A comparative Assessment of Nonlinear State Estimation Methods for Structural Health Monitoring 1

Majdi Mansouri1, Onur Avci2 , Hazem Nounou1 and Mohamed Nounou4 Electrical and Computer Engineering Program, Texas A&M University at Qatar, Doha, Qatar, 2 Civil & Architectural Engineering Department, Qatar University, Doha, Qatar, 4 Chemical Engineering Program, Texas A&M University at Qatar, Doha, Qatar.

Abstract Researchers have been studying the uncertainties unique to civil infrastructure such as redundancy; nonlinearity; interaction with surrounding; heterogeneity; boundaries and support conditions; structural continuity, stability, integrity; life cycle performance expectations and so on. For incorporating such uncertainties, filtering techniques accounting for stochasticity can be implemented employing collected data from the structures. In this paper, an Iterated Square Root Unscented Kalman Filter (ISRUKF) method is proposed for the estimation of the nonlinear state variables of nonlinear structural systems, idealized herein in simplified spring-mass-dashpot. Various conventional and state-of-the-art state estimation methods are compared for the estimation performance, namely the Unscented Kalman Filter (UKF), the Square-Root Unscented Kalman Filter (SRUKF), the Iterated Unscented Kalman Filter (IUKF) and the Iterated Square Root Unscented Kalman Filter (ISRUKF) methods. The comparison reveals that the ISRUKF method provides a better estimation accuracy than the IUKF method; while both methods provide improved accuracy over the UKF and SRUKF methods. The benefit of the ISRUKF method lies in its ability to provide accuracy related advantages over other estimation methods since it re-linearizes the measurement equation by iterating an approximate maximum a posteriori (MAP) estimate around the updated state, instead of relying on the predicted state.

Keywords—Iterated Square Root, Unscented Kalman Filter, State Estimation, Structural Health Monitoring.

I. I NTRODUCTION It has been discussed in the SHM literature that the structural identification methods proven for manufactured (mechanical) systems were unsuccessfully used for constructed (civil) systems over the years. Many characteristic attributes of the civil structures were ignored in the process since the developed methods were more applicable to manufactured systems. Even very refined three dimensional models of civil structures cannot mitigate the influences of bias sources of uncertainty. Uncertainties unique to civil structures are; local and global redundancy; material, dynamic and geometric nonlinearity; interaction with surrounding forces; heterogeneity of the members and materials; boundaries and support conditions; structural continuity, stability and integrity; life cycle performance expectations. In accounting for these for the uncertainties, there is a need for improvement of the model predictions using data collected from structures. In the widely established approach of modal parameter estimation, the system can be ill-conditioned due to uncertainties in the measurements. The potential errors in structural model updating combined with aleatory uncertainty in modal parameter estimation often results in inconsistencies between the real structural behavior and the finite element model predictions. However, in most of the published research in SHM, the ill-conditioned state and the presence of uncertainty are not considered. For nonlinear state estimations, various state estimation techniques (the Extended Kalman filter (EKF) [1]–[3], Unscented Kalman Filter (UKF) [4], [5], Central Difference Kalman Filter (CDKF) ([6]), and Square-Root Unscented Kalman Filter (SRUKF) [7]) have been developed and used by researchers. In the extended Kalman filter method, in order to approximate the covariance matrix of the state vector, the model describing the system is linearized at every time sample. However, for highly nonlinear or complex models, the EKF results are not always successful. Instead of linearizing the model, the UKF method utilizes the unscented transformation to approximate the mean and the covariance matrix of the

state vector. In the unscented transformation process, a set of samples (sigma points) are selected and propagated through the nonlinear model, providing more accurate approximations of the mean and covariance matrix of the state vector. One drawback of the UKF method is that the number of sigma points is often not very large and may not adequately represent relatively complicated distributions. As an alternative to these methods, the square-root unscented Kalman filter, and the central difference Kalman filter have been developed. The advantage of these filters is that evaluating the quasi log-likelihood distribution only takes a fraction of a second. The iterated square-root unscented Kalman filter has been recently suggested by Wu et al. ([8]) for target tracking using TDOA measurements. The ISRUKF employs an iterative procedure within a single measurement update step by resampling the sigma points till a termination criterion, based on the minimization of the maximum likelihood estimate, is satisfied. Furthermore, the ISRUKF method propagates and updates the square root of the state covariance iteratively and directly in Cholesky factored form. In addition to providing reduction in the computational complexity, ISURKF has as increased numerical stability and better (or at least equal) performance when compared to the other algorithms. The organization of the paper is as follows. In Section II, the state estimation problem is presented. Then, in Section III, the developed iterated square-root unscented Kalman filter is described. After that, in Section IV, the performance of various state estimation methods are compared for the state variables of a three degree of freedom spring-mass-dashpot system. Conclusions are presented in Section V. II. S TATE E STIMATION P ROBLEM The formulation of the state estimation problem is presented here in this section. A. Problem Formulation The state estimation problem is formulated for a general system model, in this section. Consider a nonlinear state space model to be described as follows [9]: x˙ = g(x, u, θ, w), y = l(x, u, θ, v),

(1)

where x ∈ Rn is a vector of the state variables, u ∈ Rp is a vector of the input variables, θ ∈ Rq is a known parameter vector, y ∈ Rm is a vector of the measured variables, w ∈ Rn and v ∈ Rm are process and measurement noise vectors, respectively, and g and l are nonlinear differentiable functions. Discretizing the state space model (1), the discrete model can be written as follows: xk = f (xk−1 , uk−1 , θk−1 , wk−1 ), yk = h(xk , uk , θk , vk ),

(2)

which describes the state variables at some time step (k) in terms of their values at a previous time step (k − 1). Let the process and measurement noise vectors have the following properties: E[wk ] = 0, E[wk wkT ] = Qk , E[vk ] = 0 and E[vk vkT ] = Rk . Let’s assume that the parameter vector is described by the following model: θk = θk−1 + γk−1 .

(3)

which means that it corresponds to a stationary process, with an identity transition matrix, driven by white noise. In order to include the parameter vector θk into the state estimation problem, let’s define a new state vector zk that augments the state vector xk and the parameter vector θk as follows: h x i h f (x , u , w , θ ) i k k−1 k−1 k−1 k−1 zk = = , (4) θk θk−1 + γk−1 where zk ∈ Rn+q . Also, defining the augmented noise vector as: h w i k−1 ǫk−1 = , γk−1

(5)

the model (2) can be written as, zk = F(zk−1 , uk−1 , ǫk−1 ),

(6)

yk = R(zk , uk , vk ),

(7)

where F and R are differentiable nonlinear functions. In this work, the objective is to estimate the state variables vector zk given the measurements vector yk , where the model parameter vector θk is assumed to be known.

III. D ESCRIPTION

OF

S TATE E STIMATION M ETHODS

A. Unscented Kalman Filter Method The EKF approximates the mean and covariance of the state vector by linearizing the nonlinear process and observation equations, which may not provide a satisfactory approximation of these moments. For better estimates of these moments, the unscented Kalman filter takes advantage of the unscented transformation. It must be noted that the unscented transformation is a method for calculating the statistics of a random variable that undergoes a nonlinear mapping. Assume that a random variable z ∈ RL with mean z¯ and covariance Pz is transformed by a nonlinear function, y = f (z). In order to find the statistics of y , define 2L + 1 sigma vectors as follows: Z0 = z

p Zi = z + ( (L + λ)Pz )i i = 1, ..., L (8) p Zi = z − ( (L + λ)Pz )i i = L + 1, ..., 2L p where λ = e2 (L + k) − L is a scaling parameter and ( (L + λ)Pz )i denotes the ith column of the matrix square root. The constant 10−4 < e < 1 determines the spread of the sigma points around z . The constant k is a secondary scaling parameter which is usually set to zero or 3 − L ([3]). Then, these sigma points are propagated through the nonlinear function, i.e., Yi = f (Zi )

i = 0, ..., 2L

(9)

and the mean and covariance matrix of y can be approximated as weighted sample mean and covariance of the transformed sigma points of Yi as follows: y ≈ Pz ≈

and

2L X

i=0 2L X i=0

(m)

Wi

Yi ,

(10)

(c)

Wi (Yi − y)(Yi − y)T ,

where the weights are given by: (m)

Wi

(c)

W0

and

(m)

Wi

λ , λ+r λ = + (1 − e2 + ξ), λ+r 1 (c) = Wi = , i = 0, ..., 2L. 2(λ + r)

=

(11)

The parameter ξ is used to incorporate prior knowledge about the distribution of z . It has been shown that for a Gaussian and non-Gaussian variables, the unscented transformation results in approximations that are accurate up to the third and second order, respectively. The prediction update equations are as follows: − zbk− = F(b zk−1 , uk−1 ),

δ Pz−k = Pzk−1 + Rk−1 ,
−1 δ Rk−1 = λRLS − 1 Pzk−1 ,  q q  √ √ − − − − Zk|k−1 = zbk zbk + r + λ Pzk zbk − r + λ Pz−k ,
Yk|k−1 = F Zk|k−1 , uk−1 ,

ybk =

2r X

(m)

Wi

Yi,k|k−1.

i=0

The prediction update equations are as follows:

(12)

Pyk Pzk ,yk

2r X

=

i=0 2r X

=

i=0

(c) 

Wi

(c) 

Wi

Yi,k|k−1 − ybk



Zi,k|k−1 − ybk

Kk = Pzk ,yk Py−1 , k

Yi,k|k−1 − ybk



T

+ Rkǫ ,

T

,

Zi,k|k−1 − ybk

zbk = zbk− + Kk (yk − ybk ),

Pzk

= Pz−k − Kk Pyk KkT .

(13)

B. Square-root Unscented Kalman Filter Method In the EKF, the covariance Pk itself is recursively calculated, while, the UKF requires instead calculation of the matrix square-root Sk STk = Pk , at each time step. In the SRUKF, Sk will be propagated directly, avoiding the computational complexity to refactorize at each time step [7]. The SRUKF is initialized with a state mean vector and the square root of a covariance. zˆ0 = E[z0 ] (14) and,

   S0 = chol E (z0 − zˆ0 )(z0 − zˆ0 )′

(15)

Ψk−1 = [ˆ zk−1 zˆk−1 + hSk−1 zˆk−1 − hSk−1 ]

(16)

The Cholesky factorization decomposes a symmetric, positive-definite matrix into the product of a lower-triangular matrix and its transpose. This matrix is used directly to obtain the sigma points: The scaling constant h is expressed as, √ (17) h = Lα2 where α is a tunable parameter less than one. The sigma points are then passed through the nonlinear process system, which predicts the current attitude based on each sigma point. Ψk|k−1 = f [Ψk−1 ]

(18)

The estimated state mean and square root covariance are calculated from the transformed sigma points using, zˆk− Sk−

= qr

Sk− (c)

q

=

2L X

(m)

Wi

Ψi,k|k−1

(19)

i=0

(c) W1 (Ψ1:2L,k|k−1

= cholupdate

n

(m)

−

Sk− , Ψ0,k

−

(m)

(c)

√

zˆk− )

R

w

(c) zˆk− , W0



o

(20) (21)

1 where, W0 = 2(1 − α2 + 12 β), W0 = 1 − α2 , Wi = Wi = 2Lα 2 β), β is a tunable parameter used to incorporate prior distribution. The transformed sigma points are then used to predict the measurements using the measurement model:

  Yk|k−1 = h Ψk|k−1

(22)

The expected measurement yˆk− and square root covariance of y˜k = yk − yˆk− (called the innovation) are given by the unscented transform expressions just as for the process model: yˆk− =

2L X i=0

(m)

Wi

yi,k|k−1

(23)

Sy˜k = qr

q

(c) W1 (Y1:2L,k|k−1

p − yˆk ) Rvk



(24)

n o (c) Sy˜k = cholupdate Sy˜k , Y0,k − yˆk , W0

(25)

To determine how much to adjust the predicted state mean and covariance based on the actual measurement, the Kalman gain matrix Kk is calculated as follows: P z k yk =

2L X i=0

(c) 

Wi

Ψi,k|k−1 − zˆk−



Yi,k|k−1 − yˆk−

T

Kk = Pzk yk /STy¯k /Sy¯k

(26) (27)

Finally, the state mean and covariance are updated using the actual measurement and the Kalman gain matrix: zˆk = zˆk− + Kk (yk − yˆk− )

(28)

U = Kk Sy¯k

(29)

 Sk = cholupdate S− k , U, −1

(30)

where, Rw is the process noise covariance, Rv is the measurement noise covariance, chol - is Cholesky method of matrix factorization, qr is QR matrix decomposition and cholupdate is a Cholesky factor updating. C. Iterated Square-Root Unscented Kalman Filter Method With the success of IUKF development [10] and the superiority of SRUKF, an improved performance would be expected if the iterates are implemented in SRUKF. Yet, with the potential problems experienced with the IUKF method, precaution should be taken for effective performance of the iterated filter [8]. The development of the ISRUKF method is due to the need to overcome this problem, using a different iteration strategy. The pseudo-code for the ISRUKF method can be summarized as follows. • Step 1: For each instant k(k > 1), evaluate the state estimate zˆk and corresponding square root covariance matrix Sk through (13) to (29). • Step 2: Let zˆk,0 = zˆk− , Sk,0 = S− ˆk,1 = zˆk . Sk,1 = Sk . Also let j = 2 k and z • Step 3 Generate new sigma points in the same way as (17): Ψk,j = [ˆ zk,j−1 zˆk,j−1 + hSk,j−1 zˆk−1,j−1 − hSk,j−1 ] •

(31)

Step 4 Recalculate (17) to (29) as follows Ψi,j = f [Ψi,j ] − zˆi,j =

S− k,j

= qr

q

2L X

(m)

Wi

(32)

Ψi,j

i=0

(c) W1 (Ψ1:2L,j

−

(33) 

− zˆk,j )

n o (c) − − S− = cholupdate S , Ψ − z ˆ , W i,j 1 k,j k,j k,j

(34) (35)

Yi,j = h [Ψi,j ] − yˆk,j

q

=

2L X

(m)

Wi

(36)

yi,j

(37)

i=0



(38)

n o (c) Sy˜k,j = cholupdate Sy˜k,j , Yi,j − yˆk,j , W0

(39)

Sy˜k,j = qr

Pzk,j yk,j =

2L X i=0

(c) W1 (Y1:2L,j

(c)

Wi

h

p − yˆk,j ) Rnk

− Ψi,j − zˆk,j

ih

− Yi,j − yˆk,j

iT

(40)

Kk = Pzk yk /STy¯k /Sy¯k

(41)

− − zˆk,j = zˆk−j + Kk,j (yk,j − yˆk,j )

(42)

U = Kk−j Sy¯k,j

(43)

n o Sk,j = cholupdate S− , U , −1 k−j

(44)

IV. S IMULATION

RESULTS

The results of state estimation methods (UKF, IUKF, SRUKF and ISRUKF) are compared in this section. They are compared through their utilization to estimate the states variables of a three degree of freedom spring-mass-dashpot system. A. State Estimations for Three Degree of Freedom Spring-Mass-Dashpot System A three degree of freedom spring-mass-dashpot system is utilized for the performance evaluation of state estimation techniques (shown in figure 1).

Figure 1. Three degree of freedom spring-mass-dashpot system. Note that the first degree of freedom is associated with a non- linear hysteretic component [11].

Table I SHM PARAMETERS AND PHYSICAL PROPERTIES Parameter m1 c1 k1 β

Value 1 0.25 9 2

Parameter m2 c2 k2 γ

Value 1 0.25 9 1

Parameter m3 c3 k3 n

Value 1 0.25 9 2

The purpose of this section is to estimate the seven state variables given displacement measurements for m1 and accelerometer measurements for m2 and m3 . The state space equations representing the system can be written as follows:      0  z1 (k) + T z5 (k) z1 (k + 1)  0  z2 (k + 1)   z2 (k) + T z6 (k)            z3 (k + 1)   z3 (k) + T z7 (k)   0         z4 (k + 1)  =  z4 (k) + T z˜4 + 0         F1 (k)  z5 (k + 1)   z5 (k) + T z˜5   T v +   1 m1      m1   z6 (k + 1)   z6 (k)   T (¨ xm2 + v2 )  z7 (k + 1) z7 (k) T (¨ xm3 + v3 ) where, z˜4 = z5 (k) − β|z5 (k)||z4 (k)|n−1 − γz5 (k)|z4 (k)|n , z˜5 = −k1 z4 (k) − k2 z1 (k) + k2 z2 (k) − (c1 + c2 )z5 (k) + c2 z6 (k).

The observation vector in discrete form is given by,         0 z1 (k) xm1 (k) v1  F2 (k)   y(k) =  x ¨m2 (k)  =  m12 z˜2 (k)  +  m  + v2  2 1 F (k) 3 v3 x ¨m3 (k) ˜3 (k) m3 z

(45)

m3

where, z˜2 = k2 z1 (k) − (k2 + k3 ) z2 (k) + k3 z3 (k) + c2 z5 (k) − (c2 + c3 (k)) + z6 (k) + c3 z7 (k), z˜3 = k3 z2 (k) − k2 + k3 z3 (k) + c3 z6 (k) − c3 z7 (k).

B. Generation of Dynamic Data For dynamic data generation from the SHM system the model (46) ([11]) is used to simulate the responses of the state as functions of time by solving the differential equations (46) using fourth order Runge Kutta Integration.     z5 x˙ 1   x˙ 2   z6       x˙ 3   z7     2−1 2     (46) z˙ =  r˙1  =  z5 − 2|z5 ||z4 | z4 − 1z5 |z4 |      x −9z − 9z + 9z − 0.5z + 0.25z + v ¨ ¨ 4 1 2 5 6 g   1    x ¨2   −9z1 − 18z2 + 9z3 − 0.25z5 + 0.5z6 + 0.25zz − v¨g  9z2 − 9z3 + 0.25z6 − 0.25z7 − v¨g x ¨3

where the state variables x1 , x2 , x3 are displacements and r1 is the hysteretic Bouc Wen parameter. It must be noted that these simulated states are assumed to be noise free. They are contaminated with zero mean Gaussian errors. The SHM parameters as well as other physical properties are shown in Table I. Figure 2(a) shows the changes in the state variable (displacement x1 ). For all simulations, the following parameters are used. The sampling frequency of the Northridge (1994) earthquake acceleration data that was used as ground excitation v¨g , is 100Hz (T = 0.01sec). The Northridge earthquake signal

(c)

(a) 1.2

1.2 Noise free Measurement

1

0.8

0.8

0.6

0.6

0.4

0.4

State x1

Data (x1)

1

0.2

0.2

0

0

−0.2

−0.2

−0.4

−0.4

−0.6 0

2

4

6

8

10 Time

12

14

16

18

Noise free ukf estimate srukf estimate iukf estimate isrukf estimate

−0.6 0

20

2

4

6

8

12

14

16

18

20

(d)

(b) 2

1.5

1

10 Time

Noise free ukf estimate srukf estimate iukf estimate isrukf estimate

Noise free ukf estimate srukf estimate iukf estimate isrukf estimate

1.5

1 State x2

State x3

0.5 0.5

0 0 −0.5

−1 0

−0.5

2

4

6

8

10 Time

12

14

16

18

20

−1 0

2

4

6

8

10 Time

12

14

16

18

20

Figure 2. Estimation of state variables using various state estimation techniques (UKF, IUKF, SRUKF and ISRUKF).

was filtered with a low frequency cutoff of 0.13Hz and a high frequency cutoff of 30Hz . A duration of 20 seconds of the earthquake record was adopted in this example. The number of sigma points is fixed to 33 for all the techniques (L = 16). The process noise of 1% RMS noise-to-signal ratio was added. The observation noise level was of 4-7% root mean square (RMS) noise to signal ratio. All the simulations performed in this paper are implemented using MATLAB version 7.1, using an Intel Pentium CPU 3.4 GHz, 1.0 G of RAM PC. C. Estimation of State Variables from Noisy Measurements The purpose of this study was to compare the estimation accuracy of UKF, IUKF, SRUKF and ISRUKF when they are utilized to estimate the seven state variables of the three degree of freedom spring-mass-dashpot system model. Hence, we consider the state vector that we wish to estimate, zk = xk = [x1 x2 x3 z1 v1 v2 v3 ]T , and the model parameters, k1 , k2 , k3 , c1 , c2 , c3 , β , γ , and n are assumed to be known. The simulation results for state estimations of seven state variables x1 , x2 , x3 , z1 , v1 , v2 and v3 using UKF, IUKF, SRUKF and ISRUKF are shown in Figures 2(a, b, c, d) and Figures 3(a, b, c, d), respectively. Also, the performance comparison of the state estimation techniques in terms of RMSE are presented in Table II (Mean RMSE (MRMSE) for UKF = 0.085, MRMSE (SRUKF) = 0.059, MRMSE (IUKF) = 0.051, and MRMSE (ISRUKF) = 0.039 and execution times respectively. It is easily observed from Figures 2 and Figures 3 as well as Table II that UKF is outperformed by the alternative techniques, albeit at the expense of a larger computational time (see Table II). The results also show that the ISRUKF achieves a better accuracy than the IUKF. Both ISRUKF and IUKF can provide improved accuracy over the UKF and SRUKF approaches. V. C ONCLUSIONS In this paper, the problem of nonlinear state estimations were addressed using the developed iterated square-root unscented Kalman filter. The ISRUKF method is compared to the unscented Kalman filter, the square-root unscented Kalman filter and the iterated unscented Kalman filter to estimate the state variables of the SHM. The comparative studies reveal that ISRUKF has better state accuracies than IUKF, and both of the methods can provide improved accuracy over UKF and SRUKF techniques. It must be emphasized that ISRUKF has a very good stability, and also a high state accuracy with low mean square errors.

(a)

(c)

1

6 Noise free ukf estimate srukf estimate iukf estimate isrukf estimate

0.5

Noise free ukf estimate srukf estimate iukf estimate isrukf estimate

5 4 3

State z1

State v1

2 1 0 0

−1 −2 −3

−0.5 0

2

4

6

8

10 Time

12

14

16

18

−4 0

20

6

8

10 Time

12

14

16

18

20

6 Noise free ukf estimate srukf estimate iukf estimate isrukf estimate

5 4

Noise free ukf estimate srukf estimate iukf estimate isrukf estimate

5 4 3 State v3

3 State v2

4

(d)

(b) 6

2 1

2 1

0

0

−1

−1 −2

−2 −3 0

2

2

4

6

8

10 Time

12

14

16

18

−3 0

20

2

4

6

8

10 Time

12

14

16

18

20

Figure 3. Estimation of state variables using various state estimation techniques (UKF, IUKF, SRUKF and ISRUKF).

Table II C OMPARISON OF S TATE E STIMATION T ECHNIQUES Technique UKF SRUKF IUKF ISRUKF

x1 0.015 0.011 0.010 0.010

x2 0.060 0.029 0.028 0.020

x3 0.106 0.065 0.042 0.023

z1 0.045 0.038 0.043 0.018

v1 0.125 0.088 0.085 0.079

v2 0.099 0.082 0.073 0.060

v3 0.147 0.099 0.076 0.067

Execution time 0.96 s 1.12 s 1.44 s 1.51 s

ACKNOWLEDGEMENT This work was made possible by NPRP grant NPRP08-148-3-051 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors. Special thanks to Dr. Eleni Chatzi for letting use of the model for dynamic data generation for 3-DOF SHM system. R EFERENCES [1] D. Simon, Optimal State Estimation: Kalman, H∞ , and Nonlinear Approaches. John Wiley and Sons, 2006. [2] M. Grewal and A. Andrews, Kalman Filtering: Theory and Practice using MATLAB. John Wiley and Sons, 2008. [3] S. Julier and J. Uhlmann, “New extension of the kalman filter to nonlinear systems,” Proceedings of SPIE, vol. 3, no. 1, pp. 182–193, 1997. [4] M. Mansouri, H. Nounou, M. Nounou, and A. A. Datta, “Modeling of nonlinear biological phenomena modeled by s-systems using bayesian method,” in Biomedical Engineering and Sciences (IECBES), 2012 IEEE EMBS Conference on. IEEE, 2012, pp. 305–310. [5] A. Romanenko and J. A. Castro, “The unscented filter as an alternative to the ekf for nonlinear state estimation: a simulation case study,” Computers & chemical engineering, vol. 28, no. 3, pp. 347–355, 2004. [6] J. Zhu, N. Zheng, Z. Yuan, Q. Zhang, X. Zhang, and Y. He, “A slam algorithm based on the central difference kalman filter,” in Intelligent Vehicles Symposium, 2009 IEEE. IEEE, 2009, pp. 123–128. [7] R. Van Der Merwe and E. Wan, “The square-root unscented kalman filter for state and parameter-estimation,” in Acoustics, Speech, and Signal Processing, 2001. Proceedings.(ICASSP’01). 2001 IEEE International Conference on, vol. 6. Ieee, 2001, pp. 3461–3464. [8] P. Wu, X. Li, and Y. Bo, “Iterated square root unscented kalman filter for maneuvering target tracking using tdoa measurements,” International Journal of Control, Automation and Systems, vol. 11, no. 4, pp. 761–767, 2013.

[9] A. Vajesta and R. Schmitz, “An experimental study of steady-state multiplicity and stability in an adiabatic stirred reactor,” AIChE Journal, vol. 3, pp. 410–419, 1970. [10] R. Zhan and J. Wan, “Iterated unscented kalman filter for passive target tracking,” Aerospace and Electronic Systems, IEEE Transactions on, vol. 43, no. 3, pp. 1155–1163, 2007. [11] E. N. Chatzi and A. W. Smyth, “Particle filter scheme with mutation for the estimation of time-invariant parameters in structural health monitoring applications,” Structural Control and Health Monitoring, vol. 20, no. 7, pp. 1081–1095, 2013.