Note on Minimax Sliding Mode Control Design for Linear Systems

The technical note studies the problem of sliding mode control design for linear systems with incomplete and noisy measurements of the output and additive/multiplicative exogenous disturbances. First, we construct a linear minimax observer to have an estimate of the system's state with minimal worst-case error. Second, we establish the optimality of the constructed observer in the class of all observers represented by measurable functionals of the output. Finally, we propose an algorithm, generating continuous and discontinuous feedbacks, which steers the observer as close as possible to a given sliding hyperplane in finite time. The optimality (sub-optimality) of the designed feedbacks is proven for the case of bounded noises and additive (multiplicative) disturbances of L-2-class. The efficacy of the proposed algorithm is illustrated by a numerical example.