Linear Systems: A Measurement Based Approach

94 Pages·2013·3.349 MB·English

by  S. P. Bhattacharyya, L.H. Keel, D.N. Mohsenizadeh

SPRINGER BRIEFS IN
APPLIED SCIENCES AND TECHNOLOGY
S. P. Bhattacharyya
L. H. Keel
D. N. Mohsenizadeh
Linear Systems
A Measurement
Based Approach
SpringerBriefs in Applied Sciences
and Technology
For furthervolumes:
http://www.springer.com/series/8884
S. P. Bhattacharyya L. H. Keel
•
D. N. Mohsenizadeh
Linear Systems
A Measurement Based Approach
123
S. P.Bhattacharyya D.N.Mohsenizadeh
Electrical EngineeringDepartment Mechanical Engineering
Texas A&MUniversity Texas A&MUniversity
College Station, TX College Station, TX
USA USA
L.H.Keel
Electrical andComputer Engineering
Tennessee State University
Nashville, TN
USA
ISSN 2191-530X ISSN 2191-5318 (electronic)
ISBN 978-81-322-1640-7 ISBN 978-81-322-1641-4 (eBook)
DOI 10.1007/978-81-322-1641-4
SpringerNewDelhiHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013947780
(cid:2)TheAuthor(s)2014
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To Gisele, for her love and support
SPB
To My Wife, Kuisook
LHK
To my mother, Shamsozzoha,
my father, Mohammad Farid,
and my brother, Mehrdad
DNM
Preface
This monograph presents the recent results obtained by us on the analysis,
synthesis and design of systems described by linear equations. As is well known,
linear equations arise in most branches of science and engineering as well as
social, biological and economic systems. The novelty of our approach lies in the
fact that no models of the system are assumed to be available, nor are they
required. Instead, we show that a few measurements made on the system can be
processed strategically to directly extract design values that meet specifications
without constructing a model of the system, implicitly or explicitly. We illustrate
thesenewconceptsbyapplyingthemtolinearD.C.andA.C.circuits,mechanical,
civilandhydraulicsystems,signalflowblockdiagramsandcontrolsystems.These
applications are preliminary and suggest many open problems. We acknowledge
many productive discussions with our colleagues A. Datta, Hazem Nounou,
Mohamed NounouandourgraduatestudentsRitwikLayekandSirishaKallakuri.
Earlierresearchbyushasshownthattherepresentationofcomplexsystemsby
high order models with many parameters may lead to fragility, that is, the drastic
change of system behaviour under infinitesimally small perturbations of these
parameters. This led to research on model-free measurement-based approaches to
design. The results presented in this monograph are our latest effort in this
direction and we hope they will lead to attractive alternatives to model-based
designofengineeringandothersystems.Wealsoanticipateapplicationstorobust,
adaptive and fault tolerant control.
College Station, USA, June 25, 2013 S. P. Bhattacharyya
L. H. Keel
D. N. Mohsenizadeh
vii
Contents
1 Linear Equations with Parameters. . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Parameterized Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Measurements and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Polynomial Models. . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.2 Rational Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Determining a General Parameterized Solution
from Measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.4.1 A Generalized Superposition Theorem. . . . . . . . . . . . . . 11
1.4.2 A Measurement Theorem. . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Notes and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 Application to DC Circuits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Current Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.2.1 Current Control Using a Single Resistor . . . . . . . . . . . . 20
2.2.2 Current Control Using Two Resistors . . . . . . . . . . . . . . 25
2.2.3 Current Control Using m Resistors . . . . . . . . . . . . . . . . 27
2.2.4 Current Control Using Gyrator Resistance . . . . . . . . . . . 29
2.2.5 Current Control Using m Independent Sources. . . . . . . . 31
2.3 Power Level Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.1 Power Level Control Using a Single Resistor. . . . . . . . . 33
2.3.2 Power Level Control Using Two Resistors. . . . . . . . . . . 34
2.3.3 Power Level Control Using Gyrator Resistance . . . . . . . 35
2.4 Examples of DC Circuit Design . . . . . . . . . . . . . . . . . . . . . . . 35
2.5 Notes and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3 Application to AC Circuits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1 Current Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.1 Current Control Using a Single Impedance . . . . . . . . . . 44
3.1.2 Current Control Using Two Impedances . . . . . . . . . . . . 46
ix
x Contents
3.1.3 Current Control Using Gyrator Resistance . . . . . . . . . . . 46
3.1.4 Current Control Using m Independent Sources. . . . . . . . 46
3.2 Power Control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2.1 Power Control Using a Single Impedance . . . . . . . . . . . 47
3.2.2 Power Control Using Two Impedances . . . . . . . . . . . . . 47
3.2.3 Power Control Using Gyrator Resistance. . . . . . . . . . . . 48
3.3 An Example of AC Circuit Design . . . . . . . . . . . . . . . . . . . . . 48
3.4 Notes and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4 Application to Mechanical Systems. . . . . . . . . . . . . . . . . . . . . . . . 53
4.1 Mass-Spring Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2 Truss Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Hydraulic Networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 Flow Rate Control Using a Single Pipe Resistance. . . . . 59
4.3.2 Flow Rate Control Using Two Pipe Resistances. . . . . . . 61
4.4 Illustrative Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4.1 An Example of Mass-Spring Systems . . . . . . . . . . . . . . 62
4.4.2 An Example of Truss Structures. . . . . . . . . . . . . . . . . . 64
4.4.3 An Example of Hydraulic Networks . . . . . . . . . . . . . . . 66
4.5 Notes and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5 Application to Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Block Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.3 SISO Control Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.3.1 Functional Dependency on a Single Controller. . . . . . . . 73
5.3.2 Determining a Desired Response . . . . . . . . . . . . . . . . . 74
5.3.3 Steps to Controller Design. . . . . . . . . . . . . . . . . . . . . . 74
5.4 An Example of Control System Design . . . . . . . . . . . . . . . . . . 75
5.5 Notes and References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
About the Authors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
About the Book. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Chapter 1
Linear Equations with Parameters
In this chapter, we describe some basic results on the solution of linear equations
containing parameters, and the nature of the parameterized solutions. We describe
how measurements can be used to extract these parameterized solutions when the
equationsormodelsareunknown.ThesearepresentedasageneralizedSuperposition
TheoremandaMeasurementTheorem.
1.1 Introduction
Considerthesystemoflinearequations
Ax=b, (1.1)
whereAisann×n matrix,andxandbaren×1vectorsallwithrealorcomplex
entries.Let|.|denotesthedeterminant.Assumingthat|A|(cid:2)=0,thereexistsaunique
solutionxand,byCramer’srule,theithcomponentx ofxisgivenby
i
(cid:2) (cid:2)
(cid:2)Ai(b)(cid:2)
x = , i =1,2,...,n (1.2)
i |A|
whereAi(b)isthematrixobtainedbyreplacingtheithcolumnofAbyb.
Inmanyphysicalproblems,Aandbcontainparametersthatneedtobechosen
ordesigned,asillustratedintheexamplebelow.
Example1.1. ConsiderthecircuitshowninFig.1.1. V istheidealvoltagesource,
I is the ideal current source, R ,R ,R are linear resistors, and R is a gyrator
1 2 3 4
resistance.Thegyratorisalineartwoportdevicewheretheinstantaneouscurrents
andtheinstantaneousvoltagesarerelatedbyV = R I andV = −R I .V is
2 4 2 1 4 3 amp
the dependent voltage of the amplifier where V = KI , and K is the amplifier
amp 1
gain. The equations of the system can be written in the following matrix form by
applyingKirchhoff’scurrentandvoltagelaws,
S.P.Bhattacharyyaetal.,LinearSystems,SpringerBriefsinAppliedSciences 1
andTechnology,DOI:10.1007/978-81-322-1641-4_1,©TheAuthor(s)2014
2 1 LinearEquationswithParameters
I I I
1 2 gyrator 3
A
R R R
1 2 R 3
4
+
+ + +
V I V
1 V
2
− − − Vamp= KI1
−
Fig.1.1 Ageneralcircuit
⎡ ⎤⎡ ⎤ ⎡ ⎤
1 −1 0 I I
1
⎣R R − R ⎦⎣I ⎦=⎣V ⎦. (1.3)
1 2 4 2
K − R R I 0
(cid:7) (cid:8)(cid:9)4 3 (cid:10)(cid:7) (cid:8)3(cid:9) (cid:10) (cid:7) (cid:8)(cid:9) (cid:10)
A x b
Tofixnotation,weintroducetheparametervectorpandthevectorofsourcesq:
⎡ ⎤ ⎡ ⎤
R p
1 1
⎢ ⎥ ⎢ ⎥ (cid:13) (cid:14) (cid:13) (cid:14)
p:=⎢⎢⎢⎣RRR23⎥⎥⎥⎦=⎢⎢⎢⎣ppp23⎥⎥⎥⎦ andq:= VI = qq21 , (1.4)
4 4
K p
5
so that (1.1) can be rewritten showing explicitly the dependence on the parameter
vectorpandthesourcevectorqas
A(p)x=b(q). (1.5)
Thus,(1.2)canalsoberewrittenexplicitlyshowingtheparameterizedsolutionas
(cid:2) (cid:2)
(cid:2)Ai(p,b(q))(cid:2) |B (p,q)|
x (p,q)= := i , i =1,2,...,n. (1.6)
i |A(p)| |A(p)|
Moregenerally,if
y(p,q)=cTx(p,q)=c x (p,q)+···+c x (p,q) (1.7)
1 1 n n
isanoutputofinterest,itfollowsthat
(cid:16) (cid:17)
(cid:15)n |B (p,q)|
y(p,q)= c i . (1.8)
i |A(p)|
i=1