Direct and Indirect Model Reference Adaptive Control for Multivariable Piecewise Affine Systems

This article proposes direct and indirect model reference adaptive control strategies for multivariable piecewise affine systems, which constitute a popular tool to model hybrid systems and approximate nonlinear systems. A chosen reference model, which can be linear or also piecewise affine, describes the desired closed-loop system behavior that is to be achieved by the adaptive controllers for unknown system dynamics. Each subsystem acquires its own set of control gains, which is tuned under careful consideration of the switching behavior. In the indirect approach, the use of dynamic gain adjustment avoids singularities in the certainty equivalence principle. It is shown for both algorithms that the state of the reference model is tracked asymptotically given a common Lyapunov function for the switched reference model is available. Furthermore, parameter convergence in both the direct and indirect approach is proven for sufficiently rich reference signals. Finally, both algorithms are evaluated in numerical simulations and their advantages and disadvantages are discussed.