Practical observer for impulsive systems (Q2870477)

Consider the perturbed impulsive system \(S\): NEWLINE\[NEWLINE\begin{aligned} \dot x(t)&=A(t)x(t)+g(t,x(t))+B(t)u(t),\, t\neq t_k,\\ \Delta x(t_k)&=D_kx(t^-_k),\\ y(t)&=C(t)x(t),\end{aligned}NEWLINE\]NEWLINE where \(t_0<t_1<...<t_k<...,\,\,\text{lim}_{k\rightarrow +\infty}\,t_k=+\infty \); \(A(t)=(a_{ij}(t))\in{\mathbb R}^{n\times n}, B(t)=(b_{ij}(t))\in{\mathbb R}^{n\times p}, C(t)=(C_{ij}(t))\in{\mathbb R}^{q\times n}\), with uniformly bounded piecewise continuous functions \(a_{ij}, b_{ij}, c_{ij}\) from \({\mathbb R}^+\) to \({\mathbb R}\) possessing only discontinuities of the first kind at \(t=t_k,\,k=1,2,...\); \(u(t)\in{\mathbb R}^p\) is the input vector composed of piecewise continuous functions; \(y(t)\in{\mathbb R}^q\) is the output vector; for all \(k\geq 0,\, D_k\in{\mathbb R}^{n\times n}\) are known constant matrices; \(\Delta x(t_k)=x(t^+_k)-x(t^-_k)\) with \(x(t^+_k)=\text{lim}_{h\rightarrow 0^+}x(t_k+h)\), \(x(t^-_k)=\text{lim}_{h\rightarrow 0^+}x(t_k-h)\) and \(x(t^-_k)=x(t_k)\); \(g(t,x)\) is piecewise continuous perturbation function in \(t\) with discontinuities of the first kind at \(t=t_k\) such that \(\text{lim}_{(t,y)\rightarrow (t_k,x)}g(t,y)\) exists, the function is globally Lipschitz with respect to \(x\) and \(g(t,0)\neq 0, \forall t\geq 0\).NEWLINENEWLINESuppose that the system \(S\) satisfies the following assumptions:NEWLINENEWLINENEWLINE{(\({\mathcal A}_1\))} The system \(S\) is uniformly completely observable. NEWLINENEWLINENEWLINE {(\({\mathcal A}_2\))} \(\parallel I+D_k\parallel\leq d_k<1\) and \(\sum_{k\geq 0}d_k<+\infty\).NEWLINENEWLINENEWLINE {(\({\mathcal A}_3\))} \(g(t,x)\) satisfies \(\parallel g(t,x)\parallel\leq\lambda (t),\, \forall t\geq 0,\,\forall x\in{\mathbb R}^n\) and \(\lambda (t)\) is integrable on \([0,\,+\infty ]\), where \(\lambda :\,{\mathbb R}\rightarrow {\mathbb R}\) is a nonnegative continuous function on \([0,+\infty]\).NEWLINENEWLINENEWLINE {(\({\mathcal A}_4\))} The system \(S\) is uniformly completely controllable. NEWLINENEWLINENEWLINE The article presents an observer design (Theorem 1), a stabilizing state feedback controller design (Theorem 2) and an observer-based controller design (Theorem 3) for the system \(S\). The results are presented within an uniform-ultimate-boundedness approach with exponential convergence to an ultimate bound. All assertions are proved.