Pareto suboptimal solutions under coalitions of disturbances

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 in the multi-objective control or filtering problems are derived. A multi-objective cost parameterized by weights from an N-dimensional simplex is introduced for the disturbances acting on all inputs simultaneously. It is established that the optimal solutions with respect to this cost are the 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.