PID Proportional Integral Derivative Proportional Action P Makes

PID (Proportional Integral Derivative) Proportional Action (P): • Makes the system more reactive (piu’ pronto) • Reduces disturbances on G: goes toward: • Can lead to overshoot for high G values Integral Action (I): • Reduces error at regime • Can lead to oscillations • Can move poles to the right (i. e. instability) Derivative Action (D): • It acts as a damper • It works against high slopes in the controller action • Can move poles to the right (i. e. instability) Mechatronics

PID Note: with only the proportional gain the control would oscillate around the set point G kp Set point + e - k. I + Plant k. D feedback H The transfer function is: Mechatronics

Kp low f limited narrow bandwidth poor performances Kp*Lag error Set point + - Low steepness PID bw low current peak f low narrow bw M Real E profile Mechatronics

Kp high wide band width high f good performances Kp*Lag error high current peak bw Lag error Set point + - High steepness PID M high f wide bw Real Profile E Mechatronics

PID (closed loop) and Feed. Forward (open loop) Mechatronics

PID time response to a step: Time continuous version: Proportional Action Integral Action Derivative Action (it does derivate only the feedback to avoid Dirac impulse) Mechatronics

Motion Profiles Mechatronics

Socapel Parallel Regulator (position lag proportional to torque) Position Velocity Torque PID Command Mechatronics

Rockwell Serial Regulator (position lag proportional to velocity) Position Velocity Torque PI PI Lag Command Mechatronics

Regulator Structure Components • • Digital Current (Torque) Regulator Safety Value limiting Digital PID Position Regulator Feed-Forward System Position Controller Feed. Forward Set. Points + PID Regulator - + + Torque Limitation Power Stage PI Current Regulator Current Signals to the Motor Feedback Signals Mechatronics

Parallel PID Regulator Description • The Input Value for the PID Regulator is the Position Lag (difference between the requested position and the real position) • The Torque Set-Point is the Output Value of the PID Regulator • The PID Regulator has three Paths (parts) – Proportional Part – Integral Part – Derivative Part PID Regulator Position Lag P Position Set-Point + S • For the Integral Part the Position Lag Value is integrated and then multiplied with the I-Gain Parameter S I + + • For the Proportional Part the Position Lag Value is multiplied with the P-Gain Parameter Torque Set-Point - + D Position Feedback • For the Derivative Part the Speed (velocity) Lag Value (same as the first derivative of the Position Lag) is multiplied with the D-Gain Parameter Mechatronics

Feed-Forward System Description • Four different Feed-Forward Components are available for the SAM Regulator – External Force – Static / Dry Friction – Viscous Friction – Inertia • The External Force Component is independent on the Motion and creates a constant Torque value (e. g. Gravity Compensation of a vertical axis) • The Static or Dry Friction Component uses only the sign of the Speed Set-Point to create a constant Torque value, whenever the speed is not zero • The Viscous Friction Component is always proportional to the Speed Set-Point • The Inertia Component is always proportional to the Acceleration Set-Point and compensates the mechanics Inertia during acceleration and deceleration (M = J * a) External force Inertia Static/Dry friction Viscous friction speed torque + time speed torque + time = S Torque Feed. Forward time Mechatronics

Measuring the External Force and Friction Values • Run the motor in positive direction with a low speed (S 1 = 10%) and measure the mean value of the needed Torque (Tp 1) for several samples • Then run the axis with the machine’s nominal speed (S 10 = 100%) and measure the Torque as before (Tp 10) • Do the same in negative rotating direction (Tn 1 and Tn 10) torque Tp 10 Ext. force = |Tp 1| -|Tn 1| 2 Tp 1 external force - |Tp 10|+|Tn 10| - |Tp 1|+|Tn 1| Tn 10 2 visc = S 10 + S 1 dry friction viscous friction speed 2 S 10 - S 1 dry = |Tp 1| + |Tn 1| - visc * S 1 2 Mechatronics

Regulator Tuning • Deactivate the Pos. Lag Error • Clear the Regulator’s Gain and Feed-Forward Parameters • Start with small but positive P, D-Gains • Enable the Drive (power_on) and run the axis at constant speed • Increase D-Gain step by step, until vibration starts or the audible noise is too high • Reduce D-Gain to 50% of the found value • Stop the axis, reset the Pos. Lag (power_off and then power_on) • Go through the last four steps for P-Gain (instead of R_DGain) trying to improve step response performances • Set the I-Gain so to reduce the Error At Regime (ear) • Set Feed Forward Gains Tune. Learn link Note: Here we refer to a parallel control; the same rule are Anyhow valid for a serial control substituting the word “velocity-P-Gain” to “D-Gain” Mechatronics

Optimise the PD Regulator Step Response • Observe (trace) the System’s Step Response (Pos. Lag) while performing a Torque Step Disturbance (with nominal motor torque) • Modify the D-Gain and P-Gain Parameters until a critically damped system is reached Step Response (Pos. Lag) for an over damped system Step Response (Pos. Lag) for an under damped system Increase P-Gain Decrease P-Gain Step Response (Pos. Lag) for a critically damped system Mechatronics

Adjust the Feed-Forward Parameters • Perform positioning movements (relative_move), using the machine’s nominal values for acceleration, deceleration and speed and observe (trace) the resulting Pos. Lag • Adjust the Inertia, Visc. Fric. Torq, Stat. Fric. Torq and Ext. Torq parameters until a minimum Range (depending on the application) for Pos. Lag is reached Speed proportional Pos. Lag Adjust Inertia Acceleration proportional Pos. Lag Properly adjusted Feed-Forward Parameters Adjust Stat. Fric. Torq Adjust Visc. Fric. Torq Sign of Speed proportional Pos. Lag Mechatronics

I-Gain • Observe (trace) the System’s Step Response (Pos. Lag) while performing a Torque Step Disturbance (with nominal motor torque) • Modify the I-Gain Parameter until a critically damped system is reached Step Response (Pos. Lag) for an under damped system Step Response (Pos. Lag) for an over damped system Increase I-Gain Decrease I-Gain Step Response (Pos. Lag) for a critically damped system Mechatronics

PD, no FF Mechatronics

PID, no FF Mechatronics

PD, with FF Mechatronics