## 1. Introduction

Dynamic state‐space models [1–3] are useful for describing data in many different areas, such as engineering [4–8], biological data [9, 10], chemical data [11, 12], and environmental data [8, 13–15]. Estimation of the state and model parameters based on measurements from the observation process is an essential task when analyzing data by state‐space models. Bayesian estimation filtering represents a solution of considerable importance for this type of problem definition as demonstrated by many existing algorithms based on the Bayesian filtering [16–25]. The Kalman filter (KF) [26–29] has been extensively utilized in several science applications, such as control, machine learning and neuroscience. The KF provides an optimum solution [28], when the model describing the system is supposed to be Gaussian and linear. However, the KF is limited when the model is considered to be nonlinear and present non‐Gaussian modeling assumptions. In order to relax these assumptions, the extended Kalman filter (EKF) [26, 27, 30–32], the unscented Kalman filter (UKF) [33–36], the central difference Kalman filter (CDKF) [37, 38], the square‐root unscented Kalman filter (SRUKF) [39, 40], the square‐root central difference Kalman filter (SRCDKF) [41], the iterated unscented Kalman filter (IUKF) [42, 43], the iterated central difference Kalman filter (ICDKF) [44, 45], the iterated square‐root unscented Kalman filter (ISRUKF) [46] and the iterated square‐root central difference Kalman filter (ISRCDKF) [47] have been developed. The EKF [26] linearizes the model describing the system to approximate the covariance matrix of the state vector. However, the EKF is not always performing especially for highly nonlinear or complex models. On behalf of linearizing the model, a class of filters called the sigma‐point Kalman filters (SPKFs) [48] uses a statistical linearization technique which linearizes a nonlinear function of a random variable via a linear regression. This regression is done between *n* points drawn from the prior distribution of the random variable, and the nonlinear functional evaluations of those points. The sigma‐point family of filters has been proposed to address the issues of the EKF by making use of a deterministic sampling approach. In this approach, the state distribution is approximated and represented by a set of chosen weighted sample points which capture the true mean and covariance of the state vector. These points are propagated through the true nonlinear system and capture the posterior mean and the covariance matrix of the state vector accurately to the third order (Taylor series expansion) for any nonlinearity. As part of the SPKF family, the UKF [26, 27, 33] has been developed. It uses the unscented transformation, in which a set of samples (sigma points) are propagated and selected by the nonlinear model, providing more accurate approximations of the covariance matrix and mean of the state vector. However, the UKF technique has the limit of the number of sigma‐points which are not so large and cannot represent complicated distributions. Another filter in the SPKF family is the central difference Kalman filter (CDKF) [37, 38]. It uses the Stirling polynomial interpolation formula. This filter has the benefit over the UKF in using only one parameter when generating the sigma‐point. To add some benefits of numerical stability, the SRUKF and the SRCDKF [41] have been developed. The advantage of these filters is that they ensured positive semidefiniteness of the state covariances. The iterated sigma‐point Kalman filter (ISPKF) methods employ an iterative procedure within a single measurement update step by resampling the sigma‐point till a termination criterion, based on the minimization of the maximum likelihood estimate, is satisfied.

The objectives of this chapter are threefold: (i) To estimate nonlinear state variables and model parameters using SPKF methods and extensions through a simulation example. (ii) To investigate the effects of practical challenges (such as measurement noise and number of estimated states/parameters) on the performances of the techniques. To study the effect of measurement noise on the estimation performances, several measurement noise levels will be considered. Then, the estimation performances of the techniques will be evaluated for different noise levels. Also, to study the effect of the number of estimated states/parameters on the estimation performances of all the techniques, the estimation performance will be studied for different numbers of estimated states and parameters. (iii) To apply the techniques to estimate the state variables as well as the model parameters of second‐order LTI system. The performances of the estimation techniques will be compared to each other by computing the execution times as well as the estimation root mean square error (RMSE) with respect to the noise‐free data.

## 2. State estimation problem

Next, we present the formulation of the state estimation problem.

### 2.1. Problem description and formulation

The state estimation problem for a system of nonlinear complex model is described as follows:

where

which describes the state variables at some time step (

This means that it corresponds to a stationary process, with an identity transition matrix, driven by white noise. In order to include the parameter vector

where

The model (2) can be written as:

where

## 3. Description of state estimation methods

### 3.1. UKF

The UKF is a SPKF that uses the unscented transformation. This transformation is a method for calculating the statistics of a random variable that undergoes a nonlinear mapping. It is built on the theory that “it is easier to approximate a probability distribution than an arbitrary nonlinear function”.

The state distribution is represented by a Gaussian random variable (GRV) and by a set of deterministically chosen points. These points capture the true mean and covariance of the GRV and also capture the posterior mean and covariance accurately to the second order for any nonlinearity and to the third order for Gaussian inputs. Suppose that GRV

where *L* is the dimension of the state *z,* *i*th column of the matrix square root. The constant

Then, these sigma‐points are propagated through the nonlinear function,

And the mean and covariance matrix of

where the weights are given by

The parameter ξ is used to integrate prior knowledge about the distribution of

The algorithm of the UKF includes two steps: prediction and update. In the prediction step, we calculate the predicted state estimate

The UKF technique is summarized in Algorithm 1.

### 3.2. CDKF method

The CDKF is another filter from the family of SPKF. This filter is based on Sterling polynomial interpolation formula instead of the unscented transformation used in UKF. The CDKF is similar to the UKF with the same or superior performance. However, it has an advantage over the UKF that it uses only one parameter instead of three parameters in the UKF. The CDKF uses a symmetric set of

where

These sigma‐points are propagated through the nonlinear function to form the set of the posterior sigma‐point,

Within the above results, the sterling approximation estimates of the mean

The set of corresponding weights for the mean

And the set of corresponding weights for the covariance

The CDKF technique is summarized in Algorithm 2.

### 3.3. SRUKF method

One drawback of the UKF is that it requires the calculation of the matrix square‐root

The SRUKF is initialized as follows:

Algorithm 1: UKF algorithm

• *Initialization step:*

• *Prediction step:*

• *Estimation (update) step:*

*Return the augmented state estimation*

Algorithm 2: CDKF algorithm

• *Initialization step:*

• *Prediction step:*

• *Estimation (update) step:*

*Return the augmented state estimation*

The Cholesky factorization decomposes a symmetric, positive‐definite matrix into the product of a lower triangular matrix and its transpose. This new matrix is utilized directly to obtain the sigma‐point: The scaling constant *h* is expressed as *α* is a tunable parameter less than one.

In order to predict the current attitude based on each sigma‐point, these sigma‐points are transformed through the nonlinear process system

Then, the state mean and the square‐root covariance are estimated and calculated through the transformed sigma‐point as follows:

where

The expected measurement

In an attempt to find out how much to adjust the predicted state mean and covariance based on the actual measurement, the Kalman gain matrix

Finally, the state mean and covariance are updated using the actual measurement and the Kalman gain matrix:

where *qr* is QR matrix decomposition and cholupdate is a Cholesky factor updating.

The SRUKF technique is summarized in Algorithm 3.

### 3.4. SRCDKF method

Like the SRUKF, the matrix square‐root

After the Cholesky factorization we obtain the sigma‐point:

The sigma‐point vector is then gone through the nonlinear process system, which predicts the current attitude based on each sigma‐point.

The estimated state mean and square‐root covariance are calculated from the transformed sigma‐point using,

where

The transformed sigma‐point vector is then used to predict the measurements using the measurement model:

The expected measurement

In an attempt to find out how much to adjust the predicted state mean and covariance based on the actual measurement, the Kalman gain matrix

Then, the state mean and covariance are updated using the actual measurement and the Kalman gain matrix is:

The SRCDKF technique is summarized in Algorithm 4.

### 3.5. ISPKF

In order to achieve superior performance of the statical linearization methods in terms of efficiency and accuracy, the ISPKFs have been developed. These filters include IUKF, ICDKF, ISRUKF and ISRCDKF. The major difference between the ISPKFs and the noniterated SPKFs is shown in the step where the updated state estimation is calculated using the predicted state and the observation. Instead of relying on the predicted state, the observation equation is relinearized over times by iterating an approximate maximum a posteriori estimate, so the state estimate will be more accurate.

#### 3.5.1. IUKF

The difference between the UKF and the IUKF consists in the iteration strategy.

After generating the prediction and the update steps, and getting both the state estimate

*,* *j* is the *j*th iterate.

In this loop, for each *j*, new sigma‐points are generated in the same way as the standard UKF

Algorithm 3: SRUKF algorithm

• *Initialization step:*

• *Prediction step:*

• *Estimation (update) step:*

*Return the augmented state estimation*

Algorithm 4: SRCDKF algorithm

• *Initialization step:*

• *Prediction step:*

• *Estimation(update) step:*

*Return the augmented state estimation*

Then the prediction step and the update step are executed as follows:

where

Those steps are repeated many times until a following inequality is not satisfied.

The IUKF is summarized in Algorithm 5.

#### 3.5.2. ICDKF

The iterated sigma‐point methods have the ability to provide accuracy over other estimation methods since it relinearizes the measurement equation by iterating an approximate maximum a posteriori estimate around the updated state, instead of relying on the predicted state.

In the ICDKF, the prediction step is calculated as the standard CDKF and we get

Then the sigma‐point in measurement updating is calculated as follows:

After that, the initialization

The algorithm of the ICDKF is summarized in Algorithm 6.

#### 3.5.3. ISRUKF

The ISRUKF has the same principle as the IUKF. After executing the standard SRUKF, an iteration loop is started. The predicted estimated state *j*=2 where *j* is the *j*th iteration.

In the iteration loop, and for each *j*, the new sigma‐point vector is generated as follows:

Then, the prediction and the update steps are executed as follows:

The equations in the iterative loop are repeated m times.

The ISRUKF algorithm is summarized in Algorithm 7.

#### 3.5.4. ISRCDKF

The ISRCDKF has the ability to provide accuracy over other SRCDKF since it relinearizes the measurement equation by iterating an approximate maximum a posteriori estimate around the updated state, instead of relying on the predicted state.

The algorithm of the ISRCDKF consists of generating the prediction step as the standard SRCDKF, then applying *m* iterations over the update step described as follows:

The ISRCDKF algorithm is summarized in Algorithm 8.

In the next section, the SPKF method performances will be assessed and compared to ISPKF methods. The performances of UKF, IUKF, CDKF, ICDKF, SRUKF, ISRUKF, SRCDKF and ISRCDKF methods will be evaluated through a simulation example with two comparative studies in terms of estimation accuracy, convergence and execution times.

## 4. Simulation results

### 4.1. State and parameter estimations for a second‐order LTI system

Consider a second‐order LTI described by the following state variable,

where

Algorithm 5: IUKF algorithm

• *Initialization step:*

• *Prediction step:*

*Generate the UKF prediction step and return*

• *Estimation (update) step:*

• *Generate the UKF update step and return*

• *Iteration: Let*

*And*

*Also let*

*Define these equations:*

*If the inequality is fulfilled*

*And* *then set* *and return to the iterated loop*.

*Otherwise set*

*And* *Return the augmented state estimation*

Algorithm 6: ICDKF algorithm

• *Initialization step:*

• *Prediction step:*

• *Estimation(update) step:*

• *Iteration: for j=0,.., m*

*End*

*Return the augmented state estimation*

Algorithm 7: ISRUKF algorithm

• *Initialization step:*

• *Prediction step:*

• *Generate the SRUKF prediction step and return* *and S _{k}*

^{−}

• *Estimation (update) step:*

*Generate the SRUKF update step and return*

• *Iteration: Let* *j*=2

*End*

*Return the augmented state estimation*

Algorithm 8: ISRCDKF algorithm

• *Initialization step:*

• *Prediction step:*

• *Estimation(update) step:*

• *Iteration: for j=0: m*

*End*

*Return the augmented state estimation*

The nonstationary observation model is given by,

where

#### 4.1.1. Generation of dynamic data

It must be noted that this simulated state is assumed to be noise‐free. They are contaminated with Gaussian noise. Given noisy observations **Figure 1** shows the changes in the state variable

Here, the number of sigma‐points is fixed to 9 for all the techniques (*L* = 4). The process noise

#### 4.1.2. Comparative study: estimation of state variables from noisy measurements

The purpose of this study is to compare the estimation accuracy of UKF, IUKF, CDKF, ICDKF, SRUKF, ISRUKF, SRCDKF and ISRCDKF methods when they are utilized to estimate the state variable of the system. Hence, it is considered that the state vector to be estimated *x _{k}* using UKF, IUKF, CDKF, ICDKF, SRUKF, ISRUKF, SRCDKF and ISRCDKF methods are shown in

**Figures 2**and

**3**, respectively. Also, the performance comparison of the state estimation techniques in terms of RMSE and execution times is presented in

**Table 1**.

It is easily observed from **Figures 2** and **3** as well as **Table 1** that the ISRCDKF method achieves a better accuracy than the other methods.

#### 4.1.3. Comparative study: simultaneous estimation of state variables and model parameters

The state variables and parameters are estimated and performed using the state estimation techniques UKF, IUKF, CDKF, ICDKF, SRUKF, ISRUKF, SRCDKF and ISRCDKF. The results of estimation for the model parameters using the estimation techniques (UKF, IUKF, CDKF, ICDKF, SRUKF, ISRUKF, SRCDKF and ISRCDKF) are shown in **Figures 4** and **5**, respectively. It can be seen from the results presented in **Figures 4** and **5** that the IUKF, CDKF, ICDKF, SRUKF, ISRUKF, SRCDKF and ISRCDKF methods outperform the UKF method, and that the ISRCDKF shows relative improvement over all other techniques. These results confirm the results obtained in the first comparative study, where only the state variable is estimated. The advantages of the ISRCDKF over the other techniques can also be seen through their abilities to estimate the model parameters.

### 4.1.3.1. Root Mean Square Error analysis

The effects of the practical challenges on the performances of the UKF, IUKF, CDKF, ICDKF, SRUKF, ISRUKF, SRCDKF and ISRCDKF for state and parameter estimation are investigated in the next section.

### 4.1.3.1.1. Effect of number of state and parameter to estimate on the estimation RMSE

To study the effect of the number of states and parameters to be estimated on the estimation performances of UKF, IUKF, CDKF, ICDKF, SRUKF, ISRUKF, SRCDKF and ISRCDKF, the estimation performance is analyzed for different numbers of estimated states and parameters. Here, we will consider two cases, which are summarized below. In all cases, it is assumed that the state

Case 1: the state

Case 2: the state

The estimation of the state variables and parameter(s) for these two cases is performed using UKF, IUKF, CDKF, ICDKF, SRUKF, ISRUKF, SRCDKF and ISRCDKF, and the simulation results for the state variables and the model parameters for the two cases are shown in **Tables 2** and **3**. For example, for case 1, **Table 2** compares the estimation RMSEs for the two state variables **Tables 2** and **3** also present similar comparisons for cases 1 and 2, respectively.

The results also show that the number of parameters to estimate affects the estimation accuracy of the state variables. In other words, for all the techniques the estimation RMSE of **Tables 2** and **3**). This observation is valid for the other state estimation techniques.

It can also be shown from **Tables 2** and **3** that, for all the techniques, estimating more model parameters affects the estimation accuracy. The ISRCDKF method, however, still provides advantages over other methods in terms of the estimation accuracy.

### 4.1.3.1.2. Effect of noise content on the estimation RMSE

It is assumed that a noise is added to the state variable. In order to show the performance of the estimation algorithms in the presence of noise, three different measurement noise values, **Table 4**.

In other words, for the estimation techniques, the estimation RMSEs of _{1} where the noise level in **Table 4**).

## 5. Conclusions

In this chapter, various SPKF‐based methods are used to estimate nonlinear state variables and model parameters. They are compared for the estimation performance in two comparative studies. In the first comparative study, the state variables are estimated from noisy measurements of these variables, and the several estimation methods are compared by estimating the RMSE with respect to the noise‐free data. In the second comparative study, of the state variables as well as that the model parameters are estimated. Comparing the performances of the several state estimation extensions, the impact of the number of estimated model parameters on the convergence and accuracy of these methods is also evaluated. The results of the second comparative study show that, for all the techniques, estimating more model parameters affects the estimation accuracy as well as the convergence of the estimated states and parameters. The iterated square‐root central difference Kalman method, however, still provides advantages over other methods in terms of the estimation accuracy, convergence and execution times.