Kalman and UFIR state estimation with coloured measurement noise using backward Euler method

IET Signal ProcessingVolume 14, Issue 2 p. 64-71 Research ArticleFree Access Kalman and UFIR state estimation with coloured measurement noise using backward Euler method Yuriy S. Shmaliy, Corresponding Author Yuriy S. Shmaliy shmaliy@ugto.mx Department of Electronics Engineering, Universidad de Guanajuato, Salamanca, 36885 Gto, MexicoSearch for more papers by this authorShunyi Zhao, Shunyi Zhao orcid.org/0000-0001-5363-8305 Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi, People's Republic of ChinaSearch for more papers by this authorChoon K. Ahn, Choon K. Ahn orcid.org/0000-0003-0993-9658 School of Electrical Engineering, Korea University, Seoul, 136-701 Republic of KoreaSearch for more papers by this author Yuriy S. Shmaliy, Corresponding Author Yuriy S. Shmaliy shmaliy@ugto.mx Department of Electronics Engineering, Universidad de Guanajuato, Salamanca, 36885 Gto, MexicoSearch for more papers by this authorShunyi Zhao, Shunyi Zhao orcid.org/0000-0001-5363-8305 Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi, People's Republic of ChinaSearch for more papers by this authorChoon K. Ahn, Choon K. Ahn orcid.org/0000-0003-0993-9658 School of Electrical Engineering, Korea University, Seoul, 136-701 Republic of KoreaSearch for more papers by this author First published: 01 April 2020 https://doi.org/10.1049/iet-spr.2019.0166Citations: 17AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Under coloured noise, known modifications of the Kalman filter (KF) exist only for discrete-time state-space models produced by the forward Euler (FE) method, which fits with feedback control. In this study, the authors modify the KF and unbiased finite impulse response (UFIR) filter using the backward Euler (BE) method for models with coloured measurement noise (CMN), which better fits systems without feedback. The FE- and BE-based models differ by time indexes in the system input and noise that is essential for time-varying and Markov jump systems. Employing measurement differencing, two KF algorithms and a unique UFIR algorithm are derived for time-correlated and de-correlated noise. An equivalence of the KF algorithms is proved analytically and confirmed by simulations. Numerical examples are given for target tracking and experimental verification is provided for visual object tracking. The high efficiency of the designed algorithms in removing CMN is demonstrated experimentally. 1 Introduction In electronic systems, white noise becomes coloured (correlated) after passed through bandlimited channels [1]. In narrowband receivers, such a noise is known as narrowband, whose statistics are well studied in the communication theory [2]. If the noise has a very short correlation time compared to time intervals of interest, it is considered as white and its colouredness are typically neglected. Otherwise, the system may experience a great noise interference [3]. Coloured noise may be injected by a local oscillator, whose phase noise has a multicolour nature with the power spectral density slope of , , where f is the Fourier frequency. Low precision and low-cost oscillators require phase locking to a more precise (master) oscillator using a phase-locking loop (PLL). That allows removing regular drifts, such as random works, but makes a local oscillator more coloured. The relevant mechanism is well explained in [3] for tracking systems, which measure range by comparing phase delays between transmitted and received signals. If the PLL bandwidth is small, it suppresses measurement noise. However, this causes a loop delay and errors in successive range samples, which are correlated when the sampling interval is shorter than the PLL time constant. Similar problems arise in visual tracking [4], when a video camera catches an object and follows it. In Fig. 1, we give examples of coloured measurement noise (CMN) (Figs. 1 a and b) and its simulation as a Markov process (Fig. 1 c). The first example (Fig. 1 a) taken from [5] demonstrates the signal strength CMN discovered in the Global Positioning System (GPS) receiver. The CMN was determined here by removing the antenna gain attenuation from the signal strength measurements and then normalising to zero about the mean value. The second example taken from [6] shows the positioning errors caused by the carrier phase multipath errors with a sinusoidal signature. Performance assessment of the noise reduction method was carried out here by comparing the multipath-filtered positioning errors with the original estimated positioning errors. Similar pictures of GPS noise can be found in other works, for example in [7]. Fig. 1Open in figure viewerPowerPoint Examples of CMN in electronic channels (a) Signal strength CMN in a GPS receiver [5], (b) GPS positioning noise due to carrier phase multipath errors [6], and (c) Markov noise simulated by for A simulated coloured noise shown in Fig. 1 c was assumed to originate from a Markov sequence with a white Gaussian driving source . Even a quick look at Fig. 1 allows thinking that all three processes most likely belong to the same class and thus, the CMN may have the Markov origin. Such a supposition was originally made by Bryson and Johansen in [8] for continuous-time and in [9] for discrete-time tracking systems. Two fundamental approaches were also proposed in [9] to modify the Kalman filter (KF) for CMN: state augmentation and measurement differencing. The first approach implies reassigning the state vector in a way such that the white sources are combined in an augmented system noise vector. This allows using the KF, but makes it ill-conditioned, because the measurement noise covariance matrix becomes singular due to zero noise in the augmented observation equation. Moreover, the increased size state vector may make an estimator unstable for certain types of system matrices. Even so, this approach was recognised in [3, 10-12] as a basic solution to address the problem caused by CMN. Issues arisen from matrix augmentation were analysed in [13] and it was shown that divergence in the KF could be avoided using Tikhonov regularisation. The measurement differencing approach implies assigning a new observation vector as a difference between the observation at a discrete-time index n and a weighted observation at . That transforms the Markov noise to white noise, which, however becomes time-correlated with the process noise. To circumvent this issue, two options were suggested: (i) rederive the Kalman gain and error covariance or (ii) de-correlate noise. A solution to the first option was originally proposed by Bryson and Henrikson in [9] and is now referred to as the Bryson algorithm. The Bryson algorithm makes the KF optimal for CMN, but appears to be redundant as requiring two estimation steps (smoothing and filtering) owing to an assumption that the prior estimate is affected by data, unlike in the standard KF. Referring to this flaw and representing past estimates via the present one to avoid smoothing, Petovello et al. have found in [14] another solution, which is now known as the Petovello algorithm. The optimal Kalman gain and error covariance were used here as derived in [15] for correlated noise. Later, Chang has shown in [16] that the Bryson, Pelovello, and yet another alternative Petovello algorithms are equivalent to time-invariant noise covariances. Investigations and applications of the above algorithms can be found in [16-28]. The second approach employs a de-correlation method described in [29], which suggests combining the state equation with a weighted difference between data and the observation model and deriving conditions under which noise becomes uncorrelated. We find relevant derivations in [18] for non-linear state-space models. For linear models, the results were obtained by Chang in [30], where it was also shown that the Gaussian approximation approach [31] gives equivalent results. Two critical observations can now be made Known versions of the KF for coloured and correlated noises [9, 12, 14, 16, 27, 29] were obtained using the forward Euler (FE) method associated with Itô calculus, which yields the prediction state equation basic in feedback control where is an input signal. For systems without feedback, a better fit is the backward Euler (BE) method [32, 33] yielding the real-time state equation which suits FIR filtering [34] as a causal equivalent (or innovations sequence) to [33]. Estimates for the FE- and BE-based models become equal when a system has no input and noise has a time-invariant covariance as considered by Bryson and Henrikson [9] and analysed in [16]. However, they differ otherwise that matters for time-varying and Markov jump systems. Because the FE- and BE-based models are generally not convertible, a derivation of BE-based state estimators is required for coloured noise. The problem of coloured noise can be viewed as robust against errors in the noise statistics. The unbiased finite impulse response (UFIR) filter [35] completely ignores zero-mean noise and maybe a better choice than the KF [36, 37] for models with coloured and correlated noises. Referring to the above, in this paper, we consider a BE-based time-varying model with CMN and derive algorithms for two different recursive schemes: optimal (KF) and unbiased (UFIR). We also prove the equivalence of the KF algorithms. By simulations and experiments, we show that the UFIR filter generally outperforms the KF under errors in the noise statistics and incomplete information about CMN. The main contributions are as follows: KF and UFIR filter algorithms for BE-based time-varying models with CMN. Note that only the KF was earlier modified mostly for time-invariant FE-based models with CMN. A proof of the equivalence of the KF algorithms for correlated and de-correlated noise sources caused by Markov–Gauss CMN. Examples of applications showing advantages of the modified algorithms against the standard ones under CMN. The rest of the paper is organised as follows. In Section 2, we discuss possible models. In Section 3, we derive the KF algorithms for time-correlated and de-correlated noise sources caused by CMN and prove their equivalence. In Section 4, we modify the UFIR algorithm for CMN. A numerical example of tracking is provided in Section 5, applications to visual object tracking are given in Section 6, and conclusions are drawn in Section 7. 2 State-space model A linear stochastic process with observation corrupted by CMN can be represented in discrete-time state-space using the BE method as (1) (2) (3)where is the state vector, is the observation vector, is the coloured Gauss–Markov noise, and time-varying matrices , , , and are supposed to be known. For the sake of simplicity, the input signal is excluded. Matrix is chosen such that the coloured noise remains stationary. Noise vectors and are supposed to be zero mean, and , and white Gaussian with the covariances and and the property for all n and k. We will be interested of the prior estimate , posterior estimate , prior estimation error , posterior estimation error , prior error covariance , and posterior error covariance . Since state estimation under CMN can be provided using either the augmented state or measurement differencing [9], we will next modify model (1)–(3) for both these cases. 2.1 State augmentation Following [9], one can unite (1) and (2) in an augmented state (4) (5)and rewrite (4) and (5) as (6) (7)where has a zero covariance and the process noise has the covariance (8)The KF and UFIR filter [36] can now be applied to (6) and (7) straightforwardly. However, a singular covariance in (7) makes the KF ill-conditioned [9]. It relates, for example to the information KF [12], which requires the inverse of this covariance. In turn, the UFIR filter becomes ill-conditioned by augmented matrices. 2.2 Measurement differencing To avoid state augmentation and CMN in , one can consider a new observation as a measurement difference [9], write (9)take and from (1), obtain (10)where (11) (12) (13)and notice that noise is white with the properties (14) (15)where . The modified model (1) and (10) has thus white and time-correlated and . 3 Kalman algorithms Referring to a new observation (10), one can consider two possible solutions requiring either (i) a new Kalman gain and error covariance for correlated noise or (ii) noise de-correlation. Although and are time-correlated, the correlation of two white Gaussian sources does not produce bias, and the prior estimate can thus be set as with the prior error covariance (16)For (10), the measurement residual becomes (17)and the innovation covariance can be found to be (18)The recursive KF estimate can now be written as (19)in which is the bias correction gain that should be optimised for correlated and or assigned as the Kalman gain for de-correlated and . Below, we will consider both these cases. 3.1 Correlated and For correlated and , the KF can be applied if to derive a new optimal Kalman gain by minimising the mean square error (MSE) represented with the trace of the error covariance [15]. In doing so, write the estimation error as (20)observe that is uncorrelated with both and , while , and are correlated as shown in (15), and find as (21a) (21b)where is given by (16) and by (18). Now, find the optimal gain by minimising . Equating to zero the derivative of with respect to as (22)gives (23)Next, substituting the first component in the last term in (21b) with (23) yields (24)A pseudo-code of the KF algorithm for the BE-based model with CMN and time-correlated and is listed in Fig. 2. The algorithm becomes the standard KF by and the reader can readily trace the difference with the Petovello's algorithm [14] derived for the FE-based model. Fig. 2Open in figure viewerPowerPoint Pseudo-code of the KF for BE-based model with CMN and correlated wn and 3.2 De-correlated and Efforts can also be made to make and uncorrelated and then apply the standard KF. To this end, we refer to [29] and combine taken from (1) with a term , where is a data vector and still has to be determined. That yields (25)where (26) (27) (28)and noise has the covariance defined by (29)Matrix making and uncorrelated can now be found by transforming the desired property as (30)that yields (31)Substituting taken from (30) into the last term in (29) removes and (29) becomes (32)The standard KF can now be applied to (25) and (10) with a pseudo-code shown in Fig. 3, where each internal variable is depicted with a bar to indicate a difference with the algorithm shown in Fig. 2. As can be seen, the algorithm (Fig. 3) requires a modified system noise covariance given by (32) and an input , where is given by (31). Note that the approach resulted in this algorithm (Fig. 3) was earlier applied in [29] to the FE-based model with time-constant noise covariances. Fig. 3Open in figure viewerPowerPoint Pseudo-code of the KF for BE-based model with CMN and correlated and 3.3 Equivalence of KF algorithms for CMN An equivalence of the algorithms developed by Bryson, Petovello, and Chang for CMN was shown in [16] assuming time-invariant noise covariances, which make no difference between the FE- and BE-based models. An equivalence of the algorithms derived for CMN with correlated and de-correlated noise was yet not addressed. The algorithms (Figs. 2 and 3) will be equivalent in the MSE sense if, for the same input data they produce the same estimates that are stated with the following lemma. Lemma 1.Given model (1)–(3) with Gauss–Markov CMN, then the algorithms (Figs. 2 and 3) are equivalent. Proof.Represent the prior estimate via and the prior error covariance via for as (33) (34)Equate the estimates taken from Figs. 2 and 3substitute taken from Fig. 3, combined with (26) and taken from Fig. 2, transform as skip the non-zero equal terms from both sides, and arrive at a relationship between the bias correction gains (35)which is the main condition for the estimates to be identical. If (35) also makes the error covariances identical, then the algorithms (Figs. 2 and 3) can be said to be equivalent.Then equate the error covariances taken from Figs. 2 and 3 as (36)substitute (35) and (34), write introduce , and come up with Rearrange the terms and transform as (37)substitute taken from Fig. 3 and go from (37) to After tedious manipulations with matrices arrive at substitute taken from , provide more transformations, and end up with (38)Because the matrix is symmetric, thus , and by (11), relation (38) becomes equality , which means that the filters produce identical errors and the proof is complete. □ 4 Unique UFIR filtering algorithm Unlike the KF, the UFIR filter [36] does not require any information about noise, except for the zero-mean assumption. Instead, it requires an optimal averaging horizon of points, from to n, to minimise the MSE. The UFIR filter cannot ignore the CMN , which violates the zero mean assumption on short horizons. However, it appears to be unique for both correlated and de-correlated and . Let us show it. Consider (25) and (10), neglect noise, combine with (26) and (27), and transform as that leads to an identity , from which we have and conclude that this relation holds for correlated and uncorrelated and . A pseudo-code of the unique UFIR algorithm for the BE-based model with CMN is listed in Fig. 4. To initialise iterations avoiding singularities, the algorithm requires a short measurement vector and matrix (39)Let us notice that is singular if and thus the generalised noise power gain (GNPG) cannot be computed on horizons shorter than K points. It also follows that, by , the algorithm (Fig. 4) becomes the standard UFIR filter [36]. Fig. 4Open in figure viewerPowerPoint Pseudo-code of the unique UFIR filtering algorithm for the BE-based model with CMN 4.1 Error covariance The error covariance for the UFIR filter can be specified by (21a) if to substitute with and recall that is symmetric [36] (40a) (40b)where is given by (16), by (18), and is computed by line 8 in the algorithm (Fig. 4). Note that recursion (40b) can be included in the algorithm (Fig. 4), although the UFIR filter does not require , unlike the KF. 4.2 Optimal horizon To optimise the UFIR estimate, efforts must be made to specify in the MSE sense. There are several ways of how to do it [38]. Of importance is that the UFIR filter is low sensitive to N: it produces acceptable errors when N varies up to 30% around [36]. Another observation is that the optimised UFIR estimate ranges very close to the KF estimate, especially when [35]. One thus may approximately characterise errors in the UFIR estimate with (22) of the KF, set , and obtain (41)where is given by (16). From line 8 in the algorithm (Fig. 4), one can also provide the backward recursion (42)Because an increase in N always results in the decreases in the trace of [36], can be computed as follows. Start with (41), compute (42) repeatedly backward by decreasing n, and calculate the number of reiterations k until becomes finally singular. Because the GNPG matrix is singular when [36], accept . This approach can serve if or there are enough data points to find . Otherwise, set at the beginning estimation stage when and is not singular. 5 Numerical examples of tracking To investigate the trade-off between the standard Kalman and UFIR filters, both ignoring noise colouredness, and the modified algorithms (Figs. 2-4) under CMN, in this section, we consider a target tracking problem for two scenarios: (i) two-state model on a short horizon and (ii) three-state model on a large horizon . We suppose that tracking is organised using distance measurement with a time step and simulate the CMN with for . 5.1 Two-state model In the two-state case, a target is disturbed by white Gaussian acceleration noise with the standard deviation of and the CMN originates from white Gaussian with . A trajectory is simulated at 2000 points for , , , , and Following [38], we determine for white noise and then update it experimentally for each at a test stage. An experiment has revealed that grows almost linearly from 20 to 30 when rises up to 0.6; it then grows faster and non-linearly to reach 100 when . We use function determined in such a way in the UFIR algorithm (Fig. 4). Note that an increase in caused by an increase in is due to the fact that larger horizons are required by a UFIR estimator to filter out non-white noise [35]. Typical root MSEs (RMSEs) produced by the KF and UFIR algorithms as functions of and averaged over multiple Monte Carlo (MC) runs are sketched in Fig. 5. The first point to notice is that the modified KF algorithms produce equal estimates and are thus equivalent, as stated by Lemma 1. The augmented state UFIR filter was out of normal operation at some points in view of singularities in the matrix (33). However, the iterative UFIR algorithm (Fig. 4) always produces stable estimates. Fig. 5Open in figure viewerPowerPoint Typical RMSEs produced by the KF and UFIR filter () as functions of the colouredness factor for the two-state tracking model under (a) Ideal conditions, , and (b) Errors in the noise statistics, and Errors sketched in Fig. 5 a fit with an ideal case of exactly known noise statistics. As expected, the standard KF has here the highest accuracy when ranges close to zero, . Otherwise, errors in the KF grow more rapidly than in the standard UFIR filter. It is also seen that the KF algorithms (Figs. 2 and 3) perform better when and that the UFIR algorithm (Fig. 3) is more accurate when . That means that a more robust UFIR filter [37] can be more preferable when noise is extremely coloured. In Fig. 5 b, we sketch filtering errors produced under incomplete information about noise, which is simulated with and , , set in the algorithms. As can be seen, even moderate error factors ( and ) make the KF less accurate than the UFIR filter. 5.2 Three-State model We next consider a three-state case, by allowing and and simulating a trajectory at 2000 points for , , and This case required a much larger horizon at and grew near linearly up to 150 when and increases rapidly to about 280 with further increase in up to 0.95. Typical filtering RMSEs are sketched in Fig. 6. Despite a visual similarity with Fig. 5, here we observe several important differences caused by a much larger . When and (Fig. 6 a), the standard UFIR filter produced much smaller errors than the standard KF. The modified filters improve the performance in a way such that the difference between the estimates becomes insignificant (Fig. 6 a). It is also seen (Fig. 6 b) that the UFIR filter, which ignores zero-mean noise, is, in fact as accurate on large as the KF, which requires all information about noise that is not always available. Again, as expected, errors in the noise statistics make the KF less accurate, that is neatly shown in Fig. 6 b. Fig. 6Open in figure viewerPowerPoint Typical RMSEs produced by the KF and UFIR filter () as functions of the colouredness factor for the three-state tracking model under (a) Ideal conditions, , and (b) Errors in the noise statistics, and 6 Visual target tracking As an example of practical applications, in this section, we consider a benchmark trajectory ‘Car4,’ which coordinates and car frames are measured by the following car using a visual tracking system [39]. Data are available from [40]. The car moves and manoeuvres smoothly on the road in an urban area. Measurements made along coordinate x are shown in Fig. 7 in grey with the brightly pronounced CMN mostly caused the camera tracking ability. In the presence of CMN, fast localisation of a temporary lost object is typically problematic and effect of CMN should be removed. In doing so, we exploit the above-discussed two-state model. Because not much information is provided in [40], we tune filters under some assumptions. Fig. 7Open in figure viewerPowerPoint Visual tracking of a moving car (‘Car4’) along coordinate x using (a) Standard KF and UFIR filter () and (b) Modified KFs (Figs. 2 and 3) and UFIR filter (Fig. 4) with We think that tracking is organised with 20 frames per second or . We also suppose that the car acceleration noise is not large, set , and accept for the measurement noise. Both these values were next examined experimentally for the standard KF to produce visually best denoising with no essential bias. The standard KF tuned in such a way followed the slow (coloured) parts of the trajectory, as shown in Fig. 7 a. We then find for the UFIR filter output to range close to the standard KF output and infer that, as expected, neither of the standard filters is able to remove the CMN (Fig. 7 a). We finally run the modified KF algorithms, realise experimentally that factor fits with the measured noise in the best way (best denoising), and notice again that both modified KF algorithms (Figs. 2 and 3) produce identical estimates, as stated by Lemma 1. A subsequent minimisation of the difference between the UFIR and KF outputs gave , which was set to the UFIR algorithm (Fig. 4). The results sketched in Fig. 7 b demonstrate a definitive advantage of the modified algorithms in CMN reduction. In fact, these filters track the trajectory much more precisely by filtering out ‘slow’ coloured noise. It can also be seen in Fig. 7 b that, like in simulations (Figs. 5 and 6), the difference between the KF and UFIR estimates is poorly distinguishable. 7 Conclusions The Kalman and UFIR filtering algorithms modified in this paper for the BE-based model with CMN differ from the earlier derived by Bryson, Petovello, and Chang for the FE-based models that are essential for time-varying and Markov jump systems. The KF algorithms derived using measurement differencing for correlated and de-correlated noise were shown, both analytically and numerically, to be equivalent with no essential advantage against one another. In turn, the UFIR algorithm has appeared to be unique for correlated and de-correlated noise. Note that the equivalence of the Bryson and Petovello algorithms was shown by Chang only for time-invariant noise covariances, which make the FE- and BE-based models equal. Examples of simulation of the two- and three-state tracking problems have confirmed the theoretical inferences. Furthermore, an experimental example of visual object tracking based on the ‘Car4’ benchmark contaminated by intensive CMN has fitted well with the simulation results. Overall, properly tuned the modified KF and UFIR algorithms have proved the ability to suppress CMN efficiently and provide state estimation with much higher accuracy than by the standard filters that can be useful in many applications. We finally notice that state-space modelling with non-white noise can be provided using the pairwise Markov model [41] or triplet Markov model introduced in [42, 43] and applied to KF as (14, in [44]). These models may give a better insight into processes with coloured and correlated noise. We work on such models now and plan to report the results in near future. 8 References 1Van Trees, H.L.: ‘ Detection, estimation, and modulation theory. Part I. 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