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Paul Scherrer Institut
Hans Jäckle
Basic powersupply control –
Digital implementation of analog controllers
PSI, 23. Mai 2017
Topics
Digital implementation of analog controllers
Part 1:
Basic Powersupply Controller
- Essential Tasks of a basic powersupply controller and how to achieve them
Part 2:
Discretization
- Paths to discrete controller
- Methods of discretization
- Effects of SamplingTime
Tasks of controller
• Achieve zero steady-state error
-> Integral action
PI Controller
• Ki = 1 / TMagnet
-> experimentally (step response in open loop)
-> Ki = RMagnet / LMagnet
• Kp can be found manually, starting value:
Kp_init = RMagnet / UDC_Link * ωCL / Ki
Tasks of controller
• Achieve zero steady-state error
-> Integral action
• keep operation of controller linear (avoid limitations)
-> Limit diRef / dt
-> Anti-Windup
dI/dt Limiter
• Keeps the controller in the linear regime by preventing actuator
saturation
U DC _ MIN  U CONV  I NOM * RMAGNET _ MAX
 dI 

 
LMAGNET
 dt  MAX
• Anti-Windup is a remedy in case of actuator saturation
Tasks of controller
• Achieve zero error
-> Integral action
• Stay in linear region
-> Limit diRef / dt
-> Anti-Windup
• Suppress dc-link voltage disturbances (e.g. 300Hz Ripple)
->feedforward the dc-link disturbances
DC-Link Feedforwards
DC-Link Feedforward
DC-Link Feedforward + Delay
Suppression of 300Hz ripple by a dc-link voltage feedforward
5
0
-5
gain [dB]
-10
-15
Example:
fPWM = 20kHz
-20
-30
PWM-Delay = 1/(4* fPWM)
Measurement delay
Control cycle
Total delay
-35
-> min. gain: -20dB
-25
= 12.5us
= 20us
= 10us
= 42.5us
-40
-45
0
100
200
300
Delay [us]
400
500
600
Tasks of controller
• Achieve zero error
-> Integral action
•Stay in linear region
-> Limit diRef / dt
-> Anti-Windup
• Suppress dc-link voltage disturbances
->feedforward the dc-link disturbances
Protect Output Filter Resistor
-> Output limiter
• Reduce measurement noise
-> Lowpass-filter for the measured value
• Reduce overshoot
-> Lowpass-filter for the reference value
Beyond PI
• Compensation of higher order plant dynamics e.g. the output filter
• 2-DOF structure to seperately tune reference tracking and disturbance rejection
K(s)
R(s)
G(s)
R(s)
K(s)
H(s)
K(s)
G(s)
G(s)
PI
Analog:
Paths to discrete controller
System to control
Sampled
Harmonic
response
Harmonic
response
Physical Model
Experimental
(Ziegler-Nichols,)
Analog controller
design
Sampling
Discretization
Digital controller
design
analog
digital
Source:
Commande numérique de systèmes dynamiques, R. Longchamp
Behavioural
Model
Types of discretizations
Area based approximations:
- Forward difference
- Backward difference
- Trapezoidal (Tustin, bilinear)
Response Invariant transforms:
- Step response (zero-order hold)
- Ramp response
- Impulse response
Pole-zero mapping
Area based approximations
Forward Euler:
Backward Euler:
Trapezoidal:
PI
Analog:
Discrete:
Approximations z <--> s
Source: Theorie der Regelungstechnik, Hugo Gassmann
Bode plot of integrators
Bode plot of discretized integrators
Bode plot of discretized integrators
10
10
analog
backward
forward
Magnitude (dB)
-10
-20
-10
-20
-30
-30
-40
45
-40
45
0
0
-45
-90
-135
-180
0
10
analog
tustin
0
Phase (deg)
Phase (deg)
Magnitude (dB)
0
-45
-90
-135
1
10
Frequency (rad/s)
2
10
-180 0
10
1
10
Frequency (rad/s)
2
10
How to choose sampling time
• too large
• too small
->
->
loss of information
loss of precision (numerical issues) / computational overload
• No absolute truth -> rules of thumb:
– 10x faster than Shannons sampling theorem
– Fs = 10 – 30x system bandwidth
– Loss of phase margin not more than 5°-15° compared to the continuous system
Summary
•
PI controller for magnet powersupplies is a good choice:
– can be manually tuned (only two parameters Kp & Ki)
– good static performance (zero error)
– reasonable dynamic performance
•
Fast enough sampling rate allows to regard the discrete PI controller as a (quasi-)continous controller.
•
High frequency behaviour not compensated (output filter resonance)
PSI, 23. Mai 2017
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