Abstract
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.
Highlights
I NCREASING complexity in technical systems lead to a growing interest in hybrid systems to efficiently handle switching behavior or approximate nonlinearities
In piecewise affine (PWA) systems, the state-input space is partitioned into convex polytopes and the system dynamics are governed by different linear subsystems in each polytope
This paper proposes a direct and an indirect algorithm for Model reference adaptive control (MRAC) of multivariable PWA systems
Summary
I NCREASING complexity in technical systems lead to a growing interest in hybrid systems to efficiently handle switching behavior or approximate nonlinearities. Wang and Zhao point out that deriving dwell-time constraints for switched systems may be infeasible without knowledge of the actual subsystem parameters [15] Di Bernardo et al proposed a hybrid MRAC strategy for multimodal PWA systems [19] in which the state-input space partitioning of the controlled system may be different form the reference system. The unique advantage of this approach is that estimates of all system parameters are obtained in parallel to the control task For both the direct and the indirect approach, the stability proofs for asymptotic state tracking rely on Lyapunov theory and dwell-time assumptions. In both cases, we consider reference system tracking separately from the additional requirement of parameter convergence.
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