Pareto suboptimal controllers in multi-objective disturbance attenuation problems

A multi-objective disturbance attenuation problem is considered as a novel framework for control and filtering problems under multiple exogenous disturbances. There are N potentially possible disturbance inputs of a system on each of which may act a disturbance from a certain class. A disturbance attenuation level is defined for each channel as an induced norm of the operator mapping signals of the corresponding class to the objective output of the system. Necessary conditions of the Pareto optimality are derived. It is established that the optimal solutions with respect to a multi-objective cost parameterized by weights from an N-dimensional simplex are Pareto suboptimal solutions and their relative losses compared to the Pareto optimal ones do not exceed 1−N∕N. These results are extended to the case when the disturbances acting on different inputs are combined into coalitions. The approach is applied to multiple classes of L2-bounded and impulsive disturbances for which the H∞∕γ0 optimal controllers as the Pareto suboptimal solutions are synthesized in terms of linear matrix inequalities (LMIs). Illustrative examples demonstrate the effectiveness of the approach proposed.