An optimal control method for linear systems with time delay

An optimal control method for linear systems with time delay is developed in this paper. In the proposed control method, the differential equation with time delay of the system dynamics is first rewritten into a form without any time delay through a particular transformation. Then, the optimal controller is designed by using the classical optimal control theory. A numerical algorithm for control implementation is presented, since the obtained expression of the optimal controller contains an integral term that is not convenient for online calculation. The time delay is considered at the very beginning of the control design, and no approximation and estimation are made in the control system. Thus the system performance and stability are prone to be guaranteed. Instability in responses might occur only if a system with time delay is controlled by the optimal controller that was designed with no consideration of time delay. The effectiveness of the proposed optimal controller is demonstrated by simulation studies.     

Optimal addition of servers for a time-dependent queueing process

Necessary conditions for optimal control of systems containing a time delay that is a function of the state of the system and of time are derived by utilizing calculus of variations. The time delay may be in the state vector and in the control vector. The state vector and the control vector can be constrained by inequality constraints. A transformation to eliminate state variable inequality constraints by increasing the dimensions of state space, developed by Jacobson for an undelayed system, is extended to a system with time delays. Necessary conditions to obtain an optimal delay are shown, and an example of finding an optimal delay is included. A gradient algorithm for systems with state dependent time delays has been developed.     

An Efficient Finite Difference Method for The Time‐Delay Optimal Control Problems With Time‐Varying Delay

In this paper, an efficient finite difference method is presented for the solution of time‐delay optimal control problems with time‐varying delay in the state. By using the Pontryagin's maximum principle, the original time‐delay optimal control problem is first transformed into a system of coupled two‐point boundary value problems involving both delay and advance terms. Then the derived system is converted into a system of linear algebraic equations by using a second‐order finite difference formula and a Hermite interpolation polynomial for the first‐order derivatives and delay terms, respectively. The convergence analysis of the proposed approach is provided. The new scheme is also successful for the optimal control of time‐delay systems affected by external persistent disturbances. Numerical examples are included to demonstrate the validity and applicability of the new technique. Some comparative results are included to illustrate the effectiveness of the proposed method.     

Optimal control problems for stochastic delay evolution equations in Banach spaces

We consider the optimal control for a Banach space valued stochastic delay evolution equation. The existence and uniqueness of the mild solution for the associated Hamilton–Jacobi–Bellman equations are obtained by means of backward stochastic differential equations. An application to optimal control of stochastic delay partial differential equations is also given.     

Optimal control of the delay linear systems with allowance for the terminal state constraints

Consideration was given to the linear problem of optimal control of one type of the delay systems where the delay appears in one equation of the system mathematical model. The terminal states of the system are bounded, the optimal control is realized by the discrete control actions obeying the geometrical constraints. Consideration was given to two types of solutions—program and positional. A dual method of calculation of the optimal programs was presented. Described was an algorithm of the optimal controller generating in real time the current values of the positional solution (optimal feedback). The results obtained were illustrated by the example of control of a system with the fourth-order delay.     

Computation of optimal control for linear systems with delay

A numerical algorithm is presented to calculate an optimal control recursively for linear multivariable systems with delay. The algorithm is based on the method of steepest descent in Hilbert space. The optimal control of a multivariable system with delay for a quadratic criterion function is given by the Riccati partial differential equations. These are simultaneous partial differential equations which are difficult to solve numerically. In most computational algorithms, errors are inevitable, a vast memory is required, and a lot of computational time is needed. This makes it impractical to use available computers for these algorithms. The algorithm presented here gives optimal control effectively without a large memory requirement. It also provides a practical computational method for obtaining the optimal control of multivariable systems with delay. Several numerical computations are performed to show how the effectiveness of the algorithm compares with other methods.     

离散时间非线性时滞系统最优控制的DISOPE算法

对于非线性时滞系统的最优控制,提出一种基于线性时滞模型和二次型性能指标问题的迭代处蒙混过关针时滞系统化为满足可尔可夫性质的增广状态系统,在模型和实际存在差异的情况下,该算法通过迭代求解时滞线性最优控制问题和参数估计问题,获得原问题的最优解,仿真实例表明该算法的有效性和实用性。     

Optimal control of continuous-time linear systems with a time-varying, random delay

We study a finite horizon optimal control problem for a class of linear systems with a randomly varying time-delay. The systems of this type may arise in embedded control applications and in certain applications in economics. The delay value is treated as an unknown variable but with known statistical properties, modelled by a Markov process with a finite number of states. The “probabilistic delay averaging” approach is employed to determine the optimal control in the form which is independent of the delay value.     

Stabilization of linear autonomous systems of differential equations with distributed delay

Consideration is given to the problem of optimal stabilization of differential equation systems with distributed delay. The optimal stabilizing control is formed according to the principle of feedback. The formulation of the problem in the functional space of states is used. It was shown that coefficients of the optimal stabilizing control are defined by algebraic and functional-differential Riccati equations. To find solutions to Riccati equations, the method of successive approximations is used. The problem for this control law and performance criterion is to find coefficients of a differential equation system with distributed delay, for which the chosen control is a control of optimal stabilization. A class of control laws for which the posed problem admits an analytic solution is described.     

Numerical solution of a conspicuous consumption model with constant control delay

We derive optimal pricing strategies for conspicuous consumption products in periods of recession. To that end, we formulate and investigate a two-stage economic optimal control problem that takes uncertainty of the recession period length and delay effects of the pricing strategy into account.This non-standard optimal control problem is difficult to solve analytically, and solutions depend on the variable model parameters. Therefore, we use a numerical result-driven approach. We propose a structure-exploiting direct method for optimal control to solve this challenging optimization problem. In particular, we discretize the uncertainties in the model formulation by using scenario trees and target the control delays by introduction of slack control functions.Numerical results illustrate the validity of our approach and show the impact of uncertainties and delay effects on optimal economic strategies. During the recession, delayed optimal prices are higher than the non-delayed ones. In the normal economic period, however, this effect is reversed and optimal prices with a delayed impact are smaller compared to the non-delayed case.     

Linear quadratic regulation for systems with time‐varying delay

In this paper we study the linear quadratic regulation (LQR) problem for discrete‐time systems with time‐varying delay in the control input channel. We assume that the time‐varying delay is of a known upper bound, then the LQR problem is transformed into the optimal control problem for systems with multiple input channels, each of which has single constant delay. The optimal controller is derived by establishing a duality between the LQR and a smoothing estimation for an associated system with a multiple delayed measurement. Copyright © 2009 John Wiley & Sons, Ltd.     

基于自适应动态规划的一类带有时滞的离散时间非线性系统的最优控制策略

针对一类状态和控制变量均带有时滞的非线性系统的带有二次性能指标函数最优控制问题, 本文提出了一种基于新的迭代自适应动态规划算法的最优控制方案. 通过引进时滞矩阵函数, 应用动态规划理论, 本文获得了最优控制的显式表达式, 然后通过自适应评判技术获得最优控制量. 本文给出了收敛性证明以保证性能指标函数收敛到最优. 为了实现所提出的算法, 本文采用神经网络近似性能指标函数、计算最优控制策略、求解时滞矩阵函数、以及给非线性系统建模. 最后本文给出了两个仿真例子说明所提出的最优策略的有效性.     

非线性时变时延系统的模糊采样最优控制

针对非线性时变时延系统,采用输入时延和自由权重矩阵方法研究模糊采样最优控制问题.应用T-S模糊系统表征非线性系统,控制器是零阶保持采样信号.由线性矩阵不等式给出最优控制准则,所设计的模糊采样控制器在闭环系统渐近稳定意义下可保证期望最优控制性能.最后,通过卡车拖车系统实验验证模糊采样控制设计方案的可行性.     

柔性系统的最小时间鲁棒时滞滤波器设计

本文提出了一种最小时间鲁棒时滞滤波器控制,来提高柔性系统的时间最优控制的 鲁棒性．这种最小时间鲁棒时滞滤波器控制方法与其扩展系统传统的时间最优控制是一致的 ,这种等价性提供了验证最小时间鲁棒时滞滤波器最优性的理论方法,即所求得的解是否满 足Pontryagin 最小值原理．仿真结果进一步表明了这种最小时间鲁棒时滞滤波器的优越性 ．     

OPTIMAL REJECTION WITH ZERO STEADY‐STATE ERROR OF SINUSOIDAL DISTURBANCES FOR TIME‐DELAY SYSTEMS

This paper studies the problem of optimal rejection with zero steady‐state error of sinusoidal disturbances for linear systems with time‐delay. Based on the internal model principle, a disturbance compensator is constructed to counterbalance the external sinusoidal disturbances, so that the original system can be transformed into an augmented system without disturbances. Then, with the introduction of a sensitivity parameter and expanding power series around it, the optimal disturbance rejection problem can be simplified to the problem of solving an infinite sum of a linear optimal control series without time‐delay or disturbance. The optimal control law for disturbance rejection with zero steady‐state error consists of accurate linear state feedback terms and a time‐delay compensating term, which is an infinite sum of an adjoint vector series. In the presented approach, iteration is required only for the time‐delay compensation series. By intercepting a finite sum of the compensation series, we obtain an approximate physically realizable optimal control law that avoids complex calculation. A numerical simulation shows that the algorithm is effective and easy to implement.     

时滞系统在高速网络下的最优扰动抑制

研究在含有控制时滞与测量时滞的系统在高速通讯网络下最优扰动抑制问题. 首先建立在高速通讯网络下含有控制时滞与测量时滞系统的离散化数学模型, 利用模型转换将时滞系统转化为形式上的无时滞系统. 然后通过求解离散Riccati方程和Stein方程设计含有状态反馈、扰动前馈和控制记忆项的最优控制律, 前馈项和控制记忆项分别补偿了扰动和控制时滞对系统性能的影响. 通过构造降维扰动状态观测器, 设计了含扰动前馈、输出反馈及 控制记忆项的动态控制律, 解决了前馈补偿器的物理不可实现问题. 仿真实例验证了所设计的最优控制律的有效性.     

Sampled-data optimal control of continuous systems for quadratic criterion function taking account of delayed control action

This paper is concerned with the sampled-data control of a continuous system in the situation where there exists delay for the control action after the measurement of the output at each sampling instant. The optimal control with the quadratic criterion function defined for the continuous system is adopted in this control problem. This paper shows that the problem can be transformed into the optimal control of a discrete system for the quadratic criterion function, and the variation of the control law to the sampling period and to the delay of the control action is clarified. The deterioration of cost performance with respect to delay is derived. A numerical example is also presented.     

含时滞倒立摆系统的最优控制研究

倒立摆本身是一个自然不稳定系统,时滞的存在更加恶化了系统性能。基于牛顿动力学方法建立了具时滞单级倒立摆系统的动力学模型,运用线性二次型最优控制策略设计了倒立摆控制器,将时滞环节采用Padé一阶对称逼近,通过Routh稳定判据,得到闭环系统的临界稳定的时滞大小。通过实例仿真进行验证,并与PID控制策略相比较,结果表明本文方法正确,所设计的最优控制器性能优于PID控制器,在临界时滞范围内有较好的控制效果,但与无时滞时相比,系统的动态性能有所降低。     

基于状态反馈的纯滞后对象的鲁棒优化控制

纯滞后对象是一类很难控制的对象;针对纯滞后对象,提出一种状态反馈优化控制方法;系统的设计方法包括：用高阶分时模型来逼近纯滞后环爷;以新型高阶Butlerworth最优传递函数为目标函数来设计状态反馈增益阵和状态观测器;仿真结果表明该方法设汁的系统具有较高的鲁棒性和抗扰性能。     

A sufficient condition for optimal control problems with time delays

This paper considers a class of systems modeled by ordinary differential equations which contain a delay in both the state and control variables. A sufficient condition for the optimal control of such systems is presented. The condition is a modification of a sufficient condition for optimal control problems without time delays. The application of the result is illustrated by examples.