Stability and Stabilizability of Linear Parameter Dependent System with Time Delay

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1 Proceedings of te International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, Marc, 2008, Hong Kong Stability and Stabilizability of Linear Parameter Dependent System wit Time Delay K. Mukdasai and P. Niamsup Abstract Tis paper presents sufficient condition for exponential stability wit a given convergence rate and asymptotically stability of linear parameter dependent (LPD) delay system and gives sufficient condition for stabilizability of LPD delay control system. We use appropriate Lyapunov functions and derive stability condition in term of linear matrix inequality (LMI). Keywords: exponential stability, asymptotically stability, linear parameter dependent (LPD) delay system, stabilizability 1 Introduction Te stability problem of linear parameter dependent (LPD) system as been investigated in many works 1]- 2] and 4]-8]. Tere are different conditions given to deal wit te stability problem for LPD system 1]-2] and 6]-7]. Now, tis problem is an increasing interest because it can apply to many engineering systems. Te LPD system is defined from uncertain linear time varying system3]. Wen te system matrices of uncertain system are formulated by a polytope of matrices. Te Lyapunov function metod is a important tool for studying LPD system stability. In tis paper, we will employ Lyaponov function for establising exponential stability condition wit a given convergence rate and asymptotically stable of linear parameter dependent (LPD) delay system. Our condition will been expressed in terms of linear matrix inequality (LMI). Ten we will extend LPD delay system to LPD delay control system for to find stabilizability condition also numerical example. We let some important notations R + te set of all non-negative real number; R n te n-dimensional space; x, y or x y te scalar product of two vector x, and y ; Supported by te Development and Promotion of Science and Tecnology Talents Project (DPST). Address: Department of Matematics, Faculty of Science, Ciang Mai University, Ciang Mai, Tailand, kanitmukdasai@otmail.com Supported by te Tailand Researc Fund (grant number RMU ). Address: Department of Matematics, Faculty of Science, Ciang Mai University, Tailand, scipnmsp@ciangmai.ac.t x te Euclidean vector norm of x; M n m te space of all (n m) matrices; A te transpose of te matrix A; A is symmetric if A = A ; C(, 0], R n ) te Banac space of all piecewise continuous vector function mapping -,0] into R n ; λ(a) te set of all eigienvalues of A; λ max (A) max {Reλ : λ λ(a)}, λ min (A) min {Reλ : λ λ(a)}; I te identity matrix. We consider te linear parameter dependent delay systems { ẋ(t) = A(α) + B(α)x(t ), t 0; (1) = φ(t), t, 0] were R n is te state,, R + is te delay, and φ(t) is a continuous vector-valued initial function. A(α) and B(α) are matrices belonging to te polytope Ω 1 using for teorem 2.1 Ω 1 := A(α), B(α)] = { α i A i, α i B i ], α i = 1, α i 0, i = 1,..., N}. A(α) and B(α) are uncertain time varying matrices belonging to te polytope Ω 2 using for teorem 2.2 Ω 2 := A(α), B(α)] = { α i (t)a i, α i (t)b i ], α i (t) = 1, α i (t) 0, i = 1,..., N}. We also assume te following bounds of te parameter values: β i > 0 : α i (t) β i, t > 0. Definition 1.1 Te system (1) is said to be β stable, if tere is a function ξ(.) : R + R + suc tat for eac φ(t) C(, 0], R n ), te solution x(t, φ) of te system satisfies x(t, φ) ξ( φ )e βt, t R +.

2 Proceedings of te International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, Marc, 2008, Hong Kong Lemma 1.2 Assume tat S R n n is a symmetric positive definite matrix. Ten for every Q R n n : 2 Qy, x Sy, y QS 1 Q x, x, x, y R n. Definition 1.3 Te equilibrium point x eq R n of (1) is asymptotically stable if it is Lyapunov stable and for every solution tat exists on 0, ) suc tat x eq as t. 2 STABILITY CONDITION From te system (1), we can cange te form of te state variable y(t) = e βt, t R +, ten te system (1) is transformed to te following delay system were ẏ(t) = A β (α)y(t) + B β (α)y(t ), t R +, (2) A β (α) = A(α) + βi, B β (α) = e β B(α), P ɛ = P + ɛi. Teorem 2.1 Te system (1) is β stable if tere exists P and Q be positive definite matrices and, β,ɛ > 0 suc tat te following condition olds. 1. A i P ɛ + 2βP ɛ + P ɛ A i + Q + e 2β P ɛ B i Q 1 B i P ɛ I, i = 1,..., N. 2. A i P ɛ + 4βP ɛ + P ɛ A i + 2Q + A j P ɛ + P ɛ A j +e 2β P ɛ B i Q 1 Bj P ɛ + e 2β P ɛ B j Q 1 Bi P ɛ 2I, i = 1,...,, j = i + 1,..., N. Proof. We define te following Lyapunov function for system (2) : V (t, y(t)) = y (t)p y(t) + ɛ y(t) 2 + y (s)qy(s)ds. t Te derivative of V along te trajectories of system (2) is given by Tus, we obtain tat V y (t)a β (α)p ɛ + P ɛ A β (α) +P ɛ B β (α)q 1 B β (α)p ɛ + Q]y(t) = y (t){a (α) + βi}p ɛ + P ɛ {A(α) + βi} +P ɛ {e β B(α)}Q 1 {e β B (α)}p ɛ + Q]y(t) = y (t){ α i A i + βi}p ɛ + P ɛ { α i A i + βi} N +P ɛ {e β α i B i }Q 1 {e β Ten, we get tat N α i B i }P ɛ + Q]y(t). V y (t) α i α i A i P ɛ + 2βP ɛ + P ɛ A i + Q]] +{ α i e β P ɛ B i Q 1 }{ α i e β Bi P ɛ }]y(t) = y (t) αi 2 A i P ɛ + 2βP ɛ + P ɛ A i + Q +e 2β P ɛ B i Q 1 Bi P ɛ ] + j=i+1 +P ɛ A i + 2Q + A j P ɛ + P ɛ A j ] + α i α j A i P ɛ + 4βP ɛ j=i+1 B i Q 1 B j P ɛ + e 2β P ɛ B j Q 1 B i P ɛ ]]y(t). Since N α i = 1 and α i A i N α i B i = α 2 A i B i + By te condition 1., 2. and since () αi 2 2 Terefore, we ave j=i+1 α i α j = j=i+1 V (t, y(t)) 0, t R +. j=i+1 Integrating bot sides of (4) from 0 to t, we find α i α j e 2β P ɛ α i α i A i B j +A j B i ]. α i α j ] 2 0. V = 2y (t)a β (α)p ɛ y(t) + 2y (t )Bβ (α)p ɛ y(t) +y (t)qy(t) y (t )Qy(t ). Using lemma 1.2, we ave 2y (t )Bβ (α)p ɛ y(t) y (t )Qy(t ) y (t)p ɛ B β (α)q 1 Bβ (α)p ɛ y(t). V (t, y(t)) V (0, y(0)) 0, t R +, and ence y (t)p y(t) + ɛ y(t) 2 + y (0)P y(0) + ɛ y(0) 2 + t 0 y (s)qy(s)ds 0 y (s)qy(s)ds

3 Proceedings of te International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, Marc, 2008, Hong Kong were P 0. Since and since 0 we ave y P y 0, y (s)qy(s)ds λ max (Q) φ 0 t y (s)qy(t)ds 0 e βs ds = λ max(q) (1 e β ) φ, β ɛ y(t) 2 λ max (P ) y(0) 2 + ɛ y(0) 2 + λ max(q) (1 e β ) φ. β Terefore, te solution y(t, φ) of te system (2) is bounded. Returning to te solution x(t, φ) of system (1) and noting tat we ave were y(0) = x(0) = φ(0) φ, x(t, φ) ξ( φ )e βt, t R +, ξ( φ ) := {ɛ 1 λ max (P ) φ 2 + φ 2 + λ max(q) (1 e β ) φ } 1 2. βɛ Tis means tat te system (1) is β stable. Te proof of te teorem is complete. Consider te system (1). Let P j, Q j, j = 1, 2,..., N be symmetric matrices and S be positive definite N M i (P j, Q j ) = β kp k + A i P ] j + P j A i + Q j P j B i Bi P, j Q j A N i,j (R, ) = i RA j R A i RB ] j + R Bi RA j + R Bi RB j R, S R 2n 2n. Teorem 2.2 Te system (1) is asymptotically stable if tere exist P j, Q j, j = 1, 2,..., N, let R be symmetric positive definite matrices and S be symmetric semi-positive definite matrix and R + wic satisfy te following matrix inequality olds. 1. M i (P i, Q i ) + N i,i (R, ) < S, i = 1,..., N. 2. M j (P i, Q i ) + M i (P i, Q i ) + N j,i (R, ) + N i,j (R, ) < 2S, i = 1,...,, j = i + 1,..., N. Proof. We define te following Lyapunov-Krasovskii function for system (1) : were V 1 := x (t)p (α), V 2 := t x (θ)q(α)x(θ)dθ and V 3 := t s ẋ (θ)rẋ(θ)dθds wit P (α) = N α i(t)p i, Q(α) = N α i(t)q i. Te derivative of V along te trajectories of system (1) is given by V = V 1 + V 2 + V 3. Terefore, V 1 = x (t) P (α) + 2ẋ (t)p (α) = x (t) P (α) + 2x (t)a (α)p (α) +2x (t )B (α)p (α) V 2 = x (t)q(α) x (t )Q(α)x(t ) and V 3 = ẋ (t)rẋ(t) t ẋ (θ)rẋ(θ)dθ Tus, using te jensen s inequality te last term can be bounded as follows: t ẋ (θ)rẋ(θ)dθ < x(t )] R x(t )] We obtain tat V < x (t) P (α) + 2x (t)a (α)p (α) +2x (t )B (α)p (α) + x (t)q(α) x (t )Q(α)x(t ) + x (t)a (α)ra(α) +2x (t )B (α)ra(α) + x (t )B (α)r B(α)x(t ) x (t) R + 2x (t) R x (t ) x (t ) R x(t ). We obtain tat V < α i (t)x (t) β k P k + 2x (t) α j (t)a j P i +2x (t ) α j (t)bj P i + x (t)q i x (t )Q i x(t ) + x (t) +2x (t ) +x (t ) α j (t)a j RA i α j (t)bj RA i α j (t)bj RB i x(t ) x (t) R +2x (t ) R x (t ) R x(t )] since N α i(t)p i N β ip i. We can rewrite as V < α i (t) α j (t){x (t) β k P k + 2x (t)a j P i +x (t)a j RA i + x (t)q i +2x (t )B j RA i + 2x (t )B j P i x (t )Q i x(t ) + x (t )B j RB i x(t ) V () = V 1 + V 2 + V 3 x (t) R + 2x (t ) R x (t ) R x(t )}].

4 Proceedings of te International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, Marc, 2008, Hong Kong Since N α i(t) = 1 and 1 = ( α i (t)) 2 = αi 2 (t) + 2 Let us set V < αi 2 (t) x (t) j=i+1 α i (t)α j (t). β k P k + 2x (t)a i P i +x (t)q i x (t )Q i x(t ) +2x (t )B i P i + x (t)a i RA i +2x (t )B i RA i x (t) R we ave () = j=i+1 αi 2 (t) 2 j=i+1 α i (t) α j (t)] 2 0. α i (t)α j (t) Tus, V < 0. Terefore, tis means tat te system (1) is asymptotically stable. Te proof of te teorem is complete. Example 2.1 Consider te following linear parameter dependent delay system : +2x (t ) R + x (t )B i RB i x(t ) x (t ) R x(t ) ] + j=i+1 α i (t)α j (t) x (t) β k P k +2x (t)a j P i + 2x (t )B j P i +x (t)q i x (t )Q i x(t ) +x (t)a j RA i + 2x (t )B j RA i +x (t )B j RB i x(t ) x (t) R +2x (t ) R x (t ) R x(t ) +x (t) β k P k + 2x (t)a i P j +2x (t )B i P j + x (t)q j +x (t)a i RA j + 2x (t )B i RA j +x (t )B i RB j x(t ) x (t) R +2x (t ) R x (t )Q j x(t ) x (t ) R x(t ) ]. Terefore, we ave V < { α 2 x(t ) i (t)m i (P i, Q i ) + N i,i (R)] + j=i+1 α i (t)α j (t)m j (P i, Q i ) +M i (P i, Q i ) + N j,i (R, ) + N i,j (R, )]} We use te following condition as V < ( α 2 x(t ) i (t)s) + 2 j=i+1 α i (t)α j (t)s] x(t ), x(t ) ẋ(t) = A(α) + B(α)x(t 1 2 ), t R+, (3) wit any initial function φ(t) C( 1 2, 0], R+ ) were A(α) = α 1 + α 1 3 2, B(α) = α 1 + α We ave = ] 1 2, N = 2. Taking ɛ = ] β = 1, we can find Q = and P = satisfy all conditions of Teorem 2.1. Terefore, te system (3) is 1- stable. Example 2.2 Consider te following linear parameter dependent delay system : ẋ(t) = A(α) + B(α)x(t 1), t R +, (4) were A(α) = α 1 (t) + α (t), B(α) = α 1 (t) + α (t). 1 2 We ave = 1, N = 2. Taking α i (t) 1 2, i = 1, ] 2 (suc as α 1 = e t 2 and α 2 = 1 e t ), Q 1 =, Q 2 = and R = ten P 1 = P 2 = 100 1, S = satisfy all conditions of Teorem 2.2. Terefore, te system (4) is asymptotically stable. 3 STABILIZABILITY CONDITION Consider te following LPD delay control system ẋ(t) = A(α) + B(α)x(t ) + C(α)u(t), t 0 (5)

5 Proceedings of te International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, Marc, 2008, Hong Kong were A(α), B(α) and C(α) are uncertain time varying matrices belonging to te polytope Ω Ω := A(α), B(α), C(α) ] = { N ] α i (t)a i, α i (t)b i, α i (t)c i, α i (t) = 1, α i (t) 0, i = 1,..., N}. Te u(t) R m is te control of te system. Definition 3.1 Te LPD delay control system (5) is said to be β stabilizable if tere exists control u(t) = K, were K R suc tat te closed loop system ẋ(t) = A(α) + KC(α)] + B(α)x(t ) is β stable. Te control u(t) = K is te feedback stabilizing control of te system. Definition 3.2 Te LPD delay control system (5) is said to be stabilizable if tere exists control u(t) = K, were K is nonnegative real number suc tat te closed loop system ẋ(t) = A(α) + KC(α)] + B(α)x(t ) is asymptotically stable. Te control u(t) = K is te feedback stabilizing control of te system. Teorem 3.1 Te system (5) is β stabilizable if tere exists P and Q be positive definite matrices and β, ɛ, K > 0 suc tat te following condition olds. 1. A i P ɛ + 2βP ɛ + P ɛ A i + Q + KC i P ɛ + KP ɛ C i +e 2β P ɛ B i Q 1 B i P ɛ I, i = 1,..., N. 2. A i P ɛ + 4βP ɛ + P ɛ A i + 2Q + A j P ɛ + P ɛ A j +KC i P ɛ + KP ɛ C j + KC i P ɛ + KP ɛ C j +e 2β P ɛ B i Q 1 Bj P ɛ + e 2β P ɛ B j Q 1 Bi P ɛ 2I, i = 1,...,, j = i + 1,..., N. Te feedback control is given by u(t) = K. Consider te system (5). Let P j, Q j, j = 1, 2,..., N be symmetric matrices and S be positive definite S R 2n 2n mij P M i (P j, Q j ) = j B i Bi P, j Q j 1 ij := A i RA j +KC i RA j +KA i RC j +KC i RC j R 2 ij := B i RA j + KB i RC j + R Teorem 3.2 Te system (5) is stabilizable if tere exist P j, Q j, j = 1, 2,..., N, let R be symmetric positive definite matrices and S be symmetric semi-positive definite matrix and, K R + wic satisfy te following matrix inequality olds. 1. M i (P i, Q i ) + N i,i (R, ) < S, i = 1,..., N. 2. M j (P i, Q i ) + M i (P i, Q i ) + N j,i (R, ), +N i,j (R, ) < 2S, i = 1,...,, j = i + 1,..., N. Te feedback control is given by 4 CONCLUSIONS u(t) = K. In tis paper, we study linear parameter dependent (LPD) delay system and LPD control delay system. We gave sufficient condition for exponential stability wit a given convergence rate and asymptotically stability of linear parameter dependent (LPD) delay system and also sufficient condition for stabilizability of LPD delay control system. We use appropriate Lyapunov functions and derive stability condition in term of linear matrix inequality (LMI). References 1] D. enrion, D. Arzelier and J.B. Lasserre, On parameter dependent Lyapunov functions for robust stability of linear systems, IEEE Conference,V1, pp , 12/04 2] E. Fridman and U. Saked, Parameter dependent stability and stabilization of uncertain time delay systems, IEEE Trans.Automat.Contr.,, V48, N5, pp , 6/03 3] F. Gouaisbaut and D. Peaucelle, Delay dependent robust stability of time delay systems, Pd.Tesis, Toulouse, France 4] G. Zai, H. Lin and P.J. Antsaklis, Quadratic stabilizability of switced linear systems wit polytopic uncertainties, Pd.Tesis, Wakayama university, Japan, ] J.C. Geromel and P. Colaneri, Robust stability of time varying polytopic systems, Systems Control m ij := Lett., V55, p , 2006 N β kp k +A i P j+p j A i +Q j +KCi P j+kp j C i, 6] P.T. Nam and V.N. Pat, Exponential stability and 1ij A N i,j (R, ) = i RB j + KCi RB ] j + R stabilization of linear uncertain polytopic time delay 2 ij Bi RB j R, systems, Pd.Tesis, Qui Non University, vietnam

6 Proceedings of te International MultiConference of Engineers and Computer Scientists 2008 Vol II IMECS 2008, Marc, 2008, Hong Kong 7] V. Jeyakumar, P.T. Nam and V.N. Pat, New conditions for stability and stabilizability of linear parameter depentdent systems, Pd.Tesis, Qui Non University, vietnam 8] V.F. Montagner and P.L.D. Peres, Stability of linear time varying troug quadratically parameter dependent Lyapunov function, XXVIII CNMAC - Congresso Nacional de Matematica Aplicada e Computacional, Vol. 1, pp.1-3, 01/05