The internal model principle of control theory states that algorithms designed to perfectly reject input signals must contain a model of that input.

Feedback control that adheres to this internal model principle has led to the development of harmonic cancellation algorithms and even more complex repetitive controllers.

How are they useful?

Periodic disturbances are commonplace in precision motion control applications; any oscillatory or rotational motion generates some periodic error in both the active and ancillary motion axes. Harmonic cancellation algorithms, when properly applied, give control system engineers an additional tool that is both effective and easily analyzed with common frequency domain techniques — to eliminate tracking errors in precision motion systems. They apply when either the servo command or disturbance is largely periodic — for example, in machine tools, data storage systems, and sensor testing.

Because controls that adhere to the internal model principle contain models of system input, harmonic cancellation algorithms contain periodic signal generators. When combined with a well-tuned conventional controller, these algorithms become useful tools for servo system designers.

Internal model principle

The internal model principle of control theory is a deceptively simple yet powerful concept. First formalized in the mid-1970s, it requires that an algorithm contain a generator (or model) of any input signal that is to be tracked with identically zero steady-state error.

Fig. 1 illustrates the concept with a block diagram. For zero error between the commanded reference and measured signals, the control algorithm must be capable of self-generating this signal in the absence of any further input.

The most familiar application of the internal model principle is the use of an integrator I term in common PID controllers.

Consider the case of a linear-motor-driven 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 so on) requires some error between the reference and measured positions 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 output to grow to a constant value as required to cancel the disturbance and achieve zero steady-state error.

Though the internal model principle is very general, specific expressions of it are frequently used in precision motion control applications. Any input (whether a command trajectory or a disturbance) that repeats with some known regularity can be addressed with a controller that contains a periodic signal generator. (We'll discuss these repetitive controllers shortly.) If inputs are frequency-limited, they can be represented as a summation of sinusoids; then they are addressed with harmonic cancellation algorithms that apply the internal model principle with a series of oscillators in the control algorithm.

Repetitive controllers

The presence of 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. Called repetitive control, these algorithms were first defined in the early 1980s, and the internal model principle was the basis for this “controller for repetitive operation.”

A delay element in the feedback loop of a continuous-time control algorithm satisfies the internal model principle for periodic inputs, but effectively contains a high (theoretically infinite) number of oscillators to replicate an arbitrary periodic input.

Originally, developers used a controller with a delay element in the feedback loop to form a periodic signal generator. However, in the continuous-time domain, the time delay element corresponded to a controller with an infinite number of marginally stable poles. (See Fig. 2.)