Proceedings of the 41st IEEE Conference on Decision and Control Las Vegas, Nevada USA, December 2002

FrP01-3

Optimal Control of Switched A u t o n o m o u s Systems X u p i n g Xu *, P a n o s J. Antsaklis ** * Dept. of Electr. and Comp. Eng. , Penn State Erie, Erie, PA 16563 USA- [email protected] ** Dept. of Electr. Eng., University of Notre Dame, Notre Dame, IN 46556 USA- a n t s a k i i s , [email protected]

Abstract

In this paper, optimal control problems for switched autonomous systems are studied. In particular, we focus on problems in which a prespecified sequence of active subsystems is given and propose an approach to finding the optimal switching instants. The approach derives the derivatives of the cost with respect to the switching instants and uses nonlinear optimization techniques to locate the optimal switching instants. The approach is then applied to general quadratic problems for switched linear autonomous systems and to reachability problems. Examples illustrate the results. 1 Introduction

A switched system is a particular kind of hybrid systern that consists of several subsystems and a switching law specifying the active subsystem at each time instant. Examples of such systems can be found in chemical processes, automotive systems, and electrical circuit systerns, etc. Recently, many results for optimal control of switched systems have appeared in the literature (e.g., [2, 6, 7, 8, 10]). Most of them consider problems which seek for the solution of both the optimal continuous input and the optimal switching sequence. Approaches to such problems include ones based on discretization of the time and state space [6, 7] and ones that are not based on discretization [8, 10]. Many of these approaches find approximations to local optimal solutions. In this paper, we focus on optimal control problems for switched autonomous systems where each subsystern is autonomous (i.e., with no continuous input). In particular, we focus on problems in which a prespecifled sequence of active subsystems is given. General autonomous subsystems and general performance costs are considered. For such problems the cost is a function of the switching instants. We propose to use constrained nonlinear optimization techniques to locate open-loop local optimal switching instants for such general problems. To apply nonlinear optimization techniques, we need to first determine the values of the derivatives of the cost with respect to the switching instants. An approach similar to that in [10] is proposed in this paper for their derivations. One of the main results of the pa1The partial support of the National Science Foundation (NSF ECS99-12458 & CCR01-13131), and of the DARPA/ITO-NEST Program (AF-F30602-01-2-0526) is gratefully acknowledged.

0-7803-7516-5/02/$17.00 ©2002 IEEE

per is Theorem 3.1 which gives us the expressions of the derivatives. Note here the approach provides us with accurate values of the derivatives as opposed to the approximate values in [10]. The approach is then applied to general quadratic problems for switched linear autonomous systems. The computation of the derivatives can be further simplified by utilizing the special structure of such problems. Finally, we apply the optimal control approach to reachability problems. Using the approach, the reachability switching instants can be determined if a final state is reachable from an initial state. Similar problems have also been looked into by other researchers. Giua et al in [4, 5] present closed-loop global optimal solutions to a special class of problems, i.e., infinite horizon problems for switched linear autonomous systems. However, we should indicate that our approach has the following advantages. First, our approach can deal with finite horizon problems with general subsysterns and costs as opposed to infinite horizon problems with linear subsystems and quadratic costs in [4, 5]. Moreover, our approach can be applied to reachability problems, while the approach in [4, 5] fits better for stability problems. In view of these, we believe our results are new and contribute to the understanding and the solution of optimal control problems of switched systems. 2 Problem

Formulation

We consider the following switched autonomous systerns, i.e., switched systems which consist of autonomous subsystems (i.e., without continuous input) ic-fi(x,t), f i ' R ~ x R - - + R ~, i C I - - { 1 , . . . ,M}. (2.1) The state trajectory evolution of such a system can be controlled by choosing appropriate switching sequences. A switching sequence in [t0,tf] is defined as - ((to, io), (tl,

il),

(t~, i~), • • • , ( t ~ , i ~ ) ) ,

(2.2)

with 0

FrP01-3

Optimal Control of Switched A u t o n o m o u s Systems X u p i n g Xu *, P a n o s J. Antsaklis ** * Dept. of Electr. and Comp. Eng. , Penn State Erie, Erie, PA 16563 USA- [email protected] ** Dept. of Electr. Eng., University of Notre Dame, Notre Dame, IN 46556 USA- a n t s a k i i s , [email protected]

Abstract

In this paper, optimal control problems for switched autonomous systems are studied. In particular, we focus on problems in which a prespecified sequence of active subsystems is given and propose an approach to finding the optimal switching instants. The approach derives the derivatives of the cost with respect to the switching instants and uses nonlinear optimization techniques to locate the optimal switching instants. The approach is then applied to general quadratic problems for switched linear autonomous systems and to reachability problems. Examples illustrate the results. 1 Introduction

A switched system is a particular kind of hybrid systern that consists of several subsystems and a switching law specifying the active subsystem at each time instant. Examples of such systems can be found in chemical processes, automotive systems, and electrical circuit systerns, etc. Recently, many results for optimal control of switched systems have appeared in the literature (e.g., [2, 6, 7, 8, 10]). Most of them consider problems which seek for the solution of both the optimal continuous input and the optimal switching sequence. Approaches to such problems include ones based on discretization of the time and state space [6, 7] and ones that are not based on discretization [8, 10]. Many of these approaches find approximations to local optimal solutions. In this paper, we focus on optimal control problems for switched autonomous systems where each subsystern is autonomous (i.e., with no continuous input). In particular, we focus on problems in which a prespecifled sequence of active subsystems is given. General autonomous subsystems and general performance costs are considered. For such problems the cost is a function of the switching instants. We propose to use constrained nonlinear optimization techniques to locate open-loop local optimal switching instants for such general problems. To apply nonlinear optimization techniques, we need to first determine the values of the derivatives of the cost with respect to the switching instants. An approach similar to that in [10] is proposed in this paper for their derivations. One of the main results of the pa1The partial support of the National Science Foundation (NSF ECS99-12458 & CCR01-13131), and of the DARPA/ITO-NEST Program (AF-F30602-01-2-0526) is gratefully acknowledged.

0-7803-7516-5/02/$17.00 ©2002 IEEE

per is Theorem 3.1 which gives us the expressions of the derivatives. Note here the approach provides us with accurate values of the derivatives as opposed to the approximate values in [10]. The approach is then applied to general quadratic problems for switched linear autonomous systems. The computation of the derivatives can be further simplified by utilizing the special structure of such problems. Finally, we apply the optimal control approach to reachability problems. Using the approach, the reachability switching instants can be determined if a final state is reachable from an initial state. Similar problems have also been looked into by other researchers. Giua et al in [4, 5] present closed-loop global optimal solutions to a special class of problems, i.e., infinite horizon problems for switched linear autonomous systems. However, we should indicate that our approach has the following advantages. First, our approach can deal with finite horizon problems with general subsysterns and costs as opposed to infinite horizon problems with linear subsystems and quadratic costs in [4, 5]. Moreover, our approach can be applied to reachability problems, while the approach in [4, 5] fits better for stability problems. In view of these, we believe our results are new and contribute to the understanding and the solution of optimal control problems of switched systems. 2 Problem

Formulation

We consider the following switched autonomous systerns, i.e., switched systems which consist of autonomous subsystems (i.e., without continuous input) ic-fi(x,t), f i ' R ~ x R - - + R ~, i C I - - { 1 , . . . ,M}. (2.1) The state trajectory evolution of such a system can be controlled by choosing appropriate switching sequences. A switching sequence in [t0,tf] is defined as - ((to, io), (tl,

il),

(t~, i~), • • • , ( t ~ , i ~ ) ) ,

(2.2)

with 0