CAD.2:5.1 - A held reference

For the constructed plant x_{k+1}=x_k+u_k, choose

u_k=(1-a)(r_k-x_k),\qquad a=\tfrac12.

With a held reference r, e_{k+1}=ae_k. Let the reference change by \Delta r_j at the start of each N-step block. The error immediately after that change obeys

e_{j+1,0}=a^Ne_{j,0}-\Delta r_j.

For |\Delta r_j|\leq\delta, the interval |e|\leq E is invariant at block starts if

E\geq a^NE+\delta,\quad\text{hence}\quad E=\frac{\delta}{1-a^N}.

It also bounds intermediate fast instants because |a|\leq1. The initialization condition is |e_{0,0}|\leq E. With N=4,\delta=0.1, E=8/75. The reference jump, not instability of the fast law, determines this worst-case envelope.