On extended state based Kalman filter for nonlinear time?varying uncertain systems with measurement bias

Xiaocheng Zhang · Wenchao Xue · Hai?Tao Fang

Abstract

Keywords Extended state observer · Kalman filter · Uncertain dynamics · Measurement bias

1 Introduction

State estimation plays an important role in many control engineering problems. As is well known, the classical Luenberger observer [1] is a popular method for the state reconstruction of linear systems with exact model information. As for the linear systems with white noise, Kalman filter (KF)provides an optimal estimation in the mean square sense [2].However, model uncertainties and disturbances, which are ubiquitous in practice, are not considered in the original KF.In the past years, various state estimation methods have been proposed for systems with different kinds of model uncertainties. Generally speaking, these model uncertainties can be divided into process errors and sensor errors.

For the process errors, it can be seen as the sum of an uncertain dynamics and a white-noise component. To deal with the influence of uncertain dynamics or external disturbances in actual systems, a number of observer design methods have been proposed. Extended state observer (ESO)[3-6] was proposed to estimate both the original state and the extended state lumping the unknown internal dynamics and external disturbances. In addition, Huang et al. [7, 8] and Zhang et al. [9, 10] have provided the analysis of nonlinear ESO and linear ESO, respectively. Moreover, the ESO-based control methods have been successfully used in several industrial sectors [11-15]. Additionally, treating the modeling uncertainties and external disturbances as a lumped term, uncertainty and disturbance estimator (UDE) based control has been shown to be effective in estimating model uncertainty and external disturbance [16, 17]. By adding the integral of the output estimation error to the observer,proportional integral observer (PIO) [18, 19] was proven to asymptotically estimate system state for a class of systems with bounded uncertainty. Disturbance observer (DOB) [20,21] and nonlinear disturbance observer (NDOB) [22-24] are also popular methods of estimating the external disturbance for linear systems and nonlinear systems, respectively. For the sake of attenuating the influence of stochastic noise, Bai et al. [25] and Xue et al. [26] have combined ESO and KF/KBF algorithms to estimate states of some stochastic systems with uncertain dynamics.

On the other hand, sensor errors are often modeled more accurately as the sum of a white-noise component and a strongly correlated component. The correlated component can, for example, be random constant bias [27]. A common technique to deal with this case is to augment the state vector of the original problem by adding additional component to represent the unknown bias [28, 29]. In an attempt to reduce the computation cost of the augmented state Kalman filter (ASKF), Friedland [30] proposed the two-stage or separate-bias estimation to decouple the augmented filter into two parallel reduced-order filters. This idea has also been expanded for systems with bias modeled by first-order Markov process and systems with nonlinear dynamics [31,32]. Moreover, Zhang et al. [33] have analyzed the observability of the time-invariant system with both uncertain dynamics and measurement bias. In addition, for the unobservable biases, a design method for Kalman filter was given in [27]. Although the estimator with bias observation has drawn much research attention, little of these literatures considered the situation that a time-varying system when both uncertain dynamics and measurement bias present.

Summarizing the above discussions, this paper will focus on the state estimation problem for a class of nonlinear timevarying stochastic systems with both uncertain dynamics and measurement bias which is more common in practice.Using the idea of extended state based Kalman filter (ESKF)in [25], a filter to estimate the original state, the uncertain dynamics and the measurement bias will be developed. The main contributions of this paper are threefold:

(i) A novel ESKF algorithm is constructed to estimate the original state, the uncertain dynamics and the measurement bias for a class of nonlinear time-varying uncertain systems. Also, the consistency and the stability of the proposed filter are rigorously analyzed.

(ii) ESKF is proven to achieve the convergence of the estimation error of measurement bias in the mean square sense, thereby eliminating the influence of the measurement bias on state estimation.

(iii) It is shown that the estimation result of ESKF asymptotically converges to the minimum variance estimation while the uncertain dynamics approaches any constant vector.

The remainder of the paper is organized as follows: The problem formulation is given in Sect. 2. Section 3 introduces the design method of ESKF in detail. Follow on, Sect. 4 analyzes the performance of ESKF. After that, an illustrative example is presented in Sect. 5, and finally some concluding remarks are given in Sect. 6.

Notation Throughout this paper, the notations used are fairly standard. ?nrepresents then-dimensional Euclidean space and ?m×nstands for the space of realm×n-matrices.Imstands for the identity matrix of sizem, and 0m×nstands for the zero matrix ofmrows andncolumns and the sub-index will be occasionally removed for notational convenience if no

2 Problem formulation

Consider the following nonlinear time-varying stochastic system with both uncertain dynamics and unknown measurement bias:

wherexk∈?nis the state to be estimated,yk∈?mis the measurement contaminated by unknown bias and noise,ξk∈?pis the uncertain dynamics,b∈?qis the unknown measurement bias,ωk∈?nandνk∈?mare zero mean white process noise with covariance matricesQkandRk,respectively.x0,b,ωkandνkare assumed independent.Ak∈?n×n,Bk∈?n×p,Ck∈?m×nandDk∈?m×qare all known matrices. System (1) can be seen as the extension of the system only with uncertain dynamics [25] and the system only with measurement bias [30].

We aim to develop a state estimation algorithm to reconstruct the system state, despite of the uncertainties in dynamic model and measurement model. Compared with the completely irregular process noise and measurement noise, the uncertain dynamicsξk(xk) and unknown biasbare likely to be estimated. Therefore, it is an intuitive and simple idea to treatξk(xk) andbas augmented states being estimated and attenuated. As a result, system (1) can be equivalently transformed to

For notational convenience, denote

Formally, system (3) appears to be structurally similar to the standard Kalman filter model. Nevertheless, it is important to point out that due to the presence ofδk, the process error Δk(·) in system (3) is highly correlated with the system state, which is far beyond the white noise hypothesis required by the standard Kalman filter. Thus, how to deal with this uncertain correlation is a fundamental problem to be solved, which will be discussed in detail in the next section.

To ensure the well-posedness of the state estimation problem, some assumptions on the system structure and the uncertain parts of the system are introduced.

Remark 1Assumption 1 is a fundamental assumption to ensure the stability of KF typed algorithm. Besides, for the time-invariant case, the relationship between the observability of the augmented system (3) and the original system (1)is given in [33]. The requirement of Assumption 2 dose hold for any system obtained by discretizing a continuous-time system with bounded system matrix. Assumption 3 requires the covariance matrix sequences of process noise and measurement noise being upper bounded which is reasonable due to the power limitation of physical plants and sensors.Assumption 4 relies on the boundedness of the increment of the uncertain dynamics. Besides,Qδkcan be chosen according to the priori information of the physical limitations on the practical systems.

3 Filter design

Based on the augmented system (3), we consider the following filtering structure:

whereandare the state prediction and state update at thekth moment, respectively. Moreover,Kkis the filter gain to be designed, such that the filter (4) can retain some basic features similar to the standard Kalman filter.

One of the most fundamental properties of an estimator is that the true estimation errors should be consistent with their predicted statistics. Owing to this, the following definition is introduced.

Definition 1 [34] Consider a random vectorx. Further, letbe an estimate ofxandPan estimate of the corresponding error covariance. Then, the pair () is said to be consistent if Consistency implies that the estimated error covariancePbe an upper bound of the true error covariance. This property becomes even more important for the filter with unknown cross correlation between process error and system state.Based on the concept of consistency, the following theorem provides a design principle for filter gainKk.

Remark 3In (5b), a new parameterηkhas been introduced.It can be selected as a positive constant for simplicity or derived from the following optimization problem:

Summarizing the above results, a complete solution to the problem of state estimation for system (1) or (3) is provided in Algorithm 1.

images/BZ_146_1876_1125_1894_1160.png

4 Filter performance analysis

In this section, the performance of ESKF is studied in three aspects: (i) the boundedness of the estimation error; (ii) the convergence of measurement bias estimation; and (iii) the asymptotic optimality of ESKF under certain condition.

Firstly, the following theorem shows that the estimation error of ESKF is bounded in the mean square sense.

5 Numerical simulation

To illustrate the effectiveness of the proposed ESKF, a harmonic oscillator system model from [37] will be considered.Specially, we consider system (1) with parameters:

5.1 Case 1

Figure 1 displays the estimation results of ESKF in Case 1. In this figure, the blue solid lines represent one sample of the system state, measurement bias and uncertain dynamics generated by system (1) with parameters(16) and the red dash lines are the corresponding estimation result of ESKF. As can be seen from this figure, the estimation results of ESKF can track the true state well. In addition, the estimation result of measurement bias converges on it true value, which validates Theorem 3. As for Figure 2, the red dash lines stand for the mean square errors (MSE) of the estimation result of ESKF obtained from 500 statistical experiments, and the blue solid lines represent the diagonal elements ofPkprovided by ESKF.It can be seen from this figure that the estimation error covariances of ESKF keep stable in the given period and the consistency remains, which validates Theorem 1 and Theorem 2. Figure 3 compares the estimation errors of Kalman filter, augmented state Kalman filter (ASKF)which treats measurement bias as an augmented state, and ESKF in Case 1. In this figure, the black solid lines, green solid lines and blue solid lines represent the maximum and minimum estimation errors obtained from 500 statistical experiments of KF, ASKF and ESKF, respectively.Moreover, the dark purple dash lines, pink dash lines and red dash lines are one sample from each 500 experiments.One can seen this figure, the estimation results of KF and ASKF both have biases affected by uncertain dynamics and measurement bias, but with the assistance of timely estimating uncertain dynamics and measurement bias,ESKF can eliminate such estimation bias. In addition,the estimation errors of ASKF are even greater than KF,which means that for the system with uncertain dynamics,the measurement bias cannot be estimated independently regardless of the uncertain dynamics.

5.2 Case 2

Fig. 1 One sample generated by system (1) with parameters (16)and the corresponding estimation result of ESKF in Case 1

6 Conclusions

This paper studied the state estimation problem for a class of nonlinear time-varying stochastic systems with both uncertain dynamics and unknown measurement bias.With the idea of timely estimating the uncertain dynamics and unknown bias, an extended state based filter structure was developed. In addition, by introducing the concept of consistency, a filter gain design method is developed. After that the stability of the proposed filter was analyzed. In addition, it is shown that the designed filter can realize the estimation of measurement bias. Moreover, the estimation result is also an asymptotically optimal estimation while the uncertain dynamics approaches any constant vector. A numerical simulation was also carried out to illustrate the effectiveness of proposed method.

Fig. 2 The mean square errors of the estimation result of ESKF and the diagonal elements of Pk provided by ESKF in Case 1

Fig. 3 The estimation errors of KF, ASKF and ESKF in Case 1

Fig. 4 The difference between UUT and

AcknowledgementsThis work was partly supported by National Key R&D Program of China (No. 2018YFA0703800), the National Nature Science Foundation of China (Nos. 11931018, 61633003-3) and the Beijing Advanced Innovation Center for Intelligent Robots and Systems(No. 2019IRS09).