2012/01/10 by Xiongping Dai, Dai, Xiongping
Engineering · Mathematics · #34H05 #37N35 #93C15 #93D20 #Advanced Differential Equations and Dynamical Systems #Classical Analysis and ODEs (math.CA) #FOS: Electrical engineering #FOS: Mathematics #Optimization and Control (math.OC) #Stability and Control of Uncertain Systems #Stability and Controllability of Differential Equations #Systems and Control (eess.SY) #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1201.1990
openalex publication_date 2012/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the stability and stabilizability of a continuous-time switched control system that consists of the time-invariant n-dimensional subsystems x=Aix+Bi(x)u (x∈ℝn, t∈ℝ+ \textrmand u∈ℝmi), \textrmwhere i∈1,...,N and a switching signal σ(\bcdot)\colonℝ+→1,...,N which orchestrates switching between these subsystems above, where Ai∈ℝn× n, n≥1, N≥2, mi≥1, and where Bi(\bcdot)\colonℝn→ℝn× mi satisfies the condition ‖Bi(x)‖≤\bbbeta‖x‖ ∀ x∈ℝn. We show that, if A1,...,AN generates a solvable Lie algebra over the field \mathbbmC of complex numbers and there exists an element \bbA in the convex hull coA1,...,AN in ℝn× n such that the affine system x=\bbA x is exponentially stable, then there is a constant \bbdelta>0 for which one can design "sufficiently many" piecewise-constant switching signals σ(t) so that the switching-control systems x(t)=Aσ(t)x(t)+Bσ(t)(x(t))u(t), x(0)∈ℝn\textrmand t∈ℝ+ are globally exponentially stable, for any measurable external inputs u(t)∈ℝ^mσ(t) with |u(t)|≤\bbdelta.