Bang–bang control model with optimistic value criterion for uncertain switched systems

This paper studies an optimal control problem for uncertain switched linear systems with subsystems perturbed by uncertainty. A model for this problem is investigated with optimistic value criterion. The goal is to jointly design a deterministic switching law and a continuous feedback to optimize an uncertain objective function. A two-stage algorithm is applied to handle such model. In the first stage, the maximum value of the objective function and the bang–bang control are obtained under fixed switching instants, and in the second stage, GA and PSO algorithm are used to get the optimal switching instants, respectively. An example is shown to validate the method.     

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.     

一类切换系统的输入-状态稳定性分析与优化控制

利用驻留时间法和 Gronwall-Bellman不等式研究了一类切换系统的输入一状态稳定性分析与优化控制问题.在保证切换系统输入-状态稳定的前提下,将切换时刻和切换次数约束条件转化为线性约束,提出了一种新的切换系统优化问题目标函数的形式.与已有的方法相比,该方法无需引入新的状态变量,无需同时满足构造输入-状态稳定控制李亚普诺夫函数和所有子系统都是输入-状态稳定的条件,为控制器的优化设计提供了便利.最后,通过算例仿真证实了文中所提方法的可行性.     

Optimally switched linear systems

In this paper we address the problem of optimal switching for switched linear systems. The uniqueness of our approach lies in describing the switching action by multiple control inputs. This allows us to embed the switched system in a larger family of systems and apply Pontryagin’s Minimum Principle for solving the optimal control problem. This approach imposes no restriction on the switching sequence or the number of switchings. This is in contrast to search based algorithms where a fixed number of switchings is set a priori. In our approach, the optimal solution can be determined by solving the ensuing two-point boundary value problem. Results of numerical simulations are provided to support the proposed method.     

Optimal switching instants for a switched-capacitor DC/DC power converter

We consider a switched-capacitor DC/DC power converter with variable switching instants. The determination of optimal switching instants giving low output ripple and strong load regulation is posed as a non-smooth dynamic optimization problem. By introducing a set of auxiliary differential equations and applying a time-scaling transformation, we formulate an equivalent optimization problem with semi-infinite constraints. Existing algorithms can be applied to solve this smooth semi-infinite optimization problem. The existence of an optimal solution is also established. For illustration, the optimal switching instants for a practical switched-capacitor DC/DC power converter are determined using this approach.     

Optimal control of continuous-time switched affine systems

This paper deals with optimal control of switched piecewise affine autonomous systems, where the objective is to minimize a performance index over an infinite time horizon. We assume that the switching sequence has a finite length, and that the decision variables are the switching instants and the sequence of operating modes. We present two different approaches for solving such an optimal control problem. The first approach iterates between a procedure that finds an optimal switching sequence of modes, and a procedure that finds the optimal switching instants. The second approach is inspired by dynamic programming and identifies the regions of the state space where an optimal mode switch should occur, therefore providing a state feedback control law.     

Adaptive Multilevel Correction Method for Finite Element Approximations of Elliptic Optimal Control Problems

In this paper we propose an adaptive multilevel correction scheme to solve optimal control problems discretized with finite element method. Different from the classical adaptive finite element method (AFEM for short) applied to optimal control which requires the solution of the optimization problem on new finite element space after each mesh refinement, with our approach we only need to solve two linear boundary value problems on current refined mesh and an optimization problem on a very low dimensional space. The linear boundary value problems can be solved with well-established multigrid method designed for elliptic equation and the optimization problems are of small scale corresponding to the space built with the coarsest space plus two enriched bases. Our approach can achieve the similar accuracy with standard AFEM but greatly reduces the computational cost. Numerical experiments demonstrate the efficiency of our proposed algorithm.     

Linear–quadratic switching control with switching cost

We study in this paper the linear–quadratic (LQ) optimal control problem of discrete-time switched systems with a constant switching cost for both finite and infinite time horizons. We reduce these problems into an auxiliary problem, which is an LQ optimal switching control problem with a cardinality constraint on the total number of switchings. Based on the solution structure derived from the dynamic programming (DP) procedure, we develop a lower bounding scheme by exploiting the monotonicity of the Riccati difference equation. Integrating such a lower bounding scheme into a branch and bound (BnB) framework, we offer an efficient numerical solution scheme for the LQ switching control problem with switching cost.     

OPTIMAL SWITCHING CONTROL VIA DIRECT SEARCH OPTIMIZATION

We demonstrate in this paper that for switched systems the optimal switching time instants and the optimal control policy can be solved readily by direct search optimization. General problem formulation and numerical procedures are presented together with two demonstrative examples previously considered in the literature using other numerical solution methods.     

A novel finite element method for two-point boundary value problems

A finite element method is presented for solving boundary value problems for ordinary differential equations in which the general solution of the differential equation is computed first, followed by a selection procedure for the particular solution of the boundary value problem from the general solution. In this method, the discrete representation of the differential equation is a singular matrix equation, which is solved by using generalized matrix inversion. The technique is applied to both linear and nonlinear boundary value problems and to boundary value problems requiring eigenvalue evaluation. The solution of several examples involving different types of two-point boundary value problems is presented.     

A numerical algorithm for optimal control of a class of hybrid systems: differential transformation based approach

A novel numerical algorithm based on differential transformation is proposed for optimal control of a class of hybrid systems with a predefined mode sequence. From the necessary conditions for optimality of hybrid systems, the hybrid optimal control problem is first converted into a two-point boundary value problem (TPBVP) with additional transverse conditions at the switching times. Then we propose a differential transformation algorithm for solving the TPBVP which may have discontinuities in the state and/or control input at the switching times. Using differential transformation, the hybrid optimal control problem reduces to a problem of solving a system of algebraic equations. The numerical solution is obtained in the form of a truncated Taylor series. By taking advantage of the special properties of the linear subsystems and a quadratic cost functional, the differential transformation algorithm can be further simplified for the switched linear quadratic optimal control problem. We analyse the error of the numerical solution computed by the differential transformation algorithm and some computational aspects are also discussed. The performance of the differential transformation algorithm is demonstrated through illustrative examples. The differential transformation algorithm has been shown to be simple to be implemented and computationally efficient.     

A moving switching sequence approach for nonlinear model predictive control of switched systems with state‐dependent switches and state jumps

In this paper, we propose an approach for real‐time implementation of nonlinear model predictive control (NMPC) for switched systems with state‐dependent switches called the moving switching sequence approach. In this approach, the switching sequence on the horizon moves to the present time at each time as well as the optimal state trajectory and the optimal control input on the horizon. We assume that the switching sequence is basically invariant until the first predicted switching time reaches the current time or a new switch enters the horizon. This assumption is reasonable in NMPC for systems with state‐dependent switches and reduces computational cost significantly compared with the direct optimization of the switching sequence all over the horizon. We update the switching sequence by checking whether an additional switch occurs or not at the last interval of the present switching sequence and whether the actual switch occurs or not between the current time and the next sampling time. We propose an algorithm consisting of two parts: (1) the local optimization of the control input and switching instants by solving the two‐point boundary‐value problem for the whole horizon under a given switching sequence and (2) the detection of an additional switch and the reconstruction of the solution taking into account the additional switch. We demonstrate the effectiveness of the proposed method through numerical simulations of a compass‐like biped walking robot, which contains state‐dependent switches and state jumps.     

Optimization problems for switched systems with impulsive control

By using Impulsive Maximum Principal and three stage optimization method , this paper discusses optimization problems for linear impulsive switched systems with hybrid controls ,which includes continuous control and impulsive control. The linear quadratic optimization problems without constraints such as optimal hybrid control , optimal stability and optimal switching instants are addressed in detail. These results are applicable to optimal control problems in economics ,mechanics ,and management .     

Optimization problems for switched systems with impulsive control

By using Impulsive Maximum Principal and three stage optimization method, this paper discusses optimization problems for linear impulsive switched systems with hybrid controls, which includes continuous control and impulsive control. The linear quadratic optimization problems without constraints such as optimal hybrid control, optimal stability and optimal switching instants are addressed in detail. These results are applicable to optimal control problems in economics, mechanics, and management.     

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

In 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.     

切换线性系统的聚合优化

分析了切换线性系统的稳定性和优化问题,提出了一个Armijo步长优化的共轭梯度算法来寻找适当代价函数下的优化切换时间点集.为了确保优化切换路径是可压缩的,提出了代价函数需要满足的受限表达式.设计了几种优化分段状态反馈切换律来搜索聚合系统的最优切换路径,同时这些切换路径就是对应原始切换线性系统的次优化切换路径.最后,一个实例演示了不同切换律下的切换策略和优化代价.     

An iterative method for suboptimal control of a class of nonlinear time-delayed systems

This work presents an approximate solution method for the infinite-horizon nonlinear time-delay optimal control problem. A variational iteration method (VIM) is applied to design feedforward and feedback optimal controllers. By using the VIM, the original optimal control is transformed into a sequence of nonhomogeneous linear two-point boundary value problems (TPBVPs). The existence and uniqueness of the optimal control law are proved. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. The feedback term is determined by solving Riccati matrix differential equation. By using the finite-step iteration of a nonlinear compensation sequence, we can obtain a suboptimal control law. Simulation results demonstrate the validity and applicability of the VIM.     

Optimal bang-bang control of linear stochastic systems with a small noise parameter

This study considers the problem of determining optimal feedback control laws for linear stochastic systems with amplitude-constrained control inputs. Two basic performance indices are considered, average time and average integral quadratic form. The optimization interval is random and defined as the first time a trajectory reaches the terminal regionR. The plant is modeled as a stochastic differential equation with an additive Wiener noise disturbance. The variance parameter of the Wiener noise process is assumed to be suitably small. A singular perturbation technique is presented for the solution of the stochastic optimization equations (second-order partial differential equation). A method for generating switching curves for the resulting optimal bang-bang control system is then developed. The results are applied to various problems associated with a second-order purely inertial system with additive noise at the control input. This problem is typical of satellite attitude control problems.     

基于切换次序参数化的时滞切换正系统最优控制

考虑时滞线性自治切换正系统,将切换次序参数化作为系统的离散输入,通过系统的等价转换,分析时滞线性自治切换正系统的最优控制问题.利用极小值原理,给出满足性能指标的最优切换次序条件,分析时滞项对性能指标和切换时刻的影响,最后给出数值例子验证结论的有效性.