Transients in quasi-controllable systems. Overshooting, stability and instability

TL;DR

This paper introduces a new method to estimate transient overshooting in quasi-controllable control systems, linking stability properties with the quasi-controllability measure, and applies to various stability analysis problems.

Contribution

It proposes a novel approach for bounding overshoot in quasi-controllable systems using the quasi-controllability measure, extending stability analysis techniques.

Findings

Relations between stability and instability are similar to linear systems.

Constructive bounds for overshooting are derived.

Applicable to classical absolute stability and desynchronized systems.

Abstract

Families of regimes for control systems are studied possessing the so called quasi-controllability property that is similar to the Kalman controllability property. A new approach is proposed to estimate the degree of transients overshooting in quasi-controllable systems. This approach is conceptually related with the principle of bounded regimes absence in the absolute stability problem. Its essence is in obtaining of constructive a priori bounds for degree of overshooting in terms of the so called quasi-controllability measure. It is shown that relations between stability, asymptotic stability and instability for quasi-controllable systems are similar to those for systems described by linear differential or difference equations in the case when the leading eigenvalue of the corresponding matrix is simple. The results are applicable for analysis of transients, classical absolute stability problem, stability problem for desynchronized systems and so on.