InTechOpen uses cookies to offer you the best online experience. By continuing to use our site, you agree to our Privacy Policy.

Engineering » Control Engineering » "Adaptive Robust Control Systems", book edited by Le Anh Tuan, ISBN 978-953-51-3797-9, Print ISBN 978-953-51-3796-2, Published: March 7, 2018 under CC BY 3.0 license. © The Author(s).

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

By Shunya Nagai, Hidetoshi Oya, Tsuyoshi Matsuki and Yoshikatsu Hoshi
DOI: 10.5772/intechopen.71733

Article top

Overview

Figure 1. Time histories of the states for ϑ=1.0×102. (a) The time histories of x1(t), (b) The time histories of x2(t).

Figure 2. Time histories of the control inputs and the Lyapunov function for ϑ=1.0×102. (a) The time histories of u(t) and u¯t, (b) The time histories of V(x, t) and Vx¯t.

Figure 3. Time histories of the states for ϑ=5.0×10−1. (a) The time histories of x1(t), (b) The time histories of x2(t).

Figure 4. Time histories of the control inputs and the Lyapunov function for ϑ=5.0×10−1. (a) The time histories of u(t) and u¯t, (b) The time histories of V(x, t) and Vx¯t.

Figure 5. Time histories of xi(t) and x¯it (i = 1, 2). (a) The time histories of x1(t) and x¯1t, (b) Time histories of x2(t) and x¯2t.

Figure 6. Time histories of x3(t), x¯3t, u(t) and u¯t. (a) Time histories of x3(t) and x¯3t, (b) Time histories of u(t) and u¯t.

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

Shunya Nagai1, Hidetoshi Oya2, Tsuyoshi Matsuki3 and Yoshikatsu Hoshi2
Show details

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 A and the inverse of one are denoted by AT and A1, respectively. The notations HeA and diagA1AN represent A+AT and a block diagonal matrix composed of matrices Ai for i=1,,N. The n-dimensional identity matrix and n×m-dimensional zero matrix are described by In and 0n×m, and for real symmetric matrices A and B, A>Bresp.AB means that AB is a positive (resp. nonnegative) definite matrix. For a vector αRn, α denotes standard Euclidian norm, and for a matrix A, A represents 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 vectors λ and ξ and the matrices G and H which have appropriate dimensions, the following relation holds:

2λTGΔtHξ2GTλHξ,

where ΔtRp×q is 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 matrix Ξ, the following arguments are equivalent:

1. Ξ=Ξ11Ξ12Ξ12TΞ22>0.

2. Ξ11>0 and Ξ22Ξ12TΞ111Ξ12>0.

3. Ξ22>0 and Ξ11Ξ12Ξ221Ξ12T>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, (1)

where xtRn and utRm 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 AB is stabilizable pair. Moreover, the matrix ΔtRn×n represents unknown time-varying parameters which satisfy ΔTtΔtδIn, and the elements of ΔtRn×n are Lebesgue measurable [1, 34]. Namely, the unknown time-varying matrix ΔtRn×n is bounded, and the parameter δ denotes 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 Δt in Eq. (1) is given by

 ddtx¯t=Ax¯t+Bu¯t. (2)

In Eq. (2), x¯tRn and u¯tRm are 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, (3)

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

 ddtx¯t=AKx¯t, (4)

where AK is 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×n is 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, (5)

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

 ddtVx¯t<−x¯TtQ+2αPx¯t<0,∀x¯t≠0. (6)

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

 YAT+AY−BWTBT−BWBT+2αY+YQY<0. (7)

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+AY−BWTBT−BWBT+2αYY⋆−Q−1<0. (8)

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

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

 ut=Δ1+θxtKxt, (9)

where θxt:Rn×RR is an adjustable time-varying parameter [32] which plays the important role for correcting the effect of uncertainties, that is, the control input utRm consists of a fixed gain matrix KRm×n and 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. (10)

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 θxtR such 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 θxtR so 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 of Eq. (1) and the adaptive gain robust control of Eq. (9) with the adjustable time-varying parameter θxtR.

For a given design parameter ϑ>0 and the known upper bound δ for the unknown parameter ΔtRn×n, if the scalar parameter γ>0 exists satisfying

 AKTP+PAK+γP2In⋆−γδ⋆In<0, (11)

the adjustable time-varying parameter θxtR is determined as

 θxt=δ⋆PxtxtW1/2BTPxt2ifxTtPBWBTPxt≥ϑxTtxt,δ⋆PxtxtϑxTtxtifxTtPBWBTPxt<ϑxTtxt. (12)

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

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

 Vxt=ΔxTtPxt, (13)

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

 ddtVxt=xTtAKTP+PAKxt+2xTtPΔtxt+2θxtxTtPBKxt. (14)

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

 ddtVxt≤xTtAKTP+PAKxt+2δ⋆Pxtxt+2θxtxTtPBKxt. (15)

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

 ddtVxt≤xTtAKTP+PAKxt+2δ⋆Pxtxt−2θxtxTtPBWBTPxt. (16)

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

 ddtVxt≤xTtAKTP+PAKxt+2δ⋆Pxtxt−2xTtPδ⋆PxtxtW1/2BTPxt2BWBTPxt≤xTtAKTP+PAKxt. (17)

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

 AKTP+PAK<0 (18)

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

 ddtVxt<0,∀xt≠0. (19)

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

 2αTβ≤ζαTα+1ζβTβ, (20)

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

 ddtVxt≤xTtAKTP+PAKxt+γxTtP2xt+1γxTtΔTtΔtxt+2θxtxTtPBKxt≤xTtAKTP+PAK+γP2+δ⋆γInxt+2θxtxTtPBKxt (21)

where γ is 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=WBTP that the last term of the right-hand side of Eq. (21) is nonpositive. Thus, if the scalar parameter γ exists satisfying

 AKTP+PAK+γP2+δ⋆γIn<0, (22)

then the following relation for the quadratic function Vxt holds:

 ddtVxt<0,∀xt≠0. (23)

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 Vxt of 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×n and introduce the additional constraints γ=δ and

 γ−1ε>0, (24)

where ε is a positive constant. From the relation of Eq. (24), we find that the minimization of the parameter ε means 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, (25)

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

 γ1.0⋆ε>0. (26)

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

 Minimizeγ>0,εεsubjectto25and26. (27)

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

 δ⋆=γ⋆. (28)

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

Theorem 2: Consider the uncertain linear system of Eq. (1) and the adaptive gain robust control of Eq. (8) with the adjustable time-varying parameter θxtR.

If the optimal solution γ of the constrained optimization problem of Eq. (27) exists, then the adjustable time-varying parameter θxtR is designed as Eq. (12), and asymptotical stability of the uncertain closed-loop system of Eq. (10) is ensured. Moreover, the upper bound δ for the unknown parameter ΔtRn×n is given by Eq. (28).

Remark 1: In this section, the uncertain linear dynamical system of Eq. (1) is considered, and the centralized adaptive gain robust controller has been proposed. Although the uncertain linear system of Eq. (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: In Theorem 1, the design problem of the proposed adaptive gain robust controller can be reduced to the feasibility of the LMI of Eq. (11). Namely, in order to design the proposed robust control system, designers have to solve the LMI of Eq. (11). If the LMI of Eq. (11) is feasible for δ>0, then one can easily see that the LMI of Eq. (11) is always satisfied for the positive scalar δ<δ. Moreover, if a positive scalar γ exists satisfying the LMI of Eq. (11) for δ+>δ, then the proposed adaptive gain robust controller can also be designed, and note that the adaptive gain robust controller for δ>0 coincides exactly with the one for δ+>δ>0. Furthermore, one can see from Theorem 2 that the resultant adaptive gain robust controller derived by solving the constrained convex optimization problem of Eq. (27) is same, because the solution of LMI of Eq. (8) or one of the constrained convex optimization problem of Eq. (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 parameter ϑ has 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 gain KRm×n can be derived by using the solution of the LMI of Eq. (8). Note that the quadratic function Vxt is a Lyapunov function for both the uncertain linear system of Eq. (1) and the nominal system of Eq. (2), that is, the Lyapunov function for the uncertain linear system of Eq. (1) and one for the nominal system of Eq. (2) have same level set. Therefore, by selecting the design parameter ϑ>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 control ut=Ksxt is considered, the quadratic function Vxt is replaced as Vsxt=xTtPsxt where PsRn×n is a Lyapunov matrix. Moreover, PsRn×n becomes 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=1.04.00.0−1.0xt+Δtxt+0.01.0ut. (29)

Firstly, we design the nominal control input u¯t=Kx¯t. By selecting the design parameters α and Q in Eq. (5) such as α=3.0 and Q=1.0×I2 and solving the LMI of Eq. (8), we obtain the following solution:

 Y=1.0855−1.5356⋆4.5318,W=2.1708×101. (30)

Thus, the following fixed gain matrix can be computed:

 K=−1.3017×101−9.2008. (31)

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

 γ=3.1612,ε=3.1633×10−1, (32)

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

 δ⋆=3.1612. (33)

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=1.0×102 and ϑ=5.0×101 is 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 x1t and x2t and the control input ut and Lyapunov function Vx.t for the proposed adaptive gain robust control, and “nominal” means the desired time response and the desired control input and Lyapunov function Vx¯t for the nominal system. In Table 1, Je means

 Je=Δ∫0∞eTtetdt, (34)

Figure 1.

Time histories of the states for ϑ=1.0×102. (a) The time histories of x1(t), (b) The time histories of x2(t).

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.

Figure 2.

Time histories of the control inputs and the Lyapunov function for ϑ=1.0×102. (a) The time histories of u(t) and u¯t, (b) The time histories of V(x, t) and Vx¯t.

Figure 3.

Time histories of the states for ϑ=5.0×101. (a) The time histories of x1(t), (b) The time histories of x2(t).

Figure 4.

Time histories of the control inputs and the Lyapunov function for ϑ=5.0×101. (a) The time histories of u(t) and u¯t, (b) The time histories of V(x, t) and Vx¯t.

Figure 5.

Time histories of xi(t) and x¯it (i = 1, 2). (a) The time histories of x1(t) and x¯1t, (b) Time histories of x2(t) and x¯2t.

Figure 6.

Time histories of x3(t), x¯3t, u(t) and u¯t. (a) Time histories of x3(t) and x¯3t, (b) Time histories of u(t) and u¯t.

where et is an error vector between the time response and the desired one generated by the nominal system, that is, et=Δxtx¯t. Namely, Je of 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=5.0×101 is 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=1.0×102 achieves more desirable performance. Additionally, one can see from Figures 2(a) and 4(a) that by selecting the design parameter ϑ the proposed adaptive gain robust controller can adjust the magnitude of the control input. In this example, the magnitude of the control input for ϑ=1.0×102 is suppressed comparing with one for ϑ=5.0×101. However, the online adjustment way of the design parameter ϑ for 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 ϑ and 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 N subsystems described as

 ddtxi(t)=Aii(t)xi(t)+∑j=1j≠iNAij(t)xj(t)+Biui(t), (35)

where xitRni and uitRmi (i=1,,N) are the vectors of the state and the control input for the ith subsystem, respectively, and xt=x1TtxNTtT is the state of the overall system. The matrices AiitRni×ni and AijtRni×nj in Eq. (35) are given by

 Aiit=Aii+Δiit,Aijt=Aij+Δijt. (36)

In Eqs. (35) and (36), the matrices AiiRni×ni, AijRni×nj, and BiRni×mi denote the nominal values of the system, and matrices ΔiitRni×ni and ΔijtRni×nj show unknown parameters which satisfy ΔiiTtΔiitρiiIni and 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=1j≠iNAijx¯j(t)+Biu¯i(t), (37)

where x¯itRni and u¯itRmi are 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, (38)

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

 ddtx¯i(t)=AKix¯i(t)+∑j=1j≠iNAijx¯j(t), (39)

where AKi=ΔAiiBiKi.

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

 Vx¯t=Δ∑i=1NVix¯it, (40)
 Vix¯it=Δx¯iTtPix¯it, (41)

as a Lyapunov function candidate. For the quadratic function Vix¯it of 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=1j≠iN2x¯iT(t)PiAijx¯j(t). (42)

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=1j≠iNμijx¯iT(t)PiAijAijTPx¯i(t)+∑j=1j≠iN1μijx¯jT(t)x¯j(t). (43)

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

 ddtV(x¯,t)≤∑i=1 Nx¯iT(t)(AKiTPi+PiAKi)x¯i(t)+∑i=1N∑j=1j≠iNμix¯iT(t)PiAijAijTPix¯i(t)+∑i=1N∑j=1j≠iN1μijx¯jT(t)x¯j(t). (44)

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

 ddtV(x¯,t)≤∑i=1Nx¯iT(t)(AKiTPi+PiAKi+∑j=1j≠iNμiPiAijAijTPi+∑j=1j≠iN1μjiIn)x¯i(t). (45)

Therefore, if the matrix inequality

 AKiTPi+PiAKi+∑j=1j≠iNμiPiAijAijTPi+∑j=1j≠iN1μjiIn<0 (46)

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

 ddtVx¯t<0,∀x¯t≠0. (47)

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=1j≠iNμiPiAijAijTPi+∑j=1j≠iN1μjiIn+Qi<0, (48)

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

We introduce symmetric positive definite matrices Yi=ΔPi1 and WiRmi×mi and define the fixed gain Ki as 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

 AiiYi−BiWiBiT+YiAiiT−BiWiTBiT+2αiYi+∑j=1j≠iNμijAijAijT+∑j=1j≠iN1μjiYiYi+YiQiYi<0. (49)

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:

 ΛiYiWiμijΘiYi⋆−Γiμij<0. (50)

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

 Λi(Yi,,Wi,,μij)=ΔAiiYi−BiWiBiT+YiAiiT−BiWiTBiT+2αiYi+∑j=1j≠iNμijAijAijT,Θi(Yi)=Δ(Yi Yi … Yi︷N),Γi(μij)=Δ diag(Qi−1,μ1iIn,μ2iIn,⋯,μi−1iIn,μi+1iIn,⋯,μNiIn). (51)

Therefore, if matrices YiRni×ni and WiRmi×mi and positive scalars μij exist, the nominal closed-loop subsystem is asymptotically stable, and the fixed gain matrix Ki is determined as Ki=WiBiTYi1.

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

 uit=Δ−1+θitKixit, (52)

where θitR1 is an adjustable time-varying parameter. From Eqs. (35) and (52), the uncertain closed-loop subsystem can be obtained as

 ddtxit=AKixit+Δiitxit+∑i=1NAij+Δijtxjt−θitBiKixit. (53)

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 θitR1 such that asymptotical stability of the overall system composed of N subsystems 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 of Eq. (35) and the control input of Eq. (52).

For a given positive constant ϑi, if positive constants ξii, σij, and εij exist which satisfy the LMIs

 ΠiξiiεijσijΞi⋆−Ωiξiiεijσij<0, (54)

the time-varying adjustable parameters θitR are determined as

 θit=Δρii⋆∥Pixit∥∥xit∥xiTtPiBiWiBiTPixitifxiTtPiBiWiBiTPixit≥ϑixiTtxit,ρii⋆∥Pixit∥∥xit∥ϑixiTtxitifxiTtPiBiWiBiTPixit<ϑixiTtxit, (55)

where matrices ΠiξiεijσijRni×ni, ΞiRni×2N1ni, and ΩiξiiεijσijR2N1ni×2N1ni are given by

 Πi(ξii,,εij,,σij)=Δ(AKiTPi+PiAKi)+ξiiPiPi+∑j=1j≠iNεijPiAijAijTPi+∑j=1j≠iNσijPiPi,Ξi=ΔInIn⋯In︷2N−1,Ωiξiiεijσij=Δdiagξiiρii⋆Inε1iInε2iIn⋯εi−1iInεi+1iIn⋯εNiInσ1iρ1i⋆Inσ2iρ2i⋆In⋯σi−1iρi−1i⋆Inσi+1iρi+1i⋆In⋯σNiρNi⋆In. (56)

Then, the overall close-loop system composed of N closed-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×ni which satisfy the LMIs of (50):

 Vxt=Δ∑i=1NVixit, (57)

where Vixit is a quadratic function given by

 Vixit=ΔxiTtPixit. (58)

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

 ddtVi(xi,t)=xiT(t)(AKiTPi+PiAKi)xi(t)+2xi(t)PiΔii(t)xi(t)+2xiT(t)Pi∑j=1j≠iN(Aij+Δij(t))xj(t)−2θi(t)xiT(t)PiBiKixi(t). (59)

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

 ddtVixit≤xiTtAKiTPi+PiAKixit+2ρii⋆Pixitxit+∑j=1j≠iNεijxiT(t)PiAijAijTPixi(t)+∑j=1j≠iN1εijxjT(t)xj(t)+∑j=1j≠iNσijxiT(t)PiPixi(t)+∑j=1j≠iNρij⋆σijxjT(t)xj(t)−2θi(t)xiT(t)PiBiKixi(t). (60)

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

 ddtVixit≤xiTtAKiTPi+PiAKixit+2ρii⋆Pixitxit+∑j=1j≠iNεijxiT(t)PiAijAijTPixi(t)+∑j=1j≠iN1εijxjT(t)xj(t)+∑j=1j≠iNσijxiT(t)PiPixi(t)+∑j=1j≠iNρij⋆σijxjT(t)xj(t)−2(ρii⋆∥Pixi(t)∥∥xi(t)∥xiT(t)PiBiWiBiTPixi(t))xiT(t)PiBiKixi(t)=xiT(t)(AKiTPi+PiAKi)xi(t)+∑j=1j≠iNεijxiT(t)PiAijAijTPixi(t)+∑j=1j≠iN1εijxjT(t)xj(t)+∑j=1j≠iNσijxiT(t)PiPixi(t)+∑j=1j≠iNρij⋆σijxjT(t)xj(t), (61)

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

 ddtV(x,t)≤∑i=1NxiT(t)(AKiTPi+PiAKi)xi(t)+∑i=1N∑j=1j≠iNεijxi(t)PiAijAijTPixi(t)+∑i=1N∑j=1j≠iN1εijxjT(t)xj(t)+∑i=1N∑j=1j≠iNσijxiT(t)PiPixi(t)+∑i=1N∑j=1j≠iNρij⋆σijxjT(t)xj(t). (62)

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

 ddtV(x,t)≤ ∑i=1NxiT(t)(AKiTPi+PiAKi+∑j=1j≠iNεijPiAijAijTPi+∑j=1j≠iN1εjiIn +∑j=1j≠iNσijPiPi+∑j=1j≠iNρji⋆σjiIn)xi(t). (63)

Therefore, if the matrix inequality

 AKiTPi+PiAKi+∑j=1j≠iNεijPiAijAijTPi+∑j=1j≠iN1εjiIn+∑j=1j≠iNσijPiPi+∑j=1j≠iNρij⋆σjiIn<0 (64)

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

 ddtVxt<0,∀xt≠0. (65)

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

 ddtVixit≤xiTtAKiTPi+PiAKixit+ξiixiTtPiPixit+ρii⋆ξiixiTtxit+∑j=1j≠iNεijxi(t)PiAijAijTPixi(t)+∑j=1j≠iN1εijxjT(t)xj(t)+∑j=1j≠iNσijxiT(t)PiPixi(t)+∑j=1j≠iNρij⋆σijxjT(t)xj(t)−2(ρii⋆∥Pixi(t)∥∥xi(t)∥ϑixiT(t)xi(t))xiT(t)PiBiKixi(t). (66)

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

 ddtVixit≤xiTtAKiTPi+PiAKixit+ξiixiTtPiPixit+ρii⋆ξiixiTtxit+∑j=1j≠iNεijxi(t)PiAijAijTPixi(t)+∑j=1j≠iN1εijxjT(t)xj(t)+∑j=1j≠iNσijxiT(t)PiPixi(t)+∑j=1j≠iNρij⋆σijxjT(t)xj(t). (67)

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

 ddtVxt≤∑i=1NxiTtAKiTPi+PiAKixit+∑i=1NξiixiTtPiPixit+∑i=1Nρii⋆ξiixiTtxit+∑i=1N∑j=1j≠iNεijxi(t)PiAijAijTPixi(t)+∑i=1N∑j=1j≠iN1εijxjT(t)xj(t)+∑i=1N∑j=1j≠iNσijxiT(t)PiPixi(t)+∑i=1N∑j=1j≠iNρij⋆σijxjT(t)xj(t) (68)

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+ρii⋆ξiiIn+∑j=1j≠iNεijPiAijAijTPi+∑j=1j≠iN1εjiIn+∑j=1j≠iNσijPiPi+∑j=1j≠iNρji⋆σjiIn)xi(t). (69)

Therefore, if the matrix inequality

 AKiTPi+PiAKi+ξiiPiPi+ρii⋆ξiiIn+∑j=1j≠iNεijPiAijAijTPi+∑j=1j≠iN1εjiIn+∑j=1j≠iNσijPiPi+∑j=1j≠iNρji⋆σjiIn<0 (70)

holds, then the relation of Eq. (65) for the time derivative of the function Vxt of 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×ni and ΔijtRni×nj. Thus, the following additional constraints are introduced:

 ρii⋆=ξii,ρij⋆=σij. (71)

From the relations of Eq. (71), one can find that the maximization of ξii and σij is equivalent to the maximization of ρii and ρij. Then, the LMIs of Eq. (54) can be rewritten as

 Πi′ξiiεijσijΞi′⋆−Ωi′εij<0, (72)
 Πi′(ξii,,εij,,σij)=ΔAKiTPi+PiAKi+ξiiPiPi+NIn+∑j=1j≠iNεijPiAijAijTPi+∑j=1j≠iNσijPiPi,Ξi′=ΔInIn⋯In︷N−1,Ωi′εij=Δdiagε1iInε2iIn⋯εi−1iInεi+1iIn⋯εNiIn. (73)

Furthermore, we introduce a positive scalar λ and a complementary matrix ΓRN2×N2 defined as

 Γ=Δdiagξ11ξ22⋯ξNNσ12σ13⋯σ1Nσ21σ23⋯σNN−1, (74)

and consider the following additional condition:

 Γ−1λIN2>0. (75)

Namely, we can replace the maximization problem of ξii and σij with 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

 ΓIN2⋆λIN2>0. (76)

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

 Minimizeξii>0,εij>0,σij>0λsubjectto72and76 (77)

should be solved.

As a result, the following theorem can be obtained:

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

If positive constants ξii, εij, σij, and λ exist which satisfy the constrained convex optimization problem of Eq. (77), the adjustable time-varying parameter θit is designed as Eq. (55). Then, the overall uncertain closed-loop system of Eq. (53) is asymptotically stable. Furthermore, by using the optimal solution ξii and σij for Eq. (77), the upper bound of unknown parameters ΔiitRni×ni and ΔijtRni×nj is given by

 ρii⋆=ξii⋆,ρij⋆=σij⋆. (78)

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=0.01.0−1.0−1.0,A33=1.00.01.0−3.0,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. (79)

Firstly, by selecting the design parameters αiR1 and QiR2×2i=123 as α1=α2=α3=1.0 and Q1=Q2=Q3=2.0×I2 and solving LMIs of Eq. (50), we have the symmetric positive definite matrices YiR2×2 and WiR1×1, and positive scalars μij can be obtained:

 Y1=1.8972−2.1976⋆8.1021×10−1,W1=3.9298,Y2=3.49414.7825⋆8.8702×10−1,W2=2.2200,Y3=4.0414×10−13.2732×10−2⋆3.2709×10−1,W3=3.2166,μ12=7.0526×10−1,μ13=4.5522×10−1,μ21=1.3986,μ23=3.2285×10−1,μ31=3.4477,μ32=2.0763. (80)

Thus, the symmetric positive definite matrices Pi=Yi1 and the fixed gain matrices Ki=WiBiTYi1 can be calculated as

 P1=7.68542.0845⋆1.7996,K1=8.19187.0723,P2=1.0923×101−5.8891⋆4.3025,K2=1.1174×101−3.5221,P3=2.4946−2.4964×10−1⋆3.0823,K3=8.0240−8.0297×10−1. (81)

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

 ξ11=3.4167×10−2,ξ22=3.5524×10−2,ξ33=1.5590×10−1,ε12=8.5122×10−1,ε13=5.9622×10−1,ε21=1.4174,ε23=3.1440×10−1,ε31=9.9709,ε32=1.9446,σ12=3.4167×10−2,σ13=3.4167×10−2,σ21=3.5524×10−2,σ23=3.5524×10−2,σ31=1.5590×10−1,σ32=1.5590×10−1,λ=1.0001. (82)

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

 ρ11⋆=3.4167×10−2,ρ22⋆=3.5524×10−2,ρ33⋆=1.5590×10−1,ρ12⋆=3.4167×10−2,ρ13⋆=3.4167×10−2,ρ21⋆=3.5524×10−2,ρ23⋆=3.5524×10−2,ρ31⋆=1.5590×10−1,ρ32⋆=1.5590×10−1. (83)

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

 Δiit=ρii⋆×sin5.0×π×t−cos2.0×π×t⋆cos5.0×π×t,Δijt=ρij⋆×−cosπ×tsin3.0×π×t⋆sinπ×t. (84)

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=1.0×101 and 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¯it denote the lth element (l=1,2) of the state xit and the control input uit for ith subsystem and one of the states x¯it and the control input u¯it for 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.

Acknowledgements

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

References

1 - Bermish BR, Corless M, Leitmann G. A new class of stabilizing controllers for uncertain dynamical systems. SIAM Journal on Control and Optimization. 1983;21(2):246-255
2 - Boyd S, El Ghaoui L, Feron E, Balakrishnan V. Linear matrix inequalities in system and control theory. SIAM Studies in Applied Mathmatics. 1994
3 - Chen YH. Decentralized robust control system Design for Large-Scale Uncertain Systems. International Journal of Control. 1988;47(5):1195-1205
4 - Choi YK, Chung MJ, Bien Z. An adaptive control scheme for robot manipulators. International Journal of Control. 1986;44(4):1185-1191
5 - Davison EJ, Tripathi NK. The optimal decentralized control of a large power system: Load and frequency control. IEEE Transactions on Automatic Control. 1978;23(2):312-325
6 - Doyle JC, Glover K, Khargonekar PP, Francis BA. State-space solutions to standard H2 and H control problems. IEEE Transactions on Automatic Control. 1989;34(8):831-847
7 - Faub PL, Wolovich WA. Decoupling in the design and synthesis of multivariable control systems. IEEE Transactions on Automatic Control. 1967;12(6):651-659
8 - Chilali M, Gahinet P, Apkarian P. Robust pole placement in LMI regions. IEEE Transactions on Automatic Control. 1999;44(12):2257-2270
9 - Gantmacher FR. The Theory of Matrices. Vol. 1. New York: Chelsea Publishing Company; 1960
10 - Gilbert EG, Kolmanovsky I. Nonlinear tracking control in the presence of state and control constraints: A generalized reference governor. Automatica. 2002;38(12):2071-2077
11 - Gong Z. Decentralized robust control of uncertain interconnected systems with prescribed degree of exponential convergence. IEEE Transactions on Automatic Control. 1995;40(4):704-707
12 - Goodwin GC, Graebe SF, Saogado ME. Control System Design. NJ, USA: Prentice-Hall Inc.; 2001
13 - Hua C, Guan X, Shi P. Decentralized robust model reference adaptive control for interconnected time-delay systems. Journal of Computational and Applied Mathematics. 2006;193:383-396
14 - Khalil HK. Nonlinear Systems. 3rd ed. New Jersey: Prentice-Hall Inc.; 2002
15 - Kalman RE. Contributions to the theory of optimal control. Boletín de la Sociedad Matemática Mexicana. 1960;(5):102-199
16 - Kalman RE. When is a linear control system optimal? Transactions of ASME Series D, Journal of Basic Engineering. 1964;86(1):51-60
17 - Kalman RE, Bucy RS. A new approach to linear filtering and prediction problems. Transactions of ASME Series D, Journal of Basic Engineering. 1960;82(1):35-48
18 - Khargonekar PP, Rotea MA. Mixed H2/H Control, A Convex Optimization Approach. IEEE Transactions on Automatic Control. 1991;36(7):824-837
19 - Kogiso K, Hirata K. Reference governor for constrained systems with time-varying references. Journal of Robotics and Autonomous System. 2009;57(3):289-295
20 - Krishnamurthi V. Correlation between Routh’s stability criterion and relative stability of linear systems. IEEE Transactions on Automatic Control. 1972;17(1):144-145
21 - Kuo BC. Automatic Control Systems. 4th ed. New Jersey: Prentice-Hall Inc.; 1982
22 - Leith DJ, Leithead WE. Survey of gain-scheduling analysis and design. International Journal of Control. 2010;73(11):1001-1025
23 - Luo JS, Johnson A, Van Den Bosch PPJ. Minimax guaranteed cost control for linear continuous-time systems with large parameter uncertainty. Automatica. 1994;30(4):719-722
24 - Mao CJ, Lin WS. Decentralized control of interconnected systems with unmodelled nonlinearity and interaction. Automatica. 1990;26(2):263-268
25 - Maki M, Hagino K. Robust control with adaptation mechanism for improving transient behaviour. International Journal of Control. 1999;72(13):1218-1226
26 - Mukaidani H, Takato Y, Tanaka Y, Mizukami K. The guaranteed cost control for uncertain large-scale interconnected systems. The Preprints of the 15th IFAC World Congress, Barcelona, SPAIN; 2002
27 - Mukaidani H, Kimoto M, Yamamoto T. Decentralized guaranteed cost control for discrete-time uncertain large-scale systems using fuzzy control. The Proceedings of 2006 IEEE World Congress on Computational Intelligence,Vancouver, CANADA; 2006. pp. 3099-3105
28 - Nyquist H. Regeneration theory. Bell System Technical Journal. 1932;11(1):126-147
29 - Oya H, Hagino K. Robust control with adaptive compensation input for linear uncertain systems. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences. 2003;E86-A(6):1517-1524
30 - Oya H, Hagino K. Adaptive robust control scheme for linear systems with structured uncertainties. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences. 2004;E87-A(8):2168-2173
31 - Oya H, Hagino K. A new adaptive robust controller avoiding chattering phenomenon for a class of uncertain linear systems. Proceedings of the 28th IASTED International Conference on Modeling, Identification and Control. Innsbruck; 2009. pp. 236-241
32 - Oya H, Uehara Y. Synthesis of variable gain controllers based on LQ optimal control for a class of uncertain linear systems. Proceedings of the UKACC International Conference on CONTROL 2012, Cardiff; 2012. pp. 87-91
33 - Oba K, Oya H, Kubo T, Matsuki T. Synthesis of adaptive robust controllers giving consideration to input saturations for a class of nonlinear systems. Mathematical Problem in Engineering. 2017;2017. Article ID:5345812, 9 pages
34 - Petersen IR. A Riccati equation approach to the design of stabilizing controllers and observers for a class of uncertain linear systems. IEEE Transactions on Automatic Control. 1985;30(9):904-907
35 - Petersen IR, McFarlane DC. Optimal guaranteed cost control and filtering for uncertain linear systems. IEEE Transactions on Automatic Control. 1994;39(9):1971-1977
36 - Petersen IR, Hollot CC. A Riccati equation approach to the stabilization of uncertain linear systems. Automatica. 1986;22(4):397-411
37 - Reza Moheimani SO, Petersen IR. Optimal guaranteed cost control of uncertain systems via static and dynamic output feedback. Automatica. 1996;32(4):575-579
38 - Šijjak DD. Decentralized Control of Complex Systems. New York: Academic Press; 1991
39 - Stoorvogel AA. The robust H2 control problem, a worst-case design. IEEE Transactions on Automatic Control. 1993;38(9):1358-1370
40 - Turner MC, Postlethwaite I, Walker DJ. Non-linear tracking control for multivariable constrained input linear systems. International Journal of Control. 2000;73(12):1160-1172
41 - Ushida S, Yamamoto S, Kimura H. Quadratic stabilization by H state feedback controllers with adjustable parameters. Proceedings of the 35th IEEE Conference Decision and Control. Kobe, JAPAN, 1996. pp. 1003-1008
42 - Wei G, Wang Z, Li W, Ma L. A survey on gain-scheduled control and filtering for parameter-varying systems. Mathematical Problem in Eng. 2014;2014. Article ID:105815, 10 pages
43 - Wonham WM. On pole assignment in multi-input controllable linear systems. IEEE Transactions on Automatic Control. 1967;12(6):660-665
44 - Xu D, Jiang B, Liu H, Shi P. Decentralized asymptotic fault tolerant control of near space vehicle with high order actuator dynamics. Journal of Franklin Institute. 2013;350:2519-2534
45 - Zhou K, Khargonekar PP. Robust stabilization of linear systems with norm-bounded time-varying uncertainty. Systems & Control Letters. 1988;10(1):17-20
46 - Zhou K, Khargonekar PP, Stoustrup J, Niemann HH. Robust performance of systems with structured uncertainties in state space. Automatica. 1995;31(2):249-255
47 - Zhou K, Doyle JC, Glover K. Robust and Optimal Control. New Jersey: Prentice-Hall Inc.; 1996
48 - Zhou K. Essentials of Robust Control. New Jersey, USA: Prentice Hall Inc.; 1998