Stability Analysis of Linear Systems with Time-varying State and Measurement Delays

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1 Proceeding of the th World Congress on Intelligent Control and Automation Shenyang, China, June 29 - July 4 24 Stability Analysis of Linear Systems with ime-varying State and Measurement Delays Liang Lu State Key Lab of Synthetical Automation for Process Industries Northeastern University, Shenyang, China. liangup@gmail.com Pedro Albertos and Pedro García Instituto de Automática e Informática Industrial Universidad Politécnica de Valencia Valencia, 4622 Spain pedro,pggil}@aii.upv.es Abstract In this paper, an LMI based analysis approach to chec the stability of closed-loop discrete-time linear systems with both time-varying state and measurement delays is presented. he internal as well as the measurement delays are assumed to be time-varying but bounded. he stability analysis procedure allows to determine the delay intervals ensuring the stability of the controlled plant. he procedure is applicable to any plant with two different state delays, as well as to diverse control applications, as illustrated in the reported example. he procedure can be conceptually extended to the general case of systems with multiple time-varying delays. I. INRODUCION he study of time delay systems has been for many years a matter of research. ime-delay system is a generic name for a great variety of situations. he simplest problem of constant single input/output delay for SISO plants, since the seminal wor of [, has received much attention and many solutions have been proposed to deal with different ind of plants, unstable and/or non-minimum phase, with long delays (see, for instance, [2 and the references here in). he extension to the case of multiple delays and MIMO plants has been also treated by many researchers, but still there are some open problems in dealing with the interactions and decoupling issues [3, [4, [5. o consider time-varying delays has been a recent requirement, mainly imposed by the study of networ-based control systems, where the transmission delay is random and unnown, [6, but bounded. he problem of internal or state delay has also received a lot of attention, mainly from a theoretical point of view, although any closed-loop controlled plant with input/output delay becomes a plant with internal delay. hese delays can be also constant or time-variant, single or multiple. A number of wors has been also published with this setting. In the review paper [7, a number of open issues dealing with different delay scenarios are outlined, and the simultaneous presence of delays in the state and/or the his wor has been partially supported by National Natural Science Foundation (Grant No. 6349) and Research Foundation for the Doctoral Program of Higher Education (Grant No ) of China; PROM- EEO project No , Consellería de Educación GV, Universidad Politécnica de Valencia PAID-6-2 and CICY Project DPI C2-, Spain. input/output is listed. he case of time-variant multiple delays has received a lot of attention, lie [8, [9 or [. In the wor of [, a deep analysis of the stability and robustness of different processes with internal delays is reported. Most of the wors in the literature refer to the stability and robustness analysis, trying to determine the maximum interval of the time delay admissible to eep the system stable [2, [3. Recently, in the wor of [4, a less conservative stability criterion for discrete-time (D) systems with interval timevarying state delay has been presented, their results being less conservative than those previously reported. he use of a predictor to counteract the effect of time delays, initially introduced in [5, [6 for constant and nown delays, was extended to the case of time-variant delays in [7, [8, [9, [2, [2, with promising results. However, this ind of predictor approach is only tractable for input or output delays. ime delays have been studied for both, continuous and discrete time systems. Although the D setting allows for an easy treatment of the problem by augmenting the state of the plant, for long delays this is not a good solution and the delay should be properly treated [22. Moreover, if it is timevarying, this approach would lead to a variant state-dimension system. On the other hand, any practical controller will be implemented in a digital computer, thus, a D version of the solution is needed. In the rest of the paper, the scenario of discrete-time linear systems with internal time-variant delay systems controlled under a networed setting, that is with both input and output time-varying delays, is considered. his was an open problem presented in J. Richard s review paper [7. An LMI based stability theorem for discrete-time systems with both state and output time delays is presented. he proof of the theorem is relied on the construction of a type of Lyapunov- Krasovsii functional which by explicitly comparing the bounds of different delays maes the result less conservative. he application of the theorem to an example with different time-varying delay patterns shows the effectiveness of this approach, allowing the comparison with previous published results /4/$3. 24 IEEE 4692

2 II. PROBLEM SAEMEN Consider the following closed-loop D linear system with both state and measurement time-varying delays x + = Ax + A d x d + Bu, () x =, min d M, τ M }, u = Kx τ (2) where x R n and u R m are respectively the system state and control input, and K in (2) is the state feedbac law, the state is assumed to be measurable but subject to an unnown time-varying delay. A, A d and B are the plant parameter matrices with appropriate dimensions, and assumed to be constant and nown. he system matrices A, A d do not need to be Hurwitz, and d and τ are the time-varying delays satisfying: d m d d M, (3) τ m τ τ M, (4) where d m, d M, τ m and τ M, are nown positive integers. he closed-loop system ()-(2) can be written as x + = Ax + A d x d + BKx τ. (5) As it can be easily shown, the closed-loop system presents two state delays, the first one d due to the internal delay and the second one τ due to the delayed state feedbac. Assume that a stabilizing state feedbac controller K has been designed under some fixed and nown time delays in both, the state and the measurement. he problem is to determine the range of admissible delays, as expressed by (3), (4), ensuring the controlled plant stability under time varying delays. Moreover, if d m and τ m are nown, we want to determine the maximum delays d M and τ M, to ensure the controlled plant stability under time varying delays within these intervals. For that purpose, following a well-now approach already used in the literature, a global Lyapunov-Krasovsii function composed by positive definite quadratic functions weighting any pair of delayed signals will be established and the conditions for the stability will be derived by using linear matrix inequalities (LMI). III. MAIN RESUL Next, we will present a sufficient condition for the stability of the closed-loop system (5). heorem : he system (5) is asymptotically stable if there exist positive definite matrices P, Z, Z 2, Z 3, Z 4, Z 5, Z dm, Z τm and positive semi-definite matrices Q d, Q dm, Q dm, Q τ, We will restrict in this paper to the case where the whole state is accessible and measurable. Q τm, Q τm and matrices S d, S d2, d, d2, S τ, S τ2, τ, τ2 satisfying the following matrix inequalities, Π δ ds d δ τ S τ δ d Sd δ d Z dm <, (6) δ τ S τ δ τ Z τm Π δ d d δ τ τ δ d d δ d Z dm <. (7) δ τ τ δ τ Z τm where Π = S d = [ Sd Sd2 d = [ d d2 S τ = [ Sτ Sτ2 τ = [ τ τ2 Π Π 3 Π 6 Π 8 Π,9 Π 22 Π 23 Π 24 Π 28 Π 2,9 Π 33 Z 4 Z 2 Π 44 Z 3 Z Π 55 Π 56 Π 57 Π 58 Π 59 Π 66 Π 77 P Z whose elements are Π = P +(δ d + )Q d + Q dm + Q dm +(δ τ + )Q τ + Q τm + Q τm Z 5 Π 3 = m τ m,, d Π 6 =, d m τ m, Π 8 = A P Π,9 = (A I) Z Π 22 = Q d + d + d d Sd Π 23 = S d Sd2 Π 24 = d + d2 Π 28 = A d P Π 2,9 = A d Z Π 33 = Q dm m τ m,, d Π 44 = Q dm Z 3 Z d2 d2 Π 55 = Q τ + τ + τ S τ Sτ Π 56 = S τ Sτ2 Π 57 = τ + τ2 Π 58 = K B P Π 59 = K B Z Z 4 Z 2 + S d2 + S d2 4693

3 Π 66 = Q τm, d m τ m, Z 4 Z 3 + S τ2 + Sτ2 Π 77 = Q τm Z 2 Z τ2 τ2 Z = ρ 2 Z 5 +(τ m d m ) 2 Z 4 +(τ m d M ) 2 Z 3 with +(τ M d m ) 2 Z 2 +(τ M d M ) 2 Z + δ d Z dm + δ τ Z τm ρ = min(τ m,d m ), δ d = d M d m, δ τ = τ M τ m. Proof. Due to space limitations, a setch of the proof is presented. Choose a Lyapunov-Krasovsii functional candidate as V ()=V ()+V 2 ()+V 3 ()+V 4 ()+V 5 () (8) with V () = x Px V 2 () = xi Q dx i + xi Q τx i i= d i= τ V 3 () = xi Q d m x i + xi Q τ m x i + xi Q d M x i i= d m i= τ m i= d M + xi Q τ M x i i= τ M V 4 () = d m τ m xi Q dx i + xi Q τx i j= d M + i=+ j j= τ M + i=+ j with V 5 () = ρ ηmz 5 η m i= ρ m=+i ρ + τ m d m ηmz 4 η m i= ρ 2 m=+i ρ 3 + τ m d M i= ρ 4 ρ 5 + τ M d m i= ρ 6 m=+i m=+i η mz 3 η m η mz 2 η m ρ 7 + τ M d M ηmz η m i= ρ 8 m=+i d m + ηmz dm η m i= d M m=+i τ m + ηmz τm η m i= τ M m=+i η = x + x, ρ = min(τ m,d m ), ρ 2 = max(τ m,d m ), ρ 3 = min(τ m,d M ), ρ 4 = max(τ m,d M ), ρ 5 = min(τ M,d m ), ρ 6 = max(τ M,d m ), ρ 7 = min(τ M,d M ), ρ 8 = max(τ M,d M ). Compute its increment, ΔV () ΔV () = x+ Px + x Px = [Ax + A d x d + BKx τ P[Ax + A d x d +BKx τ x Px ΔV 2 () = x Q dx x d Q d x d + +x Q τx x τ Q τ x τ + x Q dx x d Q d x d + +x Q τx x τ Q τ x τ + d i= d + + τ i= τ + + d m i= d M + τ m i= τ M + x i Q d x i x i Q τ x i x i Q d x i x i Q τ x i ΔV 3 () = x (Q d m + Q τm + Q dm + Q τm )x x d m Q dm x dm x τ m Q τm x τm x d M Q dm x dm x τ M Q τm x τm ΔV 4 () = (d M d m )x Q dx +(τ M τ m )x Q τx + d m i= d M + x i Q d x i + τ m i= τ M + x i Q τ x i ΔV 5 () = η (ρ2 Z 5 +(τ m d m ) 2 Z 4 +(τ m d M ) 2 Z 3 +(τ M d m ) 2 Z 2 +(τ M d M ) 2 Z +δ d Z dm +δ τ Z τm )η ρ ρ ηi Z 5 η i τ m d m ηi Z 4 η i i= ρ i= ρ 2 ρ 3 ρ 5 τ m d M ηi Z 3 η i τ M d m ηi Z 2 η i i= ρ 4 i= ρ 6 ρ 7 d m τ M d M ηi Z η i ηi Z dm η i i= ρ 8 i= d d τ m τ ηi Z dm η i ηi Z τm η i ηi Z τm η i i= d M i= τ i= τ M and establish the requirements to get ΔV (). Denoting by δ d = d M d m and δ τ = τ M τ m, and defining a staced state vector μ = [ x x d x dm x dm x τ x τm x τm, for some matrices S d = [ Sd Sd2 d = [ d d2 S τ = [ Sτ S τ2 τ = [ τ τ2 4694

4 with appropriate dimensions, we have ΔV () μ ()Ωμ()+(d d m )μ () S d ZdM S d μ() +(d M d )μ () d ZdM d μ() +(τ τ m )μ () S τ ZτM S τ μ() +(τ M τ )μ () τ ZτM τ μ() where and Ω Ω Ω 2 Ω 3 Ω 5 Ω 6 Ω 22 S d S d2 d + d2 Ω 33 Z 4 Z 2 Ω = Ω 44 Z 3 Z Ω 55 S τ S τ2 τ + τ2 Ω 66 Ω 77 = A PA P +(δ d + )Q d + Q dm + Q dm +(δ τ + )Q τ +Q τm + Q τm +(A I) Z(A I) Z 5 Ω 2 = A PA d +(A I) ZA d Ω 3 = m τ m,, d Ω 5 = A PBK +(A I) ZBK Ω 6 =, d m τ m, Ω 22 = A d PA d Q d + A d ZA d + d + d S d Sd Ω 33 = Q dm m τ m, Z, d 4 Z 2 + S d2 + Sd2 Ω 44 = Q dm Z 3 Z d2 d2 Ω 55 = K B PBK Q τ + K B ZBK + τ + τ S τ Sτ Ω 66 = Q τm Z, d m τ m, 4 Z 3 + S τ2 + Sτ2 Ω 77 = Q τm Z 2 Z τ2 τ2 Z = ρ 2 Z 5 +(τ m d m ) 2 Z 4 +(τ m d M ) 2 Z 3 +(τ M d m ) 2 Z 2 +(τ M d M ) 2 Z + δ d Z dm + δ τ Z τm. After some proper manipulations and Schur complements, the inequalities (6) and (7) are obtained. IV. ILLUSRAIVE EXAMPLE In this section an example taen from the literature, under different time-delay patterns, is evaluated and the results are compared with those reported in the previous proposals. Let us considered the controlled plant (5), being open-loop unstable. he following parameters are assumed: A = [ B = [. ; u = Kx τ [. ; A d =.2. and d and τ are delays satisfying (3), (4). ; For a given state feedbac control law, K, the stability of the closed-loop system can be determined by exploring the LMI condition in heorem. In what follows, different scenarios will be considered: a) Assume that τ =, and K = [ 8 hen, the closed-loop system will be: x + = (A + BK)x + A d x d (9) where the matrices (A + BK) and A d [.8 (A + BK) =.5.9 ;A d = [..2. are the same as defined in [22, [4 (example in both papers). It can be realized that this system is decoupled in two subsystems (A and A d are triangular and the upper part of B is null) and it should be more interesting to consider an interactive system. Nevertheless, the same parameters are used in order to compare both approaches. Additional experiments, not included here for brevity, show similar results if coupled plants are considered. By heorem 3 in [22, the system (9), is asymptotically stable for d m = 2 and it is shown that the stability is ensured in the interval 2 d 3. It is also stable if the delay is in the interval 2 d 7. For heorem in this paper, for the intervals 2 d 6, and 2 d 2, the stability is proved. o illustrate the effectiveness of the approach let us review some simulation results for the first time delay interval. Assume that the initial conditions in the system in the example are x =,} for. In addition, let assume that the delay d changes randomly between d m = 2 and d M = 6, which is shown in figure. d Fig.. ime-variant internal delay, 2 d 6. hen, the state response of the system is given in figure 2. It can be seen from this figure that the system is asymptotically stable. On the other hand, if the time delay d will reach larger ; 4695

5 values (d M > 7) the system stability cannot be guaranteed and, in fact, for some time-variant delay pattern the system becomes unstable. hus, our results are less conservative x() x d Fig. 3. ime-variant internal delay, 3 d 35. Fig. 2. State evolution from initial conditions..5 x Note that the stability condition in this paper is quite similar to that previously stated in the literature, but the main novelty in the proposed theorem is that it can be used to evaluate (explore) other scenarios by using the same stability condition (that is, the same theorem is suitable to chec different scenarios), being applicable to simultaneously time varying delays in both, the state and the measurement. For larger delays, and compared with the proposal in heorem in [4, for 3 d 35, our results are similar. Again we can illustrate the effectiveness of the approach. Assume that the initial condition in the system in the example are x =,} for. In addition, let assume that the delay d changes randomly between d m = 3 and d M = 35, which is shown in Fig. 3 hen, the state response of the system is given in Fig. 4. It can be seen from this figure that the system is asymptotically stable, but one of the states is highly oscillatory. If the time delay d will reach larger values (d M = 36) the system stability will not be guaranteed. b) Let us assume that d = τ. hen, the closed-loop systems is: x() Fig. 4. State evolution from initial conditions. Note that even the A-matrix is not Hurwitz, the closed-loop system (), in (), is stable, as a result of the heorem. c) In the general case, when d =τ, the closed-loop system being: x + = Ax +(A d + BK)x d () with a single time delay. Again, the stability can be obtained by exploring the LMI condition in heorem. For example, for the same closed-loop system (5) and the same controlled plant parameters A, A d, B and K, with d m =, an upper bound of d M = 4 is obtained. he simulation results, for d 4 are plotted in Figure 5 with x =,}, 4 () x + = Ax + A d x d + BKx τ (2) it is also possible to obtained an upper bound for the delay range. For example, for d 6, the stability condition on heorem is fulfilled for 2 τ 3. he simulation results assuming these ranges of time delay variations, for are shown in Figure 6. x =,}, 6 (3) 4696

6 x() x() Fig. 6. response. Fig. 5. State response with a common double delay State and measurement time varying delay, closed-loop state V. CONCLUSIONS In this paper, an LMI-based general procedure to determine the stability of a plant with two time-varying delays in the state space model has been presented. he procedure has been compared with existing results in the literature and the intervals of the delays to eep the system stable has been shown to be less conservative. his LMI can be generalized to investigate about the stability of systems with multiple timevarying delays. his approach can be also used to analyze the closed-loop stability of delayed state feedbac control of a system with internal state delay. Some simulation results on different scenarios illustrate the applicability of this procedure. x x REFERENCES [ O. Smith, Closer control of loops with dead time, Chemical Engineering Progress, vol. 53, no. 5, pp , 957. [2 J. E. Normey-Rico and E. F. Camacho, Control of Dead-time Processes. London: Springer-Verlag, 27. [3 P. García and P. Albertos, Dead-time-compensator for unstable mimo systems with multiple time delays, Journal of Process Control, vol. 2, no. 7, pp , 2. [4 P. Albertos and P. García, Predictor observer-based control of systems with multiple input/output delays, Journal of Process Control, vol. 22, no. 7, pp , 22. [5. L. Santos, R. C. Flesch, and J. E. Normey-Rico, On the filtered smith predictor for mimo processes with multiple time delays, Journal of Process Control, vol. 24, no. 4, pp , 24. [6 Y. Pan, H. Marquez, and. Chen, Remote stabilization of networed control systems with unnown time varying delays by LMI techniques, in 44th IEEE Conference on Decision and Control, 25 and 25 European Control Conference. CDC-ECC 5, 25, pp [7 J. Richard, ime-delay systems: an overview of some recent advances and open problems, automatica, vol. 39, no., pp , 23. [8 E. Fridman, Stability of systems with uncertain delays: a new complete lyapunov-rasovsii functional, IEEE ransactions on Automatic Control, vol. 5, no. 5, pp , 26. [9 E.-K. Bouas, Discrete-time systems with time-varying time delay: stability and stabilizability, Mathematical Problems in Engineering, vol. 26, 26. [ X.-G. Liu, R. R. Martin, M. Wu, and M.-L. ang, Delay-dependent robust stabilisation of discrete-time systems with time-varying delay, in Control heory and Applications, IEE Proceedings, vol. 53, no. 6. IE, 26, pp [ M. Wu, Y. He, and J.-H. She, Stability Analysis and Robust Control of ime-delay Systems. Berlin: Springer, 2. [2 P. Garcia, A. Gonzalez, P. Castillo, R. Lozano, P. Albertos et al., Robustness of a discrete-time predictor-based controller for timevarying measurement delay, in Proceedings on the 9th IFAC worshop on time delay systems. Prague, 2. [3 I. A. L. Mirin, On the robustness of sampled-data systems to uncertainty in continuous-time delays, IEEE ransactions on Automatic Control, vol. 56, no. 3, 2. [4 Y. Guo and S. Li, New stability criterion for discrete-time systems with interval time-varying state delay, in Proceedings of the 48th IEEE Conference on Decision and Control and the 29 28th Chinese Control Conference. CDC/CCC 29. IEEE, 29, pp [5 R. Lozano, P. Castillo, P. Garcia, and A. Dzul, Robust prediction-based control for unstable delay systems: Application to the yaw control of a mini-helicopter, Automatica, vol. 4, no. 4, pp , 24. [6 P. Garcia, P. Castillo, R. Lozano, and P. Albertos, Robustness with respect to delay uncertainties of a predictor-observer based discretetime controller, in Decision and Control, 26 45th IEEE Conference on. IEEE, 26, pp [7 A. Gonzalez, A. Sala, P. Garcia, and P. Albertos, Robustness analysis of discrete predictor-based controllers for input-delay systems, International Journal of Systems Science, vol. 44, no. 2, pp , 23. [8 A. Gonzalez, P. Garcia, P. Albertos, P. Castillo, and R. Lozano, Robustness of a discrete-time predictor-based controller for time-varying measurement delay, Control Engineering Practice, vol. 2, no. 2, pp. 2, 22. [9 A. González, A. Sala, and R. Sanchis, LK stability analysis of predictor-based controllers for discrete-time systems with time-varying actuator delay, Systems & Control Letters, vol. 62, no. 9, pp , 23. [2 A. González, Robust stabilization of linear discrete-time systems with time-varying input delay, Automatica, vol. 49, no. 9, pp , 23. [2 I. Karafyllis and M. Krstic, Delay-robustness of linear predictor feedbac without restriction on delay rate, Automatica, vol. 49, no. 6, pp , 23. [22 H. Gao and. Chen, New results on stability of discrete-time systems with time-varying state delay, IEEE ransactions on Automatic Control, vol. 52, no. 2, pp ,