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In repetitive control, the Bode Integral Theorem [10,21,22,49,69,138], dictates a fundamental trade-off between improved closed-loop periodic performance, degraded nonperiodic performance and transient response time. Relying on the design methodology of Chapter 2, Chapter 5 develops an optimal repetitive controller design, where optimality is translated into an optimal trade-off between performance indices γp and γnp, which respectively quantify the closed-loop periodic and nonperiodic performance. Index γp explicitly accounts for period-time uncertainty, while transient response time, the third issue involved in the performance trade-off is determined by the repetitive controller order μ. For a given order μ, the remaining trade-off between periodic and nonperiodic performance is translated into a trade-off curve between γp and γnp (see Section 5.4.1).

To meet the continual quest for higher tracking accuracy in engineering practice, feedforward controllers have become an essential complement to feedback control. The ideal feedforward controller inverts the (closed-loop) system and as it yields perfect tracking for any reference input, specialized feedforward design for periodic inputs seems superfluous. However, the ideal feedforward controller suffers from two deficiencies, hereby raising applications where exploiting the input periodicity is beneficial.

In the early 1980s, Inoue et al. [75,76] and Hara et al. [59,60] laid the foundation of repetitive control. Facing the control problem of asymptotically rejecting periodic inputs of which only the period Tp is known, they relied on the Internal Model Principle [34,45,46,47,48] and included the signal generator shown in Figure 5.1(a) in the controller. This is the most general periodic signal generator as, determined by its initial conditions, it can generate any signal with period Tp. Although the first repetitive controllers were described in continuous-time, the discrete-time repetitive controller design got the upper hand [145,146]. The preference for the discrete-time design stems from the digital controller implementation and the advantage that the discrete-time counterpart of the infinite-dimensional continuous-time signal generator of Figure 5.1(a) is of finite dimension, provided that Tp contains an integer number N of sample periods Ts. Figure 5.1(b) shows the corresponding discrete-time signal generator.

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 Tp, 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 ωp = 2π/(NTs) 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 Tp 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.

The appeal of feedforward control is to a large extent related to the following properties: (i) the only stability concern in a feedforward controller design is its own stability whereas a feedback controller must additionally guarantee stability of the closed-loop system; (ii) the effect of a feedforward controller is restricted to the input channel to which it is added, not affecting other inputs and the related performance; and (iii) many applications can cope with limited noncausality of a feedforward controller, whereas a feedback controller must be causal. Unfortunately, feedforward control only applies to reference inputs or measurable disturbances, while unmeasurable disturbances, on the other hand, are widespread in engineering practice. Extending the advantages of feedforward control to unmeasurable disturbances is the rationale for disturbance observers, where the observer developed by Ohnishi [103] is most popular [82,95,104,116,155]. Disturbance observers provide an estimate of the unmeasurable disturbance, which can be used to compute an appropriate control input similar to feedforward control.

This chapter presents the general design methodology for controllers facing periodic input signals and elaborates on its fundamental concepts.The subsequent chapters apply this methodology to specific control strategies from the literature and hereby yield a more concrete impression of its functionalities and potential.