O ( = η T (t)(e 1 e 2 ) T R 1 (e 1 e 2 )η(t) where. O η T (t)π 1 η(t) where. Π 1 = (e 1 e 2 ) T R 1 (e 2 e 1 )+(e 2 e 3 ) T R 2 (e 3 e 2 )

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1 Proceedings of the International MultiConference of Engineers and Computer Scientists 01 Vol II IMECS 01 March 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 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 & NIPA-011C 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 R Sung Hyun Kim is with the Department of Electrical Engineering University of Ulsan UOU Ulsan Korea Bum Yong Park is with the Department of Electrical Engineering POSTECH Pohang Korea PooGyeon Park is with the WCUDivision of ITCE POSTECH Pohang Korea ISBN: ISSN: Print; ISSN: Online IMECS 01

2 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 = 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

3 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 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 = 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 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 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 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 γ I 36

4 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 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

5 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 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: A 1 = A = 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 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 [] X. Fang and J. Wang Sampled-data H control for networked systems with random packet dropouts Asian J. Control vol. 14 pp [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 [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 [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 [6] P.-G. Park and D. J. Choi LPV controller design for the nonlinear RTAC system Int. J. Robust Nonlinear Control vol pp [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 [8] Gu K. Kharitonov V. L. and Chen J. Stability of time-delay systems Birkhuser 003. Int. J. Robust Nonlinear Control vol. 08 pp [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 [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 [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 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.