Heuristic Determination of Resolving Controls for Exact and Approximate Controllability of Nonlinear Dynamic Systems

Asatur Zh Khurshudyan

doi:10.1155/2018/9496371

Abstract

Dealing with practical control systems, it is equally important to establish the controllability of the system under study and to find corresponding control functions explicitly. The most challenging problem in this path is the rigorous analysis of the state constraints, which can be especially sophisticated in the case of nonlinear systems. However, some heuristic considerations related to physical, mechanical, or other aspects of the problem may allow coming up with specific hierarchic controls containing a set of free parameters. Such an approach allows reducing the computational complexity of the problem by reducing the nonlinear state constraints to nonlinear algebraic equations with respect to the free parameters. This paper is devoted to heuristic determination of control functions providing exact and approximate controllability of dynamic systems with nonlinear state constraints. Using the recently developed approach based on Green’s function method, the controllability analysis of nonlinear dynamic systems, in general, is reduced to nonlinear integral constraints with respect to the control function. We construct parametric families of control functions having certain physical meanings, which reduce the nonlinear integral constraints to a system of nonlinear algebraic equations. Regimes such as time-harmonic, switching, impulsive, and optimal stopping ones are considered. Two concrete examples arising from engineering help to reveal advantages and drawbacks of the technique.

Highlights

The ability of a controlled system to accommodate a required state at a given instant by means of attached controllers is called controllability
If by a specific choice of admissible controls the system can be transmitted from a given state to a required state within a finite amount of time exactly, it is called exactly controllable
If the system is not exactly controllable by any choice of admissible control, but its state implemented at the required instant for at least one admissible control is “sufficiently” close to the designated terminal state, it is called approximately controllable

Summary

Introduction

The ability of a controlled system to accommodate a required state at a given instant by means of attached controllers is called controllability. The analysis of controllability for a particular control system can be quite sophisticated and can require burdensome computational costs This may happen in the case of systems with singularities/discontinuities, uncertain systems, systems with nonlinear state constraints, and so on. In this paper we develop a systematic algorithm for heuristic determination of explicit expressions for resolving controls providing exact or approximate controllability at the required instant T (i.e., (2) or (3)) of systems with nonlinear state constraints. We provide a demonstration of how the derived explicit expressions of resolving controls can be used to ensure exact or approximate controllability of particular dynamic systems

Existing Approaches

Heuristic Determination of Resolving Controls

Comparison with the Moments Problem Approach

Examples

Conclusions