Positive filter synthesis for sliding-mode control

The synthesis of externally positive filters that can be used to modulate control signals of output-feedback sliding mode controllers is presented. These filters are useful to estimate the norm of unmeasured signals in systems with parametric uncertainties. These norm observers are a robust alternative to state observers, that may be difficult to design for uncertain systems. The first step of the synthesis is the search of the envelope function, which is an upper bound for the norm of each admissible impulse response of the uncertain system. After this step, the approximation of a positive filter with an impulse response that is an upper bound for the envelope function is carried out. The search of the envelope function and the synthesis of the approximation filters are formulated as non-convex optimisation problems that can be solved by genetic algorithms. It is important to design norm observers that are not too conservative, because the modulation signals should be kept small in order to reduce the amplitude of the control signals, power losses, and undesirable chattering effects that may occur in sliding-mode control. The proposed methods are illustrated by numerical examples.