Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 483568, 12 pages

doi:10.1155/2012/483568

Research Article

Finite-Horizon Optimal Control of Discrete-Time Switched Linear Systems

Qixin Zhu1,2,3 and Guangming Xie1,2

¹The Center for System and Control, College of Engineering, Peking University, Beijing 100871, China

²School of Electrical and Electronic Engineering, East China Jiaotong University, Nanchang 330013, China

³Department of Mechanical and Electrical Engineering, Suzhou University of Science and Technology, Suzhou 215011, China

Received 1 May 2012; Revised 2 July 2012; Accepted 10 July 2012

Academic Editor: Soohee Han

Copyright © 2012 Qixin Zhu and Guangming Xie. 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.

Finite-horizon optimal control problems for discrete-time switched linear control systems are investigated in this paper. Two kinds of quadratic cost functions are considered. The weight matrices are different. One is subsystem dependent; the other is time dependent. For a switched linear control system, not only the control input but also the switching signals are control factors and are needed to be designed in order to minimize cost function. As a result, optimal design for switched linear control systems is more complicated than that of non-switched ones. By using the principle of dynamic programming, the optimal control laws including both the optimal switching signal and the optimal control inputs are obtained for the two problems. Two examples are given to verify the theory results in this paper.

1. Introduction

A switched system usually consists of a family of subsystems described by differential or difference equations and a logical rule that dominates the switching among them. Such systems arise in many engineering fields, such as power electronics, embedded systems, manufacturing, and communication networks. In the past decade or so, the analysis and synthesis of switched linear control systems have been extensively studied [1–28]. Compared with the traditional optimal control problems, not only the control input but also the switching signals needed to be designed to minimize the cost function.

The first focus of this paper is on the finite-horizon optimal regulation for discrete-time switched linear systems. The goal of this paper is to develop a set of optimal control strategies that minimizes the given quadratic cost function. The problem is of fundamental importance in both theory and practice and has challenged researchers for many years. The bottleneck is mostly on the determination of the optimal switching strategy. Many methods have been proposed to tackle this problem. Algorithms for optimizing the switching instants with a fixed mode sequence have been derived for general switched systems in [29] and for autonomous switched systems in [30].

The finite-horizon optimal control problems for discrete-time switched linear control systems are investigated in [31]. Motivated by this work, two kinds of quadratic cost functions are considered in this paper. The former is introduced in [31], where the state and input weight matrices are subsystem dependent. We form the later by ourselves, where the weight matrices are time dependent. According to these two kinds of cost functions, we formulate two finite-horizon optimal control problems. As a result, two novel Riccati mappings are built up. They are equivalent to that in [31]. Actually, the optimal quadratic regulation for discrete-time switched linear systems has been discussed in [31]. However, there are at least one difference between this paper and [31]. That is to say the control strategies proposed in this paper are not the same as that of [31].

This paper is organized into six sections including the introduction. Section 2 presents the problem formulation. Section 3 presents the optimal control of discrete-time switched linear system. Two examples are given in Section 4. Section 5 summaries this paper.

Notations. Notations in this paper are quite standard. The superscript “” stands for the transpose of a matrix. and denote the dimensional Euclidean space and the set of all real matrices, respectively. The notation means the matrix is positive definite ( is semipositive definite).

2. Problem Formulation

Consider the discrete-time switched linear system defined as

where is the state, is the control input, and is the switching signal to be designed. For each and are constant matrices of appropriate dimension, and the pair is called a subsystem of (2.1). This switched linear system is time invariant in the sense that the set of available subsystems is independent of time . We assume that there is no internal forced switching, that is, the system can stay at or switch to any mode at any time instant. It is assumed that the initial state of the system is a constant.

Due to the switching signal, different from the traditional optimal control problem for linear time-invariant systems, two kinds of cost function for finite-horizon optimal control of discrete-time switched linear systems are introduced. The first one is

where is the terminal state weight matrix, and are running weight matrices for the state and the input for subsystem .

The second one is

where is the terminal state weight matrix, and are running weight matrices for the state and the input at the time instant .

Remark 2.1. The cost function , is introduced in [31]. In the weight matrices are subsystem dependent. The cost function is introduced by us. In this case, the weight matrices are time dependent.

The goal of this paper is to solve the following two finite-horizon optimal control problems for switched linear systems.

Problem 1. Find the and that minimize subject to the system (2.1).

Problem 2. Find the and that minimize subject to the system (2.1).

3. Optimal Solutions

3.1. Solutions to Problem 1

To drive the minimum value of the cost function subject to system (2.1), we define the Riccati mapping for each subsystem and weight matrices and ,

where

Let be a set consisting of only one matrix . Define the set for iteratively as

Now we give the main result of this paper.

Theorem 3.1. The minimum value of the cost function in Problem 1 is

Furthermore, for , if one defines

then the optimal switching signal and the optimal control input at time instant are

where is defined by (3.2).

Proof. For the cost function , by applying the principle of dynamic programming, we obtain the following Bellman equation when :

and the terminal condition

Now we will prove that the solution of the Bellman equation (3.8) and (3.9) may be written as

We use mathematical induction to prove that (3.10) holds for .

(i)It is easy to see that (3.10) holds for .(ii)We assume that (3.10) holds for , that is,

By (3.8), we have

Let

By simple calculation, we have

Since is unconstrained, its optimal value must satisfy .

It follows that

It follows that

Then the optimal switching signal and the optimal control input at time are and , respectively. It means that (3.10) still holds for . This completes the proof.

Remark 3.2. In [31], the optimal control input at time is , which is different with our result in (3.7).

Remark 3.3. In [31], another Riccati mapping is given by

It is easy to verify that (3.17) and (3.1) are equivalent to each other. It should be strengthen that there is a matrix inverse operation in (3.17), while, in (3.1) is not. Thus, our result is more convenient for real application.

Remark 3.4. When , the switched system (2.1) becomes a constant linear system . In this case, the cost function becomes

The Riccati mapping reduces to a discrete-time Riccati equation

where

It is easy to verify that this novel discrete-time Riccati equation (3.20) is also equivalent to the traditional ones, such as

3.2. Solutions to Problem 2

To drive the minimum value of the cost function subject to system (2.1), we define the Riccati mapping for each subsystem and weight matrices and ,

where

Let be a set consisting of only one matrix . Define the set for iteratively as

Then we give the following theorem.

Theorem 3.5. The minimum value of the cost function in Problem 2 is

Furthermore, for , if one defines

then the optimal switching signal and the optimal control input at time instant are

where is defined by (3.23).

The proof is similar to that of Theorem 3.1.

Proof. For the cost function , by applying the principle of dynamic programming, we obtain the following Bellman equation:

and the terminal condition

Now we will prove that the solution of the Bellman equation (3.29) (3.30) may be written as

We use mathematical induction to prove that (3.31) holds for .

(i)It is easy to verify (3.31) holds for .(ii)We assume (3.31) holds for , that is,

By (3.29), we have

Let

By simple calculation, we have

Since is unconstrained, its optimal value must satisfy .

It follows that

It follows that

Then the optimal switching signal and the optimal control input at time are and , respectively. It means that (3.31) still holds for . This completes the proof.

4. Examples

Example 4.1. Let us consider the following discrete-time switched linear system:

where

The parameters in simulations are as follows:

We design the controllers with the approach in Theorem 3.1, at the initial state of the system; the state response of closed-loop discrete-time switched linear system is as in Figure 1.

Figure 1: The state response of closed-loop system.

Example 4.2. Let us consider the following discrete-time switched linear system borrowed from [32]:

where

The parameters in simulations are as follows:

We design the controller in with the approach in Theorem 3.1, at the initial state of the system; the closed-loop state response of discrete-time switched linear system is as in Figure 2.

Figure 2: The state response of closed-loop system.

5. Conclusions

Based on dynamic programming, finite-horizon optimal quadratic regulations are studied for discrete-time switched linear systems. The finite-horizon optimal quadratic control strategies minimizing the cost function are given for discrete-time switched linear systems, including optimal continuous controller and discrete-time controller. The infinite-horizon optimal quadratic regulations of discrete-time switched linear system will be investigated in the future.

Acknowledgments

The authors wish to acknowledge the reviewer for his comments and suggestions, which are invaluable for significant improvements of the readability and quality of the paper. The authors would like to acknowledge the National Nature Science Foundation of China for its support under Grant no. 60964004, 60736022, 61164013 and 61164014, China Postdoctoral Science Foundation for its support under Grant no. 20100480131, Young Scientist Raise Object Foundation of Jiangxi Province, China for its support under Grant no. 2010DQ01700, and Science and Technology Support Project Plan of Jiangxi Province, China for its support under Grant no. 2010BGB00607.

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