Open access peer-reviewed chapter

# Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

Written By

Shunya Nagai, Hidetoshi Oya, Tsuyoshi Matsuki and Yoshikatsu Hoshi

Submitted: June 8th, 2017 Reviewed: October 17th, 2017 Published: December 20th, 2017

DOI: 10.5772/intechopen.71733

From the Edited Volume

## Adaptive Robust Control Systems

Edited by Le Anh Tuan

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## Abstract

In this chapter, adaptive gain robust control strategies for uncertain dynamical systems are presented. Firstly, synthesis of centralized adaptive gain robust controllers for a class of uncertain linear systems is shown. The design problem of the centralized controller is reduced to the constrained convex optimization problem, and allowable perturbation regions of unknown parameters are discussed. Next, the result for the centralized robust controller is extended to uncertain large-scale interconnected systems, that is, an LMI-based design approach for decentralized adaptive gain robust controllers is suggested.

### Keywords

• adaptive gain robust control
• adjustable time-varying parameter
• allowable perturbation regions of unknown parameters
• LMIs

## 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 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, 15, 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 state-space 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]).

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 “large-scale 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, 4, 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 time-varying 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:

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.

 R The set of the real number Rn The set of n-dimensional vectors Rn×m The set of n×m-dimensional matrices C The set of complex numbers

Other than the above, we use the following notation and terms: For a matrix A, the transpose of matrix Aand the inverse of one are denoted by ATand A1, respectively. The notations HeAand diagA1ANrepresent A+ATand a block diagonal matrix composed of matrices Aifor i=1,,N. The n-dimensional identity matrix and n×m-dimensional zero matrix are described by Inand 0n×m,and for real symmetric matrices Aand B, A>Bresp.ABmeans that ABis a positive (resp. nonnegative) definite matrix. For a vector αRn, αdenotes standard Euclidian norm, and for a matrix A, Arepresents its induced norm. The real part of a complex number s(i.e., sC) is denoted by Res, and the symbols “=Δ” 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 vectorsλandξand the matricesGandHwhich have appropriate dimensions, the following relation holds:

2λTGΔtHξ2GTλHξ,

where ΔtRp×qis a time-varying unknown matrix satisfying Δt1.

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 matrixΞ, the following arguments are equivalent:

1. Ξ=Ξ11Ξ12Ξ12TΞ22>0.

2. Ξ11>0andΞ22Ξ12TΞ111Ξ12>0.

3. Ξ22>0andΞ11Ξ12Ξ221Ξ12T>0.

Proof.See Boyd et al. [2].

## 2. Synthesis of centralized adaptive gain robust controllers

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 discussed.

### 2.1. Problem statement

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

ddtxt=A+Δtxt+But,E1

where xtRnand utRmare the vectors of the state (assumed to be available for feedback) and the control input, respectively. In Eq. (1) the constant matrices Aand Bmean the nominal values of the system, and ABis stabilizable pair. Moreover, the matrix ΔtRn×nrepresents unknown time-varying parameters which satisfy ΔTtΔtδIn, and the elements of ΔtRn×nare Lebesgue measurable [1, 34]. Namely, the unknown time-varying matrix ΔtRn×nis bounded, and the parameter δdenotes the upper bound of the perturbation region for the unknown parameter ΔtRn×n. Additionally, we suppose that the nominal system which can be obtained by ignoring the unknown parameter Δtin Eq. (1) is given by

ddtx¯t=Ax¯t+Bu¯t.E2

In Eq. (2), x¯tRnand u¯tRmare the vectors of the state and the control input for the nominal system, respectively.

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

u¯t=Kx¯t,E3

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

ddtx¯t=AKx¯t,E4

where AKis a matrix given by AK=ΔA+BK. Note that the standard LQ control theory for the nominal system of Eq. (2) for designing the fixed feedback gain KRm×nis adopted in the existing result [32]. In this section, for the nominal system of Eq. (2), we derive a state feedback controller with pole placement constraints [8]. Note that for simplicity the sector constraints are introduced only in this chapter, and of course, one can adopt some other design constraints or another controller design approach for designing the fixed gain matrix KRm×n. Therefore, we consider the matrix inequality condition:

AK+αInTP+PAK+αIn+Q<0,E5

where PRn×nand QRn×nare a symmetric positive definite matrix and a symmetric semi-positive definite matrix, respectively, and the matrix QRn×nis selected by designers. If the symmetric positive definite matrix PRn×nsatisfying the matrix inequality of Eq. (5) exists, then poles for the nominal closed-loop system of Eq. (4) are located into the subspace Sα=sResαin the complex plane. Namely, the nominal closed-loop system of Eq. (4) is asymptotically stable, and the quadratic function Vx¯t=Δx¯TtPx¯tbecomes a Lyapunov function for the nominal closed-loop system of Eq. (4), because the time derivative of the quadratic function Vx¯tcan be expressed as

ddtVx¯t<x¯TtQ+2αPx¯t<0,x¯t0.E6

Now, we introduce complementary matrices YRn×nand WRm×mwhich satisfy the relations Y=ΔP1, K=WBTP, and W=WT>0, respectively. Then, some algebraic manipulations gives

YAT+AYBWTBTBWBT+2αY+YQY<0.E7

Additionally, applying Lemma 1.2(Schur complement) to Eq. (7), one can easily see that the matrix inequality condition of Eq. (7) is equivalent to

YAT+AYBWTBTBWBT+2αYYQ1<0.E8

Thus, the control gain matrix KRm×nis determined as K=WBTP=WBTY1.

Now, for the uncertain linear system of Eq. (1), we define the following control input [37]:

ut=Δ1+θxtKxt,E9

where θxt:Rn×RRis an adjustable time-varying parameter [32] which plays the important role for correcting the effect of uncertainties, that is, the control input utRmconsists of a fixed gain matrix KRm×nand θxtR. Note that, the robust control input of the form of Eq. (9) is called “adaptive gain robust control” in this chapter. Thus, from Eqs. (1) and (9), the uncertain closed-loop system can be written as

ddtxt=AKxt+Δtxt+θxtBKxt.E10

From the above, the control objective in this section is to design the adaptive gain robust control which achieves satisfactory transient behavior. Namely, the control problem is to derive the adjustable time-varying parameter θxtRsuch that the closed-loop system of Eq. (10) can achieve the desired transient response. In addition, we evaluate the allowable perturbation region of the unknown parameter ΔtRn×n.

### 2.2. Synthesis of centralized adaptive gain robust state feedback controllers

In this subsection, we deal with design problems for the adjustable time-varying parameter θxtRso that the satisfactory transient response for the uncertain linear system of Eq. (1) can be achieved. For the proposed adaptive gain robust control, the following theorem gives an LMI-based design synthesis.

Theorem 1:Consider the uncertain linear system ofEq. (1) and the adaptive gain robust control ofEq. (9) with the adjustable time-varying parameterθxtR.

For a given design parameterϑ>0and the known upper boundδfor the unknown parameterΔtRn×n, if the scalar parameterγ>0exists satisfying

AKTP+PAK+γP2InγδIn<0,E11

the adjustable time-varying parameterθxtRis determined as

θxt=δPxtxtW1/2BTPxt2ifxTtPBWBTPxtϑxTtxt,δPxtxtϑxTtxtifxTtPBWBTPxt<ϑxTtxt.E12

Then, the uncertain closed-loop system ofEq. (10) is asymptotically stable.

Proof.In order to prove Theorem 1, by using symmetric positive definite matrix PRn×nwhich satisfies the standard Riccati equation of Eq. (4), we introduce the quadratic function

Vxt=ΔxTtPxt,E13

as a Lyapunov function candidate. Let xtbe the solution of the uncertain closed-loop system of Eq. (10) for tt0, and then the time derivative of the quadratic function Vxtalong the trajectory of the uncertain closed-loop system of Eq. (10) can be written as

ddtVxt=xTtAKTP+PAKxt+2xTtPΔtxt+2θxtxTtPBKxt.E14

Firstly, the case of xTtPBWBTPxtϑxTtxtis considered. In this case, one can see from the relation Δtδ, Eq. (14), and Lemma 1.1that the following inequality holds:

ddtVxtxTtAKTP+PAKxt+2δPxtxt+2θxtxTtPBKxt.E15

Moreover, since the relation K=WBTPholds, the inequality of Eq. (15) can be rewritten as

ddtVxtxTtAKTP+PAKxt+2δPxtxt2θxtxTtPBWBTPxt.E16

Substituting the adjustable time-varying parameter θxtof Eq. (12) into Eq. (16) gives

ddtVxtxTtAKTP+PAKxt+2δPxtxt2xTtPδPxtxtW1/2BTPxt2BWBTPxtxTtAKTP+PAKxt.E17

If the solution of the LMI of Eq. (11) exists, then the inequality

AKTP+PAK<0E18

is satisfied. Thus, one can see that the following relation holds:

ddtVxt<0,xt0.E19

Next, we consider the case of xTtPBWBTPxt<ϑxTtxt. By using the well-known inequality for any vectors αand βwith appropriate dimensions and a positive scalar ζ

2αTβζαTα+1ζβTβ,E20

we see from Eq. (14) that some algebraic manipulations give

ddtVxtxTtAKTP+PAKxt+γxTtP2xt+1γxTtΔTtΔtxt+2θxtxTtPBKxtxTtAKTP+PAK+γP2+δγInxt+2θxtxTtPBKxtE21

where γis a positive constant.

Let us consider the last term of the right-hand side of Eq. (21). We see from Eq. (12) and the relation K=WBTPthat the last term of the right-hand side of Eq. (21) is nonpositive. Thus, if the scalar parameter γexists satisfying

AKTP+PAK+γP2+δγIn<0,E22

then the following relation for the quadratic function Vxtholds:

ddtVxt<0,xt0.E23

Furthermore, applying Lemma 1.2(Schur complement) to Eq. (22), we find that the matrix inequality condition of Eq. (22) can be transformed into the LMI of Eq. (11). Namely, the quadratic function Vxtof Eq. (13) becomes a Lyapunov function of the uncertain closed-loop system of Eq. (10) with the adjustable time-varying parameter of Eq. (12), that is, asymptotical stability of the uncertain closed-loop system of Eq. (10) is ensured. It follows that the result of this theorem is true.

From the above, we show an LMI-based design strategy for the proposed adaptive gain robust control. Namely, the design problem of the proposed adaptive gain robust controller can be reduced to the feasibility of the LMI of Eq. (11). Note that the LMI of Eq. (11) defines a convex solution set of γ, and therefore one can easily see that various efficient convex optimization algorithms can be used to test whether the LMI is solvable and to generate particular solution. Furthermore, the LMI of Eq. (11) can also be exploited to design the proposed adaptive gain robust controller with some additional requirements. Thus, in this paper, we consider the allowable region of the unknown parameter ΔtRn×nand introduce the additional constraints γ=δand

γ1ε>0,E24

where εis a positive constant. From the relation of Eq. (24), we find that the minimization of the parameter εmeans the maximization of the upper bound δ. Then, by using Lemma 2 (Schur complement), we find that the LMI of Eq. (11) is equivalent to

AKTP+PAK+γP2+In<0,E25

and the constraint of Eq. (24) can be transformed into

γ1.0ε>0.E26

From the above, we consider the following constrained optimization problem:

Minimizeγ>0,εεsubjectto25and26.E27

If the optimal solution of the constrained optimization problem of Eq. (27) exists, in which are denoted by γand ε, the proposed adaptive gain robust controller can be done, and the allowable upper bound of the unknown parameter ΔtRn×nis given by

δ=γ.E28

Consequently, the following theorem for the proposed adaptive gain robust control with guaranteed allowable region of unknown parameter ΔtRn×nis developed.

Theorem 2:Consider the uncertain linear system ofEq. (1) and the adaptive gain robust control ofEq. (8) with the adjustable time-varying parameterθxtR.

If the optimal solutionγof the constrained optimization problem ofEq. (27) exists, then the adjustable time-varying parameterθxtRis designed asEq. (12), and asymptotical stability of the uncertain closed-loop system ofEq. (10) is ensured. Moreover, the upper boundδfor the unknown parameterΔtRn×nis given byEq. (28).

Remark 1:In this section, the uncertain linear dynamical system ofEq. (1) is considered, and the centralized adaptive gain robust controller has been proposed. Although the uncertain linear system ofEq. (1) has uncertainties in the state matrix only, the proposed adaptive gain robust controller can also be applied to the case that the uncertainties are included in both the system matrix and the input one. Namely, by introducing additional actuator dynamics and constituting an augmented system, unknown parameters in the input matrix are embedded in the system matrix of the augmented system [45]. As a result, the proposed controller design procedure can be applied to such case.

Remark 2:InTheorem 1, the design problem of the proposed adaptive gain robust controller can be reduced to the feasibility of the LMI ofEq. (11). Namely, in order to design the proposed robust control system, designers have to solve the LMI ofEq. (11). If the LMI ofEq. (11) is feasible forδ>0, then one can easily see that the LMI ofEq. (11) is always satisfied for the positive scalarδ<δ. Moreover, if a positive scalarγexists satisfying the LMI ofEq. (11) forδ+>δ, then the proposed adaptive gain robust controller can also be designed, and note that the adaptive gain robust controller forδ>0coincides exactly with the one forδ+>δ>0. Furthermore, one can see fromTheorem 2 that the resultant adaptive gain robust controller derived by solving the constrained convex optimization problem ofEq. (27) is same, because the solution of LMI ofEq. (8) or one of the constrained convex optimization problem ofEq. (27) cannot be reflected the resultant controller. Note that in the general controller design strategies for the conventional fixed gain robust control, the solution of the some constraints can be applied to the resultant robust controller. This is a fascinating fact for the proposed controller design strategy.

Remark 3:The proposed adaptive gain robust controller with the adjustable time-varying parameter has some advantages as follows: the proposed controller design approach is very simple, and by selecting the design parameter, the proposed adaptive gain robust control system can achieve good transient performance which is close to the nominal one or avoid the excessive control input (see [32]). Besides, the structure of the proposed control system is also simple compared with the existing results for robust controllers with adjustable parameters (e.g., [29, 30]). However, the online adjustment strategy for the design parameterϑhas not been established, and this problem is one of our future research subjects.

Remark 4:In this section, firstly the nominal control input is designed by adopting pole placement constraints, and the fixed gainKRm×ncan be derived by using the solution of the LMI ofEq. (8). Note that the quadratic functionVxtis a Lyapunov function for both the uncertain linear system ofEq. (1) and the nominal system ofEq. (2), that is, the Lyapunov function for the uncertain linear system ofEq. (1) and one for the nominal system ofEq. (2) have same level set. Therefore, by selecting the design parameterϑ>0, the proposed adaptive gain robust control system can achieve good transient performance which is close to the nominal one or avoid the excessive control input.

On the other hand, if the design problem for a state feedback controlut=Ksxtis considered, the quadratic functionVxtis replaced asVsxt=xTtPsxtwherePsRn×nis a Lyapunov matrix. Moreover,PsRn×nbecomes a variable for resultant LMI conditions, and the standard techniques for the quadratic stabilization can also be used.

### 2.3. Illustrative examples

In order to demonstrate the efficiency of the proposed control strategy, we have run a simple example.

Consider the following linear system with unknown parameter ΔtR2×2:

ddtxt=1.04.00.01.0xt+Δtxt+0.01.0ut.E29

Firstly, we design the nominal control input u¯t=Kx¯t. By selecting the design parameters αand Qin Eq. (5) such as α=3.0and Q=1.0×I2and solving the LMI of Eq. (8), we obtain the following solution:

Y=1.08551.53564.5318,W=2.1708×101.E30

Thus, the following fixed gain matrix can be computed:

K=1.3017×1019.2008.E31

Next, we solve the constrained optimization problem of Eq. (27), then the solutions

γ=3.1612,ε=3.1633×101,E32

can be derived, and therefore the allowable upper bound of unknown parameter is given as

δ=3.1612.E33

In this example, we consider the following two cases for the unknown parameter ΔtR2×2:

• Case 1) Δt=δ×0.00.07.22896.8530×101.

• Case 2) Δt=δ×sin5.0×π×tcos5.0×π×tsin5.0×π×t.

Note that the unknown parameter of Case 1 satisfies the matching condition [45]. In addition, for the design parameter ϑ, the numerical simulation for two cases such as ϑ=1.0×102and ϑ=5.0×101is run. Moreover, the initial values of the uncertain system of Eq. (29) and the nominal system are selected as x0=x¯0=(1.02.0)T. The results of the simulation of this example are shown in Figures 14 and Table 1. In these figures, “Case 1)” and “Case 2)” represent the time histories of the state variables x1tand x2tand the control input utand Lyapunov function Vx.tfor the proposed adaptive gain robust control, and “nominal” means the desired time response and the desired control input and Lyapunov function Vx¯tfor the nominal system. In Table 1, Jemeans

Je=Δ0eTtetdt,E34
ϑ=1.0×102ϑ=5.0×101
Case 1)4.2584×1021.0160×102
Case 2)9.7403×1021.0038×101

### Table 1.

The performance index Ie.

where etis an error vector between the time response and the desired one generated by the nominal system, that is, et=Δxtx¯t. Namely, Jeof Eq. (34) is a performance index so as to evaluate the transient performance.

From Figures 14 the proposed adaptive gain robust state feedback controller stabilizes the uncertain linear system of Eq. (29) in spite of uncertainties. Furthermore, we also find that the proposed adaptive gain robust controller achieves the good transient performance close to the nominal system.

For Case 1 in this example, one can see from Table 1 that the adaptive gain robust controller for ϑ=5.0×101is more desirable comparing with one for ϑ=1.0×102, that is, the error between the time response and the desired one generated by the nominal system (“nominal” in figures) is small. But for the result of Case 2), we find that the robust controller with the parameter ϑ=1.0×102achieves more desirable performance. Additionally, one can see from Figures 2(a) and 4(a) that by selecting the design parameter ϑthe proposed adaptive gain robust controller can adjust the magnitude of the control input. In this example, the magnitude of the control input for ϑ=1.0×102is suppressed comparing with one for ϑ=5.0×101. However, the online adjustment way of the design parameter ϑfor the purpose of improving transient behavior and avoiding excessive control input cannot to developed, and thus it is an important problem of our research subjects.

Therefore, the effectiveness of the proposed adaptive gain robust controller is shown.

### 2.4. Summary

In this section, an LMI-based design scheme of the centralized adaptive gain robust state feedback controller for a class of uncertain linear systems has been proposed, and by simple numerical simulations, the effectiveness of the proposed robust control strategy has been presented. Since the proposed adaptive gain robust controller can easily be obtained by solving the constrained convex optimization problem, the proposed design approach is simple. Moreover, by selecting the design parameter, the proposed adaptive gain robust controller can achieve good transient performance and/or avoid excessive control input. Note that there are trade-offs between achieving good transient performance and avoiding excessive control input.

The future research subject is the extension of proposed robust control scheme to such a broad class of systems as linear systems with state delays, uncertain systems with some constraints, and so on. Additionally, we will discuss the online adjustment for the design parameter ϑand the design problem for output feedback control systems.

## 3. Synthesis of decentralized adaptive gain robust controllers

In this section, on the basis of the result derived in Section 2, an LMI-based design method of decentralized adaptive gain robust state feedback controllers for a class of uncertain large-scale interconnected systems is suggested. The design problem of the decentralized adaptive gain robust controller under consideration can also be reduced to the feasibility of LMIs, and the allowable perturbation region of uncertainties is also discussed.

### 3.1. Problem statement

Consider the uncertain large-scale interconnected system composed of Nsubsystems described as

ddtxi(t)=Aii(t)xi(t)+j=1jiNAij(t)xj(t)+Biui(t),E35

where xitRniand uitRmi(i=1,,N) are the vectors of the state and the control input for the ith subsystem, respectively, and xt=x1TtxNTtTis the state of the overall system. The matrices AiitRni×niand AijtRni×njin Eq. (35) are given by

Aiit=Aii+Δiit,Aijt=Aij+Δijt.E36

In Eqs. (35) and (36), the matrices AiiRni×ni, AijRni×nj, and BiRni×midenote the nominal values of the system, and matrices ΔiitRni×niand ΔijtRni×njshow unknown parameters which satisfy ΔiiTtΔiitρiiIniand ΔijTtΔijtρijInj, respectively. Note that the elements of these unknown parameters are Lebesgue measurable [1, 34]. For Eq. (35), the nominal subsystem, ignoring the unknown parameters, is given by

ddtx¯i(t)=Aiix¯i(t)+j=1jiNAijx¯j(t)+Biu¯i(t),E37

where x¯itRniand u¯itRmiare the vectors of the state and the control input for the ith nominal subsystem, respectively. Furthermore, the control input for the nominal subsystem of Eq. (37) is determined as

u¯it=Kix¯it,E38

where KiRmi×niis a fixed gain matrix. From Eqs. (37) and (38), the following nominal closed-loop subsystem is obtained:

ddtx¯i(t)=AKix¯i(t)+j=1jiNAijx¯j(t),E39

where AKi=ΔAiiBiKi.

Now, by using symmetric positive definite matrices PiRni×ni, we consider the quadratic function

Vx¯t=Δi=1NVix¯it,E40
Vix¯it=Δx¯iTtPix¯it,E41

as a Lyapunov function candidate. For the quadratic function Vix¯itof Eq. (41), its time derivative along the trajectory of the nominal closed-loop subsystem of Eq. (39) is given by

ddtVi(x¯i,t)=x¯iT(t)(AKiTPi+PiAKi)x¯i(t)+j=1jiN2x¯iT(t)PiAijx¯j(t).E42

For the second term on the right side of Eq. (42), by using the well-known relation of Eq. (20), we can obtain the following relation:

ddtVi(x¯i,t)x¯iT(t)(AKiTPi+PiAKi)x¯i(t)+j=1jiNμijx¯iT(t)PiAijAijTPx¯i(t)+j=1jiN1μijx¯jT(t)x¯j(t).E43

From Eqs. (40) and (43), we have

ddtV(x¯,t)i=1Nx¯iT(t)(AKiTPi+PiAKi)x¯i(t)+i=1Nj=1jiNμix¯iT(t)PiAijAijTPix¯i(t)+i=1Nj=1jiN1μijx¯jT(t)x¯j(t).E44

The inequality of Eq. (44) can also be rewritten as

ddtV(x¯,t)i=1Nx¯iT(t)(AKiTPi+PiAKi+j=1jiNμiPiAijAijTPi+j=1jiN1μjiIn)x¯i(t).E45

Therefore, if the matrix inequality

AKiTPi+PiAKi+j=1jiNμiPiAijAijTPi+j=1jiN1μjiIn<0E46

holds, then the following relation for the time derivative of Vx¯tis satisfied:

ddtVx¯t<0,x¯t0.E47

Now, as with Section 2, we derive a decentralized controller with pole placement constraints for the nominal subsystem of Eq. (37). Namely, from Eq. (46), the matrix inequality

(AKi+αiIn)TPi+Pi(AKi+αiIn)+j=1jiNμiPiAijAijTPi+j=1jiN1μjiIn+Qi<0,E48

is considered. In Eq. (48), αiRis a positive scalar and is selected by designers.

We introduce symmetric positive definite matrices Yi=ΔPi1and WiRmi×miand define the fixed gain Kias Ki=ΔWiBiTPi. Then for the matrix inequality of Eq. (48), by pre- and post-multiplying both sides of the matrix inequality of Eq. (48) by Yi, it can be obtained that

AiiYiBiWiBiT+YiAiiTBiWiTBiT+2αiYi+j=1jiNμijAijAijT+j=1jiN1μjiYiYi+YiQiYi<0.E49

Thus, by applying Lemma 1.2(Schur complement) to Eq. (49), we find that the matrix inequality of Eq. (49) is equivalent to the following LMI:

ΛiYiWiμijΘiYiΓiμij<0.E50

In Eq. (50), matrices ΛiYiWiμijRni×ni, ΘiYiRni×Nni, and ΓiμijRNni×Nniare given by

Λi(Yi,,Wi,,μij)=ΔAiiYiBiWiBiT+YiAiiTBiWiTBiT+2αiYi+j=1jiNμijAijAijT,Θi(Yi)=Δ(Yi Yi  YiN),Γi(μij)=Δdiag(Qi1,μ1iIn,μ2iIn,,μi1iIn,μi+1iIn,,μNiIn).E51

Therefore, if matrices YiRni×niand WiRmi×miand positive scalars μijexist, the nominal closed-loop subsystem is asymptotically stable, and the fixed gain matrix Kiis determined as Ki=WiBiTYi1.

Now, by using the fixed gain matrix KiRmi×niwhich is designed for the nominal subsystem, we define the control input

uit=Δ1+θitKixit,E52

where θitR1is an adjustable time-varying parameter. From Eqs. (35) and (52), the uncertain closed-loop subsystem can be obtained as

ddtxit=AKixit+Δiitxit+i=1NAij+ΔijtxjtθitBiKixit.E53

From the above discussion, the designed objective in this section is to determine the decentralized robust control of Eq. (52) such that the resultant overall system achieves robust stability. That is to design the adjustable time-varying parameter θitR1such that asymptotical stability of the overall system composed of Nsubsystems of Eq. (53) is guaranteed.

### 3.2. Decentralized variable gain controllers

The following theorem shows sufficient conditions for the existence of the proposed decentralized adaptive gain robust control system.

Theorem 3:Consider the uncertain large-scale interconnected system ofEq. (35) and the control input ofEq. (52).

For a given positive constantϑi, if positive constantsξii, σij, andεijexist which satisfy the LMIs

ΠiξiiεijσijΞiΩiξiiεijσij<0,E54

the time-varying adjustable parametersθitRare determined as

θit=ΔρiiPixitxitxiTtPiBiWiBiTPixitifxiTtPiBiWiBiTPixitϑixiTtxit,ρiiPixitxitϑixiTtxitifxiTtPiBiWiBiTPixit<ϑixiTtxit,E55

where matricesΠiξiεijσijRni×ni, ΞiRni×2N1ni, andΩiξiiεijσijR2N1ni×2N1niare given by

Πi(ξii,,εij,,σij)=Δ(AKiTPi+PiAKi)+ξiiPiPi+j=1jiNεijPiAijAijTPi+j=1jiNσijPiPi,Ξi=ΔInInIn2N1,Ωiξiiεijσij=ΔdiagξiiρiiInε1iInε2iInεi1iInεi+1iInεNiInσ1iρ1iInσ2iρ2iInσi1iρi1iInσi+1iρi+1iInσNiρNiIn.E56

Then, the overall close-loop system composed ofNclosed-loop subsystems is asymptotically stable.

Proof.In order to prove Theorem 3, the following Lyapunov function candidate is introduced by using symmetric positive definite matrices PiRni×niwhich satisfy the LMIs of (50):

Vxt=Δi=1NVixit,E57

where Vixitis a quadratic function given by

Vixit=ΔxiTtPixit.E58

We can obtain the following relation for the time derivative of the quadratic function Vixitof Eq. (58):

ddtVi(xi,t)=xiT(t)(AKiTPi+PiAKi)xi(t)+2xi(t)PiΔii(t)xi(t)+2xiT(t)Pij=1jiN(Aij+Δij(t))xj(t)2θi(t)xiT(t)PiBiKixi(t).E59

Firstly, we consider the case of xiTtPiBiWiBiTPixitϑixiTtxit. In this case, one can see from the relations ΔiiTtΔiitρiiIniand ΔijTtΔijtρijInj, the well-known inequality of Eq. (20), and Lemma 1.1that the following relation for the quadratic function Vixitof Eq. (58) can be obtained:

ddtVixitxiTtAKiTPi+PiAKixit+2ρiiPixitxit+j=1jiNεijxiT(t)PiAijAijTPixi(t)+j=1jiN1εijxjT(t)xj(t)+j=1jiNσijxiT(t)PiPixi(t)+j=1jiNρijσijxjT(t)xj(t)2θi(t)xiT(t)PiBiKixi(t).E60

Substituting the adjustable time-varying parameter θitof Eq. (55) into Eq. (60) gives

ddtVixitxiTtAKiTPi+PiAKixit+2ρiiPixitxit+j=1jiNεijxiT(t)PiAijAijTPixi(t)+j=1jiN1εijxjT(t)xj(t)+j=1jiNσijxiT(t)PiPixi(t)+j=1jiNρijσijxjT(t)xj(t)2(ρiiPixi(t)xi(t)xiT(t)PiBiWiBiTPixi(t))xiT(t)PiBiKixi(t)=xiT(t)(AKiTPi+PiAKi)xi(t)+j=1jiNεijxiT(t)PiAijAijTPixi(t)+j=1jiN1εijxjT(t)xj(t)+j=1jiNσijxiT(t)PiPixi(t)+j=1jiNρijσijxjT(t)xj(t),E61

and, thus, we have the following inequality for the function Vxtof Eq. (57):

ddtV(x,t)i=1NxiT(t)(AKiTPi+PiAKi)xi(t)+i=1Nj=1jiNεijxi(t)PiAijAijTPixi(t)+i=1Nj=1jiN1εijxjT(t)xj(t)+i=1Nj=1jiNσijxiT(t)PiPixi(t)+i=1Nj=1jiNρijσijxjT(t)xj(t).E62

Furthermore, the inequality of Eq. (62) can be rewritten as

ddtV(x,t)i=1NxiT(t)(AKiTPi+PiAKi+j=1jiNεijPiAijAijTPi+j=1jiN1εjiIn+j=1jiNσijPiPi+j=1jiNρjiσjiIn)xi(t).E63

Therefore, if the matrix inequality

AKiTPi+PiAKi+j=1jiNεijPiAijAijTPi+j=1jiN1εjiIn+j=1jiNσijPiPi+j=1jiNρijσjiIn<0E64

holds, then the following relation for the time derivative of Vxtis satisfied:

ddtVxt<0,xt0.E65

Next, we consider the case of xiTtPiBiWiBiTPixit<ϑixiTtxit. In this case, by using the relations ΔiiTtΔiitρiiIniand ΔijTtΔijtρijInj, and Eq. (20) and substituting the adjustable time-varying parameter θitof Eq. (55) into Eq. (59), we have

ddtVixitxiTtAKiTPi+PiAKixit+ξiixiTtPiPixit+ρiiξiixiTtxit+j=1jiNεijxi(t)PiAijAijTPixi(t)+j=1jiN1εijxjT(t)xj(t)+j=1jiNσijxiT(t)PiPixi(t)+j=1jiNρijσijxjT(t)xj(t)2(ρiiPixi(t)xi(t)ϑixiT(t)xi(t))xiT(t)PiBiKixi(t).E66

The last term on the right side of Eq. (66) is less than 0 because the matrix KiRmi×niis defined as Ki=WiBiTPiand θitis a positive scalar function. Therefore, we find that the following relation for the quadratic function Vixitis satisfied:

ddtVixitxiTtAKiTPi+PiAKixit+ξiixiTtPiPixit+ρiiξiixiTtxit+j=1jiNεijxi(t)PiAijAijTPixi(t)+j=1jiN1εijxjT(t)xj(t)+j=1jiNσijxiT(t)PiPixi(t)+j=1jiNρijσijxjT(t)xj(t).E67

Therefore, we see from Eqs. (57) and (67) that the following inequality:

ddtVxti=1NxiTtAKiTPi+PiAKixit+i=1NξiixiTtPiPixit+i=1NρiiξiixiTtxit+i=1Nj=1jiNεijxi(t)PiAijAijTPixi(t)+i=1Nj=1jiN1εijxjT(t)xj(t)+i=1Nj=1jiNσijxiT(t)PiPixi(t)+i=1Nj=1jiNρijσijxjT(t)xj(t)E68

can be derived. Moreover, one can easily see that the inequality of Eq. (68) can be rewritten as

ddtV(x,t)i=1NxiT(t)(AKiTPi+PiAKi+ξiiPiPi+ρiiξiiIn+j=1jiNεijPiAijAijTPi+j=1jiN1εjiIn+j=1jiNσijPiPi+j=1jiNρjiσjiIn)xi(t).E69

Therefore, if the matrix inequality

AKiTPi+PiAKi+ξiiPiPi+ρiiξiiIn+j=1jiNεijPiAijAijTPi+j=1jiN1εjiIn+j=1jiNσijPiPi+j=1jiNρjiσjiIn<0E70

holds, then the relation of Eq. (65) for the time derivative of the function Vxtof Eq. (57) is satisfied. Due to the 3rd and 4th terms on the left side of Eq. (70) which are positive definite, if the inequality of Eq. (70) is satisfied, then the inequality of Eq. (64) is also constantly satisfied.

For the matrix inequality of Eq. (70), by applying Lemma 1.2(Schur complement), one can find that the matrix inequalities of Eq. (70) are equivalent to the LMIs of Eq. (54). Therefore, by solving the LMIs of Eq. (54), the adjustable time-varying parameter is given by Eq. (55), and proposed control input of Eq. (52) stabilizes the overall system of Eq. (35). Thus, the proof of Theorem 3 is completed.

Next, as mentioned in Section 2, we discuss the allowable region of the unknown parameters ΔiitRni×niand ΔijtRni×nj. Thus, the following additional constraints are introduced:

ρii=ξii,ρij=σij.E71

From the relations of Eq. (71), one can find that the maximization of ξiiand σijis equivalent to the maximization of ρiiand ρij. Then, the LMIs of Eq. (54) can be rewritten as

ΠiξiiεijσijΞiΩiεij<0,E72
Πi(ξii,,εij,,σij)=ΔAKiTPi+PiAKi+ξiiPiPi+NIn+j=1jiNεijPiAijAijTPi+j=1jiNσijPiPi,Ξi=ΔInInInN1,Ωiεij=Δdiagε1iInε2iInεi1iInεi+1iInεNiIn.E73

Furthermore, we introduce a positive scalar λand a complementary matrix ΓRN2×N2defined as

Γ=Δdiagξ11ξ22ξNNσ12σ13σ1Nσ21σ23σNN1,E74

and consider the following additional condition:

Γ1λIN2>0.E75

Namely, we can replace the maximization problem of ξiiand σijwith the minimization problem of λ. From Eq. (75) and Lemma 1.2(Schur complement), one can easily see that the constraint of Eq. (75) can be transformed into

ΓIN2λIN2>0.E76

Thus, in order to design the proposed decentralized adaptive gain robust controller, the constrained convex optimization problem

Minimizeξii>0,εij>0,σij>0λsubjectto72and76E77

should be solved.

As a result, the following theorem can be obtained:

Theorem 4:Consider the uncertain large-scale interconnected system ofEq. (35) and the control input ofEq. (52).

If positive constantsξii, εij, σij, andλexist which satisfy the constrained convex optimization problem ofEq. (77), the adjustable time-varying parameterθitis designed asEq. (55). Then, the overall uncertain closed-loop system ofEq. (53) is asymptotically stable. Furthermore, by using the optimal solutionξiiandσijforEq. (77), the upper bound of unknown parametersΔiitRni×niandΔijtRni×njis given by

ρii=ξii,ρij=σij.E78

### 3.3. Illustrative examples

To demonstrate the efficiency of the proposed decentralized robust controller, an illustrative example is provided. In this example, we consider the uncertain large-scale interconnected system consisting of three two-dimensional subsystems, that is, N=3. The system parameters are given as follows:

A11=1.01.00.01.0,A22=0.01.01.01.0,A33=1.00.01.03.0,B1=0.01.0,B2=1.01.0,B3=1.00.0,A12=0.50.00.01.0,A13=0.00.50.00.0,A21=0.00.00.00.5,A23=0.00.51.00.0,A31=0.50.00.00.0,A32=0.00.50.00.5.E79

Firstly, by selecting the design parameters αiR1and QiR2×2i=123as α1=α2=α3=1.0and Q1=Q2=Q3=2.0×I2and solving LMIs of Eq. (50), we have the symmetric positive definite matrices YiR2×2and WiR1×1, and positive scalars μijcan be obtained:

Y1=1.89722.19768.1021×101,W1=3.9298,Y2=3.49414.78258.8702×101,W2=2.2200,Y3=4.0414×1013.2732×1023.2709×101,W3=3.2166,μ12=7.0526×101,μ13=4.5522×101,μ21=1.3986,μ23=3.2285×101,μ31=3.4477,μ32=2.0763.E80

Thus, the symmetric positive definite matrices Pi=Yi1and the fixed gain matrices Ki=WiBiTYi1can be calculated as

P1=7.68542.08451.7996,K1=8.19187.0723,P2=1.0923×1015.88914.3025,K2=1.1174×1013.5221,P3=2.49462.4964×1013.0823,K3=8.02408.0297×101.E81

Next, by solving the constrained convex optimization problem of Eq. (77), the following solution can be obtained:

ξ11=3.4167×102,ξ22=3.5524×102,ξ33=1.5590×101,ε12=8.5122×101,ε13=5.9622×101,ε21=1.4174,ε23=3.1440×101,ε31=9.9709,ε32=1.9446,σ12=3.4167×102,σ13=3.4167×102,σ21=3.5524×102,σ23=3.5524×102,σ31=1.5590×101,σ32=1.5590×101,λ=1.0001.E82

Therefore, the allowable upper bound of unknown parameters is given as

ρ11=3.4167×102,ρ22=3.5524×102,ρ33=1.5590×101,ρ12=3.4167×102,ρ13=3.4167×102,ρ21=3.5524×102,ρ23=3.5524×102,ρ31=1.5590×101,ρ32=1.5590×101.E83

In this example, unknown parameters ΔiitR2×2and ΔijtR2×2are chosen as

Δiit=ρii×sin5.0×π×tcos2.0×π×tcos5.0×π×t,Δijt=ρij×cosπ×tsin3.0×π×tsinπ×t.E84

Moreover, the design parameters ϑii=123, the initial value of the uncertain large-scale system with system parameters of Eq. (79), and one of the nominal systems are selected as ϑ1=ϑ2=ϑ3=1.0×101and x0=x¯0=1.51.01.05.0×1012.01.0T.

The result of this example is shown in Figures 5 and 6. In these figures, xilt, uit, x¯ilt, and u¯itdenote the lth element (l=1,2) of the state xitand the control input uitfor ith subsystem and one of the states x¯itand the control input u¯itfor ith nominal subsystem.

From these figures, the proposed decentralized adaptive gain robust controller stabilizes the uncertain large-scale interconnected system with system parameters of Eq. (79). Furthermore, one can see that each subsystem achieves good transient behavior close to nominal subsystems by the proposed decentralized robust controller. Thus, the effectiveness of the proposed robust control strategy is shown.

### 3.4. Summary

In this section, on the basis of the result of Section 2, we have suggested the decentralized adaptive gain robust controller for the large-scale interconnected system with uncertainties. Furthermore, the effectiveness of the proposed controller has been shown via an illustrative example. The proposed adaptive gain robust controller can be easily designed by solving a constrained convex optimization problem and adjust the magnitude of the control input for each subsystem. Therefore, we find that the proposed decentralized robust controller design method is very useful.

Future research subjects include analysis of conservatism for the proposed controller design approach and extension of the proposed adaptive gain robust control strategies to uncertain systems with time delay, decentralized output/observer-based control systems, and so on.

## 4. Conclusions and future works

In this chapter, firstly the centralized adaptive gain robust controller for a class of uncertain linear systems has been proposed, and through a simple numerical example, we have shown the effectiveness/usefulness for the proposed adaptive gain robust control strategy. Next, for a class of uncertain large-scale interconnected systems, we have presented an LMI-based design method of decentralized adaptive gain robust controllers. In the proposed controller robust synthesis, advantages are as follows: the proposed adaptive gain robust controller can achieve satisfactory transient behavior and/or avoid the excessive control input, that is, the proposed robust controller with adjustable time-varying parameters is more flexible and adaptive than the conventional robust controller with a fixed gain which is derived by the worst-case design for the unknown parameter variations. Moreover, in this chapter we have derived the allowable perturbation region of unknown parameters, and the proposed robust controller can be obtained by solving constrained convex optimization problems. Although the solution of the some matrix inequalities can be applied to the resultant robust controller in the general controller design strategies for the conventional fixed gain robust control, the solutions of the constrained convex optimization problem derived in this chapter cannot be reflected to the resultant robust controller. Note that the proposed controller design strategy includes this fascinating fact.

In Section 2 for a class of uncertain linear systems, we have dealt with a design problem of centralized adaptive gain robust state feedback controllers. Although the standard LQ regulator theory for the purpose of generating the desired response is adopted in the existing result [32], the nominal control input is designed by using pole placement constraints. By using the controller gain for the nominal system, the proposed robust control with adjustable time-varying parameter has been designed by solving LMIs. Additionally, based on the derived LMI-based conditions, the constrained convex optimization problem has been obtained for the purpose of the maximization of the allowable perturbation region of uncertainties included in the controlled system. Section 3 extends the result for the centralized adaptive gain robust state feedback controller given in Section 2 to decentralized adaptive gain robust state feedback controllers for a class of uncertain large-scale interconnected systems. In this section, an LMI-based controller synthesis of decentralized adaptive gain robust state feedback control has also been presented. Furthermore, in order to maximize the allowable region of uncertainties, the design problem of the decentralized adaptive gain robust controller for the uncertain large-scale interconnected system has been reduced to the constrained convex optimization problem.

In the future research, an extension of the proposed adaptive gain robust state feedback controller to output feedback control systems or observer-based control ones is considered. Moreover, the problem for the extension to such a broad class of systems as uncertain time-delay systems, uncertain discrete-time systems, and so on should be tackled. Furthermore, we will examine the conservativeness of the proposed adaptive gain robust control strategy and online adjustment way of the design parameter which plays important roles such as avoiding the excessive control input.

On the other hand, it is well known that the design of control systems is often complicated by the presence of physical constraints: temperatures, pressures, saturating actuators, within safety margins, and so on. If such constraints are violated, serious consequences may ensue. For example, physical components will suffer damage from violating some constraints, or saturations for state/input constraints may cause a loss of closed-loop stability. In particular, input saturation is a common feature of control systems, and the stabilization problems of linear systems with control input saturation have been studied (e.g., [33, 40]). Additionally, some researchers have investigated analysis of constrained systems and reference managing for linear systems subject to input and state constraints (e.g., [10, 19]). Therefore, the future research subjects include the constrained robust controller design reducing the effect of unknown parameters.

## Acknowledgments

The authors would like to thank the associate editor for his valuable and helpful comments that greatly contributed to this chapter.

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Written By

Shunya Nagai, Hidetoshi Oya, Tsuyoshi Matsuki and Yoshikatsu Hoshi

Submitted: June 8th, 2017 Reviewed: October 17th, 2017 Published: December 20th, 2017