Introduction to Robust Control
Dr Abraham T Mathew
What is a Control System?
Is it the physical system?
Is it the mathematical system?
ENVIRONMENT
SYSTEM
BOUNDARY
INPUT
INPUT
CONTROL SYSTEM
OUTPUT
INPUT
SUCCESS IN CONTROL DESIGN IS SAID TO BE BASED ON THE
SUCCESS IN IDENTIFYING THE SYSTEM BOUNDARY,
INPUTS,OUTPUTS & THE ENVIRONMENT
Model Based Control Design- Issues
 Analytical or computational models cannot truly characterize and
emulate the phenomenon.
 A model, no matter how detailed, is never a completely accurate
representation of a real physical system
Control Design-classical way
 Normally, in the conventional control design for SISO
system, the stability margin is specified to ensure stability in
the presence of model uncertainties
 But, the uncertainties or perturbations are not quantified,
nor performance was not taken into account in terms of
disturbance, noise etc.
 For MIMO systems, many of the SISO methods cannot be
scaled up
Robust Control
Design a controller such that some level of performance of the controlled system is
guaranteed irrespective of the changes in the plant dynamics/process
dynamics within a predefined class and
 the stability is guaranteed
Control design targets
 Stability
 Disturbance rejection
 Sensor(measurement) noise rejection
 Avoidance of actuator saturation
 Robustness- the process/plant performance should not
deteriorate to unacceptable level if there occurs the changes due
to the uncertainties
All these targets cannot be achieved simultaneously and perfectly. So
there has to be some compromise or tradeoffs, because of various
reasons
Modeling in the context of robust control
We consider a simple example !!
Modeling a DC Servo
 We consider a DC servo mechanism consisting of a DC
motor, gear train, and the load shaft
 It is required to control the angular displacement and speed
using a voltage signal applied across the armature
Motor
Load
Linear Model of the DC Servo-Physical
Equations of dynamics



 
0
 
0

L


 

 i 

 
0

1
0
 NK m
La


 
0
0     0 
  1
NK m     
   0 v(t )   TL 


1
Je
Je 

 
 Ra   i   L 
 0 
 a
La 
 
  1 0 0  
   0 1 0  
  
 i 
 
TF FORM
 (s)
NK m
J e La
a0

 2
2
2
v( s )
 2 Ra
N K m  s s  b1s  b2

s s 
s
La
J e La 



Nominal Model
 Km=0.05 Nm/A, Ra=1.2 ohms, La=0.05H
 Jm=8x10-4 kgm2 , J=0.020 kgm2
 N=12
 Je=J+N2Jm=0.1352 kgm2
Uncertainty
 Let the parameters are subject to changes as follows
0.04≤Km ≤0.06
6x10-4 ≤Jm ≤ 10-3
0.01≤J ≤0.05
Model with Uncertainty
(as an Interval System)
[74.22, 99.58]
G( s)  2
s s  12s  [47.8, 53.4]


Abstracting a
Control System Structure
Control System Structure
Disturbance
w(t)
wm
Noise v(t)
Sensor
S1
Input yd
Controller
C
u
Plant/Process
P
y
Sensor
S
Output ym
System Equations
 If the Plant is LTI the zero state linearity dictates that y is a linear
combination of effects of the two plant inputs u and w
 That is
y(s)  P(s)u(s)  P (s)w(s)
(1)
 Quite often it is convenient to work with the disturbance d(s) at the
plant output given as
w
d (s)  P (s)w(s)
 Then, we have
w
y(s)  P(s)u(s)  d (s)
(2)
(3)
System Equations
 Sensor is assumed to have two inputs, plant output y and the
measurement noise v. So, we have
y ( s )  P ( s ) y ( s )  v( s )
 Ideally Ps(s)=1 and v(t)=0 so that ym=y
m
s
(4)
(this is achieved if sensor bandwidth is larger than system
bandwidth or we say the sensor is fast and accurate)
Now, look at the Controller
Disturbance
w(t)
wm
Noise v(t)
Sensor
S1
yd
Controller
C
u
Plant/Process
P
y
Sensor
S
ym
Contd…
 Controller gets three inputs ym, yd and wm
 Here wm is the disturbance measured using suitable sensor
 Let the controller be LTI. Then
s) all
Fdthe
(s) ythree
( s) 
Fm (s)need
ym (sto
) be
Fwused
(s)wmalways
(s) here. Several
(5)
d
u(Not
inputs
control structures are defined according to whether ym, yd or wm
is used to produce u or not . Accordingly we will have different
schemes of control
1. Single Degree of Freedom controller
 When Fm=-Fd and Fw=0, we have
u(s)  F ( s)[ y ( s)  y (s)]
d
d
m
 The figure shows 1DoF Control Structure realizing this
equation
yd
+
ym
Fd
u
Two Degree of Freedom Controller
 If we have a structure of the form given below, designer will
have freedom to independently select Fm and Fd we will have
the TDoF Feedback Controller structure
yd
Fd
+
+
u
Fm
ym
Feedback Control Scheme
v
w
Fw
Pw
yd
Fd
+
+
d
u
P
+
+
Fm
+
y
Ps
+
+
ym
Problem formulation
 System enclosed in the dotted box is seen to have three
inputs and one output
 By assuming linearity, we can say that plant output y(t) is
produced as a superposition of the effects of these three
signals coming to the output port through three transfer
channels
 That is
y(s)  H (s) y (s)  H (s)w(s)  H (s)v(s)
d
d
w
v
Tracking problem
 Let the error e(t) be defined as e(t)=yd-y
 That is,
e( s)  y ( s )  H ( s) y ( s )  H ( s) w( s)  H ( s )v( s)
d
d
d
w
v
Or
e( s)  1  H ( s)y ( s)  H ( s) w( s )  H ( s )v( s)
d
d
w
v
 The central design problem is to obtain Hd, Hw, and Hv with
desirable properties using appropriate methods or criteria
Look at it again
v
w
Fw
Pw
yd
Fd
+
+
d
u
P
+
+
Fm
+
y
Ps
+
+
ym
Emphasis for output disturbance
 In cases where it is desirable or convenient to work with the
output disturbance d rather than w, we have
y(s)  H (s) y (s)  H (s)d (s)  H (s)v(s)
d
d
wd
v
e( s )  y ( s )  H ( s ) y ( s )  H ( s ) d ( s )  H ( s ) v ( s )
d
d
d
wd
v
Or
e( s)  1  H ( s)y ( s )  H ( s)d ( s )  H ( s)v( s)
d
d
wd
v
Tracking performance
 For the system to ideally track the reference, the error must
be zero
 To achieve this for all possible yd,v,w and d, we would
require Hd(s)=1 and Hw(s)= Hwd(s)=Hv(s)=0
 In the practical setting, as we see more in detail, we can see
that this condition cannot be satisfied perfectly for the entire
bandwidth or entire region of system perturbations
 Some design tradeoffs, optimality conditions and so on would
have to be called for as we have already noted.
Admissible/acceptable designs
 In order to do the adjustment/tradeoff for obtaining an
admissible or acceptable design and discriminate between
acceptable and unacceptable departures from the ideal
performance, we need to have the specifications
 These specifications give rise to different control structures
like open loop, feedforward, feedback, etc.
 We may differentiate between SISO and MIMO and start
with SISO and generalize the notations for MIMO,
subsequently
Control System Performance
 From a system’s perspective, the performance specification
for control system starts with “Stability”
 Followed by Sensitivity, Disturbance Rejection, Noise
Rejection etc. where needed.
Stability
 When it comes to stability, in the modern settings of design,
we consider two classes of stability, namely
 Input-output stability
 Internal stability
 Internal stability is of paramount importance in the MIMO
system framework, both in Matrix Transfer function form
and State variable/transfer function forms
Internal Stability
 A system to be internally stable means all the transfer functions
associated with all the transfer channels connecting exogenous
input to the output(including set point, disturbance & noise) shall
be stable
 In reality, it is possible for a system to be internally unstable and
yet to have a stable “set point to output” channel transfer functions
 Under this circumstance, we say that system has unstable hidden
modes
 Therefore, internal stability must be ensured before the transfer
function that define the response to the system inputs are
considered
Design Model
be a set of all plants that each member of set
P is an admissible model, given the uncertainty
region (interval)
 Let P
 P0 in
P is one model with the nominal value of the
parameters
 If P0 is used for the robust designs, then let us call P0 as Design
model (for the sake of convenience!!)
Model Uncertainty & Internal stability
 If the plant is expected to deviate from the design
model(nominal model), it is better represented by a set of
models centered on the design model(nominal model)
 For a control system to be acceptable, the design must be
internally stable for every model in the set
 This property is known as robust stability
 Once stability & robustness are assured, we can shift the
attention to “response”
Summary
 A model of the physical system is only an approximation of
the real phenomenon/process
 Control system output is the measurement showing the
status or effectiveness of control
 Inputs, in a general framework will include set point,
disturbance and measurement noise
Summary contd…
 Models are subjected to various uncertainties
 Nominal model in the set of uncertain models can be used as
Design model
 Internal Stability and robust stability are starting points for
good control system design
 Once stability is assured, other performance measures can be
specified
Design Dilemma
 It will not usually be possible(which we will see in detail) to
have good set point tracking, and disturbance rejection and
noise rejection uniformly effectively for all functions of yd, v,
w and d
 Also, emphasis on sensitivity on one may negatively affect the
other
Robust Control System
 A system is said to be robust when
 It is durable, hardy and resilient
 It has low sensitivities in the system passband
 It is stable over the range of parameter variations
 The performance continues to meet the specifications in the
presence of a set of changes in the system parameters
Robustness is the sensitivity to the effects that are not
considered in the analysis and designfor example,
the disturbances,
measurement noise, and
unmodeled dynamics
Sensitivity & Sensitivity Analysis
Sensitivity
 It is the percentage change in system transmission or
response or some quantity of interest with respect to the
percentage change in another quantity
 In control theory we use
 Parameter Sensitivity
 System Sensitivity
 Root Sensitivity
 Eigenvalue Sensitivity
Parameter Sensitivity
 Let T be the system function which depends on a parameter

 Then, the parameter sensitivity ST of T with respect to  s
defined as
T
T
 ln T
T 
T

S 

 ln  
T


System Sensitivity
 Let T be the system closed loop transfer function which
depends on the open loop transfer function G
 Then sensitivity of T w.r.t G is given as
T
T
 ln T
T 
T

S 
G
 ln G G
G
G
T
G
Root Sensitivity
 Let T be the system closed loop transfer function with the ith
root given as i and the parameter of interest is say K
 Root sensitivity is the sensitivity in terms of the position of
the roots of the characteristic equation on the (, j)
plane(root locus plane)
Significance of Root Sensitivity
 Roots of the characteristic equation represents the
dominant(visible) modes of the transient response
 The effect of parameter variation on the position of the root
and the direction of shift of the root are important and useful
measures to say about the sensitivity
 Can be combined with Root Locus Method for Control
Designs
Definition of Root Sensitivity
 The root sensitivity of the system T(s) is defined as


S 

 ln K K
i
i
i
K
 Let
K
m
T ( s) 
K ( s  z )
1
j
j 1
n
( s   )
i 1
i
Contd…
 Let K be a parameter that influences the location of the roots i
and the gain K1
 Then the root sensitivity is related to the system sensitivity to K
and is given as(if zeros of T(s) are not dependent)
 ln K

1

S 
 ln K
 ln K ( s   )
n
T
1
K
i
i 1
i
 In the event of gain K1 independent of K, we have

1
1

S  
  S
 ln K ( s   )
(s   )
T
K
n
n
i
i
i 1
i 1
i
K
i
Eigenvalue Sensitivity
 Let us assume that we have the relation(A is from the state
space equation)
A   
i
i
i
 Differentiating with respect to the element akj of A we will
have
A
 

 A
 

a
a a
a
i
i
i
kj
i
i
kj
kj
i
kj
Contd…
 Premultiplying with i , the left eigenvector we have ii=1
and i (A-i I)=0
 Then, we get

A

 
a
a
i
i
i
kj
kj
Contd…
 All elements in
which will be 1
 Therefore we get
A
will
a be zero except the (k,j)th element,
kj

 
a
i
ik
kj
 This is the eigenvalue sensitivity
ji
Sensitivity Analysis of transfer
functions
 Consider a closed loop system as shown in Figure
yd
+
u
-
T
G
1 G
T
 ln T
T  G T  1
S 

 ln G G
T G 1  G
G
T
G
G
y
Waterbed effect
 Now, add T and S
We get T+S =1
System with cascade compensator
 We consider the following system
yd
+
K
u
G
y
-
T
GK
1  GK
T
 ln T
T  G T  1
S 

 ln G G
T G 1  GK
G
T
G
Check T+S
System with feedback compensator
 Consider the following system
yd
+
u
G
y
-
H
T
G
1  GH
T
 ln T
T  G T  1
S 

 ln G G
T G 1  GH
G
T
G
Check T+S
Sensitivity & Complimentary Sensitivity Functions
 In the Robust Control Literature, Sensitivity Function plays a
crucial role
 Let S(s) be the Sensitivity Function
 Then T(s) is the Complimentary Sensitivity Function such
that S+T=1 for SISO and S+T=I for MIMO
Open Loop Control
Open Loop Control
 It is the simplest control structure
 Limited in performance
 Usually reserved for special applications where feedback
control is either impossible or unnecessary
 It is a good starting point for control design
 It helps to appreciate the advantages of feedback control
 Stability, performance etc are relatively in simpler forms to
understand
Open Loop Structure
d
yd
F
u
+
P
+
y
-
e
+
Input-Output Relations
 In open loop control input yd is usually a synthesized signal
for the given application and u is derived from that as shown
 Open loop control requires no measurements.
 Now, from Figure above, we write as
y  FPy  d
d
and e  (1  FP) y  d
d
H ( s)  F ( s) P( s)
d
and H ( s)  1
wd
Tracking Performance
 Perfect tracking of yd occurs if
H ( s )  F ( s ) P( s )  1
d
 That is, if
F ( s )  P( s )
1
 The practical objective is to make
in the system passband
F ( j ) P( j )  1
Disturbance rejection
 Since
open
H ( s) 
1 loop control does nothing to attenuate
wd
the effects of disturbance inputs nor does it amplify them
either
Sensitivity
 The sensitivity of
H with
(s) respect to P(s) is calculated as
d
follows
H  F ( P  P)  FP  FP
0
d
FP
S 
H
P
P
0
FP
0
1
P
0
 A sensitivity 1 implies that a given percent change in P
translates into the equal percent change in the transmission
function
H d ( j )
 Open loop control does not affect sensitivity
Stability Conditions
 We modify the block diagram of the Open loop control
system as shown here
v
yd
+
F
+
z
u
P
y
Analysis
 In any system, any addition or deletion of some of the input lines




or some output lines won’t alter the internal stability
We shall add inputs and outputs and view this as injecting test
inputs into the system and taking extra measurements, neither of
which is expected to change the stability properties of the system
The test inputs and and outputs are chosen so that the resulting
system is controllable and observable
For such a fully controllable and observable system there shall not
be any hidden modes
So, internal stability is then guaranteed by input-output stability
Fig.1
yd
F
u
y
P
v
Fig.2
yd
+
F
+
u
P
z
The system, in Fig 1 and Fig 2 are same but with additional input v
and one additional output z in Fig 2
y
Controllability/Observability/Stability
 System in Fig.2 is controllable and observable if both F(s) and P(s)
are controllable and observable
 System in Fig 2 is internally stable if and only if the both F(s) and
P(s) are stable. See below
Y ( s )  FPy ( s )  Pv( s )
d
z ( s )  Fy ( s )
d
Or
 y( s )  FP P   y ( s )
 z ( s )    F 0   v( s ) 

 


d
Analysis contd…
 Because the realization is controllable and observable, it is
internally stable if, and only if, it is input-output stable.
 That is, if all elements of the matrix transfer function above
are stable
 Thus F(s), P(s) and F(s)P(s) must have only LHP poles
 If P is of non-minimum phase type, then F cannot be used to
cancel the RHP zeros of P, because then F will become
unstable.
Feedforward Control
 Feedforward control is a variation of open loop control.
 It is applicable when the disturbance input is measured
 The open lop controller F is chosen, to make the output to follow the
reference, in spite of the disturbance
w
Pw
u
+
P
y’ +
z
d
y
w
Pw
Pw
d
F
u
Here, to realize Feedforward control:
+
P
y’ +
z
1. d has to be obtained by proper measurements
2. F is chosen such that y’ is close to –d
3. Or FP is almost unity
d
y
Closed loop control-1 DoF
Closed loop control-1 DoF
 Consider the following system
d
e
yd +
F
u
+
P
+
+
-
y
e
ym
Ps
+
+
v
Analysis
 We have
FP
1
y( s ) 
y (s) 
d (s)
1  FP
1  FP
d
1
1
e( s )  y ( s )  y( s ) 
y (s) 
d (s)
1  FP
1  FP
d
d
With Sensor noise/Measurement Noise
 If yd =d=0 and v0, then
y( s )   FP( y  v )
 y( s )  
FP
v( s )  T ( s )v( s )
1  FP
and
e( s )  y  y( s )  T ( s )v( s )
d
Norms are Performance Measures
Signal forms and Signal Norms
 Norm based approach for control design gives a sound
platform for robust control designs
 Different types of norms are used in control systems
 Use would be depending on the mathematical approaches
used to define the norm
Norms of signals and systems
 Euclidean Norm or l2 norm for vector x is given as
 
n
x  x
2
1
2
i
i 1
2
 ( x x)
T
1
2
 For a vector signal x(t), l2 norm is
x   x (t ) x(t )dt 

2
T
1
2

 This norm is the square root of the energy in each
component of the vector
 If norm exists x(t)  l2
Norms of signals and systems
 For power signals, we may use the root mean square
value(rms) norm
1


rms( x )   lim  x (t ) x(t )dt 
2T


T
T 
T
T
1
2
Frobenius Norm
 For an mxr matrix A, the Frobenius norm is defines as

m
r
A  a
2
 It can be shown that
2
i 1 j 1

1
2
i, j
2
A 2  tr ( A A)  tr ( AA )
T
T
System Norm
 LTI systems are generalization of matrices-
 A matrix operates on a vector to produce another vector
 An LTI system operates on a signal to produce another signal
 So, analogous to Frobenius norm, we can define the system
norm
L2 Norm for LTI systems
 Let G(s) be an mxr matrix transfer function
 Then the L2 norm for G(s) is defined as
 1

G    tr G (  j)G( j)d 
 2


2
1
2
T

 ||G||2 exists if an only if each element of G(s) is strictly
proper. For SISO we have a scalar TF which need to be
strictly proper. There should not any poles on the imaginary
axis for either case.
 Then we say G L2
G(s) plane in H2
 When G L2 we can write the norm with respect to
complex s plane as
1
G 
 tr (G (  s )G( s ))ds
2j
2

2

T
1

 tr (G (  s )G( s ))ds
2j
T
 Contour of integration for the last integral is along the
entire imaginary axis and the infinite semicircle in the LHP
or RHP
 Since G(s) is strictly proper, it is easily shown that the
integral vanishes over the semicircle
 If G L2 and in addition, G is stable, then we say that G H2
 H2 is the Hardy Space defined with the 2-norm
Exercise
 Calculate the L2 norm of G(s) given as:
( s  3 )  ( s  2 ) 
1
G( s ) 


s  3s  2   2
( s  2) 
2
Answer
 3s  21
tr G (  s )G( s ) 
( s  1)( s  2)(  s  1)(  s  2)
2
T
Every term in G(s) is strictly proper
Contour is Imaginary axis + LHP semicircle with radius 
L2 norm of G(s) =(3/2)
Induced norm
 Induced norm is a different type of norm which applies to
operators and is essentially a type of “maximum gain”
 For a matrix, the induced Euclidean norm is
A  max Ad
2i
 =sqrt(eigen(ATA))
d
2
1
  ( A)
 is the max( ) and is min(  )
2
Induced norm for LTI system
 To obtain induced norm for an LTI system, consider first a
stable, strictly proper SISO system
 Then, if the input u(.)  l2 , then the output y(.)  l2
 By Parseval’s theorem
1
y 
 G( j) u( j) d
2
2

2

2
2
(A)
 Clearly
1
y  sup G( j)
 u( j) d
2
2

2
2
2


 Or
y  sup G( j) u
2
2
2

2
2
(B)
 We argue that the RHS of the inequality in (B) can be
reached arbitrarily closely for a fixed value of ||u||2 that is
chosen to be 1 with no loss of generality
 Suppose |u(j)|2 approach an impulse of weight 2 in the
frequency domain at = 0
 Then the integral of Eq(A)
1
y 
 G( j) u( j) d
2
will approach
G( j )
2

2

2
2
2
0
(A)
 If
G( j)has a maximum at some finite value of , we may
choose 0 to be that frequency
 If not, then G( j
must
) approach a supremum as
.
 We can make 0 as large as we like and
will
G( jbe
0as)
close to the supremum as we wish
 The RHS of inequality in (B) can be reached arbitrarily
closely and we get
sup y  sup G( j)
u 2 1
2

Hinfinity Norm
 The norm calculated last is also the infinity norm given by

G  lim(  G( j) )

p
p
1
p

 The infinity norm of G(s) exists if and only if G is proper
with no poles on the j axis
 In that case we write G L
 If in addition, G is stable, then we say G H
H is the Hardy Space defined with the -norm
Norms for Multivariable systems
H norm for Multivariable systems
 For multivariable systems, we have
1
y 
 G( j)u( j) d
2

2
2
2

 This can be written as
1
y 
 [ (G( j))] u( j) d
2
2
2

2
2

 Further, we may write as
1
y  sup (G( j)
 u( j) d
2
2
2
 Or

2




y  sup [G( j)] u
2
2
2

2
2
2
2
Contd…
 The factor ||u(j)||2 in the integrand refers to the 2-norm
of the vector u(j)
 In SISO, the equivalent term refers to the 2-norm of a signal
 We argue that the RHS of the last inequality


y 2  sup
j)] by
u propoer
[G( closely,
can be approached
arbitrarily
choice of
2

u(j)
2
2
2
 Essentially we pick u(j) to be the eigenvector of
G*(j)G(j) corresponding to the largest eigenvalue, and we
concentrate the spectrum of u(j) at the frequency where
is the largest (or for some frequencythat is arbitrarily large,
if has no maximum, but a supremum. Therefore

sup y  sup [G( j)]
u 2 1
2

MIMO H norm
As a continuation of the development, we define
G  sup [G( j)]


Disturbance Rejection
Disturbance Rejection
 Disturbance rejection is a performance measure
 Effect of disturbance is studied in two ways
 Input disturbance
 Output disturbance
Rejection of Input disturbance
d
yd
+
+
u
G
-
H
y
Analysis
T (s ) 
yd
G
1  GH
and
T (s ) 
d
G
1  GH
To suppress disturbance, we want |Td|<<1
For this we need |G|<<1
Keep |G(j)| small where d(t) contains stronger
components in the spectrum
Rejection of Output Disturbance
d
yd
+
u
+
G
-
H
+
y
Analysis
 We have
G
T (s ) 
1  GH
and
yd
T (s ) 
d
1
1  GH
 To suppress disturbance, we want |Td|<<1
 For this we need |G|>>1
 Keep |G(j)| large where d(t) contains stronger components in
the spectrum
Contradiction
 The requirements to suppress disturbance at the input is
opposite to that needed for suppressing disturbance at the
output
 If the disturbance is present both at input and output we need
to use some innovative ways to suppress both the
disturbances
Noise Rejection
yd
u
+
G
y
+
H
+
n
Analysis
T (s ) 
yd
G
1  GH
and
T (s ) 
n
GH
1  GH
To suppress noise, we want |Tn|<<1
For this, we need |G|<<1for a given H
Keep |G(j)| small where n(t) contains stronger
components in the spectrum
Exercise
yd
u
+
K
G
y
-
H
Derive the Sensitivity and Complimentary Sensitivity Functions
with respect to of the system given as G(s). G(s) is containing Uncertainty
Modeling the Uncertain Systems
Modeling the Uncertainties/perturbations
Uncertainties occur in control systems occur
due to variety of reasons
Actually, the purpose of control system itself is
to deal with uncertainties
Purpose of robust control is to render stability &
acceptable performance if the uncertainties of certain
class occur
Structured Uncertainty
 Interval Models
 State Space model
 Transfer function model
Unstructured Uncertainty
 Unstructured uncertainty is modeled, using the perturbation
approach, rather than representing the parameters using the
intervals
 There are different formulations that give the uncertain
models, mostly use the norm bounds and the perturbations
in the additive or multiplicative forms
General Basis
 Given a set of plants P with uncertainty in the parameters. A plant





transfer function P(,s)P is a transfer function admissible to
represent the uncertain system being considered.
P0(0,s) P is one such plant with nominal values of the
parameters, where 0 stands for the nominal value of the
parameter set(vector) 
0 could be the mean value of  in the interval [min, max], which
is intuitively appealing
Uncertainty could then be given as = 0[1+]
0 =(1/2)(min+max) & = (min -max)/ (min+max)
||1 is the perturbation
Unmodeled dynamics
 Uncertainty due to neglected and unmodeled dynamics is
more difficult to quantify
 The frequency domain is well suited for representing this class
of uncertainty through complex perturbations, which are
normalized such that ||||1 where |||| is the
H norm of =
sup ( j )

Classification of unstructured
uncertainty-SISO
 Additive Uncertainty
 Multiplicative Uncertainty
 Inverse Multiplicative Uncertainty
 Division Uncertainty
Use of the Uncertainty is depending on the problem being
considered and the designer’s skill.
For MIMO systems, the constraints of pre and post
multiplication gives rise to more classes of uncertainty
Additive Uncertainty
 Let us sue the property
 P0(0,s) P is one such plant with nominal values of the parameters,
where 0 stands for the nominal value of the parameter set(vector) 
 Let P(,s)= P0(0,s) +P(s)
 P(s) is the complex perturbation applied to obtain the class of
uncertain plants P(,s) and is stable
 Then P(,s) is given in the Additive Uncertainty form
 Usually, this is written as
P:Gp(s)=G(s)+wa(s) a(s) with ||||1
Example
 Consider the system
P: Gp(s)=AG (s). The uncertainty is in the Gain A and is
given as A[Amin ,Amax]
Let A0 =(1/2)(Amin+Amax)
A= (Amin -Amax)/ (Amin+Amax)
A= A0[1+ A ]
Gp(s)= A0[1+ A ] G (s)=A0 G (s)+ A0 A G (s)
Multiplicative Uncertainty
 Let P(,s)= P0(0,s) + P0(0,s) P(s)
P(,s)= P0(0,s)(1+ P(s))
 Or P(,s)= P0(0,s)[1+ wm(s) m(s)]
 Or
||||1
Example
P: Gp(s)=AG (s). The uncertainty is in the Gain A and is given
as A[Amin ,Amax]
Let A0 =(1/2)(Amin+Amax)
A= (Amin -Amax)/ (Amin+Amax)
A= A0[1+ A ]
Gp(s)= A0 [1+ A ] G (s)=A0G (s) [1+ A ]
General method to find the Additive &
Multiplicative Uncertainty Model
 Examples have shown the derivation of unstructured
uncertainty from parametric uncertainty
 This is simple for simple cases but
 Tough for high order systems with uncertainty in many
parameters, because
 Assumption about model and parameters may be inexact
 The exact model structure is indispensable
 Unmodeled dynamics cannot be then handled
Method
 Given a model with uncertainties
 Choose a nominal model(or lower order or delay free or a model
of mean parameters or the central plant obtained from Nyquist
plot corresponding to all plants in the given set)
 For Additive uncertainty, find the smallest radius
l a() which includes all possible plants such that
l a() =|Gp(j)-G (j)|
Find a rational lower order transfer function wa(s) which is the
uncertainty weight such that
|wa (j)| l a()  
The uncertain additive plants Gp(s)=G(s)+wa (s) a(s)
Contd…
 In the case of multiplicative uncertainty, find the smallest
radius l a() such that for all possible plants
 l a()=
G ( j )  G ( j )
max
G p P
p
G ( j )
 For a chosen rational weight wm(s), there must be
|wm (j)| l m()  
Then Gp(s)=G(s)(1+wm(s) m(s))
Block diagram forms of uncertainty
Additive Uncertainty Model
a (s)
wa(s)
+
G(s)
+
Multiplicative Uncertainty
wm(s)
m (s)
+
G(s)
+
Inverse Multiplicative
im (s)
+
wim(s)
+
G(s)
Division Uncertainty
 Consider the
1
G (s) 
s  s  1
p
2
with 0.4    0.8
 It is easy to see that =0.6+0.3 with ||1
G p ( s) 
1
s 2  0.6s  1
G p ( s)  G( s)[1  wd ( s)G( s)]1
 1
w ( s )  0.2s
d
Robust Control
Robust Control
 Normally Robust control design considers two aspects
 Robust Stability(RS)
 Robust Performance(RP)
 As a bottom line we need
 Nominal stability(NS) and
 Nominal performance(NP)
Robust Stability?
 How far the uncertainty can be, without violating the
stability, if the nominal system is stable?
Im
(-1,j0)
?
|1+G(j)|
G(j)
Nyquist Plot
Re
Robust Stability with Multiplicative Uncertainty
Wm(s)
+
yd
K(s)
-
Gp(s)
m(s)
+
u
+
G(s)
y
Analysis
 We have
G ( s )  G( s )(1  w ( s ) ( s ))
p
m
m
 G(s)  w ( s )G( s ) ( s )
m
m

m 
 Assume that the nominal plant is stable
 Using Nyquist stability condition, we need
w ( s )G( s )  1  G( s ) 
m
 Or
w ( s )G( s )
 1 
1  G( s )
m
1
 We have
S ( s )  [1  K ( s )G( s )]
1
T ( s )  K ( s )G( s )[1  K ( s )G( s )]
1
S (s)  T (s)  1
 For robust stability, we want
w ( s ) K ( s )G( s )
 1 
1  K ( s )G( s )
m
 using H-inf we have
 Or w ( s)T ( s )  1, and
m
w ( s )T ( s )  1
m

Robust Performance
 We find the bounds on the Sensitivity Function S and/or
Complimentary Sensitivity Function T for the given bounds
on Disturbance or Measurement noise
Doyle’s Theorem
 A necessary and sufficient condition for robust performance
is to satisfy the condition
W1S

W
T
2


1
Books
 Prabha Kundur “Power System Stability & Control” Tata McGrawHill,




1994|2012
Richard C Dorf & Robert H Bishop, “Modern Control Systems” Addison
Wesley, 1999
Pierre R. Belanger, “Control Engineering: A Modern Approach” Saunders
College Publishing, 1995
John Dorsey, “Continuous & Discrete Time Control Systems”, McGrawHill
International, 2002
Vladimir Zakian, “Control Systems Design-A new Framework”, Springer 2005