A generalization of the Popov and Zames-Falb multiplier stability criteria is presented for systems with one or more groups of identical MIMO nonlinear subsystems. For repeated positive n-dimensional nonlinearities, the largest class of convolution operators (stability multipliers) that preserve positivity is derived. Results are presented for both MIMO incrementally positive and MIMO monotone nonlinearities. Both the discrete time and continuous time cases are treated, and previously known nonlinear stability criteria for more restrictive classes of nonlinearities are found to emerge as special cases, including the Popov and Zames-Falb criteria. A SISO specialization permits less conservative stability analysis for systems with multiple dead-zone, relay or saturation nonlinearities and for multi-loop feedback control systems with integrator anti-windup compensation. The general repeated MIMO nonlinearity form of the stability criterion allows one to compute tighter stability robustness bounds for complex systems with several identical MIMO nonlinear elements, as for example elastic mechanical structures having two or more identical nonlinear subsystems (e.g., space structure truss elements or solar panels, multi-link robot arms, and inflatable vehicle systems like airbags or blimps).
Back to Control Seminars Page