Synthesis of Decentralized Variable Gain Robust Controllers with Guaranteed <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.199599pt" id="M1" height="16.0985pt" version="1.1" viewBox="-0.0657574 -11.8989 22.9551 16.0985" width="22.9551pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g198-13" d="M608 643C551 682 486 687 443 687C299 687 188 608 188 504C188 361 341 312 465 312C483 312 501 313 518 315C454 195 366 93 314 59C255 82 201 109 141 109C73 109 33 83 33 52C33 13 76 -9 159 -9C218 -9 267 -1 314 15C354 0 402 -15 478 -15C575 -15 671 36 759 141L740 155C657 63 571 16 475 16C448 16 411 26 372 39C458 74 565 199 627 333C806 381 931 499 931 587C931 634 908 669 848 669C767 669 671 582 620 506C591 462 569 411 534 347C503 343 490 342 471 342C330 342 229 399 229 500C229 603 346 657 451 657C486 657 540 657 596 624L608 643ZM890 588C890 519 822 457 747 414C712 395 678 378 643 369C683 484 760 632 841 632C873 632 890 615 890 588ZM266 35C235 24 200 21 161 21C103 21 74 32 74 54C74 75 122 79 141 79C172 79 209 69 266 36V35Z"/></g><g transform="matrix(.012,0,0,-0.012,16.233,4.134)"><path id="g50-51" d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"/></g></svg> Gain Performance for a Class of Uncertain Large-Scale Interconnected Systems

Nagai, Shunya; Oya, Hidetoshi

doi:https://doi.org/10.1155/2015/342867

Journal of Control Science and Engineering

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Abstract Introduction Preliminaries Conclusions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2015 | Article ID 342867 | https://doi.org/10.1155/2015/342867

Synthesis of Decentralized Variable Gain Robust Controllers with Guaranteed Gain Performance for a Class of Uncertain Large-Scale Interconnected Systems

Shunya Nagai¹and Hidetoshi Oya²

Academic Editor: Shengwei Mei

Received02 Dec 2014

Accepted13 May 2015

Published20 Dec 2015

Abstract

We consider a design problem of a decentralized variable gain robust controller with guaranteed gain performance for a class of uncertain large-scale interconnected systems. For the uncertain large-scale interconnected system, the uncertainties and the interactions satisfy the matching condition. In this paper, we show that sufficient conditions for the existence of the proposed decentralized variable gain robust controller with guaranteed gain performance are given in terms of linear matrix inequalities (LMIs). Finally, simple illustrative examples are shown.

1. Introduction

Due to the rapid development of industry in recent years, the controlled systems become more complex and such complex systems should be considered as large-scale interconnected systems (e.g., traffic systems and electric systems). As is well known, it is difficult to apply centralized control strategy to such systems because of physical constraints, calculation amount, and so on. Therefore, decentralized control problems for large-scale interconnected systems have been widely studied (see [1, 2] and references therein for details). The major problem of large-scale interconnected systems is how to deal with the interactions among subsystems.

On the other hand, robust control problem is one of the most important topics, and there is a large number of results for robust controller design and robust stability analysis (e.g., see [3, 4] and references therein). In particular, for uncertain linear systems, several quadratic stabilizing controllers have been suggested [5, 6] and the so-called robust control problem has also been considered [7]. In most of the existing results, the proposed robust control systems have fixed gain controllers which are derived by considering worst case variations of uncertainties. In contrast with these, several design methods of variable gain controllers for uncertain continuous-time systems have been shown (e.g., [8–11]). These robust controllers are composed of a fixed gain controller and a variable gain one which are tuned by updating laws. In particular, in Oya and Hagino [11], the variable gain robust output feedback controller which achieves not only robust stability but also a specified gain performance for a class of Lipschitz uncertain nonlinear systems has been proposed.

For large-scale interconnected systems with uncertainties, there are many existing results for decentralized robust control (e.g., [12–17]). In the work of Mao and Lin [12], for large-scale interconnected systems with unmodelled interaction, the aggregative derivation is tracked by using a model following technique with on-line improvement, and a sufficient condition for which the overall system when controlled by the completely decentralized control is asymptotically stable has been established. Furthermore, Gong [14] has proposed a decentralized robust controller which guarantees robust stability with prescribed degree of exponential convergence. Mukaidani et al. [15, 16] have also proposed decentralized guaranteed cost controllers for uncertain large-scale interconnected systems. In addition, we have suggested a decentralized variable gain robust controller which achieves not only robust stability but also satisfactory transient behavior for a class of uncertain large-scale interconnected systems [18, 19].

In this paper, on the basis of the existing results [11, 18, 19], we propose a decentralized variable gain robust controller with guaranteed gain performance for a class of uncertain large-scale interconnected systems; that is, this study is an extension of our previous studies in this field. For the uncertain large-scale interconnected systems, uncertainties and interactions satisfy the matching condition. The proposed decentralized robust controller consists of a fixed gain matrix and a variable one determined by a parameter adjustment law. In this paper, LMI-based sufficient conditions for the existence of the proposed decentralized variable gain robust controller are derived. To put it in the concrete, the decentralized variable gain robust controller, that is, parameter adjustment law, is designed so that the effects of uncertainties and interactions are reduced.

This paper is organized as follows. In Section 2, we show the notations and useful lemmas used in this paper. In Section 3, the class of uncertain large-scale interconnected systems under consideration is presented. Section 4 contains the main results; that is, LMI-based sufficient conditions for the existence of the proposed decentralized variable gain robust controller with guaranteed gain performance are derived. Finally, simple illustrative examples are included to show the effectiveness of the proposed decentralized robust controller.

2. Preliminaries

In this section, notations and useful and well-known lemmas (see [11, 20, 21] for details) which are used in this paper are shown.

In this paper, the following notations are adopted. For a matrix , the inverse of matrix and the transpose of one are denoted by and , respectively. Additionally, and mean and -dimensional identity matrix, respectively, and the notation represents a block diagonal matrix composed of matrices for . For real symmetric matrices and , , means that is positive (resp., nonnegative) definite matrix. For a vector , denotes standard Euclidian norm and, for a matrix , represents its induced norm. The symbols “” and “” mean equality by definition and symmetric blocks in matrix inequalities, respectively. Besides, is -space (i.e., the collection of all square integrable functions) defined on , and, for a signal , denotes its -norm.

Lemma 1. For arbitrary vectors and and the matrices and which have appropriate dimensions, the following inequality holds:where with appropriate dimension is a time-varying unknown matrix satisfying .

Lemma 2 (Schur complement). For a given constant real symmetric matrix , the following items are equivalent: (1);(2) and ;(3) and .

3. Problem Formulation

Let us consider the uncertain large-scale interconnected system composed of subsystems represented bywhere , , , and () are the vectors of the state, the control input, the controlled output, and the disturbance input for the th subsystem, respectively. Besides, the disturbance input is assumed to be square integrable; that is, . The matrices and in (2) are given bythat is, the uncertainties and the interaction terms satisfy the matching condition [18, 19]. In (2) and (3), the matrices , , , , , and are known system parameters and the matrices , , and with appropriate dimensions represent the structure of interactions or uncertainties. Besides, the matrices and denote unknown time-varying parameters satisfying the relations and , respectively.

Since the th subsystem is given by (2), we find that the overall system can be written as where , , , and are the state, the control input, the controlled output, and the disturbance input of the overall system. Besides, the matrices , , , , and are given bywhere , , , and are given by , , , and , respectively.

Now for the th subsystem of (2), we define the following control input [18, 19]:In (6), and are the fixed compensation gain matrix and the variable one for the th subsystem of (2). From (2), (3), and (6), the following closed-loop subsystem can be derived:

Now we will give a definition of the decentralized variable gain robust control with guaranteed gain performance.

Definition 3. For the uncertain large-scale interconnected system of (2), the control input of (6) is said to be a decentralized variable gain robust control with guaranteed gain performance if the resultant closed-loop system of (7) is internally stable, and -norm of the transfer function from the disturbance input to the controlled output is less than or equal to a positive constant .

By using symmetric positive definite matrices , we consider the following quadratic function: where is Besides, we introduce the following Hamiltonian:Then we have the following lemma for the decentralized variable gain robust control with guaranteed gain performance for the uncertain large-scale interconnected system of (2) and the control input of (6).

Lemma 4. Consider the uncertain large-scale interconnected system of (2) and the control input of (6).
For the quadratic function and the signals and , if there exist symmetric positive definite matrices and positive scalars which satisfy the inequality then the control input of (6) is a decentralized variable gain robust control with guaranteed gain performance , where is given by

Proof. By integrating both sides of the inequality of (11) from to with , we obtain the following inequality from . Consider We see from the inequality of (13) that the uncertain closed-loop subsystem of (7) is robustly stable (internally stable). Namely, robust stability of the uncertain closed-loop subsystem with is guaranteed and the -norm of the transfer function from the disturbance input to the controlled output is less than a positive constant because the inequality of (13) means the following relation:Thus the proof of Lemma 4 is accomplished.

From the above discussion, our design objective in this paper is to determine the decentralized variable gain robust control input of (6) such that the overall system achieves not only internal stability but also guaranteed gain performance . That is to derive the symmetric positive definite matrices , positive constants , the fixed gain matrices , and the variable one satisfying the inequality of (11) for all admissible uncertainties and and the disturbance input .

4. Decentralized Variable Gain Controllers

In this section, we show a design method of the decentralized variable gain robust controller with guaranteed gain performance.

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

Theorem 5. Let one consider the large-scale interconnected system of (2) and the control input of (6).

By using symmetric positive definite matrices , the matrices , and positive scalars and which satisfy the LMIs, the fixed gain matrix and the variable one are determined asIn (15) and (16), matrices and and scalar functions and are given byNote that in (16) is given by [8].

Then the control input of (6) is the decentralized variable gain robust control with guaranteed gain performance .

Proof. In order to prove Theorem 5, we consider the quadratic function of (8), the Hamiltonian of (10), and the inequality of (11).
For the quadratic function of (9), its time derivative along the trajectory of the resultant closed-loop subsystem of (7) can be computed as Besides, by using Lemma 1 and the well-known inequality for any vectors and with appropriate dimensions and a positive scalar , we have the following relation for the function :Firstly, we consider the case of . In this case, substituting the variable gain matrix of (16) into (20) and some algebraic manipulations derive the following inequality:Additionally, one can see from (2) that the following relation holds:where . Therefore, from (8), (10), (21), and (22), we can obtain the following relation for the Hamiltonian :The inequality of (23) can also be rewritten as Furthermore, some algebraic manipulations for (24) give the following inequality: where is given by Hence, if the matrix inequality holds, then the inequality of (11) for the Hamiltonian is satisfied.
Next we consider the case of . In this case, one can see from (18) and (22) and the definition of the control input of (6) and the variable gain matrix (16) that if the matrix inequality of (27) holds, then the inequality of (11) is also satisfied.
Finally, we consider the matrix inequality of (27). By introducing the matrices and and pre- and postmultiplying both sides of the matrix inequality of (27) by , we have the following inequality: where is given by Thus by applying Lemma 2 (Schur complement) to (28) we find that the matrix inequalities of (28) are equivalent to the LMIs of (15). In the LMIs of (15), scalar variables and can arbitrarily be selected. Therefore, we find that the LMIs of (15) are always feasible; that is, there always exists the solution of the LMIs of (15). Namely, by solving the LMIs of (15), the fixed gain matrix is determined as and the variable one is given by (16) and the proposed control input of (6) becomes a decentralized variable gain robust control with guaranteed gain performance of (12). Therefore, the proof of Theorem 5 is accomplished.

Remark 6. Although the uncertainties included in the large-scale interconnected system of (2) are structured ones, the proposed design method can also be applied to the systems with the parameter structured uncertainties [19]. Besides, in [19], the nominal system is introduced so as to generate the desired trajectory of the state and the control input. The proposed design method in this paper can be easily extended to such control problem.

Remark 7. The proposed decentralized variable gain robust controller can be obtained by solving LMIs of (15). Since LMIs of (15) define convex solution sets of , thus various efficient convex optimization algorithms can be applied to test whether these LMIs are solvable and to generate particular solutions [22, 23]. In addition, these solutions parametrize the set of decentralized variable gain robust controllers with gain performance. Namely, one can see that the result in Theorem 5 can easily be extended to the decentralized variable gain robust controller with suboptimal gain performance (see Appendices for details).

Remark 8. The decentralized robust controller synthesis proposed in this paper is adaptable when some assumptions are satisfied. Namely, if the matching condition for uncertainties and interactions is satisfied, then the proposed decentralized variable gain robust controller is applicable; that is, the LMIs of (15) are always feasible (see [19]). On the other hand, for decentralized robust controllers with fixed gain matrices based on the existing results [14–16], the size of LMIs to be solved is . However, the size of LMIs in the proposed design is equal to . Moreover, the number of variables for LMIs of (15) is smaller than that of the decentralized robust controllers with fixed gain matrices. Therefore, one can see that the proposed decentralized robust controller design method is very useful (see also Remark 5 in [19]).

5. Numerical Examples

In order to demonstrate the efficiency of the proposed robust controller, we have run a simple example. The control problems considered here are not necessary practical. However, the simulation results stated below illustrate the distinct feature of the proposed decentralized robust controller.

In this example, we consider the uncertain large-scale interconnected systems consisting of three two-dimensional subsystems; that is, . The system parameters are given as follows:

Firstly, by using Theorem 5, we design the proposed decentralized variable gain robust controller. By solving LMIs of (15), we have positive definite matrices , matrices , and positive scalars and given byThus the fixed gain matrices can be computed asFurthermore, the positive scalars can be obtained asTherefore, the guaranteed gain performance of (12) for the proposed controller is given by

The simulation result of this numerical example is shown in Figures 1–4. In this example, the initial value of the uncertain large-scale system with system parameters of (30) is selected as follows:

In these figures, denotes the th element of the state for the th subsystem, respectively. Furthermore, unknown parameters and disturbance input are given as

From these figures, the proposed decentralized variable gain controller stabilizes the uncertain large-scale systems with system parameters of (30) in spite of uncertainties, interactions, and disturbance inputs. Therefore, the effectiveness of the proposed decentralized robust control system has been shown.

6. Conclusions

In this paper, on the basis of the result of [11, 18, 19], we have proposed a decentralized variable gain robust controller with guaranteed gain performance for a large-scale interconnected system with uncertainties. For the uncertain large-scale interconnected systems, we have presented an LMI-based design method of the proposed decentralized variable gain robust controller. In addition, the effectiveness of the proposed decentralized robust controller has been shown by simple numerical examples. One can easily see that the result in this paper is an extension of the existing results [18, 19].

In the future research, we will extend the proposed controller synthesis to such a broad class of systems as large-scale systems with general uncertainties, uncertain large-scale systems with time delays, and so on.

Appendices

In this appendix, a decentralized variable gain robust controller with suboptimal gain performance and the conventional fixed gain decentralized robust controller with gain performance are presented.

A. Suboptimal Guaranteed Gain Performance

In this section, we show a design method of a decentralized variable gain robust controller with suboptimal gain performance.

Since the LMIs of (15) define a convex solution set, we consider minimizing the parameter because our interest is in establishing gain performance. Furthermore, in the LMIs of (15), has no correlation with . Thus our design problem can be replaced with the following constrained convex optimization problem (see [22, 23]):If the optimal solution , , and of the constrained optimization problem of (A.1) is obtained, then the control input of (6) with the fixed gain matrix and variable one of (16) is the decentralized variable gain robust control with suboptimal gain performance of (12).

Consequently, the following theorem for the decentralized variable gain robust controller with suboptimal gain performance is obtained.

Theorem A.1. Let one consider the uncertain large-scale interconnected system composed of subsystems of (2) and the control input of (6).

The control input of (6) is the decentralized variable gain robust control with suboptimal gain performance of (12) provided that the constrained convex optimization problem of (A.1) is feasible.

One can easily see that although the solution of LMIs of (15) is not unique, the decentralized robust controller obtained by solving the convex optimization problem of (A.1) is unique.

Note that the constrained optimization problem of (A.1) can be solved by software such as MATLAB’s LMI Control Toolbox and Scilab’s LMITOOL.

B. An LMI-Based Design Method for the Conventional Fixed Gain Decentralized Robust Controller with Gain Performance

In this section, the conventional fixed gain decentralized robust controller with gain performance is presented.

Consider the uncertain large-scale interconnected system composed of subsystems of (2). For the th subsystem of (2), we define the following control input: where is the fixed gain matrix for the th subsystem of (2). From (2), (3), and (B.1), the closed-loop subsystem of (B.2) can be obtained as Namely, the control objective in this section is to design the fixed gain matrices of (B.1) such that the overall closed-loop system composed of subsystems of (B.2) achieves not only internal stability but also guaranteed gain performance .

In order to design the fixed gain matrices , the quadratic function of (8) and the Hamiltonian of (10) are defined. Additionally, we consider the inequality of (11). The time derivative of the quadratic function of (9) along the trajectory of the closed-loop subsystem of (B.2) can be written as Besides, by using the well-known inequality of (19), we have the following relation for the function :where and are positive constants. Moreover, one can see that the following relation holds:Therefore from (B.4) and (22) we find the following relation for the Hamiltonian :Furthermore, some algebraic manipulations for (B.6) give the following inequality:where is given byHence, if the following inequality holds, then Additionally, by using and and pre- and postmultiplying both sides of the matrix inequality of (B.9) by , one can obtain Moreover, by applying Lemma 2 (Schur complement) to (B.10), we obtain Namely, by solving the LMIs of (B.13), the fixed gain matrix is determined as .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank Professor Shengwei Mei, Tsinghua University, and the anonymous reviewers for their valuable and helpful comments that greatly contributed to this paper.

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Copyright © 2015 Shunya Nagai and Hidetoshi Oya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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