**1. Introduction**

The simultaneous *H*∞ control problem concerns with designing a single controller which simultaneously renders a set of systems being internally stable and satisfying an *L*2-gain specification. In the last decades, there have been some researchers studying the simultaneous *H*∞ control problem in linear case, see references [1–6]. In references [1] and [2], necessary and sufficient conditions for the simultaneous *H*∞ control via nonlinear digital output feedback controllers were derived by using the dynamic programming approach. In reference [3], a numerical design method was proposed for designing simultaneous *H*∞ controllers. In reference [4], it was shown that the simultaneous *H*∞ control problem is equivalent to a strong *H*∞ control problem. In reference [5], linear periodically time-varying controllers were employed for simultaneous *H*∞ control. In reference [6], a simultaneous *H*∞ control problem was solved via the chain scattering framework.

© 2016 The Author(s). Licensee InTech. This chapter is 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. © 2016 The Author(s). Licensee InTech. This chapter is 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.

All the results mentioned earlier are derived for linear systems case. Till now, only few results have been reported about simultaneous *H*∞ control of nonlinear systems, see references [7, 8]. In reference [7], a control storage function (CSF) method was developed for designing simultaneous *H*∞ state feedback controllers for a collection of single-input nonlinear systems. Necessary and sufficient conditions for the existence of simultaneous *H*∞ controllers were derived. Moreover, an explicit formula for constructing simultaneous *H*∞ feedback controllers was proposed. The CSF approach was first introduced in reference [9]. It is motivated by the control Lyapunov function (CLF) method (please see references [10–18]) for designing stabilizing controllers of nonlinear control systems. One difficulty in applying CSFs/CLFs for solving control problems is that how to derive CSFs/CLFs for nonlinear systems is an open problem unless they are in some particular forms. No systematic methods for constructing CSFs have been proposed in reference [7]. It is important to identify those nonlinear systems whose corresponding CSFs/CLFs exist and can be constructed systematically. In reference [8], the CSF approach was applied to design simultaneous *H*∞ controllers for a collection of nonlinear control systems in canonical form. It was shown that under mild assumptions, CSFs can be constructed systematically for nonlinear systems in canonical form; and simultaneous *H*∞ control for such systems can be easily achieved. In this chapter, we further study the simultaneous *H*∞ control problem for nonlinear systems in strict-feedback form. It is known that the strict-feedback form is more general than the canonical form. Moreover, a restrictive assumption made in reference [8] is relaxed in this chapter. Based on the CSF approach and by using the backstepping technique, we develop a systematic method for constructing simultaneous *H*∞ state feedback controllers. The proposed results in reference [8] are special cases of the results presented in this chapter.

## **2. Problem formulation and preliminaries**

In this section, the simultaneous *H*∞ control problem to be solved will be formulated and some preliminaries will be presented. For simplifying the expressions, we use the same notations *x, u, w*, and *z* to denote the states, control inputs, exogenous inputs, and the controlled outputs of all the considered systems.

#### **2.1. Problem formulation**

Consider a collection of nonlinear control systems:

$$\begin{aligned} \dot{\mathbf{x}} &= f\_{\iota}(\mathbf{x}) + \mathbf{g}\_{\iota \iota}(\mathbf{x})\mathbf{w} + \mathbf{g}\_{\omega \iota}(\mathbf{x})\mathbf{u} \\ \dot{\mathbf{z}} &= h\_{\iota \iota}(\mathbf{x}) + k\_{11\iota}(\mathbf{x})\mathbf{w}, \quad \dot{\mathbf{i}} = \mathbf{l}, \dots, q, \end{aligned} \tag{1}$$

where = 1, , ⋯, ∈ is the state, *w* ∈ *Rm* is the disturbance input, *u* ∈ *R* is the control input, *z*∈*Rr* is the controlled output, : , 1: ×, 2: , ℎ1: , and 11: ×, = 1, …, , are smooth functions. Here we denote the *i-*th system in Eq. (1) as system *Si* . For all *i*=1,…,*q*, suppose that *f*<sup>i</sup> (0) = 0 and *h*1*<sup>i</sup>* (0) = 0. For convenience, define = [1, 2, ⋯, ] ∈ , = 1, …, . Suppose that *fi* (*x*), *g*1*<sup>i</sup>* (*x*), and *g*2*<sup>i</sup>* (*x*), *i*=1,…,*q*, have the following forms:

$$f\_i(\mathbf{x}) = \begin{bmatrix} \mathbf{x}\_2 + \theta\_{il}(\mathbf{x}\_1) \\ \vdots \\ \mathbf{x}\_{j+1} + \theta\_{jl}(\overline{\mathbf{x}}\_j) \\ \vdots \\ \mathbf{x}\_n + \theta\_{il(n-1)}(\overline{\mathbf{x}}\_{n-1}) \\ \theta\_{il}(\mathbf{x}) \end{bmatrix}, \text{ g}\_{il}(\mathbf{x}) = \begin{bmatrix} 0 \\ \vdots \\ 0 \\ 0 \\ \vdots \\ 0 \\ \rho\_i(\mathbf{x}) \end{bmatrix}, \text{ g}\_{21}(\mathbf{x}) = \begin{bmatrix} 0 \\ \vdots \\ 0 \\ 0 \\ \vdots \\ 0 \\ \eta\_i(\mathbf{x}) \end{bmatrix}. \tag{2}$$

where : , : 1 × , and : , = 1, 2, …, , = 1, …, , are smooth functions with θ*ij*(0) = 0 and () ≠ 0 for each *<sup>x</sup>* <sup>∈</sup> *Rn*. Assume that all functions (), = 1, …, , have the same sign. Without loss of generality, suppose that () > 0, = 1, …, . By Eq. (2), the *q* possible models can be explicitly expressed as

$$\begin{aligned} \dot{\mathbf{x}}\_{1} &= \mathbf{x}\_{2} + \boldsymbol{\Theta}\_{\boldsymbol{u}}(\mathbf{x}\_{1}) \\ &\vdots\\ \dot{\mathbf{x}}\_{j} &= \mathbf{x}\_{j+1} + \boldsymbol{\Theta}\_{\boldsymbol{y}}(\overline{\mathbf{x}}\_{j}) \\ &\vdots\\ \dot{\mathbf{x}}\_{n-1} &= \mathbf{x}\_{n} + \boldsymbol{\Theta}\_{\boldsymbol{u}(n-1)}(\overline{\mathbf{x}}\_{n-1}) \\ \dot{\mathbf{x}}\_{n} &= \boldsymbol{\Theta}\_{\boldsymbol{u}}(\mathbf{x}) + \boldsymbol{\uprho}\_{i}(\mathbf{x})\boldsymbol{w} + \eta\_{i}(\mathbf{x})\boldsymbol{u}, \\ \boldsymbol{z} &= h\_{\boldsymbol{u}}(\mathbf{x}) + h\_{1\boldsymbol{u}}(\mathbf{x})\boldsymbol{w}, \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \end{aligned} \tag{3}$$

Suppose that the following assumption holds.

All the results mentioned earlier are derived for linear systems case. Till now, only few results have been reported about simultaneous *H*∞ control of nonlinear systems, see references [7, 8]. In reference [7], a control storage function (CSF) method was developed for designing simultaneous *H*∞ state feedback controllers for a collection of single-input nonlinear systems. Necessary and sufficient conditions for the existence of simultaneous *H*∞ controllers were derived. Moreover, an explicit formula for constructing simultaneous *H*∞ feedback controllers was proposed. The CSF approach was first introduced in reference [9]. It is motivated by the control Lyapunov function (CLF) method (please see references [10–18]) for designing stabilizing controllers of nonlinear control systems. One difficulty in applying CSFs/CLFs for solving control problems is that how to derive CSFs/CLFs for nonlinear systems is an open problem unless they are in some particular forms. No systematic methods for constructing CSFs have been proposed in reference [7]. It is important to identify those nonlinear systems whose corresponding CSFs/CLFs exist and can be constructed systematically. In reference [8], the CSF approach was applied to design simultaneous *H*∞ controllers for a collection of nonlinear control systems in canonical form. It was shown that under mild assumptions, CSFs can be constructed systematically for nonlinear systems in canonical form; and simultaneous *H*∞ control for such systems can be easily achieved. In this chapter, we further study the simultaneous *H*∞ control problem for nonlinear systems in strict-feedback form. It is known that the strict-feedback form is more general than the canonical form. Moreover, a restrictive assumption made in reference [8] is relaxed in this chapter. Based on the CSF approach and by using the backstepping technique, we develop a systematic method for constructing simultaneous *H*∞ state feedback controllers. The proposed results in reference [8] are special

In this section, the simultaneous *H*∞ control problem to be solved will be formulated and some preliminaries will be presented. For simplifying the expressions, we use the same notations *x, u, w*, and *z* to denote the states, control inputs, exogenous inputs, and the controlled outputs

1 2

& (1)

: , 1: ×, 2: , ℎ1:

,

∈ is the state, *w* ∈ *Rm* is the disturbance input, *u* ∈ *R* is the control

() () () () () , i 1, , ,

= + =¼

*ii i*

*x f x g xw g xu z h x k xw q*

1 11

*i i*

=+ +

cases of the results presented in this chapter.

228 Nonlinear Systems - Design, Analysis, Estimation and Control

of all the considered systems.

**2.1. Problem formulation**

where = 1, , ⋯,

input, *z*∈*Rr*

**2. Problem formulation and preliminaries**

Consider a collection of nonlinear control systems:

is the controlled output,

$$\text{Assumption 1: } \mathbf{y}^2 \mathbf{I} - \mathbf{k}\_{11l}^T \mathbf{(x)} \mathbf{k}\_{11l} \mathbf{(x)} > 0. \text{ } \forall \mathbf{x} \in \mathbb{R}^n \text{ and } \forall i \in \{1, ..., q\}.$$

It is clear that we can always find a positive (semi)definite function *U*(*x*) such that, for all *i* ∈ {1,…*q*},

$$h\_{li}^{\operatorname{r}}(\mathbf{x})h\_{li}(\mathbf{x}) + h\_{li}^{\operatorname{r}}(\mathbf{x})k\_{11i}(\mathbf{x})\Big(\boldsymbol{\gamma}^{\operatorname{2}}\boldsymbol{I} - k\_{11i}^{\operatorname{r}}(\mathbf{x})k\_{11i}(\mathbf{x})\Big)^{-1}k\_{11i}^{\operatorname{r}}(\mathbf{x})h\_{li}(\mathbf{x}) \le U(\mathbf{x}), \ \forall \mathbf{x} \in \boldsymbol{R}^{\boldsymbol{n}}.$$

The objective of this chapter is to find a continuous function : such that the state feedback controller

$$
\mu = p(\mathbf{x})\tag{4}
$$

internally stabilizes the systems in Eq. (3) simultaneously; and, for each *T* > 0 and for each *w*<sup>i</sup> ∈ *L*2[0, *T*], all closed-loop systems, starting from the initial state *x*(0) = 0, satisfy (for a given γ > 0)

$$\int\_{0}^{\mathcal{T}} z^{\mathcal{T}}(t)z(t)dt \le \hat{\boldsymbol{\gamma}}^{2} \int\_{0}^{\mathcal{T}} \boldsymbol{\eta}^{\mathcal{T}}(t)\boldsymbol{\nu}(t)dt \quad \text{for some } \hat{\boldsymbol{\gamma}} < \boldsymbol{\gamma}. \tag{5}$$

#### **2.2. Control storage functions**

Here we review some important concepts about the CSF method introduced in references [7, 9]. *Definition 1* [7, 9]: Consider the system *Si* in Eq. (1). A smooth, proper, and positive definite function : is a CSF of *Si* if, for each \ <sup>0</sup> and each *w* ∈ *Rm*,

$$\left[\inf\_{\boldsymbol{\omega}\in\mathcal{R}}\left\{\frac{\partial V\_{i}(\mathbf{x})}{\partial\mathbf{x}}\Big(f\_{i}(\mathbf{x})+\mathbf{g}\_{\scriptscriptstyle\rm II}(\mathbf{x})\mathbf{w}+\mathbf{g}\_{\scriptscriptstyle\rm II}(\mathbf{x})\mathbf{u}\Big)+\Big(h\_{\scriptscriptstyle\rm II}(\mathbf{x})+k\_{\scriptscriptstyle\rm II\dot{\rm I}}(\mathbf{x})\mathbf{w}\Big)^{\intercal}\Big(h\_{\scriptscriptstyle\rm II}(\mathbf{x})+k\_{\scriptscriptstyle\rm II\dot{\rm I}}(\mathbf{x})\mathbf{w}\Big)-\boldsymbol{\gamma}^{2}\mathbf{w}^{\scriptscriptstyle\rm I}\mathbf{w}\right\}<\mathbf{0}.\right]$$

For ensuring the continuity of the obtained simultaneous *H*∞ controllers, the *L2-gain small control property* (*L2-gain SCP*) has been introduced in reference [7].

*Definition 2* [7]: A CSF : of *Si* satisfies the *L2-*gain SCP if for each ε > 0, there is a δ1 > 0 and a δ2 > 0 such that, if *x* ≠ 0 satisfies < 1 and *w* satisfies < 2, there is some *u* with |*u*| < ε satisfying

$$\frac{\partial V\_{i}(\mathbf{x})}{\partial \mathbf{x}} \Big( f\_{i}(\mathbf{x}) + \mathbf{g}\_{i\iota}(\mathbf{x})\mathbf{w} + \mathbf{g}\_{2\iota}(\mathbf{x})\mathbf{u} \Big) + \left( h\_{\iota\iota}(\mathbf{x}) + k\_{\iota\iota\iota}(\mathbf{x})\mathbf{w} \right)^{\mathsf{T}} \Big( h\_{\iota\iota}(\mathbf{x}) + k\_{\iota\iota\iota}(\mathbf{x})\mathbf{w} \Big) - \gamma^{2} \mathbf{w}^{\mathsf{T}} \mathbf{w} < 0. \,\mathrm{d}\mathbf{x}$$

### **3. Main results**

For a single system, it has been shown in reference [7] that the existence of CSFs is a necessary and sufficient condition for the existence of *H*∞ controllers. Therefore, for the existence of simultaneous *H*∞ controllers for the systems in Eq. (3), the existence of CSFs for these systems is necessary. In references [7] and [9], no systematic methods have been proposed for constructing CSFs. Here, based on the backstepping method, we first derive CSFs explicitly for the systems in Eq. (3).

Let

Simultaneous *H* <sup>∞</sup> Control for a Collection of Nonlinear Systems in Strict-Feedback Form http://dx.doi.org/10.5772/64105 231

$$\begin{aligned} \mathbf{s}\_1(\mathbf{x}\_1) &= \mathbf{x}\_1, \\\\ \hat{V}\_1(\mathbf{x}\_1) &= \frac{1}{2} \mathbf{s}\_1^2(\mathbf{x}\_1). \end{aligned}$$

It is easy to show that we can find a function 1: and a positive definite function 1: such that

$$\pm \mathbf{x}\_1 \cdot \left( \boldsymbol{\upphi}\_1(\boldsymbol{\upchi}\_1) + \boldsymbol{\uptheta}\_{i1}(\boldsymbol{\upchi}\_1) \right) \le -\boldsymbol{\upmu}\_1(\boldsymbol{\upchi}\_1) \text{, for all } i = 1, \dots, q.$$

For *j*=2,…,*n*, let

*u px* = ( ) (4)

in Eq. (1). A smooth, proper, and positive definite

satisfies the *L2-*gain SCP if for each ε > 0, there is a δ1 >

g

g

g g

ò ò (5)

internally stabilizes the systems in Eq. (3) simultaneously; and, for each *T* > 0 and for each *w*<sup>i</sup> ∈ *L*2[0, *T*], all closed-loop systems, starting from the initial state *x*(0) = 0, satisfy (for a given γ

Here we review some important concepts about the CSF method introduced in references [7, 9].

if, for each \ <sup>0</sup> and each *w* ∈ *Rm*,

( ) ( ) ( ) <sup>2</sup> 1 2 1 11 1 11 ( ) inf () () () () () () () 0. <sup>Î</sup> <sup>é</sup> ì ü ¶ <sup>ù</sup> í ý + + ++ + - < <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> î þ ¶ <sup>û</sup> *i T T*

*V x f x g xw g xu h x k xw h x k xw ww <sup>x</sup>*

For ensuring the continuity of the obtained simultaneous *H*∞ controllers, the *L2-gain small control*

0 and a δ2 > 0 such that, if *x* ≠ 0 satisfies < 1 and *w* satisfies < 2, there is some *u* with

( ) ( ) ( ) <sup>2</sup> 1 2 1 11 1 11 ( ) () () () () () () () 0. ¶

For a single system, it has been shown in reference [7] that the existence of CSFs is a necessary and sufficient condition for the existence of *H*∞ controllers. Therefore, for the existence of simultaneous *H*∞ controllers for the systems in Eq. (3), the existence of CSFs for these systems is necessary. In references [7] and [9], no systematic methods have been proposed for constructing CSFs. Here, based on the backstepping method, we first derive CSFs explicitly for

*i T T ii i i i i i V x f x g xw g xu h x k xw h x k xw ww <sup>x</sup>*

+ + ++ + - <

<sup>0</sup> <sup>0</sup> () () ˆ ˆ ( ) ( ) for some . *T T T T z t z t dt w t w t dt* £ <

2

*property* (*L2-gain SCP*) has been introduced in reference [7].

: of *Si*

**2.2. Control storage functions**

*Definition 2* [7]: A CSF


¶

**3. Main results**

the systems in Eq. (3).

Let

*Definition 1* [7, 9]: Consider the system *Si*

230 Nonlinear Systems - Design, Analysis, Estimation and Control

: is a CSF of *Si*

g

*u R i i i i i i i*

> 0)

function

$$s\_j(\overline{\mathfrak{x}}\_j) = \mathfrak{x}\_j - \varrho \mathfrak{o}\_{j-1}(\overline{\mathfrak{x}}\_{j-1})\_\*$$

$$
\hat{V}\_{\boldsymbol{\beta}}(\overline{\boldsymbol{x}}\_{\boldsymbol{\beta}}) = \hat{V}\_{\boldsymbol{\beta}^{-1}}(\overline{\boldsymbol{x}}\_{\boldsymbol{\beta}^{-1}}) + \frac{1}{2}\mathbf{s}\_{\boldsymbol{\beta}}^{2}(\overline{\boldsymbol{x}}\_{\boldsymbol{\beta}}) .
$$

Similarly, we can find functions : , = 2, …, <sup>1</sup>, and positive definite function : , = 2, …, <sup>1</sup>, such that

$$\sum\_{l=1}^{l-1} \frac{\partial \hat{V}\_j(\overline{\mathbf{x}}\_j)}{\partial \mathbf{x}\_l} (\mathbf{x}\_{l+1} + \theta\_{il}(\overline{\mathbf{x}}\_l)) + \frac{\partial \hat{V}\_j(\overline{\mathbf{x}}\_j)}{\partial \mathbf{x}\_j} (\boldsymbol{\varphi}\_j(\overline{\mathbf{x}}\_j) + \theta\_{jl}(\overline{\mathbf{x}}\_j)) \leq -\sum\_{l=1}^{l} \boldsymbol{\mu}\_l(\overline{\mathbf{x}}\_l), \quad \text{for all } i = 1, \dots, p.$$

Then, it is clear that the function

$$\hat{V}(\mathbf{x}) \equiv \frac{1}{2} \sum\_{j=1}^{n} s\_j^2(\overline{\mathbf{x}}\_j)$$

is positive definite, and radially unbounded.

Now, we discuss the existence of common CSFs for the systems in Eq. (3). For convenience, we say that a continuous function ( ) is dominated by a continuous function ( ) if there exists a constant *c* > 0 such that ( ) < ( ) for all ≠ 0.

*Theorem 1***:** Consider the systems in Eq. (3). Suppose that *Assumption 1* holds. If the functions : , *j*=1,…,*n*-1, are such that () ()=0 is dominated by ∑ = 1 1 ( ), then there exists a common *CSF* that satisfies the *L*2-*gain SCP* for all the systems in Eq. (3).

*Proof:* Let () = (), where *K* > 0 will be specified later. For system *Si* , define the corresponding Hamiltonian function as

$$H\_{\boldsymbol{\cdot}}(\mathbf{x}, \boldsymbol{w}, \boldsymbol{\mu}) \equiv \dot{\boldsymbol{V}}(\mathbf{x}) + \left(h\_{1\boldsymbol{\imath}}(\mathbf{x}) + k\_{1\boldsymbol{\imath}\boldsymbol{\imath}}(\mathbf{x})\boldsymbol{w}\right)^{\boldsymbol{\imath}} \left(h\_{1\boldsymbol{\imath}}(\mathbf{x}) + k\_{1\boldsymbol{\imath}\boldsymbol{\imath}}(\mathbf{x})\boldsymbol{w}\right) - \boldsymbol{\gamma}^{\boldsymbol{\imath}}\boldsymbol{w}^{\boldsymbol{\imath}}\boldsymbol{w}.$$

By the backstepping method, we can show that

( ) ( ) ( ) 2 1 11 1 11 1 1 1 1 1 1 1 1 1 1 11 11 (, ,) ( ) ( ) () () () () ( ) ( ) () ( ) () () () ( ) () () () *i <sup>n</sup> <sup>T</sup> <sup>T</sup> jj jj i i i i j n n n n j j n n n in i i l il l i l l T T ii i H xwu K s x s x h x k xw h x k xw ww <sup>x</sup> K x Ks x s x x x w x u x x x h xh x h xk* g f m qr h q = - - - - - - + = = = ×+ + + æ ö ¶ £- + ç ÷ ++ + - × + ¶ è ø + + å å å & ( ) <sup>2</sup> <sup>1</sup> 11 1 11 11 () () () () () . *T T T T <sup>i</sup> i i i i xw wk xh x w I k xk x w* + - g

After some manipulations, we have

$$\begin{split} H\_i(\mathbf{x}, \mathbf{w}, \boldsymbol{\mu}) &\leq a\_i(\mathbf{x}) + b\_i(\mathbf{x}) \cdot \boldsymbol{u} - (\boldsymbol{w} - \boldsymbol{w}\_{\boldsymbol{\nu}^\*}(\mathbf{x}))^T \left(\boldsymbol{\gamma}^2 I - k\_{\text{1i}}^T(\mathbf{x}) k\_{\text{1i}}(\mathbf{x})\right) (\boldsymbol{w} - \boldsymbol{w}\_{\boldsymbol{\nu}^\*}(\mathbf{x})) \\ &\leq a\_i(\mathbf{x}) + b\_i(\mathbf{x}) \cdot \boldsymbol{u}, \end{split} \tag{6}$$

where

( ) ( ) 1 1 1 1 1 1 1 1 1 1 1 <sup>1</sup> <sup>2</sup> 11 1 11 11 11 1 ( ) () ( ) () ( ) () ( ) () () () () () () () () () () () () 2 2 ( ) *n n n n T i j j n n n in l il l i i j l l T T T T T T ni ii i i ni ii i i <sup>x</sup> a x K x Ks x s x x x x h xh x <sup>x</sup> K K s x x k xh x I k xk x s x x k xh x bx K* j mq q rg r h - - - - - - + = = æ ö ¶ =- + ç ÷ + - ×+ + ¶ è ø æ öæ ö ++ - <sup>+</sup> ç ÷ç ÷ è øè ø = å å ( ) <sup>1</sup> <sup>2</sup> \* 11 11 11 1 () () () () () () () () () <sup>2</sup> *n T T T i i i ni ii xs x <sup>K</sup> w x I k xk x s x x k xh x* g r- æ ö <sup>=</sup> - + ç ÷ è ø

Therefore, () = () is a CSF of *Si* if

$$\forall \boldsymbol{\chi} \neq 0 \quad \text{such that} \quad b\_i(\boldsymbol{\chi}) = 0 \implies a\_i(\boldsymbol{\chi}) < 0.$$

As () ()=0 is dominated by <sup>∑</sup> = 1 1 ( ), we can choose a *K* > 0 such that

$$\left. U(\mathbf{x}) \right|\_{s\_\*(\mathbf{x}) = 0 \text{ and } \mathbf{x} \neq 0} - K \sum\_{j=1}^{n-1} \mu\_j(\overline{\mathbf{x}}\_j) < 0.$$

Notice that *bi* (*x*) = 0 if and only if *sn*(*x*) = 0. Therefore,

*Theorem 1***:** Consider the systems in Eq. (3). Suppose that *Assumption 1* holds. If the functions

exists a common *CSF* that satisfies the *L*2-*gain SCP* for all the systems in Eq. (3).

*Proof:* Let () = (), where *K* > 0 will be specified later. For system *Si*

()=0 is dominated by ∑ = 1

( ) ( ) <sup>2</sup>

1 11 1 11 (, ,) () () () () () . º+ + + -

& *<sup>T</sup> <sup>T</sup> H xwu V x h x k xw h x k xw ww <sup>i</sup> ii ii*

( ) ( )

( ) ( ) () ( ) () () () ( )

qr

\* 11 11 \* ( , , ) ( ) ( ) ( ( )) ( ) ( ) ( ( ))

*i ii i ii i*

*H xwu a x b x u w w x I k xk x w w x*

*<sup>x</sup> K x Ks x s x x x w x u x x*

æ ö ¶ £- + ç ÷ ++ + - × +

<sup>1</sup> 11 1 11 11 () () () () () . *T T T T <sup>i</sup> i i i i xw wk xh x w I k xk x w* + - g

g

( )

*K K s x x k xh x I k xk x s x x k xh x*

æ öæ ö ++ - <sup>+</sup> ç ÷ç ÷ è øè ø

*T T T T T ni ii i i ni ii*

() () () () () () () () () () 2 2

( ) () ( ) () ( ) () ( ) () ()

*i j j n n n in l il l i i*

=- + ç ÷ + - ×+ +

è ø

*<sup>x</sup> a x K x Ks x s x x x x h xh x <sup>x</sup>*

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


11 1 11 11 11 1

j

æ ö ¶

¶ è ø

*T T*

1 11 1 11

*i l l*

1 1


*n n*

*<sup>n</sup> <sup>T</sup> <sup>T</sup>*

*K s x s x h x k xw h x k xw ww*

1 1

= =

£ + ×- - -

1 1


*n n*

mq

rg

\* 11 11 11 1

*<sup>K</sup> w x I k xk x s x x k xh x*

() () () () () () () <sup>2</sup>

*T T T i i i ni ii*


1 1

= =

å å

*j l l T*

> r

if

å å

 1 (

g

1 1


*n n*

f

( )

 q

*n n T*

 r *x*

2

g

1 1 1

( ) <sup>2</sup>

*ax bx u* (6)


1 1 1 1 1 1 1



¶ è ø

( ) <sup>2</sup>

 h


*j j n n n in i i l il l*

), then there

, define the corre-

( )

q

: , *j*=1,…,*n*-1, are such that ()

232 Nonlinear Systems - Design, Analysis, Estimation and Control

By the backstepping method, we can show that

( ) ( ) () () () ()

*jj jj i i i i*

= ×+ + + -

sponding Hamiltonian function as

1

=

å

*j*

(, ,)

*H xwu*

*i*

where

( )

*bx K*

*i i*

=

h

g

11 11

After some manipulations, we have

() () ,

( ) <sup>1</sup> <sup>2</sup>

Therefore, () = () is a CSF of *Si*

() ()

*xs x*

*n*

£ +×

*i i*

&

*h xh x h xk*

() () ()

+ +

m

*T T ii i*

$$\left. \left( a\_l(\mathbf{x}) \right) \right|\_{b\_l(\mathbf{x}) = 0 \text{ and } \mathbf{x} \neq 0} \leq -K \sum\_{j=1}^{n-1} \mu\_j(\overline{\mathbf{x}}\_j) + U(\mathbf{x})|\_{s\_n(\mathbf{x}) = 0 \text{ and } \mathbf{x} \neq 0} < 0. \tag{7}$$

This shows that *V*(*x*) is a CSF for the *i*-th system in Eq. (3). Since Eq. (7) holds for all *i* ∈ {1,…,*q*}, *V*(*x*) is a common CSF for all the systems in Eq. (3).

Now we prove that *V*(*x*) satisfies the *L2-*gain SCP. Note that if we can find a continuous stabilizing feedback law *di* (*x*) with *di* (0) = 0 such that (, , ()) < 0 for each \ <sup>0</sup> and each *w* ∈ *Rm*, then *V*(*x*) satisfies the *L2-*gain SCP. Let

$$\begin{split} d\_{i}(\mathbf{x}) &= -\frac{1}{\eta\_{i}(\mathbf{x})} \Big( \mathbf{s}\_{n-1}(\overline{\mathbf{x}}\_{n-1}) + \theta\_{in}(\mathbf{x}) - \sum\_{l=1}^{n-1} \frac{\partial \rho\_{n-1}(\overline{\mathbf{x}}\_{n-1})}{\partial \mathbf{x}\_{l}} \cdot \left( \mathbf{x}\_{l+1} + \theta\_{il}(\overline{\mathbf{x}}\_{l}) \right) \\ &+ \rho\_{i}(\mathbf{x}) \Big( \boldsymbol{\nu}^{\mathrm{T}} \boldsymbol{I} - k\_{\mathrm{11}i}^{\mathrm{T}}(\mathbf{x}) k\_{\mathrm{11}i}(\mathbf{x}) \Big)^{-1} \Big( \frac{K}{4} \boldsymbol{s}\_{\mathrm{n}}(\mathbf{x}) \rho\_{i}^{\mathrm{T}}(\mathbf{x}) + k\_{\mathrm{11}i}^{\mathrm{T}}(\mathbf{x}) h\_{\mathrm{1}i}(\mathbf{x}) \Big) \Big) - \hat{\mu}\_{\mathrm{n}}(\mathbf{x}) \end{split}$$

where the continuous function () with (0) = 0 is such that ()()>0 if *sn*(*x*) ≠ 0, and

$$-K\sum\_{j=1}^{n-1} \mu\_j(\overline{\mathbf{x}}\_j) - K\eta\_i(\mathbf{x})\mathbf{s}\_n(\mathbf{x})\hat{\mu}\_n(\mathbf{x}) + U(\mathbf{x}) < 0 \quad \forall \mathbf{x} \neq \mathbf{0}.$$

Note that such () always exists since () ()=0 is dominated by <sup>∑</sup> = 1 1 ( ). Clearly, *di* (*x*) is continuous in *Rn* and *di* (0) = 0. By Eq. (6), we have

$$\begin{split} H\_i(\mathbf{x}, w, d\_i(\mathbf{x})) &\leq a\_i(\mathbf{x}) + b\_i(\mathbf{x}) d\_i(\mathbf{x}) \\ &= -K \sum\_{j=1}^{n-1} \mu\_j(\overline{\mathbf{x}}\_j) - K \eta\_i(\mathbf{x}) \mathbf{s}\_n(\mathbf{x}) \hat{\boldsymbol{\mu}}\_n(\mathbf{x}) \\ &\quad + h\_{1i}^T(\mathbf{x}) h\_{1i}(\mathbf{x}) + h\_{1i}^T(\mathbf{x}) k\_{11i}(\mathbf{x}) \big( \boldsymbol{\gamma}\_i^2 I - k\_{11i}^T(\mathbf{x}) k\_{11i}(\mathbf{x}) \big)^{-1} k\_{11i}^T(\mathbf{x}) h\_{1i}(\mathbf{x}) \\ &\leq -K \sum\_{j=1}^{n-1} \mu\_j(\overline{\mathbf{x}}\_j) - K \eta\_i(\mathbf{x}) \hat{\boldsymbol{\mu}}\_n(\mathbf{x}) \hat{\boldsymbol{\mu}}\_n(\mathbf{x}) + L(\mathbf{x}) < 0, \ \forall \mathbf{x} \neq \mathbf{0}, \ \forall w. \end{split}$$

This implies that *V*(*x*) satisfies the *L*2-gain SCP and completes the proof.

To derive simultaneous *H*∞ controllers, define (for *i*=1,…,*q*)

$$p\_i(\mathbf{x}) \equiv \begin{cases} -\frac{a\_i(\mathbf{x}) + \sqrt{a\_i^2(\mathbf{x}) + \beta\_i b\_i^4(\mathbf{x})}}{b\_i(\mathbf{x})}, & \text{if } s\_n(\mathbf{x}) \neq 0 \\\ 0, & \text{if } s\_n(\mathbf{x}) = 0 \end{cases}$$

where β*<sup>i</sup>* > 0, *i*=1,…,*q*, are given constants. Since *V*(*x*) satisfies the *L*2-gain SCP, the functions *pi* (*x*), *i*=1,…,*q*, are continuous in *Rn* [16]. We have the following results.

*Theorem 2:* Consider the collection of systems in Eq. (3). Suppose that *Assumption 1* holds. If the functions : , = 1, …, <sup>1</sup>, are such that () ()=0 is dominated by ∑ = 1 1 ( ), then a continuous function : exists such that the feedback law defined in Eq. (4) internally stabilizes the collection of systems in Eq. (3) simultaneously; and moreover, all the closed-loop systems satisfy the *L*2-gain requirement specified in Eq. (5). In this case,

$$u = p(\mathbf{x}) \equiv \begin{cases} \min\_{\{\boldsymbol{\alpha} \in \{1, 2, \dots, q\} \mid \boldsymbol{\alpha}\}} \text{, if } \mathbf{s}\_{\boldsymbol{\alpha}}(\mathbf{x}) > 0 \\\ \mathbf{0}, & \text{if } \mathbf{s}\_{\boldsymbol{\alpha}}(\mathbf{x}) = \mathbf{0} \\\ \max\_{\{\boldsymbol{\alpha} \in \{1, 2, \dots, q\} \mid \boldsymbol{\alpha}\}} \{p\_i(\mathbf{x})\}, & \text{if } \mathbf{s}\_{\boldsymbol{\alpha}}(\mathbf{x}) < \mathbf{0} \end{cases} \tag{8}$$

is a simultaneous *H*∞ controller for all the systems in Eq. (3).

*Proof*: Since the functions *pi* (*x*), *i*=1,2,…,*q*, are continuous in *Rn*, from the definition of *p*(*x*), its continuity is obvious. In the following, we first prove the achievement of *L*2-gain requirement [Eq. (5)], and then the internal stability of all the closed-loop systems.

A. *L*2-gain requirement

Since (, , ) ≤ () + (), if we can show that

$$a\_i(\mathbf{x}) + b\_i(\mathbf{x})p(\mathbf{x}) < 0 \quad \forall \mathbf{x} \neq 0, i = 1, \ldots, q,\tag{9}$$

Then, with the controller defined in Eq. (8), all the closed-loop systems satisfy the *L2*-gain requirement specified in Eq. (5).

**1.** *sn*(*x*) = 0 and *x* ≠ 0.

In this case, *u* = *p*(*x*) = 0 and *bi* (*x*) = 0. Then, by Eq. (7),

$$a\_i(\mathbf{x}) + b\_i(\mathbf{x}) \cdot p(\mathbf{x}) = a\_i(\mathbf{x}) < 0, \quad i = 1, \dots, q.$$

**2.** () > 0.

This implies that *V*(*x*) satisfies the *L*2-gain SCP and completes the proof.

2 4 () () () , if () 0 ( ) ( )

*Theorem 2:* Consider the collection of systems in Eq. (3). Suppose that *Assumption 1* holds. If

Eq. (4) internally stabilizes the collection of systems in Eq. (3) simultaneously; and moreover, all the closed-loop systems satisfy the *L*2-gain requirement specified in Eq. (5). In this case,

( ) 0, if () 0

*i n i q*

*i n i q*

<sup>ï</sup> <sup>=</sup> º = <sup>í</sup>

min { ( )}, if () 0

*px s x*

ì >

<sup>ï</sup> <sup>&</sup>lt; ïî

continuity is obvious. In the following, we first prove the achievement of *L*2-gain requirement

max { ( )}, if () 0

*px s x*

*n*

(*x*), *i*=1,2,…,*q*, are continuous in *Rn*, from the definition of *p*(*x*), its

: , = 1, …, <sup>1</sup>, are such that ()

{1,2,..., }

*u px s x*

Î

ï

Î

[Eq. (5)], and then the internal stability of all the closed-loop systems.

is a simultaneous *H*∞ controller for all the systems in Eq. (3).

{1,2,..., }

(), if we can show that

*i i ii*

*ax a x b x s x p x b x*

ì + +

0, if () 0

> 0, *i*=1,…,*q*, are given constants. Since *V*(*x*) satisfies the *L*2-gain SCP, the functions

), then a continuous function : exists such that the feedback law defined in

î =

bï- ¹ <sup>º</sup> <sup>í</sup>

*n*

*n*

*s x*

()=0 is dominated by

(8)

To derive simultaneous *H*∞ controllers, define (for *i*=1,…,*q*)

234 Nonlinear Systems - Design, Analysis, Estimation and Control

ï

where β*<sup>i</sup>*

the functions

*Proof*: Since the functions *pi*

(, , ) ≤

() +

A. *L*2-gain requirement

Since

*pi*

∑ = 1

 1 ( *i i*

(*x*), *i*=1,…,*q*, are continuous in *Rn* [16]. We have the following results.

In this case, since *bi* (*x*) > 0, we have

$$\begin{aligned} a\_i(\mathbf{x}) + b\_i(\mathbf{x})p(\mathbf{x}) &= a\_i(\mathbf{x}) + b\_i(\mathbf{x}) \cdot \min\_{j \in \{1, 2, \dots, q\}} \{ p\_j(\mathbf{x}) \} \\ &\le a\_i(\mathbf{x}) + b\_i(\mathbf{x}) \cdot p\_i(\mathbf{x}) \\ &= -\sqrt{a\_i^2(\mathbf{x}) + \beta\_i b\_i^4(\mathbf{x})} < 0, \quad i = 1, \dots, q. \end{aligned}$$

**3.** *sn*(*x*) < 0

Similarly, in this case we can show that

$$a\_i(\mathbf{x}) + b\_i(\mathbf{x})p(\mathbf{x}) < 0, \quad i = 1, \dots, q.$$

These discussions imply that Eq. (9) holds. That is, all the possible closed-loop systems satisfy the *L*2-gain requirement specified in Eq. (5).

B. Internal stability

To prove internal stability, notice that Eq. (6) implies that, along the trajectories of system *Si* under *w* = 0,

$$\begin{aligned} H\_i(\mathbf{x}, 0, p(\mathbf{x})) &= \frac{\partial V(\mathbf{x})}{\partial \mathbf{x}} \Big( f\_i(\mathbf{x}) + \mathbf{g}\_{2i}(\mathbf{x}) p(\mathbf{x}) \Big) + h\_{\text{il}}(\mathbf{x})^T h\_{\text{il}}(\mathbf{x}) \\ &\le a\_i(\mathbf{x}) + b\_i(\mathbf{x}) p(\mathbf{x}) < 0 \quad \forall \mathbf{x} \neq \mathbf{0}. \end{aligned}$$

That is, for each *i* ∈ {1,…,*q*}, along the trajectories of system *Si* , we have

$$\dot{V}(\mathbf{x}) = \frac{\partial V(\mathbf{x})}{\partial \mathbf{x}} (f\_i(\mathbf{x}) + \mathbf{g}\_{2i}(\mathbf{x}) p(\mathbf{x})) < 0 \quad \forall \mathbf{x} \neq \mathbf{0}.$$

This shows that all the closed-loop systems are internally stable.

*Remark 1***:** The systems considered in reference [8] are special cases of the systems considered in this chapter. If we let ( ) = 0, = 1, 2, …, , and *j*=1,2,…,*n*-1, the systems in Eq. (3) will reduce to the systems considered in reference [8]. On the other hand, in reference [17], it is assumed that *U*(*s*) is in quadratic form. In this chapter, we relax this restrictive assumption.

*Remark 2***:** In this chapter, we consider the case that the controlled output *z* is independent of the control input *u*. In this situation, a much simpler formula (not a special case of the formula in reference [7]) is proposed for constructing simultaneous *H*∞ controllers. In the case that the controlled output *z* depends on *u*, necessary and sufficient conditions for the existence of simultaneous *H*∞ controllers and a formula for constructing simultaneous *H*∞ controllers can be derived by the results in reference [7].

## **4. An illustrative example**

Consider the following nonlinear systems:

$$S\_i: \begin{cases} \dot{\mathbf{x}}\_1 = \mathbf{x}\_2 + \theta\_{i1}(\mathbf{x})\\ \dot{\mathbf{x}}\_2 = \theta\_{i2}(\mathbf{x}) + \rho\_i(\mathbf{x})\mathbf{w} + \eta\_i(\mathbf{x})u\\ z = h\_{i1}(\mathbf{x}) + k\_{11}(\mathbf{x})\mathbf{w}, & i = 1, 2, \text{and } 3 \end{cases} \tag{10}$$

where

$$\begin{aligned} \theta\_{11}(\mathbf{x}) &= \mathbf{x}\_{1}, \theta\_{21}(\mathbf{x}) = \mathbf{x}\_{1} \sin(\mathbf{x}\_{1}), \ \theta\_{31}(\mathbf{x}) = -\mathbf{x}\_{1} \cos(\mathbf{x}\_{1}), \\ \theta\_{12}(\mathbf{x}) &= \mathbf{x}\_{1}^{2} + \mathbf{x}\_{2}^{3}, \theta\_{22}(\mathbf{x}) = \mathbf{x}\_{1}(1 - 2\mathbf{x}\_{2}), \ \theta\_{32}(\mathbf{x}) = \mathbf{x}\_{1} \cos(\mathbf{x}\_{2}) + 2\mathbf{x}\_{2} \sin(5\mathbf{x}\_{1}), \\ \rho\_{1}(\mathbf{x}) &= -1 + \mathbf{x}\_{1}, \ \rho\_{2}(\mathbf{x}) = \mathbf{x}\_{1}\mathbf{x}\_{2}, \ \rho\_{3}(\mathbf{x}) = \mathbf{x}\_{1} - \mathbf{x}\_{2}^{2}, \\ \eta\_{1}(\mathbf{x}) &= 1 + (\mathbf{x}\_{1} + \mathbf{x}\_{2})^{2}, \ \eta\_{2}(\mathbf{x}) = 2 - \cos(\mathbf{x}\_{1}), \ \eta\_{3}(\mathbf{x}) = 2 + \mathbf{x}\_{2}^{2}, \\ h\_{11}(\mathbf{x}) &= \mathbf{x}\_{1} \cos(\mathbf{x}\_{2}^{2}), \ h\_{12}(\mathbf{x}) = -\mathbf{x}\_{1} \sin(\mathbf{x}\_{1}), \ h\_{13}(\mathbf{x}) = \mathbf{x}\_{2}, \\ k\_{111}(\mathbf{x}) &= -1 + \cos(\mathbf{x}\_{1}), \ k\_{112}(\mathbf{x}) = 1, \ k\_{113}(\mathbf{x}) = 1 + \sin(5\mathbf{x}\_{2}). \end{aligned}$$

It can be shown that

Simultaneous *H* <sup>∞</sup> Control for a Collection of Nonlinear Systems in Strict-Feedback Form http://dx.doi.org/10.5772/64105 237

$$h\_{\mathrm{li}}^{\mathrm{T}}(\mathbf{x})h\_{\mathrm{li}}(\mathbf{x}) + h\_{\mathrm{li}}^{\mathrm{T}}(\mathbf{x})k\_{\mathrm{1i}}(\mathbf{x})\Big(\boldsymbol{\gamma}^{2} - k\_{\mathrm{1i}}^{\mathrm{T}}(\mathbf{x})k\_{\mathrm{1i}}(\mathbf{x})\Big)^{-1}k\_{\mathrm{1i}}^{\mathrm{T}}(\mathbf{x})h\_{\mathrm{li}}(\mathbf{x}) \leq U(\mathbf{x}), \quad i = \mathrm{l, 2, and } \mathfrak{R}$$

with

( <sup>2</sup> ) ( ) ( ) ( ) ( ) ( ) 0 0. ¶ = + < "¹

*Remark 1***:** The systems considered in reference [8] are special cases of the systems consid-

will reduce to the systems considered in reference [8]. On the other hand, in reference [17], it is assumed that *U*(*s*) is in quadratic form. In this chapter, we relax this restrictive assump-

*Remark 2***:** In this chapter, we consider the case that the controlled output *z* is independent of the control input *u*. In this situation, a much simpler formula (not a special case of the formula in reference [7]) is proposed for constructing simultaneous *H*∞ controllers. In the case that the controlled output *z* depends on *u*, necessary and sufficient conditions for the existence of simultaneous *H*∞ controllers and a formula for constructing simultaneous *H*∞ controllers can

() () , 1,2,and 3

 h

12 1 2 22 1 2 32 1 2 2 1

=+ = - = +

hh

2 12 1 1 13 2

), ( ) sin( ), ( ) ,

( ) , ( ) (1 2 ), ( ) cos( ) 2 sin(5 ),

 q

*x xx xx x xx x x x*

2 2

= - =

2

) = 0, = 1, 2, …, , and *j*=1,2,…,*n*-1, the systems in Eq. (3)

*V x V x f x g xpx x <sup>x</sup>*

¶ & *i i*

This shows that all the closed-loop systems are internally stable.

ered in this chapter. If we let (

236 Nonlinear Systems - Design, Analysis, Estimation and Control

be derived by the results in reference [7].

Consider the following nonlinear systems:

12 1 2 2

*xx x*

ì = + ï

ï

& &

2 3

11 1

qq

*hx x*

r

h

It can be shown that

( ) cos(

= <sup>2</sup>

qq

1 11

q

qr

í =+ +

*S x x xw xu*

11 1 21 1 1 31 1 1

= = = -

( ) , ( ) sin( ), ( ) cos( ),

*x x xx x x x x*

1 1 2 12 3 1 2

() 1 , () , () ,

rr

=- + = = -

*x x x xx x x x*

1 12 2 13 2

111 1 112 113 2

=- + = = +

*kx xkx kx x*

( ) 1 ( ) , ( ) 2 cos( ), ( ) 2 ,

( ) 1 cos( ), ( ) 1, ( ) 1 sin(5 ). *x hx x x hx x*

=+ + = - = +

 q

*x xx x x x x*

*i i ii i i i*

( ) : () () ()

î = + =

*z h x k xw i*

**4. An illustrative example**

tion.

where

$$U(\mathbf{x}) = \frac{9}{\mathcal{S}} \mathbf{x}\_1^2 + \frac{9}{\mathcal{S}} \mathbf{x}\_2^2.$$

Let γ = 3. It can be verified that *Assumption 1* holds. Let

$$\begin{aligned} s\_1(\overline{\mathbf{x}}\_1) &= \mathbf{x}\_1 \\ \varphi\_1(\mathbf{x}\_1) &= -\mathbf{2}\mathbf{x}\_1 \\ \mu\_1(\mathbf{x}\_1) &= \mathbf{x}\_1^2 \\ s\_2(\overline{\mathbf{x}}\_2) &= \mathbf{x}\_2 - \varphi\_1(\overline{\mathbf{x}}\_1) = \mathbf{2}\mathbf{x}\_1 + \mathbf{x}\_2. \end{aligned}$$

Then,

$$\hat{V}(\mathbf{x}) = \frac{1}{2} \left( \mathbf{s}\_1^2(\mathbf{x}\_1) + \mathbf{s}\_2^2(\overline{\mathbf{x}}\_2) \right)$$

is positive, definite, and radially unbounded. By choosing *K* = 10, it can be shown that

$$-K\mu\_1(\mathfrak{x}\_1) + U(\mathfrak{x})\Big|\_{s\_2(\mathfrak{T}\_2) = 0} < \mathbf{0}, \quad \forall \mathfrak{x}\_1 \neq \mathbf{0}.$$

Therefore,

(10)

$$V(\boldsymbol{\chi}) = K\hat{V}(\boldsymbol{\chi}) = \mathfrak{S}\left(\mathbf{s}\_1^2(\boldsymbol{\chi}\_1) + \mathbf{s}\_2^2(\overline{\boldsymbol{\chi}}\_2)\right).$$

is a common CSF for the three systems in Eq. (10). For *i* = 1, 2, and 3, define

$$\begin{split} a\_{l}(\mathbf{x}) &= -K\mu\_{1}(\mathbf{x}) + K\varepsilon\_{2}(\mathbf{x}) \bigg( s\_{1}(\mathbf{x}\_{l}) + \theta\_{l2}(\mathbf{x}) - \frac{\widehat{\alpha}\rho\_{l}(\mathbf{x}\_{l})}{\widehat{\alpha}\mathbf{x}\_{l}}(\mathbf{x}\_{2} + \theta\_{l1}(\mathbf{x})) \bigg) + h\_{l1}^{T}(\mathbf{x})h\_{ll}(\mathbf{x}) \\ &+ \Big( \frac{K}{2}s\_{n}(\mathbf{x})\rho\_{l}^{T}(\mathbf{x}) + k\_{11l}^{T}(\mathbf{x})h\_{ll}(\mathbf{x}) \bigg)^{T} \bigg( \gamma^{2}I - k\_{11l}^{T}(\mathbf{x})k\_{11l}(\mathbf{x}) \bigg)^{-1} \bigg( \frac{K}{2}s\_{n}(\mathbf{x})\rho\_{l}^{T}(\mathbf{x}) + k\_{11l}^{T}(\mathbf{x})h\_{ll}(\mathbf{x}) \bigg)^{-1} \\ b\_{l}(\mathbf{x}) &= K\eta\_{l}(\mathbf{x})s\_{2}(\mathbf{x}) \end{split}$$

and (with β1 = β2 = β3 = 0.1)

$$p\_i(\mathbf{x}) \equiv \begin{cases} -\frac{a\_i(\mathbf{x}) + \sqrt{a\_i^2(\mathbf{x}) + \beta\_i b\_i^4(\mathbf{x})}}{b\_i(\mathbf{x})}, & \text{if } s\_2(\mathbf{x}) \neq 0 \\\ 0, & \text{if } s\_2(\mathbf{x}) = 0. \end{cases}$$

From *Theorem 2*, the following controller

$$u = p(\mathbf{x}) \equiv \begin{cases} \min\{p\_1(\mathbf{x}), p\_2(\mathbf{x}), p\_3(\mathbf{x})\}, & \text{if } 2\mathbf{x}\_1 + \mathbf{x}\_2 > 0 \\ 0, & \text{if } 2\mathbf{x}\_1 + \mathbf{x}\_2 = 0 \\ \max\{p\_1(\mathbf{x}), p\_2(\mathbf{x}), p\_3(\mathbf{x})\}, & \text{if } 2\mathbf{x}\_1 + \mathbf{x}\_2 < 0 \end{cases} \tag{11}$$

is a simultaneous *H*∞ controller for the three systems in Eq. (10). With arbitrarily chosen disturbance inputs, **Figures 1**–**3** show the states, control inputs, disturbance inputs, and controlled outputs of these three systems starting at different initial states with the same controller defined in Eq. (11). It can be seen that all the three closed-loop systems are internally stable and satisfy the required *L*2-gain specification. That is, the controller defined in Eq. (11) is indeed a simultaneous *H*∞ controller for the three systems in Eq. (10).

**Figure 1.** Responses of the system *S*1 controlled by the controller defined in Eq. (11).

**Figure 2.** Responses of the system *S*2 controlled by the controller defined in Eq. (11).

2 4

0, if ( ) 0.

î =

123 1 2

min{ ( ), ( ), ( )}, if 2 0

ì + >

*px px px x x*

( ) 0, if 2 0

*u px x x*

is indeed a simultaneous *H*∞ controller for the three systems in Eq. (10).

**Figure 1.** Responses of the system *S*1 controlled by the controller defined in Eq. (11).

<sup>ï</sup> = º <sup>í</sup> + =

123 1 2

max{ ( ), ( ), ( )}, if 2 0

<sup>ï</sup> + < <sup>î</sup>

is a simultaneous *H*∞ controller for the three systems in Eq. (10). With arbitrarily chosen disturbance inputs, **Figures 1**–**3** show the states, control inputs, disturbance inputs, and controlled outputs of these three systems starting at different initial states with the same controller defined in Eq. (11). It can be seen that all the three closed-loop systems are internally stable and satisfy the required *L*2-gain specification. That is, the controller defined in Eq. (11)

*px px px x x*

b<sup>ï</sup> - ¹ <sup>º</sup> <sup>í</sup>

() () () , if () 0 ( ) ( )

*i i ii*

ì + +

*ax a x b x s x p x b x*

*i i*

ï

From *Theorem 2*, the following controller

238 Nonlinear Systems - Design, Analysis, Estimation and Control

2

2

1 2

(11)

*s x*

**Figure 3.** Responses of the system *S*3 controlled by the controller defined in Eq. (11).
