Hierarchical Control of Nondeterministic Linear Systems Using ($\gamma,\delta,\rho$)-Abstraction
Armin Pirastehzad, Arjan J. van der Schaft, Bart Besselink · IEEE Transactions on Automatic Control · 2025
We compensate for scalability issues in control synthesis by developing a hierarchical scheme that splits the control of a high-dimensional non-deterministic linear ‘concrete’ system into three consecutive steps. First, a low-dimensional ‘abstract’ model of the concrete system is obtained. A controller is then designed for this abstract system, which is computationally efficient due to the low-dimensionality of the abstract system. Finally, through the use of an ‘interface,’ the designed abstract controller is refined into the concrete one. We enable this by introducing and characterizing the notion of ($\gamma,\delta,\rho$)-abstraction, which measures to what extent the controlled input-output behavior of a concrete system is similar to that of its abstraction in an$\mathcal {L}_{2}$approximation metric. We then exploit this to characterize the existence of an abstract system and utilize this characterization to propose a step-by-step procedure to construct the abstract system and the corresponding interface.