Towards Formal Proofs of Feedback Control Theory

Control theory can establish properties of systems which hold with all signals within the system and hence cannot be proven by simulation. The most basic of such property is the stability of a control subsystem or the overall system. Other examples are statements on robust control performance in the face of dynamical uncertainties and disturbances in sensing and actuation. Until now these theories were developed and checked for their correctness by control scientist manually using their mathematical knowledge. With the emergence of formal methods, there is now the possibility to derive and prove robust control theory by symbolic computation on computers. There is a demand for this approach from industry for the verification of practical control systems with concrete numerical values where the applicability of a control theorem is specialised to an application with given numerical boundaries of parameter variations. The paper gives an overview of the challenges of the area and illustrates them on a computer-based formal proof of the Small-gain theorem and conclusions are drawn from these initial experiences.