Internal Model Control - Ferdowsi University of Mashhadprofsite.um.ac.ir/~karimpor/multi/imc.pdf ·...

Outline

Part 1: Continuous SISO systems

Introduction

Control Objectives in the Presence of Uncertainty

Modeling Uncertainty

Nominal Stability & Performance

Robust Stability

Robust Performance

IMC Structure

Stability and Performance

Prefect Control

IMC Design Procedure

Stable systems

Example1: System with time delay uncertainty

2

Outline

IMC based PID

Introduction

General Relationships

Example 2: PID Design

Example 3: Model with high uncertainty

Part 2: Continuous MIMO systems

IMC Structure - MIMO Case

Internal stability for stable plants

General Internal stability

IMC Design Procedure - MIMO Case

Inner-Outer factorization

Nominal Performance

Robust Stability & Performance

Exercises 3


Introduction




Robust Stability

Robust Performance

IMC Structure


Prefect Control


Stable systems


4

Introduction

IMC is an effective method for designing and implementing

robust controllers.

IMC structure is an alternative to the classic feedback structure.

Its main advantages are:

Simple and easy to understand designing procedure

On-line tuning of IMC controllers are very convenient

It can easily control plants in presence of actuator

constraints.

use uncertainty information in design procedure.

5


Introduction




Robust Stability

Robust Performance

IMC Structure


Prefect Control


Stable systems


6

)()(

)()(~)(

wliwl

iwliwpiwp

aa

a

)()(

))(1)((~)(

wliwl

iwliwpiwp

mm

m

)()(~)(: wliwpiwpp a

Uncertainty usually increases with frequency.

)(wlm

Actual

Plant

Nominal

model 7

Control Objectives

Nominal Stability: The system is stabe with no model

uncertainty.

Nominal Performance: The system satisfies the performance

specifications with no model uncertainty.

Robust Stability: The system is stabe for all perturbed plants,

here means for all family plants.

Robust Performance: The system satisfies the performance

specifications for all family plants.

Controller Model

)(wla

8

Remember: Nominal Performance

nsTdsSrsTsy )()()()(

the sensitivity function S is a very good indicator of closed-loop

performance

1)()(,)(

1)(

jSjw

jwjS P

P

)(min : solivingby obtained is controller optimal KNHK

c r

d

n

y p~

9

Robust Stability

1~

sup~

ifonly and If

)(~)(~

:

familyfor stable is system Then the .p~plant nominal

thestabilizes c controller that Assume :1 Theorem

mw

m

m

lTlT

wlp

iwppp

dominate. totends

yuncertaint modelby imposed constraint eusually thbut

T~

of magnitudeon bound a impose to tendsalso Noise

.T~

on bound a imposesy uncertaint Model

Controller Model

)(wla

10

Robustness / Performance Tradeoff 1

~sup

~

mw

m lTlT

nsTdsSrsTsy )()()()(

design.clever by removed becannot

and controlfeedback in inherent ic problem This .robustnessfor

~ and eperformanc nominalfor

~ minimize want towe

mp lTwS

11

ROBUST PERFORMANCE

wwSlT

pjSjw

pm

P

1~~

ifonly and if

1)()(ion specificat eperformanc the

meet willsystem loop-closed then the,p~plant nominal

thestabilizes c controller that Assume :2 Theorem

pmwc

wSlT~~

supmin

solve which controller a find tois objectiveour practiceIn

It is a difficult problem to solve. For general MIMO case no

reliable solution are available

IMC easily provides a good approximation to the optimal

solution

Controller Model

)(wla

12


Introduction




Robust Stability

Robust Performance

IMC Structure


Prefect Control


Stable systems


13

IMC Structure

Controller Plant

Model

r

d y

Because in addition to the controller, It includes the plant model

explicitly this feedback configuration is called

internal model control (IMC)

e).disturbanc andy uncertaint (model

yuncertaint theexpresses d~ signalfeedback The

14

IMC vs. Classic Feedback

C

cp

cq

qp

qc ~1

,~1

So Why IMC has so much advantages over classic feedback?!!

15

Stabilizing Controllers qp

qc ~1

Assume plant is stable, so the only requirement for nominal stability is stability of q

dppq

qpr

ppq

pqy

)~(1

~1

)~(1

qpT

qpS

~~

~1~

cp

cpT ~1

~~

p~

16

Prefect Control

dppq

qpr

ppq

pqy

)~(1

~1

)~(1

But there are three reasons that make the prefect control impossible:

1) Nonminimum-phase(NMP) plants: q become noncasual or

unstable, and nominal plant become unstable.

2) Strictly proper plants: q become improper, but this can be solved

by adding some far poles.

3) Model Uncertainty:

1)~

1(1~

mm lSlT

1~

0 mlS

)(wlm

.controlprefect

achievecan weso0,S and 1T then p~

1qput weIf

17


Introduction




Robust Stability

Robust Performance

IMC Structure


Prefect Control


Stable systems


18

IMC Design For Stable Systems

pmwc

wSlT~~

supmin :Objective Final

2~2~2~)~~1(min

~minmin

yuncertaint model and sconstraintfor

regard without eperformanc good a yield toselected is ~

wqpwSe

q

qqq

IMC design procedure consists of two steps:

Step 1: Nominal Performance

Step 2: Robust Stability and Performance

19

Step 1:Nominal Performance

omitted. are ~ of poles theinvolving termsall operand the

of expansionfraction partial aafter that denotes .operator theWhere

~)p~(q~

:isequation above solves which q~ controller The, 1)(p~

and p~ of delays and zeros RHP theall includes p~ that so

.~~~ ,p~portion

MP a and p~portion allpassan into input w and p~factor

,stable is p~ assume :control) optimal (H Theorem3

1

*11

M

A

A

AM

A

2

A

mAm

M

p

wpw

wiw

ppp

20

1~

sup~

m

wm lTlT

2~

~min wS

q

proper.not in q~ controller optimal thegeneralIn

.1

1

2

611

1

1

2

2

1~

1,

2

2~~ :Example

s

s

sss

s

s

s

s

s

s

sq

s

sv

s

spp mA

21

Step2: Robust Stability

and Performance

qpT

qpS

~~

~1~

fqq ~

cp

cpT ~1

~~

k.design tas hesimplify t to

ffilter pass-low aby augmented is q~stability robust For

Low-pass filter can improve robustness.

What about performance?

22

intuitive. andeasy so tuningline-on themakes IMC

.performace thedecreaseit but robustness increase

will Increasing ,parametersfilter adjustable theis

)1(

1)( 0lim and 1)0( :2 Type

)1(

1)( 1)0( :Type1

0

ns

n

s

snsf

ds

dff

ssff

.)1(

1)1...( ,parametersfilter of

number small aover search and structurefilter fix the we

11

1 n

mm

sssf

Here n is selected large enough for q to be proper.To have

zero steady state error, the condition on f to be satisfied are:

23

Example 1: System with time delay uncertainty

15.0

2ep~ :Model,

15.0

2ep~ :Plant

s2-s-

ss

)15.0(5.0~~,~

:ePerformanc Nominal :Step1

12 spqep ms

A

5.0

1)( :so 5.0 , 2

iw

m eiwl

1

1

:ePerformanc andstability Robust :Step2

sf

Input: Step

Uncertainty: Time delay uncertainty

22)(

21)( 5.0

wwl

wewl

m

iwm

24

10-1

100

101

102

0

0.5

1

1.5

2

w

lm

10-1

100

101

102

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Tlm

10-1

100

101

102

0

0.2

0.4

0.6

0.8

1~

sup~

m

wm lTlT

6.0

2.0

Example 1

25

15.0

2ep : isplant real assume

s45.2-

s

Step Response

Time (sec)

Am

plit

ude

0 10 20 30 40 50 60-30

-20

-10

0

10

20

30

Step Responselambda 0.2

Step Response

Time (sec)

Am

plit

ude

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

1.4Step Responce, Lambda 0.5

ses

s

qp

qc

216.0

)15.0(5.0~1

How to implement?!!!!

2.0 6.0

Example 1

26

IMC based PID

Introduction










Nominal Performance


Exercises

27

By far the most widely used controllers in the industry

processes are the two-term PI and the three-term PID

controller.

Because IMC is clearly more general and therefore more

powerful it is worthwhile to explore the relationships between

IMC, PI and PID in order to gain insight into the tuning of

these simple controllers, their performance and robustness.

Introduction

28

Remember: General relationship between the classic feedback

controller c and the IMC controller q are:

The order of q obtained in this way is generally higher than the

order of the plant model. So The complexity of the equivalent

classic controller c is determined by the complexity of the

model.

General relationships

,1 1

q cc q

pq pc

Simple models simple controllers

m odel. theof com plexity by the de term ined is c controlle r c lassic equiva lent theof com plexity The So . m odelplant theoforder n thehigher thagenera lly is w ay in th is obta ined q oforder The

29

In the previous example , our model was as fellows:

By the use of “Pade” approximation our model is obtained as

follows:

Step 1: Nominal Performance


15.0

2~2

s

ep

s

)105(.

2.

1

1~

ss

sp

Ap~Mp~

*

11 ~)~(~mAmM wpwpq

30

1

*

2 1 1 1 0.5 1. .

.05 1 1 2

s sq

s s s s

Step 2: Robust stability and performance:

So:

0.5 1 1.

2 1

sq

s

0.25 0.5

( 1)(0.25 0.5)11 0.25 0.51 ( 2 )

1 .(.05 1)( 1) 1

s

q s sscs spq s s

s s s

q qf

Example 2

31

Structure of the PID controller is defined as follows:

1( ) ( ).

1

IP D

F

kK s k k s

s s

The controller is so simple and come to the form of PID controller:

)12

()2(

5.075.025.0 2

ss

ssc

Example 2

By equating , PID parameters is obtained from the above

IMC controller. As we see, all of the parameters are just depend

on .

K c

0.75 0.5 0.25, , ,

2 2 2 2P I D Fk k k

32

10

1

Example 2

0.33

2.0

achieved. is eperformanc and robustnessBoth

plotted. is of valuesdifferent for response Step

33

The IMC controllers shown in next slide Table, is designed via

the standard procedure developed in the previous example.

IMC leads to PID controllers for virtually all models common

in industrial practice. Note that the table includes systems with

pure integrators and RHP zeros. Occasionally, the PID

controllers are augmented by a first-order lag.

IMC based PID for different plants

2

2

21 1(1 ) ,

1c D

I F

c k ss s

34


35


36

IMC based PID

Introduction










Nominal Performance


Exercises

37

• Checking the uncertainty

-2.45s2eAcutal plant :

0.5 1

1Plant model :

( 5)

ps

ps s

Nyquist Diagram for p Nyquist Diagram for p


38

Step 1 (Nominal Performance):

Step 2 (Robust Stability & Performance):

1 1M

* q (p ) ( 5)m A mw p w s s

2 2

1 ( 5)f = , q = qf =

( 1) ( 1)

s s

s s

2

( 5)

1 ( 1) 1

q s sc

pq s

Example3

39

2.5 3.3

5 10

Example 3

achieve.not did eperformanc goody,uncertainthigh of Because

plotted. were of valuesdifferent for response step figures, in these seeyou As

achieve.not did eperformanc good y,uncertainthigh of Because plotted.

were of valuesdifferent for response step figures, in these seeyou As

40

IMC based PID

Introduction










Nominal Performance


Exercises

41

In MIMO case, we deals with transfer function matrices.

IMC Structure- MIMO Case

1

1

( )

( )

C Q I PQ

Q C I PC

Q

Controller Plant

Model

r d

y

d

42

Sensitivity function and Complementary sensitivity function

If the model is perfect ( )

IMC Structure- MIMO Case

1

1

( )( ( ) )

( ( ) )

S I PQ I P P Q

T PQ I P P Q

y Tr Sd Q

P P

QPTQPS~~

,~

1~

43

Theorem: Assume that the model is perfect ( ),

Then the IMC system is Internally stable IFF both the plant P

and controller Q are stable.

Internal Stability for stable plants

Q

P P

44

Remember: All elements of the below matrix have to be stable

In IMC Structure for

By substituting

We have

1( )C Q I PQ

P P

1 1

1 1

( ) ( )

( ) ( )

PC I PC I PC P

C I PC C I PC P

( )PQ I PQ P

Q QP

Q

General Internal Stability

45

This implies that Q has to be stable and that in the elements of

above matrix the factor Q and I-PQ have to cancel any unstable

poles of P. Thus both Q and I-PQsmust have RHP zeros at the

plant RHP poles. Special care has to be taken to cancel these

common RHP zeros when the controller is

constructed. Minimal or balanced realization software can be

used to accomplish that.

General Internal Stability

( )PQ I PQ P

Q QP

1( )C Q I PQ

46

IMC based PID

Introduction










Nominal Performance


Exercises

47

Theorem: let be a minimal realization of

the square transfer matrix G(s) , and let G(s) have no zeros on the

iw-axis including infinity. Then we have

where N,M are stable and , and

1( ) ( )G s C sI A B D

1( ) ( ) ( )G s N s M s

( ) ( )HN i N i I

1 1 1

1 1 1

( ) ( )( ( ))

( ) ( ) ( )T T

N s C QF sI A BR F BR Q

M s F sI A B R where F Q C BR X


with X the stabilizing [i.e., it makes stable] real symmetric solution

of the following algebraic Riccati equation (ARE):

1( )A BR F

1 1 1 1

1

( ) ( ) ( )( ) 0

:

T T T T

A M

A BR Q C X X A BR Q C X BR BR X

So that

P N and P M

48

IMC design procedure consists of two steps

Step1: Nominal Performance

is selected to yield a good performance for inputs, without regard for

constraint and model uncertainty.

Step2: Robust Stability & Performance

The obtained in step 1 is detuned to satisfy the robustness requirements.

For that purpose, is augmented by a filter F of fixed structure.

IMC Design – MIMO case

Q

Q

Q

( , ) . :Q Q Q F i e Q QF

49

o The plant P can be factored as

is stable allpass portion

is MP portion

The procedure for carrying out this factorization is:

“Inner – Outer Factorization”


A MP P P

AP

MP

AM

i

n

VVV

sv

svsvsvsV

:Similarly

input.i ofcontent frequency

anddirection expected describesat vector tha is )(

))(,),(),(()(

} vinputsn ofV(s){set matrix square theDefine

th

21

i

50

To achieve nominal performance, the controller is:


1 1 1 1

*M A M MQ P W W P V V

om itted. a re SSS of poles theinvolving te rm sall operand, theofexpansion frac tion partia l aafte r tha t denotes opera tor the W here

omitted. are

of poles theinvolving termsall operand, theofexpansion

fraction partial aafter that denotes .operator theWhere

1

*

AP

51

The controller is to be detuned through a lowpass filter

F , to satisfy robust stability and performance. So the tuned

controller is:

The filter F(s) is chosen to be a diagonal rational function that

satisfy:

o The controller must be proper

o Internal stability

o Asymptotic setpoint tracking/disturbance rejection.


Q

Q QF

1( ) ( ) , , ( )nF s diag f s f s

Q QF

52

Experience has shown that the following structure is

reasonable:

Where

v is pole-zero excess.

K is the number of open RHP poles of

is the largest multiplicity of such pole in any element of the lth row

of V.


1

1

11, 1, 0,

1( )

( 1) l

vv l l l

l v v

a s a s af s

s

0l lv m k

P

0lm

1( ) ( ) , , ( )nF s diag f s f s

53

The numerator coefficients can be computed from solving a

system of linear equations with unknowns.


1

1

11, 1, 0,

1( )

( 1) l

vv l l l

l v v

a s a s af s

s

lv lv

0

0

0

( ) 1, 0,1, ,

( ) 0, 1, , 1

:

( 1,..., )

0

l i

j

l lj

s

i

f i k

df s j m

ds

where

i k are open RHP poles of P

54

IMC based PID

Introduction










Nominal Performance


Exercises

55

Exercise 1: Robust controller design

why??robustness and eperformanc

locatoin, polefilter between iprelationsh theis what 3.

timerise find and response stepplot shoot.over 10% achieve order toin find 2.

system? thestabilize that range )(parameter filter 1.find

21

32:Plant

:isplant real that theAssume

model.plant on based controller IMCan Design

Step :Input

21

32~ : model

6.4

4

ss

sep

ss

sep

s

s

56

Exercise 2: Poor modeling impact

tion?identifica systembetter of advantages theare what results, thecompare

and modelsbetter propose 1, Exercise with results compare 3.

time.rise find and response stepplot

shoot.over 10% achieve order toin find 2.

system. thestabilize that rangeparameter filter find 1.

1)1)(0.3s(0.2s

1.5-p~ : plant with thismodeled have that weAssume

step :Input

21

32: isplant real that theAssume

6.4

ss

sep

s

57

Exercise 3: PID design

why?unstable? system themakes any does

10for response stepplot

plant, nominalon controller PID theInstall -2

. of in terms parameters

r compensato lag and PID find then c, q ,q~ findfirst -1

Step :Input

)12.0)(14.0(

2~ :ModelPlant

: IMC through controller PID aDesign

ssp

58

[1] C. E. Garcia and M. Morari, "Internal model control. A unifying review and some new results,"

Industrial & Engineering Chemistry Process Design and Development, vol. 21, pp. 308-323,

1982.

[2] C. E. Garcia and M. Morari, "Internal model control. 2. Design procedure for multivariable

systems," Industrial & Engineering Chemistry Process Design and Development, vol. 24, pp.

472-484, 1985.

[3] C. E. Garcia and M. Morari, "Internal model control. 3. Multivariable control law computation

and tuning guidelines," Industrial & Engineering Chemistry Process Design and Development,

vol. 24, pp. 484-494, 1985.

[4] M. Morari and E. Zafiriou, Robust Process Control. New Jersey: Prentice-Hall, Inc., 1989.

[5] A. Porwal and V. Vyas, "Internal model control (IMC) and IMC based PID controller,"

Bachelor of Technology, Department of Electronics & Communication Engineering, National

Institute of Technology, Rourkela, 2010.

[6] D. E. Rivera, Internal Model Control: A Comprehensive View. Tempe, Arizona: Arizona State

University, 1999.

[7] D. E. Rivera, et al., "Internal model control: PID controller design," Industrial & Engineering

Chemistry Process Design and Development, vol. 25, pp. 252-265, 1986.

References

59

Thank You For

Your Attention

60

Internal Model Control - Ferdowsi University of Mashhadprofsite.um.ac.ir/~karimpor/multi/imc.pdf ·...

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