Decentralized switched model-based predictive control for distributed large-scale systems with topology switching

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Abstract

This paper proposes a decentralized switched model-based predictive control (DeSwMPC) for handling coupling among subsystems in a distributed switched large-scale system composed of physically interconnected subsystems. In the distributed switched large-scale systems, interactions among subsystems vary over time according to an exogenous input signal named switching signal. The proposed controller aims at stabilizing the origin of the whole closed-loop system while guaranteeing the satisfaction of constraints in the presence of a switching signal. In the DeSwMPC, to consider switching signal effect in variation of network topology, a robust tube-based switched model-based predictive control (SwMPC) is employed as local controller. The SwMPC controllers with switch-robust control invariant (switch-RCI) set as its target set are robust to unknown mode switching. In the employed decentralized model-based predictive control (DeMPC), by assuming interconnections as the additive disturbances, the effect of switch is only reflected on local constraint sets of the nominal subsystems. Simulations are performed on two typical examples. In the first case, the switching times are unknown a priori but the neighborhood sets after switch are known a priori. In the second case, both of them are assumed to be unknown a priori. The obtained results demonstrate that the proposed DeSwMPC satisfies the input and state constraints at all times. They also validate that the closed-loop system converges to the origin.

Introduction

In the recent decades motivated by the economical and technological demands, many large-scale systems and plants, named as distributed large-scale systems in this study, such as electric power systems, manufacturing systems, process plants, communication networks, swarms of robots have appeared which are often composed of several interacting subsystems. In the distributed large-scale systems, the subsystems can interact with neighboring subsystems by both inputs and states, and they can have their local and/or global constraints and goals. A group of this kind of systems are the systems with switched topology. Switched systems can be employed for modeling systems that are subject to known or unknown abrupt parameter variations such as synchronously switched linear systems, networks with periodically varying switchings, and sudden change in the system structures due to the failures and faults. Hence, switching among different system structures is an essential feature of a wide variety of engineering and practical real-world systems [1], especially in large-scale systems due to existence of many actuators, sensors and communication networks. In the distributed switched large-scale systems, studied in this research, interactions among subsystems vary over time according to a known or unknown switching signal. They also can be modeled as networked control systems with time-varying network topology. The topology switching problem is a practical issue in the control systems that could be caused by physical faults or the cyber-attack [2]. An example of distributed switched large-scale systems is related to the smart grid systems. The use of transmission assets as controllable assets will inevitably lead to the topology switching in a power grid [3]. The change of topology in modern industrial systems, which could be induced by the possible cyber or physical attacks is another example [4]. How to design a distributed control algorithm for a class of large-scale networked control systems with a general topology switching phenomenon is a challenging work [2].

For the distributed large-scale systems, a centralized control implementation is typically not realistic. Some of disadvantages of centralized controllers such as the less flexibility, the worse error tolerance, the large computational burden, and the huge network communication propel people to decompose them into several relevant small-scaled controllers in a decentralized or distributed fashion [5]. In a decentralized control system, no information is exchanged among the controllers and every regulator makes its own decision, but in the distributed control systems [6], [7], [8], [9], regulators are allowed to share their information with local controllers. This paper focuses on the decentralized strategy.

The decentralized controllers are adjusted to work with each other independently without any information about state and input trajectories of other controllers even when the corresponding controlled subsystems interact with each other. They have the advantages of simple structure, less computational burden, better error tolerance, good flexibility, and easy designing and implementation [5]. Also, when a subsystem has not any means to collect information about plans of its neighborhood subsystems, decentralized control is unavoidable. However, in the cases of existence of the strong enough coupling among subsystems, the achievable performance due to absence of communication and coordination among decentralized controllers is weak.

Model-based predictive control (MPC) is an advanced control technique in which the control input is computed by solving online, at each sampling time, a finite horizon optimal control problem in which the current state of the process is considered as the initial stat. Solving the optimization problem leads to a finite control sequence, and the first control element in this sequence is applied to the system [10]. The online solving of an optimization problem at each time instant increases the computational burden. So, the standard centralized MPC cannot be implemented for systems with fast dynamics and/or a large number of control variables. An common way to solve this problem is to decompose the system into smaller subsystems to be controlled by small-sized local controllers [11]. So, this can be considered as another advantage of decentralized MPC (DeMPC) than its centralized version. Motivated by the aforementioned, the research employs a DeMPC for distributed switched large-scale systems.

For a decentralized implementation of MPC, a variety of ideas have been proposed in the literature. The main idea behind the DeMPC is to model the interactions among the subsystems as local disturbances and then to employ local robust MPC (RMPC) controllers to handle them. For example, a stabilizing decentralized MPC for discrete-time nonlinear systems with uncertainty proposed in [12]. By considering the interconnections among subsystems as perturbation terms and with robustness properties of local MPC control laws, they showed the input-to-state stability of the overall system. Also, in DeMPC technique presented in [13], each controller views the coupling signals from other subsystems as disturbance inputs in its local model. Each controller optimizes a local performance with respect to worst-case disturbances by solving a min–max problem on each iteration. The current study employs a version of DeMPC proposed in [14]. They proposed a tube-based DeMPC scheme for physically coupled linear subsystems to ensure closed-loop asymptotic stability of origin and constraints satisfaction in a large-scale system. The interconnection effect was modeled as an additive disturbance and the tube-based MPC was employed as local controllers to guarantee robustness with respect to physical interactions among subsystems.

Also, [15] proposed a decentralized state-feedback MPC algorithm for nonlinear discrete-time systems subject to decaying disturbances. Inserting a contractive constraint in the optimal control problem yields to the closed-loop stability of the system. By inserting this constraint, the state trajectories of the subsystem are forced to move toward the origin despite the perturbing effect of the coupling terms and of the disturbances. [16] proposed a Lyapunov-based DeMPC named almost DeMPC to stabilize a network of discrete-time nonlinear systems with coupled and local state/input constraints. “almost decentralized” means that each local controller can use the states of interacted subsystems for feedback, whereas it is not allowed to employ iterations among the subsystems to compute the control action. The stability was guaranteed decentralized using a set of structured Control Lyapunov functions (CLFs) for which the maximum over all the functions in the set is a CLF for the global network of systems. In all aforementioned DeMPC methods, an MPC or an RMPC has been employed as a local controller for each subsystem. In the systems studied in this research, i.e., distributed switched large-scale systems, due to the existence of switch in the communication network, it is needed to use a switched MPC controller for each subsystem.

How to apply the MPC to switched systems has been discussed in different literature. [17] proposed a Lyapunov-based predictive controller for the constrained stabilization of switched nonlinear systems with an a priori known switching signal. The stability of the closed-loop system was performed using a variable prediction horizon. Incorporation of constraints satisfaction in the MPC design ensures that: (i) the state of the closed-loop system, at the time of the transition, resides in the stability region of the mode that the system is switched into, and (ii) the Lyapunov function for each mode is nonincreasing wherever the mode is reactivated. Extension of this work was carried out in [18] to the case where the switching times are unknown but only lie in a known prior interval. They proposed an MPC for switched nonlinear systems required to satisfy a prescribed switching sequence with uncertainty in the switching times, parametric uncertainty and time-varying exogenous disturbances in the dynamics of the constituent modes, and state and control constraints. In [19], an MPC was proposed for stabilization of switched nonlinear systems whose switching times were not known a priori but under restrictions of average dwell-time and detection quickly enough. By assuming that there exists a stabilizing MPC for each of the subsystems, they showed that asymptotic stability of the closed-loop switched system could be established if a certain average dwell time condition was satisfied. [20] proposed a switched MPC (SwMPC) for a class of discrete-time switched linear systems with mode-dependent dwell time (MDT). The minimum admissible MDT guarantees the persistent feasibility of MPC design. Some stronger conditions beside of quadratic stage cost and specific terminal cost were also developed to ensure asymptotic stability.

This research employs an SwMPC proposed recently in [21]. They derived necessary and sufficient conditions for guaranteeing constraint satisfaction for control of constrained switched systems with unknown mode switching where the dwell-time and admissible mode switches are restricted. These conditions were derived using a new type of control invariant sets, called switch-robust​ control invariant (switch-RCI) sets, that are constraint admissible control invariant set with the additional property that constraints are enforced during the transit after a mode switch. The switch-RCI sets were employed for designing persistently feasible MPC controllers.

Motivated by the above discussion, the contribution of the paper is to employ a decentralized SwMPC (DeSwMPC) for the distributed switched constrained large-scale systems with input and state constraints. To the best of our knowledge, this is for the first time that an SwMPC is employed in a decentralized approach. The single published research on distributed control of this kind of system is related to [2]. They investigated the distributed control of large-scale systems with communication constraints and topology switching. Based on the Lyapunov direct method and the switched system approach, a sufficient condition was proposed such that the closed-loop system is exponentially stable in the mean-square sense and achieved a prescribed H disturbance attenuation level. Their simulation study was done on the network-based chemical reactors consisting of two subsystems. Also, the robust distributed control of large-scale systems under the uncertain topology but without topology switching was discussed in [22]. The state estimation for a class of complex networks with topology switching was investigated in [23]. But they only considered the decentralized state estimation problem, and the topology switching is assumed to satisfy a Markovian process.

The rest of the paper is organized as follows. In the next section, distributed switched large-scale systems are formulated. In Section 3, the employed DeMPC strategy is presented. Section 4 contains the employed MPC algorithm for switched constrained systems. The proposed switched DeMPC is presented in Section 5. The simulation results are presented and analyzed in Section 6. Section 7 concludes the paper.

The symbol and denote the Minkowski sum and difference of sets, respectively, i.e. C=AB={c=a+b:for all aA,bB} and C=AB if c+BA  , cC. A set O is said to be control invariant for the system x+=f(x,u), if for any xO, there exists an admissible control law uU implies that f(x,u)O  for all subsequent times. It is said to be positive invariant for x+=f(x), if any xO implies that f(x)O  for all subsequent times. A necessary and sufficient condition for control invariance and also positive invariance is OPre(O) where Pre(S) represents the predecessor operator that is the set of states x that can be steered into the set S under dynamics of the system x+=f(x,u), i.e. Pre(S)={x|u,f(x,u)S}. The set XRnx is robust positively invariant (RPI) set for the system x+=f(x,w) where wWRnw is a disturbance vector, if any xX implies that x+X for wW and for all subsequent times. The RPI set X_ is minimal if every other RPI X verifies X_X.

Section snippets

Problem statement

Large-scale systems were described in [24] as follow: “A system which is composed of a number of smaller constituents, which serve particular functions, share common resources, are governed by interrelated goals and constraints and, consequently, require more than one controllers”. “Large-scale systems have traditionally been characterized by large numbers of variables, structure of interconnected subsystems, and other features that complicate the control models such as nonlinearities, time

Decentralized tube-based model-based predictive control

This section describes the decentralized tube-based model-based predictive control proposed in [14] for the distributed large-scale systems. For such a system decentralization of the control structure is often desired since it is typically not practical to assume that all output measurements can be transmitted to each local control agent [26]. In the decentralized approach, due to the distribution of the computation required for obtaining the control law of the whole system among several

Switched model-based predictive control

Danielson et al. [21] to guarantee input and state constraints satisfaction in a MPC, proposed the switch-RCI sets for control of switched constrained systems. They are a new class of control invariant sets to be robust to unknown mode switching. The switch-RCI sets are employed to derive necessary and sufficient conditions to have a control-law that guarantees constraint satisfaction. It was assumed that a minimum dwell-time and allowable mode transitions are known in prior but the switching

Decentralized switched model-based predictive control

In this section, The DeSwMPC based on a combination of the DeMPC described in Section 3 and switched MPC explained in Section 4 is proposed. The DeSwMPC is applied on the distributed switched large-scale systems analyzed in Section 2 and guarantees asymptotic closed-loop stability of the origin of the system and constraints satisfaction. Consider the dynamic of a distributed switched large-scale system, S shown in (9). σ(t) is an unknown switching signal that switches network topology.

Simulations and results

In this section, two typical examples are given to show the effectiveness of the proposed DeSwMPC. Consider a discrete-time distributed switched large scale system composed of four subsystems. S1:x1(t+1)=1101x1(t)+0.51u1(t)+jN1σ(t)A1jxj(t)A12=0.081001,A13=0.041001,A14=0.06100111x1,1(t)x1,1(t)11,0.5u1(t)0.5S2:x2(t+1)=20.9610x2(t)+10u2(t)+jN2σ(t)A2jxj(t)A21=0.051001,A23=0.051001,A24=0.07100111x2,1(t)x2,2(t)11,0.5u2(t)0.5S3:x3(t+1)=1.20.510.11x3(t)+0.51u3(t)+jN3σ(t)A3jxj(t)A31=

Conclusions

This paper proposed a DeSwMPC for handling coupling among subsystems in a distributed switched large-scale system. In the distributed switched large-scale systems, interactions among subsystems vary over time according to a switching signal. So, these kinds of systems can be considered the networked control systems with time-varying network topology which a switching signal determines the neighborhood of each subsystem over time. The proposed DeSwMPC aimed at stabilizing the origin of the whole

CRediT authorship contribution statement

Morteza Alinia Ahandani: Conceptualization, Methodology, Software, Formal analysis, Writing - original draft, Writing - review & editing. Hamed Kharrati: Supervision, Visualization, Investigation. Farzad Hashemzadeh: Supervision, Visualization, Investigation. Mahdi Baradarannia: Visualization, Investigation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or

not-for-profit sectors.

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