Classification of linear planar systems with hybrid feedback control

Abstract

Continuing the authors' studies of hybrid dynamical systems, i.e. differential equations governed by finite automata, an efficient and complete classification of control linear systems in the plane is offered. The set of all such systems is divided into equivalence classes which are explicitly characterized by some quantitative invariants. The canonical representatives in each class are determined. It is shown how to use this classification to find out whether a given system is stabilizable or not.

Comparison between a switching controller and two LTI controllers for a class of LTI plants

“…During the preparation of this manuscript, it was pointed out that the result of Santarelli et al [12] is similar to the results of [18] and [19], with the exception that the framework of [12] employs continuous-time nonlinear feedback laws, while [18] and [19] employ hybrid feedback automata. The reader who is interested in exploring the differences between the switched feedback approach we take here and the hybrid feedback approach of [18] and [19] is encouraged to explore these two references.…”

“…If we, again, let b = √ a where ∈[−1, 1], then the 1% settling time of the switching architecture T s of Equation (19) can be reparameterized to be of the form…”

“…The reader who is interested in exploring the differences between the switched feedback approach we take here and the hybrid feedback approach of [18] and [19] is encouraged to explore these two references.…”

Efficient criteria for the stabilization of planar linear systems by hybrid feedback controls

“…In [2,3], the following result is obtained for B and C being nonzero matrices of rank 1: system (1.1) is stabilizable by a linear hybrid feedback control (LHFC) if and only if for at least one α ∈ R, the matrix A + αBC does not have nonnegative real eigenvalues. This result gives a necessary and sufficient stabilization condition, and it is straightforward that making use of hybrid feedback controls provides a better stabilization criterion compared to any one we can obtain exploiting ordinary feedback controls.…”

“…In contrast to [2,3], the present paper aims at (1) finding verifiable criteria for LHFC stabilization of system (1.1), (2) constructing efficient algorithms (which should also be "computer-friendly"), which can easily test a specific system (1.1) in terms of the input matrices (A,B,C) to find out whether the zero solution to (1.1) can be stabilized by an ordinary feedback linear control or by an LHFC.…”