1. Introduction

It is well known that control systems can be found in abundance in all sectors of industry such as robotics, power systems, transportation systems space technologies, and many others, and thus control theory has been well studied. In order to design control systems, designers have to derive mathematical models for dynamical systems, and there are mainly two types of representations for mathematical models, that is, transfer functions and state equations. In other words, control theory is divided into "classical control" and "modern control" (e.g., see [12]).

Classical control means an analytical theory based on transfer function representations and frequency responses, and for classical control theory, we can find a large number of useful and typical results such as Routh-Hurwitz stability criterion [20] based on characteristic equations in the nineteenth century, Nyquist criterion [28] in the 1930s, and so on. Moreover, by using

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classical control ideas, some design methods of controllers such as proportional, derivative, and integral (PID) controllers and phase lead-lag compensators have also been presented [21]. In classical control, controlled systems are mainly linear and time-invariant and have a single input and a single output only. Furthermore, it is well known that design approaches based on classical control theory need experiences and trial and error. On the other hand, in the 1960s, state variables and state equations (i.e., state-space representations) have been introduced by Kalman as system representations, and he has proposed an optimal regulator theory [14–16] and an optimal filtering one [17]. Namely, controlled systems are represented by state equations, and controller design problems are reduced to optimization problems based on the concept of state variables. Such controller design approach based on the statespace representation has been established as "modern control theory." Modern control is a theory of time domain, and whereas the transfer function and the frequency response are of limited applicability to nonlinear systems, state equations and state variables are equally appropriate to linear multi-input and multi-output systems or nonlinear one. Therefore, many existing results based on the state-space representation for controller design problems have been suggested (e.g., [7, 43]).

compared with the conventional robust controllers with fixed gains only, and one can easily see that these robust controllers with adjustable parameters differ from gain-scheduling control techniques [22, 41, 42]. Additionally, these robust controllers with time-varying adjustable parameters may also be referred to as "variable gain robust controller" or "adaptive gain

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187

In recent years, a great number of control systems are brought about by present technologies and environmental and societal processes which are highly complex and large in dimension, and such systems are referred to as "large-scale complex systems" or "largescale interconnected systems." Namely, large-scale and complex systems are progressing due to the rapid development of industry, and large-scale interconnected systems can be seen in diverse fields such as economic systems, electrical systems, and so on. For such large-scale interconnected systems, it is difficult to apply centralized control strategies because of calculation amount, physical communication constraints, and so on. Namely, a notable characteristic of the most large-scale interconnected systems is that centrality fails to hold due to either the lack of centralized computing capability of or centralized information. Moreover, large-scale interconnected systems are controlled by more than one controller or decision-maker involving decentralized computation. In the decentralized control strategy, large-scale interconnected systems are divided into several subsystems, and various types of decentralized control problems have been widely studied [13, 38, 44]. The major problem of large-scale interconnected systems is how to deal with the interactions among subsystems. A large number of results in decentralized control systems can be seen in the work of Šijjak [38]. Moreover, a framework for decentralized fault-tolerant control has also been studied [44]. Additionally, decentralized robust control strategies for uncertain large-scale interconnected systems have also attracted the attention of many researchers (e.g., [3–5, 11]). Moreover, in the work of Mao and Lin [24], for large-scale interconnected systems with unmodeled interaction, the aggregative derivation is tracked by using a model following the technique with online improvement, and a sufficient condition for which the overall system when controlled by the completely decentralized control is asymptotically stable has been established. Furthermore, decentralized guaranteed cost controllers for uncertain large-scale interconnected systems have also been

In this chapter, for a class of uncertain linear systems, we show LMI-based design strategies for adaptive gain robust controllers for a class of uncertain dynamical systems. The adaptive gain robust controllers consist of fixed gains and adaptive gains which are tuned by timevarying adjustable parameters. The proposed adaptive gain robust controller can achieve asymptotical stability but also improving transient behavior of the resulting closed-loop system. Moreover, by adjusting design parameters, the excessive control input is avoided [32]. In this chapter, firstly, a design method of the centralized adaptive gain robust stabilizing controllers for a class of uncertain linear systems has been shown, and the maximum allowable perturbation region of uncertainties is discussed. Namely, the proposed adaptive gain robust controllers can achieve robustness for the derived perturbation regions for unknown parameters. Additionally, the result for the centralized adaptive gain robust stabilizing controllers is extended to the design problem of decentralized robust control systems.

robust controller."

suggested [26, 27].

Now, as mentioned above, in order to design control systems, the derivation of a mathematical model for controlled system based on state-space representation is needed. If the mathematical model describes the controlled system with sufficient accuracy, a satisfactory control performance is achievable by using various controller design methods. However, there inevitably exists some gaps between the controlled system and its mathematical model, and the gaps are referred to as "uncertainties." The uncertainties in the mathematical model may cause deterioration of control performance or instability of the control system. From this viewpoint, robust control for dynamical systems with uncertainties has been well studied, and a large number of existing results for robust stability analysis and robust stabilization have been obtained [34, 36, 47, 48]. One can see that quadratic stabilization based on Lyapunov stability criterion and H<sup>∞</sup> control is a typical robust controller (e.g., [1, 6]). Furthermore, some researchers investigated quadratic stabilizing control with an achievable performance level in Ref. to such as a quadratic cost function [23, 28, 35, 37], robust H<sup>2</sup> control [18, 39], and robust H<sup>∞</sup>-type disturbance attenuation [46]. However, these approaches result in worst-case design, and, therefore, these controllers with a fixed feedback gain which is designed by considering the worst-case variations of uncertainties/unknown parameters become cautious when the perturbation region of uncertainties has been estimated larger than the proper region. In contrast with the conventional robust control with fixed gains, several design methods of some robust controllers with time-varying adjustable parameters have also been proposed (e.g., [3, 24, 36]). In the work of Maki and Hagino [25], by introducing time-varying adjustable parameters, adaptation mechanisms for improving transient behavior have been suggested. Moreover, robust controllers with adaptive compensation inputs have also been shown [29–31]. In particular, for linear systems with matched uncertainties, Oya and Hagino [29] have introduced an adaptive compensation input which is determined so as to reduce the effect of unknown parameters. Furthermore, a design method of a variable gain robust controller based on LQ optimal control for a class of uncertain linear system has also been shown [32]. These robust controllers have fixed gains and variable ones tuned by updating laws and are more flexible and adaptive compared with the conventional robust controllers with fixed gains only, and one can easily see that these robust controllers with adjustable parameters differ from gain-scheduling control techniques [22, 41, 42]. Additionally, these robust controllers with time-varying adjustable parameters may also be referred to as "variable gain robust controller" or "adaptive gain robust controller."

classical control ideas, some design methods of controllers such as proportional, derivative, and integral (PID) controllers and phase lead-lag compensators have also been presented [21]. In classical control, controlled systems are mainly linear and time-invariant and have a single input and a single output only. Furthermore, it is well known that design approaches based on classical control theory need experiences and trial and error. On the other hand, in the 1960s, state variables and state equations (i.e., state-space representations) have been introduced by Kalman as system representations, and he has proposed an optimal regulator theory [14–16] and an optimal filtering one [17]. Namely, controlled systems are represented by state equations, and controller design problems are reduced to optimization problems based on the concept of state variables. Such controller design approach based on the statespace representation has been established as "modern control theory." Modern control is a theory of time domain, and whereas the transfer function and the frequency response are of limited applicability to nonlinear systems, state equations and state variables are equally appropriate to linear multi-input and multi-output systems or nonlinear one. Therefore, many existing results based on the state-space representation for controller design problems

Now, as mentioned above, in order to design control systems, the derivation of a mathematical model for controlled system based on state-space representation is needed. If the mathematical model describes the controlled system with sufficient accuracy, a satisfactory control performance is achievable by using various controller design methods. However, there inevitably exists some gaps between the controlled system and its mathematical model, and the gaps are referred to as "uncertainties." The uncertainties in the mathematical model may cause deterioration of control performance or instability of the control system. From this viewpoint, robust control for dynamical systems with uncertainties has been well studied, and a large number of existing results for robust stability analysis and robust stabilization have been obtained [34, 36, 47, 48]. One can see that quadratic stabilization based on Lyapunov stability criterion and H<sup>∞</sup> control is a typical robust controller (e.g., [1, 6]). Furthermore, some researchers investigated quadratic stabilizing control with an achievable performance level in Ref. to such as a quadratic cost function [23, 28, 35, 37], robust H<sup>2</sup> control [18, 39], and robust H<sup>∞</sup>-type disturbance attenuation [46]. However, these approaches result in worst-case design, and, therefore, these controllers with a fixed feedback gain which is designed by considering the worst-case variations of uncertainties/unknown parameters become cautious when the perturbation region of uncertainties has been estimated larger than the proper region. In contrast with the conventional robust control with fixed gains, several design methods of some robust controllers with time-varying adjustable parameters have also been proposed (e.g., [3, 24, 36]). In the work of Maki and Hagino [25], by introducing time-varying adjustable parameters, adaptation mechanisms for improving transient behavior have been suggested. Moreover, robust controllers with adaptive compensation inputs have also been shown [29–31]. In particular, for linear systems with matched uncertainties, Oya and Hagino [29] have introduced an adaptive compensation input which is determined so as to reduce the effect of unknown parameters. Furthermore, a design method of a variable gain robust controller based on LQ optimal control for a class of uncertain linear system has also been shown [32]. These robust controllers have fixed gains and variable ones tuned by updating laws and are more flexible and adaptive

have been suggested (e.g., [7, 43]).

186 Adaptive Robust Control Systems

In recent years, a great number of control systems are brought about by present technologies and environmental and societal processes which are highly complex and large in dimension, and such systems are referred to as "large-scale complex systems" or "largescale interconnected systems." Namely, large-scale and complex systems are progressing due to the rapid development of industry, and large-scale interconnected systems can be seen in diverse fields such as economic systems, electrical systems, and so on. For such large-scale interconnected systems, it is difficult to apply centralized control strategies because of calculation amount, physical communication constraints, and so on. Namely, a notable characteristic of the most large-scale interconnected systems is that centrality fails to hold due to either the lack of centralized computing capability of or centralized information. Moreover, large-scale interconnected systems are controlled by more than one controller or decision-maker involving decentralized computation. In the decentralized control strategy, large-scale interconnected systems are divided into several subsystems, and various types of decentralized control problems have been widely studied [13, 38, 44]. The major problem of large-scale interconnected systems is how to deal with the interactions among subsystems. A large number of results in decentralized control systems can be seen in the work of Šijjak [38]. Moreover, a framework for decentralized fault-tolerant control has also been studied [44]. Additionally, decentralized robust control strategies for uncertain large-scale interconnected systems have also attracted the attention of many researchers (e.g., [3–5, 11]). Moreover, in the work of Mao and Lin [24], for large-scale interconnected systems with unmodeled interaction, the aggregative derivation is tracked by using a model following the technique with online improvement, and a sufficient condition for which the overall system when controlled by the completely decentralized control is asymptotically stable has been established. Furthermore, decentralized guaranteed cost controllers for uncertain large-scale interconnected systems have also been suggested [26, 27].

In this chapter, for a class of uncertain linear systems, we show LMI-based design strategies for adaptive gain robust controllers for a class of uncertain dynamical systems. The adaptive gain robust controllers consist of fixed gains and adaptive gains which are tuned by timevarying adjustable parameters. The proposed adaptive gain robust controller can achieve asymptotical stability but also improving transient behavior of the resulting closed-loop system. Moreover, by adjusting design parameters, the excessive control input is avoided [32]. In this chapter, firstly, a design method of the centralized adaptive gain robust stabilizing controllers for a class of uncertain linear systems has been shown, and the maximum allowable perturbation region of uncertainties is discussed. Namely, the proposed adaptive gain robust controllers can achieve robustness for the derived perturbation regions for unknown parameters. Additionally, the result for the centralized adaptive gain robust stabilizing controllers is extended to the design problem of decentralized robust control systems.

The contents of this chapter are as follows, where the item numbers in the list accord with the section numbers:

(iii) <sup>Ξ</sup><sup>22</sup> <sup>&</sup>gt; <sup>0</sup> and <sup>Ξ</sup><sup>11</sup> � <sup>Ξ</sup>12Ξ�<sup>1</sup>

Proof. See Boyd et al. [2].

discussed.

2.1. Problem statement

<sup>22</sup> Ξ<sup>T</sup> <sup>12</sup> > 0.

2. Synthesis of centralized adaptive gain robust controllers

d

region for the unknown parameter <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup>

Eq. (1). Namely, the nominal control input is given as

and thus the following nominal closed-loop system is obtained:

d

nominal system, respectively.

A centralized adaptive gain robust state feedback control scheme for a class of uncertain linear systems is proposed in this section. The adaptive gain robust controller under consideration is composed of a state feedback with a fixed gain matrix and a time-varying adjustable parameter. In this section, we show an LMI-based design method of the adaptive gain robust state feedback controller, and the allowable perturbation region of unknown parameters is

Consider the uncertain linear system described by the following state-space representation:

where xð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup> and u tð Þ<sup>∈</sup> <sup>R</sup><sup>m</sup> are the vectors of the state (assumed to be available for feedback) and the control input, respectively. In Eq. (1) the constant matrices A and B mean the nominal values of the system, and ð Þ <sup>A</sup>; <sup>B</sup> is stabilizable pair. Moreover, the matrix <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> represents unknown time-varying parameters which satisfy <sup>Δ</sup><sup>T</sup>ð Þ<sup>t</sup> <sup>Δ</sup>ð Þ<sup>t</sup> <sup>≤</sup> <sup>δ</sup><sup>⋆</sup>In , and the elements of <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> are Lebesgue measurable [1, 34]. Namely, the unknown time-varying matrix <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>�<sup>n</sup> is bounded, and the parameter <sup>δ</sup><sup>⋆</sup> denotes the upper bound of the perturbation

system which can be obtained by ignoring the unknown parameter Δð Þt in Eq. (1) is given by

In Eq. (2), <sup>x</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup> and u tð Þ<sup>∈</sup> <sup>R</sup><sup>m</sup> are the vectors of the state and the control input for the

First of all, we design the state feedback control for the nominal system of Eq. (2) so as to generate the desirable transient behavior in time response for the uncertain linear system of

d

dt <sup>x</sup>ðÞ¼ <sup>t</sup> ð Þ <sup>A</sup> <sup>þ</sup> <sup>Δ</sup>ð Þ<sup>t</sup> <sup>x</sup>ð Þþ <sup>t</sup> Bu tð Þ, (1)

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189

. Additionally, we suppose that the nominal

dt <sup>x</sup>ðÞ¼ <sup>t</sup> <sup>A</sup>xð Þþ <sup>t</sup> Bu tð Þ: (2)

u tðÞ¼ Kxð Þt , (3)

dt <sup>x</sup>ðÞ¼ <sup>t</sup> AKxð Þ<sup>t</sup> , (4)

2. Synthesis of centralized adaptive gain robust controllers.

3. Synthesis of decentralized adaptive gain robust controllers.

4. Conclusions and future works.

The basic symbols are listed below.


Other than the above, we use the following notation and terms: For a matrix A, the transpose of matrix A and the inverse of one are denoted by A<sup>T</sup> and A�<sup>1</sup> , respectively. The notations Hef g A and diagð Þ <sup>A</sup>1; <sup>⋯</sup>; AN represent <sup>A</sup> <sup>þ</sup> <sup>A</sup><sup>T</sup> and a block diagonal matrix composed of matrices <sup>A</sup><sup>i</sup> for i ¼ 1, ⋯, N . The n-dimensional identity matrix and n � m-dimensional zero matrix are described by In and 0<sup>n</sup>�m, and for real symmetric matrices <sup>A</sup> and <sup>B</sup>, <sup>A</sup> <sup>&</sup>gt; <sup>B</sup> resp: <sup>A</sup> <sup>≥</sup> <sup>B</sup> means that <sup>A</sup> � <sup>B</sup> is a positive (resp. nonnegative) definite matrix. For a vector <sup>α</sup><sup>∈</sup> <sup>R</sup>n, j j j j <sup>α</sup> denotes standard Euclidian norm, and for a matrix A, j j j j A represents its induced norm. The real part of a complex number s (i.e., s ∈ C) is denoted by Ref gs , and the symbols "¼ <sup>Δ</sup> " and "⋆" mean equality by definition and symmetric blocks in matrix inequalities, respectively.

Furthermore, the following useful lemmas are used in this chapter.

Lemma 1.1. For arbitrary vectors λ and ξ and the matrices G and H which have appropriate dimensions, the following relation holds:

$$2\lambda^T \mathcal{G} \Lambda(t) \mathcal{H} \xi \le 2||\mathcal{G}^T \lambda|| ||\mathcal{H}\xi||\_{\prime}$$

where <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>p</sup>�<sup>q</sup> is a time-varying unknown matrix satisfying k k <sup>Δ</sup>ð Þ<sup>t</sup> <sup>≤</sup> 1.

Proof. The above relation can be easily obtained by Schwartz's inequality (see [9]).

Lemma 1.2. (Schur complement) For a given constant real symmetric matrix Ξ, the following arguments are equivalent:

$$\begin{aligned} \text{(i)}\ \Xi &= \begin{pmatrix} \Xi\_{11} & \Xi\_{12} \\ \Xi\_{12}^T & \Xi\_{22} \end{pmatrix} > 0. \\\\ \text{(ii)}\ \Xi\_{11} &> 0 \text{ and } \Xi\_{22} - \Xi\_{12}^T \Xi\_{11}^{-1} \Xi\_{12} > 0. \end{aligned}$$

(iii) <sup>Ξ</sup><sup>22</sup> <sup>&</sup>gt; <sup>0</sup> and <sup>Ξ</sup><sup>11</sup> � <sup>Ξ</sup>12Ξ�<sup>1</sup> <sup>22</sup> Ξ<sup>T</sup> <sup>12</sup> > 0.

Proof. See Boyd et al. [2].

The contents of this chapter are as follows, where the item numbers in the list accord with the

Other than the above, we use the following notation and terms: For a matrix A, the transpose of

and diagð Þ <sup>A</sup>1; <sup>⋯</sup>; AN represent <sup>A</sup> <sup>þ</sup> <sup>A</sup><sup>T</sup> and a block diagonal matrix composed of matrices <sup>A</sup><sup>i</sup> for i ¼ 1, ⋯, N . The n-dimensional identity matrix and n � m-dimensional zero matrix are described by In and 0<sup>n</sup>�m, and for real symmetric matrices <sup>A</sup> and <sup>B</sup>, <sup>A</sup> <sup>&</sup>gt; <sup>B</sup> resp: <sup>A</sup> <sup>≥</sup> <sup>B</sup> means that <sup>A</sup> � <sup>B</sup> is a positive (resp. nonnegative) definite matrix. For a vector <sup>α</sup><sup>∈</sup> <sup>R</sup>n, j j j j <sup>α</sup> denotes standard Euclidian norm, and for a matrix A, j j j j A represents its induced norm. The real part of a complex

Lemma 1.1. For arbitrary vectors λ and ξ and the matrices G and H which have appropriate

Lemma 1.2. (Schur complement) For a given constant real symmetric matrix Ξ, the following argu-

k k Hξ ,

<sup>2</sup>λ<sup>T</sup>GΔð Þ<sup>t</sup> <sup>H</sup><sup>ξ</sup> <sup>≤</sup> <sup>2</sup> <sup>G</sup><sup>T</sup><sup>λ</sup>

Proof. The above relation can be easily obtained by Schwartz's inequality (see [9]).

where <sup>Δ</sup>ð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>p</sup>�<sup>q</sup> is a time-varying unknown matrix satisfying k k <sup>Δ</sup>ð Þ<sup>t</sup> <sup>≤</sup> 1.

, respectively. The notations Hef g A

<sup>Δ</sup> " and "⋆" mean equality by

2. Synthesis of centralized adaptive gain robust controllers.

3. Synthesis of decentralized adaptive gain robust controllers.

R The set of the real number R<sup>n</sup> The set of n-dimensional vectors <sup>R</sup><sup>n</sup>�<sup>m</sup> The set of <sup>n</sup> � <sup>m</sup>-dimensional matrices

C The set of complex numbers

matrix A and the inverse of one are denoted by A<sup>T</sup> and A�<sup>1</sup>

number s (i.e., s ∈ C) is denoted by Ref gs , and the symbols "¼

definition and symmetric blocks in matrix inequalities, respectively. Furthermore, the following useful lemmas are used in this chapter.

section numbers:

188 Adaptive Robust Control Systems

4. Conclusions and future works. The basic symbols are listed below.

dimensions, the following relation holds:

> 0.

12Ξ�<sup>1</sup>

<sup>11</sup> Ξ<sup>12</sup> > 0.

ments are equivalent:

(i) <sup>Ξ</sup> <sup>¼</sup> <sup>Ξ</sup><sup>11</sup> <sup>Ξ</sup><sup>12</sup> ΞT <sup>12</sup> Ξ<sup>22</sup> 

(ii) <sup>Ξ</sup><sup>11</sup> <sup>&</sup>gt; <sup>0</sup> and <sup>Ξ</sup><sup>22</sup> � <sup>Ξ</sup><sup>T</sup>
