On the Necessity of “ Block Invariance ” for the Convergence of Adaptive Pole - Placement Algorithm with Persistently Exciting Input

A parameter adaptive control stmcture for linear multivariable systems,'' IEEE Trans. Abstract-Global convergence of a number of discrete-time adaptive control algorithms with persistently exciting inputs has been established in the literature. A key element of the convergence proof is the " block-invariance " feature of the algorithm. Namely, while the parameters are estimated at every sample, the controller parameters remain invariant throughout a " block " of samples and are updated only at the end of each such block. The necessity of this " block-invariance'' feature to guarantee convergence is established in this note by a counterexample.