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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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