Elsevier

Automatica

Volume 64, February 2016, Pages 76-85
Automatica

Brief paper
Predictive control of nonlinear continuous networked control systems with large time-varying transmission delays and transmission protocols

https://doi.org/10.1016/j.automatica.2015.11.001Get rights and content

Abstract

In this paper, we consider a class of globally Lipschitz nonlinear continuous networked control systems (NCS) incorporating large time-varying transmission delays and transmission protocols of communication networks with periodic sampling. To stabilize the NCS, we propose a new predictive control design scheme with plant outputs as the only available data. With uniformly globally exponentially stable (UGES) protocols, input-to-state stability of the entire NCS is ensured by small gain theory. In particular, the predictive controller can compensate transmission delays with any finite upper bound under the constraint that the sampling periods of the plant output and the observer output are small enough as well as the constraint that the predictor is accurate enough. The scheme is applied to a benchmark example to illustrate the effectiveness of our proposed method.

Introduction

It is well known that transmitting feedback data via a communication network brings a great deal of benefits and has attracted a lot of attention (Donkers et al., 2011, Heemels et al., 2010, Li et al., 2014, Nesic and Teel, 2004, Sun et al., 2014, Walsh et al., 2002). For instance, it enables low cost as well as easy installation and maintenance in practical applications. However, network-induced communication imperfections are also simultaneously introduced, for instance, time-varying transmission delays, quantization effects, communication constraints caused by sharing the common transmission channel with multiple nodes due to the consideration of transmission protocols or schedules, as well as packet dropouts (Donkers et al., 2011, Heemels et al., 2010), which may degrade the performance and even destabilize the NCS.

It is noted that a large number of excellent results on delayed systems have been reported in the existing literature, e.g. Cloosterman, van de Wouw, Heemels, and Nijmeijer (2009), Fridman, Seuret, and Richard (2004), Gu, Kharitonov, and Chen (2003), Niculescu (2001) and Pepe and Jiang (2006). An important approach to deal with delays is based on the compensation principle (Goodwin et al., 2014, Karafyllis, 2011, Karafyllis and Krstic, 2012, Karafyllis and Krstic, 2013, Krstic, 2009, Krstic, 2010, Liu et al., 2007, Mirkin and Raskin, 2003, Sun et al., 2014, Zhou et al., 2012). One of its main advantages is that a much larger upper bound of transmission delays can be allowed in the network at the cost of more computation in the controller. In Krstic, 2009, Krstic, 2010, Krstic proposes a prediction algorithm for continuous-time system with input delay. Then in the presence of a network, Karafyllis and Krstic introduce a new dynamic sampled-data predictive algorithm for a class of continuous-time nonlinear systems that can accommodate large constant time delays in Karafyllis and Krstic (2013). The predictive control algorithms for the linear NCS with transmission delays were reported in Liu et al. (2007) and Sun et al. (2014).

Transmission constraints related to the transmission protocol or schedules are one of the fundamental communication imperfections in the network summarized in Heemels et al. (2010). Stability of NCS with variable transmission intervals and transmission protocols is firstly considered in Walsh, Beldiman, and Bushnell (2001) and Walsh et al. (2002) for linear and nonlinear systems respectively, where a conservative maximum allowable transmission interval (MATI) is obtained with the transmission protocol being the Round Robin (RR) protocol or Try-Once-Discard (TOD) protocol. Then in Nesic and Teel (2004), Nesic and Teel model the NCS with transmission protocols as an impulsive system and derive the stability conditions by small gain theorem. The MATI obtained with this approach is less conservative than Walsh et al. (2001). Recently, Carnevale et al. construct a Lyapunov function for a hybrid system model (Carnevale, Teel, & Nesic, 2007) and establish a much less conservative result than Nesic and Teel (2004).

Combining variable transmission delays and transmission protocols into a framework is reported in Bauer, Maas, and Heemels (2012), Chaillet and Bicchi (2008), Donkers et al. (2011) and Heemels et al. (2010). Different approaches are employed to obtain less conservative MATI and maximal allowable delay (MAD). However, all these results rely on the assumption that the transmission delays cannot be larger than the transmission intervals. Large delay case is reported in Sun, Jiang, Liu, and Wang (2013), where the problem is dealt with based on the emulation approach. But the results related to the MATI and MAD are rather conservative. Therefore, it is a natural consideration to introduce delay compensation algorithm. Chaillet and Bicchi in Chaillet and Bicchi (2008) propose a model-based prediction scheme and the result can admit larger upper bound of delays. However, the predictor is assumed to be ideal and the RR protocol is precluded in the network. The work of Greco et al. in Greco, Chaillet, and Bicchi (2012) extends this approach to more relaxed cases including both the dynamic protocols and static protocols though only state feedback is dealt with.

In this paper, stabilization problem of NCS with the effects of large transmission delays and transmission protocols is considered under the case of periodic sampling. The output of the nonlinear plant, with its nonlinear function satisfying a global Lipschitz condition, is transmitted via the network with communication constraints and large time-varying transmission delays. External disturbances are also considered. The NCS architecture in this paper is given in Fig. 1, where, for simplicity, only networks existing in the channel from sensors to controllers are considered, which can be relaxed as discussed in Remark 10 of this paper. Such a model can also be found in many practical engineering problems, such as the vibration control (Li et al., 2014) and the active vibration control (Yu et al., 2015).

Compared with Greco et al. (2012), output feedback is considered and a novel network predictive control framework is established based on an explicit iteration algorithm for estimating states in the paper. A timestamp technique is adopted to transform the time-varying transmission delays into constant ones such that the iteration algorithm for the state estimate can be realized. Compared with Karafyllis and Krstic (2013), due to the effect of communication constraints, an additional error dynamic from protocols must be introduced and thus the whole NCS can be regarded as a large interconnected hybrid delayed system and has more complex behaviors, which may result in interactions among plant, error subsystem from protocols, error subsystem from observers and error subsystem from predictors. The following techniques are adopted to deal with the prediction problems of such a complex system. Firstly, we show the existence of the solution of the system. Secondly, for the error subsystem from protocols, we construct a novel Lyapunov function and conclude the input-to-state (ISS) property for it with the transmission protocol being UGES. Thirdly, stability properties of other subsystems are also established. Finally, by applying small gain technique, it is shown that, with the proposed predictive controllers, ISS property of the entire NCS is achieved with relatively larger transmission delays while the predictor numerical implementation could lead to small sampling periods and high requirement of accuracy.

This paper partly solves the open problem proposed in Heemels et al. (2010) since large transmission delays and transmission constraints are simultaneously considered. We show from theory that by adopting the proposed techniques of the paper, any finite transmission delays can be compensated with the constraints of smaller sampling intervals and higher prediction accuracy under the UGES protocols. Through a benchmark example, under the restriction of periodic sampling, we illustrate that the delay bound can be many times larger than the sampling period at the cost of more computation in the traditional ZOH-type observer while Heemels et al. (2010) is restricted to the small delay case which requires the delay must be not larger than the sampling period. Moreover, the delay bound can be allowed to be any finite positive number in the case of prediction-type observer. This also shows the advantages of the predictive control techniques in the paper compared with the emulation method in Heemels et al. (2010).

Throughout this paper, the following notations are used:

Symbols R and Z denote real set and integer set, respectively. R0[0,) and Z0{0,1,2,}. For a vector xRn, |x| denotes its Euclidean norm. For matrix ARm×n, |A| denotes its spectral norm. For xR, x denotes the minimum integer greater than or equal to x; [x] denotes the maximum integer less than or equal to x. C0([a,b];Rn) denotes the set of continuous functions x:[a,b]Rn. For function x:[ad,b)Rn with b>a0 and d0, Td(t)x denotes the “history” of x from td to t, that is (Td(t)x)(θ)x(t+θ),θ[d,0] for t[a,b). Similarly, we can define (T̆d(t)x)(θ)x(t+θ),θ[d,0). Let IR be an interval. L(I;U) denotes the space of measurable and bounded functions u() defined on I and taking values in URm. For xL([0,d];Rn), define |x|dsup0sd|x(s)|.

Section snippets

Model description

Consider the following nonlinear system ẋ(t)=Ax(t)+Bu(t)+f(x(t))+g(x(t),u(t))w(t)y(t)=Cx(t) where xRn, uRp and yRm are respectively the system state, the input and the output, w is the uncertain disturbance, the function f:RnRn is continuous with f(0)=0, satisfying Lipschitz condition |f(x1)f(x2)|Lf|x1x2| for any x1,x2Rn with Lf0 and g(x,u) is bounded with |g(x,u)|G for any xRn, uRp and G0.

Rewrite the dynamics of system (1) as follows ẋ(t)=(A+BK)x(t)+f(x(t))+B(u(t)Kx(t))+g(x(t),u

Design of predictive controller

In this section, we will design a predictive controller for system (1) to compensate for the delay caused by the network imperfections.

Main results

In the paper, for the most recently transmitted plant output on the controller side, we will give two types of implementations, that is, the traditional ZOH type and the prediction type in Postoyan and Nesic (2012), which implies two corresponding observers: ZOH-type observer and prediction-type observer. In this section, under the proposed predictive controller in the previous section, we will give stability analysis for the considered NCS based on the two types of observers, respectively.

Study of the batch reactor example

Consider the batch chemical reactor example, which is widely used in the literature, for example, Donkers et al. (2011), Heemels et al. (2010), Nesic and Teel (2004) and Walsh et al. (2002), ẋp(t)=Apxp(t)+Bpu(t)y(t)=Cpxp(t), where the matrices Ap,Bp and Cp are the same as that in Heemels et al. (2010) and Nesic and Teel (2004). By pole assignment with MATLAB toolbox, the controller gain and observer gain can be chosen as follows K=[0.17560.48460.03270.49911.42170.68080.33380.3991]L=[7.2236

Conclusion

In this paper, a network predictive control framework including observer and predictor is established to stabilize NCS with transmission protocols and large varying transmission delays. With the proposed predictive controllers, ISS property of the entire NCS is ensured subject to any finite upper bound of transmission delays under UGES protocols if the sampling periods for the plant output and the observer output are small enough and the predictor is accurate enough. It will be one of

Acknowledgments

We would like to express our sincere appreciation to Professor I. Karafyllis, the reviewers, the Associate Editor and the Editor for their valuable comments and suggestions to improve this paper.

Xi-Ming Sun received the M.S. degree in Applied Mathematics from Bohai University, China, in 2003, and the Ph.D. degree in Control Theory and Control Engineering from the Northeastern University, China, in 2006. From August 2006 to December 2008, he worked as a Research Fellow in the Faculty of Advanced Technology, University of Glamorgan, UK. He then visited the School of Electrical & Electronic Engineering, Melbourne University, Australia in 2009, and Polytechnic Institute of New York

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    Xi-Ming Sun received the M.S. degree in Applied Mathematics from Bohai University, China, in 2003, and the Ph.D. degree in Control Theory and Control Engineering from the Northeastern University, China, in 2006. From August 2006 to December 2008, he worked as a Research Fellow in the Faculty of Advanced Technology, University of Glamorgan, UK. He then visited the School of Electrical & Electronic Engineering, Melbourne University, Australia in 2009, and Polytechnic Institute of New York University in 2011, respectively. He is currently a Professor in the School of Control Science and Engineering, Dalian University of Technology, China. He was awarded the Most Cited Article 2006–2010 from the journal of Automatica in 2011. His research interests include switched delay systems, networked control systems, and nonlinear systems.

    Kun-Zhi Liu received the Bachelor degree in Automation from Harbin University of Science and Technology, Harbin, China, in 2011, and the Master degree in Control Theory and Control Engineering from Dalian University of Technology, Dalian, China, in 2013. He is pursuing his Ph.D. degree in Control Theory and Control Engineering from Dalian University of Technology. His main research interests are focused on Hybrid systems, delay systems and the Cyber–Physical Systems.

    Changyun Wen received the B.Eng. degree from Xi’an Jiaotong University, Xi’an, China, in 1983 and the Ph.D. degree from the University of Newcastle, Newcastle, Australia in 1990. From August 1989 to August 1991, he was a Research Associate and then Postdoctoral Fellow at University of Adelaide, Adelaide, Australia. Since August 1991, he has been with School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a Full Professor. His main research activities are in the areas of control systems and applications, intelligent power management system, smart grids, cyber–physical systems, complex systems and networks, model based online learning and system identification, signal and image processing.

    He is an Associate Editor of a number of journals including Automatica, IEEE Transactions on Industrial Electronics and IEEE Control Systems Magazine. He is the Executive Editor-in-Chief of Journal of Control and Decision. He served the IEEE Transactions on Automatic Control as an Associate Editor from January 2000 to December 2002. He has been actively involved in organizing international conferences playing the roles of General Chair, General Co-Chair, Technical Program Committee Chair, Program Committee Member, General Advisor, Publicity Chair and so on. He received the IES Prestigious Engineering Achievement Award 2005 from the Institution of Engineers, Singapore (IES) in 2005.

    He is a Fellow of IEEE, was a member of IEEE Fellow Committee from January 2011 to December 2013 and a Distinguished Lecturer of IEEE Control Systems Society from February 2010 to February 2013.

    Wei Wang obtained the B.S., M.S., and Ph.D. Degree in Industrial Automation from Northeastern University, Shenyang, China, in 1982, 1986, and 1988, respectively. He is currently a Professor of the Department of Automation and Director of the Research Center of Information and Control at the Dalian University of Technology, Dalian, China. His research interests are in adaptive control, predictive control, nonlinear system control, robotics, computer-integrated manufacturing systems, and computer control of industrial process. He has been an Associate Editor of Control Engineering Practice since 2007 and of Information Sciences since 2010. He has published over 200 papers in international and domestic journals.

    He was the Industrial Process Control (IPC) Chairman of the 6th World Congress on Intelligent Control and Automation in 2006 and the IPC Chairman of the International Federation of Automatic Control (IFAC) Symposium on Cost Oriented Automation in 2007. He was the Chairman of the IFAC Technical Committee (4.4) of Cost Oriented Automation (2005–2008) and has been a member of the IFAC Technical Committee of Mining, Mineral and Metal Processing since 1999.

    This work was supported by the National Natural Science Foundation of China under Grant 61174058. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Nicolas Petit under the direction of Editor Miroslav Krstic.

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