Sensitivity of Automatic Control Systems

Efim Rozenwasser, Rafael Yusupov

• Publisher: CRC Press
• Year: 2019
• eBook ISBN: 9781351835572
• Topic: Technology & Engineering
• Subtopic: Electrical Engineering & Telecommunications
• Index: Technology & Engineering

Although it arose much earlier in a variety of contexts, sensitivity theory became an independent branch of science in the sixties. Since then, researchers from around the world have continued to make great strides in both the theory and its applications. However, much of the work of Russian scientific schools and specialists remain unknown in the West. Sensitivity of Control Systems summarizes the results of the authors and their disciples in sensitivity theory, addressing the basic notions of the theory and the problem of selecting technical parameters of systems. The authors formulate problems for actual technical systems and their models, and establish relations between sensitivity theory and classical stability problems. They offer a significant, general theory for investigating the sensitivity of boundary problems and use elements of this theory for sensitivity analysis of solutions to nonlinear programming and variational calculus problems, as well as oscillatory processes. The book also presents general investigation methods for discontinuous systems, including those described by operator models. Full of powerful new methods and results, this book offers a unique opportunity for those in theoretical investigation and in the design, testing, and exploitation of various control systems to explore the work of Russia's leading researchers in sensitivity theory. Furthermore, its techniques for parametric perturbation investigation, Sensitivity of Control Systems will prove useful in fields outside of control theory, including oscillation theory, motion dynamics, and mathematical economy.

Chapter 1

Parametric Models of Control Systems and Statement of the Problem in Sensitivity Theory

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1.1 State Variables and Control Systems Parameters

1.1.1 General Principles of Control

The notion of control always implies a plant to which the control is applied, and a target that is to be attained using this control. In practice, control process is realized by means of specially formed exogenous actions applied to the plant. These actions are intended to change some properties of the latter. This idea is illustrated in Figure 1.1, where O is a multivariable plant, Y is a vector of generalized coordinates that define properties of the plant, and U is a vector of control actions.

Figure 1.1
Block-diagram of control process

Denote by M_Y the set of values of Y for which properties (state) of the plant satisfy the requirements, and by R_Y the set of values of Y that can really take place during plant operation. Then, the target of the control can be defined by the relation

$R_Y\subset M_Y.(1.1)$

If equation (1.1) is satisfied for the plant without any interference from outside, control process becomes superfluous. Control is necessary if the plant is affected by factors (we will call them disturbances) that lead to violation of (1.1)¹.

Figure 1.2
General block-diagram of control

In practice control action U is realized by a system called controller that is, in the general case, functionally connected with the state of the plant and disturbances applied to it. As a result, we obtain a more detailed control scheme (Figure 1.2), where P is a controller and V_o and V_p are disturbances applied to the plant and control, respectively. According to Figure 1.2, the controller P is to provide equation (1.1) in the presence of disturbances V₀ applied to the plant and V_p infringing controller actions. The block-diagram shown in Figure 1.2 is the most general and, therefore, the least substantial. Schemes of actual control systems, and still more their realizations, are very diversified and are determined by a number of factors. The mam factors are: physical nature of the plant; the choice of the vector Y, which gives the state of the plant; the set M_Y of admissible states of the plant; availability of information about variations of plant properties during the control process; availability of information about vectors V_o and V_p, etc.

If the information about the plant and corresponding disturbances is sufficiently complete, we can, as a rule, employ the classic block-diagram of combined control shown in Figure 1.3, where $N_{V_o}$, $N_{V_p}$, a...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Preface
• Table of Contents
• 1 Parametric Models
• 2 Finite-Dimensional Continuous Systems
• 3 Finite-Dimensional Discontinuous Systems
• 4 Discontinuous Systems Given by Operator Models
• 5 Non-Time Characteristics
• 6 Sensitivity Invariants
• 7 Sensitivity of Mathematical Programming and Variational Calculus Problems
• 8 Applied Sensitivity Problems
• Bibliography
• Index