Elsevier

Automatica

Volume 92, June 2018, Pages 9-17
Automatica

Brief paper
Stochastic self-triggered model predictive control for linear systems with probabilistic constraints

https://doi.org/10.1016/j.automatica.2018.02.017Get rights and content

Abstract

A stochastic self-triggered model predictive control (SSMPC) algorithm is proposed for linear systems subject to exogenous disturbances and probabilistic constraints. The main idea behind the self-triggered framework is that at each sampling instant, an optimization problem is solved to determine both the next sampling instant and the control inputs to be applied between the two sampling instants. Although the self-triggered implementation achieves communication reduction, the control commands are necessarily applied in open-loop between sampling instants. To guarantee probabilistic constraint satisfaction, necessary and sufficient conditions are derived on the nominal systems by using the information on the distribution of the disturbances explicitly. Moreover, based on a tailored terminal set, a multi-step open-loop MPC optimization problem with infinite prediction horizon is transformed into a tractable quadratic programming problem with guaranteed recursive feasibility. The closed-loop system is shown to be stable. Numerical examples illustrate the efficacy of the proposed scheme in terms of performance, constraint satisfaction, and reduction of both control updates and communications with a conventional time-triggered scheme.

Introduction

Networked control systems are usually subject to constraints and uncertainties. The constraints include not only the traditional system constraints, such as state constraints, but also communication constraints, such as a limited bandwidth in wireless communication networks. For such systems, an integrative model predictive control (MPC) and event-based control approach is a natural idea which could ensure the system constraint satisfaction and trade off the performance of control systems and the usage of communication resources. Thus, the research of event-based MPC is of great interest.

Two specific types of event-based control are event-triggered and self-triggered control. Different from event-triggered control which requires the continuous monitoring of system states, self-triggered control determines the next update time in advance based on the information at the current sampling instant. Also, self-triggered control allows the shut-down of the sensors between two updates, resulting in a lower sampling frequency to prolong the lifespan of sensors powered by batteries. Please refer to Heemels, Johansson, and Tabuada (2012) and Hetel et al. (2017) for an overview of event-based control.

This paper considers a self-triggered implementation of stochastic MPC (SMPC) for linear systems with stochastic disturbances. One main feature of SMPC is the presence of probabilistic constraints, which require the constraints to be satisfied with given probability thresholds. Such constraints can mitigate the conservativeness introduced by hard constraints of robust MPC (RMPC). SMPC has found applications in diverse fields, e.g., building climate control (Long, Liu, Xie, & Johansson, 2014) or chemical processes (Qin & Badgwell, 2003). To the best of our knowledge, stochastic self-triggered MPC (SSMPC) has not been explored up to now. One remarkable challenge is how to characterize the ‘propagation’ of uncertainties during two sampling instants and formulate a computationally tractable optimization problem for determining sampling instants and control design.

Some developments of self-triggered MPC are available. Many of these results are proposed for systems without uncertainties Barradas Berglind et al. (2012), Hashimoto et al. (2017), Henriksson et al. (2012). For systems with uncertainties, most results account for the synthesis of self-triggered control and RMPC which aims to guarantee robust constraint satisfaction. The interested reader can refer to Aydiner, Brunner, and Heemels (2015) and Brunner et al. (2014), Brunner et al. (2016). By maximizing the inter-sampling time subject to constraints on the cost function, a robust self-triggered MPC (RSMPC) algorithm is presented for constrained linear systems with bounded additive disturbances in Brunner et al. (2014), which employs the robust Tube MPC method in Mayne, Seron, and Raković (2005) to guarantee constraint satisfaction. In Brunner et al. (2014), all constraint parameters are determined by fixing the maximal inter-sampling time, which has the drawback of leading to a conservative region of attraction. To alleviate the conservatism, a RSMPC algorithm based on a more advanced Tube method (Raković, Kouvaritakis, Findeisen, & Cannon, 2012) is proposed in Aydiner et al. (2015), where the cost function is defined depending on the length of the inter-sampling time such that the constraint parameters are not affected by the maximal sampling interval. By combining with the self-triggering mechanism in Aydiner et al. (2015), a recent RSMPC method is presented in Brunner et al. (2016) with the focus of extending the Tube method in Chisci, Rossiter, and Zappa (2001) to evaluate the effect of the uncertainty on the prediction of the self-triggered setup.

Inspired by Aydiner et al. (2015) and Brunner et al. (2016), we design a self-triggered strategy for SMPC. Notice that inherent differences between SMPC and RMPC make our SSMPC algorithm largely different from the ones presented in Aydiner et al. (2015) and Brunner et al. (2016). Following the ideas of Tube MPC (Kouvaritakis, Cannon, Raković, & Cheng, 2010), we construct stochastic tubes as tight as possible by explicitly using the distributions of the disturbances. Since a crucial assumption of feedback at every time step in Kouvaritakis et al. (2010) is not satisfied in the self-triggered setting (which allows open-loop operations between sampling instants), some appropriate and non-trivial modifications are needed: (i) by considering the multi-step open-loop operation between control updates, three predicted controllers are defined for different phases of the prediction horizon, making it more complex than (Kouvaritakis et al., 2010) to evaluate the effect of the uncertainty on predictions and construct equivalent deterministic constraints; (ii) the inter-sampling time as an optimizing variable is included in the cost function and a tuning parameter is introduced to provide a trade-off between performance and communication; (iii) an improved terminal set, which is adapted to different inter-sampling times, is designed to make the constraints recursively feasible.

The present paper is the first work on SSMPC, which extends the existing literatures on MPC considerably. The main contributions are summarized in the following. (i) Our joint design of the self-triggering mechanism and the SMPC controller effectively reduces the amount of communication, while guaranteeing control performance with specific level of trade-off. (ii) The MPC optimization problem is transformed into a tractable quadratic programming problem by using information on the disturbance distribution. (iii) For the self-triggering mechanism, the probability of constraint violation can be tight to the specified limit. (iv) Both recursive feasibility and closed-loop stability are guaranteed. To illustrate the effectiveness of the algorithm, numerical experiments are carried out to compare the proposed SSMPC with a periodically-triggered SMPC (PSMPC), RSMPC, and unconstrained MPC (LQR).

The remainder of this paper is structured as follows. Problem formulation is set up in Section 2. In Section 3, a multi-step open-loop MPC optimization problem is formulated incorporating probabilistic constraints and specific terminal sets. In Section 4, a SSMPC algorithm is developed and main results are established. Section 5 presents numerical simulations and Section 6 concludes.

Notation 1.1

Let N{0,1,}. For some q,sN and q<s, let Nq, N>q, Nq, N<q, and N[q,s] denote the sets {rNrq}, {rNr>q}, {rNrq}, {rNr<q}, and {rNqrs}, respectively. Let I and 0 denote an identity matrix and a zero matrix or zero vector of appropriate dimension. When , , <, >, and || are applied to vectors, they are interpreted element-wise. For WRn×n, W0 means that W is symmetric and positive definite. For xRn and W0, xW2xTWx. For xiRn, iN, define i=abxi=0 if a>b. Pr denotes the probability, E the expectation, Ek the conditional expectation of a random variable given the state at time k, and (k+i|k) a prediction of a variable i steps ahead from time k.

Section snippets

Problem formulation

The self-triggered MPC framework of this paper is shown in Fig. 1, in which the notations are introduced below. Consider a linear time-invariant system x(k+1)=Ax(k)+Bu(k)+w(k),kN,where x(k)RNx is the state, u(k)RNu the control input, w(k)RNw the stochastic disturbance, and (A,B) a stabilizable pair. Notice that Nx=Nw. Assume that w(k), kN, are independent and identically distributed (i.i.d.) and the elements of w(k) have zero mean. The distribution Fi of the ith element of w(k) is assumed

Optimization problem formulation

In this section, the problem described in Section 2 is formulated to a computationally tractable MPC optimization problem with a fixed inter-sampling time MN[1,N1]. Define the prototype optimization problem PoM(c(kj)) as follows. minc(kj)JM(c(kj))1αi=0M1Ekj[x(kj+i|kj)Q2+u(kj+i|kj)R2ss]+i=MEkj[x(kj+i|kj)Q2+u(kj+i|kj)R2ss]subject to (4), (5), (6) with Mj=M, z(kj|kj)=x(kj), and iNM2:z(kj+i+1|kj)=Az(kj+i|kj)+Bu(kj+i|kj)iN:x(kj+i+1|kj)=Ax(kj+i|kj)+Bu(kj+i|kj)+w(kj+i)iN1:Pr

Stochastic self-triggered MPC

Using the above MPC optimization problem as a basis, a SSMPC algorithm is designed in this section. In the self-triggered setup, the goal at each sampling instant kj is to decide not only c(kj) but also the next sampling instant kj+1. To reduce the computation and communication, we need to find the largest Mj such that PMj(c(kj)) is feasible for some c(kj)FMj(x(kj)) while still maintaining certain performance of the closed-loop system. Define the self-triggered MPC problem S(x(kj)) as Mjmax{M

Numerical example

Simulation studies are provided to show the effectiveness and the advantages of the proposed SSMPC in comparison with PSMPC (by setting Mj=1), RSMPC (by setting p=1), and the unconstrained LQR control. Consider a linearized DC–DC converter system as in Lorenzen, Dabbene, Tempo, and Allgöwer (2017), x(k+1)=10.00750.1430.996x(k)+4.7980.115u(k)+w(k)subject to Pr{[10]x(k)2}0.8. Elements of w(k) are assumed to be i.i.d. truncated Gaussian random variables with zero mean, variance 0.042, and

Conclusion

We proposed a SSMPC strategy for the stabilization of systems with additive disturbances and probabilistic constraints. It was shown that the required amount of communication was reduced while simultaneously guaranteeing a specific performance loss when compared with a periodically-triggered scheme. By taking the disturbances occurring during the inter-sampling period into account and making use of their probability distribution, a set of deterministic constraints and terminal sets were

Acknowledgments

This work of Li Dai and Yuanqing Xia was supported by the National Natural Science Foundation of China under Grant 61603041, the Beijing Natural Science Foundation under Grant 4161001, the National Natural Science Foundation Projects of International Cooperation and Exchanges under Grant 61720106010, and Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant 61621063. The work of Yulong Gao and Karl Henrik Johansson was supported in part by the

Li Dai was born in Beijing, China, in 1988. She received the B.S. degree in Information and Computing Science in 2010 and the Ph.D. degree in Control Science and Engineering in 2016 from Beijing Institute of Technology, Beijing, China. Now she is an assistant professor at the School of Automation of Beijing Institute of Technology. Her research interests include model predictive control, distributed control, data-driven control, stochastic systems, and networked control systems.

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Li Dai was born in Beijing, China, in 1988. She received the B.S. degree in Information and Computing Science in 2010 and the Ph.D. degree in Control Science and Engineering in 2016 from Beijing Institute of Technology, Beijing, China. Now she is an assistant professor at the School of Automation of Beijing Institute of Technology. Her research interests include model predictive control, distributed control, data-driven control, stochastic systems, and networked control systems.

Yulong Gao received the B.S. degree in Automation in 2013 and the M.S. degree in Control Science and Engineering in 2016 from Beijing Institute of Technology, Beijing, China. Now he is a Ph.D. student at the School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology. His research interests include model predictive control and stochastic control with application in traffic systems and networked control systems.

Lihua Xie received the B.E. and M.E. degrees in electrical engineering from Nanjing University of Science and Technology in 1983 and 1986, respectively, and the Ph.D. degree in electrical engineering from the University of Newcastle, Australia, in 1992. Since 1992, he has been with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, where he is currently a professor and Director, Delta-NTU Corporate Laboratory for Cyber-Physical Systems. He served as the Head of Division of Control and Instrumentation from July 2011 to June 2014. He held teaching appointments in the Department of Automatic Control, Nanjing University of Science and Technology from 1986 to 1989.

Dr. Xie’s research interests include robust control and estimation, networked control systems, multi-agent networks, localization and unmanned systems. He is an Editor-in-Chief for Unmanned Systems and an Associate Editor for IEEE Transactions on Network Control Systems. He has served as an editor of IET Book Series in Control and an Associate Editor of a number of journals including IEEE Transactions on Automatic Control, Automatica, IEEE Transactions on Control Systems Technology, and IEEE Transactions on Circuits and Systems-II. He is an elected member of Board of Governors, IEEE Control System Society (Jan 2016–Dec 2018). Dr. Xie is a Fellow of IEEE and Fellow of IFAC.

Karl Henrik Johansson is Director of the Stockholm Strategic Research Area ICT The Next Generation and Professor at the School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology. He received M.Sc. and Ph.D. degrees in Electrical Engineering from Lund University. He has held visiting positions at UC Berkeley, Caltech, NTU, HKUST Institute of Advanced Studies, and NTNU. His research interests are in networked control systems, cyber-physical systems, and applications in transportation, energy, and automation. He is a member of the IEEE Control Systems Society Board of Governors and the European Control Association Council. He has received several best paper awards and other distinctions, including a ten-year Wallenberg Scholar Grant, a Senior Researcher Position with the Swedish Research Council, the Future Research Leader Award from the Swedish Foundation for Strategic Research, and the triennial Young Author Prize from IFAC. He is member of the Royal Swedish Academy of Engineering Sciences, Fellow of the IEEE, and IEEE Distinguished Lecturer.

Yuanqing Xia was born in Anhui Province, China, in 1971. He graduated from the Department of Mathematics, Chuzhou University, Chuzhou, China, in 1991. He received the M.S. degree in Fundamental Mathematics from Anhui University, Anhui, China, in 1998 and the Ph.D. degree in Control Theory and Control Engineering from the Beijing University of Aeronautics and Astronautics, Beijing, China, in 2001. From 1991 to 1995, he was with Tongcheng Middle-School, Anhui, where he worked as a Teacher. During January 2002–November 2003, he was a Postdoctoral Research Associate with the Institute of Systems Science, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, China, where he worked on navigation, guidance and control. From November 2003 to February 2004, he was with the National University of Singapore as a Research Fellow, where he worked on variable structure control. From February 2004 to February 2006, he was with the University of Glamorgan, Pontypridd, UK, as a Research Fellow, where he worked on networked control systems. From February 2007 to June 2008, he was a Guest Professor with Innsbruck Medical University, Innsbruck, Austria, where he worked on biomedical signal processing. Since 2004, he has been with the Department of Automatic Control, Beijing Institute of Technology, Beijing, first as an Associate Professor, then, since 2008, as a Professor. In 2012, he was appointed as Xu Teli Distinguished Professor at the Beijing Institute of Technology and obtained a National Science Foundation for Distinguished Young Scholars of China. His current research interests are in the fields of networked control systems, robust control and signal processing, active disturbance rejection control and flight control. He has published eight monographs with Springer and Wiley, and more than 100 papers in journals. He has obtained Second Award of the Beijing Municipal Science and Technology (No. 1) in 2010, Second National Award for Science and Technology (No. 2) in 2011, and Second Natural Science Award of The Ministry of Education (No. 1) in 2012. He is a Deputy Editor of the Journal of the Beijing Institute of Technology, Associate Editor of Acta Automatica Sinica, Control Theory and Applications, the International Journal of Innovative Computing, Information and Control, and the International Journal of Automation and Computing.

The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Riccardo Scattolini under the direction of Editor Ian R. Petersen.

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Both authors contributed equally to this work.

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