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

Distributed large-scale systems such as electric power systems, manufacturing systems, process plants, communication networks and swarms of robots are those of high-dimensional systems composed of interconnections of several lower-dimensional subsystems. In the distributed large-scale systems, the subsystems can interact with neighboring subsystems by both inputs, states, and outputs and they can have their local and/or global constraints and goals. A group of this kind of system is the systems with switched topology. Switched systems as a sub-class of hybrid dynamic systems can be used to model 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 of system structures due to various reasons. They appear naturally in the study of multi-rate sampled-data systems. Also, the switched systems can also be employed to describe the overall system of a single process controlled utilizing multi-controller switching such as the hybrid control scheme for nonholonomic systems, which are not stabilizable by means of any individual continuous state feedback controller [1]. Hence, switching among different system structures is an essential feature of many engineering and practical real-world systems [2], especially in large-scale systems due to the existence of a wide variety of 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 switching signal. They also can be modeled as networked control systems with time-varying network topology.

A centralized control implementation for distributed large-scale systems is typically not realistic. In a decentralized control system, no information is exchanged between the controllers and every controller makes its own decision. Still in the distributed control systems, controllers are allowed to share their information with local controllers. This paper focuses on DMPC strategy. In a DMPC control structure, the overall system under control is divided into many interacted subsystems, where each of them is controlled using a separate local controller based on model-based predictive control (MPC) strategy. This class of controllers combines the advantage of a decentralized control structure, e.g., high flexibility and good error tolerance, and benefits of a centralized control structure, e.g., good global performance [3]. The online solving an optimization problem at each sampling time in MPC increases the computational burden. So, the standard centralized MPC cannot be implemented for plants with fast dynamics and/or a large number of control variables. A natural solution to this problem is to decompose the plant into smaller subsystems and then to design local controllers [4]. This can be considered as another advantage of the distributed implementation of MPC than its centralized version. Motivated by those mentioned above, the current research employs a distributed MPC (DMPC) for control of distributed switched large-scale systems.

The different versions of DMPC proposed in the literature [5], [6], [7] and [8] can be classified according to various criteria. The DMPC can be typically categorized based on the protocol of information exchange into non-iterative or iterative algorithms. In the DMPC with a non-iterative-based algorithm, each local MPC controller communicates only once per time-step with other local MPC controllers and solves the finite horizon optimal control problem once in a control period [9], [10] and [11]. In the DMPC with an iterative-based algorithm, each local MPC controller communicates with other neighboring agents several times per time-step [12], [13] and [14].

Also, they can be categorized based on the type of cost function, which is optimized per time-step into cooperative or non-cooperative algorithms. In the non-cooperative version of DMPC, each local MPC controller optimizes the cost function of its corresponding subsystem [15], [16], [17] and [18] . In its cooperative version, to obtain a better performance, each local controller optimizes a global cost function [12] and [19]. So, in this version, each MPC agent needs a network to exchange information with all other subsystems. In another version of cooperative DMPC, to provide a trade-off between performance and communication cost, a novel coordination strategy was proposed in [13] and [20]. Here each MPC agent minimizes the cost function of its own subsystem and the cost function of the subsystems directly impacts them. Another classification of DMPC can be done in terms of the topology of the communication network into fully connected or partially connected networks. In the DMPC with a fully connected structure [14], [21], [22] and [23] , exchange of information can be done among of all local controllers. In contrast, in DMPC with a partially connected structure [15], [20], [24] and [25], it is required that send and receive of information are only done among neighboring subsystems [5].

For a distributed implementation of MPC regulators, various approaches have been proposed in the literature. The most widely employed idea is to consider the interactions between subsystems as local disturbances and then to employ the local robust MPC (RMPC) controllers to handle them. [15] and [26] employed a non-cooperative, non-iterative DMPC for control of a system composed of linear discrete-time dynamically interconnected subsystems. In their proposed DMPC method, an RMPC has been employed as a local controller for each subsystem. The satisfaction of state and input constraints, guaranteeing convergence of the closed-loop system, obtaining the local control inputs of each subsystem without needing the dynamical models of the other subsystems and to be the limited transmission of information are the main traits of their proposed DMPC. Also, some realization issues for practical use of their proposed method, such as the automatic selection of some tuning parameters, the initialization of the algorithm, or its response to unexpected disturbances, have been discussed and solved in [27]. The current research employs their proposed DMPC for control of distributed switched large-scale systems. For the reason of the existence of switch in the communication network of this kind of system, it is required to use a switched RMPC controller instead of a simple RMPC for each subsystem.

Recently, some researches have been applied the MPC to switched systems. A controller with a predictive framework has been proposed in [28] for the constrained stabilization of switched nonlinear systems with an apriori known switching signal. [29] extended this work for switched non-linear systems in which the switching times are unknown but only lie in a prior known interval. [30] presented an MPC for asymptotic stabilization of switched nonlinear systems whose switching times were a priori unknown but under restrictions of average dwell-time and detection fast enough. In [31] a switched MPC (SwMPC) of a class of discrete-time switched linear systems with idea of mode-dependent dwell-time (MDT) of variable lengths was investigated. This research uses a SwMPC proposed recently in [32]. By defining a new type of control invariant sets, called switch–robust control invariant (switch-RCI) sets, which are robust to unknown mode switching and under restrictions of minimum dwell-time and admissible mode transitions, it derived necessary and sufficient conditions for guaranteeing constraint satisfaction to control of constrained switched systems. The switch-RCI sets were used in the framework of a SwMPC to ensure constraint satisfaction in the presence of unknown mode switching with known minimum dwell-time.

Motivated by the above discussion, We employ a distributed SwMPC (DSwMPC) for the distributed switched constrained large-scale systems with input and state constraints.

Large-scale systems have traditionally been characterized by large numbers of variables, the structure of interconnected subsystems, and other features that complicate the control models such as nonlinearities, time delays, and uncertainties. The decomposition of this kind of system into interconnections of a set of smaller-scale and more manageable subsystems allowed for implementing effective decentralization and coordination mechanisms [43]. This decomposition or partitioning is often performed for either computational or practical reasons. In order to design a control structure based on robust control for large-scale systems, this research is used the modeling proposed in [3]. This paper considers the constrained discrete-time LTI distributed large-scale system, , described by the following state space model:

where $x(t)\in{\mathcal{R}}^{n_{x}}$ and $u(t)\in{\mathcal{R}}^{n_{u}}$ are the state and input of system, respectively. ${\mathcal{X}}$ and ${\mathcal{U}}$ are state and control constraint sets. It is assumed that the distributed large-scale system ${\mathcal{S}}$ can be decomposed of $M$ discrete-time linear subsystems ${\mathcal{S}_{i}}$ , $i\in{\mathcal{P}=\{1,\ldots M}\}$ each of which is controlled by controllers ${\mathcal{C}_{i}}$. Then, the subsystem ${\mathcal{S}_{i}}$ can be expressed as

where $x_{i}(t)\in{\mathcal{R}}^{n_{x_{i}}}$ and $u_{i}(t)\in{\mathcal{R}}^{n_{u_{i}}}$ are the state and input vectors of i-th subsystem, respectively, $A_{ii}\in{\mathcal{R}}^{{n_{x_{i}}}\times{n_{x_{i}}}}$ , $B_{ii}\in{\mathcal{R}}^{{n_{u_{i}}}\times{n_{u_{i}}}}$ , $A_{ij}\in{\mathcal{R}}^{{n_{x_{i}}}\times{n_{x_{j}}}}$ and $B_{ii}\in{\mathcal{R}}^{{n_{u_{i}}}\times{u_{x_{j}}}}$ . ${\mathcal{N}}_{i}$ is the set of neighbors of ${\mathcal{S}}_{i}$ defined as: ${\mathcal{N}}_{i}=\{j\in{\mathcal{P}}:A_{ij}\neq 0\hskip 2.84544ptand/or\hskip 2.84544ptB_{ij}\neq 0,i\neq j\}$ . It is said that subsystem ${\mathcal{S}}_{j}$ is a dynamic neighbor of subsystem ${\mathcal{S}}_{i}$ if only if at least one of the matrices $A_{ij}$ or $B_{ij}$ is nonzero, so ${\mathcal{N}}_{i}$ will be non-empty. The subsystem ${\mathcal{S}}_{i}$ includes the local (uncoupled) state and control constraints

So

Also, $n_{x}=\sum_{i\in{\mathcal{P}}}n_{x_{i}}$ , $n_{u}=\sum_{i\in{\mathcal{P}}}n_{u_{i}}$ ,${\mathcal{X}}=\Pi_{i=1}^{M}{\mathcal{X}}_{i}$ and ${\mathcal{U}}=\Pi_{i=1}^{M}{\mathcal{U}}_{i}$. Also the interacted subsystems can be subject to global (coupled) constraints described in collective form by

where the matrices $\psi_{i}^{x}\in{\mathcal{R}}^{p\times{n_{x_{i}}}}$ and $\psi_{i}^{u}\in{\mathcal{R}}^{p\times{n_{u_{i}}}}$ define the global constraint for subsystem ${\mathcal{S}}_{i}$ and $1_{p}$ is a $p$ -vector of all ones. By definition of the interaction vector $w_{i}$ , named as perturbation or disturbance vector, Eq. (2) is rewritten as follows:

where

In the distributed switched large-scale systems, a switching signal changes the network topology over time. So this kind of system is described by the following state space model:

where the switching sequence $\sigma:\mathcal{N}\rightarrow\mathcal{I}$ is a known or unknown exogenous input that switches network topology between a finite number of modes $\mathcal{I}\subset\mathcal{N}$ . From the viewpoint of the dynamic of the $i$ -the subsystem, due to switching signal varies the interactions among subsystems, it affects only the interaction vector $w_{i}$ . In other words, a switch can only change the set of neighbors of ${\mathcal{S}}_{i}$ , i.e. ${\mathcal{N}}_{i}$. So the dynamic equation of subsystem ${\mathcal{S}}_{i}$ in such a system can be expressed as

where

Eqs. (10) and (11) clearly show that the local constraints of subsystems have not been affected by switch in network topology but the global or interaction constraints can be varied by a switching signal [1]. So, the system employed in this research to be controlled is a distributed switched large-scale system described by the state space model of (9). This system includes $M$ coupled subsystems with dynamics shown in (10) and (11). In this networked modeling with time-varying topology, the role of switching signal $\sigma(t)$ is that it can provide a possibility to vary the set of neighbors of subsystems. This paper, based on this modeling, presents a distributed control structure to be applied on such systems by considering the switched interaction term among the subsystems as disturbances to be rejected.

This section describes the distributed robust tube-based MPC proposed in [15], [26] and [27] for control of large-scale constrained discrete-time linear systems consisting dynamically coupled subsystems. This DMPC in combination with SwMPC will be adopted to switch in the network topology of the system (see Sect. 5).

The DMPC presented in [15], [26] and [27] guarantees stability and convergence properties of the system (1) and constraints satisfaction under mild assumptions. The dynamic equations of subsystems, in (2), can be subjected to local and global state and control constraints. In their proposed DMPC, at each time instant, the subsystem ${\mathcal{S}}_{i}$ sends information about its future state $\tilde{x}_{i}$ and input $\tilde{u}_{i}$ reference trajectories, calculated based on the solution of optimal control problem in the previous time instant, to its interconnecting subsystems. The DMPC solves the optimal control problem of current time instant as its actual trajectories $x_{i}$ and $u_{i}$ lie within certain regions in the neighborhood of the reference ones. The DMPC limits the error between reference trajectories, i.e. $\tilde{x}_{i}$ and input $\tilde{u}_{i}$ , and predictive states calculated in the current sampling time within certain regions by adding some new constraints to the optimal control problem of each subsystem. So the DMPC considers the differences $x_{i}-\hat{x}_{i}$ and input $u_{i}-\hat{u}_{i}$ as unknown bounded disturbances and it employs a tube-based RMPC controller motivated by the discussion in [44] for each subsystem to reject them.

Consider the dynamic of a subsystem ${\mathcal{S}}_{i}$ as (2) subjects to constraints of (3) where ${\mathcal{X}}_{i}$ and ${\mathcal{U}}_{i}$ are convex neighborhoods of the origin. Also, for the system (2), it can be considered the global static constraints as follows

where $s=1,\ldots,n_{c}$ . The subsystem ${\mathcal{S}}_{i}$ is subjected to constraint $H_{s}$ if $x_{i}$ and/or $u_{i}$ are arguments of $H_{s}$ , while ${\mathcal{D}}_{i}=\{s\hskip 2.84544pt\in\hskip 2.84544pt\{1,\ldots,n_{c}\}:H_{c}\hskip 4.83691ptis\hskip 4.83691ptconstraint\hskip 4.83691pton\hskip 5.69046pti\}$ denotes the set of constraints on ${\mathcal{S}}_{i}$ . The subsystem ${\mathcal{S}}_{j}$ is a constraint neighbor of subsystem ${\mathcal{S}}_{i}$ if there exists $\hskip 2.84544pt\bar{s}\in{\mathcal{D}}_{i}$ such that $x_{j}$ and/or $u_{j}$ are arguments of $H_{\bar{s}}$ , while ${\mathcal{H}}_{i}$ denotes the set of the constraint neighbors of ${\mathcal{S}}_{i}$ [15], [26] and [27]. In general, ${\mathcal{S}}_{j}$ is called a neighbor of ${\mathcal{S}}_{i}$ if $j\in{\mathcal{P}}_{i}$ where ${\mathcal{P}}_{i}={\mathcal{N}}_{i}\cup{\mathcal{H}}_{i}$ . Concerning system (1) and its partition, the following primary assumption on decentralized stabilizability is introduced.

Assumption 1.

(i) The matrices $F_{ii}=A_{ii}+B_{ii}K_{i}$ , $i\in{\mathcal{P}}_{i}$ are Schur.

(ii) The matrix $F=A+BK$ is Schur.

where $K=diag\{K_{1},\ldots,K_{M}\}$ .

Let $\tilde{x}_{i}(t+k)$ and $\tilde{u}_{i}(t+k)$ are future state and input reference trajectories transmitted by ${\mathcal{S}}_{i}$ to its neighbor subsystems. In DMPC some added constraints to finite-horizon optimal control problem ensure that real trajectories of each subsystem lie in the specified time-invariant neighborhoods of their reference trajectories, i.e., $x_{i}(t)\in\tilde{x}(t)+\mathcal{E}_{i}$ and $u_{i}(t)\in\tilde{u}(t)+\mathcal{E}_{i}^{u}$ , where $0\in\mathcal{E}_{i}$ and $0\in\mathcal{E}_{i}^{u}$ . So the dynamic equation of the subsystem ${\mathcal{S}}_{i}$ in (2) can be expressed as

where

and ${\mathcal{W}_{i}}=\bigoplus_{j\in{\mathcal{N}_{i}}}(A_{ij}\mathcal{E}_{j}+B_{ij}\mathcal{E}_{j}^{u})$ . In DMPC, a local robust MPC is employed for each subsystem with the dynamic of (13), where the term $\sum_{j\in{\mathcal{N}_{i}}}A_{ij}\tilde{x}_{j}(t+k)+\sum_{j\in{\mathcal{N}_{i}}}B_{ij}\tilde{u}_{j}(t+k)$ is a prior known input over the prediction horizon $k=0,\ldots N_{i}-1$ and $w_{i}^{\varepsilon}(t)$ is a bounded disturbance to be rejected. In order to produce predictions, a nominal model for subsystem ${\mathcal{S}_{i}}$ is obtained by removing the disturbance term of $w_{i}^{\varepsilon}(t)$ in (13):

So the local controller ${\mathcal{C}_{i}}$ when the current state of subsystem ${\mathcal{S}_{i}}$ is $x_{i}(t)$ , is defined in the form of an implicit MPC law as follows:

where $K_{i}\in{\mathcal{R}}^{n_{u_{i}}\times n_{x_{i}}}$ , $i\in{\mathcal{P}}_{i}$ must be chosen to satisfy Assumption 1. A local MPC controller is employed to obtain two unknown variables of $\hat{u}_{i}(t)$ and $\hat{x}_{i}(t)$ .

Given (13), (15) and (16), and since $F_{ii}=A_{ii}+B_{ii}K_{i}$ is Schur and $w_{i}^{\varepsilon}\in{\mathcal{W}}_{i}$ is bounded, they defined the nonempty RPI sets , ${\mathcal{Z}}_{i}$ (see [15]). Based on ${\mathcal{Z}}_{i}$ , two new sets, neighborhoods of origin, $\Delta\mathcal{E}_{i}$ and $\Delta{\mathcal{U}}_{i}$ for $i=1,\ldots,M$ are defined in a way that $\Delta\mathcal{E}_{i}\oplus{\mathcal{Z}}_{i}\subseteq\mathcal{E}_{i}$ and $\Delta\mathcal{U}_{i}\oplus K_{i}{\mathcal{Z}}_{i}\subseteq\mathcal{E}_{i}^{u}$ , respectively [44].

By assuming that each subsystem $\mathcal{S}_{i}$ knows future reference trajectories of its neighbors over the entire prediction horizon, i.e. $\tilde{x}_{j}(t+k)$ and $\tilde{u}_{j}(t+k)$, $k=0,\ldots,N_{j}-1$ and $j\in\mathcal{N}_{i}\cup\mathcal{H}_{i}\cup\{i\}$ , the unknown variables of (16) are computed by online solving the MPC local performance index of subsystem $\mathcal{S}_{i}$ described in (17) at each sampling time $t$

where $N_{i}\in\mathcal{N}$ is the prediction horizon, $l_{i}(\hat{x}_{i}(k|t),\hat{u}_{i}(k|t))=(\|\hat{x}_{i}(k|t)\|^{2}_{Q_{i}^{0}}+\|\hat{u}_{i}(k|t)\|^{2}_{R_{i}^{0}})$ is stage cost, $V_{f_{i}}(\hat{x}_{i}(t+N_{i}|t))=\|\hat{x}_{i}(N_{i}|t)\|^{2}_{P_{i}^{0}}$ is terminal cost and $\hat{\mathcal{X}}_{f_{i}}$ is terminal region.

The local state and input constraints for the nominal system (15) are $\hat{\mathcal{X}}_{i}$ and $\hat{\mathcal{U}}_{i}$ , respectively. These constraints, used in (17e), can be obtained after computing the RPI set ${\mathcal{Z}}_{i}$ as

The positive-definite stage and terminal matrices in (17), i.e. $Q_{i}^{0}$ , $R_{i}^{0}$ and $P_{i}^{0}$ , are chosen based on algorithms discussed in [26] to ensure the stability of MPC. Also, other design parameters employed in DMPC i.e. the sets $\mathcal{Z}_{i}$ , $\Delta\mathcal{E}_{i}$ and $\Delta\mathcal{E}_{i}^{u}$ , $\hat{\mathcal{X}}_{f_{i}}$ and the gain matrices $K_{i}$ are calculated as proposed in [26].

Danielson et al. [32] studied the control of switched constrained linear systems by defining a new kind of control invariant sets for switched constrained systems robust to unknown mode switching, named switch-RCI sets. These switch-RCI sets are used to derive necessary and sufficient conditions for the existence of a control-law to ensure constraint satisfaction in the presence of unknown mode switching with known minimum dwell-time. The switch-RCI sets are also employed to design a recursively feasible MPC that enforces closed-loop constraint satisfaction for switched constrained systems. This section briefly describes the SwMPC proposed by [32].

Consider a switched constrained system described as follows:

where $x(t)\in{\mathcal{R}}^{n_{x}}$ is the state and $u(t)\in{\mathcal{R}}^{n^{\sigma(t)}_{u}}$ is the input. The switching sequence $\sigma:\mathcal{N}\rightarrow\mathcal{I}$ is an unknown exogenous input that switches the dynamics $f_{i}:{\mathcal{R}}^{n_{x}}\times{\mathcal{R}}^{n_{u}}\rightarrow{\mathcal{R}}^{n_{x}}$ and the constraint sets $\mathcal{X}\in{\mathcal{R}}^{n_{x}}$ and $\mathcal{U}\in{\mathcal{R}}^{n^{i}_{u}}$ between a finite number of modes $\mathcal{I}\subset\mathcal{N}$. The number of inputs $n^{i}_{u}$ may depend on the mode $i\in\mathcal{I}$ . [32] implemented two common constraints on the switching sequence, i.e. dwell-time and mode transition restrictions. The set of switching signals $\sigma$ that satisfies these restrictions is described as follows:

where

and $dwell_{i}(\sigma)$ is dwell-time of a mode $i\in\mathcal{I}$, $\tau_{s}\in\mathcal{N}$ are the discrete-times at which the mode changes $\sigma(\tau_{s})\neq\sigma(\tau_{s}-1)$ and $\mathcal{G}=(\mathcal{I},E)$ is a directed graph in which the graph nodes $\mathcal{I}$ denote the modes of the switched system (20) and each directed edge $(i,j)\in E$ indicates that a switch from mode $\sigma(\tau_{s})=i$ to mode $\sigma(\tau_{s+1})=j$ is allowed. For brevity, we will use the short-hand $\sum=\sum(d,\mathcal{G})$ later.

The system (20) has a property of the time-varying depend on the remaining amount of dwell-time denoted by $\delta(t)$ , where

The objective of [32] is to determine whether there exists a control-law $u=k(x,\sigma,\delta)$ that ensures constraints satisfaction $u(t)\in\mathcal{U}_{\sigma(t)}$ and $x(t)\in\mathcal{X}_{\sigma(t)}$ for all allowable switching signals $\sigma(t)\in\sum$  and subsequent times $t\geq 0$ provided that initial states, modes, and remaining dwell-times are $x_{0}$ , $\sigma_{0}$ and $\delta_{0}$ , respectively.

They defined a new class of control invariant sets for switched constrained systems (20) named switch-RCI sets. Switch-RCI sets $\{\mathcal{C}_{i}\}_{i\in\mathcal{I}}$ are constraint admissible control invariant sets that are robust to unknown mode switching provided that switching sequences are contained in (21). They are defined in Definition 1.

Definition 1 (Switch-RCI sets). A collection of sets $\mathcal{C}_{i}\subseteq\mathcal{X}_{i}$ for $i\in\mathcal{I}$ is named switch-RCI if they are control invariant, i.e. $\mathcal{C}_{i}\subseteq Pre_{i}^{1}(\mathcal{C}_{i})$ , for each mode $i\in\mathcal{I}$ and mutually reachable $\mathcal{C}_{i}\subseteq Pre_{j}^{d_{j}}(\mathcal{C}_{j})\subseteq\mathcal{X}_{j}$ within the dwell-time $d_{j}$ for each allowable mode transition $(i,j)\in E$.

Then they by defining maximal switch-RCI sets in Definition 2, combined the sufficient and necessary conditions for the existence of a control-law to ensure constraint satisfaction.

Definition 2 (Maximal Switch-RCI Sets). A collection of sets $\{\mathcal{C}_{i}\}_{i\in\mathcal{I}}^{\infty}$ is called maximal switch-RCI if they are switch-RCI and, for any group of switch-RCI sets $\{\mathcal{C}_{i}\}_{i\in\mathcal{I}}$ , we have $\mathcal{C}_{i}\subseteq\mathcal{C}_{i}^{\infty}$ for each mode $i\in\mathcal{I}$.

Theorem 1. A control-law can be defined as $k(x,\sigma,\delta)$ to guarantee constraint satisfaction for all times $t\in\mathcal{N}$ and for all allowable switching signals $\sigma\in\sum$ if and only if the initial state $x(t_{0})$ , mode $\sigma(t_{0})$ , and remaining dwell-time $\delta(t_{0})$ are contained in the following initial conditions set for the maximal switch-RCI sets $\{\mathcal{C}_{i}\}_{i\in\mathcal{I}}^{\infty}$ :

Pseudocode of Fig. 1 computes the maximal switch-RCI sets $\{\mathcal{C}_{i}\}_{i\in\mathcal{I}}^{\infty}$ [32].

[32] employed the maximal switch-RCI sets $\{\mathcal{C}_{i}\}_{i\in\mathcal{I}}^{\infty}$ to obtain terminal constraint set in the SwMPC to generate control inputs that satisfy state and input constraints. They formulated the SwMPC where (i) the prediction horizon is longer than the dwell-time of each mode $i\in\mathcal{I}$ (i.e. $N\geq d_{i}$), named long-horizon SwMPC, (ii) the prediction horizon is shorter than the dwell-time of each mode $i\in\mathcal{I}$ (i.e. $N\leq d_{i}$), named short-horizon SwMPC. In this section, we concentrate on long-horizon SwMPC. The following proposed relations can be extended for short-horizon SwMPC with some modifications based on [32].

For the long-horizon SwMPC, the control input $u(t)$ is computed by solving the following open-loop constrained finite horizon optimal control problem:

in which $x(0|t)=x(t)$ is the initial condition of the switched system defined in (20), $x(t+k|t)$ is the predicted state trajectory of the system driven by $u(t+k|t)$ over the prediction horizon $N\geq d_{i}$ , $\sigma(0|t)=\sigma(t)$ is the current mode of the system, $\delta(0|t)=\delta(t)$ is the remaining dwell-time, and $\mathcal{T}^{\sigma(0|t)}$ is the switch-RCI set. By solving (25), the obtained control input guarantees the constraints satisfaction and the terminal cost, $V^{\sigma(0|t)}_{f}(\cdot)$ , and the stage cost, $l^{\sigma(0|t)}(\cdot,\cdot)$ , are assigned to satisfy secondary control objectives such as stability or reference tracking for the individual modes. In solving optimization problem (25), it is assumed that the mode $\sigma(t)\in\mathcal{I}$ is constant $\sigma(k|t)=\sigma(0|t)$ over the entire horizon $k=0,\ldots,N-1$ . The control input is the first element $u^{*}(0|t)$ of the optimal open-loop input sequence $u^{*}(0|t)$,…, $u^{*}(N-1|t)$ i.e.,

In contrast to traditional MPC, due to being depend on the current state $x(t)$ , mode $\sigma(t)$, and the remaining dwell-time $\delta(t)$, the controller (26) is mode-varying and time-varying [32]. Also for more details about stability and the recursively feasible of (25) see [32].

In order to adopt the DMPC of Sec. 3 for the large-scale systems with switched topology, this section presents the DSwMPC. In this method, the local RSwMPC controllers are used to handle the existence of switch in the network topology, instead of employing the RMPC regulators. The DSwMPC preserves all characteristics of DMPC such as asymptotic stability of the closed-loop system and constraints satisfaction. Because of employing the switch-RCI sets, the DSwMPC is robust to unknown mode switching. In other words, the DSwMPC is robust to unknown switching times and unknown next topologies, but as mentioned for SwMPC, it is assumed that a minimum dwell-time and allowable mode transitions are known in prior. Restriction of mode transitions for the distributed switched large-scale systems means that it is not necessary to know neighbors of a subsystem in prior but a set of allowable neighborhoods has to be known in prior. Let the dynamic equation of a distributed switched large-scale system, $\mathcal{S}$ , as defined in (9). Suppose $\sigma(t)$ is an unknown switching signal that changes the network topology. Based on modeling proposed in (10), (11) and (13) for dynamic of $i$ -th subsystem, dynamic equation of subsystem $\mathcal{S}_{i}$ can be expressed as:

where

where $w_{i}^{\varepsilon,\sigma(t)}(t)\in\mathcal{W}_{i}^{\sigma}(t)=\bigoplus_{j\in{\mathcal{N}}_{i}^{\sigma(t)}}(A_{ij}\mathcal{E}_{j}+B_{ij}\mathcal{E}_{j}^{u})$ . Observe that according to (27) and (28), the effect of a switch in the network topology appears in the terms of future state and input reference trajectories and also the term of additive disturbance in dynamic equation of $\mathcal{S}_{i}$ . Also, we know that when a switch occurs, due to the new possible neighborhood set, $\mathcal{Z}_{i}$ ,$i\in{\mathcal{P}}$ , must be updated based on the new network topology. On the other side, based on (17b), (17c), (17d), (18) and (19), $\mathcal{S}_{i}$ is utilized for computation of feasible regions of $\hat{x}_{i}(0)$ , $\hat{x}_{i}(t)$ , $\hat{u}_{i}(t)$ and also tightened constraints $\hat{\mathcal{X}}_{i}$ and $\hat{\mathcal{U}}_{i}$ . So a switch in the network topology affects both the dynamic equation and constraints of the nominal subsystem in the structure of the proposed DSwMPC and therefore its effect appears in the optimization problem of (17). From the model defined in (27) for the dynamic equation of subsystem $\mathcal{S}_{i}$ , its corresponding nominal subsystem is considered as follows:

where

and $\hat{{\mathcal{Z}}}_{i}^{\sigma(t)}$ is the RPI set of ${\mathcal{S}}_{i}$ under switching signal of $\sigma(t)$ .

In the DMPC proposed in [15], the controller ${\mathcal{C}}_{i}$ by assumption that the current state of ${\mathcal{S}}_{i}$ is $x_{i}$ , is obtained as proposed in (16) where two unknown variables of $\hat{x}_{i}(t)$ and $\hat{u}_{i}(t)$ are calculated using the MPC controller for the model defined in (5). The variables $\hat{x}_{i}(t)$ and $\hat{u}_{i}(t)$ are equal to optimization parameters of $\hat{x}_{i}(0)$ and $\hat{u}_{i}(0)$ , respectively. These parameters, for the long-horizon MPC, are obtained by optimizing the following MPC performance index:

Note that the effect of the switch is reflected in all constraints of the abovementioned optimal control problem. Constraint (32g) is a combination of constraint (17f) for the optimal control problem of DMPC and constraint (25d) for the optimal control problem of SwMPC. It ensures that before a switch occurs, the states of each subsystem in the current topology will reach a feasible region for the next topology. To guarantee stability and also feasibility when a switch occurs, a modified switch-RCI set of for the current topology in (32g), $\mathcal{T}_{i}^{\sigma(0|t)}$ , must be computed as it satisfies the required conditions of a terminal region and also a switch-RCI set. This is performed by revising stage 2 of algorithm proposed in Fig. 1, i.e. by considering $\Omega_{i}^{0}\hat{\mathcal{X}}_{f_{i}}$ , where $\hat{\mathcal{X}}_{f_{i}}$ is a terminal region of the nominal subsystem (5) computed as presented in [15], [26] and [27]. It should be noticed that when a switch occurs, all matrices include stage and terminal matrices in (32), i.e. $Q_{i}^{0}$ , $R_{i}^{0}$ and $P_{i}^{0}$ and all sets in (32) include $\mathcal{Z}_{i}$ , $\Delta\mathcal{E}_{i}$ and $\Delta\mathcal{E}_{i}^{u}$ , and the gain matrices $K_{i}$ of the DSwMPC have to be updated based on the new topology. Note that for the case where the next network topology is unknown in prior, $\mathcal{Z}_{i}^{\sigma(t)}$ is updated based on the maximum applicable disturbance. This value can be obtained with the participation of all neighbors in the allowable neighborhood set of each subsystem.