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

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.

Exact and approximate controllability of nonlinear dynamic systems in infinite time: The Green's function approach

“…For explicit determination of resolving controls from moments problem or heuristic approaches can be involved. As it is shown in [], in the case when is linear in u , then under proper assumptions these two approaches are equivalent. Nevertheless, the heuristic approach seems to be more convenient in the general case of nonlinear .…”

“…One of computationally efficient analytical methods for explicit determination of resolving controls is the heuristic approach . The idea of this approach is in construction of particular admissible control functions containing a set of free parameters which are determined to satisfy the controllability constraints exactly (exact controllability) or with a required accuracy (approximate controllability).…”

“…For explicit determination of the resolving controls from , it is possible to involve the control regimes obtained in [] heuristically. Consider the following truncation of the orthogonal expansion (c.f. )…”

Archives of Control Sciences

“…which implies (9). Thus, the problem of the exact controllability is reduced to characterization of the set of exactly resolving controls…”

“…The existence of the explicit L 2 -optimal solution of ( 8) is among its advantages. The method of heuristic determination of control also can be applied efficiently (see [8,9] for details and for a proof of equivalency of these two methods).…”