Output integral sliding mode with application to the LQ - Optimal control

The problem of the realization of integral sliding mode controllers, based on output information only, is discussed. In the case when the number of inputs is more or equal than a number of outputs, the closed system, describing the integral sliding mode dynamics, is shown to lose observability. For any other case, a hierarchical sliding mode observer is proposed, allowing to reconstruct the system states practically just after the initial moment (theoretically). A low pass filter is suggested to be used for the observer realization. Taking the sampling step (during this filter realization) small enough, the convergence time and the observation error can be made arbitrary without the observer parameters adjustment. This allows to implement an output integral sliding mode concept ensuring the insensitivity (robustness) of the state trajectory with respect to the matched uncertainties. The results obtained are illustrated by numerical simulations.