Unfalsified Adaptive Control: Multi-objective cost-detectable cost functions

The notions of Pareto optimality for vector-valued cost functions reflecting multiple objectives of a control system are examined within the framework of Unfalsified Adaptive Control. The set of conditions under which vector-valued cost functions are cost-detectable is discussed. A sampled data/discrete-time Level-Set controller switching algorithm is investigated which allows for the relaxation of the assumption that the controller cost function be monotonically nondecreasing in time. This opens up the possibility of the use of fading memory cost functions which are nonmonotone. When an active controller is falsified at the current threshold cost level, the Level-Set switching algorithm solves the multi-objective cost minimization problem and replaces the active controller by another as yet unfalsified Pareto optimal controller. Theoretical results for convergence and stability of the adaptive system are given. Simulation results validate the use of cost-detectable multi-objective cost functions. An example of a cost-detectable cost function which uses fading memory norm of the fictitious tracking error as a performance measure is shown. This allows for computation of performance of nonactive controllers with respect to a reference model.