2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009
FrB17.2
Output Tracking Control for Discrete-Time Networked Control Systems Yu-Long Wang and Guang-Hong Yang
Abstract— This paper studies the problem of H∞ output tracking control for discrete-time networked control systems (NCSs) with and without parameter uncertainty. By defining new Lyapunov function and using the discrete Jensen inequality, linear matrix inequality (LMI)-based H∞ output tracking performance analysis and controller design are presented, and the merit of the proposed methods lies in their less conservativeness, which is achieved by decoupling the Lyapunov and the system matrices. The designed controllers can guarantee asymptotic tracking of prescribed reference outputs while rejecting disturbances. The simulation results illustrate the effectiveness of the proposed H∞ output tracking performance analysis and controller design.
I. INTRODUCTION Nowadays, networks play important roles in many industrial control applications, especially for example, complex control systems and remote control systems. Advantages of NCSs include low cost, high reliability, ease of diagnosis and maintenance, small volume of wiring, etc. Many researchers have studied stability/stabilization for NCSs [1]-[3]. For other methods dealing with time delay and packet dropout, see also [4]-[10]. Recently, there have been considerable research efforts on H∞ control for systems with delay [11]-[14]. In [15], the random delays were modeled as a linear function of the stochastic variable satisfying Bernoulli random binary distribution, and the observer-based controller was also designed to exponentially stabilize the networked system in the sense of mean square. In [16], a new robust stability condition for uncertain discrete-time systems with convex polytopic uncertainty was given, and the condition exhibited a kind of decoupling between the Lyapunov and the system matrices which may be explored for control synthesis purposes. Synthesizing feedback controllers to achieve asymptotic tracking of prescribed reference outputs while rejecting disturbances is a fundamental problem in control. The main objective of tracking control is to make the output of the This work was supported in part by Program for New Century Excellent Talents in University (NCET-04-0283), the Funds for Creative Research Groups of China (No. 60521003), Program for Changjiang Scholars and Innovative Research Team in University (No. IRT0421), the State Key Program of National Natural Science of China (Grant No. 60534010), the Funds of National Science of China (Grant No. 60674021, 60804024) and the Funds of PhD program of MOE, China (Grant No. 20060145019), the 111 Project (B08015). Yu-Long Wang is with the School of Electronics and Information, Jiangsu University of Science and Technology, Zhenjiang 212003, China.
plant, via a controller, track the output of a given reference model as close as possible. For the problem of output tracking control, [17] studied the reliable robust tracking controller design problem against actuator faults and control surface impairment for aircraft. [18] solved the tracking and disturbance rejection problem for infinite-dimensional linear systems, with reference and disturbance signals that were finite superpositions of sinusoids. For other results on output tracking control, see also [19] and the references therein. The study on NCSs and output tracking control keeps attracting considerable attention due to the demands from practical dynamic processes in industry. To the best of our knowledge, however, few works pay attention to the problem of output tracking control for networked control systems except in [20]-[22], and output tracking control for discretetime NCSs is not studied in the literature, which motivates the present study. In this paper, the problem of H∞ output tracking control for discrete-time NCSs with time delay and packet dropout is studied. By defining new Lyapunov function and using the discrete Jensen inequality, LMI-based H∞ output tracking performance analysis and controller design are presented for nominal NCSs, and the LMIs do not involve any product of the Lyapunov matrices and the system dynamic matrices, which may introduce extra degree of freedom. The H∞ output tracking performance analysis is also presented for NCSs with parameter uncertainty. The proposed methods can guarantee asymptotic tracking of prescribed reference outputs while rejecting disturbances. This paper is organized as follows. Section 2 presents the closed-loop model of discrete-time output tracking NCSs. Section 3 is dedicated to H∞ output tracking performance analysis for nominal and uncertain NCSs with time delay and packet dropout. Section 4 presents H∞ output tracking controller design. The results of numerical simulation are presented in Section 5. Conclusions are stated in Section 6. Notation. Throughout this paper, M T represents the transpose of matrix M. I and 0 represent identity matrices and zero matrices with appropriate dimensions, respectively. ∗ denotes the entries of matrices implied by symmetry. Matrices, if not explicitly stated, are assumed to have appropriate dimensions. II. PRELIMINARIES AND PROBLEM STATEMENT Consider a linear time-invariant plant described by
[email protected] Guang-Hong Yang is with the College of Information Science and Engineering, Northeastern University, Shenyang 110004, China. He is also with the Key Laboratory of Integrated Automation of Process Industry, Ministry of Education, Northeastern University, Shenyang 110004, China. Corresponding author:
[email protected]
978-1-4244-4524-0/09/$25.00 ©2009 AACC
xk+1 = Axk + B1 uk−Lk + B2 ωk yk = Cxk + Dωk
(1)
where xk ∈ Rn , uk ∈ Rm , yk ∈ Rr , ωk ∈ Rq are the state vector, control input vector, measured output, and disturbance input,
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respectively, and ωk is assumed to belong to L2 [0, ∞). A, B1 , B2 , C, D are known constant matrices of appropriate dimensions, B1 is full column rank. The main objective of this paper is to make the output yk of the plant, via a controller, track the output of a given reference model as close as possible. Suppose the reference signal yˆk is generated by xˆk+1 = J xˆk + rk yˆk = H xˆk
Lemma 1 [23]. For any positive definite matrix M ∈ Rs∗s , integers β2 > β1 , vector function C : {β1 , β1 +1, · · · , β2 } → Rs such that the sums in the following are well defined, then β2
− (β2 − β1 + 1)
∑ C T (i)MC (i)
i=β1
´T ³ β 2 ´ ³ β2 ≤ − ∑ C (i) M ∑ C (i)
(2)
i=β1
(5)
i=β1
where xˆk ∈ R p and rk ∈ R p are the reference state and the energy bounded reference input, respectively, yˆk has the same dimension as yk . J and H are known constant matrices of appropriate dimensions with J stable. The state feedback controller takes the following form
Lemma 2 [5]. Suppose D, E, and F are real matrices of appropriate dimensions with ||F|| ≤ 1, for any scalar ε > 0, the following inequality holds
uk = K1 xk + K2 xˆk
Suppose Γ1 , Γ2 , Γ3 , Γ4 , and Γ5 are matrices of appropriate dimensions, and Γ1 is symmetric negative definite, then we have the following lemma which will be used in Section 3. Lemma 3. For given scalars LM and Lm , if there exist symmetric positive definite matrices P, Z1 , Z2 , Z3 , and matrix G, such that the following LMI holds Γ1 ΓT2 GT ΓT3 GT ΓT4 GT ΓT5 GT ∗ Γ22 0 0 0 ∗ ∗ Γ33 0 0 (6) 0, such that the following LMI holds eT Λ11 0 Z3 Z1 0 Ψ 1 ∗ Λ22 Z2 Z2 0 0 ∗ ∗ Λ33 0 0 0 ∗ ∗ ∗ Λ 0 0 44 ∗ eT ∗ ∗ ∗ −γ I Ψ 2 ∗ ∗ ∗ ∗ ∗ −γ I ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ΨT1 GT W1 W1 W1 W2 W2 W2 W2 0 0 0 0 0 0 0 0 ΨT3 GT ΨT3 GT ΨT3 GT ΨT3 GT 0, if there exist symmetric positive definite matrices P, Q1 , Q2 , Q3 , Z1 , Z2 , Z3 , matrix G, scalar γ > 0, such that the following LMI holds Ω εW U T ∗ −ε I 0 0, such that (14) and (15) Z1 Z2 0 Λ44 ∗ ∗ ∗ ∗ ∗ ∗
W1 ST ΨT2 0 0 ΨT3 G 0 0 0 Λ99 ∗
0 0 0 0 −γ I ∗ ∗ ∗ ∗ ∗
eT Ψ 1 0 0 0 eT Ψ 2 −γ I ∗ ∗ ∗ ∗
W1 ST ΨT2 0 0 ΨT3 G 0. Hence, if (14) is satisfied, we have rank(V ) ≥ rank(Ψ2V ) = rank(GΨ2 ) ≥ rank(Ψ2 ) = m (18) which implies that the matrix V must be nonsingular. Therefore, (16) is obtained from (17), this completes the proof. For the sake of eliminating the matrix equation constraint in (15), we assume that the extra matrix G is symmetric positive definite. For the matrix Ψ2 of full column rank, there
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always exist two orthogonal matrices X ∈ R(n+p)∗(n+p) and Y ∈ Rm∗m such that · ¸ · ¸ X Σ XΨ2Y = 1 Ψ2Y = (19) X2 0
y
k
0.08
yrk
0.06 0.04 0.02
where X1 ∈ Rm∗(n+p) , X2 ∈ R(n+p−m)∗(n+p) , and Σ = diag(σ1 , · · · , σm ), where σi (i = 1, · · · , m) are nonzero singular values of Ψ2 . Lemma 4 [24]. Let the matrix Ψ2 be of full column rank. If the matrix G has the following structure ¸ · G11 0 G = XT X = X1T G11 X1 + X2T G22 X2 (20) 0 G22
0 −0.02 −0.04 −0.06 −0.08 −0.1 −0.12
where G11 and G22 are symmetric positive definite matrices with appropriate dimensions, X1 and X2 are defined in (19), then there exists a nonsingular matrix V such that GΨ2 = Ψ2V . Based on Lemma 4, we can get the following theorem. Theorem 4. For given scalars LM , Lm , if there exist symmetric positive definite matrices P, Q1 , Q2 , Q3 , Z1 , Z2 , Z3 , G11 , G22 , matrix S, scalar γ > 0, such that (14) holds, where G = X1T G11 X1 + X2T G22 X2 , then with the control law T uk = K ξk , K = Y Σ−1 G−1 11 ΣY S
0
10
20
Fig. 1.
30 40 No. of samples, k
50
60
Outputs yk and yˆk
0.1 y
k
0.08
yrk
0.06 0.04 0.02 0
(21)
−0.02 −0.04
the augmented closed-loop system described by (4) is asymptotically stable with H∞ output tracking performance γ . Proof: The proof is similar to Theorem 4 of [15], it is omitted here. In the following, we will illustrate the effectiveness of the proposed methods by two examples.
−0.06 −0.08 −0.1
0
10
20
Fig. 2.
30 40 No. of samples, k
50
60
Outputs yk and yˆk
V. SIMULATION RESULTS AND DISCUSSION Example 1. To illustrate the effectiveness of the proposed H∞ output tracking performance analysis for nominal NCSs, we present the following system · ¸ · ¸ 1.1740 −0.1053 0.4735 xk+1 = xk + u 0.0719 0.2676 −0.3325 k−Lk · ¸ −0.0078 (22) ωk + −0.3332 £ ¤ yk = 0.2253 0.1501 xk − 0.1552ωk The reference model is described as follows xˆk+1 = 0.58xˆk + rk yˆk = 0.36xˆk
(23)
In this example, we suppose LM = 2, Lm =0, the initial state of the augmented system (4) is ξ0 = [0 0 0]T . For simplicity, suppose the packets at the instants 0, 2, 4, · · · are transferred to the actuator successfully, that is one packet are dropped among every two packets. By using Matlab and from (19), we can get £ ¤ X1 = ·−0.8184 0.5747 0 ¸ 0.5747 0.8184 0 X2 = 0 0 1.0000 (24) Y = −1 Σ = 0.5786
Solving the LMI presented in Theorem 4, we obtain the minimum H∞ output tracking performance γ = 1.0890 with the controller gain K = [K1 K2 ] = [−0.4913 − 0.0330 0.0165]. Suppose the input signals are given as follows 0, 0 ≤ k ≤ 11 −0.32, 12 < k ≤ 15 ωk = 0.39, 16 < k ≤ 17 0, 18 < k ≤ 59 (25) 0, 0 ≤ k ≤ 11 0.09, 12 < k ≤ 15 rk = −0.21, 16 < k ≤ 17 0, 18 < k ≤ 59 The output yk of the system (22) and yˆk (denoted as yrk ) of the reference model (23) are pictured in Fig. 1. From Fig. 1, we can see that the output yk of the system (22) tracks the reference signal yˆk generated by the reference model (23) well in the H∞ sense, which illustrates the effectiveness of the proposed H∞ output tracking controller design. Example 2. To illustrate the effectiveness of the proposed H∞ output tracking performance analysis for the augmented closed-loop system (12), we suppose the system matrices A,
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B1 , B2 , C, D are the same as the ones presented in (22), J and H are the same as the ones presented in (23), and · ¸ 0.0010 M1 = , M2 = −0.1246, (26) £ −0.0122 ¤ N1 = 0.0083 0.1181 , N2 = −0.0783
In this example, we suppose LM = 2, Lm = 0, the initial state of the augmented system (12) is ξ0 = [0 0 0]T , the controller gain K = [−0.4913 − 0.0330 0.0165], the scalar ε = 160. For simplicity, suppose the packets at the instants 0, 2, 4, · · · are transferred to the actuator successfully, that is one packet are dropped among every two packets. Solving the LMI presented in Theorem 2, we obtain the minimum H∞ output tracking performance γ = 3.1420. Suppose the input signals are given as follows 0, 0 ≤ k ≤ 11 12 < k ≤ 15 −0.25, 0.24, 16 < k ≤ 19 ωk = −0.1, 20 < k ≤ 21 0, 21 < k ≤ 59 0, 0 ≤ k ≤ 11 12 < k ≤ 15 0.07, −0.09, 16 < k ≤ 19 rk = (27) 0.07, 20 < k ≤ 21 21 < k ≤ 59 0, 0, 0 ≤ k ≤ 11 12 < k ≤ 15 1/(k + 1), 0, 16 < k ≤ 19 F(k) = 0, 20 < k ≤ 21 0, 21 < k ≤ 59
The output yk and yˆk (denoted as yrk ) of the system (12) are pictured in Fig. 2. From Examples 1 and 2, one can see that the H∞ output tracking performance of uncertain NCSs is a little worse than that of nominal NCSs. Fig. 2 also illustrates the effectiveness of the proposed H∞ output tracking performance analysis for uncertain NCSs. VI. CONCLUSIONS In this paper, the problems of H∞ output tracking performance analysis and controller design for discrete-time NCSs with time delay and packet dropout have been investigated. By defining new Lyapunov function and using the discrete Jensen inequality, LMI-based H∞ output tracking performance analysis and controller design for nominal NCSs are presented. The results are also extended to uncertain NCSs. The merit of the proposed methods lies in their less conservativeness, which is achieved by decoupling the Lyapunov and the system matrices. Numerical examples have illustrated the effectiveness of the proposed H∞ output tracking performance analysis and controller design.
[2] G. C. Walsh, H. Ye, and L. Bushnell, Stability analysis of networked control systems, IEEE Transactions on Control Systems Technology, vol. 10, no. 3, 2002, pp. 438-446. [3] W. Zhang, M. S. Branicky, and S. M. Phillips, Stability of networked control systems, IEEE Control Systems Magazine, vol. 21, no. 1, 2001, pp. 84-99. [4] H. J. Gao and T. W. Chen, New results on stability of discretetime systems with time-varying state delay, IEEE Transactions on Automatic Control, vol. 52, no. 2, 2007, pp. 328-334. [5] X. Li and C. E. de Souza, Delay-dependent robust stability and stabilization of uncertain linear delay systems: a linear matrix inequality approach, IEEE Transactions on Automatic Control, vol. 42, no. 8, 1997, pp. 1144-1148. [6] Y. L. Wang and G. H. Yang, Multiple communication channelsbased packet dropout compensation for networked control systems, IET Control Theory Applications, vol. 2, no. 8, 2008, pp. 717-727. [7] M. Wu, Y. He, J. H. She, and G. P. Liu, Delay-dependent criteria for robust stability of time-varying delay systems, Automatica, 40, 2004, pp. 1435-1439. [8] X. L. Zhu and G. H. Yang, Jensen integral inequality approach to stability analysis of continuous-time systems with time-varying delay, IET Control Theory Applications, vol. 2, no. 6, 2008, pp. 524-534. [9] J. L. Xiong and J. Lam, Stabilization of linear systems over networks with bounded packet loss, Automatica, 43, 2007, pp. 80-87. [10] H. Gao, J. Lam, C. Wang, and Y. Wang, Delay-dependent outputfeedback stabilisation of discrete-time systems with time-varying state delay, IEE Proc. Control Theory Applications, vol. 151, no. 6, 2004, pp. 691-698. [11] S. Y. Xu and T. W. Chen, H∞ output feedback control for uncertain stochastic systems with time-varying delays, Automatica, 40, 2004, pp. 2091-2098. [12] D. Yue, Q. L. Han, and J. Lam, Network-based robust H∞ control of systems with uncertainty, Automatica, 41, 2005, pp. 999-1007. [13] H. Ishii, H∞ control with limited communication and message losses, Systems & Control Letters, 57, 2008, pp. 322-331. [14] D. K. Kim, P. G. Park, and J. W. Ko, Output-feedback H∞ control of systems over communication networks using a deterministic switching system approach, Automatica, 40, 2004, pp. 1205-1212. [15] F. W. Yang, Z. D. Wang, Y. S. Hung, and M. Gani, H∞ control for networked systems with random communication delays, IEEE Transactions on Automatic Control, vol. 51, no. 3, 2006, pp. 511-518. [16] M. C. de Oliveira, J. Bernussou, J. C. Geromel, A new discrete-time robust stability condition, Systems & Control Letters, 37, 1999, pp. 261-265. [17] F. Liao, J. L. Wang, and G. H. Yang, Reliable robust flight tracking control: an LMI approach, IEEE Transactions on Control Systems Technology, vol. 10, no. 1, 2002, pp. 76-89. [18] R. Rebarber and G. Weiss, Internal model based tracking and disturbance rejection for stable well-posed systems, Automatica, 39, 2003, pp. 1555-1569. [19] N. C. Shieh, K. Liang, and C. Mao, Robust output tracking control of an uncertain linear system via a modified optimal linear-quadratic method, Journal of Optimization Theory and Applications, vol. 117, no. 3, 2003, pp. 649-659. [20] Y. L. Wang and G. H. Yang, Output tracking control for networked control systems with time delay and packet dropout, International Journal of Control, vol. 81, no. 11, 2008, pp. 1709-1719. [21] N. van de Wouw, P. Naghshtabrizi, M. Cloosterman, J. P. Hespanha, Tracking control for networked control systems, Proc. of the 46th IEEE Conference on Decision and Control, 2007, pp. 4441-4446. [22] H. J. Gao and T. W. Chen, Network-based H∞ output tracking control, IEEE Transactions on Automatic Control, vol. 53, no. 3, 2008, pp. 655-667. [23] X. F. Jiang, Q. L. Han, and X. H. Yu, Stability criteria for linear discrete-time systems with interval-like time-varying delay, Proc. of the American Control Conference, 2005, pp. 2817-2822. [24] D. W. C. Ho and G. Lu, Robust stabilization for a class of discrete-time nonlinear systems via output feedback: The unified LMI approach, International Journal of Control, vol. 76, no. 2, 2003, pp. 105-115.
R EFERENCES [1] H. Ishii and B. A. Francis, Stabilization with control networks, Automatica, 38, 2002, pp. 1745-1751.
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