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Aircraft Structural Health Monitoring Using Transmissibility Identification
Proceedings,18th IFAC Symposium on System Identification Proceedings,18th IFAC Symposium on System Identification July 9-11, 2018. Stockholm, Sweden on Proceedings,18th IFAC Symposium Symposium on System System Identification Identification Proceedings,18th IFAC online at www.sciencedirect.com July 9-11, 2018. Stockholm, Sweden Available Proceedings,18th IFAC Symposium July Sweden July 9-11, 9-11, 2018. 2018. Stockholm, Stockholm, Sweden on System Identification July 9-11, 2018. Stockholm, Sweden
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IFAC PapersOnLine 51-15 (2018) 969–974
Aircraft Structural Health Monitoring Aircraft Structural Health Monitoring Aircraft Structural Health Monitoring Using Transmissibility Identification Aircraft Structural Health Monitoring Using Transmissibility Identification Using Transmissibility Identification Using Transmissibility Identification Abdelrahman Khalil ∗∗ Khaled F. Aljanaideh ∗∗ ∗∗
Abdelrahman ∗∗ Abdelrahman Khalil Khalil ∗∗ Khaled Khaled F. F. Aljanaideh Aljanaideh ∗∗ Abdelrahman Khalil ∗ Khaled F. Aljanaideh ∗∗ Abdelrahman Khalil Khaled F. Aljanaideh ∗ ∗ Jordan University of Science and Technology, Irbid, Jordan 22110 ∗ Jordan University of Science and Technology, Irbid, Jordan 22110 ∗ Jordan University of and (e-mail: [email protected]) of Science Science and Technology, Technology, Irbid, Irbid, Jordan Jordan 22110 22110 ∗ Jordan University (e-mail: [email protected]) ∗∗ Jordan University of Science and Technology, Irbid, Jordan 22110 (e-mail: [email protected]) Jordan University of Science and Technology, Irbid, Jordan 22110 (e-mail: [email protected]) ∗∗ Jordan University of Science and Technology, Irbid, Jordan ∗∗ (e-mail: [email protected]) ∗∗ Jordan University University of Science Science and Technology, Technology, Irbid, Irbid, Jordan Jordan 22110 22110 (e-mail: [email protected]) Jordan of and 22110 ∗∗ (e-mail: [email protected]) Jordan University of Science and Technology, Irbid, Jordan 22110 (e-mail: [email protected]) (e-mail: [email protected]) (e-mail: [email protected]) Abstract: A transmissibility is an output-only relationship that relates one subset of outputs of Abstract: A transmissibility is an output-only relationship that relates one subset of outputs of Abstract: A transmissibility is an output-only relationship that relates one subset of outputs of a system to another subset of outputs of the same system. In this paper, we use transmissibility Abstract: A transmissibility is an output-only relationship that relates one subset of outputs of aAbstract: system to another subset of outputs of the same system. In this paper, we use transmissibility A transmissibility is an output-only relationship that relates one subset of outputs ofa a system to another subset of outputs of the same system. In this paper, we use transmissibility operators for fault detection and health monitoring of aircraft structures. We use ANSYS, a system to another subset of outputs of the same system. In this paper, we use transmissibility operators fault and health monitoring of aircraft structures. We use aa a system tofor subset ofsoftware, outputs of the same system. In this paper, wetwo use transmissibility operators foranother fault detection detection and health monitoring of aircraft structures. Weforces use ANSYS, ANSYS, finite-element-analysis based to simulate the aircraft wing, where are acting operators for fault detection and health monitoring of aircraft structures. We use ANSYS, a finite-element-analysis based software, to simulate the aircraft wing, where two forces are acting operators for fault detection and health monitoring of aircraft structures. We use ANSYS, a finite-element-analysis basedFive software, toare simulate theto aircraft wing, where two two forces are acting on the wing simultaneously. sensors attached the wing to measure the wing deflection finite-element-analysis based software, to simulate the aircraft wing, where forces are acting on the wing simultaneously. Five sensors are attached to the wing to measure the wing deflection finite-element-analysis based software, to simulate the aircraft wing, where two forces are acting on the wing simultaneously. Five sensors are attached to the wing to measure the wing deflection at locations. Noncausal FIR models are used with identify transmissibilities on their the wing simultaneously. Five sensors are attached toleast the squares wing to to measure the wing deflection at their locations. Noncausal FIR models are used with squares identify transmissibilities on the wing simultaneously. Five sensors are attached toleast the wing to to measure the wing deflection at their locations. Noncausal FIR models are used with least squares to identify transmissibilities between the five sensors measurements. We consider the case where a crack occurs in the wing at their locations. Noncausal FIR models are used with least squares to identify transmissibilities between the five measurements. We consider case where crack occurs in the wing at their locations. Noncausal FIRshow models aretransmissibilities used withthe least squares toaaaused identify transmissibilities between the five sensors sensors measurements. We consider the case where crack occurs in the wing structure during operation. We that can be to detect the change between the five sensors measurements. We consider the case where crack occurs in the wing structure during operation. We show that transmissibilities can be used to detect the change between the five sensors measurements. We consider the case where a crack occurs in the wing structure during operation. We without show that that transmissibilities can be used to detect detect the change change in the wing structure dynamics the need for a model of the underlying system or the structure during operation. We show transmissibilities can be used to the in the wing structure dynamics without the need for aa model of the underlying system or the structure during operation. We show that transmissibilities can be used to detect the change in the wing structure dynamics without the need for model of the underlying system or the excitation signal. in the wing structure dynamics without the need for a model of the underlying system or the excitation signal. in the wing structure dynamics without the need for a model of the underlying system or the excitation signal. excitation signal. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. excitation signal. Keywords: Structural identification. Keywords: Structural fault fault detection, detection, transmissibility, transmissibility, output-only output-only identification. Keywords: identification. Keywords: Structural Structural fault fault detection, detection, transmissibility, transmissibility, output-only output-only identification. Keywords: Structural fault detection, transmissibility, output-only identification. 1. INTRODUCTION INTRODUCTION describe the the structural structural response response of of the the vehicle vehicle (Shearer (Shearer 1. describe 1. describe the structural response of the vehicle (Shearer and Cesnik, 2007). A measurement of this excitation is not not 1. INTRODUCTION INTRODUCTION describe the structural response of the vehicle (Shearer and Cesnik, 2007). A measurement this excitation is 1. INTRODUCTION describe the structural response ofof the vehicle (Shearer and Cesnik, 2007). A measurement of this excitation is not available as it is a result of several sources of excitation and Cesnik, 2007). A measurement of this excitation is not A failure failure in in one one of of the the aircraft’s aircraft’s components components such such as as airair- available as it is aa Aresult of several of excitation and Cesnik, 2007). measurement ofsources this excitation is and not A available as it is result of several sources of excitation such as wind disturbance, propulsion system noise, available as it is a result of several sources of excitation A failure in one of the aircraft’s components such as aircraft sensors, actuators, or structural components can have A failure in one of the aircraft’s components such ashave air- such as wind propulsion system and available as it disturbance, is a result ofact several sources of noise, excitation craft sensors, actuators, or structural components can such as wind disturbance, propulsion system noise, and weather conditions, which on the underlying system A failure in one of the aircraft’s components such as airsuch as wind disturbance, propulsion system noise, and craft actuators, components catastrophic consequences on both both human human lives can andhave air- weather craft sensors, sensors, consequences actuators, or or structural structural components can have conditions, which act on the underlying system such as wind disturbance, propulsion system and catastrophic on lives and airweather conditions, which on system simultaneously. Therefore, excitation-free faultnoise, detection craft sensors, actuators, or structural components can have weather conditions, which act act on the the underlying underlying system catastrophic consequences on both human lives and aircraft other components. A sensor failure can, for example, catastrophic consequences on both human lives and airsimultaneously. Therefore, excitation-free fault detection weather conditions, which act on the underlying system craft other components. A sensor failure can, for example, simultaneously. Therefore, excitation-free fault detection techniques must Therefore, be considered considered in this this case. case.fault detection catastrophic consequences on both human lives and air- techniques simultaneously. excitation-free craft A sensor failure can, for destabilize closed-loop control system, or provide provide wrong craft other other aacomponents. components. A sensor failure can, for example, example, must be in simultaneously. Therefore, excitation-free detection destabilize closed-loop control system, or wrong techniques must be considered in this case. craft other aacomponents. A sensor failure actions can, for from example, techniques must be considered in this case.fault destabilize closed-loop control system, or provide wrong measurements that can lead to wrong the destabilize closed-loop control system, or provide wrong Transmissibilities are mathematical models that characcharactechniques must beare considered in thismodels case. that measurements that can lead to wrong actions from the Transmissibilities mathematical destabilize a closed-loop control system, or provide wrong measurements that lead to actions from pilot. A failure failure in in onecan of the the aircraft structural components measurements that can lead to wrong wrong actions from the the Transmissibilities are mathematical models that characterize the relationship between outputs of an underlying Transmissibilities are mathematical models that characpilot. A one of aircraft structural components the relationship between outputs of an underlying measurements that can lead actions from the pilot. in one of aircraft structural Transmissibilities are mathematical models that characor oneA offailure the control control surfaces oftothe thewrong aircraft cancomponents change the terize pilot. Aof failure in onesurfaces of the the aircraft structural components terize the relationship between outputs of an underlying system, that is, the input and output of a transmissibility terize the relationship between outputs of an underlying or one the of aircraft can change the system, that is, the input and output of aaoftransmissibility pilot. Aof failure in one of theoraircraft structural components or one the control surfaces of the aircraft can change the terize the relationship between outputs an underlying dynamics of the aircraft yield a designed closed-loop or one of the control surfaces of the aircraft can change the system, that is, input and output of transmissibility are outputs of the underlying system. Transmissibilities system, that is, input and output of a transmissibility dynamics of the aircraft or yield designed closed-loop outputs underlying system. Transmissibilities or one ofsystem the surfaces the a change the are dynamics of the aircraft or yield aaaircraft designed closed-loop system, that of is, the theininput and domain output of athe transmissibility control invalid. These problems cancan lead to losing losing dynamics of control theinvalid. aircraft or of yield designed closed-loop are outputs of underlying system. can be represented the time time or Transmissibilities frequency dodoare be outputs of the underlying system. Transmissibilities control system These problems can lead to can represented in the domain or the frequency dynamics of the aircraft or yield a designed closed-loop control system invalid. These problems can lead to losing are outputs of the underlying system. Transmissibilities both human lives and expensive components on board. control system invalid. These problems can lead to losing can be represented in the time domain or the frequency domain. Although most studies in the literature have considcan be represented in the time domain or the frequency doboth human lives and expensive components on board. Although most studies the literature have considcontrol system invalid. These problems can lead to losing main. both lives and components on can befrequency-domain represented in the timein domain or the frequency doboth human human lives and expensive expensive components on board. board. main. Although most studies in the literature have considered transmissibilities (Devriendt and main. Although most studies in the literature have considFault detection techniques include model-based techered frequency-domain transmissibilities (Devriendt and both human lives techniques and expensive components on board. main. Although most studies in2001; the literature have considFault detection include model-based techered frequency-domain transmissibilities (Devriendt and Guillaume, 2007; Maia et al., Ribeiro et al., 2000; ered frequency-domain transmissibilities (Devriendt and Fault detection techniques include model-based techniques and excitation-free techniques (Hwang et al., al., 2010). 2010). Fault and detection techniques include(Hwang model-based tech- Guillaume, 2007; Maia et al., 2001; Ribeiro et al., 2000; ered frequency-domain transmissibilities (Devriendt and niques excitation-free techniques et al., Ribeiro et 2000; Zhou et al., al., 2007; 2016),Maia it was waset shown that under under nonzero initial Fault detection techniques include model-based tech- Guillaume, Guillaume, 2007; Maia et al., 2001; 2001; Ribeiro et al., al., initial 2000; niques and excitation-free techniques (Hwang et al., 2010). Model-based techniques such as analytical redundancy, niques and excitation-free techniques (Hwang et al., 2010). Zhou et 2016), it shown that nonzero Guillaume, 2007; Maia et al., 2001; Ribeiro et al., 2000; Model-based techniques such as analytical redundancy, Zhou et al., 2016), it was shown that under nonzero initial conditions frequency-domain transmissibilities depend on niques and excitation-free techniques (Hwang et al.,require 2010). conditions Zhou et al.,frequency-domain 2016), it was shown that under nonzero initial Model-based techniques such as redundancy, optimization-based, and observer-based observer-based techniques Model-based techniques such as analytical analytical redundancy, transmissibilities depend on Zhou et al., 2016), it was shown thatinput underthat nonzero initial optimization-based, and techniques require conditions frequency-domain transmissibilities depend on both the initial conditions and the excites the Model-based techniques such as analytical redundancy, conditions frequency-domain transmissibilities depend on optimization-based, and observer-based techniques require knowledge of a model of the system and the excitation optimization-based, and observer-based techniques require both the initial conditions and the input that excites conditions frequency-domain transmissibilities depend the on knowledge of a the system and the excitation both initial conditions and the that the system, which renders frequency-domain transmissibilities optimization-based, andof observer-based techniques require both the thewhich initial conditions and the input inputtransmissibilities that excites excites the knowledge of aa model model of the system and the excitation signal that acts on the system (Edwards and Tan, 2006; knowledge of model of the system and the excitation system, renders frequency-domain both the initial conditions and the input that excites the signal that acts on the system (Edwards and Tan, 2006; system, which renders frequency-domain transmissibilities invalid under nonzero initial conditions conditions transmissibilities (Aljanaideh and and knowledge of a model the system andand the Tan, excitation system, under which nonzero renders frequency-domain signal that acts on the system (Edwards 2006; Chow and Willsky, 1984; Staroswiecki and Comtet-Varga, signal and thatWillsky, acts on 1984; the of system (Edwards and Tan, 2006; invalid initial (Aljanaideh system, which rendersOn frequency-domain transmissibilities Chow Staroswiecki and Comtet-Varga, invalid under nonzero initial conditions (Aljanaideh and Bernstein, 2015c,b). the other hand, transmissibility signal that acts on the system (Edwards and Tan, 2006; invalid under nonzero initial conditions (Aljanaideh and Chow and Willsky, 1984; Staroswiecki and Comtet-Varga, 2001; Gertler, 1991; Patton and Chen, 1997). On the other Chow Gertler, and Willsky, 1984; Staroswiecki and Comtet-Varga, Bernstein, 2015c,b). Oninitial the other hand, transmissibility invalid under nonzero conditions (Aljanaideh and 2001; 1991; Patton and Chen, 1997). On the other Bernstein, 2015c,b). the hand, transmissibility operators, which are On sensor-to-sensor time-domain modChow and Willsky, 1984; Staroswiecki and Comtet-Varga, Bernstein, 2015c,b). On the other other hand, transmissibility 2001; Gertler, 1991; Patton and Chen, 1997). On the other hand, excitation-free techniques such as hardware redun2001; Gertler, 1991; Patton and Chen, 1997). On the other operators, which are sensor-to-sensor time-domain modBernstein, 2015c,b). On the otherofhand, transmissibility hand, excitation-free techniques such as hardware redunoperators, which are sensor-to-sensor time-domain models, are shown to be independent both nonzero initial 2001; Gertler, 1991; Patton and Chen, 1997). On the other operators, which are sensor-to-sensor time-domain modhand, such as hardware dancy techniques and and techniques methods based based on transmissibilities hand, excitation-free excitation-free techniques such on as transmissibilities hardware redunredun- els, are shown to be independent of both nonzero initial operators, which sensor-to-sensor time-domain moddancy techniques methods els, to be independent of nonzero initial conditions of the the underlying system and the excitation excitation hand, excitation-free techniques such of asthe hardware redunels, are are shown shown to are be independent of both both nonzero initial dancy techniques and methods based on transmissibilities do not require knowledge of a model system or the dancy techniques and methods based on transmissibilities conditions of underlying system and the els, are shown to be independent of both nonzero initial do not require knowledge of a model of the system or the conditions of the underlying system and the excitation signal that acts on the system (Aljanaideh and Bernstein, dancy andetmethods based on transmissibilities conditions the and and the Bernstein, excitation do require knowledge aa model the system or excitation signal (Lo al.,of 2013, 2016;of Chesn´ and DeraeDeraedo not not techniques require knowledge of2013, model of the system or the the signal that of acts on underlying the system system (Aljanaideh conditions of underlying system and and the excitation excitation signal (Lo et al., 2016; eee and acts the (Aljanaideh Bernstein, 2015c,b). This ison important because using transmissibilido not require knowledge of2013, aGuillaume, model ofChesn´ the system or the signal signal that thatThis actsthe onimportant the system systembecause (Aljanaideh and Bernstein, excitation signal (Lo et al., 2016; Chesn´ and Deraemaeker, 2013; Devriendt and 2007; Aljanaideh excitation signal (Lo et al., 2013, 2016; Chesn´ e and Derae2015c,b). is using transmissibilisignal that acts on the system (Aljanaideh and Bernstein, maeker, 2013; Devriendt and Guillaume, 2007; Aljanaideh This is because using transmissibilities for fault fault detection and fault fault tolerant control requires excitation signal (Lo et2016). al., 2013, 2016; Chesn´ Derae- 2015c,b). 2015c,b). Thisdetection is important important because usingcontrol transmissibilimaeker, 2013; Devriendt and Guillaume, 2007; Aljanaideh and Bernstein, 2015a, maeker, 2013; Devriendt and Guillaume, 2007;e and Aljanaideh ties for and tolerant requires 2015c,b). This is in important because using transmissibiliand Bernstein, 2015a, 2016). ties for fault detection and fault tolerant control requires identifying them the presence of an unknown ambient maeker, 2013; Devriendt and Guillaume, 2007; Aljanaideh ties for fault detection and fault tolerant control requires and Bernstein, 2015a, 2016). andsome Bernstein, 2015a,such 2016). them in the presence of an unknown ties for fault detection and fault tolerant control ambient requires In applications, as structural structural fault fault detection, detection, the the identifying identifying them in the presence of an unknown ambient excitation, and then using the identified transmissibilities andsome Bernstein, 2015a,such 2016). identifying them in the presence of an unknown ambient In applications, as excitation, and using the identified transmissibilities identifying themthen in the presence of an unknown ambient In some applications, such as structural fault detection, the dynamics of the system of interest can be complicated, and excitation, and then using the identified transmissibilities In some applications, such as structural fault detection, the under a different excitation. excitation, and then using the identified transmissibilities dynamics of the system of interest can be complicated, and a different excitation. some applications, as structural fault detection, the under excitation, and then using the identified transmissibilities dynamics of system of can complicated, and aIn measurement of thesuch excitation that acts on the system system under excitation. dynamics of the the of system of interest interest can be be complicated, and under aa different different excitation. a measurement the excitation that acts on the Transmissibilities were used for for aircraft aircraft sensor sensor fault fault dededynamics ofbethe system ofConsider, interest that can be complicated, and Transmissibilities under a different excitation. amight measurement of the excitation acts on the system not available. for example, the struca measurement of the excitation that acts on the system were used might not be available. Consider, for example, the strucTransmissibilities were used for aircraft sensor fault detection in (Aljanaideh and Bernstein, 2015a). Moreover, a measurement of the excitation that acts on the system Transmissibilities were used for aircraft sensor fault demight be Consider, for the structure ofnot a lightweight lightweight flexible aircraft, where the the dynammightof not be available. available. Consider, for example, example, thedynamstruc- tection in (Aljanaideh and 2015a). Moreover, Transmissibilities were considered usedBernstein, for aircraft sensor fault and deture a flexible aircraft, where tection in (Aljanaideh and Bernstein, 2015a). Moreover, transmissibilities were for fault detection might not be available. Consider, for example, the structection in (Aljanaideh and Bernstein, 2015a). Moreover, ture of a lightweight flexible aircraft, where the dynamics ofofthe the aircraft are are flexible described by nonlinear nonlinear differential tureof a lightweight aircraft, where the dynam- transmissibilities were considered for fault detection and tection in (Aljanaideh and Bernstein, 2015a). Moreover, ics aircraft described by differential transmissibilities were considered for fault detection and fault localization were in acoustic acoustic systems in (Aljanaideh (Aljanaideh ture a lightweight flexible aircraft, where the dynamtransmissibilities considered for fault detection and and ics aircraft described by differential equations coupled with the areoelasticity areoelasticity equations that fault ics of ofofthe the aircraft are are described by nonlinear nonlinear differential localization in systems in transmissibilities considered for fault detection and equations with the equations that fault localization in systems in ics of the coupled aircraft are by nonlinear differential localization were in acoustic acoustic systems in (Aljanaideh (Aljanaideh and equations coupled with the equations that equations coupled withdescribed the areoelasticity areoelasticity equations that fault fault localization in acoustic systems in (Aljanaideh and equations with the areoelasticity 2405-8963 ©coupled 2018, IFAC (International Federation ofequations Automatic that Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 969 Copyright © under 2018 IFAC 969 Control. Peer review responsibility of International Federation of Automatic Copyright © 969 Copyright © 2018 2018 IFAC IFAC 969 10.1016/j.ifacol.2018.09.068 Copyright © 2018 IFAC 969
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Bernstein, 2018a). In this paper, we use transmissibilities for fault detection of aircraft structural components. We consider the aircraft wing structure with several sensors that measure the vertical deflection of the wing structure at their locations. We use least squares with a noncausal finite impulse response (FIR) model structure to identify the transmissibilities between the available sensor measurements. Then, the identified transmissibilities are used along with the available sensor measurements to detect faults in the wing structure. The paper is organized as follows: In Section 2, we review transmissibility operators and transmissibility functions and the differences between them. In Section 3, we use least squares with a noncausal FIR model structure to identify transmissibility operators. In Section 4, the identified transmissibilities are used for health monitoring of the aircraft wing structure. Conclusions and future work are given in Section 5. 2. TRANSMISSIBILITY FUNCTIONS AND TRANSMISSIBILITY OPERATORS Consider the state space model x(t) ˙ = Ax(t) + Bu(t), (1) x(0) = x0 , (2) (3) yi (t) = Ci x(t) + Di u(t) ∈ Rm , yo (t) = Co x(t) + Do u(t) ∈ Rp−m , (4) where u ∈ Rm , p is the number of measurements, m is the number of pseudo inputs, and p − m is the number of pseudo outputs, A ∈ Rn×n , B ∈ Rn×m , Ci ∈ Rm×n , Co ∈ R(p−m)×n , Di ∈ Rm×m , and Do ∈ R(p−m)×m .
Let yi and yo be the input and output of the transmissibility, respectively. Suppose that m = 1 and p = 2, then transforming (3) and (4) to the Laplace domain yields
yˆi (s) = Ci (sI − A)−1 x0 + [Ci (sI − A)−1 B + Di ]ˆ u(s), (5) −1 −1 yˆo (s) = Co (sI − A) x0 + [Co (sI − A) B + Do ]ˆ u(s), (6) respectively. Then, the Laplace transform of the transmissibility from yi to yo is given by yˆo (s) Tˆ (s) = (7) yˆi (s) Co (sI − A)−1 x0 + [Co (sI − A)−1 B + Do ]ˆ u(s) . = Ci (sI − A)−1 x0 + [Ci (sI − A)−1 B + Di ]ˆ u(s) Replacing s by ω in (7) yields the Fourier transform of the transmissibility from yi to yo , that is, yˆo (ω) (8) Tˆ (ω) = yˆi (ω) Co (ωI − A)−1 x0 + [Co (ωI − A)−1 B + Do ]ˆ u(ω) . = Ci (ωI − A)−1 x0 + [Ci (ωI − A)−1 B + Di ]ˆ u(ω) The relationships (7) and (8) are called transmissibility functions (Maia et al., 2001; Chesn´e and Deraemaeker, 2013). Note that, if x0 is zero in (7) and (8), then u ˆ(s) and u ˆ(ω) can be cancelled in (7) and (8), respectively, and the Laplace and Fourier transforms of yi and yo are related by a transmissibility that is independent of the input. However, if x0 is not zero, then a transmissibility 970
function that is independent of the input and the initial condition of the underlying system cannot be obtained. Considering a time-domain approach using the differentiation operator p d/dt, define Γi (p) Ci adj(pI − A)B + Di δ(p) ∈ Rm×m [p],
Γo (p) Co adj(pI − A)B + Do δ(p) ∈ R
(p−m)×m
(9) [p], (10)
δ(p) det(pI − A) ∈ R[p]. (11) Then, the transmissibility operator from yi to yo is given by (Aljanaideh and Bernstein, 2015c) T (p) = Γo (p)Γi −1 (p), (12) which is independent of both the initial condition x0 and the input u. Note that (13) yo (t) = T (p)yi (t) represents the differential equation det Γi (p)yo (t) = Γo (p) [adj Γi (p)] yi (t). (14) Unlike the complex Laplace variable s, the time-domain operator p in (14) accounts for nonzero initial conditions (Aljanaideh and Bernstein, 2015c, 2018b). Since sensor measurements are obtained in discrete time, we consider discrete-time transmissibility operators in the forward-shift operator q, that is, we replace p in (14) by the forward shift operator q (Middleton and Goodwin, 1990). 3. IDENTIFICATION OF TRANSMISSIBILITIES 3.1 Frequency-domain identification of transmissibilities Frequency-domain identification of transmissibilities can be performed by computing the ratio of the discrete Fourier transform of the pseudo-output to the discrete Fourier transform of the pseudo-input, that is, yˆo (eθ ) (15) TˆN (eθ ) = N θ , yˆiN (e ) where for all N ≥ 1 and for all θ ∈ [−π, π), yˆoN (eθ )
N −1 k=0
yo (k)e−θk ,
yˆiN (eθ )
N −1
yi (k)e−θk ,
k=0
(16) are the discrete Fourier transforms of yo and yi , respectively. Other parametric and nonparametric frequency domain identification methods such as least squares and methods based on spectral analysis can be also considered. 3.2 Time-domain identification of transmissibilities Consider (13) with p replaced by q in the transmissibility operator, m = 1, and p = 2, that is, (17) yo = T (q)yi , where Γo (q) T (q) = . (18) Γi (q) Note that if Γi has a nonminimum phase (unstable) zero, then T will be unstable. Also, if Γo has more zeros than Γi , then T will be noncausal. Moreover, transmissibilities are
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usually identified using the output measurements with no information about the dynamics of the system, and thus the order of the transmissibility is unknown. Therefore, to identify transmissibilities we need to consider a model structure that can approximate noncausal and unstable transmissibilities with unknown order. In this paper we consider noncausal FIR models, which are truncations of the Laurent expansion in an analytic annulus that contains the unit circle (Aljanaideh and Bernstein, 2017). Identification using noncausal FIR models: FIR model of T is given by r T (q, ΘFIR ) = Hi q−i , r,d
A noncausal (19)
i=−d
where r, d denote the order of the causal and noncausal parts of the FIR model of T , respectively, Hi is the ith coefficient of the Laurent expansion of T in the annulus T that contains the unit circle, and ΘFIR r,d [H−d , . . . , Hr ] . FIR FIR ˆ Then, the least squares estimate Θ r,d, of Θr,d is given by FIR T −1 ˆ Θ = (Φr,d, Φ ) Φr,d, Ψy , , (20) r,d,
r,d,
o
where is the number of samples, T
Ψyo , [ yo (r) · · · yo ( − d) ] ,
(21)
Φr,d, [ φr,d (r) · · · φr,d ( − d) ] ,
T
φr,d (k) [ yi (k + d) · · · yi (k − r) ] .
(22) (23)
The residual of the identified transmissibility obtained using least squares with a noncausal FIR model at time k is given by ˆ FIR ) = yo (k) − yˆo (k|Θ ˆ FIR ), er,d (k|Θ (24) r,d, r,d, where r ˆ FIR ) = T (q, Θ ˆ FIR )yi (k) = ˆ i, yi (k − i), yˆo (k|Θ H r,d, r,d, i=−d
(25)
ˆ FIR ) = T (q, Θ r,d,
r
ˆ i, q−i , H
(26)
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4. NUMERICAL APPLICATION TO AIRCRAFT STRUCTURAL HEALTH MONITORING We consider the structure of the aircraft wing shown in Figure 1 where the end near point A is fixed to the fuselage, and thus the wing acts as a cantilever beam. A finite-element-analysis based software (ANSYS) is used to simulate the wing structure. The wing is made of Aluminum and the wing span is l = 12 m. We assume that two transverse forces F1 and F2 are applied at points B and C, respectively, in the x-direction as shown in Figure 1. Moreover, we assume that five sensors P1–P5 are attached to the wing and they provide measurements of vertical deflections at their locations, where the locations of these sensors are as shown in Figure 1. A model of the structure and the time history of the applied forces F1 and F2 are assumed to be unknown. In this section, we use the available measurements from the sensors P1–P5 to identify transmissibility operators between them under healthy conditions of the wing structure. Then, we assume that a crack occurs in the wing, and the transmissibilities identified under healthy conditions are used along with the available sensor measurements to detect the fault in the structure. This is performed by monitoring the norm of the residual of the transmissibility over time. The forces F1 and F2 acting on the wing structure are given by 11 sin(ωi Ts k), (27) F1 (Ts k) = 2000 i=1
and
F2 (Ts k) =
11
Ai sin(ωi Ts k + ψi ),
(28)
i=1
where Ts = 0.1 sec and for all i = 1, . . . , 11, ωi = 5i rad/sec, and Ai = 1000 and ψi = 1.15 rad for odd i and Ai = 2000 and ψi = −1.15 rad for even i. Figure 2 shows the time history of F1 and F2 .
i=−d
ˆ −d, , . . . , H ˆ r, ]T . ˆ FIR = [H and Θ r,d, 3.3 Estimating the number of independent excitation signals Note from (3) that the dimension of yi is equal to the dimension of u. Therefore, constructing a transmissibility operator requires knowledge of the number of independent excitation signals m acting on the system. We use the following procedure to estimate the number of excitation signals m. We identify a transmissibility operator with m ˆ pseudo inputs and pˆ pseudo outputs, where m ˆ ∈ {1, . . . , p − 1} and pˆ ∈ {1, . . . , p − m}. ˆ For each identified transmissibility operator we compute the residual using (24), and the norm of the residual. The estimated number of independent external excitation signals is the value of m ˆ at which a sharp drop occurs in the norm of the residual. If a sharp drop is not obvious, then the estimated number of external excitation signals is the smallest value of m ˆ for which no sizable improvement is obtained for larger values of m. ˆ Redundant sensors can then be removed or retained for possible benefit in terms of the accuracy of the identified transmissibility operators. 971
Fig. 1. Basic aircraft wing. The end near point A is fixed to the fuselage, and thus the wing acts as a cantilever beam. Transverse forces F1 and f2 are applied at points B and C as shown. Moreover, measurements from five sensors P1–P5 attached to the wing are available.
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4.1 Estimating the number of independent excitation signals For all i = 1, . . . , 5, let yi denote the measurement obtained from the ith sensor, that is, Pi. Consider the case where F1 is operating and F2 is not operating. We use least squares with a noncausal FIR model with r = 25 and d = 25 to identify the transmissibility whose pseudo input is Yi [y1 · · · yi ] and whose pseudo output is y5 , where i = 1, . . . , 4. Figure 3 shows the norm of the residual of the identified transmissibility with 1, 2, 3, and 4 pseudo inputs. Note from Figure 3 that the benefits obtained from increasing the number of pseudo inputs from 1 to 4 are not significant. Therefore we correctly conclude that the number of inputs acting on the system is 1. Figure 4 shows the estimated Markov parameters of the transmissibility whose input is y1 and whose output is y5 . The estimated Markov parameters are obtained using least squares with a noncausal FIR model with r = 25 and d = 25. Figure 5 shows the measured output y5 and the predicted output yˆ5 obtained by applying y1 to the identified transmissibility whose Markov parameters are shown in Figure 4. Note from Figure 5 that the measured and predicted outputs are very close to each other. Next, consider the case where F1 and F2 are operating simultaneously. We add zero-mean, Gaussian white noise with a signal-to-noise ratio of 10 to the measurements of y1 –y5 . We use least squares with a noncausal FIR model with r = 25 and d = 25 to identify the transmissibility whose pseudo input is Yi [y1 · · · yi ]T and whose pseudo output is y5 , where i = 1, . . . , 4,. Figure 6 shows the norm of the residual of the identified transmissibility with 1, 2, 3, and 4 pseudo inputs. Note from Figure 6 that increasing the number of pseudo inputs from 1 to 2 yields a sizable improvement. On the other hand, the improvement obtained from using 3 and 4 pseudo inputs are not significant. Therefore we correctly conclude that the number of inputs acting on the system is 2. Figure 7 shows the estimated Markov parameters of the transmissibility whose inputs are y1 and y2 and whose output is
y5 . The estimated Markov parameters are obtained using least squares with a noncausal FIR model with r = 25 and d = 25. Figure 8 shows the measured output y5 and the predicted output yˆ5 obtained by applying y1 and y2 to the identified transmissibility whose Markov parameters are shown in Figure 7. Note from Figure 8 that the measured and predicted outputs are very close to each other. 4.2 Health Monitoring of the Wing Structure We assume that both F1 and F2 are operating simultaneously. We use the transmissibility identified under healthy conditions whose Markov parameters are shown in Figure 7 for health monitoring of the wing structure. We compute the residual of the identified transmissibility using (24), that is, for all k ≥ 0, r ˆ FIR ) = yo (k) − ˆ i, yi (k − i), e(k|Θ (29) H r,d, i=−d
T
where yo = y5 , yi = [y1 y2 ] .
Next, define the norm of the residual over a window of data of width w to be w+k ˆ FIR ). ˆ FIR , w) e(i|Θ (30) E(k|Θ r,d,
r,d,
i=k
We assume that at time t = 100 sec, a crack occurs in the structure of the wing. Figure 9 shows a plot of ˆ FIR , w) for w = 100 steps. Note from Figure 9 that E(k|Θ r,d, the level of the residual changed at t = 100 sec, which indicates that a fault has occurred in the wing structure. 5. CONCLUSIONS AND FUTURE WORK A transmissibility is a sensor-to-sensor relationship that relates one subset of sensors to another subset. In this 1.2
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Fig. 3. Norm of the residual of the identified transmissibility with 1, 2, 3, and 4 pseudo inputs. In this case F1 is operating and F2 is not operating. Note that the benefits obtained from increasing the number of pseudo inputs from 1 to 4 are not significant. Therefore we correctly conclude that the number of inputs acting on the system is 1.
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paper, we used transmissibility operators for fault detection and health monitoring of aircraft structures. We used the ANSYS software to simulate the aircraft wing, where two forces were acting on the structure of the wing. Five sensors were attached to the wing, where they measure the wing vertical deflection at their locations. Transmissibilities were identified using least squares with a noncausal FIR model structure. We showed that transmissibilities can be used to detect changes in the dynamics of the wing structure without the need to know a model of the structure or the excitation signal. This approach can be used to detect changes in the wing structure before reaching the fracture stage.
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tion error methods, instrumental variables methods, and subspace methods will be considered in future research. REFERENCES Aljanaideh, K.F. and Bernstein, D.S. (2015a). Aircraft sensor health monitoring based on transmissibility operators. Journal of Guidance, Control, and Dynamics, 38(8), 1492–1495. Aljanaideh, K.F. and Bernstein, D.S. (2015b). Timedomain analysis of motion transmissibilities in force0.16
Future work will consider comparing the accuracy of timedomain transmissibilities to frequency-domain transmissibilities in fault detection of aircraft structural components. Moreover, several identification methods such as predic-
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Fig. 6. Norm of the residual of the identified transmissibility with 1, 2, 3, and 4 pseudo inputs. Note that increasing the number of pseudo inputs from 1 to 2 yields a sizable improvement in the residual. On the other hand, the improvement obtained from using 3 and 4 pseudo inputs are not significant. Therefore we correctly conclude that the number of inputs acting on the system is 2. 1 0.8 Markov Parameter Value
Fig. 4. Estimated Markov parameters of the transmissibility whose input is y1 and whose output is y5 . The estimated Markov parameters are obtained using least squares with a noncausal FIR model with r = 25 and d = 25. 0.3 0.2
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Fig. 7. Estimated Markov parameters of the transmissibility whose inputs are y1 and y2 and whose output is y5 . The estimated Markov parameters are obtained using least squares with a noncausal FIR model with r = 25 and d = 25.
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Fig. 8. Measured output y5 and predicted output yˆ5 obtained by applying y1 and y2 to the identified transmissibility whose Markov parameters are shown in Figure 7. Note that the measured and predicted outputs are very close to each other. 0.023 0.022 0.021 0.02 0.019 0.018 0.017 0.016 0
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Fig. 9. Transmissibility residual computed using (30) with w = 100 steps. Note that at t = 100 sec the level of the residual changed due to a crack in the wing structure. driven and displacement-driven structures. Journal of Sound and Vibration, 347, 169–183. Aljanaideh, K.F. and Bernstein, D.S. (2015c). Timedomain analysis of sensor-to-sensor transmissibility operators. Automatica, 53, 312–319. Aljanaideh, K.F. and Bernstein, D.S. (2016). Experimental application of transmissibility operators to fault detection. In American Control Conference (ACC), 2016, 6833–6838. IEEE. Aljanaideh, K.F. and Bernstein, D.S. (2017). Closed-loop identification of unstable systems using noncausal FIR models. International Journal of Control, 90(2), 168– 185. Aljanaideh, K.F. and Bernstein, D.S. (2018a). Experimental application of time-domain transmissibility identification to fault detection and localization in acoustic systems. Journal of Vibration and Acoustics, 140(2). Aljanaideh, K.F. and Bernstein, D.S. (2018b). Initial conditions in time-and frequency-domain system identification: Implications of the shift operator versus the Z 974
and discrete fourier transforms. IEEE Control Systems, 38(2), 80–93. Chesn´e, S. and Deraemaeker, A. (2013). Damage localization using transmissibility functions: A critical review. Mechanical systems and signal processing, 38(2), 569– 584. Chow, E. and Willsky, A. (1984). Analytical redundancy and the design of robust failure detection systems. IEEE Transactions on Automatic Control, 29(7), 603–614. Devriendt, C. and Guillaume, P. (2007). The use of transmissibility measurements in output-only modal analysis. Mechanical Systems and Signal Processing, 21(7), 2689– 2696. Edwards, C. and Tan, C.P. (2006). Sensor fault tolerant control using sliding mode observers. Control Engineering Practice, 14(8), 897–908. Gertler, J. (1991). Analytical redundancy methods in fault detection and isolation. In Preprints of IFAC/IMACS Symposium on Fault Detection, Supervision and Safety for Technical Processes SAFEPROCESS91, 9–21. Hwang, I., Kim, S., Kim, Y., and Seah, C.E. (2010). A survey of fault detection, isolation, and reconfiguration methods. IEEE Transactions on Control Systems Technology, 18(3), 636–653. Lo, C., Lynch, J.P., and Liu, M. (2013). Distributed reference-free fault detection method for autonomous wireless sensor networks. IEEE Sensors Journal, 13(5), 2009–2019. Lo, C., Lynch, J.P., and Liu, M. (2016). Distributed model-based nonlinear sensor fault diagnosis in wireless sensor networks. Mechanical Systems and Signal Processing, 66, 470–484. Maia, N., Silva, J., and Ribeiro, A. (2001). The transmissibility concept in multi-degree-of-freedom systems. Mechanical Systems and Signal Processing, 15(1), 129– 137. Middleton, R.H. and Goodwin, G.C. (1990). Digital Control and Estimation: A Unified Approach. Prentice Hall PTR, USA. Patton, R. and Chen, J. (1997). Observer-based fault detection and isolation: Robustness and applications. Control Engineering Practice, 5(5), 671–682. Ribeiro, A., Silva, J., and Maia, N. (2000). On the generalisation of the transmissibility concept. Mechanical Systems and Signal Processing, 14(1), 29–35. Shearer, C.M. and Cesnik, C.E. (2007). Nonlinear flight dynamics of very flexible aircraft. Journal of Aircraft, 44(5), 1528–1545. Staroswiecki, M. and Comtet-Varga, G. (2001). Analytical redundancy relations for fault detection and isolation in algebraic dynamic systems. Automatica, 37(5), 687–699. Zhou, Y.L., Yang, X., and Abdel Wahab, M. (2016). Structural health monitoring using transmissibility coherence. In 5th International Conference on Fracture Fatigue and Wear, volume 4, 200–203. Labo Soete, Universiteit Gent.
ScienceDirect
IFAC PapersOnLine 51-15 (2018) 969–974
Aircraft Structural Health Monitoring Aircraft Structural Health Monitoring Aircraft Structural Health Monitoring Using Transmissibility Identification Aircraft Structural Health Monitoring Using Transmissibility Identification Using Transmissibility Identification Using Transmissibility Identification Abdelrahman Khalil ∗∗ Khaled F. Aljanaideh ∗∗ ∗∗
Abdelrahman ∗∗ Abdelrahman Khalil Khalil ∗∗ Khaled Khaled F. F. Aljanaideh Aljanaideh ∗∗ Abdelrahman Khalil ∗ Khaled F. Aljanaideh ∗∗ Abdelrahman Khalil Khaled F. Aljanaideh ∗ ∗ Jordan University of Science and Technology, Irbid, Jordan 22110 ∗ Jordan University of Science and Technology, Irbid, Jordan 22110 ∗ Jordan University of and (e-mail: [email protected]) of Science Science and Technology, Technology, Irbid, Irbid, Jordan Jordan 22110 22110 ∗ Jordan University (e-mail: [email protected]) ∗∗ Jordan University of Science and Technology, Irbid, Jordan 22110 (e-mail: [email protected]) Jordan University of Science and Technology, Irbid, Jordan 22110 (e-mail: [email protected]) ∗∗ Jordan University of Science and Technology, Irbid, Jordan ∗∗ (e-mail: [email protected]) ∗∗ Jordan University University of Science Science and Technology, Technology, Irbid, Irbid, Jordan Jordan 22110 22110 (e-mail: [email protected]) Jordan of and 22110 ∗∗ (e-mail: [email protected]) Jordan University of Science and Technology, Irbid, Jordan 22110 (e-mail: [email protected]) (e-mail: [email protected]) (e-mail: [email protected]) Abstract: A transmissibility is an output-only relationship that relates one subset of outputs of Abstract: A transmissibility is an output-only relationship that relates one subset of outputs of Abstract: A transmissibility is an output-only relationship that relates one subset of outputs of a system to another subset of outputs of the same system. In this paper, we use transmissibility Abstract: A transmissibility is an output-only relationship that relates one subset of outputs of aAbstract: system to another subset of outputs of the same system. In this paper, we use transmissibility A transmissibility is an output-only relationship that relates one subset of outputs ofa a system to another subset of outputs of the same system. In this paper, we use transmissibility operators for fault detection and health monitoring of aircraft structures. We use ANSYS, a system to another subset of outputs of the same system. In this paper, we use transmissibility operators fault and health monitoring of aircraft structures. We use aa a system tofor subset ofsoftware, outputs of the same system. In this paper, wetwo use transmissibility operators foranother fault detection detection and health monitoring of aircraft structures. Weforces use ANSYS, ANSYS, finite-element-analysis based to simulate the aircraft wing, where are acting operators for fault detection and health monitoring of aircraft structures. We use ANSYS, a finite-element-analysis based software, to simulate the aircraft wing, where two forces are acting operators for fault detection and health monitoring of aircraft structures. We use ANSYS, a finite-element-analysis basedFive software, toare simulate theto aircraft wing, where two two forces are acting on the wing simultaneously. sensors attached the wing to measure the wing deflection finite-element-analysis based software, to simulate the aircraft wing, where forces are acting on the wing simultaneously. Five sensors are attached to the wing to measure the wing deflection finite-element-analysis based software, to simulate the aircraft wing, where two forces are acting on the wing simultaneously. Five sensors are attached to the wing to measure the wing deflection at locations. Noncausal FIR models are used with identify transmissibilities on their the wing simultaneously. Five sensors are attached toleast the squares wing to to measure the wing deflection at their locations. Noncausal FIR models are used with squares identify transmissibilities on the wing simultaneously. Five sensors are attached toleast the wing to to measure the wing deflection at their locations. Noncausal FIR models are used with least squares to identify transmissibilities between the five sensors measurements. We consider the case where a crack occurs in the wing at their locations. Noncausal FIR models are used with least squares to identify transmissibilities between the five measurements. We consider case where crack occurs in the wing at their locations. Noncausal FIRshow models aretransmissibilities used withthe least squares toaaaused identify transmissibilities between the five sensors sensors measurements. We consider the case where crack occurs in the wing structure during operation. We that can be to detect the change between the five sensors measurements. We consider the case where crack occurs in the wing structure during operation. We show that transmissibilities can be used to detect the change between the five sensors measurements. We consider the case where a crack occurs in the wing structure during operation. We without show that that transmissibilities can be used to detect detect the change change in the wing structure dynamics the need for a model of the underlying system or the structure during operation. We show transmissibilities can be used to the in the wing structure dynamics without the need for aa model of the underlying system or the structure during operation. We show that transmissibilities can be used to detect the change in the wing structure dynamics without the need for model of the underlying system or the excitation signal. in the wing structure dynamics without the need for a model of the underlying system or the excitation signal. in the wing structure dynamics without the need for a model of the underlying system or the excitation signal. excitation signal. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. excitation signal. Keywords: Structural identification. Keywords: Structural fault fault detection, detection, transmissibility, transmissibility, output-only output-only identification. Keywords: identification. Keywords: Structural Structural fault fault detection, detection, transmissibility, transmissibility, output-only output-only identification. Keywords: Structural fault detection, transmissibility, output-only identification. 1. INTRODUCTION INTRODUCTION describe the the structural structural response response of of the the vehicle vehicle (Shearer (Shearer 1. describe 1. describe the structural response of the vehicle (Shearer and Cesnik, 2007). A measurement of this excitation is not not 1. INTRODUCTION INTRODUCTION describe the structural response of the vehicle (Shearer and Cesnik, 2007). A measurement this excitation is 1. INTRODUCTION describe the structural response ofof the vehicle (Shearer and Cesnik, 2007). A measurement of this excitation is not available as it is a result of several sources of excitation and Cesnik, 2007). A measurement of this excitation is not A failure failure in in one one of of the the aircraft’s aircraft’s components components such such as as airair- available as it is aa Aresult of several of excitation and Cesnik, 2007). measurement ofsources this excitation is and not A available as it is result of several sources of excitation such as wind disturbance, propulsion system noise, available as it is a result of several sources of excitation A failure in one of the aircraft’s components such as aircraft sensors, actuators, or structural components can have A failure in one of the aircraft’s components such ashave air- such as wind propulsion system and available as it disturbance, is a result ofact several sources of noise, excitation craft sensors, actuators, or structural components can such as wind disturbance, propulsion system noise, and weather conditions, which on the underlying system A failure in one of the aircraft’s components such as airsuch as wind disturbance, propulsion system noise, and craft actuators, components catastrophic consequences on both both human human lives can andhave air- weather craft sensors, sensors, consequences actuators, or or structural structural components can have conditions, which act on the underlying system such as wind disturbance, propulsion system and catastrophic on lives and airweather conditions, which on system simultaneously. Therefore, excitation-free faultnoise, detection craft sensors, actuators, or structural components can have weather conditions, which act act on the the underlying underlying system catastrophic consequences on both human lives and aircraft other components. A sensor failure can, for example, catastrophic consequences on both human lives and airsimultaneously. Therefore, excitation-free fault detection weather conditions, which act on the underlying system craft other components. A sensor failure can, for example, simultaneously. Therefore, excitation-free fault detection techniques must Therefore, be considered considered in this this case. case.fault detection catastrophic consequences on both human lives and air- techniques simultaneously. excitation-free craft A sensor failure can, for destabilize closed-loop control system, or provide provide wrong craft other other aacomponents. components. A sensor failure can, for example, example, must be in simultaneously. Therefore, excitation-free detection destabilize closed-loop control system, or wrong techniques must be considered in this case. craft other aacomponents. A sensor failure actions can, for from example, techniques must be considered in this case.fault destabilize closed-loop control system, or provide wrong measurements that can lead to wrong the destabilize closed-loop control system, or provide wrong Transmissibilities are mathematical models that characcharactechniques must beare considered in thismodels case. that measurements that can lead to wrong actions from the Transmissibilities mathematical destabilize a closed-loop control system, or provide wrong measurements that lead to actions from pilot. A failure failure in in onecan of the the aircraft structural components measurements that can lead to wrong wrong actions from the the Transmissibilities are mathematical models that characterize the relationship between outputs of an underlying Transmissibilities are mathematical models that characpilot. A one of aircraft structural components the relationship between outputs of an underlying measurements that can lead actions from the pilot. in one of aircraft structural Transmissibilities are mathematical models that characor oneA offailure the control control surfaces oftothe thewrong aircraft cancomponents change the terize pilot. Aof failure in onesurfaces of the the aircraft structural components terize the relationship between outputs of an underlying system, that is, the input and output of a transmissibility terize the relationship between outputs of an underlying or one the of aircraft can change the system, that is, the input and output of aaoftransmissibility pilot. Aof failure in one of theoraircraft structural components or one the control surfaces of the aircraft can change the terize the relationship between outputs an underlying dynamics of the aircraft yield a designed closed-loop or one of the control surfaces of the aircraft can change the system, that is, input and output of transmissibility are outputs of the underlying system. Transmissibilities system, that is, input and output of a transmissibility dynamics of the aircraft or yield designed closed-loop outputs underlying system. Transmissibilities or one ofsystem the surfaces the a change the are dynamics of the aircraft or yield aaaircraft designed closed-loop system, that of is, the theininput and domain output of athe transmissibility control invalid. These problems cancan lead to losing losing dynamics of control theinvalid. aircraft or of yield designed closed-loop are outputs of underlying system. can be represented the time time or Transmissibilities frequency dodoare be outputs of the underlying system. Transmissibilities control system These problems can lead to can represented in the domain or the frequency dynamics of the aircraft or yield a designed closed-loop control system invalid. These problems can lead to losing are outputs of the underlying system. Transmissibilities both human lives and expensive components on board. control system invalid. These problems can lead to losing can be represented in the time domain or the frequency domain. Although most studies in the literature have considcan be represented in the time domain or the frequency doboth human lives and expensive components on board. Although most studies the literature have considcontrol system invalid. These problems can lead to losing main. both lives and components on can befrequency-domain represented in the timein domain or the frequency doboth human human lives and expensive expensive components on board. board. main. Although most studies in the literature have considered transmissibilities (Devriendt and main. Although most studies in the literature have considFault detection techniques include model-based techered frequency-domain transmissibilities (Devriendt and both human lives techniques and expensive components on board. main. Although most studies in2001; the literature have considFault detection include model-based techered frequency-domain transmissibilities (Devriendt and Guillaume, 2007; Maia et al., Ribeiro et al., 2000; ered frequency-domain transmissibilities (Devriendt and Fault detection techniques include model-based techniques and excitation-free techniques (Hwang et al., al., 2010). 2010). Fault and detection techniques include(Hwang model-based tech- Guillaume, 2007; Maia et al., 2001; Ribeiro et al., 2000; ered frequency-domain transmissibilities (Devriendt and niques excitation-free techniques et al., Ribeiro et 2000; Zhou et al., al., 2007; 2016),Maia it was waset shown that under under nonzero initial Fault detection techniques include model-based tech- Guillaume, Guillaume, 2007; Maia et al., 2001; 2001; Ribeiro et al., al., initial 2000; niques and excitation-free techniques (Hwang et al., 2010). Model-based techniques such as analytical redundancy, niques and excitation-free techniques (Hwang et al., 2010). Zhou et 2016), it shown that nonzero Guillaume, 2007; Maia et al., 2001; Ribeiro et al., 2000; Model-based techniques such as analytical redundancy, Zhou et al., 2016), it was shown that under nonzero initial conditions frequency-domain transmissibilities depend on niques and excitation-free techniques (Hwang et al.,require 2010). conditions Zhou et al.,frequency-domain 2016), it was shown that under nonzero initial Model-based techniques such as redundancy, optimization-based, and observer-based observer-based techniques Model-based techniques such as analytical analytical redundancy, transmissibilities depend on Zhou et al., 2016), it was shown thatinput underthat nonzero initial optimization-based, and techniques require conditions frequency-domain transmissibilities depend on both the initial conditions and the excites the Model-based techniques such as analytical redundancy, conditions frequency-domain transmissibilities depend on optimization-based, and observer-based techniques require knowledge of a model of the system and the excitation optimization-based, and observer-based techniques require both the initial conditions and the input that excites conditions frequency-domain transmissibilities depend the on knowledge of a the system and the excitation both initial conditions and the that the system, which renders frequency-domain transmissibilities optimization-based, andof observer-based techniques require both the thewhich initial conditions and the input inputtransmissibilities that excites excites the knowledge of aa model model of the system and the excitation signal that acts on the system (Edwards and Tan, 2006; knowledge of model of the system and the excitation system, renders frequency-domain both the initial conditions and the input that excites the signal that acts on the system (Edwards and Tan, 2006; system, which renders frequency-domain transmissibilities invalid under nonzero initial conditions conditions transmissibilities (Aljanaideh and and knowledge of a model the system andand the Tan, excitation system, under which nonzero renders frequency-domain signal that acts on the system (Edwards 2006; Chow and Willsky, 1984; Staroswiecki and Comtet-Varga, signal and thatWillsky, acts on 1984; the of system (Edwards and Tan, 2006; invalid initial (Aljanaideh system, which rendersOn frequency-domain transmissibilities Chow Staroswiecki and Comtet-Varga, invalid under nonzero initial conditions (Aljanaideh and Bernstein, 2015c,b). the other hand, transmissibility signal that acts on the system (Edwards and Tan, 2006; invalid under nonzero initial conditions (Aljanaideh and Chow and Willsky, 1984; Staroswiecki and Comtet-Varga, 2001; Gertler, 1991; Patton and Chen, 1997). On the other Chow Gertler, and Willsky, 1984; Staroswiecki and Comtet-Varga, Bernstein, 2015c,b). Oninitial the other hand, transmissibility invalid under nonzero conditions (Aljanaideh and 2001; 1991; Patton and Chen, 1997). On the other Bernstein, 2015c,b). the hand, transmissibility operators, which are On sensor-to-sensor time-domain modChow and Willsky, 1984; Staroswiecki and Comtet-Varga, Bernstein, 2015c,b). On the other other hand, transmissibility 2001; Gertler, 1991; Patton and Chen, 1997). On the other hand, excitation-free techniques such as hardware redun2001; Gertler, 1991; Patton and Chen, 1997). On the other operators, which are sensor-to-sensor time-domain modBernstein, 2015c,b). On the otherofhand, transmissibility hand, excitation-free techniques such as hardware redunoperators, which are sensor-to-sensor time-domain models, are shown to be independent both nonzero initial 2001; Gertler, 1991; Patton and Chen, 1997). On the other operators, which are sensor-to-sensor time-domain modhand, such as hardware dancy techniques and and techniques methods based based on transmissibilities hand, excitation-free excitation-free techniques such on as transmissibilities hardware redunredun- els, are shown to be independent of both nonzero initial operators, which sensor-to-sensor time-domain moddancy techniques methods els, to be independent of nonzero initial conditions of the the underlying system and the excitation excitation hand, excitation-free techniques such of asthe hardware redunels, are are shown shown to are be independent of both both nonzero initial dancy techniques and methods based on transmissibilities do not require knowledge of a model system or the dancy techniques and methods based on transmissibilities conditions of underlying system and the els, are shown to be independent of both nonzero initial do not require knowledge of a model of the system or the conditions of the underlying system and the excitation signal that acts on the system (Aljanaideh and Bernstein, dancy andetmethods based on transmissibilities conditions the and and the Bernstein, excitation do require knowledge aa model the system or excitation signal (Lo al.,of 2013, 2016;of Chesn´ and DeraeDeraedo not not techniques require knowledge of2013, model of the system or the the signal that of acts on underlying the system system (Aljanaideh conditions of underlying system and and the excitation excitation signal (Lo et al., 2016; eee and acts the (Aljanaideh Bernstein, 2015c,b). This ison important because using transmissibilido not require knowledge of2013, aGuillaume, model ofChesn´ the system or the signal signal that thatThis actsthe onimportant the system systembecause (Aljanaideh and Bernstein, excitation signal (Lo et al., 2016; Chesn´ and Deraemaeker, 2013; Devriendt and 2007; Aljanaideh excitation signal (Lo et al., 2013, 2016; Chesn´ e and Derae2015c,b). is using transmissibilisignal that acts on the system (Aljanaideh and Bernstein, maeker, 2013; Devriendt and Guillaume, 2007; Aljanaideh This is because using transmissibilities for fault fault detection and fault fault tolerant control requires excitation signal (Lo et2016). al., 2013, 2016; Chesn´ Derae- 2015c,b). 2015c,b). Thisdetection is important important because usingcontrol transmissibilimaeker, 2013; Devriendt and Guillaume, 2007; Aljanaideh and Bernstein, 2015a, maeker, 2013; Devriendt and Guillaume, 2007;e and Aljanaideh ties for and tolerant requires 2015c,b). This is in important because using transmissibiliand Bernstein, 2015a, 2016). ties for fault detection and fault tolerant control requires identifying them the presence of an unknown ambient maeker, 2013; Devriendt and Guillaume, 2007; Aljanaideh ties for fault detection and fault tolerant control requires and Bernstein, 2015a, 2016). andsome Bernstein, 2015a,such 2016). them in the presence of an unknown ties for fault detection and fault tolerant control ambient requires In applications, as structural structural fault fault detection, detection, the the identifying identifying them in the presence of an unknown ambient excitation, and then using the identified transmissibilities andsome Bernstein, 2015a,such 2016). identifying them in the presence of an unknown ambient In applications, as excitation, and using the identified transmissibilities identifying themthen in the presence of an unknown ambient In some applications, such as structural fault detection, the dynamics of the system of interest can be complicated, and excitation, and then using the identified transmissibilities In some applications, such as structural fault detection, the under a different excitation. excitation, and then using the identified transmissibilities dynamics of the system of interest can be complicated, and a different excitation. some applications, as structural fault detection, the under excitation, and then using the identified transmissibilities dynamics of system of can complicated, and aIn measurement of thesuch excitation that acts on the system system under excitation. dynamics of the the of system of interest interest can be be complicated, and under aa different different excitation. a measurement the excitation that acts on the Transmissibilities were used for for aircraft aircraft sensor sensor fault fault dededynamics ofbethe system ofConsider, interest that can be complicated, and Transmissibilities under a different excitation. amight measurement of the excitation acts on the system not available. for example, the struca measurement of the excitation that acts on the system were used might not be available. Consider, for example, the strucTransmissibilities were used for aircraft sensor fault detection in (Aljanaideh and Bernstein, 2015a). Moreover, a measurement of the excitation that acts on the system Transmissibilities were used for aircraft sensor fault demight be Consider, for the structure ofnot a lightweight lightweight flexible aircraft, where the the dynammightof not be available. available. Consider, for example, example, thedynamstruc- tection in (Aljanaideh and 2015a). Moreover, Transmissibilities were considered usedBernstein, for aircraft sensor fault and deture a flexible aircraft, where tection in (Aljanaideh and Bernstein, 2015a). Moreover, transmissibilities were for fault detection might not be available. Consider, for example, the structection in (Aljanaideh and Bernstein, 2015a). Moreover, ture of a lightweight flexible aircraft, where the dynamics ofofthe the aircraft are are flexible described by nonlinear nonlinear differential tureof a lightweight aircraft, where the dynam- transmissibilities were considered for fault detection and tection in (Aljanaideh and Bernstein, 2015a). Moreover, ics aircraft described by differential transmissibilities were considered for fault detection and fault localization were in acoustic acoustic systems in (Aljanaideh (Aljanaideh ture a lightweight flexible aircraft, where the dynamtransmissibilities considered for fault detection and and ics aircraft described by differential equations coupled with the areoelasticity areoelasticity equations that fault ics of ofofthe the aircraft are are described by nonlinear nonlinear differential localization in systems in transmissibilities considered for fault detection and equations with the equations that fault localization in systems in ics of the coupled aircraft are by nonlinear differential localization were in acoustic acoustic systems in (Aljanaideh (Aljanaideh and equations coupled with the equations that equations coupled withdescribed the areoelasticity areoelasticity equations that fault fault localization in acoustic systems in (Aljanaideh and equations with the areoelasticity 2405-8963 ©coupled 2018, IFAC (International Federation ofequations Automatic that Control) Hosting by Elsevier Ltd. All rights reserved.
Copyright © 2018 IFAC 969 Copyright © under 2018 IFAC 969 Control. Peer review responsibility of International Federation of Automatic Copyright © 969 Copyright © 2018 2018 IFAC IFAC 969 10.1016/j.ifacol.2018.09.068 Copyright © 2018 IFAC 969
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Bernstein, 2018a). In this paper, we use transmissibilities for fault detection of aircraft structural components. We consider the aircraft wing structure with several sensors that measure the vertical deflection of the wing structure at their locations. We use least squares with a noncausal finite impulse response (FIR) model structure to identify the transmissibilities between the available sensor measurements. Then, the identified transmissibilities are used along with the available sensor measurements to detect faults in the wing structure. The paper is organized as follows: In Section 2, we review transmissibility operators and transmissibility functions and the differences between them. In Section 3, we use least squares with a noncausal FIR model structure to identify transmissibility operators. In Section 4, the identified transmissibilities are used for health monitoring of the aircraft wing structure. Conclusions and future work are given in Section 5. 2. TRANSMISSIBILITY FUNCTIONS AND TRANSMISSIBILITY OPERATORS Consider the state space model x(t) ˙ = Ax(t) + Bu(t), (1) x(0) = x0 , (2) (3) yi (t) = Ci x(t) + Di u(t) ∈ Rm , yo (t) = Co x(t) + Do u(t) ∈ Rp−m , (4) where u ∈ Rm , p is the number of measurements, m is the number of pseudo inputs, and p − m is the number of pseudo outputs, A ∈ Rn×n , B ∈ Rn×m , Ci ∈ Rm×n , Co ∈ R(p−m)×n , Di ∈ Rm×m , and Do ∈ R(p−m)×m .
Let yi and yo be the input and output of the transmissibility, respectively. Suppose that m = 1 and p = 2, then transforming (3) and (4) to the Laplace domain yields
yˆi (s) = Ci (sI − A)−1 x0 + [Ci (sI − A)−1 B + Di ]ˆ u(s), (5) −1 −1 yˆo (s) = Co (sI − A) x0 + [Co (sI − A) B + Do ]ˆ u(s), (6) respectively. Then, the Laplace transform of the transmissibility from yi to yo is given by yˆo (s) Tˆ (s) = (7) yˆi (s) Co (sI − A)−1 x0 + [Co (sI − A)−1 B + Do ]ˆ u(s) . = Ci (sI − A)−1 x0 + [Ci (sI − A)−1 B + Di ]ˆ u(s) Replacing s by ω in (7) yields the Fourier transform of the transmissibility from yi to yo , that is, yˆo (ω) (8) Tˆ (ω) = yˆi (ω) Co (ωI − A)−1 x0 + [Co (ωI − A)−1 B + Do ]ˆ u(ω) . = Ci (ωI − A)−1 x0 + [Ci (ωI − A)−1 B + Di ]ˆ u(ω) The relationships (7) and (8) are called transmissibility functions (Maia et al., 2001; Chesn´e and Deraemaeker, 2013). Note that, if x0 is zero in (7) and (8), then u ˆ(s) and u ˆ(ω) can be cancelled in (7) and (8), respectively, and the Laplace and Fourier transforms of yi and yo are related by a transmissibility that is independent of the input. However, if x0 is not zero, then a transmissibility 970
function that is independent of the input and the initial condition of the underlying system cannot be obtained. Considering a time-domain approach using the differentiation operator p d/dt, define Γi (p) Ci adj(pI − A)B + Di δ(p) ∈ Rm×m [p],
Γo (p) Co adj(pI − A)B + Do δ(p) ∈ R
(p−m)×m
(9) [p], (10)
δ(p) det(pI − A) ∈ R[p]. (11) Then, the transmissibility operator from yi to yo is given by (Aljanaideh and Bernstein, 2015c) T (p) = Γo (p)Γi −1 (p), (12) which is independent of both the initial condition x0 and the input u. Note that (13) yo (t) = T (p)yi (t) represents the differential equation det Γi (p)yo (t) = Γo (p) [adj Γi (p)] yi (t). (14) Unlike the complex Laplace variable s, the time-domain operator p in (14) accounts for nonzero initial conditions (Aljanaideh and Bernstein, 2015c, 2018b). Since sensor measurements are obtained in discrete time, we consider discrete-time transmissibility operators in the forward-shift operator q, that is, we replace p in (14) by the forward shift operator q (Middleton and Goodwin, 1990). 3. IDENTIFICATION OF TRANSMISSIBILITIES 3.1 Frequency-domain identification of transmissibilities Frequency-domain identification of transmissibilities can be performed by computing the ratio of the discrete Fourier transform of the pseudo-output to the discrete Fourier transform of the pseudo-input, that is, yˆo (eθ ) (15) TˆN (eθ ) = N θ , yˆiN (e ) where for all N ≥ 1 and for all θ ∈ [−π, π), yˆoN (eθ )
N −1 k=0
yo (k)e−θk ,
yˆiN (eθ )
N −1
yi (k)e−θk ,
k=0
(16) are the discrete Fourier transforms of yo and yi , respectively. Other parametric and nonparametric frequency domain identification methods such as least squares and methods based on spectral analysis can be also considered. 3.2 Time-domain identification of transmissibilities Consider (13) with p replaced by q in the transmissibility operator, m = 1, and p = 2, that is, (17) yo = T (q)yi , where Γo (q) T (q) = . (18) Γi (q) Note that if Γi has a nonminimum phase (unstable) zero, then T will be unstable. Also, if Γo has more zeros than Γi , then T will be noncausal. Moreover, transmissibilities are
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usually identified using the output measurements with no information about the dynamics of the system, and thus the order of the transmissibility is unknown. Therefore, to identify transmissibilities we need to consider a model structure that can approximate noncausal and unstable transmissibilities with unknown order. In this paper we consider noncausal FIR models, which are truncations of the Laurent expansion in an analytic annulus that contains the unit circle (Aljanaideh and Bernstein, 2017). Identification using noncausal FIR models: FIR model of T is given by r T (q, ΘFIR ) = Hi q−i , r,d
A noncausal (19)
i=−d
where r, d denote the order of the causal and noncausal parts of the FIR model of T , respectively, Hi is the ith coefficient of the Laurent expansion of T in the annulus T that contains the unit circle, and ΘFIR r,d [H−d , . . . , Hr ] . FIR FIR ˆ Then, the least squares estimate Θ r,d, of Θr,d is given by FIR T −1 ˆ Θ = (Φr,d, Φ ) Φr,d, Ψy , , (20) r,d,
r,d,
o
where is the number of samples, T
Ψyo , [ yo (r) · · · yo ( − d) ] ,
(21)
Φr,d, [ φr,d (r) · · · φr,d ( − d) ] ,
T
φr,d (k) [ yi (k + d) · · · yi (k − r) ] .
(22) (23)
The residual of the identified transmissibility obtained using least squares with a noncausal FIR model at time k is given by ˆ FIR ) = yo (k) − yˆo (k|Θ ˆ FIR ), er,d (k|Θ (24) r,d, r,d, where r ˆ FIR ) = T (q, Θ ˆ FIR )yi (k) = ˆ i, yi (k − i), yˆo (k|Θ H r,d, r,d, i=−d
(25)
ˆ FIR ) = T (q, Θ r,d,
r
ˆ i, q−i , H
(26)
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4. NUMERICAL APPLICATION TO AIRCRAFT STRUCTURAL HEALTH MONITORING We consider the structure of the aircraft wing shown in Figure 1 where the end near point A is fixed to the fuselage, and thus the wing acts as a cantilever beam. A finite-element-analysis based software (ANSYS) is used to simulate the wing structure. The wing is made of Aluminum and the wing span is l = 12 m. We assume that two transverse forces F1 and F2 are applied at points B and C, respectively, in the x-direction as shown in Figure 1. Moreover, we assume that five sensors P1–P5 are attached to the wing and they provide measurements of vertical deflections at their locations, where the locations of these sensors are as shown in Figure 1. A model of the structure and the time history of the applied forces F1 and F2 are assumed to be unknown. In this section, we use the available measurements from the sensors P1–P5 to identify transmissibility operators between them under healthy conditions of the wing structure. Then, we assume that a crack occurs in the wing, and the transmissibilities identified under healthy conditions are used along with the available sensor measurements to detect the fault in the structure. This is performed by monitoring the norm of the residual of the transmissibility over time. The forces F1 and F2 acting on the wing structure are given by 11 sin(ωi Ts k), (27) F1 (Ts k) = 2000 i=1
and
F2 (Ts k) =
11
Ai sin(ωi Ts k + ψi ),
(28)
i=1
where Ts = 0.1 sec and for all i = 1, . . . , 11, ωi = 5i rad/sec, and Ai = 1000 and ψi = 1.15 rad for odd i and Ai = 2000 and ψi = −1.15 rad for even i. Figure 2 shows the time history of F1 and F2 .
i=−d
ˆ −d, , . . . , H ˆ r, ]T . ˆ FIR = [H and Θ r,d, 3.3 Estimating the number of independent excitation signals Note from (3) that the dimension of yi is equal to the dimension of u. Therefore, constructing a transmissibility operator requires knowledge of the number of independent excitation signals m acting on the system. We use the following procedure to estimate the number of excitation signals m. We identify a transmissibility operator with m ˆ pseudo inputs and pˆ pseudo outputs, where m ˆ ∈ {1, . . . , p − 1} and pˆ ∈ {1, . . . , p − m}. ˆ For each identified transmissibility operator we compute the residual using (24), and the norm of the residual. The estimated number of independent external excitation signals is the value of m ˆ at which a sharp drop occurs in the norm of the residual. If a sharp drop is not obvious, then the estimated number of external excitation signals is the smallest value of m ˆ for which no sizable improvement is obtained for larger values of m. ˆ Redundant sensors can then be removed or retained for possible benefit in terms of the accuracy of the identified transmissibility operators. 971
Fig. 1. Basic aircraft wing. The end near point A is fixed to the fuselage, and thus the wing acts as a cantilever beam. Transverse forces F1 and f2 are applied at points B and C as shown. Moreover, measurements from five sensors P1–P5 attached to the wing are available.
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4.1 Estimating the number of independent excitation signals For all i = 1, . . . , 5, let yi denote the measurement obtained from the ith sensor, that is, Pi. Consider the case where F1 is operating and F2 is not operating. We use least squares with a noncausal FIR model with r = 25 and d = 25 to identify the transmissibility whose pseudo input is Yi [y1 · · · yi ] and whose pseudo output is y5 , where i = 1, . . . , 4. Figure 3 shows the norm of the residual of the identified transmissibility with 1, 2, 3, and 4 pseudo inputs. Note from Figure 3 that the benefits obtained from increasing the number of pseudo inputs from 1 to 4 are not significant. Therefore we correctly conclude that the number of inputs acting on the system is 1. Figure 4 shows the estimated Markov parameters of the transmissibility whose input is y1 and whose output is y5 . The estimated Markov parameters are obtained using least squares with a noncausal FIR model with r = 25 and d = 25. Figure 5 shows the measured output y5 and the predicted output yˆ5 obtained by applying y1 to the identified transmissibility whose Markov parameters are shown in Figure 4. Note from Figure 5 that the measured and predicted outputs are very close to each other. Next, consider the case where F1 and F2 are operating simultaneously. We add zero-mean, Gaussian white noise with a signal-to-noise ratio of 10 to the measurements of y1 –y5 . We use least squares with a noncausal FIR model with r = 25 and d = 25 to identify the transmissibility whose pseudo input is Yi [y1 · · · yi ]T and whose pseudo output is y5 , where i = 1, . . . , 4,. Figure 6 shows the norm of the residual of the identified transmissibility with 1, 2, 3, and 4 pseudo inputs. Note from Figure 6 that increasing the number of pseudo inputs from 1 to 2 yields a sizable improvement. On the other hand, the improvement obtained from using 3 and 4 pseudo inputs are not significant. Therefore we correctly conclude that the number of inputs acting on the system is 2. Figure 7 shows the estimated Markov parameters of the transmissibility whose inputs are y1 and y2 and whose output is
y5 . The estimated Markov parameters are obtained using least squares with a noncausal FIR model with r = 25 and d = 25. Figure 8 shows the measured output y5 and the predicted output yˆ5 obtained by applying y1 and y2 to the identified transmissibility whose Markov parameters are shown in Figure 7. Note from Figure 8 that the measured and predicted outputs are very close to each other. 4.2 Health Monitoring of the Wing Structure We assume that both F1 and F2 are operating simultaneously. We use the transmissibility identified under healthy conditions whose Markov parameters are shown in Figure 7 for health monitoring of the wing structure. We compute the residual of the identified transmissibility using (24), that is, for all k ≥ 0, r ˆ FIR ) = yo (k) − ˆ i, yi (k − i), e(k|Θ (29) H r,d, i=−d
T
where yo = y5 , yi = [y1 y2 ] .
Next, define the norm of the residual over a window of data of width w to be w+k ˆ FIR ). ˆ FIR , w) e(i|Θ (30) E(k|Θ r,d,
r,d,
i=k
We assume that at time t = 100 sec, a crack occurs in the structure of the wing. Figure 9 shows a plot of ˆ FIR , w) for w = 100 steps. Note from Figure 9 that E(k|Θ r,d, the level of the residual changed at t = 100 sec, which indicates that a fault has occurred in the wing structure. 5. CONCLUSIONS AND FUTURE WORK A transmissibility is a sensor-to-sensor relationship that relates one subset of sensors to another subset. In this 1.2
10 -3
1 10 4
2
0.8
0
0.6
0.4
-2 0
20
40
60
80
100
10 4
2
0.2
0
0
1
2
3
4
-2 -4 0
20
40 60 Time (sec)
80
100
Fig. 2. Time history of the forces F1 and F2 applied at points B and C, respectively, in Figure 1, where F1 and F2 are given by (27) and (28), respectively. 972
Fig. 3. Norm of the residual of the identified transmissibility with 1, 2, 3, and 4 pseudo inputs. In this case F1 is operating and F2 is not operating. Note that the benefits obtained from increasing the number of pseudo inputs from 1 to 4 are not significant. Therefore we correctly conclude that the number of inputs acting on the system is 1.
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paper, we used transmissibility operators for fault detection and health monitoring of aircraft structures. We used the ANSYS software to simulate the aircraft wing, where two forces were acting on the structure of the wing. Five sensors were attached to the wing, where they measure the wing vertical deflection at their locations. Transmissibilities were identified using least squares with a noncausal FIR model structure. We showed that transmissibilities can be used to detect changes in the dynamics of the wing structure without the need to know a model of the structure or the excitation signal. This approach can be used to detect changes in the wing structure before reaching the fracture stage.
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tion error methods, instrumental variables methods, and subspace methods will be considered in future research. REFERENCES Aljanaideh, K.F. and Bernstein, D.S. (2015a). Aircraft sensor health monitoring based on transmissibility operators. Journal of Guidance, Control, and Dynamics, 38(8), 1492–1495. Aljanaideh, K.F. and Bernstein, D.S. (2015b). Timedomain analysis of motion transmissibilities in force0.16
Future work will consider comparing the accuracy of timedomain transmissibilities to frequency-domain transmissibilities in fault detection of aircraft structural components. Moreover, several identification methods such as predic-
0.14 0.12 0.1
Markov Parameter Value
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0.08
0.4
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0.02
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0 1
3
4
0 -0.1 -0.2 -0.3 -20
-10 0 10 Markov Parameter Index
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Fig. 6. Norm of the residual of the identified transmissibility with 1, 2, 3, and 4 pseudo inputs. Note that increasing the number of pseudo inputs from 1 to 2 yields a sizable improvement in the residual. On the other hand, the improvement obtained from using 3 and 4 pseudo inputs are not significant. Therefore we correctly conclude that the number of inputs acting on the system is 2. 1 0.8 Markov Parameter Value
Fig. 4. Estimated Markov parameters of the transmissibility whose input is y1 and whose output is y5 . The estimated Markov parameters are obtained using least squares with a noncausal FIR model with r = 25 and d = 25. 0.3 0.2
Deflection (m)
2
0.1 0
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Fig. 5. Measured output y5 and predicted output yˆ5 obtained by applying y1 to the identified transmissibility whose Markov parameters are shown in Figure 4. Note that the measured and predicted outputs are very close to each other. 973
-10 0 10 Markov Parameter Index
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Fig. 7. Estimated Markov parameters of the transmissibility whose inputs are y1 and y2 and whose output is y5 . The estimated Markov parameters are obtained using least squares with a noncausal FIR model with r = 25 and d = 25.
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0.3
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24 26 Time (sec)
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Fig. 8. Measured output y5 and predicted output yˆ5 obtained by applying y1 and y2 to the identified transmissibility whose Markov parameters are shown in Figure 7. Note that the measured and predicted outputs are very close to each other. 0.023 0.022 0.021 0.02 0.019 0.018 0.017 0.016 0
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