Matlab 鲁棒控制宝典.pdf

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Building Uncertain Models

Introduction to Uncertain Atoms

Uncertain Real Parameters

Uncertain LTI Dynamics Atoms

Complex Parameter Atoms

Complex Matrix Atoms

Unstructured Uncertain Dynamic Systems

Uncertain Matrices

Creating Uncertain Matrices from Uncertain Atoms

Accessing Properties of a umat

Row and Column Referencing

Matrix Operation on umat Objects

Substituting for Uncertain Atoms

Uncertain State-Space Systems (uss)

Creating Uncertain Systems

Properties of uss Objects

Sampling Uncertain Systems

Feedback Around an Uncertain Plant

Interpreting Uncertainty in Discrete Time

Lifting a ss to a uss

Handling Delays in uss

Uncertain frd

Creating Uncertain Frequency Response Objects

Properties of ufrd Objects

Interpreting Uncertainty in Discrete Time

Lifting an frd to a ufrd

Handling Delays in ufrd

Basic Control System Toolbox™ and MATLAB® Interconnections

Simplifying Representation of Uncertain Objects

Effect of the Autosimplify Property

Direct Use of simplify

Sampling Uncertain Objects

Generating One Sample

Generating Many Samples

Sampling ultidyn Atoms

Substitution by usubs

Specifying the Substitution with Structures

Nominal and Random Values

Array Management for Uncertain Objects

Referencing Arrays

Creating Arrays with stack and cat Functions

Creating Arrays by Assignment

Binary Operations with Arrays

Creating Arrays with usample

Creating Arrays with usubs

Creating Arrays with gridureal

Creating Arrays with repmat

Creating Arrays with repsys

Using permute and ipermute

Decomposing Uncertain Objects (for Advanced Users)

Normalizing Functions for Uncertain Atoms

Properties of the Decomposition

Syntax of lftdata

Advanced Syntax of lftdata

Generalized Robustness Analysis

Introduction to Generalized Robustness Analysis

Robust Stability Margin

Robust Performance Margin

Worst-Case Gain Measure

Introduction to Linear Matrix Inequalities

Linear Matrix Inequalities

LMI Features

LMIs and LMI Problems

Three Generic LMI Problems

Further Mathematical Background

References

LMI Lab

Introduction

Some Terminology

Overview of the LMI Lab

Specifying a System of LMIs

A Simple Example

Initializing the LMI System

Specifying the LMI Variables

Specifying Individual LMIs

Specifying LMIs with the LMI Editor

How It All Works

Querying the LMI System Description

lmiinfo

lminbr and matnbr

LMI Solvers

From Decision to Matrix Variables and Vice Versa

Validating Results

Modifying a System of LMIs

Deleting an LMI

Deleting a Matrix Variable

Instantiating a Matrix Variable

Advanced Topics

Structured Matrix Variables

Complex-Valued LMIs

Specifying cTx Objectives for mincx

Feasibility Radius

Well-Posedness Issues

Semi-Definite B(x) in gevp Problems

Efficiency and Complexity Issues

Solving M + PTXQ + QTXTP < 0

References

Function Reference

Functions — By Category

Uncertain Elements

Uncertain Matrices and Systems

Manipulation of Uncertain Models

Interconnection of Uncertain Models

Model Order Reduction

Robustness and Worst-Case Analysis

Robustness Analysis for Parameter-Dependent Systems (P-Systems)

Controller Synthesis

m-Synthesis

Sampled-Data Systems

Gain Scheduling

Frequency-Response Data (FRD) Models

Supporting Utilities

LMIs

Functions — Alphabetical List

actual2normalized

aff2pol

augw

balancmr

bilin

bstmr

complexify

cmsclsyn

cpmargin

dcgainmr

decay

decinfo

decnbr

dec2mat

defcx

dellmi

delmvar

diag

dkitopt

dksyn

dmplot

drawmag

evallmi

feasp

fitfrd

fitmagfrd

gapmetric

genphase

getlmis

gevp

gridureal

h2hinfsyn

h2syn

hankelmr

hankelsv

hinfgs

hinfsyn

icomplexify

iconnect

icsignal

imp2exp

imp2ss

ispsys

isuncertain

lftdata

ltiarray2uss

lmiedit

lmiinfo

lminbr

lmireg

lmiterm

lmivar

frd/loglog

loopmargin

loopsens

loopsyn

ltiarray2uss

ltrsyn

matnbr

mat2dec

mincx

mixsyn

mkfilter

mktito

modreal

msfsyn

mussv

mussvextract

ncfmargin

ncfmr

ncfsyn

newlmi

normalized2actual

pdlstab

pdsimul

polydec

popov

psinfo

psys

pvec

pvinfo

quadperf

quadstab

randatom

randumat

randuss

frd/rcond

reduce

repmat

robopt

robustperf

robuststab

frd/schur

schurmr

sdhinfnorm

sdhinfsyn

sdlsim

sectf

frd/semilogx

setlmis

setmvar

showlmi

simplify

skewdec

slowfast

squeeze

uss/ssbal

stack

frd/svd

symdec

sysic

ucomplex

ucomplexm

udyn

ufrd

ultidyn

umat

uplot

ureal

usample

usimfill

usiminfo

usimsamp

uss

usubs

wcgain

wcgopt

wcmargin

wcnorm

wcsens

Block Reference

MultiPlot Graph

USS System

Index

Robust Control Toolbox™ 3 User’s Guide Gary Balas Richard Chiang Andy Packard Michael Safonov

How to Contact The MathWorks Web Newsgroup www.mathworks.com comp.soft-sys.matlab www.mathworks.com/contact_TS.html Technical support suggest@mathworks.com bugs@mathworks.com doc@mathworks.com service@mathworks.com info@mathworks.com Product enhancement suggestions Bug reports Documentation error reports Order status, license renewals, passcodes Sales, pricing, and general information 508-647-7000 (Phone) 508-647-7001 (Fax) The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 For contact information about worldwide offices, see the MathWorks Web site. Robust Control Toolbox™ User’s Guide © COPYRIGHT 2005–2009 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. No part of this manual may be photocopied or repro- duced in any form without prior written consent from The MathWorks, Inc. FEDERAL ACQUISITION: This provision applies to all acquisitions of the Program and Documentation by, for, or through the federal government of the United States. By accepting delivery of the Program or Documentation, the government hereby agrees that this software or documentation qualifies as commercial computer software or commercial computer software documentation as such terms are used or defined in FAR 12.212, DFARS Part 227.72, and DFARS 252.227-7014. Accordingly, the terms and conditions of this Agreement and only those rights specified in this Agreement, shall pertain to and govern the use, modification, reproduction, release, performance, display, and disclosure of the Program and Documentation by the federal government (or other entity acquiring for or through the federal government) and shall supersede any conflicting contractual terms or conditions. If this License fails to meet the government's needs or is inconsistent in any respect with federal procurement law, the government agrees to return the Program and Documentation, unused, to The MathWorks, Inc. Trademarks MATLAB and Simulink are registered trademarks of The MathWorks, Inc. See www.mathworks.com/trademarks for a list of additional trademarks. Other product or brand names may be trademarks or registered trademarks of their respective holders. Patents The MathWorks products are protected by one or more U.S. patents. Please see www.mathworks.com/patents for more information.

Revision History September 2005 March 2006 September 2006 March 2007 September 2007 March 2008 October 2008 March 2009 Online only Online only Online only Online only Online only Online only Online only Online only New for Version 3.0.2 (Release 14SP3) Revised for Version 3.1 (Release 2006a) Revised for Version 3.1.1 (Release 2006b) Revised for Version 3.2 (Release 2007a) Revised for Version 3.3 (Release 2007b) Revised for Version 3.3.1 (Release 2008a) Revised for Version 3.3.2 (Release 2008b) Revised for Version 3.3.3 (Release 2009a)

1 Building Uncertain Models Introduction to Uncertain Atoms . . . . . . . . . . . . . . . . . . . . . . . 1-2 Uncertain Real Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-3 Uncertain LTI Dynamics Atoms . . . . . . . . . . . . . . . . . . . . . . . . 1-10 Complex Parameter Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-13 Complex Matrix Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-15 Unstructured Uncertain Dynamic Systems . . . . . . . . . . . . . . . 1-17 Uncertain Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-19 Creating Uncertain Matrices from Uncertain Atoms . . . . . . . 1-19 Accessing Properties of a umat . . . . . . . . . . . . . . . . . . . . . . . . . 1-20 Row and Column Referencing . . . . . . . . . . . . . . . . . . . . . . . . . . 1-21 Matrix Operation on umat Objects . . . . . . . . . . . . . . . . . . . . . . 1-22 Substituting for Uncertain Atoms . . . . . . . . . . . . . . . . . . . . . . 1-23 Uncertain State-Space Systems (uss) . . . . . . . . . . . . . . . . . . 1-25 Creating Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-25 Properties of uss Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-26 Sampling Uncertain Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 1-27 Feedback Around an Uncertain Plant . . . . . . . . . . . . . . . . . . . 1-28 Interpreting Uncertainty in Discrete Time . . . . . . . . . . . . . . . 1-30 Lifting a ss to a uss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-31 Handling Delays in uss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-31 Uncertain frd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-33 Creating Uncertain Frequency Response Objects . . . . . . . . . . 1-33 Properties of ufrd Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-33 Interpreting Uncertainty in Discrete Time . . . . . . . . . . . . . . . 1-36 Lifting an frd to a ufrd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-36 Handling Delays in ufrd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-36 Basic Control System Toolbox™ and MATLAB® Interconnections . . . . . . . . . . . . . . . . . . . . . . . . . . 1-37 Simplifying Representation of Uncertain Objects . . . . . . . 1-38 Contents v

Effect of the Autosimplify Property . . . . . . . . . . . . . . . . . . . . . 1-39 Direct Use of simplify . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-41 Sampling Uncertain Objects . . . . . . . . . . . . . . . . . . . . . . . . . . 1-42 Generating One Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-42 Generating Many Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-42 Sampling ultidyn Atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-43 Substitution by usubs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-46 Specifying the Substitution with Structures . . . . . . . . . . . . . . 1-47 Nominal and Random Values . . . . . . . . . . . . . . . . . . . . . . . . . . 1-47 Array Management for Uncertain Objects . . . . . . . . . . . . . . 1-49 Referencing Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-49 Creating Arrays with stack and cat Functions . . . . . . . . . . . . 1-50 Creating Arrays by Assignment . . . . . . . . . . . . . . . . . . . . . . . . 1-52 Binary Operations with Arrays . . . . . . . . . . . . . . . . . . . . . . . . . 1-53 Creating Arrays with usample . . . . . . . . . . . . . . . . . . . . . . . . . 1-53 Creating Arrays with usubs . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-55 Creating Arrays with gridureal . . . . . . . . . . . . . . . . . . . . . . . . . 1-56 Creating Arrays with repmat . . . . . . . . . . . . . . . . . . . . . . . . . . 1-57 Creating Arrays with repsys . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-58 Using permute and ipermute . . . . . . . . . . . . . . . . . . . . . . . . . . 1-58 Decomposing Uncertain Objects (for Advanced Users) . . 1-60 Normalizing Functions for Uncertain Atoms . . . . . . . . . . . . . . 1-60 Properties of the Decomposition . . . . . . . . . . . . . . . . . . . . . . . . 1-61 Syntax of lftdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-62 Advanced Syntax of lftdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-64 2 Generalized Robustness Analysis Introduction to Generalized Robustness Analysis . . . . . . . . 2-2 Robust Stability Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-4 vi Contents

3 4 Robust Performance Margin . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-5 Worst-Case Gain Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2-6 Introduction to Linear Matrix Inequalities Linear Matrix Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2 LMI Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-2 LMIs and LMI Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-4 Three Generic LMI Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-5 Further Mathematical Background . . . . . . . . . . . . . . . . . . . . . 3-9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3-10 LMI Lab Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 Some Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-2 Overview of the LMI Lab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-5 Specifying a System of LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-7 A Simple Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-8 Initializing the LMI System . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10 Specifying the LMI Variables . . . . . . . . . . . . . . . . . . . . . . . . . . 4-10 Specifying Individual LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-13 Specifying LMIs with the LMI Editor . . . . . . . . . . . . . . . . . . . . 4-15 How It All Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-19 Querying the LMI System Description . . . . . . . . . . . . . . . . . 4-20 lmiinfo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-20 vii

lminbr and matnbr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-20 LMI Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-21 From Decision to Matrix Variables and Vice Versa . . . . . . 4-27 Validating Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-28 Modifying a System of LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-29 Deleting an LMI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-29 Deleting a Matrix Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-29 Instantiating a Matrix Variable . . . . . . . . . . . . . . . . . . . . . . . . 4-30 Advanced Topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-32 Structured Matrix Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-32 Complex-Valued LMIs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-34 Specifying cTx Objectives for mincx . . . . . . . . . . . . . . . . . . . . . 4-37 Feasibility Radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-38 Well-Posedness Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-39 Semi-Definite B(x) in gevp Problems . . . . . . . . . . . . . . . . . . . . 4-40 Efficiency and Complexity Issues . . . . . . . . . . . . . . . . . . . . . . . 4-41 Solving M + PTXQ + QTXTP < 0 . . . . . . . . . . . . . . . . . . . . . . . . 4-41 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4-43 5 Function Reference Functions — By Category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-2 Uncertain Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-4 Uncertain Matrices and Systems . . . . . . . . . . . . . . . . . . . . . . . . 5-4 Manipulation of Uncertain Models . . . . . . . . . . . . . . . . . . . . . . . 5-4 Interconnection of Uncertain Models . . . . . . . . . . . . . . . . . . . . . 5-6 Model Order Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-6 Robustness and Worst-Case Analysis . . . . . . . . . . . . . . . . . . . . . 5-7 Robustness Analysis for Parameter-Dependent Systems (P-Systems) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5-8 viii Contents

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