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This paper investigates the optimal control problem of a particular class of hybrid dynamical systems with controlled switching. Given a prespecified sequence of active subsystems, the objective is to seek both the continuous control input and the discrete switching instants that minimize a performance index over a finite time horizon. Based on the use of the calculus of variations, necessary conditions for optimality are derived. An efficient algorithm, based on nonlinear optimization techniques and numerical methods, is proposed to solve the boundary-value ordinary differential equations. In the case of linear quadratic problems, the two-point boundary-value problems can be avoided which reduces the computational effort. Illustrative examples are provided and stress the relevance of the proposed nonlinear optimization algorithm.

The optimization of hybrid dynamical systems has been widely investigated in the last years [

The focus of this paper is on the optimal control of a particular class of hybrid dynamical systems called switched systems. The behaviour of interest of such systems is described by a set of time-driven continuous-state subsystems and a switching law specifying the active subsystem at each time instant. A switching happens when an event signal is received. This signal may be an external generated signal or an internal signal generated if a condition on the time evolution, the states, and/or the inputs is satisfied. Consequently, we call a switching triggered by an external event an externally forced switching. So, according to the nature of the switching signal, switched systems may be classified into switched systems with externally forced switching or switched systems with internally forced switching.

In the last decade the switched systems have been extensively studied [

Modeling switched system depends on the different dynamics of its subsystems described by indexed differential or difference equations. Otherwise, if there is no external control influence on the system, we call it an autonomous switched system. To address the optimal control problem of autonomous switched systems, we have to focus on deriving the optimal sequence of switching times, and even if this sequence is the sole control influence, its determination remains a challenging task. In previous work [

For nonautonomous switched systems, it is necessary to consider the continuous input together with switching times and sequences. Similar to the optimization for autonomous switched systems, optimization techniques are needed to find the optimal solutions for nonautonomous switched systems. But, despite the relevant contributions to find numerical solutions to such problems by employing the established theoretical conditions, effective algorithms still remain to be developed.

Inspired by what is cited above, the main contribution of this paper is to solve the optimal control problem for nonautonomous switched systems with controlled switching. So

This paper is organized as follows. In Section

In this paper, we focus on continuous-time switched systems whose dynamics are described for

For such switched systems, we can control the state trajectory evolution by appropriately choosing the continuous control input

The optimal control problem considered in the present paper can be formulated as follows.

Given a continuous-time switched system whose dynamics are governed by (

In order to solve this problem, we will resort to the calculus of variations.

Based on the optimal control problem of continuous systems and the calculus of variations, the variational problem can be stated as follows. Let

In order to solve the above problem, we need to express the variation of the cost functional

Perturbations of trajectory.

The variation of

In this section, we will consider the case of a single switching, but the proposed approach and used methods can be straightforwardly applied to the case of several subsystems and more than one switching. The first problem is then reduced to the following problem.

A continuous-time switched system is given whose dynamics are governed by

Find the continuous control

To solve this problem, we introduce a costate variable

Numerical methods for solving boundary-value ordinary differential equations: the Hamiltonian system consists of two boundary-value ordinary differential equations whose solutions must satisfy conditions specified at the boundaries of the time intervals

Nonlinear optimization algorithms: to locate the optimal switching instant

See Algorithm

(i) Solve the Hamiltonian system using the shooting method

(a) Guess the unspecified initial conditions

and

(b) Integrate the Hamiltonian system (

to

(c) Using the resulting values of

the error function:

(d) Adjust the value of

numerical method for solving nonlinear equations, to bring

the function

(ii) Calculate the continuous optimal control (

technique;

Note that, at each iteration

For the case of linear switched systems with quadratic performance index, the present work will show that dealing with two-point boundary-value problems can be avoided, and therefore the computational effort can be reduced.

In this section, we consider the problem of minimizing a quadratic criterion subject to switched linear subsystems. For this special class, we can obtain a closed-loop continuous control within each time interval

According to (

By differentiating (

Since

(a) Solve the matrix Riccati equations (

backward from

(b) Solve the auxiliary equations (

from

(c) Solve the state equation (

(d) Calculate

(e) Calculate the continuous optimal control (

To illustrate the validity of the proposed result and the efficiency of the algorithms, two examples are considered in this section. The former concerns the optimization of a nonlinear switched system. The latter deals with quadratic optimal control problem for linear switched system. The computation was performed using MATLAB 6.5 on a Celeron 2 GHz PC with 256 Mo of RAM.

Consider a nonlinear switched system described by

Subsystem 1

Subsystem 2

Subsystem 3

The objective is to find the continuous control

The optimal control and the corresponding state trajectory are shown in Figure

Continuous control input and optimal state trajectory.

Cost

Let us consider a linear switched system, described by three subsystems as

Subsystem 1

Subsystem 2

Subsystem 3

The problem is to find the continuous control

Control input and optimal state trajectory.

Cost

By examining Figures

Based on nonlinear optimization techniques and numerical methods for solving boundary-value ordinary differential equations, we proposed an algorithm for solving optimal control problems for switched systems with externally forced switching. We assumed that the switching sequence is fixed, and therefore the control variables are only the continuous control input and the discrete switching instants. The effectiveness of the presented algorithm was demonstrated through simulation results.

The obtained results will be extended to the optimal control of interconnected switched systems. Otherwise parametric uncertainties and input disturbances are often present in real-life applications. So, analysis procedures and control synthesis algorithm for hybrid systems if additive disturbances and/or parametric uncertainties are present are topics that are starting to deserve the attention of researchers [

The author declares that there are no conflicts of interest regarding the publication of this paper.