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Mechatronics 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 Can move poles to the right (i.e. instability)

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Mechatronics Shannon Sampling Theorem 45Hz < signal bw 55Hz > signal bw 50Hz f sample = 100Hz = 2f signal f sample = 110Hz > 2f signal f sample = 90Hz < 2f signal

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Mechatronics Nyquist Criterion I 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.

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Mechatronics 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.

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Mechatronics PID Note: with only the proportional gain the control would oscillate around the set point The transfer function is: kpkp kDkD kIkI + Plant H + - Set point e feedback G

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Mechatronics 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.

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Mechatronics kDkD kpkp kIkI Matlab Commands: tf, bode, nyquist, rlocus, …

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Mechatronics PI PID with P and I dominant and k I >>k D Example: Root Locus: Matlab Commands: tf, bode, nyquist, rlocus, …

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

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Mechatronics PD PID control with P and D dominant and k D >>k I Example: Root Locus: Matlab Commands: tf, bode, nyquist, rlocus, …

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

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Mechatronics 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

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

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Mechatronics Notch Filter Characteristics: 2 nd order recursive digital filter it does remove a pair of poles not well dampened and it does substitute them with a pair adequately dampened b0b0 a2a2 a1a1 b1b1 b2b2 Z x y + These high (with respect to the denominator) make the system more reactive but oscillating with high k 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)

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Mechatronics z -2 b 2 z -1 b 1 b0b0 z -2 a 2 z -1 a 1 x y With these blocks its a IIR (Infinite Impulse Response) Without its a FIR (Finite Impulse Response, doent use its past outputs to determine actual output ) Basic Filtering Equation:

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Mechatronics Matlab Commands: tf, bode, nyquist, rlocus, …

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

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Mechatronics PID (closed loop) and FeedForward (open loop)

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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)

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Mechatronics Digital-Analog border Digital World

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Mechatronics Motion Profiles

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

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

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Mechatronics Torque step response

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Mechatronics Motion Profile Implementation

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Mechatronics Position Regulator

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Mechatronics Controller Architecture

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Mechatronics 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/Os –Fibre Optic Interface (EasyBus) –Brake Control Option –Safety Functions –Power Stage Service Interface –RS232 Connection to PC –Test Board Option Drive Architecture

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Mechatronics Drive Architecture (contd)

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Mechatronics Drive - Power Side - Architecture

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Mechatronics HW Layout

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Mechatronics Position Controller Power Stage Components Regulator Structure 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

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Mechatronics Parallel PID Regulator Description For the Proportional Part the Position Lag Value is multiplied with the P-Gain Parameter 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 For the Integral Part the Position Lag Value is integrated and then multiplied with the I-Gain Parameter P + PID Regulator - Position Feedback Torque Set-Point + Position Set-Point Position Lag + I D + 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

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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 External force time speed torque time speed torque + Static/Dry frictionViscous friction time speed torque ++ Inertia time speed torque Torque Feed- Forward = The Inertia Component is always proportional to the Acceleration Set-Point and compensates the mechanics Inertia during acceleration and deceleration (M = J *

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Mechatronics Measuring the External Force and Friction Values speed Ext.force = |Tp1| -|Tn1| 2 visc = |Tp10|+|Tn10| - |Tp1|+|Tn1| 22 S10 - S1 dry = |Tp1| + |Tn1| 2 - visc * S1 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 machines nominal speed (S10 = 100%) and measure the Torque as before (Tp10) viscous friction torque + - dry friction external force S10 Tp10 Tn10 Tn1 Do the same in negative rotating direction (Tn1 and Tn10)

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Mechatronics Parallel PID Regulator Details 1 0 AMPLI max min POS_REF POS_MES POS_LAGTORQ_REGTORQ_QUIET TORQ_REF R_MaxPosLag R_PGain R_MaxTorq BIT 0 StatusB AMPLI 1 0 VEL_REF VEL_MES VEL_LAG R_MaxVelLag BIT 0 StatusB R_DGain -F-F AMP LI ACC_REF FLOW TORQ_FLOW TORQ_FEEDF R_StatFriqTorq R_ViscFriqTorq R_ExtTorq R_Inertia +F+F AMPLI R_IGain dt

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Mechatronics Regulator Tuning Deactivate the PosLag Error Clear the Regulators 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 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 TuneLearn link

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Mechatronics Optimise the PD Regulator Step Response 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 Observe (trace) the Systems 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

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Mechatronics Speed Adjust the Feed-Forward Parameters Adjust Inertia Adjust StatFricTorq Adjust ViscFricTorq Acceleration proportional PosLag Speed proportional PosLag Sign of Speed proportional PosLag Properly adjusted Feed-Forward Parameters Perform positioning movements (relative_move), using the machines 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

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Mechatronics I-Gain Step Response (PosLag) for an over damped system Step Response (PosLag) for an under damped system Step Response (PosLag) for a critically damped system Decrease I-Gain Increase I-Gain Observe (trace) the Systems 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

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Mechatronics PD, no FF

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Mechatronics PID, no FF

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Mechatronics PD, with FF

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Mechatronics 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.

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