On the Development of Batch Stable Inverse Indirect Adaptive Control of Systems with Unstable Discrete-Time Inverse

Indirect discrete time adaptive control creates a control law that promises to converge to zero tracking error for any commanded output. A major limitation is that the theory only guarantees convergence if the system inverse is asymptotically stable. Because it is necessarily discrete time, the continuous time governing equation must be converted to the equivalent difference equation, and this conversion normally introduces zeros. For a majority of desired applications, there are zeros introduced outside the unit circle producing instability of the inverse. The basic indirect adaptive control relies on the one-step ahead control law followed by the projection algorithm (or similar algorithm) to update the model based on the data currently available. This paper presents generalized versions of these building blocks that address the unstable inverse issue. The one-step ahead control is replaced by a batch stable inverse control law updating p steps with each batch. A new batch inverse theory for discrete time systems is used. The value of p is chosen to produce stability of the inverse control action produced in the sequence of batch updates. The needed computations to form the projection algorithm in this new formulation are presented. This paper presents an approach to obtaining a stable discrete time adaptive control theory for discrete time systems with unstable inverse; it remains to develop proofs of converges.