Convex combination of multiple models for discrete-time adaptive control

The principal drawback of traditional adaptive systems is the possibility of rather poor transient response. In this regard, the multiple-model methodology has been beneficial over the past couple of decades. However, the application of this methodology requires proven indirect model reference adaptive control. Such a gap exists in the case of discrete-time linear time-invariant systems with unknown parameters. The principal contribution of this paper is to fill this void by arriving at proof of global stability of discrete-time adaptive systems that use the prediction error rather than the control error to update the parameters of the prediction model. This proof is based on the theory of Lyapunov and the properties of square-summable sequences. An analysis of the available literature reveals that such a result is not available even for linear time-invariant systems. Therefore, the proof is first provided for adaptive systems with a single prediction model wherein the transfer function of the plant has an arbitrary relative degree. Subsequently, this proof is extended to the case of multiple prediction models in the context of multiple-model methodology with second-level adaptation. Two parameter update algorithms are considered. Simulation studies are included to demonstrate the improvement in transient performance.