Robust stabilization of a class of nonlinear time-delay systems
The authors consider the controlled system \[ \dot{x} = Ax + Bu(t) + B\sum_{1}^r\eta_j(x(t-h_j(t)),t). \] Two feedback control laws are proposed: a) the memoryless adaptive controller \[ u = -(\mu/2)B^TPx - \overline{\theta}B^T(\partial V/\partial x)^T \] with \(P>0\) solution of the matrix Riccati equation \[ A^TP + PA + Q = \mu PBB^TP \] where \(Q>0\) and \(V(x) = \sum_{1}^s (1/i)(x^TPx)^i\); the adaptation law is given by \[ \dot{\overline{\theta}} = k| (\partial V/\partial x)B| ^2 - kl\overline{\theta} \] with \(k>0,l>0\) adjustable; b) the nonlinear feedback controller \[ u = - (1/2)(\mu+\nu)B^TPx - \rho(x)(B^TPx\rho(x))/(| B^TPx\rho(x)| +\vartheta\exp\{-\varepsilon t\}) \] \(\mu, P\) from the Riccati equation, \(\nu>0\) adjustable, \[ \rho(x) = \sum_{1}^s(\nu_i+\zeta_i| x(t)| )| x(t)| ^i \] and \(\nu_i>0\), \(\zeta_i>0\), \(\vartheta>0\) adjustable.
scientific article; zbMATH DE number 2107378