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

Morteza Alinia Ahandani<EMAIL_ADDRESS>Hamed Kharrati
<EMAIL_ADDRESS>Farzad Hashemzadeh<EMAIL_ADDRESS>Mahdi
Baradarannia<EMAIL_ADDRESS>Department of Control Engineering,
Faculty of Electrical and Computer Engineering, University of Tabriz, Tabriz,
Iran

###### Abstract

Distributed switched large-scale systems are composed by dynamically coupled
subsystems, in which interactions among subsystems vary over time according to
an exogenous input signal named switching signal. In this paper, we present a
distributed robust switched model-based predictive control (DSwMPC) to control
such systems that are subject to local state/input constraints. The proposed
method guarantees stabilizing the origin of the whole closed-loop system and
ensures the constraints satisfaction in the presence of an unknown switching
signal. In the distributed model-based predictive control (DMPC) used in this
work, by considering the interactions among subsystems as an additive
disturbance, the effect of the switch is reflected on the dynamic equation,
local, and consistency constraint sets of the nominal subsystems. In the
DSwMPC, to compensate the effect of switching signal which creates a time-
varying network topology, a robust tube-based switched model-based predictive
control (RSwMPC) with switch–robust control invariant (switch-RCI) set as the
target set robust to unknown mode switching is used as local controller. The
scheme performance is assessed using three typical examples. In the first
example, the switching times are unknown in prior, but the next neighborhood
sets are assumed to be known in prior. In the second and third cases, both of
them are supposed to be unknown in prior. The simulation results show that the
input and state constraints are satisfied by the proposed DSwMPC at all times.
They also validate that the closed-loop system converges to the origin. Also,
to demonstrate the performance of the proposed DSwMPC more clearly, a
comparison of the DSwMPC with a centralized SwMPC (CSwMPC) and a decentralized
SwMPC (DeSwMPC) shows that the DSwMPC outperforms the DecSwMPC and also the
shapes of response curves under the CSwMPC are very similar to those obtained
by the DSwMPC.

###### keywords:

Distributed robust switched model-based predictive control , distributed
switched large-scale system , switching signal, network topology.

††journal: Nonlinear Analysis: Hybrid Systems

, , , ,

The main contribution of this paper is to apply the Robust switched MPC
controllers in a distributed fashion on the distributed large-scale systems
with switched topology, input and state constraints, in which the subsystems
interact with each other by states and inputs.

The proposed method guarantees stabilizing the origin of the whole closed-loop
system and ensures the constraints satisfaction in the presence of an unknown
switching signal.

By considering the interactions among subsystems as an additive disturbance in
distributed switched large-scale systems, the effect of the switch in network
topology is reflected on the dynamic equation, local, and consistency
constraint sets of the nominal subsystems.

In the proposed distributed robust switched model-based predictive control,
the robust tube-based switched model-based predictive control with
switch–robust control invariant set as the target set robust to unknown mode
switching is used as local controller.

The simulation results show that the input and state constraints are satisfied
by the proposed DSwMPC at all times. They also validate that the closed-loop
system converges to the origin.

##  1 Introduction

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.

###  1.1 Switch and large-scale systems: a literature review

The published works about the relation of the switch and large-scale systems
can be categorized into two main groups. The first one includes those of
researches in which the switch exists in the network topology of the system.
One of the published studies on distributed control of distributed switched
large-scale system is related to [33]. They investigated sensor-network-based
distributed control of networked control systems, in which the communication
constraint and topology switching problems were addressed. Based on the
Lyapunov direct method and the switched system approach, a sufficient
condition was established to ensure the exponentially stability of the closed-
loop system. Also, Schiffer et al. [34] studied large-scale systems with the
dynamic communication topology. In this research, first they derived a strict
Lyapunov function for a nominal distributed averaging-based integral (DAI)
-controlled power system model without communication uncertainties. Then, they
extended this strict Lyapunov function to a common Lyapunov–Krasovskii
functional to provide sufficient delay-dependent conditions for robust
stability of a DAI-controlled power system with dynamic communication topology
as well as heterogeneous constant and fast-varying delays. The single
published research in which the MPC was employed for the distributed large-
scale systems with switched topology is related to Shi et al. [35]. They
proposed a controller with a hierarchical structure for a class of distributed
networked control systems with quantization and switching topology. An upper
control layer receives the system-wide information and solves the MPC
optimization problem in a centralized fashion, while the local controllers are
implemented in a distributed way. Based on the switched control system theory
and the Lyapunov direct method, the desired controller is designed by solving
a constrained LMI optimization problem at each time step.

The second group includes the researches in which the switch has been inserted
in the network topology of controllers to provide a reconfiguration
capability. In this group, the topology of the applied system does not include
an online switching. Baldivieso et al. [36] studied the nesting properties of
feasible regions and robust positive invariant sets for different partitions
of the large-scale system. It investigated how subsystems may be grouped
together into coalitions. A coalition of subsystems is a non-empty subset of
all subsystems. Their idea is that each coalition of subsystems operates and
is controlled as a single entity. A coalitional controller replaces local
subsystem controllers and may achieve better performance, albeit at a higher
cost of complexity and communication. Also a coalitional control framework
based on a switching MPC architecture aimed at large-scale dynamically-coupled
systems was employed by Fele et al. [37]. Their proposed game theoretical
framework provides a dynamic establishment of cooperation in the control of a
multi-agent system. The obtained results showed that the reconfiguration
capabilities provided to the system through the proposed framework were suited
for fault-tolerance needs or plug-and-play settings.

A switching mechanism for communication between local MPC controllers for
large-scale systems was employed in [38]. Nunez et al. [38] proposed a time-
varying scheme for noncentralized MPC (NC-MPC) of large-scale systems. Their
control strategy is implemented by clustering dynamically the local MPC
controllers. In this strategy, several possible control structures for the
communication between subsystems are considered and the hierarchical control
system implements the one with the best performance according to a set of
given objectives. Barreiro-Gomez et al. [39] studied two fundamental
components of the design of distributed optimization-based controllers for
large-scale systems, i.e., system partitioning and distributed optimization
algorithms. They combined the DMPC with the distributed partitioning algorithm
with static and dynamical system partitioning. The obtained results of these
two DMPC controllers demonstrated the effectiveness of both the density-
dependent population game approach and the partitioning for a large-scale
system.

Also, in some researches, the MPC is employed to handle plug and play
operations in the structure of large-scale systems. Plug-and-play operation
means that we exchange a subsystem in the overall system with another,
possibly different, one or we add or remove a subsystem to the overall system.
Lucia et al. [40] proposed a DMPC to control coupled, possibly large-scale,
systems for the case of plug-and-play operations. They explicitly consider the
case in which a subsystem of the network is exchanged by a new one, but, the
same physical interconnections. In their proposed method, when an exchange
request is received, it is checked if the new subsystem can fulfill the
contracts of the old subsystem, given the contracts of its future neighbors
and its own dynamics and degrees of freedom. If this is possible, there exists
no need to perform any redesign of the controllers. If the new subsystem
cannot comply directly with the existing contracts, the controllers of all the
neighbors can be virtually redesigned/tuned at the time which the request
takes place. If the virtual redesign is infeasible, the exchange operation is
rejected; otherwise it is performed. [41] designed a DMPC for networked
systems where certain subsystems may be removed or inserted. In their proposed
DMPC for this kind of reconfigurable systems, once the switch of topology is
commanded, some additional optimization problems required to be solved in the
related subsystems’ controllers to ensure the feasibility of removal or
plugging-in operation. [42] investigated the DMPC of linear systems whose
network topologies are changeable by the way of inserting new subsystems,
disconnecting existing subsystems, or merely modifying the couplings between
different subsystems. They firstly presented a distributed reconfiguration
control strategy applying to any reconfigurable requirements. Then, based on
the alternating direction method of multipliers (ADMM) algorithm, the way to
redesign the reconfigured DMPC controller was provided and the iterative
formulas employed in solving the reconfiguration optimization problem via ADMM
algorithm were derived.

Based on categories mentioned above, in the current research, a switch exists
in both the network topologies of controllers and system. The contribution of
the paper is to apply the RSwMPC controllers in a distributed fashion on the
distributed large-scale systems with switched topology, input and state
constraints, in which the subsystems interact with each other by states and
inputs. In comparison with [35], this work considers a more general system
that includes both input and state constraints and both input and state
interactions. Also, the structure of the controller in [35], is hierarchical
and the MPC optimization problem has to be solved in a centralized fashion. So
its computational complexity increases when the modes of switching signal and
the number of the subsystems increase. In our proposed DSwMPC, the MPC
optimization problems are locally solved for each controller, so the
computational burden is considerably less than a centralized structure. Also,
we already successfully applied a decentralized SwMPC (DeSwMPC) on distributed
switched large-scale systems in [1]. In comparison with [1], structure of the
controller proposed in the current research is distributed and the
interactions among subsystems in [1] were only based on states, whereas the
proposed approach in the current research can handle states and input
interactions.

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

###  1.2 Notations and Definitions

The Minkowski addition and subtraction of sets are denoted by the symbols
$\oplus$ and $\ominus$ , respectively, and they are defined as follows:
$C=A\oplus B=\\{c=a+b:\hskip 2.84544ptfor\hskip 2.84544ptall\hskip
2.84544pta\in A,\hskip 2.84544ptb\in B\\}$ and $C=A\ominus B=\\{c:c+b\in
A\hskip 2.84544pt,for\hskip 2.84544ptall\hskip 2.84544ptb\in B\\}$ . A set
${\mathcal{O}}$ is a control invariant set for the system $x(t+1)=f(x,u)$ if
there exists an admissible control law $u(x)\in{\mathcal{U}}$ such that for
all $x\in{\mathcal{O}}$ , $f(x,u)\in{\mathcal{O}}$ and for all $t\geq 0$ .
Also, ${\mathcal{O}}$ is positive invariant set for $x(t+1)=f(x)$ if for all
$x\in{\mathcal{O}}$ , $f(x)\in{\mathcal{O}}$ and for all $t\geq 0$ . A
necessary and sufficient condition to be control invariance and as-well-as
positive invariance is ${\mathcal{O}}\subseteq Pre({\mathcal{O}})$ where
$Pre({\mathcal{S}})$ is the predecessor operator that is the set of states
that can be directed to the target set ${\mathcal{S}}$ under the dynamics of
the system $x(t+1)=f(x,u)$ , i.e. $Pre({\mathcal{S}})=\\{x|\exists
u,f(x,u)\in{\mathcal{S}}\\}$ without violating constraints. Also the set of
$Pre^{k}({\mathcal{S}})\subseteq{\mathcal{X}}$ is the set of states
$x\in{\mathcal{X}}$ that can be directed to the target set ${\mathcal{S}}$
under the dynamics of the system in $k$ discrete-time instances without
violating the constraints. The set $\Omega\subseteq{\mathcal{R}}^{n_{x}}$ is
robust positively invariant (RPI) set for the system $x(t+1)=f(x,w)$ where
$w\in{\mathcal{W}}\subseteq{\mathcal{R}}^{n_{x}}$ is a disturbance vector, if
$f(x,w)\subseteq\Omega$ for all $x\in\Omega$ and all $w\in{\mathcal{W}}$ and
for all $t\geq 0$, i.e. if only if $f(\Omega,{\mathcal{W}})\subseteq\Omega$.
The RPI set $\underline{\Omega}$ is minimal if it is contained in every other
RPI $\Omega$ verifies $\underline{\Omega}\subseteq\Omega$ .

##  2 Problem Statement

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:

$\displaystyle x(t+1)=Ax(t)+Bu(t)$ (1) $\displaystyle
x(t)\in{\mathcal{X}},u(t)\in\hskip 1.42271pt{\mathcal{U}}$

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

$x_{i}(t+1)=A_{ii}x_{i}(t)+B_{ii}u_{i}(t)+\sum_{j\in{\mathcal{N}_{i}}}A_{ij}x_{j}(t)\\\
+\sum_{j\in{\mathcal{N}_{i}}}B_{ij}u_{j}(t)$ (2)

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

$x_{i}(t)\in\hskip
1.42271pt{\mathcal{X}}_{i}\subseteq{\mathcal{R}}^{n_{x_{i}}},\hskip
2.84544ptu_{i}(t)\in\hskip
1.42271pt{\mathcal{U}}_{i}\subseteq{\mathcal{R}}^{n_{u_{i}}}$ (3)

So

$x(t)=[x_{1}^{T}(t),x_{2}^{T}(t),\ldots,x_{M}^{T}(t)]\in{\mathcal{X}}\subseteq{\mathcal{R}}^{n_{x}}$
(4)
$u(t)=[u_{1}^{T}(t),u_{2}^{T}(t),\ldots,u_{M}^{T}(t)]\in{\mathcal{U}}\subseteq{\mathcal{R}}^{n_{u}}$
(5)

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

$\sum_{i=1}^{M}(\psi_{i}^{x}x_{i}(t)+\psi_{i}^{u}u_{i}(t))\leq 1_{p}$ (6)

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:

$x_{i}(t+1)=A_{ii}x_{i}(t)+B_{ii}u_{i}(t)+w_{i}(t)$ (7)

where

$w_{i}(t)=\sum_{j\in{\mathcal{N}_{i}}}A_{ij}x_{j}(t)+\sum_{j\in{\mathcal{N}_{i}}}B_{ij}u_{j}(t)$
(8)

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:

$\displaystyle x(t+1)=A^{\sigma(t)}x(t)+B^{\sigma(t)}u(t)$ (9) $\displaystyle
x(t)\in{\mathcal{X}}^{\sigma(t)},u(t)\in{\mathcal{U}}^{\sigma(t)}$

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

$x_{i}(t+1)=A_{ii}x_{i}(t)+B_{ii}u_{i}(t)+w_{i}^{\sigma(t)}(t)$ (10)

where

$w_{i}^{\sigma(t)}(t)=\sum_{j\in{\mathcal{N}_{i}^{\sigma(t)}}}A_{ij}x_{j}(t)+\sum_{j\in{\mathcal{N}_{i}^{\sigma(t)}}}B_{ij}u_{j}(t)$
(11)

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.

##  3 Distributed robust tube-based MPC

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

$H_{s}(x(t),u(t))\leq 0$ (12)

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

$\displaystyle
x_{i}(t+1)=A_{ii}x_{i}(t)+B_{ii}u_{i}(t)+\sum_{j\in{\mathcal{N}_{i}}}A_{ij}\tilde{x}_{j}(t)+\sum_{j\in{\mathcal{N}_{i}}}B_{ij}\tilde{u}_{j}(t)+w_{i}^{\varepsilon}(t)$
(13)

where

$w_{i}^{\varepsilon}(t)=\sum_{j\in{\mathcal{N}_{i}}}A_{ij}({x}_{j}(t)-\tilde{x}_{j}(t))+\sum_{j\in{\mathcal{N}_{i}}}B_{ij}({u}_{j}(t)-\tilde{u}_{j}(t))\in{\mathcal{W}_{i}}$
(14)

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):

$\displaystyle
x_{i}(t+1)=A_{ii}x_{i}(t)+B_{ii}u_{i}(t)+\sum_{j\in{\mathcal{N}_{i}}}A_{ij}\tilde{x}_{j}(t)+\sum_{j\in{\mathcal{N}_{i}}}B_{ij}\tilde{u}_{j}(t)$
(15)

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:

$u_{i}(t)=\hat{u}_{i}(t)+K_{i}(x_{i}(t)-\hat{x}_{i}(t))$ (16)

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 $t$

$\textbf{min}_{\begin{subarray}{c}\hat{x}_{i}(0)\\\
\hat{u}_{i}(0:N_{i}-1)\end{subarray}}\sum_{k=0}^{N_{i}-1}(\|\hat{x}_{i}(k|t)\|^{2}_{Q_{i}^{0}}+\|\hat{u}_{i}(k|t)\|^{2}_{R_{i}^{0}})+\|\hat{x}_{i}(N_{i}|t)\|^{2}_{P_{i}^{0}}$
(17a) subject to (13), $x_{i}(t)-\hat{x}_{i}(t)\in\mathcal{Z}_{i}\hskip
28.45274pt$ (17b)
$\hat{x}_{i}(t+k|t)-\tilde{x}_{i}(t+k|t)\in\Delta\mathcal{E}_{i}\hskip
22.76228ptk=0,\ldots,N_{i}-1$ (17c)
$\hat{u}_{i}(t+k|t)-\tilde{u}_{i}(t+k|t)\in\Delta\mathcal{U}_{i}\hskip
19.91684ptk=0,\ldots,N_{i}-1$ (17d)
$\hat{x}_{i}(t+k|t)\in\hat{\mathcal{X}}_{i}\hskip 2.84544pt,\hskip
2.84544pt\hat{u}_{i}(t+k|t)\in\hat{\mathcal{U}}_{i}\hskip 2.84544pt\hskip
17.07182ptk=0,\ldots,N_{i}-1$ (17e)
$\hat{x}_{i}(t+N_{i}|t)\in\hat{\mathcal{X}}_{f_{i}}$ (17f)

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

$\hat{\mathcal{X}}_{i}={\mathcal{X}}_{i}\ominus{\mathcal{Z}}_{i}$ (18)
$\hat{\mathcal{U}}_{i}={\mathcal{U}}_{i}\ominus K_{i}\mathcal{Z}_{i}$ (19)

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].

##  4 Switched model-based predictive control

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:

$\displaystyle x(t+1)=f_{\sigma(t)}(x(t),u(t))$ (20) $\displaystyle
x(t)\in{\mathcal{X}}_{\sigma(t)},u(t)\in{\mathcal{U}}_{\sigma(t)}$

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:

$\displaystyle\sum(d,\mathcal{G})=\\{\sigma:\mathcal{N}\rightarrow\mathcal{I}:dwell_{i}(\sigma)\geq
d_{i},(\sigma(\tau_{s}),\sigma(\tau_{s+1}))\in E,\forall s\in\mathcal{N}\\}$
(21)

where

$\displaystyle
dwell_{i}(\sigma)=min\\{\tau_{s+1}-\tau_{s}:\sigma(\tau_{s})=i,s\in\mathcal{N}\\}$
(22)

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

$\delta(t+1)=\left\\{\begin{aligned} &max\\{\delta(t)-1,0\\}&&\text{if}\
\sigma(t+1)=\sigma(t)\\\
&d_{\sigma(t+1)},&&\text{otherwise}\end{aligned}\right.$ (23)

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 $\\{\mathcal{C}_{i}\\}_{i\in\mathcal{I}}^{\infty}$ :

$IC=\\{(x(t_{0}),\sigma(t_{0}),\delta(t_{0})):(x(t_{0})\in
Pre_{\sigma(t_{0})}^{\delta(t_{0})}(\mathcal{C}_{\sigma(t_{0})})\\}$ (24)

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].

1:for each mode $i\in\mathcal{I}$ do

2: $\Omega_{i}^{0}=\mathcal{X}_{i}$

3:end for

4:repeat

5: for each mode $i\in\mathcal{I}$ do

6: update sets

7: $\Omega_{i}^{k+1}=\Omega_{i}^{k}\bigcap
Pre_{i}(\Omega_{i}^{k})\bigcap_{(i,j)\in E}Pre_{j}^{d_{j}}(\Omega_{j}^{k})$

8: end for

9:until $\Omega_{i}^{k+1}=\Omega_{i}^{k}$ for all $i\in\mathcal{I}$

10:$\mathcal{C}_{i}^{\infty}=\Omega_{i}^{k}$ for all $i\in\mathcal{I}$

Figure 1: Pseudocode for computation of maximal switch-RCI sets [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:

$\textbf{min}_{u(0:N-1)}\sum_{k=0}^{N-1}l^{\sigma(0|t)}(x(k|t),u(k|t))+V^{\sigma(0|t)}_{f}(x(N|t))$
(25a) $x(t+k+1|t)=f_{{\sigma(0|t)}}(x(t+k|t),u(t+k|t))\hskip
8.5359ptk=0,\ldots,N-1$ (25b) $x(t+k|t)\in{\mathcal{X}}^{\sigma(0|t)}\hskip
2.84544pt,\hskip 2.84544ptu(t+k|t)\in{\mathcal{U}}^{\sigma(0|t)}\hskip
17.07182ptk=0,\ldots,N-1$ (25c) $x(t+k|t)\in\mathcal{T}^{\sigma(0|t)}\hskip
17.07182ptfor\hskip 5.69046ptk\geq\delta(0|t)$ (25d)

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.,

$u(t)=u^{*}(0|t)$ (26)

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].

##  5 Distributed switched model-based predictive control

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:

$\displaystyle
x_{i}(t+1)=A_{ii}x_{i}(t)+B_{ii}u_{i}(t)+\sum_{j\in{\mathcal{N}_{i}^{\sigma(t)}}}A_{ij}\tilde{x}_{j}(t)+\sum_{j\in{\mathcal{N}_{i}^{\sigma(t)}}}B_{ij}\tilde{u}_{j}(t)+w_{i}^{\varepsilon,\sigma(t)}(t)$
(27) $\displaystyle
x(t)\in{\mathcal{X}}_{\sigma(t)},u(t)\in{\mathcal{U}}_{\sigma(t)}$

where

$\displaystyle
w_{i}^{\varepsilon,\sigma(t)}(t)=\sum_{j\in{\mathcal{N}_{i}}}A_{ij}({x}_{j}(t)-\tilde{x}_{j}(t))+\sum_{j\in{\mathcal{N}_{i}}}B_{ij}({u}_{j}(t)-\tilde{u}_{j}(t))\in{\mathcal{W}_{i}^{\sigma(t)}}$
(28)

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:

$\displaystyle\hat{x}_{i}(t+1)=A_{ii}\hat{x}_{i}(t)+B_{ii}\hat{u}_{i}(t)+\sum_{j\in{\mathcal{N}_{i}^{\sigma(t)})}}A_{ij}\tilde{x}_{j}(t)+\sum_{j\in{\mathcal{N}_{i}^{\sigma(t)}}}B_{ij}\tilde{u}_{j}(t)$
$\displaystyle\hat{x}(t)\in\hskip
2.84544pt\hat{{\mathcal{X}}}_{i}^{\sigma(t)},\hat{u}(t)\in\hat{{\mathcal{U}}}_{i}^{\sigma(t)}$
(29)

where

$\hat{{\mathcal{X}}}_{i}^{\sigma(t)}={\mathcal{X}}_{i}\ominus\hat{{\mathcal{Z}}}_{i}^{\sigma(t)}$
(30) $\hat{{\mathcal{U}}}_{i}^{\sigma(t)}={\mathcal{U}}_{i}\ominus
K_{i}^{\sigma(t)}\hat{{\mathcal{Z}}}_{i}^{\sigma(t)}$ (31)

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:

$\textbf{min}_{\begin{subarray}{c}\hat{x}_{i}(0)\\\
\hat{u}_{i}(0:N_{i}-1)\end{subarray}}\sum_{k=0}^{N_{i}-1}(\|\hat{x}_{i}(k|t)\|^{2}_{Q_{i}^{0}}+\|\hat{u}_{i}(k|t)\|^{2}_{R_{i}^{0}})+\|\hat{x}_{i}(N_{i}|t)\|^{2}_{P_{i}^{0}}$
(32a)
$\displaystyle\hat{x}_{i}(t+1)=A_{ii}\hat{x}_{i}(t)+B_{ii}\hat{u}_{i}(t)+\sum_{j\in{\mathcal{N}_{i}^{\sigma(t)})}}A_{ij}\tilde{x}_{j}(t)$
(32b)
$\displaystyle+\sum_{j\in{\mathcal{N}_{i}^{\sigma(t)}}}B_{ij}\tilde{u}_{j}(t)\hskip
14.22636ptk=0,\ldots,N_{i}-1$
$x_{i}(t)-\hat{x}_{i}(t)\in\mathcal{Z}_{i}^{\sigma(t)}\hskip 28.45274pt$ (32c)
$\hat{x}_{i}(t+k|t)-\tilde{x}_{i}(t+k|t)\in\Delta\mathcal{E}_{i}^{\sigma(t)}\hskip
22.76228ptk=0,\ldots,N_{i}-1$ (32d)
$\hat{u}_{i}(t+k|t)-\tilde{u}_{i}(t+k|t)\in\Delta\mathcal{U}_{i}^{\sigma(t)}\hskip
19.91684ptk=0,\ldots,N_{i}-1$ (32e)
$\hat{x}_{i}(t+k|t)\in\hat{\mathcal{X}}_{i}^{\sigma(t)}\hskip 2.84544pt,\hskip
2.84544pt\hat{u}_{i}(t+k|t)\in\hat{\mathcal{U}}_{i}^{\sigma(t)}\hskip
2.84544pt\hskip 17.07182ptk=0,\ldots,N_{i}-1$ (32f)
$\hat{x}(t+k|t)\in\mathcal{T}_{i}^{\sigma(0|t)}\hskip 17.07182ptfor\hskip
5.69046ptk\geq\delta_{i}^{\sigma(0|t)}(0|t)$ (32g)

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.

As proposed in [15], [26] and [27], their DMPC guarantees asymptotic closed-
loop stability of the origin of the distributed large-scale system and
constraints satisfaction by employing the RMPC controllers as local
regulators. The current research for control of distributed switched large-
scale systems in which a switching signal changes the network topology
proposes the DSwMPC in which the RSwMPC is employed as local regulator. The
RSwMPC controllers use an implicit MPC law (16) as control input, and employ a
SwMPC for a dynamic equation of nominal subsystems. As proposed in [32], the
SwMPC by defining the switch-RCI sets, determines necessary and sufficient
conditions for the existence of a control-law that guarantees constraint
satisfaction in the presence of unknown mode switching. To ensure stability,
the switch-RCI sets of SwMPC are modified as they satisfy the required
conditions of a terminal region and also a switch-RCI set. In an overall view,
the DSwMPC does not impose any new constraints than the original DMPC. The
DSwMPC modifies the DMPC to be adopted to switch in network topology by
replacing the MPC with a SwMPC controller. The SwMPC is similar to MPC only
with a change in determining a terminal region. The switch-RCI set in SwMPC by
modification of $\Omega_{i}^{0}\hat{\mathcal{X}}_{f_{i}}$ , on one side, it
has all necessary conditions of a terminal region, and the other side it can
ensure a feasible switching among different modes of system. So, all
discussions and results about convergence, stability and constraints
satisfaction in the DMPC and the SwMPC can be generalized for the DSwMPC. In
other words, by solving (32), the asymptotic stability of the closed-loop
distributed switched large-scale system and constraints satisfaction are
guaranteed.

##  6 Results and discussions

In this section, the proposed DSwMPC is tested by being applied on a discrete-
time distributed switched large-scale system, $\mathcal{S}$, consisting of
four interacting subsystems. The models of these four subsystems are given by

$\displaystyle\mathcal{S}_{1}:{x}_{1}(t+1)=\begin{pmatrix}1&1\\\
0&1\end{pmatrix}{x}_{1}(t)+\begin{pmatrix}0.5\\\
1\end{pmatrix}{u}_{1}(t)+\sum_{j\in\mathcal{N}_{1}^{\sigma(t)}}A_{1j}{x}_{j}(t)$
$\displaystyle A_{12}=0.08\begin{pmatrix}1&0\\\
0&1\end{pmatrix}A_{13}=0.05\begin{pmatrix}1&0\\\
0&1\end{pmatrix}A_{14}=0.06\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$
$\displaystyle\begin{pmatrix}-1.2\\\
-1.2\end{pmatrix}\leq\begin{pmatrix}x_{11}(t)\\\
x_{12}(t)\end{pmatrix}\leq\begin{pmatrix}1\\\ 1\end{pmatrix}\hskip
1.42271pt-0.5\leq u_{1}(t)\leq 0.5$
$\displaystyle\mathcal{S}_{2}:{x}_{2}(t+1)=\begin{pmatrix}2&-0.96\\\
1&0\end{pmatrix}{x}_{2}(t)+\begin{pmatrix}1\\\
0\end{pmatrix}{u}_{2}(t)+\sum_{j\in\mathcal{N}_{2}^{\sigma(t)}}A_{2j}{x}_{j}(t)$
$\displaystyle A_{21}=0.05\begin{pmatrix}1&0\\\
0&1\end{pmatrix}A_{23}=0.04\begin{pmatrix}1&0\\\
0&1\end{pmatrix}A_{24}=0.04\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$
$\displaystyle\begin{pmatrix}-1\\\
-1\end{pmatrix}\leq\begin{pmatrix}x_{21}(t)\\\
x_{22}(t)\end{pmatrix}\leq\begin{pmatrix}1\\\ 1\end{pmatrix}\hskip
1.42271pt-0.6\leq u_{2}(t)\leq 0.6$
$\displaystyle\mathcal{S}_{3}:{x}_{3}(t+1)=\begin{pmatrix}1.2&0.51\\\
0.1&1\end{pmatrix}{x}_{3}(t)+\begin{pmatrix}0.5\\\
1\end{pmatrix}{u}_{3}(t)+\sum_{j\in\mathcal{N}_{3}^{\sigma(t)}}A_{3j}{x}_{j}(t)$
$\displaystyle A_{31}=0.04\begin{pmatrix}1&0\\\
0&1\end{pmatrix}A_{32}=0.04\begin{pmatrix}1&0\\\
0&1\end{pmatrix}A_{34}=0.1\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$
$\displaystyle\begin{pmatrix}-1.3\\\
-1.3\end{pmatrix}\leq\begin{pmatrix}x_{31}(t)\\\
x_{32}(t)\end{pmatrix}\leq\begin{pmatrix}1.3\\\ 1.3\end{pmatrix}\hskip
1.42271pt-1.1\leq u_{3}(t)\leq 1.1$
$\displaystyle\mathcal{S}_{4}:{x}_{4}(t+1)=\begin{pmatrix}1.1&2\\\
0&0.95\end{pmatrix}{x}_{4}(t)+\begin{pmatrix}0\\\
0.7787\end{pmatrix}{u}_{4}(t)+\sum_{j\in\mathcal{N}_{4}^{\sigma(t)}}A_{4j}{x}_{j}(t)$
$\displaystyle A_{41}=0.03\begin{pmatrix}1&0\\\
0&1\end{pmatrix}A_{42}=0.03\begin{pmatrix}1&0\\\
0&1\end{pmatrix}A_{43}=0.05\begin{pmatrix}1&0\\\ 0&1\end{pmatrix}$
$\displaystyle\begin{pmatrix}-1.4\\\
-1.4\end{pmatrix}\leq\begin{pmatrix}x_{41}(t)\\\
x_{42}(t)\end{pmatrix}\leq\begin{pmatrix}1.4\\\ 1.4\end{pmatrix}\hskip
1.42271pt-1\leq u_{4}(t)\leq 1$

In the aforementioned system, after switching times, it is required that
$N_{i}^{\sigma(t)}$ is updated based on the new network topology. Simulation
time is considered equals to 15 time instances and it is assumed that the
switching signal changes $N_{i}^{\sigma(t)}$ in $t=4$ and $t=9$ . The initial
conditions are chosen to be

$\displaystyle x_{1}(0)=\begin{pmatrix}-0.55\\\
0.9\end{pmatrix},x_{2}(0)=\begin{pmatrix}-0.8\\\
-0.9\end{pmatrix},x_{3}(0)=\begin{pmatrix}-0.8\\\
-0.6\end{pmatrix},x_{4}(0)=\begin{pmatrix}-0.7\\\ 0.8\end{pmatrix}$

The constrained finite-time optimal control problem of (32) is solved for each
subsystem using the fmincon function in the MATLAB environment. The chosen
control horizon $N_{i}$ and minimum dwell-time $d_{i}$ are 5 and 3 samples,
respectively [1]. It can be noticed that the dimension of overall system does
not impose any restrictions on application of the DSwMPC. But it is natural
that a problem with a large dimension leads to an increase in computational
time. Our simulation study is divided into two parts. In the first part, in
order to test of the DSwMPC under different work conditions and to provide the
comprehensive simulations, three examples are designed based on system (6)-(6)
under various scenarios and the performance of DSwMPC is tested on them. In
the second part, the effectiveness of DSwMPC is verified by comparing its
results with those obtained by a centralized SwMPC (CSwMPC) and a DeSwMPC [1].

###  6.1 Illustrative examples

Example 1. Consider a distributed switched system as illustrated in (6)-(6).
In this scenario, let the switching times are unknown in prior but the next
neighborhood sets, $N_{i}^{\sigma(t)}$ , are known in prior. The neighborhood
set of each subsystem in three considered topologies are shown in Table 1 and
Fig. 2. It can be noticed that it is not necessary to have the same number of
neighbors for all the subsystems and size of $N_{i}^{\sigma(t)}$ in each
subsystem can be different. The size of $N_{i}^{\sigma(t)}$ can be restricted
by (30) and (31) where a big neighborhood set imposes a big
$\mathcal{Z}_{i}^{\sigma(t)}$ and so it is possible that the tightened
constraints of $\hat{\mathcal{X}}_{i}^{\sigma(t)}$ and
$\hat{\mathcal{U}}_{i}^{\sigma(t)}$ be the empty sets.

Table 1: Neighborhood set of each subsystem in three considered topologies. Time interval | $N_{1}$ | $N_{2}$ | $N_{3}$ | $N_{4}$
---|---|---|---|---
0-4 | {2} | {1} | {4} | {3}
5-9 | {2,3} | {3,4} | {3,4} | {1,3}
10-14 | {4} | {4} | {2} | {1}
Figure 2: Three considered network topologies. (a) Initial topology, (b)
Second topology, (c) Third topology

The state responses and the inputs of the four subsystems are presented in
Fig. 4 and Fig. 6. It can be observed that constraints on states and inputs
are always fulfilled by the proposed DSwMPC. The simulation results also
confirm that the closed-loop system successfully converges to the origin.

Example 2. In Example 2, let both switching times and the next neighboring
sets to be unknown in prior. All simulation parameters and characteristics are
the same with Example 1. Only the possible neighbors of different subsystems
are restricted to be: $N_{1}^{\sigma(t)}=\\{2,3\\}$ ,
$N_{2}^{\sigma(t)}=\\{1,3,4\\}$ , $N_{3}^{\sigma(t)}=\\{1,2\\}$ and
$N_{4}^{\sigma(t)}=\\{1,3\\}$ where for example $N_{1}^{\sigma(t)}=\\{2,3\\}$
means that the neighbors of subsystem 1 under the switching signal can be
subsystem 2 or 3 or both of them. Since the neighborhood sets have been
supposed to be unknown in prior, the maximum RPI set $\mathcal{Z}_{i}$ ,
$i=1,\ldots,4$ , of each subsystem must be computed corresponding the maximum
applicable disturbance to it in all possible topologies. For example for
$\mathcal{S}_{1}$ , the maximum RPI set $\mathcal{Z}_{1}$ is computed with
this assumption that its neighborhood set is composed of both subsystems 2 and
3. It can be computed off-line and is used to provide a feasible switch among
topologies.

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Figure 4: The response curves of the states trajectories in Example 1.

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Figure 6: The response curves of the input trajectories in Example 1.

The evolutions of states and control input of the four subsystems for Example
2 are depicted in Fig. 8 and Fig. 10, respectively. The achieved results show
that under this control strategy the closed-loop system converges to the
origin. In other words, they confirm that the DSwMPC successfully stabilizes
closed-loop system. Clearly, it can be observed that the constraints on the
input and states are satisfied by the DSwMPC at all times. However, in
comparison with Example 1, the convergence speed of curves is slower than the
corresponding curves in Fig. 4 and Fig. 6. These observations as mentioned
above, clearly verify the robustness characteristics of the DSwMPC against
unknown switching times and especially against unknown next topology.

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Figure 8: The response curves of the states trajectories in Example 2.

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Figure 10: The response curves of the input trajectories in Example 2.

Example 3. In this example such Example 2, it is assumed that both of
switching time and the neighborhood sets, $N_{i}^{\sigma(t)}$ , after switch
are unknown in prior. The neighborhood set of each subsystem in every topology
and switching times are the same as Example 1. In comparison with Example 2,
the sets of possible neighborhoods are unknown in this example and the
controller have no information about the next possible topologies. In other
words, it is assumed that $N_{1}^{\sigma(t)}=\\{2,3,4\\}$ ,
$N_{2}^{\sigma(t)}=\\{1,3,4\\}$ , $N_{3}^{\sigma(t)}=\\{1,2,4\\}$ and
$N_{4}^{\sigma(t)}=\\{1,2,3\\}$ where for example
$N_{1}^{\sigma(t)}=\\{2,3,4\\}$ means that the neighbors of subsystem 1 under
switching signal $\sigma(t)$ can be subsystem 2 or 3 or 4 or a combination of
two or three neighbors. So such Example 2, the maximum RPI set
$\mathcal{Z}_{i}$ , $i=1,\ldots,4$ , of each subsystem must be computed
corresponding the maximum applicable disturbance to it in all possible
topologies. It means that the maximum applicable disturbance to each subsystem
can be computed by assumption that it is impacted by all of the other
subsystems, e.g. for subsystem 1 we have
$\mathcal{W}_{1}^{\sigma}(t)=\bigoplus_{j\in{\mathcal{N}}_{1}^{\sigma(t)}}(A_{1j}\mathcal{E}_{j}+B_{1j}\mathcal{E}_{j}^{u})$
. The simulation results for evolutions of states and control input on Example
3 are depicted in Fig. 12 and Fig. 14, respectively. It can be observed that
despite considering the maximum applicable disturbance on each subsystem, the
constraints on the input and states are addresses by the DSwMPC at all times.
Moreover, as again shown by these figures, the closed-loop system converges to
the origin. However, the convergence speed of curves is less than the
corresponding curves in Fig. 4 to Fig. 10. These figures in comparison with
Example 2, clearly show the robustness characteristics of the DSwMPC against
strong interactions among subsystems. Also Table 2 summarizes the numerical
comparisons accrording to the state square errors of close-loop system under
the control of the DSWMPC for Examples 1-3. As the obtained results of this
table show, the value of the sum of square errors increases from Example 1 to
Example 3.

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Figure 12: The response curves of the states trajectories in Example 3.

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Figure 14: The response curves of the input trajectories in Example 3. Table
2: State square errors of DSwMPC for Examples 1-3.

Example |  | Sub1 |  | Sub2 |  | Sub3 |  | Sub4 |  | Sum
---|---|---|---|---|---|---|---|---|---|---
|  | $x_{1}$ | $x_{2}$ |  | $x_{1}$ | $x_{2}$ |  | $x_{1}$ | $x_{2}$ |  | $x_{1}$ | $x_{2}$ |  |
Example 1 |  | 0.5694 | 0.9693 |  | 0.8921 | 1.2341 |  | 1.3781 | 1.1156 |  | 1.2964 | 0.8580 |  | 8.3129
Example 2 |  | 0.5574 | 0.9470 |  | 0.9124 | 1.2493 |  | 1.2198 | 1.2752 |  | 1.4370 | 0.8346 |  | 8.4326
Example 3 |  | 0.8510 | 1.0143 |  | 1.0298 | 1.3331 |  | 1.3771 | 1.1054 |  | 1.2866 | 0.8607 |  | 8.8579

###  6.2 Comparison study

To demonstrate the performance of the proposed DSwMPC more clearly, a
comparison of the DSwMPC and results of DeSwMPC [1] and the CSwMPC is also
provided. The CSwMPC has been designed as proposed in Section 4. All parameter
values of CSwMPC are considered equal to parameters of the local SwMPC. This
comparison study is carried out using Example 1 to Example 3. To apply the
CSwMPC of Section 4 on a system, it is necessary that the operating modes of
system to be known in advance (see algorithm of Fig. 1). Since in Examples 2
and 3 the subsequent topology is not known in advance, so subsequent operating
modes of the overall system are not known and this version of CSwMPC cannot be
applied on such a system. For the DeSwMPC, all parameters were tuned based on
those proposed in [1].

The trajectories of the control inputs and the states associated with the four
subsystems are depicted in Fig. 18 to Fig. 26. Also, the sum of square errors
of close-loop system for the DSwMPC, DeSwMPC and CSwMPC are shown in Table. 3.
It can be observed that the shapes of the state response curves under the
CSwMPC are very similar to those obtained by the DSwMPC. Only the states of
closed loop system under CSwMPC are a bit faster with respect to the DSwMPC,
especially for states of subsystems 2 and 3. The achieved results of Table. 3
highlight the effectiveness of DSwMPC. However, the CSwMPC has the best
quality in Example 1 but in addition to overall problems of centralized
control strategies, it can not be applied on systems with scenarios such as
Example 2 and Example 3. Also about the DecSwMPC, it is observed that however
it achieves a faster convergence in Example 1, but its performance is
dramatically decreased in Examples 2 and 3 when number of agents in the sets
of possible neighborhoods is increased so that this method can not find any
feasible control actions on Example 3. From these results we can see that,
compared to the CSwMPC and the DeSwMPC proposed in [1], the DSwMPC proposed in
this article has a considerable flexibility to be applied on distributed
switched systems in which the system topology changes under various scenarios.

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Figure 18: The comparison results of DSwMPC, DeSwMPC and CSwMPC in Example 1
in terms of the states trajectories.

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Figure 20: The comparison results of DSwMPC, DeSwMPC and CSwMPC in Example 2
in terms of the input trajectories.

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Figure 24: The comparison results of DSwMPC and DeSwMPC in Example 2 in terms
of the states trajectories.

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Figure 26: The comparison results of DSwMPC and DeSwMPC in Example 1 in terms of the input trajectories.  Table 3: Sum state square errors of CSwMPC, DeSwMPC and DSwMPC on Examples 1-3. Method |  | Example 1 |  | Example 2 |  | Example 3
---|---|---|---|---|---|---
CSwMPC |  | 8.1124 |  | - |  | -
DeSwMPC |  | 8.2829 |  | 8.5480 |  | -
DSwMPC |  | 8.3129 |  | 8.4326 |  | 8.8579

##  7 Conclutions

A DSwMPC was presented in this study for handling dynamic interconnections
among subsystems in a distributed switched large-scale system subjected to
state/input constraints. In a distributed switched large-scale system,
coupling among subsystems varies over time according to a switching signal.
Such systems are described as the networked control systems with time-varying
network topology in which a switching signal determines the neighborhood set
of each subsystem over time. The proposed DSwMPC ensured the stability of the
origin of the whole closed-loop system and as well constraints satisfaction.
In the employed DMPC, by modeling the coupling terms of subsystems as the
additive disturbance, their effect appeared in the dynamic equation, local
states and input control constraints of the nominal subsystem. The local
controllers exploited tube-based MPC to guarantee robustness than
interconnections among subsystems. To consider the effect of switching signal
which creates a time-varying network topology, the DSwMPC employed a robust
tube-based SwMPC with the switch-RCI set as the target set robust to unknown
mode switching as local regulators. Since the switch of network topology
changes interactions among subsystems, it affects the terms of future state
and input reference trajectories and additive disturbance in the dynamic of
each subsystem, so the switching signal affects the dynamic equation and
constraints of nominal subsystems. Three typical examples were employed to
illustrate the merits and effectiveness of DSwMPC. In all of them, it was
supposed that the switching times are unknown in prior, but the next
neighborhood sets were assumed to be known in prior in the first example and
it was assumed to be unknown in the second and third ones. The simulation
results demonstrated that the proposed DSwMPC satisfied the input and state
constraints at all times. They also validated that the closed-loop system
converges to the origin. It can be seen from the comparison results
that,compared to the CSwMPC and the DeSwMPC, the DSwMPC proposed in this
article has a flexible structure to be applied on distributed switched systems
in which the system topology changes under various scenarios.

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