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

Abstract

We suggest some criteria for the stabilization of planar linear systems via linear hybrid feedback controls. The results are formulated in terms of the input matrices. For instance, this enables us to work out an algorithm which is directly suitable for a computer realization. At the same time, this algorithm helps to check easily if a given linear 2 × 2 system can be stabilized (a) by a linear ordinary feedback control or (b) by a linear hybrid feedback control.

Classification of linear planar systems with hybrid feedback control

“…The present paper continues and in some sense completes the study started in [4], [5]. The following are the main results of the paper: 1) A set of invariants is identified, which characterize the GT -equivalence classes, where GT is a certain group of transformations of the triples (A, B, C).…”

“…c) Using the canonical triples, Theorem 3.1 and the results from [5], [6], [4], and [7], the functionΨ can explicitly be calculated in all cases. In other words, for any GTequivalence classΩ ∈ Σ/GT it is found which of the seven properties P k , k = 1, ..., 7 are valid for the representatives Ω of the classΩ.…”

“…If, on the contrary, the pair (A, B) is not stabilizable, or the pair (A, C) is not detectable (for example, if B = 0 or C = 0), then system (1) can be stabilized neither by linear static output feedback controls, nor by LH-controls. Thus, the assumption posed on system (1) that B, C are nonzero vectors is indeed essential and provides no loss of generality: only in this case LH-controls can do the stabilization job better than linear static output feedback controls (see [5], [4] for the details).…”

“…However, this condition is not always easy to check. In [4] the "GTclassification" was exploited to construct verifiable criteria for LH-stabilization of system (1). An efficient algorithm was designed which tests specific systems (1) using the input data (A, B, C).…”

“…It is possible to check explicitly [4] whether a triple Ω ∈ Σ, related to system (1), has one or several following properties: P 1 . The pair (A, B) is controllable, P 2 .…”