RST Digital Controls
POPCA3 Desy Hamburg 20 to 23rd may 2012
Fulvio Boattini
CERN TE\EPC
Bibliography
• “Digital Control Systems”: Ioan D. Landau; Gianluca Zito
• “Computer Controlled Systems. Theory and Design”: Karl J. Astrom; Bjorn
Wittenmark
• “Advanced PID Control”: Karl J. Astrom; Tore Hagglund;
• “Elementi di automatica”: Paolo Bolzern
POPCA3 Desy Hamburg 20 to 23rd may 2012
SUMMARY
SUMMARY
•RST Digital control: structure and calculation
•RST equivalent for PID controllers
•RS for regulation, T for tracking
•Systems with delays
•RST at work with POPS
•Vout Controller
•Imag Controller
•Bfield Controller
•Conclusions
POPCA3 Desy Hamburg 20 to 23rd may 2012
RST Digital control: structure and calculation
POPCA3 Desy Hamburg 20 to 23rd may 2012
RST calculation: control structure
R  r0  r1  z 1  r2  z 2 ...  rn  z  n
S  s0  s1  z 1  s2  z  2 ...  sn  z  n
T  t0  t1  z 1  t 2  z  2 ...  t n  z  n
A combination of FFW and FBK actions
that can be tuned separately
S u  T  r  R  y
T
Hff 
S
R
Hfb 
S
REGULATION
y
A S

d A S  B  R
A( z 1 )  S ( z 1 )  B( z 1 )  R( z 1 )  Pdes ( z 1 )
TRACKING
POPCA3 Desy Hamburg 20 to 23rd may 2012
Pdes ( z 1 )
T  B (1)
Pdes (1)
B (1)
y
B T

r A S  B  R
RST calculation: Diophantine Equation
Getting the desired polynomial
2
s2  2    s   2
  2   100
  0.3
0.001949 z -1 + 0.001924  z -2
1 - 1.959 z -1 + 0.963  z -2
Sample Ts=100us
Pdes  1 - 1.959 z -1 + 0.963  z -2
Calculating R and S: Diophantine Equation
A( z 1 )  S ( z 1 )  B( z 1 )  R( z 1 )  Pdes ( z 1 )  M  x  p
Matrix form:
nB+d
xT  1, s1 ,..., snS , r0 , r1 ,..., rnR 
POPCA3 Desy Hamburg 20 to 23rd may 2012
0
b1
b2
bnB
0
0









0 bnB 
... ...
0
0
0
0
b1
b2
nA+nB+d
pT  1, p1 ,..., pnP ,0,...,0
 1 0 ... 0
a
0
 1
 a2
1

M 
a1
anA
a2

0
 0 ... 0 a
nA

nA
RST calculation: fixed polynomials
Controller TF is:
R( z 1 )
Hfb 
S ( z 1 )
Add integrator
R( z 1 )
1  z 1  S * ( z 1 )





Add 2 zeros more on S A  1  z 1  1  1  z 1   2  z 2  S *
y
A S

A S  B  R
d A S  B  R
Integrator active on step reference and step disturbance.
Attenuation of a 300Hz disturbance
Calculating R and S: Diophantine Equation
A( z 1 )  Hs ( z 1 )  S ( z 1 )  B( z 1 )  Hr ( z 1 )  R( z 1 )  Pdes ( z 1 )
Hs ( z 1 )  fixed part of S
Hr ( z 1 )  fixed part of R
POPCA3 Desy Hamburg 20 to 23rd may 2012
RST equivalent for PID controllers
POPCA3 Desy Hamburg 20 to 23rd may 2012
RST equivalent of PID controller: continuous PID design
Consider a II order system:
B
2
 2
A s  2    s   2
  2   100
  0.3
Pole Placement for
continuous PID
1


PID  Kp  1 
 s  Td 
 s  Ti

HclPIDc 

Kp
Ti
Kd  Kp  Td
Ki 
PID  Hsy
 HclPIDc .Den  1  PID  Hsy 
1  PID  Hsy




s 3  2      Kd   2  s 2  Kp   2   2  s  Ki   2  s   0  0   s 2  2   0  0  s  0
POPCA3 Desy Hamburg 20 to 23rd may 2012
2

RST equivalent of PID controller: continuous PID design
Consider a II order system:
B
2
 2
A s  2    s   2
  2   100
  0.3
02
Kp  1  2   0   0   2  1

03
Ki   0  2

  2   0   0  2    
Kd  0
2

1
s  Td 

PIDf  Kp  1 

s  Ti 1  s  Td 
N

POPCA3 Desy Hamburg 20 to 23rd may 2012
Kp
Ti
Kd  Kp  Td
Ki 
RST equivalent of PID: s to z substitution
All control actions on error
T ( z 1 )  R( z 1 )
Proportional on error;
Int+deriv on output
T ( z 1 )  R(1)
R( z 1 ) r0  r1  z 1  r2  z 2
1
Choose R and S coeffs such that the 2 TF


PIDd
(
z
)
are equal
S ( z 1 ) s0  s1  z 1  s2  z 2


 Ts   1     z 1
 R r 0  r1 z 1  r 2  z 2
N  Td  1  z 1
PIDd  Kp  1  

 
1
1 
Td  N  Ts     N  Ts  1     Td  z  S s0  s1 z 1  s2  z 2
1 z
 Ti
 = 0 forward Euler
 = 1 backward Euler
POPCA3 Desy Hamburg 20 to 23rd may 2012
 = 0.5 Tustin
RST equivalent of PID: pole placement in z
Choose desired poles
2
s2  2    s   2
  2   150
  0.8
Sample Ts=100us
0.001949 z -1 + 0.001924  z -2
1 - 1.959 z -1 + 0.963  z -2
Pdes  1 - 1.959 z -1 + 0.963  z -2
Choose fixed parts for R and S
Hs  1  z 1
Hr  1
Calculating R and S: Diophantine Equation
1
1
*
1
A( z )  Hs ( z )  S ( z )  B( z 1 )  Hr ( z 1 )  R* ( z 1 )  Pdes ( z 1 )
POPCA3 Desy Hamburg 20 to 23rd may 2012
RST equivalent of PID: pole placement in z
The 3 regulators behave very
similarly
Increasing Kd
Manual Tuning with Ki, Kd and
Kp is still possible
POPCA3 Desy Hamburg 20 to 23rd may 2012
RS for regulation, T for Tracking
POPCA3 Desy Hamburg 20 to 23rd may 2012
RS for regulation, T for tracking
REGULATION

Pdes  s   0  0  s 2  2   0  0  s  0
Diophantin e Equation : A  Hs  S *  B  R  Pdes  Paux
=942r/s
=0.9
=31400r/s
=0.004
Hol 
 0  1.5 0  940r / s  0  0.9
=1260r/s
=1
T
B  R 0.022626 z^-1 (1 + 0.9875z^-1) (1 - 1.929z^-1 + 0.9322z^-2)

A S
(1 - z^-1) (1 - 0.6494z^-1) (1 - 1.959z^-1 + 0.963z^-2)
Closed Loop TF without T
Hcl 
Pure delay
Hcl 

After the D. Eq solved we get the following open loop TF:
Hs  [1  1] integrator
TRACKING
2
=1410r/s
=1
Pdes  Paux
B(1)
T B
0.50314 z^-1 (1 + 0.9875z^-1)

A S  B  R
1
T polynomial compensate most of
the system dynamic
POPCA3 Desy Hamburg 20 to 23rd may 2012
B
0.0019487 z^-1 (1 + 0.9875z^-1)

A  S  B  R (1 - 0.8819z^-1) (1 - 0.8682z^-1) (1 - 1.836z^-1 + 0.844z^-2)
Systems with delays
POPCA3 Desy Hamburg 20 to 23rd may 2012
Systems with delays
II order system with pure delay
B
 2  e  s
 2
A s  2    s   2
  2   100r / s
  0.3
  3ms
Sample Ts=1ms
0.1692 z -4 + 0.149 z -5
1 - 1.368 z -1 + 0.6859 z -2
Continuous time
PID is much
slower than before
POPCA3 Desy Hamburg 20 to 23rd may 2012
Systems with delays
Predictive controls
Diophantine Equation
A( z 1 )  S ( z 1 )  z  d  B0 ( z 1 )  R( z 1 )  A( z 1 )  Paux ( z 1 )
Choose fixed parts for R and S
Hs  1  z
1
Hr  1
POPCA3 Desy Hamburg 20 to 23rd may 2012
RST at work with POPS
POPCA3 Desy Hamburg 20 to 23rd may 2012
RST at work with POPS
Vout Controller
Imag or Bfield control
AC/DC converter - AFE
DC/DC converter - charger module
DC/DC converter - flying module
Ppk=60MW
Ipk=6kA
Vpk=10kV
18KV AC
Scc=600MVA
MV7308
MV7308
DC
DC
AC
AC
AC
AC
CC1
CC2
MAGNETS
OF1
DC
OF2
+
CF1
DC
CF2
Lw2
-
DC
RF1
DC3
+
RF2
CF11
DC
CF21
DC4
Crwb1
DC
Crwb2
+
-
TW2
TW1
-
DC
DC
+
DC1
DC
DC2
Lw1
CF12
DC
CF22
-
+
DC
MAGNETS
DC5
POPCA3 Desy Hamburg 20 to 23rd may 2012
+
-
DC
DC
DC6
RST at work with POPS: Vout Control
Lfilt
Im ag
Iind
C dm p
Vout
Vin
Icdmp
Icf
Cf
III order output filter
R dm p
Decide desired dynamics
2
s2  2    s   2
  2   150 c 2 d Ts 1ms 0.1431 z -1 + 0.1043  z -2
 
  0.85
1 - 1.142  z -1 + 0.3897  z -2
Pdes  1 - 1.142  z -1 + 0.3897  z -2
Eliminate process well dumped zeros
Pzeros  z - 0.8053
y
B T

r A S  B  R
Solve Diophantine Equation
A( z 1 )  S ( z 1 )  B( z 1 )  R( z 1 )  Pdes ( z 1 )  Pzeros ( z 1 )
Calculate T to eliminate all dynamics
Pdes ( z 1 )
T (z ) 
B(1)
1
POPCA3 Desy Hamburg 20 to 23rd may 2012
HF zero
responsible for
oscillations
RST at work with POPS: Vout Control
Well… not very nice performance….
There must be something odd !!!
Identification of output filter with a step
Put this back in the RST
calculation sheet
POPCA3 Desy Hamburg 20 to 23rd may 2012
RST at work with POPS: Vout Control
In reality the response is a bit less nice… but still very good.
Performance to date (identified with initial step
response):
Ref following: 130Hz
Disturbance rejection: 110Hz
POPCA3 Desy Hamburg 20 to 23rd may 2012
Ref following -----Dist rejection ------
RST at work with POPS: Imag Control
PS magnets deeply saturate:
Magnet transfer function for Imag:
I mag
Vmag

1
s  Lmag  Rmag
Lmag  0.96 H
Rmag  0.32
The RST controller was badly oscillating at the flat
top because the gain of the system was changed
26Gev without Sat compensation
POPCA3 Desy Hamburg 20 to 23rd may 2012
RST at work with POPS: Imag Control
26Gev with Sat compensation
POPCA3 Desy Hamburg 20 to 23rd may 2012
RST at work with POPS: Bfield Control
Tsampl=3ms.
Ref following: 48Hz
Disturbance rejection: 27Hz
Magnet transfer function for Bfield:
Bmag
Vmag

s  K mag 
1
Rmag
Lmag
Lmag  0.96 H
Rmag  0.32
 K mag
K mag  2.5
Error<0.4Gauss
Error<1Gauss
POPCA3 Desy Hamburg 20 to 23rd may 2012
RST control: Conclusions
•RST structure can be used for “basic” PID controllers and
conserve the possibility to manual tune the performances
•It has a 2 DOF structure so that Tracking and Regulation
can be tuned independently
•It include “naturally” the possibility to control systems with
pure delays acting as a sort of predictor.
•When system to be controlled is complex, identification is
necessary to refine the performances (no manual tuning is
available).
•A lot more…. But time is over !
POPCA3 Desy Hamburg 20 to 23rd may 2012
Thanks for the attention
Questions?
POPCA3 Desy Hamburg 20 to 23rd may 2012
Towards more complex systems
(test it before !!!)
POPCA3 Desy Hamburg 20 to 23rd may 2012
Unstable filter+magnet+delay
150Hz
Ts=1ms
1.2487 (z+3.125) (z+0.2484)
-------------------------------------z^3 (z-0.7261) (z^2 - 1.077z + 0.8282)
POPCA3 Desy Hamburg 20 to 23rd may 2012
Unstable filter+magnet+delay
Choose Pdes as 2nd order system 100Hz well dumped
Aux Poles for Robusteness lowered
the freq to about 50Hz @-3dB
(not Optimized)
POPCA3 Desy Hamburg 20 to 23rd may 2012