Low Friction System – Effect of Derivative Term
In a low or no friction air bearing system, the response has a tendency to become unstable quickly, even with low gains.
To remedy this instability, derivative action is required. Because the D term differentiates the position error which changes fast in case of instability it helps suppress the large changes by creating an opposing torque. In the system above, with
the same proportional gain and with some derivative gain following response is obtained.
Large overshoot now is is eliminated. However, the overall response is also slower (meaning the time required to reach the target).
Frictional System – Effect of Integral Term
As noted above, by just using a proportional term, a stable response can be obtained, however the final target is not reached.
With added integral term, the longer the error exists, the larger the integral will become, resulting in a correction torque. In the system above, with the same proportional gain, and with added integral gain, position error (steady state error) is reduced to zero, even in the presence of friction.
So far we have looked at the effect of each term on the response curve, in order to show how they contribute. From the response curves one can clearly see that in addition to the shape of the response, the response time is dramatically affected. Response curves have a few attributes that help quantify the response:
Overshoot: by how much is the target position exceeded.
Step response time: after how much time is 67% of the final target reached.
Settling time: after how much time is the position settled within some % of the target.
As all the gains are gradually increased to obtain the desired response, some additional effects may occur:
-Saturation: one can run into current, torque or voltage limitation, which creates a strong nonlinearity typically resulting in overshoot and instability.
-Resonance: as the system gets excited at higher frequencies and power, system resonance may affect system response in unpredictable ways.
-Jitter: higher gains will cause small changes in the feedback to cause large jumps in currents. This leads to jitter (small oscillations at standstill).
For example, below is the system response as we keep increasing the gains to obtain faster response:
The current limit in our system is reached at 4A. Reaching this limit causes saturation and an ensuing oscillation (which does die out after coming out of saturation).
This means that motor and drive sizing should not only be based on the torque and speed requirements, but also on the required system response.
Mechanical transmission components should also be carefully selected to avoid system resonance (see previous article on the effect of coupling stiffness).
Lastly the feedback resolution should not just be based on positioning resolution requirements but also on desired system response.
Servo loop tuning is not a trivial task. Hands on experience is invaluable and above information will, hopefully, allow to achieve the desired response.
Although the behavior of the PID algorithm is well understood, there are many implementation details and system parameters that influence the response.
By utilizing a systematic approach of increasing the proper gains gradually while monitoring the response, stable behavior can be more quickly obtained.
In parallel, one should also observe current and torque to determine how much power is applied, or to avoid saturation. If the system is marginally sized, it may be necessary to adjust performance expectations.
Stay well servo-tuned !
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