Loop shaping Robust Control Free ebook download

Loop-shaping Robust Control

Loop-shaping

Robust Control

Philippe Feyel

Series Editor

Bernard Dubuisson First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

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ISBN: 978-1-84821-465-1

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Table of Contents

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Chapter 1. The Loop-shaping Approach . . . . . . . . . . . . . . . . . . . . . .

2.1.3. Reconstruction of a transfer function from its coprime factors . . .

2.1.2. Practical calculation of normalized coprime factorizations . . . . .

2.1.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

33 2.1. The formalism of coprime factorizations . . . . . . . . . . . . . . . . . .

30 Chapter 2. Loop-shaping H   Synthesis . . . . . . . . . . . . . . . . . . . . . . .

29 1.5. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4.4. Example 4: stability robustness in relation to

system uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4.3. Example 3: issue of flexible modes and high-

frequency disturbances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4.2. Example 2: reference tracking and friction rejection . . . . . . . . .

1 1.1. Principle of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.4.1. Example 1: sinusoidal disturbance rejection . . . . . . . . . . . . . .

17 1.4. Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16 1.3. Limitations inherent to bandwidth . . . . . . . . . . . . . . . . . . . . . .

1.2.2. Phase and gain margins at the model’s input: . . . . . . . . . . . . .

1.2.1. Phase and gain margins at the model’s output . . . . . . . . . . . . .

8 1.2. Generalized phase and gain margins . . . . . . . . . . . . . . . . . . . . .

1.1.4. Declination of the robustness objectives . . . . . . . . . . . . . . . .

1.1.3. Declination of performance objectives . . . . . . . . . . . . . . . . .

1.1.2. Sensitivity functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi Loop-shaping Robust Control

2.1.4. Set of stabilizing controllers – Youla parameterization

of stabilizing controllers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37 2.2. Robustness of normalized coprime factor plant descriptions . . . . . . .

2.2.1. Taking account of modeling uncertainties . . . . . . . . . . . . . . .

2.2.2. Stability robustness for a coprime factor plant description. . . . . .

2.2.3. Property of the equivalent “weighted mixed sensitivity” form . . .

2.2.4. Expression of the synthesis criterion in “4-blocks”

equivalent form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.3. Explicit solution of the problem of robust stabilization of coprime factor plant descriptions . . . . . . . . . . . . . . . . . . . . . . . .

2.3.1. Expression of the problem by the Youla parameterization. . . . . .

2.3.2. Explicit resolution of the robust stabilization problem . . . . . . . .

57 2.4. Robustness and -gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4.1. -gap and ball of plants . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4.2. Robustness results associated with the -gap . . . . . . . . . . . . .

79 2.5. Loop-shaping synthesis approach. . . . . . . . . . . . . . . . . . . . . . .

2.5.1. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.2. Loop-shaping H  synthesis . . . . . . . . . . . . . . . . . . . . . . . .

2.5.3. Associated fundamental robustness result . . . . . . . . . . . . . . .

2.5.4. Phase margin and gain margin . . . . . . . . . . . . . . . . . . . . . .

2.5.5. 4-blocks interpretation of the method . . . . . . . . . . . . . . . . . .

2.5.6. Practical implementation . . . . . . . . . . . . . . . . . . . . . . . . .

2.5.7. Examples of implementation . . . . . . . . . . . . . . . . . . . . . . . 100

2.6. Discrete approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

2.6.1. Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

2.6.2. Discrete approach to loop-shaping H  synthesis . . . . . . . . . . . 121

2.6.3. Example of implementation. . . . . . . . . . . . . . . . . . . . . . . . 127

Chapter 3. Two Degrees-of-Freedom Controllers . . . . . . . . . . . . . . . . 135

3.1. Principle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

3.1.1. Reference tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

3.1.2. Parameterization of 2-d.o.f. controllers . . . . . . . . . . . . . . . . . 141

3.2. Two-step approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

3.2.1. General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

3.2.2. Simplification of the problem by the Youla parameterization. . . . 145

3.2.3. Extension. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

3.2.4. Setting of the weighting functions . . . . . . . . . . . . . . . . . . . . 152

3.2.5. Associated performance robustness result . . . . . . . . . . . . . . . 154

3.3. One-step approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

3.3.1. General formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

3.3.2. Expression of the problem by Youla parameterization . . . . . . . . 158

3.3.3. Associated performance robustness result . . . . . . . . . . . . . . . 161

Table of Contents vii

3.3.4. Connection between the approach and loop-shaping synthesis . . . 163

3.4. Comparison of the two approaches . . . . . . . . . . . . . . . . . . . . . . 165

3.5. Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

3.5.1. Optimization of an existing controller (continued) – scanning . . . 166

3.6. Compensation for a measurable disturbance at the model’s output . . . 174

3.6.1. Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

3.6.2. Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

Chapter 4. Extensions and Optimizations . . . . . . . . . . . . . . . . . . . . . 187

4.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

4.2. Fixed-order synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

4.2.1. Fixed-order robust stabilization of a coprime factor plant description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

4.2.2. Optimization of the order of the final controller. . . . . . . . . . . . 197

4.2.3. Example: fixed-order robust multivariable synthesis . . . . . . . . . 214

4.3. Optimal setting of the weighting functions . . . . . . . . . . . . . . . . . 220

4.3.1. Weight setting on the basis of a frequency specification . . . . . . . 220

4.3.2. Optimal weight tuning using stochastic optimization and metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

4.4. Towards a new approach to loop-shaping fixed-order controller synthesis, etc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

4.4.1. Taking account of objectives of stability robustness . . . . . . . . . 243

4.4.2. Taking account of objectives of performance robustness . . . . . . 244 A

PPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

Appendix 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

Appendix 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Introduction

I.1 Presentation of the book

In an increasingly competitive industrial context, an automation engineer has to

apply servo-loops in accordance with ever more complex sets of functional

specifications, associated with increasingly broad conditions of usage. In addition to

this, the product is often destined for large-scale production. Thus, the engineer has

to be able to implement a robust servo-loop on a so-called “prototype”, whilst taking

account of this broad spectrum in its entirety, at the very earliest stage of design.

An example of such a system, upon which most of the examples given in this

book are based, is a mass-produced viewfinder, for which the automation engineer

has to inertially stabilize the line of sight, whose usage conditions may be extremely

varied – indeed there are often as many potential applications as there are types of

carriers (aircraft, ships, etc.). In addition, the viewfinder is required to deliver

increasingly high-end functionalities – e.g. target tracking, guidance, etc. In order to

moderate and reduce development costs, there is a growing tendency to carry out

so-called “generic” stabilizations. This is possible only if the servo-loop designed

has a certain degree of robustness, which needs to be taken into account as an a

priori constraint on synthesis.

In the 1990s, automation engineering made a great leap forward, with the emergence of H -based controller synthesis techniques: ∞

– Firstly, it became possible to obey a complex set of frequency specifications by
using frequency weighting functions on exogenous inputs and on monitored signals,

and then minimizing the H transfer norm between those signals by using a

∞

stabilizing controller whose state-space representation was explicitly formulated in

[DOY 89], inspired by a dichotomy in the solution of Riccati equations (the so-called “γ-iteration”) and based on the following standard form: x Loop-shaping Robust Control

z e

P(s)

u y

K(s)

Figure I.1. Standard form for control

where e represents the exogenous inputs (reference points, disturbances, etc.),

z represents the signals being monitored (error signals, commands, etc.) and

y represents the measurements used by the controller to calculate the command u.

– Secondly, the small-gain theorem gives us a necessary and sufficient condition
for the stability of the loop obtained for any uncertainty Δ(s) such that

−

1 ( ) s < . This is stable if and only if (iff) T s , and in this Δ γ ( ) < γ ez

∞ ∞

knowledge, we can take account of objectives of robustness during the synthesis

process.

Δ (s)

w v

T(s)

e z

Figure I.2. Standard form for robustness analysis

Thus, with the standard approach to robust control, the complexity of controller

calculus – hitherto usually based on examination of the open loop – is now reflected

in the complexity of determining the set of relevant frequency weights, which make

a crucially important contribution to the performances of the final controller. Owing

to the difficulty in calculating these weights, the know-how that this operation

requires and the conceptual difference from conventional frequency automation

engineering, certain engineers are deterred from using the standard approach to

robust control, preferring to employ more conventional open-loop concepts.

Introduction xi

However, at the same time, the world witnessed the publication of the explicit

solution to the robust stabilization of normalized coprime factor plant descriptions

[MCF 90], based on the following form. v

1 v

Δ (s) Δ M (s) N

N(s) M(s)

u y

K(s)

Figure I.3. Robust coprime factor plant description stabilization

– This method, which is highly attractive because of its simplicity, consists of
solving two LQG-type Riccati equations. In its 4-blocks equivalent representation, it

is a particular case of the standard H ∞ approach to robust control. Noting that we can

model the direct and complementary sensitivity functions by modeling the open-

loop response, and seeing that any loop transfer is proportional to those sensitivity

functions, it is therefore possible to model any loop transfer by working on a single

transfer – the open-loop response. This is the principle upon which loop-shaping

synthesis is founded. Drawing inspiration from frequency-shaped LQG synthesis,

we shape the singular values of the open-loop response using weighting functions on

the input and output of the system, thereby creating a loop-shape for which a

stabilizing controller can be calculated. This is the definition of H ∞ loop-shaping
synthesis.
– However, thanks to the notion of the gap metric (which expresses a distance
between two systems in mathematical terms) as well as the small-gain theorem, the

stability of the loop can be evaluated even before the controller has been explicitly
formulated.

There is a growing interest in H ∞ loop-shaping synthesis. Obviously, it is less

general than the standard H ∞ approach, because the number of degrees of freedom is

constrained by the dimensions of the system. However, the adjustment of the input

and output weighting functions on the basis of the concepts of conventional

frequency automation makes the loop-shaping technique extremely attractive and

easy to access – all the more so as it has the qualities of robustness which are inherent to H ∞ techniques. xii Loop-shaping Robust Control

In Chapter 1, we introduce the loop-shaping approach by showing how to obtain

a specification on the open-loop response of the servo-loop from a complex

frequency specification on multiple loop transfers. Chapter 2 introduces the robust

stabilization of a normalized coprime factor plant description. Along with the notion

of the gap metric which we then introduce, it constitutes the basis for robust H ∞

loop-shaping synthesis. Chapter 3 relates to two-degrees-of-freedom controllers

(2 d.o.f controllers), and two techniques that are closely linked to H ∞ loop-shaping

synthesis are presented, thus greatly extending the possibilities for the use of the

method. Finally, Chapter 4 opens up avenues for future work: it discusses the main

drawbacks to loop-shaping synthesis, and how to solve these issues using modern

optimization techniques.

I.2. Notations and definitions Below, we review a number of fundamental notions and notations that are frequently employed in the various chapters of this book.

I.2.1. Linear Time-Invariant Systems (LTISs)

I.2.1.1. Representation of LTISs An n-order linear time-invariant system with m inputs and p outputs is described

by a state-space representation defined by the following system of differential

equations: dx

= Ax t ( ) Bu t ( ), ( ) x t = x + dt

y t ( ) = Cx t ( ) Du t ( )

1 where : n

– x t ( ) ∈ R is the state of the system;
– x t ( ) is the initial condition;

– u t ( ) ∈ R is the system input;

– y t ( ) ∈ R is the system output;

n n × A R ∈ is the state matrix; – 1 The set of real numbers is denoted as R; the set of complex numbers is denoted as C.

– n m
– p n
– p m

−   = − +      

1 ( ) H s C sI A B D

− = − +

For the sake of convenience, we represent this as:

( )

1 :

A B C sI A B D C D

or:

eigenvalue of A. For a zero initial condition, the input/output transfer matrix of the system is defined in Laplace form by:

[ ] ( )

1 , , , : A B C D C sI A B D

− = − +

When H(∞) is bounded, H is said to be “proper”

. When H(∞)=0, then the system is said to be “strictly proper”, and D = 0.

( )

A λ is the i th

Introduction xiii

( ) ( ) ( ) t A t t A t t x t e x t e Bu d y t Cx t Du t

B R ×

∈ is the control matrix;

C R ×

∈ is the observation matrix;

D R × ∈ is the direct transfer matrix.

For a given initial condition x(t ), the evolution of the system’s state and its output is given by: ( ) ( ) ( ) ( ) ( )

τ τ τ

( ) i

− − = + = +

 The system is stable (in the sense that it has bounded input/bounded output) if the eigenvalues of A all have a strictly negative real part, i.e. if:

[ ] ( ) ( ) 1,...

max Re

i i n A λ

∈

< where

2 In the case of a SISO transfer, this means that the degree of the numerator is less than or equal to the degree of the denominator. xiv Loop-shaping Robust Control Finally, for the same transfer matrix, there are an infinite number of possible

n n ×

state-space representations. Indeed, consider the linear transformation ∈ ,

T R

where T is invertible, such that:

x Tx =

In this case, the initial state-space representation becomes:

−

= ( ) ( )

dx TAT x t TBu t

−

1 y t ( ) = CT x t ( ) + Du t ( )

The corresponding transfer function is:

−

1 − − −

1 CT sI TAT − TB D C sI A = − B D H s = ( ) + + ( )

( )

I.2.1.2. Controllability and observability of LTISs The system H or the pair (A,B) is said to be controllable if, for any initial condition x(t ) = x , for any t

1 > 0 and for any final state x 1 , there is a piecewise

continuous command u(.) which can change the state of the system to x(t

1 ) = x 1 .

We determine controllability by checking that for any value of t > t , the controllability Gramian W ( t) is positive definite:

c _T τ τ t A T A

W t ( ) : = e BB e d τ c

 t

2 n −

1 An equivalent condition is that the matrix must be full B AB A B  A B

( ) n.

row rank, i.e

H or the pair (C, A) is observable if, for any value of t

The system

1 > 0, the initial x(t x u(t) and of

state ) = can be determined by the past values of the control signal

y(t) in the interval [t t

the output , 1 ].

t > t

We determine observability by checking that, for any value of , the

W t) is positive definite:

observability Gramian ( _T o

t A τ T A τ W t ( ) : = e C Ce d τ o

 t

Introduction xv An equivalent condition is that the matrix:

 C   

  

2  CA

   

    −

1  n

  must be full column rank, i.e. n.

I.2.1.3. Elementary operations on LTISs Consider

H, the transfer system:  A B 

−

1 : = C sI A − B D

 

( )

C D    

The transpose of H is defined by the system:

T T

 

−

1 A C T T T T T

 

H s B sI A C D

( ) = − : =

T T

( )

 

B D

  The conjugate of H is defined by the system:

T T

 

−

− A − C

T T T T T

 

H s ( ) = H ( ) − = s B − − sI A C D : = ( )

T T

 

B D

  If D is invertible, the inverse of H is defined by the system:

− 1 −

 

A BD C BD

− −

−

 

H = − 1 −

 

D C D

  Now consider two systems H

1 and H 2 , whose respective state representations are:  A B   A B 

H = H =    

2 C D C D



1 1  

2 2      xvi Loop-shaping Robust Control The serial connection of H

1 with H 2 (or the product of H 1 by H 2 ) gives us the

system:

^{2 H}

¹  A B   A B 

2 H H =    

1 2 C D C D



1 1  

2 2     

   

A B C B D A B 1 1 2 1 2

   

= A B = B C A B D

   

2 1 2

1 1 2

   

C D C D D D C C D D 1 1 2 1 2 1 2 1 1 2

   

The parallel connection (or addition) of H

1 to H 2 gives us the following system:  A B   A B 

2 = + H H    

1 2 C D C D



1 1  

2 2       

A B

1  

A B =  

2  

C C D D +

2  

The looping of H

2 with feedback from H 1 gives us the system: H

1 H

Introduction xvii

− 1 − 1 −

 

A B D R C B R C B R

− −

1 1 2 12 1 1 21 2 1 21

 

−

1 − 1 − 1 −

 

I H H H = B R C A − B D R C + B D R

( 1 2 ) 1 2 12

1 2 2 1 21 2 2 1 21

 

− 1 − 1 −

 

R C − R D C D R

1 12 1 2 1 21

  where R = +

I D D and R = +

I D D .

12 1 2 21 2 1 Many notions about linear time invariant systems are explained in [ZHO 96].

I.2.2. Singular values

I.2.2.1. Definition The singular values of a transfer matrix H(s) of dimensions p×m are defined as

H(jω) by

the square roots of the eigenvalues of the product of its frequency response its conjugate:

T T H j = H j H − j = H − j H j

σ ω λ ω ω λ ω ω

( ) ( ) ( ) ( ) ( ) i ( ) i i

( ) ( ) i m p 

=  1, , min  ,

( )

 

The singular values are positive or null real numbers and can be classified. The largest singular value, also called the maximum singular value, is denoted as

σ H , and the smallest, also called the minimum singular value, is denoted as

( ) σ H .

( ) σ H j ω = σ H j ω ≥ σ H j ω ≥  ≥ σ H j ω

( ) ( ) ( ) ( )

( ) 1 ( ) 2 ( ) ( )

In the case of a monovariable system (i.e. m = p =1), the unique singular value is equal to the gain of the frequency response:

σ H j ω = σ H j ω = H j ω ( ) ( ) ( )

( ) ( )

Hence, the singular values extend the notion of gain established with monovariable systems to multivariable systems. We say that H is high-gain if

σ H is large and is low-gain if σ H is small.

( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

In the case of two serial systems, an important property is:

 

   

 

    ≤ ≤  

σ σ σ

  ≤ +  

H H H H H H H H H H H H H H

σ σ σ

σ σ σ σ σ

max , 2 max ,

max ,

σ σ σ σ σ σ σ σ σ σ

2 H H H H H H H H H H H H H H H H H H H H H H H H σ σ σ σ σ σ σ σ σ σ

1 2 1 2

2 or i i i i i i

1 2 1 2

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 2 1 2

In particular, we shall use the following specific cases: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

≤ ≤ ≤ ≤

H H H H H H H H H H H H σ σ σ σ σ σ σ σ σ σ

xviii Loop-shaping Robust Control I.2.2.2.

T i i x C x x C x

1 max max ^m _m

if exists,

H x σ σ σ σ σ σ α α σ

( )

( ) ( ) ( ) ( )

( ) ( )

In this book, we make abundant use of the following properties:

Properties

H H H H H H H H H H H H H Hx H x Hx

σ σ σ σ

σ σ

− − − ∈ ≠ ∈ ≠

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

In the case of two parallel systems, we use the following properties:

= = = =

= =

= ⇔ = =

≤ ≤ ≤ ≤ ≤ ≤ ≤ ≤

− ≤ + ≤ + or indeed:

( )

( ) ( ) ( )

− − ≤ ≤ +

σ σ σ

1 I H 1 H H

which lead us to: ( ) ( ) ( )

− ≤ + ≤ + − ≤ + ≤ +

2 H H H H H H H H H H H H σ σ σ σ σ σ σ σ σ σ

H H 1 I H σ σ σ

Introduction xix

In the case of the sum of two systems, an important property is: ( ) ( ) ( ) ( ) ( )

2 i i i

H H H H H H σ σ σ σ σ − ≤ + ≤ + In particular, we shall use the following two specific cases:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

Finally, we use the following property:
>

2 H H H H

I.2.3. Subspace RH ∞ ∞ and H ∞ ∞ norm

ω ω

in C

is the subspace of the analytical and bounded functions of L ∞

H ∞ n

2 is the Euclidean norm.

where

∞ = < +∞

2 sup f f j

( ) ( ) ( )

( ) xx Loop-shaping Robust Control

to represent the set of vectorial functions f(s), s ∈ C of dimension n and bounded on the imaginary axis, i.e. which satisfy:

I.2.3.1. Definition We use the notation L ∞

The interested reader can find further discussion about inequalities on singular values in [MER 04].

σ σ σ < 

p×m RL ∞ is the subspace of rational proper transfer matrices of dimensions p×m with real coefficients and without pole on the imaginary axis.

3 p×m

RH ∞ is the subspace of rational stable proper transfer matrices of dimensions p×m with real coefficients. p×m

For any system H ∈ RH ∞ , the H ∞ norm of H is defined by:

H H j = sup σ ω

( ) ( )

∞ ω ∈ R Hence, this is the highest value of the system’s gain for the set of pulsations.

I.2.3.2. Properties The set of properties valid for the maximum singular value is also valid for the H ∞ norm.

In particular, in this book, we shall very frequently make use of the following properties:

H s H s ( ) ( ) ≤ H s ( ) H s ( )

2 ∞ ∞ ∞

 H s ( ) 

sup H s ( ) , H s ( )

2 ≤  

( ∞ ∞ )

H s ( )



2  ∞

sup H s ( ) , H s ( ) ≤ H s ( ) H s ( )

1 2 (

2 ) ( ∞ ∞ )

∞

Notably, this implies that: 

H s ( ) ≤

1 ∞



H s ( ) ≤

 γ

H s ( ) H s ( )

3 ∞

≤   γ

H s H s

( ) ( ) H s ( ) ≤

2 4 γ



3 ∞ ∞



H s ( ) ≤

 ∞ Finally:

 H s ( ) 

≤ + H s ( ) H s ( )  

∞ ∞ H s ( )



2  ∞

3 This means that they do not have a pole in C+.

I.2.4. Linear fractional transformation (LFT)

P P P C P P

I.2.4.1. Definition Consider a complex matrix P divided as follows: ₁ ₂ ₁ ₂

11 12 ( ) ( )

Introduction xxi

  = ∈

× +

22 p p q q

1 11 u

1 l z w P P w P y u P P u u y Δ

        = =

               

= Assuming that

( )

I P Δ

−

− exists, the upper LFT is defined by:

( ) ( )

u u u u F P P P

I P P Δ Δ Δ −

= + − which corresponds to the following block diagram, where the matrix Δ

re-loops P “from above”:

   

( ) ( )

Consider two other complex matrices ² ²

q p

∈

and ¹ ¹

q p u C

Δ ×

∈ .

Assuming that

( )

1 22 l

I P Δ

−

− exists, the lower linear fractional transformation (LFT) is defined by:

1 w

l l l l F P P P

I P P

Δ Δ Δ

−

= + − This corresponds to the following block diagram where the matrix Δ

re-loops P “from below”:

Δ l

1 z

1 y

    −     = =     − − −    

Consider M and Q, divided as follows:

I.2.4.2. Properties A fundamental property of LFTs is that the combination of several LFTs remains an LFT.

H H H H H H H H H

M N H H H H H H H H H − − − − − − − −

4 2 4 3 1 4 3 ,

= = 

     

I I

I I N M M M

11 M M

= where:

Δ Δ

u u F M F N

, ,

( ) ( )

The upper and lower LFTs are linked by the following equality:

       

    = =

M M Q Q M Q M M Q Q

xxii Loop-shaping Robust Control

2 u y u P P u P z w P P w u y

Loop shaping Robust Control Free ebook download

UK USA

ISBN: 978-1-84821-465-1

I.1 Presentation of the book

2 C D C D

I.2.2. Singular values

2 H H H H

I.2.3. Subspace RH ∞ ∞ and H ∞ ∞ norm

I.2.4. Linear fractional transformation (LFT)

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Loop shaping Robust Control Free ebook download

UK USA

ISBN: 978-1-84821-465-1

I.1 Presentation of the book

2 C D C D

I.2.2. Singular values

2 H H H H

I.2.3. Subspace RH ∞ ∞ and H ∞ ∞ norm

I.2.4. Linear fractional transformation (LFT)

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