Optimal Feedback Control for Linear Systems with Input Delays Revisited

The design problem of optimal feedback control for linear systems with input delays is very important in many engineering applications. Usually, the linear systems with input delays are firstly converted into linear systems without delays, and then all the design procedures are based on the delay-free linear systems. In this way, the feedback controllers are not designed in terms of the original states. This paper presents some new closed-form formula in terms of the original states for the delayed optimal feedback control of linear systems with input delays. We firstly reveal the essential role of the input delay in the optimal control design of the linear system with a single input delay: the input delay postpones the action of the optimal control only. Based on this fact, we calculate the delayed optimal control and find that the optimal state can be represented by a simple closed-form formula, so that the delayed optimal feedback control can be obtained in a simple way. We show that the delayed feedback gain matrix can be “smaller” than that for the controlled system with zero input delay, which implies that the input delay can be considered as a positive factor. In addition, we give a general formula for the delayed optimal feedback control of time-variant linear systems with multiple input delays. To show the effectiveness and advantages of the main results, we present five illustrative examples with detailed numerical simulation and comparison.     

Stochastic control problems with delay

We consider optimal control problems for systems described by stochastic differential equations with delay. We state conditions for certain classes of such systems under which the stochastic control problems become finite-dimensional. These conditions are illustrated with three applications. First, we solve some linear quadratic problems with delay. Then we find the optimal consumption rate in a financial market with delay. Finally, we solve explicitly a deterministic fluid problem with delay which arises from admission control in ATM communication networks.     

非线性退化时滞微分控制系统的函数能控性

本文讨论非线性退化时滞微分控制系统．首先就非线性退化时滞微分控制系统的一阶变分系统给出变易公式,然后就非线性退化时滞微分控制系统的一阶变分系统的函数能控性给出一些判据,最后给出关于非线性退化时滞微分控制系统的函数能控性的判据．     

Synthesis of optimal feedback and the stabilization of systems with a delay in the control

The linear problem of the optimal control of systems in which the input signals contain a time delay is considered. The method of realizing optimal feedback control that is proposed is based on a special procedure for correcting the current optimal programme controls, realized by an optimal controller using a dual linear programming method. The results are used to construct two types of stabilizer of systems with a delay in the control.     

在模型和实际存在差异时的非线性离散时滞系统最优控制算法

研究非线性分布时滞系统最优控制,提出一种基于线性分布时滞模型和二次型性能指标问题的迭代算法,将分布时滞系统化为满足马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分布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。给出该算法收敛于实际最优解的充分条件。     

An Indefinite Stochastic Linear Quadratic Optimal Control Problem with Delay and Related Forward–Backward Stochastic Differential Equations

In this paper, we will study an indefinite stochastic linear quadratic optimal control problem, where the controlled system is described by a stochastic differential equation with delay. By introducing the relaxed compensator as a novel method, we obtain the well-posedness of this linear quadratic problem for indefinite case. And then, we discuss the uniqueness and existence of the solutions for a kind of anticipated forward–backward stochastic differential delayed equations. Based on this, we derive the solvability of the corresponding stochastic Hamiltonian systems, and give the explicit representation of the optimal control for the linear quadratic problem with delay in an open-loop form. The theoretical results are validated as well on the control problems of engineering and economics under indefinite condition.     

OPTIMAL CONTROL OF A CLASS OF DISTRIBUTED PARAMETER DELAY SYSTEMS

In this paper we consider the optimal control problem for a class of infinite dimensional delay evolution systems whose principal operator is the infinitesimal generator of an analytic semigroup. We give an existence result of α - solutions of the controlled systems and prove the existence of solutions for an extremal problem subject to such systems. In particular, the necessary conditions of optimality for the same problem are presented.     

Real-time calculation of current optimal feedbacks for a delay system

A linear optimal control problem for a nonstationary system with a single delay state variable is examined. A fast implementation of the dual method is proposed in which a key role is played by a quasi-reduction of the fundamental matrices of solutions to the homogeneous part of the delay models under analysis. As a result, an iteration step of the dual method involves only the integration of auxiliary systems of ordinary differential equations over short time intervals. A real-time algorithm is described for calculating optimal feedback controls. The results are illustrated by the optimal control problem for a second-order stationary system with a fixed delay.     

Optimal control of switched systems with time delay

In this note, we develop a computational method for solving an optimal control problem which is governed by a switched dynamical system with time delay. Our approach is to parameterize the switching instants as a new parameter vector to be optimized. Then, we derive the required gradient of the cost function which is obtained via solving a number of delay differential equations forward in time. On this basis, the optimal control problem can be solved as a mathematical programming problem.     

移动通信系统中的最优功率控制算法

本文研究了移动通信系统中的功率最优控制问题.我们首先将这一个工程问题转化为最大可满足线性不等式组问题的一个特殊情形,然后通过对这个组合优化问题的最优解的性质研究,给出了求解该问题的多项式时间算法.     

Delay systems and optimal control

0. IntroductionIt is well known that evolution systems are an important class of distributed parametersystems and the optimal control of infinite dimensional systems is a remarkable subjectin control theory. Some authors including us investigated the existence of solutions orperiodic solutions (see [l--4] or [5,6] ). To the knowledge of the authors, optimal controls fordistributed parameter delay systems have not been extensively studied. Li and Yao obtaina maximum principle (see [7]). Here w…     

Optimal synchronization of two different in‐commensurate fractional‐order chaotic systems with fractional cost function

This article investigates the optimal synchronization of two different fractional‐order chaotic systems with two kinds of cost function. We use calculus of variations for minimizing cost function subject to synchronization error dynamics. We introduce optimal control problem to solve fractional Euler–Lagrange equations. Optimal control signal and minimum time of synchronization are obtained by proposed method. Examples show the optimal synchronization of two different systems with two different cost functions. First, we use an ordinary integer cost function then we use a fractional‐order cost function and comparing the results. Finally, we suggest a cost function which has the optimal solution of this problem, and we can extend this solution to solve other synchronization problems. © 2016 Wiley Periodicals, Inc. Complexity 21: 401–416, 2016     

Banach空间中非光滑指标奇异最优控制

本文对于Ｂａｎａｃｈ空间中分布参数系统讨论非光滑指标奇异最优控制问题，利用Ｍｏｏｒｓ－Ｐｅｎｒｏｓｅ广义逆与Ｃｌａｒｋｅ广义梯度证得奇异最优控制的存在性，并给出广义一阶必要条件，推广了Ｌｉｏｎｓ［３］的相应结果．     

Geometry of neighborhoods of singular trajectories in problems with multidimensional control

It is shown that the order of a singular trajectory in problems with multidimensional control is described by a flag of linear subspaces in the control space. In terms of this flag, we construct necessary conditions for the junction of a nonsingular trajectory with a singular one in affine control systems. We also give examples of multidimensional problems in which the optimal control has the form of an irrational winding of a torus that is passed in finite time.     

A unified treatment of dividend payment problems under fixed cost and implementation delays

In this paper we study the dividend optimization problem for a corporation or a financial institution when the management faces (regulatory) implementation delays. We consider several cash reservoir models for the firm including two mean-reverting processes, Ornstein–Uhlenbeck and square-root processes. Since the cash flow structure of different companies have different qualitative behaviors, it makes sense to use different diffusions to model them. The delay causes significant difficulties to the optimization problem since the cash reservoir fluctuates during the delay period. We provide a uniform mathematical framework to analyze all the models and provide optimal threshold strategies at which the management initiates actions, i.e., declaration and payment of dividends. Our solution depends on a new characterization of the value function for one-dimensional diffusions and provide easily implementable algorithms to find the optimal control and the value function.     

Impulsive control for differential systems with delay

This paper deals with the impulsive control for a class of differential systems with delay. Using Lyapunov functions and the comparison principle, we present some sufficient conditions for the asymptotic stability and exponential stability of impulsive control systems with delay. Moreover, we give an estimate of the upper bound of impulse interval. The results in this paper extend and improve the earlier publications. Copyright © 2012 John Wiley & Sons, Ltd.     

Stability of time-varying switched systems with time-varying delay

In this paper, we investigate the stability of time-varying switched systems with time-varying delay. We first give a generalization of Halanay’s inequality and then use this inequality to obtain sufficient conditions for the stability of switched systems.     

一类概周期时滞捕食-食饵系统的概周期解

本文讨论一类概周期时滞捕食－食饵系统的一致持久性,通过构造一个Ｌｉａｐｕｎｏｖ函数得到该系统有界解的唯一性,并且给出正概周期解的存在唯一性定理。     

退化时滞差分系统的通解

讨论具有时滞的退化差分系统，给出其初始函数的相容性条件，并给出其通解，最后举例说明之．     

Solvability of mixed-type optimal control problems for distributed systems

In this paper we consider optimal control problems for distributed systems which are not solved with respect to the time derivative, where performance functionals take a quadratic form. We concurrently use the distributed and starting controls.