ISSS National Conference on MEMS, Smart Materials, Structures and Systems September 28-30, 2016, Kanpur, India
OPTIMAL INPUT LOCATION FOR PARAMETER ESTIMATION IN VIBRATION BASED STRUCTURAL IDENTIFICATION Debasish Janaa, Suparno Mukhopadhyay b, Samit Ray-Chaudhuric Department of Civil Engineering, Indian Institute of Technology, Kanpur, Uttar Pradesh-208016 Email:
[email protected],
[email protected],
[email protected] Abstract: Structural identification is the inverse problem to estimate physical parameters of a structural system from its vibration response measurements. Accuracy of structural system identification is dependent on location of forcing function. For real application, particular locations are required where identification of system parameters will be estimated with minimum error. So an algorithm is proposed to find out optimal location of forcing function, using some optimality criterion (D, A and T) to Fisher Information Matrix (FIM). A modification of FIM is proposed, where number of unknown parameters is more than number of measurements. Numerical Simulations are performed on a truss. Extended Kalman Filter (EKF) Algorithm is used as structural system identification algorithm for numerical verification of this proposed algorithm. Keywords: System Identification, Optimal Input Location, Fisher Information Matrix, Optimality Criterion
1.
INTRODUCTION
In vibration based system identification, forcing function is necessary to induce vibration. Vibration can be generated either from ambient vibration or from any external artificial source like actuator or mass shaker or impulse hammer. Ambient vibration may not produce sufficiently high Signal to Noise (SNR) responses. So often an input device is used to generate vibration. The location of the forcing function plays important role in system identification in terms of error in estimation of physical parameters. So if any external vibration source is used, then it is necessary to know where to apply that forcing function. To perform this task, numerical trials can be done by changing the location of forcing function with suitable system identification technique. This is time consuming and accuracy depends on the estimator type which is used to solve the inverse problem. So there should be a general approach, so that force locations can be finalized which will provide the best estimate for system unknown parameters. In this article an approach is discussed which eases substantial computational time as well as effort. Several researchers have addressed the problem about optimal sensors location in structural system identification. Shah1 presented a sensor placement technique based on the Covariance Matrix Norm which reduces Fisher Information Matrix (FIM). Optimal sensor locations are chosen by maximizing the trace or the determinant (i.e. minimization of the covariance matrix of the estimate error which gives best of the structural response) of FIM. For structural identification, exhaustive search technique can be used for identification of the stiffness matrix by minimizing the trace of the covariance matrix2. Papadimitriou3 introduced information entropy (depends on the determinant of FIM) to minimize the uncertainty in the model parameter estimates over the set of possible sensor configuration using a genetic algorithm. In this article, optimal input location for structural system identification is proposed to best estimate the system
parameters. Udwadia2 has presented a methodology for optimal localization of sensors in a dynamic system, so that responses acquired from those locations will yield the best identification of the unknown parameters to be identified. Here the optimal sensor location methodology decouples the optimization problem from the identification problem which means the optimal sensor location is not dependable to the estimator. Similar concept is being utilized in this article. Udwadia2 optimizes the location matrix of measurement and in this approach the location matrix of forcing function has been optimized by Fisher information matrix and CramerRao lower bound. The ranking for best location for placing the actuator is done by different optimality criterion which is based on different norms of Fisher information matrix. In this article, optimal location of forcing function is done by proposed method for a simple plane truss. Structural system identification is performed by changing the location of forcing function in different suitable with a suitable system identifier Extended Kalman Filter (EKF)4. 2.
ALGORITHM FOR OPTIMAL LOCATION OF FORCE
2.1 Model Formulation The equation of motion of a linear structural system under external excitation can be written as ̈( )
̇( )
( )
( )
( )
Where, ( ) ̇ ( ) and ̈ ( ) are vectors of displacement, velocity and acceleration response respectively. and are the mass, damping and stiffness matrices, respectively; and is the influence matrix associated with external excitation vector ( ). This system is to be identified by any suitable vibration based system identification algorithm. External forcing function is required for generating the vibration. To best estimate the unknown parameters, the forcing function should be located in those dynamic degree of freedom, acceleration response of which provides most information about the
unknown system parameters. To accomplish the task, all unknown parameters are arranged in a vector, * + ; where , and are related to and matrices. The state vector of the system is defined as * ̇ + . Now Equation.(1) can be written as ̇ 2 3 ̈ ̇
{
( )
,
̇
-} ̇
( )
For performing any experiment regarding vibration, accelerometers are mainly used and when limited number of accelerometer is used in structures to measure acceleration responses, then the measurement equation can be presented as , ̈
( )
̇
)
Where, ( )
( ) ,
( ) ̇
-
( )
2.2 Fisher Information Matrix An evaluator is required which can extract maximum amount of information about unknown parameters from the response time history. Fisher Information is a scheme to compute the amount of information about unknown parameters carried by observable random variable. Here, these unknown parameters of a distribution represent this observable random variable. In this problem, observable random variable is the acceleration time histories of different degrees of freedom and the unknown parameters are the unknown system parameters. Fisher information is used in this article to find the optimized matrix in Equation.(5). Here a methodology is developed to locate actuator so that, when starting from a close nearby guess, the unknown parameters are best identified. This method results in the conditional estimation problem for which the covariance of the vector parameter estimates satisfy the relation 5. 0(
̂ )(
̂) 1
( )
( )
Where, ( )
∫
.
/ .
/
Add As is measurement noise covariance which is constant, so it will not affect the optimal actuator location, so it can be factored out. So Equation.(7) can be transformed to ( )
∫ (
) (
̅( )
)
( )
As ̅ ( ) is a matrix, to obtain an idea of the information content, suitable scalar norm ‖ ̅ ( )‖ should be done. One most commonly used norm7 is as follows D-optimality: Minimize the determinant of ̅ ( ) or equivalently maximizing the determinant of ̅ ( )
(3)
Where, is the observation vector (measured acceleration responses) at time . Here is the sampling time step, and ̇ are the corresponding discretized values at time , is the matrix associated with accelerometer location and is the measured Gaussian noise vector with zero mean and a covariance matrix [ ] , where is the Kronecker delta. The measurement equation in Equation.(3) is also a nonlinear function of the extended state vector. So the discretized observation equation can be expressed as (
2.3 Choice of Matrix Norm
( )
( ), in Equation.(6), Here, ̂ denotes the estimate of . is called Cramer Rao Lower Bound (CRLB). The expression in Equation.7 is called as Fisher Information Matrix2. By maximizing a certain norm of this Fisher Information Matrix would yield the minimum possible value of the covariance of the estimation error and hence least uncertainty in the identified model6.
( ̅ )-)
,
(
(
[∏
Where, is the eigenvalue of Fisher information matrix ̅ ( ) which has total ' ' number of eigenvalues. As the basis of fisher information matrix in this vibration based system identification method is the responses and the order of acceleration response of a real structure of its dynamic degree of freedom is very small, so formula (Equation after ‘=’ sign with ‘log’) of D-optimality can be used. 2.4 Modification in Fisher Information Matrix If number of measurements are ' ' and number of unknown (
)
(
)
So dimension of ̅ ( ) becomes ( ). If , then ̅ ( ) possess rank deficiency of ( ), as the ( ) dimensional ̅ ( ) matrix has rank of ' '. This rank (̅) deficiency causes So optimal input location cannot be found out by D optimality as for every location of forcing function, D-optimality comes as ‘zero’. So comparison is not possible. To circumvent this problem, a modification of fisher information matrix is proposed. If number of unknown parameters is greater than number of measurements then optimality criterion should be executed on modified fisher information matrix. The expression for Modified Fisher Information Matrix (MFIM) is as follows ̿( )
∫ (
)( 3.
)
( )
NUMERICAL SIMULATIONS
3.1 Structural Parameters Numerical simulation is shown for a simple truss structure. A plane warren truss is supported at two ends on hinge joints, which is shown in Figure.(1). In this figure, ‘dof-n’ represents the DOF. Sensors are mounted in all dynamic degrees of freedoms. represents stiffness values of member. Truss material is having Young modulus of GPa and mass density of kg/m3. So the stiffness of the diagonal and horizontal members are 26.59 MN/m and 18.82 MN/m respectively. Masses are assumed to be lumped. First and last modal frequency of this truss is 0.85 Hz and 7.23 Hz respectively. Finite Element Method is used to formulate global mass and stiffness
])
matrices. Rayleigh Damping of first two modal damping ratio 3% is assumed to compose global damping matrix. In this problem, the acceleration responses of all the dynamic degrees of freedoms are taken. Now, the question is which will be the optimal input location for estimating all the unknown stiffness properties with minimum error.
After enumerating the errors in the unknown parameters for 100 candidate models, standard deviation is calculated. Subsequently, norm is calculated to obtain an idea about the estimated error in total structure, which is shown in Figure.(3). This shows that the best input location as DOF 10. DOF 8 also shows one of the best input locations.
Figure 1. Plane truss with both end hinged (Dimensions are in m) Figure 3. Optimal Input Location by Brute Force Approach
3.2 Forcing Function Random white noise is chosen for this simulation. Random white noise of zero mean and standard deviation 100 N is used, which is applied for 4 seconds. Time history is shown in Figure.(2) This is filtered with a low-pass Butterworth filter of order 6 and cut-off frequency 20 Hz. Now the problem is to find the best DOF to mount this force. Dynamic responses of the structure are calculated by RungeKutta order method and numerically generated noise is added for simulating practical case.
5.
OPTIMAL INPUT LOCATION BY PROPOSED ALGORITHM
Optimal input location algorithm needs prior knowledge about the unknown parameters. For showing a sample calculation, exact values are considered for calculation of optimal input location of this truss which is mentioned in Section.(3.1). As this truss is consists of 10 dynamic degrees of freedoms and 11 unknown stiffness parameters, so optimal input location has been calculated by optimizing the Modified Fisher Information Matrix Norms. Optimal input location, which is found out by the proposed algorithm is shown in Figure.(4). Here, the optimal input location found out to be DOF 8 or DOF 10. Both these DOFs contains similar amount of information about the unknown structural parameters, which is obvious because of the symmetric nature of the truss structure.
Figure 2. Forcing Function Time History
4.
OPTIMAL INPUT LOCATION BY BRUTE FORCE APPROACH
As the stiffness parameters are unknown, but the range where the stiffness parameters may belong is known, so the model space can be generated. Generally, stiffness properties are related to mass properties of the structures. Stiffness and mass is a direct function of Young Modulus and Density, respectively. So, here the Young Modulus is assumed to be in the range of 150 to 250 GPa and similarly, Density is in the range of to kg/m3. Now when the ranges are known, the model space are generated by using Latin Hypercube Sampling Technique. This model space consists of 100 candidate models. In the model space, stiffness of diagonal members belongs in the range 19.94 MN/m to 33.23 MN/m and similarly stiffness of horizontal members belong in the range 14.10 MN/m to 23.50 MN/m. After generation of all these candidate models, the system parameters are identified for all these samples. Extended Kalman Filter4 is used as the system identification algorithm.
Figure 4. Optimal Input Location by Proposed Algorithm
6.
COMPARISON BETWEEN BRUTE FORCE APPROACH AND PROPOSED ALGORITHM
To show the efficiency of the proposed algorithm, the results obtained from brute force approach and proposed algorithm are shown together in Figure.(5). This figure shows that more the information, lesser the standard deviation of error, which signifies lesser the uncertainty in the estimated parameters of the structure. So this is clear that, the proposed algorithm emulates the uncertainty estimation of unknown structural parameters by brute force approach, but in substantially computationally efficient way.
Judging from the computationally efficient point of view, this proposed algorithm can be a powerful tool in the field of system identification.
REFERENCES
Figure 5. Comparison between Proposed Algorithm and Brute Force Approach
7.
CONCLUSION
This proposed algorithm finds out the best location for forcing function so that structural system parameters are estimated with minimum amount of error. It saves lots of computational time. If any structural system identification algorithm is used to find the best input location, it would be time consuming and it would be dependent of that system identification algorithm. This proposed algorithm is not dependent on any system identification algorithm. This is a model based algorithm, so the knowledge of structural model is very important. As the parameters cannot be guessed properly by the overview, parametric sensitivity analysis by any sampling algorithm (Monte Carlo Simulation or Latin Hypercube Sampling) can be performed prior to the field application. In the field, a possible range of system parameters are estimated and sampling techniques generates samples in that range. For all those samples, this proposed algorithm can be used and optimal location can be found out. Then from that sample, DOF having majority of top rank is chosen as optimal location for forcing function.
[1] Shah, P. C., and F. E. Udwadia. "A methodology for optimal sensor locations for identification of dynamic systems." Journal of Applied Mechanics 45.1 (1978): 188-196. [2] Udwadia, Firdaus E. "Methodology for optimum sensor locations for parameter identification in dynamic systems." Journal of Engineering Mechanics 120.2 (1994): 368-390. [3] Papadimitriou, Costas, James L. Beck, and Siu-Kui Au. "Entropy-based optimal sensor location for structural model updating." Journal of Vibration and Control 6.5 (2000): 781-800. [4] Kalman, Rudolph E., and Richard S. Bucy. "New results in linear filtering and prediction theory." Journal of basic engineering 83.1 (1961): 95-108. [5] Nahi, Nasser E. Estimation theory and applications. New York: Wiley, 1969. [6] Jazwinski, Andrew H. Stochastic processes and filtering theory. Courier Corporation, 2007. [7] Mehra, Raman K., and Dimitri G. Lainiotis. System identification advances and case studies. Vol. 126. Academic press, 1977.
