On the stability of bimodal systems in ℝ³

Stability of Bimodal Planar Switched Linear Systems with Both Stable and Unstable Subsystems

“…In each of these sections, we find expressions for increasing functions S or T and compute τ 1,2 (F) and τ 2,1 (F), as discussed already. Using ( 6), (7), (8), (9) and Remark 1.2, the switched system (1) is stable for all signals σ ∈ S[τ 1,2 (F), F] S[τ 2,1 (F), F] and asymptotically stable for all signals σ ∈ S [τ 1,2 (F), F] S [τ 2,1 (F), F]. Hence if τ (F) = min{τ 1,2 (F), τ 2,1 (F)}, then the switched system (1) is stable for each σ ∈ S[τ (F), F] and asymptotically stable for each σ ∈ S [τ (F), F].…”

“…The issues of stability, stabilizability and controllability (with or without the presence of control) have also been studied specifically for bimodal setting. A necessary and sufficient condition for stability under arbitrary switching is given in [4] for bimodal planar linear systems, and in [8] for bimodal linear systems in R 3 . In [27], we study bimodal linear switched systems, with both its subsystems stable, to find a dwell time expression as a function of eigenvalues and eigenvectors of the subsystem matrices.…”

Stability of bimodal planar linear switched systems

“…In the literature, the issues of stability, stabilizability and controllability (in the presence of control) have been studied specifically for bimodal setting. A necessary and sufficient condition for stability under arbitrary switching is given in [4] for bimodal planar systems, and in [9] for bimodal linear systems in R 3 . For an overview of results on stabilizability and controllability of bimodal planar linear switched systems, see [27].…”

“…It is possible to choose matrices P 1 , P 2 such that the determinant of the transition matrix M is either 1 or −1. Thus, without loss of generality, we will assume that the determinant of M is either 1 or −1 since it does not affect the left hand side of the above inequalities (9). Moreover, we choose the scaling matrices such that det(D 1 ), det(D 2 ) = ±1 for the same reason.…”

The effect of coupling conditions on the stability of bimodal systems inR3

“…This issue is investigated extensively in [8][9][10][11]. BPLS is also investigated in the context of stability, stabilizability and control in [12][13][14][15][16][17]7]. Moreover, BPLS also provide convenient tools to investigate nonlinear phenomenon encountered in biological, physical or engineering processes as demonstrated in [18,19].…”