This paper considers system theoretic conditions for the solvability of the so-called Constrained Lyapunov Problem for non-square systems. These problems commonly appear in the control systems literature. Both a static output feedback problem and an observer problem are considered. The basis for the work described here is a new canonical form which simplifies the analysis and deals with the equality constraint in a simple way.
This paper introduces a new approach to discretization of non-linear control laws with a Lipschitz property. The sampling time is defined as a parameter, which must be selected sufficiently small so that the closed loop system is stable. In contrast to similar results, the stabilizing effect of the control is taken into account. This can result in less conservative constraints on the minimum sampling frequency. The discretization techniques are explained on a general non-linear model and applied to the discretization of a novel non-linear, robust sliding-mode-like control law. Similar robustness features as for continuous control are demonstrated. Non-smooth Lyapunov functions are used for the discretized sliding-mode-like control introducing cone shaped regions of the state space. One of these cone shaped regions coincides with a cone shaped layer around the sliding mode defined by the continuous slidingmode like control. A stability theorem using non-smooth Lyapunov functions is provided.
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