National Instruments 370760B-01 User Manual

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XmathTM Xµ Manual

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April 2004 Edition

Part Number 370760B-01

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Contents

1 Introduction 1

1.1 Notation..................................... 1

1.2 Manual Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 How to avoid really reading this Manual . . . . . . . . . . . . . . . . . . . 3

2 Overview of the Underlying Theory 5

2.1 Introduction................................... 5

2.1.1 Notation................................. 6

2.1.2 AnIntroductiontoNorms....................... 8

2.2 Modeling Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.1 Perturbation Models for Robust Control . . . . . . . . . . . . . . . 13

2.2.2 Linear Fractional Transformations . . . . . . . . . . . . . . . . . . 17

2.2.3 Assumptions on P ,∆,andtheunknownsignals .......... 22

2.2.4 Additional Perturbation Structures . . . . . . . . . . . . . . . . . . 23

iii

iv CONTENTS

2.2.5 Obtaining Robust Control Models for Physical Systems . . . . . . 28

2.3 H

and H2DesignMethodologies ...................... 29

∞

2.3.1 H

2.3.2 Assumptions for the H

DesignOverview ......................... 31

∞

DesignProblem .............. 32

∞

2.3.3 A Brief Review of the Algebraic Riccati Equation . . . . . . . . . . 33

2.3.4 Solving the H

2.3.5 Further Notes on the H

2.3.6 H

DesignOverview.......................... 40

2

2.3.7 Details of the H

Design Problem for a Special Case . . . . . . . . . 36

∞

Design Algorithm . . . . . . . . . . . . . 38

∞

DesignProcedure ................. 40

2

2.4 µ Analysis.................................... 42

2.4.1 MeasuresofPerformance ....................... 42

2.4.2 Robust Stability and µ ......................... 44

2.4.3 RobustPerformance .......................... 46

2.4.4 Properties of µ ............................. 47

2.4.5 TheMainLoopTheorem ....................... 49

2.4.6 State-spaceRobustnessAnalysisTests................ 51

2.4.7 Analysis with both Real and Complex Perturbations . . . . . . . . 58

2.5 µ Synthesis and D-K Iteration ........................ 58

2.5.1 µ-Synthesis ............................... 58

2.5.2 The D-K Iteration Algorithm . . . . . . . . . . . . . . . . . . . . . 60

CONTENTS v

2.6 ModelReduction................................ 64

2.6.1 Truncation and Residualization . . . . . . . . . . . . . . . . . . . . 65

2.6.2 BalancedTruncation.......................... 65

2.6.3 HankelNormApproximation ..................... 68

3 Functional Description of Xµ 71

3.1 Introduction................................... 71

3.2 DataObjects .................................. 71

3.2.1 Dynamic Systems .......................... 72

3.2.2 pdms................................... 74

3.2.3 Subblocks: selecting input & outputs . . . . . . . . . . . . . . . . . 77

3.2.4 BasicFunctions............................. 78

3.2.5 Continuous to Discrete Transformations . . . . . . . . . . . . . . . 81

3.3 MatrixInformation,DisplayandPlotting .................. 81

3.3.1 Information Functions for Data Objects . . . . . . . . . . . . . . . 81

3.3.2 FormattedDisplayFunctions ..................... 82

3.3.3 PlottingFunctions ........................... 82

3.4 SystemResponseFunctions .......................... 85

3.4.1 CreatingTimeDomainSignals .................... 85

3.4.2 Dynamic System TimeResponses ................. 85

3.4.3 FrequencyResponses.......................... 88

vi CONTENTS

3.5 SystemInterconnection ............................ 91

3.6 H

and H∞AnalysisandSynthesis...................... 95

2

3.6.1 ControllerSynthesis .......................... 95

3.6.2 SystemNormCalculations ...................... 105

3.7 Structured Singular Value (µ) Analysis and Synthesis . . . . . . . . . . . 107

3.7.1 Calculation of µ ............................ 107

3.7.2 The D-K Iteration........................... 110

3.7.3 Fitting D Scales ............................ 112

3.7.4 Constructing Rational Perturbations . . . . . . . . . . . . . . . . . 120

3.7.5 BlockStructuredNormCalculations................. 121

3.8 ModelReduction................................ 121

3.8.1 Truncation and Residualization . . . . . . . . . . . . . . . . . . . . 122

3.8.2 BalancedRealizations ......................... 123

3.8.3 Hankel Singular Value Approximation . . . . . . . . . . . . . . . . 125

4 Demonstration Examples 127

4.1 TheHimatExample .............................. 127

4.1.1 ProblemDescription.......................... 127

4.1.2 State-spaceModelofHimat...................... 128

4.1.3 Creating a Weighted Interconnection Structure for Design . . . . . 131

4.1.4 H

Design ............................... 133

∞

CONTENTS vii

4.1.5 µ Analysis of the H

Controller ................... 138

∞

4.1.6 Fitting D-scales for the D-K Iteration................ 140

4.1.7 DesignIteration#2 .......................... 143

4.1.8 Simulation Comparison with a Loopshaping Controller . . . . . . . 146

4.2 A Simple Flexible Structure Example . . . . . . . . . . . . . . . . . . . . 153

4.2.1 TheControlDesignProblem ..................... 153

4.2.2 Creating the Weighted Design Interconnection Structure . . . . . . 155

4.2.3 Design of an H

Controller...................... 162

∞

4.2.4 RobustnessAnalysis .......................... 165

4.2.5 D-K Iteration ............................. 168

4.2.6 ASimulationStudy .......................... 173

5 Bibliography 192

6 Function Reference 201

6.1 Xµ Functions .................................. 201

6.2 Xµ Subroutines and Utilities . . . . . . . . . . . . . . . . . . . . . . . . . 377

Appendices 391

A Translation Between Matlab µ-Tools and Xµ ............... 391

A.1 DataObjects .............................. 392

A.2 Matrix Information, Display and Plotting . . . . . . . . . . . . . . 397

viii CONTENTS

A.3 SystemResponseFunctions...................... 398

A.4 SystemInterconnection ........................ 399

A.5 ModelReduction............................ 399

A.6 H

and H∞AnalysisandSynthesis ................. 399

2

A.7 Structured Singular Value (µ) Analysis and Synthesis . . . . . . . 400

Chapter 1

Introduction

Xµ is a suite of Xmath functions for the modeling, analysis and synthesis of linear
robust control systems. Robust control theory has developed rapidly during the last
decade to the point where a useful set of computational tools can be used to solve a wide
range of control problems. This theory has already been applied to a wide range of
practical problems.

This manual describes the Xµ functions and presents a demonstration of their
application. The underlying theory is outlined here and further theoretical details can
be found in the many references provided.

It is assumed that the reader is familiar with the use of Xmath; the Xmath Basics
manual and the on-line demos are a good way of getting started with Xmath. A good
knowledge of control theory and application is also assumed. The more that is known
about robust control theory the better as the details are not all covered here.

1.1 Notation

Several font types or capitalization styles are used to distinguish between data objects.
The following table lists the various meanings.

1

2 CHAPTER 1. INTRODUCTION

Notation Meaning

pdm Xmath parameter dependent matrix data object
Dynamic System Xmath dynamic system data object

Code examples and function names are set in typewriter font to distinguish them from
narrative text.

1.2 Manual Outline

Chapter 2 outlines the applicable robust control theory. Perturbation models and linear
fractional transformations form the basis of the modeling framework. The discussion is
aimed at an introductory level and not all of the subtleties are covered. The theory
continues with an overview of the H
elsewhere for detail of the theory. The robust control methodology covered here is based
on the analysis of systems with perturbations. This is covered in some detail as such an
understanding is required for effective use of this software. Repeated analysis can be
used to improve upon the synthesis; this takes us from the standard H
to the more sophisticated µ-synthesis techniques.

design technique. Again the reader is referred

∞

design method

∞

The translation between the theoretical concepts and the use of the software is made in
Chapter 3. The means of performing typical robust control calculations are discussed in
some detail. This chapter also serves to introduce the Xµ functions. The discussion is
extended to include some of the relevant Xmath functions. A prior reading of Chapter 2
is helpful for putting this material in context.

The best means of getting an idea of the use of the software is to study completed design
examples, given in Chapter 4. These currently includes a design study for an aerospace
application. Typical modeling, analysis, synthesis, and simulation studies are illustrated.
These studies can be used as initial templates for the user’s application.

Chapter 6 is a function reference guide containing a formal description of each function.
This is similar to that given via the on-line help capability. Functions are listed in
relevant groupings at the start of the chapter. This gives an overview of some of the
software capabilities.

1.3. HOW TO AVOID REALLY READING THIS MANUAL 3

1.3 How to avoid really reading this Manual

The layout of the manual proceeds from introduction to background to syntax detail to
application descriptions. This may be tediously theoretical for some. If you are one of
those that considers reading the manual as the option of last resort
the applications (Chapter 4). If you have no prior Xmath experience then skimming
through Chapter 3 is essential. After running the demos and getting a feel for what the
software can do look briefly through the theory section.

1

then go directly to

1

And it seems that you are now exercising that option

Chapter 2

Overview of the Underlying
Theory

2.1 Introduction

The material covered here is taken from a variety of sources. The basic approach is
described by Doyle [1, 2], and further elaborated upon by Packard [3]. Summaries have
also appeared in work by Smith [4] and others.

Motivating background can be found in the early paper by Doyle and Stein [5]. An
overview of the robust control approach, particularly for process control systems, is
given by Morari and Zafiriou [6]. The reader can also find a description of the H
synthesis robust control approach in [7].

/µ

∞

There are a number of descriptions of this approach to practical problems. In the last
few years a significant number of these have been described in the proceedings of the
American Control Conference (ACC) and the IEEE Control and Decision Conference
(CDC). Only some of the early illustrative examples are cited here.

Application of µ synthesis to a shuttle control subsystem is given by Doyle et al. [8].
Examples of flexible structure control are described by Balas and
coworkers [9, 10, 11, 12] and Smith, Fanson and Chu [13, 14]. There have also been

5

6 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY

several studies involving process control applications, particularly high purity distillation
columns. These are detailed by Skogestad and Morari in [15, 16, 17, 18]

Section 2.2 introduces robust control perturbation models and linear fractional
transformations. Weighted H

design is covered in Section 2.3. The analysis of closed

∞

loop systems with the structured singular value (µ) is overviewed in Section 2.4.
Section 2.5 discusses µ synthesis and the D-K iteration. Model reduction is often used
to reduce the controller order prior to implementation and this is covered in Section 2.6.

2.1.1 Notation

We will use some fairly standard notation and this is given here for reference.

R set of real numbers
C set of complex numbers

n

R

n

C

n×m

R

n×m

C

I

n

0 matrix (or vector or scalar) of zeros of appropriate dimension

set of real valued vectors of dimension n × 1
set of complex valued vectors of dimension n × 1
set of real valued matrices of dimension n × m
set of complex valued matrices of dimension n × m
identity matrix of dimension n × n

The following apply to a matrix, M ∈C

M
M

T

∗

transpose of M
complex conjugate transpose of M

n×m

.

|M| absolute value of each element of M (also applies if M is a vector or scalar)
Re{M} real part of M
Im{M} imaginary part of M
dim(M) dimensions of M

σ

(M) maximum singular value of M

max

(M) minimum singular value of M

σ

min

M

ij

element of M in row i, column j. (also used for the i,j partition of a previously defined

partition of M)

(M) an eigenvalue of M

λ

i

ρ(M) spectral radius (max

i|λi

(M)|)

kMk norm of M (see section 2.1.2 for more details)

2.1. INTRODUCTION 7

-

z v

 

y

P

P

11

21



P

P

12

22



u

Figure 2.1: The generic robust control model structure

P

n

Trace(M) trace of M (

i=1

Mii)

Block diagrams will be used to represent interconnections of systems. Consider the
example feedback interconnection shown in Fig. 2.1. Notice that P has been partitioned
into four parts. This diagram represents the equations,

z = P
y = P

v + P12u

11

v + P22u

21

v =∆z.

This type of diagram (and the associated equations) will be used whenever the objects

P , z, y, etc., are well defined and compatible. For example P could be a matrix and z,
y, etc., would be vectors. If P represented a dynamic system then z, y, etc., would be

signals and

y = P

v + P22u,

21

8 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY

is interpreted to mean that the signal y is the sum of the response of system P
input signal v and system P

to input signal u. In general, we will not be specific about

22

to

21

the representation of the system P . If we do need to be more specific about P ,then
P(s) is the Laplace representation and p(t) is the impulse response.

Note that Figure 2.1 is drawn from right to left. We use this form of diagram because it
more closely represents the order in which the systems are written in the corresponding
mathematical equations. We will later see that the particular block diagram shown in
Figure 2.1 is used as a generic description of a robust control system.

In the case where we are considering a state-space representation, the following notation
is also used. Given P(s), with state-space representation,

sx(s)=Ax(s)+Bu(s)

y(s)=Cx(s)+Du(s),

we associate this description with the notation,



P (s)=

A

B

C D



.

The motivation for this notation comes from the example presented in Section 2.2.4. We
will also use this notation to for state-space representation of discrete time systems
(where s in the above is replaced by z). The usage will be clear from the context of the
discussion.

2.1.2 An Introduction to Norms

A norm is simply a measure of the size of a vector, matrix, signal, or system. We will
define and concentrate on particular norms for each of these entities. This gives us a
formal way of assessing whether or not the size of a signal is large or small enough. It
allows us to quantify the performance of a system in terms of the size of the input and
output signals.

Unless stated otherwise, when talking of the size of a vector, we will be using the

2.1. INTRODUCTION 9

Euclidean norm. Given,





x

1





.

x =

.



,



.

x

n

the Euclidean (or 2-norm) of x, denoted by kxk, is defined by,

!

|xi|

1/2

.

n

kxk =

X

i=1

Many other norms are also options; more detail on the easily calculated norms can be
found in the on-line help for the norm function. The term spatial-norm is often applied
when we are looking at norms over the components of a vector.

Now consider a vector valued signal,

x(t)=






x

x

1

n

(t)
.

.
.

(t)






.

As well as the issue of the spatial norm, we now have the issue of a time norm. In the
theory given here, we concentrate on the 2-norm in the time domain. In otherwords,

kx

(t)k =

i



Z

∞

−∞

|xi(t)|2dt



1/2

.

This is simply the energy of the signal. This norm is sometimes denoted by a subscript
of two, i.e. kx

(t)k2. Parseval’s relationship means that we can also express this norm in

i

the Laplace domain as follows,

kx

(s)k =

i



Z

∞

1

2π

|xi(ω)|2dω

−∞



1/2

.

10 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY

For persistent signals, where the above norm is unbounded, we can define a power norm,

!

1/2

. (2.1)

(t)k = lim

kx

i

T →∞

2T

Z

T

1

|xi(t)|2dt

−T

The above norms have been defined in terms of a single component, x

(t), of a vector

i

valued signal, x(t). The choice of spatial norm determines how we combine these
components to calculate kx(t)k. We can mix and match the spatial and time parts of the
norm of a signal. In practice it usually turns out that the choice of the time norm is
more important in terms of system analysis. Unless stated otherwise, kx(t)k implies the
Euclidean norm spatially and the 2-norm in the time direction.

Certain signal spaces can be defined in terms of their norms. For example, the set of
signals x(t), with kx(t)k




L

=x(t)

2



kx(t)k < ∞.



finite is denoted by L2. The formal definition is,

2

A similar approach can be taken in the discrete-time domain. Consider a sequence,

∞

{x(k)}

, with 2-norm given by,

k=0

∞

X

kx(k)k

=

2

|x(k)|

k=0

!

1/2

2

.

A lower case notation is used to indicate the discrete-time domain. All signals with finite
2-norm are therefore,




l

=x(k),k =0,...,∞

2

We can essentially split the space L
elements of L

which are analytic in the right-half plane. This can be thought of as

2



kx(k)k2<∞.



into two pieces, H2and H

2

⊥

. H2is the set of

2

those which have their poles strictly in the left half plane; i.e. all stable signals.

⊥

Similarly, H

are all signal with their poles in the left half plane; all strictly unstable

2

2.1. INTRODUCTION 11

signals. Strictly speaking, signals in H

2

or H

are not defined on the ω axis. However

2

⊥

we usually consider them to be by taking a limit as we approach the axis.

A slightly more specialized set is RL
strictly proper functions with no poles on the imaginary axis. Similarly we can consider

as strictly proper stable functions and RH

RH

2

poles in Re(s)< 0. The distinction between RL

, the set of real rational functions in L2. These are

2

⊥

as strictly proper functions with no

2

and L2is of little consequence for the

2

sorts of analysis we will do here.

The concept of a unit ball will also come up in the following sections. This is simply the
set of all signals (or vectors, matrices or systems) with norm less than or equal to one.
The unit ball of L

=x(t)

BL

2

, denoted by BL2, is therefore defined as,

2





kx(t)k2< 1.



Now let’s move onto norms of matrices and systems. As expected the norm of a matrix
gives a measure of its size. We will again emphasize only the norms which we will
consider in the following sections. Consider defining a norm in terms of the maximum
gain of a matrix or system. This is what is known as an induced norm. Consider a
matrix, M , and vectors, u and y,where

y=Mu.

Define, kM k,by

kyk

kMk=max

u,kuk<∞

kuk

.

Because M is obviously linear this is equivalent to,

kMk =max

u,kuk=1

kyk.

The properties of kMk will depend on how we define the norms for the vectors u and y.
If we choose our usual default of the Euclidean norm then kMk is given by,

kMk = σ

max

(M),

12 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY

where σ

denotes the maximum singular value. Not all matrix norms are induced

max

from vector norms. The Froebenius norm (square root of the sum of the squares of all
matrix elements) is one such example.

Now consider the case where P(s) is a dynamic system and we define an induced norm
from L

to L2as follows. In this case, y(s) is the output of P (s)u(s)and

2

ky(s)k

kP(s)k=max

u(s)∈L

2

ku(s)k

2

.

2

Again, for a linear system, this is equivalent to,

kP (s)k =max

u(s)∈BL

This norm is called the ∞-norm, usually denoted by kP (s)k

ky(s)k2.

2

. In the single-input,

∞

single-output case, this is equivalent to,

kP (s)k

= ess supω|P (ω)|.

∞

This formal definition uses the term ess sup, meaning essential supremum. The
“essential” part means that we drop all isolated points from consideration. We will
always be considering continuous systems so this technical point makes no difference to
us here. The “supremum” is conceptually the same as a maximum. The difference is
that the supremum also includes the case where we need to use a limiting series to
approach the value of interest. The same is true of the terms “infimum” (abbreviated to
“inf”) and “minimum.” For practical purposes, the reader can think instead in terms of
maximum and minimum.

Actually we could restrict u(s) ∈H

in the above and the answer would be the same. In

2

other words, we can look over all stable input signals u(s) and measure the 2-norm of
the output signal, y(s). The subscript, ∞, comes from the fact that we are looking for
the supremum of the function on the ω axis. Mathematicians sometimes refer to this
norm as the “induced 2-norm.” Beware of the possible confusion when reading some of
the mathematical literature on this topic.

If we were using the power norm above (Equation 2.1) for the input and output norms,
the induced norm is still kP (s)k

.

∞

2.2. MODELING UNCERTAIN SYSTEMS 13

The set of all systems with bounded ∞-norm is denoted by L
into stable and unstable parts. H
finite for all Re(s)> 0. This is where the name “H
often call this norm the H
functions, so RL
Similary, RH

is the set of proper transfer functions with no poles on the ω axis.

∞

is the set of proper, stable transfer functions.

∞

-norm. Again we can restrict ourselves to real rational

∞

denotes the stable part; those systems with |P (s)|

∞

control theory” originates, and we

∞

. We can again split this

∞

Again, we are free to choose a spatial norm for the input and output signals u(s)and
y(s). In keeping with our above choices we will choose the Euclidean norm. So if P (s)is

a MIMO system, then,

kP (s)k

=supωσ

∞

max

[P(ω)].

There is another choice of system norm that will arise in the following sections. This is
the H

-norm for systems, defined as,

2

kP (s)k

where P (ω)



Z

∞

1

=

2

2π

∗

denotes the conjugate transpose of P (ω) and the trace of a matrix is the

Trace[P(ω)∗P (ω)]dω

−∞



1/2

,

sum of its diagonal elements. This norm will come up when we are considering linear
quadratic Gaussian (LQG) problems.

2.2 Modeling Uncertain Systems

2.2.1 Perturbation Models for Robust Control

A simple example will be used to illustrate the idea of a perturbation model. We are
interested in describing a system by a set of models, rather than just a nominal model.
Our uncertainty about the physical system will be represented in an unknown
component of the model. This unknown component is a perturbation, ∆, about which
we make as few assumptions as possible; maximum size, linearity, time-invariance, etc..

Every different perturbation, ∆, gives a slightly different system model. The complete

14 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY

robust control model is therefore a set description and we hope that some members of
this set capture some of the uncertain or unmodeled aspects of our physical system.

For example, consider the “uncertain” model illustrated in Figure 2.2. This picture is
equivalent to the input-output relationship,

y =[(I+∆W

y

  

m)Pnom

] u. (2.2)



W

m

s

P

nom

u

+

?

m



Figure 2.2: Generic output multiplicative perturbation model

In this figure, ∆, W

m

and P

theory can be stated with these blocks as elements of H

are dynamic systems. The most general form for the

nom

. For the purposes of

∞

calculation we will be dealing with Xmath Dynamic Systems, and in keeping with this
we will tend to restrict the theoretical discussion to RH

, stable, proper real rational

∞

transfer function matrices.

The only thing that we know about the perturbation, ∆, is that k∆k
with k∆k

≤ 1 gives a different transfer function between u and y. The set of all

∞

≤ 1. Each ∆,

∞

possible transfer functions, generated in this manner, is called P. More formally,





k∆k

P =(I +∆W

m)Pnom



≤ 1. (2.3)

∞

Now we are looking at a set of possible transfer functions,

y(s)=P(s)u(s),

where P (s) ∈P.

Equation 2.2 represents what is known as a multiplicative output perturbation
structure. This is perhaps one of the easiest to look at initially as W (s) can be viewed

2.2. MODELING UNCERTAIN SYSTEMS 15

as specifying a maximum percentage error between P
The system P

(s) is the element of P that comes from ∆ = 0 and is called the

nom

and every other element of P.

nom

nominal system. In otherwords, for ∆ = 0, the input-output relationship is
y(s)=P
nominal system is multiplied by (I +∆W

(s) u(s). As ∆ deviates from zero (but remains bounded in size), the

nom

(s)). Wm(s) is a frequency weighting function

m

which allows us the specify the maximum effect of the perturbation for each frequency.
Including W
normalization of ∆ is simply included in W

(s) allows us to model P with ∆ being bounded by one. Any

m

(s).

m

We often assume that ∆ is also linear and time-invariant. This means that ∆(ω)is
simply an unknown, complex valued matrix at each frequency, ω.Ifk∆k
each frequency, σ

(∆(ω)) ≤ 1. Section 2.2.3 gives a further discussion on the pros

max

≤1, then, at

∞

and cons of considering ∆ to be linear, time-invariant.

Now consider an example of this approach from a Nyquist point of view. A simple first
order SISO system with multiplicative output uncertainty is modeled as

y(s)=(I+W

(s)∆)P

m

nom

(s)u(s),

where

P

nom

(s)=

1+0.05s

1+s

and W

m

(s)=

0.1+0.2s
1+0.05s

.

Figure 2.3 illustrates the set of systems generated by a linear time-invariant ∆,

k∆k

≤ 1.

∞

At each frequency, ω, the transfer function of every element of P, lies within a circle,
centered at P

(ω), of radius |P

nom

(ω)Wm(ω)|. Note that for certain frequencies the

nom

disks enclose the origin. This allows us to consider perturbed systems that are
non-minimum phase even though the nominal system is not.

It is worth pointing out that P is still a model; in this case a set of regions in the
Nyquist plane. This is model set is now able to describe a larger set of system behaviors
than a single nominal model. There is still an inevitable mismatch between any model
(robust control model set or otherwise) and the behaviors of a physical system.

16 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY

1

0.5

0

Imaginary

-0.5

-1
0 0.5 1-0.5 1.5

Real

Figure 2.3: Nyquist diagram of the set of systems, P

2.2. MODELING UNCERTAIN SYSTEMS 17





W

a

y



?

j

u u j



+

P

0

Figure 2.4: Unity gain negative feedback for the example system, P0+∆W

u r



+

,

6

a

2.2.2 Linear Fractional Transformations

A model is considered to be an interconnection of lumped components and perturbation
blocks. In this discussion we will denote the input to the model by u, which can be a
vector valued signal representing input signals such as control inputs, disturbances, and
noise. The outputs signal, denoted in this discussion by y, are also vector valued and can
represent system outputs and other variables of interest.

In order to treat large systems of interconnected components, it is necessary to use a
model formulation that is general enough to handle interconnections of systems. To
illustrate this point consider an affine model description:

y =(P

where u is the input and y is the output. ∆ again represents an unknown but bounded
perturbation. This form of perturbed model is known as an additive perturbation
description. While such a description could be applied to a large class of linear systems,
it is not general enough to describe the interconnection of models. More specifically, an
interconnection of affine models is not necessarily affine. To see this, consider unity gain
positive feedback around the above system. This is illustrated in Figure 2.4.

+∆Wa)u, k∆k∞≤ 1, (2.4)

0

The new input-output transfer function is

y =(P

+∆Wa)[I +(P0+∆Wa)]−1r. (2.5)

0

It is not possible to represent the new system with an affine model. Note that stability
questions arise from the consideration of the invertibility of [I +(P

+∆Wa)].

0

18 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY



1

-

z v

.

.

.



m

P

11

 

y

P

21

P

P

12

22



u

Figure 2.5: Generic LFT model structure including perturbations,∆

A generic model structure, referred to as a linear fractional transformation (LFT),
overcomes the difficulties outlined above. The LFT model is equivalent to the
relationship,



y =P

∆(I − P11∆)−1P12+ P

21

u, (2.6)

22

where the ∆ is the norm bounded perturbation. Figure 2.5 shows a block diagram
equivalent to the system described by Equation 2.6. Because this form of interconnection
is widely used, we will give it a specific notation. Equation 2.6 is abbreviated to,

y = F

(P, ∆)u.

u

The subscript, u, indicates that the ∆ is closed in the upper loop. We will also use

(., .) when the lower loop is closed.

F

l

In this figure, the signals, u, y, z and v can all be vector valued, meaning that the
partitioned parts of P ,(P

, etc.) can themselves be matrices of transfer functions.

11

To make this clear we will look at the perturbed system example, given in Equation 2.4,

2.2. MODELING UNCERTAIN SYSTEMS 19

in an LFT format. The open-loop system is described by,

u(Polp

, ∆)u,

y = F

where

0 W

IP

0



a

.



=

P

olp

The unity gain, negative feedback configuration, illustrated in Figure 2.4 (and given in
Equation 2.5) can be described by,

u(Gclp

, ∆)r,

y = F

where



(I + P0)−1Wa(I + P0)

−W

=

G

clp

a

(I + P0)

−1

P0(I + P0)

−1

−1



Figure 2.5 also shows the perturbation, ∆ as block structured. In otherwords,

∆ = diag(∆

,...,∆m). (2.7)

1

This allows us to consider different perturbation blocks in a complex interconnected
system. If we interconnect two systems, each with a ∆ perturbation, then the result can
always be expressed as an LFT with a single, structured perturbation. This is a very
general formulation as we can always rearrange the inputs and outputs of P to make ∆
block diagonal.

The distinction between perturbations and noise in the model can be seen from both
Equation 2.6 and Figure 2.5. Additive noise will enter the model as a component of u.
The ∆ block represents the unknown but bounded perturbations. It is possible that for
some ∆, (I − P

∆) is not invertible. This type of model can describe nominally stable

11

systems which can be destabilized by perturbations. Attributing unmodeled effects
purely to additive noise will not have this characteristic.

20 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY

The issue of the invertibility of (I − P

∆) is fundamental to the study of the stability of

11

a system under perturbations. We will return to this question in much more detail in
Section 2.4. It forms the basis of the µ analysis approach.

Note that Equation 2.7 indicates that we have m blocks, ∆

, in our model. For

i

notational purposes we will assume that each of these blocks is square. This is actually
without loss of generality as in all of the analysis we will do here we can square up P by
adding rows or columns of zeros. This squaring up will not affect any of the analysis
results. The software actually deals with the non-square ∆ case; we must
specify the input and output dimensions of each block.

The block structure is a m-tuple of integers, (k

block. It is convenient to define a set, denoted here by ∆, with the appropriate block

∆

i

,...,km), giving the dimensions of each

1

structure representing all possible ∆ blocks, consistent with that described above. By
this it is meant that each member of the set of ∆ be of the appropriate type (complex
matrices, real matrices, or operators, for example) and have the appropriate dimensions.
In Figure 2.5 the elements P

. For consistency the sum of the column dimensions of the ∆imust equal the row

∆

i

dimension of P

∆ =ndiag (∆

. Now define ∆ as

11

,...,∆m)

1

It is assumed that each ∆
norm bound is one. If the input to ∆

and P12are not shown partitioned with respect to the

11




dim(∆i)=ki×k



is norm bounded. Scaling P allows the assumption that the

i

is ziand the output is vi,then

i

o

.

i

k = k∆izik≤kzik.

kv

i

It will be convenient to denote the unit ball of ∆, the subset of ∆ norm bounded by
one, by B∆. More formally



B∆ =n∆ ∈ ∆



k∆k≤1o.



Putting all of this together gives the following abbreviated representation of the
perturbed model,

y = F

(P,∆)u, ∆ ∈ B∆. (2.8)

u

2.2. MODELING UNCERTAIN SYSTEMS 21

w

?

W

n



W



u

y

+

?

j

?

j



+

u

P

nom

u



Figure 2.6: Example model: multiplicative output perturbation with weighted output
noise

References to a robust control model will imply a description of the form given in
Equation 2.8.

As a example, consider one of the most common perturbation model descriptions,
illustrated in Figure 2.6. This model represents a perturbed system with bounded noise
at the output.

The example model is given by,

y = W

n

The system W

w +(I+∆Wu)P

is a frequency dependent weight on the noise signal, w. This allows us

n

nom

u.

to use a normalized representation for w. In other words the model includes the
assumption that kwk

≤ 1. Similarly, we assume that k∆k∞≤ 1andWuis a frequency

∞

dependent weight which specifies the contribution of the perturbation at each frequency.
In a typical model W
nominal output) and W

will be small (assuming that the noise is small compared to the

n

will increase at high frequencies (to capture the likely case that

u

we know less about the model at higher frequencies). The LFT representation of this
model is,

where

y = F

P =

h

i

(P, ∆)

u

w

,

u

"

00W

IW

n

P

P

u

nom

nom

#

22 CHAPTER 2. OVERVIEW OF THE UNDERLYING THEORY

Robust control models are therefore set descriptions. In the analysis of such models it is
also assumed that the unknown inputs belong to some bounded set. Several choices of
set for the unknown signals can be made, leading to different mathematical problems for
the analysis. Unfortunately not all of them are tractable. The following section discusses
the assumptions typically applied to the robust control models.

2.2.3 Assumptions on P , ∆, and the unknown signals

It will be assumed that the elements of P are either real-rational transfer function
matrices or complex valued matrices. The second case arises in the frequency by
frequency analysis of systems.

In modeling a system, P

defines the nominal model. Input/output effects not

22

described by the nominal model can be attributed to either unknown signals which are
components of the model input (w in the previous example), or the perturbation ∆.
Unmodeled effects which can destabilize a system should be accounted for in ∆. The ∆
can loosely be considered as accounting for the following. This list is by no means
definitive and is only included to illustrate some of the physical effects better suited to
description with ∆.

• Unmodeled dynamics. Certain dynamics may be difficult to identify and there

comes a point when further identification does not yield significant design
performance improvement.

• Known dynamics which have been bounded and included in ∆ to simplify the

model. As the controller complexity depends on the order of the nominal model a
designer may not wish to explicitly include all of the known dynamics.

• Parameter variations in a differential equation model. For example linearization

constants which can vary over operating ranges.

• Nonlinear or inconsistent effects. At some point a linear model will no longer

account for the residual differences between the behaviors of the model and the
physical system.

Several assumptions on ∆ are possible. In the most general case ∆ is a bounded
operator. Alternatively ∆ can be considered as a linear time varying multiplier. This
assumption can be used to capture nonlinear effects which shift energy between
frequencies. Analysis and synthesis are possible with this assumption; Doyle and