Spatially invariant systems: Identification and adaptation

We present a distributed projection algorithm for system identification of spatially invariant systems. Each subsystem communicates only with its immediate neighbor to share its current estimate along with a cumulative improvement index. Based on the cumulative improvement index, the best estimate available is picked in order to carry out the next iterate. For small estimation error, the scheme switches over to a "smart" averaging routine. The proposed algorithm guarantees to bring the local estimates arbitrarily close to one another. Based on this we present a general class of indirect adaptive controllers for spatially invariant systems. The control design is based on certainty-equivalence approach, where at each step system parameters are estimated and the controller is implemented using the estimated parameters. At each estimation stage a modeling error is committed which affects the output of the plant. We show that under suitable assumptions on the rates of variation of the estimated plant, which follow from utilizing the distributed projection algorithm, a globally stable adaptive scheme can be guaranteed.