SN 4 Uncertainties

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ROBUST CONTROLSources of uncertainties and its

representation

Dr. S. Ushakumari

Associate Professor

Department of Electrical EngineeringCollege of Engineering Trivandrum

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Introduction

The problem of Robust Control is to design a fixedcontroller that guarantees acceptable performance

norms in the presence of plant and input

uncertainty.

The performance specification may include

properties such as stability, disturbance rejection,

reference tracking, control energy reduction, etc.

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Model uncertainty and its

representation

Origins of Model Uncertainty

1) Parameters in a linear model, which are approximately

known or are simply in error.

2) Parameters, which may vary due to nonlinearities orchanges in operating conditions.

3) Neglected time delays and diffusion processes.

4) Imperfect measurement devices.

5) Reduced (low-order) models of a plant, which are

commonly used in practice, instead of very detailedmodels of higher order.

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Contd..

6) Ignorance of the structure and the model order at high

frequencies

7) Controller order reduction issues and implementation

inaccuracies.

The above sources of model uncertainties are grouped intothree main categories:

Parametri c or St ruct ured Uncert aint y

In this case the structure of the model and its order is known,

but some of the parameters are uncertain and vary in a subset

of the parameter space

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Contd..

Neglect ed and Unmodeled Dynami cs uncert aint y

In this case the model is in error because of missing

dynamics, most likely due to lack of understanding of the

physical process.

Lumped Uncertainty or Unstructured Uncertainty

In this case uncertainty represents several sources of

parametric and/or unmodeled dynamics uncertainty

combined into a single lumped perturbation of prespecified

structure.

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Modeling Uncertainties

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Need for modeling

We may consider the modeling of

structured and unstructured uncertainties

in the true System model so that the

normed algebra can be applied for the

robust control designs.

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Four forms for Uncertainty representation

Multiplicative Uncertainty

Inverse Multiplicative Uncertainty

Division Uncertainty Additive Uncertainty

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Modeling structured parametric

uncertainties as unstructured uncertainties In this case, the model will be given as a

transfer function. Some parameters of thismodel will be having uncertainties, with

known range of variations (bounds)

For such cases, proper substitution canconvert the model into one of the four

uncertainty model forms seen earlier. Let us see this through examples.

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Representation of uncertainty

Parametric uncertainty will be quantified by assuming that

each uncertain parameter is bounded with some region

[min,max ].In other words, there are parameters sets of the

form

p=m(1+r)

where m is the mean parameter value,

r

=(max-min)/(max+min) is the relative parametric

uncertainty.

is any scalar satisfying

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Multiplicative Uncertainty

++

Wm(s) m(s)

G(s)

Gp(s)

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Contd..

++

Wm(s) m(s)

G(s)

Gp(s)

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Contd..

+ -

Wm(s) m(s)

G(s)

Gp(s)

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Contd..

+ -

Wm(s) m(s)Gp(s)

G(s)

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Additive uncertainty

Wa(s) a(s)

G(s)

++

Gp(s)

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Contd..

+ -

Gp(s)Wa(s) a(s)

G(s)

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Case 1: Multiplicative uncertainty

Given

where p is an uncertain gain and G0(s) is atransfer function without uncertainty.

Determine the multiplicative uncertainty

description for this family of uncertain

systems.

maxmin)()( p0pp sGsG

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Solution: Write p as follows

Where

m Average gain & r - Relative magnitude of the gain uncertainty

1)1( rmp

minmax

minmax

minmax

r

2m

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Therefore, the given model description of

uncertain systems as

G(s) is the Nominal Plant without

uncertainty

Gp(s) is the TRUE Plant or Real-life Plant

1r1sG

r1sGsG 0mp

])[(

])[()(

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Case 2: Uncertain zero

A set of possible plants is given by

Where G0(s) is assumed to have nouncertainty.

Determine the multiplicative uncertainty

description

maxmin0 ),()1()( zzzsGszsG ppp

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Solution:

Let

1r1zz zmp )(

minmax

minmax

minmax

zz

zzr

2zzz

z

m

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Then we have

])()[(

)()()()()(

)()()(

)(][)(

sw1sG

sGsz1sz1

srzsGsz1

sGsrzsGsz1

sGsrzsz1sG

m

0m

m

zm0m

0zm0m

0zmmp

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Case 3: Inverse Multiplicative Uncertainty

Given

Determine the Inverse multiplicative

uncertainty description for this family of

uncertain systems

maxmin)()(

p0

p

p sG1

1sG

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Let 1r1mp )(

minmax

minmax

minmax

r

2m

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The given model is written as

1

im

1

im

m

0

mm

0p

sw1sG

sw1s1

sG

srs1

sGsG

])()[(

])([)(

)()(

This is the Inverse multiplicative uncertainty form

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Case 4: Division Uncertainty

Given

Determine the Division Uncertaintydescription for this set of uncertain systems

80401ss

1sG

2p..)(

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Here max=0.8, min=0.4

21

40

r

602

m

.

.

.

minmax

minmax

minmax

121

406060r1mp

:0.20.6)

.

...()(

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We have

.)(:)]()()[(

)()(

.

..

1

..

1

)..()(

s20swsGsw1sG

sGsw1

1

G(s)

1s60s

s201

1

1s60s

s201s60s

1s2060s

1

sG

d

1

d

d

2

2

2

2p

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Additive & Multiplicative forms of Uncertainty

Additive:

Multiplicative:

)()()(:)()()( sGsGsssGsG paap

)(

)()()(

)()](1[)(

sG

sGsGs

sGssG

p

m

mp

)(

)()(sGss am

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Multiplicative uncertainty

Multiplicative Uncertainty factor m(s) is expressedas a relative gain error with respect to the

Nominal Plant Model.

The Multiplicative Uncertainty form is the most

popular form of uncertainty description in the Robust

Control literature.

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Case 5: Flexible structures

Command

Controller+ _Rigid

Feedback

Flexible

+

+ Output

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Example:

Rigid: Nominal Plant -

Flexible: perturbation -

Feedback: Sensor Dynamics -

2

2

s

1ss

12

)( 4s

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Case 5- contd

For this case, the Uncertain Plant family is

The Nominal Plant is

)(

)(

1sss

2s2ssG

22

2

p

2s

2sG )(

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Additive Form

1ss

1s

2a

)(

G(s)

a(s)

+

+

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Multiplicative Form

)()()()()(

1ss2s

sGsGsGs

2

2p

m

G(s)

m(s)

+

+

G(s)

m(s)

+

+

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Unstructured uncertainty in Frequency domain

Let us see the quantification of unstructureduncertainty using frequency domain analysis

What we have seen previously is the conversion ofparametric uncertainty to unstructured forms

The structured parametric descriptions call fordeeper knowledge on plant behaviour, which isdifficult to obtain

Therefore, unstructured formalisms based onfrequency domain analysis are more practical toobtain from the experiments or simulations

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Advantages of Unstructured Uncertainty

Descriptions

The unstructured uncertainty descriptions assumes

less knowledge of the system.

We only assume that the frequency response of the

plant lies within certain bounds.

This approach can cover both structured

uncertainty as well as unmodelled dynamics in the

convenient rational transfer function formats

amenable for a standard design.

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Development of Unstructured

Uncertainty Descriptions

Let P be the family of uncertain plants.

G(s) P is the nominal plant. a(s), m(s) etc stands for the unstructured

perturbations in terms of stable rational

transfer functions.

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Methodology

Step 1: Choose a Nominal plant model G(s)

through one of the following ways

Lower order delay free model

Model with mean parameter values

Central plant obtained from Nyquist Plots

corresponding to all of the plants of the given set P

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Step 2- For additive forms

Find the smallest radius la() which includesall possible plants

la()= )()(max jGjGpPG ap

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Additive uncertainty

In most cases we look for a rational

transfer function weight wa(s) for additive

uncertainty.

This weight must be chosen such that

And must be selected to be of low order to

simplify the design of controllers.

)( jwa la()

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Multiplicative ~

In the case of multiplicative uncertainty, find

the smallest radius lm() which includes allpossible plants

lm() =)(

)()(max

jG

jGjGpPG mp

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Multiplicative ~

For a chosen rational weight wm(s) there

must be

)( jwm

lm()

And wm(s) must be selected to be of low order

to simplify the design of controllers.

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Case Study Example

Consider the family of plants with parametric

uncertainty given by

6b23,a1:P

bass

ssG

2

p )(

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Choose the Nominal Plant

Choose 9 combinations of values a=1,2.5,3 and

b=2,5,6 and obtain the Bode magnitude plots for

the errors for the 9 member plants

4s2s

ssG

2 )(

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Let the plots be like this

Bode Diagram

Frequency (rad/sec)

Phase(deg)

Magnitude(dB)

-40

-35

-30

-25

-20

-15

-10

-5

0

10-1

100

101

-90

-45

0

45

90

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First Trial for the bound

Bode Diagram

Frequency (rad/sec)

Phase(deg)

Magnitude(dB)

-40

-35

-30

-25

-20

-15

-10

-5

0

10-1

100

101

102

-90

-45

0

45

90

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Second trial

Bode Diagram

Frequency (rad/sec)

P

hase(deg)

Magnitude(dB)

-40

-30

-20

-10

0

10

10-1

100

101

102

-90

-45

0

45

90

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Third trial

Bode Diagram

Frequency (rad/sec)

Pha

se(deg)

Magnitude(dB)

-40

-30

-20

-10

0

10

10-1

100

101

102

-90

-45

0

45

90

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How to find the bound?

First we plot the

for all the plant family members.

There may be one case which is the worstcase error.

Try to fit a lower order transfer functionwhich will be of lowest magnitude to limit

from above the worst case error plot for thecomplete range of frequencies.

)(

)()(

jG

jGjGp

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Select the TF as the bound

In the example considered we find that onefirst order TF and one second order TF can befound out for this purpose. These are:

20s37swp

)(

20s21s

20s43sw

2p

)(

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Thus we have

6b23,a1:P

bass

ssG

2

p )(

Uncertain Plant family

4s2s

ssG

2 )(

Nominal Plant

20s

37swp

)(

20s21s

20s43sw

2p

)(

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Conclusions

Uncertainty can be described in both

structured & unstructured forms

Unstructured forms can cover both

structured uncertainty as well as unmodelleddynamics in the convenient rational transfer

function formats amenable for a standard

design.

Unstructured forms can be obtained throughexperiments/simulation

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SN 4 Uncertainties