Concept of the Best Achievable Control
This chapter introduces a special decomposition of the sensitivity function which makes possible to reformulate the classical paradigm: which is the best achievable control. The three decomposed parts are the design loss, the realizability loss, and the modeling loss. This decomposition helps to formulate the optimization of these contributing losses independently. The optimization of the design loss results in selecting the fastest reference models under the amplitude constraints of the actuators. It is possible to develop an iterative redesign technique for the selection of best reachable reference model. For the optimization of the realizability loss, new (energy and supremum) norms E 2 E 2 and E ∞ E ∞ are applied instead of the classical norms H 2 H 2 and H ∞ H ∞ . Unfortunately, these last norms are not applicable for such—mostly integrating—regulator classes, which are important for industrial applications. The norms E 2 E 2 and E ∞ E ∞ are certain generalizations of the classical norms H 2 H 2 and H ∞ H ∞ . The optimization of E 2 E 2 can be usually solved by the application of certain Diophantine equations. The optimization of E ∞ E ∞ can be usually solved by the iterative solution of a nonlinear equation system. The optimization of the modeling loss results in a special binary form of the reference signal, where the switching condition can be computed via a specially formed filter.