Hierarchical controller synthesis using ($$\upgamma ,\updelta $$)-Similarity

Abstract We compensate for the scalability issues in controller synthesis by developing a hierarchical control scheme within the framework of ( $$\upgamma ,\updelta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:math> )-similarity, which measures to what extent a potentially non-deterministic system satisfies specifications expressed as solution trajectories of a dynamical ‘specification’ system. This scheme synthesizes a controller for a non-deterministic ‘concrete’ system in three hierarchical steps. First, an ‘abstract’ system, which is a low-dimensional model of the concrete system, is obtained. Then, a controller is designed for the abstract system. At last, the abstract controller is refined into the concrete controller through an ‘interface’. To enable this, we introduce and characterize the notion of ( $$\upgamma ,\updelta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:math> )-abstraction that utilizes an $$\mathcal {L}_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> approximation metric to measure the behavioral similarity of the concrete system and its abstraction in the presence of the interface. We utilize this characterization to propose a step-by-step procedure to construct the interface. We then synthesize the abstract controller and refine it into a concrete one.