Event-triggered decentralized robust model predictive control for constrained large-scale interconnected systems

Abstract This paper considers the problem of event-triggered decentralized model predictive control (MPC) for constrained large-scale linear systems subject to additive bounded disturbances. The constraint tightening method is utilized to formulate the MPC optimization problem. The local predictive control law for each subsystem is determined aperiodically by relevant triggering rule which allows a considerable reduction of the computational load. And then, the robust feasibility and closed-loop stability are proved and it is shown that every subsystem state will be driven into a robust invariant set. Finally, the effectiveness of the proposed approach is illustrated via numerical simulations.


Introduction
A class of complex large-scale systems composed of several interconnected subsystems has been receiving an increasing attention due to its various practical applications, e.g. power systems, chemical processes, and transportation systems (Hua, Leng, & Guan, 2012;Yan, Edwards, Spurgeon, & ABOUT THE AUTHORS Ling Lu is currently pursuing her masters degree in East China University of Science and Technology. Her research interests include large scale system, event-triggered control, model predictive control, and their applications. Yuanyuan Zou is an Associate Professor in East China University of Science and Technology. Her research interests include predictive control, network-based control systems, and distributed control systems.
Yugang Niu is a Professor and Vice-Dean in the East China University of Science and Technology. His research interests include stochastic systems, sliding mode control, wireless sensor network, congestion control, and smart grid.

PUBLIC INTEREST STATEMENT
Model predictive control (MPC) is a popular and effective control method to handle the uncertainties and hard constraints on states and controls in the process industry. To deal with the computational complexity in complex large-scale systems, decentralized MPC strategy has been developed. Not only does it maintain the superior properties of MPC method, but it also provides some advantages such as easier maintenance, greater reliability, and less computational effort. However, the time-triggered control scheme in traditional decentralized MPC algorithms will consume redundant computation resources. To overcome this problem, this paper considers the problem of event-triggered decentralized model predictive control for constrained large-scale linear systems subject to additive bounded disturbances. The proposed strategy can not only reduce the on-line computation load, but also achieve the alleviation of computational complexity. , 2004;Zhang & Liu, 2013;Zhang, Zhang, & Wang, 2014). In the control of large-scale systems, decentralized control structure is often the most appropriate control method for handling the computational complexity. Also, it has the advantages such as easier maintenance, greater reliability, and less computational effort (see e.g. Keviczky, Borrelli, & Balas, 2006;Riverso, Farina, & Trecate, 2013;Yan, Lam, Li, & Chen, 2000, and the references therein).

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On the other hand, as a popular control technique, model predictive control (MPC) strategy can effectively handle the uncertainties and hard constraints on states and controls in the process industry. In recent years, many MPC synthesis algorithms that ensure closed-loop stability and robust convergence have been proposed (see e.g. Alessio, Barcelli, & Bemporad, 2011;Magni & Scattolini, 2006;Mayne, Rawlings, Rao, & Scokaert, 2000;Zou & Niu, 2013). Especially, the study of decentralized MPC algorithm for large-scale systems has attracted much attention (Mayne, 2014;(Raimondo, Magni, & Scattolini, 2007;Tran & Ha, 2014). Among them, decentralized MPC design was introduced in Tran and Ha (2014) for networks of linear systems with bounded coupling delay. The stability condition was derived for the constrained optimization problem and the issues of input and state constraints had been addressed by adopting decentralized MPC method. In Magni and Scattolini (2006), a stabilizing decentralized MPC algorithm was presented for nonlinear, discrete time systems under the assumption that no information can be exchanged between local control laws. The closed-loop stability was achieved based on the inclusion of a contractive constraint in the optimization problem. Alessio et al. (2011) proposed a decentralized MPC algorithm for constrained large-scale linear system and analyzed the asymptotic stability of closed-loop system. In particular, the decentralized MPC strategy for large-scale nonlinear system with bounded disturbances was considered in Raimondo et al. (2007), where each subsystem was locally controlled with a MPC algorithm ensuring the robust stability. However, it should be pointed out that the main mechanism in the aforementioned decentralized MPC works was based on time-triggered control scheme. That is, at each sampling instant, a finite horizon local optimization problem was solved on-line to determine the local optimal control sequence, in which only the first control signal would be applied to the subsystem. Apparently, this will consume redundant computational and communication resources, and even affect its applications for a case with limited resources and insufficient communication bandwidth. This motivates the research on event-triggered decentralized MPC algorithms.
The key feature of event-triggered control schemes is that the decision for the execution of control laws is not made periodically, but depending on the detailed system behaviors, such as the system state or the performance index (Dimarogonas, Frazzoli, & Johansson, 2012). At present, many developments have been reported on the event-triggered schemes (Dong, Wang, Alsaadi, & Ahmad, 2015;Dong, Wang, Ding, & Gao, 2015;Liu & Hao, 2013). In Liu and Hao (2013), a decentralized eventtriggered scheme is proposed for networked control systems in order to reduce network traffic and computation resource. In Dong, Wang, Alsaadi, et al. (2015), an event-triggered robust distributed state estimation problem for sensor networks was studied, and in Dong, Wang, Ding, et al. (2015), an event-triggered H-infinity filter algorithm was presented to alleviate the unnecessary waste of communication resources. For event-triggered MPC, some related works can be found in Eqtamin, Dimarogonas, and Kyriakopoulos (2010), Lehmann, Henriksson, and Hohansson (2013), Eqtami, Dimarogonas, and Kyriakopoulos (2011a), Li and Shi (2014). In Eqtamin et al. (2010), an event-triggered MPC algorithm for discrete-time systems was presented, where the optimization problem was solved only when the triggering condition was violated. Eqtami et al. (2011a) considered the eventtriggered robust MPC for both continuous and discrete-time uncertain nonlinear systems with additive disturbances, and derived the triggering rule according to the input-to-state stability (ISS) property. More recently, a class of interconnected large-scale system with bounded disturbances was considered in Eqtami, Dimarogonas, and Kyriakopoulos (2011b), whose key idea was that each subsystem was controlled by a local event-triggered robust model predictive controller. However, it is worthy to note that although the method in Eqtami et al. (2011b) can achieve the reduction on the number of the optimal control updating, there still exists high computational complexity in the optimization problem due to the uncertainties.
In this paper, we investigate the event-triggered decentralized predictive control problem based on constraint tightening approach to reduce both the times of solving optimization problem and computational complexity. By constructing a candidate control sequence and ISS stability, the event-triggered conditions are derived to determine whether the local predictive control optimization problem is solved. Moreover, the robust feasibility and closed-loop stability are proved to show the convergence of subsystem states.
The remainder of the paper is organized as follows. In Section 2, the problem statement for the large-scale system is presented. In Section 3, the main results, including the event-triggered decentralized model predictive controller and the proof of robust feasibility and robust stability are presented. Section 4 provides a numerical example to show the efficiency of the proposed algorithm.
Notations: ℝ n denotes the real n dimensional Euclidean space, ℝ + denotes the positive real number. Given two vectors x, y ∈ ℝ n , x ≥ y ⇔ x i ≥ y i , i = 1, 2, … , n. For any vector x ∈ ℝ n and matrix Q, ||x|| 2 Q = x T Qx. max (⋅) represents the maximum eigenvalue of a real matrix. Given any two sets A, B of ℝ n , the operator "∼" denotes the Pontryagin set difference, i.e.

System description
Consider the linear discrete-time interconnected system composed of M local subsystems The output, input, and disturbance of the ith subsystem are assumed to satisfy the following constraints Define the following augmented vectors The whole system can be written as (1)

Decentralized MPC formulation
In the sequel, we present the decentralized MPC scheme based on the constraint tightened approach, in which each local MPC optimization control problem (OCP) is formulated based on the nominal subsystem corresponding to (1). Moreover, we take the sum of the interaction term ỹ i (k) and the additive disturbance v i (k) as the perturbation Thus, we obtain the following nominal subsystem The following finite horizon optimization problem for the uncertain subsystem (1) The constraint sets U i (j) in (14) are defined by a tightening recursion Similarly, the constraint sets Y i (j) in (13) are The matrices K i (j) and L i (j) denote the associated state transmission matrices under the following candidate policy (7) x x i (k|k) = x i (k), The terminal constraint set X iF in (15) (Richards & How, 2006) is adopted such that L i (N i ) = 0. Hence, (23-24) can be rewritten as Since the condition (25) and (26) do not involve the disturbance, it is much simpler for identifying a suitable set X iF .
Remark 2 In this work, a constraint tightened strategy is applied to each uncertain subsystem.
Since only a nominal prediction model is used in the OCP and the effect of disturbances is considered by resorting to suitable restrictions of the constraints, the resultant computational complexity is avoided effectively.
In this work, our objective is to propose an event-triggered decentralized MPC algorithm based on the constraint tightened strategy such that system resources can be saved and each local OCP computational complexity caused by uncertainties can be reduced.

Event-triggered decentralized robust model predictive controller
In the traditional decentralized MPC strategy, the local optimal control law is usually applied to each subsystem at each sampling instant by solving on-line the local OCP. In this work, we propose an event-triggered decentralized MPC strategy, which determines the updating of control inputs according to a certain triggering condition. In other words, the optimal control law is applied to each subsystem only at its triggered time instant k t i . During the interval step k t i + m i , m i ∈ {1, 2, …, N i } of any two successive triggering events k t i and k t+1 i , a candidate control sequence Ū i (k t i + m i ) based on the optimal control sequence U * i (k t i ) at time k t i is applied to the ith subsystem. Note that k t i is the prior triggering step. Hence, it is important to provide an appropriate control sequence Ū i (k t i + m i ) which satisfies specific constraints at time k t i + m i . Based on the analysis of feasibility and robust stability, we further obtain the triggering condition for each subsystem.

Robust feasibility
In this case, the robust feasibility of the local constrained OCP is analyzed in the following theorem.
Theorem 1 Suppose that the local OCP has the optimal control sequence U * i (k t i ) at the triggered time k t i . The local OCP with the candidate control sequence L i (0) = I, Proof Assume that the local OCP is successfully solved on time k t i , the optimal control sequence U * i (k t i ) satisfying (14) is obtained with the corresponding optimal outputs y * i (k t i + 1 + j|k t i ), j ∈ {0, 1, … , N i − 1 satisfying (13). The feasibility of the local OCP at time k t i + m i , m i ∈ {1, 2, … , N i } is ensured if the constraints (10-15) are satisfied.
Firstly, the feasibility of the local OCP at time k t i + 1 is proved. The following candidate control se- With the candidate control sequence (28), we have The feasibility at In the sequel, we prove the candidate control sequence Ū i (k t i + 1) can satisfy constraints (13-15).
The optimal outputs y * i (k t i + 1 + j|k t i ) satisfies (13) at k t i , so we obtain According to (18-19), we have Combining (33) and (34), it can be further written as Considering that U * i (k t i ) is the optimal control sequence at k t i , it holds that According to (16-17), we have From (37) and (38), it can be obtained that (23), the subsequent state must satisfy By the definition of X iF in (22), we have which implies that the terminal constraint (15) at k t i + 1 is satisfied.
From the above it shows that the candidate control law Ū i (k t i + 1) at time instant k t i + 1 can satisfy the constraints (10-15) and the local OCP is feasible at step k t i + 1. By means of similar arguments, the feasibility of the local OCP at subsequent time k t i + m i , m i ∈ {2, 3, … , N i } can be recursively proved. This completes the proof. □

ISS and triggering condition
We choose the cost function in (9) as a candidate Lyapunov function for the ith subsystem, and define the difference of the feasible cost function as Before we discuss the ISS stability and the triggering condition, the following results are presented.
Theorem 2 Consider the subsystem (1) subject to (3)(4)(5) and the control law (27), and suppose the matrix P i in (9) The difference of the feasible cost functions between the time k t i + m i and k t i + m i − 1 is bounded by Proof For m i = 1, we have For m i = 2, the difference (41) is By adopting similar procedures as in (47) and (48)

Conclusions
In this work, we have provided an event-triggered decentralized robust model predictive controller for a class of constrained linear discrete-time system with additive bounded disturbances. The proposed strategy can not only reduce the on-line computation load, but also achieve the alleviation of computational complexity. It should be pointed out that the systems under consideration in this work are assumed to have full knowledge of states. Actually, it is often difficult to measure the system state in practical application; the event-triggered output feedback MPC strategy will be further considered in future research.

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