# Robustness Bound of LQ Guaranteed Cost Control Systems for Parameter Uncertainty.

код для вставкиСкачатьDev. Chem Eng. Mineral Process., 9(ID),pp.199-204, 2001. Robustness Bound of LQ Guaranteed Cost Control Systems for Parameter Uncertainty Anke Xue", Yao Chen and Youxian Sun National Laboratory of Industrial Control Technology, Institute of Industrial Process Control, Zhejiang University, Hangzhou, P R. China ~ ~~ The problem of robustness bound for parameter uncertain systems with LQ guaranteed cost controller is considered in this paper. A parameter dependent robustness bound for parameter uncertainties is derived. Then, an optimization problem of improving the robustness of parameter uncertain systems is proposed. And also, the responding Optimization technique is given in the paper. Introduction During the past few years, many results for robust stability bounds of systems with parameter uncertainties have been presented using a variety of methods [l-31. Most of those proposed robustness bounds only guarantee the uncertain systems remaining stable, i.e. robust stability bounds on uncertain systems. Although some results are derived based upon the robust LQ problem, conservatism may obviously be introduced in order to minimize the value of performance index by using a quadratic Lyapunov function. In fact, those robustness bounds, which try to ensure fvred cost function is minimal, will certainly make the stable domain of the closed-loop system very small. On the other hand, these results may present high norm gains, since they consider the maximization of the stability robustness of the closed-loop system without considering the trade-off between performance and control effort [4]. Therefore, robustness bounds considering the trade-off between stability and performance would be quite useful in the robust control system design. A possible way to deal with above problems is to adopt an idea of minimizing a certain upper bound (not minimal value) for the quadratic criterion of the LQ control problem for the uncertain closed-loop system. Such idea results from the guaranteed cost control problem (GCCP) [ 5 ] . The GCCP has been one of the most active areas of research because the guaranteed cost controllers guarantee the closed-loop system designed quadratic stability and a certain level of LQ performance for all admissible values of the uncertainty. Many results for GCCP have been presented [6-91. However, no general procedure for buildmg a guaranteed cost controller for parametric uncertainties is available in the literature because of the linear or norm bound restriction for the uncertainty [7,8], and the difficulty of solving the three or more coupled design equations with some parameters. Practically, solving these nonlinear equations is not an easy task. The numerical solution still remains open [9, * Authorfor correspondence (akxue@iipc..zju.edu.cn). 199 A. Xue, Y. Chen and Y.Sun 101. A difficult thing here is how to determine the robustness bounds of GCCP. We shall refer to this as robustness bounds of guaranteed cost control (RBGCC). Problem Formulations We consider an uncertain continuous-time system of the form i ( t )= ( A , +AA)x(t) + Bu(t), x(0) = X, (1) where x ( t ) E R" is the state. u(t) E R m is the control input, A, and B are real constant matrices of appropriate dimensions that describe the nominal system, and AA is a real valued matrix representing the plant uncertainty as follows: AA I = CkiAi i=l where ki are real uncertain parameters and A, are given real constant matrices. Associated with this system is the cost function. J = 10"(x * (t)Qx(t) + u (t)Ru(t))dt (3) where, Q 2 0, R > 0. Definition 1: The uncertain system (1) is said to be quadratically stable if there exists a positive definite symmetric ma& such that (A,+AA)*~+~(A,+AA)cO (4) Furthermore, the uncertain system (1) is said to be quadratically stabilizable via linear state feedback if there exists a h e a r state feedback law of the form u(t) = f i ( t ) , K €Rmxn such that #he resulting closed-loop system i ( t ) = ( A + A&(t) (6) is quadratically stable, where A = A, + BK . Definition 2: A linear state feedback controller ( 5 ) is said to be a quadratic guaranteed cost controller with associated cost matrix for the uncertain system (1) and cost function (3) if ( A + AA)* P + P ( A + AA) + Q c 0 (7) for all existing uncertainty AA . The positive-definite matrix P satisfjmg (7) is said to be a quadratic guaranteed cost matrix, and the resulting uncertain closed-loop system (6) is then said to be a quadratic guaranteed cost control (QGCC) system. Lemma 1: If there exists a matrix P > 0 satisfjmg (7) for the uncertain system (1) and cost function (3), then the system is quadratically stabilizable and the cost function satisfies the bound 200 Robustness Bound of LQ Guaranteed Cost Control Systems J IxoT Px, (8) Proof: In fact, the Lemma 1 is an immediate consequence of the Definition 1,2. Remark 1: Lemma 1 shows that a QGCC system is a quadratic stable system. Robustness Bound of LQ Guaranteed Cost Control Leta,,(X) (d,(X)) and a-(X) ( d ~ n ( X ) denote ) the maximum and minimum singular values (eigenvalues) of matrix, respectively, and d( X) denote any eigenvalue of matrix X. Define Hi as i=1,2, ...1 H~ = A'P+ PA, (9) Note that Hi 's are real and symmetric (Hermitian) matrices. Theorem 1 The uncertain closed-loop system (6) is a QGCC system. if I c Jki10- ( H i ) < 1- i=l (-( A P + PA + Q)) (10) for any parameter uncertainty A4 satisfying (2). Proof: From (2) and (9), we have s = ( A + AA)' I P + P ( A + AA)+ Q = z k , ~+ ,A' P + P A + Q (11) i=l For the system (6) to be a QGCC system, S must be negative, or equivalently, the all eigenvalues of S are negative, i.e. 1 n(s) = n(x k i ~+iA ~ P P+A + Q ) < o i=l (12) Note that for any Hermitian matrices A and B A ( A + B ) IA(, A ) + d,(B) (13) then I k i d , ( H i )- d ~( -"( A TP + PA + Q)) d(S) I 14) i=l Now, since A , (Hi ) = CT, (Hi ) for any Hermitian matrices Hi ,it follows 1 A(S)I c I ~ ~ J ~ , ( H ~ ) - ~ . , ~ ( PA - ( +Q)) A~P+ i=l In view of (12), the proof of this theorem is completed. Remark 2: The significance of the Theorem 1 is that the theorem relates the weighting and feedback matrices to the parameter uncertainties of the system. Thus, the robustness of QGCC m a y be improved by choosing the weighting matrices and 201 A. Xue, Y. Chen and Y. Sun solving state feedback law. Now, we give a design approach for QGCC system with parmeter uncertainty. Using Cauchy- Schwarz inequality, we obtain I I lZlkilcma.x(Hi)/ i= 1 2 1 5 ClkiI *C&x(Hi) i=l i=1 (16) From the Definition 2, we have M ~ P + PU C - ( A ' P + P A + Q ) (17) Note that identity X T Y + Y T X IX T X + Y T Y (18) is hold for any real matrices X and Y. Now using (10) into (9) we obtain AA~AAI-(A~P+PA+Q+P~) (19) From (ll), it is easy to know that A r P + PA + Q c 0 , hence 1- ( - ( A T P + PA + Q)) = 0- ( - ( A T P+ PA + Q)) Clearly, (10) is satisfied if I c IkiI2 c i-1 I ( 0 : " ( - ( ~ f pPA + + QN~/C 015,(H,) i=l Thus, the robust stability of QGCC system can be improved by optimizing the condition (21). The corresponding optimization problem is shown as follows Obviously, solving this optimization problem is not an easy task in general. One possible approach is to first choose a proper feedback gain matrix K , which makes the closed-loop system quadratically stable and minimizes the cost function J ( M ,K) , and then to apply optimization technique to (22). However, no general procedure for choosing such a matrix K is available in the literature. As stated so far this is still an analytically untractable problem. Many methods for finding such a matrix K for some class of uncertainties have been presented [6-91. Note that those methods involve solving several coupled nonlinear equations. Therefore some numerical difficulties arise certainly in using these methods. In fact, even for the case where the system is free of uncertain parameters, the numerical solution for the resulting equations is still an open problem. And also, the question of when these equations have a solution remains open [9].In this paper, we propose a most practical algorithm for this problem for the structured parameter uncertainty with the form of (2). To remove the dependence of the upper cost bound (8) on the initial condition xo , 202 Robustness Bound of LQ Guaranteed Cost Control Systems we consider the initial condition to be a zero-mean random variable with covariance matrix E { xox: } = I where E (.} denotes the expectation operator. Considering the cost function Thus a bound on the closed-loop cost function is found to be Now, with above assumptions, the problem of finding a feedback matrix K so as to minimize J(AA,K) is equivalent to finding a feedback matrix K that minimizes the bound given by (24) on the cost function (23) for the uncertain linear system (1). To make the problem tractable and numerically solvable we introduce the following assumption for the uncertainty of the system (1). Assumption 1 Assume that there exists a linear upper bound p(P) for the uncertainty (2) such that p(P)2AATP+PAA A I (25) , 1 where p( P) = C (1 / y P + y AT PA ) = 2 d + C y AT PA, , where j=1 1 ' a =2 l/y, ,y > 0 , ( j = 1J) i are arbitrary real positive numbers. j=l Thus, with the Assumption 1, it is easy to venfy that the positive definte matrix P of (24) is the solution of the parameter dependent algebraic Riccati equation ( A , + B K ) ~ P + P ( A+, B K ) + Q + K ~ R K + ~ ( P ) = o (26) Furthermore, using the results given in [8], we can find a desired feedback matrix K such that the bound f'{P} of the cost function (23) is minimized. Theorem 2 [8] If there exists a ma& KO and P > 0 ,such that 1 ( A , - B K , ) P + P ( A , - B K , , ) ~ + ~ ~ ~ A ~< Po A ; j-1 where A, is defined by A, = A, + aI then the desired K = R-' BT , where the unique positive definte solution of the generalized algebraic Riccati equation is I A , P + PA: +Q- P B R - ' B ~ P +C Y ~ A ; P A=~o j=I Using the resulting feedback gain matrix K , the robustness bounds of QGCC system can be obtained by solving the optimizationproblem (22). A possible design procedure for QGCC system with structured parameter 203 A. Xue. Y. Chen and Y.Sun uncertainties may be introduced as follows Step 1 Calculate the Hermitian matrices H i as defined in (9). Step 2 Choose a proper matrix KO satisfjmg (27). Step 3 Find a feedback matrix K by solving Riccati-like equation (28). Step 4 Solve the optimization problem (22) to obtain weighting matrices. Step 5 Apply (10) of Theorem 1 to obtain the robustness bounds of QGCC system. It is clear that the feedback gain K obtained above is optimal. In t h s case, the linear state-feedback controller ( 5 ) is a robust optimal guaranteed cost controller. Conclusions In this paper, a robustness bound of optimal LQ guaranteed cost control systems with parameter uncertainties is proposed. As the resulting robustness bounds are parameter and variable dependent, therefore the robustness of controlled systems can be improved by solving optimization problems. On the other hand, the proposed results may reduce the conservatism of the robustness bounds because of no norm bounded restriction on uncertainties. A design procedure of optimal LQ guaranteed cost controller for parameter uncertain systems is presented in the paper. This procedure consists of finding an optimal feedback matrix K and solving an optimization problem. Acknowledgments The work was supported by the National Natural Science Foundation of China under grants No. 69874036,the Chinese Postdoctoral Science Foundation, and Zhejiang Provincial Natural Science Key Foundation of China under grants No.ZD9905. References 1. Patel, R. V. et. al. 1977. Robusmess of linear quadratic state feedback designs in the presence of 2. system uncertainty. E E E Trans.Auto. Contr., 22(6), 945-949. Zhou, K. et al. 1987. Stability robusmess bounds for linear state-space models 3. with stnrctured uncertainty. E E E Trans. Auto Contr., 32(7), 621-623. Ezzine, J. 1995.Robust stability bounds for sampleddata systems.Int. J. System Sci., 26(10), 19511966. Fischman, A. et. al. 19% A linear matrix inequality approach for guaranteed cost control. Pmc.of the 13th IFAC Congress, 197-202. 5. Chang,S. S. L. et. al. 1972.Adaptive guaranteed cost control of systems with uncertain parameters. IEEE Trans. Auto. contr., 17(4), 474-483. 6. Petersen, I. 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