Mathematical Problems in Engineering

Volume 2018, Article ID 3834601, 16 pages

https://doi.org/10.1155/2018/3834601

## Stability Analysis for a Type of Multiswitching System with Parallel Structure

^{1}School of Electric Power, South China University of Technology, Guangzhou, Guangdong, China^{2}Centre of Faculty Development and Educational Technology, Guangdong University of Finance and Economics, Guangzhou, Guangdong, China

Correspondence should be addressed to Liu Yongqiang; nc.ude.tucs@uilqype

Received 28 March 2018; Revised 28 June 2018; Accepted 25 July 2018; Published 8 August 2018

Academic Editor: Asier Ibeas

Copyright © 2018 Zhang Yan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper proposes a new type of multiswitching system and a subsystems-group as a basic switching unit that obeys the law. Unlike traditional switching systems, the system selects multiple subsystems instead of one on each time interval. Thus, a framework of parallel structure organizes the subsystems as a group. A multiswitched system is widely used in engineering for modelling and control; this system reflects the actual industrial dynamical process. Thus, the stability of the system is studied. Assuming that these continuous and discrete-time subsystems are Hurwitz and Schur stable, the subsystems-groups matrices commute each other based on the subsystems matrices pairwise commutative. Then, the multiswitched system is exponentially stable under arbitrary switching, and there exists a common Lyapunov function for these subsystems. The main result is extended to a parallel-like structure; therefore, some stability results are gained under some reasonable assumption. At last, a numeral example is given to illustrate the structure and the stability of this system.

#### 1. Introduction

A switched system is a particular kind of hybrid dynamical system that consist of a finite number of subsystems and a switching signal that switches between these subsystems in an orderly manner. The set of subsystems can be continuous-time and discrete-time; the mathematical models are differential equations and difference equations. According to the dynamics behaviours, it can be classified as continuous-time switched system, discrete-time switched system, or mixed switched system (both them) [1], linear or nonlinear and so on [2]. Actually, many fields of complex industrial processes include typical switching systems, which reflect the hybrid dynamical characteristics that are widely used in many fields [3–8], such as in aircraft control, the automotive industry, communication systems, electrical engineering, chemical processes, and so on.

Multiswitched systems are a novel type of system, which can be composed of a set of continuous-time subsystems-groupsand a set of discrete-time subsystems-groupswhere* x(t)*,* x(k)* are the states,* t* is the time scalar, and is positive integer of the sample period.* E*_{ci} and* E*_{dj} are nonnull constant matrices with appropriate dimensions, which are combined by some constant matrices* A*_{cp} and* A*_{dq}, respectively. Subsystems-groups (1) and (2) are composed of continuous-time subsystems with and discrete-time subsystems with , respectively. The system switches between matrices belonging to the set* E; *, , and . belong to the set , belong to the set , and . and are finite sets.

The major difference between a multiswitched system and a traditional switched system is that the former selects any amount of subsystems, and the latter selects only one each time. There is a structure for organizing the subsystems as a subsystems-group on each interval that includes mutual independence between subsystems. Many different structures can be considered, such as parallel, tandem, and other mixed modes. Equations (2) and (4) show the parallel structure of the matrices. However, the traditional switching system has only one matrix without any structure.

A multiswitched system is widely used in engineering cases, such as in the chilled water system in central air conditioning. The dynamical pumping process is a kind of typical multiswitch system, and each different pump as a subsystem with switching and variable frequency behaviours. For variable flow control, there are several different pumps working together as a subsystem-group at each time period. According to the water pipe network, the pumps are parallel in structure. Thus, this physical model illustrates the proposed system. In this paper, only parallel and parallel-like structures are researched.

Modelling [9], performance analysis [10] and optimal control [11, 12] are the key issues of switched systems. Obviously, stability analysis is a very important research branch that has attracted the attention of researchers globally. Stability is the most basic property and the primary issue to be solved by the control system. The stability problems of switched systems are reduced to the following three basic issues [13]: finding conditions to guarantee asymptotical stability for any switching signal; identifying some switching signals for asymptotical stability; and constructing a switching signal to make the system asymptotically stable. Accordingly, some excellent theoretical methods are proposed for solving those problems, such as common Lyapunov functions [14, 15], multiple Lyapunov functions [16–18], dwell time and average dwell time [19, 20], and piecewise quadratic Lyapunov functions [16].

It is worth noting that stability under arbitrary switching is a fundamental in the design and analysis of switched systems [21]. It is known that all the subsystems that are exponentially stable are not sufficient to guarantee stability under arbitrary switching, expect for some reasonable assumptions, such as the state matrices of subsystems commute pairwise [2, 14] (e.g., , for all , ; , for all , c and d denote continuous-time and discrete-time subsystems, respectively); the state matrices of subsystems are symmetric [22] (e.g., , for all ); the state matrices of subsystems are normal [23] (e.g., , for all and , for all ,* c* and* d* denote the same substances above). Thus, some researchers present a sufficient condition in terms of the Lie algebra. If this Lie algebra is solvable, the exponential stability of the switched system for arbitrary switching can be gained [20]. A more general result is that the semisimple subalgebra* S* is a compact Lie algebra by considered Levi decomposition, and the exponentially stability can be ensured [24]. Previous research shows that the pairwise commutation of the vector fields is a sufficient condition for the stability [25]. By using Hurwitz stable matrix pencils and , a common quadratic Lyapunov function exists that can guarantee stability under arbitrary switching [26]. Based on previous research, the main emphasis is placed on the conditions of stability analysis and a common Lyapunov function is constructed under an assumption.

In this paper, a type of linear multiswitched system with parallel structure is put forward for the first time. The composition of the system is expounded, and the structure and form of the subsystems-group are described. From the switching law and the physical meaning, we illustrate the difference between the system and the traditional switching system. A case of chilled water system pumps in central air conditioning shows the engineering significance of the system. Next, stability under arbitrary switching for this type of system is studied. The inference of properties of state matrices of the subsystems-group is derived from the subsystems. Then, based on the conditions of Hurwitz stable and Schur stable for continuous-time subsystems and discrete-time subsystems, respectively, the stability of subsystems-groups can be obtained. The matrices and of the subsystems-groups commute pairwise can be ensured based on the assumptions of the state matrices and [2, 14]. Thus, the stability under arbitrary switching is guaranteed, and a common Lyapunov function is given for all subsystems-groups and subsystems. Finally, the result is extended to a parallel-like structure, which reduces conservativeness in stability analysis.

The body of this paper is organized as follows: in Section 2, the system description, the example of the multiswitching system, and the preliminaries are presented. In Section 3, the main stability results in the continuous-time system and the mixed system are expressed; two examples are also given to illustrate the results. In Section 4, the studies extended to parallel-like structure is presented. Section 5 concludes the paper.

#### 2. System Description and Mathematical Preliminaries

##### 2.1. System Description and Examples

Throughout, the following notation is adopted: and denote the fields of real and complex number, respectively; denotes the* n*-dimensional real Euclidean space; denotes the space of matrices with real entries;* x*_{i} and* x*_{j} denote the* i*^{th} and* j*^{th} components of the vector* x* in , respectively; and denote the entry in the () and () position of the matrices* A* or* E* in , respectively.

There are three examples to show the multiswitched system and the parallel structure.

*Example 1. *Consider a set of a switched system, which are the constant matrices of 3 subsystems, respectively. Assume in the classical switched system that there are 3 subsystems. The system switches between , , and *.* Assume in the multiswitched system that there are 7 subsystems-groups. The system switches between* E,*

Here, and .

*Remark 2. *A continuous-time subsystems-group composed of a subsystem or all the subsystems. Obviously, there are subsystems, but the value is null. It is same with a discrete-time subsystems-group.

*Remark 3. *Continuous-time subsystems cannot mix with discrete-time subsystems to be a subsystems-group. Continuous-time subsystems-groups and discrete-time subsystems-groups must be distinguished.

*Remark 4. *When* E*_{ci} is a singular matrix, zero rows or zero columns remain with the aim of uniform description. In Example 1, , but the uniform description is kept to show , with , . Thus, all* E*_{ci} values look the same, . The same is true of .

*Remark 5. *Equations (2) and (4) are standard parallel structures. A parallel-like structure will be considered in Section 4, which has a coupling phenomenon based on a parallel structure between some subsystems. Assumptions are made for reducing conservativeness to obtain ideal stability.

*Example 6. *In a chilled water system of a central air conditioning system, there are three pumps driving the chilled water from the evaporator to the air conditioning unit (see Figure 1). All the pumps can be switched ON/OFF and the variable frequency obeying a range of 50%-100% rated frequency. For setting an energy-saving control strategy with variable water volume technology, the whole volume cannot be less than 50% of the rated volume. Assuming in a time interval that the cold load changes, the switching, and pump water volume are shown in Figures 2 and 3. The parameters and symbols are shown in Table 2.

It is assumed that the cold load in a certain area requires air conditioning to meet the cold demand within 5 minutes to achieve a balanced state. Then, in the first 5 minutes, 3 pumps work at the rated frequency. After reaching the preset value, the cold load is stable with small fluctuations. In this period, pump* A* and* B* exit; only pump* C* works in the form of frequency conversion, maintaining the driving chilled water and transferring cooling capacity. In the last 3 minutes, due to the cold load increase, a pump is not enough to meet the demand, and the system puts pump* A* into work with pump* C* together with frequency conversion.

From the three time intervals (, , and ), we find that there are three, one, and two subsystems working, respectively. According to the framework in this paper, the three pumps denote different continuous-time subsystems, and the work combinations of the pumps denote the different continuous-time subsystems-groups. Assuming the pump* A*,* B*, and* C* denote the subsystem , , and , respectively; and the combination* A/B/C*,* C* and* A/C* we just said denote the subsystems-group , , and , respectively.