Observer-based approximate optimal tracking control for time-delay systems with external disturbances

This paper proposes a successive approximation design approach of observer-based optimal tracking controllers for time-delay systems with external disturbances. To solve a two-point boundary value problem with time-delay and time-advance terms and obtain the optimal tracking control law, two sequences of vector differential equations are constructed first. Second, the convergence of the sequences of the vector differential equations is proved to guarantee the existence and uniqueness of the control law. Third, a design algorithm of the optimal tracking control law is presented and the physically realisable problem is addressed by designing a disturbance state observer and a reference input state observer. An example of an industrial electric heater is given to demonstrate the efficiency of the proposed approach.     

一种高效的快速近似控制向量参数化方法

控制向量参数化(Control vector parameterization, CVP) 方法是目前求解流程工业中最优操作问题的主流数值方法,然而,该方法的主要缺点之一是 计算效率较低,这是因为在求解生成的非线性规划(Nonlinear programming, NLP) 问题时,需要随着控制参数的调整,反复不断地求解相关的微分方程组,这也是CVP 方法中最耗时的部分.为了提高CVP 方法的计算效率,本文提出一种新颖的快速近似方法,能够有效减少微分方程组、函数值以及 梯度的计算量.最后,两个经典的最优控制问题上的测试结果及与国外成熟的最优控制 软件的比较研究表明:本文提出的快速近似CVP 方法在精度和效率上兼有良好的表现.     

Direct approach for the optimal control of linear time-delay systems via shifted Legend re polynomials

The present paper proposes a direct approach to the optimal control of linear time-delay systems. In the first step, both control and state are approximated by shifted Legendre polynomials, and then the Legendre-coefficients vector of states can be expressed in terms of the Legend re-coefficients vector of controls. In the second step, the relation between Legendre-coefficients vectors of states and controls, which satisfies the state equations approximately, is introduced into the performance index. Next, the variational principle is applied to find the Legendre-coefficients vector of near-optimal control that minimizes the performance index directly. In the present approach, the Legendre-coefficients vector of sub-optimal control can be obtained directly from a set of algebraic equations. Thus, both difficulties in solving simultaneous partial differential equations of large dimension, or in solving n sets of ordinary differential equations, are avoided.     

Optimal control of quasi-linear systems of the diffusion type under incomplete information on the state

The problem of optimal control for a stochastic state-linear control system with coefficients of diffusion depending on the vector of state and control with quadratic criterion of the control quality is investigated. We assume that only a part of components of the state vector of the system are measured. On the basis of the Lyapunov-Lagrange method, we propose a method for searching for an optimal control strategy depending on known components of the state vector. The problem of synthesis of a control is reduced to solving a boundary-value problem for a system of ordinary differential equations of the Riccati type. Several model examples with different awareness about the system state are analyzed.     

Global existence and uniqueness of positive solutions and optimal control for a novel model of pest control

In the present paper, a novel model of pest control governed by coupled sets of partial and ordinary differential equations is investigated in detail. Global existence and uniqueness of positive solutions to the above model are obtained. Also, the existence of optimal control is proved, and the first-order and second-order optimality conditions satisfied by optimal control are verified. In particular, the optimal control is demonstrated to be of Bang-Bang form.     

Optimal feedback control of non-linear systems

A method is proposed for computing optimal nonlinear feedback control laws. It is shown that the feedback loop satisfies a system of quasi-linear partial differential equations. This system is of first order when the dimensions of the state and the control vector are the same. Hereby, it degenerates into algebraic equations when the performance index does not depend on the control. These important results offer new ways for determining optimal feedback laws. Connections with the Volterra series and the Hamilton-Jacobi-Bellman equation are treated. Some examples are given to illustrate the advantages of this 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 problem for stochastic evolution equations in Hilbert spaces

In this article, we consider an optimal control problem in which the controlled state dynamics is governed by a stochastic evolution equation in Hilbert spaces and the cost functional has a quadratic growth. The existence and uniqueness of the optimal control are obtained by the means of an associated backward stochastic differential equations with a quadratic growth and an unbounded terminal value. As an application, an optimal control of stochastic partial differential equations with dynamical boundary conditions is also given to illustrate our results.     

Delay optimal control and viscosity solutions to associated Hamilton–Jacobi–Bellman equations

In this article, optimal control problems of differential equations with delays are investigated for which the associated Hamilton–Jacobi–Bellman (HJB) equations are nonlinear partial differential equations with delays. This type of HJB equation has not been previously studied and is difficult to solve because the state equations do not possess smoothing properties. We introduce a new notion of viscosity solutions and identify the value functional of the optimal control problems as the unique solution to the associated HJB equations. An analytical example is given as application.     

线性二次最优控制的精细积分法

LQ控制虽然是最优控制的最基本问题,但其数值求解仍有很多问题.黎卡提微分 方程的精细积分法利用黎卡提方程的解析特点,求出计算机上高度精密的解,并已证明误差 在计算机倍精度数的误差范围之外.这对于Kalman-Bucy滤波,LQG问题以及H∞控制及滤 波等都可运用,精细积分还求解了反馈后的状态微分方程.数例验证了其高精度特性.     

On solving hybrid optimal control problems with higher index DAEs

The paper deals with hybrid optimal control problems described by higher index differential–algebraic equations (DAEs). We introduce a numerical procedure for solving these problems. The procedure has the following features: it is based on the appropriately defined adjoint equations formulated for the discretized equations being the result of the numerical integration of systems equations by an implicit Runge–Kutta method; the consistent initialization procedure is applied whenever control functions jumps, or state variables transition occurs. The procedure can cope with hybrid optimal control problems which are defined by DAEs with the index not exceeding three. Our approach does not require differentiation of some system equations in order to transform higher index DAEs to the underlying ordinary differential equations (ODEs). The presented numerical examples show that the proposed approach can be used to solve efficiently hybrid optimal control problems with higher index DAEs.     

Kalman filter for controlled hybrid systems

The linear partially observed discrete-continuous (hybrid) stochastic controllable system described by differential equations with measures is considered. The optimal filtering equations in the form of generalized Kalman filter are obtained in the case of non-anticipating control. This result could be a theoretical basis for the optimal control in stochastic hybrid systems with incomplete information.     

Maximum principle for the stochastic optimal control problem with delay and application

In this paper, we consider an optimal control problem for the stochastic system described by stochastic differential equations with delay. We obtain the maximum principle for the optimal control of this problem by virtue of the duality method and the anticipated backward stochastic differential equations. Our results can be applied to a production and consumption choice problem. The explicit optimal consumption rate is obtained.     

Optimal control laws for a class of constrained linear-quadratic problems

In general it seems impossible to specify the form of the solution of the linear-quadratic optimal control problem when there are control constraints. This paper deals with a restricted class of quadratic performance criteria, for which the optimal feedback control law is shown to be linear-in-half-spaces when the control is constrained to take values in a closed cone $\text{Γ ?}\text{R}{}^{\mathrm{<mtext>m</mtext>}}$. The time-varying vector quantities which are required for implementation of the optimal control law may be pre-computed by solving two initial-value problems for certain nonlinear first-order ordinary differential equations, and maximizing a quadratic function over the cone Γ at each time. The instants at which the solutions of these initial-value problems blow up are precisely the ‘escape times’ at which the performance criterion becomes unbounded below on some half-space. Analogues of the familiar Riccati equation are also presented.     

Linear‐Quadratic Optimal Control Problems for Mean‐Field Stochastic Differential Equations with Jumps

In this paper, we study a linear‐quadratic optimal control problem for mean‐field stochastic differential equations driven by a Poisson random martingale measure and a one‐dimensional Brownian motion. Firstly, the existence and uniqueness of the optimal control is obtained by the classic convex variation principle. Secondly, by the duality method, the optimality system, also called the stochastic Hamilton system which turns out to be a linear fully coupled mean‐field forward‐backward stochastic differential equation with jumps, is derived to characterize the optimal control. Thirdly, applying a decoupling technique, we establish the connection between two Riccati equations and the stochastic Hamilton system and then prove the optimal control has a state feedback representation.     

Probability‐weighted nonlinear stochastic optimal control strategy of quasi‐integrable Hamiltonian systems with uncertain parameters

The nonlinear stochastic optimal control problem of quasi‐integrable Hamiltonian systems with uncertain parameters is investigated. The uncertain parameters are described by using a random vector with λ probability density function. First, the partially averaged Itô stochastic differential equations are derived by using the stochastic averaging method for quasi‐integrable Hamiltonian systems. Then, the dynamical programming equation is established based on stochastic dynamical programming principle. By minimizing the dynamical programming equation with respect to control forces, the optimal control forces can be derived, which are functions of the uncertain parameters. The final optimal control forces are then determined by probability‐weighted average of the obtained control forces with the probability density of the uncertain parameters as weighting function. The mean control effectiveness and mean control efficiency are used to evaluate the proposed control strategy. The robustness of the proposed control is measured by using the ratios of the variation coefficients of mean control effectiveness and mean control efficiency to the variation coefficients of uncertain parameters. Finally, two examples are given to illustrate the proposed control strategy and its effectiveness and robustness. Copyright © 2014 John Wiley & Sons, Ltd.     

Solution of optimal control problem in class of piecewise-constant functions

A numerical method of solution of problems of optimal control of objects that may be described by systems of ordinary differential equations in the class of piecewise-constant controls is proposed in the article. At the same time, both the piecewise-constant values of the controls as well as the constancy intervals of these values are optimized. Analytic formulas of the gradient of the functional with respect to the optimized parameters are obtained. The formulas for the gradient of the functional obtained for an initial continuous optimal control problem and the corresponding discretized optimal control problem are compared and results of numerical experiments are presented.     

The Boltzmann–Hamel equations for the optimal control of mechanical systems with nonholonomic constraints

In this paper, we generalize the Boltzmann–Hamel equations for nonholonomic mechanics to a form suited for the kinematic or dynamic optimal control of mechanical systems subject to nonholonomic constraints. In solving these equations one is able to eliminate the controls and compute the optimal trajectory from a set of coupled first‐order differential equations with boundary values. By using an appropriate choice of quasi‐velocities, one is able to reduce the required number of differential equations by m and 3m for the kinematic and dynamic optimal control problems, respectively, where m is the number of nonholonomic constraints. In particular we derive a set of differential equations that yields the optimal reorientation path of a free rigid body. In the special case of a sphere, we show that the optimal trajectory coincides with the cubic splines on SO(3). Copyright © 2010 John Wiley & Sons, Ltd.     

线性时滞系统基于观测器的最优输出跟踪控制器近似设计

研究线性时滞系统的最优输出跟踪控制器近似设计过程.首先根据原系统最优输出跟踪控制问题构造了两个分别具有已知初始条件和终端条件的微分方程迭代序列,并证明它们一致收敛于原问题的最优解.然后通过对该解序列的有限次迭代,得到最优输出跟踪控制问题的一个近似解,进一步给出一个计算近似最优输出跟踪控制律的算法.最后通过构造降维参考输入观测器解决了最优输出跟踪控制器中前馈项的物理可实现问题.仿真结果表明该方法是有效的,且易于实现.     

不确定拟哈密顿系统的随机最优控制

本文提出了不确定拟哈密顿系统、基于随机平均法、随机极大值原理和随机微分对策理论的一种随机极大极小最优控制策略.首先,运用拟哈密顿系统的随机平均法,将系统状态从速度和位移的快变量形式转化为能量的慢变量形式,得到部分平均的It随机微分方程;其次,给定控制性能指标,对于不确定拟哈密顿系统的随机最优控制,根据随机微分对策理论,将其转化为一个极小极大控制问题;再根据随机极大值原理,建立关于系统与伴随过程的前向-后向随机微分方程,随机最优控制表达为哈密顿控制函数的极大极小条件,由此得到最坏情形下的扰动参数与极大极小最优控制;然后,将最坏扰动参数与最优控制代入部分平均的It随机微分方程并完成平均,求解与完全平均的It随机微分方程相应的Fokker-Planck-Kolmogorov(FPK)方程,可得受控系统的响应量并计算控制效果;最后,将上述不确定拟哈密顿系统的随机最优控制策略应用于一个两自由度非线性系统,通过数值结果说明该随机极大极小控制策略的控制效果.