RST Digital Controls - indico.cern.ch · Kd Kp Td Ti Kp Ki s Td s Ti PID Kp Pole Placement for...

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


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

SUMMARY


RST Digital control: structure and calculation


RST calculation: control structure

n

n

n

n

n

n

ztztzttT

zszszssS

zrzrzrrR

...

...

...

2

2

1

10

2

2

1

10

2

2

1

10

S

RHfb

S

THff

yRrTuS

A combination of FFW and FBK actions

that can be tuned separately

)()()()()( 11111 zPzRzBzSzA des

REGULATION

RBSA

SA

d

y

RBSA

TB

r

y

TRACKING

)1(

)1(

)1(

)( 1

B

P

B

zP

Tdes

des


RST calculation: Diophantine Equation

3.0

1002

2 22

2

ss

Sample Ts=100us 2-1-

-2-1

z0.963 + z 1.959 - 1

z0.001924 + z 0.001949

-2-1 z0.963 + z 1.959 - 1 desP

Getting the desired polynomial

pxMzPzRzBzSzA des )()()()()( 11111

Calculating R and S: Diophantine Equation

nBnA

nBnA

ba

bbaa

ba

ba

ba

M

0000...0

00

01

00

0......00...01

22

11

22

11

nB+d nA

nA

+nB

+d

Matrix form:

0,...,0,,...,,1

,...,,,,...,,1

1

101

nP

T

nRnS

T

ppp

rrrssx


RST calculation: fixed polynomials

Controller TF is:

)(

)(1

1

zS

zRHfb

)(1

)(1*1

1

zSz

zR

RBSA

SA

d

y

Add integrator

Add 2 zeros more on S RBSA

SzzzA

*2

2

1

1

1 11

R ofpart fixed)(

S ofpart fixed)(

)()()()()()()(

1

1

1111111

zHr

zHs

zPzRzHrzBzSzHszA des


Integrator active on step reference and step disturbance.

Attenuation of a 300Hz disturbance


RST equivalent for PID controllers


RST equivalent of PID controller: continuous PID design

3.0

1002

2 22

2

ssA

B

Consider a II order system:

2

000

2

00

222223 22

1.1

11

sssKisKpsKds

HsyPIDDenHclPIDcHsyPID

HsyPIDHclPIDc

TdKpKdTi

KpKi

TdsTis

KpPID

Pole Placement for

continuous PID


RST equivalent of PID controller: continuous PID design

3.0

1002

2 22

2

ssA

B

Consider a II order system:

TdKpKdTi

KpKi

NTds

Tds

TisKpPIDf

1

11

2

000

2

3

00

2

2

000

22

121

Kd

Ki

Kp


RST equivalent of PID: s to z substitution

Choose R and S coeffs such that the 2 TF

are equal )(

)(

)( 1

2

2

1

10

2

2

1

10

1

1

zPIDd

zszss

zrzrr

zS

zR

21

21

1

1

1

1

210

210

1

1

1

11

zszss

zrzrr

S

R

zTdTsNTsNTd

zTdN

z

z

Ti

TsKpPIDd

= 0 forward Euler = 1 backward Euler = 0.5 Tustin

)()( 11 zRzT

)1()( 1 RzT

All control actions on error

Proportional on error;

Int+deriv on output


RST equivalent of PID: pole placement in z

Choose desired poles

8.0

1502

2 22

2

ss

Sample Ts=100us 2-1-

-2-1

z0.963 + z 1.959 - 1

z0.001924 + z 0.001949

-2-1 z0.963 + z 1.959 - 1 desP

1 1 1 HrzHs

Choose fixed parts for R and S

)()()()()()()( 11*111*11 zPzRzHrzBzSzHszA des



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


RS for regulation, T for Tracking


RS for regulation, T for tracking

9.0/9405.12 000

2

000

2

00 srsssPdes

PauxPdesRBSHsA

Hs

* :Equation eDiophantin

integrator ]11[

0.963z -2) + 1.959z -1 - (1 )0.6494z -1-(1 z -1)-(1

)0.9322z -2 + 1.929z -1 - (1 )0.9875z -1+(1 z -1 0.022626

SA

RBHol

After the D. Eq solved we get the following open loop TF:

=942r/s

=0.9

=1410r/s

=1 =1260r/s

=1

Pure delay

=31400r/s

=0.004

Closed Loop TF without T

0.844z -2) + 1.836z -1 - (1 )0.8682z -1-(1 )0.8819z -1-(1

)0.9875z -1+(1 z -1 0.0019487

RBSA

BHcl

)1(B

PauxPdesT

REGULATION

TRACKING

1

)0.9875z^-1+(1 z^-1 0.50314

RBSA

BTHcl

T polynomial compensate most of

the system dynamic


Systems with delays


Systems with delays

II order system with pure delay

ms

sr

ss

e

A

B s

3

3.0/1002

2 22

2

Sample Ts=1ms 2-1-

-5-4

z 0.6859 + z 1.368 - 1

z 0.149 + z 0.1692

Continuous time

PID is much

slower than before


Systems with delays

Predictive controls

Diophantine Equation

)()()()()()( 1111

0

11 zPzAzRzBzzSzA aux

d

1 1 1 HrzHs

Choose fixed parts for R and S


RST at work with POPS


RST at work with POPS

DC3

DC

DC +

-

DC1

DC

DC

DC5

DC

DC +

-

DC4

DC

DC-

+

DC2

DC

DC-

+

DC6

DC

DC

MAGNETS

+

-

+

-

CF11

CF12

CF1

CF21

CF22

CF2

AC

CC1

DC

MV7308

AC AC

CC2

DC

MV7308

AC

18KV AC

Scc=600MVA

OF1 OF2

RF1 RF2

TW2

Crwb2

TW1

Lw1

Crwb1

Lw2

MAGNETS

AC/DC converter - AFE

DC/DC converter - charger module

DC/DC converter - flying module

Vout Controller Imag or Bfield control

Ppk=60MW

Ipk=6kA

Vpk=10kV


RST at work with POPS: Vout Control

Lfilt

C fCdm p

Rdm p

Vin

Vout

Iind Im ag

Icf

Icdm

p

III order output filter

HF zero

responsible for

oscillations

2-1-

2-1-

-2-11msTs 2

22

2

z0.3897 + z 1.142 - 1

z0.3897 + z 1.142 - 1

z0.1043 + z0.1431

85.0

1502

2

des

dc

P

ss

Decide desired dynamics

)()()()()()( 111111 zPzPzRzBzSzA zerosdes

Solve Diophantine Equation

Calculate T to eliminate all dynamics

)1(

)()(

11

B

zPzT des

RBSA

TB

r

y

Eliminate process well dumped zeros

0.8053-zzerosP



Identification of output filter with a step

Well… not very nice performance….

There must be something odd !!!

Put this back in the RST

calculation sheet


Ref following ------

Dist rejection ------

Performance to date (identified with initial step

response):

Ref following: 130Hz

Disturbance rejection: 110Hz


In reality the response is a bit less nice… but still very good.


RST at work with POPS: Imag Control

Magnet transfer function for Imag:

32.0

96.01

mag

mag

magmagmag

mag

R

HL

RLsV

I

PS magnets deeply saturate:

26Gev without Sat compensation

The RST controller was badly oscillating at the flat

top because the gain of the system was changed


26Gev with Sat compensation

RST at work with POPS: Imag Control


RST at work with POPS: Bfield Control

Magnet transfer function for Bfield:

5.2

32.0

96.01

mag

mag

mag

mag

mag

mag

magmag

mag

K

R

HL

KL

RKs

V

B

Tsampl=3ms.

Ref following: 48Hz

Disturbance rejection: 27Hz

Error<0.4Gauss

Error<1Gauss


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 !


Thanks for the attention

Questions?


Towards more complex systems

(test it before !!!)


Unstable filter+magnet+delay

1.2487 (z+3.125) (z+0.2484)

--------------------------------------

z^3 (z-0.7261) (z^2 - 1.077z + 0.8282)

150Hz

Ts=1ms


Unstable filter+magnet+delay

Aux Poles for Robusteness lowered

the freq to about 50Hz @-3dB

(not Optimized)

Choose Pdes as 2nd order system 100Hz well dumped

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