1 research outputs found
Criteria of stabilizability for switching-control systems with solvable linear approximations
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 \sigma(\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_{\sigma(t)}x(t)+B_{\sigma(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.Comment: 32 pages; accepted by SIAM J Control & Opti