National Instruments 370757C-01 User manual

NI MATRIXx

TM

Xmath

TM

Robust Control Module

MATRIXx Xmath Robust Control Module

April 2007

370757C-01

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Contents

Chapter 1

Introduction

Using This Manual.........................................................................................................1-1

Document Organization...................................................................................1-1

Bibliographic References ................................................................................1-2

Commonly-Used Nomenclature......................................................................1-2

Related Publications ........................................................................................1-2

MATRIXx Help...............................................................................................1-3

Overview........................................................................................................................1-3

Chapter 2

Robustness Analysis

Modeling Uncertain Systems.........................................................................................2-1

Stability Margin (smargin).............................................................................................2-3

smargin( ).........................................................................................................2-4

Worst-Case Performance Degradation (wcbode) ..........................................................2-8

wcbode( ).........................................................................................................2-9

Advanced Topics ...........................................................................................................2-10

Stability Margin...............................................................................................2-10

Stability Margin and Structured Singular

Values (µ) .......................2-10

Stability Margin Bounds Using Singular Values ..............................2-11

Approximation with Scaled Singular Values....................................2-12

ssv( ) ................................................................................................................2-15

osscale( )..........................................................................................................2-16

pfscale( ) ..........................................................................................................2-16

optscale( ) ........................................................................................................2-17

Reducibility....................................................................................................................2-17

Worst-Case Performance Degradation (wcgain) ...........................................................2-18

Conversion to a Stability Margin Problem......................................................2-18

wcgain( )..........................................................................................................2-19

Contents

MATRIXx Xmath Robust Control Module vi ni.com

Chapter 3

System Evaluation

Singular Value Bode Plots............................................................................................. 3-1

L Infinity Norm (linfnorm)............................................................................................ 3-3

linfnorm( ) ....................................................................................................... 3-4

Singular Value Bode Plots of Subsystems .................................................................... 3-7

perfplots( )....................................................................................................... 3-7

clsys( ) ............................................................................................................. 3-10

Chapter 4

Controller Synthesis

H-Infinity Control Synthesis ......................................................................................... 4-1

Problem Definition.......................................................................................... 4-1

Extended Transfer Matrix ............................................................................... 4-2

Building the Plant Model ................................................................................ 4-3

Weight Selection ............................................................................................. 4-5

Restrictions on the Extended Plant ................................................................. 4-7

hinfcontr( ) ...................................................................................................... 4-8

singriccati( ) .................................................................................................... 4-13

Linear-Quadratic-Gaussian Control Synthesis .............................................................. 4-14

LQG Frequency Shaping ................................................................................ 4-14

fsregu( ) ........................................................................................................... 4-14

fsesti( )............................................................................................................. 4-16

fslqgcomp( ) .................................................................................................... 4-17

Frequency-Shaped Control Design Commands.............................................. 4-17

Loop Transfer Recovery (lqgltr) ................................................................................... 4-22

lqgltr( ) ............................................................................................................ 4-23

Appendix A

Bibliography

Appendix B

Technical Support and Professional Services

Index

1

Introduction

The Xmath Robust Control Module (RCM) provides a collection of

analysis and synthesis tools that assist in the design of robust control

systems.

This chapter starts with an outline of the manual and some use notes. It

continues with an overview of the Xmath Robust Control Module (RCM)

functions.

Using This Manual

This manual provides complete documentation for all the RCM functions

along with their associated theoretical background, references, and

examples.

Document Organization

This manual includes the following chapters:

• Chapter 1, Introduction, describes the Robust Control Module (RCM)

and shows the RCM function structure.

• Chapter 2, Robustness Analysis, covers the robustness analysis

tools and introduces the concepts of uncertainty, robustness, and

performance degradation in the framework of closed-loop systems.

The Modeling Uncertain Systems section should be read by all those

interested in robustness analysis or performance degradation, which

are explained in the Stability Margin (smargin) section and the

Worst-Case Performance Degradation (wcbode) section. The

Advanced Topics section provides additional information but this

material is not prerequisite to the use of RCM functions.

• Chapter 3, System Evaluation, describes system analysis functions that

create singular value Bode plots, performance plots, and calculate the

L

∞

norm of a linear system. This chapter should be of interest to all

users.

• Chapter 4, Controller Synthesis, discusses synthesis tools in two

categories, H

∞

and H

2

. This manual does not attempt to explain all

of the theory of H

∞

, LQG/LTR, and frequency shaped LQG design

Chapter 1 Introduction

MATRIXx Xmath Robust Control Module 1-2 ni.com

techniques. The general problem setup is explained together with

known limitations; the rest is left to the references.

Bibliographic References

Throughout this document, bibliographic references are cited with

bracketed entries. For example, a reference to [DoS81] corresponds

to a document published by Doyle and Stein in 1981. For a table of

bibliographic references, refer to Appendix A, Bibliography.

Commonly-Used Nomenclature

This manual uses the following general nomenclature:

• Matrix variables are generally denoted with capital letters; vectors are

represented in lowercase.

• G(s) is used to denote a transfer function of a system where s is the

Laplace variable. G(q) is used when both continuous and discrete

systems are allowed.

• H(s) is used to denote the frequency response, over some range of

frequencies of a system where s is the Laplace variable. H(q) is used to

indicate that the system can be continuous or discrete.

• A single apostrophe following a matrix variable, for example, x

'

,

denotes the transpose of that variable. An asterisk following a matrix

variable (for example, A*) indicates the complex conjugate, or

Hermitian, transpose of that variable.

Related Publications

For a complete list of MATRIXx publications, refer to Chapter 2,

MATRIXx Publications, Help, and Online Support, of the MATRIXx

Getting Started Guide. The following documents are particularly useful

for topics covered in this manual:

• MATRIXx Getting Started Guide

• Xmath User Guide

• Xmath Control Design Module

• Xmath Interactive Control Design Module

• Xmath Interactive System Identification Module, Part 1

• Xmath Interactive System Identification Module, Part 2

• Xmath Model Reduction Module

Chapter 1 Introduction

• Xmath Optimization Module

• Xmath Robust Control Module

• Xmath X

μ

Module

MATRIXx Help

Robust Control Module function reference information is available in the

MATRIXx Help. The MATRIXx Help includes all Robust Control functions.

Each topic explains a function’s inputs, outputs, and keywords in detail.

Refer to Chapter 2, MATRIXx Publications, Help, and Online Support, of

the MATRIXx Getting Started Guide for complete instructions on using the

Help feature.

Overview

RCM functionality is structured as shown in Figure 1-1.

Chapter 1 Introduction

MATRIXx Xmath Robust Control Module 1-4 ni.com

Figure 1-1. RCM Function Structure

Many RCM functions are based on state-of-the-art algorithms implemented

in cooperation with researchers at Stanford University. The robustness

analysis functions are based on structured singular value calculations.

The synthesis tools expand on existing LQG (H

2

) techniques (LQG/LTR

and frequency shaping) while adding new H

∞

design functions.

perfplotslinfnorm

Utility Functions

Synthesis Functions

lqgltr

fslqgcomp

fsesti

fsregu

singriccati clsys

hinfcontr

Analysis Functions

smargin

wcbode

wcgain

ssv

pfscale optscale osscale

2

Robustness Analysis

This chapter describes RCM tools used for analyzing the robustness

of a closed-loop system. The chapter assumes that a controller has been

designed for a nominal plant and that the closed-loop performance of

this nominal system is acceptable. The goal of robustness analysis is to

determine whether the performance will remain acceptable if the plant

differs from the nominal plant.

Modeling Uncertain Systems

This section describes the method RCM uses to model an uncertain system.

The closed-loop system is modeled as a known or nominal closed-loop

system with input w and output z, together with k unknown or uncertain

transfer functions δ

1

(jω), …, δ

k

(jω), as shown in Figure 2-1.

Figure 2-1. Model of an Uncertain System

The following transfer functions are assumed to be stable:

(2-1)

where the l

i

are given non-negative functions of frequency. This type of

uncertainty model is known as structured nonparametric uncertainties.

To describe this model, you also must describe the nominal closed-loop

Uncertain Transfer Function

Known Nominal

System

w

z

q

1

r

1

δ

1

q

2

r

2

δ

2

δ

i

jω()l

i

jω()≤

Chapter 2 Robustness Analysis

MATRIXx Xmath Robust Control Module 2-2 ni.com

system, including how the uncertain transfer functions are connected to the

system and the magnitude bound functions l

i

(w).

To do this, extract the uncertain transfer functions and collect them into a

k-input, k-output transfer matrix Δ, where:

(2-2)

The resulting closed-loop system can be viewed as a feedback connection

of the nominal closed-loop system with transfer matrix H(jω) and the

uncertain transfer matrix Δ(jω). You describe your nominal closed-loop

system as a linear system with

input and output .

Note The signals r and q are not really inputs and outputs of the nominal system; r and q

show how the uncertain transfer functions connect to your nominal system. The signals r

and q each have k components.

You will partition H into the four submatrices,

so that H

zw

is the nominal transfer matrix from w to z, H

zr

is the nominal

transfer matrix from r to z, H

qw

is the nominal transfer matrix from w to q,

and H

qr

is the nominal transfer matrix from r to q.

The magnitude bound functions l

i

(jω) from Equation 2-1 are described

with the PDM

delb:

Thus, a complete description of your system requires the system

SysH

to represent H

jw

and the response delb to represent the bounds.

Δ jω() diagonal δ

1

jω(),...,δ

k

jω()()=

w

r

z

q

H

zw

H

zr

H

qw

H

qr

=

DELB

ω

1

:

ω

m

l

1

ω

1

()…l

k

ω

1

()

::

l

1

ω

m

()…l

k

ω

m

()

,

=

Chapter 2 Robustness Analysis

Stability Margin (smargin)

Assume that the nominal closed-loop system is stable. That belief raises a

question: Does the system remain stable for all possible uncertain transfer

functions that satisfy the magnitude bounds (Equation 2-1)? If so, the

system is said to be robustly stable. If the magnitude bounds are small

enough, the uncertainties will not destabilize the system; your system will

be robustly stable.

Roughly speaking, the stability margin of your system is defined as the

factor by which you can increase all the magnitude bounds l

i

and still

maintain stability for all possible uncertain transfer functions δ

i

. If this

number is larger than one (0 dB), then you know that there are no uncertain

transfer functions that satisfy the magnitude bound and destabilize your

system. Moreover, the number tells you how much more uncertainty your

system could tolerate than the given bounds l

i

(ω). If the margin is less than

one, then there are uncertain transfer functions that satisfy the magnitude

bound (Equation 2-1) and result in an unstable system. In this case, the

margin tells you how much you must reduce the magnitude bounds before

you have robust stability.

More precisely, the stability margin at frequency ω is defined as the

smallest α such that the system can have a pole at jω, with the uncertain

transfer functions satisfying |δ

i

(jω)| ≤ αl

i

(ω):

margin(w) = min{ α| systems can have a pole at jω with magnitude bounds αl

i

(jω)}

The stability margin also can be expressed as:

margin(w) = min{ α| det I – H

qr

jωΔ ≠ 0 such that |Δ

ii

|≤αl

i

(α)}

Note The stability margin only depends on H

qr

.

The margin often is expressed in dB. If the margin is greater than zero for

all frequencies, then your system is robustly stable. If the margin is less

than zero for some frequencies, then your system is not robustly stable.

In particular, there are uncertain transfer functions that satisfy the

magnitude bound (Equation 2-1) and cause the system to have a pole at

those frequencies where the margin is negative. This does not mean that any

δ

i

values that satisfy the magnitude bound will destabilize the system: it

means that there are some bad δ

i

values that satisfy the magnitude bounds

and destabilize the system.

Chapter 2 Robustness Analysis

MATRIXx Xmath Robust Control Module 2-4 ni.com

smargin( )

marg = smargin(SysH, delb {scaling, graph})

The smargin( ) function plots an approximation to the stability margin

of the system as a function of frequency. For a full discussion of

smargin( ) syntax, refer to the MATRIXx Help. The approximation is

exact if the number of uncertain transfer functions is less than four and

scaling="OPT" (optimum scaling).

In other cases, the approximation is generally considered to be extremely

good. Refer to the Approximation with Scaled Singular Values section. The

approximation is always conservative.

smargin( ) always will report a

margin that is less than or equal to the actual margin.

The

smargin( ) function counts the columns in delb to calculate the

number of uncertainties k. It then assumes that the last k inputs of

SysH are

signal r in Figure 2-2, and the last k outputs are signal q. To create a Nominal

System, refer to the Creating a Nominal System section.

Figure 2-2. Nominal Closed-Loop System

Creating a Nominal System

To better understand how to create H(s) in Figure 2-3, you will examine

a SISO tracking system with three uncertainties. δ

1

is a multiplicative

actuator uncertainty, while δ

2

and δ

3

are multiplicative sensor uncertainties.

Known Closed-Loop System

w

r

z

q

H(s)

size

k

size

k

Chapter 2 Robustness Analysis

Figure 2-3. SISO Tracking System with Three Uncertainties

The H system will have the reference input as input1 and the error output

as output1 (w and z, respectively, in Figure 2-2). Removing the δ values will

create inputs 2 through 4 and outputs 2 through 4 (r and q, respectively, in

Figure 2-2).

1. The A, B, C, D matrices of the state-space system representing H are

as follows:

A=[-4,-8;1,0];

B=[8,1,-4,-8;zeros(1,4)];

C=[0,-1;-4,-8;1,0;0,1];

D=[1,0,0,0;8,0,-4,-8;zeros(2,4)];

H = system(A,B,C,D,{inputNames=["reference",

"r1","r2","r3"],outputNames=["error",

"q1","q2","q3"],stateNames=["x1","x2"]});

2. Specify the uncertainty bounds.

The sensor uncertainty δ

3

is known to be bounded by l

3

(w), according

to Equation 2-1. Because the position x

2

sensor model is known to be

accurate to 10% up to one radian per second, and very inaccurate at

high frequencies, the l

3

shown in Figure 2-4 is selected.

reference

error

8

K

1

= 4

K

2

= 8

1

s

x

1

x

2

+

1

+

1

+

2

+

––

+

–

1

s

Chapter 2 Robustness Analysis

MATRIXx Xmath Robust Control Module 2-6 ni.com

Figure 2-4. Bound for Sensor Uncertainty

Note

A value of l

3

at one radian per second of –20 dB indicates that modeling

uncertainties of up to 10% (–20 dB = 0.1) are allowed.

The actuator and sensor uncertainties δ

1

and δ

2

are bounded by –20 dB

at all frequencies. You will use these values to interpolate to obtain l

3

.

First, create the bound for δ

3

in Hz.

L3 = pdm([-20,-20,10,10],[0.1,1,30,100]/2/pi);

3. Now interpolate to obtain 30 points:

L3 = interpolate(L3,logspace(0.01,10,30),{xlog});

4. Create L1 and L2 (bounds for and ):

L1=-20*ones(L3); L2 = L1;

delb = [L1,L2,L3];

5. Calculate the stability margin:

marg=smargin(H,delb);

smargin --> Scaling algorithm is type: PF

smargin --> Margin computation 10% complete

smargin --> Margin computation 50% complete

smargin --> Margin computation 90% complete

The output indicates that Perron-Frobenius scaling (the default) is

used. Refer to the Approximation with Scaled Singular Values section.

The stability margin plot is shown in Figure 2-5. The minimum margin

is about 8 dB at about 1/2 Hz. This implies that all three l

1

values

(uncertainty bounds) could be increased (relaxed) simultaneously

by 8 dB, and the system would still remain robustly stable.

0.1

1

30 100

10

0

–20

Frequency, Radian/Second

dB

δ

1

δ

2

Chapter 2 Robustness Analysis

Figure 2-5. Stability Margin

Now examine the effect on the stability margin of discretizing H(s) at

100 Hz.

dt = 0.01;

Hd = discretize(H,dt);

margD = smargin(Hd,delb);

smargin --> Scaling algorithm is type: PF

smargin --> Margin computation 10% complete

smargin --> Margin computation 50% complete

smargin --> Margin computation 90% complete

100 Hz is a high discretization frequency for H, so the stability margin

is unchanged in the discrete-time case. The new plot is not much

different from Figure 2-6. Again, minimum margin is about 8 dB

at about 1/2 Hz.

Chapter 2 Robustness Analysis

MATRIXx Xmath Robust Control Module 2-8 ni.com

Worst-Case Performance Degradation (wcbode)

Even if a system is robustly stable, the uncertain transfer functions still can

have a great effect on performance. Consider the transfer function from the

qth input, w

q

, to the pth output, z

p

. With δ

1

= ... = ...δ

k

= 0, you have the

nominal system, and this transfer function is the p,q entry of H

zw

. This is

called the nominal transfer function.

When the δ values are not zero, the transfer function from w

q

to z

p

is the p,q

entry of H

pert

given by the formula:

This is referred to as the perturbed transfer function. The perturbed transfer

function depends on the particular δ

1

, …, δ

k

.

If the magnitude bounds are small enough, then you expect the perturbed

transfer function H

pert

to be close to the nominal transfer function. Roughly

speaking, small perturbations should not significantly alter the closed-loop

transfer function from w

q

to z

p

.

The worst-case gain is defined as the largest magnitude of the perturbed

transfer function, considering all δ values that satisfy the magnitude bound.

More precisely:

(2-3)

wcgain(ω) is always larger than the nominal gain, |H

zw,pq

(jω)|. This is not

because the uncertain transfer functions only can increase the magnitude of

the transfer function from w

q

to z

q

. In fact, it is possible that for a lucky

choice of the δ values, the perturbed transfer function actually can be

smaller than the nominal transfer function over all frequencies. But in the

worst-case gain, you consider only the worst possible δ values, and these

always increase the perturbed gain over the nominal gain.

Intuitively, if the stability margin is large, then the uncertain transfer

functions should not greatly effect the gain from w

q

to z

p

, so that wcgain(ω)

should be not much larger than the nominal gain |H

zw,pq

(jω)|. If the stability

margin is small, however, wcgain(ω) could be much larger than the nominal

gain. An extreme case occurs if the stability margin is negative (in dB) at

the frequency δ. Then you have wcgain(ω) = ∞, although

wcbode( ) clips

the worst-case gain curve so that it never exceeds (the maximum nominal

gain) * 100, or +20 dB. Of course, instability is an extreme form of

performance degradation.

H

pert

H

zw

H

zr

Δ IH

qr

Δ–()

1–

H

qw

+=

wcgain ω() max H

pert,pq

Δ = diagonal δ

1

,...,δ

k

()δ

i

l

i

ω()≤,{}=

Chapter 2 Robustness Analysis

wcbode( )

[WCMAG, NOMMAG] = wcbode (SysH, delb, {input, output,

graph})

The wcbode( ) function computes and plots the worst-case gain of a

closed-loop transfer function.

This function is useful for checking a system that already has been verified

to be robustly stable using

smargin( ). For example, a system can have a

minimum stability margin of 4 dB, so it is robustly stable. If the worst-case

gain from a function input to the output it commands has a 20 dB peak, then

even though the system is robustly stable, the design is unacceptable. On

the other hand, if you verify that the perturbed closed-loop transfer function

increases only 2 dB over the nominal, then the design is probably

acceptable.

The

wcbode( ) function computes and plots an approximation to

wcgain(ω), the largest possible magnitude of a perturbed closed-loop

transfer function that can be caused by uncertain transfer functions that

satisfy the magnitude bound. The

wcbode( ) function is conservative:

it does not under-report the maximum of the perturbed transfer function.

A large value of

wcbode( ) indicates instability: wcgain(ω) = ∞. In this

case,

wcbode( ) returns a maximum value of ten times the maximum of

the nominal transfer function over all frequencies. Consequently, the

window is clipped at 20 dB above the maximum of the nominal transfer

function over all frequencies.

The wcbode( ) function also plots the

nominal transfer function for reference.

Using wcbode( ) to Analyze Performance Degradation

The wcbode( ) function can be used to analyze performance degradation

for the system you have been using (Figure 2-3). The transfer function,

which should be small, is from reference to error (input 1 to output 1).

Figure 2-6 shows the results of the following function call:

[NOMMAG,WCMAG]=wcbode(H,delb,{input=1,output=1});

The performance degradation due to the uncertainties is small but not

negligible.

Chapter 2 Robustness Analysis

MATRIXx Xmath Robust Control Module 2-10 ni.com

Figure 2-6. Performance Degradation of the SISO Tracking System

Advanced Topics

This section describes the theoretical background on robustness analysis

and performance degradation.

Stability Margin

This section discusses advanced aspects of computing the stability margin

and the related scaling algorithms.

Stability Margin and Structured Singular Values (μ)

The stability margin was first defined by Safonov in [Saf82]. If you let

then you can express the margin at frequency d as

MH

qr

diagonal l

1

w()...,l

k

w(),()=

margin ω() max= α det I MΔ–()0≠{

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