Adaptive control of a class of multilinearly parameterized systems by using noncertainty equivalence control

We propose in this work a new adaptive controller for a class of non-linearly parameterized systems with multi-linear parametrization dependent on the state containing an arbitrary number l of parameters. The proposed controller is non-certainty equivalence based and only the original parameters are adapted. Moreover, since the parameterization depends on the state, a trivial solution by over parameterization is not possible. The main idea of the paper is to combine two parameterizations, one to convexify and another to concavify the nonlinear function of the unknown parameters, because this permits the use of gradient search based convergence in the entire state-space. This leads however to discontinuous Lyapunov functions, and the use of a non-certainty equivalence based control eliminates any possibility of discontinuity, allowing then a standard Lyapunov function to be used. The stability proof relies on the concavity proof of some special functions, that requires in turn the proof of the positive semi-definiteness of an l-dimensional matrix, that is a challenge by itself.