RST Digital Controls POPCA3 Desy Hamburg 20 to 23 rd may 2012

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

nn

nn

nn

ztztzttT

zszszssS

zrzrzrrR

...

...

...

22

110

22

110

22

110

SRHfb

STHff

yRrTuS

A combination of FFW and FBK actions that can be tuned separately

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

REGULATIONRBSA

SAdy

RBSATB

ry

TRACKING

)1()1(

)1()( 1

BPBzP

Tdes

des


RST calculation: Diophantine Equation

3.01002

2 22

2

ssSample Ts=100us

2-1-

-2-1

z0.963 + z 1.959 - 1z0.001924 + 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

bbaaba

baba

M

0000...000

01000......00...01

22

11

22

11

nB+d nA

nA+nB

+d

Matrix form:

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

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

1

101

nPT

nRnST

ppp

rrrssx


RST calculation: fixed polynomials

Controller TF is:

)()(

1

1

zSzRHfb )(1

)(1*1

1

zSzzR

RBSASA

dy

Add integrator

Add 2 zeros more on S RBSA

SzzzA

*22

11

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

3.01002

2 22

2

ssAB

Consider a II order system:

2000

200

222223 22

1.1

11

sssKisKpsKds

HsyPIDDenHclPIDcHsyPID

HsyPIDHclPIDc

TdKpKdTiKpKiTds

TisKpPID

Pole Placement for continuous PID


RST equivalent of PID controller: continuous PID design

3.01002

2 22

2

ssAB

Consider a II order system:

TdKpKdTiKpKi

NTdsTds

TisKpPIDf

1

11

2

000

2

30

0

2

20

00

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

22

110

22

110

1

1

zPIDdzszsszrzrr

zSzR

21

21

1

1

1

1

210210

11

111

zszsszrzrr

SR

zTdTsNTsNTdzTdN

zz

TiTsKpPIDd

= 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

8.01502

2 22

2

ssSample Ts=100us

2-1-

-2-1

z0.963 + z 1.959 - 1z0.001924 + z 0.001949

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

1 1 1 HrzHsChoose 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

2000

200 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

SARBHol

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(BPauxPdesT

REGULATION

TRACKING

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

RBSABTHcl

T polynomial compensate most of the system dynamic


Systems with delays


Systems with delays

II order system with pure delay

ms

sr

sse

AB s

33.0

/1002

2 22

2

Sample Ts=1ms

2-1-

-5-4

z 0.6859 + z 1.368 - 1z 0.149 + z 0.1692

Continuous time PID is much slower than before


Systems with delaysPredictive controls

Diophantine Equation

)()()()()()( 11110

11 zPzAzRzBzzSzA auxd

1 1 1 HrzHsChoose 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 ACScc=600MVA

OF1 OF2

RF1 RF2

TW2Crwb2

TW1

Lw1

Crwb1

Lw2

MAGNETS

AC/DC converter - AFEDC/DC converter - charger moduleDC/DC converter - flying module

Vout Controller Imag or Bfield control

Ppk=60MWIpk=6kAVpk=10kV


RST at work with POPS: Vout ControlLfilt

C fCdm p

Rdm p

Vin

Vou

t

Iind Im ag

Icf

Icdmp

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 - 1z0.1043 + z0.1431

85.01502

2

des

dc

P

ss

Decide desired dynamics

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

Solve Diophantine Equation

Calculate T to eliminate all dynamics

)1()()(

11

BzPzT des

RBSATB

ry

Eliminate process well dumped zeros 0.8053-zzerosP


RST at work with POPS: Vout Control

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: 130HzDisturbance rejection: 110Hz

RST at work with POPS: Vout Control

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:

32.096.01

mag

mag

magmagmag

mag

RHL

RLsVI

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 ControlMagnet transfer function for Bfield:

5.2

32.096.0

1

mag

mag

mag

magmag

magmag

mag

mag

K

RHL

KLR

KsVB

Tsampl=3ms.Ref following: 48HzDisturbance 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

RST Digital Controls POPCA3 Desy Hamburg 20 to 23 rd may 2012

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