Robust semiglobally practical stabilization for nonlinear singularly perturbed systems (Q1006706)

The paper is devoted to the construction of a state feedback control which provides robust semi-global, practical stabilization for a MIMO nonlinear singularly perturbed system with uncertain variables of the form \[ \dot x= f_1(x,\theta)+ Q_1(x,\theta)z+ g_1(x,\theta)u, \] \[ \varepsilon\dot z= f_2(x,\theta)+ Q_2(x,\theta)z+ g_2(x,\theta) u, \] where \(x\) is the slow variable, \(z\) the fast variable, \(\theta\) the time-varying uncertain variable (\(\theta\) takes value in a compact set \(\Theta\)). The authors assume that \(Q_2\) is invertible, that the fast subsystem can be stabilized by a linear feedback \(u= k^t(x)z\) for each \((x,\theta)\), that the slow subsystem has a relative degree and that it has an asymptotically stable zero-dynamics. The construction of the feedback depends on two quadratic Lyapunov functions (with coefficients dependent on \((x,\theta)\).