Proceedings of the International MultiConference of Engineers and Computer Scientists 01 Vol II IMECS 01 March 14-16 01 Hong Kong This research was supported by Basic Science Research Program through the National Research Foundation of KoreaNRF funded by the Ministry of Education Science and Technology 010-0009743. This research was supported by the MKEThe Ministry of Knowledge Economy Korea under the ITRCInformation Technology Research Center support program supervised by the NIPANational IT Industry Promotion Agency NIPA-011-C-1090-111-0006 & NIPA-011C1090-1111-0011. This research was supported by World Class University program funded by the Ministry of Education Science and Technology through the National Research Foundation of Korea R31-10100. Sung Hyun Kim is with the Department of Electrical Engineering University of Ulsan UOU Ulsan 680-749 Korea e-mail: shnkim@ulsan.ac.kr. Bum Yong Park is with the Department of Electrical Engineering POSTECH Pohang 790-784 Korea e-mail: merongyong@postech.ac.kr. PooGyeon Park is with the WCUDivision of ITCE POSTECH Pohang 790-784 Korea e-mail:ppg@postech.ac.kr ISBN: 978-988-1951-9-0 ISSN: 078-0958 Print; ISSN: 078-0966 Online IMECS 01
ζt = coletx r t wt = colwtrt AΘt AΘ ÃΘ t = t A r 7 0 A r Bu Θ Ã d Θ t = t FΘ t 0 8 Bw Θ B w Θ t = t I 0 I CΘ t = [ CΘ t 0 ] Lemma.1: For real matrices X Y and S > 0 with appropriate dimensions it is satisfied that 0 X SY T S 1 X SY and hence the following inequality holds: Y T SY X T Y Y T X X T S 1 X. Further if X = µi then Y T SY µy µy T µ S 1 µ is a scalar. On the other hand if S < 0 then it is assured that Y T SY µy µy T µ S 1 A. PLMI-type condition III. MAIN RESULTS Choose a Lyapunov-Krasovskii functional Vt V 1 tv tv 3 t 9 V 1 t =ζt T Pζt V t = V 3 t =d 1 0 ζ T αq 1 ζαdα d 1 d 1 d1 tβ d ζ T αr 1 ζαdαdβ tβ t d ζ T αq ζαdα ζ T αr ζαdαdβ P Q 1 Q R 1 and R are positive definite with appropriate dimensions. Let us define an augmented state ηt = colζtζt d 1 ζt dtζt d wt and the corresponding block entry matrices as e 1 [ I ] e [ 0 I 0 ] e 3 [ I ] e 4 [ 0 I 0 ] e 5 [ I ] Φ t ÃΘ te 1 ÃdΘ t e 3 B w Θ t e 5 Time derivative of V i t V 1 t = η T thee T 1PΦ t ηt 10 V t = η T t e T 1Q 1 Q e 1 e T Q 1 e e T 4Q e 4 ηt 11 V 3 t = η T tφ T t d 1 R 1 d 1R Φt ηto 1 O= d 1 ζt αr 1 ζαdα d1 dt d ζt 1 αr ζαdα t d d1 t dt ζ T αr ζαdα Vt = η T tπ 0 ηto 13 Π 0 =Hee T 1 PΦ te T 1 Q 1 Q e 1 e T Q 1e e T 4 Q e 4 d 1 R 1 d 1R Φt 14 Φ T t By employing the Jensen inequality [8] and the lower bounds lemma [9] we can obtain a upper bound of O as follows O ζαdα T R 1 t ρ t d1 t dt dt ζαdα T R ζαdα d1 t dt dt ζαdα ζαdα T R ζαdα t d t d = η T te 1 e T R 1 e 1 e ηt t ηt te e 3 T R e e 3 ηt ρ t ηt te 3 e 4 T R 15 t = d /d 1 0ρ t = d dt/d 1 0 tρ t = 1 16 15 is represented as the following inequality: O η T tπ 1 ηt [ ρ e e 3 ηt ρ ] T [ ρ ρ Π 1 e e 3 ηt ρ Π 1 = e 1 e T R 1 e e 1 e e 3 T R e 3 e ] 17 e 3 e 4 T R e 4 e 3 Hee e 3 T Se 3 e 4 18 R S Π = S T 19 R Vt is upper-bounded by Vt η T tπ 0 Π 1 ηt [ ρ e e 3 ηt ρ ] T Π [ ρ ] e e 3 ηt ρ 0 Note: Stability criterion in the H sense Vt z T tzt γ w T t wt < 0 0 > Π 0 Π 1 Π 3 0 Π 1 Π 3 = e T 1 C T Θ t CΘ t e 1 γ e T 5e 5 Set P = diagp 1 P and P = P 1 = diag P 1 P P 1 = P1 1 and P = P 1. Then PÃΘ P1 AΘ t = t P 1 AΘ t P 1 A r 0 P A r P1 B PÃdΘt = u Θ t FΘ t P 1 0 P B P1 B ω Θ t = ω Θ t P 1 0 P FΘ t = FΘ t P 1
The H stabilization condition is given as follows: PΦ t = XĀΘ txe 1 XĀdΘ t Xe 3 X B ω Θ t e 5 3 X = diagp 1 I 4 AΘt P ĀΘ t = 1 AΘ t A r 5 0 P A r Bu Θ Ā d Θ t = t FΘ t 0 6 Bω Θ B ω Θ t = t I 7 0 P Theorem 1: Let µ 1 > 0 µ > 0 be prescribed. Suppose that there exist a scalar γ > 0; matrices F and S; and symmetric positive definite matrices P 1 P Q1 Q R1 and R such that 0 > ñ R 0 S T 11 0 d 1 ĀΘ t 0 0 d 1 ĀΘ t 0 d 1 Ā d Θ t d 1 Ā d Θ t 33 R1 Ā d Θ t 44 R S 55 0 CΘt X 0 d 1 Bω Θ t 0 0 d 1 Bω Θ t 0 0 Bω Θ t X CT Θ t S R S 6 6 0 γ I 0 I S R ô 11 = µ 1 R 1 diag µ 1 P1 µ 1 P = µ R diag µ P1 µ P 33 = HeĀΘ t Q 1 Q R 1 44 = Q 1 R 1 R 8 9 55 = R He S66 = Q R AΘt P ĀΘ t = 1 AΘ t A r 0 P A r Bu Θ Ā d Θ t = t FΘ t 0 B Bω Θ ω Θ t = t I 0 P CΘ t = [ CΘ t 0 ] Then the closed-loop systems 6 is asymptotically stable and satisfies z < γ w for all nonzero wt L [0 and for any time-varying delay dt satisfying d 1 dt d. Moreover the minimized H performance can be achieved by the following optimization problem: min γ subject to 8 and 9. Here the control and observer gain matrices can be reconstructed as FΘ t = FΘ t P 1 1. 30 0 Π 31 0 > Π 0 Π 1 Π 3 = Hee T 1 PΦ te T 1 Q 1 Q e 1 e T Q 1e e T 4 Q e 4 d 1 R 1 d 1 R Φt e 1 e T R 1 e e 1 Φ T t e e 3 T R e 3 e e 3 e 4 T R e 4 e 3 Hee e 3 T Se 3 e 4 e T 1 C T Θ t CΘ t e 1 γ e T 5e 5 3 By letting R 1 = X PR 1 PX and R = X PR PX. Then the condition 3 can be converted by 3 into 0>Hee T 1XĀΘ txe 1 e T 1XĀdΘ t Xe 3 e T 1X B ω Θ t e 5 e T 1Q 1 Q e 1 e T Q 1 e e T 4Q e 4 XĀΘ txe 1 XĀdΘ t Xe 3 X B ω Θ t e 5 T X d 1 R1 d 1 R X XĀΘ txe 1 XĀdΘ t Xe 3 X B ω Θ t e 5 e 1 e T P X R 1 XPe e 1 e e 3 T P X R XPe3 e e 3 e 4 T P X R XPe4 e 3 Hee e 3 T Se 3 e 4 e T 1 C T Θ t CΘ t e 1 γ e T 5e 5 33 X = X 1. Futher since diag X X X XIe T i = e T i X for i = 134 and diag X X X XIe T 5 = e T 5 pre- and post-multiplying both sides of 33 by diag X X X XI and its transpose yields 0 >ΨĀΘ te 1 ĀdΘ t e 3 B ω Θ t e 5 T d 1 R 1 d 1 R ĀΘ te 1 ĀdΘ t e 3 B ω Θ t e 5 34 Ψ = Hee T 1ĀΘ te 1 e T 1ĀdΘ t e 3 e T B 1 ω Θ t e 5 e T 1 Q 1 Q e 1 e T Q 1 e e T Q 4 e 4 e 1 e T R1 e e 1 e e 3 T R e 3 e e 3 e 4 T R e 4 e 3 Hee e 3 T Se3 e 4 e T X C 1 T Θ t CΘ t Xe 1 γ e T 5 e 5 in which X = XP X = diag P 1 P > 0 Q1 = XQ 1 X Q = XQ X R1 = X R 1 X R = X R X and S = XS X. That is by applying the Schur complement to 34 we can obtain R 1 1 0 d 1 ĀΘ te 1 ĀdΘ t e 3 B ω Θ t e 5 0 > 0 R 1 d 1 ĀΘ te 1 ĀdΘ t e 3 B ω Θ t e 5 Ψ 35 R 1 1 1 1 Here since 1 = X R 1 X and R = X R X it follows from lemma.1 that R 1 1 µ 1 X µ 1 1 R1 and R µ X µ R. In this sense it is clear that 35 holds if 11 0 d 1 ĀΘ t 0 d 1 Ā d Θ t 0 d 1 Bω Θ t 0 d 1 ĀΘ t 0 d 1 Ā d Θ t 0 d 1 Bω Θ t 33 R1 Ā d Θ t 0 Bω Θ t 0 > 44 R S S 0 55 R S 0 0 66 0 0 γ I 36
11 = µ 1 R 1 diag µ 1 P1 µ 1 P = µ R diag µ P1 µ P 33 = HeĀΘ t Q 1 Q R 1 X C T Θ t CΘ t X 44 = Q 1 R 1 R 55 = R He S66 = Q R There exist a non-convex term in 33. Thus to deal with the term we can obtain 8 using the Schur complement. Let us pre- and post-multiply both sides of 31 by diag X X and its transpose. Then we can obtain ñ ô XR X S 0 S T 37 XR X B. LMI-type condition Another representation for 8 is given as follows: 0 >LΘt L 0 θ i t L i L T i θitl ii j=i1 θ i tθ j tl ij θ i tθ j tl T ij 11 0 d 1 Ā 0 d 1 Ā HeĀ0 Q 1 Q R 1 R1 L 0 44 0 0 C0 X 0 d 1 Ā d d 1 Bω d 1 Ā d d 1 Bω Ā d Bω0 X CT 0 R S S 55 R S 66 γ I 0 0 I ï A0 P1 A Ā 0 = 0 A r Bu0 F Ād0 = 0 P A r Bω0 I B ω0 = 0 P C 0 = [ C ] d 1 Ā i 0 d 1 Ā di 0 d 1 Bωi 0 d 1 Ā i 0 d 1 Ā di 0 d 1 Bωi 0 Ā i 0 Ā di 0 Bωi X CT i L i 38 ò Ai P1 A Ā i = i A r Bu0 Fi B Ādi = ui F 0 P A r Bωi I B ωi = 0 P C i = [ C i 0 ] d 1 Ā dii 0 d 1 Ā dii 0 Ā dii 0 L ii 0 Bui Fi 0 Ā dii = d 1 Ā dij 0 d 1 Ā dij 0 Ā dij 0 L ij 0 Bui Fj 0 Ā dii = By the S-procedure 0 > LΘtNΘt 39 0 NΘt is given by r θ it = 1 a i θ i t b i 0 θ i tθ j t NΘt =C 1 C T 1 C i Λ i Λ T i j i C 3ij Σ ij Σ T ij T I I I θ 1 ti I [ ] θ 1 ti 0 =C 1 W0 W 1 W r... θ r ti I θ r ti 0 C i θ ita i b i θ i t a i b i 0 C 3ij θ i tθ j t for0 < Λ i Λ T i and0<σ ij Σ T ij. With some algebraic manipulations the constraint 0 NΘt can be represented as follows: 0 NΘt =N 0 θ i tn i N T i θi tn ii j=i1 θ i tθ j tn ij θ i tθ j tn T ij 40 N 0 = W 0 W0 T r a i b i Λ i Λ T i N i = a i b i Λ i W 0 W i N ii = Λ i Λ T i W i Wi T and N ij = W i W j Σ ij Σ ji. Hence the condition 39
becomes 0 >Γ 0 θ i tγ i Γ T i θi t i j=i1 θ i tθ j tφ ij θ i tθ j tφ T ij 41 Γ 0 = L 0 N 0 = L 0 W 0 W0 T a i b i Λ i Λ T i Γ i = L i N i = L i a i b i Λ i W 0 W i i = L ii N ii = L ii Λ i Λ T i W i W T i Φ ij = L ij N ij = L ij W i W j Σ ij Σ ji The condition 41 boils down to 0 > [ I θ 1 ti θ r ti ] L[ I θ1 ti θ r ti ] T 4 Γ 0 Γ 1 Γ Γ r 1 Φ 1 Φ 1r L..... 43........ Φr 1r r IV. NUMERICAL EXAMPLE In this section Consider the following plant [11]: ẋ 1 t = x t ẋ t = x 3 1 t 0.1x t1costut 44 The LPV representation of the above system is as followings: 0 1 0 1 A 1 = A 0 0.1 = 5 0.1 ï ï 0 1 B u0 = B 1ò ω0 = A 1ò r = 3 A 0 = 0C 0 = 0B ui = 0B ωi = 0i = 1...r θ 1 t = 1 x 1 t 5 θ t = x 1 t 5 µ 1 = µ = 1a 1 = a = 0b 1 = b = 1 0 x 1 t [ 5 5]rt = ωt = 1cost 4sint x0 = [ 1] T x r 0 = [ 0.5 1] T Fig. 1..5 1.5 1 0.5 0 0.5 1 1.5 0 5 10 15 0 5 30 Time sec Simulation Results ACKNOWLEDGMENT REFERENCES x1 xr1 [1] W.-W. Che Y.-P. Li and Y.-L. Wang H Tracking control for NCS with packet losses in multiple channels case Int. J. Innov. Comp. Inf. Control vol. 711 pp. 6507 651 011. [] X. Fang and J. Wang Sampled-data H control for networked systems with random packet dropouts Asian J. Control vol. 14 pp. 549 557 010. [3] D. Xie Y. Wu and X. Chen Stabilization of discrete-time switched systems with input time delay and its applications in networked control systems Circuits Syst. Signal Process. vol. 84 pp. 595 607 009. [4] D. Yue E. Tian Z. Wang and J. Lam Stabilization of systems with probabilistic interval input delays and its applications to networked control systems IEEE Trans. Syst. Man Cybern. Paart A-Syst. Hum. vol. 394 pp. 939 945 009. [5] X. Jiang Q.-L. Han S. Liu and A. Xue A new H stabilization criterion for networked control systems IEEE Trans. Autom. Control vol. 534 pp.105 103 008. [6] P.-G. Park and D. J. Choi LPV controller design for the nonlinear RTAC system Int. J. Robust Nonlinear Control vol. 1114 pp. 1343 1363 001. [7] S. W. Yun Y. J. Choi and P.-G. Park State-feedback disturbance attenuation for polytopic LPV systems with input saturation Int. J. Robust Nonlinear Control vol. 08 pp. 899 9 010. [8] Gu K. Kharitonov V. L. and Chen J. Stability of time-delay systems Birkhuser 003. Int. J. Robust Nonlinear Control vol. 08 pp. 899 9 010. [9] P.-G. Park J. W. Ko and C. Jeong Reciprocally convex approach to stability of systems with time-varying delays Automatica vol. 47 pp. 35-38 011. [10] S. H. Kim and P.-G. Park H state-feedback control design for fuzzy systems using Lyapunov functions with quadratic dependence on fuzzy weighting functions EEE Trans. Syst. Man Cybern. B Cybern vol. 166 pp. 1655 1663 008. [11] X. Jia D. Zhang X. Hao and N. Zheng Fuzzy H Tracking Control for Nonlinear Networked Control Systems in T-S Fuzzy Model IEEE Trans. Syst. Man Cybern. B Cybern vol. 394 pp. 1073 1079 009. Fig. 1 displays the first state behaviors. The simulation result shows that the H tracking controller on the LPV systems over a communication network has a good performance to track the reference signal. V. CONCLUSION Our future work is directed to proposing a method of addressing the practical case the parameters in the LPV system and control part are asynchronous/mismatched each other.