Comparison of state estimation techniques.
In this chapter, iterated sigma‐point Kalman filter (ISPKF) methods are used for nonlinear state variable and model parameter estimation. Different conventional state estimation methods, namely the unscented Kalman filter (UKF), the central difference Kalman filter (CDKF), the square‐root unscented Kalman filter (SRUKF), the square‐root central difference Kalman filter (SRCDKF), the iterated unscented Kalman filter (IUKF), the iterated central difference Kalman filter (ICDKF), the iterated square‐root unscented Kalman filter (ISRUKF) and the iterated square‐root central difference Kalman filter (ISRCDKF) are evaluated through a simulation example with two comparative studies in terms of state accuracies, estimation errors and convergence. The state variables are estimated in the first comparative study, from noisy measurements with the several estimation methods. Then, in the next comparative study, both of states and parameters are estimated, and are compared by calculating the estimation root mean square error (RMSE) with the noise‐free data. The impacts of the practical challenges (measurement noise and number of estimated states/parameters) on the performances of the estimation techniques are investigated. The results of both comparative studies reveal that the ISRCDKF method provides better estimation accuracy than the IUKF, ICDKF and ISRUKF. Also the previous methods provide better accuracy than the UKF, CDKF, SRUKF and SRCDKF techniques. The ISRCDKF method provides accuracy over the other different estimation techniques; by iterating maximum a posteriori estimate around the updated state, it re‐linearizes the measurement equation instead of depending on the predicted state. The results also represent that estimating more parameters impacts the estimation accuracy as well as the convergence of the estimated parameters and states. The ISRCDKF provides improved state accuracies than the other techniques even with abrupt changes in estimated states.
- Kalman filter
- sigma point
- state estimation
- parameter estimation
- nonlinear system
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 , 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) , 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)  and the iterated square‐root central difference Kalman filter (ISRCDKF)  have been developed. The EKF  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)  uses a statistical linearization technique which linearizes a nonlinear function of a random variable via a linear regression. This regression is done between
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 is the state variable vector, is the measurement vector, is the unknown vector, is the input variable vector, and are respectively process and measurement noise vectors, and and are nonlinear differentiable functions. The discretization of the model (1) is presented as follows:
which describes the state variables at some time step () in terms of their values at a previous time step (). Since we are interested to estimate the state vector , as well as the parameter vector , the parameter vector is assumed to be presented as follows:
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 into the state estimation problem, let us define a new state vector that augments the state vector and the parameter vector as follows:
where . Also, defining the augmented noise vector as:
The model (2) can be written as:
where and are differentiable nonlinear functions. Thus, the objective here is to estimate the augmented state vector , given the measurement vector .
3. Description of state estimation methods
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 characterized by a mean and covariance is used in the model. This variable is transformed by a nonlinear function . To reach the statistics of , a sigma vector is defined as follows:
Then, these sigma‐points are propagated through the nonlinear function,
And the mean and covariance matrix of can be approximated as weighted sample mean and covariance of the transformed sigma‐point of as follows:
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 and the predicted estimate covariance . In the update step, we calculate the updated state estimate and the updated estimate covariance after calculating the innovation residual and the optimal Kalman gain .
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 sigma‐point which are calculated as follows,
where is the dimension of the state , is a scaling parameter (the optimal value is ) and indicates the column of the matrix.
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 , covariance and cross covariance are obtained through a linear regression of weighted point,
The set of corresponding weights for the mean which are used to compute the posterior mean is defined as:
And the set of corresponding weights for the covariance which is used to recover the covariance and the cross‐covariance is defined as,
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 , at each time step. That is why a square‐root form of the UKF has been developed to reduce the computational complexity. In this new method the covariance matrix will be propagated directly, avoiding to refactorize at each time step .
The SRUKF is initialized as follows:
Algorithm 1: UKF algorithm
Algorithm 2: CDKF algorithm
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
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 is a tunable parameter used to include prior distribution. The transformed sigma‐point vector is then used to predict the measurements using the measurement model:
The expected measurement and square‐root covariance of (called the innovation) are given by the unscented transform expressions just as for the process model:
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 is calculated as follows:
Finally, the state mean and covariance are updated using the actual measurement and the Kalman gain matrix:
where is the process noise covariance, is the measurement noise covariance, chol is Cholesky method of matrix factorization,
The SRUKF technique is summarized in Algorithm 3.
3.4. SRCDKF method
Like the SRUKF, the matrix square‐root will be propagated directly, avoiding the computational complexity to refactorize at each time step in the CDKF. The SRCDKF is initialized with a state mean vector and the square root of a covariance.
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 next step, the sigma‐point for measurement update is calculated as,
The transformed sigma‐point vector is then used to predict the measurements using the measurement model:
The expected measurement and square‐root covariance of (called the innovation) are given by expressions just as for the process model:
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 is calculated as follows:
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.
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.
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 and the covariance matrix , an iteration loop is set up with the following initializations:
In this loop, for each
Algorithm 3: SRUKF algorithm
Algorithm 4: SRCDKF algorithm
Then the prediction step and the update step are executed as follows:
where represents the ith component of
Those steps are repeated many times until a following inequality is not satisfied.
The IUKF is summarized in Algorithm 5.
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 is set up and then the iteration step is executed, so the following equations are repeated m times.
The algorithm of the ICDKF is summarized in Algorithm 6.
The ISRUKF has the same principle as the IUKF. After executing the standard SRUKF, an iteration loop is started. The predicted estimated state and the predicted and estimated covariance matrix obtained through the prediction and the update steps will be the initialization inputs for the iteration loop (and Also let
In the iteration loop, and for each
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.
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
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 is a Gaussian process noise , and is a matrix with scalar parameter .
Algorithm 5: IUKF algorithm
Algorithm 6: ICDKF algorithm
Algorithm 7: ISRUKF algorithm
Algorithm 8: ISRCDKF algorithm
The nonstationary observation model is given by,
where and . The observation noise is a Gaussian noise . Given only the noisy observations , the different filters were used to estimate the underlying clean state sequence for .
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 , the various KFs were used to estimate the clean state sequence for k = 1...100. Figure 1 shows the changes in the state variable .
Here, the number of sigma‐points is fixed to 9 for all the techniques (
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 and the model parameters are assumed to be known. The simulation results for state estimations of state variable
|Technique||Time execution(s)||Technique||Time execution(s)|
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.
22.214.171.124. 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.
126.96.36.199.1. 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 is measured. Case 1: the state along with the first parameter will be estimated. Case 2: the state along with the two parameters and will be estimated.
Case 1: the state along with the first parameter will be estimated.
Case 2: the state along with the two parameters and will be estimated.
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 (with respect to the noise‐free data) and the mean of the estimated parameter at steady state (i.e., after convergence of parameter(s)) using the estimation methods. 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 increases from the first comparative study (where only the state variables are estimated) to case 1 (where the states and one parameter is estimated) to case 2 (where the states and two parameters, and , are estimated). For example, the RMSEs obtained using ISRCDKF for in the first comparative study and cases 1–2 of the second comparative study are 0.3121, 0.3737 and 0.3846, respectively, which increase as the number of estimated parameters increases (see 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.
188.8.131.52.2. 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, and , are considered. The simulation results of estimating the state using the UKF, IUKF, CDKF, ICDKF, SRUKF, ISRUKF, SRCDKF and ISRCDKF methods when the noise levels vary in are shown in Table 4.
In other words, for the estimation techniques, the estimation RMSEs of increase from the first comparative study (noise value ) to case (where the noise value ). For example, the RMSEs obtained using ISRCDKF for x1 where the noise level in are 0.3121, 0.1066 and 0.0376, which increase as the noise variance increases (refer to Table 4).
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.
This work was made possible by NPRP grant NPRP7‐1172‐2‐439 from the Qatar National Research Fund (a member of Qatar Foundation). The statements made herein are solely the responsibility of the authors.