DEERMINISIC SOCHASIC SBSPACE IDENIFICAION FOR BRIDGES H. hai, V. DeBrunner, L. S. DeBrunner, J. P. Havlicek 2, K. Mish 3, K. Ford 2, A. Medda Deartment o Electrical and Comuter Engineering Florida State niversity 2 School o Electrical and Comuter Engineering 3 School o Civil Engineering and Environmental Science niversity o Oklahoma ABSRAC System identiication or structural engineering has received signiicant attention in the last thirty years. With the ever increasing caacity o comuting technology, system identiication has been alied to imortant structures such as bridges and aircrat. In the case o bridges, the outut can easily be measured by accelerometers. Considerable research in system identiication on bridges has been done using outut only models. O course, it is diicult to measure the inuts on an in service bridge. In this aer, we see how the inuts can be estimated rom the outut measurements. We then use an inut outut model to develo an imroved system identiication technique or identiying bridges. We show that the roosed method using the estimated inuts yields suerior identiication in a simulated case (i.e., where everything is controlled). We then use the method on the in-service Walnut Creek Bridge located on the north-bound lanes o.s. Interstate I 35 between Dallas, X and Oklahoma City, OK. Index erms system identiication. INRODCION System identiication o large structures such as bridges, aircrat and buildings has received considerable attention. hrough identiication, we can determine whether the roerties o the observed structures are changing over time. his is an essential comonent o the structural health monitoring roblem. Structural engineers monitor structural resonse in order to detect damage. hey are also interested in the modal (natural) requencies o the system because any matching between inut requencies and modal requencies can cause the structures to resonate, causing signiicant structural resonse with attendant damage. Previously, the stochastic subsace identiication algorithm was alied to identiy bridges using an outut only model []. his algorithm assumes that the inuts are white rocesses, which is never the case. In system identiication, the more inormation we ossess and use roerly, the better our roduced estimation. In this aer, we will show that the deterministic stochastic subsace identiication method using an inut outut system model imroves the identiication results over the stochastic subsace identiication using the outut only system model. 2.. System Modeling 2. MEHODOLOGY he comlete system modeling algorithm is ound in [2]. In this section, we only summarize the algorithmic base that is used in this aer. A bridge system under a moving load can be modeled using the ollowing inut outut state sace model. x( k ) Ax( k) Bu( k) ( k) yk ( ) Cxk ( ) Duk ( ) vk ( ) () Q S E q vq ( q) v S R where x(k) is the state vector, u(k) is the vector o inut orces o the moving load and y(k) is the vector o outut measurements. he system is assumed to be stationary. he rocess noise (k) is caused by traic, variations in the moving load, and modeling error. he measurement noise o the sensors is accounted or in the rocess v(k). We assume (k) and v(k) are zero mean and white sequences with the covariance given in (). 2.2. Algorithm Block Hankel matrices can be built rom the inut and outut sequences. he inut block Hankel matrix is ound in equation (2), with each element being a column vector o the inut sequence. u() u() u(2) u( j) u() u(2) u(3) u( j) (2) i 2i ui ( ) ui ( ) ui ( ) ui ( j2) i 2i ui ( ) ui ( ) ui ( 2) ui ( j) u(2i) u(2 i) u(2i) u(2i j 2) he value o i must be greater than the maximum order o the system that we want to identiy. he inut block Hankel matrix can be divided into two arts, and, which denote ast and uture, resectively. he outut block Hankel matrix can be derived similarly. Notice that we have to choose j such that 2i + j 2 is within the observed inut and outut sequences. he number o columns j must also be much larger than the number o block rows i or the algorithm to work accurately. A combination matrix o inut and outut can be deined -4244-98-X/7/$25. 7 IEEE 749 SSP 7
W Y (3) An oblique rojection Ob i o the row sace o Y along on W is comuted as in (4). he oblique rojection removes all o the uture inut inormation rom the uture outut using only the ast inormation. WW C W (4) Obi Y / W Y WC W W C irst r columns he denotes the Moore Penrose seudo-inverse discussed by Ben-Israel in [3]. he main deterministic stochastic subsace identiication theorem states that the oblique rojection Ob i is equal to the roduct o the extended observability matrix i and the state sequence X i, i.e. i Obi ixi C CA CA x() i x( i ) x( i j ) (5) he extended observability matrix and state sequence can be extracted using the singular value decomosition o Ob i. S V Obi SV 2 SV (6) V2 and V are unitary matrices, and S is a diagonal matrix;, V and S are similar to, V and S without their zero singular values. he extended observability matrix i is derived as ollowing: / 2 i S (7) he state sequence X i is the other hal o the decomosition or can be calculated as in (8) 2 Xi S V (8) he irst block rows o the uture block Hankel matrices can be moved into the ast block Hankel matrices to yield we the Hankel matrices,, Y, Y and the inut outut matrix W. Another oblique rojection is deined. Ob i Y / W (9) he main theorem also holds or the new oblique rojection. Obi i X () i he state sequence X i can be derived rom the new oblique rojection. Xi iob () i he system matrices (A, B, C, D) can be determined in a least square sense in the ollowing equation system. Xi A B Xi (2) Y ii C D ii v he above system matrices are the true system matrices u to a similarity transormation. 2.3. Inut estimation In the case o real bridges, it is diicult to measure the inuts due to hardware and access limitations. In our exerience insecting the bridge accelerations, we have ound that only traic that signiicantly imacts the bridge is heavy trucks. hereore, we roose to estimate the truck inuts using our estimated outut (the easily measured bridge accelerations). In our aroach, we want to ind the initial osition o the truck (the osition o the truck when we start collecting data) and the velocity o the truck. From the measured outut accelerations, we determine that the vibration amlitudes are strongest when the vehicle is directly above the accelerometer. At other times, there are only small changes in accelerations. Consequently, a straightorward estimation rocess or our required inuts can be develoed. First, we sum the energy rom the oututs o the bridge as the truck crosses. We determine the amount o time in this observation that divides that energy into two equal ortions. Now we estimate the time required by the truck to reach the sensors. Now, since we know the osition o the sensors on the bridge and the time that the truck required to reach each sensor, we can determine the initial osition and velocity o the truck using the least square method. he resulting estimates can now be lugged into a bridge-load model to generate the inuts required in the system identiication algorithm. 2.4. Modal Analysis A mode o a structure is a attern o deormation o the structure at a seciic harmonic requency [4]. Modes are o concern to structural engineering because signiicant damage due to resonance can result i a structure is excited by an external orce having a requency that coincides with one o the modes. he requencies o the structural modes can be ound by taking the imaginary arts o the eigenvalues o the continuous time state matrix A [4]. hese natural requencies may be determined as arg( i ) s i (3) 2 where i is the i th eigenvalue o the discrete-time state matrix A, s is the samling requency, and i is in Hz. 2.5. Order Determination heoretically, ater all the zero singular values are removed rom the diagonal matrix S, the dimension o S is the order o the system. In real systems, all signals are subject to noise and so there are no identically zero singular values. It is hard to determine the correct order when noise is added to the signals. One o the tools or order determination is the stabilization diagram described in [5]. he basic idea o the stabilization diagram is that we can ick the n highest singular values rom the S matrix and comute the natural requencies o the structure. We iterate n rom 2 to the maximum order that we want to identiy or the structure and comare the natural requencies o the current order with those o one order lower. I the requencies are within a given tolerance, we will accet them as stable estimates o natural requency. In this aer, we choose the tolerance to be 2%, so that the criterion or acceting a new mode is given by ( n) ( n) 2% (4) ( n) Although the stabilization diagram can be used to identiy and retain the hysical oles, it does not show us the energy or the strength o those modes. For our uroses, we want to ind the most imortant modes and get rid o the insigniicant ones. he whole system can be decouled into individual modes. Each mode can be reresented by a state sace equation. x( k) ix( k) bu i ( k) (5) yk ( ) ci ( k) where i is the i th eigenvalue o the identiied state matrix A, b i is the i th row o the inut matrix B and c i is the i th column o the outut matrix C. B and C are obtained rom the identiied matrices B and C by a similarity transormation such that the 7
matrix A becomes a diagonal matrix. he eed through matrix D can be ignored since the inuts are the same or all the modes. he transer unction o each mode is deined as: cb i i Gi ( z) (6) z i We comute the norms o the transer unctions o the modes to determine which modes are signiicant in the system. In this aer, we use the H 2 norm described in [4] 2 2 * j j Gi trgi ( e ) Gi( e ) d (7) 2 Where the tr() denotes the sum o the diagonal elements and * is the conjugate transose. here are several aroaches to determine the model order o the system. One aroach is that we truncate at the oint that has the biggest ga in H 2 norms. he disadvantage o this method is that there may be no clear ga between the modes. A better aroach aears to be based on thresholds. For examle, a hard threshold o 9% would mean that we retain only those largest norm modes that contribute 9% o the total energy comuted, and thus remove those smallest norm modes contributing only % o the total. he 9% threshold corresonds to a 9.54 db SNR where the signal is the large norm comonents and the noise is the small norm comonents. Ater analyzing the identiied system through the stabilization and the H 2 norm, we can determine the order o the system consisting o signiicant modes. 3. EXPERIMENAL CASES 3.. Bridge Load Simulation An eective, ractical bridge load simulation was thoroughly described in [6]. his simulation consists o two arts: bridge simulation and load simulation. he bridge is simulated using a inite element analysis or a beam. he load is simulated by a three orce constant seed load such as would be ound by a truck wheel loading. he inut orces and outut accelerations are collected or identiication. he true modal requencies can be comuted and comared with the results o the system identiication methods. able shows the irst six true modal requencies o the bridge simulation. Mode Frequency.8244 Hz 2 2.2667 Hz 3 4.54 Hz 4 7.289 Hz 5.888 Hz 6 4.84 Hz able : rue modal requencies o the bridge simulation he test case in this aer has 39 db inut noise and 57 db outut noise. Fig. illustrates the stabilization diagram o the bridge load simulation using our roosed deterministic stochastic subsace identiication (inut outut model). We observe that it can detect the irst six modes. Modes 2 and 3 are more suscetible to the noise than are the other modes. Fig. 2 illustrates the stabilization diagram o the bridge load simulation using stochastic subsace identiication (outut only model) which was used in []. We also use the same set o data as in Fig. or a air comarison. Modes 2 and 3 cannot be detected by this method. So, our roosed method outerorms the existing one. 2 3 4 5 6 7 8 9 2 3 4 5 Fig. : Stabilization diagram using deterministic-stochastic subsace identiication on bridge-load simulation. 2 3 4 5 6 7 8 9 2 3 4 5 Fig. 2: Stabilization diagram using stochastic subsace identiication on bridge-load simulation. More analyses can be ound in [6] which show that the stochastic subsace identiication sometimes introduces surious modes. his is exlained by the act that stochastic subsace identiication assumes white noise inuts, which is rarely ever the true case in ractice. hereore, a better system model can be obtained by estimating the true system inuts. H2 norm 2 7 2 2 3 4 5 6 7 8 9 2 3 4 5 Fig. 3: H 2 norms o the modes o the bridge simulation. 75
he H 2 norm diagram o the modes is shown in Fig. 3. he modal energies are concentrated in the lower requency modes o the system, as exected by structural engineers. 3.2. Walnut Creek Bridge 2 8 X= 9.78 Y=.6443 Walnut Creek Bridge (Purcell, Oklahoma) is the bridge on Interstate highway I 35, near mile marker 9 in the northbound direction (see Fig. 4). he bridge is t long. It consists o 4 sans; each san is t. long. he test san is chosen to be the third san. he oututs were collected rom the accelerometers laced on the underside o the bridge along the girders. he inuts were estimated by the inut estimation method discussed above. Fig. 5 illustrates the stabilization diagram ater identiication using our algorithm. We ind 3 modes with requencies below Hz. Walnut Creek Bridge on I-35 near mile marker 9 built in 97 2 Lanes o South to North continuous girders (5) Piers at intervals 22m ( t) overall length;.6m (38 t) wide Four san steel girder bridge Concrete deck carries two lanes o traic 45 skew to accommodate creek below Avg. Daily Vehicle raic 8, Avg. Daily Heavy ruck raic 2, Fig. 4: Walnut Creek Bridge 2 3 4 5 6 7 8 9 Fig. 5: Stabilization diagram using deterministic-stochastic subsace identiication on Walnut Creek Bridge. Fig. 6 shows the strength o the modes. We observe that the energy is concentrated in the low requency modes excet or the modes at 74.25 Hz and.42 Hz. H2 Norm X= 7.6953 6 Y= 5.2389 X= 3.52 Y= 4.24 4 X= 3.625 Y= 2.4565 2 X= 8.627 Y= 2.998 X= 74.2539 Y= 3.226 X=.4269 Y= 4.722 Fig. 6: H 2 norms o the modes o the Walnut Creek Bridge It is diicult to veriy the results with the true modes o the bridge because the bridge has been in service or more than years. Visual insection o the bridge conirms extensive structural damage to the structure (e.g. a twisted girder is resent in the observed san o the bridge). However, we can comare our results with revious (imerect) identiication on this bridge. he imulse resonse method [7] was alied by using a droed hammer on the Walnut Creek Bridge in 996, and the stochastic subsace identiication method was used in 4 []. deterministic stochastic imulse stochastic.29 Hz - - 2.665 Hz - - - 2. Hz - 2.8 Hz 2.9579 Hz - 3.25 Hz - - - 3.324 Hz 3.52 Hz 3. Hz - 3.62 Hz 3.625 Hz - - 3.875 Hz - 4.29 Hz 4.2 Hz - - 4.375 Hz 4.3435 Hz - 4.7 Hz - - 5. Hz - - - 5.246 Hz 5.459 Hz - - - 5.625 Hz - 6.4455 Hz - - 7.8 Hz - - - - 7. Hz 7.6953 Hz 7.625 Hz - - 8. Hz 8.368 Hz able 2: Comarison among deterministic-stochastic, stochastic subsace identiication and imulse methods Insecting Fig. 6 and able 2, we observe that the high energy modes in our deterministic stochastic subsace identiication match with the imulse resonse method (at 3.52Hz, 3.625 Hz, 4.29 Hz and 7.6953 Hz). he stochastic subsace identiication does not roduce results that closely match with either o the two 752
other methods. his is robably exlained by the act that that algorithm assumes the inuts are white while they are, in act, not white. he high requency modes that we detected can be exlained by either the characteristic o the moving load or the uncertainty that was created when we estimated the inuts. 4. BRIDGE MONIORING SYSEM he Bridge Monitoring System (BMS) has been develoed at the Dynamic Structures Sensing and Control (DySSC) Center at the niversity o Oklahoma with the intention to acquire bridge data continuously over long eriods o time. he BMS has our major comonents: central comuter, sensor rototyes, network devices and ower system. he sensor nodes collect outut data (mainly acceleration). he central comuter acquires the data rom the sensor rototyes and archives them. he network devices maintain the connection between the central comuter and sensor rototyes. he ower system gets energy rom solar anels. It rovides the ower or the whole system to ensure that the BMS can oerate continuously. sing the CP/IP rotocol, the BMS can theoretically handle u to our million sensors in one local network with high seed communication. he rotocol also allows us to connect to the system remotely. he detailed imlementation o the system is described in [6] and [8]. he system rovides the real data or us to identiy bridge systems. he system will also be used or other uroses such as bridge health monitoring/damage detection and real time eedback control on moving trucks. Currently, the system has been successully imlemented at the Walnut Creek Bridge, and a similar system with imroved electronics was successully deloyed at the Canadian River Bridge on the southbound lanes o Interstate I-35 in Oklahoma (Fig. 7). A test system was also imlemented at the Walnut Creek Bridge. Additional analysis rom the Walnut Creek Bridge data can also be ound in [6]. Fig. 7: he Bridge Monitoring System at Canadian River Bridge aroriately handle the unknown inuts comared to the stochastic algorithm that is built on the imlicit erroneous assumtion that the inuts are white. he identiication results can be used by structural engineers to monitor the health conditions o the bridges and detect the damage by tracking any changes in the structural modes over time. hey can also be used to alter the requencies o the truck chassis in order to attenuate any resonance o the truck chassis with the bridge. Our identiication results rom Walnut Creek Bridge show several high energy modes at high requencies which could be rom the truck chassis, rom estimation errors that cree into the results when we estimate the inuts o the trucks, or hysically resent in the structure. his must be clariied in the uture. For uture research, we can incororate the Kalman ilter state into the state sace model to imrove the results. We can also study several bridge and truck models to have better identiication results. Furthermore, measuring the initial osition, velocity and weight o the trucks by hardware will hel to reduce the errors on estimation o the inuts. he subsace identiication also introduces the surious mode at DC which can also be investigated. 6. REFERENCES [] V. DeBrunner, L.S. DeBrunner, P. Wang, J. David Baldwin, A. Medda and H. hai, Stochastic sub-sace identiication methods or Bridges, Proceedings o the 39 th Asilomar Conerence on Signals, Systems and Comuters,. 53-57, 5. [2] P. Van Overschee and B. De Moor, Subsace Identiication or Linear Systems: heory-imlementation-alications, Dordrecht, he Netherlands, 996. [3] A. Ben-Israel, he Moore o the Moore-Penrose inverse, Electronic Journal o Linear Algebra, vol. 9,. -57, 2. [4] W. K. Gawronski, Advanced Structural Dynamics and Active Control o Structures, Sringer-Verlag, New York, 4. [5] H. Van der Auweraer and B. Peeters, Discriminating hysical oles rom mathematical oles in high order systems: use and automation o the stabilization diagram, Instrumentation and Measurement echnology Conerence IMC 4, vol. 3,.293-298, 4. [6] H. hai, System Identiication o Bridges under a Moving Load and Imlementation o the Bridge Monitoring System, Master hesis, he School o Electrical and Comuter Engineering, niversity o Oklahoma, 7. [7] J. Pang, Modeling and Exerimental Modal Analysis o Highway Bridges, PhD hesis, he School o Aerosace and Mechanical Engineering, niversity o Oklahoma, 996. [8] K. Ford, Comuter Hardware or Vibration Mitigation and Monitoring, Master hesis, he School o Electrical and Comuter Engineering, niversity o Oklahoma, 7. 5. CONCLSIONS AND FRE WORK he deterministic-stochastic subsace identiication with estimated inuts signiicantly imroves the results o identiication o bridge systems comared to the stochastic outut-only algorithm. Our inut-outut system model is able to more 753