The first approach is to use finite horizon cost function. However, by applying this idea to a finite horizon cost function the overall Youla-Kucera predictive controller (YKPC) becomes time-variant. The stability analysis of such a system is performed. Stability proofs lead to minimum possible horizon lenghts. These are given as number of unstable poles (control horizon) and number of unstable zeros (output horizon) plus the degree of optimised YK polynomials.

Another possibilty how to overcome stability problem is to use infinite horizon cost function. In that case the controller is time-invariant when constraints are inactive. Constrained stability is as in the previous case assured if the corresponding optimisation problem is feasible.

The third approach to stability is based on the following idea. Let
us assume that the loop is already closed and controlled by a controller.
The question is what is the minimum number of control moves, or control
error moves that still lead to stable closed-loop behaviour and to a time-invariant
controller. Again, the minimum number if steps is directly related to the
number of unstable poles and zeros.

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