Positive invariant set estimation of L₁ adaptive controller in the presence of input saturation

For a class of linear dynamical systems with constant unknown parameters, an L₁ adaptive control scheme is developed that provides stable adaptation in the presence of input magnitude constraints. Whereas for open-loop stable systems the results are global, for open-loop unstable systems, the problem of nonconservative estimation of the nonempty positive invariant set is cast into an LMI framework, which can be efficiently solved numerically via convex optimization. To achieve this, a standard result toward invariant set characterization is appropriately extended to accommodate bounded disturbance and model uncertainties. In addition to closed-loop stability, performance bounds of the L1 adaptive closed-loop system are analyzed, and the degradation due to the possible control deficiency is quantified. Simulation examples of aerospace applications are included to illustrate the proposed method.