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DEPARTMENT OF THE AIR FORCEAIR UNIVERSITYAIR FORCE INSTITUTE OF TECHNOLOGY

Wright-Patterson Air Force Base, Ohio


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AFIT/GA/ENY/9 1M-01 /

'y

EFFECTS OF COLOCATION AND NON-COLOCATEONOF SENSORS AND ACTUATORSON FLEXIBLE STRUCTURES

THESISW. C. LEE, Captain, USAF

AFIT/GA/ENY/9 IM-01

Approved for public release; distribution unlimited.

91-05743 "~~~ ~~~~~~'lli ~lll 11 l


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March 1991 Master ThesisEffect of Colocation and

Non-colocation of Sensors and Actuatorson Flexible Structures

W. C. Lee, Capt, USAF

Air Force Institute of Technology AFIT/GA/ENY/91M-01WPAFB, OH 45433-6583

Approved for public release;distributionunlimited

This thesis compares the performance and robustnesof colocated and non-colocated sensor/actuator pairs. Linear quadratiGaussian and loop transfer recovery (LQG/LTR) procedures are used todesign optimum controllers for three example problems. The firstexample is a single motion degree-of-freedom system having motion onlyabout one axis. This type of system does not have any right half planzeros for the non-colocated case;however, the thesis shows it behavesa non-minimum phase system. The second example is a Bernoulli-Eulerbeam which does have RHP zeros. An optimum LQG/LTR controller can notimprove the robustness of non-minimum phase systems;whereas, colocatedsystems are aisy stabilize and make robust with LTR. However, thisthesis shows that non-colocated systems can achieve higher bandwidthand performance. So, each system has its own advantage.

control theory flexible qtructurcs, actuators, LQG/LTR

unclassified unclassified uncl .;sified unlimited


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AFIT/GA/ENY/91M-01

EFFECTS OF COLOCATION AND NON-COLOCATIONOF SENSORS AND ACTUATORSON FLEXIBLE STRUCTURES

THESIS

Presented to the Faculty of the School of Engineeringof the Air Force Institute of Technology

Air UniversityIn Partial Fulfillment of theRequirements for the Degree of

Master of Science in Astronautical Engineering

W. C. Lee, B.S.C.E. /Captain, USAF .

March 1991

Approv'ed for public release; distribution unlimited.


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Acknowledqements

I wish to thank Mr Dan Malwitz of Contraves USA forprovidinq the finite element model used in this thesis. Hewas very helpful in clarifying questions I had concerningfinite element modeling and providing supporting materials.Another group who deserves much thanks is the people atEngineering Methods, Inc. of West Lafayette, Indiana. FromLuanna Hellerger, Paul Hunckler and Dave Phillips to Dr.Charles Hunckler, I can not appreciate the amount of time andeffort they took to assist me in getting ANSYS up and running.Also, Mr Bob Jurek and Mr Larry Falb at ASD's Scientific andEngineering Applications Division were very helpful in gettingan account set up so I could use the commercial version ofANSYS for some of my computer runs. I also appreciate thecontributions of the in-house people here at AFIT's computersupport and Capt Howard Gans in getting ANSYS set up in ourown laboratory. Finally, I need to thank Dr Patrick Sweeneyand Mr Tom Davis from University of Dayton who provided theinitial support in my using ANSYS as a research tool.

My deepest appreciation goes toward my thesis advisor, DrBrad Liebst, for his motivation and support during this thesiseffort. His patience and dedication to higher learningallowed me to leave AFIT with a feeling of accomplishment in

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having learned a little about the new and exciLing field ofcontrol engineering. I also appreciate the help of my thesiscommittee members, Capt Brett Ridgely, Lt Col Ron Bagley andDr Curtis Spenny for the-- helpful advice and support infinishing my thesis. Of course, my wife, Myong Sook, deservesthe most gratitude. Her endless patience during my long hoursat the computer and in the laboratory is something few peoplecould give. Her love and support got me through this projectand AFIT.

W. C. Lee

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Table of Contents

Acknowledgements................ . .. .. . .....List of Figures....................viList of Tables...................ixAbstract.......................x

I. Introduction.....................1-1

II. Background.....................2-12.1 Pole/Zero Patterns for the Colocated Systems 2-22.2 Pole/Zero Patterns for theNon-colocated Systems.............2-62.3 Three Element Lumped Mass Model...........2-82.3.1 Colocation..................2-132.3.2 Non-crlocation...............2-17

III. Linear Quadratic Gaussian/Loop Traznsfer Recovery 3-13.1 LQG/LTR Theory..................3-13.2 Linear Quadrati.c Regulator...........3-23.3 Linear Quadratic Estimator............3-43.4 Linear Quadratic Gaussian............3-73.5 LQG Design On Three Element Model. ...... 3-103.6 Loop Transfer Recovery.............3-16

3.6.1 Root Mean Square Response ........ 3-193.6.2 LTR With Non-minimum Phase Systems . 3-20

3.7 LTR On Three Element Model...........3-203.8 Step Response...................3-253.9 Conclusion....................3-26

IV. A Two Motion Degree of Freedom Syste1........4-14.1 Introduction..................4-14.2 Exact Solution of a Transversr-, Beam.... 4-1

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4.2.1 Modal Truncation and ItsEffect on Zero Location ..... . . 4-5

4.3 Approximate Solution byFinite Element Method ... ........... 4-54.4 Ten Element Transverse Beam Example .... 4-84.4.1 Model Reduction to Four Modes . . 4-12

4.5 LQG/LTR Control Design on the Beam .... 4-194.5.1 Step Response ... ........... 4-25

4.6 Conclusion ..... .............. .. 4-26

V. Finite Element Model of Contraves P52F Gimbals . 5-15.1 Finite Element Modeling ... .......... 5-15.2 LQG/LTR Control Design on the Gimbals 5-85.3 Step Response ..... ............... 5-135.4 Conclusion ...... ................ 5-14

VI. Conclusion and Recommendations .. ......... 6-16.1 Summary ........ .. .................. 6-16.2 Contribution ........ ............... 6-46.3 Recommendations for Further Study ..... 6-5

APPENDIX A: LQG/LTR Design Results .. ......... . A-IAPPENDIX B: ANSYS FEM Files .... ........... . B-iAPPENDIX C: MATLAB Programs ... ............ C-1

Bibliography ........ ................... BIB-iVita ......... ..................... V-i

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List of Figures

Figure Page2.1 Alternating Poles and Zeros .. .......... 2-52.2 Three Element Lumped Mass Model . ........ 2-82.3 Bode Plots of Colocated and Non-colocated

Three Element Model ..... .............. 2-122.4 Closed Loop System with Unity Feedback

and Proportional Gain .... ............. 2-132.5 Colocation of Sensors and Actuators ...... 2-152.6 Colocation with Lead Compensation . ....... 2-162.7 Non-colocation under Proportional Feedback with

Sensor Located at Mass J2 ... ........... 2-172.8 Non-colocation under Proportional Feedback wth

Sensor Located at the Opposite End at Mass J3 2-182.9 Non-colocation with a Lead Compensator ..... . 2-193.1 Block Diagram of the Full-state Controller . . 3-43.2 Block Diagram of the Kalman Filter ........ 3-53.3 Block Diagram of the LQG System . ........ 3-93.4 Closed Loop Bandwidths for the

Three Element Model ..... .............. 3-123.5 Root Locus Plot for the LQG Design Without LTRfor the Colocated System ... ............ 3-143.6 Root Locus Plot for the LQG Design Without LTRfor the Non-colocated System .. .......... 3-153.7 Loop Transfer Recovery for the Colocated System 3-233.8 Root Locus for the LQG Design with LTR

for the Colocated System ... ............ 3-23

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3.9 LTR for the Non-colocated System ......... 3-243.10 Foot Locus for the LQG Design With LTR

for the Non-colocated System .... ......... 3-243.il Step Response Three Element Model ........ . 3-264.1 Free-free Transverse Beam ... ........... 4-24.2 Exact Poles/Zeros of a Colocated System . . .. 4-44.3 Exact Poles/Zeros of a Non-colocated System 4-44.4 Ten Element Transverse Beam ... .......... 4-94.5 DOF for the Two Dimensional Element ...... 4-94.6 Bode Plot Comparing Frequency Response of 11 Modesto the Reduced 4 Modes for the Colocated Beam . 4-144.7 Bode Plot Comparing Frequency Response of 11 Modesto the Reduced 4 Modes for the Non-colocated Beam 4-144.8 Four Mode Shapes of the Ten Element Beam . . . . 4-154.9 Bode Plots of the Colocated and Non-colocated

Ten Element Beam ...... ................ 4-184.10 Closed Loop Bandwidths for the

Ten Element Beam ...... ............... 4-204.11 Root Locus for the LQG Design Without LTR

for the Colocated System ... ........... 4-214.12 Root Locus for the LQG Design Without LTRfor the Non-colocated System .. ......... 4-224.13 LTR for the Colocated System .. ......... 4-234.14 Root Locus for the LQG Design with LT..for the Colocated System ... ........... 4-234.15 LTR for the Non-Colocated System . ....... 4-244.16 Root Locus for the LQG Design with LTR

for the Non-colocated System .. ......... 4-244.17 Step Response of the Ten Element Beam ..... . 4-25

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5.1 Contraves P52F Gimbal .... ............. 5-25.2 First Mode Showing the Rigid Boy Rotation 5-45.3 Bode Plots of the Colocated and Non-colocated

Contraves Model ...... ................ 5-55.4 Truncated Model Compared to the Frequency Response

for the Colocated Case .... ............. 5-65.5 Truncated Model Compared to the Frequency Response

for the Non-colocated Case ... ........... 5-65.6 8th Ord-- Curve Fit Using Ansys.m to the

Frequency Response Data ... ............ 5-75.7 Closed Loop Bandwidths for the Contraves Model 5-95.8 LTR for te Colocated Systems .. ......... 5-116.9 Root Locus for the TQG Design with LTR for theColocated System ...... ................ 5-1]5.10 LTR for the Non-colocated Systems ........ .. 5-125.11 Root Locus for the LQG Design with LTR for theNon-colocated System .................... . 5-5.12 Step Response of the Contraves Girbals . . . . 5-13

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List of Tables

Tables Page2.1 Poles and Zeros of the Transfer Function,

Eqn. 2.11 ....... ................... 2-123.1 LTR for Colocated and Non-colocated Three

Element Model ....... ................. 3-214.1 Eigenvalues from ANSYS and the Closed

Form Solution ....... ................. 4-104.2 Poles and Zeros of the Ten Element Beam . . . . 4-17

5.1 Poles and Zeros of the Contraves Model . . . . 5-36.1 Comparison of Colocated and Non-colocated Systems 6-4

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AFIT/GA/ENY/9 M-01

Abstract

This thesis examines the effects of colocation and non-colocation of sensors and actuators on flexible structures.Colocated structures are easy to stablize and make robustbecause they have alternating poles and zeros on the imaginaryaxis. On the other hand, non-colocated systems are difficultto make robust because they have right half plane (RHP) zeros.Systems with RHP zeros are defined as non-minimum phasesystems and have traditionally been avoided in control design.

The thesis uses linear quadratic Gaussian and looptransfer recovery (LQG/LTR) procedures to design optimumcontrollers for three example problems. The first examplestudied is a three element single motion degree of freedom(DOF) system analogous to torsional or longitudinal vibrationmodels. Because single motion DOF systems have motion aboutjust one axis, they have no RHP zeros. However, this thesiswill show that indeed the non-colocated single motion degree-of-freedomd system behaves like non-minimum phase systems.

To model structures with RHP zeros, a finite elementmodel of a Bernoulli-Euler beam is investigated. LQG/LTRcontrollers are designed for the colocated and non-colocated

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systems. Then the closed loop systems are compared forperformance and robustness. Finally, a three dimensionalfinite element model of a Contraves P52F gimbal structure is

analyzed. This third example has significantly more complexplant characteristics than the previous two examples.LQG/LTR controllers were designed for this commerciallyavailable structure. The results from the different desiniterations show that non-colocated structures although theymay not be robust can achieve higher performance.

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AFIT/GA/ENY/91M-01


THESISW. C. LEE, Captain, USAF

AFIT/GA/ENY/91M-01

Approved for public release; distribution unlimited.


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I. INTRODUCTION

Active control systems will be required in the design ofLarge Flexible Space Structures (LFSS). The future spacestation is a good example because it will likely consist of arigid central hub with flexible beam appendages surroundingit. These beams will have many low vibrational modes as aresult of maneuvers, motion of mass within the structure anddocking procedures [12:521]. Because such low densitystructures have little damping, the control system must dampenout these vibrational modes. Furthermore, the control systemcan help increase the size by allowing lighter and moreflexible structures to be built. However, the low mass andlow frequency modes of such structures can cause problems inkeeping it stable.

Not only is the low vibrational modes of the structure aproblem, but physical limitations will govern the location ofthe sensor and the actuator pairs. Ideally, colocated systemsare desired because stability is easy to achieve using simplecontrol laws as shown by Gevarter in 1970 [3]. Since thattime, most of the control theories and designs have been based


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on using colocated systems. But, frequently on spacestructures, physical limitations prevent colocation. A recentexample is the Galileo spacecraft launched by NASA in 1989.It has a separated actuator and sensor pair which made itdifficult to control a television camera and other instruments(sensors) located at the end of a flexible beam [4:1]. Thisbeam had four low frequency modes below the requiredbandwidth. To meet the performance specifications, thecontroller designed by the Jet Propulsion Laboratory had to besupplemented with a slewing manuever to minimize vibratorymotion. When possible, the sensor and the actuator should becolocated for ease of control design.

In some cases, as stated by Thomas and Schmidt, it isbetter to have non-colocated sensors and actuators [12:521].Sometimes the sensor must be placed at a node which gives themost information, but where an actuator can not be placed. Agood example (given in Reference 12) is placing a sensor atthe tip end of a cantilever beam to measure the largestdisplacements with its actuator located at the wall. Non-colocated systems also allow greater design flexibility ifprecise stable control can be exerted at a distance throughthe flexible structure.

In design of these control systems, whether they arecolocated or non-colocated, an accurate mathematical model ofthe structure is required. When the structure is modeled asa continuous systems, it is described by partial differential

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equations. But, for large problems these may be impracticalto solve. Instead, computer based Finite Element Method (FEM)is the preferred way to model structures. These models toocan become very difficult to solve simply because of its size.Each element has nodes associated with it and each node canhave up to six Degrees of Freedom (DOF) for translations androtations (the DOF represents the unknowns in the problem).Large finite element models are usually truncated to smallernumber of modes. But, the truncation introduces moreuncertainties and modeling errors to the system. Inparticular, the truncated models introduce RHP zeros whichmakes the system non-minimum phase. Traditionally, non-minimum phase systems have been avoided because they aredifficult to control. Therefore, the objective of thecontroller is to a make the system robust in the face ofuncertainties and model variations.

This thesis examines the effects between colocated andnon-colocated sensors and actuators on flexible structures.To truly model the complexities of a flexible system, acommercially available finite element model of a ContravesP52F gimbal structure will be used. First, a simple threeelement lumped spring-mass system is used to illustrate basicprinciples of applying control theory to a structure.However, this single motion DOF model does not exhibit any RHP

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zeros of a non-minimum phase system usually associated withnon-colocation. A two motion degree of freedom system, atransverse beam, is examined to deal with RHP zeros in controldesign.

The optimal control design techniques of linear quadraticGaussian and loop transfer recovery (LQG/LTR) are used tocontrol these systems. The design techniques developed forthe beam are then applied to the Contraves P52F gimbal. Thisallows the LQG/LTR design methodology to be used on astructure which exhibits more complex plant characteristicsthan a transverse beam. Then a thorough design analysis canbe evaluated for performance and robustness characteristicsbetween the two types of systems.

The following sections of the thesis first covers thetransfer function between the input (actuator) and the output(sensor) for the flexible systems in Chapter II. It will beshown that the poles of the transfer function are the system'snatural frequencies; whereas, the zeros are functions of thesensor and the actuator locations. Chapter II also introducesthe single motion DOF three element lumped mass model.Chapter III introduces the LQG/LTR theory used to design thecontrollers for the colocated and the non-colocated systems.The three element model is used to illustrate the concepts ofthe LQG/LTR theory. Chapter IV presents the transverse beamhaving RHP zeros for the non-colocated case. Modal equationsof motion (EOM) for finite element models is used to derive

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the state-space equations which are needed for control theory.Then the LQG/LTR theory is used to design controllers for thefinite element transverse beam. Chapter V investigates thegimbal structure which because of its large number of modes istruncated. The control design is compared between the twosystems and evaluated for performance and robustness.Finally, Chapter VI summarizes the results and givesrecommendations for further study.

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II. BACKGROUND

This section will present a thorough background on thepoles and zeros associated with colocated and non-colocatedsystems. Before analyzing the more complex Contraves P52Fgimbal structure, a simpler model will first be examined toillustrate basic principles. A three element model was chosento simulate the torsional effects of the motor on the gimbal.This model is analogous to studying the longitudinal or thetorsional vibration of a system. Because these systems haveonly motion about one axis, the term single motion DOF will beused in this thesis.

This elementary model can show the effects of colocationand non-colocation of the actuator and the sensor. Withcolocation it is much simpler to stabilize a system becausethe transfer function exhibits the desirable property ofpole/zero alternation on the imaginary axis. In this case thepoles go to the neutrally stable zeros during high gaincompensated feedback. Non-colocated systems may have thepoles and zeros alternating on the imaginary axis, but theyusually have right half plane zeros as well. As discovered byCannon and Rosenthal [1], non-colocated systems are sensitiveto parameter variations and may have "zero flipping".

The three element model for the non-colocated system

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examined in this chapter has no zeros. But, it can be shownto behave as a non-minimum phase system. Therefore, one cannot necessarily make the assumption that the system is minimumphase because it has no RHP zeros when modeling structures asa single motion DOF system (i.e. longitudinal and torsionalvibration models).

2.1 POLE/ZERO PATTERNS FOR COLOCATED SYSTEMS

First, the theory on pole/zero alternation is presentedbelow as proved by Martin in Reference 7. Shown below is thetransfer function with finite number of modes for a positionoutput sensor, y(x,s), and a control force input actuator,u(s), for a flexible structure

H(s) = y(xs) - 2 (2.1)U(S) i S2 + 2Ci~is + Wi

where Wi = i'th mode natural frequency,Ci = i'th mode damping

s is the frequency domain notation afterLaplace transformation

The quantities g, and h, are functions of the location ofthe sensor and the actuator, respectively, with respect to the

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i'th mode shape. The zeros which are a function of hig havethe following properties as proved by Martin.

Theorem:A rational function of the form

NN-X ) a, + a2 + + (2.2)TN(X) D (x) x-b I x-b 2 x-bN

with a i * 0; b i > b2 > ... > bN ; a,, b i E R;

will have alternating poles and zeros on the real axis if, andonly if

sign(aj) = sign(ak) for all j, k.

Collorary:

In Eqn 2.2, ifsign(a) = sign(ai 1 )

then TN(x) has a real zero between the poles b, and b,+l .

The significance of the alternating poles and zeros asshown by the theorems will be developed in the followingsections. The transfer function for a flexible mechanicalsystem given in Eqn 2.1 can be related to that of Eqn 2.2 by

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setting x = s, = O, b i = _2, and a. = hig i. Thus anundamped system with poles at

s -2 = _2, 1 i N

will have zeros at

s 2 = Zk 2 , 1 k N

with wk < Z k < Wk+12if and only if

sign(hjg,) = sign(hkgk) for all j,k.

Eqn 2.1 for the colocated sensor/actuator pair withoutdamping for position output can be expressed as

N a,H(s) = y ,S) - (2.3)u (s) i= s2 , 223

where a = hg i is th e residue of th e i'th mode shape. Usingthe mode shape for a longitudinal vibration of a rod as anexample (this is also analogous to torsional vibration)[9:158, 15:155], the actuator and the sensor mode shapes arethe following:

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hi F2cos{ iw (0) }, actuator located at x = 0,gi= F2cos( ir (x1 }, sensor located at distance x

When the sk-sor and the actuator are colocated at : = 0,the dot product of h. and gi becomes /2cos(0)*/2cos(0) = 2.The residues ai will always equal 2 and is positive;therefore,the zeros and poles alternate on the imaginary axis as shownin Figure 2.1.

Y

Figure 2.1 AlternatingPoles and Zeros

To stabilize this type of system, the double poles at theorigin require a lead compensator to shift them away from theorigin. Then the remaining flexible poles will go to thezeros with an angle of departure greater than 90" tc remain inthe left half plane (LHP). In section 2.3.1, an examplc ofthis will be shown.

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2.2 POLE/ZERO PATTERNS FOR NON-COLOCATED SYSTEMS

Using Eqn 2.3 when the sensor (gi) is located at theopposite end (x = 1), the finite transfer function is

H(S) - y(l,s) _ 1 + (-l)llau~s)~ Y2 ++ (2.4)(S) S 2 j=1 S2 + W2

h i = /2cos{ i7r(O ) = /-g i = F4cos{ iir(l) } = +,/2 for even number i

- -,/ for odd number i

Clearly, the residues do not have the same sign for all themodes and thus do not have alternating pole/zero pattern onthe imaginary axis. Bryson and Wie show that the transferfunction Eqn 2.4 for infinite number of modes do not havezeros except at infinity [15:159]. In Reference 15, theyderive the product expansion form of Eqn 2.4 from thetranscendental transfer functions of a wave equation forsingle motion DOF systems. Eqn 2.5 shows the zero locationsfor the three different cases of sensor location used inSection 2.3 for the three element model.

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_____~ - 25

H(s) = ______ _ _ __

Wk,

where Zk 2 7rk7r

For colocation when x =0, the zeros,

Zk = (k - 1/2) 7r

alternates between the poles. For non-colocation whenx= 1/2, the zeros,

Zk = (2k - 1)7r

are canceled by the poles. For non-colocation when x =1,the zeros,

z k

so that s 2/co -0, and has no zeros.

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2.3 THREE ELEMENT LUMPED MASS MODEL

This section will examine a single motion DOF systemabout a single axis for a three element lumped spring-massmodel to illustrate the principles used in this thesis. Thismodel is in essence a simple finite element model of atorsional rod. The diagram below shows the lumped spring-massmodel and the governing equations of motion.

ei 62

Ji J2 J3Figure 2.2 Three ElementLumped Mass Model

Ji01 = Tm - K 1 (0 1 - Oe)J 2 0 2 = K, (0 1 -0 2 ) - K2(02 - 03) (2.6)J303 = K2(02 - 03 )

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whereJi are the moments of inertia,ei are the angle of rotation,Tm is the motor torque applied at mass i1Ki are the spring constants.

Taking the values for the moments of inertia and thespring constants as unity, the ecquation simplifies to thefollowing:

1= -01 + 02 + Tm

02 = 01 - 202 + 03 (2.7)03 = 02 - 03

The output of the measurements of the torsional rotationare the following:

Y= 01Y2 02'Y3= 03

These three second order linear equations can be put intothe tollowing state space form:

= Ax + Bu (2.8)y = Cx

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Defining the following as the state and control vectors

010203

x= 01 U = Tm = input0203

you have the following state space form for the equations ofmotion

000100 x1 0000010 X2 0000001 X3 0X + U-110000 X4 11-2 1000 0 (2.9)01-1000 0X6 I100000]

y= 010000 x0 0 1 0 0 0.

Eqn 2.8, the state space equation, can be represented inthe frequency domain by the following familiar form of atransfer function.

H(s) = C[sI - A]B (2.10)2 - 10


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For the lumped spring-mass system, the SISO transfer functionsare the following:'

S4+ 3 S2 + 1S 2 + 1 (2.11)1H(s) 2 (4 + 42 + 3 )

The numerator represents the zeros for the three outputlocations. Note the first polynomial in the numerator is forthe colocated case and the second one is the non-colocatedcase for Y2 = 02" The third polynomial which is zeroth orderrepresents the case where the sensor/actuator pair are locatedon opposite ends. The denominator of the transfer function isthe characteristic equation of the model. Colving for theroots of the characteristic equation gives the system poles(structure's natural frequencies). These poles are the sameas taking the eigenvalues of the A matrix. The poles and thezeros are listed below in Table 2.1. The table agrees withEqn 2.5 which shows colocation has alternating zeros and non-colocation does not have zeros.

Matlab was used, using [num,den] = ss2tf(A,B, C, D,l,w)2 - 11


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Table 2.1 Poles and Zeros of the Transfer Function, Ean 2.11

poles zeros, y. zeros, v2 zeros, Y30,0 1.000i .618i 1.000i none 1.732i 1.618i

The following are the Bode plots of the magnitude versusfrequency for the colocated (yl as output) and the non-colocated (Y3 as output) transfer functions. The dipsrepresent the zeros on the imaginary axis and the rises arethe poles.

BODE PLOT

104

102

10'

100

10-2

10o- 2 KEY \o-3 ,o',-colocoted10-4

10-2 10 - 1 100 10' 102FREO (Rod/See)

Figure 2.3 For Colocated, yl, and Non-colocated, Y3

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2.3.1 COLOCATION

The zeros of the transfer function depend on the sensorand the actuator location. One can see from Eqn 2.11 thatthere are th'ree polynomial equations each representing one ofthe output cases. For the colocated case (y, 01), the secondorder polynomial, s4 + 3s2 + 1, has zeros(roots of thenumerator) as shown in Table 2.1. The poles and zeros areshown to be alternating. Using Eqn 2.2 reveals the transferfunction after partial-fraction expansion has all its residueshaving the same sign. The root locus plot of Figure 2.4 showsthe alternating pattern of poles and zeros when the residueshave all the same sign as Martin's theorem shows.

To see the robust properties of the colocated system, theroot locus for open loop plant for the system in Figure 2.4will be plotted.

.,, TG(s)

Figure 2.4 Closed Loop System with UnityFeedback and Proportional Gain

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The following equation represents the closed loop transferfunction of the plant under unity feedback with proportionalgain.

GCL (s) = kG(s) (2.12)1 + kG(s)where G(s) represents the transfer function of the open loop

plant.k = proportional feedback of the output with nocompensation

The root locus plot of Figure 2.5 contains the poles ofthe solution to the characteristic equation of Eqn 2.12 asgain k increases. The plot reveals one of the rigid bodypoles (there are two at the origin) are destabilizedimmediately while the flexible poles are marginally stable onthe imaginary axis.

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ROOT LOCUS2

1.5

u, 0.5

-0.5-1

-1.5

-5 -4 -3 -2 -1 0 1 2 3 4 5REAL AXIS

Figure 2.5 Colocation of Sensors and Actuators

The system is clearly unstable for proportional compensation.But, a simple lead compensator can stabilize the plant asshown in Figure 2.6.

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RO O T LOCUS6

4

20

-2

-4

-6-25 -20 -15 -10 -5 0REAL AXIS

Figure 2.6 Colocation with Lead Compensation

The compensator used is the following:

GCL =-32( s + 8 ) (2.11)

As depicted in Figure 2.6, the colocated system remains stableeven under high gain feedback. The gain value was increasedfrom .5 to 20.

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2.3.2 NON-COLOCATION

For the case where the sensor is located in the middle(Y2 = 02), there is an excess pole which causes th e root locusplot to go unstable during proportional feedback as shown inFigure 2.7.

ROOT LOCUS2.52

1.51

0.5

S-0.5

-1-1.5

-2.5-2 -1.5 -1 -0.5 0 0.5 1 1.5 2REAL AXIS

Figure 2.7 Non-colocation Under ProportionalFeedback with the Sensor Located at Mass J2

Partial-fraction expansion reveals this non-colocatedsystem does not all have the same sign for its residues, so,the system does not have alternating poles and zeros. This isclear by examining the poles and zeros in Table 2.1. In fact,

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this system has the iindesirable property of being unobservablebecause the zeros cancel one of the poles. For the third casewhere the sensor and the actuator are located on opposite endsof the structure, there are no zeros. Therefore the system isclearly unstable during proportional feedback as shown inFigure 2.7.

) 'TOCUS2

0 0.5

-0.5

-1,5

-1.5 -1 -0.5 0 0.5 1.5REAL AX;S

Figure 2.8 Non-colocated Case Under i7-oportionalFeedback with Sensor at the Opposite End at Mass J3

Using a simple lead compensator as in the colocated case,the first flexible pole is immediately destabilized as shownin the root locus plot in Figure '.8. The transfer functionfor the lead compensator used is

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GcL 3 s__.2

In the n'>i-colocated case, a lead compensator is ineffective.To ach'.;-ve stability, a higher order compensator is clearlyneeded. For this thesis, th e Linaar Quadratic Gaussian (LQG)desig:, techniques will be used to design the controller.

ROOT LOCUS

...........................................................(A 0.5

-05-

-1 -0.8 -0.5 -0.4 -0.2 0 0.2 0.4 0. 0.5PEA1. Axis

Figure 2.9 Non-colocation With a Lead Compensator

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III. LINEAR QUADRATIC GAUSSIAN / LOOP TRANSFER RECOVERY

3.1 LQG/LTR THEORY

Linear Quadratic Gaussian/Loop Transfer Recovery(LQG/LTR) theory is used to design the controllers to increasethe performance and robustness of the control systemspresented in this thesis. The first step to the LQG/LTRmethod is to design the Linear Quadratic Gaussian (LQG)controller. A quadratic performance index is used to definethe LQG controller; therefore, it produces an optimal controldesign stabilizes the closed loop system for the given designconditions. The LQG controller is comprised of the LinearQuadratic Regulator (LQR) and the Linear Quadratic Estimator(LQE). The LQR is a full-state feedback controller and theLQE is an estimator, commonly called the Kalman filter.

The LQG/LTR design method modifies the standard LQE andrecovers the guaranteed robustness of the full-stateregulator. This results in a closed loop system (the combinedplant and the compensator) that is robust to uncertainties,disturbances and gain increases. The LQG/LTR methodology isalso ideal for mu?.ti-input-multi-output (MIMO) systems. Thefollowing section will first introduce the basics of the full-state feedback control theory. Then, the estimator and the

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LQG controller will be presented. Finally, loop transferrecovery is developed. These sections briefly introduce theoptimal control theory which is presented in more detail inmany other references. In particular the technical report,Introduction to Robust Multivariable Control, by Ridqely andBanda is extensively used by the author to develop the nextfour sections [10].

3.2 LINEAR QUADRATIC REGULATOR

The linear quadratic regulator as taken from chapter 6 ofRidgely and Banda is the following. The LQR relies onminimizing the quadratic performance index

j = f [XI( ) Qc)x(t) + uT(t) Rcu(t) ] dt (3.1)

for the state-space Eqn 2.9.The weighting matrices Qc and RC are chosen by the

designer depending on the importance of the state or thecontrol vectors. If the (A, B] pair is stabilizable, then theoptimal control law is

u = -KCx (3.2)

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The gain matrix, K., is given by

Kc = KC-IB P (3.3)

where P is the symmetric positive semi-definite matrixsolution to the algebraic Riccati equation

0 = ATp + PA -PBRlCBTp + Q (3.4)

where selecting Qc = Q > 0 (postive semi-definite) is asufficient condition for asymptotic stability of the closed-loop system. Rc must be postive definite. Substituting thecontrol law, Eqn 3.2, into Eqn 2.9 yields the closed loopsystem.

x= Ax + B[-Kcx]= [A - BKC]x (3.5)

Figure 3.1 depicts the state space system of Eqn 3.5.

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G(s) PlantUo

Figure 3.1 Block Diagram of the Full-state Controller

3.3 LINEAR QUADRATIC ESTIMATOR

The one drawback of using the full-state regulator is itassumes all the states can be measured. It is frequentlyimpossible or very expensive to measure all the states (manysensors may be required on a space structure) ;therefore, anestimator must be used to model the system. This estimatormust also take into account disturbances in the system. Thefamiliar state-space equation now including disturbances is

x = Ax + Bu + rw (3.6)y = Cx + v (3.7)

where w and v are input and output disturbances, respectively.

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The disturbances are assumed zero-mean Gaussian stochasticprocesses which are uncorrelated in time and have thecovariances

E[ w(t)wT(7) ] = Q 06(t-T) 0,E[ v(t)vT(7) ] = Rf6(t-r) > 0

The covariance matrix Q0 is positive semi-definite;whereas, Rfis positive definite. The correlation between w and v will beassumed to be zero

E[ w(t)vT(t) j = 0

Equation 3.6 and the Kalman filter are represented by theblock diagram below.

B T

Figure 3.2 Block Diagram of the KalmanFilter

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The equation for the estimator is given by

Xe = Axe + Bu + Kf(y - CXe) (3.8)

where xe is the estimate of the states.The following development of the Kalman filter is taken

directly from Ridgely and Banda. First, an assumption must bemade: the input and output disturbances, w and v, are time-invariant and wide-sense stationary. Then the matrices Q0 andRf are constant. This is a valid assumption as long as theobservation of the output is much greater than the dominanttime constants of the system. The Kalman filter gain matrix,Kf, minimizing the mean-square error

E[eTe] (3.9)where

e = x - xe (3.10)

isKf = ZCTRf " (3.11)

Z, the variance of the error function, is found by solving thealgebraic variance Riccati equation

0 = AZ + ZAT - ZCTRf ICZ + Qf (3.12)

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with Qf = rQoQT . Frequently, F = I (meaning each state has itsown distinct process noise).

The algebraic variance Riccati equation shown in Eqn 3.12has several solutions, for which the correct solution isunique and positive definite. A sufficient condition for Z toexist is that the pair [A, C] be completely observable. Thiscondition may be relaxed to detectability, in which case it isnecessary and sufficient and Z may be positive semidefinite.Given that Z exists, the error dynamics of the filter are

=X - e (3.13)= [Ax + Bu + rw] - [Axe + Bu +Kf{y - CXe}]= [A - KfC]e + [F -Kf][w v]'

Therefore, the poles of [A - KfC] are the poles of theestimator error response. The poles must be asymptoticallystable( the error must be getting smaller) if and only if thepair [A, F] is stabilizable.

3.4 LINEAR QUADRATIC GAUSSIAN

Combining the results of LQR and LQE gives the followingexpression for the LQG compensator

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xe(t) = Ax(t) - BKcXe(t) + Kf[y(t) - CXe(t)] (3.14)= (A - BK c + KfC)Xe(t) + Kfy(t)

Taking the Laplace transform yields the following

Xe(S) = [sI - A + BK c + KfC]- Kfy(s) (3.15)

Now, substituting this into the control law (dropping thes notation)

u = -KcX e (3.16)

yields the expression for the LQG compensator

u = -Kc[sI - A + BK c + KfC]IKfy (3.17)

substituting the plant transfer function without the noises,w and v,

y = C(sI - A) 'Bu (3.18)gives

u = -Kc[sI - A + BKc + KfC] 'KfC(sI - A)-Bu (3.19)

The following figure represents the Eqn 3.19 for the LQGcompensator and the plant.

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K's)__Compsar

Figure 3.3. Block diagram of the LQG system.

The poles of the LQG compensator are seen to be solutions of

detfsI -A + BK- + KfCE = 0

Although the poles of the regulator, the estimator and theclosed loop system are all stable, the LQG compensator polesare not always stable. The eigenvalues of the closed-loopsystem are evaluated to prove the stability. Taking the twoequations we have for the state

X = Ax + B[-KcXe] + Tw (3.20a)Xe = (A - BK C + KfC)X e + KfCX + Kfv (3.20b)

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Remembering that

= X - Xe (3.13)

We can put Eqn 3.20a and Eqn 3.20b into its state space form

[xA BKA BK [ ] (3.21)= 0 A - BKf Cj~e ] r -EUsing Schur's formula, the above equation can be broken intoits determinant form

det[sI - A + BKc].det[sI - A + KfC] = 0 (3.22)

Eqn 3.22 shows that the LQG compensator is composed of theregulator and the estimator poles which are stable.Therefore, the closed loop LQG systems are always stable,although the compensator itself may not be stable. An exampleof this unstable compensator will be shown in the nextsection.

3.5 LOG DESIGN ON THREE ELEMENT MODEL

The Linear Quadratic Gaussian (LQG) design techniquedeveloped in section 3.4 will be used to develop a sixth order

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compensator (same order as the plant) for both the colocatedand non-colocated cases. To make a fair design comparison,the closed loop bandwidths for the two systems were madeequal. This bandwidth was pushed as close as possible to thelowest frequency zero. This ensures the flexible poles willhave an effect the plant and the controller.

Briefly, the LQ Regulator is designed for the highestbandwidth it can achieve by varying the state space weightingmatrix, QC" The bandwidth achievable is limited by the firstimaginary zero which causes the closed loop gain to go belowthe -3 dB line quickly. Then, the LQ Estimator is calculatedby estimating the noise matrices, Q0 and Rf. This thesis usesan approximate noise intensity for Q0 as ten percent of theinput force. The regulator and the estimator are thencombined to construct the LQG controller. In each case, thedesign is iterated by varying the QC weighting matrix for theLQR until they have approximately the same bandwidth. SeeFigure 3.4.

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BODE PLOT30 . . . .. . . . . . . .

20100

1; -10

-20\-30\

-40- KEY

-5 0 non-olocoted

-6010-2 10-1 100 101 102FREQ (Rad/Sec)

Figure 3.4 Closed Loop Bandwidths for the ThreeElement Model

The procedure leading to the LQG design was facilitatedby using a MATLAB script file, LQGD.M (See Appendix C for theprogram). The weighting matrices used for the LQR for thecolocated case is

QC = diag(10 10 10' 10' 10' 10')

and for the non-colocated case is

Qc = diag(10 10 10 10 102.5 102.5)

For both cases, the noise intensity at the input for the LQEis chosen as

Q0 = 102.

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After designing the LQG controller the -oot locus of theclosea-loop system is plotted to check the stabilityrobustness. The transfer function of the LQG controller forthe colocated case is

KLOG = (s + .177) (s + .19.892i) (s + .371.637i)(s + 316.2) (s + 3.125) (s + .155.838i) (s + .351.62i)

Figure 3.5 shows the rc3t locus plot of the closed loopcolocated system. The compensated plant rewains stable forall gains in the range 0.5 to 30.0.

ROOT LOCUS3

2

( 0 -z

-2 -.

-4 -3 -2 -2REAL AXIS

Figure 3.5 Root Locus for the LQG Design withoutLTR for the Colocated System

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The transfer function of the LQG controller for the non-colocated case is

KLOG " (s + .12) (s + 1.72) (s - 7.5) (s - .2081.07i)(s + 1.304) (s + 3.1) (s 1.331.56i) (s + .9522.li)

Figure 3.6 shows the root locus of the non-colocatedclosed loop system. With increasing gain, the closed looppoles of the compensator and the plant move into the righthalf plane after a gain of k=1.75. This is a seventy-fivepercent gain increase from the nominal design, but it is notas robust as the colocated system which remains stable forgains up to k = 30. One can see that LQG does not guaranteestability in the face of uncertainties. In fact, for the non-colocated system, it goes unstable rather easily. In otherwords, the controller is not robust and LTR techniques areneeded.

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ROOT LOCUS3

xx-2-

xx

-1

-4 -3 -2 -1G 2REAL AXIS

Figure 3.6 Root Locus for the LQG Design without LTR forthe Non-colocated System

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3.6 LOOP TRANSFER RECOVERY

As mentioned in section 3.5, the disadvantage of the LQGcompensator is that it is generally not robust. The threeelement non-colocated example went unstable after a small gainincrease. One reason is that the estimator inserted into theLQG loop no longer guarantees the stability margins from thefull-state feedback regulator. Loop recovery technique mustbe used by adjusting the Kalman filter gains.

In looking at Figure 3.3 of the LQG block diagram, thereturn ratio (the characteristic equation of the SISO transferfunction) at point marked 1 determines the robustness andperformance properties for the plant, G(s). The return ratioat point 1 is

K(s)G(s) = -Kc[sI - A + BKC + KfC]'KfC(sI - A) IB (3.23)

If you break the loop at point 1, the return ratio at point 2is the full state feedback controller

KLQR(S) = Kc(sI - A)_1B (3.24)

The reason is when you break the loop at point 1, the signalto the Kalman filter matrix Kf is zero. This assumes that theA, B, and C matrices of the filter match those of the plantexactly. Thus, the gist of the LTR theory is to make the

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return ratios at points 1 and 2 to be the same. The followingis taken from Maciejowski (6:232].

Let * = (sI - A) and rewrite Eqn 3.23 without thefrequency domain notation (s) for brevity.

K(s)G(s) = -Kc[t + BK c + KfC]-KfC-B (3.25)

Without going into the formal proof which is shown inRidgely and Banda [10:ch. 8] and Maciejowski [6:232-234], onecan see how Eqn 3.25 becomes 3.24 by observing what happens asKf is increased toward infinity. At low frequenies (s = jw),the term inside the bracket, t + BKc, becomes much smallerthan KfC. Thus, the bracketed term approaches [KfC] I . Thisleads to

K(s)G(s) = -Kc [ KfC] 1 KfC t-1 B (3.26)K(s)G(s) = -K c I V1BK(s)G(s) = -KcVIB

Intuitively, Eqn 3.26 shows that at low frequencies andhigh estimator gains the LQG controller and the plantapproaches the behavior of a full state regulator. In termsof poles and zeros, the effect of increasing the gain Kfcauses the closed loop poles of the Kalman filter to notch outthe plant zeros. The plant poles then move toward the

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compensator zeros. In turn, these compensator zeros approachthe closed loop pole locations of the full-state regulator andthus achieves the robustness guaranteed by the LQR. One canthen increase the robustness over large a bandwidth byincreasing the gain Kf. See the LTR plots of Figure 3.7 andFigure 3.9 in Section 3.7 for examples.

However, there is a design trade off because increasingthe filter gains also increases the system noise response.One thing to keep in mind is that increasing the gain nolonger makes it a true Kalman filter. The noise intensity ischanging from the initial assumed Qf = Q0. See the equationbelow when the loop transfer recovery is applied at the plantinput.

Qf = Q, + q 2BVBTRf = Ro

V is any postive definite symmetric matrix and is usuallytaken as identity. Q0 and Ro are the actual noise intensities.The second term with q2 is the additional fictitious outputnoise associated with LTR. When q = 0, it becomes thestandard Kalman filter.

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3.6.1 ROOT MEAN SQUARE RESPONSE

The effect of the incredsing the fictitious noise on thesystem can be measured by the Root Mean Square (RMS) responsefor the LQG/LTR closed loop system. The covariance of theoutput must be found by first solving the Lyapuno,, equationfor the state variance [5:104].

AQ x + QXAT + BVBT = 0 (3.27)where

Qx E(xxT)V= IA and B are from state space equation

The output covariance is then

Qy = E(yyT)

which after combining with Eqn 3.27 becomes

Qy = CQxCT (3.28)

The RMS response is found by simply taking the square root ofthe diagonal of the covariance in Eqn 3.28.

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3.6.2 LTR WITH NON-MINIMUM PHASE SYSTEMS

LTR theory is not guaranteed to work with non-minimumphase systems. Maciejowski states that some of the Kalmanfilter's eigenvalues or poles from the LQG design moves towardthe plant's zeros [6:231]. This implies LTR can only be usedfor minimum phase plants since the eigenvalues of the closed-loop regulator and the estimator cannot be in the left halfplane. However, LTR may still work if the right half planezeros are above the cross-over frequency.

3.7 LOOP TRANSFER RECOVERY ON THE THREE ELEMENT MODEL

For the three element model, the colocated system showsexcellent loop recovery properties throughout its frequencyrange (10-2 to 102 rad/sec). Examination of the systemeigenvalues shows that the poles of the plant and thecompensator moving toward the poles of the full-statecontroller. Figure 3.7 shows clearly the open loop systemapproaching the target loop of the full-state. The highfictitious noise of 1015 was able to get full recovery (asevident by the loop matching the full state). The root locusof the closed loop K(s)G(s) system shows it retains excellentrobustness with the poles going unstable after a gain increaseof k = 28.5 at q2 = 1015. This is almost a three-thousand

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percent gain variation that the colocated system can handle!See Figure 3.8 for the root locus plot.

The non-colocated system does improve its robustnessslightly with LTR, although loop recovery is not as easy toachieve with the non-colocated system (indication that thistype of system is non minimum phase). See Figure 3.9 forq2 = [ 0 1018 o1020 . Notice the good recovery at lowfrequencies, but not at high frequencies. In fact there aresmall spikes in the loops where it diverges from the fullstate's open loop. Using a large fictitious noise value of1020 did get good recovery and damped out the spike. It alsoimproved the robustness of the system to gain variations up tok = 4. See Table 3.1

Table 3.1 LTR for Colocated and Non-colocated Three ElementModel

2 colocated non-colocated

0 30 1.7510 30 1.7510 17.0 1. 5109 6.5 1.501012 8.0 1.501015 28.5 2.01018 30 2.751020 30 4.01024 30 7.0

The table shows a decrease in the colocated system whenq2 = 19 and 1012. Starting at these fictitious noise values,

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the colocated RMS response becomes greater. Then the RMSresponse stabilizes after q2 = 1015. See Table A.l in AppendixA. Thus, the increase in the noise of the colocated systemappears to have an effect on the robustness of the system toloop transfer recovery.

In looking at the LTR plots and the Bode plots, the non-colocated system has the advantage of having the highfrequency roll-off. This is desirable to prevent the noisefrom amplifying and driving the system unstable. The non-colocated system does have a major drawback in that it behaveslike a non-minimum phase system with RHP zeros. Figure 3.10shows the root locus plot using the fictitious noise value of1018. Note the poles moving into the right half plane as ifthere were RHP transmission zeros.

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LOG .1th LTR140 !U.t r' P

8000

40020000

-200O-400 0

-- 00-1 0 - 00 1-600 -4000 -200 0 1 200RRCIEALYXIS.

Fiue3LRfor the Colocated System

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LQG .1th LTR

1220

q401202

-0 q!l

10-2 10-' 100 10' 10'FREOUENCY ( Rod/Sec

Figure 3.9 LTR for the Non-colocated System

ROOT LOCUS

30-

20

10-

-20-

-30

-35 -30 -25 -20 -15 -10 -5 0 5 10 15REAL AXIS

Figure 3.10 Root locus for the LQG desing withoutLTR for the non-colocated system

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3.8 STEP RESPONSE

The unit step input will now be applied to the closedloop full state regulator. The unit step input response isuseful for analyzing the performance characteristics of asystem. The time response is useful in revealing the risetime, settling time, and the steady state value [9:40].

The RHP zero has an interesting influence on the stepresponse. Reid states that for non-minimum phase systems theinitial response is negative. This means the system moves inthe opposite direction in relation to the final steady-statevalue. If the system has no zeros (i.e. the non-colocatedthree element mass model), the initial response will be flat.The plot of the step response for the colocated and non-colocated systems are shown in Figure 3.11. The non-colocatedsystem shows a flat response initially since it has no zeros.The colocated step response shows it fluctuates as it comes toits steady-state value. This is the result of the imaginaryzeros and because the poles are lightly damped.

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STEP RESPONSE1.2

0.2

0.2 //// KEY

./ colocoted " -...---- non-colocoted

-0.2 10 2 4 6 8 10 12 14 16

TIME (sec)

Figure 3.11. Step Response of the Three ElementModel

3.9 CONCLUSION

The colocated LQG/LTR system is very robust to gain

variations. However, the robustness decreasesfor q2 = 109

through 1012. The RMS response at these fictitious noisevalues show that the colocated system is much higher than thenon-colocated system. While the non-colocated case has lowRMS response, it is not robust in comparison to the colocatedsystem. It does improve slightly for very large q2 values.See Table A.I in Appendix A. Looking at the closed loopbandwidths show that the non-colocated system has very gooddamping and has the desirable high frequency roll off. Thecolocated system has little damping of its flexible poles. Inthis regard the non-colocated case is better.

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The step response to the unit input also shows the non-colocated system has better damped poles and a smoother risetime curve. However, the system exhibits non-minimum phasebehavior of having no initial response to the input for thefirst two seconds. See Figure 3.11. This maj be anunacceptable response. Recall this single motion DOF modeldid not have RHP zeros to cause a negative response typical ofnon-minimum phase systems. This leads to the next chapterwhich shows that non-colocated systems have RHP zeros for two-motion DOF models.

Observing the closed loop eigenvalues of the LQG/LTRcontroller showed how it becomes robust. LQG design is helpedby LTR because as the fictitious noise is increased, thecompensator poles move toward the open loop planK zeros tonotch them out. The Kalman filter accomplishes this tasksince its closed loop poles (A - KfC) at high q2 values movesright onto the plant zeros. On the other hand, thecompensator zeros move toward the stable poles of the closedloop regulator (A-EKC). Thus, the plant poles duringcompensated feedback must move toward the compensator zeroswhich guarantees the stability and the robustness of theclosed loop full-state regulator. In essence, the LQGcompensator inverts the nominal plant and substitutes thedesired LQR dynamics.

To summarize for the three element model using an initialnoise intensity of Q0 = 102, the colocated system can be made

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robust by just using the LQG compensator; whereas, the non-colocated system can be improved slightly with LTR. To getbetter rise time curve and damping for the colocated systemrequires a lower bandwidth away from its low frequencyimaginary zeros and therefore a drop in performance. On theother hand, the non-colocated system can achieve a higherbandwidth and better damping since it has no imaginary zerosaffecting the crossover frequency it can attain.

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IV. A TWO MOTION DEGREE OF FREEDOM SYSTEM

4.1 Introduction

The first example presented in this thesis of the threeelement lumped mass system showed the non-colocated system wastechnically minimum phase (i.e. no zeros in the right halfplane). However, the system showed non-minimum phasecharacteristics. For instance, the step response had initialflat resronse characteristic of a non-minimum phase systemwith no zeros. Loop transfer recovery to the full-state loopshape was limited, especially in comparison to the colocatedloop recovery shapes. At high frequencies, it exhibited thecharacteristic steep roll-off of a non-minimum phase system.

4.2 EXACT SOLUTION OF A TRANSVERSE BEAM

The three element lumped mass model is a single motionDOF system since it only has plane rotation about one axis.The numerator of the non-colocated transfer function waszeroth order and had no roots (refer to section 2.3.2). Thus,a single motion DOF system will not yield any zeros in theRHP. Therefore, a two motion DOF system will be examined to

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demonstrate the non-minimum phase behavior for non-colocatedstructures. A two-dimensional beam which can translate androtate will be needed. As shown in Reference 14 by Bryson andWie, a Bernoulli-Euler beam with transverse bending vibrationswill have zeros in the RHP [14:165].

7El , L,E(

Figure 4.1 Free-free Transverse Beam

The equation of motion for the transverse beam is

y(IV, (x, t) + (x, t) = 0(4.1)

where y(IV) a and a y(x4 at 2

and y(x,t) is the transverse displacement.The displacement variables x and y are in units of L,

time in units of (cL4/EI) /2. a is the mass density per unitlength. Taking the Laplace transforms of Eqn 4.1 yields

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y(iV) (x,s) + s2 y(x,s) = 0 (4.2)

Thus the transcendental transfer functions for free-freeboundary conditions for the two cases of colocation and non-colocation yields the following

v(0,s) = sinhXcosX - coshlsinX (4.3a)u(s) A3 ( 1 - coslcoshA )

y(ls) = sinhA - sinX (4.3b)u(s) A3 ( 1 - coslcoshA )

where ;,= - s2 or s = jw (in the frequency domain)

Solving for Eqn 4.3a directly shows the poles and zerosalternating on the imaginary axis of the complex plane asshown in 1igure 4.2. The non-colocated case, Eqn 4.3b, hasinfinite number of reflexive zeros on the real axis as shownin Figure 4.3.

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120.90A

61.67 C49.96

22.37x 15.41

Figure 4.2 Exact Poles/Zeros of aColocated System

Figure 4.3 Exact Poles/Zeros of aNon-colocated System

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4.2.1 MODAL TRUNCATION AND ITS EFFECT ON ZERO LOCATION

Eqn 4.3 for the closed form solution to the Bernoulli-Euler beam shows that non-colocated structures have RHP zeroson the real axis. When the model is truncated, the finitedimensional form of the transfer function derived from Eqn 2.1shows that there can be complex zeros as well.

y(l,s) , 2 I 4 (-l)i!+a(- + E (4.2)U(S) S2 S=2 +

Reference 14 shows that Eqn 4.4 yields zeros on the real axisof the RHP when N = 1. When N = 2, Eqn 4.4 yields complexzeros in the RHP. Thus, a truncated model will notnecessarily have all its zeros on the real axis. But, it willhave RHP zeros.

4.3 APPROXIMATE SOLUTION BY FINITE ELEMENT METHOD

Using partial differential equation representation forstructures can be difficult to solve; therefore, the FiniteElement Method (FLM), will be used to analyze a two motion DOFbeam. The author used the FEM computer program, ANSYS, tosolve for the eigenvectors and the eigenvalues needed to get

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the equations of motion (EOM).Using the finite element approach as presented by

Meirovitch [8:ch.4], the general EOM for an undamped system is

[m](q(t)) + [k](q(t)) = (Q(t)) (4.5)

where [M] and [K] are constant mass and stiffnessmatrices of size n (number of modes used)(q) and (Q(t)) are the column matrices ofgeneralized coordinates and generalized forces,respectively.

Eqn 4.4 is difficult to solve because of the coupling inthe mass and the stiffness matrices. To solve these linear2nd order equations, the linear transformation

(q(t)) = [u](r(t)} (4.6)where (r(t)} is a new set of generalized coordinates,

[u] is a transformation matrix or a modal matrixcomprised of the eigenvectors,

is used to decouple the mass and stiffness matrices so thatthey are diagonalized. The EOM using the new generalizedcoordinates is

[m][u] ( (t)) + [k][u](?7(t)} = (Q(t)} (4.7)

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Pre-multiplying by [u]T results in the modal EOM

[M]{(r(t)) + [K](n(t)) = (N(t)} (4.8)

where [M] = [u]T[m][u] is the modal mass matrix,[K] = [u]T[k][u] is the modal stiffness matrix.

Orthogonality of the modal matrices diagonalize the massand the stiffness matrices (given that they are constant andsymmetric). It is common to normalize the mass matrix to anidentity matrix. Multiplying by the inverse of the identitymass matrix makes the stiffness matrix equal to theeigenvalues since w = K/M. The final form of the EOM is then

[I] {u(t) ) + [2] (77(t) ) = {N(t) ) (4.9)

This form of the EOM is readily available from finite elementcomputer programs such as ANSYS. Eqn 4.8 can be put into thestate-space form of Eqn 2.8

= [ 2 ] + UT (4.10)

where u is the input control force,

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Linear transformation must also be applied to y = Cx to getthe output matrix in the state space equation form.

y = .[u] [7 ]'T (4.11)

Thus, the state space equation for output becomes

y = [ C[u] : 0 ]x

The matrix C is padded with zeros to match the size of thestate vector x (since no velocity measurements are taken).The C matrix in Eqn 4.11 for a single output case will be arow matrix of size 1 x n with ones at the nodes of interest.

4.4 TEN ELEMENT TRANSVERSE BEAM EXAMPLE

To show that the FEM can produce the [A], [B], and [C]matrices required for the state-space equations, a ten elementtransverse beam is modelled in ANSYS. See Appendix B for theANSYS data input file.

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y Xx F

node 1 rode 1

Figure 4.4 Ten Element Transverse Beam

The two dimensional beam element (STIF 3) is an uniaxialelement with tension-compression and bending capabilities. Ithas three DOF at each node: translation in the x and ydirection; and rotation about the z axis.

Uyl Uy2

I &x

STIF e1enznt

Figure 4.5 DOF for the TwoDimensional Element

For this example problem because bending is the primarymode of interest, x translational DOFs were neglected. Theten element beam has a total of 40 DOF (4 DOF/element x 10elements). Eliminating the DOFs at the internal nodes due tothe constraints results in total of 22 unkown DOF. Solvingthis free-free beam using modal analysis in ANSYS produces a

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22x22 modal matrix, [u], assoicated with the DOF

UylOz1

DOF = 1 !uy11Ozl

The A matrix is formed from the mass and stiffness matrices,but ANSYS has normalized the mass matrix to identity so thatA can be derived by Eqn 4.9. The A matrix is

r 0 22x22 I22x221A [_W[ 2 2x 2 2 0 22X22

ANSYS outputs the eigenvalues in natural frequencies in unitsof cycles per time and must be converted to radians per timebefore using in the A matrix. Listed in Table 4.1 are theeigenvalues from Ansys compared to the closed form solutionfor free-free transverse beams [14:275].

Table 4.1 Eiqenvalues from Ansys and the Closed Form SolutionAnsys Closed Form Solution Error0.0 Hertz 0.0 0.0 %0.0 0.0 0.04.072 4.088 0.411.196 11.259 0.621.884 22.080 0.9

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Note the first two modes are the rigid body modes (X = 0.0).The error is less than one percent for the first five modes,which shows the accuracy of the model with the full twenty-twomodes. Later section will compare the eigenvalue solution forjust four modes. The Q force matrix is

00Q= 0

1

where 1 has been placed at the 21st row representing node 11in the y direction. Therefore the B matrix is

[0

where UTQ is the 21st row of the transposed modal matrix.The output matrix is

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where the ones represent a displacement sensor at node 1 (non-colocated) and node 11 (colocated). Remembering

C = [c[u](2x22) o(2X22)]

where C[u] becomes the 1st row and the 21st row of the modalmatrix, [u].

4.4.1 MODEL REDUCTION TO FOUR MODES

Because of the large size of the A matrix (44x44), themodel was reduced to only the first four modes. For the tenelement beam, going from twenty-two to four modes stillproduced accurate results for the first two flexible poles.This is evident by examining Figures 4.6 and 4.7. The Bodeplot of the reduced beam with four modes is compared with thefrequency response spectrum from ANSYS (KAN 6 Analysis). Inthis way, two different methods are used to find the transferfunction between input and output for a beam.

The reduction produced a 8x8 A matrix containing the modeshapes of interest. The first four eigenvalues are still veryaccurate to the closed form solution shown in Table 4.1.

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0.0 Hertz0. 87!e-64.23212.727

The second mode is the rigid body rotation, so the value wasmade zero. The first flexible pole is accurate to within 3.5percent of the closed form solution. See Figure 4.8 for themode shapes associated with the eigenvalues.

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BODE PLOTIC'

MATLAEB'1C - 51 A',SY S

me 10' 10 2 1C,3FREQ (Red/Sec)

Figure 4.6 Bode Plot Comparing Frequenc,%,, Responserf 11 Modes to the Reduced 4 Modes for theColocated Beam

muDE PLOT

1- ,

FR[E (e i, S~eFigure 4.7 Bode Plot Comparing the Frecjuernc,Response of 11 Modes to the Reduced 4 Modes for theNon-colocated Beam

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y]

Figure 4.8 The Four Mode Shapes of th e TenElement Beam

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The final A matrix is

0 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1

A= 0 0 0 0 0 0 0 00 0 0 0 0 0 0 00 0 -7.06x10 2 0 0 0 0 00 0 0 -6.39x10 3 0 0 0 0

The B matrix is0000B=

3.288').7273.3934.128]

The C matrix is

[ 3.288 0.727 3.393 4.128 0 0 0 0'C = [-2.272 2.485 3.654 -3.011 0 0 0 0,

The eigenvalues of the A matrix are the system poles andthe transmission zeros are found using the tzero command inMATLAB. They are listed in Table 4.2.

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Table 4.2 Poles and Zeros of the Ten Element Beam

Poles Colocated Zeros Non-colocated Zeros0,0,0,0 r/s 0,0 0,026.59i 18.32i 32.5279.97i 61.89i 65.20

Note the extra pair of zeros for the second rigid body motionis canceled by the pair of zeros for the colocated and non-colocated zeros. This pole/zero cancellation did not causeobservability problems. Shown below is the Bode plot of theopen loop plant for both the colocated and non-colocatedsystems. Note the dips in the colocated plot indicating ithas imaginary zeros;whereas, the non-colocated has no dipsbecause its zeros are on the real axis. The non-colocated lowfrequency gain is little less than the colocatedgain;therefore, a pre-filter gain of 2 is required to theinput of the B matrix to make the two systems equal forcomparison purposes.

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I,

10'

Figure 4.9. Bode Plots of the Colocated and Non-colocated Ten Element Beam

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4.5 LOG/LTR CONTROL DESIGN ON THE BEAM

The control design on the transverse beam producedinteresting results. The control design without LTR (thefictitious noise parameter q = 0 ), resulted in the colocatedsystem being robust to large gain variations (from 0.5 to 43).The non-colocated system did not improve with LTR as expectedof a non-minimum phase system.

Both systems were designed to have the same bandwidth ofabout 8 rad/sec. Figure 4.10 represents the Bode plot of theclosed loop bandwidths for both cases. The weighting matricesused for the LQR for the colocated case is

Qc = diag(10 5 1 0 5 10 5 10 5 10 1 0 103 L03),

and for the non-colocated case is

Qc = diag(104-5 104.1 104 104 103 103 103 103)

For both cases, the noise intensity at the input for the LQEwas selected as

Q4 = 101

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BODE PLOT60

40

20

C3 0

S-20

-40

-0 - c :o- oted - - -

10 ' 102 10,3

Figure 4.10 Closed Loop Bandwidths for the TenElement Beam

The root locus plots show the poles of the closed loopsystem as gain is varied. The colocated system went unstableat gain of k = 43. See Figure 4.11. This system became lessrobust after LTR for q2 = 109" Then the system started toincrease in robustness until the gain value exceeded 100.Observing the LTR plots of Figure 4.13 clearly shows that thesystem gained the robustness of a full-state regulator. SeeTable A.2 in Appendix A for the results from the designiterations. Figures 4.14 shows the root locus plot forq2 = 1012. Note the highest frequency pole is the one thatgoes unstable. So, this system is very robust for the lowerfrequency poles.

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The non-colocated case was not able to get full looprecovery no matter how much fictitious noise was added. Infact, gains after q = 1012 made no improvement on the lowfrequency recovery. See Figure 4.15 for the LTR plots. Thepoles became unstable after a gain of k = 2 for all q2 values.See Table A.2 in Appendix A. The root locus plot for q2 = 1012is shown in Figure 4.16. This plot clearly shows the polesmoving toward the right half plane zeros.

POOT LOCUS80 ...........................

S200 ,,..

-4 0 "x

-0 -25 -20 -15 -10 -5 0REAL AXIS

Figure 4.11 Root Locus for the LQG Design Without LTR forthe Colocated System

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ROOT LOCUS80

60-

40

_ 20 o

- -a0

-40 "

-60-

-80 __ __ _ __ _ __ __ _ __ _ __ _ __ __ _ __ _ __ __ _ __ _ __ _-60 -40 -20 0 20 40 60REAL AYIS

Figure 4.12 Root Locus for th e LQG Design Without LT R forthe Non-colocated System.

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LQ G WIth LTR

80

60

+0. 220

- 20 q1

-40

-00 -21-01 -1102 -503FRQEALY AXIS

Fgr43LR for the Colocated System

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LOC with LTR

10 101122 0

40

-401

-60O

-10-0 20 12 0 3FRQEALY RAIS )

Fgr 45LRfor the Non-colocated System

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4.5.2 STEP RESPONSE

The step response plot clearly shows which system is non-minimum phase. The non-colocated system has an initialnegative response to the step input. Otherwise, this systemhas much better damping than the colocated system which hasringing.

STEP RESPONSE1.2

06

0.4

I/ KEY/ cOIccOted

- 20 01 02 03 04 G5 06 0.7 S0 LI 171ME (Se':)

Figure 4.17 Step Response of the Ten Element Beam

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4.6 CONCLUSION

For the ten element beam after large amounts offictitious noise was added, the colocated system became veryrobust as it approached the target full-state regulator. Onthe other hand, the non-colocated system did not get anyrecovery for any value of fictitious noise. The RMS responseof the colocated system was also much less than the non-colocated system. See Table A.2 in Appendix A.

The step response for both systems had similar results asthe three element lumped mass example. The colocated systemhad ringing in its rise curve due to the lightly damped polesand the influence of the zeros on the imaginary axis. Thenon-colocated rise curve had the initial negative riseassociated with the RHP zero.

Reviewing the closed loop pole locations show thatLQG/LTR designs agree with theory. As q2 is increased forthe colocated system, the Kalman filter's closed loop polesnotches out the plant zeros. Meanwhile, the LQG/LTRcompensator zeros move toward the closed loop poles of thefull-state regulator. In turn, the root locus shows theplant's poles moving toward the compensator zeros duringfeedback. This is clear in Figure 4.15 of the non-colocatedsystem which shows the filter poles notching out the plantzeros in the LHP. Thus, the compensated feedback systemachieves the robustness of a full-state regulator. The non-

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colocated system can not achieve loop recovery since it hasRHP zeros which prevent the Kalman filter poles from notchingout the zeros. However, the colocated system has higher noise

which seems to cause the design trade-off in improving systemrobustness with LTR.

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V. FINITE ELEMENT MODEL OF CONTRAVES P52F GIMBALS

5.1 FINITE ELEMENT MODELING

The gimbal structure used in this investigation is atarget motion simulator built by Contraves USA. This gimbalmust be robust to parameter variations because it mustaG.omodate many different types of missiles for testing. Thecontrol design must be able to handle the changes in the massand inertia of the different payloads.

The gimbal has an inner (azimuth axis) and an outer(elevation axis) gimbal arrangement. The gimbal is driven byhydraulic actuators that provide torque and uses position andtorque transducers (sensors) The torque for the finiteelement model is applied on a node on the inner gimbal atpoint A in Figure 5.1. Point A is also the location of thecolocated sensor and Point B is the location for the non-colocated sensor. The two gimbals are connected together bya pipe element (STIF 12) with soft beam elements (STIF 3)representing the bearing connection that simulates therotation of the inner gimbal. For this thesis, the outergimbal was fixed against rotation. The model is threedimensional in that it has all six DOF (xyz translation androtation) associated with the nodes.

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The ANSYS model in Figure 5.1 has over one thousandelements. Because of the large number of modes available, themodal analysis used only the first five modes.

Figure 5.1 Contraves P52F Gimbal

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This truncated model produced the following eigenvalues in itsnatural frequency form.

S=1.418 Hz13.73661.4603940.8614761.030

The first mode was found to be the rigid body mode of theinner gimbal. So, this frequency was taken as zero. The modeshape plot of Figure 5.2 shows the rotation (the solid shape)and the dotted line representing the original shape. Thesecond mode caused a pole/zero cancelation and was omittedfrom the analysis. The last two eigenvalues are at very highfrequencies. Thus, the only effective flexible pole is thethird mode at 61.46 Hz or 386.2 rad/sec.

Table 5.1 Poles and Zeros of the Contraves Model

Poles Colocated Zeros Non-colocated Zeros0,0386.2i 173.6i 229.724761.Oi 2247.8i 8667.129914.Oi 26915.Oi 0.0 11631.Oi

Note Table 5.1 shows the non-colocated case has an imaginaryzero. This is not that unusual and is the result of modaltruncation. The bode plot shows the poles and zeros in Figure5.3.

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Figure 5.2 The First Mode Showing the Rigid BodyRotation

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BODE PLOT

10-'

102/ IEY

1 D- 4 OIOC tea

103 102 103FEEC (ReciSe:

Figure 5.3 Bode Plots of the Colocated and Non-colocated Contraves Model

The accuracy of the truncation compared to the frequencyresponse of the structure shows the dominant pole at 386rad/sec was matched very well. But, the frequency responseshows there is also another pole/zero pair near 250 rad/sec

that the truncated model does not have. See Figures 5.4 and5.5 on the next page. Considering the fact this model has allsix DOF and our sensor is only taking the rotationalmeasurement, the author considers this to be an acceptablesince the closed loop bandwidth is at 100 rad/sec which isbelow this first pole at 250 rad/sec. A position sensorlocated elsewhere on the model may sense the pole missed bythe sensor located at points A and B in Figure 5.1. Then thecontrol problem becomes single input and multi-output; but,this thesis is only examining the SISO case.

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BODE P .CT

102 1- 14,FRC-Pc/Sc

Fiur 5. Trnae MdlCmard t hFrqec Resons fo th----1td as

Figure 5.4 Truncated Model Compared to theFrequency Response for the Col-ocated Case

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Further investigation was done to better match thefr-quency response curve by using a MATLAB script file ANSYS.r!written by Professor Bradley Liebst to do a least squares fit.This method produces a transfer function in polynomial form.The curve fit modeled the first two flexible poles well andproduced an eighth order polynomial. See Figure 5.6.

SoId - True Trans. Fur c. I oshed - Best Fit Trans. Func.10 1 I. . .. . .

100

10-2K

Q-210 - 3

10--10-5

1O-e I. 1.100 10' 102 103

Frequency (Hz)

Figure 5.6 8th Order Curve Fit Using Ansys.m tothe Frequency Response Data

However, converting from transfer function to state spaceproduced a poorly conditioned (A, B, C, D) representation. Itappears the most numerically reliable method for state spacedesign is using the FEM modal analysis presented in Chapter 4.

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5.2 LOG/LTR DESIGN ON THE GIMBAL

The final A matrix after truncation and reduction is8 x 8 in size. The state weighting selected for khe colocatedsystem is

QC = diag(10 6 106 1 03 103 102 102 10 10)

and for the non-collocated sytem is

K diag(101-5 105"5 101 1 03 10? 102 10 10)For both cases, the noise intensity at the input was model2das

Qo = lG

The closed loop bandwidth for the LQG designs are shownin Figure 5.6 where the crossover frequency is defined as -3dBbelow the 0dB axis.

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ODE PLOT40

20

-20

-40

-60- otocctednorl-COlocoted -- -

-80-101 102 103 104 105 Joe

FREO (Pod/Sec)

Figure 5.7 Closed Loop Bandwidths for theContraves Model

The LQG design without LTR for the colocated caseproduced a system that went unstable at gain k = 5.5. Thenon-colocated system went unstable at k = 2.5. Interestingly,the LQG design for both systems placed compensator zeros inthe RHP. It was not until the fictitious noise value becameq2 = 109 did the all the colocated compensator zeros move intothe LHP. At this time, the system became very robust and itwpnt unstable at k = 25. The non-colocated case never gainedany robustness after LTR. Its compensator zeros remained inthe RHP for all fictitious noise values. See the results inTable A.3 in appendix A. The loop transfer recovery plots and

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the root locus plots for the colocated and non-colocated casesare shown on the following pages.

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L.C3 -t" LTP

50'-

10,J2 103 10. IC ,PE. ,NCZ' ( Rod,'Sec

Figure 5.8 LTR for the Crn1ocated System

08G R007 !.OC1L'S4

3-

- I-

-5 -4 - 2 -

REAL AxIS X10,

Figure 5.9 Root Locus for the LQG Design with LTRfor the Colocated System

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FP .GE-C. " Se

Figure 5.10 LTR for the Non-colocated System

ROCT LOCUS

-21

REAL AXIS

Figure 5.11 Root Locus for the LQG Desiqi with LTRfor thie lNcn-colocated System


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5.3 STEP RESPONSE

The step response of Figure 5.12 shows the colocated andnon-colocated systems having similar rise times. Both systemshave good damping of its poles with the colocated showing asmall fluctuation due to its zero on the imaginary axis. Thenon-colocated system because it has RHP zeros has the initialnegative rise in its step reponse.

K.K'

T

Figure 5.12 Step Response of the Contraves Gimbal

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5.4 CONCLUSION

The non-colocated system since it was non-minimum phasecould not recover any robustness with LTR. On the other hand,the colocated system after sufficient fictitious noise wasadded at the plant input increased its robustness. See TableA.3 in Appendix A.

The RMS noise responses for the colocated system wasgenerally larger than the non-colocated system due to thelarger QC weighting matrix used. The unit step input responserevealed the negative rise expected from non-minimum phasesystems for the non-colocated case. Both rise times were thesimilar and both had good damping.

Some modeling inaccuracies were introduced by modaltruncation of the very large system to only the first fivemodes. Even then, one of the modes proved to be unobservable

and had to be removed from the analysis. More analysis withhigher order models might be needed to more accurately modelthe system.

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VI. CONCLUSIONS AND RECOMMENDATIONS

6.1 SUMMARY

Chapter II introduced the three element lumped mass modelto illustrate basic principles of pole/zero patternsassociated with different cases of sensor and actuatorlocations. The colocated system (sensors and actuators at thesame location) has the desirable property of alternating polesand zeros on the imaginary axis. This pole/zero pattern makesit easy to stabilize the system with a simple le&dcompensator. The non-colocated system could not be stabilize:with a lead compensator. Because the three element model isa single motion DOF system, it did not have any RHP zeros.But, it exhibited non-minimum phase like behavior. The non-colocated system required a higher order controller tostabilize it.

Chapter III introduced the LQG/LTR theory used to designthe higher order controller for the non-colocated system.Despite large fictitious noise added, loop recovery waslimited and marginally successful only at low frequencies. Itmaintained its high frequency roll-off characteristic of anon-minimum phase system.

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To show RHP zeros of a non-minimum phase system, ChapterIV introduced the FEM and the ten element transverse beam.This model exhibited the RHP zeros for the non-colocated case.

As expected of a non-minimum phase system, LTR had littleeffect on improving the robustness in comparison to thecolocated system. Because the LTR theory relies on placingthe poles of the Kalman filter at the plant's zeros, the RHPzeros make this impossible to accomplish. For the colocatedsystem, LQG design without LTR was sufficient to make it veryrobust. However, LTR made the system much more robust as itapproached the robustness of the target full-state loop.

Chapter V designed the LQG/LTR controllers for theContraves gimbal. The final results were similar to the tenelement beam. LTR made the colocated system more robust,although the robustness did not increase linearly. Of course,the non-colocated system which had RHP zeros had unsuccessfulLTR.

This finite element model was much more complex than thebeam;therefore, modal truncation did not model one of thesystem's flexible poles at 250 rad/sec. However, thisflexible pole was above the design bandwidth of 100 rad/sec.The author considered the system to be accurate enough fordesign comparisons.

Interesting observations were made concerning the LQG/LTRcontrollers. LQG design is helped by LTR because as thefictitious noise is increased, the compensator poles move

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toward the open loop plant zeros to notch them out. TheKalman filter accomplishes this task since its closed looppoles (A - KfC) at high q2 values moves to the plant zeros.On the other hand, the compensator zeros move toward thestable poles of the closed loop regulator (A-BKc). Thus,during compensated feedback the plant poles move to thecompensator zeros which recovers the robustness of the full-state regulator.

The two designs are compared in Table 6.1. The followingwere used as criteria:

A. Robustness1. Gain at which the root locus goes into the RHP2. Whether full recovery to the target full state loopis achieved3. Alternating poles and zeros indicating whether asystem can be easily stabilized

B. Performance1. Bandwidth2. Step response

- settling time- ringing- initial negative response or no response

3. Damping4. RMS noise response

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C. Designability1. RHP compensator poles2. High frequency roll off3. State weighting required

Table 6.1 Comparison of Colocated and Non-colocated Systems

COLOCATED DESIGN NON-COLOCATED DESIGN

A.1: Robust to gain variations Not robust to gainvariations

A.2: LTR is possible over a LTR is not achieved,large bandwidth especially at high

frequencyA.3: Has alternating poles/zeros No alternating pattern

B.1: Lower bandwidth achievable Higher bandwidth achievableB.2: - Faster settling time - Slower settling time- Ringing - No ringing- n/a - Initial negative responseB.3: Poorly damped poles Poles damped wellB.4: Higher RMS response Lower RMS responseC.I: Compensator can have RHP poles for both systemsC.2: No high frequency roll-off High frequency roll-offC.3: Larger state weighting Smaller state weighting

matrix required to matchbandwidth

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6.2 CONTRIBUTION

It appears each system has its own advantage. Thecolocated system is very robust and can handle large gainvariations. But, the colocated system for a given stateweighting matrix can not achieve the same bandwidth the non-colocated system can. If performance was more important thanrobustness, the non-colocated system may be desired. Ofcourse, one has to live with the initial negative response ofa non-minimum phase system. Therefore, the control engineercan determine whether to go with colocation or non-colocationdepending on the specifications that are required.

6.3 RECOMMENDATION FOR FURTHER STUDY

To continue to the research on the effects of colocationand non-colocation of sensors and actuators, several otherfactors need to be evaluated. The selection of the stateweighting matrix, QC, seems to have a major influence on thebandwidth, damping, and the noise. Several references shouldbe investigated further in selecting

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