Approximate hierarchies of linear control systems

Recently, a hierarchical control approach based on the notion of approximate simulation relations has been introduced. The proposed hierarchical control architecture consists of a precise model (the concrete system) of the plant to be controlled and of a rough model (the abstract system) of the plant that is used for control synthesis. In this paper, we consider the problem of computing approximate simulation relations for linear control systems. For stabilizable systems, we give an effective characterization of simulation functions (i.e. functions whose level sets define approximate simulation relations) and compute the associated interfaces which allow the refinement of the control inputs of the abstract system into control inputs of the concrete system. Then, we propose a procedure for the computation of abstractions of linear control systems that can be used in our hierachical control framework. Finally, as an example of application, we use our approach to synthesize hierarchical controllers of a class of high-order systems.