When Is a Linear Control System Optimal?

1964 ◽  
Vol 86 (1) ◽  
pp. 51-60 ◽  
Author(s):  
R. E. Kalman
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Optimal Control Theory ◽  
Control Variable ◽  
Point Of View ◽  
Complete Analysis ◽  
Control Law ◽  
Quadratic Loss ◽  
Mathematical Form ◽  
State Variables

The purpose of this paper is to formulate, study, and (in certain cases) resolve the Inverse Problem of Optimal Control Theory, which is the following: Given a control law, find all performance indices for which this control law is optimal. Under the assumptions of (a) linear constant plant, (b) linear constant control law, (c) measurable state variables, (d) quadratic loss functions with constant coefficients, (e) single control variable, we give a complete analysis of this problem and obtain various explicit conditions for the optimality of a given control law. An interesting feature of the analysis is the central role of frequency-domain concepts, which have been ignored in optimal control theory until very recently. The discussion is presented in rigorous mathematical form. The central conclusion is the following (Theorem 6): A stable control law is optimal if and only if the absolute value of the corresponding return difference is at least equal to one at all frequencies. This provides a beautifully simple connecting link between modern control theory and the classical point of view which regards feedback as a means of reducing component variations.

Optimal Bounded Control of Linear Sampled-Data Systems With Quadratic Loss

1965 ◽  
Vol 87 (1) ◽  
pp. 135-141 ◽  
Author(s):  
G. W. Deley ◽  
G. F. Franklin
Keyword(s):  
Optimal Control ◽  
Piecewise Linear ◽  
Control Variable ◽  
Piecewise Linear Function ◽  
Control Law ◽  
Quadratic Loss ◽  
Data Systems ◽  
Sampled Data ◽  
Bounded Control ◽  
Unbounded Control

A method is presented for the computation of optimal control for linear sampled-data systems when the control variable is a bounded scalar. It is shown that for this problem the optimal control is a piecewise linear function of the state and may be computed by piecewise iteration of suitable recurrence relations. The optimal control is presented in terms of the control coefficients (matrices) and the regions to which they apply. No solution other than computer storage is suggested for the synthesis of these controls. In the second section of the paper, it is shown that the method applies with trivial modification to the random-input, random-observation noise case. The optimal control law has the same form as the deterministic case with the conditional expectation used in the control law in place of the stale itself. A simple deterministic example computed on an IBM 1620 is presented. As might be expected, the computer capacity required for the problem is intermediate between the unbounded control case, where the control is linear, and more general problems.

scholarly journals Application of optimal control theory on optimal advertising expenditure in monopoly

2020 ◽  
Vol 83 ◽  
pp. 01017
Author(s):  
Nora Grisáková ◽  
Peter Štetka
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Market Share ◽  
Optimal Control Theory ◽  
Control Variable ◽  
New Product ◽  
Advertising Expenditure ◽  
Advertising Rate ◽  
State Variable ◽  
Economic Application

Presented paper is being focused on Optimal control theory, Variation Calculus and its economic application. Aim of this research paper is to shortly describe Optimal control and Variation Calculus and to present how can we deal with these type of issues. The last part of this paper is presenting possible economic application of Optimal control, based on the maximization of profit in monopoly while introducing new product on the market. Our control variable is the advertising rate, which affects the profit of monopoly through advertising expenditures and as a state variable was the market share defined.

Optimal Time Control for a Magnetic Levitation Linear Servo System

2013 ◽  
Vol 416-417 ◽  
pp. 670-675
Author(s):  
Bao Quan Kou ◽  
Feng Xing ◽  
Chao Ning Zhang ◽  
Lu Zhang
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Optimal Control Theory ◽  
Magnetic Levitation ◽  
Control Variable ◽  
Target Function ◽  
Servo System ◽  
Optimal Time ◽  
Feed Speed ◽  
Time Control

The paper presents a new way to control the feed speed of the magnetic levitation linear servo system by adopting optimal time control theory. The optimal switch curve is obtained by building the system model and determining the control variable, control condition and target function. According to the optimal switch curve, the system can realize the optimal control of the speed. The validity of the optimal control theory in this servo system has been proven by modeling and simulating in Matlab/Simulink. The simulation result shows that the execution efficiency of the magnetic linear servo system can be improved by applying the optimal control theory.

Development of Methods of Active Influence on the Processes of Precipitation Formation in Clouds in Order to Solve Environmental and Economic Problems

2018 ◽  
Vol 931 ◽  
pp. 1025-1030
Author(s):  
Boris A. Ashabokov ◽  
Alexander V. Shapovalov ◽  
Alla A. Tashilova
Keyword(s):  
Optimal Control ◽  
Numerical Modeling ◽  
Control Theory ◽  
Optimal Control Theory ◽  
Point Of View ◽  
High Mountain ◽  
Geophysical Institute ◽  
Economic Problems ◽  
Natural Development ◽  
Precipitation Formation

The paper discusses some approaches to the development of methods of active influence on clouds, which develop in the High-Mountain geophysical institute. We considered the problems of determination in the cloud the region in which to make the particles of the reactants, concentration of these particles, the beginning and completion of seeding. Such questions should be solved on the basis of numerical modeling of clouds. The most common approach to the development of methods of active influence on clouds is a simulation of different variants of any particles of reagent in the cloud and choose the most effective one from the point of view of achieving the objectives of impact compared to a natural development of the cloud. Another approach to the development of the method of sedimentation control in the clouds is to consider this problem in the framework of optimal control theory. Despite the difficulties in implementing this approach, it is very effective and produces optimal results.

Flight Control Law Clearance Using Optimal Control Theory

2016 ◽  
Vol 53 (2) ◽  
pp. 515-529 ◽  
Author(s):  
Avriel A. Herrmann ◽  
Joseph Z. Ben-Asher
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Flight Control ◽  
Optimal Control Theory ◽  
Control Law

Dispersion of nanoparticles with optimal control theory based on destabilization

2014 ◽  
Vol 2 ◽  
pp. 86-86
Author(s):  
Miki U. Kobayashi ◽  
Nobuaki Aoki ◽  
Noriyoshi Manabe ◽  
Tadafumi Adschiri
Keyword(s):  
Optimal Control ◽  
Control Theory ◽  
Optimal Control Theory ◽  
Dispersion Of Nanoparticles
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