Xioxue FENG, Shuhui LI,*, Feng PAN,b

a School of Automation, Beijing Institute of Technology, Beijing 100081, China

b Kunming-BIT Industry Technology Research Institute INC, Kunming 650106, China

KEYWORDS Adaptive filter;Decoupling filter;State estimate;Stochastic system;Unknown interference input

Abstract Stochastic system state estimation subject to the unknown interference input widely exists in many fields, such as the control, communication, signal processing, and fault diagnosis.However,the research results are mostly limited to the stochastic system in which only the dynamic state model or the measurement model concerns the individual unknown interference input,and the state model and the measurement model are both with the same unknown interference input.State estimate of the stochastic systems where the state model and the measurement model contain dual Unknown Interference inputs (dual-UI) with different physical meanings and mathematical definitions is concerned here. Firstly, the decoupling condition with the Unknown Interference input in the State model(S-UI)is shown,which introduces the decoupled system with the adjacent Measurement concerned Unknown Interference inputs (M-UI) appearing in the state model and the measurement model. Then, through defining the Differential term of the adjacent M-UI (M-UID),the equivalent system with only M-UID in the state model is obtained. Finally, considering the design freedom of the equivalent system, the decoupling filter in the minimum mean square error sense and the adaptive minimum upper filter with different applicable conditions are represented to obtain the optimal and sub-optimal state estimate, respectively. Two simulation cases verify the effectiveness and superiority compared with the traditional methods.

1. Introduction

State estimate has attracted considerable research interests during the past several decades, since it has widespread applications in numerous fields involving the industry monitoring,object tracking, fault diagnosis, body sensor network, etc.The accurate state estimation is related to the precise mathematical model representation of the system behavior. However, in many applications, the system behavior cannot be accurately described due to various kinds of uncertainties,for example, linearization errors, unmodeled dynamics, and unmeasurable disturbances which cannot be satisfactorily described as stochastic signals with known statistics.Modeling uncertainties of this kind in system model and measurement model can be treated as the so-called dual Unknown Interference inputs(dual-UI).Once the dual-UI are not properly modeled and handled, the state estimation performance will significantly deteriorate. Up to now, the state estimation with UIs has been widely investigated in various fields,including the target tracking, inertial navigation, fault diagnosis and industry robots.1-4The state estimate for the dynamic stochastic system, where the State model contains the Unknown Interference input(S-UI)or the Measurement model also contains the same Unknown Interference input(M-UI),has drawn much attention recently.And the corresponding state estimate algorithms can be divided into three categories, the unbiased minimum-variance filter, the robust filter and the adaptive filter.

The unbiased minimum-variance filter aims at obtaining the globally optimal estimate. Thereinto, the state augmentation technology is a common and effective method to deal with UIs, which are often treated as stochastic processes with known statistics. In this way, an augmented state vector including the system state and UIs is estimated by the Augmented State Kalman Filter (ASKF). To this end, an optimal two-stage Kalman filter has been proposed, in which the augmented filter was parallelly decoupled into the reduced-order state filter and UIs filter.5However, the ASKF methods are computationally expensive. Later, alternative derivations of the ASKF method have been proposed to alleviate this issue.6-11But these filters easily suffer from the filter degradation problem caused by the state augmentation.The robust filter aims at limiting and mitigating the effect of the UI, whose performance can be guaranteed to some extent, if the actual disturbance deviates from the nominal one in a reasonable way.12-14For example, an augmented robust three-stage Kalman filter, for discrete linear systems subject to unknown disturbances and actuator and sensor faults,has been proposed to solve the joint state and fault estimate problem,12while the unknown disturbance was modeled as the wide stationary random walk noise.It should be noted that the robust filter design may be relatively conservative, for the derived filter mainly focuses on the worst case analysis. Different from the above mentioned filters,the adaptive filter mainly adopts the methodology of joint optimal state estimate and online disturbance identification.15-19For example, through modeling the UI as a randomly switching parameter obeying a Markov chain,the estimate problem with both the state model and the measurement model interfered by the same UI is hence transformed into the simultaneous implementation of model-based filtering and model identification.17-18However, the multimodel estimate, based on the Bayesian reasoning, needs enough priori information to guarantee the model completeness, and is sensitive to the unmodeled error such as the linearized error.

However,the above-mentioned approaches do not consider the state estimate problem for the stochastic systems concerning the dual-UI,namely,the S-UI and M-UI terms with different physical meanings and mathematical definitions simultaneously appeared in the dynamic model and the measurement model, though such situation generally exists. It should be noted that state estimate for the stochastic system concerning dual-UI is a newly challenging issue,which is absolutely different from the state estimate problem for the stochastic system with the same S-UI and M-UI term mentioned above. The S-UI and M-UI terms of the dual-UI are individually related with the dynamic model and the measurement model respectively, which originate from different physical phenomena and cannot be replaced by each other. For example, in tracking a maneuvering target, the target may abruptly maneuver (defined as S-UI for the general constant acceleration motion model) accompanied by its Electronic Counter-Measure(ECM) (defined as M-UI).20,21In longitudinal flight control system, the aircraft may suffer atmospheric disturbance (defined as S-UI) during flight, while the inertial sensors output the attitude information subject to fault signal(defined as M-UI).22The attitude systems of a rigid satellite may be subjected to actuator faults (defined as S-UI) and sensor faults(defined as M-UI).23It is more suitable to model the above physical system behaviors by introducing both the S-UI and the M-UI term,i.e.,the dual-UI.In conclusion,the introduction of dual-UI is expected to describe the system behavior more accurately, and further improve the state estimate accuracy.

Up to now, little attention has been paid to the state estimate for the stochastic systems with dual-UI, which mainly focus on the robust filter. For example, a mixed H2and H∞filter has been designed for the discrete-time systems affected by unknown finite energy disturbances and white noise disturbances.24A novel distributed filter was constructed to practically reflect the impact from both cyber-attacks and gain perturbations.25A secure filtering problem was investigated for a class of uncertain stochastic non-linear systems,in which multiple channel attack model was established to account for the missing measurements and random deception attacks.26A distributed set-membership filtering for a class of general discrete-time nonlinear systems was designed based on the Lagrange remainder and the event-based media-access condition to improve the filter performance.27Most of them devote to the state estimate with unknown but bounded UIs (or randomly UIs)in the worst cases,which are strictly dependent on the constraints of the linear matrix inequality, and can hardly achieve the optimal estimation results. Besides, few adaptive filters or optimal filters in the unbiased minimum variance sense have been proposed.For example,a multi-model estimator was designed specifically for a system containing independent unknown or randomly switching inputs and measurement biases.But a series of appropriate model sets for the unknown or randomly switching inputs have been set in advance, which brings in the huge computation burden.Once the model sets of the UIs deviate from the true physical system,it will lead to the filter degradation.28A two-stage estimator was designed for the systems with unknown inputs and constant biases,29which extended the two-stage Kalman filter5based on the state augmentation. Although this method can obtain the unbiased minimum variance estimation, the dual-UI are limited to the constant value.

Based on the above discussion, the state estimation problem for stochastic systems with dual-UI is considered in this paper.This paper first shows the S-UI term decoupling condition, producing the decoupled system with adjacent M-UIs appearing in the state model and the measurement model.Through introducing the M-UID term, the equivalent system with only M-UID in the dynamic model is then obtained.Finally, the decoupling filter in the minimum mean square error sense and the adaptive minimum upper filter with different applicable conditions are represented to obtain the optimal and sub-optimal state estimate, respectively. Two simulation cases verified the effectiveness and superiority compared with the traditional Kalman filter method,the least square method,the three-stage Kalman filter and the robust Kalman filters.The rest of this paper is organized as follows. Section 2 presents the problem formulation. The globally optimal filter and sub-optimal filter for the discrete-time stochastic system in the presence of dual-UI are presented in Section 3. Numerical simulations for performance comparison are presented in Section 4, and a conclusion is drawn in Section 5.

2. Problem formulation

Consider the following discrete-time linear stochastic system with dual-UI:

where xk∈Rnis the system state vector,yk∈Rmis the measurement vector,μk∈Rlis the control input,dk∈Rsis the unknown disturbance input in the state model, and bk∈Rois the unknown disturbance input in the measurement model.Ek∈Rn×sand Dk∈Rm×oare S-UI and M-UI coefficient matrices of full column rank, respectively. Ak∈Rn×n, Bk∈Rn×l,Ck∈Rm×nare the known state matrix, control input matrix,and measurement matrix with proper dimensions,respectively.The process noise ζk∈Rnand measurement noise ηk∈Rmare zero mean white noise with known covariance Qkand Rk,respectively. The initial state x0is Gaussian distribution with known mean x-0and associated covariance P0|0.Gaussian noise ζk,ηkand the initial state x0are independent mutually.It should be noted that the introduced S-UI and M-UI terms satisfy

The term Ekdkcan be used to describe not only additive disturbance, but also a number of different kinds of modelling uncertainties, for example, interconnecting term in the large scale systems, nonlinear terms in system dynamics, linearization and model reduction errors, and parameter variations.30-31Meanwhile, the term Dkbkcan represent the sensor measurement bias, communication interference, and deception jamming.21-22Obviously, the S-UI and M-UI terms stem from different physical sources, which should be individually defined as different items in the dynamic system model.

The S-UI and M-UI terms are deeply coupled and affect each other during the state estimate process. The accurate measurement value is the precondition of state estimation.Once the measurement value is interfered by the M-UI without appropriate handling, the measurement update process will definitely be affected and the state estimate result will be biased. Besides, the precise state estimate after S-UI decoupling is helpful for the measurement prediction to estimate and compensate the M-DI term further. Once the S-UI term cannot be handled well, the inaccurate state prediction makes the measurement prediction deviate from the true measurement value and results in worse M-UI compensation.The measurement prediction of the next time instant will be implicated recursively, leading to the divergence of the estimate result. In conclusion, the deeply coupled relationship between the S-UI and M-UI term makes the state estimate problem for the stochastic system with the dual-UI be a new challenge and difficulty (somewhat like a double-edged sword).

The existing methods are not applicable to this challenging problem.First,it is not feasible to directly design the robust H2or H∞filters for the stochastic systems with dual-UI, for the strong uncertainty and the conservation design of the inequality derivation may severely reduce the filtering performance.Second, applying the Multiple Model (MM) algorithms to handle the dual-UI via establishing possible model set is undesirable,for the prior information or themodel dictionary of the dual-UI is required. Finally, utilizing the two-stage or threestage Kalman filter technique is neither adoptable, for the assumption of the dual-UI as a stochastic process with given wide-sense presentation is prerequisite. Based on the aforementioned considerations, directly performing the state estimate for the stochastic system with dual-UI is rather difficult and unsolvable, and thus a feasible and promising solution is to divide this uroborus problem into two phases. Decouple one term of the dual-UI from the stochastic system first, eliminating the influence of the deep coupling on each other and reducing the uncertainty of the stochastic system, and then construct the appropriate filter to give the state estimate result for the decoupled system.

3. State estimate of dual-UI stochastic system

To estimate the unknown state of the stochastic system with dual-UI described in Eqs.(1)and(2),the augmented state Kalman filter is the straightforward technique,which can satisfactorily handle the interaction between the S-UI and M-UI terms through treating xk,bkand dktogether as the augmented state.However,due to the strong uncertainty(i.e.,dual-UI),the computation burden and the numerical errors of the augmented state Kalman filter increase drastically with the augmented state dimension. Therefore, the proposed approach in this paper first decouples one term of the dual-UI from the stochastic system, reducing the system uncertainty, which is followed by the state augmented filter design. Based on the S-UI term decoupling,the decoupled system only affected by the adjacent M-UI terms is derived. Through introducing the differential term of the adjacent M-UI terms and augmenting the state vector with the M-DI term as the new state,the equivalent system with only M-UID in the dynamic model is then constructed,in which the remaining design freedom can be utilized to obtain the state estimation with the required performance. Finally,two different applicable conditions to design the globally optimal decoupling filter in the minimum mean square error sense and the sub-optimal adaptive minimum upper bounded filters for the equivalent system are presented. The block diagram of the proposed scheme is shown in Fig. 1.

3.1. Equivalent system derivation based on S-UI decoupling

Before we give the necessary and sufficient conditions to decouple the S-UI term,the following Lemma 1 is introduced.

Fig. 1 Block diagram of proposed scheme.

3.2. M-UID decoupled MMSE filter design for equivalent system

In order to obtain the state estimate of the equivalent system interfered by the unknown disturbance term M-UID, the straightforward solution is to further decouple the M-UID term to obtain the decoupled system free from the unknown inputs, and then the unknown disturbance decoupled filter can be designed. Similar with Theorem 1, the sufficient and necessary condition for decoupling the M-UID term is given,which is followed by the state estimate filter in the minimum mean square error sense.

Proof. See Appendix C.

The main advantage of the M-UID Decoupled Minimum Mean Square Error (MMSE) filter based on S-UI Decoupling(DD-MMSE) is that its performance is not affected by the SUI term and the M-UI term,especially when the dual-UI have unknown statistics.It should be noted that this is also the original intention of our design. Besides, the M-UID decoupling based MMSE method is globally optimal.However,the applicable conditions for the M-UID decoupling based MMSE filter are somewhat strict, for the equivalent system after S-UI decoupling may not have the enough freedom to guarantee the filter performance. Thus, in the next section, the suboptimal filter design with relaxed applicable conditions is shown.

3.3. Minimum upper bound filter design for equivalent system

Apart from the M-UID decoupling based MMSE technique for the equivalent system Eq. (4), once again augmenting the state Xkand the M-UID term b～kto perform filter design is another alternative way. However, the augmented state is unsolvable from the collected measurements up to time k,for Ckis not full rank. Besides, the correlation between the M-UID term b～kand the M-UI term bkin the state vector Xkmakes the filter design more complicated due to the existence of the cross-covariance matrix.In this section,the robust state estimate technique is given through constructing the upper bounds of filter performance and minimizing such bounds in pursuit of the desirable filtering parameters to obtain the Minimum Upper Bound Filter, which is promising to realize the robust state estimate for the equivalent system after the S-UI term Decoupling (D-MUBF).

Consider the general linear filter for the equivalent system shown in Eq. (4),

Fig. 2 Flowchart of proposed algorithm.

Table 1 Methods performance comparison (Case 1).

The adjustment factor αk+1is introduced, which is determined so that the filter can produce an unbiased minimum variance estimate. Thus the following recursive upper bound requires that the equivalent system Eq.(4)should provide enough state estimate information for M-UID compensation. The third constraint of Theorem 3 means that the fading factor αkshould be large enough to guarantee the existence of Eq. (21).

The D-MUBF is a robust filter because it pursuits the best solution in the‘‘worst”possible case.At the same time,the DMUBF method is obviously not globally optimal given such conservation design of the inequality derivation. However,the D-MUBF method needs more relaxed conditions for the filter design compared with the M-UID decoupling based MMSE filter,because the M-UID term in the D-MUBF design is compensated rather than decoupled in the M-UID decoupling based MMSE method, which does not lose freedom information.

3.4. Complexity analysis of proposed algorithm

Algorithm 1. Pseudo codes of proposed algorithm

4. Simulation experiments

In the simulation, the aerodynamic coefficients perturbation is 50%,i.e.,Δalj=-0.5alj，Δbj=0.5bj.The sensor parameter perturbations are chosen on the basis of measurement accuracy,and the segmented step fault signal is chosen without loss of generality. To be specific, the measurement perturbation is set as Δclj=0.01, and the sensor fault subject to step signal is considered in the measurement model with

Fig. 3 State estimate results (Case 1).

In order to show the performance of the DD-MMSE method and the D-MUBF method, two cases are given in the simulation section. In Case 1, only the conditions in Theorem 1 and Theorem 3 are satisfied. In Case 2, besides Theorem 1 and Theorem 3, the conditions in Theorem 2 are also satisfied.Once the corresponding applicable conditions are satisfied, the DD-MMSE filter is optimal, while the D-MUBF method is sub-optimal. In the two cases, the state estimate RMSE of the proposed DD-MMSE method and the DMUBF method based on 100 Monte-Carlo simulations are compared with the following five methods: the Least Square(LS) method, the Kalman Filter (KF) method, the threestage KF,the robust three-stage KF and the Variational Bayesian based Adaptive Kalman Filter(VBAKF).The three-stage KF method is essentially the state augmented method, which decouple the augmented Kalman filter equations into state,fault and UI three sub-filters via UV transformation. On the precondition that the dynamical evolutions of the fault and the unknown inputs are available, the three-stage KF method is optimal.10The robust three-stage KF method is robust against the fault and UI through updating the fault and UI subfilter just with the measurement information and omitting the history estimation information.10The VBAKF method has satisfying robustness to resist the inaccurate and slowly varying process and measurement noise covariance matrix through choosing inverse Wishart priors for the predicted error covariance matrix and measurement noise covariance matrix.37

4.1. Case 1

In this subsection, we consider that the numerical system only satisfies the conditions in Theorem 1 and Theorem 3. An example of the state disturbance matrix and the measurement disturbance matrix are given as

In Fig.3,the state estimate results of the D-MUBF method and the DD-MMSE filter together with the true state and the sensor measurement are shown. It is clearly that the state estimate of the DD-MMSE method is disappointing, for the remaining freedom of the equivalent system after S-UI decoupling is decreased and cannot guarantee further decoupling the M-UID term.However,the D-MUBF method obtains the satisfying state estimate result of the dual-UI interfered system,which performs robust state estimate via the M-UID term compensation rather than decoupling using the remaining freedom.

Fig. 4 State estimate RMSE comparison (Case 1).

In Fig. 4 the state estimate RMSE compared with the LS method, the KF method, the three-stage KF, the robust three-stage KF and the VBAKF method are given. It can be seen that the LS method only utilizes the whole measurement data to obtain the state estimate result, which can obtain the optimal estimate when the measurement is free from the M-UI term as shown in Fig.4(c).Once the measurement suffers from the M-UI term, the estimate result is catastrophic as shown in Figs.4(a)and 4(b).Even though the KF method is the optimal linear minimum mean square error estimator, whose performance decreases greatly when the modeling errors caused by the parameter perturbations in dynamic model and the measurement model are ignored. Considering the precondition that the dynamical evolutions of the S-UI and M-UI terms are available, the three-stage KF method here assumes that the S-UI and M-UI terms are constant values,which obviously violates the actual situation, and thus the estimate result is frustrating. The robust three-stage KF method simplistically performs the measurement update for the S-UI and M-UI terms estimate and omits the history evolution information,which relatively causes that the state estimate result is disastrous, once the M-UI term exists as shown in Figs. 4(a) and(b).The VBAKF method can obtain the robust estimate result through treating the unknown S-UI and M-UI terms as the uncertainty in the process noise and measurement noise covariance matrix; however the estimate result is rather disappointment once the S-UI and M-UI terms significantly vary as shown in Fig. 4(a). The DD-MMSE method fails to realize the M-UID decoupling and the corresponding state estimate result is not satisfied, for the decoupling condition in Theorem 2 is not satisfied.Even the D-MUBF method has an overshoot once the measurement changes, the state estimate result immediately converges after two simulation times. Table 1 shows the estimate performance and the calculation cost of the seven candidate methods. Compared with the other six methods, the D-MUBF method obtains the best state estimation performance at the cost of the increased calculation time to some extent, which is mainly caused by solving the optimization result of the desirable filter parameters. The calculation cost of the DD-MMSE method is much lower than that of the D-MUBF method, and almost the same as the two robust filters.

4.2. Case 2

In this subsection, we consider that the numerical system satisfies all the conditions in Theorem 1, Theorem 2 and Theorem 3. An example of the state disturbance matrix and the measurement disturbance matrix is given as

Fig. 5 State estimate result (Case 2).

Fig. 6 State estimate RMSE comparison (Case 2).

The state estimate results of the D-MUBF method and the DD-MMSE filter together with the true state and the sensor measurement are shown in Fig. 5. It can be seen that both the DD-MMSE method and the D-MUBF method can realize the satisfying state estimate with dual-UI, while the state estimate result of the DD-MMSE filter is much superior than the D-MUBF method. Fig. 6 represent the state estimate RMSE of the seven comparison methods, and the methods performance comparison is given in Table 2. The three-stage KF and the robust three-stage KF methods cannot obtain the satisfying estimate results in dimension 2 and dimension 3 as shown in Fig. 6(b) and (c), for the true measurement values are interfered by the M-UI term.The estimate result in dimension 1 of the VBAKF method is rather disappointment for it cannot handle the significantly varying issue of the S-UI term.It can be seen that the state estimate RMSE in dimension 1 of the DD-MMSE filter equals that of the LS method. For the measurement of dimension 1 is free of M-UI, the LS method can obtain the optimal state estimate result,which also reflects that the state estimate result of the DD-MMSE method is globally optimal.It can be concluded that under the conditions of Theorem 2, the DD-MMSE filter can obtain the best estimate performance compared with the sub-optimal D-MUBF method.For the batch processing,the LS method has the least calculation cost.For the real-time iterative processing,the calculation cost of the KF method is also satisfying.The calculation cost of the DD-MMSE filter is much lower than that of the D-MUBF method, and almost approximates with that of the robust filters. The calculation burden of the three-stage KF method seriously depends on the QR factorization, which is disappointing. In conclusion, the DD-MMSE filter is favorable both in the estimate performance and the calculation cost,once the applicable conditions are satisfied.

5. Conclusions

In this paper, state estimate of the dynamic discrete system in which the dynamic state model and the measurement model contain different unknown interference inputs is concerned here. This paper first shows the S-UI term decoupling condition, producing the decoupled system with adjacent M-UIs appearing in the state model and the measurement model.Through introducing the differential term of the adjacent measurement concerned unknown inputs, the equivalent system with only M-UID in the dynamic model is then obtained.Finally, the decoupling filter in the minimum mean square error sense and the adaptive minimum upper filter with different applicable conditions are represented to obtain the optimal and sub-optimal state estimate, respectively. Two simulation cases verify the effectiveness and superiority compared with the traditional Kalman filter method,the least square method,the three-stage Kalman filter and the robust Kalman filters.In the future,we will study the state estimation for the stochastic system with the dual-UI coupled with the unknown noise covariance, for the true noise is always extremely complex and unknown to us.Besides,we will focus on the more relaxed decoupling condition to make the proposed methods more applicable.

Table 2 Methods performance comparison (Case 2).

Acknowledgements

This work has been supported by the National Natural Science Foundation of China(Nos. 61603040 and 61433003),Yunnan Applied Basic Research Project of China (No.201701CF00037), Guangdong Province Science and Technology Innovation Strategy Special Fund Project,China(No.skjtdzxrwqd2018001) and Yunnan Provincial Science and Technology Department Key Research Program (Engineering), China (No. 2018BA070).

Appendix A. Proof of Lemma 1

Appendix B. Proof of Theorem 1