Sequence Of Active Subsystems Research Articles

Optimal control for a class of impulsive switched systems

In this paper, an optimal switch-time control problem is solved for a class of impulsive switched autonomous systems. The considered systems jump at the switching times, and the sequence of active subsystems is pre-specified. The control variables consist of the impulse times and a set of scalars which determine the jump amplitudes. Moreover, the subsystems do not require a refractory period, which can bring more generality. Using the calculus of variation, the partial derivatives of the cost with respect to the control variables are derived, based on which the optimality conditions are given. Meanwhile, the obtained formulas can be used in some gradient descent algorithms to locate the optimal control variables. Finally, the viability of the proposed method is illustrated through two numerical examples.

Calculus of Variations and Nonlinear Optimization Based Algorithm for Optimal Control of Hybrid Systems with Controlled Switching

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.

Optimal control of switching time in switched stochastic systems with multi-switching times and different costs

ABSTRACTIn this paper, we solve an optimal control problem for a class of time-invariant switched stochastic systems with multi-switching times, where the objective is to minimise a cost functional with different costs defined on the states. In particular, we focus on problems in which a pre-specified sequence of active subsystems is given and the switching times are the only control variables. Based on the calculus of variation, we derive the gradient of the cost functional with respect to the switching times on an especially simple form, which can be directly used in gradient descent algorithms to locate the optimal switching instants. Finally, a numerical example is given, highlighting the validity of the proposed methodology.

Time optimisation problem for switched stochastic systems with multi‐switching times

In this study, a time optimisation problem is solved for a class of linear time-invariant switched stochastic systems with multi-switching times using the calculus of variations. The objective of authors’ study is to minimise a cost functional defined on the system state, where the sequence of active subsystems is pre-specified and the switching times are the only control variables. The gradient of the cost functional with respect to the switching times is derived, which has an especially simple form and can be directly used in gradient descent algorithms to locate the optimal switching instants. The proposed method is applied on a two-tank system, which is a practical example of switched systems. Combining with another numerical example with three switching times case, the viability of the proposed method is further demonstrated.

Optimal Control of Switching Times in Switched Stochastic Systems

AbstractIn this paper, we solve an optimal control problem for switched stochastic systems using calculus of variations, where the objective is to minimize a cost functional defined on the state and the switching times are the sole control variables. In particular, we focus on the problem in which a pre‐specified sequence of active subsystems is given. For one switching time case, the derivative of the cost functional with respect to the switching time is derived, which has an especially simple form and can be directly used in gradient descent algorithms to locate the optimal switching instant. Then, we propose an approach to deal with the problem with multi‐switching times case. Finally, two numerical examples are given, highlighting the viability and advantages of the proposed methodology.

Optimal timing control of discrete-time linear switched stochastic systems

Optimal switch-time control is the study that investigates how best to switch between different modes. In this paper, we investigate the optimal switch-time control problem for discrete-time linear switched stochastic systems. In particular, under the assumption that the sequence of active subsystems is pre-specified, we focus on the problem where the objective is to minimize a cost functional defined on the states and the switching times are the only control variables. For systems with one switching time, using calculus of variations, we firstly derive the difference formulae of the cost functional with respect to the switching time, which can be directly used to find the optimal switching instant. Then, a method is presented to deal with the problem with multiple switching times case. Finally, the viability of the proposed method is illustrated through two numerical examples.

Optimal control of switched systems and its parallel optimization algorithm

In this paper, a class of optimal switching control problems is considered in which the mode sequence of active subsystems and the number of mode switchings are not pre-specified, and both the switching sequence and the control inputs are to be chosen such that the cost functional is minimized. For solving this problem, we discuss the necessary conditions for optimality and the construction method for suboptimal solutions, and develop a few sufficient conditions of judgment on the optimal or suboptimal solutions of the switched system. According to the sufficient conditions, a parallel computational algorithm is constructed to find optimal or suboptimal solutions. For illustration, two examples are solved using the proposed algorithm.

Constrained optimal control of switched systems and its application

In this paper, we consider an optimal co with state and control constraints, where a prespecified sequence of active subsystems is given. Both the switching instants and the control function are to be chosen such that the cost functional is minimized. Firstly, we convert the problem into a conventional optimal control problem based on parameterization of the switching instants. Next, through discretizing the control space, applying a time-scaling transformation and using the smoothing technique, the optimal control problem is approximated by a non-linear optimization problem with linear constraints and bounds on the variables. Furthermore, an algorithm that finds a suboptimal solution to the original problem is proposed. One major advantage with the new algorithm implementation is the only need to solve a non-linear optimization problem with linear constraints and bounds on the variables. The gradients of these linear constraints and the objective function can be easily obtained. Therefore, the original optimization problem can be solved efficiently using any gradient-based method, such as sequential quadratic programming algorithm. Finally, a flight planning example illustrates the efficiency of the proposed approach.

Optimal control of switched systems via non-linear optimization based on direct differentiations of value functions

This paper presents an approach for solving optimal control problems of switched systems. In general, in such problems one needs to find both optimal continuous inputs and optimal switching sequences, since the system dynamics vary before and after every switching instant. After formulating a general optimal control problem, we propose a two stage optimization methodology. Since many practical problems only concern optimization where the number of switchings and the sequence of active subsystems are given, we concentrate on such problems and propose a method which uses nonlinear optimization and is based on direct differentiations of value functions. The method is then applied to general switched linear quadratic (GSLQ) problems. Examples illustrate the results.

AN APPROACH TO SWITCHED SYSTEMS OPTIMAL CONTROL BASED ON PARAMETERIZATION OF THE SWITCHING INSTANTS

This paper provides a viable general approach to switched systems optimal control. Such optimal control problems require the solutions of not only optimal continuous inputs but also optimal switching sequences. Many practical problems only involve optimization where the number of switchings and the sequence of active subsystems are given. This is stage 1 of the two stage optimization methodology proposed by the authors in previous papers. In order to solve stage 1 problems, the derivatives of the optimal cost with respect to the switching instants need to be known. In (Xu and Antsaklis, 2001), we proposed an approach for solving a special class of such problems, namely, general switched linear quadratic problems. In this paper, the idea of (Xu and Antsaklis, 2001) is extended to general switched systems optimal control problems and an approach is proposed for solving them. The approach first transcribes a stage 1 problem into an equivalent problem parameterized by the switching instants and then the values of the derivatives are obtained based on the solution of a two point boundary value differential algebraic equation formed by the state, costate, stationarity equations, the boundary and continuity conditions and their differentiations. Examples are shown to illustrate the results in the paper.