Criteria of stabilizability for switching-control systems with solvable linear approximations
arXiv:1201.1990
Abstract
We study the stability and stabilizability of a continuous-time switched control system that consists of the time-invariant -dimensional subsystems \dot{x}=A_ix+B_i(x)u\quad (x\in\mathbb{R}^n, t\in\mathbb{R}_+ \textrm{and} u\in\mathbb{R}^{m_i}),\qquad \textrm{where} i\in{1,...,N} and a switching signal $σ(\bcdot)\colon\mathbb{R}_+\rightarrow{1,...,N}$ which orchestrates switching between these subsystems above, where , and where $B_i(\bcdot)\colon\mathbb{R}^n\rightarrow\mathbb{R}^{n\times m_i}$ satisfies the condition $\|B_i(x)\|\le\bbbeta\|x\|\;\forall x\in\mathbb{R}^n$. We show that, if generates a solvable Lie algebra over the field $\mathbbm{C}$ of complex numbers and there exists an element $\bbA$ in the convex hull in such that the affine system $\dot{x}=\bbA x$ is exponentially stable, then there is a constant $\bbdelta>0$ for which one can design "sufficiently many" piecewise-constant switching signals so that the switching-control systems \dot{x}(t)=A_{σ(t)}x(t)+B_{σ(t)}(x(t))u(t),\quad x(0)\in\mathbb{R}^n\textrm{and} t\in\mathbb{R}_+ are globally exponentially stable, for any measurable external inputs with $|u(t)|\le\bbdelta$.
32 pages; accepted by SIAM J Control & Optim