A User’s Guide to Repetitive Control Systems and the Internal Model Principle

Repetitive control refers to a family of algorithms that are particularly useful when either the servo command or disturbance is largely periodic. Examples are very common in machine tools, data storage systems, and sensor testing. Any oscillatory or rotational motion generates some periodic error in both the active and ancillary motion axes. The Internal Model Principle of control theory states that algorithms designed to perfectly reject input signals must contain a model of that input, and thus a repetitive controller contains a periodic signal generator. The most general cases of repetitive control can be difficult to stabilize. If however the algorithms are limited to operate within the existing servo system bandwidth, the tuning becomes much more straightforward and the algorithms become useful tools for a servo system designer. THE INTERNAL MODEL PRINCIPLE The Internal Model Principle of control theory is a deceptively simple, but powerful concept. First formalized in the mid-1970’s [1], it can be loosely stated as requiring that an algorithm contain a generator (or model) of any input signal if that input is to be tracked with identically zero steady state error. Figure 1 illustrates this concept with a block diagram. For there to be zero error between the commanded reference and measured signals, then the control algorithm must be able to self-generate this signal in the absence of any further input. A familiar example of the Internal Model Principle applied in practice is through the use of an integrator (I) term in the common PID controller. Consider the case of a linear-motordriven positioning stage modeled as a free mass with a control force applied to it. Proportional and Derivative control alone are sufficient to stabilize the system, but any constant disturbance force (due to the process, gravity, cables, and etc.) requires some error between the reference and measured positions in order for the spring-like proportional control term to generate an output. A constant disturbance is modeled as a step input with a Laplace transform of 1/s. Adding this term, an integrator, to the control algorithm allows the output to grow to a constant value as required to cancel the disturbance and achieve zero steady-state error. The Internal Model Principle is very general, but specific realizations of it appear frequently in precision motion control applications. Any input (whether a command trajectory or a disturbance) that repeats with some known period can be addressed with a controller that contains a periodic signal generator. These are the repetitive controllers that will be discussed in the next section. If these inputs are frequencylimited, they can be represented as a summation of sinusoids. In that case we approach them with harmonic cancellation algorithms that apply the Internal Model Principle with a series of oscillators in the control algorithm. FIGURE 1. The Internal Model Principle requires that the controller contain a model of the input signals in order to be able to generate the appropriate output in the absence of any steadystate forcing error. REPETITIVE CONTROLLERS A periodic signal generator in the feedback control algorithm satisfies the Internal Model Principle and allows for perfect tracking of periodic commands and perfect rejection of periodic disturbances. The class of control algorithms that collectively addresses this problem is called repetitive control. These algorithms first appeared in the literature in the early 1980’s with a paper by Inoue [2] who used the Internal Model Principle as the basis for a “controller for repetitive operation”. The authors used a controller with a delay element in the feedback loop to form a periodic signal generator. In the continuous-time domain P(s) C(s) R(s) W(s)