1. PID (Proportional Integral Derivative)
Proportional Action (P):
Makes the system more reactive (piu’ pronto)
Reduces disturbances on G: goes toward: for high G values
Can lead to overshoot
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

2. Shannon Sampling Theorem
45Hz < signal bw
55Hz > signal bw
50Hz
fsample = 100Hz = 2fsignal
fsample = 110Hz > 2fsignal
fsample = 90Hz < 2fsignal

3. Nyquist Criterion I theorem: II theorem:
Hp) The GH open loop transfer function has every pole with negative real part (I.e. stable) with the exception of one or two poles in the origin (O).
Th) For asynthotic stability of G/(1+GH) is necessary and sufficient that the polar diagram of GH(jω) does not go around or touch the point (-1,j0) leaving it to the left (increasing the frequencies).
II theorem:
Hp) The GH open loop transfer function does not have pure imaginary poles with the exception of one or two in the origin (O).
Th) For asynthotic stability of G/(1+GH) is necessary and sufficient that the polar diagram of GH(jω) does go around counter-clockwise the point (-1,j0) as many times as the number of poles with positive real part.

4. Polar Diagram: representation of GH with respect to ω.
Root Locus (Luogo delle radici): path of the roots (solutions) of 1+GH with varying k (i.e. the PID) of GH.

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

6. The PID gain is:
H(s) introduces poles and, sometimes, zeroes.
Representing the root locus
For the characteristic equation (1+GH), for different PIDs we can observe the different gains effect.

7. Matlab Commands: tf, bode, nyquist, rlocus, …kD kp kI
Matlab Commands: tf, bode, nyquist, rlocus, …

8. Matlab Commands: tf, bode, nyquist, rlocus, …PI
PID with P and I dominant and kI>>kD
Example:
Root Locus:
Matlab Commands: tf, bode, nyquist, rlocus, …

9. PI: Nyquist Diagram: (integral 1/jω delays)
Matlab Commands: tf, bode, nyquist, rlocus, …

10. Matlab Commands: tf, bode, nyquist, rlocus, …PD
PID control with P and D dominant and kD>>kI
Example:
Root Locus:
Matlab Commands: tf, bode, nyquist, rlocus, …

11. PD: Nyquist Diagram: (derivative jω anticipates)
Forward
(anticipo)
Matlab Commands: tf, bode, nyquist, rlocus, …

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

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

14. Notch Filter x y Characteristics: 2nd order recursive digital filter
it does remove a pair of poles not well dampened and it does substitute them with a pair adequately dampened
These high (with respect to the denominator) make the system more reactive but oscillating with high k b0 a2 a1 b1 b2
Z-1 + - x y
These high (with respect to the numerator) make the system more oscillating (and high k it dampen it) but more reactive (high k slows it)

15. Basic Filtering Equation:
z-2b2
z-1b1 b0
z-2a2
z-1a1 x y + - +
With these blocks it’s a IIR (Infinite Impulse Response)
Without it’s a FIR
(Finite Impulse Response, doen’t use its past outputs to determine actual output)
Basic Filtering Equation:

16. Matlab Commands: tf, bode, nyquist, rlocus, …

17. Error at Regime (ear)
Number of GH poles in O

18. PID (closed loop) and FeedForward (open loop)

19. 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)

20. Digital-Analog border
Digital World

21. Motion Profiles

22. Socapel Parallel Regulator (position lag proportional to torque)
Command
Velocity
Command
PID
Torque
Command

23. Rockwell Serial Regulator (position lag proportional to velocity)PI
Velocity
Command
Torque

24. Torque step response

25. Motion Profile Implementation

26. Position Regulator

27. Controller Architecture

28. Drive Architecture Drive Architecture Main Processor Intel i960
Interface handling
Data conversions
Motion Functions
Application Handler
Digital Signal Processor (DSP)
Regulation Loops (position, speed, current)
Feedback Measurement (position, speed)
Power Stage Set-Points
Hardware
User Interface (Display)
I/O’s
Fibre Optic Interface (EasyBus)
Brake Control Option
Safety Functions
Power Stage
Service Interface
RS232 Connection to PC
Test Board Option

29. Drive Architecture (cont’d)

30. Drive - Power Side - Architecture

31. HW Layout

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

33. 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 P +
PID Regulator S -
Position Feedback
Torque
Set-Point
Position
Lag I D
The PID Regulator has three Paths (parts)
Proportional Part
Integral Part
Derivative Part
For the Proportional Part the Position Lag Value is multiplied with the P-Gain Parameter
For the Integral Part the Position Lag Value is integrated and then multiplied with the I-Gain Parameter
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

34. S Feed-Forward System Description External Force Static / Dry Friction
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
time
speed
torque +
Static/Dry friction
Viscous friction S
Inertia
Torque
Feed-Forward =

35. Measuring the External Force and Friction Values
Run the motor in positive direction with a low speed (S1 = 10%) and measure the mean value of the needed Torque (Tp1) for several samples S1
Tp1
Then run the axis with the machine’s nominal speed (S10 = 100%) and measure the Torque as before (Tp10)
Do the same in negative rotating direction (Tn1 and Tn10)
viscous friction
torque + -
dry friction
external force
S10
Tp10
Tn10
Tn1
Ext.force =
|Tp1| -|Tn1| 2
visc =
|Tp10|+|Tn10| - |Tp1|+|Tn1|
S10 - S1
dry =
|Tp1| + |Tn1|
- visc * S1
speed

36. Parallel PID Regulator Details1
AMPLI
max
min
POS_REF
POS_MES
POS_LAG
TORQ_REG
TORQ_QUIET
TORQ_REF
R_MaxPosLag
R_PGain
R_MaxTorq
BIT 0 StatusB
VEL_REF
VEL_MES
VEL_LAG
R_MaxVelLag
R_DGain -F
ACC_REF
FLOW
TORQ_FLOW
TORQ_FEEDF
R_StatFriqTorq
R_ViscFriqTorq
R_ExtTorq
R_Inertia +F
R_IGain
dt

37. Regulator Tuning Deactivate the PosLag 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 PosLag (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
TuneLearn 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”

38. Optimise the PD Regulator Step Response
Observe (trace) the System’s Step Response (PosLag) 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 (PosLag) for a critically damped system
Step Response (PosLag) for an over damped system
Step Response (PosLag) for an under damped system
Increase P-Gain
Decrease P-Gain

39. 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 PosLag
Adjust the Inertia, ViscFricTorq, StatFricTorq and ExtTorq parameters until a minimum Range (depending on the application) for PosLag is reached
Speed
Speed proportional PosLag
Adjust Inertia
Acceleration proportional PosLag
Adjust ViscFricTorq
Properly adjusted Feed-Forward Parameters
Adjust StatFricTorq
Sign of Speed proportional PosLag

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

41. PD, no FF

42. PID, no FF

43. PD, with FF

44. Notes on D-Gain
In practice, Kd is limited by several factors including the
Drive current loop bandwidth which is directly related to the
PWM frequency selected for the drive (standard: 8 kHz; optional: 4 kHz)
resonance frequency of the axis mechanics,
signal-to-noise-ratio of the speed measurement (to avoid to derivate white noise)
In practice Kd is increased to the point where
the axis goes into sustained oscillation,
then reduced by a factor of 2 to obtain
good speed loop stability.
Kp is then determined with the
step response performance.