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. Low-frequency resonance is common in industry. It differs from the less common but more often studied problem of high-frequency resonance. High-frequency resonance causes instability at the natural frequency of the mechanical system, typically between 500 and 1,200 Hz. Low-frequency 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 high-gain 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 lighter-weight 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*

*. Were the load rigidly coupled, the plant would be an ideal integrator. However, compliance causes attenuation at and around*

_{AR}*F*

*, and amplification at, around, and above*

_{AR}*F*

*.*

_{R}The key problem in low-frequency resonance is the increase in gain at frequencies above *F** _{R}*. Below

*F*

*, the system behaves like a simple integrator. The gain falls at 20 dB/decade and the phase is approximately –90*

_{AR}**°**. It also behaves like an integrator above

*F*

_{R}, but with a gain substantially increased compared to the gain well below

*F*

*. Above*

_{AR}*F*

*, the load is effectively disconnected from the motor so that the gain of the plant is the inertia of the motor.*

_{R}The open-loop 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.

High-frequency resonance is different. It occurs in lightly damped mechanisms when the natural frequency of the mechanical system (*F** _{R}*) is well above the first-phase crossover. Here, the gain near

*F*

*forms a strong peak. While both types of resonance are caused by compliance, the relationship of F*

_{R }*R*and the first-phase crossover changes the remedy substantially; cures for highfrequency resonance can exacerbate problems with lowfrequency resonance. The mechanical structures that cause high-frequency resonance (stiff transmission components and low damping) are typical of high-end servo machines such as machine tools. Smaller and more cost-sensitive general-purpose machines in such industries as packaging, textiles, plotting, and medical, typically have less rigid transmissions and higher damping so that low-frequency resonance is more common.

**Cures for low-frequency resonance **Numerous methods are available to eliminate lowfrequency resonance. The most common in industry and the simplest to implement is the low-pass filter. A biquadratic filter is another choice. A biquad filter has two poles and two zeros; it can be thought of as a high-pass filter in series with a low-pass filter. Another method is acceleration feedback, where the acceleration is provided by an observer.

Acceleration feedback applied together with a bi-quad 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 closed-loop 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 bi-quad filter provides dramatic improvement for systems suffering from low-frequency resonance. Compared to the traditional solution of a single-pole low-pass filter, the combination of acceleration feedback and the bi-quad 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.

The Bode plot shows the key problem in lowfrequency resonance; increase in gain at frequencies above the resonant frequency, *F _{R}*.

Below *F _{R}*, the system acts like a simple integrator, with gain falling off at 20 dB/decade and phase at –90°. Above

*F*, the loop gain is raised significantly, reducing margins of stability.

_{R}In a typical velocity-control loop, the velocity error, *V _{E}*, is processed by a control law and filters. The current command,

*I*, is connected to the current controller which produces current,

_{C}*I*, in the motor. The motor/load plant is connected to an encoder. An observer, fed by the feedback current and position, produces an observed acceleration,

_{F}*A*.

_{MO}The effective gain increase that results from the compliant load reduces the gain margin. Here, the gain margin is just over 10 dB. The ideal system gain margin would've been about 30 dB.

A Luenberger observer takes input from the motor current and the encoder, adds the two, and feeds the sum to a motor model. The model then produces observed position and compares this with actual position. The PID observer compensator drives out most error up to the observer bandwidth, which is usually between 200 and 500 Hz. One by-product of the observer is an acceleration signal, which represents acceleration much better than double-differentiating the position-feedback signal. The step response of the baseline system shows a settling time of about 60 msec with some overshoot and a system bandwidth of 23 Hz.

The step response of the system with acceleration feedback shows a settling time of about 12 msec with less overshoot. System bandwidth increased to 77 Hz.

George Ellix, Senior ScientistKollmorgen, Danaher Motion

Radford, Va.