Application to Feedback Control

Abstract

This chapter deals with the general design of a feedback controller in the presence of periodic inputs. Repetitive controllers, discussed in the previous chapter, are the most popular type of feedback controllers for periodic inputs (see references in Section 5.1), where this popularity is to a large extent related to their simple structure, shown in Figure 5.2(b), and intuitive design. Moreover, a repetitive controller design only requires knowledge of the input period T_p, and its implementation is easily made adaptive for varying periods by adjusting \(N=\textrm{int}(T_{\mathrm{p}}/T_{\mathrm{s}})\) [42,139]. On the other hand, its particular structure impedes a repetitive controller to properly account for additional information on the periodic input spectrum as well as uncertainty on the input period. These disadvantages are caused by the period delays \(\mathcal{Z}^{-N}\) embedded in the repetitive controller structure. The FRF of \(\mathcal{Z}^{-N}\) is periodic with ω_p = 2π/(NT_s) and this FRF periodicity enforces an equal treatment of all harmonics l. Hence, a repetitive controller: (i) also accounts for harmonics l not present in the input (\(l \notin \mathcal{L}\)); (ii) cannot attribute different weights W _l to the harmonics \(l\in \mathcal{L}\); and (iii) a robust design for uncertainty on T_p is usually overly robust at the lower harmonic frequencies, since the uncertainty intervals Ω _l (2.4) grow linearly with harmonic number l. These disadvantages restrict the ability of a repetitive controller to optimize closed-loop periodic performance, and the quantification of the resulting performance loss is valuable for a control engineer who has to decide whether the advantages of the repetitive controller structure pay off its disadvantages.