On the necessity of `block-invariance´ for the convergence of adaptive pole-placement algorithm with persistently exciting input

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. 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 for guaranteeing convergence is established here by a counterexample