Implementation of stabilizing control laws - How many controller blocks are neede for a universally good implementation?

In this article we investigated three questions. The first question deals with the existence of a universally good implementation of the control law u = C_rr - C_yy. This question is answered in the literature, where it is shown that there does exist a universally good three-block implementation. We then turned to the second question, asking whether there exists a universally good implementation that has fewer than three controller blocks. We considered a broad collection of one- and two-block implementations and determined that none of them are good implementations. Consequently, none of the implementations that we considered are universally good. The final question asks whether, for a given plant and control law u = C_rr - C_yy, there necessarily exists a good implementation that has fewer than three blocks. Again, based on the example (9)-(12), we conjecture that the answer is no. The overall conclusion of the article is therefore a conjecture, namely, that it is not only sufficient, but sometimes necessary, to use three controller blocks in a good implementation of u = C_rr - C_yy