Optimal control of switched systems based on parameterization of the switching instants

This paper presents a new approach for solving optimal control problems for switched systems. We focus on problems in which a prespecified sequence of active subsystems is given. For such problems, we need to seek both the optimal switching instants and the optimal continuous inputs. In order to search for the optimal switching instants, the derivatives of the optimal cost with respect to the switching instants need to be known. The most important contribution of the paper is a method which first transcribes an optimal control problem into an equivalent problem parameterized by the switching instants and then obtains the values of the derivatives 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, along with their differentiations. This method is applied to general switched linear quadratic problems and an efficient method based on the solution of an initial value ordinary differential equation is developed. An extension of the method is also applied to problems with internally forced switching. Examples are shown to illustrate the results in the paper.     

Features of motion of dynamic systems with disturbances in the neighborhood of manifolds of switchings

The method of computation of control in real time of a linear system with disturbance is suggested. The system of linear algebraic equations is obtained, which links the deviations of phase coordinates to the deviations of initial conditions of the normalized conjugate system and to the deviation of the finite moment. The calculations reduce to the sequence of the solutions of systems of linear algebraic equations and the integration of a matrix differential equation over transfer intervals of the control switching moments and the finite moment of time. The correction of switching moments and the finite moment of control in the accompaniment of the phase trajectory of motion of a controllable object is considered. Simple constructive conditions of the origin of the sliding mode, motions of the representative point over manifolds of switchings, and changes of the control structure in accompanying the phase trajectory of the system motion are obtained. The convergence of the computational method is proved.     

Shortest correlation control strategy

A new stochastic control strategy is developed for discrete linear constant systems. The strategy is optimal in the sense of yielding shortest correlation in the error sequence, i.e., the resulting error sequence is as white as possible. This is a stochastic counterpart of the deterministic discrete-time time-optimal control problem. Algebraic manipulations are used to reduce the synthesis procedure to solving a linear polynomial equation.     

Particle swarm optimization for time-optimal control design

In this paper,a particle swarm optimization(PSO)based method is proposed to obtain the time-optimal bang-bang control law for both linear and nonlinear systems.By introducing a penalty function,the method can be modified to deal with systems with constraints.Compared with existing computational methods,the proposed method can be implemented in a straightforward manner.The convergent solutions can be achieved by selecting suitable PSO parameters regardless of the initial guess of the switching times.A double integrator and a third-order nonlinear system are used to demonstrate the effectiveness and robustness of the proposed method.The method is applied to obtain the time-optimal control law for a high performance linear motion positioning system.The results show the practicality of the proposed algorithm.     

Time-Optimal Control of T--S Fuzzy Models via Lie Algebra

This paper investigates a geometric property of time-optimal problem in the Takagi–Sugeno (T--S) fuzzy model via Lie algebra. We will focus on the existence of a time-optimal solution, singularity of switching function, and number of switching. These inherent problems are considered because of their rich geometric properties. The sufficient condition for the existence of a time-optimal solution reveals the controllability of T--S fuzzy model that can be found by the generalized rank condition. The time-optimal controller can be found as the bang–bang type with a finite number of switching by applying the maximum principle. In the study of the singularity problem, we will focus on the switching function whenever it vanishes over a finite time interval. Finally, we show that the bounded number of switching can be found if the T--S model (also a nonlinear system) is solvable.      

Time-optimal control of multiple cooperating manipulators with trajectory and motion program constraints

Two types of problems associated with time-optimal control of multiple manipulators moving a commonly held object along specified trajectories are studied. The first problem involves finding the minimum traveling time and the optimal control torques for any desired motion programs of the given trajectory. The second problem involves finding the optimal velocity distribution along the trajectory such that the motion can be completed in the minimum time. To solve these problems, a parametric form of the generalized dynamic equation is derived. An iterative search procedure is developed for solving the first problem. During the search, the lower bound of the traveling time at any point of the given trajectory is determined by using the linear programming technique. The second problem is solved by integrating the parametric dynamic equation along the given trajectory based on the phase-plane switching curve approach. The maximum acceleration and the upper bound of the operation speed at each integration instance are determined from two linear programs. The proposed methods are applicable to various complex multi-robot systems and can handle nonlinear torque-speed characteristics of the joint actuators. © 1996 John Wiley & Sons, Inc.     

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.     

车型机器人的时间最优轨迹

基于几何最优控制理论, 本文提出了一种新的几何推理方法, 这种方法能够有效地推导出车型机器人给定两点间的时间最优轨迹类型的充分集合. 同时也为此类非线性问题提供了一种新的思路. 首先根据旁氏极大值原理和李代数推导出切换函数的结构特性, 根据这种结构特性建立一个切换坐标系并引入一个新的向量, 该向量在此坐标系中的旋转轨迹与时间最优轨迹具有一一对应关系. 进而得到一个结论: 如果该向量的始末旋转位置和方向一致, 那么将唯一的确定一条最优轨迹. 这是第一次得到一个能够直接应用于计算一条精确的最优轨迹的结论.     

Exact solutions to some minimum time problems with inequality state constraints

A heuristic method is developed for generating exact solutions to certain minimum time problems, with inequality state and control constraints. The control equation is linear and autonomous, with scalar-valued control. The state constraints are also linear inequalities. Assuming knowledge of a finite sequence, in which state and/or control constraints become active along an optimal path, the maximum principle is reduced to a set of equations and inequalities in a finite number of unknowns. A solution to the equations and inequalities determines both the solution path and a proof of its optimality. Certain types of constraint sequences lead to overdetermined equation systems, and this fact is interpreted in terms of the qualitative behavior of solutions to these problems. Two path-planning problems are solved, as illustrations of the solution technique.     

事件驱动的网络化系统最优控制

针对随机事件驱动的网络化控制系统, 研究其中的有限时域和无限时域内最优控制器的设计问题. 首先, 根据执行器介质访问机制将网络化控制系统建模为具有多个状态的马尔科夫跳变系统; 然后, 基于动态规划和马尔科夫跳变线性系统理论设计满足二次型性能指标的最优控制序列, 通过求解耦合黎卡提方程的镇定解, 给出最优控制律的计算方法, 使得网络化控制系统均方指数稳定; 最后, 通过仿真实验表明了所提出方法的有效性.

     

LQR optimization of linear system switching

Considers offline optimization of a switching sequence for a given finite set of linear control systems, together with joint optimization of control laws. A linear quadratic full information criterion is optimized and dynamic programming is used to find an optimal switching sequence and control law. The main result is a method for efficient pruning of the search tree to avoid combinatoric explosion. A method to prove optimality of a found candidate switch sequence and corresponding control laws is presented.     

近似时间最优的舵机多模位置控制策略

高超声速飞行器要求其伺服作动系统具有高动态高精度的特性,针对传统控制方法难以兼顾系统动态和精度的难题,本文设计了一种近似时间最优的舵机多模位置控制策略.首先建立了以一阶惯性环节串联积分器表征舵机输入输出特性的特征模型.然后以相平面为分析工具,给出了其近似时间最优控制的切换区;在切换区外以快速性为目标而采用bang-bang最优控制,在切换区内以避免振荡和超调为目标而采用bang-bang次优控制;为提高稳态性能,在小误差时时采用线性控制.实验表明,该方法响应快速,避免了振荡和超调,很好地满足了对伺服作动系统高动态高精度的要求.     

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 .     

On the method of dynamic programming for linear-quadratic problems of optimal control in hybrid systems

For a sufficiently wide class of the linear hybrid systems, an algorithm of optimal feedback control was proposed. Consideration was given to the hybrid control systems with autonomous switching, as well as the corresponding problems of the hybrid linear-quadratic optimal control based on the recently suggested principle of maximum. Interrelations between the hybrid principle of maximum and the method of dynamic programming for the systems of this class were discussed. The classical formalism was extended, the corresponding Riccati equations were obtained, and discontinuity of the “hybrid” Riccati matrix was proved. The computational aspects of the established theoretical results were considered.     

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.     

A time-optimal control algorithm for digital computer control

A time-optimal control algorithm for digital computer control allowing bounds on control variables and state variables is presented. Through linear programming techniques a time-optimal sequence is computed by using a linear discrete model of the process. For real-time applications feedback control is achieved by recalculating the control sequence each sampling period. In addition, an adaptive control strategy based on on-line estimation of the state variables and the parameters of the system is introduced. In order to be able to apply the algorithm in a real-time environment, computational efficiency is emphasized. An application of the algorithm to a sixth-order multivariable system is given.     

Generalized bang-bang control for feedforward constrained regulation

In the behavioral framework for continuous-time linear scalar systems, simple sufficient conditions for the solution of the minimum-time rest-to-rest feedforward constrained control problem are provided. The investigation of the time-optimal input-output pair reveals that the input or the output saturates on the assigned constraints at all times except for a set of zero measure. The resulting optimal input is composed of sequences of bang-bang functions and linear combinations of the modes associated to the zero dynamics. This signal behavior constitutes a generalized bang-bang control that can be fruitfully exploited for feedforward constrained regulation. Using discretization, an arbitrarily good approximation of the optimal generalized bang-bang control is found by solving a sequence of linear programming problems. Numerical examples are included.     

一类不受控离散事件驱动不确定线性切换系统优化控制

考虑一类不受控离散事件驱动不确定约束线性切换系统, 提出具有任意切换特性的约束优化控制策略. 引入有限时域性能函数, 定义约束最优控制问题. 为降低控制器的在线计算量, 通过参数化方法压缩最优控制问题决策变量的维数. 应用公共Lyapunov 函数和控制Lyapunov 函数理论建立的闭环切换系统, 具有对不确定扰动和不受控切换信号的渐近稳定性和逆最优性. 最后通过算例仿真验证了结果的有效性.

     

A computational method for convex optimal control problems involving linear hereditary systems†

We consider a class of convex optimal control problems involving a linear hereditary system. The main aim of the paper is to devise a computational algorithm for generating a minimizing sequence of controls such that the sequence converges to the optimal control in both the weak? topology of L∞ and the almost-everywhere topology.