持续激励线性系的镇定
On the stabilization of persistently excited linear systems
摘要：
我们考虑$\dot x = A x +α(t)bu$类型的控制系统，其中$u\in\R$，$(A,b)$是可控对，$α$是未知时变信号，其中$[0,1]$中的值满足持续激励条件，即，每个$t\geq 0$的$\int_t^{t+T}\al(s)ds\geq μ$，$0<μ\leq T$独立于$t$。证明了当$A$的特征值具有非正实部时，系统的线性反馈可镇定性仅依赖于对$(T,μ)$.我们还表明，稳定性不持有任意矩阵$A$。此外，系统是否能以任意大的收敛率稳定的问题引起了依赖于参数$μ/T$的分叉现象。
英文摘要：
We consider control systems of the type $\dot x = A x +α(t)bu$, where $u\in\R$, $(A,b)$ is a controllable pair and $α$ is an unknown time-varying signal with values in $[0,1]$ satisfying a persistent excitation condition i.e., $\int_t^{t+T}\al(s)ds\geq μ$ for every $t\geq 0$, with $0<μ\leq T$ independent on $t$. We prove that such a system is stabilizable with a linear feedback depending only on the pair $(T,μ)$ if the eigenvalues of $A$ have non-positive real part. We also show that stabilizability does not hold for arbitrary matrices $A$. Moreover, the question of whether the system can be stabilized or not with an arbitrarily large rate of convergence gives rise to a bifurcation phenomenon in dependence of the parameter $μ/T$.
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