Recursive performance ranking of Kalman filter with mismatched noise covariances

IET Control Theory & ApplicationsVolume 13, Issue 4 p. 459-466 Research ArticleFree Access Recursive performance ranking of Kalman filter with mismatched noise covariances Teng Shao, Teng Shao Center for Information Engineering Science Research, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049 People's Republic of ChinaSearch for more papers by this authorZhansheng Duan, Zhansheng Duan Center for Information Engineering Science Research, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049 People's Republic of ChinaSearch for more papers by this authorQuanbo Ge, Corresponding Author Quanbo Ge qbge@hdu.edu.cn Institute of Systems Science and Control Engineering, School of Automation, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310018 People's Republic of ChinaSearch for more papers by this authorHuaping Liu, Huaping Liu Department of Computer Science and Technology, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this author Teng Shao, Teng Shao Center for Information Engineering Science Research, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049 People's Republic of ChinaSearch for more papers by this authorZhansheng Duan, Zhansheng Duan Center for Information Engineering Science Research, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049 People's Republic of ChinaSearch for more papers by this authorQuanbo Ge, Corresponding Author Quanbo Ge qbge@hdu.edu.cn Institute of Systems Science and Control Engineering, School of Automation, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310018 People's Republic of ChinaSearch for more papers by this authorHuaping Liu, Huaping Liu Department of Computer Science and Technology, Tsinghua University, Beijing, 100084 People's Republic of ChinaSearch for more papers by this author First published: 28 January 2019 https://doi.org/10.1049/iet-cta.2018.5064Citations: 4AboutSectionsPDF 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 The Kalman filter is a powerful recursive state estimator and has been widely used in many applications. To guarantee its optimality, the noise covariances should to be exactly known. In reality, however, for most practical applications, it is difficult or unrealistic to obtain the noise covariances. A typical practice is to use some pre-determined alternatives for unknown noise covariances. The main issue concerning this is how the pre-determined alternatives will affect the performance of the Kalman filter. In this study, the authors study recursive performance ranking of Kalman filter with mismatched noise covariances. For this purpose, three types of mean squared errors (MSEs) have been used, i.e., the ideal MSE (IMSE), the filter calculated MSE (FMSE), and the true MSE (TMSE). This study considers the recursive ranking of these three types of MSEs at each time step. It is found that for the case with positive semi-definite deviation from the truth, they have FMSE TMSE IMSE at each time step recursively. On the contrary, for the case with negative semi-definite deviation, they have TMSE IMSE FMSE at each time step recursively. Target tracking examples further verify these results. 1 Introduction The Kalman filter [1, 2] is an effective tool to deal with state estimation and has successful applications in many fields [3], e.g. orbit determination [4], target tracking [5, 6], statistical signal processing [7], information fusion [8], weather forecasting [9], navigation [10], and robotics [11]. Under the Gaussian assumption for the initial state and all noises entering into the system, the Kalman filter is the optimal minimum mean squared error (MSE) state estimator. If they are not Gaussian and one has only their first two moments, then the Kalman filter is the best linear state estimator [3]. Also, because of its favourable recursive nature, it has attracted extensive attention. Yet, the success of the Kalman filter heavily relies on the linear Gaussian assumption associated with it. In reality, however, for most practical applications, it is difficult or unrealistic to exact accurate model parameters for the Kalman filter, especially for noise covariances. Numerical methods have been used to deal with the unknown model parameters identification problems, e.g. hierarchical identification methods for non-uniformly sampled systems [12], for general dual-rate systems [13], for pseudo-linear systems [14] and for bilinear systems [15], a generalised projection identification method for the time-varying systems [16] and a least squares parameter estimation method for the state-space model [17, 18]. However, these parameters identification methods render the used identified models generally mismatch the true ones [19–21]. As a result of this mismatch, most properties and conclusions on the performance of the Kalman filter for the ideal case with matched models, e.g. MSE, will not hold any more. It is therefore necessary to study the performance of the Kalman filter when the system parameters are inaccurate, so that some useful guidelines as to how to determine the unknown noise covariances can be provided to practitioners when the Kalman filter is applied. There have been some work on the analysis of model parameters, e.g. initial prior, transition matrix, measurement matrix, noise covariances, on the performance of the Kalman filter. The effect of errors in initial prior for a batch case was studied in [22] and the calculated and true posteriori variance with respect to the ideal minimum variance were obtained. However, no recursive estimation was performed and no system noise was assumed therein. The same problem was considered in [23] for a recursive case and a more general dynamic system with system noise was considered. The relations among the three types of covariance matrices, i.e. the ideal, calculated, and true covariance, were obtained. [24] extended the results to continuous systems. Zhao et al. [25] addressed the problem as to how the initial mean and variance will affect the estimates and how much performance will be sacrificed by comparing the calculated and ideal covariance if incorrect alternatives are used. The effect of errors from the initial state, and noise in the erroneous models was examined in [26]. The ranking of the ideal, calculated, and true covariance was given. However, the ranking was obtained only through numerical examples. The transition and measurement matrix errors in the Kalman filter were considered in [27]. The upper and lower bounds for the estimation performance of the suboptimal filters with transition and measurement matrix errors were obtained. The bounds require knowledge of the model matrices and the range for the errors of these matrices. Sangsuk-Iam and Bullock [28] studied the effect of the Kalman filter under incorrect noise covariances. The filter performance was measured by one step ahead predictor error covariance. It was found that the relationship between filter calculated covariance and true covariance depends on the signs of the noise covariance deviations. In this paper, we consider the modelling uncertainty on noise covariances. That is, either the process noise covariance (PNC), or measurement noise covariance (MNC), or both PNC and MNCs (PMNCs) are unknown. There are two main theoretical approaches to handle state estimation for such cases. The first type is adaptive filtering, e.g. strong tracking filter [29, 30], Sage-Husa adaptive filter [31], variational Bayesian filter [32–34]. The idea of the first type is to use the current measurement innovation to alleviate the effect of unknown noise covariances or to estimate noise covariances online using the measurement recursively. The second type is robust Kalman filter [35–37]. The objective of a robust Kalman filter is to ensure a minimum possible upper bound on the estimation error covariance. In practice, however, these theoretical approaches are not that widely used. The main reasons are as follows. For adaptive filtering, its computational burden is increased by much compared with the standard Kalman filter due to the use of adaptation strategy. For robust Kalman filter, the estimation performance is optimised for the worst case. In real applications, however, most cases are normal cases but not the worst case. So there is a gap between the optimality criterion used by the robust Kalman filter and what we expect. For practitioners, a typical practice is to use some pre-determined alternatives for unknown noise covariances. The main issue concerning this is how the pre-determined alternatives will degrade the performance of the Kalman filter at each time step. In [38], by assuming the three types of MSEs at the previous time step are the same, it is found that for positive semi-definite deviation from the truth, the ranking of these three types of MSEs at the current time is FMSE TMSE IMSE. For negative semi-definite deviation, the ranking of these three types of MSEs at the current time is TMSE IMSE FMSE. However, whether this ranking can still be maintained in the future time is unclear. In this paper, given optimal initialisation, we show that for positive semi-definite deviation from the truth, we do have FMSE TMSE IMSE recursively true. And for negative semi-definite deviation from the truth, we do have TMSE IMSE FMSE recursively true. This is achieved using mathematical induction. The advantages of this work can be summarised as follows: (1) To be in accordance with the recursive nature of the Kalman filter, recursive performance ranking (RPR) of Kalman filter with mismatched noise covariances has been obtained in this work. Comparatively, the same ranking of these three types of MSEs was obtained only at a given arbitrary time step non-recursively in [38]. (2) It is found that the ranking FMSE TMSE IMSE is recursively true for the case with positive semi-definite deviation and the ranking TMSE IMSE FMSE is recursively true for the case with negative semi-definite deviation. (3) From the above recursively ranking, we can see that the filter with mismatched noise covariance is pessimistic for positive semi-definite deviation and optimistic for negative semi-definite deviation. Therefore, we suggest to render noise covariance deviations positive semi-definite in practical applications. This can serve as a guideline for practitioners as to how to determine pre-determined alternatives for unknown noise covariances. In this work, we have considered the problem of Kalman filter using mismatched noise covariances with complete information. The extension to filtering for complex dynamic systems with incomplete information, similar to [39, 40], will be considered in our future work. The remaining of this paper is organised as follows. Section 2 formulates the problem. In Sections 3–5, the RPR of Kalman filter with mismatch on process noise only, on measurement noise only, as well as on both PMNCs, are established. Section 6 provides numerical examples in target tracking system to further verify the results. Finally, we conclude this paper in Section 7. 2 Problem formulation Consider the following generic linear dynamic system: (1)where k is the discrete time index and is the system state; is the state transition matrix and is the process noise, assumed to be zero-mean white Gaussian with covariance . The associated linear measurement equation is (2)where is the measurement of ; is the measurement matrix and is the measurement noise, assumed to be zero-mean white Gaussian with covariance . The initial state satisfies and . The two noises and are assumed mutually independent. When and are exactly known, one cycle of the (ideal) Kalman filter [1] can be summarised as follows: Time update (for ) (3) (4) Measurement update (for ) (5) (6) (7)or (8)with the optimal initialisation . As well known to all, when and are exactly known, the ideal Kalman filter is the best linear unbiased estimator for the linear system given by (1) and (2) and we term the MSE of the ideal Kalman filter ideal MSE (IMSE). In practical applications, however, it is usually hard or even unrealistic to know them exactly. Thus, the assumption of the ideal Kalman filter will be not satisfied. A widely used practice in real applications for such cases is to use pre-determined alternatives for or . Because of their deviations from the truth, the estimation performance will be different from that of the ideal Kalman filter. For example, we will have filter calculated MSE (FMSE) and true MSE (TMSE). The FMSEs are the MSEs provided by the Kalman filter and the TMSEs are the true MSEs of the predicted and updated estimates when pre-determined alternatives for or are used. They are in general different from the IMSE. In this paper, we study the recursive ranking of these three types of MSEs of the Kalman filter with mismatched noise covariances. For this purpose, three cases for noise covariances are considered, i.e. with mismatched PNC only, with mismatched MNC only, and with both mismatched PNC and MNC. 3 RPR of Kalman filter with mismatched PNC 3.1 MESs with mismatched PNC To obtain RPR of the Kalman filter with mismatched PNC, the corresponding FMSE and TMSE are defined next first. Denote the pre-determined as follows: (9)where is the true PNC, is its practically used mismatched alternative (pre-determined) which is indicated by the superscript 'u ' and represents the deviation. When is used in place of , one cycle of the Kalman filter can be written as follows: (10) (11) (12) (13) (14)where the superscript 'f ' is used to indicate filter calculated quantities. However, the FMSE does not provide a true measure of the performance of the actual filter estimate at time k. When is used, the TMSEs associated with and are actually (15) (16)where the superscript t is used to denote the TMSEs. For the case in which the covariance of the process noise is mismatched, it is very important to know the relations between the IMSE matrix , the FMSE matrix and the TMSE matrix . This is because it is of great help to provide some guidelines for practitioners as to how to determine the unknown noise covariances. Thereby, we will study the RPR when the PNC is mismatched in the Kalman filter. 3.2 Ranking of three MSEs with mismatched PNC Theorem 1.For the optimal initialisation and , at any time , we always have for ; and for . Proof: Basis of induction.When , from (4) (11) (15) and , we have for ; and for . Next, we discuss the ranking of , , and .First, we discuss the interrelationship between and .Using the information form in (8), the difference between and is (17)It can be easily seen that (1) For , we have . (2) For , we have . Second, we discuss the interrelationship between and .From (14) and (16), the difference between and is (18)It can be easily seen that (1) For , we have . (2) For , we have . Third, we discuss he interrelationship between and .Combining (7) and (16), it follows that (19)The right side of (19) is positive semi-definite. Therefore, we always have regardless of or .So, we have for ; and for .Inductive hypothesis: Suppose that at time , we have for ; and for .Inductive step: Next, we prove for ; and for .When , according to the inductive hypothesis and (4) (11) (15), we have for ; and for . Next, we discuss the ranking of , , and .First, we discuss the interrelationship between and .Using the information form in (8), the difference between and is (20)It can be easily seen that (1) For , one has . (2) For , one has . Second, we discuss the interrelationship between and .From (14) and (16), the difference between and is (21)It can be easily seen that (1) For , one has . (2) For , one has . Third, we discuss the interrelationship between and .Combining (7) and (16), it follows that (22)The right side of (22) is positive semi-definite. Therefore, we always have regardless of or .Thus, for the inductive step, one has for ; and for .Hence, the ranking holds for n and thus by the principle of mathematical induction for all positive integers k.This completes the proof. □ 4 RPR of Kalman filter with mismatched MNC 4.1 MESs with mismatched MNC To obtain RPR of the Kalman filter with mismatched MNC, the corresponding FMSE and TMSE are defined next first. Denote the pre-determined as (23)where is the true MNC, is its practically used mismatched alternative (pre-determined), and is the corresponding deviation. When is used in place of , one cycle of the practical Kalman filter is similar to that with mismatched process noise covariance in (10)–(14), except that (11), (12), and (14) are replaced by the following equations: (24) (25) (26) However, when is used, the TMSEs associated with and are actually given by (15) and (16), respectively. Next, we will study the RPR when the MNC is mismatched in the Kalman filter. 4.2 Ranking of three MSEs with mismatched MNC Theorem 2.For the optimal initialisation and , at any time , we always have for ; and for . Proof: Basis of induction.When , from (4), (15), (24) and , we have . Next, we discuss the ranking of , , and .First, we discuss the interrelationship between and .Using the information form in (8), the difference between and is (27)It can be easily seen that (1) For , we have . Thus, one has . (2) For , we have . Thus, one has . Second, we discuss the interrelationship between and .From (16) and (26), the difference between and is (28)It can be easily seen that (1) For , we have . (2) For , we have . Third, we discuss the interrelationship between and .Combining (7) and (16), it follows that (29)The right side of (29) is positive semi-definite. Therefore, we always have regardless of or .So, we have for ; and for .Inductive hypothesis: Suppose that at time , we have for ; and for .Inductive step: Next, we prove for ; and for .When , according to the inductive hypothesis and (4), (15), (24), we have for ; and for . Next, we discuss the ranking of , , and .First, we discuss the interrelationship between and .Using the information form in (8), the difference between and is (30)It can be easily seen that (1) For , we have and . Thus, . (2) For , we have and . Thus, . Second, we discuss the interrelationship between and .From (16) and (26), the difference between and is (31)It can be easily seen that (1) For , we have . Thus, . (2) For , we have . Thus, . Third, we discuss the interrelationship between and .Combining (7) and (16), it follows that (32)The right side of (32) is positive semi-definite. Therefore, we always have regardless of or .Thus, for the inductive step, we have for ; and for .Hence, the ranking holds for n and thus by the principle of mathematical induction for all positive integers k.This completes the proof. □ 5 RPR of Kalman filter with mismatched PNC and MNC 5.1 MESs with mismatched PNC and MNC To obtain RPR of the Kalman filter with mismatched both PNC and MNCs (PMNCs), the corresponding FMSE and TMSE are defined next first. Denote the pre-determined and as follows: (33) When and are used in place of and , one cycle of the Kalman filter can be given by (10), (11), (13), (25) and (26). However, when and are used, the true MSEs associated with and are actually given by (15) and (16), respectively. Next, we will study the RPR when the PMNCs are mismatched in the Kalman filter. 5.2 Ranking of three MSEs with mismatched PMNCs Theorem 3.For the optimal initialisation and , at any time , we always have for , ; and for , . Proof: Basis of induction.When , from (4), (11), (15) and at time , we have for ; and for . Next, we discuss the ranking of , , and .First, we discuss the interrelationship between and .Using the information form in (8), the difference between and is (34)It can be easily seen that (1) For , , we have and . Thus, one has . (2) For , , we have and . Thus, one has . (3) For , ; or , , the interrelationship between and cannot be uniquely determined. Second, we discuss the interrelationship between and .From (16) and (26), the difference between and is (35)It can be easily seen that (1) For , , we have . (2) For , , we have . (3) For , ; or , , the interrelationship between and cannot be uniquely determined. Third, we discuss the interrelationship between and .Combining (7) and (16), it follows that (36)The right side of (36) is positive semi-definite. Therefore, we always have regardless of or and or .So, we have for , ; and for , .Inductive hypothesis: Suppose that at time , we have for , ; and for , .Inductive step. Next, we prove for , ; and for , .When , according to the inductive hypothesis and (4), (11), (15), we have for ; and for . Next, we discuss the ranking of , , and .First, we discuss the interrelationship between and .Using the information form in (8), the difference between and is (37)It can be easily seen that (1) For , , we have and . Thus, . (2) For , , we have and . Thus, . (3) For , ; or , , the interrelationship between and cannot be uniquely determined. Second, we discuss the interrelationship between and .From (16) and (26), the difference between and is (38)It can be easily seen that (1) For , , we have . Thus, . (2) For , , we have . Thus, . (3) For , ; or , , the interrelationship between and cannot be uniquely determined. Third, we discuss the interrelationship between and .Combining (7) and (16), it follows that (39)The right side of (39) is positive semi-definite. Therefore, we always have regardless of or and or .Thus, for the inductive step, we have for , ; and for , .Hence, the ranking holds for n and thus by the principle of mathematical induction for all positive integers k.This completes the proof. □ Remark 1.Only either positive or negative semi-definite noise covariance deviations are considered in this paper. The main reason is that the summation of the true noise covariance and the deviation, i.e. the used mismatched noise covariance, is required to be still positive semi-definite. For positive or negative semi-definite noise covariance deviations, this requirement can be very easily satisfied. If the noise covariance deviations are not semi-definite, then more general conditions need to be figured out for the above requirement to be true. Remark 2.From the above theorems, it can be seen that when positive semi-definite noise covariance deviations are used, we have . That is, the filter with mismatched noise covariances is pessimistic [41]. Also, and have the same trend to measure the estimation performance. However, when negative semi-definite noise covariance deviations are used, we have . That is, the filter with mismatched noise covariances is optimistic. Also, and have opposite trends. In view of all these, if noise covariances are unknown, we suggest to render noise covariance deviations positive semi-definite. Thus we can at least guarantee the trend of the MSEs to be consistent, i.e. to be pessimistic. 6 Illustrative examples In this section, three numerical examples are provided to verify the conclusions obtained above. To investigate whether the rankings of three types of MSEs are recursively true, a metric called consensus count , is defined as follows: (40)where . The following examples describe the constant velocity motion of a target. The target motion state is , where is target displacement from the origin and indicates its velocity. The target motion model and sensor measurement model parameters are given as follows: where s is the sampling period. The first two moments of the initial state are 6.1 RPR with mismatched PNC only This example is used to validate the results in Theorem 1. Let The mismatched PNCs are specified as Case 1.1: positive deviations (41)Case 1.2: negative deviations (42)where is the total number of time steps and uniformly distributed random number on the interval . From Figs. 1 a and b, we can see that , for the case with positive definite mismatched PNC deviation. From Figs. 2 a and b, we can see that , for the case with negative definite mismatched PNC deviation. Fig. 1Open in figure viewerPowerPoint Case with positive definite mismatched PNC deviation (a) Traces of MSEs, (b) Consensus count Fig. 2Open in figure viewerPowerPoint Case with negative definite mismatched PNC deviation (a) Traces of MSEs, (b) Consensus count 6.2 RPR with mismatched MNC only This example is used to validate the results in Theorem 2. Let The mismatched MNCs are specified as Case 2.1: positive deviations (43)Case 2.2: negative deviations (44) From Figs. 3 a and b, we can see that , for the case with positive definite mismatched MNC deviation. From Figs. 4 a and b, we can see that , for the case with negative definite mismatched MNC deviation. Fig. 3Open in figure viewerPowerPoint Case with positive definite mismatched MNC deviation (a) Traces of MSEs, (b) Consensus count Fig. 4Open in figure viewerPowerPoint Case with negative definite mismatched MNC deviation (a) Traces of MSEs, (b) Consensus count 6.3 RPR with mismatched PMNCs This example is used to validate the results in Theorem 3. Let The mismatched PMNCs are specified as Case 3.1: positive deviations is as in Case 1.1 and is as in Case 2.1 Case 3.2: negative deviations is as in Case 1.2 and is as in Case 2.2 This example is to validate Theorem 3 and the results are shown in Figs. 5 and 6. Fig. 5Open in figure viewerPowerPoint Case with positive definite mismatched PMNCs deviation (a) Traces of MSEs, (b) Consensus count Fig. 6Open in figure viewerPowerPoint Case with negative definite mismatched PMNCs deviation (a) Traces of MSEs, (b) Consensus count From Figs. 5 a and b, we can see that , for the case with positive definite mismatched PMNCs deviation. From Figs. 6 a and b, we can see that , for the case with negative definite mismatched PMNCs deviation. Through all above three examples, we have verified that the rankings are really recursively true for three types of MSEs. 7 Conclusions The RPR of Kalman filter with mismatched noise covariances have been studied in this paper. Three types of MSEs, i.e. the IMSE, the FMSE, and the TMSE of the Kalman filter have been considered when there is mismatch on process noise covariance only, or on MNC only, or on both PMNCs. It is found that for the case with positive semi-definite deviation from the truth, FMSE TMSE IMSE is recursively true at each time step. For the case with negative semi-definite deviation from the truth, TMSE IMSE FMSE is recursively true at each time step. Comparatively, the same ranking of these three types of MSEs was obtained only at an arbitrary time step non-recursively in [38] where a key requirement is that the three types of MSEs have to be the same at its previous time step. A target tracking system provided has validated these results. 8 Acknowledgments This work was supported in part by the National Natural Science Foundation of China (NSFC) through Grants 61673317, 61673313, 61773147, by the Fundamental Research Funds for the Central Universities of China, by Zhejiang Provincial Nature Science Foundation of China through grant LR17F030005. 9 References 1Kalman R.E.: 'A new approach to linear filtering and prediction problems', ASME J. 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