## Table of Content

How Are Harmonic Cancelation Algorithms Useful?

Internal Model Principle

Harmonic Cancellation

Application Examples

## How Are Harmonic Cancelation Algorithms Useful?

Periodic disturbances are very common in precision motion control applications because any rotational or oscillatory motion produces some periodic error in the active and ancillary motion axes. When correctly applied, harmonic cancelation algorithms provide control system engineers with an additional tool that is effective and can be easily analyzed with standard frequency domain techniques to prevent tracking errors in precision motion systems.

These algorithms can be applied when either the disturbance or servo command is largely periodic, for instance, in sensor testing, machine tools, and data storage systems.

Harmonic cancelation algorithms contain periodic signal generators because controls that follow the internal model principle include system input. These algorithms, when combined with a well-tuned conventional controller, become very useful for servo system designers.

## Internal Model Principle

The internal model principle of control theory is a simple but powerful concept. It was first formalized in the mid-1970s. This principle demands that an algorithm should contain a generator (or model) of some input signal that needs to be tracked with identically zero steady-state error.

**Figure 1. **The internal model principle requires that the controller contain a model of the input signals to generate the appropriate output in the absence of any steady-state forcing error.

Figure 1 shows the concept with a block diagram. The control algorithm should be able to self-generate the signal in the absence of any additional input to retain zero error between measured signals and commanded reference.

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

Consider the case of a linear-motor-driven positioning stage modeled as a free mass together with a control force applied to it: Derivative and proportional control sufficient enough to stabilize the system; however, any constant disturbance force that occurs due to the process, cables, gravity, etc. needs some error between the measured and reference signals for the spring-like proportional control term to produce an output.

A constant disturbance is modeled as a step input using a Laplace transform of 1/s. Adding the term, an integrator, to the control algorithm enables output to increase to a constant value needed to cancel the disturbance and attain zero steady-state error.

Specific expressions of the internal model principle, which is very general, are commonly used in precision motion control applications. Any input (whether it is a command trajectory or a disturbance) that repeats with some known regularity can be handled with a controller containing a periodic signal generator.

If the inputs are frequency-limited, they can be shown as a summation of sinusoids; then they are addressed with harmonic cancelation algorithms that use the internal model principle with a chain of oscillators in the control algorithm.

## Harmonic Cancelation

Consider the case of repetitive control applied to a restricted number of discrete frequencies as harmonic cancelation. These cases are very common in precision motion control applications and consist of:

- Unbalanced payloads on rotary axes
- Torque and force ripple
- Link-style cable carrier systems
- Screw lead and gear pitch
- Cyclic command profiles

It must be noted that some disturbances can be periodic on displacement and others can be periodic in time, and as a result the specific frequency can differ.

In the following analysis, lets assume that we have a system under constant-speed operation with a known frequency.

**Figure 2. **The harmonic cancelation algorithm C(s) is implemented in a plug-in architecture. This leaves the PID controller unchanged and allows ready enabling and disabling of the harmonic cancellation algorithms.

When required, harmonic cancelation algorithms can be implemented in a “plug-in” style for easy enabling and disabling. The representative block diagram shown in Figure 2 includes standard PID controller, the harmonic cancelation algorithm, and the plant. The harmonic cancelation algorithm, in line with the internal model principle, includes parallel oscillators for each frequency included within the disturbance signal.

The effect of the harmonic cancelation algorithm in the frequency domain can be seen by observing the case of a single-frequency disturbance. In the algorithm, each individual oscillator has a constant-time Laplace transform representation:

**Figure 3. **Bode plots show very high (infinite) magnitude at the oscillator frequency.

In Figure 3, a frequency response plot of the harmonic cancelation algorithm can be seen as the gain term moves from zero (disabling the oscillator) to higher values. It must be noted that the magnitude is infinite at the oscillator frequency.

Now, if we recall how the familiar integrator term provides zero steady-state error to constant disturbances, we can easily interpret the harmonic cancelation block as an integrator at a non-zero frequency.

The tracking error that occurs due to disturbances is identically zero at the oscillator frequency. This can be seen by referring back to the block diagram shown in Figure 4, and calculating that the tracking error due to a disturbance becomes:

Evaluating this expression at the oscillator frequency:

There is zero steady-state error to disturbances at the frequency of interest. If a periodic profile has to be tracked, a similar analysis reveals unity response with zero phase shift between the actual and commanded position profiles.

In summary, an oscillator in the control algorithm functions as an “integrator” term to signals at the specific oscillator frequency. The application of multiple oscillators in parallel allows more complex waveforms to be canceled, and the general case of full-scale repetitive controllers can be approached. These controllers are implemented in a “plug-in” style that leaves the standard PID control gains unchanged.

Familiar frequency-domain tuning tools were used to discover stability margins (phase margin, gain margin, and crossover frequency) when applying harmonic cancelation algorithms.

However, due to the very limited frequency range over which the harmonic cancelation algorithm is most active, these systems can be easily tuned, provided the correction frequency remains well below the system crossover frequency.

**Figure 4. **Loop gain increases sharply at the cancelation frequencies of 10 and 20 Hz. Even so, they have minimal influence on the response near the 55 Hz crossover frequency.

Figure 4 displays a real open-loop frequency response of a system with an active harmonic cancelation algorithm. Although dominant peaks in the loop gain below the system crossover frequency are clearly visible, their effect is sufficiently localized in a way that gain and phase at the crossover frequency remains relatively unchanged.

**Figure 5. **Harmonic cancellation algorithms applied to a horizontally mounted rotary stage reduce the root mean square tracking error by approximately 12 x at 60 rpm. The dominant errors in the case plotted here were the payload unbalance and torque variations at the motor pole period.

## Application Examples

Once the overall concepts of the internal model principle, harmonic cancellation, and repetitive control are understood, they can be widely applied.

**Example 1: **Cconsider the control of the read-write arm of a disk drive. While the disk does not spin on a perfectly true axis, the repetitive control applied to the synchronous portion of the error motion enhances the capability of the head to track the motion.

**Example 2:** Fast tool servomechanisms employed in asymmetric turning operations also gain from repetitive control. While turning the surface of a toric shape, such as the mold for a contact lens that rectifies astigmatism, the cutting tool essentially returns to the same point with each spindle revolution. This periodic toolpath can be decomposed into Fourier series coefficients by applying harmonic cancelation oscillators to each of these coefficients.

**Example 3:** One simple application is a system with a horizontally mounted rotary stage. From such a system, a customer required improved velocity stability. The dominant terms were nine motor pole pitches per revolution and one unbalance per revolution. This is shown in Figure 5, which shows the position error measured during the rotation of the stage at 60 rpm. At these frequencies, harmonic cancelation algorithms can be applied to reduce the root mean square tracking error from 33 to 1.7 arc-sec - a 12× reduction.

**Figure 6. **Motion Simulator software imports and generates motion profiling, and allows I/O monitoring and control (above) – plus has a frequency-response mode (left) for detailed motion evaluation.

This information has been sourced, reviewed and adapted from materials provided by Aerotech, Inc.

For more information on this source, please visit Aerotech, Inc.