with respect to *w*ˇ 1,*<sup>b</sup>* and *w*ˇ 2,*<sup>b</sup>* to eliminate their effect on the squared L<sup>2</sup> norm of the tracking error. Moreover, the asymptotic tracking is achieved even if the measured disturbances are only uniformly bounded without requiring them to be of finite energy.

**7. Conclusions**

**Author details**

Emmanuel Fernandez

37(8): 1175–1188.

**8. References**

*Critical Care R&D Engineering, Carefusion Corporation, CA, 92887, USA*

*School of Electronics & Computing Systems, University of Cincinnati, OH, 45221-0030, USA*

[1] Arslan, G. & Ba¸sar, T. [2001]. Disturbance attenuating controller design for strict-feedback systems with structurally unknown dynamics, *Automatica*

Sheng Zeng

	- [20] Zhao, Q., Pan, Z. & Fernandez, E. [2009]. Convergence analysis for reduced-order adaptive controller design of uncertain siso linear systems with noisy output measurements, *International Journal of Control* 82(11): 1971–1990.

© 2013 Garcia-Diaz and Lee, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

© 2013 Garcia-Diaz and Lee, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

distribution, and reproduction in any medium, provided the original work is properly cited.

**Models for Highway Cost Allocation** 

Historically, *equity* has been one of the most important principles applied to formulate tax policy. It has been considered when raising revenues and allocating funds for maintenance, capital improvements, operating programs, and services to the public. The problem of determining how the total cost of a shared facility or service should be divided fairly and rationally is common both in public and private enterprises. The theory of cooperative games is widely used for allocating these costs. Examples of this include but are not limited to public utilities providing telephone services, electricity, water, and transport; public works projects designed to serve different constituencies; access fees or user charges for airports, highways, bridges or waterways; internal accounting rules to allocate overhead

The purpose of a *Highway Cost Allocation* (HCA) study is to determine the fair share that each class of road user (vehicle class) should pay for the construction, maintenance, operation, improvement, and related costs of highways, roads, bridges, and streets in a highway network, such as those managed by state Departments of Transportation in the U.S.A. Particular emphasis should be placed on criteria and methods for allocating costs among vehicle classes using a common highway facility (road or bridge, for example) in a just, equitable, fair, and reasonable manner. Cost allocation is ultimately concerned with fairness. Through a comparison of revenues (user fees paid) and cost responsibilities, this study will estimate current equity and recommend alternatives to bring about a closer

A significant objective of HCA studies is to analyze highway-related costs attributable to different highway users as a basis for evaluating the equity and efficiency of user charges. Ideally, the costs incurred by the various user groups should be in proportion to the damage they contribute to the highway system. The cost of supporting a highway infrastructure may be deemed fair if there is an equitable distribution of costs and revenues among the various groups of highway users. With this assumption, equity is achieved when each group's

match between payments and cost responsibilities for each vehicle class.

Alberto Garcia-Diaz and Dong-Ju Lee

http://dx.doi.org/10.5772/53927

costs in private companies [1-4].

**1. Introduction** 

Additional information is available at the end of the chapter
