Global exponential stabilization for a class of uncertain nonlinear systems with time-varying delay arguments and input deadzone nonlinearities
Stabilization results are obtained for a controlled \(n\)-dimensional differential equation system with delay, \[ \dot x(t)=A_0x(t)+ \sum^m_{i=1}A_jx(t-h_i(t))+ \Delta\Phi_2(z(t),t)+B\Delta\Phi_1(z(t),t)+B\phi(u),\quad t\geq0, \] with ``initial data'' \(x(t)=\varphi(t)\), \(-H\leq t\leq 0\), where \(z(t)=[x(t),x(t-h_1(t),\dots]\), the pair \((A_0,B)\) is completely controllable; all delays \(h_i(t)\) are bounded; \(\Delta\Phi_{1,2}\) are unknown continuous functions of different nature: \[ |\Delta\Phi_1(z,t)|\leq a_0|x(t)|+ \sum^m_{i=1}a_i|x(t-h_i(t))|, \] and \(|\Phi_2(z,t)|\leq f(z,t)\) for some continuous function \(f\). A dead-zone nonlinearity \(\phi\) has a form \[ \phi(u)= d_2(u-\overline r)1(u\geq\overline r)+d_2(u+\underline r)1(u\leq-\underline r) \] with some \(0\leq \underline r\), \(\overline r\leq d_1\). The main result states a global exponential stability under hypotheses on the pair \((A_0,B)\): \(\exists \alpha>0\) s.t. \[ \alpha\lambda_{\min}(P)/\lambda_{\max}(P)>a_0+ \sum^m_{i=1}(|A_i|+a_i), \] where \(P\) is a positive definite and symmetric solution of the Riccati equation \[ 2r_1d_2PBB^TP-(A_0+ \alpha I)^TP- P(A_0+\alpha I)= Q,\;Q=EE^T \] for some \(E\) s.t. \((A_0,E)\) is completely observable. In the case \(\text{rank }B=n\) this condition is always satisfied with any \(\alpha>0\) and any convergence rate may be attained.
scientific article; zbMATH DE number 923728