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

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.


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 discretetime 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 .

Problem Formulation
Consider the discrete-time switched linear system defined as 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 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 x 0 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 0 are running weight matrices for the state and the input for subsystem i ∈ M.
The second one is where i > 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 J 1 , is introduced in 31 .In J 1 the weight matrices are subsystem dependent.The cost function J 2 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 J 1 u, r subject to the system 2.1 .
Problem 2. Find the u j and r j that minimize J 2 u, r subject to the system 2.1 .

Solutions to Problem 1
To drive the minimum value of the cost function J 1 subject to system 2.1 , we define the Riccati mapping f i : Y → Y for each subsystem A i , B i and weight matrices where Now we give the main result of this paper.
Theorem 3.1.The minimum value of the cost function J 1 in Problem 1 is Furthermore, for k ≥ 0, if one defines then the optimal switching signal and the optimal control input at time instant k are where Proof.For the cost function J 1 , by applying the principle of dynamic programming, we obtain the following Bellman equation when k 0, 1, . . ., N − 1: and the terminal condition 3.9 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 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,

3.11
By 3.8 , we have

3.12
Let By simple calculation, we have Since u k is unconstrained, its optimal value u * i k must satisfy ∂H i u /∂u 0. It follows that It follows that

Mathematical Problems in Engineering
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 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 A 1 , B 1 A, B .In this case, the cost function J 1 becomes x T j Qx j u T j Ru j .

3.18
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

3.21
Mathematical Problems in Engineering 7

Solutions to Problem 2
To drive the minimum value of the cost function J 2 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 : where

3.24
Then we give the following theorem.
Theorem 3.5.The minimum value of the cost function J 2 in Problem 2 is Furthermore, for k ≥ 0, if one defines then the optimal switching signal and the optimal control input at time instant k are where The proof is similar to that of Theorem 3.1.
Proof.For the cost function J 2 , by applying the principle of dynamic programming, we obtain the following Bellman equation: and the terminal condition 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 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, By 3.29 , we have

3.34
By simple calculation, we have

3.35
Since u k is unconstrained, its optimal value u * i k must satisfy ∂S i u /∂u 0. It follows that

3.36
Mathematical Problems in Engineering 9 It follows that

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.

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 x 0 1 − 1 T of the system; the state response of closed-loop discrete-time switched linear system is as in Figure 1.
Example 4.2.Let us consider the following discrete-time switched linear system borrowed from 32 : The parameters in simulations are as follows: We design the controller in with the approach in Theorem 3.1, at the initial state x 0 1 −2 T of the system; the closed-loop state response of discrete-time switched linear system is as in Figure 2.

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.

Figure 1 :Figure 2 :
Figure 1: The state response of closed-loop system.