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ROBUST CONTROL

Traditional Control

Does not address plant/model mismatch issues in a systematic manner.

Performance initially may be satisfactory, i.e. performance good for nominal model, but

it may deteriorate or even become unstable when process dynamics vary with time.

Why do dynamics change?

Process throughput change

Feed quality change

Ambient temperature

Equipment efficiency

etc

t

y

t

u

Response obtained

during sunny days

Response obtained

during rainstorm

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What is Robustness?

Robustness is the ability to control under changing operating conditions.

As plant drifts from current conditions and process dynamics change, a robust controllerwill provide the best performance under different conditions.

Robustness vs. Performance

PID

Ad Hoc

Z-N Tuning C-C Tuning Direct

Synthesis

Robust Control

Model

Information partly partly invertible

parts

a set of models

Sensitivity

Information uncertainty

description

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Feedback Principle

GK

GKy

GKdGK

GK

r +

=

+=

+=

11

1

1

)()( sKsG : loop-gain

)(1

1sS

GK=

+ : sensitivity function

)(1

sTGK

GK=

+ : complementary sensitivity function

1=+

TS

+

+

+

+r+

K G

d

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1. Set-Point Tracking

Ifs

Ar=

)0()0()0()0(1

)0()0()(

1KGA

KG

KGtyT

GK

GK

r

+==

+= should be large orT(0) = 1

Note PI control => no steady-state offset => G(0)K(0)=?

If tr sin=

[ ]))((sin)()( jTtjTAty +=

1)( jT and 0))(( jwT when )()( jjG

High loop gainis required for good set-point performance,

or )( jT = 1 over a large frequency range.time

y

2b

1b

2

1

)( jT

Tr

12

y

2

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Example 23

1

2++

= ssG , ),

1

1(5.11 sK += )

1

1(75.02 sK +=

0 5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

10-3

10-2

10-1

100

101

0

0.2

0.4

0.6

0.8

1

frequency time

T

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2. Disturbance Rejection

SGKd

y=

+=

1

1

For a step disturbance, 0)('

==ty => large )0()0( KG orsmall )0(S

Note PI control => no steady-state offset => S(0)= ?

If td sin=

[ ]))((sin)()( jStjSAty +=

It is clear that 1)()( >> jKjG implies that 0)( =>jS .

Again,high loop gainis required for good disturbance performance,

or 0)( =jS over a large frequency range.

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Summary of Performance Requirements

(a) Good disturbance rejection 0)(1

1)(

+=

jGKjS or )( jGK >> 1,

(b) Good set-point tracking 1

)(1

)()(

+

=

jGK

jGKjT or )( jGK >> 1,

where is the frequency range that covers the frequency content of disturbances and set-

point changes. Typically band-width of the control system is used, i.e.= [ ]b0 . In general,the larger b is, the better control performance is. Any trade-off between (a) and (b) ?

3.Measurement Noise )(1

sTGK

GKy=

+=

1)( jT (for good performance) => 1

y, i.e. no suppression of noise

On the other end,00 == T

y

(what is the implication on performance ?)

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The performance weight is normally chosen as

Aws

wMssw

B

Bp

+

+=

/)(

= ?

= ?

Bw = ?

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Various sources of model uncertainty may be grouped into three classes:

1. Parametric uncertainty.The model structure is known, but its parameters are uncertain.

1,,2

),1(

),()(

minmax

minmaxminmax

maxmin

+

=

+=+=

=

rkrk

p

kk

kkr

kkkrkk

kkkskGsG

2. Neglected and unmodelled dynamics.The modelling errors occur either through deliberate

neglect or because of a lack of understanding of the physical process. This class of uncertainty

is normally described as the complex perturbations in the frequency domain, which is

normalized as 1 .

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The disc-shaped uncertainty can be described by multiplicative uncertainty description:

( ) ( )[ ] ( ) 1,1)()( += jwsslsGsG mmmmp

or

( ) ( ) ( ) ( )jwljwGjwjwljwGjwGjwG mmmmmmp )()()( =

Nyquist Plot Nyquist Band

Gm(jw1)Gm(jw2)

|Gm(jw1)|

|Gm(jw2)|

Gm(jw1)

Gm(jw2)

|Gm(jw1)lm(jw1)|

Gp(jw1)

|Gm(jw2)lm(jw2)|

Gp(jw2)

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Example12

1

+=

sGm ,

)12)(1(

1

++

+=

ss

sGp

, 11

1

2)(

12

1

)1(1|)(|

+==>

+=

+

=

s

ssl

s

s

s

sMax

G

GMaxjl m

m

p

m

Now, apply RS condition to controller designs 10=K and 2=

.

10-2

10-1

100

101

102

10-2

10-1

100

101

102

10-2

10-1

100

101

102

10-2

10-1

100

101

102

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Performance Robust Stability

1Tviolates RS condition

2T satisfies RS condition by detuning, i.e. at the expense of performance loss

This problem is inherent in feedback control (why ?) and cannot be overcome by any clever

controller design.

1T

1pM

)(11 jT

pM

1)( jT

)(ml 1

2

T

1

2T

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Example ,12

1

+=

sGm ,

)12)(1(

1

++

+=

ss

sGp

11

The weights are 21

2

+

=ml (from previous discussion) ands

sWp

7

1725.0 +

=

Case 1: .2=K

S

1

pW

1T

10-3

10-2

10-1

100

101

0

1

2

3

4

5

6

10-3

10-2

10-1

100

101

0

2

4

6

8

10

12

frequency frequency

NP+

RS

RP

ml

2 1.5

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Case 2:s

K1

5.1

11+=

1T

10-3

10-2

10-1

100

101

0

1

2

3

4

5

6

10-3

10-2

10-1

100

101

0

0.2

0.4

0.6

0.8

1

0 20 40-1

0

1

0 20 40-0.5

0

0.5

1

0 20 40-0.5

0

0.5

1

1.5

0 20 40-0.5

0

0.5

1

1.5

frequency frequency

NP

+

RSRP

)12)(1(

1

++

+=

ss

sGp

)12)(1(

15.0

++

+=

ss

sGp

)12)(1(

1

++=

ssGp

12

1

+=

sGp

1

pW

ml

S

50 2

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Case 3:

+=

sK

1

5.1

112

ml

1

pW

)12)(1(

1

++

+=

ss

s

Gp

10-3

10-2

10-1

100

101

0

1

2

3

4

5

6

10-3

10-2

10-1

100

101

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 20 40-50

0

0 20 40-1

0

1

0 20 40-0.5

0

0.5

1

1.5

0 20 40-0.5

0

0.5

1

1.5

frequency frequency

NP

+

RS

RP

)12)(1(

15.0

++

+=

ss

s

Gp

)12)(1(

1

++=

ssGp

12

1

+=

sGp

S

1T

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