research article online structural health monitoring and

Research ArticleOnline Structural Health Monitoring andParameter Estimation for Vibrating Active CantileverBeams Using Low-Priced Microcontrollers

Gergely Takaacutecs Jaacuten Vachaacutelek and Boris Rohalrsquo-Ilkiv

Institute of Automation Measurement and Applied Informatics Faculty of Mechanical EngineeringSlovak University of Technology in Bratislava Nam Slobody 17 812 31 Bratislava 1 Slovakia

Correspondence should be addressed to Gergely Takacs gergelytakacsstubask

Received 28 November 2014 Accepted 23 April 2015

Academic Editor Xinjie Zhang

Copyright copy 2015 Gergely Takacs et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents a structural health monitoring and parameter estimation system for vibrating active cantilever beams usinglow-cost embedded computing hardware The actuator input and the measured position are used in an augmented nonlinearmodel to observe the dynamic states and parameters of the beam by the continuous-discrete extended Kalman filter (EKF) Thepresence of undesirable structural change is detected by variations of the first resonance estimate computed from the observedequivalent mass stiffness damping and voltage-force conversion coefficients A fault signal is generated upon its departure froma predetermined nominal tolerance band The algorithm is implemented using automatically generated and deployed machinecode on an electronics prototyping platform featuring an economically feasible 8-bit microcontroller unit (MCU) The validationexperiments demonstrate the viability of the proposed system to detect sudden or gradual mechanical changes in real-time whilethe functionality on low-cost miniaturized hardware suggests a strong potential for mass-production and structural integrationThemodest computing power of the microcontroller and automated code generation designates the proposed system only for veryflexible structures with a first dominant resonant frequency under 4Hz however a code-optimized version certainly allows muchstiffer structures or more complicated models on the same hardware

1 Introduction

Recent engineering trends foreshadow a tendency wheremechanical structures are becoming more vibration proneThe design approach in which lighter materials and moreslender shapes are used may simply have an aesthetic justifi-cation but the savings in material costs are an important fac-tor aswell Aircraftwing surfaceswith progressive designmaybenefit from reduced fuel andmaintenance costs or increasedmaneuverability while robotic arms and manipulators arefaced with increasing demands on working speeds Oftenthese improvements are plagued by the curse of lowered stiff-ness and a susceptibility to demonstrate undesired vibrationresponse

Smart materials and the availability of cheap computingtechnology enable designers to balance or even completely

eliminate the undesirable vibration response of modernstructures by active vibration control (AVC) measures [1 2]Microcontroller units (MCU) are essentially miniaturizedcomputing platforms that are capable of independent oper-ation and pose the built in peripherals necessary to readsensors or drive actuators As both the price and the sizeof MCUs are decreasing [3] it is now possible to integratevibration control and diagnostics systems into mechanicalstructures on a mass produced scale even with the pos-sibility to use several detached systems in a decentralizedmanner

In addition to the control functionality of structures withAVC safety-critical applications demand highly integratedonline diagnostics functionality [4] Most engineering struc-tures may experience changes in properties and responsedynamics due to unexpected outside influences such as

Hindawi Publishing CorporationShock and VibrationVolume 2015 Article ID 506430 14 pageshttpdxdoiorg1011552015506430

2 Shock and Vibration

sensor failures or degradation material buildup fatiguecracks icing and a plethora of other factors In addition tothese process variables such as changing load or workingconditions may induce substantial changes in the structureproviding a good reason to use structural health monitoringand diagnostic systems

The requirements for diagnostic systems can range fromsimple fault indicators to complex parameter estimationprocedures Online parameter estimation allows for a moredetailed assessment of the properties of the monitoredstructures representing unexpected changes as variations inphysically interpretable parameters Moreover the estimatedparameters can be used to adjust the controllers to counteractstructural changes thereby creating an adaptive vibrationcontrol system [5]

Several exciting fault and damage detection strategieshave appeared for cantilever beams and other structures inthe recent literature Localization of the damaged section ispossible by employing progressive optimization-based algo-rithms such as genetic algorithms (GA) [6 7] fuzzy-geneticalgorithms [8] particle swarm optimization [9] wavelettransform [10 11] and artificial neural networks [12] Cracklocation and depth were detected by Vakil Baghmisheh et al[13] using a hybrid particle swarm-Nelder-Mead algorithmand modal testing Although most of these methods offermuch more than a simple status signal on the presenceor absence of a fault in the structure they require a largecomputational effort suggest but do not demonstrate onlineoperation or are simply not suitable for online applicationdue to design and technical limitations [7ndash13]

This paper introduces a prototype of an online diag-nostic system intended to detect and monitor changes inthe mechanical properties of actively controlled structuresresembling the dynamic characteristics of thin cantileverbeams In addition to monitoring the occurrence of changescaused by unexpected outside influences the system basedon a nonlinear joint state and parameter observer is capableof providing real-time estimates that can be used to createadaptive mechanical structures The proposed estimationscheme is implemented on a low-cost electronic prototypingplatform using automatized code generation tools

The approach presented here assumes that the dynamicsof active mechanical structures resembling the propertiesthin cantilever beams can be sufficiently approximated usinga single degree of freedom (SDOF) system [14]This assump-tion holds well in the case of flexible structures with singledigit firstmdashfundamentalmdashresonant frequencies dominatingthe overall response in comparison with the contributionof the higher modes The state-space model of the drivenmass-spring-damper is augmented by themonitored physicalparameters namely the unknown equivalent stiffness damp-ing mass and force conversion coefficients

The introduction of unknown parameters in the lin-ear SDOF dynamic system renders the augmented modelnonlinear This paper uses the nonlinear expansion ofthe acknowledged Kalman filter [15ndash17]mdashthe well-knownextended Kalman filter (EKF) [18 19]mdashto observe the statesand parameters of this augmented system Though it ispossible to directly compute and output the equivalent

Piezoceramicactuators

Laser CPU

Poweramplifier

Laser head

USB

Aluminumbeam

Nominal andadded weight

Excitation signal

Positionmeasurement

signal

PC withMatlabSimulink

ArduinoMega 2560Signal generator

Base

Fault signal

AI0

AI1

DO9 LED

Figure 1 Simplified hardware connection scheme of the experimen-tal system

Power

USB

Atmel ATmega 2560 MCU

Deflection output z(k)

Fault indication e(k)

Voltage input u(k)

Figure 2 Embedded microcontroller unit and the electronicsprototyping board

physical parameters for the needs of adaptive control ordetailed monitoring here the parameters estimated by theEKF are used to assess the first damped resonant frequencyof the active structure It is assumed that this remainsbounded if there is no change in the structure but willdiverge from the nominal levels as soon as a mechanicalfailure or an undesired change of physical configurationoccurs [20] The proposed structural monitoring system istested in a laboratory setting The test structure uses theclassical example of the clamped cantilever (see Figure 1)that is often employed to represent the general dynamicbehavior of real-life structures resembling flexible thin beams[14] such as fixed and rotary aerodynamic surfaces [21 22]antennae [23] manipulators [24] and others [25] The thinaluminum cantilever is equipped with two large double-layerpiezoceramic transducers mounted on opposing sides closeto the clamping mechanism The transducers are driven bya power amplifier receiving a pseudorandom-binary (PRBS)signal The motion of the beam is measured at a singlelocation near the end of the structureThe actuator inputs andthe measurements are fed to a low-cost microcontroller andelectronics prototyping platform featuring a 8-bit MCU (seeFigure 2)

The experimental validation procedure of the proposedparameter monitoring and fault detection scheme involvessupplying the beam with the PRBS signal emulating thecontrol input to the actuator In the first stage of the exper-iments the beam is in its nominal configuration without

Shock and Vibration 3

any changes and the parameters converge to nominal levelsAt the beginning of the second phase of the validationexperiments different weights are added to the end of thebeam in order to emulate the occurrence of a mechanicalfault The algorithm running on the MCU estimates theparameters and calculates the first resonant frequency of thebeam in real-time based on the measured actuator input andposition As soon as the frequency leaves its user determinedtolerance band the fault is detected and a warning signalis sent The experiments featured in this paper demonstratethe estimates of the dynamic states and parameters for thedifferent levels of mechanical faults and indicate the timewhen the error is detected

The use of the extended Kalman filter for the estimationof vibration dynamics is with a few exceptions limited tooffline a posteriori system identification procedures Real-time application of EKF for diagnostics and monitoringis presented by Lourens et al [26] for disturbance forceestimation and by Mu et al [27] for the monitoring ofstructures under seismic loads stiffness and damping coef-ficient estimation in automotive applications [28] and forthe adaptive control of torsional vibration in a two-massrotating drive system [29 30] While Williams suggestedthe use of unscented EKF to assess damage and damagelocation based on accelerometermeasurements [31] theworkis limited to offline numerical studies Stress and strainfields are observed experimentally using a Kalman filter-based methods by Erazo and Hernandez [32] but the workis again limited to an a posteriori test The EKF was usedin previous works to provide state and parameter estimatesfor an adaptive vibration control system for thin beams [533 34] however this was implemented on an expensiverapid control prototyping platform offering extensive com-putational power The main contribution of this paper istherefore not only the proposition to use the EKF for theestimation of vibration dynamics but also the online andreal-time implementation and validation of an EKF-basedscheme on a low-cost platform with a potential for mass-production and close structural integration

Having stated the contributions of the paper it is equallyimportant to realize what is not the ambition of this workThe proposed monitoring system is capable of providingreal-time estimates of the parameters however this paperdoes not attempt to solve the question of the usage ofthis online information The observed parameters may beemployed to generate simple warning signals drive fail-safe mechanisms or serve as a basis for adaptive vibrationcontrol systems Likewise the paper does not attempt tomake recommendations on the mechanical integration of thesystem into structures since it implements the algorithmon a prototyping platform nevertheless the miniaturizedMCU used on the board suggests that this is a likelypossibility

It is also important to note that structural failure detectionin this paper is only solved on the level of its occurrenceLocalization or quantification of the structural failure is notwithin the scope of this work and the proposed method willonly provide information on the presence or absence of acertain change

2 Materials and Methods

21 Theoretical Foundations

211 Dynamic Model of the Beam Due to the flexible natureof the structure the dynamics of the actively controlledthin beam is dominated by its first resonant frequency [1435] Even though from the perspective of mechanics theSDOF assumption may seem simplistic it is sufficient forour purposes and is often used in combination with EKF toestimate vibration motion [36ndash38]

Naturally a first order model and the resulting EKF maycover only structures with a single and significantly dominantresonant mode Nonetheless a better representation of astructure yielding models that are still feasible for computa-tional implementation can be obtained from larger modelsby reduced order modelling techniques or for exampleby the use of the proper order decomposition method Amodel covering approximately 3 modes of vibration andaugmented by unknown parameters is plausible for EKF andits implementation on a low-costMCU however beyond thatone may encounter difficulties with algorithm tuning andcomputational feasibility

Let us assume now that from the viewpoint of theproposed fault detection algorithm the cantilever beam canbe approximated by the second order differential equationdescribing the motion of a mass-spring-damper system withan external driving force [39]

119898 119902 (119905) + 119887 119902 (119905) + 119896119902 (119905) = 119865 (119905) (1)

where the driving actuator force

119865 (119905) = 119888119906 (119905) (2)

is a linear multiple of the input voltage to the actuators 119906(119905)(V) while 119888 (NV) is the voltage-force conversion coefficientThe mass of the structure 119898 (kg) is a sum of the mass of thebeam and the actuators with an additional weight attached atits end The damping coefficient is given by 119887 (Nsm) while119896 (Nm) is the stiffness coefficient The true position of thebeam at the free end is denoted by 119902(119905) (m) while its velocityis 119902(119905) (msminus1) and acceleration is 119902(119905) (msminus2)

The second order differential equation in (1) may bedeconstructed into two first order differential equations tocreate a state-space representation of the beam dynamics[40] The choice of position 119909

1

(119905) = 119902(119905) and velocity1199092

(119905) = 119902(119905) as dynamic states in the representation will yieldthe familiar continuous state-space equations for the mass-spring-damper [41]

x (119905) = Fx (119905) + G119906 (119905)

119911 (119905) = Hx (119905) (3)

where 119911(119905) (m) is the position measured at a single locationclose to the end of the active beamTheposition output can beobtained from the state vector x(119905) = [119909

1

(119905) 1199092

(119905)]

119879 by using


H = [1 0] as theR2times1 output matrix Furthermore F isin R2times2

is the dynamic matrix given by

F =[

[

0 1

minus

119896

119898

minus

119887

119898

]

]

(4)

and G isin R2times1 is the input matrix given by

G =[

[

0

119888

119898

]

]

(5)

212 Augmented Dynamic Model The dynamic states ofthe structuremdashposition and velocity as measured at the freeendmdashare contained in the state variable x(119905) This statevariable can be augmented by a vector incorporating theunknown and potentially changing parameters to obtain

x119886

(119905) = [x (119905) p (119905)]

119879

(6)

The system dynamics given by (3) is clearly linear howeverby including the unknown parameters in the augmented stateformulation this new estimation problem will be nonlinearandwill requiremethods designed for nonlinear systems [42ndash44] In the case of the active beam none of the parameters areknown beforehand and all of them are likely to vary duringthe operation of the structure so the parameter vector will be

p (119905) = [119896 (119905) 119887 (119905) 119888 (119905)

1

119898 (119905)

]

119879

(7)

The process noise and the measurement noise areassumed to be sequentially uncorrelated and have a Gaussiandistribution with zero mean This white noise disturbanceassumption comes from the theoretical formulation of theEKF [44] but is also a common premise in operational modalanalysis (OMA) If the process noise for themotion dynamicscomponents is denoted by w

119904

(119905) and the process noise forthe parameters is w

119901

(119905) the nonlinear continuous functionexpressing the dynamics of the augmented model with noisewill be

x (119905) = 119891119888

(x (119905) p (119905) 119906 (119905)) + w119904

(119905)

p (119905) = 0 + w119901

(119905)

(8)

for themotion andparameter dynamicswhich are assumed toremain constant between the individual samples Combiningthe motion dynamics and parameter noise into a single term119908(119905) = [119908

119904

(119905) 119908119901

(119905)]

119879 enables us to express the augmenteddynamics compactly by

x119886

(119905) = 119891119888

(x119886

(119905) 119906 (119905)) + w (119905)

119911 (119905) = ℎ119888

(x119886

(119905) 119906 (119905)) + V (119905) (9)

where the continuous function 119891119888

defines the augmentednonlinear system dynamics ℎ

119888

is a continuous measurementfunction and V(119905) is the measurement noise The properties

of the process noise are defined using the covariance matrix[42]

119908 (119905) sim 119873 (0Q119905

) (10)

where the discretized analogy of the process noise is obtainedby the relation

Q =

Q119905

119879119904

(11)

where 119879119904

is the measurement sampling period Similar to theprocess noise covariance the properties of the measurementnoise are given by [42]

V (119905) sim 119873 (0R) (12)

This work assumes the covariance matrices Q119905

and R toremain constant during the operation of the fault detectionsystem This implies that the statistical characteristics of theprocess noise of the dynamic states and parameters andthe measurement noise are assumed to remain unchangedWhile it is possible to conceive a modification to the EKFwith robust rescalable and possibly adaptive process andmeasurement noise covariance matrices [45ndash47] this isbeyond the scope of this paper

Modern control systems operate on a discrete basis Thenonlinear estimation procedure introduced in the followingsystem will be also implemented on an embedded microcon-troller unit performing measurements at discrete intervals119905 = 119879(119896) 119896 = 1 2 3 ( ) The continuous augmenteddynamics in (9) will be propagated using the Eulerrsquos methodby the system

x119886(119896+1)

= 119891119889

(x119886(119896)

119906(119896)

) + w(119896)

119911(119896)

= ℎ119889

(x119886(119896)

119906(119896)

) + V(119896)

(13)

213 Joint State and Parameter Estimation The extendedKalman filter is proposed in this work to estimate thenonlinear augmented states and parameters of the activestructure [43 44] The algorithm considered here is oftenreferred to as the continuous-discrete or hybrid formulationin literature [42ndash44] since the EKF takes its measurementsand yields new state estimates at discrete sample times whilethe system model is evaluated by numerical integration Theobserver algorithm is initialized by a first state estimate at zerotime

x+1198860

= 119864 [x1198860

] (14)

that is determined by the user as an initial guess of the statesand parameters Similarly the initial covariance matrix of theerror of this state estimate is given by the user as

P+0

= 119864 [(x1198860

minus x+1198860

) (x1198860

minus x+1198860

)

119879

] (15)

The primary estimatemdashor a priori estimate as it is oftendenoted in literaturemdashof the augmented state is obtainedusing simulation The dynamics of the driven point-mass-damper with the unknown parameter vector is described by


the continuous augmented function 119891(x+119886(119896minus1)

119906(119896)

) More-over another assumption is that the parameters p(119905) =

[119896(119905) 119887(119905) 119888(119905) 1119898(119905)]

119879 remain constant between the sam-ples The continuous state is then propagated using

xminus119886(119896)

= 119891119888

(x+119886(119896minus1)

119906(119896)

) (16)

between samples 119905 = (119896 minus 1)119879119904

to 119905 = (119896)119879119904

to obtain apriori estimates denoted here by the minus subscript Thoughseveral methods exist to solve ordinary differential equationsnumerically this work utilizes the first order Euler methodThe continuous augmented function 119891

119888

(x+119886(119896minus1)

119906(119896)

) can besafely considered as

119902 (119905) = 1

(119905) = 1199092

(119905)

119902 (119905) = 2

(119905)

= minus1199091

(119905) 1199093

(119905) 1199096

(119905) minus 1199092

(119905) 1199094

(119905) 1199096

(119905)

+ 1199095

(119905) 1199096

(119905) 119906 (119905)

119896 (119905) =

3

(119905) = 0

119887 (119905) =

4

(119905) = 0

119888 (119905) = 5

(119905) = 0

1

(119905)

= 6

(119905) = 0

(17)

for flexible active structures with dominant first resonantfrequencies

In addition to the computation of the a priori stateestimate it is also necessary to evaluate the a priori updateof the covariance matrix of the state estimates

P (119905) = Z (x119886

(119905))P (119905) + P (119905)Z119879 (x119886

(119905)) +Q119905

(18)

where Z(x119886

(119905)) is the Jacobian of (17) with respect to theaugmented state

Z (x119886

(119905)) =

120597119891119888

(x119886

(119905))

120597x119886

(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=x119886(119905)

(19)

which for the formerly described augmented beam model itis given by

Z (x119886

(119905)) =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0 1 0 0 0 0

minus1199096

(119905) 1199093

(119905) minus1199096

(119905) 1199094

(119905) minus1199096

(119905) 1199091

(119905) minus1199096

(119905) 1199092

(119905) minus1199096

(119905) 119906 (119905)

minus1199091

(119905) 1199093

(119905) minus

minus1199092

(119905) 1199094

(119905) minus

minus119906 (119905) 1199095

(119905)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(20)

Similar to the numerical propagation of the augmented statethe covariance matrix is obtained by simulating (18) fromsample 119905 = (119896 minus 1)119879

119904

to 119905 = (119896)119879119904

The discrete part of the EKF uses the current measure-

ment to update the a priori state to get a revised a posterioristate estimate which is denoted by the + subscript The firststep is to compute the Kalman gain K

(119896)

using

K(119896)

= Pminus(119896)

L119879 [LPminus(119896)

L119879 +MRM119879]minus1

(21)

The a priori state estimate is then revised by the mea-surement at time (119896) with its relative importance weightedto the simulation estimate by the Kalman gain K

(119896)

whichis computed at the next step The a posteriori update of theaugmented state is

x+119886(119896)

= xminus119886(119896)

+ K(119896)

[119911(119896)

minus ℎ (xminus119886(119896)

119906(119896)

)] (22)

The state error covariance matrix is updated by

P+(119896)

= [I minus K(119896)

L]Pminus(119896)

[I minus K(119896)

L]119879

+ K(119896)

MRM119879K119879(119896)

(23)

where the expression L is the Jacobian of the measurementequation ℎ

119888

(x119886(119905)

) with respect to the state x119886(119905)

L =

120597ℎ119888

(x119886(119905)

)

120597x119886(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=xminus119886(119905)

=

[

[

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0

1199096

(119905) 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

]

]

]

]

]

]

]

]

]

]

]

]

(24)


and matrix M is the Jacobian of ℎ119888

(x119886(119905)

) with respect to themeasurement noise term k

119886(119905)

M =

120597ℎ119888

(x119886(119905)

)

120597k119886(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=xminus119886(119905)

= 1 (25)

The a posteriori augmented state estimate x+119886(119896)

and covari-ance matrix estimate P+

(119896)

are passed onto the next computa-tion step where they become the basis for the next a prioriestimates

214 Resonance Estimates The first dominant resonant fre-quency of the beam is approximated on the basis of theestimated parameters p(119896) outside the observer algorithmGiven the parameter estimates

119896(119896)119887(119896) (119896) at 119905 = 119896119879

119904

and assuming that the damped natural frequency of thedynamically analogous SDOF system in (1) is equal to thedominant resonance frequency of the structure [14] thefrequency is

119891(119896)

=

radic4119896(119896)

(119896)

minus1198872

(119896)

4120587(119896)

(26)

The algorithm monitors this frequency estimate throughoutthe operation of the system As it has been noted by Jassimet al [20] and others monitoring the change of naturalfrequency is a feasible tool to indicate damage occurrence Incase that the estimate is outside the tolerance band

119890(119896)

=

0 if (119891119899

minus 119891119905

) le119891(119896)

ge (119891119899

+ 119891119905

)

1 if (119891119899

minus 119891119905

) ge119891(119896)

1 if (119891119899

+ 119891119905

) le119891(119896)

(27)

a fault detection signal 119890(119896)

is produced and retained unlesscleared The resonance of the nominal structure is denotedby 119891119899

and 119891119905

is a symmetric tolerance band

215 Algorithm Summary The resulting real-time faultdetection strategy with joint state and parameter estimationcan be summarized in the following algorithm

Algorithm 1 Perform the following operations at each sam-pling instant (119896)

(1) Propagate (16) and (18) by numerical simulation usingthe Euler method Obtain the a priori estimates of theaugmented state xminus

119886(119896)

and covariance matrix Pminus(119896)

(2) Sample the actual deflection 119911

(119896)

at the end of thecantilever beam

(3) Compute the Kalman gain K119891(119896)

using (21)(4) In order to compute the a posteriori state estimate

x+119886(119896)

use the new measurement sample 119911(119896)

andKalman gain update K

119891(119896)

to revise the a priori stateestimate through (22)

(5) Compute the a posteriori update of the covariancematrix P+

(119896)

in (23)

(6) Retain the a posteriori state and covariance estimatesx+119886(119896)

and P+(119896)

in the memory for the next estimationstep

(7) Approximate the damped natural frequency 119891(119896)

ofthe beam according to (26)

(8) If the frequency estimate 119891(119896)

is outside the toleranceband in (27) enable the fault detection signal 119890

(119896)

(9) At the next sample restart the procedure from 1 usethe estimates from Step 6

22 Experimental Hardware and Software

221 Experimental Hardware The proposed fault detectionsystem was tested on a thin flexible cantilever beam withpiezoceramic actuation The test hardware composition willbe described only briefly as the test bed was previouslyintroduced in detail in previous work including [5 35 48]The schematic setup of the experimental hardware includingthe active thin structure and the prototyping system isillustrated in Figure 1

The cantilever is a 550 times 40 times 3mm beam fixed at oneend and freely moving at the other As it is indicated inthe side view of the beam in Figure 1 it undergoes flexuralvibration in the direction of its thicknessThebeam ismade ofcommercially pure aluminium havingmechanical propertieswhich render the structure more flexible than it would bepossible with regularly available aluminium alloys

To further decrease its first resonant frequency a heavynut bolt and a set of four washers are mounted at the freeend of the beam The nominal weight installed at the endis necessary to shift its original resonance (817Hz) to thearea where is possible to run the proposed fault detectionsystem in real-time on the hardware with limited computingpower and automatically deployed code The weight of thealuminum cantilever beam was 119898

119887

= 0182 kg and the totalweight of the nut-washer-bolt assembly is 119898

119908

= 0305 kgmaking the total physical weight of 119898 = 0487 kg This baseconfiguration assumed here therefore consists of the beamwith a weight with a first nominal resonance decreased to226Hz This nominal configuration is taken as the initialconfiguration in each experiment which is then furthermodified by the experiments

An industrial laser triangulation device is mounted nearthe free end of the beam to provide position measurementsfor the fault detection algorithm The Keyence LK-G 32 lasersensor head is pointed at a single location at the beam endwhile its signal is preprocessed with a low pass filter inthe Keyence LK-G3001V processing unit The Keyence LK-G3001V unit provides an analog signal scaling the plusmn5mmmaximal measurement range into a 0ndash5V signal compatiblewith the analogmeasurement range of the prototyping boardThe laser processing unit is directly connected to the AI1analog input port of MCU The measurements are rescaledand shifted to readings in meters within the initial portion ofthe estimation algorithm

The actuation of the active thin structure is realizedwith a set of two double layer piezoceramic transducers


The MIDE QuickPack QP45n double layer actuators aremounted on both fixed sides of the beam and receive thesame electric signal with opposing polarity The high voltagedriving signal is provided through a MIDE 1225 capacitiveprecision amplifier with a 20timesVV gain on the signal inputThe active control signal is emulated by a pseudorandombinary voltage signal having its low level at 0V and high levelat 4V resulting in 80V impulses to the transducersThePRBSsignal has a 10 bits base sample (in order to excite the firstresonance of the beam) and is generated by an Agilent 3351Asignal generator unit In addition to the measured positioninput 119911

(119896)

the extendedKalmanfilter requires the input signal119906(119896)

as well therefore the output from the signal generatoris directly connected to the AI1 analog input port of theprototyping board and then rescaled to the amplified levelswithin the EKF algorithm

The low-cost electronics prototyping board employedin this work is an Arduino Mega 2560 Revision 3 devicefeaturing a simple 8-bit Atmel ATmega 2560 microcontrollerunit (Figure 2) The MCU used in this board has a clockspeed of 16MHz and a 256 kB large flash memory for storinguser algorithms [49] enabling its online external use withthe Simulink product range The detected fault signal 119890

(119896)

ispassed on to the DO9 digital output of the board with an LEDconnected to it for signal indication Although this is only abinary information relating to the occurrence of amechanicalfault the parameter estimates may also be output using apulse-width modulated signal or encoded into several digitaloutputs The board is powered through the USB port whichalso serves as the means for programming and online datatransfer for logging and storage Programming of the boardand data logging is realized using a laptop computer with anIntel Core i3 CPU running at 24GHz and 4GB operatingmemory The computing power of the design computer hasno effect on the speed of the proposed algorithm as it isrunning in a stand alone mode on the Arduino Mega 2560prototyping board

222 Experimental Software The proposed estimation andfault detection algorithm was developed within the Math-works Matlab and Simulink framework The software wasimplemented in Matlab and Simulink 80 (R2012b) runningunder the Microsoft Windows 7 times 64 operating system Inaddition to the programming role of this software imple-mentation setup the prototyping board was also providingthe estimates and other signals through the USB connectionin external mode There was an approximately 4 s com-munication delay between the prototyping board and theSimulink data logging and visualizing interface Even thoughthis portion of the experiments is not shown on figures herethe proposed algorithmwas still fully functional on theMCUduring this initialization period

The simplified Simulink scheme of the software imple-mentation is featured in Figure 3 Communication betweenthe analog input and digital output ports of the ArduinoMega 2560 board and the real-time logging capability wasensured using the Simulink Target for use with ArduinoHardware (v 20) suite providing access to the peripheralsof the MCU and a possibility to deploy Simulink schemes

Pin 0Position

Pin 1Voltage

Pin 9Fault

USBLoggingConversion

Conversion Observer

EKF

z(k)

u(k) e(k)

x (k)intgtdouble

intgtdouble

Figure 3 Block scheme of the algorithm running on the microcon-troller unit

onto the hardware The 10-bit in-processor AD converter[49] provides a decimal integer reading of the voltages whichis converted using type conversion blocks The fault signalis transferred into the digital port using the correspondingoutput block Internal state estimates are displayed and savedusing a scope block with its logging capability enabled

In addition to the readily available standardized Simulinkblocks the core of the algorithm featured previously wasdeveloped in the Matlab M-script language and then auto-matically transcribed into a CC++ version for the embed-ded MCU The custom algorithm block noted as EKF inFigure 3 contains means to manipulate voltage inputs tophysical units the extended Kalman filter providing state andparameter estimates and the simple algorithm generating thefault signal based on the calculated damped first resonantfrequency This M-script code is automatically recompiledinto embedded CC++ code and downloaded to the targethardware using the Embedded Coder (v 63) and SimulinkCoder (v 83) The compilation itself is performed by theMicrosoft Visual C++ 2010 compiler

23 Experiment Design and Settings

231 Experiment Design and Procedures The experimentsfeatured in this paper were designed to demonstrate thefunctionality of the proposed estimation and fault detectionalgorithm and the viability of its real-time implementationon low-cost microcontroller hardware Experiments werelasting 80 s with the pseudorandom binary driving signalsupplied to the actuators during the entire duration of the testAfter initializing the experiments with completely identicaltuning and settings the estimated dynamic states and theparameters converged to the values in agreement with themechanical properties of the nominal beam The majority ofthis starting convergence is not indicated in this paper dueto the previously mentioned communication issue betweenthe MCU and the laptop computer which is performing datalogging and visualization

In addition to a nominal beam configuration with thebeam itself and the heavy bolt-washer-nut combination 6types of different weights ranging from 16 g to 345 g wereadded to the end of the beam approximately at the sim40 s timemark (see Figure 4) The added weights emulated the effectsof mechanical failures and changes with different severityAs these weights were placed manually there may be a plusmn1-2 s variance in this time between the individual tests The


MeasuredEstimated

0

20

40

60

80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

u(V

)times10minus3

5

0

minus5

005

0

minus005

CalculatedEstimated

z=q1x

1(m

m)

Δzasympq2x

2(m

sminus1 )

Figure 4 Estimated dynamic states for the experiment with the246 g added weight

weights and individual experiment types are summarizedin Table 1 The added weights changed the overall dynamicbehavior of the beam and yielded parameters which divergedfrom the nominal values If the approximate of the firstdamped resonance

119891(119896)

left the 119891119905

= plusmn01Hz toleranceband of the nominal first resonance at 119891

119899

= 227Hz thealarm signal 119890

(119896)

was switched on and this fact was loggedalong with the rest of the parameters The first resonantfrequency of the thin structure in its nominal configurationwas determined by supplying the piezoceramic transducerswith a PRBS signal and performing a position measurementThis time domain measurement was then analysed using thefast Fourier transform (FFT) to extract the first fundamentalresonant frequency

232 Experimental Parameters and Settings All experimentswere initialized with the initial state estimate of

x+1198860

= [0 0 100 1 1119864 minus 4

1

05

]

119879

(28)

denoted in the experiments with an upside down blacktriangle located at zero time Likewise the initial covariance

Table 1 Summary of experimental scenarios

Number Weight Change Nominal Color1 0 g 0 Blue2 16 g 3 M Red3 50 g 10 M Green4 99 g 20 M Black5 195 g 40 M Magenta6 246 g 50 M Orange7 345 g 70 M Grey

estimates remained the same throughout the experimentswith

P+0

=

[

[

[

[

[

[

[

[

[

[

[

[

1119864 minus 5 0 0 0 0 0

0 1119864 minus 4 0 0 0 0

0 0 1119864 + 5 0 0 0

0 0 0 1119864 minus 2 0 0

0 0 0 0 1119864 minus 5 0

0 0 0 0 0 1119864 + 0

]

]

]

]

]

]

]

]

]

]

]

]

(29)

The values in the P+0

matrix provide a compromise between arealistic guess of the variance in the initial state estimate errorand an incremental tuning of the EKF algorithm

The covariance of the single measurement signal wasestimated based on the square of the maximal amplituderesolution of the AD input of the prototyping board inmeters yielding sim25 V at 10 bits [49] or in the terms of themeasurement covariance

R = 119877 = 1119864 minus 5 (30)

The greatest freedom in tuning the behavior of the algorithmcan be achieved by manipulating the continuous-time pro-cess noise covariance matrix The process noise matrix wastuned for best performance based on incremental testing andthe expected range of changes in the physical variables andthen fixed for all experiments featured in this paper as

Q119905

=

[

[

[

[

[

[

[

[

[

5119864 minus 3 0 0 0 0

0 1119864 + 1 0 0 0

0 0 1119864 minus 1 0 0

0 0 0 1119864 minus 12 0

0 0 0 0 3119864 minus 1

]

]

]

]

]

]

]

]

]

(31)

with its discrete-time analogy recomputed by Q = Q119905

119879119904

Global sampling times were set as 119879

119904

= 01 s while thecontinuousmodel was numerically integrated using the Eulermethod with a 119889119905 = 002 s time step

3 Results and Discussion

31 Dynamic State Estimates A typical experiment is shownin Figure 4 where the nominal beam configuration is altered


Table 2 Summary of estimated results (mean estimates at 119905 = 50ndash60 s)

Number Weight Detected 119891 (Hz) (kg)

119896 (Nmminus1) 119887 (Nsmminus1) 119888 (NkVminus1)

1 0 g M 226 042 838 17 052 16 g 765 s 223 042 832 18 053 50 g 492 s 220 044 838 18 044 99 g 461 s 211 047 829 17 045 195 g 444 s 198 056 869 19 046 246 g 417 s 191 060 865 22 037 345 g 399 s 181 061 790 19 02

by adding the 246 g weight to the free end at time 119905 = 414 sthereby simulating a mechanical failure of the structure Theresults logged during the experiments (from top to bottom)are the estimated and measured position 119909

1

119911 asymp 119902 theestimated and calculated velocity 119909

2

Δ119911 asymp 119902 and themeasured voltage input to the transducers 119906 The PRBS inputsignal is supplied constantly regardless of the change

Both dynamic state estimates start from the commoninitial state of x+

0

= [0 0]

119879 shown as black triangles inthe figure As it has been previously mentioned the initialportion of the measurement cannot be logged due to theproperties of the hardware implementation The differencein the position and velocity estimates before and after theemulated failure is visually observable The amplitude of theestimated position and speed signals becomes smaller afterthe failure while due to the decrease in the first resonantfrequency the oscillations are less frequent which is alsovisually noticeable in the figure

The estimated position 1199091(119896)

is compared to the measuredsignal 119911

(119896)

As it is evident from the figure the estimates agreewith the measurement very well The estimate error 119909

1

=

|1199091(119896)

minus 119911(119896)

| was calculated to be 1199091

= 38119864 minus 5 plusmn 35119864 minus 5mfor this demonstration case As direct speed measurementsare not available the estimated speed 119909

2(119896)

is compared to aspeed signal calculated from the positionmeasurement using[38]

Δ119911(119896)

asymp

119911(119896+1)

minus 119911(119896minus1)

2119879119904

(32)

The difference between the computed and estimated velocityis larger than that in the case of the position The EKFalgorithm tends to overestimate the velocity in comparisonwith the calculated values However one has to note thatthe calculated velocity signal is hardly exact and cannot beconsidered as the conventionally true velocity It is not lesslikely that the estimates are actually closer to the true velocitythan the calculated values Keeping this in perspective thevelocity estimate error 119909

2

= |1199092(119896)

minus Δ119911(119896)

| was calculated tobe 1199092

= 49119864 minus 3 plusmn 41119864 minus 3msminus1 for this demonstration case

32 Parameter Estimates and Failure Detection The esti-mated first resonance frequency with the instances of faultdetection (vertical dashed lines) for the different failureemulation cases is illustrated in Figure 5 along with theestimated model parameters The figure shows (from topto bottom) the estimates of the first damped resonance

frequency 119891 the dynamic mass of the structure = 119909

6

the dynamic equivalent stiffness coefficient

119896 = 1199093

thedynamic equivalent damping

119887 = 1199094

and the voltage-force conversion coefficient 119888 = 119909

5

Again the hardwarecommunication with the Arduino board prevents the first38ndash45 s of the measurement to be logged however the real-time algorithm running on theMCU is fully functional at thistime and the parameters estimates are converging from theinitial estimates (triangles at 119905 = 0 s) to the nominal values

After a brief convergence period the frequency estimatestend to the neighborhood of the nominal frequency withinplusmn0015Hz (maximal mean value between 119905 = 20ndash30 s) wellwithin the tolerance band of the failure detection limits Thisis also true for the estimated parameters the nominal massstiffness damping and conversion coefficients are located inthe same relative region for all experimental scenarios Themost variation in the nominal case is visible in the conversioncoefficient which is harder to estimate in general [33]

After adding the different weights to emulate the failurethe parameters start to diverge from the nominal levels andso do the estimated resonances As soon as the estimatedresonance frequency leaves the preset tolerance band thefault signal is activated The time of detection dependsentirely on the severity of the fault and varies between 0 s forthe largest weight to sim365 s for the smallest (see Table 2)Theestimated frequency naturally corresponds to the amount ofweight added and it varies between the nominal frequency119891 = 226Hz to an average of

119891 = 181Hz for the largestweight (see Table 2)

The estimated parameters react to the change in mechan-ical properties on various levels (see Table 2) Computedas mean values between the times 119905 = 50ndash80 s the massparameter varies the most as one would expect ranging from = 042 kg to = 061 kg for the different fault emulationcases The stiffness and damping coefficients do not vary asmuch for the individual cases average stiffness estimates 119896stay within 10 of the nominal stiffness while the shift inthe damping coefficient is also moderate The effect of theactuators tends to diminish as greater and greater weightsare placed at the end of the beam This is demonstrated inFigure 5 where the estimated voltage conversion parameter 119888starts to decrease with the increasing weight as the actuatorsare capable to provide less force with more weight

While the first resonant frequency estimates are stablein all experimental scenarios the parameter estimates revealpossible stability issues with the heaviest weight (Experimentnumber 7 grey) Although the initial convergence after the


18

2

22

24

26

04

05

06

07

0

20

40

60

80

100

120

0

1

2

3

0

02

04

06

08

1times10minus3

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

f(H

z)m

(kg)

0g16 g50g99g

195g246g345g

c(N

Vminus

1 )b

(Ns m

minus1 )

k(N

mminus

1 )

Figure 5 Estimated first resonance and model parameters

change is meaningful the parameters begin to decline to lev-els that are not representative for the change This may causenumerical instability in the estimation algorithm whichmdashin case only a binary fault signal is requiredmdashdoes not posea problem as it is likely that the change will be detectedmuch earlier than the instability occurs This divergence of

parameters is largely due to the fact that the conversion coef-ficient 119888 will be understandably compromised if extensivemechanical faults are present

It is important to note that the identified physical param-eters are meaningful for the dynamic model of the structureand are not necessarily identical to the parameters one would


Table 3 Verification experimental results (mean EKF estimates at 119905 = 40ndash80 s)

Experimental setting 0 g 16 g 50 g 99 g 195 g 246 g 345 gPeriodogram estimate (Hz) 2289 2239 2214 2139 2015 1940 1841EKF estimate (Hz) 2261 2238 2209 2131 2014 1945 1835Abs difference (Hz) 0028 0001 0005 0008 0001 0005 0006Percentual difference () 122 004 0023 037 005 026 033

obtain using exact first-principle measurements Althoughthe estimated parameters have a physical meaning and sensethey are evaluated in a way to match the measured positionto the behavior of the model in accordance with the generalEKF formulation For example a change in the estimatedmass is rather a dynamic change and one cannot expect toobtain the exact weight change The nominal beam-weightconfiguration with no additional change is 119898

0

= 0487 kgwhich is estimated to a lower value of = 420 kg This isto be expected and normal as the algorithm deals with theparameters only on the level of the dynamic behavior of thestructure Also if only one physical parameter is changedmdashasit is the case in this workmdashits effect is translated to a changein all parameters

The global sampling times used in this experimentwere too fast for the external mode Simulink simulationand caused a computation overflow meaning that the timenecessary to evaluate the algorithm along with the communi-cation and logging procedures was longer than the allocatedsampling time The overflow detection was enabled in thesimulation and indicated by lighting an LED connected tothe output port DO13 of the ArduinoMega 2560 prototypingplatform This overflow posed only a problem in the casewhere the algorithm was running with the data loggingfacilities enabled In case the fault detection scheme ran onthe MCU in a stand-alone mode the sampling periods werestrictly enforced Data logging is only necessary to present theexperimental results in this work therefore it is not consid-ered to be a serious limitation of the application Moreoverthe estimated parameters can be output in computationallymore efficient ways than the use of USB communicationTheproposed fault detection algorithm was able to run down toa sampling rate of 119879

119904

= 008 s on the Atmel ATmega 2560MCU Increasing the number of integration steps has thegreatest effect on computation load as a 119889119905 = 119879

119904

20 s wasunable to run without overflows with a 119879

119904

= 01 sampling

33 Verification of the Frequency Estimates The verifica-tion of the mean EKF frequency estimates was carriedout by comparing the mean observed first resonance

119891 ineach experimental scenario to the first resonant frequencyobtained from the power spectral density (PSD) 119878

119909119909

(119891) of themeasured displacement signal The mean frequency estimatewas computed from the EKF results after the additionalweights have been placed on the beam that is between 40ndash80 s of the measurements The power spectral density ofthe displacement measurement signal 119911

(119896)

was computed onthe same interval The spectral estimate was acquired using

1 12 14 16 18 2 22 24 26 28 3

minus35

minus40

minus45

minus50

minus55

minus60

minus65

minus70

minus75

minus80

Frequency f (Hz)Po

wer

S xx(f

)(d

BH

z)Nominal16 g50g99g

195g246g345g

Figure 6 Comparison of power spectral density frequency esti-mates with average EKF estimates

the Yule-Walker parametric autoregressive analysis of thedisplacement signal assuming an 8th order autoregressivemodel of the PSDThe results of this comparison are demon-strated in Figure 6 where the PSD from the displacementis marked by the thicker solid lines while the vertical thindashed lines indicate the average EKF estimates From thisit is clear that the difference between the EKF estimates andthe first resonances judged by the raw signal itself match verywell

To quantify this verification procedure in more detailTable 3 lists the periodogram estimates EKF estimates andtheir absolute and percentual difference The absolute differ-ence of the frequencies is well below |Δ119891| = 005Hz in allexperimental cases meaning a difference of lt15 in all thelisted cases

Note that comparing the direct coefficient estimates(

119887119896 119888) to the true or assumed values would be inconse-

quential in this case While the frequency estimates of theEKF and the real resonant frequency of the structure canbe directly related and verified the simple model assumedin (17) does not allow a direct assessment of the observeraccuracy As it has been previously noted the coefficients ofthe model represent the dynamic equivalent of the systemnot a physically accurate mechanical system The estimationof the dynamically equivalent

(119896)

119887(119896)

119896(119896)

and 119888(119896)

is usefulfor structural detection and monitoring or as online iden-tification in adaptive control but naturally it is absurd toexpect them to fully represent the beam from the mechanicalstandpoint


4 Conclusions

This paper proposed an algorithm framework for detectingmechanical faults in thin active cantilever beams and othersmart structures with dominant first resonance frequenciesThis structural health monitoring algorithm was then imple-mented on a low-cost embedded microcontroller unit

The presented experimental results verify the feasibil-ity of the hardware implementation and demonstrate thefunctionality of the fault detection algorithm The emulatedmechanical faults were detected in a matter of seconds Inaddition to the digital fault signal the algorithm is capableof identifying the equivalent parameters of the single degreeof freedom driven mass-spring-damper representing thedynamics of the active structure

Upon contrasting the estimated first resonance frequen-cies it has been found that they match very well to the truefrequencies assessed by the power spectral density estimate ofthe displacement measurement signal

The embedded 8-bit microcontroller unit considered inthis work enables sampling speeds down to 119879

119904

= 008 s Thissuggests that the use of the proposed fault detection methodwith the Atmel ATmega 2560 MCU is limited to thin activestructures with a largely dominant first resonance frequencyof up to sim3-4Hz Of course this work used a machine-transcribed real-time code for the experiments therefore thisis the worst case scenario for the usability of the method onlow-priced hardware

Commonly available 8-bit microcontrollers like theAtmel ATmega328 [50] or Texas Instruments C2000 [51]series with an even lower price range are likely to run theproposed algorithmwithout issues In case the EKF algorithmis directly transcribed to a lower-level language (C++ orassembler) it is safe to assume that either the complexityof the algorithm can be increased in order to estimate morethan one resonant frequency or the sampling speed may beincreased for stiffer structures

Excluding the cost of other componentsmdashsuch as aMEMS semiconductor accelerometer instead of a preciselaser system as used heremdashthe unit price of the AtmelATmega328 embedded microcontroller is currently under2 EUR for quantities over 100 pieces [52] rendering theproposed fault detection system viable for mass productionand deployment

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to gratefully acknowledge the finan-cial support granted by the SlovakResearch andDevelopmentAgency (APVV) under the Contracts APVV-0090-10 andAPVV-0131-10 and by the Scientific Grant Agency (VEGA)of the Ministry of Education Science Research and Sport ofthe Slovak Republic under the Contract 1014415

References

[1] C R Fuller ldquoActive control of sound transmissionradiationfrom elastic plates by vibration inputs I Analysisrdquo Journal ofSound and Vibration vol 136 no 1 pp 1ndash15 1990

[2] A Preumont and K Seto Active Control of Structures JohnWiley amp Sons Chichester UK 3rd edition 2008

[3] C Edwards ldquoThe 8-bit strikes backrdquo IETElectronics Systems andSoftware vol 5 no 2 pp 36ndash39 2007

[4] D C Hyland and L D Davis ldquoToward self-reliant control foradaptive structuresrdquo Acta Astronautica vol 51 no 1ndash9 pp 89ndash99 2002

[5] G Takacs T Poloni and B Rohalrsquo-Ilkiv ldquoAdaptive modelpredictive vibration control of a cantilever beam with real-timeparameter estimationrdquo Shock and Vibration vol 2014 ArticleID 741765 15 pages 2014

[6] M-TVakil-BaghmishehM PeimaniMH Sadeghi andMMEttefagh ldquoCrack detection in beam-like structures using geneticalgorithmsrdquo Applied Soft Computing Journal vol 8 no 2 pp1150ndash1160 2008

[7] F S BuezasM B Rosales andC P Filipich ldquoDamage detectionwith genetic algorithms taking into account a crack contactmodelrdquo Engineering Fracture Mechanics vol 78 no 4 pp 695ndash712 2011

[8] P M Pawar and R Ganguli ldquoGenetic fuzzy system for onlinestructural health monitoring of composite helicopter rotorbladesrdquo Mechanical Systems and Signal Processing vol 21 no5 pp 2212ndash2236 2007

[9] F Kang J-J Li and Q Xu ldquoDamage detection based onimproved particle swarm optimization using vibration datardquoApplied Soft Computing Journal vol 12 no 8 pp 2329ndash23352012

[10] R O Curadelli J D Riera D Ambrosini and M G AmanildquoDamage detection by means of structural damping identifi-cationrdquo Engineering Structures vol 30 no 12 pp 3497ndash35042008

[11] W L Bayissa N Haritos and S Thelandersson ldquoVibration-based structural damage identification using wavelet trans-formrdquo Mechanical Systems and Signal Processing vol 22 no 5pp 1194ndash1215 2008

[12] M B Rosales C P Filipich and F S Buezas ldquoCrack detectionin beam-like structuresrdquo Engineering Structures vol 31 no 10pp 2257ndash2264 2009

[13] M T Vakil Baghmisheh M Peimani M H Sadeghi M MEttefagh and A F Tabrizi ldquoA hybrid particle swarm-Nelder-Mead optimization method for crack detection in cantileverbeamsrdquo Applied Soft Computing Journal vol 12 no 8 pp 2217ndash2226 2012

[14] R Y Chiang and M G Safonov ldquoDesign of119867infin

controller for alightly damped system using a bilinear pole shifting transformrdquoin Proceedings of the American Control Conference pp 1927ndash1928 June 1991

[15] R E Kalman ldquoContributions to the theory of optimal controlrdquoBoletın de la Sociedad Matematica Mexicana vol 2 no 5 pp102ndash119 1960

[16] R E Kalman ldquoA new approach to linear filtering and predictionproblemsrdquo Transactions of the ASME Series D Journal of BasicEngineering vol 82 pp 35ndash45 1960

[17] R E Kalman and R S Bucy ldquoNew results in linear filtering andprediction theoryrdquo Journal of Basic Engineering Series D vol 83pp 95ndash107 1961


[18] G L Smith S F Schmidt and L A McGee ldquoApplica-tion of statistical filter theory to the optimal estimationof position and velocity on board a circumlunar vehi-clerdquo Tech Rep NASA TR R-135 National Aeronautics andSpace Administration (NASA) Moffet Field Calif USA 1962httpsarchiveorgdetailsnasa techdoc 19620006857

[19] BAMcElhoe ldquoAn assessment of the navigation and course cor-rections for amannedflyby ofmars or venusrdquo IEEETransactionson Aerospace and Electronic Systems vol 2 no 4 pp 613ndash6231966

[20] Z A Jassim N N Ali F Mustapha and N A Abdul Jalil ldquoAreview on the vibration analysis for a damage occurrence of acantilever beamrdquo Engineering Failure Analysis vol 31 pp 442ndash461 2013

[21] J Richelot J B Guibe and V P Budinger ldquoActive control of aclamped beam equipped with piezoelectric actuator and sensorusing generalized predictive controlrdquo in Proceedings of the IEEEInternational Symposium on Industrial Electronics (ISlE rsquo04) vol1 pp 583ndash588 May 2004

[22] H Lu and G Meng ldquoAn experimental and analytical investiga-tion of the dynamic characteristics of a flexible sandwich platefilled with electrorheological fluidrdquoThe International Journal ofAdvancedManufacturing Technology vol 28 no 11-12 pp 1049ndash1055 2006

[23] B N Agrawal and H Bang ldquoAdaptive structures for largeprecision antennasrdquo Acta Astronautica vol 38 no 3 pp 175ndash183 1996

[24] M Sabatini P Gasbarri R Monti and G B PalmerinildquoVibration control of a flexible space manipulator during onorbit operationsrdquo Acta Astronautica vol 73 pp 109ndash121 2012

[25] Q Hu ldquoA composite control scheme for attitude maneuveringand elastic mode stabilization of flexible spacecraft with mea-surable output feedbackrdquoAerospace Science and Technology vol13 no 2-3 pp 81ndash91 2009

[26] E Lourens E Reynders G De Roeck G Degrande and GLombaert ldquoAn augmented Kalman filter for force identificationin structural dynamicsrdquoMechanical Systems and Signal Process-ing vol 27 no 1 pp 446ndash460 2012

[27] T F Mu L Zhou and J N Yang ldquoAdaptive extended Kalmanfilter for parameter tracking of base-isolated structure underunknown seismic inputrdquo in Proceedings of the 10th Interna-tional Bhurban Conference on Applied Sciences and Technology(IBCAST rsquo13) pp 84ndash88 IEEE Islamabad Pakistan January2013

[28] A Turnip K-S Hong and S Park ldquoModeling of a hydraulicengine mount for active pneumatic engine vibration controlusing the extended Kalman filterrdquo Journal of Mechanical Scienceand Technology vol 23 no 1 pp 229ndash236 2009

[29] K Szabat and T Orlowska-Kowalska ldquoAdaptive control oftwo-mass system using nonlinear extended Kalman Filterrdquo inProceedings of the 32nd Annual Conference on IEEE IndustrialElectronics (IECON rsquo06) pp 1539ndash1544 November 2006

[30] K Szabat and T Orlowska-Kowalska ldquoApplication of theextended Kalman filter in advanced control structure of adrive system with elastic jointrdquo in Proceedings of the IEEEInternational Conference on Industrial Technology (ICIT rsquo08) pp1ndash6 April 2008

[31] P Williams ldquoStructural damage detection from transientresponses using square-root unscented filteringrdquo Acta Astro-nautica vol 63 no 11-12 pp 1259ndash1272 2008

[32] K Erazo and E M Hernandez ldquoA model-based observer forstate and stress estimation in structural and mechanical sys-tems experimental validationrdquo Mechanical Systems and SignalProcessing vol 43 no 1-2 pp 141ndash152 2014

[33] G Takacs T Poloni and B Rohalrsquo-Ilkiv ldquoAdaptive modelpredictive vibration controlwith state and parameter estimationusing extended kalman filteringrdquo in Proceedings of the 20thInternational Congress on Sound and Vibration (ICSV rsquo13) pp6111ndash6118 Bangkok Thailand July 2013

[34] G Takacs T Poloni and B Rohal-Ilkiv ldquoAdaptive predictivecontrol of transient vibrations in cantilevers with changingweightrdquo in Proceedings of the 19th World Congress of theInternational Federation of Automatic Control (IFAC rsquo14) pp4739ndash4747 Cape Town South Africa August 2014

[35] G Takacs and B Rohal-Ilkiv Model Predictive Vibration Con-trol Efficient Constrained MPC Vibration Control for LightlyDamped Mechanical Systems Springer London UK 2012

[36] N P Jones T Shi J H Ellis and R H Scanlan ldquoSystem-identification procedure for system and input parameters inambient vibration surveysrdquo Journal of Wind Engineering andIndustrial Aerodynamics vol 54-55 pp 91ndash99 1995 ThirdAsian-Pacific Symposium onWind Engineering

[37] VNamdeo andC SManohar ldquoNonlinear structural dynamicalsystem identification using adaptive particle filtersrdquo Journal ofSound and Vibration vol 306 no 3ndash5 pp 524ndash563 2007

[38] T Poloni A A Eielsen T A Johansen and B Rohal-IlkivldquoAdaptivemodel estimation of vibrationmotion for a nanoposi-tioner with moving horizon optimized extended Kalman filterrdquoJournal of Dynamic Systems Measurement and Control vol 135no 4 Article ID 041019 2013

[39] H Benaroya andM LNagurkaMechanical Vibration AnalysisUncertainities and Control CRC Press Boca Raton Fla USA3rd edition 2010

[40] G F Franklin J D Powell andM LWorkmanDigital Controlof Dynamic Systems Addison-Wesley Boston Mass USA 3rdedition 1997

[41] D J Inman Vibration with Control John Wiley amp SonsChichester UK 2006

[42] P S Maybeck Stochastic Models Estimation and Control vol141 ofMathematics in Science and Engineering Academic PressNew York NY USA 1st edition 1979

[43] A Gelb Applied Optimal Estimation The MIT Press Cam-bridge Mass USA 1974

[44] D SimonOptimal State Estimation Kalman H1 and NonlinearApproaches Wiley-Interscience Hoboken NJ USA 1st edition2006

[45] Z Duan C Han and H Dang ldquoAn adaptive Kalman filter withdynamic resealing of process noiserdquo in Proceedings of the 6thInternational Conference of Information Fusion vol 2 pp 1310ndash1315 IEEE Cairns Australia July 2003

[46] A Khitwongwattana and T Maneewarn ldquoExtended kalmanfilter with adaptive measurement noise characteristics for posi-tion estimation of an autonomous vehiclerdquo in Proceedings ofthe IEEEASME International Conference on Mechtronic andEmbedded Systems and Applications (MESA rsquo08) pp 505ndash509IEEE Beijing China October 2008

[47] V A Bavdekar A P Deshpande and S C PatwardhanldquoIdentification of process and measurement noise covariancefor state and parameter estimation using extended Kalmanfilterrdquo Journal of Process Control vol 21 no 4 pp 585ndash601 2011


[48] G Takacs andB Rohal-Ilkiv ldquoReal-time diagnostics ofmechan-ical failure for thin active cantilever beams using low-costhardwarerdquo in Proceedings of the 7th Forum Acusticum pp 1ndash6 EuropeanAcoustics Association Krakow Poland September2014

[49] Atmel Corporation ldquoAtmel ATmega640V-1280V-1281V-2560V-2561V 8-bit atmel microcontroller with 163264KBrdquoin System Programmable Flash Atmel Corporation San JoseCalif USA 2549q-avr-022014 edition 2014 Datasheet

[50] Atmel Corporation ATmega48APA88APA168APA328PComplete Atmel 8-Bit Microcontroller with 481632KBytesIn-System Programmable Flash 8271g-avr-022013 editionDatasheet Atmel Corporation San Jose Calif USA 2013

[51] Texas Instruments C2000119879119872 Real-Time Microcontrollerssprb176s edition Datasheet Texas Instruments Dallas TexUSA 2014

[52] Digi-Key Corporation ATMEGA328P-PU Atmel |

ATMEGA328P-PU | DigiKey April 2014 httpwwwdigikeycomproduct-detailenATMEGA328P-PUATMEGA328P-PU-ND1914589

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



sensor failures or degradation material buildup fatiguecracks icing and a plethora of other factors In addition tothese process variables such as changing load or workingconditions may induce substantial changes in the structureproviding a good reason to use structural health monitoringand diagnostic systems

The requirements for diagnostic systems can range fromsimple fault indicators to complex parameter estimationprocedures Online parameter estimation allows for a moredetailed assessment of the properties of the monitoredstructures representing unexpected changes as variations inphysically interpretable parameters Moreover the estimatedparameters can be used to adjust the controllers to counteractstructural changes thereby creating an adaptive vibrationcontrol system [5]

Several exciting fault and damage detection strategieshave appeared for cantilever beams and other structures inthe recent literature Localization of the damaged section ispossible by employing progressive optimization-based algo-rithms such as genetic algorithms (GA) [6 7] fuzzy-geneticalgorithms [8] particle swarm optimization [9] wavelettransform [10 11] and artificial neural networks [12] Cracklocation and depth were detected by Vakil Baghmisheh et al[13] using a hybrid particle swarm-Nelder-Mead algorithmand modal testing Although most of these methods offermuch more than a simple status signal on the presenceor absence of a fault in the structure they require a largecomputational effort suggest but do not demonstrate onlineoperation or are simply not suitable for online applicationdue to design and technical limitations [7ndash13]

This paper introduces a prototype of an online diag-nostic system intended to detect and monitor changes inthe mechanical properties of actively controlled structuresresembling the dynamic characteristics of thin cantileverbeams In addition to monitoring the occurrence of changescaused by unexpected outside influences the system basedon a nonlinear joint state and parameter observer is capableof providing real-time estimates that can be used to createadaptive mechanical structures The proposed estimationscheme is implemented on a low-cost electronic prototypingplatform using automatized code generation tools

The approach presented here assumes that the dynamicsof active mechanical structures resembling the propertiesthin cantilever beams can be sufficiently approximated usinga single degree of freedom (SDOF) system [14]This assump-tion holds well in the case of flexible structures with singledigit firstmdashfundamentalmdashresonant frequencies dominatingthe overall response in comparison with the contributionof the higher modes The state-space model of the drivenmass-spring-damper is augmented by themonitored physicalparameters namely the unknown equivalent stiffness damp-ing mass and force conversion coefficients

The introduction of unknown parameters in the lin-ear SDOF dynamic system renders the augmented modelnonlinear This paper uses the nonlinear expansion ofthe acknowledged Kalman filter [15ndash17]mdashthe well-knownextended Kalman filter (EKF) [18 19]mdashto observe the statesand parameters of this augmented system Though it ispossible to directly compute and output the equivalent

Piezoceramicactuators

Laser CPU

Poweramplifier

Laser head

USB

Aluminumbeam

Nominal andadded weight

Excitation signal

Positionmeasurement

signal

PC withMatlabSimulink

ArduinoMega 2560Signal generator

Base

Fault signal

AI0

AI1

DO9 LED

Figure 1 Simplified hardware connection scheme of the experimen-tal system

Power

USB

Atmel ATmega 2560 MCU

Deflection output z(k)

Fault indication e(k)

Voltage input u(k)

Figure 2 Embedded microcontroller unit and the electronicsprototyping board

physical parameters for the needs of adaptive control ordetailed monitoring here the parameters estimated by theEKF are used to assess the first damped resonant frequencyof the active structure It is assumed that this remainsbounded if there is no change in the structure but willdiverge from the nominal levels as soon as a mechanicalfailure or an undesired change of physical configurationoccurs [20] The proposed structural monitoring system istested in a laboratory setting The test structure uses theclassical example of the clamped cantilever (see Figure 1)that is often employed to represent the general dynamicbehavior of real-life structures resembling flexible thin beams[14] such as fixed and rotary aerodynamic surfaces [21 22]antennae [23] manipulators [24] and others [25] The thinaluminum cantilever is equipped with two large double-layerpiezoceramic transducers mounted on opposing sides closeto the clamping mechanism The transducers are driven bya power amplifier receiving a pseudorandom-binary (PRBS)signal The motion of the beam is measured at a singlelocation near the end of the structureThe actuator inputs andthe measurements are fed to a low-cost microcontroller andelectronics prototyping platform featuring a 8-bit MCU (seeFigure 2)

The experimental validation procedure of the proposedparameter monitoring and fault detection scheme involvessupplying the beam with the PRBS signal emulating thecontrol input to the actuator In the first stage of the exper-iments the beam is in its nominal configuration without











119898 119902 (119905) + 119887 119902 (119905) + 119896119902 (119905) = 119865 (119905) (1)


119865 (119905) = 119888119906 (119905) (2)



1

(119905) = 119902(119905) and velocity1199092


x (119905) = Fx (119905) + G119906 (119905)

119911 (119905) = Hx (119905) (3)


1

(119905) 1199092

(119905)]

119879 by using




F =[

[

0 1

minus

119896

119898

minus

119887

119898

]

]

(4)


G =[

[

0

119888

119898

]

]

(5)


x119886

(119905) = [x (119905) p (119905)]

119879

(6)


p (119905) = [119896 (119905) 119887 (119905) 119888 (119905)

1

119898 (119905)

]

119879

(7)


119904


119901


x (119905) = 119891119888

(x (119905) p (119905) 119906 (119905)) + w119904

(119905)

p (119905) = 0 + w119901

(119905)

(8)


119904

(119905) 119908119901

(119905)]


x119886

(119905) = 119891119888

(x119886

(119905) 119906 (119905)) + w (119905)

119911 (119905) = ℎ119888

(x119886

(119905) 119906 (119905)) + V (119905) (9)



119888



119908 (119905) sim 119873 (0Q119905

) (10)


Q =

Q119905

119879119904

(11)

where 119879119904


V (119905) sim 119873 (0R) (12)




x119886(119896+1)

= 119891119889

(x119886(119896)

119906(119896)

) + w(119896)

119911(119896)

= ℎ119889

(x119886(119896)

119906(119896)

) + V(119896)

(13)


x+1198860

= 119864 [x1198860

] (14)


P+0

= 119864 [(x1198860

minus x+1198860

) (x1198860

minus x+1198860

)

119879

] (15)




119906(119896)


[119896(119905) 119887(119905) 119888(119905) 1119898(119905)]


xminus119886(119896)

= 119891119888

(x+119886(119896minus1)

119906(119896)

) (16)


to 119905 = (119896)119879119904


119888

(x+119886(119896minus1)

119906(119896)


119902 (119905) = 1

(119905) = 1199092

(119905)

119902 (119905) = 2

(119905)

= minus1199091

(119905) 1199093

(119905) 1199096

(119905) minus 1199092

(119905) 1199094

(119905) 1199096

(119905)

+ 1199095

(119905) 1199096

(119905) 119906 (119905)

119896 (119905) =

3

(119905) = 0

119887 (119905) =

4

(119905) = 0

119888 (119905) = 5

(119905) = 0

1

(119905)

= 6

(119905) = 0

(17)



P (119905) = Z (x119886

(119905))P (119905) + P (119905)Z119879 (x119886

(119905)) +Q119905

(18)

where Z(x119886


Z (x119886

(119905)) =

120597119891119888

(x119886

(119905))

120597x119886

(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=x119886(119905)

(19)


Z (x119886

(119905)) =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0 1 0 0 0 0

minus1199096

(119905) 1199093

(119905) minus1199096

(119905) 1199094

(119905) minus1199096

(119905) 1199091

(119905) minus1199096

(119905) 1199092

(119905) minus1199096

(119905) 119906 (119905)

minus1199091

(119905) 1199093

(119905) minus

minus1199092

(119905) 1199094

(119905) minus

minus119906 (119905) 1199095

(119905)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(20)


119904

to 119905 = (119896)119879119904



(119896)

using

K(119896)

= Pminus(119896)

L119879 [LPminus(119896)

L119879 +MRM119879]minus1

(21)


(119896)


x+119886(119896)

= xminus119886(119896)

+ K(119896)

[119911(119896)


119906(119896)

)] (22)


P+(119896)

= [I minus K(119896)

L]Pminus(119896)

[I minus K(119896)

L]119879

+ K(119896)

MRM119879K119879(119896)

(23)


119888

(x119886(119905)


L =

120597ℎ119888

(x119886(119905)

)

120597x119886(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=xminus119886(119905)

=

[

[

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0

1199096

(119905) 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

]

]

]

]

]

]

]

]

]

]

]

]

(24)



(x119886(119905)


119886(119905)

M =

120597ℎ119888

(x119886(119905)

)

120597k119886(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=xminus119886(119905)

= 1 (25)



(119896)



119896(119896)119887(119896) (119896) at 119905 = 119896119879

119904


119891(119896)

=

radic4119896(119896)

(119896)

minus1198872

(119896)

4120587(119896)

(26)


119890(119896)

=

0 if (119891119899

minus 119891119905

) le119891(119896)

ge (119891119899

+ 119891119905

)

1 if (119891119899

minus 119891119905

) ge119891(119896)

1 if (119891119899

+ 119891119905

) le119891(119896)

(27)



and 119891119905





119886(119896)



(119896)




x+119886(119896)



119891(119896)



(119896)

in (23)


and P+(119896)






(119896)






119887


119908






(119896)




(119896)




Pin 0Position

Pin 1Voltage

Pin 9Fault


Conversion Observer

EKF

z(k)

u(k) e(k)

x (k)intgtdouble

intgtdouble








MeasuredEstimated

0

20

40

60

80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

u(V

)times10minus3

5

0

minus5

005

0

minus005

CalculatedEstimated

z=q1x

1(m

m)

Δzasympq2x

2(m

sminus1 )



119891(119896)

left the 119891119905


119899


(119896)



x+1198860

= [0 0 100 1 1119864 minus 4

1

05

]

119879

(28)





P+0

=

[

[

[

[

[

[

[

[

[

[

[

[

1119864 minus 5 0 0 0 0 0

0 1119864 minus 4 0 0 0 0

0 0 1119864 + 5 0 0 0

0 0 0 1119864 minus 2 0 0

0 0 0 0 1119864 minus 5 0

0 0 0 0 0 1119864 + 0

]

]

]

]

]

]

]

]

]

]

]

]

(29)




R = 119877 = 1119864 minus 5 (30)


Q119905

=

[

[

[

[

[

[

[

[

[

5119864 minus 3 0 0 0 0

0 1119864 + 1 0 0 0

0 0 1119864 minus 1 0 0

0 0 0 1119864 minus 12 0

0 0 0 0 3119864 minus 1

]

]

]

]

]

]

]

]

]

(31)


119879119904


119904








1 0 g M 226 042 838 17 052 16 g 765 s 223 042 832 18 053 50 g 492 s 220 044 838 18 044 99 g 461 s 211 047 829 17 045 195 g 444 s 198 056 869 19 046 246 g 417 s 191 060 865 22 037 345 g 399 s 181 061 790 19 02


1


2



0

= [0 0]




(119896)


1

=

|1199091(119896)

minus 119911(119896)



2(119896)


Δ119911(119896)

asymp

119911(119896+1)

minus 119911(119896minus1)

2119879119904

(32)


2

= |1199092(119896)

minus Δ119911(119896)





6


119896 = 1199093


119887 = 1199094


5








18

2

22

24

26

04

05

06

07

0

20

40

60

80

100

120

0

1

2

3

0

02

04

06

08

1times10minus3

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

f(H

z)m

(kg)

0g16 g50g99g

195g246g345g

c(N

Vminus

1 )b

(Ns m

minus1 )

k(N

mminus

1 )









0



119904


119904


119904

= 01 sampling



119909119909


(119896)


1 12 14 16 18 2 22 24 26 28 3

minus35

minus40

minus45

minus50

minus55

minus60

minus65

minus70

minus75

minus80

Frequency f (Hz)Po

wer

S xx(f

)(d

BH

z)Nominal16 g50g99g

195g246g345g







(119896)

119887(119896)

119896(119896)

and 119888(119896)



4 Conclusions





119904






Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation



















119898 119902 (119905) + 119887 119902 (119905) + 119896119902 (119905) = 119865 (119905) (1)


119865 (119905) = 119888119906 (119905) (2)



1

(119905) = 119902(119905) and velocity1199092


x (119905) = Fx (119905) + G119906 (119905)

119911 (119905) = Hx (119905) (3)


1

(119905) 1199092

(119905)]

119879 by using




F =[

[

0 1

minus

119896

119898

minus

119887

119898

]

]

(4)


G =[

[

0

119888

119898

]

]

(5)


x119886

(119905) = [x (119905) p (119905)]

119879

(6)


p (119905) = [119896 (119905) 119887 (119905) 119888 (119905)

1

119898 (119905)

]

119879

(7)


119904


119901


x (119905) = 119891119888

(x (119905) p (119905) 119906 (119905)) + w119904

(119905)

p (119905) = 0 + w119901

(119905)

(8)


119904

(119905) 119908119901

(119905)]


x119886

(119905) = 119891119888

(x119886

(119905) 119906 (119905)) + w (119905)

119911 (119905) = ℎ119888

(x119886

(119905) 119906 (119905)) + V (119905) (9)



119888



119908 (119905) sim 119873 (0Q119905

) (10)


Q =

Q119905

119879119904

(11)

where 119879119904


V (119905) sim 119873 (0R) (12)




x119886(119896+1)

= 119891119889

(x119886(119896)

119906(119896)

) + w(119896)

119911(119896)

= ℎ119889

(x119886(119896)

119906(119896)

) + V(119896)

(13)


x+1198860

= 119864 [x1198860

] (14)


P+0

= 119864 [(x1198860

minus x+1198860

) (x1198860

minus x+1198860

)

119879

] (15)




119906(119896)


[119896(119905) 119887(119905) 119888(119905) 1119898(119905)]


xminus119886(119896)

= 119891119888

(x+119886(119896minus1)

119906(119896)

) (16)


to 119905 = (119896)119879119904


119888

(x+119886(119896minus1)

119906(119896)


119902 (119905) = 1

(119905) = 1199092

(119905)

119902 (119905) = 2

(119905)

= minus1199091

(119905) 1199093

(119905) 1199096

(119905) minus 1199092

(119905) 1199094

(119905) 1199096

(119905)

+ 1199095

(119905) 1199096

(119905) 119906 (119905)

119896 (119905) =

3

(119905) = 0

119887 (119905) =

4

(119905) = 0

119888 (119905) = 5

(119905) = 0

1

(119905)

= 6

(119905) = 0

(17)



P (119905) = Z (x119886

(119905))P (119905) + P (119905)Z119879 (x119886

(119905)) +Q119905

(18)

where Z(x119886


Z (x119886

(119905)) =

120597119891119888

(x119886

(119905))

120597x119886

(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=x119886(119905)

(19)


Z (x119886

(119905)) =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0 1 0 0 0 0

minus1199096

(119905) 1199093

(119905) minus1199096

(119905) 1199094

(119905) minus1199096

(119905) 1199091

(119905) minus1199096

(119905) 1199092

(119905) minus1199096

(119905) 119906 (119905)

minus1199091

(119905) 1199093

(119905) minus

minus1199092

(119905) 1199094

(119905) minus

minus119906 (119905) 1199095

(119905)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(20)


119904

to 119905 = (119896)119879119904



(119896)

using

K(119896)

= Pminus(119896)

L119879 [LPminus(119896)

L119879 +MRM119879]minus1

(21)


(119896)


x+119886(119896)

= xminus119886(119896)

+ K(119896)

[119911(119896)


119906(119896)

)] (22)


P+(119896)

= [I minus K(119896)

L]Pminus(119896)

[I minus K(119896)

L]119879

+ K(119896)

MRM119879K119879(119896)

(23)


119888

(x119886(119905)


L =

120597ℎ119888

(x119886(119905)

)

120597x119886(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=xminus119886(119905)

=

[

[

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0

1199096

(119905) 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

]

]

]

]

]

]

]

]

]

]

]

]

(24)



(x119886(119905)


119886(119905)

M =

120597ℎ119888

(x119886(119905)

)

120597k119886(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=xminus119886(119905)

= 1 (25)



(119896)



119896(119896)119887(119896) (119896) at 119905 = 119896119879

119904


119891(119896)

=

radic4119896(119896)

(119896)

minus1198872

(119896)

4120587(119896)

(26)


119890(119896)

=

0 if (119891119899

minus 119891119905

) le119891(119896)

ge (119891119899

+ 119891119905

)

1 if (119891119899

minus 119891119905

) ge119891(119896)

1 if (119891119899

+ 119891119905

) le119891(119896)

(27)



and 119891119905





119886(119896)



(119896)




x+119886(119896)



119891(119896)



(119896)

in (23)


and P+(119896)






(119896)






119887


119908






(119896)




(119896)




Pin 0Position

Pin 1Voltage

Pin 9Fault


Conversion Observer

EKF

z(k)

u(k) e(k)

x (k)intgtdouble

intgtdouble








MeasuredEstimated

0

20

40

60

80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

u(V

)times10minus3

5

0

minus5

005

0

minus005

CalculatedEstimated

z=q1x

1(m

m)

Δzasympq2x

2(m

sminus1 )



119891(119896)

left the 119891119905


119899


(119896)



x+1198860

= [0 0 100 1 1119864 minus 4

1

05

]

119879

(28)





P+0

=

[

[

[

[

[

[

[

[

[

[

[

[

1119864 minus 5 0 0 0 0 0

0 1119864 minus 4 0 0 0 0

0 0 1119864 + 5 0 0 0

0 0 0 1119864 minus 2 0 0

0 0 0 0 1119864 minus 5 0

0 0 0 0 0 1119864 + 0

]

]

]

]

]

]

]

]

]

]

]

]

(29)




R = 119877 = 1119864 minus 5 (30)


Q119905

=

[

[

[

[

[

[

[

[

[

5119864 minus 3 0 0 0 0

0 1119864 + 1 0 0 0

0 0 1119864 minus 1 0 0

0 0 0 1119864 minus 12 0

0 0 0 0 3119864 minus 1

]

]

]

]

]

]

]

]

]

(31)


119879119904


119904








1 0 g M 226 042 838 17 052 16 g 765 s 223 042 832 18 053 50 g 492 s 220 044 838 18 044 99 g 461 s 211 047 829 17 045 195 g 444 s 198 056 869 19 046 246 g 417 s 191 060 865 22 037 345 g 399 s 181 061 790 19 02


1


2



0

= [0 0]




(119896)


1

=

|1199091(119896)

minus 119911(119896)



2(119896)


Δ119911(119896)

asymp

119911(119896+1)

minus 119911(119896minus1)

2119879119904

(32)


2

= |1199092(119896)

minus Δ119911(119896)





6


119896 = 1199093


119887 = 1199094


5








18

2

22

24

26

04

05

06

07

0

20

40

60

80

100

120

0

1

2

3

0

02

04

06

08

1times10minus3

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

f(H

z)m

(kg)

0g16 g50g99g

195g246g345g

c(N

Vminus

1 )b

(Ns m

minus1 )

k(N

mminus

1 )









0



119904


119904


119904

= 01 sampling



119909119909


(119896)


1 12 14 16 18 2 22 24 26 28 3

minus35

minus40

minus45

minus50

minus55

minus60

minus65

minus70

minus75

minus80

Frequency f (Hz)Po

wer

S xx(f

)(d

BH

z)Nominal16 g50g99g

195g246g345g







(119896)

119887(119896)

119896(119896)

and 119888(119896)



4 Conclusions





119904






Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












F =[

[

0 1

minus

119896

119898

minus

119887

119898

]

]

(4)


G =[

[

0

119888

119898

]

]

(5)


x119886

(119905) = [x (119905) p (119905)]

119879

(6)


p (119905) = [119896 (119905) 119887 (119905) 119888 (119905)

1

119898 (119905)

]

119879

(7)


119904


119901


x (119905) = 119891119888

(x (119905) p (119905) 119906 (119905)) + w119904

(119905)

p (119905) = 0 + w119901

(119905)

(8)


119904

(119905) 119908119901

(119905)]


x119886

(119905) = 119891119888

(x119886

(119905) 119906 (119905)) + w (119905)

119911 (119905) = ℎ119888

(x119886

(119905) 119906 (119905)) + V (119905) (9)



119888



119908 (119905) sim 119873 (0Q119905

) (10)


Q =

Q119905

119879119904

(11)

where 119879119904


V (119905) sim 119873 (0R) (12)




x119886(119896+1)

= 119891119889

(x119886(119896)

119906(119896)

) + w(119896)

119911(119896)

= ℎ119889

(x119886(119896)

119906(119896)

) + V(119896)

(13)


x+1198860

= 119864 [x1198860

] (14)


P+0

= 119864 [(x1198860

minus x+1198860

) (x1198860

minus x+1198860

)

119879

] (15)




119906(119896)


[119896(119905) 119887(119905) 119888(119905) 1119898(119905)]


xminus119886(119896)

= 119891119888

(x+119886(119896minus1)

119906(119896)

) (16)


to 119905 = (119896)119879119904


119888

(x+119886(119896minus1)

119906(119896)


119902 (119905) = 1

(119905) = 1199092

(119905)

119902 (119905) = 2

(119905)

= minus1199091

(119905) 1199093

(119905) 1199096

(119905) minus 1199092

(119905) 1199094

(119905) 1199096

(119905)

+ 1199095

(119905) 1199096

(119905) 119906 (119905)

119896 (119905) =

3

(119905) = 0

119887 (119905) =

4

(119905) = 0

119888 (119905) = 5

(119905) = 0

1

(119905)

= 6

(119905) = 0

(17)



P (119905) = Z (x119886

(119905))P (119905) + P (119905)Z119879 (x119886

(119905)) +Q119905

(18)

where Z(x119886


Z (x119886

(119905)) =

120597119891119888

(x119886

(119905))

120597x119886

(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=x119886(119905)

(19)


Z (x119886

(119905)) =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0 1 0 0 0 0

minus1199096

(119905) 1199093

(119905) minus1199096

(119905) 1199094

(119905) minus1199096

(119905) 1199091

(119905) minus1199096

(119905) 1199092

(119905) minus1199096

(119905) 119906 (119905)

minus1199091

(119905) 1199093

(119905) minus

minus1199092

(119905) 1199094

(119905) minus

minus119906 (119905) 1199095

(119905)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(20)


119904

to 119905 = (119896)119879119904



(119896)

using

K(119896)

= Pminus(119896)

L119879 [LPminus(119896)

L119879 +MRM119879]minus1

(21)


(119896)


x+119886(119896)

= xminus119886(119896)

+ K(119896)

[119911(119896)


119906(119896)

)] (22)


P+(119896)

= [I minus K(119896)

L]Pminus(119896)

[I minus K(119896)

L]119879

+ K(119896)

MRM119879K119879(119896)

(23)


119888

(x119886(119905)


L =

120597ℎ119888

(x119886(119905)

)

120597x119886(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=xminus119886(119905)

=

[

[

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0

1199096

(119905) 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

]

]

]

]

]

]

]

]

]

]

]

]

(24)



(x119886(119905)


119886(119905)

M =

120597ℎ119888

(x119886(119905)

)

120597k119886(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=xminus119886(119905)

= 1 (25)



(119896)



119896(119896)119887(119896) (119896) at 119905 = 119896119879

119904


119891(119896)

=

radic4119896(119896)

(119896)

minus1198872

(119896)

4120587(119896)

(26)


119890(119896)

=

0 if (119891119899

minus 119891119905

) le119891(119896)

ge (119891119899

+ 119891119905

)

1 if (119891119899

minus 119891119905

) ge119891(119896)

1 if (119891119899

+ 119891119905

) le119891(119896)

(27)



and 119891119905





119886(119896)



(119896)




x+119886(119896)



119891(119896)



(119896)

in (23)


and P+(119896)






(119896)






119887


119908






(119896)




(119896)




Pin 0Position

Pin 1Voltage

Pin 9Fault


Conversion Observer

EKF

z(k)

u(k) e(k)

x (k)intgtdouble

intgtdouble








MeasuredEstimated

0

20

40

60

80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

u(V

)times10minus3

5

0

minus5

005

0

minus005

CalculatedEstimated

z=q1x

1(m

m)

Δzasympq2x

2(m

sminus1 )



119891(119896)

left the 119891119905


119899


(119896)



x+1198860

= [0 0 100 1 1119864 minus 4

1

05

]

119879

(28)





P+0

=

[

[

[

[

[

[

[

[

[

[

[

[

1119864 minus 5 0 0 0 0 0

0 1119864 minus 4 0 0 0 0

0 0 1119864 + 5 0 0 0

0 0 0 1119864 minus 2 0 0

0 0 0 0 1119864 minus 5 0

0 0 0 0 0 1119864 + 0

]

]

]

]

]

]

]

]

]

]

]

]

(29)




R = 119877 = 1119864 minus 5 (30)


Q119905

=

[

[

[

[

[

[

[

[

[

5119864 minus 3 0 0 0 0

0 1119864 + 1 0 0 0

0 0 1119864 minus 1 0 0

0 0 0 1119864 minus 12 0

0 0 0 0 3119864 minus 1

]

]

]

]

]

]

]

]

]

(31)


119879119904


119904








1 0 g M 226 042 838 17 052 16 g 765 s 223 042 832 18 053 50 g 492 s 220 044 838 18 044 99 g 461 s 211 047 829 17 045 195 g 444 s 198 056 869 19 046 246 g 417 s 191 060 865 22 037 345 g 399 s 181 061 790 19 02


1


2



0

= [0 0]




(119896)


1

=

|1199091(119896)

minus 119911(119896)



2(119896)


Δ119911(119896)

asymp

119911(119896+1)

minus 119911(119896minus1)

2119879119904

(32)


2

= |1199092(119896)

minus Δ119911(119896)





6


119896 = 1199093


119887 = 1199094


5








18

2

22

24

26

04

05

06

07

0

20

40

60

80

100

120

0

1

2

3

0

02

04

06

08

1times10minus3

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

f(H

z)m

(kg)

0g16 g50g99g

195g246g345g

c(N

Vminus

1 )b

(Ns m

minus1 )

k(N

mminus

1 )









0



119904


119904


119904

= 01 sampling



119909119909


(119896)


1 12 14 16 18 2 22 24 26 28 3

minus35

minus40

minus45

minus50

minus55

minus60

minus65

minus70

minus75

minus80

Frequency f (Hz)Po

wer

S xx(f

)(d

BH

z)Nominal16 g50g99g

195g246g345g







(119896)

119887(119896)

119896(119896)

and 119888(119896)



4 Conclusions





119904






Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119906(119896)


[119896(119905) 119887(119905) 119888(119905) 1119898(119905)]


xminus119886(119896)

= 119891119888

(x+119886(119896minus1)

119906(119896)

) (16)


to 119905 = (119896)119879119904


119888

(x+119886(119896minus1)

119906(119896)


119902 (119905) = 1

(119905) = 1199092

(119905)

119902 (119905) = 2

(119905)

= minus1199091

(119905) 1199093

(119905) 1199096

(119905) minus 1199092

(119905) 1199094

(119905) 1199096

(119905)

+ 1199095

(119905) 1199096

(119905) 119906 (119905)

119896 (119905) =

3

(119905) = 0

119887 (119905) =

4

(119905) = 0

119888 (119905) = 5

(119905) = 0

1

(119905)

= 6

(119905) = 0

(17)



P (119905) = Z (x119886

(119905))P (119905) + P (119905)Z119879 (x119886

(119905)) +Q119905

(18)

where Z(x119886


Z (x119886

(119905)) =

120597119891119888

(x119886

(119905))

120597x119886

(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=x119886(119905)

(19)


Z (x119886

(119905)) =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

0 1 0 0 0 0

minus1199096

(119905) 1199093

(119905) minus1199096

(119905) 1199094

(119905) minus1199096

(119905) 1199091

(119905) minus1199096

(119905) 1199092

(119905) minus1199096

(119905) 119906 (119905)

minus1199091

(119905) 1199093

(119905) minus

minus1199092

(119905) 1199094

(119905) minus

minus119906 (119905) 1199095

(119905)

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

(20)


119904

to 119905 = (119896)119879119904



(119896)

using

K(119896)

= Pminus(119896)

L119879 [LPminus(119896)

L119879 +MRM119879]minus1

(21)


(119896)


x+119886(119896)

= xminus119886(119896)

+ K(119896)

[119911(119896)


119906(119896)

)] (22)


P+(119896)

= [I minus K(119896)

L]Pminus(119896)

[I minus K(119896)

L]119879

+ K(119896)

MRM119879K119879(119896)

(23)


119888

(x119886(119905)


L =

120597ℎ119888

(x119886(119905)

)

120597x119886(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=xminus119886(119905)

=

[

[

[

[

[

[

[

[

[

[

[

[

0 0 0 0 0

1199096

(119905) 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 1

]

]

]

]

]

]

]

]

]

]

]

]

(24)



(x119886(119905)


119886(119905)

M =

120597ℎ119888

(x119886(119905)

)

120597k119886(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=xminus119886(119905)

= 1 (25)



(119896)



119896(119896)119887(119896) (119896) at 119905 = 119896119879

119904


119891(119896)

=

radic4119896(119896)

(119896)

minus1198872

(119896)

4120587(119896)

(26)


119890(119896)

=

0 if (119891119899

minus 119891119905

) le119891(119896)

ge (119891119899

+ 119891119905

)

1 if (119891119899

minus 119891119905

) ge119891(119896)

1 if (119891119899

+ 119891119905

) le119891(119896)

(27)



and 119891119905





119886(119896)



(119896)




x+119886(119896)



119891(119896)



(119896)

in (23)


and P+(119896)






(119896)






119887


119908






(119896)




(119896)




Pin 0Position

Pin 1Voltage

Pin 9Fault


Conversion Observer

EKF

z(k)

u(k) e(k)

x (k)intgtdouble

intgtdouble








MeasuredEstimated

0

20

40

60

80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

u(V

)times10minus3

5

0

minus5

005

0

minus005

CalculatedEstimated

z=q1x

1(m

m)

Δzasympq2x

2(m

sminus1 )



119891(119896)

left the 119891119905


119899


(119896)



x+1198860

= [0 0 100 1 1119864 minus 4

1

05

]

119879

(28)





P+0

=

[

[

[

[

[

[

[

[

[

[

[

[

1119864 minus 5 0 0 0 0 0

0 1119864 minus 4 0 0 0 0

0 0 1119864 + 5 0 0 0

0 0 0 1119864 minus 2 0 0

0 0 0 0 1119864 minus 5 0

0 0 0 0 0 1119864 + 0

]

]

]

]

]

]

]

]

]

]

]

]

(29)




R = 119877 = 1119864 minus 5 (30)


Q119905

=

[

[

[

[

[

[

[

[

[

5119864 minus 3 0 0 0 0

0 1119864 + 1 0 0 0

0 0 1119864 minus 1 0 0

0 0 0 1119864 minus 12 0

0 0 0 0 3119864 minus 1

]

]

]

]

]

]

]

]

]

(31)


119879119904


119904








1 0 g M 226 042 838 17 052 16 g 765 s 223 042 832 18 053 50 g 492 s 220 044 838 18 044 99 g 461 s 211 047 829 17 045 195 g 444 s 198 056 869 19 046 246 g 417 s 191 060 865 22 037 345 g 399 s 181 061 790 19 02


1


2



0

= [0 0]




(119896)


1

=

|1199091(119896)

minus 119911(119896)



2(119896)


Δ119911(119896)

asymp

119911(119896+1)

minus 119911(119896minus1)

2119879119904

(32)


2

= |1199092(119896)

minus Δ119911(119896)





6


119896 = 1199093


119887 = 1199094


5








18

2

22

24

26

04

05

06

07

0

20

40

60

80

100

120

0

1

2

3

0

02

04

06

08

1times10minus3

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

f(H

z)m

(kg)

0g16 g50g99g

195g246g345g

c(N

Vminus

1 )b

(Ns m

minus1 )

k(N

mminus

1 )









0



119904


119904


119904

= 01 sampling



119909119909


(119896)


1 12 14 16 18 2 22 24 26 28 3

minus35

minus40

minus45

minus50

minus55

minus60

minus65

minus70

minus75

minus80

Frequency f (Hz)Po

wer

S xx(f

)(d

BH

z)Nominal16 g50g99g

195g246g345g







(119896)

119887(119896)

119896(119896)

and 119888(119896)



4 Conclusions





119904






Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(x119886(119905)


119886(119905)

M =

120597ℎ119888

(x119886(119905)

)

120597k119886(119905)

100381610038161003816100381610038161003816100381610038161003816x119886(119905)=xminus119886(119905)

= 1 (25)



(119896)



119896(119896)119887(119896) (119896) at 119905 = 119896119879

119904


119891(119896)

=

radic4119896(119896)

(119896)

minus1198872

(119896)

4120587(119896)

(26)


119890(119896)

=

0 if (119891119899

minus 119891119905

) le119891(119896)

ge (119891119899

+ 119891119905

)

1 if (119891119899

minus 119891119905

) ge119891(119896)

1 if (119891119899

+ 119891119905

) le119891(119896)

(27)



and 119891119905





119886(119896)



(119896)




x+119886(119896)



119891(119896)



(119896)

in (23)


and P+(119896)






(119896)






119887


119908






(119896)




(119896)




Pin 0Position

Pin 1Voltage

Pin 9Fault


Conversion Observer

EKF

z(k)

u(k) e(k)

x (k)intgtdouble

intgtdouble








MeasuredEstimated

0

20

40

60

80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

u(V

)times10minus3

5

0

minus5

005

0

minus005

CalculatedEstimated

z=q1x

1(m

m)

Δzasympq2x

2(m

sminus1 )



119891(119896)

left the 119891119905


119899


(119896)



x+1198860

= [0 0 100 1 1119864 minus 4

1

05

]

119879

(28)





P+0

=

[

[

[

[

[

[

[

[

[

[

[

[

1119864 minus 5 0 0 0 0 0

0 1119864 minus 4 0 0 0 0

0 0 1119864 + 5 0 0 0

0 0 0 1119864 minus 2 0 0

0 0 0 0 1119864 minus 5 0

0 0 0 0 0 1119864 + 0

]

]

]

]

]

]

]

]

]

]

]

]

(29)




R = 119877 = 1119864 minus 5 (30)


Q119905

=

[

[

[

[

[

[

[

[

[

5119864 minus 3 0 0 0 0

0 1119864 + 1 0 0 0

0 0 1119864 minus 1 0 0

0 0 0 1119864 minus 12 0

0 0 0 0 3119864 minus 1

]

]

]

]

]

]

]

]

]

(31)


119879119904


119904








1 0 g M 226 042 838 17 052 16 g 765 s 223 042 832 18 053 50 g 492 s 220 044 838 18 044 99 g 461 s 211 047 829 17 045 195 g 444 s 198 056 869 19 046 246 g 417 s 191 060 865 22 037 345 g 399 s 181 061 790 19 02


1


2



0

= [0 0]




(119896)


1

=

|1199091(119896)

minus 119911(119896)



2(119896)


Δ119911(119896)

asymp

119911(119896+1)

minus 119911(119896minus1)

2119879119904

(32)


2

= |1199092(119896)

minus Δ119911(119896)





6


119896 = 1199093


119887 = 1199094


5








18

2

22

24

26

04

05

06

07

0

20

40

60

80

100

120

0

1

2

3

0

02

04

06

08

1times10minus3

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

f(H

z)m

(kg)

0g16 g50g99g

195g246g345g

c(N

Vminus

1 )b

(Ns m

minus1 )

k(N

mminus

1 )









0



119904


119904


119904

= 01 sampling



119909119909


(119896)


1 12 14 16 18 2 22 24 26 28 3

minus35

minus40

minus45

minus50

minus55

minus60

minus65

minus70

minus75

minus80

Frequency f (Hz)Po

wer

S xx(f

)(d

BH

z)Nominal16 g50g99g

195g246g345g







(119896)

119887(119896)

119896(119896)

and 119888(119896)



4 Conclusions





119904






Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











(119896)




(119896)




Pin 0Position

Pin 1Voltage

Pin 9Fault


Conversion Observer

EKF

z(k)

u(k) e(k)

x (k)intgtdouble

intgtdouble








MeasuredEstimated

0

20

40

60

80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

u(V

)times10minus3

5

0

minus5

005

0

minus005

CalculatedEstimated

z=q1x

1(m

m)

Δzasympq2x

2(m

sminus1 )



119891(119896)

left the 119891119905


119899


(119896)



x+1198860

= [0 0 100 1 1119864 minus 4

1

05

]

119879

(28)





P+0

=

[

[

[

[

[

[

[

[

[

[

[

[

1119864 minus 5 0 0 0 0 0

0 1119864 minus 4 0 0 0 0

0 0 1119864 + 5 0 0 0

0 0 0 1119864 minus 2 0 0

0 0 0 0 1119864 minus 5 0

0 0 0 0 0 1119864 + 0

]

]

]

]

]

]

]

]

]

]

]

]

(29)




R = 119877 = 1119864 minus 5 (30)


Q119905

=

[

[

[

[

[

[

[

[

[

5119864 minus 3 0 0 0 0

0 1119864 + 1 0 0 0

0 0 1119864 minus 1 0 0

0 0 0 1119864 minus 12 0

0 0 0 0 3119864 minus 1

]

]

]

]

]

]

]

]

]

(31)


119879119904


119904








1 0 g M 226 042 838 17 052 16 g 765 s 223 042 832 18 053 50 g 492 s 220 044 838 18 044 99 g 461 s 211 047 829 17 045 195 g 444 s 198 056 869 19 046 246 g 417 s 191 060 865 22 037 345 g 399 s 181 061 790 19 02


1


2



0

= [0 0]




(119896)


1

=

|1199091(119896)

minus 119911(119896)



2(119896)


Δ119911(119896)

asymp

119911(119896+1)

minus 119911(119896minus1)

2119879119904

(32)


2

= |1199092(119896)

minus Δ119911(119896)





6


119896 = 1199093


119887 = 1199094


5








18

2

22

24

26

04

05

06

07

0

20

40

60

80

100

120

0

1

2

3

0

02

04

06

08

1times10minus3

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

f(H

z)m

(kg)

0g16 g50g99g

195g246g345g

c(N

Vminus

1 )b

(Ns m

minus1 )

k(N

mminus

1 )









0



119904


119904


119904

= 01 sampling



119909119909


(119896)


1 12 14 16 18 2 22 24 26 28 3

minus35

minus40

minus45

minus50

minus55

minus60

minus65

minus70

minus75

minus80

Frequency f (Hz)Po

wer

S xx(f

)(d

BH

z)Nominal16 g50g99g

195g246g345g







(119896)

119887(119896)

119896(119896)

and 119888(119896)



4 Conclusions





119904






Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










MeasuredEstimated

0

20

40

60

80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

Time (s)0 10 20 30 40 50 60 70 80

u(V

)times10minus3

5

0

minus5

005

0

minus005

CalculatedEstimated

z=q1x

1(m

m)

Δzasympq2x

2(m

sminus1 )



119891(119896)

left the 119891119905


119899


(119896)



x+1198860

= [0 0 100 1 1119864 minus 4

1

05

]

119879

(28)





P+0

=

[

[

[

[

[

[

[

[

[

[

[

[

1119864 minus 5 0 0 0 0 0

0 1119864 minus 4 0 0 0 0

0 0 1119864 + 5 0 0 0

0 0 0 1119864 minus 2 0 0

0 0 0 0 1119864 minus 5 0

0 0 0 0 0 1119864 + 0

]

]

]

]

]

]

]

]

]

]

]

]

(29)




R = 119877 = 1119864 minus 5 (30)


Q119905

=

[

[

[

[

[

[

[

[

[

5119864 minus 3 0 0 0 0

0 1119864 + 1 0 0 0

0 0 1119864 minus 1 0 0

0 0 0 1119864 minus 12 0

0 0 0 0 3119864 minus 1

]

]

]

]

]

]

]

]

]

(31)


119879119904


119904








1 0 g M 226 042 838 17 052 16 g 765 s 223 042 832 18 053 50 g 492 s 220 044 838 18 044 99 g 461 s 211 047 829 17 045 195 g 444 s 198 056 869 19 046 246 g 417 s 191 060 865 22 037 345 g 399 s 181 061 790 19 02


1


2



0

= [0 0]




(119896)


1

=

|1199091(119896)

minus 119911(119896)



2(119896)


Δ119911(119896)

asymp

119911(119896+1)

minus 119911(119896minus1)

2119879119904

(32)


2

= |1199092(119896)

minus Δ119911(119896)





6


119896 = 1199093


119887 = 1199094


5








18

2

22

24

26

04

05

06

07

0

20

40

60

80

100

120

0

1

2

3

0

02

04

06

08

1times10minus3

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

f(H

z)m

(kg)

0g16 g50g99g

195g246g345g

c(N

Vminus

1 )b

(Ns m

minus1 )

k(N

mminus

1 )









0



119904


119904


119904

= 01 sampling



119909119909


(119896)


1 12 14 16 18 2 22 24 26 28 3

minus35

minus40

minus45

minus50

minus55

minus60

minus65

minus70

minus75

minus80

Frequency f (Hz)Po

wer

S xx(f

)(d

BH

z)Nominal16 g50g99g

195g246g345g







(119896)

119887(119896)

119896(119896)

and 119888(119896)



4 Conclusions





119904






Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













1 0 g M 226 042 838 17 052 16 g 765 s 223 042 832 18 053 50 g 492 s 220 044 838 18 044 99 g 461 s 211 047 829 17 045 195 g 444 s 198 056 869 19 046 246 g 417 s 191 060 865 22 037 345 g 399 s 181 061 790 19 02


1


2



0

= [0 0]




(119896)


1

=

|1199091(119896)

minus 119911(119896)



2(119896)


Δ119911(119896)

asymp

119911(119896+1)

minus 119911(119896minus1)

2119879119904

(32)


2

= |1199092(119896)

minus Δ119911(119896)





6


119896 = 1199093


119887 = 1199094


5








18

2

22

24

26

04

05

06

07

0

20

40

60

80

100

120

0

1

2

3

0

02

04

06

08

1times10minus3

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

f(H

z)m

(kg)

0g16 g50g99g

195g246g345g

c(N

Vminus

1 )b

(Ns m

minus1 )

k(N

mminus

1 )









0



119904


119904


119904

= 01 sampling



119909119909


(119896)


1 12 14 16 18 2 22 24 26 28 3

minus35

minus40

minus45

minus50

minus55

minus60

minus65

minus70

minus75

minus80

Frequency f (Hz)Po

wer

S xx(f

)(d

BH

z)Nominal16 g50g99g

195g246g345g







(119896)

119887(119896)

119896(119896)

and 119888(119896)



4 Conclusions





119904






Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










18

2

22

24

26

04

05

06

07

0

20

40

60

80

100

120

0

1

2

3

0

02

04

06

08

1times10minus3

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

0 10 20 30 40 50 60 70 80

Time (s)

f(H

z)m

(kg)

0g16 g50g99g

195g246g345g

c(N

Vminus

1 )b

(Ns m

minus1 )

k(N

mminus

1 )









0



119904


119904


119904

= 01 sampling



119909119909


(119896)


1 12 14 16 18 2 22 24 26 28 3

minus35

minus40

minus45

minus50

minus55

minus60

minus65

minus70

minus75

minus80

Frequency f (Hz)Po

wer

S xx(f

)(d

BH

z)Nominal16 g50g99g

195g246g345g







(119896)

119887(119896)

119896(119896)

and 119888(119896)



4 Conclusions





119904






Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













0



119904


119904


119904

= 01 sampling



119909119909


(119896)


1 12 14 16 18 2 22 24 26 28 3

minus35

minus40

minus45

minus50

minus55

minus60

minus65

minus70

minus75

minus80

Frequency f (Hz)Po

wer

S xx(f

)(d

BH

z)Nominal16 g50g99g

195g246g345g







(119896)

119887(119896)

119896(119896)

and 119888(119896)



4 Conclusions





119904






Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










4 Conclusions





119904






Acknowledgments


References



























































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















































RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









research article online structural health monitoring and

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