On dimensionality reduction and the stability of a class of switched descriptor systems

Stability of a class of switched descriptor systems

“…With A invertible, the index of system (1) or the pair (E, A) is the smallest integer k * ≤ n for which…”

“…Now we consider switching between index-one and index-two descriptor systems. Our results in this subsection are based on an order reduction result from [1]. They result from an application of full rank decomposition to a switched descriptor system in the form of (10).…”

“…Theorem 4 (Order reduction of linear switching descriptor systems [1]): Consider a switching descriptor system described by (10) and switching conditions (38),(39) when σ switches from i to j and suppose that (X i ,Y i ) is a decomposition of E i with Y i ∈ R n×r for i = 1, · · · N. Then, there exist matrices T 1 , · · · , T N such that the following holds. A function x(·) is a solution to system (10) with (38),(39) if and only if…”

“…S Our approach in this paper differs from that given in the above papers. Our results in this note are based on a fundamental result derived in [1] to recursively reduce the dimension of a switched descriptor system using full rank decomposition. This approach allows us to obtain conditions which can also be checked without resorting to complicated linear algebraic manipulations.…”

Stabilization of Switched Linear Differential Algebraic Equations and Periodic Switching

“…[2], [3], [4]. In [5] a result is presented that reduces the dimension of the stability problem and [6] considers stabilization via statefeedback. To the best of the authors' knowledge this is the first paper on the stabilization of switched DAEs via timedependent switching.…”