Stabilizing ℓ-semioptimal fractional controller for discrete non-minimum phase system under unknown-but boundeddisturbance

We consider the problem of optimal stabilizing controller synthesis for a discrete non-minimum phase dynamic plant described by a linear difference equation with an additive unknown-but-bounded disturbance. When considering the ’worst’ case of disturbance, solving this optimization problem has combinatorial complexity. However, by choosing an appropriate sufficiently high sampling rate, it becomes possible to achieve an arbitrarily small level of suboptimality using a noncombinatorial algorithm. In this article, we propose using fractional delays to achieve a small level of suboptimality without significantly increasing the sampling rate. We approximate fractional delays by minimizing the ℓ1-norm of the objective function. The proposed approximation of the fractional delay allows obtaining zero additional error for many non-integer solutions. Furthermore, it is shown that with a non-zero approximation error, the resulting controller may have a smaller additional error than the controller obtained using integer optimization. The theoretical results are illustrated by simulation examples with non-minimum-phase plants of the second and third orders.