New Forms of Extended Kalman Filter Via Transversal Linearization And

Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 www.elsevier.com/locate/cma

New forms of extended Kalman ﬁlter via transversal linearization and applications to structural system identiﬁcation

Shuva J. Ghosh, D. Roy *, C.S. Manohar

Structures Lab, Department of Civil Engineering, Indian Institute of Science, Bangalore 560 012, India

Received 28 September 2006; received in revised form 2 July 2007; accepted 5 July 2007 Available online 25 July 2007

Abstract

Two novel forms of the extended Kalman filter (EKF) are proposed for parameter estimation in the context of problems of interest in structural mechanics. These filters are based on variants of the derivative-free locally transversal linearization (LTL) and multi-step transversal linearization (MTrL) schemes. Thus, unlike the conventional EKF, the proposed filters do not need computing Jacobian matrices at any stage. While the LTL-based filter provides a single-step procedure, the MTrL-based filter works over multiple time-steps and finds the system transition matrix of the conditionally linearized vector field through Magnus’ expansion. Other major advantages of the new filters over the conventional EKF are in their superior numerical accuracy and considerably less sensitivity to the choice of time- step sizes. A host of numerical illustrations, covering single- as well as multi-degree-of-freedom oscillators with time-invariant parameters and those with discontinuous temporal variations, are presented to confirm the numerical advantages of the novel forms of EKF over the conventional one. 2007 Elsevier B.V. All rights reserved.

Keywords: Extended Kalman ﬁlter; Transversal linearization; Magnus’ expansion; State estimation; Structural system identiﬁcation

1. Introduction filter. More recent developments have encompassed studies on evaluation of posterior pdf using Monte-Carlo simula- Dynamic state estimation techniques offer an effective tion procedures. These methods, styled as particle filters, framework for time-domain identification of structural sys- are computationally intensive but have wide-ranging capa- tems. These methods have their origin in the area of control bilities in terms treating nonlinearity and non-Gaussian of dynamical systems with the Kalman filter [19] being one features (see, for example, [9,34]). of the well-known tools of this genre. The filter provides In the area of structural engineering problems, dynamic the exact posterior probability density function (pdf) of state estimation methods have been developed and applied the states of linear dynamical systems with linear measure- in the context of structural system identification [49,15,10], ment model, with additive Gaussian noise processes. input identification [18,26], and active structural control Recent research in this area is focused on extending the (see, for example, [40]). In these applications, the process state estimation capabilities to cover nonlinear dynamical equations are typically derived based on the finite element systems and non-Gaussian additive/multiplicative noises. method (FEM) of discretizing structural systems. These Early efforts in dealing with nonlinear dynamical systems equations represent the condition of dynamic equilibrium were focused on application of linearization techniques and often constitute a set of ordinary differential equations which have led to the development of the extended Kalman (ODEs). The measurements are typically made on displacements, velocities, accelerations, strains and/or support reactions. Furthermore, the process and measurement * Corresponding author. Tel.: +91 8022933129; fax: +91 8023600404. equations are often taken to be contaminated by additive E-mail address: [email protected] (D. Roy). Gaussian noise processes. These noises themselves are

0045-7825/$ - see front matter 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2007.07.004 5064 S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 modeled as a vector of white-noise processes or, more gen- stage. In addition to the laborious and possibly inaccurate erally, as a vector of filtered white-noise processes. In this nature of such an exercise in the context of finite element context the classical Kalman filter itself has limited applica- applications, this assumes importance in systems having tions since interest is often focused on systems with nonlin- vector fields with C0 continuity. Also, it has been validated ear mechanical properties. Consequently, studies dealing by the authors [12] that, due to less propagation of local with application of EKF methods to structural system errors, these methods work for considerably higher step identification have been undertaken [49,15,10]. Though in sizes. Moreover, a certain class of these methods (MTrL- most of the studies, the parameters to be estimated are I, [12,13]) qualifies as one of geometric integrators and essentially time-invariant, time varying parameter identifi- hence may be elegantly applied to preserve the invariants cation problems have also been studied [43,24,48,7]. One of motion. Thus, they offer an effective solution for a more of the important issues in the identification of structural general class of problems than methods based on tangential parameters is the interplay of the various uncertainties in (path-wise) linearization. Consequently, it is of consider- modeling and measurements and their effect on the vari- able interest to explore the application of these integration ability of the identified parameters. The variance of the sys- schemes in the context of deriving discrete equations for tem parameters, conditioned on measurements, essentially process equations in dynamic state estimation problems captures this feature. If this variance itself is computed in arising in structural system identification. an approximate manner, as is the case, for instance, when The present work accordingly proposes two alternative EKF or any suboptimal filtering strategy are used, satisfac- forms of the extended Kalman filter on the basis of the the- tory interpretation of the parameter variability becomes ories of local and multi-step transversal linearizations. The difficult. The works of Koh and See [22], Kurita and Mat- LTL method, originally proposed as a numeric-analytic sui [23] and Sanayei et al. [39] contain discussions on principle for solutions of nonlinear ODEs, can be consid- related issues. Recently applications of particle filter tech- ered as a bridge between the classical integration tools niques to structural system identification have been and the geometric methods. The essence of the LTL reported [29,7]) and control [38]. method is to exactly reproduce the vector fields of the gov- One of the main ingredients of filtering techniques erning nonlinear ODEs at a chosen set of grid points and applied to structural dynamic system is the discretization then to determine the discretized solutions at the grid of the governing equations of motion into a set of discrete points through a transversality argument. The condition- maps. In this context, it is of interest to note that the solu- ally linearized vector fields may be non-uniquely chosen tion of equilibrium equations in the finite element frame- in such a way that their form precisely matches the given work has been widely researched (see, for example, [3]). nonlinear vector field and are transversal to this field at Various methods such as central difference, Wilson-theta, least at the grid points. The recent multi-step extension of Newmark-beta, etc., have been developed and are routinely the LTL, referred to as the MTrL method, uses the solu- available in commercial finite element packages. Questions tions of transversally linearized ODEs in closed-form and on spectral stability and accuracy of these schemes have hence distinguishable from other multi-step, purely numer- thus been discussed. Most studies that combine finite ele- ical, implicit methods. Owing to the condition of transver- ment structural modeling with dynamic state estimation sality, the need for any differentiation of the original vector methods employ either forward or central difference-based field is eminently avoided in this class of methods. schemes for discretizing the governing equations (see, for It is theoretically impossible to derive a time-invariant example [6]). Alternatively, in the context of nonlinear linear dynamical system whose evolution will, in general, structural mechanics problems, a closed-form expression precisely match that of a given nonlinear system over any of the state transition matrix (STM) is employed on the lin- finite time interval. Over a sufficiently small time interval, earized equations of motion to obtain a discrete map for fil- however, the nonlinear equations may be conditionally lin- tering applications (see, for example [49]). Thus it appears earized which is presently accomplished using the still- that the relationship between the methods of integrating unknown, instantaneous estimates to be determined at governing equation of motion and the formulation of dis- the grid points over the interval. Once these conditionally crete maps for dynamic state estimation techniques has linearized equations are derived, a discrete Kalman filtering not been adequately explored in structural engineering algorithm may be applied to get estimates of states of the applications. The present work is in the broad domain of system. It should be understood that, the LTL-based Kal- addressing this issue. man filter, dealing with conditionally linearized equations, In the recent years, the present authors have been results in implicit and nonlinear algebraic equations involved in research related to the development of thereby necessitating the use of either a Newton–Raphson numeric-analytic solution techniques for nonlinear ODEs or a fixed-point iteration scheme to solve for the instanta- and stochastic differential equations (SDE) based on the neous estimates. The basic idea is that the constructed con- concept of transversality, namely, locally transversal linear- ditionally linearized estimate manifold is made to ization (LTL) [37,35] and multi-step transversal lineariza- transversally intersect the nonlinear (true) estimate mani- tion (MTrL) [36,12,13] methods. These methods fold at time instants where the estimate vectors need to completely avoid gradient (Jacobian) calculations at any be determined. While choices for conditionally linearized S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 5065

LTL vector fields are non-unique (i.e., infinitely many), the external (non-parametric and deterministic) force vec- solutions are, more or less, insensitive to the specific chosen tor. The components of the process noise vector m forms. The condition of intersection of manifolds at the nsðtÞ2R are taken to be independently evolving Gaussian grid points provides a set of constraint equations whose white noise processes. Gb is an n · m noise distribution ma- zeros are the instantaneous estimates. For the MTrL-based trix. The noise term in Eq. (1) represents the effects of mod- Kalman Filter, the linearization is implicitly achieved using eling errors and errors in applying the test signals. an interpolating expansion of part(s) of the nonlinear vec- Presently, we assume that elements of QðX ; X_ ; tÞ and FbðtÞ tor functions in terms of the unknown estimates at p (for an are Ck functions, k 2 Zþ,inX and t with k P 1. Define order p MTrL) grid points. We emphasize that this scheme the new state vector as yields the estimates at p points simultaneously (via a single T T T linearization step) and, to the authors’ knowledge, it is not ZðtÞ¼fZsðtÞ ZpðtÞ g ; ð2Þ an aspect that has been explored earlier. However, the pro- where ZðtÞ2R2nþl is the extended state vector at time t, posed filtering algorithms still use a sequential linear esti- Z (t) is the vector of states of the original system, i.e., mation theory in the form of the Kalman filter and thus s Z ðtÞ¼fX TX_ TgT, Z 2 Rl is the vector of model parame- complete information about the posterior probability den- s p ters (e.g., stiffness, damping or any nonlinear stiffness or sity function of the states is not obtainable. We have used a damping parameters to be identified). Furthermore, n is weighted global iteration procedure (WGI) [41] with the the number of degrees of freedom (DOF) and l is the total proposed filters replacing the conventional EKF. Several number of model parameters that need to be identified. For numerical examples, covering both linear and nonlinear a general time-invariant parameter identification problem, structural behavior, are considered to verify the potential the state space equations for the dynamical system in Eq. and relative merits of the method. Along with a linear (1) has the following representation: multi-degree-of-freedom (MDOF) shear building, a hardening Duffing oscillator and a hysteric oscillator (Bouc– _ e e ZsðtÞ¼g~ðZsðtÞ; ZpðtÞ; tÞþF ðtÞþGðtÞnsðtÞ; ð3Þ Wen hysteresis model) have been used for numerical study. _ In each case, the forward system model is simulated on a ZpðtÞ¼0 þ npðtÞ; ð4Þ computer using the stochastic Heun’s method to generate where g~ : R2n Rl R ! R2n, Fe : R ! R2n and Ge 2 R2nm synthetic noisy measurements, which are fed to the filtering are respectively the augmented state-dependent drift vec- algorithm to arrive at estimates of the model parameters. tor, forcing vector and process noise distribution matrix with the last two being appropriately padded by zeros 2. Process and measurement equations wherever needed. Note that the model parameters are ex- pressed in terms of elements of Zp(t), which are declared l Identification of any dynamical system requires, in gen- as the additional states of the system. npðtÞ2R is a vector eral, two models. Firstly, a mathematical model describing of independent white noise processes that are small random the evolution of state with time, i.e., the process equations, disturbances to states and may be interpreted as an artifi- to idealize the dynamics of the physical system, and, sec- cial evolution of the model parameters in time [8,25].It ondly, measurement (or observation) equations, relating should be noted that once the system identification prob- the measured quantities to the state variables, should be lem is formulated by declaring the unknown model param- available. Both these models include noise terms that serve eters as additional states of the system, even a problem to account for unmodeled dynamics in the process equa- without any mechanical nonlinearity is transferred to the tions and measurement errors in the observation equations. domain of nonlinear filtering. It is common practice to model the noise terms as Gaussian The two Eqs. (3) and (4) may be combined to obtain white noise processes. In time-domain system identification _ problems, the parameters of the mathematical model are ZðtÞ¼gðZðtÞ; tÞþF ðtÞþGðtÞnðtÞ; ð5Þ declared as additional states so that any acceptable algo- 2nþl ð2nþlÞ ð2nþlÞ where g : R R ! R ; F : ½t0; T R ! R ; rithm for state estimation becomes applicable for these 2n l G(t)isa(2n + l) · (2n + l) matrix and nðtÞ2R þ . The problems as well. In the general context of engineering observation equation may be written as dynamical systems, one generally starts with the following set of second order, nonlinear ODEs, derivable via a spatial Y ðtÞ¼h~ðZðtÞ; tÞþgðtÞ: ð6Þ projection technique (such as the Petrov–Galerkin) applied Here Y ðtÞ2Rr; h~ : Rð2nþlÞ R ! Rr; gðtÞ2Rr is a vector to the governing PDEs: of independent white noise processes and r 6 2n. The noise € _ _ b b term g(t) in Eq. (6) is used to model measurement errors MX þ CX þ KX þ QðX ; X ; tÞ¼F ðtÞþGðtÞnsðtÞð1Þ and possible errors in relating the observed quantities with n with X 2 R , t 2½t0; T R, M being the mass matrix, C the state variables. The above representation of the obser- the viscous damping matrix, K the stiffness matrix, vation equation is formal and observations, in practice, QðX ; X_ ; tÞ : Rn Rn R ! Rn the nonlinear and/or para- would be made at a set of discrete time instants so the mea- b n metric part of the vector field, and F : ½t0; T R ! R surement equation may be realistically represented as 5066 S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083

~ Y k ¼ Y ðtkÞ¼hðZðtkÞ; tkÞþgðtkÞ; ð7Þ where Rk+1 is the covariance matrix of the measurement noise where tk = kDt, Dt being the sampling interval and g(tk) "# represents a sequence of r independent Gaussian random oh~ðZ; tÞ variables with covariance matrix R . As has been men- H kþ1 ¼ : ð15Þ k oZ b tioned in Section 1, the objective of the present study is ZðtÞ¼Z kþ1jk ;tkþ1 to develop two new forms of EKF based on transversal 4. Obtain the new estimate Zb of state and the associ- intersection concepts, namely the LTL-based EKF and kþ1jkþ1 ated error covariance matrix P as the MTrL-based EKF. As a prelude to that we briefly out- k+1jk+1 b b ~ b line the steps involved in the estimation of unknown struc- Z kþ1jkþ1 ¼ Z kþ1jk þ Kkþ1½Y kþ1 hðZ kþ1jk; tkÞ; ð16Þ tural parameters by the conventional EKF. T P kþ1jkþ1 ¼½I Kkþ1H kþ1P kþ1jk½I Kkþ1H kþ1 T 3. Structural system identification by EKF þ Kkþ1Rkþ1Kkþ1: ð17Þ Estimation of model parameters by EKF has been widely We begin by considering the process Eq. (1) and the studied. Asymptotic convergence results are also available measurement Eq. (7) and summarize the recursive algo- [27]. Since EKF is based on a linearization of the state rithm involved in implementing the EKF [49]. Given an ini- b and observation equations over each time-step (only up tial estimate Z 0j0 of the state and the error covariance P0j0, to first-order terms in the Taylor expansion are retained) start the loop. and a subsequent usage of the linear estimation theory in For k ¼ 0; 1; 2; 3 ...; m and ðt 2ð0; tmÞ, perform the fol- the form of Kalman filter, the deviation of the estimated lowing steps: trajectory form the true trajectory increases with time b and the significance of higher order terms in the Taylor 1. Begin with the filter estimate Z kjk and its error covari- expansion also increases. Due to the accumulation and ance matrix Pk k. j b propagation of local errors, there is a distinct possibility 2. Evaluate the predicted state Z kþ1jk and the predicted of divergence, especially if the EKF starts off with poor ini- error covariance matrix Pk+1jk by Z tial estimates (see, for instance, [44,14,4]). This also implies t no b b kþ1 b that an increased sensitivity of numerical solutions to step- Z kþ1jk ¼ Z kjk þ gðZ tjkðtÞÞ þ F ðtÞ dt; ð8Þ sizes can potentially lead to biased estimates [6]. tk T P kþ1jk ¼ Uðtkþ1; tkÞP kjkU ðtkþ1; tkÞþQkþ1; ð9Þ

where Uðtkþ1; tkÞ is the state transition matrix of the lin- 4. Formulations of LTL- and MTrl-based EKFs earized system and is given by We will presently employ the concepts of transversal Uðt ; t Þ¼exp½ðr Þðt t Þ: ð10Þ kþ1 k k kþ1 k intersection to develop two new forms of the EKF, which Here bear a resemblance to a collocation method in the sense ogðZ; tÞ that the filter estimates will be equal to the ‘optimal’ esti- r ¼ ð11Þ mates at the points of discretization and not necessarily k oZ b ZðtÞ¼Z kjk ;tk over the entire time interval. Nevertheless, we note that is the (2n + l) · (2n + l) system Jacobian matrix (that the estimate for a general nonlinear system via the pro- obtains the first-order Taylor increment of the nonlinear posed as well a conventional forms of the EKF is not function g) and the mean square estimate, as obtainable (in principle) Z Z through a strictly nonlinear filter (such as a Monte-Carlo tkþ1 tkþ1 T filter). Indeed the ‘optimal’ solution presently refers to a Qkþ1 ¼ Uðtkþ1; uÞGðuÞE½nðuÞn ðvÞ tk tk minimum mean square estimator (MMSE) based on the T T G ðvÞU ðtkþ1; vÞdudv; ð12Þ assumption of a Gaussian posterior probability density function. where Qk+1 is the process error covariance matrix (E(Æ) denotes the expectation operator). Note that, following 4.1. The LTL-based EKF linearization over the time interval ðtk; tkþ1, the approximate solution to Eq. (5) is given by Z The general idea of the LTL-based EKF for system t b identification is to replace the nonlinear vector field of ZðtÞ¼Uðt; tkÞZ kjk þ Uðtk; uÞ½F ðuÞþGðuÞnðuÞdu: the process equation by a time-invariant conditionally lin- t k earized vector field over a time-step and an application of 13 ð Þ Kalman filter based on the estimates at the right end of 3. The Kalman gain is given by the time-step. Consider Eq. (5) corresponding to a nonlinear dynamical system. Let the time interval of interest T T 1 Kkþ1 ¼ P kþ1jkH kþ1 H kþ1P kþ1jkH kþ1 þ Rkþ1 ; ð14Þ ð0; T R be divided into m sub-intervals with S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 5067

0 ¼ t0 < t1 < t2 < ti < < tm ¼ T and hi = ti+1 ti, gðZ; tÞ¼BðZ; tÞZ; ð22Þ i Zþ. It is now required to replace the process equation 2 h~ðZ; tÞ¼HðZ; tÞZ ð23Þ by a suitably chosen set of m transversally linearized vector equations, wherein the solution to the i-th element of the with BðZ; tÞ and HðZ; tÞ being finite quantities. Eqs. (20) set should be, in a sense, a representative of the nonlinear and (21) may be viewed as being conditionally linear with flow over the i-th time-step. Note that when the measure- constant coefficients, given the value of the still-unknown m b b ment relationship given by Eq. (6) is nonlinear, addi- estimated quantities Z i ¼ ZðtiÞ. Now the proposed filter tional sets of conditionally linearized algebraic vector may be summarized as follows: equations have also to be chosen. We define T : (t ,t ]. b i ¼ i i+1 Given initial estimates of state Z 0j0 and the error covari- For a given set of measurements with the process equations ance matrix P0j0, loop over k ¼ 0; 1; 2; 3 ...; m for precisely known, the optimal estimate of the vector state ðt 2ð0; tmÞ using the following steps: t(Æ) (the sense of optimality has been enunciated earlier) may be thought of as a flow evolving on a compact mani- b 1. Start with filter estimates Z kjk and its error covariance fold M such that matrix Pk k. j b 2. Evaluate the predicted state Z kþ1jk as the LTL-based t : M R ! M ð18Þ solution of the following nonlinear differential equation k k b is C (k P 0) on M. Then, for any t, t(Z)isaC diffeomor- at t = tk+1 subject to the initial condition Z kjk at t = tk: phism M ! M. Let ðZi; tiÞ and ðZiþ1; tiþ1Þ2M denote a pair of points on the estimate manifold (with a local struc- b_ b Z tjtk ¼ gðZ tjtk ; tÞþF ðtÞ; ð24Þ 2nþlþ1 ture of R ). Now one canP constructP a couple of 2n + l- dimensional cross-sections i and iþ1 transverse to the which is the process Eq. (5) without noise. This is be- vectorP field at ðZi; tiÞPand ðZiþ1; tiþ1Þ, respectively, such that cause the prediction stage basically provides the ex- Zi 2 i and Ziþ1 2 iþ1. At this point we can view the pected values of the state at the forward time-points P P and hence it is reasonable to ignore the additive zero- Kalman filter as a map Ke : Rr R R ! such i iþ1 mean white noise at this stage. that 3. Write down the exact solution of the conditionally line- e Ziþ1 ¼ KðZi; Y iþ1; ti; hiÞ; ð19Þ arized equation as Z where Y 2 Rr is the observation vector at the (i + 1)th t iþ1 ZðtÞ¼Uðt; t ; Zb ÞZb þ UðZb ; t ; uÞ time instant. Ke may be thought of as a Poincare (strobo- k kþ1jkþ1 kjk kþ1jkþ1 k tk scopic) map as in the case of a nonlinear flow. The principle ½F ðuÞþGðuÞnðuÞdu; ð25Þ of an LTL-based EKF is thus to generate a set of conditionally linear equations which, when used with a Kalman where t 2ðt ; t and Uðt; t ; Zb Þ is the state transi- filter, gives rise to a map Kb that should be (ideally) identical k kþ1 k kþ1jkþ1 e tion matrix (or the fundamental solution matrix) of the to K for the same initial condition ðZi; tiÞ and the same linearized system. The latter is given by interval hi. It is conceivable that the estimate manifold of the proposed filter has to transversally intersect the optimal b b Uðt; tk; Z kþ1jkþ1Þ¼exp½ðBðZ kþ1jkþ1; tiÞðt tkÞ: ð26Þ estimate manifold at least at the grid points in 4. Estimate the predicted error covariance matrix P kþ1jk by S ¼ftiji ¼ 0; 1; 2; 3; ...g and not necessarily elsewhere. Remark 1: From a straightforward application of the P ðZb Þ¼Uðt ; t ; Zb ÞP UTðt ; t ; Zb Þ weak transversality theorem [2], it follows that mappings kþ1jk kþ1jkþ1 kþ1 k kþ1jkþ1 kjk kþ1 k kþ1jkþ1 b transversal to M at a given instant ti form an open and þ Qkþ1ðZ kþ1jkþ1Þ: everywhere dense set in the space of smooth maps ð27Þ f i : M ! M i, where M i denotes the set of all transversally intersecting smooth manifolds. Hence the derivation of a Here the process error covariance matrix Qkþ1 is given by conditionally linear system and therefore, the estimates of Z Z the proposed filter are non-unique and infinitely many. tkþ1 tkþ1 b T For instance, Eqs. (5) and (6) may give rise to the fol- Qkþ1 ¼ Uðtkþ1; u; Z kþ1jkþ1ÞGðuÞE½nðuÞn ðvÞ t t lowing LTL system (with conditionally constant coeffi- k k T T b G ðvÞU ðtkþ1; v; Z kþ1jkþ1Þdudv: ð28Þ cients) over the interval ðti; tiþ1 for purposes of state estimation using the Kalman filter: 5. The Kalman gain is given by _ b b b T b Z ¼ BðZ i; tiÞZ þ F ðtÞþGðtÞnðtÞ; ð20Þ Kkþ1ðZ kþ1jkþ1Þ¼P kþ1jkðZ kþ1jkþ1ÞH kþ1ðZ kþ1jkþ1; tkþ1Þ b b b Y ¼ HðZ i; tiÞZ þ gðtÞð21Þ ½P kþ1jkðZ kþ1jkþ1ÞH kþ1ðZ kþ1jkþ1; tkþ1ÞP kþ1jk ðZb ÞH T ðZb ; t ÞþR 1; provided that the vector functions g and h~ are decompos- kþ1jkþ1 kþ1 kþ1jkþ1 kþ1 kþ1 able as ð29Þ 5068 S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083

where Now consider the sub-interval I1 and let it be ordered into p b smaller intervals as Y kþ1 ¼ H kþ1ðZ kþ1jkþ1; tkþ1ÞZ þ gðtÞ: ð30Þ b ð0 ¼ t0; t1; ðt1; t2; ...; ðtp1; tp ¼ T 1 6. Obtain the new estimate of the state Z kþ1jkþ1 via with the time-step h = t t , where i 2 Zþ; i 6 p. Simi- b b ~ b i i+1 i Z kþ1jkþ1 ¼ Z kþ1jk þ Kkþ1½Y kþ1 hðZ kþ1jk; tkÞ: ð31Þ larly each succeeding interval is subdivided into p sub- Now the conditionally linearized estimate and the opti- intervals and so ordered that mal estimate will become instantaneously identical at ð0; T ¼fðt0; t1; t2; ...; tp; ðtp; tpþ1; tpþ2; ...; t2p;...; t if a root ðZb Þ of Eq. (31) can be found such k+1 kþ1jkþ1 ðt ; t ; t ; ...; t g; ð36Þ that ðM1Þp ðM1Þpþ1 ðM1Þpþ2 Mp b b where tjp ¼ T j; j ¼ 1; 2; 3; ...; M. Z kþ1jkþ1 ¼ Z kþ1jkþ1: ð32Þ Let Si ¼fði 1Þp; ði 1Þp þ 1; ði 1Þp þ 2; ...; ipg b Zþ. The idea is to derive a linearized system for each In other words,T the constraint condition i.e. Z kþ1jkþ1; b sub-interval Ii,16 i 6 M. Moreover, following the concept Z kþ1jkþ1 2 M M i is automatically satisfied if a real and physically admissible root of the following transcen- of transversal linearization, it is intended that the state esti- dental equation can be found: mate computed through an application of the Kalman filter to the linearized equations and the ‘best’ estimates remain b b ~ b Z kþ1jkþ1 Z kþ1jk þ Kkþ1½Y kþ1 hðZ kþ1jk; tkÞ ¼ 0: ð33Þ identical at all points of discretization in that interval, viz. at t ; t ; t ; ...; t 2 I . This can be done through any standard root-finding ði1Þp ði1Þpþ1 ði1Þpþ2 ip i For the problem of parameter estimation at hand based algorithm as the fixed-point iteration, Newton–Raphson on Eqs. (5) and (7), it is naturally sought to derive the lin- method, etc. earized system such that it is also (2n + l)-dimensional and 7. Once the true estimate is obtained as the solution of Eq. is obtainable from the given nonlinear system with simple (33), the error covariance matrix P associated k+1jk+1 and least alterations. Firstly, the part of time-axis, over with the new state may be estimated as which the estimates are sought, needs to be discretized T P kþ1jkþ1 ¼½I Kkþ1H kþ1P kþ1jk½I Kkþ1H kþ1 and ordered as mentioned above. The integer M denotes the number of times the MTrL-based linearization proce- þ K R KT ; ð34Þ kþ1 kþ1 kþ1 dure has to be applied to get the estimates over the entire where the quantities (without bar) are computed by time interval of interest (0,T]. Each application of the substituting in the corresponding quantities (with bar) MTrL procedure produces the estimates of (2n + l)p dis- the true value of the estimates obtained from Eq. (33). cretized (scalar) state variables at p grid points over each sub-interval Ii. Thus it is a multi-step procedure for the esti- 4.2. The MTrL-based EKF mation problem at p points over each linearization step. The estimate at the right end of the interval Ii will provide The second class of the EKF, referred to as the MTrL- an initial estimate for the linearized system over Ii+1. based EKF, may be viewed as an extension and further For further elaboration of the methodology, Eqs. (5) generalization of the LTL-based EKF. The proposed filter and (6) are considered again and linearized over the inter- is based on the new and more flexible MTrL method [12] val Ii using the unknown estimates at p + 1 points as instead of the earlier version of the MTL method [36] follows: wherein the only way to treat the nonlinear terms was to Z_ ¼ BðZ; tÞZ þ F ðtÞþGðtÞnðtÞ; ð37Þ convert them into conditional forcing functions. With the latest version of the MTrL applied to nonlinear structural Y ¼ HðZ; tÞZ þ gðtÞð38Þ dynamics, the nonlinear damping and stiffness terms in provided that the vector functions g and h~ are decompos- the original system appear respectively as conditionally able as determinable damping and stiffness coefficients in the linearized system as well. The fundamental solution matrix gðZ; tÞ¼BðZ; tÞZ; ð39Þ (FSM) or the state transition matrix of the linearized sys- h~ðZ; tÞ¼HðZ; tÞZ ð40Þ tem may be obtained through Magnus’ expansion [28], which is derivable based on the assumption that the linear- with BðZ; tÞ and HðZ; tÞ bounded over Ii. In the present ized system matrix is a sufficiently smooth Lie element. work Z is obtained through an interpolation over the b For the purpose of implementation, let the subset of the known initial vector estimate fZ i 1 p i 1 pg and unknown b ð Þ jð Þ time-axis over ð0; T be divided into M closed–open sub- vectors fZ kjkjk 2 Si and k 6¼ði 1Þpg at tk : k 2 Si. This intervals as expansion leads to an approximation of BðZ; tÞ and HðZ; tÞ over I and hence a conditionally linearized set of ð0; T ¼fI ; I ; I ; ...; I g i 1 2 3 1 equations over the i-th time interval. For instance, if one ¼fð0 ¼ T 0; T 1; ðT 1; T 2; ...; ðT N1; T N ¼ T g: ð35Þ chooses these interpolating functions to be Lagrangian S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 5069 polynomials of degree p, Zb is approximated in a vector Recast, for notational convenience, Eqs. (37) and (38) as space Vp of polynomials spanned by the Lagrange polynomials such that the dimension of this approximating space Z_ ¼ WðtÞZ þ F ðtÞþGðtÞnðtÞ; ð43Þ is p. The set of interpolating Lagrange polynomials Y ¼ WH Z þ gðtÞ; ð44Þ fP kjk ¼ 0; 1; 2; ...; pg over Ii are given by Yip where WðtÞ¼BðZ; tÞ and WH ¼ HðZ; tÞ. Also let t tj P kðtÞ¼ ð41Þ tk tj b b b b j¼ði1Þp Z I ¼fZ ði1Þpþ1jði1Þpþ1; Z ði1Þpþ2jði1Þpþ2; ...; Z ipjipgð45Þ and one may write denote the set of unknown estimates over Ii. Given initial b estimates of state Z 0j0 and the error covariance matrix Xip P , use a loop over i ¼ 1; 2; 3 ...; M,withðt 2ð0; T Þ,to b 0j0 Z ¼ P kðtÞZ kjk: ð42Þ implement the following steps: k¼ði1Þp b 1. Start with filter estimates Z ði1Þpjði1Þp and its error One is not restricted in choosing only Lagrangian polyno- covariance matrix P . mials to arrive at the vector function Z. For instance, one (i1)pj(i1)p 2. Evaluate the predicted states fZb jj 2 S ; j 6¼ði 1Þ may use distributed approximating functionals [47] or jjði1Þp i pg as solution of the following nonlinear differential wavelet-based [42] interpolating functions to obtain the equation at {tjjj 2 Si; k 5 (i 1)p} subject to the initial function. However, irrespective of the specific form of condition Zb at t = t interpolation, the MTrL-based linearized system corre- ði1Þpjði1Þp (i1)p sponding to the nonlinear system (5) and (6) takes the form b_ b Z tjt i 1 p ¼ gðZ tjt i 1 p ; tÞþF ðtÞ; ð46Þ given by Eqs. (37) and (38). It must however be recorded ð Þ ð Þ that the use of such expansions is not quite consistent with which is the process Eq. (5) without noise. This is feasi- the nature of strong solutions of a stochastic differential ble as the prediction stage basically gives the expected equation under white noise. It is well-known that the solu- values of the estimate at the forward time-points and tion of an SDE is a semi-martingale and admits an expan- hence the additive zero-mean white noise (vector) term sion through a stochastic Taylor expansion (STE) (see, for is left out. The simultaneous prediction at p points instance, [21,30]). For instance, an STE over a time-step Dt may be achieved using the class II MTrL [12]. The class has terms with fractional powers of Dt and such an expan- II MTrL avoids the computationally cumbersome Mag- sion is therefore not spanned by the monomials 1, t, t2 and nus’ expansion by treating the nonlinear functions in the so on. However, we are presently interested in obtaining a process equation as a conditional forcing function and reasonably accurate and numerically stable state estimate, thereafter solves for the discretized state variables ensur- which is itself non-stochastic. Indeed, as substantiated with ing a transversal intersection of the original and the con- extensive numerical simulations, the errors in the computed ditionally linearized flows at the grid points through estimates owing to the relaxation of the mathematical rigor appropriate constraints. are not substantial as long as the additive noise intensity is 3. Discretize the conditionally linearized equation as in Eq. not too high. The rest of the procedure is quite similar to (25) and derive p sets of STMs and process noise covari- the LTL-based EKF with the exception that the state tran- ance matrices. The STMs are given by sition matrix for the linearized system with time dependent U :¼ Uðt ; t ; Zb ; Zb Þ¼expðXðtÞÞ; ð47Þ coefficients has to computed in a different manner (see j ði1Þp j ði1Þpjði1Þp I

Appendix A) in the form of a series expansion following where {j 2 Si; j 5 (i 1)p} and are obtainable by substi- Magnus [28]. It should be noted that the STM U(t) for tuting W(t) for A(t) in Eq. (A.4), the limits of integration the linearized system over the interval Ii is obtained from being t(i 1)p to tj in the jth case. Denoting the error b b the originally nonlinear system of equations through condi- covariance matrix as Qj :¼ QjðZ ði1Þpjði1Þp; Z I Þ may be tional linearization using the state estimates at discrete obtained in an identical manner like Eq. (28) by using b b time-points i.e. Z ði1Þp (known) and fZ kjk 2 Si; k 6¼ the STMs as obtained above.

ði 1Þpg (unknown) through Eq. (A.5) and thus the 4. Estimate the predicted error covariance matrices P jjði1Þp, STM is a function of these unknown estimates at the grid where {j 2 Si; j 5 (i 1)p}, by points. Following a similar line of transversality argument b b T as in the LTL-based EKF, it is evident that for a given set P j :¼ P jjði1ÞpðZ ði1Þpjði1Þp; Z I Þ¼UjP ði1Þpjði1ÞpUj þ Qj: of observations, the ‘best’ estimate manifold and the esti- ð48Þ mate manifold through the application of the Kalman filter to the linearized system may be made instantaneously iden- 5. The Kalman gains at {tjjj 2 Si; j 5 (i 1)p} are given by tical at the p grid points over I through enforcement of 1 i K :¼ K ðZb ; Zb Þ¼P WT P W P WT þ R : appropriate constraint equations (algebraic). The whole j j ði1Þpjði1Þp I j H j H j H j procedure can be summarized as follows. ð49Þ 5070 S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 b 6. Obtain new estimates of states fZ jjjjj 2 Si; j 6¼ði 1Þpg 5.1. Example-1 – a hardening, single-well Duffing oscillator via: Consider a hardening, single-well Duffing oscillator sub- b b b b b ~ b Z jjj :¼ Z jjjðZ ði1Þpjði1Þp; Z I Þ¼Z jjði1Þp þ Kj½Y j hðZ jjði1Þp;tjÞ: jected to sinusoidal driving force corrupted by a white- ð50Þ Gaussian noise: 3 Now the conditionally linearized estimates and the opti- m€x þ cx_ þ kx þ ax ¼ F cosðktÞþrpnðtÞ: ð54Þ mal estimates would become instantaneously identical at The problem at hand consists of estimating the parameters {t jj 2 S ; j 5 (i 1) p} if and only if zeros of the Eq. j i fc; k; ag from a set of observations related directly to the (31) can be found such that states. Accordingly, these are declared as the additional b states corresponding to an augmented system. The mass Z ¼ Zb s:t: fj 2 S ; j 6¼ði 1Þpg: ð51Þ jjj jjj i m has been set to 1 kg and hence does not appear in the In other words, the above constraint condition is auto- expressions to follow. For the present problem, unless men- matically satisfied if p sets of (2n + p) dimensional real tioned otherwise, the measured component (Y) will be the and physically admissible vector root of the following displacement (x) of the oscillator. Writing Eq. (54) in a (2n + l)p dimensional transcendental equation can be state space form and including the measurement model, found: the mathematical model for the estimation problem is provided by the equations: b b ~ b Z jjj Z jjði1Þp þ Kj½Y j hðZ jjði1Þp; tjÞ ¼ 0 z_1 ¼ z2; s:t: fj 2 Si; j 6¼ði 1Þpg: ð52Þ 3 z_2 ¼z4x1 z3z2 z5z1 þ F cosðktÞþrpnpðtÞ;

In order to avoid taking derivatives of the vector func- z_3 ¼ 0 þ rcncðtÞ; ð55Þ tions at all stages, we have presently used ﬁxed-point z_4 ¼ 0 þ rkn ðtÞ; iteration in preference to the Newton–Raphson method. k 7. Once the true estimate is obtained as the solution of Eq. z_5 ¼ 0 þ ranaðtÞ (52), the associated error covariance matrix Pipjip associ- and ated with the new state at the last point of discretization in Ii can be estimated via Y ¼ z1 þ rY nY ðtÞ: ð56Þ

T T Here z1 = x and z2 ¼ x_ are the displacement and the veloc- P ipjip ¼½I KipWH P ipjip½I KipWH þ KipRipKip; ð53Þ ity states, respectively, z3, z4, z5 are the states in the ex- where the quantities (without bar) are computed by tended state space corresponding to fc; k; ag. rp is the substituting in the corresponding quantities (with bar) enveloping factor for the original process noise np(t) the true value of the estimate obtained from Eq. (53). whereas rc, rk, ra are the enveloping factors for the artificial evolution of the respective parameters. rY is the enveloping factor for the measurement noise. As has been 5. Numerical examples mentioned earlier, Eq. (56) is just a representative form and in practice the measurement can be represented as in Numerical illustrations on single and multi-DOF sys- Eq. (7). The specific forms of the conditionally linearized tems are provided in this section. The synthetic measure- equations used for different cases will be indicated at ment data have been generated on a computer by the appropriate places. However, the solution procedure by integrating the process SDE using the stochastic Heun’s the conventional EKF is well known and will not be elab- method. With these measurements and the associated sys- orated presently. tem models, the problem of parameter identification has For the LTL-based EKF, over the time-interval ðti1; ti, been taken up by the three kinds of filters, viz. the conven- the conditionally-linearized system has the following state tional EKF, the LTL-based EKF and the MTrL-based space representation: 8 9 2 38 9 EKF. While the first problem involving a hardening Duf- > _ > > > >z1 > 01 0 0 0 >z1 > fing oscillator is extensively studied for a number of sub- > > 6 3 7> > <z_2 = 6 00z z fz g 7<z2 = cases, a hysteretic as well as an MDOF oscillator have 6 2 1 1 7 z_3 ¼ 6 00 0 0 0 7 z3 been incorporated to demonstrate the universality of the > > 4 5> > >z_ > 00 0 0 0 >z4 > proposed methods and its extensibility to problems of : 4 ; : ; z_ 00 0 0 0 z higher dimensionality. Henceforth, unless mentioned 5 8 9 8 9 5 > > > > otherwise, all noises refer to zero-mean Gaussian white- > 0 > > 0 > <> => <> => noises. The desired standard deviations have been incorpo- F cosðktÞ rpnpðtÞ rated through appropriate enveloping factors. Wherever þ > 0 > þ > rcncðtÞ >; ð57Þ > > > > used the value of the weighing factor for the WGI scheme :> 0 ;> :> rknkðtÞ ;> has been fixed at 100. 0 ranaðtÞ S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 5071

where z1 and z2 are the estimates of the states correspond- 5.1.1. Results ing to still-unknown displacement and velocity, respec- Subcase-1: The simulation parameters are taken to be tively at the i-th grid point. The measurement equation at m = 1 kg, k = 7 N/m, c = 0.9 N s/m and a = 3 N/m3. The the i-th time-point (linear in this case) following Eq. (7) is test signal is sinusoidal with an amplitude F = 4 N and fre- presently written as quency k = 3.5 rad/s. While the enveloping factor for the 8 9 process noise rp has been taken to be 0.001 N, that for i >z1 > the evolution of the additional states has been taken as > > > i > 3 <z2 = 0.00002 (units of rc, rk, ra are N s/m, N/m and N/m , zi i respectively). The standard deviation of the observation Y i ¼ ½10000> 3 > þ rY nY : ð58Þ > > noise is about 5% of the root mean square value of the >zi > :> 4 ;> response, i.e., the approximate solution of the process i z5 equation via the stochastic Heun’s method (rY = 0.030 m). The order of MTrL (p) used in this exam- For the MTrL-based Kalman ﬁlter, the linearized system ple is 2. Note that, owing to the fact that the strong solu- involves a known value of the estimates at the initial point tion of the process SDE is a semi-martingale [31] and and unknown estimates at multiple grid points of the time- therefore does not strictly admit a polynomial-type expan- interval corresponding to a single linearization step. For an sion, the order p has presently nothing to do with the for- MTrL system of order p, the set of unknown estimates mal order of accuracy of the MTrL-based EKF. comprises of the vector estimates at p distinct grid points. Figs. 1–6 show the results for a time-step of 0.01 s. For

In the present case, for the time-interval I i ¼ðtði1Þp; the MTrL-based filter the time-step over which lineariza- tði1Þpþ1; ...; tip, the linearized process equation may be tion takes place is even higher, i.e., 0.02 s. The results show written as that the estimated parameter values are close to their refer- 8 9 2 38 9 ence values for all the three filters. > _ > > > >z1 > 01 0 0 0 >z1 > Subcase-2: Consider a forced Duffing oscillator in the <> _ => 6 3 7<> => z2 6 00Wz2 ðtÞWz1 ðtÞfWz1 ðtÞg 7 z2 chaotic regime [35]. The unknown parameter values have _ 6 7 >z3 > ¼ 6 00 0 0 0 7>z3 > been set to k = 39.4784 N/m, c = 1.5708 N s/m and > _ > 4 5> > 3 :>z4 ;> 00 0 0 0 :>z4 ;> a = 39.4784 N/m . The test signal is sinusoidal with an z_ 00 0 0 0 z5 amplitude F = 1658.094 N and frequency k =2p rad/s. 5 8 9 8 9 The standard deviation of the observation noise is about > 0 > > 0 > > > > > 5% of the root mean square value of the response, i.e., < F cosðktÞ = < rpnðtÞ = the approximate solution of the process equation via the þ 0 þ rcnðtÞ ; > > > > stochastic Heun’s method (r = 0.025 m). All other quan- > 0 > > rknðtÞ > Y : ; : ; tities are the same as in subcase-1. The time-step used in 0 ranðtÞ this problem is 0.01 s. Figs. 7–9 show that among the three 59 ð Þ filters, only the LTL- and MTrL-based filter estimates are close to their reference values while estimate from the other where filter is not so. However, for all the three filters, the estimate values are close to the reference values when the Xip b 1 Wz1 ðtÞ¼ P kðtÞZ kjk; ð60Þ global iteration vs k1 k¼ði1Þp 8 Xip b 2 7 Wz2 ðtÞ¼ P kðtÞZ kjk; ð61Þ k¼ði1Þp 6

Q ), c (Ns/m)

ip ttj 3 5 where P kðtÞ¼ is the set of Lagrange interpo- j¼ði1Þp tk tj b1 b 2 4 lating polynomials and Z kjk and Z kjk denotes the first and second components of the augmented state vector. The 3 set of measurement equations at the discrete grid points over Ii is presently represented as 2 8 9 K (N/m), alpha (N/m j >z > 1 > 1 > > i > <z2 = 0 Y ¼ ½10000 j þ r nj such that fj 2 S ; j 6¼ði 1Þpg: 1 2 3 4 5 6 j >z3 > Y Y i global iteration no +1 > j > >z > :> 4 ;> zj Fig. 1. Estimated parameter values in different global iterations for the 5 Duffing oscillator of example-1 via EKF for subcase-1 (h = 0.01 s), ð62Þ reference values of parameters: c = 0.9 N s/m; k = 7 N/m; a = 3 N/m3. 5072 S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083

8 8

7 7

6 6 K ), c (N/m) ), c (Ns/m) 3 5 3 5 alpha k c 4 alpha 4 c 3 3

2 2 K (N/m), alpha (N/m 1 K (N/m), alpha (N/m 1

0 0 0 5 10 15 20 1 2 3 4 5 6 time (s) global iteration no +1

Fig. 2. Time evolution of estimated parameters in the last global iteration Fig. 5. Estimated parameter values in different global iterations for the for the Duffing oscillator of example-1 via EKF for subcase-1 (h = 0.01 s), Duffing oscillator of example-1 problem via MTrL-based EKF for 3 reference values of parameters: c = 0.9 N s/m; k = 7 N/m; a = 3 N/m . subcase-1 (h = 0.01 s), reference values of parameters: c = 0.9 N s/m; k = 7 N/m; a = 3 N/m3.

7 8 )

6 K 3 7

), c (Ns/m) alpha 3 5 6 c K alpha 4 5 c 3 4

2 3

K (N/m), alpha (N/m 1 2

0 K (N/m), c (Ns/m), alpha (N/m 1 1 2 3 4 5 global iteration no +1 0 0 5 10 15 20 Fig. 3. Estimated parameter values in different global iterations for the time (s) Duffing oscillator of example-1 problem via LTL-based EKF for subcase- 1(h = 0.01 s), reference values of parameters: c = 0.9 N s/m; k = 7 N/m; Fig. 6. Time evolution of estimated parameters in the last global iteration a = 3 N/m3. for the Duffing oscillator of example-1 via MTrL-based EKF for subcase- 1(h = 0.01 s), reference values of parameters: c = 0.9 N s/m; k = 7 N/m; a = 3 N/m3.

7 4.5 MTrL 6 4 LTL K EKF ), c (Ns/m) 3 5 alpha 3.5 c 4 3

3 2.5

2 c (Ns/m) 2

K (N/m), alpha (N/m 1 1.5

0 0 5 10 15 20 1 time (s) 0.5 0 5 10 15 Fig. 4. Time evolution of estimated parameters in the last global iteration time (s) for the Duffing oscillator of example-1 via LTL-based EKF for subcase-1 (h = 0.01 s), reference values of parameters: c = 0.9 N s/m; k = 7 N/m; Fig. 7. Time evolution of damping parameter c for the Duffing oscillator of a = 3 N/m3. example-1 for subcase-2 (h = 0.01 s), reference value of c = 1.5708 N s/m. time-step used is lower e.g., h = 0.005 s, 0.002 s or 0.001 s. value of the stiffness parameter (k). From Fig. 13 it could Figs. 10–12 show the effect of time-step on the estimated be observed that for smaller values of the time-step, the S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 5073

70 60 60 50 50 MTrL LTL 40 40 EKF 30 K (N/m)

K (N/m) 30 reference value 20 h = 0.01 s 20 h = 0.005 s h =0.002 s 10 10 h =0.001 s

0 0 0 5 10 15 0 5 10 15 time (s) time (s)

Fig. 8. Time evolution of stiffness parameter k for the Duffing oscillator of Fig. 11. Time evolution of estimated parameter k for the Duffing example-1 for subcase-2 (h = 0.01 s), reference value of k = 39.4784 N s/m. oscillator of example-1 via LTL-based EKF for subcase-2 for various time-steps, reference value: 39.4784 N/m.

45 60

40 50 35 ) 3 30 40

25 MTrL 30 20 K (N/m) alpha (N/m LTL reference value 15 EKF 20 h = 0.01 s h = 0.005 s 10 h = 0.002 s 10 5 h = 0.001 s

0 0 0 5 10 15 0 5 10 15 time (s) time (s)

Fig. 9. Time evolution of the hardening parameter a for the Duﬃng Fig. 12. Time evolution of estimated parameter k for the Duﬃng oscillator of example-1 for subcase-2 (h = 0.01 s), reference value of oscillator of example-1 via MTrL-based EKF for subcase-2 for various 3 a = 39.4784 N/m . time-steps, reference value: 39.4784 N/m.

70 60 65 MTrL LTL 60 50 EKF 55 reference value 40 50 45 30 40 K (N/m) K (N/m) reference value 35 20 h = 0.01 s h =0.005 s 30 h =0.002 s 25 10 h =0.001 s 20 0 15 0 5 10 15 0.001 0.002 0.005 0.01 time (s) h (s)

Fig. 10. Time evolution of estimated parameter k for the Duffing Fig. 13. Converged estimate of k for different time-steps for the Duffing oscillator of example-1 via EKF for subcase-2 for various time-steps, oscillator of example-1 via EKF, LTL-based EKF and MTrL-based EKF reference value: 39.4784 N/m. for subcase-2. estimates of k as per the three methods lie close to the ref- mates move away from the reference value. Beyond erence value. As the step-size is increased from 0.001 s to h = 0.005 s, the EKF results deteriorate rapidly, while the 0.005 s, the results of the LTL- and MTrL-based filters LTL- and MTrL-based filters perform relatively better. remain insensitive to the changes in h, while the EKF esti- Fig. 14 shows the time-evolution of the variance associated 5074 S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083

1 10 8 estimated k EKF 6 reference k LTLEKF 0 MTRLEKF 10 4

2 –1 0 5 10 15 20 25 10 10 ) 2 estimated α ), k (N/m) 3 /m

2 –2 reference α 10 5 (N 2 (N/m k α σ

–3 0 10 0 5 10 15 20 25 2

c (Ns/m), estimated c –4 10 1.5 reference c

1 –5 10 0.5 0 5 10 15 0 5 10 15 20 25 time (s) time (s)

Fig. 14. Time evolution of the variance of the stiffness parameter k for the Fig. 16. Time evolution of estimated parameters, c; k; a, for the Duffing Duffing oscillator of example-1 via EKF, LTL-based EKF and MTrL- oscillator of example-1 via LTL-based EKF for subcase-3 (h = 0.01 s). based EKF for h = 0.01 s in subcase-2.

10 with the estimates of the stiffness parameter k for estimated k h = 0.01 s. As has been mentioned earlier, the actual vari- reference k 5 ability of the identified parameter is not captured accurately within the framework of extended Kalman 0 filtering. Though a further investigation into the real nature 0 5 10 15 20 25 6 of the uncertainty associated with the identified parameters α

), k (N/m) reference is pending, it can be observed that of the three filtering 3 4 estimated α (N/m schemes studied, the results from MTrL-based EKF show 2 α faster numerical convergence as the measurements are 0 assimilated sequentially. Quantitatively, however, the pre- 0 5 10 15 20 25 2 dicted variances from the three methods show notably c (Ns/m), estimated c large differences. This clearly points towards the need for reference c further research into the convergence properties of the 1 algorithms. 0 Subcase-3: Although most of the problems considered in 0 5 10 15 20 25 the present work do have essentially time-invariant param- time (s) eters, a very limited study for a time-varying parameter Fig. 17. Time evolution of estimated parameters, c; k; a, for the Duffing oscillator of example-1 via MTrL-based EKF for subcase-3 (h = 0.01 s).

10 estimation will now be taken up. Here the unknown 5 estimated k parameters k and a are chosen such that there is a sudden reference k change in the parameter values after some time in the 0 0 5 10 15 20 25 observation period. Thus, during the ﬁrst 10 s of simula- 2 tion, the values of k and a are set to 6 N/m and 2 N/m3, estimated c reference c respectively. At t = 10 s, a discontinuous change (jump) 1 of these two parameters are assumed to happen and the ), c (Ns/m), k (N/m) 3 new values of k and a are set to 3.75 N/m and 2.3 N/m3, 0

(N/m 0 5 10 15 20 25 respectively and the latter values remain unchanged till α 5 the end of the observation period (i.e., 25 s). The value of all other simulation quantities are taken to be same as in 0 subcase-1. The time-step used is 0.01 s. Figs. 15–17 report estimated α reference α time-evolutions of the parameters. Only the MTrL-based –5 0 5 10 15 20 25 EKF is successful in tracking the temporal variation of time (s) parameters. We emphasize that no adaptive tracking tech- Fig. 15. Time evolution of estimated parameters, c; k; a, for the Duﬃng niques [48] have been incorporated in any of the three oscillator of example-1 via EKF for subcase-3 (h = 0.01 s). ﬁlters. S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 5075

5.2. One-and-a-half DOF oscillator with nonlinear hysteretic In the above equations, z1, z2 and z3 are the estimates of damping states corresponding to displacement, velocity and restoring force, respectively at the i-th grid point and they are In the current example, we consider a one-and-a-half still-unknown. The measurement equation (linear in this DOF oscillator with the Bouc–Wen hysteretic model. Hys- case), following Eq. (7), at the i-th time-point may be writ- teresis is a mechanical phenomenon which is useful to ten as 8 9 model structures under earthquake loads. Hence it has i >z1 > been studied widely: for instance the recent work by Gha- ()> > () < i = i nem and Ferro [11] which considers the identification of a Y 1 100000 z2 r n i ¼ þ Y 1 Y 1 : structure with hysteretic behavior modeled by the Bouc– 2 010000> . > i Y i > . > rY 2 nY 2 Wen hysteretic model using the ensemble Kalman filter. :> ;> i The governing equation of motion is given by z6 m€x þ 2cx_ þ k½ax þð1 aÞr¼F cosðktÞþrpnpðtÞ; ð63Þ ð67Þ n n1 r_ ¼ Ax_ bx_jrj cjx_jrjrj þ rH nH ðtÞ: ð64Þ For the MTrL-based Kalman filter, the linearized system Here m is the mass of the system (set to unity in the present involves the known values of the estimates at the initial case), a is a scalar parameter and r is the restoring force. time-point of the time-interval for one linearization step The present problem consists of estimating the hysteretic and unknown estimates at multiple time-points over the parameters A, b, c. n is assumed to be known in the present same interval. For an MTrL system of order p (not the for- case. The measured quantities are the displacement and mal order of accuracy of the filter), the set of unknown esti- velocity histories of the oscillator. Again declaring these mates comprises the vector estimates at p distinct grid unknown parameters as additional states of the system, points. In the present case, for the time-interval the extended state space equations corresponding to the Ii ¼ðtði1Þp; tði1Þpþ1; ...; tip, the linearized process equation process and the measurement models may be written as can be written as 8 9 >z_ > z_1 ¼ z2; > 1 > > > >z2 > z_2 ¼akz1 2cz2 ð1 aÞkz3 þ F cosðktÞþrpnðtÞ; <> => z_3 n n1 > > z_3 ¼ z4z2 z6z2jz3j z5jz2jz3jz3j þ rH nH ðtÞ; >z_4 > ð65Þ > > >z_ > z_4 ¼ 0 þ rAnAðtÞ; :> 5 ;> z_ 26 3 z_5 ¼ 0 þ rcncðtÞ; 01 0 0 0 0 6 7 z_ ¼ 0 þ r n ðtÞ 6 ak 2c ða 1Þk 00 07 6 b b 6 7 6 n1 n 7 6 00 0 Wz2 ðtÞjWz2 ðtÞjWz3 ðtÞjWz2 ðtÞj Wz2 ðtÞjWz3 ðtÞj 7 and ¼ 6 7 6 00 0 0 0 0 7 6 7 z1 rY nY 1ðtÞ 4 5 Y ¼ þ 1 ; ð66Þ 00 0 0 0 0 00 0 0 0 0 z2 rY 2 nY 2ðtÞ 8 9 8 9 8 9 >z1 > > 0 > > 0 > where z1 = x and z2 ¼ x_ are the displacement and the veloc- > > > > > > >z > > F cosðktÞ > > rpn ðtÞ > > 2 > > > > p > ity states, respectively, z4, z5, z6 are the states in the ex- < = < = < = z3 0 rH nH ðtÞ tended state space corresponding to A; c; b . r and r > > þ > > þ > >; ð68Þ f g p H >z4 > > 0 > > rAnAðtÞ > > > > > > > are the enveloping factors for the original process noises >z > > 0 > > r n ðtÞ > :> 5 ;> :> ;> :> c c ;> np(t) and nH(t), respectively whereas rA, rc, rb are the z6 0 rbnbðtÞ enveloping factors for the artificial evolution of the respec- where tive parameters. For the LTL-based EKF, over the time- Xip interval ðti1; ti, the conditionally linearized system has t P t Zb 1 the following state space representation: Wz1 ð Þ¼ kð Þ kjk ð69Þ 8 9 2 3 k¼ði1Þp >z_ > 01 0 0 0 0 > 1 > > > 6 ak c a k 7 and >z2 > 6 2 ð 1Þ 00 07 < = 6 n1 n 7 z_ 6 00 0 z z z z z z 7 ip 3 6 2 j 2j 3j 3j 2j 3j 7 X q ¼ b 2 >z_ > 6 00 0 0 0 07 W ðtÞ¼ P ðtÞZ ð70Þ > 4 > 6 7 z2 k kjk > _ > 4 5 k¼ði1Þp :>z5 ;> 00 0 0 0 0 z_ 00 0 0 0 0 68 9 8 9 8 9 and >z1 > > 0 > > 0 > > > > > > > Xip >z2 > > F cosðktÞ > > rpnpðtÞ > <> => <> => <> => b 3 Wz3 ðtÞ¼ P kðtÞZ kjk; ð71Þ z3 0 rH nH ðtÞ > > þ > > þ > >: k¼ði1Þp >z4 > > 0 > > rAnAðtÞ > Q > > > > > > ip ttj >z5 > > 0 > > rcncðtÞ > where P kðtÞ¼ is the set of Lagrange interpo- : ; : ; : ; j¼ði1Þp tk tj z6 0 rbnbðtÞ b1 b 2 b 3 lating polynomials and Z kjk; Z kjk; Z kjk are respectively the 5076 S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083

ﬁrst, second and third components of the extended state 4 vector. The set of measurement equations at the discrete A 3.5 gamma time-points over I can be represented as beta i 8 9 > j > 3 >z1 > ()> > 1 <zj = 2.5 Y j 100000 2 ¼ Y 2 010000> . > 2 j > . > :> ;> j 1.5 z A, beta and gamma () 6 j 1 rY 1 nY 1 þ j such that fj 2 Si; j 6¼ði 1Þpg: rY 2 nY 2 0.5 0 5 10 15 ð72Þ time (s)

Fig. 19. Time evolution of estimated parameters in the last global 5.2.1. Results iteration for the hysteretic oscillator via EKF of example-2, h = 0.01 s, The simulation parameters have been taken to be reference values of parameters: A =1;b = 0.5; c = 0.5. m = 1 kg, k = 7 N/m, c = 0.02 N s/m, a = 0.25 and n =2. The test signal is sinusoidal with an amplitude F =4N 5 A and frequency k = 3 rad/s. rp is taken as 0.02 N and rH 4.5 gamma beta is also taken as 0.02 m/s, while the enveloping factors for 4 the evolution of the additional states are each equal to 3.5 0.00002 (rA is non-dimensional while units of both rc, rb 3 are m2). The values of the parameters to be estimated have been taken to be A =1, b = 0.5 m2 and 2.5 c = 0.5 m2. The standard deviations of the observation 2 1.5 noises is about 3% of the root mean square value of the A, beta and gamma responses, i.e., the approximate solution of the process 1 0.5 equation via the stochastic Heun’s method (rY 1 ¼ 0:008 m and rY ¼ 0:025 m=s). The order of MTrL (p) used in this 0 2 1 2 3 4 5 example is 2. global iteration no +1 All the three filtering algorithms have been used to solve for the unknown parameters with a time-step of 0.01 s. The Fig. 20. Estimated parameter values in different global iterations for the hysteretic oscillator via LTL-based EKF of example-2, h = 0.01 s, converged estimates for all the three filters are close to their reference values of parameters: A =1;b = 0.5; c = 0.5. reference values. Figs. 18–23 show the results of the parameter estimation problem by the three filters. 4 A 3.5 gamma beta 3

2.5

5 2 A 4.5 gamma 1.5 beta 4 A, beta and gamma 1 3.5

3 0.5

2.5 0 0 5 10 15 2 time (s) 1.5 A, beta and gamma Fig. 21. Time evolution of estimated parameters in the last global 1 iteration for the hysteretic oscillator via LTL-based EKF of example-2, 0.5 h = 0.01 s, reference values of parameters: A =1;b = 0.5; c = 0.5.

0 1 2 3 4 5 global iteration no +1 5.3. Planar three-story linear shear building Fig. 18. Estimated parameter values in diﬀerent global iterations for the hysteretic oscillator via EKF of example-2, h = 0.01 s, reference values of Consider an idealized planar three DOF shear building parameters: A =1;b = 0.5; c = 0.5. system with known masses ðm1; m2; m3Þ. The inter-story S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 5077

5 Declaring k1, k2, k3, c1, c2, c3 as additional states A T 4.5 gamma fz7; z8; z9; z10; z11; z12g , the extended state space equations beta for the identiﬁcation model can be written as 4 8 9 > z_ > 3.5 > 1 > 2 3 > > > z_2 > z4 3 > > > > 6 7 <> z_3 => 6 z5 7 2.5 6 7 z_ 6 z 7 > 4 > ¼ 6 6 7 2 > > 6 1 T 1 T 7 > z_5 > 4 M KZ f z1 z2 z3 g M fCZ gf z4 z5 z6 g 5 A, beta and gamma > > 1.5 > > > z_6 > ½0 :> ;> ð66Þ 1 fz_ g P ð61Þ 8 9 8 9 > 0 > > 0 > 0.5 > > > > 1 2 3 4 5 > > > > global iteration no +1 <> 0 => <> 0 => þ 0 þ 0 : ð74Þ Fig. 22. Estimated parameter values in diﬀerent global iterations for the > > > > > 1 > > > > M F > > rpnpðtÞ > hysteretic MTrL-based EKF of example-2, h = 0.01 s, reference values of :> ;> :> ;> parameters: A =1;b = 0.5; c = 0.5. f0gð61Þ frP nP ðtÞgð61

Here KZ and CZ are obtained from the stiffness and damp- 4 ing matrices, respectively by replacing the element stiff- A nesses and dampings by the corresponding additional 3.5 gamma T beta states zP ¼fz7; z8; z9; z10; z11; z12g ; and {rPnP(t)}(6·1) de- 3 notes the enveloping factors associated with the artificial

2.5 evolution of parameters. For notational convenience we compress Eq. (74) and 2 rewrite it following Eq. (5) as 1.5 z_ ¼ gðzÞþf ðtÞþnðtÞ:

A, gamma and beta 1 It is evident that in spite of the linear mechanics of the 0.5 problem, the (augmented) identiﬁcation model is nonlinear

0 in the states of the system. As before, the linearization pro- 0 5 10 15 time (s) cedure for the two proposed filters will be indicated. In the LTL-based EKF, the coefficient matrix of the linearized Fig. 23. Time evolution of estimated parameters in the last global process equation over the time-interval ðti1; ti following iteration for the hysteretic oscillator via MTrL-based EKF of example-2, Eq. (20) can be written as h = 0.01 s, reference values of parameters: A =1;b = 0.5; c = 0.5. 2 3 ½B1 ½B2 6 36 36 7 b 4 B B 5 stiffnesses ðk1; k2; k3Þ are assumed to be unknown along BðZ i; tiÞ¼ ½ 336 ½ 436 ; with the inter-story viscous damping coefficients c , c , c ½B ½B 1 2 3 25 66 6 66 3 [46,33]. The linear equation of motion of this system sub- 000100 ject to floor level excitations F(t) is given by 6 7 where ½B1¼4 0000105;

M€x þ Cx_ þ Kx ¼ F ðtÞþNpðtÞ: ð73Þ 000001 ½B 3 2 3 Here M, C, K are the mass, damping and stiffness matrices, ðz7Þ ðz7Þ ðz10Þ ðz10Þ T a m a m 0 a m a m 0 respectively. X 1 :¼ x ¼fx1; x2; x3g is the displacement vec- 6 1 1 1 1 7 6 ðzÞ ðzþzÞ ðzÞ ðz Þ ðz þz Þ ðz Þ 7 T ¼ 6 a 7 a 7 8 a 8 a 10 a 10 11 a 11 7; tor, X 2 :¼ x_ ¼fx_ 1; x_ 2; x_ 3g is the velocity vector, Np(t) is the 4 m2 m2 m2 m2 m2 m2 5 ðzÞ ðzþzÞ ðz þz Þ process noise vector (3 · 1), Z denotes the vector corre- 0 a 8 a 8 9 0 a ðz11 Þ a 11 12 sponding to the augmented state space and its first 6 ele- m3 m3 m3 m3 T T T T ð75Þ ments are obtained as f z1 z2 z6 g ¼fX 1 ; X 2 g 2 3 ðzzÞ ðzzÞ 2 3 2 3 b 2 1 00b 5 4 00 m1 m1 m1 00 k1 k1 0 6 7 6 z z z z 7 6 7 6 7 ðz1z2Þ ð 3 2Þ ðz4z5Þ ð 6 5Þ ½B4¼6 b b 0 b b 0 7 M ¼ 4 0 m2 0 5; K ¼ 4 k1 k1 þ k2 k2 5; 4 m2 m2 m2 m2 5 ðzzÞ z ðzzÞ z 00m3 0 k2 k2 þ k3 0 b 2 3 b 3 0 b 5 6 b 6 2 3 m3 m3 m3 m3 c1 c1 0 6 7 ð76Þ C ¼ 4 c1 c1 þ c2 c2 5: and B2 is a (3 · 6) matrix of zeros while B5 and B6 are 0 c2 c2 þ c3 (6 · 6) matrices of zeros. Also z1; z2; ...; z12 are the 5078 S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 8 9 unknown estimates at the i-th grid point and a, b 2½0; 1 2 3> zj > > 1 > are real parameter such that a + b = 1. These parameters, 100000000000<> j => 6 7 z2 somewhat similar to those in the Newmark method of 4 5 Y j ¼ 010000000000> . > numerical integration of ODEs, allow a splitting of terms, > . > 001000000000> . > as indicated above and below, of the system coefficient ma- : j ; z trix corresponding to the linearized (augmented) process 8 9 12 > r n1 > equation. In this way, the linearized coefficient matrix as <> Y 1 j => well as the FSM may be made to be more populated there- þ r n2 such that fj 2 S ; j 6¼ði 1Þpg: ð80Þ > Y 2 j > i by increasing the accuracy as well as convergence of the :> ;> r n3 modified EKF algorithm. The measured quantities in this Y 3 j case are the floor displacements and the measurement equation at the i-th time instant can be represented as 8 9 5.3.1. Results 2 3> zi > The simulation parameters are presently taken to be > 1 > 100000000000< zi = k = 5000 N/m, k = 5000 N/m, k = 5000 N/m, 4 5 2 1 2 3 Y i ¼ 010000000000 . c = 200 N s/m, c = 200 N s/m, c = 250 N/m. The values > . > 1 2 3 001000000000:> . ;> i of the masses, assumed to be known, are set to z12 8 9 m1 = 100 kg, m2 = 200 kg and m3 = 300 kg, respectively. < 1 = rY 1 ni Accordingly, the natural frequencies of the system to be 2 þ rY 2 ni identified are 0.3949 Hz, 0.9798 Hz and 1.5038 Hz. The test : 3 ; rY 3 ni signal is in the form of a sample realization of a zero-mean ð77Þ white Gaussian noise vector with a standard deviation of For an MTrL-based EKF of order p, the set of unknown 350 estimates comprises the vector estimates at p distinct grid points. In the present case, for the time interval 300 Ii ¼ðtði1Þp; tði1Þpþ1; ...; tip; the coefficient matrix of the linearized process equation following (43) can be written as 250

2 3 c1

(Ns/m) c ½WB1 36 ½WB2 36 3 2 6 7 200 c 4 5 3 WðtÞ¼ ½WB3 36 ½WB4 36 ; and c 2 150

½WB5 ½WB6 , c 626 66 3 1 000100 c 6 7 100 4 5 where ½WB1 ¼ 000010; 50 000001 1 2 3 4 5 6 global iteration no +1

½WB3 2 3 Fig. 24. Estimated damping values in diﬀerent global iterations via EKF ðWz7 Þ ðWz7 Þ ðWz10 Þ ðWz10 Þ a m a m 0 a m a m 0 6 1 1 1 1 7 for example-3, h = 0.1 s, reference values of parameters: c1 = 200 N s/m; 6 ðW Þ ðW þW Þ ðW Þ ðW Þ ðW þW Þ ðW Þ 7 ¼ 6 a z7 a z7 z8 a z8 a z10 a z10 z11 a z11 7; c = 200 N s/m; c = 250 N s/m. 4 m2 m2 m2 m2 m2 m2 5 2 3 ðW Þ ðWz þw Þ ðW Þ ðW þW Þ 0 a z8 a 8 z9 0 a z1 1 a z11 z12 m3 m3 m3 m3 ð78Þ 7000

k1 ½WB4 k 2 3 2 ðWz w Þ ðW W Þ k b 2 z1 00b z5 z4 00 6500 3 6 m1 m1 7 6 ðW W Þ ðW W Þ ðW W Þ ðW W Þ 7 6 z1 z2 z3 z2 z4 z5 z6 z5 7; ¼ b m b m 0 b m b m 0 4 2 2 2 2 5 6000

ðW W Þ W ðW W Þ W (N/m) 0 b z2 z3 b z3 0 b z5 z6 b z6 3 m3 m3 m3 m3

ð79Þ and k 5500 2 , k 1 k where WB2 is a (3 · 6) matrix of zeros whileP WB5 and WB6 are l ip b l 5000 (6 · 6) matrices of zeros and WzðtÞ¼ k¼ði1ÞpP kðtÞZ kjk and l = 1 to 12 denoting the various states of the augmented 4500 system. Also, as before, a and b 2½0; 1 are theQ splitting 1 2 3 4 5 6 ip ttj parameters such that a + b = 1 and P kðtÞ¼ global iteration no + 1 j¼ði1Þp tk tj is the set of Lagrange interpolating polynomials. The set Fig. 25. Estimated stiﬀness values in diﬀerent global iterations via EKF of measurement equations at the grid points over Ii can for example-3, h = 0.1 s, reference values of parameters: k1 = 5000 N/m; be represented as k2 = 5000 N/m; k3 = 5000 N/m. S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 5079

350 7000

k1 k2 300 k 6500 3

250 6000 (N/m) (Ns/m) 3 3 200

c1 and k and c 5500 2 2 c2 , k , c 150 c 1 1 3 k c 5000 100

50 4500 0 10 20 30 40 50 1 2 3 4 5 6 time (s) global iteration no +1

Fig. 26. Time evolution of estimated damping parameters in the last Fig. 29. Estimated stiﬀness values in diﬀerent global iterations via LTL- global iteration of example-3 via EKF (h = 0.1 s), reference values of based EKF for example-3, h = 0.1 s, reference values of parameters: parameters: c1 = 200 N s/m; c2 = 200 N s/m; c3 = 250 N s/m. k1 = 5000 N/m; k2 = 5000 N/m; k3 = 5000 N/m.

6000 c1 k 350 1 c2 5800 k 2 c3 k 5600 3 300

5400 250 (Ns/m)

5200 3 (N/m) 3 5000 200 and c 2 and k 4800 2 , c

1 150 c , k

1 4600 k 4400 100

4200 50 4000 0 10 20 30 40 50 0 10 20 30 40 50 time (s) time (s) Fig. 30. Time evolution of estimated damping parameters in the last Fig. 27. Time evolution of estimated stiﬀness parameters in the last global global iteration of example-3 via LTL-based EKF (h = 0.1 s), reference iteration of example-3 via EKF (h = 0.1 s), reference values of parameters: values of parameters: c1 = 200 N s/m; c2 = 200 N s/m; c3 = 250 N s/m. k1 = 5000 N/m; k2 = 5000 N/m; k3 = 5000 N/m.

6000 250 k 5800 1 k2 k 5600 3

200 5400

5200 (N/m) 3 (Ns/m)

3 5000 150

and k 4800 2 and c 2 , k

1 4600 , c k 1 c 100 4400 c 1 4200 c2 c 3 4000 50 0 10 20 30 40 50 1 2 3 4 5 6 time (s) global iteration no +1 Fig. 31. Time evolution of estimated stiﬀness parameters in the last global Fig. 28. Estimated damping values in diﬀerent global iterations via LTL- iteration of example-3 via LTL-based EKF (h = 0.1 s), reference values of based EKF for example-3, h = 0.1 s, reference values of parameters: parameters: k1 = 5000 N/m; k2 = 5000 N/m; k3 = 5000 N/m. c1 = 200 N s/m; c2 = 200 N s/m; c3 = 250 N s/m.

5 N. The enveloping factor for each of the process noise are taken as 0.00002 (units corresponding to damping components have been taken to be 0.001 N, while the states and stiﬀness states are N s/m and N/m, respectively). enveloping factors for the evolution of the additional states The standard deviation of the observation noise is about 5080 S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083

250 6000 k 5800 1 k2 k 5600 3 200 5400

5200 (Ns/m) (N/m) 3 3 150 5000 and c and k 4800 2 2 , c , k 1 1 4600 c 100 k 4400 c1 c 2 4200 c3 50 4000 1 2 3 4 5 6 0 10 20 30 40 50 global iteration no +1 time (s)

Fig. 32. Estimated damping values in diﬀerent global iterations via Fig. 35. Time evolution of estimated stiﬀness parameters in the last global MTrL-based EKF for example-3, h = 0.1 s, reference values of parame- iteration of example-3 via MTrL-based EKF (h = 0.1 s), reference values ters: c1 = 200 N s/m; c2 = 200 N s/m; c3 = 250 N s/m. of parameters: k1 = 5000 N/m; k2 = 5000 N/m; k3 = 5000 N/m.

7000 values of a and b are taken to be 0.5, partly inspired by the k 1 commonly used values of these splitting parameters in the k2 k Newmark method of numerical integration. 6500 3 All the three ﬁltering algorithms have been used to solve for the unknown parameters with a time-step of 0.1 s. 6000 (N/m)

3 Though all the ﬁlters perform well in the estimation of the stiﬀness parameters, the damping parameter estimates

and k 5500 2 with the conventional EKF are away from their reference ,k 1 values (as also reported in [44,48]) while the two proposed k 5000 filters estimate the damping terms with significant accuracy. Figs. 24–35 shows the results of the three filters.

4500 1 2 3 4 5 6 global iteration no +1 6. Conclusions Fig. 33. Estimated stiffness values in different global iterations via MTrL- based EKF for example-3, h = 0.1 s, reference values of parameters: Two new forms of EKF are proposed with emphasis on k1 = 5000 N/m; k2 = 5000 N/m; k3 = 5000 N/m. nonlinear system identification of interest in structural engineering. The derivations of these filters are based on the concept of transversal linearization and, in the process, 450 these filters do not need computing derivatives of the non- c1 400 c2 linear vector field and they (in particular, the MTrL-based c 3 EKF) are found to be less sensitive to the choice of time- 350 step sizes vis-a´-vis the conventional EKF. Though a formal 300 convergence analysis is still to be performed, a reasonably (Ns/m) 3 250 large class of numerical examples are solved to demonstrate the versatility and advantages of the proposed filter- and c

2 200

, c ing algorithms in identiﬁcation of model parameters. The 1 c 150 ﬁlters perform satisfactorily in the chaotic regime and in

100 the case of jumps in parameter values. The performance of the conventional EKF is known to be poor in the last 50 0 10 20 30 40 50 case. Discontinuous changes in the temporal variations of time (s) parameters assume importance in view of the fact that, Fig. 34. Time evolution of estimated damping parameters in the last for health monitoring of structures, it is important to iden- global iteration of example-3 via MTrL-based EKF (h = 0.1 s), reference tify the damage when it occurs. Pending a more thorough values of parameters: c1 = 200 N s/m; c2 = 200 N s/m; c3 = 250 N s/m. investigation into convergence and confidence analysis, the superior performance of the MTrL-based EKF is tenta- 3% of the root mean square value of the response, i.e., the tively attributable to a more accurate computation of the approximate solution of the process equation via the sto- state transition matrix within a Lie algebra framework. chastic Heun’s method ðrY 1 ; rY 2 ; rY 3 ¼ 0:008 mÞ. The order However, it must as well be emphasized that the usage of of MTrL (p) used in this example is 2. In both the cases the polynomials (such as Lagrange interpolating polynomials) S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 5081 in achieving the MTrL-based linearization is not entirely Y_ ¼ AðtÞY ðA:1Þ consistent with the nature of solutions of the (augmented) process equations, which are presently SDEs under addi- such that t P 0 and Y(0) = Y0 2 G where G is a Lie group, tive noises. Indeed, Taylor expansions of solutions of such A : Rþ ! g is Lipschitz continuous and g is the Lie algebra SDEs contain fractional exponents of the time-step Dt. of G. It is well known that solution of Eq. (A.1) stays on G Accordingly, the formal order of accuracy of the MTrL- for all t P 0. The important geometric structural features based EKF of so-called ‘order’ p will be less than Dtp,an of the differential equation (A.1) should ideally be retained order that is readily achievable for deterministic dynamical under numerical discretization or approximation. This is- systems. However, we emphasize that if the effect of sue is not explored further in this paper. However, it is imposed white noise is so predominant as to induce a bifur- interesting to note that [17] an appealing feature of a Mag- cation and thus to affect the dynamical characteristics of nus numerical method is that, whenever the exact solution the process equation, then the very purpose of system iden- of Eq. (A.1) evolves in a Lie Group, so does the numerical tification is defeated and we possibly end up estimating a solution. wrong value of the parameter. In other words, despite the Recall that a Lie group is a differentiable manifold noise terms in the process and measurement equations, it equipped with a group structure, which is continuous with is generally important to maintain a correspondence with respect to underlying topology of the manifold. The Lie the deterministic dynamical response (corresponding to algebra g is a tangent space of G. The tangent space con- zero noise terms). Given that the MTrL is well-suited in sists of all c(t), where cðtÞ : R ! G is a smooth curve lying accurately capturing the deterministic response, it should on the manifold and c(0) = Idm, the identity of G. The Lie be reasonable to ignore the effects (in the sense of expecta- algebra g is a linear space closed under a binary operation tions) of the stochastic terms on the linearized expansion of ½; : g g ! g. This binary operation is bilinear, antisym- the vector field as well in Magnus’ expansion. Note that the metric and subjected to the Jacobi identity, i.e. LTL- or MTrL-based schemes, as applied to SDEs, are drift-implicit methods. Owing to their superior stability ½a; b¼½a; b; characteristics, drift-implicit methods are widely used in ½a; ½b; c þ ½b; ½c; a þ ½c; ½a; b ¼ 0; where a; b; c 2 g: the literature for strong or weak solutions of SDEs [5]. ðA:2Þ The effect of interpolating polynomials on the MTrL- based EKF needs further study. Particularly for problems Although the present discussion is based on matrix Lie with jumps in parameter values, it is anticipated that use groups only, the restriction to matrix Lie groups is not se- of more sophisticated basis functions e.g., B-splines or vere, since Ado’s theorem guarantees that any analytical NURBS should lead to much more accurate tracking of group is locally isomorphic to a matrix Lie groups [45]. such jumps. Moreover, the proposed filters may be viewed The binary operation of Lie algebra g is called the Lie as a first step toward explorations of implicit techniques bracket, which is given by ½a; b¼ab ba where a,b 2 g. that offer higher numerical accuracy and stability of esti- The Lie algebra corresponding to matrix Lie groups are mation procedures. It can be mentioned in this context that well known [32]. Thus, to summarize the internal opera- techniques to tackle divergence problems, e.g., adaptive tions in each space for instance, while we are allowed to two-stage Kalman filters [20], have been proposed. Com- proceed by multiplication in G, it is forbidden to multiply bining these techniques with the present form of EKF con- elements of the algebra g. However, the algebra g is a vec- stitutes an interesting element of future study. Applicability tor space and we can add and commute elements. The of the present approach to very large dimensional problems essential feature of g is that it can be mapped to G through is closely related to the applicability of the classical form of the exponential map. Thus we are allowed to move from the EKF to such problems and the identifiability of the the algebra to the group. Given A 2 g and t 2 Rþ, e(At) 2 G inverse problem at hand. Thus, the problem of coupling is the flow generated by the infinitesimal generator A (see these methods as well as any related tools (such as the [32]). two-stage Kalman filter [1], fading Kalman filter [20] and The solution of Eq. (A.1) is well researched starting with even inverse approaches based on non-stochastic optimiza- the work of Magnus [28]. As shown by Magnus, the solution and regularization) with commercial FE codes for lar- tion may be locally represented in the form ger problems is of immense practical importance. Studies on such issues have been taken up by the authors and the Y ðtÞ¼expðXðtÞÞY 0; ðA:3Þ findings will be reported elsewhere. where XðtÞ : Rþ ! g is an infinite sum of elements in g, Appendix A. An adaptation of the Magnus expansion for the known as the Magnus expansion. Xt is expressible as a spe- MTrL-based EKF cific expansion in integrals of Lie bracket commutators. Iserles and Norsett [16] have proved the convergence of For the purpose of deriving the state transition matrix of Magnus’ expansion. For instance, terms containing up to the linearized system, let us consider the following differen- three integrals in Magnus’ expansion (the continuous ana- tial equation: logue of Baker Housdorff formula) may be written as 5082 S.J. Ghosh et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 5063–5083 Z Z Z t 1 t s [18] C.C. Ji, Chao Liang, A study on an estimation method for applied XðtÞ¼ AðsÞds þ AðsÞ; Aðs Þds ds 2 1 1 force on the rod, Comput. Methods Appl. Mech. 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