Parameter-dependent Input Normalization: Direct-Adaptive control with Uncertain Control Direction

In this work, we propose a direct-adaptive MRAC for relative-degree-unity SISO systems with unknown control direction. The proposed scheme, employing an original construction of the control law and the use of an adaptive observer, achieves the long-searched objective of injecting, through the input, the unmeasurable derivative of the output error. The output derivative injection is performed by a smart construction of the control input that features a Parameter-dependent Input Normalization (PIN). The PIN scheme does not make use of Nussbaum functions usually invoked in the direct-adaptive setting, does not require persistence of excitation of indirect adaptive schemes, does not require switching between multiple models, does not suffer from singularities and does not require to know a-priori bounds on the norm of the high-frequency gain and on the parameters. Effectiveness of the algorithm is illustrated by a numerical example.