International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

Force identification by comparing likelihood function using Bayesian filtering methods Radhika, B. Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  1 Force Identification by Comparing Likelihood Function Using Bayesian Filtering Methods Radhika B. DST-INSPIRE Faculty, Dept. of Civil Engg., Indian Institute of Technology Kharagpur, India ABSTRACT: The paper addresses the problem of force identification in randomly excited dynamical systems using Bayesian filtering methods. These are recursive algorithms comprising of the prediction and updating steps operating on specified models for the dynamical system and measurements. The Bayes‘ theorem is adopted in the updating step where the updated estimate is obtained as a product of normalization constant, likelihood function and the estimate in the prediction step. It is assumed that a model for the dynamical system is specified either in the form of a governing equation or a finite element model and measurements of system response to the unknown input are also available. In the proposed method the force to be identified is modeled as a white noise process with an unknown standard deviation. A procedure based on comparing the likelihood function in the updating step is proposed to select the noise parameter adaptively. This adaptive choice is proposed to identify forces which are realizations of a nonstationary random process model. The same will be demonstrated using illustrations on linear and nonlinear dynamical systems.  1. INTRODUCTION The load acting on an existing structure is often, difficult to directly measure, given the random nature of the loads and how they interact with the system. But the system response to the loads, like acceleration, displacements and strains can be recorded by instrumenting the structure, and these can be analysed to estimate the corresponding loads. Here the structure can be perceived as a sensor to measure the applied loads. Traditionally, this is the problem of force identification pertaining to the broader class of inverse problems.  Algorithms to identify different types of forces like impact (Martin and Doyle, 1996), wind (Law et al., 2005 and Hwang et al., 2011) and wave forces (Worden et al., 1994) have been proposed in literature. In the present paper a Bayesian filtering based solution is proposed for the force identification problem, hence the review of literature will focus on Bayesian estimation tools applied to address identification problems. The filtering tools are recursive algorithms used for state estimation where the states could be system response, system parameters, forces and sensitivities. The linear filtering based methods were demonstrated as identification tools in the works of Yun and Shinozuka (1980) and Imai et al., (1989). After the successful implementation of the nonlinear filtering tools (Gordon et al., 1993 and Doucet et al., 2000) in the 1990s the application of these to time invariant parameter identification was explored in the works of Ching et al., (2006), Manohar and Roy (2006), and Nasrellah and Manohar (2011).  The unknown parameters were taken as additional states with white noise priors.  In contrast to the time invariant system parameter identification problem, the force identification problem essentially involves identifying a time varying dynamical input to a system like earthquake loads, vehicle axle loads, wind forces and wave forces. In this context Kalman filtering based methods have been proposed to address the problem in linear and nonlinear systems by Ma et al., (2003) and Ma and Ho (2004). The methods involved least squared optimization to identify the forces. Alternatively, the possibility of assuming 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  2 unknown force as an additional state, as in parameter identification has been explore by Lourens (2012) in linear systems, where the covariance of the force is obtained by regularization. Radhika and Manohar (2013) have applied particle filtering based methods to nonlinear systems and demonstrated force identification for multi-axial and spatially varying inputs by assuming a white noise prior for the force with constant noise variance. In addressing the force identification problem with force as an additional state the choice of the noise parameter becomes critical. Identifying the noise variance by taking the extended state vector approach makes it a problem of parameter identification. Since the parameter here should vary with time to capture the time variation in the force, the current work proposes an adaptive method using the likelihood of the measurement, computed in the filtering steps to identify the noise variance.  To begin with a formal description of the force identification problem in the filtering framework is presented in section 2. Subsequently, the proposed algorithm is discussed in section 3 followed by numerical illustrations on linear and nonlinear dynamical systems and a discussion on the method in sections 4 and 5, respectively. Finally, a summary of the paper is presented in section 6. 2. PROBLEM FORMULATION Consider an n-degree of freedom (dof) dynamical system governed by    ̈( )   (   ̇  )   ( )   ( ) (1)  where     ̇  ̈  are     displacement, velocity and acceleration responses, respectively,  (   ̇  )  is a     nonlinear function,  ( )  is     applied load and  ( )  is a     white noise process accounting for error in the modeling with  , ( )-    and  , ( )  (   )-      ( );  ( ) is the Dirac delta function. Re-writing Eq.(1) in the state space form with  ̃( )  * ( )  ̇( )+ will result in a stochastic differential equation (SDE) of the form   ̃( )   , ̃( )  -    , ̃( )  -  ( )   ̃( )   ̃        (2)  where  ̃( )  is      state vector,  , ̃( )  -  is the      drift vector,  , ̃( )  -  is      diffusion matrix and   ( )  is     vector of increments of Brownian motion process with  ,  ( )-    and  ,  ( )   ( )-         In  order for Eq.(2) to be amenable for use within the Bayesian framework to be discussed in the next section, it is discretized using Taylor series based methods (Kloeden and Platen, 1992) to obtain   ̃     ̃ ( ̃ )   ̃   ̃ ( ̃ ) ̃    ̃            (   )         (3)  where  ̃     ̃ ( ̃ )  ̃    ̃ ( ̃ ) ̃ and  ̃  are      vectors with the subscript   denoting discretization in time.   is the time length and   the time step considered. Furthermore, measurements on systems response are available and are given by        ( ̃ )              (   )   (4)  where     is      measurement vector,   ( ̃ ) is      vector relating the measurement with system states and    is the measurement noise with  ,  -    and  [     ]                   (   ) which accounts for errors in measurement model and data acquisition. In the above formulation it may be noted that  ̃     and  ̃  are all independent for         (   ). In this setting, the problem is to identify the input  ( )  acting on the system conditioned on the available measurements      *     +  superscript   denotes the matrix transposition operation. Mathematically, the quantity of interest is the conditional probability density function (pdf)  , (  )     -          (   ) . This is a problem of estimating the filtering density in the dynamic state estimation (Bayesian filtering) framework. The problem of force identification 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  3 stated above will be addressed using Bayesian filtering method as described in the next section. 3. FORCE IDENTIFICATION USING BAYESIAN FILTERING The Bayesian filtering framework is employed to address the problem of force identification. To begin with, the unknown force is defined as an additional state to obtain the extended state vector  ( )  * ( )  ( )̇  ( )+   Furthermore, the model for the force is assumed to be a white noise process   ( ), given by   ̇( )    ( );     [  ( )]   ; [  ( )   (   )]       ( )(5)  which after discretization will have the form                        (6)  where     denotes the third element in the state vector   ( )  (i.e. force) at the     time instant and    is a Gaussian random variable with distribution  (   ) , where  (   )  denotes a Gaussian pdf with mean   and variance  .  Considering Eqs. (3) and (5) as process and measurement equations respectively, the filtering problem is to find the conditional joint pdf  , (  )  ̇(  )  (  )     -          (   )  of which the marginal pdf  , (  )     -  is the problem of force identification. There are methods in Bayesian estimation which provide a solution to this filtering problem depending on nature of process and measurement equation. For linear process and measurement equations with Gaussian additive noises, the filtering problem is exactly solved to be a Gaussian pdf whose mean and covariance are given by the Kalman filter (Kalman, 1960). For nonlinear and non-Gaussian models, methods based on Monte Carlo simulations (MCS) called particle filters or sequential Monte Carlo filters (Doucet et al., 2001) provide an approximate solution to the filtering problem. The efficiency of the proposed methods depends on the value of     chosen in Eq.(6). In an earlier work by the author (Radhika and Manohar 2013) involved using a constant value for    . In the present paper, a recursive method to estimate     at each time instant is proposed. The method involves updating the parameter    using the likelihood of the measurement evaluated in the Bayes‘ theorem. The steps of the proposed force identification algorithm are as follows: 1. Initiate N samples of {     }     distributed uniformly and  (  )  2. Set      ; where         (   )  3. For each sample of       , estimate the filtering pdf using Kalman filter or particle filter. 4. The filter estimate is used to compute the likelihood of the measurement for the given value of       denoted by  .        /. This is the ordinate of the updated filtering pdf corresponding to    . 5. The likelihood of the measurement is used to obtain an update for     as  .        / ∑ .        /∑  .        /         .         /  6. The updated pdf is used to generate samples of {     }    which are used in the estimation of the updated pdf at the next time instant. 7. The      corresponding to      ,  -| ̃(  )   ̃(    )|  is taken as the optimal value of the noise parameter and corresponding state estimate is the updated value at the    time instant. Here   ̃( ) denotes the value of the updated pdf evaluated at  . 8. If   (   ) go to step 2, else end.  12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  4 The steps delineated above are adopted to identify input in linear and nonlinear systems as discussed in the next section. 4. NUMERICAL ILLUSTRATION 4.1. Linear single degree of freedom system A single degree of freedom (sdof) system subject to El Centro support motion (PEER Ground Motion Database 2000) is considered, to illustrate the method proposed in this paper. The governing equation of the dynamical system is given by    ̈( )      ̇( )     ( )     ̇ ( )      ( )    ( )                       ( )     ̇( )      (7)  where   and   are respectively the damping coefficient and natural frequency of the system,  ̇ ( )  and   ( )  are the support velocity and displacement which are the input to the system and  ( ) is the white noise process accounting for the modeling errors. Eqn.(7) is represented in the state space with state vector  ( )  * ( )  ̇( )   ( )  ̇ ( )+ and the resulting SDE is discretized to obtain                                                                                                                                                         where                                                                             (   )     (    )     (  )     (   );                                       (    )         (   )                       (8) In arriving at Eq.(8) the prior model for the support acceleration is assumed as  ̈ ( )    ( ) with with  ,  ( )-    and  ,  ( )   (   )-      ( ) The measu-       -rement is simulated by subjecting the system to an El Centro support motion. The resulting displacement response is taken to be the measured quantity related to system states as                   (   )    (9)  where    is the white noise process accounting for the measurement error. In this setting the problem of force identification is to estimate the quantity  * ̇ (  )   (  )     +         .  /     Figure 1: Estimated support displacement for the system considered in section 4.1.   Figure 2: Estimated support velocity for the system considered in section 4.1. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  5 For the purpose of simulation                    ,                      and           are assumed. The filtering based method discussed in section 3, involving the Kalman filter is adopted to estimate the applied force. The values of noise variance are adaptively chosen between 0.01 and 1, by computing the likelihood function. The identified support displacement and velocity are shown in Figures.1 and 2, respectively. The estimated quantities are compared with the actual applied loads and a satisfactory match is observed. Figures 3 and 4 show the system response estimates compared to the measurement (in case of displacement response) and the noise free quantities.   Figure 3: Estimated displacement response for the system considered in section 4.1.   Figure 4: Estimated velocity response for the system considered in section 4.1. 4.2. Hysteretic nonlinear system The Bouc-Wen (Wen, 1989) hysteretic system, subjected to El Centro support excitation is considered, to illustrate the method on nonlinear systems. The governing equation of the dynamical system is given by   ̈( )     * ̇( )   ̇ ( )+     * ( )    ( )+     ( )(   )    ( )  ̇( )      ̇( )   ̇ ( )  ( )  ( )        * ̇( )   ̇ ( )+  ( )    * ̇( )   ̇ ( )+    ( ); ( )     ̇( )     ( )                (10)  The measurement is again taken to be on the displacement response with assumed system parameter values                              Figure 5: Estimated support displacement for the system considered in section 4.2.   Figure 6: Estimated support velocity for the system considered in section 4.2. 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  6  Figure 7: Estimated displacement response for the system considered in section 4.2.                                                                   and              Since    the    problem    of   force identification here involves a nonlinear system with linear measurement the sequential importance sampling particle filter (Doucet et al., 2000)  is adopted for the state estimation in the method discussed in section 3. The results of the identified support motion are shown in Figures 5 and 6 for the El Centro support displacement and velocities, respectively. The estimates have been compared with applied excitation and are found to compare reasonably for the support displacement and slightly less for the support velocity. This may be overcome by modifying the prior model or also by increasing the number    Figure 8: Estimated velocity response for the system considered in section 4.2.  of samples of the noise parameter. The estimates of the system responses: displacement and velocity are also given in Figures 7 and 8 respectively. They have been compared with their respective noise free estimates and also with the measurement in the case of displacement response (Figure 7).  A discussion on the implementation issues and salient features of the proposed method are discussed in the next section. 5. DISCUSSION The proposed filtering based force identification algorithm assumes a white noise model for the unknown force. This is justified given the lack of any direct measurement on the force and the only available information being the system model and the measurements on the system response. The classical system identification is the inverse problem of determining time invariant system parameters governing the response. The force identification can also be perceived to be a similar problem but with time varying quantities. In order to capture the time variation the parameter of the assumed white noise model for the force is adaptively chosen based on the likelihood of the measurement given a particular value for the parameter, as explained in section 3. Following are a few observations specific to the force identification problem and the proposed method. 1. The range of values for the noise parameter and number of samples ‗N‘ is taken to be small in number order to ensure identification of the actual force. It may be noted that the comparison of the measured and estimated response (displacement in the examples considered) is used in determining the values of the noise parameter by computing the likelihood value. It is observed that the measured quantity is estimated accurately even for larger values of the parameter, but the corresponding estimates of the force are dominated by the noise owing to the choice of white noise model as prior. To overcome 12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12 Vancouver, Canada, July 12-15, 2015  7 this, the values are limited to a maximum of 0.05 to 1. 2. The relatively small values of ‗N‘ above would lead to loss of sample diversity which is encountered in particle filtering methods. This also attributes to the choice of large number of samples in the time invariant identification problems using these methods. In the current work, loss in sample diversity is not encountered in the filtering steps, but is observed in determining the value of noise variance. This is overcome by introducing a jitter to the samples in step 6 of the procedure discussed in section 3. 3. Furthermore, the present study refrains from appending the noise parameter as additional state, since this curtails the application of the method when the system information is in the form of a finite element model residing in commercial software like ANSYS or Abaqus. 4. In addition, the proposed methodology accommodates the parallelization of the estimation at each time instant corresponding to each value of the noise parameter. This ability of an algorithm would be particularly of importance from a computation point of view when analyzing higher order structural systems. 6. SUMMARY The paper proposes a filtering based force identification strategy where a white noise prior for the unknown force is assumed. The time variation of the force is accounted for by choosing adaptively the noise parameter based on the likelihood of the measurement. Issues like application of the method to multiple degree of freedom systems with multi-axial excitations involving finite element models for systems, is intended as future research work of the author. 7. REFERENCES Ching, J., Beck, J. L. and Porter, K.A. (2006). ―Bayesian state and parameter estimation of uncertain dynamical systems‖ Probabilistic Engineering Mechanics, 21, 81-96. Doucet, A., de Freitas, N. and Gordon, N. (2001). Sequential Monte Carlo Methods in Practice, Springer, New York. Doucet, A.,  Godsill, S. and Andrieu, C. (2000).  ―On sequential Monte Carlo sampling methods for Bayesian filtering‖ Statistics and Computing, 10, 197-208. Gordon, N. J., Salmond, D. J. and Smith, A. F. (1993). ―Novel approach to nonlinear/non-Gaussian Bayesian state estimation‖ IEE Proceedings-F, 140, 107-113. Hwang, J. –S., Kareem, A. and Kim, H. 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Radhika, B and Manohar, C. S. (2013). ―Dynamic states estimation for identifying earthquake support motions in instrumented structures‖ Earthquakes and Structures, 5(3), 359-378. Wen, Y. K. (1989). ―Methods of random vibration for inelastic structures‖ ASME Applied Mechanics Reviews, 42(2), 39-52. Worden, K., Stansby, P. K. and Tomlinson, G. R. (1994). ―Identification of nonlinear wave forces‖ Journal of Fluids and Structures, 8, 19-71. Yun, C. B. and Shinozuka, M. (1980). ―Identification of nonlinear structural dynamic systems‖ Journal of Structural Mechanics, 8, 187-203.     


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