Necessary optimality conditions for quasisingular controls in a step control problem

Necessary second-order optimality conditions are obtained for one class of optimal control problems for step systems. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 101–115, January–February 2008.     

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.     

Optimality of affine control system of several species in competition on a sequential batch reactor

In this paper, we analyse the optimality of affine control system of several species in competition for a single substrate on a sequential batch reactor, with the objective being to reach a given (low) level of the substrate. We allow controls to be bounded measurable functions of time plus possible impulses. A suitable modification of the dynamics leads to a slightly different optimal control problem, without impulsive controls, for which we apply different optimality conditions derived from Pontryagin principle and the Hamilton–Jacobi–Bellman equation. We thus characterise the singular trajectories of our problem as the extremal trajectories keeping the substrate at a constant level. We also establish conditions for which an immediate one impulse (IOI) strategy is optimal. Some numerical experiences are then included in order to illustrate our study and show that those conditions are also necessary to ensure the optimality of the IOI strategy.     

Regional quadratic control problem for distributed bilinear systems with bounded controls

The aim of this paper is to study regional quadratic control problem for a distributed bilinear system evolving in a spatial domain Ω. The question is to obtain a bounded feedback control that drives such a system from an initial state to a desired one in finite time, only on a subregion ω of Ω, and minimises a quadratic functional cost. Our purpose is to prove that an optimal control exists, and characterised as solution of an optimality system. The approach is successfully illustrated by simulations.     

Cheap control problem of a linear uniform rank system: Design by composite control

Cheap control problems where a small parameter μ² multiplies the control cost are considered. Due to the cheapness of control, a strong control action in the form of high-gain feedback forces the given system to have slow and fast, low and high amplitude variations. For a class of linear systems (uniform rank systems), a systematic procedure of amplitude scaling and time-scale decomposition which normalizes high and low amplitude variations and which separates slow and fast time scales is presented. The method permits the explicit characterization of all the limiting properties of the considered cheap control problem as μ → 0. Methods of calculating singular controls and how non-uniqueness can arise in them are discussed. Above all, several suboptimal composite control schemes are developed based on the decomposition of the given optimal design into two lower order subsystem designs.     

The time-invariant linear-quadratic optimal control problem

This paper provides a review of one of the basic problems of systems theory—the general time-invariant optimal control problem involving linear systems and quadratic costs. The problem includes on one hand the regulator problem of optimal control and on the other, the theory of linear dissipative systems, itself central to network theory and to the stability theory of feedback systems. The theory is developed using simple properties of dynamical systems and involves a minimum of ‘hard’ analysis or algebra. It includes a full existence theory of the matrix quadratic equation, of interest in its own right.     

一类状态时滞广义串级控制系统的H∞镇定

针对一类具有状态时滞的广义串级控制系统,研究其H∞控制问题.考虑到当非零扰动作用于系统时采用静态反馈控制对于被控对象的输出具有不足性,从而将串级控制应用于具有状态时滞的广义系统中,并且建立了闭环广义串级控制系统新的模型.提出的方法着重在于串级控制在广义系统中的应用.利用线性矩阵不等式方法得到了相应系统正则、无脉冲和稳定的条件,并进一步利用得到的条件给出了相应的H∞稳定控制器设计方法.最后通过仿真实例说明了所提方法的有效性.     

Realization of a bounded feedback in a nonlinear control problem

We substantiate a method for design of a limited feedback to control a system whose mathematical model includes a nonlinear state function. Designing such a feedback is based on a positional solution of auxiliary optimal control problems which provides the transition from the vicinity of one equilibrium state to the vicinity of a new equilibrium state and stabilization of the system in the new state. Preassigned additional constrains on the trajectory of the control system are satisfied. Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 108–116, January–February 2009.     

奇异时滞系统对时滞参数的自适应控制

针对一类状态时滞奇异系统,研究状态反馈控制器的设计问题。基于Lyapunov稳定性理论和线性矩阵不等式（LMI）工具,当时滞常数精确已知时,设计带有记忆的状态反馈控制器,使得相应的闭环系统渐近稳定;当时滞常数不能精确已知时,通过求解相应的线性矩阵不等式,得到满足设计要求的对时滞参数的自适应控制器,使得时滞系统镇定。最后,仿真实例表明此方法的有效性。     

Optimal Control for a Class of Complex Singular System Based on Adaptive Dynamic Programming

This paper presents a new design approach to achieve decentralized optimal control of high-dimension complex singular systems with dynamic uncertainties. Based on robust adaptive dynamic programming (robust ADP) method, controllers for solving the singular systems optimal control problem are designed. The proposed algorithm can work well when the system model is not exactly known but the input and output data can be measured. The policy iteration of each controller only uses their own states and input information for learning, and do not need to know the whole system dynamics. Simulation results on the New England 10-machine 39-bus test system show the effectiveness of the designed controller.      

Regional optimal control problem of a class of infinite-dimensional bi-linear systems

The purpose of this paper is to study regional optimal control of an infinite-dimensional bi-linear system, excited by unbounded or bounded controls. Then we prove the existence of an optimal control that minimises a functional cost and we give the characterisation of such a control. Also we give the condition that ensures the uniqueness of the optimal control. The obtained results lead to an algorithm that we illustrate by simulations for a bi-dimensional diffusion system excited by a zone actuator.     

Optimal control solutions to the maximum volume isoperimetric pillars problem

This paper extends the isoperimetric problem. The problem is to find an enclosed cross-sectional/base region of a pillar defined by a simple closed curve of fixed perimeter such that the volume of the constructed pillar, bounded above by a relatively smooth ceiling, is maximized. Green’s Theorem is applied in the formulation of the problem such that the problem can be transformed into canonical form handled by MISER3. For the case of multiple pillars, a novel elliptic separation technique is developed for multiple pillars constructions. This technique is used to ensure that the cross-sectional regions of any pillars are separated. Illustrative examples are provided to demonstrate the effectiveness of the technique developed.     

Stabilisation and state trajectory tracking problem for nonlinear control systems in the presence of disturbances

In this paper, we consider the problem of stabilisation and tracking of desired state trajectory for a wide range of nonlinear control problems with disturbances. We present the sufficient conditions for the existence of C ^k state-feedback controllers and the process of their mathematical designing is described.     

一类不确定时滞奇异系统的鲁棒非脆弱控制

针对不确定奇异时滞系统，研究了鲁棒非脆弱控制问题。假定不确定性范数有界，设计了基于观测器的控制器，使得闭环系统在不确定性影响下保持鲁棒稳定。仅通过求解相应线性矩阵不等式就可得到鲁棒状态反馈控制器，该控制器不仅对所有容许的系统不确定性使相应的闭环系统渐近稳定，而且满足控制器本身是非脆弱的，达到抑制干扰的效果。     

Symplectic algorithms with mesh refinement for a hypersensitive optimal control problem

A symplectic algorithm with nonuniform grids is proposed for solving the hypersensitive optimal control problem using the density function. The proposed method satisfies the first-order necessary conditions for the optimal control problem that can preserve the structure of the original Hamiltonian systems. Furthermore, the explicit Jacobi matrix with sparse symmetric character is derived to speed up the convergence rate of the resulting nonlinear equations. Numerical simulations highlight the features of the proposed method and show that the symplectic algorithm with nonuniform grids is more computationally efficient and accuracy compared with uniform grid implementations. Besides, the symplectic algorithm has obvious advantages on optimality and convergence accuracy compared with the direct collocation methods using the same density function for mesh refinement.     

Optimality condition of a nonsmooth switching control system

In this paper, a survey and refinement of recent results in discrete optimal control theory are presented. The step control problem depending on a parameter is investigated. Nonsmoothness of the cost function φ is assumed and new versions of the discrete maximum principle for the step control problem are derived. The text was submitted by the author in English.     

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.     

Discrete-time interval optimal control problem

This paper proposes a new approach for the discrete-time interval optimal control problem. The optimal interval solution of discrete-time interval optimal control problem is obtained using the single-level constrained interval arithmetic coupled with a dynamic programming technique. Moreover, the optimal interval solution contains the real-valued optimal solutions. To illustrate an optimal solution for the interval case, we use the minimax regret criterion in the interval inventory control problem we present.     

Study of the discrete singularly perturbed linear-quadratic control problem by a bilinear transformation

This paper presents a new approach in the study of the linear quadratic control problem of singularly perturbed discrete systems. By applying a bilinear transformation, the algebraic discrete Riccati equation is converted into a continuous one, which can be solved by using the reduced-order recursive method already documented in the control literature. This method produces the reduced-order near-optimal solution up to an arbitrary order of accuracy and reduces the size of required computations. The method is very suitable for parallel programming. A real world example, an F-8 aircraft, demonstrates the efficiency of the proposed method.     

Reach control problem for affine multi-agent dynamic systems on simplices by affine feedback

Aiming at the reach control problem for a multi-agent system composed of affine coupling systems, an affine controller is proposed. Different from affine single-agent systems, additional difficulties are faced for multi-agent systems on the reach control because of the coupling between agents. In this paper, we use the compactness and convexity of the simplex to overcome the above difficulties, and propose a sufficient condition to solve the reach control problem for multi-agent systems, which aims to detect an affine feedback control such that each agent in the system first leaves a given simplex through a prespecified facet rather than a certain restricted facet in a finite time. In addition, necessary conditions for the solvability of the problem are obtained. Finally, the validity of the results is verified by a three-dimensional numerical example.