Stability Analysis and H Synthesis for Linear Systems With Time-Varying Delays

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1 Stability Analysis and H Synthesis for Linear Systems With Time-Varying Delays Anke Xue Yong-Yan Cao and Daoying Pi Abstract This paper is devoted to stability analysis and synthesis of the linear systems with time-varying delays Some new stability conditions are developed for the systems based on Lyapunov-Razumikhin theorem Then a design method for the general state feedback controllers is proposed for the systems with multiple delays by the LMI optimization based approach Numerical examples are used to demonstrate the effectiveness of the proposed design technique I INTRODUCTION An ever-growing number of internet-connected devices is now accessible to a multitude of users Being a ubiquitous communication means the internet could allow any user to reach and command any device connected to the network The internet has become the preferred form of interactive communication with new applications in multiplayer games teleconferencing and telerobotics being developed and tested every day [11 Internet based control has become an important means in the control systems design A very important problem in this kind of systems is the delays due to the signal communication Actually nonlinear systems with time-delay constitute basic mathematical models of real phenomena for instance in circuits theory economics and mechanics Not only dynamical systems with timedelay are common in chemical processes and long transmission lines in pneumatic hydraulic or rolling mill systems but computer controlled systems requiring numerical computation have time-delays in control loops The presence of time-delays in control loops usually degrades system performance and complicates the analysis and design of feedback controllers Stability analysis and synthesis of retarded systems is an important issue addressed by many authors and for which surveys can be found in several monographs (see eg [7 [6) Most of these previous work on time delays has focused on unknown but constant time delays In practical systems the time delays are variable due to the motion of the slave systems for example space-based or underwater telerobotic applications involve moving vehicles and thus experience changing transmission times to and from the stationary operator Moreover some systems possess rapidly and This work was partially supported by National Natural Science of China under grant Anke Xue is with Dept of Automation Hangzhou Institute of Electronics Engineering Hangzhou 3137 China akxue@hzieeeducn Yong-Yan Cao is with Dept of Electrical & Computer Engineering University of Virginia Charlottesville VA 2293 yycao@virginiaedu Daoying Pi is with Dept of Control Zhejiang University Hangzhou 3127 China dypi@iipczjueducn possibly randomly varying transmission delays A obvious example is the satellite-based transmission through varying relay sites In the internet which has frequently been used as a means for creating teleoperation systems between a variety of remote sites information is transmitted in small packets and is routed in real-time through a possibly large number of intermediate stops Although average latencies may be low the instantaneous delays may increase suddenly due to rerouting or other network traffic In the extreme the connection may be temporarily blocked It could be expected that the above mentioned methods are applicable even in the case of random time-varying delay by designing the control for the maximum value of the delay (worst case controller) However it is shown in [8 that a control algorithm designed for a fixed maximum delay T may not stabilize the system when the delay varies between and T Niemeyer and Slotine [1 also showed some stability problems due to internet transmission Stability criteria for linear system with time delays can be classified into two categories considering their dependence from time delays Delay-independent stability conditions are independent of the size of the delays (ie the time delays are allowed to be arbitrarily large) and thus in general are conservative especially in situations delays are small It can be used to study the systems without any information on the time delays Delay-dependent results [2 [1 [3 are usually used to determine a maximum value for the time delays which guarantees stability They are expected to be less conservative To facilitate the computation process the linear matrix inequality (LMI) approach is employed in the development Some work has been devoted to extend these results to the systems with time-varying delays However the time-varying information of the delay is required and the stability conditions always require the upper bound of the time derivative of delays less than 1 [2 [3 [4 It is well known that the choice of an appropriate Lyapunov functional is the key-point for deriving of stability criteria In this paper we will present a new method for the stability analysis of the systems with time-varying delays by the Lyapunov-Razumikhin function approach We will propose some delay-independent conditions and delaydependent conditions which will be used to test the stability for the various control systems without time-varying information of delays The requirement that the upper bound of the time derivative of delays is less than 1 in the above mentioned papers will be removed in these conditions The paper is organized as follows Section 2 gives the problem description Delay-independent and delay-

2 dependnet stability analysis and design will be addressed in Sections 3 and 4 respectively H control will be studied in Section 5 We will give some numerical examples to show the feasibility of the result in Section 6 Finally Section 7 will conclude the paper II PROBLEM STATEMENT Consider linear systems with time-varying delays ẋ(t) = A x(t)+ A i x(t d i (t)) + B u(t) (1) x(t) = ψ(t) t [ τ (2) x R n is the state u R m the control input <d i (t) τ i τ the time-varying delays and A A i and B are appropriately dimensioned real-valued matrices Assume that the initial condition ψ is a continuous vectorvalued function ie ψ C nτ We use x t C nτ to denote the restriction of x(t) to the interval [t τt translated to [ τ that is x t (θ) =x(t + θ) θ [ τ In this paper we will be interested in the stability analysis and design for the system (1)-(2) We will also consider the control of the system (1) by the instantaneous state feedback u(t) =F x(t) (3) and the delayed state feedback u(t) = F i x(t d i (t)) (4) F F i R m n But in what follows we will combine the instantaneous state feedback (3) and the delayed state feedback (4) as u(t) = F i x(t d i (t)) (5) i= with d (t) = Then the closed-loop system under the above state feedback (5) can be written as ẋ(t) = (A i + B F i )x(t d i (t)) (6) i= III DELAY-INDEPENDENT STABILITY ANALYSIS In this section we will first give methods on delayindependent stability for the system (1)-(2) with u = We will then present a state feedback controller design such that the closed-loop system is delay-independent stable Define y(t) =ẋ(t) We can then rewrite system (1) with the following form A x(t) y(t)+ A i x(t d i (t)) = (7) For simplicity in what follows we will use the following notation: x d = [ x T (t d 1 (t)) x T (t d r (t)) T x(t) = x T (t) y T (t) x T T d A d = A 1 A r Then (7) can be rewritten as Hence we have A x(t) y(t)+a d x d = 2(P 1 x + P 2 y + P d x d ) T (A x y + A d x d )= (8) for any weighting matrices P 1 P 2 and P d with compatible dimensions A Delay-independent stability condition Theorem 1: Consider system (1)-(2) with u if there exist matrices P > P 1 P 2 P d and Q> of compatible dimensions such that P 1 T A + A T P 1 + rp P2 T A P 1 + P P2 T P 2 < (9) Pd T A + A T d P 1 A T d P 2 Pd T Λ 33 Q diag{p P P } < (1) Λ 33 = P T d A d + A T d P d Q then the solution x(t) is delay-independently asymptotically stable for any time-varying time-delays d i (t) > for i =1 2r Proof: Given P > consider a quadratic Lyapunov function candidate as First we have V (x(t)) = x T (t)p x(t) ε 1 x 2 V (x) ε 2 x 2 ε 1 = λ min (P ) ε 2 = λ max (P ) The derivative of V (x(t)) along the solutions of (1)-(2) is V (x(t)) = 2x T (t)p y(t) By (8) we have V (x(t)) = x T (t)ω x(t) Ω= P 1 T A + A T P 1 P2 T A P 1 + P P2 T P 2 Pd T A + A T d P 1 A T d P 2 Pd T Pd T A d + A T d P d Define Note that ξ(t) = [ x(t) y(t) [ P T Γ= 1 A d + A T P d P2 T A d P d 2x T d Γ T ξ(t) ξ T (t)γ Q 1 Γ T ξ(t)+x T d Qx d Q = Q (Pd T A d + A T d P d) > We then have { V } (x(t)) ξ T (t) Λ+Γ Q 1 Γ T ξ(t)+x T d Qx d { } ξ T (t) Λ+Γ Q 1 Γ T ξ(t) + x T (t d i )P x(t d i )

3 if Q<diag{P P P } [ P T Λ= 1 A + A T P 1 A T P 2 P1 T + P P2 T A P 1 + P P2 T P 2 By using Razumikhin theorem we assume that there exists a real v>1 such that V (x(t θ)) <vv(x(t)) for θ [τ then ( V ) (x(t)) ξ T (t) Λ+Γ Q 1 Γ T ξ(t)+vrv (x(t)) = ξ T (t)(ω + vr P +Γ Q 1 Γ T )ξ(t) P = [ P Obviously if (9) holds by continuity we can always find a v>1 such that P 1 T A + A T P 1 + vrp P2 T A P 1 + P P2 T P 2 < Pd T A + A T d P 1 A T d P 2 Pd T Λ 33 By Schur complements we have V (x(t)) < for x This completes the proof Note that the condition of Theorem 1 does not include any information of time-delay ie the theorem provides a delay-independent condition for stability of linear timedelay systems with time-varying delays in terms of the solvability of several linear matrix inequalities Remark 1: In some of the early references by the Lyapunov-Krasovskii functional approach for example [2 [4 the upper bound of the derivative information of the time-varying delay is required to be known and then some LMI conditions involving this upper bound were derived Based on the Lyapunov-Razumikhin theorem and the special equality (8) we obtained a new LMI condition on the stability of the systems with time-varying time-delays which also works in the case that the time-delays are timevarying and the time-varying information is not available B Controller design With the state feedback (5) the closed-loop system can be described by (6) By Theorem 1 the closed-loop system is delay-independently stable for all time-delays if there exist matrices P > P 1 P 2 P d and Q i > of compatible dimensions satisfying (1) and P T 1  + ÂT P 1 + rp P2 T  P 1 + P P2 T P 2 < Pd T  + ÂT d P 1  T d P 2 Pd T Λ 33 (11) Λ 33 = Pd T Âd + ÂT d P d Q ie P T  + ÂT P + rγ 2 P Γ T 2 P T Γ 1  d + ÂT Γ 1 P d  T d ΓT 1 P + P d T ΓT 1  PT d Âd + ÂT d P < d Q  d =  1  2  r P d = P d1 P d2 P dr I P  = P =  I P 1 P 2 I Γ 1 = Γ I 2 = Let X = P 1 X = X X 1 X di = P 1 di 2 X d = diag{x d1 X d2 X dr } Γ 3 = P d X d = [ I I I Θ = diag{x X d } Right- and left-multiplying (11) by Θ and Θ T respectively we obtain X 1 + X T 1 + rx P X  X X 1 + X2 T X 2 X2 T < Γ T 3 (ÂX X 1 ) Xd T Âd Γ T 3 X 2 Ξ 33 Ξ 33 =Γ T 3 ÂdX d + Xd T ÂT d Γ 3 Xd T QX d By substituting X di = X Y = F P 1 and Y i = F i P 1 it is easy to obtain the following result Theorem 2: Consider system (1)-(2) if there exist matrices X > X 1 X 2 Q > Y and Y i 2r of compatible dimensions such that X 1 + X T 1 + rx A X + BY X 1 + X T 2 X 2 X T 2 Γ T 3 (A X + BY X 1 ) Ξ 32 Ξ33 < Q diag{x X X } < Ξ 32 = X T A T d + Yd T B T Γ T 3 X 2 Ξ 33 =Γ T 3 (A dx + BY d )+(X T AT d + Y d T BT )Γ 3 Q Y d = Y 1 Y 2 Y r then the delayed state feedback (5) with F i = Y i X 1 i = 1 r stabilizes the system (1)-(2) for any time-delays d i (t) > IV DELAY-DEPENDENT STABILITY ANALYSIS To reduce conservativeness in the analysis when the size on the delay is available in this section we will establish a new delay-dependent stability condition for the system (1)- (2) Recently a new descriptor model transformation and a corresponding Lyapunov-Krasovskii functional have been introduced for stability of systems with constant delays [3 The advantage of this transformation is to transform the original system to an equivalent descriptor form representation and will not introduce additional dynamics in the sense defined in [5 In this section based on the Lyapunov- Razumikhin theorem we will present a new method for the stability analysis of the systems with time-varying delays which also does not introduce any additional dynamics

4 A Delay-dependent stability condition Rewrite (1) with the following equivalent formulation y(t) = A x(t)+ A i x(t d i (t)) (12) Since x(t) is continuously differentiable for t using the Leibniz-Newton formula one can write x(t d i (t)) = x(t) t d i(t) y(s)ds for t d i (t) Then equation (12) can be represented in the following form with discrete and distributed delays in y = y(t)+ax(t) A i y(s)ds (13) t d i(t) A = A + r A i In what follows we will denote M i = M 11i M 12i M 13i M12i T M 22i M 23i M13i T M23i T M 33i N i = N 1i N 2i N 3i Then (13) can be rewritten as Ax(t) y(t)+a d y d = [ y d = t d yt 1(t) (s)ds t d r(t) yt (s)ds T Theorem 3: Consider system (1)-(2) with u if [ there exist constants τ i and matrices P > P = P1 P 2 P d Mi > Ni Q>Q 2i > i = 1 2 r of compatible dimensions satisfying T 11 + rp T 12 T 13 T12 T T 22 + r τ iq 2i T 23 < (14) T13 T T23 T T 33 Q T Mi N i > (15) N i Q 2i Q diag{p P P } < (16) T 11 = A T P 1 + P T 1 A + (τ i M 11i + N1i T + N 1i ) T 12 = P P T 1 + A T P 2 + (τ i M 12i + N 2i ) T 22 = P 2 P2 T + τ i M 22i T 13 = P T 1 A d N T 1 + AT P d + T 23 = P T 2 A d N T 2 P d + (τ i M 13i + N 3i ) τ i M 23i T 33 = A T d P d + P T d A d N 3 N T 3 + τ i M 33i N 1 = N11 T N12 T N1r T T N 2 = N21 T N22 T N2r T T N 3 = N31 T N32 T N3r T T then the solution x(t) is asymptotically stable for all time-varying delays d i (t) τ i Proof: We choose the Lyapunov candidate as It is easy to see V (x t) V (x(t)t) = x T (t)p x(t) =2x T P y +2(P 1 x + P 2 y + P d x d ) T (Ax y A d y d ) = x T (A T P 1 + P1 T A)x + yt ( P 2 P2 T )y +2x T (P P1 T + A T P 2 )y +2x T d Pd T Ax 2x T d Pd T y 2 x T P T ( A i y(s)ds) Note that [9 t d i(t) 2 x T T P A i y(s)ds t d i τ i x T M i x +2 x T ( N i T P T A i ) y(s)ds t d i t + y T (s)q 2i y(s)ds t d i τ i x T M i x +2 x T ( N i T P T A i )(x x(t d i )) + y T (s)q 2i y(s)ds t τ i Then we have ) V (x t) x (Θ+ T τ i M i x + y T (s)q 2i y(s)ds t τ i Θ= P 1 T A + A T P 1 + r (N 1i T + N 1i) P + P2 T A P 1 + r N 2i T Λ 22 A T d P 1 N 1 + Pd T A + r N 3i T Λ 32 Λ 33 and Λ 22 = P 2 P T 2 Λ 32 = A T d P 2 N 2 Pd T Λ 33 = A T d P d + Pd T A d N 3 N3 T To apply Razumikhin theorem we assume that there exist a constant µ 1 > 1 such that V (x(t + θ)t + θ) µ 1 V (x(t)t) θ [ τ ie x T (t + θ)p x(t + θ) µ 1 x T (t)p x(t) θ [ τ From (1) there exists a constant µ 2 > 1 such that y T (t + θ)q 2i y(t + θ) µ 2 y T (t)q 2i y(t) θ [ τ

5 Let ν =maxµ 1 µ 2 we then have ν>1 such that x T (t + θ)p x(t + θ) νx T (t)p x(t) y T (t + θ)q 2i y(t + θ) νy T (t)q 2i y(t) θ [ τ The remain proof is similar to Theorem 1 B Controller design Theorem 4: Given system (1)-(2) if there exist constants τ i and matrices X > X 1 X 2 Q>Q 2i > M i > N i Y i i= 1r satisfying ˆT 11 + rx ˆT12 ˆT13 X T ˆT 1 14 ˆT 12 T ˆT 22 ˆT23 X2 T ˆT 14 ˆT 13 T ˆT 23 T ˆT 33 Q < (17) ˆT 14 T X T 1 ˆT 14 X 2 ˆT44 T Mi N i N i X T > (18) + X Q 2i Q diag{x X X } < (19) ˆT 11 = X 1 + X T 1 + ( τ i M 11i + N 1i + N1i T ) ˆT 12 =(A X + BY X 1 ) T + X 2 + ˆT 22 = X 2 X2 T + τ i M 22i (τ i M 12i + N 1i ) ˆT 13 =(A X + BY X 1 ) T Γ 3 N T 1 + ˆT 23 = A d X + BY d X T 2 Γ 3 N T 2 + τ i M 13i τ i M 23i ˆT 33 =Γ T 3 (A d X 1 + BY d )+(X1 T A T d + Yd T B T )Γ 3 N 3 N3 T + τ i M 33i ˆT 14 = [ τ 1 I τ 2 I τ r I ˆT 44 = diag{τ 1 Q 21 τ 2 Q 22 τ r Q 2r } then the state feedback (5) with F i = Y i X 1 i = 1 r stabilizes the system (1)-(2) for all time-delays d i (t) τ i Proof: By Theorem 3 the closed-loop system with state feedback (5) is stable for time-delays d i (t) τ i if (15) and  T P 1 + P T 1  + rp P2 T  P 1 + P P 2 P2 T Pd T  + ÂT d P 1  T d P 2 Pd T Λ 33 + (τ i M i +Π 1 Ni + N i T Π T 1 + τ i Π 2 Q 2i Π T 2 ) Π 3 N N T Π T 3 < (2) Λ 33 = ÂT d P d + Pd T Âd Q Π 1 = [ I T Π2 = [ I T Π 3 = [ I T N = T T T T N 1 N 2 N r Right- and left-multiplying the above inequality by Θ and Θ T respectively and substituting M i Θ T M i Θ Ni X Ni ΘQ 2i Q 1 2i and Q X dqx d it is easy to find that (2) and (14) are equivalent to (17) and T Mi N i N i X T Q 1 2i X > (21) respectively Note that for any matrix X i > we have X T Q 1 2i X X T + X Q 2i Hence (21) holds if (18) holds V STATE FEEDBACK H CONTROLLER DESIGN Now we consider the H controller design of the following system Let us first consider linear systems with time-varying delays ẋ(t) = A i x(t d i )+Bu(t)+B 1 w(t) (22) z(t) = i= C i x(t d i )+Du(t)+D 1 w(t) (23) i= x R n is the state u R m the control input w L q 2 [ the exogenous disturbance and z(t) R p the output to be attenuated For H control we always consider the following performance index J = (z T z γ 2 w T w)ds (24) under zero initial condition γ>is a prescribed constant In what follows we will consider the design of the state feedback controller (5) such that inequality J< holds for all nonzero w L q 2 [ Theorem 5: Given autonomous system (22)-(23) with u for a prescribed γ> J< holds for all nonzero w L q 2 [ and all time-varying delays d i(t) τ i if there exist matrices P > P = P 1 P 2 P d Mi > Ni Q> Q 2i > i = 1 2r of compatible dimensions satisfying LMI (14) (16) and T 11 + rp T 12 T 13 P T 1 B 1 C T T T 12 T 22 + Q 2i T 23 P T 2 B 1 T T 13 T T 23 T 33 Q C T d B T 1 P 1 B T 1 P 2 γ 2 I D T 1 C C d D 1 I Q 2i = τ i Q 2i C d = C 1 C 2 C r < (25)

6 Theorem 6: Given system (22)-(23) and a prescribed γ > J < holds for all nonzero w L q 2 [ and all time-varying delays d i (t) τ i if there exist matrices X > X 1 X 2 Y Y i M i > Ni Q > Q 2i > i = 1 2 r of compatible dimensions satisfying (18) (19) and ˆT 11 + rx ˆT12 ˆT13 ˆT 12 T ˆT 22 ˆT23 ˆT 13 T ˆT 23 T ˆT 33 Q ˆT 14 T X T < 1 ˆT 14 X 2 ˆT44 B1 T γ 2 I ˆT 61 T T ˆT 63 I (26) ˆT T 61 = C X + DY ˆT T 63 = [ C 1 X + DY 1 C r X + DY r then the state feedback (5) with F i = Y i X 1 i = 1 r stabilizes the system (22)-(23) with J< VI NUMERICAL EXAMPLES Example 1 Consider the example given in [4 The system matrices are A = [ 2 9 [ 1 A 1 = 1 1 This example is also discussed in many other references The reference [4 gave a very detailed comparison with the known results By Theorem 3 we find the upper bound of delay for the stability is τ τ =4472 which is slightly better than the result of [4 Moreover our theory shows that the system is still stable even when the time-delay is timevarying for any d(t) τ =4472 However the result of [4 can only guarantee the stability for the system with constant delay τ 447 When time-delay is time-varying the upper bound of the derivative of time-delay is involved in [4 and it has to be assumed to be less than 1 This example is also discussed in many other references The reference [4 gave a very detailed comparison with the known results By Theorem 3 we find the upper bound of delay for the stability is τ τ =4472 which is slightly better than the result of [4 Moreover our theory shows that the system is still stable even when the time-delay is timevarying for any d(t) τ =4472 which does not involve the size of derivation of the time-delay However the result of [4 can only guarantee the stability for the system with constant delay τ 447 When time-delay is time-varying the upper bound of the derivative of time-delay is involved in [4 and it has to be assumed to be less than 1 Example 2 Now we consider the following example 1 1 A = A 1 1 = 9 1 B = B 1 1 = 1 1 C = C 1 = D = D 1 1 = In [4 the stabilizability bounds was studied for the above system The obtained maximum value in [4 is τ =148 for the system with constant delay By Theorem 6 an upper bound τ =1684 is obtained This implies that the above system with any time-varying delay d(t) 1684 can be stabilized by a memoryless state feedback Actually when τ =1684 we obtain the state feedback gain F = [ At this time the minimum H performance bound is γ = 5646 which is obtained by Theorem 5 VII CONCLUSIONS In this paper both the delay-independent and delaydependent stability are discussed by applying Lyapunov- Razumikhin theorem for the time-delay system with timevarying time delays This paper also improved the stability conditions of the known references We also discussed the stabilizability problem and the H control design for this class of systems with time-varying delays Numerical examples are also proposed to show the effectiveness and the less conservativeness of the proposed method REFERENCES [1 Y-Y Cao Y-X Sun and C Cheng Delay-dependent robust stabilization of uncertain systems with multiple state delays IEEE Trans Automat Control 43(11): [2 Y-Y Cao Y-X Sun and J Lam Delay-dependent robust H control for uncertain systems with time-varying delays IEE Proc D: Contr Theory and Appl 145(3): [3 Fridman E New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems Systems Control Lett 43: [4 Fridman E and U Shaked Delay-dependent stability and H control: constant and time-varying delays Int J Control 76: [5 K Gu and S I Niculescu Additional dynamics in transformed timedelay systems IEEE Trans Automat Control 45(3): [6 K Gu and S I Niculescu Survey on recent results in the stability abd control of time-delay systems J Dyn Syst-T ASME 125(2): [7 J Hale Theory of Functional Differential Equations Springer New York 1977 [8 K Hirai and Y Satoh Stability of systems with variable time delay IEEE Trans Automat Contr 25: June 198 [9 YS Moon P Park WH Kwon and YS Lee Delay-dependent robust stabilization of uncertain state-delayed systems Int J Control 74: [1 G Niemeyer and J J E Slotine Towards force-reflecting teleoperation over the internet Proc IEEE Int Conf on Robot & Autom pages [11 R Oboe Web-interfaced force-reflecting teleoperation systems IEEE Trans Industrial Electronics 48(6): Dec 21