Elsevier

Automatica

Volume 43, Issue 5, May 2007, Pages 903-911
Automatica

Brief paper
Sampled-data control of networked linear control systems

https://doi.org/10.1016/j.automatica.2006.11.015Get rights and content

Abstract

In this paper, the problem of synthesis and analysis for the networked control systems (NCSs) with time-driven digital controllers and event-driven holders is considered. The NCS is modelled as a sampled-data system with time-delay in its discrete-time subsystem. This model is able to capture many network-induced features, for example, time-delay and packet dropout. Moreover, the model allows different combinations of the time-driven or event-driven mode of the devices, including the samplers, the controllers and the holders. By transforming time-delay in the discrete-time subsystem into its continuous-time subsystem of the sampled-data system, we have also obtained a less conservative time-delay dependent stability result for the NCSs, using a new Lyapunov function and a relaxed condition. Some limitations of the existing literatures on network-induced time-delay and sampling period are removed in the proposed framework. Furthermore, a sampled-data control design procedure is developed for the NCSs. Linear matrix inequality approach has been employed to solve the stability and control design problems. Finally, numerical examples are included to demonstrate the effectiveness of the proposed stability result and the potential of the proposed techniques.

Introduction

Nowadays, networks play important roles in many industrial control applications, especially for example, complex control systems and remote control systems. Comparing with the conventional point-to-point control systems, the networked control systems (NCSs) show some nice features, such as, flexibility of operation, ease of diagnosis and maintenance, small volume of wiring, low cost, etc. In the NCSs, all devices (the samplers, the digital controllers and the holders) are distributed, and connected by the networks. Because of the limitation of the network resource, time-delay caused by data transmission and/or packet drop will inevitably degrade control performance of the NCSs, or even cause the control systems instable (Branicky, Phillips, & Zhang, 2002; Halevi & Ray, 1988; Nilsson, 1998; Walsh, Ye, & Bushnell, 2002; Zhang, Branicky, & Phillips, 2001). Recently, the network-induced time-delay has been received widely attentions. In Halevi and Ray (1988) and Nilsson (1998), the authors systematically considered synthesis and analysis of the NCSs with synchronization, message rejection and vacant sampling. In the case of periodic time-delay, the sufficient and necessary condition for the systems with uniformly asymptotic stability was established. Using stochastic Lyapunov function, in Krtolica, Oguner, and Chan (1994), the sufficient and necessary condition of mean-square exponential stability for the linear discrete-time systems with a time-invariant controller in the place and time-varying time-delay modelled as a Markov chain was set up. An exponential stability condition for the systems with random one-step time-delay was also obtained using the deterministic structural perturbation approach. In Nilsson (1998) and references therein, the authors systematically investigated the modelling and analysis problems for NCSs under an assumption that the time-delay from sensor to actuator is less than one sampling period. In the work, the author also considered the NCSs with stochastic time-delay.

Besides the network-induced time-delay, the data transmission mechanism is also an important issue for NCSs. Considering two different network protocols, the try-once-discard/maximum-error-first and the statically scheduled protocols, two globally exponential stability conditions for the NCSs with respect to multiple-packet and one-packet transmission cases were presented in Walsh et al. (2002). In the paper, the authors presented a conservative bound of the allowable maximum transfer interval using the perturbation theory. In Zhang et al. (2001), the stability problems for the NCSs modelled as hybrid or asynchronous dynamical systems with network-induced delay, packet dropout and multi-packet transmission between sensor and controller were considered. Considering that the power spectral density of the output is a function of the dropout probability, the upper bound of the packet dropout rates was obtained under which the NCSs are stochastically asymptotically stable in Ling and Lemmon (2002).

Basically, NCS is hybrid, which involves a continuous plant and event-driven or time-driven devices (digital controller, holder and sampler) and networks. Hybrid nature of NCSs makes the synthesis and analysis problems for NCSs difficult. This kind of the systems is usually referred as the sampled-data control systems which simultaneously contain continuous-time and discrete-time signals (Boukas, 2005; Boukas & Al-Muthairi, 2006; Chen & Francis, 1995; Chen, Lam, & Xu, 2006; Shi, 1998; Shi, Boukas, Agarwal, & Shue, 1998; Shi, de Souza, & Xie, 1997; Shi, Fu, & de Souza, 1996; Xie, Shi, & de Souza, 1993). Sampled-data control system provides a direct approach to design digital controller for the continuous-time systems without approximations, see for example, Hu, Lam, Cao, and Shao (2003). Sampled-data control system formulation has been recognized a modelling method for NCSs for years, see Walsh et al. (2002) and Zhang et al. (2001), for example. In Lian, Moyne, and Tilbury (2003), the authors considered a modelling problem of NCSs for multivariable linear systems with distributed asynchronous sampling. In Zhang et al. (2001), the authors considered an analysis problem for the NCSs under hybrid system framework. In Walsh et al. (2002) and Zhang et al. (2001), the authors set up a stability result for the NCSs with only one time-delay. In the paper (Halevi & Ray, 1988), the authors proved that two delays cannot be lumped together for the cases referring as the so-called message rejection and vacant sampling even though feedback controller is time-invariant. Conventionally, the lifting technique (Chen & Francis, 1995) and the traditional discretized method are used as tools to consider the synthesis and analysis problems for NCSs (Walsh et al., 2002). However, the lifting technique only works for LTI systems, and the traditional discretized method has approximation. In our previous works, a hybrid system synthesis approach was set up for the sampled-data systems. Moreover, a series works on stability, robust control, H2 control, H control (filtering) and model predictive control problems for the sampled-data control systems (with time-delay) have been investigated (Hu, Cao, & Shao, 2002; Hu & Huang, 2005; Hu, Huang, & Cao, 2004; Hu et al., 2003; Hu, Shi, & Frank, 2006). However, the methods developed (Hu et al., 2002, Hu et al., 2003, Hu et al., 2004, Hu et al., 2006; Hu & Huang, 2005) do not work for the sampled-data systems with time-delay in its discrete-time subsystems. In a new work (Fridman, Seuret, & Richard, 2004), this kind of sampled-data systems was considered. In that paper, the authors proposed a robust control method by transforming the sampled-data system into a continuous-time system with control time-delay. However, this transformation is under a condition: the sampling period is infinite small.

In this paper, a sampled-data system with time-delay in its discrete-time subsystem is proposed to model the NCSs with time-driven digital controllers and event-driven holder devices. The configuration shows that this model is quite general, it can capture many network-induced features, for example, time-delay, packet dropout, multi-packet transmission. The sampled-data system formulation also provides a nature way to describe the so-called “communication sequence”. Moreover, it allows different combinations of the time-driven or event-driven mode of the devices, including the samplers, the controllers and the holders. It is also able to remove the assumption that the time-delay from sensor to actuator is less than one sampling period used in Nilsson (1998). By transforming time-delay in the discrete-time subsystem into its continuous-time subsystem of the sampled-data system, a less conservative time-delay dependent stability result is obtained for the NCSs by using the new Lyapunov function and a special relaxed condition, which removes the limitation on the sampling period imposed in Fridman et al. (2004). Linear matrix inequality (LMI) formulation of a sampled-data control design procedure is then presented for the NCSs, which is solvable using Toolbox for Matlab. Finally, numerical examples are given to show the effectiveness of the proposed method.

Notations: Throughout this paper, for symmetric matrix X and Y, the notation XY (respectively, X>Y) means that the matrix X-Y is positive semi-definite (respectively, positive definite). I is the identity matrix with appropriate dimension. {tk,k=1,2,,} are the sampling instants satisfying tk+1-tk=Ts, where Ts denotes the sampling period with which outputs of a plant are synchronously measured by ideal samplers. u(·) and u˜[·] are used to denote the continuous-time and the discrete-time signal of the associated variable, respectively.

Section snippets

Modeling and problem formulation

Consider an NCS shown as in Fig. 1, in which a continuous plant P is controlled by a digital controller C. In the system, the output signals x(t) of the plant are synchronously measured with ideal samplers S at a sampling rate 1/Ts. The digital controller uses the information of the plant transmitted through the networks to generate a digital control action u˜[tk]. The control action u˜[tk] is then transmitted via the networks again at the rate 1/Ts, held by a zero-order holder H to drive the

Stability analysis

In what follows in this section, we will present a delay-dependent Lyapunov function to ensure the stability of NCS (9), (10) with F=0 and possible maximum time-delay d0 satisfying dd0.

Theorem 2

The NCS(9), (10)withF=0is stable for delaydd0, if there exist matricesP>0, Q1>0, Q2>0, Θ>0, T=[T1,T2,T3,T4], N such that the following matrix inequalities hold:ΘNNTQ20,M+TTA^+A^TT+ΓTNT+NΓ+dΘ<0,ΞA¯TPA¯-P<0,whereM=Q10P00-Q100P0dQ200000,A^=A˜B˜-I0,Γ=I-I0-I.

Proof

Recall the fact that the control action u(t) is only

Control design

In this section, we present a method to design a control (2) such that the NCS (1), (2), (3) stable.

Theorem 4

If there exist matricesP>0, Y andY1, F¯, T=[T1,T2,T3,T4], N satisfying the following matrix inequalities:-P+A¯TPA¯A¯TPB¯C¯TF¯TC¯TF¯TB¯TPA¯-Y00F¯C¯0-Y1-Y1T+Y0F¯C¯00-Y1-Y1T+B¯TPB¯<0and(11), (12), then the NCS(1), (2), (3)is stabilized via a state feedback controller, where the gain can be calculated byF=Y1-1F¯whenever the loop delay satisfiesd<d0.

Proof

Eq. (13) can be expressed as -P+(A¯+B¯FC¯)TP(A¯+B¯FC

Numerical example

Example 1

Consider the following system described (Zhang, 2001)x˙(t)=010-0.1x(t)+00.1u(t),y(t)=01x(t),which is controlled by a controller u(t)=-[3.7511.5]x(t). If this controller is implemented with networks, the allowable maximum time-delay in the control loop is 2.7×10-4s given by Walsh, Beldiman, and Bushnell (1999). The results given by Zhang (2001) are 4.5×10-4s and 0.0593s. However, using Theorem 2, we obtain the allowable maximum time-delay guaranteeing stability is 0.699s. This result is much

Conclusion

In this paper, a networked-control system (NCS) with a continuous-time plant, a time-driven digital controller and a event-driven holder device is considered. This system is modelled as a sampled-data one with discrete-time delay. The model shows some nice features to describe many cases of the NCS. By transferring time-delay in the discrete-time subsystem to its continuous counterpart, an equivalent sampled-data system with continuous-time delay is obtained with no limitation on the sampling

Acknowledgments

The authors are grateful to the anonymous reviewers for their valuable comments and suggestions that helped improve the presentation of the paper. This work was partially supported by Natural Science Foundation of China (60474020 and 60574049), and Natural Science Foundation of Shan Xi Province (20051020). Peng Shi also gratefully acknowledges the support from Harbin Institute of Technology; Nanjing University of Aeronautics and Astronautics; and the LCSIS, Institute of Automation, Chinese

Li-Sheng Hu received his B.Sc. degree in engineering science, and his Ph.D. degree in industrial automation, both from Zhejiang University, China in 1986 and 1988, respectively.

He was a postdoctoral fellow in Automation Engineering, Shanghai Jiao Tong University, Shanghai, China, from 1998 to 2000, and in Chemical Engineering, University of Alberta, Canada, from 2002 to 2003. He is now a Professor in the Department of Automation, Shanghai Jiao Tong University. His current research interests are

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    Li-Sheng Hu received his B.Sc. degree in engineering science, and his Ph.D. degree in industrial automation, both from Zhejiang University, China in 1986 and 1988, respectively.

    He was a postdoctoral fellow in Automation Engineering, Shanghai Jiao Tong University, Shanghai, China, from 1998 to 2000, and in Chemical Engineering, University of Alberta, Canada, from 2002 to 2003. He is now a Professor in the Department of Automation, Shanghai Jiao Tong University. His current research interests are in the areas of robust model prediction control, process monitoring, control performance limitation and performance assessment, and sampled-data control.

    Tao Bai received her M.S. degree in industry automation, and her Ph.D. degree in control engineering from Shanghai Jiao Tong University in 2001 and 2005, respectively. She was with the Department of Automation, Tai Yuan Heavy Machinery Institute, and is now with the Department of Automation, Shanghai Jiao Tong University. Her research interests include industrial networked system modeling, control and optimization.

    Peng Shi received his B.Sc. degree in mathematics from Harbin Institute of Technology, China in 1982, the M.E. degree in control theory from Harbin Engineering University, China in 1985, the Ph.D. degree in electrical engineering from the University of Newcastle, Australia in 1994, and the PhD degree in mathematics from the University of South Australia in 1998. He was also awarded the degree of Doctor of Science by the University of Glamorgan, UK in 2000.

    From 1985 to 1989, Dr. Shi was a lecturer in Heilongjiang University. He held visiting fellow position in the University of Newcastle, Australia from 1989 to 1990. He was postdoctorate from 1995 to 1997, and lecturer from 1997 to 1999, in the University of South Australia. He worked in the Defence Science and Technology Organisation, Department of Defence, Australia from 1999 to 2003, as research scientist, senior research scientist and task manager. In 2004, he joined the University of Glamorgan, United Kingdom, as professor. Dr. Shi's research interests include fault detection and tolerant control, intelligent systems and information processing, robust control and filtering, and operations research. He has published a number of papers in these areas. He is a co-author (with W. Assawinchaichote and S. Nguang) of the book Fuzzy Control and Filtering Design for Uncertain Fuzzy Systems (Berlin, Springer, 2006), and a co-author (with M. Mahmoud) of the book Methodologies for Control of Jump Time-Delay Systems (Boston, Kluwer, 2003).

    Dr. Shi serves as Editor-in-Chief of International Journal of Innovative Computing, Information and Control, and as Regional Editor of International Journal of Nonlinear Dynamics and Systems Theory. He is also an Associate Editor for several other journals, such as IEEE Transactions on Systems, Man and Cybernetics-B, and IEEE Transactions on Fuzzy Systems. Dr. Shi is a fellow of Institute of Mathematics and its Applications (UK), and a senior member of the IEEE.

    Zhiming Wu served as a full professor in the Department of Automation, Shanghai Jiaotong University from 1985. His main research interests include discrete event system, artificial intelligence, soft-computing and their application in manufacturing engineering.

    This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Ioannis Paschalidis under the direction of Editor Ian Petersen.

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