UNIVERSITY OF CALIFORNIA, SAN DIEGODamage Detection in Dynamic Systems with Nonlinearities

A thesis submitted in partial satisfaction of therequirements for the degree Masters of Sciencein Engineering Sciences (Applied Mechanics)by

William Bruce DunbarCommittee in charge:Professor John B. Kosmatka, ChairpersonProfessor Raymond de CallafonProfessor Miroslav Krstic

1999

The thesis of William Bruce Dunbar is approved:ChairUniversity of California, San Diego1999

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TABLE OF CONTENTSSignature Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivNomenclature Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42. Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Mathematical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101. Methods of Parametric Identi�cation of Nonlinear Structural System Models forDamage Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102. Methods of Fault Detection and Isolation in Nonlinear Electro-Mechanical Systems 163 Alternate Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211. Selection of a Structural Fault Estimation Method . . . . . . . . . . . . . . . . . . 212. Selection of a Dry Friction Fault Detection and Isolation Method . . . . . . . . . 254 Parameter Estimation of a Damaged Structure Using a Quasilinearization Approach 301. Procedure for Parameter Estimation of a Nonlinear Structural Model . . . . . . . 312. Analytic Model of a Nonlinear Space Antenna Structure . . . . . . . . . . . . . . 313. Quasilinearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331. Step 1 : Linearization of Equations and Solution Form . . . . . . . . . . . . . 332. Step 2 : Generation of Recursive Solution . . . . . . . . . . . . . . . . . . . . . 353. Step 3 : Minimization of Cost Function and Estimate Generation . . . . . . . 354. Application to Modi�ed Kabe Model . . . . . . . . . . . . . . . . . . . . . . . . . 365. Application Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381. Discussion of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416. Conclusions and Extension to Damage Detection . . . . . . . . . . . . . . . . . . 451. Extension of Application of Quasilinearization to Damage Detection . . . . . . 475 Detection and Isolation of a Dry Friction Fault in a Pneumatically Actuated Air Bear-ing Mass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 561. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562. The Pneumatic Actuation of an Air Bearing Mass . . . . . . . . . . . . . . . . . . 573. Procedure for Dry Friction Fault Detection . . . . . . . . . . . . . . . . . . . . . . 624. Dynamic Modeling of the Pneumatically Actuated Air Bearing Mass . . . . . . . 641. Analytic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 642. Experimental Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665. Friction Detection by Monitoring Acceleration . . . . . . . . . . . . . . . . . . . . 751. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752. Dynamic Relation Between Friction Force and Measured Acceleration . . . . . 75iv

6. Experimentally Based Modeling Approach . . . . . . . . . . . . . . . . . . . . . . 761. Assume a Friction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762. Identi�cation of Dynamic Friction Signal Filter . . . . . . . . . . . . . . . . . . 773. Incorporation of Friction Model with Model Based Filter . . . . . . . . . . . . 774. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 787. Dry Friction Fault Detection Results . . . . . . . . . . . . . . . . . . . . . . . . . 801. Numerical Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 812. Results for Application to the Precision Positioning Experiment . . . . . . . . 838. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107A Selection of Nonlinear Structural Spring by Modal Participation . . . . . . . . . . . . 109B Friction FDI Scheme Source Code and Numerical Analysis Model . . . . . . . . . . . 112C Linear Transfer Function Model Performance in the Precision Positioning Apparatus 119Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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NOMENCLATURE TABLESChapter 2 Nomenclaturea Unknown shear frame parameter vectorA Matrix in algebraic equationsbA Augmented state matrix for state vector �xkb Vector in algebraic equationsc Unknown initial condition vector(u(0))�c Incremental estimate of cC Damping matrix of an o�shore tower structureCd Drag coe�cient matrix of wave forcesCm Inertia coe�cient matrix of wave forcesF Input force to inverted pendulum cartg(u) Vector function in vector u state space modelgu Jacobian of g with respect to ug (X;uin; t) Vector function in vector X state space modelG(z�1) Discrete linear transfer functionh(x; _x;a) Elastic and inelastic restoring forces in the shear frameH State to observation matrix for state vector XJ(a) Direct approach least-squares functionalJ(u) Gauss-Newton approach least-squares functionalK0 Sti�ness matrix of an o�shore tower structureK(k + 1) Kalman gain matrixl Length of the pendulumM Mass of the cart and pendulumM0 Mass matrix of an o�shore tower structureMp Mass of the pendulump(t) External force applied to the shear frameP (k + 1=k) State error covariance matrix for predicted state X(k + 1=k)RARMA ARMA parity equation residualRMA MA parity equation residualu Input variablevi

Chapter 2 Nomenclature, Continuedu0 Input set point value�u Input erroru Augmented state vector in Gauss-Newton approachu0 Best a priori guess of unknown initial condition u(0)uin Input for vector X state equationsv(k) Discretized observational noise vector_v; �v Horizontal wave particle velocity, acceleration vectorsw(k) Discretized system noise vectorW ; V State, output noise covariance matricesx Shear frame position�x0 Shear frame base acceleration excitationxc Translational position of an inverted pendulum cartxk Discretized state vector of the linearized pendulum/cart dynamics�xk State vector xk augmented with unknown system and observation inputsx̂k Filter estimation of augmented state vector x̂kX Augmented state vector in extended Kalman �lter approachX(k + 1=k) Filtered predicted state vectory Output variabley0 Output set point value�y Output errory Output for vector X state equationsyk Output for vector xk state equationsz Horizontal nodal displacement vector of an o�shore tower structure�k Unknown system inputs�k Unknown observation inputs� Linear observation error vector� Linear observation matrix� Symmetric nonsingular matrix of weighting factorsrv; rw Observational, system noise covariance matrices! Linear observation vector� Azimuth angle of an inverted pendulum� Polar mass moment of inertia of the pendulumvii

Chapter 3 NomenclatureA; B; Cf First and second order friction model matricesC Structural damping matrixfk Kinetic dry friction force valuefs Maximum or static dry friction force valuef Forcing vectorF (u(t)) First and second order model friction forcesFf Coulomb and viscous model friction forceg(u) Vector function in vector u state space modelgu Jacobian of g with respect to ug (X;f ; t) Vector function in vector X state space modelG Nonlinear structural matrixG(z�1) Discrete linear transfer functionJ(a) Quadratic cost functionalJ(u) Gauss-Newton approach least-squares functionalK Structural sti�ness matrixmi Nodal mass iM Structural mass matrixn Number of nodal degrees-of-freedom in the structurer ARMA residualse Elastic microdisplacement or break away distance of F (u(t)) from restsp Plastic displacement or distance above which F (u(t)) is within 5% of fkT Observation interval lengthu(t) Relative position of the sliding contact friction surfacesu Augmented state vector in Gauss-Newton approach�u Input errorw System noise vector_x Relative velocity between to sliding surfacesx; _x; �x Vectors of nodal displacements, velocities, accelerationsxf First and second order friction model vectorX Augmented state vector in extended Kalman �lter approach�y Output errorviii

Chapter 3 Nomenclature, Continued� Coulombic friction term parameter� Viscous friction term parameter� Linear observation error vector� Linear observation matrix� Symmetric nonsingular matrix of weighting factors! Linear observation vector�i State variable function vector i

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Chapter 4 Nomenclaturea Vector of unknown constant parameters in the nonlinear equations of motiona0 True values for the unknown vector ab Linear algebraic vector from cost functional minimizationc Initial condition for the state vector x(t)C Augmented vector of initial conditions for the state vector X(t)D Linear algebraic matrix from cost functional minimizatione(t) Error between observed and generated state vector signalsf Applied forcing vectorf(x;a) General nonlinear di�erential equation functionF (X;f ) Nonlinear di�erential equation function for modi�ed Kabe modelH(n)i (t) Homogeneous part of the recursive vector and associated with estimated parameter iJ(x(n)) Jacobian matrix of f (x;a) with respect to x(t)ki Sti�ness coe�cient iK Structural symmetric sti�ness matrixm Length of discretized observation signalsmi Nodal mass iM Structural diagonal mass matrixNa Dimension of the unknown parameter vector aNx Dimension of the state vector x(t)Nxa Combined Dimension of state and parameter vectorsP (n)(t) Particular part of the recursive vector x(n)(t)rj Residual between observed and generated position signals at mass iT Length of continuous observation signalsx; �x Vectors of nodal displacements, accelerationsx(t) General continuous state vectorx(n)(t) Recursive estimation of the state vector x(t)X(t) Augmented state vector for modi�ed Kabe model� Nonlinear (cubic) sti�ness parameter Recursively estimated parameter vector� Least-squares cost functionalx

Chapter 5 NomenclatureA Cross-sectional area of chamber portsAi Cross-sectional area of chamber iAie E�ective cross-sectional area of chamber port iB(q�1) Numerator of the output error modelCd Chamber port discharge coe�cient (Cd = 0:85)Ci Reduced analytic model coe�cientse(t) Error between acceleration residual and �ltered friction signalf1 Compressible ori�ce ow functionF (q�1) Denominator of the output error modelFf Dry friction force (�Ff positive to the right)G(!) Frequency response from voltage input to load position outputGe(q�1) Dynamic �lter with normalized Coulombic friction model parameterbG(z�1) Discrete linear transfer functioneGe(q�1) Dynamic �lter or Ff to �xr output error modeli Control volume or chamber index(i = 1; 2)m Combined mass of piston rod and air bearingnb Order of B(q�1)nf Order of F (q�1)Pd Absolute downstream pressurePi0 Absolute initial pressure of chamber iPui Absolute up stream pressure of chamber iPe Absolute exhaust and atmospheric pressure (Pe = 101 kPa)q Delay operatorrcrit Critical pressure ratioR Ideal gas constantL Piston stroke lengthTi Absolute temperature of chamber iTi0 Absolute initial temperature of chamber iTiu Absolute up stream temperature of chamber iVi0 Initial Volume of chamber i, including dead airVin Servo valve voltage input signalxi

Chapter 5 Nomenclature, ContinuedW (!) Least-squares model �t weighting functionx; _x; �x Position, velocity, acceleration of the load massx̂; �̂x Non friction modeled load position, acceleration of the load massxf Friction model signal with normalized Coulombic friction model parameter�xr Load acceleration residual� Coulombic friction term parameter� Viscous friction term parameter Ratio of speci�c heats( = 1:4 for air) f Relative level of viscous to Coulombic friction during the friction fault�1; �2 Compressible ori�ce ow constants�i Leakage coe�cient of chamber i� Parameterization of the transfer function bG(z�1)�1(t) Time-varying Coulombic friction term coe�cient�2(t) Time-varying viscous friction term coe�cient

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LIST OF FIGURES1.1 Two-stage structure of the FDI process . . . . . . . . . . . . . . . . . . . . . . . 32.1 One Degree-of-freedom Shear Frame Model . . . . . . . . . . . . . . . . . . . . . 113.1 Qualitative Behavior of Friction Force F versus Position u . . . . . . . . . . . . . 274.1 Kabe model of Space Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Measurement of x6(t) for Conditions of Table 4.7 and of No Added Noise . . . . 424.3 Measurement of x6(t) for Conditions of Table 4.7 and 10 % Added Noise . . . . . 434.4 Measurement of x6(t) for Conditions of Table 4.7 and 20 % Added Noise . . . . . 444.5 Measurement of x6(t) for Conditions of Table 4.7 and 50 % Added Noise . . . . . 454.6 Continuity of Non Convergent Step Input Case in Table 4.5 . . . . . . . . . . . . 464.7 Continuity of Convergent Step Input Case in Table 4.5 . . . . . . . . . . . . . . . 474.8 Continuity of Initial Convergent Case in Table 4.4 . . . . . . . . . . . . . . . . . 484.9 Continuity of Less Convergent Case in Table 4.4 . . . . . . . . . . . . . . . . . . 494.10 Continuity of Non Convergent Case in Table 4.4 . . . . . . . . . . . . . . . . . . 504.11 Higher Frequency for Non Convergent Step Input Case of Table 4.5 . . . . . . . . 514.12 Lower Frequency for Convergent Sinusoidal Input Case of Table 4.5 . . . . . . . . 524.13 Residual Measurements for Conditions of Case 1 of Table 4.1 and Fault in k4 Overthe Entire Observed Responses (x̂1(t); :::; x̂8(t)). . . . . . . . . . . . . . . . . . . . 534.14 Residual Measurements for Conditions of Case 1 of Table 4.1 and Fault in k4 att = 5 sec in the Observed Responses. . . . . . . . . . . . . . . . . . . . . . . . . . 545.1 Experimental Apparatus of an Air Bearing Mass with Pneumatic Actuation. . . 575.2 Engineering Drawing of the Top Section of the Air Bearing Mass. . . . . . . . . . 595.3 Engineering Drawing of the Bottom Section of the Air Bearing Mass. . . . . . . . 605.4 Wide Frontal View of Experimental Precision Positioning Set Up. . . . . . . . . . 615.5 Close Up of Frontal View of Experimental Precision Positioning Set Up. . . . . . 615.6 Top Close Up View of Air Bearing Mass. . . . . . . . . . . . . . . . . . . . . . . . 625.7 Amplitude and Phase Bode Plot of Voltage Input to Air Bearing Load PositionFrequency Response G(!) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.8 Amplitude and Phase Bode Plot of the Second Derivative of G(!) . . . . . . . . 685.9 Frequency Response Match Between Measured Voltage to Acceleration and theSecond Derivative of G(!) for a Chirp Input of Amplitude One Volts. . . . . . . 695.10 Voltage Input to Up Stream Pressure Frequency Response . . . . . . . . . . . . . 705.11 Change in Frequency Response of Voltage to Position for Chirp Signal Input ofVarying Amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.12 Change in Frequency Response of Voltage to Acceleration for Chirp Signal Inputof Varying Amplitude. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 725.13 Amplitude and Phase Bode Plot of Voltage Input to Air Bearing Load PositionFrequency Response G(!) (dashed) and fourth-order model �tted on the databG(z�1) (solid) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 735.14 Amplitude and Phase Bode Plot of the Second Derivative of G(!) (dashed) andbG(z�1) (solid) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.15 Fault Sensitive Acceleration for Simulated Test Case 1 . . . . . . . . . . . . . . . 825.16 Parameter Estimates, Thresholds and Signatures for Simulated Test Case 1 . . . 835.17 Fault Sensitive Acceleration for Simulated Test Case 2 . . . . . . . . . . . . . . . 845.18 Parameter Estimates, Thresholds and Signatures for Simulated Test Case 2 . . . 865.19 Fault Sensitive Acceleration for Simulated Test Case 3 . . . . . . . . . . . . . . . 875.20 Parameter Estimates, Thresholds and Signatures for Simulated Test Case 3 . . . 88xiii

5.21 Measured Acceleration for Test Case 1 . . . . . . . . . . . . . . . . . . . . . . . . 895.22 Measured (dashed) and Modeled (solid) Accelerations, Pre- and Post-Fault forTest Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.23 Parameter Estimates, Thresholds and Signatures for Test Case 1 . . . . . . . . . 915.24 Measured Acceleration for Test Case 2 . . . . . . . . . . . . . . . . . . . . . . . . 925.25 Parameter Estimates, Thresholds and Signatures for Test Case 2 . . . . . . . . . 935.26 Measured Acceleration for Test Case 3 . . . . . . . . . . . . . . . . . . . . . . . . 945.27 Parameter Estimates, Thresholds and Signatures for Test Case 3 . . . . . . . . . 955.28 Acceleration Residual for Test Case 3 . . . . . . . . . . . . . . . . . . . . . . . . 965.29 Acceleration Residual (dashed) and Filtered Friction Model Signal (solid) for TestCase 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.30 Measured Acceleration for Test Case 4 . . . . . . . . . . . . . . . . . . . . . . . . 985.31 Parameter Estimates, Thresholds and Signatures for Test Case 4 . . . . . . . . . 995.32 Measured Acceleration for Test Case 5 . . . . . . . . . . . . . . . . . . . . . . . . 1005.33 Parameter Estimates, Thresholds and Signatures for Test Case 5 . . . . . . . . . 1015.34 Measured Acceleration for Test Case 6 . . . . . . . . . . . . . . . . . . . . . . . . 1025.35 Parameter Estimates, Thresholds and Signatures for Test Case 6 . . . . . . . . . 1035.36 Measured Acceleration for Test Case 7 . . . . . . . . . . . . . . . . . . . . . . . . 1045.37 Parameter Estimates, Thresholds and Signatures for Test Case 7 . . . . . . . . . 1055.38 Flow Diagram of Scheme for Fault Detection and Isolation of Dry Friction in aPrecision Positioning Device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106A.1 Percentage of strain energy per spring per mode . . . . . . . . . . . . . . . . . . 110A.2 Modal percentage of strain energy in k1 at m2;m3;m6 and m7 . . . . . . . . . . 110B.1 Second Order Models for the Numerical Simulation of the FDI Scheme . . . . . . 113C.1 Plot of Modeled (solid) and Measured (dashed) Mass Position Signals for a GivenInput . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120LIST OF TABLES4.1 Convergence Results for Variable � with f5 = 1000 . . . . . . . . . . . . . . . . . 384.2 Convergence Results for Variable � with f5 = 4000 . . . . . . . . . . . . . . . . . 384.3 Convergence Results for Variable � with f6 = 1000 . . . . . . . . . . . . . . . . . 394.4 Decrease in Time Step with f6 = 1000; � = 0:1 . . . . . . . . . . . . . . . . . . . 394.5 Variable Time Step, f6 Input with I.G. = (150; 750; 7500); � = 10 . . . . . . . 404.6 Variable Time Step and Initial Guess with f6 = 4000; � = 0:1 . . . . . . . . . . . 404.7 Measurements (x3, x6, x8) Contaminated by Random Noise for f5 = 1000; � = 0:1 415.1 Dimensions and Properties of Components and Hardware . . . . . . . . . . . . . 585.2 Dry Friction Fault Detection Test Cases . . . . . . . . . . . . . . . . . . . . . . . 855.3 Dry Friction Fault Detection Test Cases . . . . . . . . . . . . . . . . . . . . . . . 85

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ACKNOWLEDGEMENTSI'd like to thank my advisor Prof. John Kosmatka for giving me the opportunity tocome to UCSD to study and for helping me grow up as a student. I'd also like to thank Prof.de Callafon for his guidance and laboratory space and equipment in pursuing the experimentalstudy of this thesis. Finally, I thank Prof. Krstic for his help in my course work and for beingon my committee.I would be remiss if I didn't thank the Achievement Rewards for Collegiate Scientists(ARCS) and the Reuben H. Fleet Foundations for awarding and providing me with a fellowship,respectively, for the duration of my studies at UCSD. Also, I thank a NASA Training Grant fromwhich my advisor augmented my �nancial support. Lastly, I'd like to again thank the ReubenH. Fleet Foundation for a scholarship awarded to me in the Spring of 1999 through the AIAAstudent chapter at UCSD.I thank Chris Roman and Dan Schickle for their time and energy in helping me designand machine the experiment, as well as countless others who took an interest in designing the\ oating hockey puck", especially Prof. de Callafon.My family deserves many thanks and their support and love has meant a great deal tome. This thesis is dedicated to the memory of Michael Patrick Finch, the bravest man I've everknown and loved.

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ABSTRACT OF THE THESISDamage Detection in Dynamic Systems with NonlinearitiesbyWilliam Bruce DunbarMaster of Science in Engineering Sciences (Applied Mechanics)University of California, San Diego, 1999Professor John B. Kosmatka, ChairIn general, damage detection can be described by the process of monitoring a physical dynamic(moving) system accompanied by con�rmation and assessment of any degradation of systemperformance. This study is concerned with systems that contain nonlinearities in the dynamicsthat either pre-exist and/or result from some form of damage. Commonly, these systems aremathematically modeled and terms that are representative of a speci�c fault are identi�ed andmonitored in a computer for detection. Health monitoring of dynamic systems for the detectionof damage is an important practice in assessing performance of those systems and has a wideapplication range varying from structural and electro-mechanical systems to bio-medical sys-tems. Although health monitoring of structural and bio-medical systems with nonlinearities hasbeen largely untouched, there exist successful tools for monitoring nonlinear electro-mechanicalsystems, most of which come from system identi�cation (SI) techniques. SI is concerned withderiving dynamic models of systems from experimentally obtained data. This thesis focuses onthe application of SI techniques to a structural and an electro-mechanical system that containnonlinearities. A numerical analysis of identifying linear terms and nonlinear terms (resultingfrom damage) in a structural system is investigated by applying a SI parameter estimation tech-nique called quasilinearization. Also, an experimental analysis is performed to isolate and detecta dry friction fault in a high precision positioning mechanism. The designed fault detection andisolation scheme involves model �tting, dynamic �ltering and recursive parameter estimation SItools. The high precision positioning mechanism is a servo pneumatic cylinder that drives atranslational air bearing apparatus, designed to permit the addition of friction on-line. In thelast several years, pneumatic cylinders have found wide application in electro-mechanical systemswith precision positioning objectives.xvi

Chapter 1IntroductionThis thesis is concerned with detecting damage in a nonlinear structural system (bynumerical analysis) and a nonlinear electro-mechanical system (by experimental analysis). Thede�nitions and motivation of detecting damage in these systems are di�erent. However, the for-mulation of the damage detection schemes applied here are common in that system identi�cation(SI) techniques are utilized in both cases.Health monitoring of structural systems for the detection of damage is a critical prac-tice in assessing the safety and service life of those structures. Yet, the application of healthmonitoring to structures that are modeled by nonlinear di�erential equations has not been ad-equately treated, even though many real structures exhibit nonlinear behavior prior to, or as aresult of, the occurrence of a fault. To detect damage in a structural system with nonlinearities,a reliable model of the system dynamics must be obtained. Fault generally refers to a reductionin the sti�ness values that parameterize the model. For a given model of a structural system,the following subsequent steps are involved in detecting damage:1. Detect a change in expected system operating conditions.2. Locate where the damage occurred in the system.3. Estimate the reduction in sti�ness in the damaged element.4. Perform a structural analysis (e.g. �nite element) and determine whether the damagedsystem is safe by design requirements or estimate the remaining life.In the areas of monitoring ground and space structures with linear models, particularlyin the aerospace �eld, the problem of detection, location, and estimation (DLE) of structuraldamage (steps 1 - 3) has been extensively treated as discussed in [1]. Modal data is often used

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2for model identi�cation and damage monitoring, where damage is commonly characterized as achange in sti�ness. However, these tools of DLE do not extend to nonlinear structures.In this study, the de�nition of damage in a structural system is expanded. Henceforth,a structural fault refers to the possibility of hardening or softening of a structural spring, whichalso may remain linear or become nonlinear as a result of the damage.Alternatively, fault detection and isolation (FDI) is the name for the subject as it relatesto electro-mechanical systems [2] where automation or control is used and linear and nonlinearmodels are employed. In this realm, fault may occur in the actuators, the components of thesystem and/or the sensors. It is the goal of FDI techniques to detect and isolate (locate) faultsand compensate for them to ensure the success of the control objective. This study does notinvestigate or employ any type of control. However, the SI techniques employed in model basedFDI schemes can be applied to nonlinear systems for fault detection and isolation without theneed for control.Traditionally, hardware redundancy FDI schemes were used to diagnose faults by imple-menting physically redundant sensors, actuators and/or components and comparing the redun-dant signals for consistency. The modern analytical redundancy FDI schemes are distinguishedfrom the hardware redundancy schemes in that they numerically generate faultless signals whichserve as the redundant signals for comparison to the measured signals. The redundant signalsare numerically generated by formulating a dynamic model of the process to be monitored andapplying the same input to the model that the process is subjected to. These schemes \are ba-sically signal processing techniques employing state estimation, parameter estimation, adaptive�ltering, variable threshold logic, statistical decision theory, and various combinatorial and logi-cal operations, all of which can be performed (in modern computers)..." ([2], p.4). The processedoutput measurements from sensors that represent a di�erence between no fault and fault casesare called residuals. The signature of the residual is the enhanced characteristic that the residualdisplays in the presence of failure. In the absence of failure, the residuals should be unbiasedand the signature is typically the presence of a residual bias. In this realm, robustness is de�nedas the minimization of sensitivity of detection performance to model errors and uncertainties.Following the generation of residuals, the statistical hypothesized faults are calculated and adecision making test is applied to these statistics. This two-stage structure of the FDI processdiscussed above is given in [3] and shown in Figure 1.1.SI techniques are concerned with deriving mathematical (dynamic) models of systemsbased on observed data from the systems [4]. As such, the tools of SI have been extensivelyused in analytical redundancy FDI schemes for detecting and isolating faults in nonlinear dy-namic systems. Within SI there are parametric and nonparametric methods of modeling [5], [6].

3sensor outputs

RESIDUAL GENERATION

CALCULATION OF DECISION STATISTICS

DECISION

residualsdecision statistics

failure decision

Robust Residual Generation

High Performance Decision ProcessFigure 1.1: Two-stage structure of the FDI processParametric methods seek to identify a pre-parameterized model of the system and the param-eter values in that model. Nonparametric methods are designed to determine a functional orqualitative representation of the signals observed from the system to be identi�ed. This studyis concerned with parametric methods of modeling as they are more widely applied for damagedetection and simpler to apply, particularly in highly nonlinear systems [7].Within parametric methods, a modeling distinction can be made between black boxand gray box modeling [4]. Black box models do not exhibit a physical interpretation of thesystem to be modeled. As such, black box models are �t to data by adjusting parameters in apre-determined parameterization that is assumed to capture the relevant dynamics of the systemto be identi�ed. Gray box models contain adjustable parameters that do have a physical interpre-tation and the parameterization is determined by the physical interpretation of the parametersto be identi�ed. The way in which the black box models are parameterized determines the nameof the model being used. This thesis applies ARX (autoregressive with exongeneous input) andOE (output error) black box models [4]. In both gray and black box modeling a model structureis obtained and the parameters in the model need to be identi�ed. The methods for parametricidenti�cation of these models include direct approaches, perturbation techniques, quasilineariza-tion, Markov modeling, and �ltering and estimation methods [8], [6], [7], [9]. Filtering andestimation techniques include least-squares, maximum likelihood, and Bayesian estimation.Most of these tools have found application in identifying parametric models of nonlinearstructural and electro-mechanical systems. More importantly, some of the techniques above havebeen or can be applied to health monitoring of structural and electro-mechanical systems for thedetection of damage.

41.1 Literature ReviewDamage detection in dynamic systems with nonlinearities requires identi�cation of amodel of the system followed by monitoring of fault sensitive measurements for detection. Theliterature related to structural systems deals either with both model identi�cation and damagedetection in linear systems or strictly model identi�cation of systems with nonlinearities, withfar more literature that is current in the �rst area. On the other hand, there are many papersthat address FDI in nonlinear electro-mechanical systems.The identi�cation and damage detection of ground and space structural systems withlinear models has been extensively treated with modal tools. Kabe [10] used measured mode datato identify and adjust the sti�ness matrix in the modeling of a severe test case space antennastructure. Potential damage can be located and estimated using a weighted sensitivity analysisthat accommodates mass and sti�ness uncertainty, as investigated by Ricles and Kosmatka [11].Papadopoulos and Garcia [1] used modal information and apply a statistical approach to iden-tify structural damage. Modal tools parametrically identify linear structures while keeping thephysically based model structure in tact. However, these tools do not have an analog in nonlin-ear systems. Natke and Yao gave a brief investigation of SI approaches for structural damageevaluation in linear civil engineering structures [12].The application of SI tools to the identi�cation of a nonlinear structure dates back asearly as 1975 [9]. In this paper by Distefano and Rath, parameters associated with a numericallygenerated nonlinear response were identi�ed for one degree-of-freedom structural systems subjectto seismic conditions. These systems were extensively treated numerically by the direct methodof estimation, a �ltering approach and a Gauss-Newton approach. Since then, the appearance ofpapers that investigate the identi�cation of nonlinear structural systems using SI techniques hasbeen sporadic.Nayfeh introduced a self-contained perturbation approach that proposes experimentaltechniques of identifying low degree-of-freedom dynamic systems that contain smooth nonlinear-ities [6]. The techniques exploit nonlinear resonances and compares the known systems behaviorto the parameterized model to be identi�ed. In this way, the model can be parameterized toqualitatively match the physical system under chosen excitations. Further, for an assumed math-ematical model of a two-degree of freedom structure that contains quadratic and cubic restoringterms, an experimental parameter identi�cation approach is given. The method requires anincreasing number of experiments, under excitation of the multiple subharmonics, for identi�ca-tion of models of increasing nonlinearity and degrees of freedom. For higher order parametersto be identi�ed, redundant measurements are required with the application of least-squares ormaximum likelihood estimations.

5Hanagud et al. [13] expanded the solution to a nonlinear one degree-of-freedom systemby perturbations (method of multiple scales) and identi�ed the parameters by requiring only themeasurement of one state variable, e.g. acceleration or position. Although good for high noiselevels, success was obtained only for small nonlinearity and the solution was accurate to the orderof �, the small quantity that parameterized the nonlinearity. For higher degrees of freedom orlevels of nonlinearity, iterative direct approaches were recommended in this study. Imai et al.gave an extensive study of parametric identi�cation of linear and nonlinear structural systemsusing least-squares, maximum likelihood and extended Kalman �ltering in 1989 [14].There exists a need therefore to extend SI techniques to the parametric identi�cationof structural systems with nonlinearities for damage assessment.Few papers discuss gray box modeling approaches for FDI in nonlinear electro-mechanicalsystems. Zell and A. Medvedev applied a model-based technique for fault detection and isolationthat was speci�c to systems that utilize rotational induction machines [15]. Shields et al. ap-plied a nonlinear fault detection method for a bilinear hydraulic system [16]. The modeling errorserved as the residual used in detection. However, only numerical success was provided and themethod was generalized for bilinear systems only. Both of these papers identi�ed analytic modelsthat were not highly nonlinear and success was given under conditions of numerical simulation.The higher the order and the greater the degree of nonlinearity in a process, the more di�cultit becomes to apply gray box modeling techniques to the process. This is because the morenonlinear a process is, the more di�cult it is in general to identify a reliable model structure torepresent the process. Even when these models are available, there remains the di�cult task ofidentifying multiple, and often obscure coe�cients that parameterize the models.A more common approach of identi�cation and fault detection is to apply black boxmodeling techniques. Ho ing and Deibert [17] combined parameter estimation techniques and theparity space approach to a class of nonlinear processes with mainly constant input signals. Theirapproach required knowledge of the dynamic model order and the static nonlinear characteristiccurve related to a owrate control with a pneumatic driven valve. The experiment validated theproposed on-line fault detection with small signal models.Other studies linearize the nonlinear equations that model an electro-mechanical sys-tem and apply an extended Kalman �lter and decision statistics. This approach was taken byNowakowski et al. [18] and Gomez and Unebehauen [19], with experimental validation in thelatter. Other methods apply the modern techniques of nonlinear control, in the form of nonlinearobserver schemes, to perform fault detection and isolation [20], [21], [22]. Caccavale and Walker[22] applied an observer-based fault detection algorithm for a robot manipulator application. Theon-line approach designs a discrete time nonlinear observer of the systems outputs, joint posi-

6tion and velocities. The issue of robustness with respect to unknown dynamics and discretizationerrors was addressed by adding linear feedback of the observer error and a delayed nonlinear com-pensation action. Numerical simulations showed good results in detecting and isolating sensorand actuator failures, even in the presence of large modeling errors.This study involves the application of a servo controlled pneumatic cylinder and a briefreview of their use in industry and the current need for friction fault detection is now given.Pneumatic cylinders are air driven actuation devices, where the air is regulated by some typeof voltage controlled valve [23]. These cylinders have been in use as early as 1969 [24] and arein increasing use in industry in electro-mechanical systems with precision positioning objectives[25], [26]. High payload-to-weight and payload-to-volume ratios, high speed and force capabil-ities give them an advantage over electric actuators. The availability and cost of supply airmakes them signi�cantly more cost e�ective. The high degree of compliance due to compress-ibility of air makes pneumatic cylinders capable of dexterous manipulations. This in term makesthem more applicable to representing human characteristics, as they do in rehabilitation andprosthetic applications. Pneumatic cylinders are used in agricultural robotic drive systems asthey are cleaner in maintenance and leakage than hydraulic actuation [27]. However, \unlikeconventional electrically powered manipulator dynamics which are generally of second-order, thepneumatically powered manipulator dynamics are characterized by high-order dynamics, typi-cally of third or fourth order" ([28], p.666). Moreover, the presence of dry Coulombic frictionin these actuators is a chief obstacle in automated systems that require precision positioning.This dry friction phenomena can interfere with the precision positioning objectives and causeproblems like overshooting and force limit-cycling [29]. Currently, \the lack of available straight-forward, well-de�ned techniques for the identi�cation of such nonlinearities (as dry friction) inactual plants (has) emerged as a tough problem," according to Hatipoglu and Ozguner ([30],p.2133). Most papers that addressed the compensation of friction in pneumatic actuator appli-cations [25], [28] do not apply identi�cation tools. According to Johnson and Lorenz, \...becauseof its nonlinear nature, friction is often neglected or inadequately compensated by conventionalcontrollers" ([31], p.1392). They used state feedback controller errors to iteratively identify thefunctional form of friction and the parameters therein in a robotic gripper application. Theunmodeled e�ects, such as friction, appear as state errors in the controller. The nonparamet-ric friction model was then used for feedforward or feedback compensation to achieve a trackingobjective and reject disturbances. Although friction can be compensated by modern control algo-rithms such as this, sudden and unpredictable changes in friction due to, for example, added wearand side loading in a precision positioner cause unacceptable behavior of the positioning mecha-

7nism. Therefore, there is a need for a friction FDI scheme in precision positioning applications,such as pneumatic cylinder positioning devices.In considering the identi�cation of friction for detection, parametric modeling techniquesin the literature are reviewed. The topic of modeling friction spans the subjects of control theory,mathematical physics, tribology, lubrication science and even extends to the science of earthquakedynamics. A most extensive treatment of friction models and types of compensation for uidlubricated metal on metal junctions was given by Armstrong-Helouvry et al. [32]. The paper byLim and Chen was geared toward the numerical study of the friction mechanism in earthquakesmodels [33].As a subset, there exist numerous investigations of modeling stick-slip behavior of dryCoulomb friction, which is of interest in precision positioning applications. In [34], Blimanand Sorine designed simple �rst and second order models for use in control applications. Bothmodels were uniquely and nonlinearly parameterized by physically meaningful terms that can beidenti�ed from experiment.The modeling in the controls literature focuses particularly on the low velocity stickingbehavior, as this is the chief obstacle to precision control. In [35], Dupont utilized the tribologyliterature and applied a stick-slip friction model that depends upon the history of motion. Thisstudy was concerned with very low velocity proportional-derivative (PD) control, for which stick-slip phenomena can plague the control objective with problems like overshooting and force limit-cycling. This was expanded on with experiments in [29]. Dupont examined a state variablemodel and a time-delay model of friction to achieve steady, low velocity motion with PD control.Both models were experimentally based, nonlinear in the parameters and identi�cation proceedsfrom steady-state friction-velocity curves. The former model was designed to represent stick-slipfriction for large velocity ranges and the latter model to represent low velocity ranges.Experimental identi�cation of friction and its compensation in precise, position con-trolled mechanisms was also investigated by Johnson and Lorenz [31]. Their approach wasdi�erent in that a parameterized model of the friction force was identi�ed from the loop errors ina state feedback motion controller. Signal processing was used to isolate the errors as functionsof the states and the physical relationship between friction and the spatial states (e.g. position,velocity) were used to formulate the model structure. The approach was experimentally validatedin a robotic gripper application.In the low velocity regime, friction has been experimentally observed to contain sti�nessand damping properties [36]. Haessig and Friedland in [37] gave a comprehensive study of �vedi�erent friction models, all compared by numerical simulations. The reset integrator model, anextension of the Karnopp approach [38] is parameterized by a nonlinear coe�cient that depends

8on velocity and a method of identi�cation was described. Also, a more physically based bristlemodel was investigated. These models require peak sticking and sliding friction values for pa-rameter identi�cation and most of the models are parametrically nonlinear. In most cases, themodels had some experimental validation.The studies by Bliman and Sorine [34] and Dupont [35] provided experimentally validfriction models. However, as these models are parametrically nonlinear, identi�cation must beperformed o�-line and by possibly multiple experimental observations. Other studies examinedfriction models for simulation of dynamic processes in which friction was a signi�cant factor [37],[38]. However, for the purpose of identi�cation and detection of friction in an on-line applicationthese models are di�cult. Most of the models are computationally ine�cient, and the nonlinearparameterization makes identi�cation more cumbersome.1.2 Thesis OutlineA need exists for the parametric identi�cation of models of structural systems withnonlinearities, where the identi�cation can be considered as damage assessment of that struc-ture. Further, the extension of health monitoring of that system for structural fault detectionis desirable and can be investigated by examining the tools of system identi�cation. Monitoringthese types of structures is critical in evaluating their safety and service life expectancy.A need has also been expressed for a dry friction fault detection technique in a precisionpositioning application, as dry friction phenomena can interfere with the precision positioningobjectives. Although friction can be compensated by modern control algorithms, sudden andunpredictable changes in friction due to, for example, wear and side loading of a pneumaticcylinder cause unacceptable behavior of the positioning mechanism. An on-line detection ofdry friction (fault) would greatly facilitate the compensation of dry friction in high precisionpositioning. Moreover, a fault detection and isolation technique for monitoring dry frictionwould help the detection of changing process conditions in the case of, for example, wear andexcessive side loading of a pneumatic cylinder.This thesis contains the following steps in addressing these needs� Review methods of parametric identi�cation of structural system models with nonlineari-ties. Also, discuss these approaches as they relate to damage assessment and detection ofnew structural faults to the system.� Select and numerically investigate a parameter estimation approach for the assessment ofstructural damage in a space antenna model with nonlinearities and discuss the use of thisapproach in identifying and estimating new structural faults.

9� Review methods of Fault Detection and Isolation (FDI) in nonlinear electro-mechanicalsystems. Discuss the modeling of friction and identi�cation of friction as a fault in precisionpositioning applications.� Design a fault detection and isolation scheme for isolating Coulombic friction and detectinga fault of this type in a precision positioning application.� Provide numerical and experimental validation of the FDI scheme.The mathematical approaches related to damage detection in structural and electro-mechanicalsystems with nonlinearities are in discussed in Chapter 2. Chapter 3 investigates these approachesas alternatives in the numerical and experimental analyses examined here.A parametric identi�cation method developed by Richard Bellman ([40], [8]), calledthe quasilinearization parameter estimation method, is applied to a nonlinear structural systemo�-line in Chapter 4. The structure is a space antenna model, studied by Kabe in [10], that ismodi�ed by a structural fault in the form of an added nonlinear spring. Multiple simulationsare performed to numerically identify three unknown sti�ness parameters; two that accompanylinear terms and one that accompanies the nonlinear term. A discussion of extending the methodto detecting additional structural faults is also given.In Chapter 5 a scheme for isolating and detecting dry friction in a precision positioningapplication is described. The scheme is validated numerically and successfully applied to adesigned precision positioning experiment. In the experiment, the precision positioner employedwas a servo pneumatic cylinder. The cylinder was used to actuate an air bearing mass apparatusthat was designed to permit the addition of friction on-line. Adding the friction represents anincrease in the translational Coulomb friction that pneumatic actuators exhibit under conditionsof an internal friction fault, i.e. added wear and side-loading of the piston and rod mechanisminside the cylinder.

Chapter 2Mathematical ApproachThree of the parametric identi�cation tools applied to nonlinear structural systems withknown models that were discussed in the literature review are mathematically expanded uponhere. Two current approaches of FDI in nonlinear electro-mechanical systems are also given morerigorous mathematical descriptions.2.1 Methods of Parametric Identi�cation of Nonlinear Struc-tural System Models for Damage DetectionDistefano and Rath investigated the parametric identi�cation of one degree-of-freedomstructural system [9]. Consider the single degree-of-freedom shear frame in Figure 2.1 and letx(t) be the displacement at the top of the shear frame. The assumed equation of motion of thisstructure by Newton's second law ism (�x+ �x0) + h(x; _x;a) = p(t); (2.1)where x0 is the displacement at the base, m is the mass, h(x; _x;a) denotes the elastic andinelastic restoring forces parameterized by the unknown vector a and p(t) is an external force.The modeling problem remains of identifying the structure of the function h. The simplest modelis a linear function of displacement and velocityh(x; _x;a) = a1x+ a2 _x: (2.2)However, this model does not capture the nonlinear responses observed in structures that aresubjected to seismic loading, which are the loading conditions of interest in the study. a simple

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Figure 2.1: One Degree-of-freedom Shear Frame Modelextension to nonlinear modeling, they pose the smooth nonlinear restoring termh(x; _x;a) = a1x+ a2x3 + a3 _x+ a4 _x3: (2.3)The objective of the study focuses on the parametric identi�cation of this model, with a subse-quent discussion that addresses the ability of such a model as this to predict observed nonlinearresponses in real structural systems.The simplest method of parametric identi�cation is the direct approach, which requiresmeasurements of acceleration and knowledge of the forcing term. An example of applying thisapproach is given for the systemm�x+ a1x+ a2x3 + a3 _x+ a4 _x3 = p(t); (2.4)where p(t) is the known forcing term and the acceleration �x is observed over the interval (0; T ).The parameters ai can be estimated by minimizing the quadratic functionalJ(a) = Z T0 �m�x+ a1x+ a2x3 + a3 _x+ a4 _x3 � p(t)�2 dt; (2.5)where T is the record length. The measured displacement x and velocity _x are obtained byintegrating the measured acceleration. Minimization of J(a) requires the solution of the linearalgebraic equations Aa = b; a = fa1; a2; a3; a4g; (2.6)where the symmetric matrix A and the vector b are given byAij = Z T0 hihjdt; bi = Z T0 (p�m�x)hidt; (2.7)and h1 = x; h2 = x3; h3 = _x; h4 = _x3: (2.8)

12The quantities aij and bi can be found by solving the di�erential equationsdAijdt = hihj ; Aij(0) = 0; (2.9)dbidt = (p�m�x)hi; bi(0) = 0: (2.10)The simulations provided revealed that although computationally e�cient, the method requiredvery accurate acceleration measurements. As the noise level on the measured accelerations in-creased, the method deteriorated.Also investigated were a �ltering approach and a Gauss-Newton approach, which belongsto the general family of the quasilinear methods. These approaches took a more general approachto the parametric identi�cation of dynamic models of the form of Equation 2.1. Writing the modelin state-space form and augmenting the state vector with the unknown parameter vector a yields_u = g(u) (2.11)where the augmented state vector u, with initial condition vector u(0), and the vector functiong are given by u = fx; _x; a1; :::; akg ; (2.12)g = � _x; � 1mh� �x0 + 1mp; 0; :::; 0� : (2.13)A linear observation error is assumed as ! = �u+ �; (2.14)where � is a rectangular matrix and � is the observation error vector. The least-squares functionaltakes the form J(u) = Z T0 (! � �u; ! � �u) dt+ (u(0)� u0; � (u(0)� u0)) ; (2.15)where u0 is the best a priori estimate of u(0) and � is a symmetric nonsingular matrix ofweighting factors that represents the degree of con�dence in such an estimate. The inner productvector operation is denoted by parentheses in these equations.The problem of �nding the vector function u(t) over the interval (0; T ) that satis�esthe di�erential constraint in Equation 2.11 and such that Equation 2.15 is a minimum is anoptimization problem. The estimation of the unknown parameters a is now contained in theoptimization, which seeks the displacement x and velocity _x as well. As the state u(t) is uniquelydetermined by Equation 2.11 throughout the interval (0; T ), the optimal state can be found atany point in time in this interval. Typically the problem is formulated to �nd that optimal stateat one of the two ends of the interval, i.e. at u(0) or at u(T ).

13The optimal value of u(T ) that satis�es the di�erential Equation 2.11 and minimizesthe error functional in Equation 2.15 is called the optimal least-squares �lter of the state u in[9]. The other formulation, i.e. estimating the optimal state u at t = 0, is a problem that can besolved by a variety of methods. The study investigated a solution to this optimization problem bythe Gauss-Newton approach, a technique that is derived from the family of quasilinear methods.This method is expanded upon here as it is simpler to apply than the �ltering approach andyielded better accuracy in the simulation results given in the study.Take u(0) = c and the problem formulation becomes the determination of c such thatEquation 2.15 is minimized and u satis�es Equation 2.11. So, u that satis�es Equation 2.11depends upon the initial condition c, i.e.u(t) = u(c; t): (2.16)The linear expansion of u(c; t) about an estimate of c, denoted as c0 givesu = u0 + uc�c; (2.17)where u0 represents u(c0; t), uc is the Jacobian matrix with elements @u0i =@cj and �c is theincrement of the estimate c0 de�ned as �c = c1 � c0; (2.18)and c1 is the improved estimated value of c. The substitution of Equation 2.17 in Equation 2.15yields J(�c) = Z T0 ��u0 + �uc�c�!; �u0 + �uc�c�!� dt+ �u0(0) + uc(c0; 0)�c� u0; � �u0(0) + uc(c0; 0)�c� u0�� : (2.19)The new quadratic functional in Equation 2.19 can be minimized to yield the desired �c. Thisminimization requires thatZ T0 ucT�T ��u0 + �uc�c�!� dt+ uc(c0; 0)T� �u0(0) + uc(c0; 0)�c� u0� = 0; (2.20)giving the linear algebraic equation in �c that can be written asA�c = b; (2.21)where A and b are given byA = Z T0 ucT�T�uc dt+ uc(c0; 0)T�uc(c0; 0); (2.22)b = Z T0 �ucT�T�u0 � ucT�T!�dt+ uc(c0; 0)T� �u0(0)� u0� : (2.23)

14Taking derivatives in the constraint di�erential Equation 2.11 yields the sensitivity equationsducdt = guuc; uc(c0; 0) = 24 I2 00 0 35 ; (2.24)where the matrix gu is the derivative of g(u) with respect to u and the Jacobian matrix uc isfound by integrating Equation 2.24. The identity matrix I2 is order two for the one degree-of-freedom problem of interest. As shown before, the matrixA and vector b can found by integratingthe di�erential equations (again found by taking derivatives)dAdt = ucT�T�uc; A(0) = uc(c0; 0)T�uc(c0; 0); (2.25)dbdt = ucT�T�u0 � ucT�T!; b(0) = uc(c0; 0)T� �u0(0)� u0� : (2.26)The solution to the original optimization problem is found by integrating the initial value systemsin Equation 2.24 - Equation 2.26 and solving the system linear algebraic equationsA(T )�c = b(T ): (2.27)One step of this process produces the new estimate for c, i.e. c1, and the process can be iterativelyperformed until convergence of the parameters a is achieved, if they converge at all.Numerical simulations of the direct and Gauss-Newton approaches were performed onthe one degree-of-freedom structural model�x = � 1m �a1x+ a2x3 + a3 _x+ a4 _x3�+ 1mp(t): (2.28)For assumed known parameter valuesa1 = 25; a2 = 2:5; a3 = 1; a4 = 0:1; (2.29)and initial conditions x(0) = 0; _x(0) = 0; (2.30)the system was excited (p(t)) using the North-South El Centro earthquake recorded accelogramsignal and di�erent levels of uniformly distributed random noise was implemented to corruptone of the measured signals x, _x or �x. The application of the Gauss-Newton approach showedgood robustness with respect to noise in the acceleration measurement when observing both theposition and velocity signals in the estimation equations. The weighting matrix � was set tounity in these results. The approach achieved higher accuracy than the other two methods forgreater computational time.

15Imai at el. [14] numerically examined an idealized nonlinear two degree-of-freedommodel of a �xed o�shore tower subject to wave forces(M0 +Cm) �z +C _z +K0z = Cm�v +Cd ( _v � _z) j _v � _zj (2.31)where z is the vector of horizontal displacement, _v and �v are the horizontal wave particle velocityand acceleration vectors, M0, C and K0 are the matrices of structural mass, damping andsti�ness, and Cm and Cd are diagonal matrices containing, respectively, the inertia and dragcoe�cients associated with the wave force acting on the structure. Simplifying this model gives�z + J _z �D f( _v � _z) j _v � _zjg+Kz = L�v; (2.32)where J = [M0 +Cm]�1C; D = [M0 +Cm]�1Cd;K = [M0 +Cm]�1K0; L = [M 0 +Cm]�1Cm: (2.33)The model is transformed into state-space form and the state vector is augmented with theunknown hydrodynamic coe�cients, given byX = fz1; z2; _z1; _z2; J11; J21; J12; J22; D11;D22; K11; K21; K12; K22; L11; L22gT : (2.34)The displacement measurements of the two discrete masses z1 and z2 are assumed available. Anextended Kalman �lter with a weighted global iteration procedure is applied to estimate theunknown parameters. The extended Kalman �lter algorithm is a recursive process for estimatingthe optimal state of a nonlinear system based on observed data for the input (excitation) andoutput (response) [4]. The design of such a �lter without the extension to weighted globaliterations can be summarized as follows. A continuous state equation with an input uin can bewritten as X(t) = g (X;uin; t) +w(t); (2.35)with a discrete observation vector de�ned at time t = k�t asy(k) =HX(k) + v(k): (2.36)The vector X(k) is the state vector at t = k�t, v(k) is the observational noise vector withcovariance rv, w(k) is the system noise vector with covariance rw and H is the state toobservation matrix. The algorithm provides a predicted state X(k+1=k) that evolves accordingto X(k + 1=k) = E fX(k + 1)jy(1); y(2); :::; y(k)g= X(k=k) + Z (k+1)�tk�t g (X(t=k);u; t) dt; (2.37)P (k + 1=k) = �(k + 1; k)P (k=k)�(k + 1; k)T +rw; (2.38)

16where P (k+1=k) is the corresponding error covariance matrix, EfAjBg is the expected value ofA conditional to B, and the state transition matrix �(k+1; k) is obtained by the approximation(for small �t) �(k + 1; k) � I +�t �@g(X(t); t)@X �X(t) =X(k=k): (2.39)Now, the �ltered state X(k + 1=k + 1) and its error covariance matrix P (k + 1=k + 1) can beestimated asX(k + 1=k + 1) = E fX(k + 1)jy(1); y(2); :::; y(k + 1)g= X(k + 1=k) +K(k + 1) [y(k + 1)�HX(k + 1=k)] ; (2.40)P (k + 1=k + 1) = [I �K(k + 1)H]P (k + 1=k) [I �K(k + 1)H]T +K(k + 1)rvK(k + 1)T ; (2.41)where K(k + 1) is the Kalman gain matrix de�ned asK(k + 1) = P (k + 1=k)HT hHP (k + 1=k)HT +rwi�1 : (2.42)The application of this �lter with weighted global iteration to the augmented state-space modelwith vector de�ned in Equation 2.34 was numerically examined. Note that in the applicationof this approach the measurement and system noise vectors and their corresponding covariancematrices are assumed known. The results of four di�erent simulations, estimating 12 parametersunder two di�erent wind speed and noise level cases, showed moderately good estimates. Half ofthe estimates were within 2 signi�cant digits of the assumed exact values and the rest were evenless accurate.2.2 Methods of Fault Detection and Isolation in NonlinearElectro-Mechanical SystemsIn this section, two current approaches of FDI in nonlinear electro-mechanical systemsare described in greater mathematical detail.Ho ing and Deibert developed estimation of parity equations in a nonlinear electro-mechanical system [17], where parity equations are a method of computing residuals (see [3]).Speci�cally, they designed an FDI approach for a class of nonlinear processes that have mainlyconstant input signals. Finally, the scheme is applied to a owrate control with a pneumaticdriven valve. An overview of this FDI scheme and it's application to the valve experiment isoverviewed here.

17The authors assume that the process inputs change merely step-wise or vary slowly in asmall band around an operating point, thereby justifying �tting a linear model to the nonlinearprocess. The process knowledge that was assumed known is an idea about the dynamic modelorder. A proper linear discrete transfer function model for known model order 3 takes the formG(z�1) = B(z�1)A(z�1) = b1z�1 + � � �+ b4z�41 + a1z�1 + � � �+ a4z�4 (2.43)where the parameter b0 was set zero to reduce the number of parameters for identi�cation andsince the lowpass character assumption presumes no jump discontinuities in the process. Least-squares was used to estimate the parameters for a given operating point and a parity space wasgenerated for that setpoint. Once the set point changes, as detected in the parity space, a newset of parameters were estimated and a new parity space was generated corresponding to the newset point for detection of more faults. So, the system toggles between parameter estimation andparity space for detection of faults, as explained in the following discussion.A static nonlinear characteristic curve of the process was also assumed to be known. Inthis way the small signal input and output variables can be generated asu = U � U00; y = Y � Y00 (2.44)where U and Y are the current input and output variables and Y00 = fNL(U00) �ts the nonlinearstatic map. These signals indicate that the set point change has occurred and are thereforesensitive to faults. These signals were used to exclude the e�ects of the fault from the post-faultparameter estimation, i.e. when u and y were \small" again after a fault, the new process modelparameters were estimated.The model parameters were estimated by least-squares over a selected window of data.When possible, each estimation is performed in one shot by applying a small excitation in theset point, e.g. two small step inputs. If the process cannot be arti�cially excited in this way,an interval at a speci�ed time after the fault and a window around that time is selected as thewindow of estimation. The time is selected as the moment at which the output Y does not leavea speci�ed settling threshold. Estimation in this case provides the poles of A(z�1) and the zerosof B(z�1) were assumed to reduce to one parameter, i.e. b1. This parameter was identi�ed usingthe estimated poles and the static gain, computed with the poles and static map.Autoregressive moving average (ARMA) and moving average (MA) parity equationswere used and compared to generate the residuals for fault detection. The ARMA residualequation takes the form RAMRA = y(z�1)�G(z�1)u(z�1); (2.45)

18and the MA residual is given byRMA = A(z�1)y(z�1)�B(z�1)u(z�1): (2.46)Recall that the small signal models in Equation 2.44 are sensitive to faults. In the presence of afault, consider similarly the perturbations on the input and output signalsu = u0 ��u; y = y0 ��y; (2.47)where (u0; y0) is the current operating point and (�u;�y) represents the input and output errors.The parity or residual equations in the event of a fault becomeRARMA = �y(z�1)�G(z�1)�u(z�1); (2.48)and RMA = A(z�1)�y(z�1)�B(z�1)�u(z�1); (2.49)where �u(z�1) and �y(z�1) are the discretized input and output error signals, respectively. Theresiduals were both normalized and detection occurred when the residuals jumped out of the �1window. Note that these two types of residuals, although sensitive to faults in the same way,react di�erently in the event of di�erent faults. To see this, observe that if the fault signals �uand �y are constant, the residuals deviate di�erently asRARMA = �y �G(1)�u; (2.50)and RMA = A(1)�y �B(1)�u: (2.51)To di�erentiate smaller unmodeled e�ects from signi�cant faults, adaptive thresholdswere applied to these residuals. The adaptive threshold generation included a lead-lag �lter thatrequired the estimation of four parameters. Further, the threshold computation was discretizedto reduce computational costs and speed up the time of detection.The scheme was experimentally applied to a third-order owrate control pilot plant con-trolled by a pneumatic driven valve. Note that the requirements for the scheme were knowledgeof the process input, output, dynamic model order and a static nonlinear map of the process.Faults in the sensors and concerning the given static gain were implemented. These faults af-fect the modeled output and modeled setpoint, respectively, but do not a�ect the experimentdirectly. Rather, the faults are simulated under experimental conditions. The application showsgood detection results but no distinction between the two di�erent fault types. Further, the use

19of adaptive thresholds seems redundant as a �1 window was applied to the residuals for detectionof the simulated faults.Nowakowski et al. [18] investigated FDI using an extended Kalman �lter with a statisti-cal test for detection. The approach was applied numerically to a nonlinear model of an invertedpendulum, given as ��� + C _� �Mp lg sin � +Mp l�xc cos � = 0 (2.52)M �xc + Fx _xc +Mp l��� cos � � ( _�)2 sin �� = F (2.53)where Mp; l are the mass and length of the pendulum, M is the mass of the cart and pendulum,xc and � are the cart translational position and pendulum azimuth angle, respectively, and F isthe input force.The nonlinear equations were linearized and discretized. In the general case, the lineardiscrete dynamical system a�ected by unknown inputs or failures is described by the equationsxk+1 = Axk +Buk + F�k +wk (2.54)yk = Hxk +G�k + vk (2.55)where xk, uk and yk are the state (of dimension n), known input and measured output vectorsrespectively and the �k and �k represent unknown inputs. As before, wk and vk are the stateand measured output noise vectors. By augmenting the state with the unknown inputs, thesystem can be written as E�xk+1 = bA�xk +Buk +wk (2.56)yk = C�xk + vk (2.57)where bA = �A ... 0 ... 0� ; E = �In ... 0 ... � F � ; (2.58)C = �H ... G ... 0� ; �xk = 26664 xk�k�k�1 37775 : (2.59)The existence condition for a unique solution to the above equations for state and failure esti-mation is posed as a rank test in the form of a theorem (see p.425, citeNBD93).When the test holds, the state estimation equations take the formx̂k+1 = P k+1ET �W + �P k�T��1 �x̂k +P k+1CTV �1yk+1 (2.60)P k+1 = �ET �W + �P k�T��1E +CTV �1C��1 ; (2.61)

20where x̂ is the estimated state with covariance P , � is the jacobian of the nonlinear system, Eand C are from the linearized state-space equations, andW and V are the assumed known stateand output noise covariances. In the linear systems, the jacobian matrix takes the place of thebA matrix.Numerically, a simple constant bias fault is applied to the sensor measuring the cartposition x. The fault was detected by applying a statistical test on the �lter innovation sequence.Once detected, the bias was estimated by computing the fault free states with the generalizedKalman �lter and comparing them with the faulty states. The authors claim correction of thefault by subtracting its now known value (bias) from the faulty sensor output. The function �is assumed zero and the function � is given. By design, the method is restricted to sensor andactuation type faults.

Chapter 3Alternate ApproachesThe di�erent parametric identi�cation tools discussed in Chapter 2.1 were applied tononlinear structural systems with known models. These tools are here discussed as alternativesfor DLE of structural faults in these types of systems. Further, the reviewed approaches of FDI inSection 2.2 that were applied to nonlinear electro-mechanical systems are considered in designinga friction fault isolation and detection scheme for a precision positioning application. As such,three models of dry friction for identi�cation are considered as well.3.1 Selection of a Structural Fault Estimation MethodThe methods discussed in Section 2.1 parametrically identi�ed given low order modelsof structural systems with nonlinearities. This thesis is concerned with identifying higher orderstructures that contain nonlinearities as a result of damage. Further it is of interest to investigateextending the identi�cation procedures to detection, location and estimation of new structuralfaults. Recall that these faults may occur in the form of structural spring hardening and/orsoftening and may or may not result in more nonlinear (smooth) spring terms. The analysesconsidered in Section 2.1 could equally be considered as identi�cation of unknown parametersthat pre-exist or result from damage, while using observed responses that contain the faultyinformation throughout the measurements. When a structural fault occurs, the more realisticexperimental situation is response measurements that contain the occurrence of the fault, i.e. themeasurements change behavior at some point in the observation time interval due to the presenceof a fault. Therefore, the extension to DLE of new structural faults considers this situation as itis more likely to occur in real structures.Henceforth, structural fault estimation refers to the identi�cation and extension pro-cedures discussed above. To relate the approaches to the general problem of structural fault21

22estimation in a higher order nonlinear model, consider the equations of motion for an n degree-of-freedom statically coupled structural system given asM �x+C _x+Kx+G (x; _x) = f (t) (3.1)where, the diagonal mass matrix is given byM = 26664 . . . mi . . . 37775 ; (3.2)the symmetric coupled damping and sti�ness matrices are C andK respectively, and the vectorsof the nodal displacements and applied forces are given byx = n x1; x2; : : : ; xn o ; f = n f1; f2; : : : ; fn o :The matrix G represents nonlinearities in the structural model. A simple example of this matrixis considered here for discussion, given by[G] = 26666664 �1x31 �2x32 . . . �nx3n37777775 ; (3.3)where these terms represent cubic restoring springs from each nodal mass mi to ground and areparameterized by the nonlinear sti�ness coe�cients �i.The methods so far have successfully identi�ed unknown parameters that may exist inthe matrices M , C, K and G when the model order n is small, i.e. one or two. The �rstrequirement of the approaches as alternatives for structural fault estimation is that they appearto be potentially successful in identifying higher order systems, as the numerical analysis of thisthesis considers identi�cation of an eight degree-of-freedom nonlinear model of a space antenna.To apply the direct approach, consider the quadratic functionalJ(a) = Z T0 ( m1�x1 + a1�1 (x; _x)T � f1(t) + (3.4)m2�x2 + a2�2 (x; _x)T � f2(t) + (3.5)� � �+mn�xn + an�n (x; _x)T � fn(t) ) dt; (3.6)where the parameter vector a = [:::;ai; ::: ] accounts for the damping, sti�ness and nonlinearcoe�cients that parameterize the state variables (�i) in the equations of motion for each nodalmass mi. The minimization of this functional yields linear algebraic equations that, when solved,produce the best �t parameters a over the interval of observation (0; T ).

23The direct approach su�ers from the disadvantage of requiring very accurate accelera-tion measurements �xi. These measurements are integrated to provide the position and velocitymeasurements at each of the n nodes. For the generalized higher order system considered here,successful identi�cation would be possible only if very low or no noise level conditions and nobias were present on the acceleration measurements. These requirements are not realizable exceptunder conditions of numerical simulation or by processing (�ltering) the measurements to extractthe noise and any bias present. However, the simplicity of the approach and its computationale�ciency indicate that the estimated parameters could provide a quick and reliable initial guessfor the iterative methods, even when the measurements are contaminated with moderate levelsof noise. This can be a useful tool for the iterative methods when intuition or information usedin making an initial guess at the unknown parameters is not available.The application of the Gauss-Newton approach to the system described by Equation3.1 generates the augmented state-space equations as_u = g(u) (3.7)where the augmented state vector u, with initial condition vector u(0), and the vector functiong are given by u = fx1; _x1; x2; _x2; :::; xn; _xn; a1; :::; ang ; (3.8)g = � _x1; � 1m1a1�1 (x; _x)T + 1m1 f1(t); :::; (3.9)_xn; � 1mnan�n (x; _x)T + 1mn fn(t); 0; :::; 0: (3.10)The linear observation error is assumed as! = �u+ �; (3.11)where � is a rectangular matrix and � is the observation error vector. The least-squares functionaltakes the form J(u) = Z T0 (! � �u; ! � �u) dt+ (u(0)� u0; � (u(0)� u0)) : (3.12)The matrix � speci�es which state variables are observed. In this formulation, position or velocity(or both) measurements may be observed at some or all of the nodes. The obvious advantage ofthis method over the direct approach is that it requires only the measurements speci�ed by theuser, given a su�cient interval of observation. Also, the results in [9] show good robustness ofestimation with respect to noise.By minimizing the functional above and following the steps of this approach laid out in

24Chapter 2, the sensitivity equations are given byducdt = guuc; uc(c0; 0) = 24 In 00 0 35 ; (3.13)where the matrix gu is the derivative of g(u) with respect to u and the Jacobian matrix uc isfound by integrating these sensitivity equations. The order of the system n does not compromisethe success of this method except when the number of unknown parameters also becomes large.The greater the number of unknown parameters, the larger the augmented state vector u becomesand the more unwieldy the computation of the matrix gu becomes.Caution should be taken in using too few measurements or too short of an observationlength in the measurements. As long as the discretized interval of observation contains moresteps than there are unknown parameters, the estimation will provide unique estimates at eachiteration. However, the parameters are not likely to settle to their true value unless there is asu�cient amount of information in the measured signals and a su�cient number of these signals.To visualize this, consider �tting the parameters of an eighth order polynomial to a line segmentthat contains eight points. A least-squares �t of the parameters cause the polynomial to liedirectly onto the line, but the polynomial captures the \dynamics" of the line only.In all, the Gauss-Newton method shows promise in estimating a low number of param-eters in a higher order nonlinear structural model.Applying the extended Kalman �lter, the continuous state equation discussed with theinput f can also be written as X(t) = g (X;f ; t) +w(t); (3.14)where system input error is introduced as w. The computation of the jacobian of g with respectto the augmented state X also creates a computational headache when the number of unknownparameters becomes large. Also, the technique requires known state and measurement errorcovariance matrices. For these reasons the extended Kalman �ltering approach is severely re-stricted in higher order systems. The results shown in [14] also reveal moderate to low accuracyin a second order system. Further, the method is more complex in its implementation than thedirect and Gauss-Newton approaches.The second stage of structural fault estimation considers the extension to DLE of newstructural faults, where the faults occur while the observations are being collected. ExtendedKalman �ltering has shown on-line success in this type of application, as shown in Section2.2. However, the approach is limited to low order systems and therefore is not investigatedfurther. The advantages of the Gauss-Newton approach over the direct approach indicate that aquasilinearization type method for the identi�cation of a higher order nonlinear structural model

25is worthy of investigation. The extension of this method to DLE of new structural faults is alsoqualitatively investigated in the Chapter 4 investigation.3.2 Selection of a Dry Friction Fault Detection and Isola-tion MethodThe desirable properties of the FDI scheme are that it be able to isolate and detect anonsmooth nonlinearity such as Coulombic friction in an already highly nonlinear system, i.e. apneumatic cylinder positioning device.The model estimation and parity space approach taken by Ho ing and Deibert discussedin Section 2.2 shows several attributes that would be of bene�t in designing a dry friction FDIscheme for implementation in a precision positioning experiment. By �tting a discrete lineartransfer function to the system dynamics, the di�cult tasks of identifying a model structure andidentifying the multiple (linear and nonlinear) coe�cients that parameterize the model are elim-inated. Although the transfer function parameters have no physical interpretation, the resultingresiduals show successful on-line detection of sensor and static process map faults for the processof low pass character that was considered [17]. The more successful ARMA residual is againgiven by r = �y(z�1)�G(z�1)�u(z�1); (3.15)where the order of G is unknown. In other studies, the model error, i.e. the di�erence betweenthe modeled and measured signals, also successfully serves as a residual that is employed fordetection directly (see e.g. [16]).There are however several limitations in the approach taken by Ho ing and Deibert,particularly when considering the problem of dry friction fault detection and isolation. Thefaults considered in their study were perturbations of the existing information, e.g. altering thestatic nonlinear map or simulating a sensor fault in the model. However, friction is consideredas an unknown input to the process dynamics and residuals are not equipped to relate anyinterpretation to this type of fault. Moreover, the residuals are valid for detection only and areunable to isolate one fault from the next.The chief limitation of their approach is the restriction to single-variable processes oflow-pass character. Not only do precision positioning devices not exhibit such character, butCoulombic friction as a fault in these devices does not exhibit low pass behavior. Therefore,blank spots in which no fault monitoring is taking place in the estimation and parity spaceapproach are unacceptable, as the process is subject to the dynamic behavior of the friction fault

26input at every moment after the occurrence of the fault.Extended Kalman �ltering, as applied in [18] and [19], assumes the model error to be azero-mean stochastic process of known covariance. This restriction is often incorrect for systemswith nonsmooth nonlinearities such as Coulombic friction [31]. The assumption also eliminatesthe option of computing residuals based on the model error directly, which is shown to be e�ectivein other studies. Also, as discussed in Section 3.1, the approach is severely restricted in higherorder systems.These alternative approaches are not directly applicable to the problem of implementinga much needed identi�cation and friction FDI scheme in precision positioning devices. Therefore,such a scheme is designed here for implementation in a precision positioning experiment. Asdescribed below, the scheme steps out of the traditional two-stage structure of the FDI process(see Figure 1.1).The designed experimental set-up consists of a precision positioning device that actuatesan air bearing mass. The positioning device is a servo pneumatic cylinder. The air bearing masspermits the addition of Coulombic friction on-line for detection. A (nonparametric) discretefourth-order linear model, or transfer function, is �t by weighted least-squares to the frequencyresponse from the servo voltage input to the measured air bearing position output for the nofriction fault case. The transfer function gives a position output for a voltage input and thesecond derivative of the position signal is computed to give the modeled, no friction acceleration.Caution should of course be taken when �tting a linear model to a highly nonlinearprocess. As will be shown in Chapter 5, this approximation is good enough for the dry frictionFDI scheme to be successful in the designed precision positioning experiment. As discussed, thesame process modeling approach was taken in [17] and several improvements to their approachcan be considered here. First, the model is �t by weighted least-squares in one shot, a simple andvery e�ective extension of least-squares alone. Also, the robustness of the discrete model �t tovariable inputs is addressed by observing the changes in the frequency response for the inputs ofinterest. Moreover, successful detection is achieved here for a wide range of time-varying inputs,rather than being restricted to constant inputs that change only step-wise as in [17].The model error between the modeled and measured acceleration serves as a residual.This residual is not used for detection directly. Instead, the friction is found to be dynamicallyrelated to the acceleration residual. A prediction error approach [4] is used to model the dynamicrelationship between the friction force and the acceleration residual as a �lter, using an outputerror optimization. This is what isolates the friction fault from other possible process, sensor oractuation faults. An assumed friction model is incorporated with this experimentally generated�lter and the procedure is implemented by recursive least-squares estimation of the friction model

27parameters. Simple threshold decision logic is employed to detect the dry friction fault on-line.The dynamic �lter here is valid for each input so that detection requires only recursiveleast-squares estimation of the friction model parameters for monitoring, rather than switchingfrom parameter estimation for model �tting to monitoring the model error based residuals asused in [17]. Also, no static curves of the process are required.Three parametric friction models are considered for implementation with the dynamic�lter. Bliman and Sorine [34] presented two state variable dry friction models with desirableproperties for control purposes. The identi�cation of the parameters is open-loop and thereforeapplicable to non control applications as well. The models are geared to be mathematically soundand simple. The main considerations were to represent kinetic and static e�ects, stick slip andother experimentally observed e�ects. These e�ects were summarized in Figure 3.1. This �gure

Figure 3.1: Qualitative Behavior of Friction Force F versus Position udisplays the qualitative relationship between the dry friction force F between a sliding block anda surface and the position u of the block. Starting from rest (u = 0) the dry friction exhibitsan elastic behavior until a maximum friction value, called the static friction fs, is reached. Theposition at this point is considered a microdisplacement, labeled se. Beyond the static value,the friction decreases until it settles to the kinetic value fk for displacements greater than sp.This behavior is of plastic type (Coulomb friction behavior) and displacements take the place

28of microdisplacements. When the velocity switches, the same curve (with a sign change) beginsfrom the reached position. This behavior is irreversible and therefore gives rise to hysteresiscycles, i.e. cycles that are rate independent in shape. This means that the transient behaviorseems to be independent from the velocity but depends upon the covered distance or positionwhen the sign of the velocity remains constant.>From this qualitative experimentally observed behavior, the desired properties can besummarized as a dry friction model that1. matches the rate independence or hysteresis property,2. models friction is dissipative,3. provides easy identi�cation of the model parameters in terms of the main experimentalobservations (fk; fs; se; sp),4. must constitute, with the equations of motion, a well-posed set of equations,5. must agree with the classical Coulomb and viscous friction model when sp ! 0, and6. be simple enough in order to be used in real-time algorithms.The models are ordinary di�erential equations de�ning the friction operator u 7! F (u) as_xf = j _uj �Axf +B _u; xf (0) = 0 (3.16)F (u(t)) = Cfxf (t) (3.17)and for �rst and second order models, respectively, the matrices A; B; Cf are de�ned asA = � 1"f ; B = f1"f ; Cf = 1; and (3.18)A = � 1"f 0@ 1� 00 1 1A ; B = 1"f 0@ f1��f2 1A ; Cf = (1 1) : (3.19)F (u) represents the dynamic friction force at the contact area between two surfaces in relativemotion. The second order model has the advantage of better representing transient behavior,speci�cally that friction force is independent of velocity and dependent upon covered distance inthe transient range. The parameters (fk; fs; se; sp) represent the kinetic and static friction val-ues, the breakaway distance of F (u(t)) from rest and the distance above which F (u(t)) is within5 % of fk, respectively. A method of observing these parameters experimentally is given and thenonlinear relation between the model parameters in Equation 3.18 and Equation 3.19 and thesemeasurements is also given. These models facilitate computer simulations of friction and a sim-ulation involving tuning a proportional-integral-derivative (PID) controller was given. Although

29simple and sound in representing the physical phenomena in dry friction, the identi�cation ofthe parameters is not suited to FDI purposes. As these models are parametrically nonlinear,identi�cation must be performed o�-line and by possibly multiple experimental observations.Haessig and Friedland [41] contend that the �ve low velocity models considered in thestudy are numerically ine�cient and if sticking is not an issue coulomb and viscous friction su�cesas a model. The model gives the sliding friction force Ff between two surfaces asFf = � _xj _xj + � _x; (3.20)where _x is relative velocity between the surfaces, � and � are the coulomb and viscous termparameters, respectively, and �Ff and _x are conventionally assumed positive to the right.The coulomb and viscous model of friction is employed in this experiment because it fa-cilitates the isolation and on-line detection of the friction fault as the parameters appear linearlyin this model. Linear parameterization of this model make it possible to employ the accurateand e�cient least-squares estimation of the model parameters recursively. Further, this modelrepresents the low stick high slip friction characteristics observed at the aluminum on aluminumcontact surface in the experimental setup. The majority of the science and engineering �elds thatinvestigate friction modeling consent that the dry slipping mechanism is accurately described bycoulomb and viscous behavior [30], [29], [41], with experimental validation in [36]. As in the liter-ature, it is also experimentally observed that the validity of this model in representing the frictionforce decreases as velocity goes to zero. The coulomb viscous model is also numerically muchmore e�cient than other models that more accurately represent friction at smaller velocities, e.g.the reset integrator and bristle models.As an aside, Johnson and Lorenz [31] also used signal processing to generate the dynamicrelationship between state errors, i.e. the di�erences between modeled and measured states, andthe (measured) states in a friction model. The scheme here does not require motion control asin [31]. The method employed in this paper does suggest that an extension of the coulomb andviscous friction model could be made by including higher order velocity and position terms. Byparameterizing these additional terms linearly, such an extension would be possible in the on-linedetection scheme designed here.

Chapter 4Parameter Estimation of aDamaged Structure Using aQuasilinearization ApproachSimulation of systems for observation and/or health monitoring for fault detection re-quires a model of the system. When modeling structural systems for observation and healthmonitoring, it is very important to consider a physically based model, as structural analysistechniques that assess these structures for safety rely upon physical interpretation. A physicallybased linear model of a exible space antenna is given by Kabe [10]. A nonlinearity is addedto this model to simulate a structural spring that becomes nonlinear as a result of damage.To identify unknown sti�ness parameters in the model, including the coe�cient of the addednonlinear spring term, a quasilinearization approach is applied. The technique successfully es-timates two linear and one nonlinear sti�ness coe�cients, o�-line, under various conditions ofloading, nonlinearity, measurement noise and under various computation time constraints. Fur-ther, the quasilinearization approach is brie y extended to the location and estimation of newstructural faults, i.e. new damage in the form of sti�ness hardening or softening and possiblyadded nonlinearities, that can occur in the system.

30

314.1 Procedure for Parameter Estimation of a NonlinearStructural ModelThe key steps in the derivation and application of the quasilinearization parameterestimation technique are� Modify the analytic linear model of the space antenna structure given by Kabe [10] toinclude a nonlinear structural spring that results from damage.� Derive the quasilinearization approach of parameter estimation.� Apply the approach to identify unknown parameters in the nonlinear space antenna modelunder varying conditions of loading, nonlinearity levels and measurement noise.� Discuss the extension of quasilinearization to detecting further damage to the structure.In Section 4.2 the linear exible model of the space antenna structure is given and modi�ed toinclude a nonlinearity as a result of damage. Speci�cally, one structural spring is assumed to gofrom linear restoring force behavior to linear and cubic behavior as a result of damage.A rigorous derivation of the quasilinearization approach of parameter estimation ofnonlinear models is given in Section 4.3. The application of this approach to the damaged spaceantenna model is in Section 4.4 and the results of numerical tests are given and discussed inSection 4.5. In Section 4.6 conclusions about the method and the extension to the detection ofmore structural faults to the nonlinear structure is discussed.4.2 Analytic Model of a Nonlinear Space Antenna Struc-tureFigure 4.1 shows the analytical test structure of a exible space antenna as given byKabe [10]. This structure can be thought of equivalently as eight interconnected masses in series,i.e. all mass's move translationally in the same direction, on a frictionless surface. Each masshas one degree-of-freedom and the 14 load paths are the springs that connect the masses, toeach other or to ground, in the same axial direction as the mass motion. As the relative sti�nessmagnitude range is large (from 1.5 to 1000) this structure represents a severe test case. There arethree unique lumped mass values and six unique sti�ness coe�cients. The equations of motionfor this eight degree-of-freedom (8 DOF) statically coupled system are given as[M ] f�xg+ [K] fxg = ffg (4.1)

32

Figure 4.1: Kabe model of Space Antennawhere, the diagonal mass matrix is given by[M ] = 26664 . . . mi . . . 37775 ; (4.2)the coupled sti�ness matrix is given by[K] = 2666666666666666664

k5 �k5 0 0 0 0 0 0�k5 k1+k2+k5 �k2 0 0 0 0 00 �k2 k1+k2+k4 0 �k4 0 0 00 0 0 k3+2k4 �k4 �k4 0 00 0 �k4 �k4 k3+2k4 0 0 00 0 0 �k4 0 k1+k2+k4+k6 �k2 �k60 0 0 0 0 �k2 k1+k2+k5 �k50 0 0 0 0 �k6 �k5 k5+k63777777777777777775;(4.3)and the vectors of the nodal displacements and applied forces are given by�x = n x1; x2; : : : ; x8 o ; �f = n f1; f2; : : : ; f8 o :One nonlinearity is added to the Kabe model above to simulate a structural springthat becomes nonlinear as a result of damage. Only one spring is altered due to computational

33limitations, discussed in Section 4.6. The spring k1 of m6 is selected to include this nonlinearityin the form of cubic dependence on position. This spring location was chosen because it a�ectsthe greatest number of modes in the undamaged linear structure. The analysis of identifyingk1 at m6 as the primary modal participant in the structure under general loading is given inAppendix A. The added nonlinearity generates Du�ng's equationm6�x6 + k1x6(1 + �x26) = ~f6; where (4.4)~f6 = k2(x7 � x6) + k4(x4 � x6) + k6(x8 � x6) + f6: (4.5)All other equations are retained in their original linear form. The damaged structuralspring k1 and k1� at m6 are unknown constants and are to be estimated. For added complexity,the adjacent k4 at m6 is also assumed unknown and requires estimation. All other sti�nessvalues and all mass values are known. It is henceforth assumed that the units in the equationsare normalized so that the equations to follow remain dimensionally correct, as is done in [8].4.3 QuasilinearizationQuasilinearization was developed as a numerical tool for solving problems de�ned bynonlinear di�erential equations and has been extended to identi�cation problems [40], [8]. Con-sidered in this study is a simpli�ed identi�cation (or optimization) problem, as the model andparameterization are given. Under certain assumptions, the technique can successfully reduce anonlinear optimization problem to a succession of operations involving the numerical solution oflinear di�erential and algebraic equations. First the equations of motion are linearized into theform of a sequence of equations, in a way analogous to the Newton-Raphson (N-R) method forroot �nding, and a linear solution to these equations is formulated and parameterized in termsof the unknown constants. Second, the resulting sequence of linear di�erential equations aresolved to generate the linear solution, comprised of particular and homogeneous parts. Third,given the response (displacement) data from the original equations of motion, a cost function isminimized in a least-squares sense which yields a set of linear algebraic equations. The solutionof these equations generates the next estimate for the unknown parameters. Steps two and threeare repeated until the parameters converge to their true values, as discussed in the followingsections.4.3.1 Step 1 : Linearization of Equations and Solution FormThe N-R method properties of monotonicity and quadratic convergence apply hereprovided the equations behave in the following way. A set of autonomous nonlinear �rst order

34di�erential equations can be expressed in the form_x = f(x); x(0) = c; t � 0 (4.6)where ( _ ) is a derivative with respect to time and c represents the initial condition vector. Asthis study is considering a dynamics problem, this set of equations is henceforth referred to as theequations of motion. Assume f is continuous in x and t and has a bounded continuous Hessian(second partial derivatives w.r.t x), for all x and t in the region of interest. Further, assume fis strictly convex in x in the region of interest. The �rst order Taylor series around a nominaltrajectory x(0)(t) is a linear di�erential equation that yields a sequence of continuous functionsdetermined by the recurrence relationdx(1)dt = f(x(0)) + @f(x(0))@x (x(1) � x(0)); x(1)(0) = c: (4.7)where x(0)(t) is used to begin the iterative process and represents the initial approximation ofthe solution to Equation 4.6.According to Bellman [8], the success of convergence of the recurrence solutions tothe true solution depends upon selecting x(0)(t) su�ciently close to the true solution, just asthe success of N-R in root �nding depends upon the initial guess. Under the bounded Hessianassumption there exists a common interval in time in which the sequence solutions x(n)(t); (n =1; 2; : : : ); are uniformly bounded. Further, the sequence of solutions can be proven to convergequadratically to the solutions of the original equations of motion, if they converge at all. Theproperty of monotonicity of the sequence requires f to be positive and convex. These propertiesare very restrictive, but are used in [8] for the derivation of the quasilinearization technique andthe examples contained therein do not require these properties (page 19, [8]). The parametersin Equation 4.6 appear linearly and this system can be augmented to include those parameterswhich are unknown. In this way the unknowns are collected into the vector a and the augmentedsystem is _x(t) = f(x;a); x(0) = c; x(t) � RNx (4.8)_a(t) = 0; a(0) = a0; a(t) � RNa (4.9)where RN represents the N -dimensional real vector space and Nx is the dimension, or moreclearly the number of degrees-of-freedom, of the equations of motion. The unknown parametervector a, of dimension Na, is equal to it's unknown initial condition a0 for all time in the intervalof interest.The vector x(n)(t) � RNxa that evolves according to Equation 4.7 now represents therecursive augmented state, where dimensionally Nxa = Nx +Na. A solution form to this aug-mented set of recursive linear di�erential state equations is assumed as a sum of particular and

35homogeneous parts (by superposition) asx(n)(t) = P (n)(t) + NaXi=1 i H(n)Nx+i(t); (4.10)where P (n) is the particular part, H(n)Nx+i make up the homogeneous parts and the initial condi-tions are P (n)(0) , P0 = [ cT ; 0TNa ]T ; where 0TNa = [::: 0; :::] � RNa ; (4.11)and H(n)Nx+i(0) , H0 = [:::; �j;Nx+i; :::]T ; j = 1; : : : ; Nxa; i = 1; : : : ; Na: (4.12)Therefore, the goal of this recursive technique is to get the estimated parameter vector � RNa to converge to a0, thereby matching the initial conditions of Equation 4.10 with theinitial conditions of the augmented system in Equation 4.8 and Equation 4.9. This solution formassumes that all initial conditions for the equations of motion, i.e. the c vector, are known.However, the method can be easily extended to identify any unknown initial conditions as well.Note that the initial approximate solution to Equation 4.8 and Equation 4.9 now requires initialguess's for the true parameter values as well as the initial guess for the solution to Equation 4.8.Henceforth in this Section, solution refers to the augmented system discussed here.4.3.2 Step 2 : Generation of Recursive SolutionPlugging Equation 4.10 into Equation 4.7 yields the recursive set of �rst order lineardi�erential equations, where the particular part of the solution evolves according todP (n)dt = f (x(n�1)) + J(x(n�1))(P (n) � x(n�1)); P (n)(0) = P0; (4.13)the homogeneous part of the solution evolves according todH(n)Nx+idt = J(x(n�1))H(n)Nx+i; H(n)Nx+i(0) =H0; i = 1; : : : ; Na; (4.14)and the jacobian matrix is given byJ(x(n�1)) = @f(x(n�1))@x ; n = 1; 2; : : : : (4.15)4.3.3 Step 3 : Minimization of Cost Function and Estimate GenerationIt is assumed that the system we are studying is observed x(t) over a �nite intervalt � [0; T ]. The least squares method provides a cost function to be minimized at each iteration,where the cost function is� = Z T0 " P (n)(t) + NaXi=1 iH(n)Nx+i(t) � x(t) #2 dt (4.16)

36Note that Equation 4.16 is the L2(0; T ) norm of the time-varying error vector, where the erroris between the linearized solution x(n)(t) and the observed solution x(t). Minimizing a leastsquares function such as this to obtain a best �t set of parameters is the most common methodof parameter estimation, although it often arises in di�erent forms (cite papers and books).It is here implied that errors superimposed upon the observed data are uniformly distributedthroughout the time interval, i.e. the weighting function (not shown) in Equation 4.16 is identityfor all time in the interval. The minimization of � with respect to the estimated parameter vector yields a system of linear algebraic equations@�@ i = 0; i = 1; :::; Na =) D = b (4.17)where Dji = Z T0 H(n+1)Nx+j (t) TH(n+1)Nx+i (t) dt; D � RNa�Na ; (4.18)and bj = Z T0 � x(t)�P (n+1)(t) ]TH(n+1)Nx+j (t) dt; b � RNa (4.19)D is a symmetric matrix and the solution to the estimation problem at each iteration becomes = D�1b, for nonsingular D. For discrete observations, the integrals in the minimizationEquation 4.17 are replaced by summations.As stated before, the success of the convergence of the approximate solutions to theobserved solutions, and therefore the success of the parameter estimates to the true parametervalues, depends upon the initial approximation of the solution to the equations of motion and thefunction properties of continuity, etc.. In a numerical example such as this, a simple and e�cientway to obtain a �rst approximate solution x(0)(t) is to integrate the original equations of motionand make a reasonable guess at the parameter values. This is feasible for our formulation as wehave assumed that all initial conditions for the equations of motion are known. This methodserves as a reliable starting point for the iterations but other methods when this is not feasibleare discussed in Chapter 14 of [8].4.4 Application to Modi�ed Kabe ModelThe estimation is performed assuming all constants in Equation 4.1, modi�ed by Equa-tion 4.4, are known except k1, � and an adjacent spring k4, all at m6 as appearing in Figure 4.1.

37Prior to linearization, we write the modi�ed equations of motion in state space form as_X = F (X;f); X(0) = C; t � 0 (4.20)where X = h x1; _x1; : : : ; x8; _x8; aT iT ; a = h a1; a2; a3 iT ; (4.21)f = h f1; : : : ; f8 iT ; F = h _x1; : : : ; �x8; 0; 0; 0 iT (4.22)and C = h x1(0); : : : ; _x8(0); a0 iT = � cT ; aT0 �T : (4.23)As before the unknown parameter vector a is equal to it's unknown initial condition a0 =h k4; k1; k1� iT for all time in the interval of interest. The X and a vectors are what theiterations are estimating in the hopes of convergence.The linear recursive solution for this system is given byX(n)(t) = P (n)(t) + 3Xk=1 kH(n)16+k(t); n = 1; 2; : : : ; where (4.24)dP (n)dt = F (X(n�1)) + J(X(n�1))(P (n) �X(n�1)); P (n)(0) = C; (4.25)dH(n)16+kdt = J(X(n�1))H(n)16+k; H(n)16+k(0) = [:::; �j;16+k; :::]T ; j = 1; : : : ; 19; (4.26)k = 1; 2; 3 ; and J(X(n�1)) = �dF (X(n�1);f)dX �T �Jij = dFidXj �: (4.27)For m discrete measurements in time of the eight mass displacements, the cost functionbecomes � = mXi=1 8Xj=1 h x(n)j (ti)� xj(ti) i2; where (4.28)x(n)j (ti) = �X(n)(ti) �T Ij ; and Ikj = �k;2j�1; k = 1; :::; 19 ; (4.29)and the minimization follows according to Equation 4.17 - Equation 4.19. The �rst approximatesolution X(0)(t) is generated from the original equations of motion and a guess at the trueparameter values is used for to begin the iterations. For all iterations that follow the initialguess stage, only the parameter values change, as generated by Equation 4.28. Recall that thestrain energy argument of selecting k1 at m6 for nonlinearization as a result of damage is basedupon the condition of general loading. The inputs to the system considered here are step loads(initiating at t = 0) of varying magnitude for f5 and f6 and a sum of sinusoids for f6, appliedseparately. The damage level, i.e. the parameter �, the time step of integration and the initialguess's for the parameters are varied to examine conditions for successful convergence. Further,the case of noise contamination on the eight mass displacement measurements is examined. Theunits are all normalized and the model is integrated with a step size of 0.01 sec, unless speci�edotherwise.

38Table 4.1: Convergence Results for Variable � with f5 = 1000� True Values Initial 1st 2nd 3rd ConvergedVariations (k4; k1; �k1) Guess Iteration Iteration Iteration Values0.1 100 50 100.5 99.991000 1300 997.1 1000. same same100 145 101.4 101.90.5 100 150 99.59 99.981000 1300 997.2 1000. same same500 320 506.1 509.72.5 100 150 99.71 100.01000 750 1002. 1000. same same2500 1700 2548. 2543.10 100 150 98.80 100.1 100.11000 750 996.0 1001. 1000. same10000 7500 10520 10130 10140Table 4.2: Convergence Results for Variable � with f5 = 4000� True Values Initial 1st 2nd 3rd ConvergedVariations (k4; k1; �k1) Guess Iteration Iteration Iteration Values0.1 100 50 100.5 100.01000 1300 998.9 1000. same same100 145 101.4 101.70.5 100 150 99.63 100.11000 1300 995.7 1000. same same500 320 507.4 507.32.5 100 150 99.37 100.11000 750 999.5 1002. same same2500 1700 2574. 2532.10 100 150 99.50 100.0 100.11000 750 1001. 1002. 1002. same10000 7500 10250 10130 101404.5 Application ResultsAs Table 4.1 and Table 4.2 show, for � variations and applied step loads at m5 off5 = 1000 and 4000 the two linear parameters (k4; k1) converged to three and four signi�cantdigits and the nonlinear parameter (�k1) converged to two signi�cant digits, all within twoiterations. In all tabulated results that follow, converged values are de�ned as those which don'tchange to four signi�cant �gures after two or more iterations.Table 4.3 shows that for an applied step load at m6 of f6 = 1000, the parameterconvergence accuracy deteriorates to failure as � is increased. For the most damaged or nonlinearcase, i.e. � = 10, the iterations remained bounded but never converged to any values. Not shownhere is that for an initial guess of (1; 1; 1), all converged cases in Table 4.1 - Table 4.3 remainedconvergent in the same number of iterations.Table 4.4 shows that convergence of the least nonlinear case of Table 4.3 is lost as the

39Table 4.3: Convergence Results for Variable � with f6 = 1000� True Values Initial 1st 2nd 3rd ConvergedVariations (k4; k1; �k1) Guess Iteration Iteration Iteration Values0.1 100 50 100.5 100.01000 1300 1007. 1003. same same100 145 95.02 99.210.5 100 150 99.41 101.3 101.41000 1300 1014. 1003. 1002. same500 320 493.9 497.7 497.82.5 100 150 99.50 55.00 100.4 99.831000 750 1005. 960.0 1076. 1070.2500 1700 2550. 2592. 2440. 2440.10 100 150 99.50 1536 68.81000 750 1066. 11260 872.5 none10000 7500 10140 -1024. 9871.Table 4.4: Decrease in Time Step with f6 = 1000; � = 0:1Time Step True Values Initial 1st 2nd 3rd Converged(k4; k1; �k1) Guess Iteration Iteration Iteration Values0.01 100 50 100.5 100.01000 1300 1007. 1003. same same100 145 95.02 99.210.03 100 50 108.1 98.9 100.7 100.31000 1300 1023. 1002 1041 1030100 145 61.40 103.3 84.7 88.90.04 100 50 58.5 111.0 109.91000 1300 321.3 1198. 549.5 none100 145 235.9 113.0 200.6time step is increased. For the non convergent case of Table 4.3, using even the true values (100,1000, 10000) as an initial guess does not yield convergence.Table 4.5 shows that by decreasing the time step by 10 orders of magnitude, convergenceis obtained in three iterations. The table also reveals that by changing the input to a sum ofsinusoids, the originally nonconvergent time step converges in three iterations. As with all othercases, the parameters eventually fail to converge as the time step is increased for this sinusoidalinput. Table 4.6 combines variations in time step as well as in initial guess for a larger inputat m6 of f6 = 4000, while showing computation time per iteration. For the extreme initial guesscase in the second row in this table, inf means that the parameters not only did not converge butbecame unbounded. The time step reduction shown in row three demonstrates that convergenceis obtainable for this extreme initial guess at the cost of larger computation time.The success of convergence depends primarily upon the time step of integration andsecondarliy upon the type of loading and the magnitude of damage in k1 present, i.e. the

40Table 4.5: Variable Time Step, f6 Input with I.G. = (150; 750; 7500); � = 10f6 Time Step True Values 1st 2nd Converged(k4; k1; �k1) Iteration Iteration Values1000 0.01 100 99.5 1536(step) 1000 1066. 11264 none10000 10139 -10241000 0.001 100 99.99 100. 100.(step) 1000 1066. 1000.66 1002.110000 10001.6 99968. 99968.0.01 100 100.3 99.63 99.63�f�6 1000 998.0 975.1 974.810000 9737. 10020 100100.015 100 15490�f�6 1000 -12800 inf10000 34820� �f6 = 1000 ( 0:35 sin(10t) + 0:3 sin(700t) + 0:25 sin(800t) + 0:2 sin(200t) )

Table 4.6: Variable Time Step and Initial Guess with f6 = 4000; � = 0:1Time True Values Initial 1st 2nd Converged ComputationStep (k4; k1; �k1) Guess Iteration Iteration Values Time/Iter0.01 100 50 100.4 99.96 99.961000 1300 1119. 1045. 1049. 2.5 sec100 145 92.42 98.21 97.970.005 100 50 24580 3584.1000 750 -1536. 400 inf 5.5 sec100 7500 1408 -5120.0005 100 50 100.0 100.01000 750 1012. 1000. same 50.1 sec100 7500 99.05 100.0

41Table 4.7: Measurements (x3, x6, x8) Contaminated by Random Noise for f5 = 1000; � = 0:1Percentage True Values Initial 1st 2nd Converged At 25Added Noise (k4; k1; �k1) Guess Iteration Iteration Values Iterations0 % 100 50 100.5 99.991000 1300 997.1 1000. same100 145 101.4 101.95 % 100 50 98.57 99.99 99.961000 1300 988.5 1000. 1000.100 145 140.5 102.3 102.110 % 100 50 96.51 99.95 99.921000 1300 983.1 1001 1001100 145 197.9 104.9 102.220 % 100 50 92.58 98.93 98.821000 1300 978.9 997.6 1002100 145 334.2 172.1 102.650 % 100 50 87.90 92.34 98.071000 1300 989.8.1 1023 none 1009100 145 595.3 349.6 170.4magnitude of �. For a small enough time step, all cases considered here converge in one iteration.The success of the quasilinearization parameter estimation approach is also investigatedfor cases where the system observations are contaminated by noise. Uniformly distributed randomnoise is added to the measured displacements x3, x6 and x8 at 5, 10, 20 and 50 % of the peakvalue of each of these signals. Uniformly distributed noise is restricted between � 1 in peak valueand a scaling is used to achieve the desired percentage of the peak values of the measurements.These cases are considered for the conditions of the �rst case given in Table 4.1 and the resultsare given in Table 4.7. To compare the noise-free and increasingly noisy measurements, Figure4.2 - Figure 4.5 show the measured displacement of mass six, i.e. x6, for the increasing levelsof added noise. Clearly, the greater the noise content in the observed signals, the poorer theconvergence performance.4.5.1 Discussion of ResultsThe tabulated results indicate that the time step of integration is the chief factor thatdetermines the success of convergence of the unknown model parameters to the true values. Asthe time step is decreased, all of the variable conditions of loading and nonlinearity (damage)levels eventually result in convergent results. This dependence of successful convergence on timestep can be translated into a dependence of the simulated signals on continuity in time. As thetime step is increased, the ability of the applied integration routine to maintain continuity intime of the observed and generated signals becomes compromised. The loss of continuity in timeof the observed signals directly corrupts the continuity of the generated recursive equations as

42

0 1 2 3 4 5 6 7 8 9 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Response of x6(t) for 0 % noise

time (sec)

x 6(t)

Figure 4.2: Measurement of x6(t) for Conditions of Table 4.7 and of No Added Noisethe observed signals force the linear recursive equations.To see the forcing terms explicitly in the recursive equations, the jacobian is computedfor the application to the modi�ed kabe model. The jacobian in Equation 4.27 yields x6(t); x6(t)3and x4(t) as the forcing functions in the iterative homogeneous equations, Equation 4.26, ap-pearing in the formdH(n)j8 (t)dt =� 1m4 (k3 + k4 + a1)Hj7 + k4m4Hj9 + a1m4Hj11 + 1m4 (x6(t)� x4(t))Hj17 ; (4.30)dH(n)j12(t)dt = a1m6Hj7 � 1m6 (k2 + k6 + a1 + a2 + 3a3r26)Hj11 + k2m6Hj13+ k6m6Hj15 + 1m6 (x4(t)� x6(t))Hj17 � x6(t)m6 Hj18 � x6(t)3m6 Hj19 ; (4.31)with initial conditions H(n)jk (0) = �jk; j = 17; 18; 19; k = 1; : : : ; 19: (4.32)

43

0 1 2 3 4 5 6 7 8 9 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Response of x6(t) for 10 % noise

x 6(t)

time (sec)Figure 4.3: Measurement of x6(t) for Conditions of Table 4.7 and 10 % Added NoiseThese terms also drive the iterative particular equation along with the varying input functions.The e�ect of losing continuity in time becomes magni�ed in the x6(t)3 term, since x6(t)itself is losing continuity. Therefore, the tabulated convergence results can be translated intoa comparison of continuity in time of these non input forcing terms. For the highly nonlinearcase investigated in line one and two of Table 4.5, Figure 4.6 shows x6(t) and x6(t)3 for thenon convergent time step and Figure 4.7 displays the improvement in the continuity in time ofthese signals for the reduced convergent time step. Similarly, the breakdown of convergenceinvestigated in Table 4.4 is displayed as the breakdown in continuity of the x6(t) and x6(t)3signals in Figure 4.8, Figure 4.9 and Figure 4.10.A comparison of Figure 4.6 to Figure 4.9 would lead one to question the idea thatincreased continuity always yields increased convergence, since the signals for the nonconvergentcase of Figure 4.6 are smoother (smaller time step) than those of the convergent case of Figure 4.9.

44

0 1 2 3 4 5 6 7 8 9 10−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

Response of x6(t) for 20 % noise

time (sec)

x 6(t)

Figure 4.4: Measurement of x6(t) for Conditions of Table 4.7 and 20 % Added NoiseThe �gures so far show that the trend of continuity increasing with convergence performance istrue for a given set of conditions. What di�erentiates the case of Figure 4.6 from Figure 4.9 is thatthe former is 100 times more nonlinear, in terms of the parameter �, resulting in higher frequency,reduced magnitude and chie y less linear responses. The approximation of linearization becomesless valid and the iterative solutions fail to converge. Table 4.3 exhibits this trend for all otherconditions being equal. By comparison, the convergence of the sinusoidal input for the third caseinvestigated in Table 4.5 has the e�ect of reducing the frequency of the x6(t) and x4(t) responses,as shown in Figure 4.11 and Figure 4.12. Further, the increase in time step for this sinusoidalcase results in loss of convergence.The coupling between time step, nonlinearity and input as they a�ect convergence is anon trivial one. There seems also to exist a relationship between convergence and the frequencyof response in the x4(t); x6(t), and x6(t)3 terms that drive the iterative linearized equations.

45

0 1 2 3 4 5 6 7 8 9 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Response of x6(t) for 50 % noise

time (sec)

x 6(t)

Figure 4.5: Measurement of x6(t) for Conditions of Table 4.7 and 50 % Added Noise4.6 Conclusions and Extension to Damage DetectionThe success of this method in parametrically identifying unknown parameters o�-linethat accompany linear and nonlinear terms and pre-exist or result from damage in the model ofthe space antenna structure is extensively proven for a wide range of loading, nonlinearity level,integration routine conditions and noise levels in the observed dynamic responses. Bellman [8]has also shown success of the method in cases where only partial information in the dynamicresponse is observed. The success of the method can also be improved in cases where weightingthe cost function information is appropriate. For example, when a section of the measured datacontains better information for identi�cation, i.e. if initial condition mismatching exists and noiseor damage contaminated the observed signals at a certain point in time, a Jabobi approximationweighting is used. Further, if the initial or �nal data in the measurements are more reliable foridenti�cation, exponential (Laguerre approximation) weighting is appropriate.

46

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1F6 = 1000, alpha = 10 and delt = 0.01

time (sec)

m6

disp

lace

men

t (x6

)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−1

−0.5

0

0.5

1

time (sec)

x63

Figure 4.6: Continuity of Non Convergent Step Input Case in Table 4.5Although the method is highly successful for the eighth-order space antenna model, thenumber of parameters is kept small since the number of di�erential and linear equations thatrequire solutions for each iteration grows rapidly for a higher number of unknown parameters. Inthe structural model considered here, consider the extreme case that all of the masses are unknownand possibly di�erent, each spring element is nonlinearized, to account for possible damage, byadding cubic dependence and all of the sti�ness values are unknown and not necessarily equal.Each iteration would require the solution of 784 di�erential equations and the inversion of a48 by 48 matrix. The observation measurements in this case must contain at least 48 samplesfor the problem to be determined and many more samples than that for any hope of successfulconvergence. The likelihood of a sign error in the enormous equation formulation and possibleill-conditioning of the matrix pose threats to the success of the method, which would also requirelarge computational time.As a tool for identifying a limited number of unknown parameters in damaged and

47

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1F6 = 1000, alpha = 10 and delt = 0.001

time (sec)

m6

disp

lace

men

t (x6

)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−1

−0.5

0

0.5

1

time (sec)

x63

Figure 4.7: Continuity of Convergent Step Input Case in Table 4.5therefore nonlinear higher order structural systems, given the model structure and post dam-age dynamic response observations, the quasilinearization approach of parameter estimation isreliable.4.6.1 Extension of Application of Quasilinearization to Damage Detec-tionThe analysis so far has considered identifying unknown parameters, that pre-exist orresult from damage, using observed responses that contain the faulty information throughout themeasurement. A more realistic situation is response measurements that contain the occurrenceof the fault, i.e. the measurements change behavior at some point in time due to the presence ofa fault. Further, the fault may result in hardening or softening of the structural spring, whichalso may remain linear or become nonlinear as a result of the damage.For the numerical exercise studied here, the model of a nonlinear space antenna struc-

48

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1

1.5

2F6 = 1000, alpha = 0.1 and delt = 0.01

time (sec)

m6

disp

lace

men

t (x6

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2

0

2

4

6

time (sec)

x63

Figure 4.8: Continuity of Initial Convergent Case in Table 4.4ture has been identi�ed with quasilinearization parameter estimation. By its formulation, thequasilinearization approach yields a solution that, when it converges, is extendable to the problemof locating and assessing, or estimating, new damage to the structure.For a set of known, non faulty dynamic response observations, quasilinearization gen-erates an iterative linearized solution to the nonlinear di�erential equations that describe thestructural system dynamics. The estimated coe�cients in the model generate a least-squares �tbetween the measured dynamic responses, e.g. the mass position measurements, and the recur-sively generated responses. The relationship between the measured and generated responses isgiven by x(t) = x(n)(t; ) + e(t) = P (n)(t) + NaXi=1 i H(n)Nx+i(t) + e(t); (4.33)where x(t) are the observed responses, x(n)(t) are the nth iterative responses parameterizedby the vector , P (n)(t) and H(n)(t) are the associated recursive particular and homogeneous

49

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1

1.5

2F6 = 1000, alpha = 0.1 and delt = 0.03

time (sec)

m6

disp

lace

men

t (x6

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2

0

2

4

6

time (sec)

x63

Figure 4.9: Continuity of Less Convergent Case in Table 4.4equations, and e(t) represents the error between the observed and generated signals. Whenconvergence of the parameters is achieved, the L2 norm of e(t) is minimized over the measureddynamic responses.In health monitoring of a system for fault detection, a residual serves as a detectiongauge as it carries information that is sensitive in some way to the occurrence of a fault. Thesignature of the residual is the quantitative or qualitative behavior that the residual displays inthe event of a fault. In structural systems where quasilinearization is employed for parametricidenti�cation, a residual r(t) in component form is given byrj(t) = x(n)j (t)� x̂j(t); (4.34)where x̂j(t) is the currently observed dynamic response, e.g. a mass position measurement, ofthe structure under certain loading conditions and x(n)j (t) is the \modeled" response that isgenerated under the same loading conditions and using the converged and known parameters in

50

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−1

−0.5

0

0.5

1

1.5

2F6 = 1000, alpha = 0.1 and delt = 0.04

time (sec)

m6

disp

lace

men

t (x6

)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−2

0

2

4

6

time (sec)

x63

Figure 4.10: Continuity of Non Convergent Case in Table 4.4the model. In the absence of a structural fault, i.e. when none of the sti�ness values have beencompromised, rj(t) = ej(t) and accordingly remains small. When a fault occurs, rj(t) becomes(generally) closer in magnitude to the order of the measured or generated responses.This qualitative description can be seen quantitatively as it applies to the nonlinearspace antenna structure. For the loading, nonlinearity and integration conditions of case 1in Table 4.1, a softening fault in k4 from 100 to 50 is induced prior to any observations andthe generated residuals are plotted in Figure 4.13. A threshold of � 0.001 is put on all of theresiduals as the signature value and the fault is detected within 0.02 sec. The previously estimatedparameters then serve as the initial guess for a new set of iterations. In this case, the parametersconverged to the new true values (50,1000,100) within 2 iterations. This exercise is essentially arepetition of the assessment procedure already examined in the previous sections.Figure 4.14 shows the generated residuals for the same conditions except that the soft-ening fault at k4 is induced at t = 5 sec for 10 sec of observed data. Naturally, di�culty arises as

51

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1f6 = 1000, alpha = 0.1, delt = 0.03

x4

time (sec)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

x6

time (sec)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3−0.5

0

0.5

1

x4

time (sec)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3−0.5

0

0.5

1

1.5

2

x6

time (sec)Figure 4.11: Higher Frequency for Non Convergent Step Input Case of Table 4.5

52

0 1 2 3 4 5 6 7 8 9 10−1

−0.5

0

0.5

1f6 = 1000, alpha = 0.1, delt = 0.03

x4

time (sec)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

2

x6

time (sec)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3−0.5

0

0.5

1

x4

time (sec)

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3−0.5

0

0.5

1

1.5

2

x6

time (sec)Figure 4.12: Lower Frequency for Convergent Sinusoidal Input Case of Table 4.5

53

0 1 2 3 4 5 6 7 8 9 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Residual measurements for fault monitoring ( rj(t) = x

j(n)(t) − x

j(t) )

mea

sure

men

t res

idua

ls (

rj(t

) )

time (sec)Figure 4.13: Residual Measurements for Conditions of Case 1 of Table 4.1 and Fault in k4 Overthe Entire Observed Responses (x̂1(t); :::; x̂8(t)).the entire observation data batch can't simply be used for estimation. Using the entire observedresponses for this example illustrates this, as the converged parameter values are (87.92, 960.9,421.5) after 10 iterations. Clearly, when damage occurs within a set of data measurements, onlythe portion of the responses that contain the damaged information can be used for successfullocation and estimation of the fault.First, consider that the fault location is restricted so that damage a�ects only thoseparameters that were previously estimated. Therefore, the equations that de�ne the algorithmremain unchanged. The problem then becomes one of detection and assessment. The location intime of the occurrence of the fault is required in this case and is simple to estimate by monitoringthe residuals de�ned by Equation 4.34.Several approaches are suggested to accommodate this situation when observations that

54

0 1 2 3 4 5 6 7 8 9 10−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Residual measurements for fault monitoring ( rj(t) = x

j(n)(t) − x

j(t) )

mea

sure

men

t res

idua

ls (

rj(t

) )

time(sec)Figure 4.14: Residual Measurements for Conditions of Case 1 of Table 4.1 and Fault in k4 att = 5 sec in the Observed Responses.contain non damaged and damaged information. An appropriate weighting, e.g. exponential,can magnify the information containing the post damage responses in the iterative estimationsand therefore the �t parameters will gravitate more towards their faulty values.Another approach would be to use only the data that was measured after the occur-rence of the fault. Di�culty arises here as the particular part of the recursive equations requiresthe initial conditions of the state variables and would therefore require the position and velocitymeasurements at the beginning of the post-fault measurements, i.e. just after the fault. Al-though the 8 position values just after the fault could be extracted from the measurements, the8 velocity values are most likely not available in real experimental situations. In this case, therecursive equations and linear algebraic equations are reformulated for the case of unknown 8initial conditions of the velocity. This greatly increases the computational load but eliminates

55the generation of weighting functions and only requires extension of the already proven quasilin-earization approach.More importantly, this unknown initial condition approach could accommodate multiplehardening and/or softening faults in time and be applied recursively to the data to estimate thecorresponding changes due to damage in the previously estimated linear and nonlinear structuralsprings. Now consider a more di�cult and likely situation. When softening or hardening damageoccurs in a linear spring whose sti�ness was not previously estimated but known, or when damageresults in more added nonlinear terms of known form, e.g. an added cubic sti�ness term, thequasilinearization approach is still successful but requires rede�ning the iterative linear di�erentialand algebraic equations. When the potential location of damage is known, this reduces thecomputation requirements as the algorithm is structured to estimate these unknowns. With noknowledge of the location of potential damage, a comparative process where sets of parametersare assumed unknown can be performed. The estimated sets are compared to their previouslyknown values for discrepancies. This would serve as alternative to assuming all parametersare possibly fault sensitive, which again is likely too computationally cumbersome and is likelycompromised by mistakes in the derivation.As a tool for identifying a limited number of unknown parameters in damaged andtherefore nonlinear higher order structural systems, given the model structure and post damagedynamic response observations, the quasilinearization approach of parameter estimation is re-liable. The approach also shows promise in the detection, location and assessment of multiplestructural faults in such a model.

Chapter 5Detection and Isolation of a DryFriction Fault in a PneumaticallyActuated Air Bearing Mass5.1 MotivationThe detection of dry, or Coulomb, friction in pneumatic linear actuators is pertinentin the increasing number of automated systems that require precision positioning. The dryfriction phenomena can interfere with the precision positioning objectives and cause problems likeovershooting and force limit-cycling. Although friction can be compensated by modern controlalgorithms, sudden and unpredictable changes in friction due to, for example, wear and sideloading of a pneumatic cylinder cause unacceptable behavior of the positioning mechanism. Anon-line detection of dry friction (fault) would greatly facilitate the compensation of dry frictionin high precision positioning. Moreover, a fault detection technique for monitoring dry frictionwould help the detection of changing process conditions in the case of, for example, wear andexcessive side loading of a pneumatic cylinder. In this chapter, an experiment is designed inwhich a servo pneumatic cylinder drives an air bearing mass load that permits the addition offriction on-line. By design, the addition of friction in the experimental apparatus is equivalentto an increase in the piston dry friction. It therefore serves to design a simple and e�ective dryfriction fault detection scheme for this system. Furthermore, a fault detection of dry friction isdeveloped that is able to monitor Coulomb friction by simply adding an acceleration sensor tothe load of the pneumatic cylinder. Dynamic models of the behavior of the load and pneumatic

56

57cylinder are obtained by system identi�cation techniques and a recursive least-squares estimationtechnique is used to monitor dry friction model parameters. Numerical validation of the FDIscheme is also provided.5.2 The Pneumatic Actuation of an Air Bearing MassA schematic of the experimental precision positioning apparatus discussed in this chap-ter is given in Figure 5.1. Table 5.1 displays the properties and dimensions of the componentsand hardware in the apparatus. The voltage supplied to the apper servo-valve Vin directs the

chamber 1 chamber 2

Flapper Servo-valve

Pneumatic Actuation of Air Bearing Mass Assembly

m

L

Pu1 Pu2Pe

flapper

Ps

Ffpiston

Air Bearing Masschamber ports

air exhaust

+Vin

x, x..

rod

Pneumatic Cylinder P1 P2

Ps

Figure 5.1: Experimental Apparatus of an Air Bearing Mass with Pneumatic Actuation. apper and hence the ow of air to the two piston chambers. When the apper creates a pressuredi�erence in the chambers, the piston and rod are forced into motion and in turn drive the airbearing mass.The voltage to the valve directly regulates the up stream pressures, Pu1 and Pu2. Bydesign the valve is always exhausting some of the ow to the atmosphere, which remains atatmospheric or exhaust pressure Pe = 101 kPa. The up stream pressures from the chambers, Pu1and Pu2, are dynamically related to the chamber pressures P1 and P2 due to the ow throughthe chamber port ori�ce, where the cross-sectional area is small compared to the chamber andtubing areas.The air bearing mass is designed to permit the addition of dry friction on-line. Themass is two hollowed out sections of aluminum block bolted together with a supply hose port in

58Table 5.1: Dimensions and Properties of Components and HardwareDyval Single Stage Pneumatic ValveModel 1-SPOperating Pressure 80 - 160 psiTemp. Range 40oF - 160oFSystem Filtration Coalescing + 25 micron particulateMaximum Input Voltage � 5 Volts (for parallel connection)Pressure Recovery 80%Phase Lag < 90o at 200 Hz, blocked portHose Output Ports 1/4 inch Quick connectDimensions 2 inch � 2.165 inch base, 2.22 inch heightTri-Star Arrow Coalescing FilterModel F55Maximum Pressure 250 psiTemp. Range 40oF - 200oFKistler Piezo Instrumentation Accelerometer and Charge Ampli�erModel 8632B50Range � 50 g (g = 9:807 m=sec2)Sensitivity at 100 Hz 3 g rms, 101.4 mV/gMounted Resonant Freq. 22.0 kHzMacro Sensors DC 750 Series DC-Operated LVDTModel DC 750-1000Input Voltage � 15 Volts DCFull Scale Output � 10 Volts DCMeasurement Range � 1.000 inchScale Factor 10 Volts DC/inchLength 8.24 inch body, 3.45 inch coreOmega PX236 Series Pressure TransducerExcitation 10 Vdc regulated, 16 Vdc maxNull O�set � 2 mVSensitivity 1 mV/psiAirpel Anit-Stiction Air CylinderModel E9, double actingBore Size 0.366 inchOperating Pressure 0.2 - 100 psiStroke 5.0 inchMass piston/rod 11.3 gramscomplete unit 79.3 gramsPressure Ports 10-32 connectionsRod Diameter 0.125 inchMount Diameter 0.375 inchOutter Diameter 0.562 inchMounting Surface (Aluminum)Dimensions 21.9 inch length, 6 inch width0.5 inch thicknessSlot Dimensions 8 inch length, 0.375 inch widthAir Bearing Mass (Aluminum)Dimensions given in Figure 5.2, Figure 5.3Mass 220.1 grams

59the top section and 36 small exhaust holes in the bottom section. The holes were made near thecorners of the bottom section of the block to promote stability of the air bearing mass when inmotion. Engineering drawings of the two sections of the air baring mass are given in Figure 5.2

Figure 5.2: Engineering Drawing of the Top Section of the Air Bearing Mass.and Figure 5.3. The supply pressure Ps to the mass controls the exhaust ow from the bottomholes and therefore the amount of dry friction contact between the mass and the surface thatthe mass rides on. At a critical Ps value the mass makes contact with the surface. By reducingPs below the critical value, the amount of dry friction increases and reaches a maximum valuewhen Ps is cut-o�.A wide frontal view of the experimental precision positioning set up is shown in Figure5.4. This image shows the pre-valve air �ltration system on the left, the Dyval pneumatic servo-valve, the Airpel anti-stiction air cylinder, the constructed air bearing mass apparatus and signalmeasuring and monitoring devices. Note that the piston rod alignment is not perfectly parallel

60

Figure 5.3: Engineering Drawing of the Bottom Section of the Air Bearing Mass.with the plane surface on which the mass rides. There exists, therefore, a preferred range ofmotion of the mass at which any side-loading of the piston rod is minimized. Binding of themass's leading edge on the piston mount side is observed for 0.5 inches of motion nearest themount. At the far end of the stroke, the piston aimed high and created downward side loading.It is also desirable to avoid bottoming out the piston, i.e. making contact between the cylinder'send and piston while in use. Therefore, the preferred range of motion is approximately 0.75 to 1inch from either end of the stroke. All of the testing is done in this preferred range to investigatethe a�ects of variable air supply and input while minimizing the afore mentioned side loadinga�ects. A close up of the assembly is in Figure 5.5, showing from the left the valve, cylinder andmass more clearly. The pressure sensors located just downstream from the valve ow outportsare used to compute the servo voltage to up stream pressure frequency response.

61

Figure 5.4: Wide Frontal View of Experimental Precision Positioning Set Up.

Figure 5.5: Close Up of Frontal View of Experimental Precision Positioning Set Up.A top close up view of the air bearing mass is shown in Figure 5.6. This �gure containsa clear view of the supply air hose to the air bearing mass and the rod of the mounted pneumaticcylinder, which threads directly into the mass. The measured signals of interest for identi�cationand detection besides the voltage input Vin to the servo-valve are the position x and acceleration

62

Figure 5.6: Top Close Up View of Air Bearing Mass.�x of the air bearing mass. Mounted on the mass are an accelerometer on the top for measuring�x and a linear variable di�erential transformer (LVDT) opposite the cylinder rod for measuringx. The mass rides above a leveled smooth aluminum surface with an exhaust channel cut in thedirection of motion to escape the air from the mass. It is observed that escaping the air owthrough this channel enables a smoother transition to friction. As the air takes the channeledpath of least resistance, it is not compressed as much by the weight of the mass, resulting inrelatively even and increasing contact as the supply pressure is reduced. The height of thecylinder is set to thread the rod into the mass when the mass is exhausting enough air to makeno contact with the level surface.5.3 Procedure for Dry Friction Fault DetectionIn order to model the dynamic behavior of the valve, pneumatic cylinder and air bearingmass load, system identi�cation techniques are used. Collecting frequency domain observationsfrom the valve input Vin to the position x and acceleration �x of the air bearing load, linearmodels of the dynamic behavior are estimated via black box modeling identi�cation techniques.The models are used to simulate the (ideal) response of the precision positioning apparatus inthe case of no-friction. Further, these models are used for the design and development of �ltersand a recursive estimation approach to monitor and detect dry friction changes. The key stepsare:

63� Investigate an analytically derived dynamic model of the pneumatic cylinder and the airbearing mass.� Experimentally obtain a dynamic model from servo-valve voltage input to acceleration ofthe air bearing mass.� Explore the dynamic relationship between the measured and modeled acceleration of themass and a model of the dry friction force.� Implement on-line estimation of the friction model parameters for monitoring and frictionfault detection.In Section 5.4 the analytic model of the system is brie y derived to reveal the empirical nonlin-earities and degree of the system. Frequency response measurements of the servo valve voltageinput to the position and acceleration of the mass, under no added friction conditions, are givenand discussed. Also, a discrete fourth-order model (transfer function) is �t to the frequency re-sponse data to simulate acceleration of the air bearing without friction for a given servo voltageinput. Section 5.5 details how friction detection is possible by monitoring acceleration. Specif-ically, it is shown in Section 5.5.2 that the friction force is dynamically related to the di�erencebetween the measured and simulated (non friction) accelerations, called the acceleration residual.In Section 5.6 an experimentally based model between friction and acceleration is designed andimplemented on-line. In Section 5.6.1 a simple viscous and coulomb friction model is employedto facilitate identi�cation and detection of the friction in terms of the model coe�cients. A pre-diction error approach [4] is used to model the dynamic relationship between the friction forceand the acceleration residual, using an output error optimization in Section 5.6.2.The assumed friction model is incorporated with this �lter in Section 5.6.3 and theprocedure is implemented by recursive least-squares estimation of the friction model parametersin Section 5.6.4. Simple threshold decision logic is employed to detect the dry friction fault on-line. Results of the scheme are given in Section 5.7. First, the �ltering and parameter estimationdetection procedure is applied as a stand alone scheme in a numerical analysis. In Section 5.7.1,the scheme is applied to a second order model generated in Matlab Simulink [42]. In Section5.7.2, experimental results are given for a wide range of voltage inputs and for varying conditionsof added dry friction are given in Section 5.7. Conclusions are discussed in Section 5.8.

645.4 Dynamic Modeling of the Pneumatically Actuated AirBearing MassA fourth-order highly nonlinear state-space model of a rodless pneumatic actuatordriven by a four-way servo-valve is derived by Drakunov et al. in [43]. This analytic modelis modi�ed to include the actuation rod and extended to the apper servo-valve and air bearingmass apparatus employed in this experiment, as shown in Figure 5.1. A discrete fourth ordermodel is �t to the transfer function from servo voltage input to mass acceleration output byweighted least-squares.5.4.1 Analytic ModelThe model developed by Drakunov evolves from the landmark text in modeling valvecontrolled pneumatic systems by Blackburn et al. [44] and is consistent in other sources [45], [23],[25], [46]. Assumptions in the pneumatic dynamics include an ideal equation of state (ideal gasequation), isentropic valve and chamber port ori�ce ow and isentropic thermodynamic controlvolume, or chamber, behavior. Air leakage between the two chambers and from the chamberto the atmosphere is assumed and included in the air bearing dynamics as a non dimensionalcoe�cient � that e�ectively reduces the cross-sectional chamber areas. The assumption in the airbearing dynamics is that the piston, rod and mass combined are one rigid body and the contactfriction between the piston and rod and cylinder casing is negligible. Combining the continuityequation, isentropic thermodynamic control volume relationships and a piecewise isentropic ori�ce ow function, the actuator state-space model is given bydP1dt = V10 +A1x h�P1A1 _x+pRT1A�2f2(�1)i (5.1)dP2dt = V20 �A2x hP2A2 _x+pRT2A�2f2(�2)i (5.2)and the air bearing mass state-space model is given bydxdt = _x (5.3)d _xdt = A1em P1g � A2em P2g � 1mFf (5.4)where the states are de�ned asP1 Absolute piston pressure in chamber 1P2 Absolute piston pressure in chamber 2x Position of the piston rod / air bearing_x Velocity of the piston rod / air bearing

65and the functions in the models are de�ned asf2(�i) = 8<: �i T�1=2iu Piu Cd f1�Pi; Piu� when �i > 0�i T�1=2i Pi Cd f1�Piu; Pi� when �i � 0 (5.5)�i = sgn(Piu � Pi) (5.6)Ti = Ti0� PiPi0 �( �1)= (5.7)f1�PdPu� = 8><>: �1=�2��Pd=Pu�2= � �Pd=Pu�( +1)= �1=2 when Pd > Pcr1 when Pd � Pcr (5.8)�1 = � 2 =( � 1)�1=2; �2 = � � 2=( + 1) �( +1)=( �1)�1=2 (5.9)Pcr = Pu rcrit = Pu� 2=( + 1)� =( �1) (5.10)Aie � Ai (1� �i) ; Pig � Pi � Pe: (5.11)The coe�cients and variables are de�ned asA Cross-sectional area of chamber portsAi Cross-sectional area of chamber iAie E�ective cross-sectional area of chamber port iCd Chamber port discharge coe�cient (Cd = 0:85)Ff Dry friction force (positive to the left)f1 Compressible ori�ce ow functioni Control volume or chamber index(i = 1; 2)m Combined mass of piston rod and air bearingPd Absolute downstream pressurePi0 Absolute initial pressure of chamber iPiu Absolute up stream pressure of chamber iPe Absolute exhaust and atmospheric pressure (Pe = 101 kPa)rcrit Critical pressure ratioR Ideal gas constantL Piston stroke lengthTi Absolute temperature of chamber iTi0 Absolute initial temperature of chamber iTiu Absolute up stream temperature of chamber iVi0 Initial Volume of chamber i, including dead air Ratio of speci�c heats( = 1:4 for air)�1; �2 Compressible ori�ce ow constants�i Leakage coe�cient of chamber i

66Over the pressure ranges and temperatures of interest for air, it can be assumed (see[25, page 736]) that the temperature of the two chamber remains constant and equal to thesupply temperature (T = 294 K). Applying Fleigener's equation, as in [45],[46], the ow functionis simpli�ed by assuming subsonic ow in both directions at the chamber ports (Pd > Pcr) andthe state equations become dP1dt = � P1x _x+ C1 f(Pu1; P1)x (5.12)dP2dt = P2L� x _x+ C2 f(Pu2; P2)L� x (5.13)where the ow function is de�ned asf(Pui; Pi) = �1Pu �PdPu �1� PdPu� �1=2 (5.14)Pu = 8<: Pi and Pd = Pui; if Pui � PiPui and Pd = Pi; if Pui > Pi (5.15)and the coe�cients are de�ned asCi = 2 �2Cd AAipRT; i = 1; 2:Equation 5.12 and Equation 5.13 reveal the complex dependence of the chamber pressures onpiston rod / mass position and velocity and up stream pressure even after assuming the mostsimpli�ed conditions. The high degree of nonlinearity in this fourth-order system indicates thedi�culty in attempting to identify all of the model parameters with a su�cient level of certainty.Attempting to analytically isolate the friction force from the other dynamics in the system wouldthen include signi�cant parametric uncertainty. However, the identi�cation and detection offriction as a fault requires an isolation of the friction signal that is robust to these large parametricuncertainties.5.4.2 Experimental ModelSystem identi�cation is used to experimentally model the dynamics from the valve inputVin to the position x and acceleration �x of the air bearing mass. A linear dynamic time invariantmodel of the system is found by �tting the frequency response from the voltage input to measuredposition. The position measurement is used as it is more reliable in the low frequency range thanpiezo accelerometer measurements. Also the use of the position signal comes at no extra costsince position sensing devices commonly accompany high precision positioning mechanisms forfeedback control purposes.

67Experimental data acquisitionAn LVDT [47] is used to measure the position of the air bearing load. The LVDTcan measure accurately to 0.001 inches and by design is virtually frictionless, as the positionsensing core is magnetically suspended in a cylindrical housing. The frequency response of the

10−2

10−1

100

101

102

10−4

10−2

100

102

log

mag

frequency (Hz)

LVDT / Vin

frequency response for Vin

= 0.7 Volts amplitude chirp signal

10−2

10−1

100

101

102

−200

−100

0

100

200

phas

e (d

eg)

frequency (Hz)Figure 5.7: Amplitude and Phase Bode Plot of Voltage Input to Air Bearing Load PositionFrequency Response G(!)voltage input to the load mass position output was generated using a chirp voltage input signalof amplitude 0.7 Volts, a frequency range up to 20 Hz and a record length or frame size of 1024samples using DSP signal processing software [48]. Figure 5.7 shows a plot of this frequencyresponse (dashed), de�ned as G(!). By inspection, this frequency response displays fourth-orderdynamics, with a single integrator near 0.2 Hz and three orders of integration around 8-9 Hz.The second derivative of G(!), i.e. [j!]2 G(!), is computed to relate voltage input toacceleration of the load and is shown in Figure 5.8. To validate that the accelerometer mea-surements and the second derivative of the LVDT measurements are consistent in the frequency

68

10−2

10−1

100

101

102

10−2

10−1

100

101

102

log

mag

frequency (Hz)

2nd derivative of LVDT / Vin

frequency response for Vin

= 0.7 Volts amplitude chirp signal

10−2

10−1

100

101

102

−200

−100

0

100

200

phas

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frequency (Hz)Figure 5.9: Frequency Response Match Between Measured Voltage to Acceleration and the Sec-ond Derivative of G(!) for a Chirp Input of Amplitude One Volts.chamber pressures, remain essentially constant. Therefore, for voltage inputs that exceed 10 Hzvery little motion in the pneumatic cylinder piston and the air bearing load is observed. Below10 Hz, the upstream pressures are directly proportional to the voltage input.To examine how the pneumatic cylinder and air bearing load system changes for di�erentinputs, Figure 5.11 and Figure 5.12 show the frequency responses measured by the LVDT andthe accelerometer for chirp input signals of varying amplitude. Although only the variations inacceleration measurements are of concern for the detection scheme, from a hardware redundancystand point it is good to check that the accelerometer measurements are consistent with theLVDT measurements under these variable input conditions. For the LVDT measurements, themaximum input voltage of 1.5 V is based on the maximum range of the LVDT of � 1 inch. Theminimum voltage of 0.7 V is the smallest voltage observed to yield a large enough response of

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frequency (Hz)Figure 5.10: Voltage Input to Up Stream Pressure Frequency Responsethe air bearing to be able to detect friction. The same voltage amplitude range is applied for theaccelerometer measurements, except that a higher voltage was used as a maximum.These �gures reveal the highly nonlinear dynamics present in the system. The mea-surements are insensitive to variations in amplitude of a chirp input for a frequency range of 1to 3 Hz and increasingly sensitive from 4 Hz up to the resonant 8-9 Hz.Description of model �tting to frequency responseThe parametric discrete time model is curve �tted by weighted least-squares to thefrequency response of the voltage input to the mass position output. The �t by estimation ofthe model parameters is done in the frequency domain model identi�cation program FREQID[49], [50]. The interface permits selection of the weighting function and model order during theestimation of a parametric model.The selected order of the discrete time �tted transfer function model is [4; 1; 1], to �t

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frequency (Hz)Figure 5.11: Change in Frequency Response of Voltage to Position for Chirp Signal Input ofVarying Amplitude.to the fourth-order frequency response in Figure 5.7. The model takes the formy(t) = bG(z�1)u(t) = B(z�1)A(z�1)u(t) = b1z�11 + a1z�1 + � � �+ a4z�4u(t); (5.16)where y(t) = x̂(t); u(t) = Vin(t); (5.17)x̂(t) is the (non friction) modeled load position and b0 is set to 0. By setting b0 to 0, the numberof parameters used to �t the model to the frequency response is reduced, which in this caseresulted in a better �t than a model order of [4; 2; 0] that includes b0 as a free parameter. Thetransfer function bG can be written bG(z�1) = bG �ej!T ;�� ; (5.18)where � is the parameterization of the transfer function (b1; :::; a4) and T is the discrete sample

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frequency (Hz)Figure 5.12: Change in Frequency Response of Voltage to Acceleration for Chirp Signal Input ofVarying Amplitude.time. The weighted least-squares �t is generated from the cost functionmin� h bG �ej!T ;���G(!)iW (!) 2; (5.19)where W (!) is the weighting function.The transfer function bG(z�1) is generated using a sample frequency fs = 256 Hz, ora sample time ts = 0:00390625 sec. Furthermore, all of the measured voltage and accelerationsignals and modeled acceleration signals captured for detection are also sampled at this frequency.Note that fs captures all of the dynamics of the system, as Figure 5.9 shows a magnitude roll-o�well below the half-sampling frequency of 128 Hz.The �tted transfer function model from the voltage input to the no friction load positionis shown in a solid line on top of the frequency response in Figure 5.13. The order of the zerosin the model, i.e. the order of B(z�1), was kept to two so that in taking the second derivative

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frequency (Hz)Figure 5.13: Amplitude and Phase Bode Plot of Voltage Input to Air Bearing Load PositionFrequency Response G(!) (dashed) and fourth-order model �tted on the data bG(z�1) (solid)of this model, the resulting transfer function from voltage input to non friction air bearing loadacceleration is causal (proper). This transfer function is shown in Figure 5.14 on top of thefrequency response given in Figure 5.8.For implementation in the detection scheme, the transfer function bG(z�1) is convertedto a discrete state-space model and entered as a block in Matlab Simulink [42] to generate x̂(t)for Vin signals. Further, two numerical (�nite-di�erence) derivative blocks are used to generate�̂x(t). By �nite-di�erence, the modeled signal delays the measured acceleration by two sampletime steps. To account for the delay, the modeled signal is shifted forward in time by two sampletime steps.While a linear model simply shifts up for an increase in voltage input amplitude, thepneumatic cylinder and air bearing load apparatus behave nonlinearly for such increases, asshown in Figure 5.11 and Figure 5.12. To match the amplitudes of the non friction measured

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frequency (Hz)Figure 5.14: Amplitude and Phase Bode Plot of the Second Derivative of G(!) (dashed) andbG(z�1) (solid)and modeled accelerations, a scaling is built into to the detection scheme. This scaling accountsfor the conversion factor ratio and the a�ect of input amplitude variations. The scaling takesplace in the time domain and after a transient mismatch, do to non matching initial conditions.A simple linear model �tted to the highly nonlinear dynamics of the pneumatic cylinderand air bearing mass load apparatus is implemented to generate a non friction accelerationsignal for a valve voltage input. Robustness in matching the measured and modeled non frictionaccelerations with respect to varying the voltage input amplitude is incorporated as a scaling inthe detection scheme. The model and scaling provide a means of comparison of fault free andfault sensitive measurements (accelerations), a prerequisite for detection of a friction fault in theexperimental precision positioning apparatus, and has the added advantage of being somewhatrobust with respect to varying inputs (voltages).

755.5 Friction Detection by Monitoring Acceleration5.5.1 MotivationAs was stated, detection of dry friction in a servo pneumatic application is desirable.In this detection scheme, only the voltage input and measured and modeled accelerations ofthe air bearing mass are required to design the dynamic �lter and monitor the nonlinear dryfriction force on-line for fault detection. The advantage of requiring the measurement of onlyacceleration is that inexpensive, reliable accelerometers are available and easily added to the loadof a precision positioning apparatus.5.5.2 Dynamic Relation Between Friction Force and Measured Accel-erationThe non friction acceleration signal �̂x(t) provides the non fault information with whichthe fault sensitive measurements, i.e. the measured acceleration �x(t), may be compared. Asfriction force is proportional to acceleration, it is intuitive that the di�erence in these signalswould capture a friction fault in some way if it occurs. The acceleration residual is de�ned as�xr , �x� �̂x: (5.20)This signal alone can capture any unmodeled faults such as dry friction. However, to achieve thefault detection and isolation objective, this signal must somehow be reduced to a signal or signalsthat are sensitive to friction faults and relatively robust to other possible sensor, actuation orprocess faults. To relate the friction force to this residual, a qualitative description about thebehavior of the air bearing dynamics with and without friction is helpful. The air bearingmass is driven by the piston of a pneumatic cylinder that has sti�ness and damping properties.As discussed, the cylinder dynamics including the sti�ness and damping properties are highlynonlinear with and without the addition of friction. In this qualitative discussion a simple linearforced mass-spring-damper system is investigated to observe that there exists a dynamic relationbetween an added friction force and an accleration residual in systems that have sti�ness anddamping properties. The linear system is given bym �x1 = F � k x1 � c _x1; (5.21)where x1 represents the position, _x1 the velocity and �x1 the acceleration of a mass m. Clearly,by no means does this model represent the pneumatic cylinder and mass load dynamics with orwithout friction, but it does provide useful insight into a friction to acceleration residual relation

76in the pneumatic cylinder experiement. When a dry friction force is present, the dynamics of thesystem in Equation 5.21 change tom �x2 = F � k x2 � c _x2 � Ff ; (5.22)where x2 represents the position, _x2 the velocity and �x2 the acceleration of the mass subjectedto the added dry friction force Ff . Subtracting Equation 5.22 from Equation 5.21 yieldsm �xe + c _xe + k xe = �Ff ; (5.23)where �xe = �x1 � �x2, and so on. The friction force Ff is positive when acting to the left and �xeis e�ectively an acceleration residual in Equation 5.23. Therefore, Equation 5.23 shows that adynamic relation exists between the dry friction force Ff and the acceleration residual �xe. Notethat the existence of this dynamic relation holds even if the assumption that the sti�ness anddamping coe�cients do not change with the added friction force is invalid. With this insight,it is left to model the dynamic relation between a measured friction force and the acclerationresidual in the experimental test.In Section 5.6.2 this dynamic relation is generated as a �lter in the highly nonlinearpneumatic cylinder and mass load dynamics with friction. This gives the freedom to select afriction model based upon the measured states of the air bearing motion, i.e. the accelerationmeasurements or computed velocity or displacement values from this measurement. For thefriction model employed here, only velocity measurements are needed. A third-order high passdigital Butterworth �lter is used to eliminate any DC content in the measured acceleration signaland a discrete time integrator is used in Matlab Simulink [42] to compute the measured velocity.5.6 Experimentally Based Modeling Approach>From the analysis performed in Section 5.5 it is clear that a dynamic relationship existsbetween the acceleration residual �xr and the dry friction force Ff . This analysis is valid onlyfor a single mass-spring-damper system as discussed in Equation 5.21 - Equation 5.23. In morecomplicated precision positioning systems, such as the experimental apparatus of Section 5.2,the relation between friction and �xr becomes more complicated.In order to discuss the more complicated relation, �rst we analyze a friction model inSection 5.6.1. The dynamic model between friction and �xr is then discussed in Section 5.6.2.5.6.1 Assume a Friction ModelFor a dynamic system undergoing slipping dry friction, i.e. dry friction that has acomparatively low sticking and high slipping characteristic, the coulomb and viscous friction

77model has been validated in [30], [29], [41], with experimental validation in [36]. The frictionmodel takes the form Ff = � _xj _xj + � _x; (5.24)where the friction force �Ff and velocity _x are assumed positive to the right. The high slip lowstick condition is achievable in this experimental precision positioning apparatus and it is alsopossible to add friction of high stick character. The success of the detection scheme under bothof these conditions is examined in Section 5.6.4. The small amount of sticking present in thehigh slip cases will be modeled within the estimated dynamic �lter. The chief advantage of thismodel is that it is linear in the parameters � and � and therefore facilitates identi�cation. Byrecursively estimating these parameters with least-squares the friction fault level in the precisionpositioner is isolated and monitored for detection.5.6.2 Identi�cation of Dynamic Friction Signal FilterThe dynamic model, or �lter, between the friction force and the acceleration residual iswritten as an output error model, given asy(t) = �B(q�1)=F (q�1)�u(t� nk) + e(t); where (5.25)y(t) = �xr(t); u(t) = Ff (t); (5.26)[ nb; nf; nk ] = � order of B(q�1); order of F (q�1); model delay order � ; (5.27)and e(t) represents the error between the acceleration residual and the �ltered friction signal.The prediction error estimate of Equation 5.25 �nds the parameters in B(q�1) and F (q�1) thatminimize the L2 norm of e(t), given y(t) and u(t). With these parameters, the dynamic �ltersimulates the acceleration residual for a given friction signal. Since friction lags the accelerationresidual, the order of B(q�1) is higher than that of F (q�1). For this problem, the orders selectedare [nb, nf , nk] = [4, 3, 0].5.6.3 Incorporation of Friction Model with Model Based FilterThe friction fault signature is de�ned here by incorporating the friction model with thedynamic �lter. Substituting Equation 5.24 into Equation 5.25,�xr(t) = eGe(q�1) �� _xj _xj + � _x�+ e(t): (5.28)Normalizing the lead coe�cient in the friction model,�xr(t) = Ge(q�1) � _xj _xj + f _x�+ e(t) = Ge(q�1) xf (t) + e(t): (5.29)

78where xf (t) represents the friction signal after normalization of the coulomb term coe�cient andthe constant f represents the relative amount of viscous to coulomb friction when the frictionfault is present. It remains to estimate f and Ge in the presence of a friction fault. A windowof data in time, comprised of faulty velocity and acceleration measurements and a modeledacceleration signal, de�nes the signals in Equation 5.29. Let the data window time interval bede�ned as t � [t1; t2]. The parameter f is iterated through the values [0:0; 10:0] in steps of 0:2and for each value, the parameters in Ge are estimated by least-squares with the cost functione(t), i.e. the estimated parameters minimize �xr(t)�Ge(q�1) xf (t) L2(t1;t2): (5.30)The set of Ge parameters and corresponding value of f that result in the smallest L2(t1; t2) normof the error e(t) de�ne, respectively, the dynamic �lter and the relative level of viscous frictionfor the given window of data. Clearly then, for each servo voltage input and correspondingmeasured and modeled signals in which a friction fault occurs, a dynamic �lter and associatedlevel of viscous friction f are generated. For this set of data, the �lter parameters are �xedand the normalized friction model coe�cients (1; f ) serve as the signature values to which therecursively estimated friction model parameters are compared.5.6.4 ImplementationIn this section a procedure is formulated for recursively estimating the friction modelparameters and monitoring them for friction fault detection. Equation 5.29 can be rewritten inthe form �xr(t) = Ge(q�1) � _xj _xj + f _x�+ e(t) (5.31)= Ge(q�1) � _xj _xj�+ f Ge(q�1) [ _x] + e(t); (5.32)where Ge and f are de�ned. The normalized friction model coe�cients (1; f ) �t the �lteredmeasured velocity signals in Equation 5.32 to the acceleration residual in a least-squares sense.The �t is generated over a time domain window of data in which a friction fault is present. To�t �ltered measured velocity signals to the acceleration residual signal on-line, these parameterscan be estimated by least-squares using current windows of data, i.e. the parameters can berecursively estimated. Now the friction model coe�cients are non-constant and are expected tovary slightly within and largely in transition between the non friction and friction regimes. To

79account for time varying coe�cients, Equation 5.32 is modi�ed byy(t) = �1(t) g1(t) + �2(t) g2(t) + e(t); (5.33)where; y(t) = �xr(t); (5.34)g1(t) = Ge(q�1) � _xj _xj� ; (5.35)g2(t) = Ge(q�1) [ _x] ; (5.36)and �1(t) and �2(t) represent the time varying coulomb friction and viscous friction coe�cients,respectively. As was stated, these parameters ought to converge to their respective signaturevalues (1; f ) when the fault is present. These values indicate the relative level of Coulombicto viscous friction, respectively. It is important to note that prior to the fault, the parametersdo not have any physical interpretation. Rather, they match the small non friction accelerationresidual and the �ltered measured signals g1(t) and g2(t). Also, as the fault is added over asmall �nite window of time, the �t parameters in this transition are expected to display tran-sient behavior that again has no physical interpretation. The physical interpretation of the twoparameters remains solely in their signature levels, exhibited when the fault is present and aftersome transient behavior. To implement the recursive estimation, Equation 5.33 is rewritten asy(t) = g(t)T�(t) + e(t); where (5.37)g(t)T = [ g1(t); g2(t) ] ; �(t)T = [ �1(t); �2(t) ] : (5.38)For a current window of y(t) and g(t) data t � [ta; tb], the well known least-squares estimate �̂(tb)of the parameter vector �(tb) at window time tb is�̂(tb) = � g(t) g(t)T ��1 g(t) y(t); t � [ta; tb]: (5.39)Alternatively, an exponentially weighted forgetting factor could be used on signals y(t)and g(t) instead of using current windows of data. However, this slows the estimation processdown as time grows since all of the data from the beginning of the test is accounted for in exponen-tial weighting, and speed of estimation is critical for on-line implementation. The detection logiccontains a non friction parameter value threshold level and compares the estimated parametersto the signature values. Simply, a fault is detected once the estimates exceed the thresholds andthe parameters are monitored for tracking to the signature values. The parameters would likelyexit the pre-fault threshold levels in the event of di�erent types of faults. Successful tracking ofthe parameters to the signature values could therefore be considered as success in detection andisolation of the dry friction fault. The threshold levels are computed as the average plus andminus three times the standard deviation of an initial set of estimated parameter values �̂ thatare known to be friction free.

80Numerical validation of the friction fault isolation and detection scheme is �rst given. Asecond order model, or mass-spring-damper model if you prefer, is generated in Matlab Simulink[42]. A copy of the model is subjected to a friction fault half way through the time interval ofobservation. In this way the simulated acceleration signals and velocity signal (from the faultsensitive model) provide a means of validating the �ltering, estimation and detection componentsof the scheme developed in the previous sections.A few words are in order about some properties of the parameter estimates. A propertyof recursive estimation of variables that are subject to random processes such as noise is thatthey exhibit variance. The variance is a function of the data batch length, the interval betweendata batches, and naturally the noise present in the batches themselves. Although the �rst twoof these elements can be controlled in the detection scheme, the noise level cannot. Moreover, thesignal noise levels are highly sensitive to varying input types and levels of friction, particularlyin the experimental case. So, it is expected to observe di�erent levels of parametric variance inthe highly nonlinear system of the experiment for test cases that investigate varying input types,levels of friction, etc.A property of convergence of recursively estimated random variables is that the esti-mates remain bounded. Therefore, convergence of the parameters refers to boundedness. Track-ing performance of the parameters to the signature values refers to the variance level presentin their estimates, prior to and after the fault occurs. When a fault occurs, it immediately in-fects the pre fault data to which the parameters were �t. Once the data batch contains solelyfaulty information, the parameters generally track to their signatures. It is expected therefore toobserve transient behavior in the parameter estimates for a period after the fault occurs.It is important to note that the recursive estimation of the �lter parameters is a non-linear process, i.e. the estimation has its own nonlinear dynamics. Moreover, the parametersare mapped to data that is also nonlinear since it contains information about the friction, oncethe fault occurs. Therefore, estimates in the transient range are likely to behave spuriously, e.g.exhibit peaks in amplitude.5.7 Dry Friction Fault Detection ResultsIn Section 5.7.1, the �ltering, parameter estimation and threshold detection steps of theFDI scheme are validated in a numerical experiment. The application of the entire scheme tothe designed experiment is then detailed in Section 5.7.2.

815.7.1 Numerical ValidationA general linear second order model, that could represent for example a liner mass-spring-damper, was generated in Matlab Simulink [42]. The model was copied and the copiedmodel, under the same loading conditions, was subjected to three di�erent friction faults. Theadded faults were viscous friction, coulombic friction, and a combination of both. The results ofthese three cases are given as test cases 1, 2 and 3 respectively. The common time length of eachtest was 20 seconds and each fault was initiated at time equal to 10 seconds. To better representexperimental conditions, uniformly distributed random noise was added to the acceleration andvelocity signals generated by the fault sensitive model. The noise amplitude range was set at�2% of the peak value of the signals as measured before the fault.In the description of the parametric values related to the simulations, units are omittedfor convenience. The input, or forcing function, applied for each test was a sinusoid of amplitude3.0 and frequency 4 Hz. The nominal sti�ness k and damping c in the models was 1.0 and2.2, respectively. In the results that follow, the fault sensitive acceleration and velocity signalsare considered as \ measured" signals and are therefore plotted in dashed lines to follow theconvention of this thesis. Similarly, the fault fault free acceleration signal is considered to bea \modeled" signal and is therefore plotted in solid lines. Any generated signal, such as theparameter estimates, is also plotted in solid lines.For test case 1, an added viscous friction fault was simulated by increasing the nominalviscous friction coe�cient c from 2.2 to 9.2. A plot of the fault sensitive acceleration signal, i.e.the signal from the model subjected to the fault, is shown in Figure 5.15. The acceleration signalchanges little in this case as a result of the fault. To see the detection of the fault, Figure 5.16shows the evolution of the recursively estimated friction model parameters �1(t) and �2(t) forthis test case. The parameters remain near zero with low variance prior to the fault and escapetheir thresholds immediately after the fault is initiated. In this case, the coulombic parameter�1(t) naturally remains small and the viscous parameter �2(t) tracks to a signature value of 10 at1.6 seconds after the fault. The signature value of 10, de�ned in the dynamic �lter generation, isnot exactly the known viscous friction coe�cient value of 9.2. So, for physical interpretation, onemust consider that the linear model(�lter) �tting will incorporate some bias into the estimatedparameters. As expected, a peak is observed in the transient range of estimation. The parametersexhibit low variance around their signatures. Note that without the noise added, it was observedthat the variance goes to zero.For test case 2, an added coulombic friction fault was simulated by adding this term(linear dependence on the sign of velocity) in the dynamics of the fault sensitive model. Thefriction coe�cient was given a value of 1. A plot of the fault sensitive acceleration signal is

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Figure 5.15: Fault Sensitive Acceleration for Simulated Test Case 1shown in Figure 5.17. The a�ect of the fault in the acceleration signal is more evident in thiscase than in test case 1. Figure 5.18 shows the estimated friction model parameters for this testcase. Again, the parameters remain near zero with low variance prior to the fault and escapetheir thresholds immediately after the fault is initiated. The viscous parameter remains small(signature of 0.2) and the coulombic parameter tracks to a signature value of 1 at 1.5 secondsafter the fault. In this case, the signature accurately represented the physical parameter in themodel. Also, the parameters again exhibit low variance around their signatures.Lastly, for test case 3, a combined coulombic and viscous friction fault was added to thedynamics of the fault sensitive model. The fault coe�cients were both set to 1. A plot of the faultsensitive acceleration signal is shown in Figure 5.19. The a�ect of the fault in the accelerationsignal is nearly the same as that observed in test case 2 (Figure 5.17). Figure 5.18 shows theestimated friction model parameters for this test case. Again, the parameters remain near zerowith low variance prior to the fault and escape their thresholds immediately after the fault isinitiated. The coulombic and viscous coe�cients track to their signature values of 1 and 1.5,respectively, at 1.5 seconds after the fault. The parameters again exhibit low variance around

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Figure 5.16: Parameter Estimates, Thresholds and Signatures for Simulated Test Case 1their signatures. In all three test cases, detection of the parameters escaping their thresholds waswithin 0.001 seconds.The friction fault isolation and detection scheme is composed of the dynamic �lter, therecursive estimator of the friction model parameters and the detection ag. The scheme has beenvalidated here under the ideal conditions of numerical simulation. In the next section, the moreimportant validation, that of an experimental analysis, is given.For reference, a copy of the Simulink model and the source code is given in Appendix B.For implementation in the experimental analysis, the m-�les are modi�ed only slightly and sup-plemented with another m-�le that relates the signal processor output to the Matlab workspace.5.7.2 Results for Application to the Precision Positioning ExperimentReported here are seven di�erent test cases that display the success of the methoddescribed in the preceding sections when applied to the pneumatic cylinder and mass load exper-iment. These cases compare di�erent levels of friction fault for a range of voltage input signals.The wide range of inputs is applied to show that the scheme is not restricted to a speci�c input

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|−−> Coulombic friction fault

Figure 5.17: Fault Sensitive Acceleration for Simulated Test Case 2for success in detection and isolation of the friction fault. The seven test cases are all displayedin Table 5.2 and Table 5.3. Speci�cally, tests 1 - 3 compare di�erent types of applied frictionfault for common inputs. Test cases 3,4 and 3,5 compare di�erent input voltages and frequencies,respectively, for common applied fault types. Test cases 6 investigates a di�erent input type forthe common fault types of cases 3 - 5. Lastly, test case 7 investigates the extreme case of twostaggered faults, with the second fault exhibiting very high stiction. This case is also subjectedto a di�erent input type than the other six cases.In the tables, air supply at the fault and the fault character describe the type of faultapplied. Air supply at the fault refers to the to the amount and speed that the air supply tothe block is reduced to induce a friction fault. The fault character refers to the size of andamount of sticking or slipping observed in the load acceleration measurements due to the frictionfault. Fault occurrence time and detection time refer the time of initiation of the fault and ofthe parameters exiting their threshold windows, respectively. Parametric performance describesthe behavior of the parameters prior to and after the fault. The parameters are \well behaved"when they track to a pre-friction level and to their signature values with little variance.

85Table 5.2: Dry Friction Fault Detection Test CasesTest Input Air SupplyNo. Voltage (V) Frequency (Hz) Type at the Fault1 1.4 8 sine gradually reduced supply2 1.4 8 sine slightly reduced supply3 1.5 8 sine cut-o� supply4 1.2 8 sine cut-o� supply5 1.5 4 sine cut-o� supply6 1.0 8 square cut-o� supply7 1.2 8 triangle reduced and cut-o� forfaults 1 and 2, resp.

Table 5.3: Dry Friction Fault Detection Test CasesTest Fault Occurrence Fault Detection Parameter (�1; �2)No. Time (sec) Character Time (sec) Performance1 1.0 - 4.5 from high slip to { both well(interval) increasing stick fault behaved2 3.05 small and 0.015 both veryhigh slip fault well behaved3 2.9 abrupt fault with 0.035 both verylow stick well behaved4 3.05 abrupt fault with 0.06 �1 behaves, �2high stick exhibits large variance5 2.9 abrupt fault with 0.25 both behavehigh stick very well6 2.8 abrupt fault with 0.04 both behave pre-fault,high stick post-fault variance7 2.7, 3.6 small high slip fault 1, 0.1 both behave for fault 1,very high stick fault 2 (fault 1) �2 poor post-fault 2

86

2 4 6 8 10 12 14 16 18 20

−3

−2.5

−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Ff = theta1*sgn(vel) + theta2*vel for Simulations

para

met

ers,

thre

shol

ds a

nd s

igna

ture

val

ues

time(sec)

x − theta 1 signature, thresholdso − theta 2 signature, thresholds

theta 1

theta 2

Figure 5.18: Parameter Estimates, Thresholds and Signatures for Simulated Test Case 2In test cases 1 - 3, all conditions are nearly equivalent except in the manner in which thefriction fault is applied. The only other variation is that cases 1 and 2 have 1.4 Volts amplitudeinputs and case 3 has a 1.5 Volts amplitude input. This variation is assumed negligible. Thechief variation is in the application of the fault. In case 1, the fault is gradually applied over aninterval of time. Cases 2 and 3 investigate small and large abrupt faults, respectively, which aremore likely types of actual faults. For this reason, all of the added faults in the remaining fourtest cases are also of the abrupt type.A plot of the measured acceleration for test case 1 is shown in Figure 5.21. In this plot,the gradual addition of the friction fault for 1 � t � 4:5 seconds can be seen as the amplitudeof the signal decreases. It is also observed that the signal is initially of sinusoidal form, as isthe input. The form becomes distorted as the added friction is increased. This is seen moreclearly in Figure 5.22, which shows the measured and modeled accelerations for pre- and post-fault intervals of time. As the modeled acceleration is generated under no friction conditions,the discrepancy between the modeled and measured accelerations is small prior to the addedfriction fault. Note that the measured acceleration shows a stiction e�ect in the peaks prior to

87

8.5 9 9.5 10 10.5 11 11.5 12 12.5 13

−3

−2

−1

0

1

2

3

4

Fault sensitive acceleration signal

time (sec)

Acc

eler

atio

n (m

/sec

2 )

|−−> Coulomb and viscous friction fault

Figure 5.19: Fault Sensitive Acceleration for Simulated Test Case 3the fault occurrence. For di�erent voltage inputs, this e�ect varies and generally decreases forincreasing input amplitude and frequency, as will be seen in the other test case results. In anycase, the added dry friction fault is distinguished from the pre-fault stiction in the dynamic �ltergeneration, thereby isolating the desired dry friction fault for detection. Although the estimatedparameters have no physical meaning except by their signature values, they converge to thispre-fault stiction behavior. A discussion about the system model performance under non frictionconditions is given in Appendix C.To see the detection of the dry friction fault, Figure 5.23 shows the evolution of therecursively estimated friction model parameters �1(t) and �2(t) for the gradually added fault ofthis test case. The parameters slowly increase to their signature values, peak before tracking tothe signatures and �nally exhibit low variance around their signatures until 6 sec. After 6 sec, theparameters divert from their signatures as another reduction in the acceleration signal amplitudeis present between 5.8 and 7.5 sec. This reduction is a result of drifting in the mass load whilethe fault was added. The drifting created added binding between the mass and surface contactarea.

88

2 4 6 8 10 12 14 16 18 20

−3

−2

−1

0

1

2

Ff = theta1*sgn(vel) + theta2*vel for Simulations

para

met

ers,

thre

shol

ds a

nd s

igna

ture

val

ues

time(sec)

theta 1

x − theta 1 signature, thresholdso − theta 2 signature, thresholds

theta 2

Figure 5.20: Parameter Estimates, Thresholds and Signatures for Simulated Test Case 3The more likely fault types, i.e. abrupt faults, are investigated in the remaining sixcases. Now, the results of test cases 2 and 3 are given to compare to test case 1 as all conditionsare nearly equivalent. The measured acceleration and estimated parameter evolution plots fortest cases 2 and 3 are shown in Figure 5.24,Figure 5.25 and Figure 5.26, Figure 5.27, respectively.The parameter estimate plots show fast tracking to the signature values with low variance forthese abrupt fault cases. The value of �1(t) shows lower variance than �2(t) in these three cases.The estimated parameters for test case 3 are well behaved and the conditions of thisadded fault, i.e. the air supply at the fault and the fault character, are common to the next threetest cases (4, 5 and 6). Prior to the investigation of these cases, it is useful to see how the pa-rameter estimates are quantitatively related to measured acceleration. As discussed, the frictionforce is dynamically related to the di�erence between the measured and modeled accelerations,i.e. the acceleration residual, when friction is present. A plot of the acceleration residual for testcase 3 is shown in Figure 5.28.As discussed in Section 5.6.2, the �lter parameters and the signature values are estimatedto �t the �ltered friction model signal to this acceleration residual after the occurrence of the

89

0 1 2 3 4 5 6 7 8

−20

−15

−10

−5

0

5

10

15

20

Measured Acceleration for Vin

= 1.4 sin(8Hz*2pi*t)

time (sec)

Acc

eler

atio

n (m

/sec

2 )|−−> Gradual Friction Fault

Figure 5.21: Measured Acceleration for Test Case 1fault. In Figure 5.29, the �t between the �ltered friction signal (solid) and the accelerationresidual (dashed) show the success of this approach. Thus, in the event of a fault, the estimatedparameters track to the parameter values that, with the estimated �lter, generate a �t such asthat exhibited in Figure 5.29.The acceleration and parameter estimation plots for test case 4 are in Figure 5.30 andFigure 5.31. Test cases 3 and 4 show the e�ect of varying the amplitude of the voltage inputunder the same added dry friction fault condition. The fault for case 3 displays less stickingbehavior as the mass has more inertia to push through the sticking e�ect when the direction ofmotion changes. The high sticking character in the friction fault and resulting greater noise inthe post-fault acceleration signal for case 4 resulted in lower post-fault performance for �2. Still,the pre-fault estimates behave well enough to detect the fault.The acceleration and parameter estimate plots for test case 5 are in Figure 5.32 and

90

0.3 0.4 0.5 0.6 0.7−20

−15

−10

−5

0

5

10

15

20Pre−Fault Accelerations

time (sec)

Acc

eler

atio

n (m

/sec

2 )

5.3 5.4 5.5 5.6 5.7−20

−15

−10

−5

0

5

10

15

20Post−Fault Accelerations

time (sec)Figure 5.22: Measured (dashed) and Modeled (solid) Accelerations, Pre- and Post-Fault for TestCase 1Figure 5.33. Test cases 3 and 5 show the e�ect of varying the frequency of the voltage input forthe same added dry friction fault. Case 3 displays less sticking behavior overall as the pneumaticsystem is sti�er at a higher frequency, particularly at or near the resonant frequency (8-9 Hz).The high pre-fault stiction character and high sticking character after the friction fault for case 5still resulted in good performance of the estimates. The threshold bounds accommodate highervariations in �2 prior to the fault and the detection is successful.The plots for cases 6 and 7 are shown in Figure 5.34, Figure 5.35 and Figure 5.36,Figure 5.37, respectively. These two test cases show the a�ect of di�erent voltage input signalforms, with a square wave input for test case 6 and a triangular waveform input for test case 7.Also, case 7 investigates a staggered fault case, where a small fault of high slip character occursat 2.7 sec and an abrupt cut-o� fault of very high stick character occurs at 3.6 sec. Althoughthere exists a high level of post fault variance in Figure 5.35, attributed to applying an input of

91

0 1 2 3 4 5 6 7 8−1

0

1

2

3

4

5

6

7

Ff = g*theta1*sgn(vel) + theta2*vel for V

in = 1.4 sin(8Hz*2pi*t)

para

met

ers,

thre

shol

ds a

nd s

igna

ture

val

ues

time(sec)

theta1

theta2o − theta2 signature

x − theta1 signature

Figure 5.23: Parameter Estimates, Thresholds and Signatures for Test Case 1higher nonlinear content, the threshold detection step is successful and tracking is still behaved.In test case 7, Figure 5.37 shows successful parametric performance and detection for the slipfault and poor detection performance for the extremely high stick fault. As discussed, theCoulomb and viscous friction model is inadequate to model friction of such high sticking content.This is evident as �1(t) is seen to track to a negative value and �2(t) becomes very large afterthe occurrence of the high stick fault. The extreme test case 7 represents a limiting case of thedetection and isolation scheme.5.8 ConclusionsThere are multiple factors that contribute to the level of and stick / slip content in thedry Coulombic friction between the air bearing mass and the level surface on which it rides. The

92

0 1 2 3 4 5 6 7 8

−20

−15

−10

−5

0

5

10

15

20

Measured Acceleration for Vin

= 1.4 sin(8Hz*2pi*t)

time (sec)

Acc

eler

atio

n (m

/sec

2 )|−−> Friction Fault

Figure 5.24: Measured Acceleration for Test Case 2supply pressure to the air bearing mass largely controls the level of friction between the bottomof the block and the level surface. For a pressure of 10 psi decreasing down to 1 psi the slidingfriction slowly increases and exhibits high slipping / low sticking character. Once the supplypressure is turned o�, the sticking characteristic becomes more present.As a friction control, however, the supply pressure is also coupled to the type, amplitudeand frequency of the voltage input signal. For decreasing amplitude and / or frequency of asinusoidal input, the sticking e�ect increases. For a higher amplitude, the block has more inertiato push through the direction changes. Further, higher frequencies make the cylinder sti�er andhence less sticking is present. The dry friction fault detection scheme is successful and displaysgood robustness with respect to variable input signals and levels of sticking in the dry frictionfault. The fault detection scheme designed here detects changing process conditions in the case

93

1 2 3 4 5 6 7−3

−2

−1

0

1

2

3

4

Ff = g*theta1*sgn(vel) + theta2*vel for V

in = 1.4 sin(8Hz*2pi*t)

para

met

ers,

thre

shol

ds a

nd s

igna

ture

val

ues

time(sec)

theta2

theta1

o − theta2 signature

x − theta1 signature

Figure 5.25: Parameter Estimates, Thresholds and Signatures for Test Case 2of wear and excessive side loading, in the form of dry friction, of a precision positioning device.A ow diagram of the implementation of this friction detection scheme is shown in Figure 5.38.The process input d represents noise present in the experimental apparatus, with or withoutfriction. All other parameters and variables are de�ned in the previous sections as follows. InSection 5.4 the frequency response measurement from the servo valve voltage input Vin to themass acceleration �x, under no added friction conditions, was given and discussed. Also, thediscrete fourth-order transfer function bG(z�1) was �t to the frequency response data to simulateacceleration �̂x(t) of the mass without friction for a given servo voltage input.It was shown in Section 5.5.2 that the friction force Ff is dynamically related to thedi�erence between the measured and simulated (non friction) accelerations, called the accelera-tion residual �xr(t). In Section 5.6.1 the viscous and coulomb friction model was employed anda prediction error approach [4] was used to model the dynamic relationship between the friction

94

0 1 2 3 4 5 6 7 8

−20

−15

−10

−5

0

5

10

15

20

Measured Acceleration for Vin

= 1.5 sin(8Hz*2pi*t)

time (sec)

Acc

eler

atio

n (m

/sec

2 )

|−−> Friction Fault

Figure 5.26: Measured Acceleration for Test Case 3force and the acceleration residual, using an output error optimization in Section 5.6.2. A param-eterized friction signal xf (t) was incorporated with the normalized �lter Ge(q�1) in Section 5.6.3and the procedure was implemented by recursive least-squares (RLS) estimation of the frictionmodel parameters �̂(t) in Section 5.6.4. Simple threshold decision logic was applied to theseestimates to detect the isolated dry friction fault on-line.By systematically reducing the data contained in the measured input and output signalsto the estimation of two friction model parameter signals, the scheme isolates the friction faultfrom other possible sensor, actuation or process faults that can occur in precision positioningdevices. Thus, the goal of detection and isolation of FDI schemes has been achieved.Experimental identi�cation of friction and its compensation in precise, position con-trolled mechanisms was also investigated by Johnson and Lorenz [31]. Their approach wasdi�erent in that a parameterized model of the friction force was identi�ed from the loop errors in

95

0 1 2 3 4 5 6 7−3

−2

−1

0

1

2

3

4

5

6

Ff = g*theta1*sgn(vel) + theta2*vel for V

in = 1.5 sin(8Hz*2pi*t)

para

met

ers,

thre

shol

ds a

nd s

igna

ture

val

ues

time(sec)

theta2

theta1

o − theta2 signature

x − theta1 signature

Figure 5.27: Parameter Estimates, Thresholds and Signatures for Test Case 3a state feedback motion controller. Signal processing was used to isolate the errors as functionsof the states and the physical relationship between friction and the spatial states (e.g. position,velocity) were used to formulate the model structure. The approach was experimentally validatedin a robotic gripper application. Their schemes parallels the FDI scheme designed here but lacksthe interpretation provided by the estimated friction model parameters that the scheme in Figure5.38 generates. It would in fact be possible to incorporate the detection scheme into their controlalgorithm. The scheme would inherently provide the information they require for compensationand at no added cost the detection of changing process conditions, that may have resulted in afault, would be possible.By design, the addition of friction in the apparatus is equivalent to an increase in thepiston dry friction. This fault represents a signi�cant problem that exists in industry whereprecision positioning devices are employed. The scheme detects a change in process conditions

96

0 1 2 3 4 5 6 7 8

−20

−15

−10

−5

0

5

10

15

20

Acceleration Residual for Vin

= 1.5 sin(8Hz*2pi*t)

time (sec)

Acc

eler

atio

n (m

/sec

2 )|−−> Friction Fault

Figure 5.28: Acceleration Residual for Test Case 3due to wear and excessive side loading that result from a friction fault in a precision positioningdevice. The controls engineer would certainly bene�t by applying this simple, reliable method ofdry friction fault detection in lieu of or adjointly with any precision compensation. In simulationanalyses, the scheme has been proven to be very e�ective and successful. Furthermore, forimplementation in an experiment, the only added hardware required is an accelerometer, whichis light-weight, inexpensive and easy to add to a positioned load.

97

4 4.2 4.4 4.6 4.8 5

−15

−10

−5

0

5

10

15

Acceleration Residual and Filtered Friction Signal for Test Case 3

time (sec)

Acc

eler

atio

n (m

/sec

2 )

Figure 5.29: Acceleration Residual (dashed) and Filtered Friction Model Signal (solid) for TestCase 3

98

0 1 2 3 4 5 6 7 8

−20

−15

−10

−5

0

5

10

15

20

Measured Acceleration for Vin

= 1.2 sin(8Hz*2pi*t)

time (sec)

Acc

eler

atio

n (m

/sec

2 )

|−−> Friction Fault

Figure 5.30: Measured Acceleration for Test Case 4

99

0 1 2 3 4 5 6 7 8−6

−4

−2

0

2

4

6

8

Ff = g*theta1*sgn(vel) + theta2*vel for V

in = 1.2 sin(8Hz*2pi*t)

para

met

ers,

thre

shol

ds a

nd s

igna

ture

val

ues

time(sec)

theta1

theta2

o − theta2 signature

x − theta1 signature

Figure 5.31: Parameter Estimates, Thresholds and Signatures for Test Case 4

100

0 1 2 3 4 5 6 7 8

−20

−15

−10

−5

0

5

10

15

20

Measured Acceleration for Vin

= 1.5 sin(4Hz*2pi*t)

time (sec)

Acc

eler

atio

n (m

/sec

2 )

|−−> Friction Fault

Figure 5.32: Measured Acceleration for Test Case 5

101

1 2 3 4 5 6 7−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Ff = g*theta1*sgn(vel) + theta2*vel for V

in = 1.5 sin(4Hz*2pi*t)

para

met

ers,

thre

shol

ds a

nd s

igna

ture

val

ues

time(sec)

theta2

theta1o − theta2 signature,threshold

x − theta1 signature,threshold

Figure 5.33: Parameter Estimates, Thresholds and Signatures for Test Case 5

102

0 1 2 3 4 5 6 7 8

−20

−15

−10

−5

0

5

10

15

20

Measured Acceleration for Vin

= 1 square(8Hz*2pi*t)

time (sec)

Acc

eler

atio

n (m

/sec

2 )

|−−> Friction Fault

Figure 5.34: Measured Acceleration for Test Case 6

103

1 2 3 4 5 6 7 8−10

−5

0

5

10

15

20

Ff = g*theta1*sgn(vel) + theta2*vel for V

in = 1 square(8Hz*2pi*t)

para

met

ers,

thre

shol

ds a

nd s

igna

ture

val

ues

time(sec)

o − theta2 signature,threshold

x − theta1 signature,threshold

theta1

theta2

Figure 5.35: Parameter Estimates, Thresholds and Signatures for Test Case 6

104

0 1 2 3 4 5 6 7 8

−20

−15

−10

−5

0

5

10

15

20

Measured Acceleration for Vin

= 1.2 tri(8Hz*2pi*t)

time (sec)

Acc

eler

atio

n (m

/sec

2 )

|−−> Friction Fault 1

|−−> Friction Fault 2

Figure 5.36: Measured Acceleration for Test Case 7

105

1 2 3 4 5 6 7 8−20

−10

0

10

20

30

40

50

60

Ff = g*theta1*sgn(vel) + theta2*vel for V

in = 1.2 tri(8Hz*2pi*t)

para

met

ers,

thre

shol

ds a

nd s

igna

ture

val

ues

time(sec)

theta2

theta1

x − theta1 signature

o − theta2 signature

Figure 5.37: Parameter Estimates, Thresholds and Signatures for Test Case 7

106

Figure 5.38: Flow Diagram of Scheme for Fault Detection and Isolation of Dry Friction in aPrecision Positioning Device.

Chapter 6ConclusionsThe success of the quasilinearization method in parametrically identifying unknown pa-rameters o�-line that accompany linear and nonlinear terms and pre-exist or result from damagein the model of the space antenna structure is extensively proven for a wide range of loading, non-linearity level, integration routine conditions and noise levels in the observed dynamic responses.Bellman [8] has also shown success of the method in cases where only partial information inthe dynamic response is observed. The success of the method can also be improved in caseswhere weighting the cost function information is appropriate. For example, when a section of themeasured data contains better information for identi�cation, i.e. if initial condition mismatchingexists and noise or damage contaminated the observed signals at a certain point in time, a Jacobiapproximation weighting is used. Further, if the initial or �nal data in the measurements aremore reliable for identi�cation, exponential (Laguerre approximation) weighting is appropriate.Although the method is highly successful for the eighth-order space antenna modelinvestigated, the number of parameters was kept small since the number of di�erential and linearequations that require solutions for each iteration grows rapidly for a higher number of unknownparameters. In the modi�ed Kabe model considered here, consider the extreme case that all ofthe masses are unknown and possibly di�erent, each spring element is nonlinearized, to accountfor possible damage, by adding cubic dependence and all of the sti�ness values are unknown andnot necessarily equal. Each iteration would require the solution of 784 di�erential equations andthe inversion of a 48 by 48 matrix. The observation measurements in this case must contain atleast 48 samples for the problem to be determined and many more samples than that for any hopeof successful convergence. The likelihood of a sign error in the enormous equation formulationand possible ill-conditioning of the matrix pose threats to the success of the method, which wouldalso require large computational time.

107

108As a tool for identifying a limited number of unknown parameters in damaged andtherefore nonlinear higher order structural systems, given the model structure and post damagedynamic response observations, the quasilinearization approach of parameter estimation is re-liable. The approach also shows promise in the detection, location and assessment of multiplestructural faults in such a model.Successful fault detection and isolation of a dry friction fault in a pneumatically actuatedair bearing mass has also been achieved. The FDI scheme systematically reduces the datacontained in the measured servo voltage input and measured mass load acceleration outputsignals. The reduction results in the estimation of two friction model parameter signals andthe scheme isolates the friction fault from other possible sensor, actuation or process faults thatcan occur in precision positioning devices. Thus, the goal of detection and isolation of FDIschemes has been achieved. Moreover, in generating an FDI scheme for friction fault detection inprecision positioning devices, a more general dry friction FDI scheme has emerged and appears tobe implementable in other dynamic systems, as in the (admittedly simple) second order systemin the numerical analysis.By design, the addition of friction in the experimental apparatus is equivalent to anincrease in the piston dry friction. This fault represents a signi�cant problem that exists inindustry where pneumatic positioners are employed. It therefore serves to design a simple ande�ective fault detection and isolation scheme. The controls engineer would certainly bene�t byapplying this simple, reliable method of dry friction fault detection prior to or in lieu of anyprecision compensation. The only added hardware required is an accelerometer, which is light-weight, inexpensive and easy to add to the positioned load. Since this technique of identi�cationand detection is successful for this highly nonlinear system and for a wide class of input signals,it is likely to bene�t the engineer interested in identifying a dry friction fault in a lower ordernonlinear electro-mechanical system.

Appendix ASelection of Nonlinear StructuralSpring by Modal ParticipationThis appendix describes the method by which k1 at m6 in the Kabe model was selectedas the structural spring that became nonlinear as a result of damage. It is assumed that thequasilinearization technique is more successful when the constants being estimated accompanyterms that participate more in the equations of motion and the linearized equations that themethod generates. Intuitively, the parameters must be observable in the dynamic response inorder to estimate them. Moreover, in selecting which sti�ness coe�cients to estimate and whichstructural spring to nonlinearize, it is necessary to rank the sti�ness coe�cients by their amountof observability. This ranking is done quantitatively by computing the participation of the sixdi�erent sti�ness values in terms of their mass normalized strain energies, U . The justi�cationfor this is that the greater the mass normalized strain energy for the greatest number of modes,the more that particular spring participates in the equations of motion under general loading.The mass normalized strain energy of spring kj in mode �i is computed asU = 12f�̂igT [ K ]jf�̂ig; (A.1)where each mass normalized mode �̂i isf�̂ig = 1pMi f�ig (A.2)where Mi is the modal mass of �i. The matrix [ K ]j represents the sti�ness matrix modi�ed bysetting all spring constants to zero except kj . For each spring, these energies and their percentagecontribution to the total strain energy of each mode are calculated. Figure A.1 reveals that springk1 participates the most in the greatest number of modes, i.e. greater than 20 percent in six109

110

Figure A.1: Percentage of strain energy per spring per modemodes and 90 percent in two modes. Further, this percentage is calculated for each of the fourmasses that k1 grounds. Figure A.2 shows that k1 at m6 participates the most in the greatestnumber of modes (note the symmetry between m3 and m6). Therefore, to nonlinearize the

Figure A.2: Modal percentage of strain energy in k1 at m2;m3;m6 and m7

111equations of motion, k1 of m6 is given linear and cubic dependence on position (see Equation4.4) to simulate damage in that structural spring.

Appendix BFriction FDI Scheme Source Codeand Numerical Analysis ModelThe Simulink model is shown in Figure B.1 and the source code follows. For theexperimental analysis, the same code was used, modi�ed slightly, with the addition of the codefor relating the signal processor to the Matlab workspace.

112

113

UNDAMAGED

DAMAGED

Acceleration Residual

Friction Force

Signals were observed and the Noise is +/− 2% the peak

value of the signal after settling

Uniform RandomNumber1

Uniform RandomNumber

accm

To Workspace4

accf

To Workspace3

vel

To Workspace2

Ff

To Workspace1

acc_res

To Workspace

Switch1

Switch

Sine Wave

Scope1

Scope

s

1

Integrator3s

1

Integrator2

s

1

Integrator1s

1

Integrator

−co

Gain5

−co

Gain4

−a

Gain3

−c

Gain2

−k

Gain1

−k

Gain

Coulomb &Viscous Friction

0

Constant1

0

Constant

Clock1

Clock

Figure B.1: Second Order Models for the Numerical Simulation of the FDI Scheme

114% Given below are the three m-files used in Matlab to implement% the dry friction FDI scheme.% This file contains the model and fault data and runs the% simulink models.% Input propertiesFo = 3.0;wo = 4*2*pi; % in rad/sec% Model propertiesk = 1.;co = 2.2;% Fault Propertiesc = 1.; %viscousa = 1.; %Coulomb% Simulation Propertiesdelt = 0.005;tend = 20;% Fault additions and exclusionsswitch1 = tend/2.; % Adds viscous friction%switch1 = tend; % Turns off viscous frictionswitch2 = tend/2.; % Adds Coulombic friction%switch2 = tend; % Turns off Coulombic friction% run the simulink modeloptions = simset('FixedStep',delt);sim('damage',tend,options);time = tout;figure,subplot(1,2,1)

115plot(time,accf,'-',time,accm),title('Pre-Fault Accelerations')xlabel('time (sec)'), ylabel('Acceleration (m/sec^2)')subplot(1,2,2)plot(time,accf,'-',time,accm),title('Post-Fault Accelerations')xlabel('time (sec)')figure,plot(time,acc_res,'-')xlabel('time (sec)'), ylabel('Acceleration (m/sec^2)')title('Simulated Acceleration Residual')%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This file creates the filter for the Clmb/Visc friction to acc_res signals.% The model fitting is done in the time domain.% Gamma is varied to find a best fit model (norm of difference criterion).% friction modelled as Coulomb and Viscous (positive to the right)I = 1;nmag(I) = 30.;nmg = 30.;for gam = 1:0.5:2 % whatever a/c is in this casesFf = -(sign(vel) + gam*vel); %positive to the right% compute filter based on added friction datasZ = [acc_res(2500:3000) sFf(2500:3000)];% design the systemsTH = oe(sZ,[3 2 0],1e8,256\1);%TH=arx(Z,[3 4 0],1e8,256\1);%simulate the systemsysim = idsim(sZ(:,2),sTH);

116%check the norm of the difference (time domain)nmag(I+1) = norm(sysim(50:500)-sZ(50:500,1));if lt(nmag(I+1),nmg)nmg = nmag(I+1);gamma = gamFf = sFf;Z = sZ;TH = sTH;ysim = sysim;endI = I+1;[a,b,c,d,k]=th2ss(sTH);dcg(I) = ddcgain(a,b,c,d)endfigure,plot(time,acc_res,'k --')axis([0 tend -23 23])grid ontitle('Acceleration Residual for V_i_n = ?') %1.5 sin(8Hz*2pi*t)')xlabel('time (sec)')ylabel('Acceleration (m/sec^2)')zoom ontm = time(2500:3000);figureplot(tm,ysim,tm,Z(:,1),'k --'),figure(gcf),zoom ongrid ontitle('Acceleration Residual and Filtered Friction Signal for Simulation')xlabel('time (sec)')ylabel('Acceleration (m/sec^2)')zoom onfigure,plot([ysim Z(:,1)]),figure(gcf),zoom on

117figure,plot([ysim-Z(:,1)]),figure(gcf),zoom on[a,b,c,d,k]=th2ss(TH);save Ff_filter3a gamma a b c d dcg delt% This plot characterizes the filter in the frequency domainsyse=ss(a,b,c,d,delt);figure,bode(syse),figure(gcf),zoom on%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% This file runs the whole schebang at once. Parameters l and c are the data% batch length and interval, respectively.damage1%load Ff_filter1 % Filter for Viscous Friction Fault%load Ff_filter2a % Filter for Coulomb Friction Faultload Ff_filter3 % Filter for Coulomb and Viscous Friction Faultfile1 = -sign(vel);file2 = -vel; %positive to the right sign convention% run the simulink model using voltage inputoptions = simset('FixedStep',delt); % sample time for modelsim('fricfilt',tend,options);[s,ss] = size(time);% Exclude initial condition and signal truncation mismatchingZ = [xone(100:s-200) xtwo(100:s-200) acc_res(100:s-200)]; %s = vector lengthl = 300; c = 20;fprintf(' the data batch length l and interval c are ')l,cpar = testrarx(Z,l,c)';

118% time vector for parameterstm = time(100+l:c:s-200);% calculate the threshold valuesmp = [mean(par(1:50,1)) mean(par(1:50,2))];sdp = [std(par(1:50,1)) std(par(1:50,2))];ths = [mp+3*sdp;mp-3*sdp];threshs = ones(size(tm))*[ths(1) ths(2) ths(3) ths(4)];% generate the signature value vectorssigtrs = ones(size(tm))*[1.0 gamma];% plot the resultfigure,plot(tm,par(:,1),tm,par(:,2),tm,threshs,'-.',tm,sigtrs)hold on,plot(tm(1:30:171),sigtrs(1:30:171,1),'x')plot(tm(15:30:171),sigtrs(15:30:171,2),'o')plot(tm(1:30:171),threshs(1:30:171,1:2),'x')plot(tm(15:30:171),threshs(15:30:171,3:4),'o')title('F_f = g*theta1*sgn(vel) + theta2*vel for Simulations')ylabel('parameters, thresholds and signature values')xlabel('time(sec)')grid onzoom on

Appendix CLinear Transfer Function ModelPerformance in the PrecisionPositioning ApparatusThe purpose of this appendix is to discuss the results of using a linear transfer functionto model the highly nonlinear pneumatic cylinder and air bearing mass load dynamics fromvoltage input to acceleration output. As discussed in Section 5.7, not all of the system dynamics,e.g. pre-fault stiction in the piston, are captured by this linear model. Moreover, as linear modelsare only good at or near the operating point for which they were generated, the inability of themodel to capture all of the system dynamics becomes accentuated when friction is added. Thisis exactly what it is supposed to do; the resulting acceleration residual provides sensitivity toa disruption to the systems operating dynamics. The dynamic �lter generated in Section 5.6 iswhat isolates the fault of interest within this disruption, namely a dry friction fault. Furthermore,the model is simple to identify and good enough for the purposes of fault detection and isolationof dry friction in a precision positioning device.As an aside, the load position measurements in this precision positioning apparatuscontain less noise than the measured load accelerations and the modeled position more accuratelymatches these measurements, as seen in Figure C.1.

119

120

10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

Displacement Signals for Vin

= 1.0 sin(7.2Hz*2pi*t)

Dis

plac

emen

t (m

)

time (sec)

Modeled (solid)

Measured (dashed)

Figure C.1: Plot of Modeled (solid) and Measured (dashed) Mass Position Signals for a GivenInput

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