IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 10, OCTOBER 2010
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H∞ Control for Networked Predictive Control Systems Based on the Switched Lyapunov Function Method Rui Wang, Guo-Ping Liu, Wei Wang, David Rees, and Yun-Bo Zhao
Abstract—This paper investigates the problem of H∞ control for a class of networked control systems (NCSs) with time-varying delay in both forward and backward channels. Combined with the switched Lyapunov function technique, an improved predictive controller design strategy is proposed to compensate for the delay and data dropout to achieve the desired control performance. Based on these methods, the controllers can be designed to guarantee that the closed-loop system is asymptotically stable with an H∞ -norm bound in terms of nonlinear matrix inequalities. An iterative algorithm is presented to solve these nonlinear matrix inequalities to obtain a suboptimal minimum disturbance attenuation level. Numerical simulations and a practical experiment are given to illustrate the effectiveness of the proposed method. Index Terms—H∞ control, networked control systems (NCSs), predictive control, switched Lyapunov function, switched systems.
I. I NTRODUCTION
R
ECENT YEARS have witnessed a rapidly growing interest in networked control systems (NCSs), in which the control loops are closed via a communication network [1]– [10]. However, the insertion of the communication network will inevitably lead to network-induced time delay and data packet dropout, which are potential sources of instability and poor performance in the NCSs. Thus, it is important to overcome the adverse influences of time delay and packet dropout. There are a number of design methods that have been proposed to deal with this problem [11]–[20], e.g., the queuing method [11], the optimal stochastic control method [12], the sampling time
Manuscript received July 9, 2009; revised October 5, 2009; accepted November 19, 2009. Date of publication December 31, 2009; date of current version September 10, 2010. This work was supported in part by the China Postdoctoral Science Foundation under Project 200902539 and in part by the National Natural Science Foundation of China under Grants 60804011 and 60934006. R. Wang is with the Research Center of Information and Control, Dalian University of Technology, Dalian 116024, China, and also with the Faculty of Advanced Technology, University of Glamorgan, Pontypridd, CF37 1DL, U.K. (e-mail:
[email protected]). G.-P. Liu is with the Faculty of Advanced Technology, University of Glamorgan, Pontypridd, CF37 1DL, U.K., and also with the CTGT Center, Harbin Institute of Technology, Harbin 150001, China. W. Wang is with the Research Center of Information and Control, Dalian University of Technology, Dalian 116024, China. D. Rees is with the Faculty of Advanced Technology, University of Glamorgan, Pontypridd, CF37 1DL, U.K. Y.-B. Zhao was with the Faculty of Advanced Technology, University of Glamorgan, Pontypridd, CF37 1DL, U.K. He is now with the University of Glasgow, Glasgow, G12 8QE, U.K. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2009.2038341
scheduling method [13], and the hybrid system stability analysis method [14]. However, in these control methods, the system just passively accepts the presence of the delay in the network rather than actively compensating for it. To overcome the negative impact of the network delay on system stability and performance, a networked predictive control (NPC) scheme, which is an active control strategy and includes a control prediction generator (CPG) at the controller side and a network delay compensator (NDC) at the actuator side, has been shown to be an effective method of addressing this problem [21]– [24]. However, there are limitations on the work reported in the publications, as follows: 1) A fixed controller gain was used so that this results in a significant conservative design because the controller gain does not reflect the range of possible delay in the network; 2) the controller gain design problem was not considered; 3) the H∞ performance index was not taken into account; and 4) a common Lyapunov function method was used, but this method is very conservative, and most switched systems do not possess a common Lyapunov function. Motivated by these limitations, some new techniques need to be developed. To overcome these limitations, an improved delay compensation strategy for NCS systems is proposed in this paper based on the switched Lyapunov function method, which is an effective method in the study of discrete switched systems under arbitrary switching [25]–[28]. In this paper, we study the design problem of the H∞ controller for a class of NCSs with network delay and data dropout, which are present in both forward and feedback channels. The switched Lyapunov function technique is combined with an improved predictive controller scheme, in which the controller gain varies with the delay to make the corresponding closed-loop system asymptotically stable with an H∞ -norm bound. In contrast with some existing results on NPCs, which are based on the fixed controller gain approach and the common Lyapunov function method, the contributions of this paper are as follows: First, the adoption of the varying feedback controller gain method and the switched Lyapunov function technique can lead to less conservative results. Second, an iterative algorithm is presented to design the desired controllers with a suboptimal minimum disturbance attenuation level. This paper is organized as follows: In Section II, preliminaries and problem formulations are introduced. Section III gives the sufficient condition of H∞ control and the controller design algorithm. Section IV provides two examples to show the effectiveness of the proposed method. Conclusions are summarized in Section V.
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 10, OCTOBER 2010
imply that the delay to the control system is upper bounded by N = N1 + N2 + L1 + L2 . 1) H∞ Control Problem: Given a real number γ > 0, system (1) is said to be stabilizable with an H∞ -norm bound γ if there exists a state feedback law ut such that two conditions are satisfied. 1) The closed-loop system (1) is asymptotically stable when wt = 0. 2) Under zero initial conditions, the controlled output zt satisfies K
Fig. 1. NPC system structure (CPG and NDC).
t=0
II. S YSTEM D ESCRIPTION AND P RELIMINARIES
K
γ 2 wtT wt
t=0
for all K and any nonzero wt ∈ L2 (0, +∞).
In this paper, ∗ denotes the symmetric block in one symmetric matrix. I denotes the identity matrix of appropriate dimension. The trace of a matrix is denoted by tr(·). The NCS system structure considered in this paper is shown in Fig. 1, where f and k are backward and forward channel delays, respectively. The plant is described in the following discrete-time state-space form: xt+1 = Axt + But + Ewt
III. H∞ C ONTROL U SING THE P REDICTIVE C ONTROLLER FOR NCS S A. Prediction of the Future Control Sequence Since the system states are normally unavailable, the following state observer is then designed: xt + But + L(yt − C x ˆt ) x ˆt+1 = Aˆ
yt = Cxt zt = Dxt
ztT zt
0, if there exist positive ¯ , such that for ∀i, j ∈ N ¯ definite matrices Pi > 0, i ∈ N T ¯ ¯ ¯TD A¯i Pj A¯i − Pi + D A¯Ti Pj E
