INTERNATIONAL JOURNAL OF INFORMATON AND SYSTEMS SCIENCES Volume 5 Number 3-4 Pages 480 487 c 2009 Institute for Scientific Computing and Information ROBUST PASSIVE OBSERVER-BASED CONTROL FOR A CLASS OF SINGULAR SYSTEMS LI YANG 1 Abstract In this paper passive uncertainties are introduced and we deal with the robust passive observer-based control problem for a class of singular systems and then provide the sufficient conditions of the passive control for the closed-loop systems of this kind of systems whose controller is based on observer Finally two numerical examples are provided to show the usefulness and the feasibility of the presented approach Key Words passive control observer-based controller passive uncertainty 1 Introduction In the past decade dissipative system theory has played the important role in the control theory Passivation is one of the most important aspects of dissipation ie the energy supply rate is taken as the product of input and output and passivation represents the attenuation characteristic of the systems under the condition of bounded input In fact passivation is the advanced abstract of stability Therefore passive control problem are focused on by a lot of scholars [1 7 And the dissipative perturbation was proposed by Xie [8 Then dissipative uncertainties were discussed in robust control theory [9 This kind of uncertainties included the other kind of uncertainties description when some matrices are taken as the special forms and the dissipative uncertainties can include both the phase information and the gain information meanwhile the energy information of uncertainties are taken into account In the paper we introduced the passive uncertainties and discuss the passive observer-based control problem for a class of singular systems Then we establish the criterion of passivation for this kind of systems which is based on observer by matrix inequalities and we get the controllers of the systems meanwhile the closedloop systems and uncertainties are all passive Finally we testify the feasibility of the theorems by the numerical examples The paper is structured as follows In the next section Section 2 we introduce notations and review some important definitions and lemmas And then in Section 3 we discuss the passive observer-based control problem and design the controller In Section 4 passive uncertainties are proposed and robust passive control problem for singular systems with passive uncertainties is discussed Section 5 contains numerical examples by which we can testify the feasibility of the theorems The final Section 6 contains conclusions Received by the editors January 15 2008 and in revised form October 9 2008 2000 Mathematics Subject Classification 35R35 49J40 60G40 480
ROBUST PASSIVE OBSERVER-BASED CONTROL FOR SINGULAR SYSTEMS 481 2 System Description Consider the following singular system (1) { Eẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) E is singular matrix A B C D are all known constant matrices of appropriate dimensions x(t) R n is state y(t) R n is output u(t) R n is input We assume that D are symmetry matrices and D is negative Next our purpose is to design passive controllers based on observer for system (1): (2) { E ˆx(t) = Aˆx(t) + Bu(t) + L(y C ˆx Du) u(t) = K ˆx(t) We introduce the error vector e(t) = x(t) ˆx(t) and the error dynamic equation is as follows: Eė(t) = (A LC)e Therefore the closed-loop system of system (1) via the controller (2) is (3) Ẽ x(t) = Ã x(t) Ẽ = and u y can be denoted as [ E 0 A LC 0 e 0 E Ã = x = BK A + BK x u = [ K K x y = [ DK C + DK x Definition 1 [10 For system (1) if there exists nonnegative storage function V (x) (V (0) = 0) so that the following inequality (4) V (t) u T y holds for any t 0 then system (1) is said to be passive In this paper we let the storage function be V (t) = x T E T P x Lemma 1 [11 S11 S For given positive symmetry matrix S = 12 S S 21 S 11 22 is r r dimensions then the following three conclusions are equivalent: (i) S < 0; (ii) S 11 < 0 S 22 S T 12S 1 11 S 12 < 0; (iii) S 22 < 0 S 11 S 12 S 1 22 ST 12 < 0
482 L YANG 3 Passive Observer-based Controller Design In this section a method is given to design passive observer-based controller for system (1) Theorem 1 For singular system (1) if the following inequality holds for any t 0 E T P = P T E 0 A (5) T P + P A T P T B 1 2 CT B T P 1 2 C 1 2 DT 1 2 D < 0 then system (1) is passive Proof: According Definition 1 we can get V u T y = (Ax + Bu) T P x + x T P T (Ax + Bu) u T (Cx + Du) = [ [ x T u T x Π u A Π = T P + P A T P T B 1 2 CT B T P 1 2 C 1 2 DT 1 2 D thus if inequality (5) holds system (1) is passivethis complete the proof For closed-loop system (3) we consider the storage function as the following V (t) = x T Ẽ T P x P = P 0 0 P Theorem 2 For system (1) if there is feasible solution X L W satisfying the following inequalities X T E T = EX 0 Ξ 11 W T B T + 1 2 (6) W T CX W T BW + 1 2 XT C T W Ξ 22 W T < 0 W W D 1 then controller (2) is passive controller for closed-loop system (3) K = W X 1 Ξ 11 = X T A T + AX L T L Ξ 22 = X T A T + AX + W T B T + BW 1 2 W T CX 1 2 XT C T W and the matrix L satisfies L = LCX Proof: For the closed-loop system
ROBUST PASSIVE OBSERVER-BASED CONTROL FOR SINGULAR SYSTEMS 483 V u T y [ A LC 0 = x T BK A + BK T [ P x + x T T A LC 0 P BK A + BK x K x T T [ DK C + DK x K T = x T [ Ξ11 Ξ12 Ξ 21 Ξ22 x Ξ 11 = A T P + P T A C T L T P P T LC K T DK Ξ 12 = K T B T P + 1 2 KT C + K T DK Ξ 21 = P T BK + 1 2 CT K + K T DK Ξ 22 = A T P + P T A + K T B T P + P T BK 1 2 KT C 1 2 CT K K T DK Therefore if Ξ X T 0 < 0 the closed-loop system is passive Then multiply 0 X T to Ξ X 0 < 0 from the left side and from the right side X = P 0 X 1 Let L = LCX we can obtain inequality (6) according to complementary Schur theorem Thus if X W L is one of feasible solutions of (6) the passive feedback controller is given by K = W X 1 and matrix L satisfies L = LCXThis complete the proof 4 Robust Passive Observer-based Controller Design Consider the following systems with uncertainties Eẋ(t) = Ax(t) + Bu(t) + L i=1 H iω i (t) (7) y(t) = Cx(t) + Du(t) z i (t) = F i x(t) + F ui u(t) + L j=1 F ωijω j (t) ω i (t) R ki and z i (t) R hi are uncertain variables H i F i F ui and F ωij are all known constant matrices of appropriate dimensions Definition 2 For system (7) if the uncertain variables ω i (t) and z i (t) satisfy the following inequalities: (8) < ω i z i > T 0 i = 1 2 L then the uncertainties of system (7) are said to be passive If we denote ω = the inequality (8) can be rewrited as ω 1 ω L z = (9) < ω z > T 0 z 1 z L
484 L YANG In order to be simple we introduce the following signals F = Ω i = F 1 F L F u = F u1 F ul F ωi = 0 0 1 2 F T i IT i 0 0 1 2 F T ui IT i 1 2 I 1 if i 2 I 1 if ui F ωi1 F ωil 2 (I if ωi + Fωi T IT i ) H = [ H 1 H L Fω = [ F ω1 F ωl I T i = [ 0 1 I hi 0 i+1 0 L i = 1 L The next purpose is to design passive controller based on observer as controller (2) for singular system (7) so that the closed-loop system and the uncertainties are passive Then the error system of system (7) via observer (2) is (10) Eė = (A LC)e + Hω and the closed-loop system of system (7) via the observer (2) is (11) Ẽ x = Ã x + Hω H = [ H H Theorem 3 For system (7) if there is feasible solution X L W satisfying the following inequalities X T E T = EX 0 Γ 11 Γ 12 2HΛ X T F T W T (12) Γ T 12 Γ 22 2HΛ X T Fu T W T 2ΛH T F X 2ΛH T F u X 2F ω Λ 2ΛFω T 0 < 0 W W 0 D 1 then controller (2) is passive for system (7) K = W X 1 Γ 11 = X T A T + AX L L Γ 12 = W T B T + 1 2 W T CX Γ 22 = X T A T + AX + W T B T + BW 1 2 W T CX 1 2 XT C T W and matrix L satisfies L = LCX
ROBUST PASSIVE OBSERVER-BASED CONTROL FOR SINGULAR SYSTEMS 485 Proof: For closed-loop system (11) V u T y [ A LC 0 = ( BK A + BK [ x + Hω) T P x + x T A LC 0 P T ( BK A + BK x + Hω) K x T T [ DK C + DK x K T = x T Θ11 Θ 12 x + ω [ Θ 21 Θ T H T P 22 H T P x + xt P T H P T ω H Θ 11 Θ 12 P T H = ζ T Θ 21 Θ 22 P T H ζ H T P H T P 0 ζ T = [ x T ω T Θ 11 = A T P + P T A C T L T P P T LC K T DK Θ 12 = K T B T P + 1 2 KT C + K T DK Θ 21 = P T BK + 1 2 CT K + K T DK Θ 22 = A T P + P T A + K T B T P + P T BK 1 2 KT C 1 2 CT K K T DK And if ζ T Ω i ζ 0 inequality (8) holds ie the uncertainties of system is passive Therefore according S-procedure [11 if there exist λ 1 0 λ L 0 inequality L (13) Θ λ i Ω i < 0 holds then i=1 (14) ζ T Θζ 0 and inequality (9) holds too ie the uncertainties of system and the closed-loop system (11) are both passive The inequalities (13) is equal to 11 12 P T H 1 2 F T λ (15) = 21 22 P T H 1 2 F u T λ < 0 H T P 1 2 λf HT P 1 2 λf u 1 2 λf ω 1 2 λf ω T 11 = A T P + P T A C T L T P P T LC K T DK 12 = K T B T P + 1 2 KT C + K T DK
486 L YANG 21 = P T BK + 1 2 CT K + K T DK 22 = A T P + P T A + K T B T P + P T BK 1 2 KT C 1 2 CT K K T DK X T 0 0 X 0 0 Then multiply 0 X T 0 to < 0 from left side and 0 X 0 from 0 0 2Λ 0 0 2Λ right side X = P 1 λ = diag{λ 1 λ L } and Λ = λ 1 Let L = LCX we can obtain inequality (12) according to complementary Schur theorem Thus if X W L is one of feasible solutions of (12) the strictly passive feedback controller is given by K = W X 1 and matrix L satisfies L = LCX This complete the proof 5 Numerical Example Example 1 The parameters of system (1) are as follows: 1 0 13 0 1 1 5 0 E = A = B = C = 0 1 0 5 1 2 0 5 D = 1/16 0 0 1/16 By using tool box of Matlab we can get X = 06789 0 W = 00173 03418 78909 18899 60326 67591 then we can obtain the controller based on observer as (2) 114821 55294 14641 00253 K = L = 83821 197756 01928 70714 Example 2 The parameters of system (7) are just as the Example 1 and the other parameters is as follows: 4 0 0 3 1 0 1 1 2 05 0 1 H = F = 1 0 0 005 1 15 1 2 F u = 0 1 2 1 F ω = 0 1 1 2 1 0 5 1 3 0 0 2 0 1 3 2 By using tool box of Matlab we can get X = 09609 0 W = 01400 03667 33192 86038 45631 73541 and λ 1 = 05670 λ 2 = 06007 then we can obtain the controller based on observer as (2) K = 00358 234606 L = 18269 200529 16269 16668 11104 119858
ROBUST PASSIVE OBSERVER-BASED CONTROL FOR SINGULAR SYSTEMS 487 6 Conclusion We discuss the robust passive observer-based problem for a class of singular systems then we give the method to design passive state feedback controllers based on observer and it may bring forward a feasible way to design observer and make the closed-loop system dissipative for singular systems but this kind of controller is designed by matrices inequalities method thus we can not gain the necessary conditions References [1 Li Zhihu Shao Huihe Wang Jingcheng Dissipative control for linear time delay systems Control Theory and Applications 200118(5): 838-842 [2 Xinzhuang Dong Qingling Zhang Passive Control of Linear Singular Systems Journal of Biomathematics 2004 19: 185-187 [3 Xinzhuang Dong Qingling Zhang Passive Control of Linear Singular Systems Journal of Biomathematics 2004 19: 185-187 [4 Xinzhuang Dong Qingling Zhang Robust passive control for singular systems with timevarying uncertainties Control Theory and Applications 2004 21: 517-520 [5 Fridman E Shaked U On Delay-Dependent Passivity IEEE Transactions Automatic on Automatic Control 2002 47: 664-669 [6 Guan Xinping Hua Changchun Duan Guangren Robust Dissipation Control of Uncertain Time-Delay Systems[J Systems Engineering and Electronics 2002 24(1):48-51 [7 Yang Li Zhang Qingling etc Robust Inpulse Dissipative Control of Singular Systems with Uncertainties Acta Automatica Sinica 2007 33(5): 554-556 [8 Shoulie Xie Lihua Xie Robust Dissipative Control for Linear Systems with Dissipative and Nonlinear Perturbation[J Systems & Control Letters 1997 29(5):255-268 [9 Liu Fei Su Hongye Chu Jian Robust Strictly Dissip ative Control for Linear Discrete Time- Delay Systems[J Acta Automatica Sinica 2002 28(6):897-903 [10 Yang Li Zhang Qingling etc Robust Dissipative Control for a Class of Singular Time-delay Systems with Uncertainties Control Theory and Application 2007 24(6): 1038-1042 [11 Yu Li Robust Control-Linear Matrix Inequality Method[M Tsinghua Press 2002 1 College of Mathematics Liaoning University Shenyang 110036 China E-mail: yangli2923@163com