Exponentially contractive invariant sets for discrete-time switching linear systems

Further Results on Diagonally Invariant Exponential Stability of Switching Linear Systems

“…The DIES concept was introduced by paper [13] for linear time-invariant systems whose discrete-and continuous-time dynamics are described by a unique matrix. Papers [15,16] extended the DIES concept to switching linear systems in accordance with the following definition.…”

“…Paper [13] shows that DIES characterizes a particular type of exponential stability, reason for which the test instruments rely on algebraic properties stronger than the eigenvalue location, namely, inequalities involving matrix norms and matrix measures. The DIES concept was extended to interval systems by [14] and to switching systems of form (1-S)//(1-H) by [15] for discrete-time and [16] for continuoustime.…”

“…The current research is founded on the DIES concept and its characterization, by equivalence, presented in our previous works [15] for discrete-time and [16] for continuous-time. The developments proposed in this article refer to novel approaches to DIES characterization that exploit mathematical instruments completely different from [15,16].…”

“…Briefly speaking, the old results, corresponding to [15,16], rely on the properties of matrix norms and matrix measures applied to the matrices A ∈ R × , = 1, . .…”

“…The exposition plan for the remainder of the text takes into consideration a short visit of the already existing results (also called old results) that are summarized by Section 2, by using a presentation style that unifies the discrete-and continuous-time cases (separately addressed by [15,16], respectively). Section 3 develops the novel results in full accordance with the background created by the previous section and by using the same unified presentation style.…”