Quantification of tolerable parametric and dynamic uncertainty for robust mrac systems

Adaptive controllers have been proposed in a number of applications characterized by large modelling uncertainties in the open loop system. Despite this attractive feature it is not always immediate to guarantee, by design, a predictable closed-loop transient. Indeed the characterization of the transient response of a closed loop adaptive control system is still a challenging problem from the verification and validation standpoint. This issue is particularly relevant in the presence of unmodelled dynamics, time delays, disturbances and unmatched uncertainties. In this paper it is proposed a practical mythology for the quantification of the tolerable parametric and dynamic uncertainty in systems controlled by a Model Reference Adaptive Controller (MRAC) in the case the parameters that characterize the parametric and the dynamic uncertain as long as the controller adaptation weights are assumed unknown but bounded within a defined box domain. Given these bounded uncertainties, the MRAC design is formalized as a H2 controller design to guarantee minimum H2 gain between the reference model states and the tracking error states. Exploiting a quadratic Lyapunov function a parameter dependent LMI condition is derived whose feasibility guarantees a specified input-output H2 gain. This LMI condition can be exploited to quantify the tolerable matched and dynamic uncertainty and to evaluate the corresponding input-output H2 gain. The proposed approach has been applied to the design of a MRAC controller and to evaluate the tolerable uncertainties of a benchmark cart-pole system.