Optimal control for nonlinear systems calculated with small computers

Using Ritz's procedure of representing the control functions of an optimal control problem by a function series with parameters to be optimized, it is shown that, from the well-known gradient procedure for dynamic problems, a simple iteration formula for the optimization of these parameters can be derived. Using an example with a technical background, the effectiveness of the program realization of this approach is demonstrated and is compared with the results of unrestricted dynamic optimization.This work was performed at the Technische Hochschule in Darmstadt, West Germany, with financial support from the DFG (Deutsche Forschungs-Gemeinschaft).     

随机非线性物价模型的最优控制

将政府对价格系统的宏观调控作为外部控制力,建立受控的随机非线性物价模型;利用拟Hamilton系统随机平均法和随机动态规划原理的非线性随机控制策略对系统实施最优控制,控制目标是实现系统的稳定性变大;并通过对比控制前后的Lyapunov指教值说明了控制的有效性.     

Optimal feedback control policies for stochastic,distributed-parameter systems

This paper considers optimal feedback control policies for a class of discrete stochastic distributed-parameter systems. The class under consideration has the property that the random variable in the dynamic systems depends only on the time and possesses the Markovian property with stationary transition probabilities. A necessary condition for optimality of a feedback control policy, which has form similar to the Hamiltonian form in the deterministic case, is derived via a dynamic programming approach.     

Robust static output feedback variable structure control for nonlinear systems

In this paper, a robust stabilization problem for nonlinearsystems is investigated using static output feedback. Matchedand mismatched uncertainties are considered and their boundingfunctions take general forms. Based on a Lyapunov analysis approachand sliding mode techniques, some conclusions are presented.Then, a robust nonlinear variable structure control scheme issynthesized to stabilize the system globally, and it is notnecessary for the nominal system to be linearizable globally.The conservatism is reduced by using uncertainty bounds in thecontrol design. Finally, a mass–spring system is employedto illustrate the effectiveness of the results.     

Optimal feedback control of bilinear systems

Feedback synthesis of optimal constrained controls for single-input bilinear systems is considered. Quadratic cost functionals (with and without quadratic control penalization) are modified by the inclusion of additional nonnegative state penalizing functions in the respective cost integrands. The latter functions are chosen so as to regularize the problems, in the sense that feedback solutions of particularly simple form are obtained. Finite and infinite time horizon problem formulations are treated, and associated aspects of feedback stabilization of bilinear systems are discussed.     

Optimal control of a nonlinear singular system with state constraints

A control system described by a nonlinear equation of parabolic type is considered in the situation where there may be no global solution. A particular optimal control problem subject to state constraints is studied. A proof of the existence of an optimal control is presented. The penalty method is used to obtain necessary conditions for optimal control. A proof of the convergence of this method is given. The successive approximation method is used to obtain an approximate solution for the conditions derived. Translated fromMatematicheskie Zametki, Vol. 60, No. 4, pp. 511–518, October, 1996.     

Stabilization of a class of nonlinear stochastic systems

The aim of this paper is to study the stabilizability for a class of multi-inputs nonlinear stochastic systems. We prove under general assumptions the existence of a linear feedback law which stabilizes in probability the system at its equilibrium.     

Optimal control in nonlinear infinite-dimensional systems with nondifferentiability of two types

We consider the optimal control problem for systems described by nonlinear equations of elliptic type. If the nonlinear term in the equation is smooth and the nonlinearity increases at a comparatively low rate of growth, then necessary conditions for optimality can be obtained by well-known methods. For small values of the nonlinearity exponent in the smooth case, we propose to approximate the state operator by a certain differentiable operator. We show that the solution of the approximate problem obtained by standard methods ensures that the optimality criterion for the initial problem is close to its minimal value. For sufficiently large values of the nonlinearity exponent, the dependence of the state function on the control is nondifferentiable even under smoothness conditions for the operator. But this dependence becomes differentiable in a certain extended sense, which is sufficient for obtaining necessary conditions for optimality. Finally, if there is no smoothness and no restrictions are imposed on the nonlinearity exponent of the equation, then a smooth approximation of the state operator is possible. Next, we obtain necessary conditions for optimality of the approximate problem using the notion of extended differentiability of the solution of the equation approximated with respect to the control, and then we show that the optimal control of the approximated extremum problem minimizes the original functional with arbitrary accuracy.     

Power-series approximations for the optimal feedback control of noisy nonlinear systems

Power-series methods are developed for designing approximately optimal state regulators for a nonlinear system subject to white Gaussian random disturbances. The performance index of the control is an ensemble average of a quadratic form. A perfect observation of the system state is assumed. When the system nonlinearity is small and it is characterized by a polynomial function of the state, a definite method is presented to construct a suboptimal feedback control of a power-series form in a small nonlinearity parameter. If the variance of noise is small, an alternative method is also applicable which yields a suboptimal control in a power series with respect to a variance parameter. A simple one-dimensional problem is examined to make comparison between controls of the two different forms.     

Optimal control computation for nonlinear time-lag systems

In this paper, a computational scheme using the technique of control parameterization is developed for solving a class of optimal control problems involving nonlinear hereditary systems with linear control constraints. Several examples have been solved to test the efficiency of the technique.     

Optimal feedback control laws using nonlinear programming

A procedure of parametrizing feedback controls when solving the optimal control problem using nonlinear programming is considered. The maximum principle is utilized to determine the forms of the parametrized feedback control. Applications are demonstrated by numerical examples.     

Finite and infinite horizon H{infty} control for stochastic nonlinear systems

In this paper, the problem of disturbance attenuation with internalstability for nonlinear systems with Markovian jumping parametersis considered. It is shown that this problem is solvable ifthere exists a sequence of smooth-positive semidefinite functionssatisfying certain Hamilton-Jacobi-Isaac equations (inequalities).Furthermore, it is shown that, if this solution exists, it representsa stochastic Lyapunov function for the closed-loop nonlinearsystem. A parametrization of the family of full-informationstate-feedback controllers satisfying the disturbance-attenuationrequirement with stochastic stability for the closed-loop systemis also given.     

Synchronization of nonlinear systems with delays via periodically nonlinear intermittent control

In this paper, the synchronization for a class of nonlinear chaotic systems with delays is proposed by using periodically intermittent nonlinear feedback control. Some synchronization criteria are derived based on Lyapunov functional theory and several differential inequalities such as Halanay inequality. As a special case, some sufficient conditions are obtained to ensure the synchronization of nonlinear systems without delays. Finally, some numerical simulations are presented to verify the theoretical results.     

Suboptimal control of nonlinear time-delay systems

An interative method for the suboptimal control of nonlinear time-delay systems is presented. Polynomial forms are assumed for the state in cases with delay in the state only and for the control in other cases. The coefficients in the approximating polynomial are chosen by a direct search method.Many discussions with Dr. S. Narayana Iyer, Professor of Electrical Engineering, College of Engineering, Trivandrum, India, are gratefully acknowledged.     

Optimal control of diffusion processes with reflection

In this paper, we consider the problem of optimally controlling a diffusion process on a closed bounded region ofR ⁿ with reflection at the boundary. Employing methods similar to Fleming (Ref. 1), we present a constructive proof that there exists an optimal Markov control that is measurable or lower semicontinuous. We prove further that the expected cost function corresponding to the optimal control is the unique solution of the quasilinear parabolic differential equation of dynamic programming with Neumann boundary conditions and that there exists a diffusion process (in the sense of Stroock and Varadhan) corresponding to the optimal control.This work was partially supported by the National Science Foundation, Grant No. GK-18339, by the Office of Naval Research, Grant No. NR-042-264, and by the National Research Council of Canada, Grant No. A3609.The author would like to thank S. R. Pliska, J. Pisa, and N. Trudinger for helpful suggestions. He is especially grateful to Professor A. F. Veinott, Jr., for help and advice in the preparation of the doctoral dissertation, on which part of this paper is based. Finally, he wishes to thank one of the referees for the careful reading and constructive comments on an earlier version of this paper.     

非线性控制系统的梯度扩张的能达性

研究了仿射非线性控制系统的梯度扩张系统.利用非线性控制系统的微分几何理论,通过计算梯度扩张系统的输出函数沿着输入向量场和系统向量场的李导数,讨论仿射非线性控制系统的梯度扩张系统的能达性分布,研究了非线性控制系统和它的梯度扩张系统的能达性之间的关系,证明了如果梯度扩张系统是能达的,则原非线性控制系统也是能达的.     

Optimal control of a facility with periodic interrupted demand

The system investigated consists of a stochastic periodic stream of raw material, a continuous processing operation with controllable deterministic service rates, and a storage facility. The arrival stream is periodically interrupted and divided into alternating on-off intervals of fixed length. The processing facility is allowed to operate during the off-interval. Superimposed on this system is a cost structure composed of processing and holding costs. Such operations may be found in manufacturing as well as service systems (for example, dry cleaners, machine shops, repair and maintenance shops, printers, information processing centers, etc). A service rate control rule that minimizes the infinite-horizon discounted expected total cost is found. Existence and uniqueness of long-term optimal cost and policy functions is shown. Since the optimal policy cannot be expressed explicitly, an approximate solution was obtained. An error bound on the optimal cost associated with this solution is exhibited. The approximate solution is characterized by a service rate control rule that is a linear function of the level of inventory at the start of each on-interval and a piecewise linear function of inventory at the start of each off-interval. The optimal discounted expected total cost is quadratic in the inventory level at the start of each interval. Computational results indicate relative cost errors in the order of 2–3 percent.This research was performed at the Sanitary Engineering Research Laboratory and Operations Research Center of the University of California, Berkeley. It was made possible by US Public Health Research Grant UI-00547 from the Environmental Control Administration-Bureau of Solid Waste Management and by National Science Foundation Grant GK-1684.The author thanks Professor C. R. Glassey for not only suggesting this research, but for his constant encouragement and suggestions throughout its duration. He also thanks Professors W. S. Jewell and P. H. McGauhey whose comments on the draft were very helpful.     

Opticon: An algorithm for the optimal control of nonlinear stochastic models

In this paper we describe the algorithm OPTCON which has been developed for the optimal control of nonlinear stochastic models. It can be applied to obtain approximate numerical solutions of control problems where the objective function is quadratic and the dynamic system is nonlinear. In addition to the usual additive uncertainty, some or all of the parameters of the model may be stochastic variables. The optimal values of the control variables are computed in an iterative fashion: First, the time-invariant nonlinear system is linearized around a reference path and approximated by a time-varying linear system. Second, this new problem is solved by applying Bellman's principle of optimality. The resulting feedback equations are used to project expected optimal state and control variables. These projections then serve as a new reference path, and the two steps are repeated until convergence is reached. The algorithm has been implemented in the statistical programming system GAUSS. We derive some mathematical results needed for the algorithm and give an overview of the structure of OPTCON. Moreover, we report on some tentative applications of OPTCON to two small macroeconometric models for Austria.     

Optimal feedback control for fractional neutral dynamical systems

In this paper, we consider feedback control systems governed by fractional neutral equations involving Riemann–Liouville derivatives. Firstly, we show the existence and uniqueness of solutions for the feedback control systems by applying Krasnoselskiis fixed point theorem. Then, we use Filippove theorem to obtain the existence of feasible pairs. Finally, an existence result of optimal control pairs for the Lagrange problem is proved.     

The Lagrange-Newton method for nonlinear optimal control problems

We investigate local convergence of the Lagrange-Newton method for nonlinear optimal control problems subject to control constraints including the situation where the terminal state is fixed. Sufficient conditions for local quadratic convergence of the method based on stability results for the solutions of nonlinear control problems are discussed.