具有可解线性近似的切换控制系统的可镇定准则
Criteria of stabilizability for switching-control systems with solvable linear approximations
AI总结：
本文研究连续时间切换控制系统在子系统矩阵生成可解李代数且存在凸包中指数稳定仿射系统时，通过设计充分多的分段常数切换信号实现全局指数稳定。
摘要：
我们研究连续时间切换控制系统的稳定性和可镇定问题，该系统由时不变n维子系统 \dot{x}=A_ix+B_i(x)u\quad (x\in\mathbb{R}^n, t\in\mathbb{R}_+ \text{和} u\in\mathbb{R}^{m_i}),\qquad \text{其中} i\in{1,...,N} 以及一个切换信号$\sigma(\bcdot)\colon\mathbb{R}_+\rightarrow{1,...,N}$组成，该信号协调上述子系统之间的切换，其中$A_i\in\mathbb{R}^{n\times n}, n\ge1, N\ge2, m_i\ge1$，并且$B_i(\bcdot)\colon\mathbb{R}^n\rightarrow\mathbb{R}^{n\times m_i}$满足条件$\|B_i(x)\|\le\bbbeta\|x\|\;\forall x\in\mathbb{R}^n$。我们证明，如果$\{A_1,...,A_N\}$在复数域$\mathbbm{C}$上生成一个可解李代数，并且存在一个元素$\bbA$属于$\mathbb{R}^{n\times n}$中凸包$\mathrm{co}\{A_1,...,A_N\}$，使得仿射系统$\dot{x}=\bbA x$是指数稳定的，那么存在一个常数$\bbdelta>0$，使得可以设计“足够多”的分段常数切换信号$\sigma(t)$，从而对于任何满足$|u(t)|\le\bbdelta$的可测外部输入$u(t)\in\mathbb{R}^{m_{\sigma(t)}}$，切换控制系统 \dot{x}(t)=A_{\sigma(t)}x(t)+B_{\sigma(t)}(x(t))u(t),\quad x(0)\in\mathbb{R}^n\text{和} t\in\mathbb{R}_+ 是全局指数稳定的。
英文摘要：
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 $σ(\bcdot)\colon\mathbb{R}_+\rightarrow{1,...,N}$ 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 $σ(t)$ 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 $u(t)\in\mathbb{R}^{m_{σ(t)}}$ with $|u(t)|\le\bbdelta$.
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