Kollmorgen, Danaher Motion
Radford, Va.






Mechanical resonance is a pervasive problem in servosystems. Most resonance problems are caused by compliance of powertransmission components. Standard servocontrol laws are structured for rigidly coupled loads. However, in practical machines some compliance is always present. This compliance often reduces stability margins, forcing servo gains down and reducing machine performance.
Mechanical resonance falls into two categories: low frequency and high frequency. Lowfrequency resonance is common in industry. It differs from the less common but more often studied problem of highfrequency resonance. Highfrequency resonance causes instability at the natural frequency of the mechanical system, typically between 500 and 1,200 Hz. Lowfrequency resonance causes oscillations at the first phase crossover, typically 200 to 400 Hz.
It is well known that servo performance is enhanced when controllaw gains are high. However, instability results when a highgain control law is applied to a compliantly coupled motor and load. Machine designers specify transmission components, such as couplings and gearboxes, to be rigid in an effort to minimize mechanical compliance. However, some compliance is unavoidable. In addition, limitations such as cost and weight force designers to choose lighterweight components than would otherwise be desirable. The resulting rigidity of the transmission is so low that instability results when servo gains are raised to levels necessary to achieve desired performance.
The ideal plant for traditional control laws is a scaled integrator. However, the ideal plant is corrupted by compliance. The compliance term has a gain peak at the resonant frequency, F_{R}, and a gain minimum at the antiresonant frequency, F_{AR}. Were the load rigidly coupled, the plant would be an ideal integrator. However, compliance causes attenuation at and around F_{AR}, and amplification at, around, and above F_{R}.
The key problem in lowfrequency resonance is the increase in gain at frequencies above F_{R}. Below F_{AR}, the system behaves like a simple integrator. The gain falls at 20 dB/decade and the phase is approximately 90°. It also behaves like an integrator above F_{R}, but with a gain substantially increased compared to the gain well below F_{AR}. Above F_{R}, the load is effectively disconnected from the motor so that the gain of the plant is the inertia of the motor.
The openloop transfer function is well known to predict stability problems using two measures: phase margin (PM) and gain margin (GM). PM is the difference of 180° and the phase of the open loop at the frequency where the gain is 0 dB. GM is the negative of the gain of the open loop at the frequency where the phase crosses through 180°.
When the resonant frequency is well below the first phase crossover (270 Hz) the effect of the compliant load is to reduce the GM. If the inertia mismatch is 5, the reduction of GM will be 6 or about 16 dB. Assuming no other remedy were available, the gain of the compliantly coupled system would have to be reduced by 16 dB, compared to the rigid system, assuming both would maintain the same GM. Such a large reduction in gain would translate to a system with much poorer command and disturbance response.
Highfrequency resonance is different. It occurs in lightly damped mechanisms when the natural frequency of the mechanical system (F_{R}) is well above the firstphase crossover. Here, the gain near F_{R} forms a strong peak. While both types of resonance are caused by compliance, the relationship of F_{R} and the firstphase crossover changes the remedy substantially; cures for highfrequency resonance can exacerbate problems with lowfrequency resonance. The mechanical structures that cause highfrequency resonance (stiff transmission components and low damping) are typical of highend servo machines such as machine tools. Smaller and more costsensitive generalpurpose machines in such industries as packaging, textiles, plotting, and medical, typically have less rigid transmissions and higher damping so that lowfrequency resonance is more common.
Cures for lowfrequency resonance
Numerous methods are available to eliminate lowfrequency resonance. The most common in industry and the simplest to implement is the lowpass filter. A biquadratic filter is another choice. A biquad filter has two poles and two zeros; it can be thought of as a highpass filter in series with a lowpass filter. Another method is acceleration feedback, where the acceleration is provided by an observer.
Acceleration feedback applied together with a biquad filter provides dramatic improvement over all other methods, with increased bandwidth (the frequency where the gain falls to 3 dB). In addition, stability margins are maintained. Peaking, the undesirable phenomenon where gain rises above 0 dB at high frequency in the closedloop response, is a reliable measure of stability. The peaking of all four configurations is about the same, with the baseline system displaying the most peaking, indicating that the cures for resonance allow higher gain while maintaining equivalent margins of stability.
Acceleration feedback in combination with the biquad filter provides dramatic improvement for systems suffering from lowfrequency resonance. Compared to the traditional solution of a singlepole lowpass filter, the combination of acceleration feedback and the biquad filter allow the settling time to be cut by a factor of three and the bandwidth to be raised by that same factor. At the same time, acceleration feedback maintains stability margin.