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

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 “𝑇” stands for the transpose of a matrix. 𝑅^𝑛 and 𝑅^𝑛×𝑚 denote the 𝑛 dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices, respectively. The notation 𝑋>0(𝑋≥0) means the matrix 𝑋 is positive definite (𝑋 is semipositive definite).

2. Problem Formulation

Consider the discrete-time switched linear system defined as 𝑥(𝑘+1)=𝐴_𝑟(𝑘)𝑥(𝑘)+𝐵_𝑟(𝑘)𝑢(𝑘),𝑘=0,1,…𝑁−1,(2.1) where 𝑥(𝑘)∈𝑅^𝑛 is the state, 𝑢(𝑘)∈𝑅^𝑝 is the control input, and 𝑟(𝑘)∈𝑀={1,2,…,𝑑} is the switching signal to be designed. For each 𝑖∈𝑀,𝐴_𝑖 and 𝐵_𝑖 are constant matrices of appropriate dimension, and the pair (𝐴_𝑖,𝐵_𝑖) is called a subsystem of (2.1). This switched linear system is time invariant in the sense that the set of available subsystems {(𝐴_𝑖,𝐵_𝑖)}^𝑑𝑖=1 is independent of time 𝑘. 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 𝑥(0)=𝑥₀ 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 𝐽₁(𝑢,𝑟)=𝑥(𝑁)^𝑇𝑄_𝑓𝑥(𝑁)+^𝑁−1_𝑗=0𝑥(𝑗)^𝑇𝑄_𝑟(𝑗)𝑥(𝑗)+𝑢(𝑗)^𝑇𝑅_𝑟(𝑗)𝑢(𝑗),(2.2) where 𝑄_𝑓=𝑄^𝑇𝑓≥0 is the terminal state weight matrix, 𝑄_𝑖=𝑄^𝑇𝑖>0 and 𝑅_𝑖=𝑅^𝑇𝑖>0 are running weight matrices for the state and the input for subsystem 𝑖∈𝑀.

The second one is 𝐽₂(𝑢,𝑟)=𝑥(𝑁)^𝑇𝑄_𝑓𝑥(𝑁)+^𝑁−1_𝑗=0𝑥(𝑗)^𝑇𝑄_𝑗𝑥(𝑗)+𝑢(𝑗)^𝑇𝑅_𝑗𝑢(𝑗),(2.3) where 𝑄_𝑓=𝑄^𝑇𝑓≥0 is the terminal state weight matrix, 𝑄_𝑖=𝑄^𝑇𝑖>0 and 𝑅_𝑖=𝑅^𝑇𝑖>0 are running weight matrices for the state and the input at the time instant 𝑗∈{0,1,…,𝑁−1}.

Remark 2.1. The cost function 𝐽₁, is introduced in [31]. In 𝐽₁ the weight matrices are subsystem dependent. The cost function 𝐽₂ 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 𝑢(𝑗) and 𝑟(𝑗) that minimize 𝐽₁(𝑢,𝑟) subject to the system (2.1).

Problem 2. Find the 𝑢(𝑗) and 𝑟(𝑗) that minimize 𝐽₂(𝑢,𝑟) subject to the system (2.1).

3. Optimal Solutions

3.1. Solutions to Problem 1

To drive the minimum value of the cost function 𝐽₁ subject to system (2.1), we define the Riccati mapping 𝑓_𝑖∶𝑌→𝑌 for each subsystem (𝐴_𝑖,𝐵_𝑖) and weight matrices 𝑄_𝑖 and 𝑅_𝑖, 𝑖∈𝑀𝑓_𝑖(𝑃)=𝐴_𝑖−𝐵_𝑖𝐾_𝑖(𝑃)^𝑇𝑃𝐴_𝑖−𝐵_𝑖𝐾_𝑖(𝑃)+𝐾^𝑇𝑖(𝑃)𝑅_𝑖𝐾_𝑖(𝑃)+𝑄_𝑖,(3.1) where 𝐾_𝑖(𝑃)=𝑅_𝑖+𝐵^𝑇𝑖𝑃𝐵_𝑖⁻¹𝐵^𝑇𝑖𝑃𝐴_𝑖.(3.2) Let 𝐻_𝑁={𝑄_𝑓} be a set consisting of only one matrix 𝑄_𝑓. Define the set 𝐻_𝑘 for 0≤𝑘<𝑁 iteratively as 𝐻_𝑘=𝑋∣𝑋=𝑓_𝑖(𝑃),∀𝑖∈𝑀,𝑃∈𝐻_𝑘+1(3.3) Now we give the main result of this paper.

Theorem 3.1. The minimum value of the cost function 𝐽₁ in Problem 1 is 𝐽^∗1(𝑢,𝑟)=min_𝑃∈𝐻0𝑥^𝑇0𝑃𝑥₀.(3.4) Furthermore, for 𝑘≥0, if one defines 𝑃^∗𝑘,𝑖^∗𝑘=argmin_{𝑃∈𝐻𝑘}𝑥(𝑘)^𝑇𝑃𝑥(𝑘)(3.5) then the optimal switching signal and the optimal control input at time instant 𝑘 are 𝑢𝑟^∗(𝑘)=𝑖^∗𝑘,(3.6)_∗(𝑘)=−𝐾_𝑖^∗𝑘𝑃^∗𝑘𝑥(𝑘),(3.7) where 𝐾_𝑖^∗𝑘(𝑃^∗𝑘) is defined by (3.2).

Proof. For the cost function 𝐽₁, by applying the principle of dynamic programming, we obtain the following Bellman equation when 𝑘=0,1,…,𝑁−1: 𝐽_1,𝑘(𝑢,𝑟)=min_{𝑖∈𝑀,𝑢∈𝑅𝑝}𝑥^𝑇(𝑘)𝑄_𝑖𝑥(𝑘)+𝑢^𝑇(𝑘)𝑅_𝑖𝑢(𝑘)+𝐽_1,𝑘+1(𝑢,𝑟)(3.8) and the terminal condition 𝐽_1,𝑁=𝑥^𝑇(𝑁)𝑄_𝑠𝑥(𝑁).(3.9) Now we will prove that the solution of the Bellman equation (3.8) and (3.9) may be written as 𝐽_1,𝑘=min_{𝑃∈𝐻𝑘}𝑥^𝑇(𝑘)𝑃𝑥(𝑘).(3.10) We use mathematical induction to prove that (3.10) holds for 𝑘=0,1,…,𝑁.(i)It is easy to see that (3.10) holds for 𝑁.(ii)We assume that (3.10) holds for 𝑘+1, that is, 𝐽_1,𝑘+1=min_{𝑃∈𝐻𝑘+1}𝑥^𝑇(𝑘+1)𝑃𝑥(𝑘+1).(3.11)By (3.8), we have 𝐽_1,𝑘(𝑢,𝑟)=min_{𝑖∈𝑀,𝑢(𝑘)∈𝑅𝑝}𝑥^𝑇=min_{𝑖∈𝑀,𝑢(𝑘)∈𝑅𝑝}(𝑘)𝑄_𝑖𝑥(𝑘)+𝑢^𝑇(𝑘)𝑅_𝑖𝑢(𝑘)+min_{𝑃∈𝐻𝑘+1}𝑥^𝑇(𝑘+1)𝑃𝑥(𝑘+1)=min_{𝑖∈𝑀,𝑢(𝑘)∈𝑅𝑝}𝑥^𝑇(𝑘)𝑄_𝑖𝑥(𝑘)+𝑢^𝑇(𝑘)𝑅_𝑖𝑢(𝑘)+𝑥^𝑇(𝑘+1)𝑃^∗𝑘+1𝑥(𝑘+1)+𝐴𝑥^𝑇(𝑘)𝑄_𝑖𝑥(𝑘)+𝑢^𝑇(𝑘)𝑅_𝑖𝑢(𝑘)_𝑖=min_{𝑖∈𝑀,𝑢(𝑘)∈𝑅𝑝}𝑥(𝑘)+𝐵_𝑖𝑢(𝑘)^𝑇𝑃^∗𝑘+1𝐴_𝑖𝑥(𝑘)+𝐵_𝑖𝑢(𝑘)+2𝑥_𝑇𝑥^𝑇(𝑘)𝑄_𝑖+𝐴^𝑇𝑖𝑃^∗𝑘+1𝐴_𝑖𝑥(𝑘)+𝑢^𝑇(𝑘)𝑅_𝑖+𝐵^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢(𝑘)(𝑘)𝐴^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢(𝑘).(3.12) Let 𝐻_𝑖(𝑢)=𝑢^𝑇𝑅_𝑖+𝐵^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢+2𝑥^𝑇(𝑘)𝐴^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢.(3.13) By simple calculation, we have 𝜕𝐻_𝑖(𝑢)𝜕𝑢=2𝑅^𝑖+𝐵^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢+2𝐵^𝑇𝑖𝑃^∗𝑘+1𝐴_𝑖𝑥(𝑘).(3.14) Since 𝑢(𝑘) is unconstrained, its optimal value 𝑢^∗𝑖(𝑘) must satisfy 𝜕𝐻_𝑖(𝑢)/𝜕𝑢=0.
It follows that 𝑢^∗𝑖(𝑘)=−𝑅_𝑖+𝐵^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖⁻¹𝐵^𝑇𝑖𝑃^∗𝑘+1𝐴_𝑖𝑥(𝑘)=−𝐾_𝑖𝑃^∗𝑘+1𝑥(𝑘)(3.15) It follows that 𝐽_1,𝑘=min+_{𝑖∈𝑀,𝑢(𝑘)∈𝑅𝑝}2𝑥𝑥^𝑇_𝑇(𝑘)𝑄_𝑖+𝐴^𝑇𝑖𝑃^∗𝑘+1𝐴_𝑖𝑥(𝑘)+𝑢^∗𝑖𝑇(𝑘)𝑅_𝑖+𝐵^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢^∗𝑖(𝑘)=min_𝑖∈𝑀(𝑘)𝐴^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢^∗𝑖(𝑘)−𝑥2^𝑇𝑥_𝑇(𝑘)𝑄_𝑖+𝐴^𝑇𝑖𝑃^∗𝑘+1𝐴_𝑖𝑥(𝑘)+𝑥^𝑇(𝑘)𝐾^𝑇𝑖𝑃^∗𝑘+1𝑅_𝑖+𝐵^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝐾_𝑖𝑃^∗𝑘+1𝑥(𝑘)=min_{𝑖∈𝑀,𝑃∈𝐻𝑘+1}(𝑘)𝐴^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝐾_𝑖𝑃^∗𝑘+1𝑥(𝑘)=min_{𝑃∈𝐻𝑘}𝑥^𝑇(𝑘)𝑓_𝑖(𝑃)𝑥(𝑘)𝑥^𝑇(𝑘)𝑃𝑥(𝑘).(3.16) Then the optimal switching signal and the optimal control input at time 𝑘 are 𝛾^∗(𝑘)=𝑖^∗𝑘 and 𝑢^∗(𝑘)=−𝐾_𝑖^∗𝑘(𝑃^∗𝑘)𝑥(𝑘), respectively. It means that (3.10) still holds for 𝑘. This completes the proof.

Remark 3.2. In [31], the optimal control input at time 𝑘 is 𝑢^∗(𝑘)=−𝐾_𝑖^∗𝑘(𝑃^∗𝑘)𝑥(0), which is different with our result in (3.7).

Remark 3.3. In [31], another Riccati mapping is given by 𝑓_𝑖(𝑃)=𝑄_𝑖+𝐴^𝑇𝑖𝑃𝐴_𝑖−𝐴^𝑇𝑖𝑃𝐵_𝑖𝑅_𝑖+𝐵^𝑇𝑖𝑃𝐵_𝑖⁻¹𝐵^𝑇𝑖𝑃𝐴_𝑖.(3.17) 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 𝑀={1}, the switched system (2.1) becomes a constant linear system (𝐴₁,𝐵₁)=(𝐴,𝐵). In this case, the cost function 𝐽₁ becomes 𝐽₁(𝑢)=𝑥^𝑇(𝑁)𝑄_𝑓𝑥(𝑁)+^𝑁−1_𝑗=0𝑥^𝑇(𝑗)𝑄𝑥(𝑗)+𝑢^𝑇(𝑗)𝑅𝑢(𝑗).(3.18) The Riccati mapping reduces to a discrete-time Riccati equation 𝑃^∗𝑘=𝐴−𝐵𝐾^∗𝑘^𝑇𝑃^∗𝑘+1𝐴−𝐵𝐾^∗𝑘+𝐾^∗𝑘^𝑇𝑅𝐾^∗𝑘+𝑄,(3.19) where 𝐾^∗𝑘=𝑅+𝐵^𝑇𝑃^∗𝑘+1𝐵⁻¹𝐵^𝑇𝑃^∗𝑘+1𝐴.(3.20) It is easy to verify that this novel discrete-time Riccati equation (3.20) is also equivalent to the traditional ones, such as 𝑃^∗𝑘=𝑄+𝑃𝐴^𝑇𝑃^∗𝑘+1𝐴−𝐴^𝑇𝑃^∗𝑘+1𝐵𝑅+𝐵^𝑇𝑃^∗𝑘+1𝐵⁻¹𝐵^𝑇𝑃^∗𝑘+1𝐴,_∗𝑘𝑃=𝑄+𝐴^𝑇𝑃^∗𝑘+1⁻¹+𝐵^𝑇𝑅⁻¹𝐵⁻¹𝐴,_∗𝑘=𝑄+𝐴^𝑇𝑃^∗𝑘+1𝐴𝐼+𝐵^𝑇𝑅⁻¹𝑃^∗𝑘+1⁻¹𝐴.(3.21)

3.2. Solutions to Problem 2

To drive the minimum value of the cost function 𝐽₂ subject to system (2.1), we define the Riccati mapping 𝑓_𝑖,𝑘∶𝑃→𝑃 for each subsystem (𝐴_𝑖,𝐵_𝑖) and weight matrices 𝑄_𝑘 and 𝑅_𝑘, 𝑖∈𝑀,𝑘=0,1,…,𝑁−1∶𝑓_𝑖,𝑘(𝑃)=𝐴_𝑖−𝐵_𝑖𝐾_𝑖(𝑃)^𝑇𝑃𝐴_𝑖−𝐵_𝑖𝐾_𝑖(𝑃)+𝐾^𝑇𝑖(𝑃)𝑅_𝑘𝐾_𝑖(𝑃)+𝑄_𝑘,(3.22) where 𝐾_𝑖(𝑃)=𝑅_𝑘+𝐵^𝑇𝑖𝑃𝐵_𝑖⁻¹𝐵^𝑇𝑖𝑃𝐴_𝑖.(3.23) Let 𝐽_𝑁={𝑄_𝑓} be a set consisting of only one matrix 𝑄_𝑓. Define the set 𝐿_𝑘 for 0≤𝑘≤𝑁 iteratively as 𝐿_𝑘=𝑋∣𝑋=𝑓_𝑖,𝑘(𝑃),∀𝑖∈𝑀,𝑃∈𝐿_𝑘+1.(3.24) Then we give the following theorem.

Theorem 3.5. The minimum value of the cost function 𝐽₂ in Problem 2 is 𝐽^∗2(𝑢,𝑟)=min_𝑃∈𝐿0𝑥^𝑇0𝑃𝑥₀.(3.25) Furthermore, for 𝑘≥0, if one defines 𝑃^∗𝑘,𝑖^∗𝑘=argmin_{𝑃∈𝐿𝑘}𝑥(𝑘)^𝑇𝑃𝑥(𝑘),(3.26) then the optimal switching signal and the optimal control input at time instant 𝑘 are 𝑢𝑟^∗(𝑘)=𝑖^∗𝑘,(3.27)_∗(𝑘)=−𝐾_𝑖^∗𝑘𝑃^∗𝑘𝑥(𝑘),(3.28) where 𝐾_𝑖^∗𝑘(𝑃^∗𝑘) is defined by (3.23).

The proof is similar to that of Theorem 3.1.

Proof. For the cost function 𝐽₂, by applying the principle of dynamic programming, we obtain the following Bellman equation: 𝐽_2,𝑘(𝑢,𝑟)=min_{𝑖∈𝑀,𝑢∈𝑅𝑝}𝑥^𝑇(𝑘)𝑄_𝑘𝑥(𝑘)+𝑢^𝑇(𝑘)𝑅_𝑘𝑢(𝑘)+𝐽_2,𝑘+1(𝑢,𝑟),𝑘=0,1,…,𝑁−1(3.29) and the terminal condition 𝐽_2,𝑁=𝑥^𝑇(𝑁)𝑄_𝑠𝑥(𝑁).(3.30) Now we will prove that the solution of the Bellman equation (3.29) (3.30) may be written as 𝐽_2,𝑘(𝑢,𝑟)=min_{𝑃∈𝐿𝑘}𝑥^𝑇(𝑘)𝑃𝑥(𝑘).(3.31) We use mathematical induction to prove that (3.31) holds for 𝑘=0,1,…,𝑁.(i)It is easy to verify (3.31) holds for 𝑘=𝑁.(ii)We assume (3.31) holds for 𝑘+1, that is, 𝐽_1,𝑘+1(𝑢,𝑟)=min_{𝑃∈𝐿𝑘+1}𝑥^𝑇(𝑘+1)𝑃𝑥(𝑘+1).(3.32)By (3.29), we have 𝐽_2,𝑘(𝑢,𝑟)=min_{𝑖∈𝑀,𝑢(𝑘)∈𝑅𝑝}𝑥^𝑇=min_{𝑖∈𝑀,𝑢(𝑘)∈𝑅𝑝}(𝑘)𝑄_𝑖𝑥(𝑘)+𝑢^𝑇(𝑘)𝑅_𝑖𝑢(𝑘)+min_{𝑃∈𝐿𝑘+1}𝑥^𝑇(𝑘+1)𝑃𝑥(𝑘+1)=min_{𝑖∈𝑀,𝑢(𝑘)∈𝑅𝑝}𝑥^𝑇(𝑘)𝑄_𝑖𝑥(𝑘)+𝑢^𝑇(𝑘)𝑅_𝑖𝑢(𝑘)+𝑥^𝑇(𝑘+1)𝑃^∗𝑘+1𝑥(𝑘+1)=min_{𝑖∈𝑀,𝑢(𝑘)∈𝑅𝑝}𝑥^𝑇(𝑘)𝑄_𝑖𝑥(𝑘)+𝑢^𝑇(𝑘)𝑅_𝑖𝑢(𝑘)+𝐴_𝑖𝑥(𝑘)+𝐵_𝑖𝑢(𝑘)^𝑇𝑃^∗𝑘+1𝐴_𝑖𝑥(𝑘)+𝐵_𝑖𝑢(𝑘)+2𝑥𝑥^𝑇_𝑇(𝑘)𝑄_𝑖+𝐴^𝑇𝑖𝑃^∗𝑘+1𝐴_𝑖𝑥(𝑘)+𝑢^𝑇(𝑘)𝑅_𝑖+𝐵^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢(𝑘)(𝑘)𝐴^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢(𝑘).(3.33) Let 𝑆_𝑖(𝑢)=𝑢^𝑇𝑅_𝑖+𝐵^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢+2𝑥^𝑇(𝑘)𝐴^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢.(3.34) By simple calculation, we have 𝜕𝑆_𝑖(𝑢)𝜕𝑢=2𝑅^𝑖+𝐵^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢+2𝐵^𝑇𝑖𝑃^∗𝑘+1𝐴_𝑖𝑥(𝑘).(3.35) Since 𝑢(𝑘) is unconstrained, its optimal value 𝑢^∗𝑖(𝑘) must satisfy 𝜕𝑆_𝑖(𝑢)/𝜕𝑢=0.
It follows that 𝑢^∗𝑖(𝑘)=−𝑅_𝑖+𝐵^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖⁻¹𝐵^𝑇𝑖𝑃^∗𝑘+1𝐴_𝑖𝑥(𝑘)=−𝐾_𝑖𝑃^∗𝑘+1𝑥(𝑘).(3.36) It follows that 𝐽_2,𝑘(𝑢,𝑟)=min+_{𝑖∈𝑀,𝑢(𝑘)∈𝑅𝑝}2𝑥𝑥^𝑇_𝑇(𝑘)𝑄_𝑖+𝐴^𝑇𝑖𝑃^∗𝑘+1𝐴_𝑖𝑥(𝑘)+𝑢^∗𝑖𝑇(𝑘)𝑅_𝑖+𝐵^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢^∗𝑖(𝑘)=min_𝑖∈𝑀(𝑘)𝐴^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝑢^∗𝑖(𝑘)−2𝑥𝑥^𝑇_𝑇(𝑘)𝑄_𝑖+𝐴^𝑇𝑖𝑃^∗𝑘+1𝐴_𝑖𝑥(𝑘)+𝑥^𝑇(𝑘)𝐾^𝑇𝑖𝑃^∗𝑘+1𝑅_𝑖+𝐵^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝐾_𝑖𝑃^∗𝑘+1𝑥(𝑘)=min_{𝑖∈𝑀,𝑃∈𝐿𝑘+1}(𝑘)𝐴^𝑇𝑖𝑃^∗𝑘+1𝐵_𝑖𝐾_𝑖𝑃^∗𝑘+1𝑥(𝑘)=min_{𝑃∈𝐿𝑘}𝑥^𝑇(𝑘)𝑓_𝑖(𝑃)𝑥(𝑘)𝑥^𝑇(𝑘)𝑃𝑥(𝑘).(3.37) Then the optimal switching signal and the optimal control input at time 𝑘 are 𝛾^∗(𝑘)=𝑖^∗𝑘 and 𝑢^∗(𝑘)=−𝐾_𝑖^∗𝑘(𝑃^∗𝑘)𝑥(𝑘), respectively. It means that (3.31) still holds for 𝑘. This completes the proof.

4. Examples

Example 4.1. Let us consider the following discrete-time switched linear system: 𝑥(𝑘+1)=𝐴_𝜎(𝑘)𝑥(𝑘)+𝐵_𝜎(𝑘)𝑢(𝑘),𝑘=0,1,…,𝑁−1,𝜎(𝑘)∈𝑀={1,2},(4.1) where 𝐴₁=diag(−1,−2),𝐴₂=diag(10,10),𝐵₁=11,𝐵₁=12.(4.2) The parameters in simulations are as follows: 𝑄₁=diag(0.1,0.1),𝑄₂=(0.2,0.2),𝑅₁=1,𝑅₂=0.1,𝑄_𝑓=diag(1,1),𝑁=400.(4.3) We design the controllers with the approach in Theorem 3.1, at the initial state 𝑥₀=[1−1]^𝑇 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]: 𝑥(𝑘+1)=𝐴_𝜎(𝑘)𝑥(𝑘)+𝐵_𝜎(𝑘)𝑢(𝑘),𝑘=0,1,…,𝑁−1,𝜎(𝑘)∈𝑀={1,2},(4.4) where 𝐴₁=00..514855−−00..463100,𝐴₂=−00.2.51555−−00..53970,𝐵₁=01.5,𝐵₁=13.(4.5) The parameters in simulations are as follows: 𝑄₁=diag(1,1),𝑄₂=diag(2,2),𝑅₁=0.1,𝑅₂=0.1,𝑄_𝑓=diag(10,10),𝑁=50.(4.6) We design the controller in with the approach in Theorem 3.1, at the initial state 𝑥₀=[¹⁻²]^𝑇of 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.