K.N.Toosi University of Technology 1

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K.N.Toosi University of Technology 2

The Interconnection of two subsystems

K.N.Toosi University of Technology 3

The Control Problems

IdentificationInstability RHP-zero

Time delay

K.N.Toosi University of Technology 4

MIMO Flow-Level

Plant

IdentificationLimitations in

designing

Design the Robust

Controller

�Decentralized

ControllerConclusionH

H

K.N.Toosi University of Technology 5

The flow System

The flow valve has integrator property( 1)

sLKeG

s Ts

Valve position Control Signal

K.N.Toosi University of Technology 6

Static Identification and The Operating points of system

§ � �

10

15

20

25

30

35

40

t � � � � § t � � § t � � § t � � § t � � § t � � § t � � § t � �

53

85

9/5

105

4/6

140

9/6

190

12/8

241

9.4

274

8/9

297

3/10

56

92

15

113

4/15

154

9/16

212

3/18

272

8/19

307

3/21

333

3/22

59

100

6/25

126

5/26

172

3/28

235

6/30

302

5/33

343

9/34

372

1/36

62

113

8/37

144

39

191

9/41

263

6/45

335

50

390

153

418

5/54

65

124

3/51

158

6/52

216

56

288

-

372

-

434

-

464

-

K.N.Toosi University of Technology 7

The Identification of a MIMO system should be achieved in MISO form If the RGA matrix become

If change to The system will

be singular

11 12

21 22

( )G

1(1 )p ij ij

ij

g g

ijg

1u

2u 1y 1u

2u 2y1Model 2Model

K.N.Toosi University of Technology 8

The most important problems for Identification The excitation signals

should be Uncorrelated Should be distinguished from output noise Should not have the frequencies much more than actuator bandwidths Should have a big range of frequency Are Chirp in our experiments

The sample time is selected by these rules ; ; or to

min / 3T T 010f f / 20sT T

/100sT T

0 100 200 300 400 500-20

0

20

40

60

Step Response

Time (sec)

Am

plit

ude

0 100 200 300 400 500-20

0

20

40

Step Response

Time (sec)

Am

plit

ude

0 100 200 300 400 5000

10

20

30

Step Response

Time (sec)

Am

plit

ude

0 100 200 300 400 500-1

0

1

2

3Step Response

Time (sec)

Am

plit

ude

Real System

model

Real System

Model

Real Syatem

Model

Real System

model

K.N.Toosi University of Technology 9

11 12

21 22

4.1 4.5 , 5 6.8

5.5 6.6 , 4.3 5.3

11 12

21 22

3.23 14.81 , 4.48 15.54

0.17 0.82 , 5.4 7.28

K K

K K

Input and output :

The transfer function

The effect of input flow on output level is small

The Disturbance model

This model has no delay

11 12

21 22

G GFlowout Flowinput

G GLevel out Level input

50 100 150 200 250 300 350 400 450 50010

15

20

25

30

35

Time

Leve

l(Cm

)

Disturbance Step Response

0

4.4

(63 1)dg

s

11 1211 12

2

21 2221 22

( 2 1)( 2.57 1)

( 42 1) (72 1)

k ks s

ss

k ks s

s s

e e

e e

K.N.Toosi University of Technology 10

Nominal model

,

, , , ,0 , min , max , min , , max( ) ( ) ,i j

i j i j i j i j

s

p p i j p i j i j i j i jG s k e G s k k k

, , 0 , ( )i jn i j i jG k G s

,

, min , max , max , min,

,

( ) / 2,

2 i j

i j i j i j i ji j k

i j

k k k kk r

k

,

,

,

, max

, max

(1 )2( )

( 1)2

i j

i j

i j

K

i j k

Ti j

rs r

W s

s

10-4

10-3

10-2

10-1

100

101

102

103

104

10-2

10-1

100

101

WT

11 12

21 22

n n

n

n n

g gG

g g

• Nominal Model for MIMO system

• The system with parametric Uncertainty :

• Converting to unstructured uncertainty

K.N.Toosi University of Technology 11

Scaling

The scaling transfer function of plant and disturbance

The requirement for performance is |e(ω)| ≤1 for |u(ω)| ≤1, |r(ω)| ≤1 and |d(ω)| ≤1

The max value of control signal 20 in flow Ch, and 30 in level Ch.

1 1ˆ ˆ,e u d e d dG D G D G D G D

K.N.Toosi University of Technology 12

MIMO Flow-Level

Plant

IdentificationLimitations in

designing

Design the Robust

Controller

�Decentralized

ControllerConclusionH

H

K.N.Toosi University of Technology 13

Decentralized Control Based on Passive Approaches

Pairing Avoid the Pairing correspond to negative RGA elements The pairing is selected with RGA closed to unity

Design controller for each subsystemThere is no specific approach for this problem

Good phase and gain margin for each subsystem of nominal model PI controller for each channel

1.0949 -0.0949

-0.0949 1.0949 RGA

K.N.Toosi University of Technology 14

Stability Of Decentralized Control

A system with decentralized controller is stable if Each subsystem is stable√ Following inequality holds

Where and is the complementary sensitivity function of

If The system is closed two triangular this condition is always hold √

( ( )) 1 ,ET j

1( )E G G G

GT

K.N.Toosi University of Technology 15

Performance Of Decentralized Control

Desired performance by disturbance input ×

is the element of CLDG Matrix:

,

Desired performance by reference input

Where is the element of PRGA matrix ×

10-2

100

102

10-6

10-5

10-4

10-3

10-2

10-1

100

101

Frequency10

-210

010

210

-8

10-6

10-4

10-2

100

102

104

Frequency

1 i diL g i

1 .i ij jL R

d dG G 1G G

dig

ij

100

10-3

10-2

10-1

100

101

Frequency(rad/sec)10

010

-3

10-2

10-1

100

101

102

103

Frequency(rad/sec)

Tracking Band

|1+L2|

Tracking Band

|1+L1|

FlowLevel

K.N.Toosi University of Technology 16

Step Response Of Decentralized Controller

Settling time after 300 (sec) in both channels × The control signal is in its range √ The inverse response of non-minimum phase zero √ Overshoot is about 50% in flow ch. × And less than 5% in level ch. √

0 100 200 300 400 500 600 700220

240

260

280

300

320

340

360

380Out flow

Time(sec)

Real sys

Model Sys

0 100 200 300 400 500 600 70034

36

38

40

42

44

46

Out level

Time(sec)

Real Sys

Model Sys

200 400 600-1

0

1

2

3

4

5

6

7Cont sig Flow

Time(sec)

0 200 400 6001

2

3

4

5

6

7Cont sig Level

Time(sec)

Model Sys

Real Sys

Model Sys

Real Sys

step response Control Signal

K.N.Toosi University of Technology 17

Disturbance Rejection of Decentralized Controller

Disturbance rejection after 170 (sec) in flow Ch. √ and 400 in level ch. × The control signal is in its range. √ The maximum peak of response about 25(lit/h) in flow ch. × and about

6.4(cm) in level ch. √

0 200 400 600210

215

220

225

230

235

240

245Out flow Dist Resp.

Time(sec)

0 200 400 60035

36

37

38

39

40

41

42

Time(sec)

Out level Disturbance Resp.

Real Sys

Model Sys

Real Sys

Model Sys

100 200 300 400 500 600 700-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4Cont. sig. Flow

Time(sec)

0 200 400 600-7

-6

-5

-4

-3

-2

-1

0Cont.sig. Level

Time(sec)

Real Sys

Model Sys

Real Sys

Model Sys

Disturbance response Control Signal

K.N.Toosi University of Technology 18

MIMO Flow-Level

Plant

IdentificationLimitations in

designing

Design the Robust

Controller

�Decentralized

ControllerConclusionH

H

K.N.Toosi University of Technology 19

Uncertainty and Robustness

Uncertainty in Input or in output The uncertainty is much more in MIMO systems Uncertainty can be parametric or unstructured

Input Uncertainty Output Uncertainty

K.N.Toosi University of Technology 20

Nominal Model and Multiplicative uncertainty

1TW 11TW

11 12

21 22

1 1 1 1

2 2 2 2

(1 ) 0 (1 ) 0

0 (1 ) 0 (1 )( )

n nT T

p n

T Tn n

T

g gW WG G

W Wg gI W G

2

9.02 10.01

(2.57 1) (2 1)

0.495 6.34

(42 1) (72 1)

n

s sG

s s

Z=1.7363

Output Multiplicative Model:

is the upper band of and 12TW

K.N.Toosi University of Technology 21

The model Validation

In MIMO systems the gain of system varies between minimum and maximum singular value

The gain of system also can vary because of uncertainty

10-3

10-2

10-1

100

101

102

103

-120

-100

-80

-60

-40

-20

0

20

40

Singular Values

Frequency (rad/sec)

Sin

gula

r V

alu

es (

dB

)

Sigular Values of Model in different Working points

Singular Values of Multiplicative Model

Sigular Values of Multiplicative Model

Nominal Model

K.N.Toosi University of Technology 22

Limitations on Performance

In MIMO systems each element has its own direction such as RHP-zero, RHP-Pole, disturbance, uncertainty and etc.

Effect of zero in MIMO system

There is no effect with input

0.9997 0.0224 2.2285 0 -0.1319 -0.9913( )

0.0224 0.9997 0 0 -0.9913 0.1319G z

2HV

2( ) 0G z v

[1 0]Tu

2

9.02 0.495[ ](2.57 1) (42 1)

Tys s

K.N.Toosi University of Technology 23

Waterbed Effect Theorem

In SISO systems

In MIMO systems

No special conclusion for MIMO system

0

ln( ( ) ) 0S j d

10 0

ln( det ( ) ) ln ( ( ) . Re( )pN

j ij i

S j d S j d p

K.N.Toosi University of Technology 24

Limitation on S and T

S+T=I

We can not decrease both S and T Simultaneously S is large if and only if T is large

|1 ( ) | ( ) |1 ( ) |S T S

|1 ( ) | ( ) |1 ( ) |T S T

K.N.Toosi University of Technology 25

Interpolation Condition

for internal stability for the plant with no RHP-Pole and one RHP-zero

This condition restricts the performance weight

( ) 0; ( )H H H

z z zy T z y S z y

( )p p

W S w z

K.N.Toosi University of Technology 26

Limitations By Time Delay

In SISO systems delay will limit the bandwidth

In MIMO systems, every output has at least the minimum delay of the elements in its own row

Delay may improve performance of MIMO systems

max

1c

K.N.Toosi University of Technology 27

By using Maximum Modules Theorem

For tight control at low frequencies

For tight control at high frequencies

Limitation By RHP zero

max ( ) ( ( )) ( )p p pW S W j S j W z

* / 2B z

* 2B z

K.N.Toosi University of Technology 28

Maximum Peak Criterion

The growing on maximum peak of sensitivity function cause to big overshoot

M more than 2 is not suitable The phase margin of system

will decrease

K.N.Toosi University of Technology 29

Limitations Caused By Uncertainty

In each channel for robust performance it is required

min{ , } 1s TW W

10-2

100

102

0.5

1

1.5

2

2.5

3

dB

Ws &WT

10-2

100

102

0.5

1

1.5

2

2.5

3

3.5Ws & WT

1s TW S W T

K.N.Toosi University of Technology 30

Limitations Caused By Disturbance

Condition number of plant and disturbance

A single disturbance with one RHP-zero

is the output direction of RHP-zero

Another condition

Hzy

( ) 1Hz dy g z

†( ) ( ) ( )d dG G G y

( ) 1H

i i dG u g

2

1d d

d

y gg

FlowLevel

K.N.Toosi University of Technology 31

The Selection of Performance Weight

Performance weight in each channel

The selection of bandwidth Restricting the overshoot

( ) Bp

B

s Ms

s A

2 ( )

iA A

1 1

2 2

11 12

21 22

p p

pp p

W s W sW S

W s W s

1 1 1 111 12 11 12 1( )p p p p piW S W s W s W s s W S

1 11 12( )S s s

K.N.Toosi University of Technology 32

The selection of Control Signal Weight

Performance Weight

0 50 100 150 200 250 300 35051

51.5

52

52.5

53

53.5

54

54.5

55

uM

1/ uW

( )KS j

bc1

1

1

/i i

i

i i

bc uu

bc

s MW

s

K.N.Toosi University of Technology 33

MIMO Flow-Level

Plant

IdentificationLimitations in

designing

Design the Robust

Controller

�Decentralized

ControllerConclusionH

H

K.N.Toosi University of Technology 34

Robust Control Design

Integrator in each element of controller The order of controller is equal to the states of generalized

plant order The controller is obtained by iterative algorithm

111 12 22 21( ) ( , )lN P P K I P K P F P K

K.N.Toosi University of Technology 35

The Generalized Plant in Our Problem

The infinity norm by considering disturbance in SISO system

Without consideration of disturbance

2 2 2 2*max( ) ( ) ( ) ( 2)P T u dN N N W S W T W U G

2 2 2*max( ) ( ) 2( )P T uN N N W S W T W U

K.N.Toosi University of Technology 36

The Step Responses

Settling time 150 (sec) in both channels without considering disturbance √ and 200 in second one. √

The control signal is in its range √ The inverse response of non-minimum phase zero √ The Overshoot is about 15% in flow ch. √ and 5% in level ch. √ without

considering disturbance and more than 40% for the controller by considering disturbance ×

0 100 200 300 400180

200

220

240

260

280

300

320

340

360

Time0 100 200 300 400

42

44

46

48

50

52

54

56

58

Time

Hinf Contr

Model sysDisturb Contr

Hinf Contr

Model sysDisturb Contr

0 100 200 300 400-10

-5

0

5

10

15

Time(sec)

0 100 200 300 400-2

0

2

4

6

8

10

12

Time(sec)

Hinf Contr.

Model sysDisturb contr.

Hinf Contr.

Model sysDisturb contr.

Hinf Contr.

Model sysDisturb contr

Step responseControl signal

K.N.Toosi University of Technology 37

Disturbance Responses

Disturbance rejection after 170 (sec) for both controllers in flow ch.×

Disturbance rejection after 200 (sec) in controller with disturbance model and 300(sec) in another controller in level ch.×

The control signal is in its range. √

The maximum peak of response 20(lit/h) in flow ch. with disturbance model× and 25(lit/h) for another controller √

The maximum peak of response 5.7(cm) √ and about 7(cm) for another controller in level ch.. √

0 100 200 300 400 500225

230

235

240

245

250

255

260

265

270

Time(sec)

0 100 200 300 400 50036

37

38

39

40

41

42

43

44

Time(sec)

Disturbance Rejection Response

Hinf Contr.

Model of Distr.Dist Contr.

Hinf Contr.Model of Distr.Dist Contr.

Hinf Contr.

Model of Distr.Disturbance Contr.

0 100 200 300 400 5000

1

2

3

4

5

6

7

Time(sec)

0 100 200 300 400 500-8

-7

-6

-5

-4

-3

-2

-1

0

1 Disturbance Rejection Control Signal

Time(sec)

Hinf contr

Dist contr

Hinf contr

Dist contr

K.N.Toosi University of Technology 38

Robust StabilityRobustDesign

RobustStability

NominalPerformance

Robust Performance

10-3

10-2

10-1

100

101

102

103

10-10

10-8

10-6

10-4

10-2

100

Frequency(rad/sec)

Magnitude

Decent Contr.

Distr ContHinf contr.

( ( )) 1M j

1TW TSufficient Condition

Necessary and Sufficient Condition

K.N.Toosi University of Technology 39

Nominal Performance

RobustDesign

RobustStability

NominalPerformance

Robust Performance

10-3

10-2

10-1

100

101

102

103

10-12

10-10

10-8

10-6

10-4

10-2

100

Frequency(rad/sec)

Magnitude

Nominal Performance

Decent Contr

Hinf Contr.Distr Contr.

1pW S

K.N.Toosi University of Technology 40

Robust Performance

RobustDesign

RobustStability

NominalPerformance

Robust Performance

ˆ( , ) 1, 1 ( ( ) ) 1,uF F N N j

0ˆ0 p

K.N.Toosi University of Technology 41

µ-Analysis for Robust Performance

Max Peak of µ: 1.2544 without

disturbance model 1.7257 in second

design Too large for

Decentralized controller

10-2

100

102

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Frequency(rad/sec)

10-2

100

102

0

50

100

150

200

250

300

350

400

Frequency(rad/sec)

Magnitude

Mu bands for Robust Performance Analysis

K.N.Toosi University of Technology 42

μ-Synthesis and DK-Iteration

Step K: Design a controller such that

Step D: Find D such that following equation

become minimum

1min ( ( ) )K DN K D

1( )DND 10

-310

-210

-110

010

110

210

30.5

1

1.5

2

Frequency(rad/sec)

Mu Bands in DK-Iteration

Iter1

Iter2Iter3

Iter4

K.N.Toosi University of Technology 43

Step Response for DK-Iteration Controller

Settling time after 150 (sec) in both channels The control signal is in its range√ The inverse response of non-minimum phase zero × The overshoot less than 15% in both outputs √ The order of controller is 16

0 100 200 300 400 500200

250

300

350

Time(sec)0 100 200 300 400 500

42

44

46

48

50

52

54 Step Response

Time(sec)

0 100 200 300 400 500-8

-6

-4

-2

0

2

4

6

8

10

12

Time(sec)

Control Signal

0 100 200 300 400 5000

1

2

3

4

5

6

7

Time(sec)

K.N.Toosi University of Technology 44

Disturbance Response for DK-Iteration Controller

Disturbance rejection after 200 in flow Ch. √ and 300 in level channel × The control signal is in its range √ The maximum peak of response 13(lit/h) in flow ch. √ and 5.7(cm) in level

ch. √

0 100 200 300 400 500 600 700215

220

225

230

235

240

245

Time(sec)0 100 200 300 400 500 600 700

44

45

46

47

48

49

50

51 Step Response

Time(sec)

0 100 200 300 400 500 600 7000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time(sec)0 100 200 300 400 500 600 700

-6

-5

-4

-3

-2

-1

0 Control Signal

Time(sec)

Step Response Control Signal

K.N.Toosi University of Technology 45

Controller Order Reduction

Coprime factorizationBalanced ResidualizationBalanced Truncation

0 100 200 300 400 500

2

4

6

8

10

12Level

Time(sec)

0 100 200 300 400 500

-20

0

20

40

60

80

100

120

140

Time(sec)

Flow

levelresid

level trunc.level coprime

Level non Reduction

flow coprime

flow residflow trunc

flow non Reduction

100

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5Balanced Truncation

Frequency(rad/sec)10

00

5

10

15

20

25

30

35

40Balanced Residualization

Frequency(rad/sec)10

00.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3Coprime Factorization

Frequency(rad/sec)

Level Flow

K.N.Toosi University of Technology 46

MIMO Flow-Level

Plant

IdentificationLimitations in

designing

Design the Robust

Controller

�Decentralized

ControllerConclusionH

H

K.N.Toosi University of Technology 47

Conclusions:

Decentralized controller for the systems with small interactions and performance is not important problem

Obtaining uncertainty for MIMO systems is hard Useful tool for analyzing the robustness of the system The induced norm can represent many properties of system High order of H∞ controller and order reduction

K.N.Toosi University of Technology 48

Suggestions:

The position of valve(3) can be rearranged to have more interaction

Designing controller by considering input uncertainty Designed Controller by considering parametric uncertainty The structure of controller can be determined by designer The QFT robust controller combined with decentralized

control approaches .

Fetching control signal in saturation

K.N.Toosi University of Technology 49

Acknowledgment