Reduction to Versal Deformations of Matrix Pencils and Matrix Pairs with Application to Control Theory

Abstract

Abstract. Matrix pencils under the strict equivalence and matrix pairs under the state feedback equivalence are considered. It is known that a matrix pencil (or a matrix pair) smoothly dependent on parameters can be reduced locally to a special typically more simple form, called the versal deformation, by a smooth change of parameters and a strict equivalence (or feedback equivalence) transformation. We suggest an explicit recurrent procedure for finding the change of parameters and equivalence transformation in the reduction of a given family of matrix pencils (or matrix pairs) to the versal deformation. As an application, this procedure is applied to the analysis of the uncontrollability set in the space of parameters for a one-input linear dynamical system. Explicit formulae for a tangent plane to the uncontrollability set at its regular point and the perturbation of the uncontrollable mode are derived. A physical example is given and studied in detail.

Uncontrollability for Linear Autonomous Multi-input Dynamical Systems Depending on Parameters

“…A local structure of the orbit O(α 0 ) near the point α 0 is determined by the range of the mapping df α0 and null-space of df * α0 as follows [8]:…”

“…Nevertheless, a reduction to the Brunovsky form generally cannot be achieved by the feedback equivalence transformation γ(p) smoothly dependent on parameters. The following theorem proved in [6,8] provides another form called a versal deformation that can be used for multiparameter families of matrix pairs. Note that the concept of versal deformation was first introduced by Arnold [1] for families of square complex matrices; see also [14] for the generalization to the case of a Lie group acting on a complex manifold.…”

“…Note that the concept of versal deformation was first introduced by Arnold [1] for families of square complex matrices; see also [14] for the generalization to the case of a Lie group acting on a complex manifold. [8]. The family of matrix pairs…”

“…As an example, an elastic mechanical system controlled by a force and dependent on three design parameters is studied. The results of the paper are based on the versal deformation theory for matrix pairs under state feedback equivalence [6,8,14].…”

Local analysis of the no-controllable set for one-input generalized linear systems

“…The triple of matrices s h ∈ M has the form Proof: Analogous to the case of pairs of matrices given in [6].…”