Criteria of stabilizability for switching-control systems with solvable linear approximations

TL;DR

This paper establishes criteria for the stabilizability of continuous-time switched control systems with linear approximations, leveraging the solvability of the Lie algebra generated by subsystem matrices and the stability of a convex combination of these matrices.

Contribution

It introduces new stabilizability conditions for switched systems where subsystem matrices generate a solvable Lie algebra, ensuring exponential stability under certain convex combination conditions.

Findings

If subsystem matrices generate a solvable Lie algebra and a convex combination is exponentially stable, then the system can be stabilized with appropriate switching.

Existence of a convex combination matrix in the convex hull that guarantees exponential stability.

Design of switching signals that ensure global exponential stability for the controlled system.

Abstract

We study the stability and stabilizability of a continuous-time switched control system that consists of the time-invariant $n$-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 $A_i\in\mathbb{R}^{n\times n}, n\ge1, N\ge2, m_i\ge1$, 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 ${A_1,...,A_N}$ generates a solvable Lie algebra over the field $\mathbbm{C}$ of complex numbers and there exists an element $\bbA$ in the convex hull $\mathrm{co}{A_1,...,A_N}$ in $\mathbb{R}^{n\times n}$ 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 $\sigma(t)$ 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 $u(t)\in\mathbb{R}^{m_{\sigma(t)}}$ with $|u(t)|\le\bbdelta$.