Robust ISS Stabilization On Disturbance For Uncertain Singularly Perturbed Systems

In this thesis, the problem of robust input-to-state stability (ISS) control for a class of linear uncertain singularly perturbed system under disturbance is presented. We first find a linear matrix inequality (LMI) condition under which the given system (original system) is made with ISS; assuming that the free system (i.e. the system without input and disturbance) is stable and by setting ε= 0, we obtain the limit system. Then the two time scale decomposition method is applied to guarantee the ISS for the slow and fast subsystems both derived from the limit system. Based on the established results, the robust ISS of linear uncertain singularly perturbed system is obtained by considering the singular perturbation parameter as an ordinary parameter; in other words the original is considered as a normal system.Second, we suppose that the free system is unstable; then the problem of de-signing a control law through feedback transformation is addressed for such linear uncertain singularly perturbed system with the input u such that the closed loop system is robust ISS with respect to disturbance w when the singular perturbation parameter is sufficiently small. To achieve this, a linear matrix inequality which put the closed loop system into the standard form and made it with ISS is proposed and proved. For the issue of simulation, a feasible solution of the linear matrix inequality criterion is found; the state feedback control gain matrix is derived by solving such a linear matrix inequality.In addition, since the proposed linear matrix inequality is not depending on the singular perturbation parameter, the upper bound for the small parameters is obtained effectively by solving an optimal problem by applying GEVP solver in LMI Control Toolbox.Finally, a numerical example shows the effectiveness of the obtained theoretical results.