A Non-Conservative Approach for the Estimation of the Region of Operation of Uncertain Adaptive Control Systems

A challenging problem for Model Reference Adaptive Control Systems is the accurate characterization of the transient response in the presence of large uncertainties. Early prior research by the authors has demonstrated that using a projection mechanism for parameters adaptation the tracking error dynamics behaves as a linear system perturbed by bounded uncertainties. This brings the benefit that the stability analysis can be cast in terms of a convex optimization problem with LMI constraints so that efficient numerical tools can be used for the adaptive controller design. A possible limitation of the approach is that the design is restricted to quadratic control Lyapunov functions that could produce a conservative estimation of the regions of operation for the actual uncertain adaptive system. In this paper this approach is extended to arbitrary high degree polynomial Lyapunov functions by translating the design and performance requirements in terms of Sum of Square (SOS) inequalities and then using SOS optimization tools for the design. In this effort the new SOS approach is introduced and compared with the previous one. A numerical example based on the short period longitudinal dynamics of the F16 aircraft is used to demonstrate the efficacy of the novel method.