Robust extended Kalman filtering for non‐linear systems with unknown input: a UBB model approach

IET Radar, Sonar & NavigationVolume 14, Issue 11 p. 1837-1844 Research ArticleFree Access Robust extended Kalman filtering for non-linear systems with unknown input: a UBB model approach Mersad Asgari, Corresponding Author Mersad Asgari asgari@ee.kntu.ac.ir Department of Systems and Control, Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, IranSearch for more papers by this authorHamid Khaloozadeh, Hamid Khaloozadeh Department of Systems and Control, Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, IranSearch for more papers by this author Mersad Asgari, Corresponding Author Mersad Asgari asgari@ee.kntu.ac.ir Department of Systems and Control, Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, IranSearch for more papers by this authorHamid Khaloozadeh, Hamid Khaloozadeh Department of Systems and Control, Faculty of Electrical Engineering, K.N. Toosi University of Technology, Tehran, IranSearch for more papers by this author First published: 28 September 2020 https://doi.org/10.1049/iet-rsn.2020.0258Citations: 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 onFacebookTwitterLinked InRedditWechat Abstract This study presents an unknown-but-bounded (UBB) model approach to robust extended Kalman filtering (REKF) problem for the simultaneous input and state estimation of non-linear systems with parametric uncertainties. An augmented non-linear state-space model is suggested to estimate the unknown input concurrently with the state variables without any delay. Due to more satisfactory physical assumptions and interpretations, the initial state vector and the disturbances are considered as UBB white processes, rather than the conventional stochastic white processes. Hence, they are modelled as ellipsoidal sets, and a recursive UBB-REKF is developed. The suggested algorithm aims to guarantee an optimal upper bound for the estimation error covariance considering the parametric uncertainties and the linearisation errors. Finally, the effectiveness and remarkable estimation accuracy of the proposed UBB-REKF is illustrated in a manoeuvring target tracking problem. 1 Introduction The Kalman filter and the extended Kalman filter (EKF) are ubiquitous in numerous practical applications, such as control, target tracking and aerospace [1-4]. There is an underlying assumption in the standard Kalman filter that the system model is exactly known. However, in practice, uncertainty in the system model seems inevitable, and there is a discrepancy between the mathematical model and the actual dynamics of the system. The model uncertainties cause the performance of the standard EKF to be adversely affected. Hence, much attention has been drawn to the robust filtering problem, including the filter [5, 6], mixed filter [7, 8] and the robust Kalman filter [9-16]. The robust EKF is a reliable solution to deal effectively with the estimation problem in non-linear systems with model uncertainties. In [9], stochastic uncertainties are considered in the system model, and a robust EKF (REKF) is developed for discrete-time non-linear systems. In [10], the authors have suggested another type of REKF to cope with parameter uncertainties in non-linear systems using a Riccati equation approach. In some cases, it is necessary to take into account more than a single type of uncertainty in the mathematical model of the system. Therefore, a recursive robust EKF is proposed in [11] for the attitude estimation problem that both unknown external disturbances and state-dependent noises are considered simultaneously. Multivariable systems constitute an important class of practical systems, and a robust Kalman filter is introduced in [14] for multivariable non-linear systems. Another robust version of the Kalman filter is discussed in [16] based on regularisation and penalty functions for rigid body attitude estimation. However, it is worth noting that most studies have tended to focus on the Bayesian model to design a robust algorithm. In the Bayesian model, the disturbances are assumed to be stochastic white processes characterised by their statistical properties. In the unknown-but-bounded (UBB) modelling approach, unlike the Bayesian model, the disturbances are assumed to belong to convex sets, and their statistical properties are not required. It is physically more reasonable to employ the UBB model rather than the Bayesian model on account of the fact that in the most actual systems the values of the noises are bounded. The solution of a UBB-based estimation algorithm is an estimate set rather than a single vector, and the estimate set always includes the true state of the system. If a vector estimate is desired, the centre of the estimate set is a reasonable choice for it. Ellipsoidal sets are more common among the convex sets owing to appropriate mathematical representation. Although the UBB-based algorithms require complex mathematical derivation, they have attracted increasing interest due to their great estimation accuracy. In [17], an ellipsoidal set-membership estimation algorithm has been introduced for linear systems. Recently, an ellipsoidal state estimation algorithm has also been presented in [18] for a class of non-linear systems with rational polynomial functions describing their models. Moreover, a UBB-based non-linear dynamic model for manoeuvring target tracking problem is given in [19]. However, all of the UBB-based filters have been developed for systems whose dynamics and measurement models are precisely known, and to the best of our knowledge, there is not any UBB-based robust algorithm for dynamic systems with modelling uncertainties. The unknown input estimation is another problem that plays a prominent role in various practical applications, such as manoeuvring target tracking problems [20] and power systems [21]. The unknown inputs can significantly degrade the filtering performance if they are not carefully considered. A variety of approaches have been put forward to solve this problem [22-27]. However, to date, all of the input estimation methods have only been suggested for the systems with completely known mathematical models, and there is still a need for an input and state estimation algorithm that is also robust against modelling uncertainties. Motivated by the above discussion, this paper proposes a robust EKF for the simultaneous input and state estimation of non-linear systems with parametric uncertainties using a novel UBB model approach. During the derivation of the Riccati equation, both the parametric uncertainties and the linearisation errors are considered. In order to estimate the unknown input simultaneously with the state variables, a new augmented non-linear state-space model is developed. When ellipsoidal sets are used, the UBB and Bayesian model equations are very similar in appearance. Owing to the mathematical intractability of the UBB model, when the modelling uncertainties are considered, we employ the expectation operator to develop a recursive UBB robust EKF (UBB-REKF) to estimate both the states and the unknown inputs. The objective of the proposed UBB-REKF is to ensure an optimal upper bound on the estimation error covariance, despite the parametric uncertainties and the linearisation errors. Finally, an application of the UBB-REKF in manoeuvring target tracking problem is presented, and its performance is evaluated in comparison with the existing approaches. 2 Problem formulation Consider the following non-linear uncertain system with an unknown input: (1) where is the state vector, is the measurement vectorand denotes the unknown input. and are non-linear functions that are assumed to be differentiable. and represent the parametric uncertainties, where , , and are known real constant matrices of appropriate dimensions. and are norm-bounded uncertainty satisfying (2) and are the process and measurement noises, respectively. The UBB white processes bear a close resemblance to stochastic white processes in that they have no time structure, but we choose the UBB processes since they are physically more satisfactory than their stochastic white processes counterparts. Hence, the initial state vector and noise signals are considered as the members of convex sets. For this purpose, ellipsoidal sets are usually used due to the facts that their detailed mathematical manipulations are appropriate enough, and their final equations are fairly similar to those of the Bayesian approach. The initial state vector and noise signals are, therefore, modelled as follows: (3) where , and are ellipsoidal sets which can vary in size and shape with time, and , and are the shaping matrices of the sets. The shaping matrices are positive definite that the orientation and lengths of the major axes of the ellipsoids are determined by them. It should also be noted that , and are assumed to be uncorrelated (4) 3 Augmented model for non-linear systems with unknown input A major difficulty in state estimation problems for non-linear systems with unknown inputs, such as manoeuvring target tracking problems, is the input detection and estimation [20]. In this section, a new augmented model for non-linear systems in the presence of parametric uncertainty and unknown input is proposed. Using this augmented model, there is no need for unknown input detection procedure. The main idea of this approach is to add to the original state vector as a new state (5) where (6) The process noise of the new augmented system is (7) Meanwhile, is still a UBB process since is bounded. Moreover, according to (4), it is clear that (8) Finally, the new augmented system equations, where there is no input term, can be rewritten as (9) where (10) Remark 1.Unlike the most input estimation techniques that delay phenomenon is the inevitable result of input detection stage, the proposed augmented model (9) enables us to estimate the unknown input simultaneously with the state variables without delay.From the augmented system (9), the shaping matrix of the process noise becomes (11) According to (2) and (6), we also have (12) Using the structure of the EKF for the augmented uncertain system (9), we aim to design the UBB-REKF of the form (13) Furthermore, one-step prediction error and the estimation error are defined as (14) and their covariance matrices can be written as (15) In the filter (13), is the Kalman gain to be determined to ensure that the covariance of the estimation error is always smaller than a specific bound. Indeed, the estimation error dynamics must satisfy (16) Moreover, it is desired to minimise the covariance bound and finally obtain the optimal filter. Substituting (9) and (13) into (14), the prediction error can be written as (17) Employing the Taylor series expansion around , we have (18) where and denote the high-order terms of the Taylor series expansion. Since the desired filter is also supposed to be robust against the linearisation errors, the linearisation of should not be confined to the first-order term. Thus, according to [11], we transform the high-order terms into an equivalent formulation that is easier to deal with (19) where is a known scaling matrix; is a tuning matrix, and is an unknown time-varying matrix that takes account of the linearisation errors of the system model and satisfies (20) Hence, from (18) and (19), the prediction error dynamics, considering the linearisation errors, can be expressed as (21) Similarly, the innovation of the filter (22) can be expressed as (23) where , is a known scaling matrix; is a tuning matrix and is an unknown time-varying matrix that accounts for the linearisation errors of the measurement model and satisfies (24) By virtue of (13), (14) and (23), the estimation error dynamics is obtained as (25) 4 Design of UBB-REKF In this section, we are going to design the UBB-REKF using the augmented model (9). Our main objective is to obtain an upper bound for and find the Kalman gain such that the upper bound is minimised. First, we define a new state vector as follows: (26) From (21) and (26), the new system is expressed as (27) where (28) The covariance matrix of the system (27) is also defined as (29) Using (15), (26) and (29), the prediction error covariance matrix can be represented by (30) We are now concerned with finding an upper bound for . Thus, we first write the shaping matrix of the system (27) as follows (31) Analysis of linear systems driven by UBB processes, leading up to the proof of (31), is summarised in Appendix 1. From the model (27), (31) can be rewritten as (32) (33) The derivation of the covariance upper bound is based on the following lemma. Lemma 1.Givenmatrices , , and with compatible dimensions such that . Let be a symmetric positive-definite matrix and be an arbitrary positive constant such that ; then, the following matrix inequality holds [28]: (34) By substituting , , , , and into (34) and applying the matrix inversion lemma to the right side of the inequality, we have (35) such that . Substituting (35) into (32) yields (36) where (37) On combining (26), (28) and (30) with (36) and using the matrix inversion lemma, we have (38) where (39) We can consider a bound like , and the tuning matrices and can be set as (40) to make the subsequent deductions easier. It is worth mentioning that setting the tuning matrices as (40) does not have an unfavourable effect on the actual uncertainties and the linearisation errors since their effect can be controlled by the scaling matrices and . From (38) and (40), we achieve the following inequality which is an upper bound for the prediction error covariance (41) Remark 2.It is worthwhile noting that the unknown matrices and do not appear in the upper bound (41), which is a valuable feature of the robust algorithm. Remark 3.The existence of the tuning parameters and is a difference between the robust algorithm and the estimation algorithm for systems without uncertainty. When the tuning parameters are chosen conservatively, the upper bound (41) is conservative. Next, we are going to find an upper bound for . For this purpose, the new state vector is defined as (42) From (42) and (25), the new system is described by (43) where (44) The covariance matrix of the system (44) is expressed as (45) According to (15), (42) and (45), is written as (46) Following similar steps leading up to the upper bound (41), it can be shown that the upper bound for is obtained as (47) where and . It should be noted that the unknown matrices and are not seen in the upper bound (47). The main result of this study, the UBB-REKF, is summarised in the following theorem. Theorem 1.Consider the non-linear uncertain system (1) and the augmented model (9). Assume that (12), (20) and (24) are fulfilled. Let , , and be positive scalars satisfying and . The ellipsoidal estimate set is described by (48) where the centre of the estimate set is , and the size and shape of the set are determined by (49) (50) Furthermore, is given by the robust filter (13), if the following Riccati equations hold (51) (52) where (53) (54) (55) with the initial value have positive-define solutions, the estimation error covariance of the filter (13) satisfies (56) Moreover, the filtering gain (57) minimises the upper bound .The proof of Theorem 1 is given in the Appendix 2. For the sake of clarity, the steps of the UBB-REKF are summarised in Table 1. Remark 4.According to Theorem 1, the UBB-REKF can be described by (13), (51), (52) and (57). It is apparent that the UBB-REKF is broadly similar to the Bayesian-based EKF. However, the UBB-REKF incorporates the tuning parameters and owing to the robustness of the filter and the parameters and due to the UBB processes. The parameter and the matrices , , , and should be set to zero when the effects of the linearisation errors and the modelling uncertainties are negligible. Consequently, the UBB-REKF reduces to the EKF in [19]. Remark 5.Although the prediction error dynamics (21) and the estimation error dynamics (25) are rather complicated, the Riccati equations (51) and (52) are straightforward and independent of the unknown matrices , , and . It is worth mentioning that designing the UBB-REKF with conservative upper bounds may lead to poor filtering performance. Remark 6.The scaling matrices , , and , which are used to represent the bounds of the parametric uncertainties and the linearisation errors, are required in the Riccati equations (51) and (52). In practice, however, the exact values of these matrices are generally unknown, and there is no systematic method to determine them. This is a major hurdle for the implementation of robust filtering algorithms (see, e.g. [9, 11]). Hence, the scaling matrices should be determined using the prior knowledge of the practical system. Table 1. Steps of the UBB-REKF Steps Required calculations at the time step step 1 one-step prediction from (13) step 2 upper bound for the prediction error covariance from (51) step 3 filtering gain from (57) step 4 optimal upper bound from (52) step 5 shaping matrix from (49) and (50) step 6 the ellipsoidal estimate set from (13) and (48) 5 Simulation In this section, the target tracking problem of [29] is employed to demonstrate the high performance of the proposed UBB-REKF. The problem is the estimation of the position and velocity of a ballistic target in the presence of unknown acceleration input. For the sake of simplicity, the effects of wind, spinning motion, lift force, Coriolis acceleration, and centrifugal acceleration are neglected. The state space model of the target motion with unknown input and modelling uncertainty is described by (58) where (59) (60) The state vector and the unknown input are (61) (62) where and denote the target position, and are the target velocity and and stand for the target unknown acceleration in the Cartesian coordinates. We also have (63) where T is the radar sampling time. is an uncertain parameter satisfying . The drag is described by (64) where g and are the gravity acceleration and ballistic coefficient, respectively. is the air density defined by where , for and , for . Since the measurement model is linear and has no parametric uncertainty, the scaling matrices and are set to zero. However, determining the values of and is not straightforward in this system model, and there are not any certain limitations on their magnitude [11]. Thus, to surmount this obstacle, a program is written that considers a range of diagonal matrices with appropriate dimensions and selects the most suitable values for multiple iterations. The simulation is performed in MATLAB R2018b using a Core i5 CPU 2.53 GHz and 4 GB RAM. The simulation time is set as 300 s. The average execution time, using 200 simulation runs, is 1.5241 s. It is assumed that the target unknown acceleration is for and for where and . The shaping matrices and the initial state vector are set as , and , respectively. To confirm the validity of the UBB-REKF, we compare its performance with the AEKF proposed in [20] and the REKF in [10]. The AEKF is suggested for input estimation in non-linear manoeuvring target tracking problem. First, we assume that there is no deterministic uncertainty in the system (58) and compare the UBB-REKF with the AEKF. In this case, since the AEKF is designed based on the nominal system and without considering modelling uncertainty, it is expected to perform better than the UBB-REKF. The root mean square error (RMSE) of the UBB-REKF and the AEKF for the nominal system are compared in Fig. 1. As expected, when there is not any modelling uncertainty, the standard AEKF performs slightly better than the UBB-REKF, and its RMSE is smaller than that of the UBB-REKF. Furthermore, as shown in Fig. 2, the UBB-REKF and AEKF are both able to estimate the unknown acceleration inputs for the nominal system. The standard AEKF, however, estimates the inputs with lower variance. Fig. 1Open in figure viewerPowerPoint Comparison of the RMSE for the nominal system Fig. 2Open in figure viewerPowerPoint Comparison of the estimated inputs for the nominal system The RMSE and the square root of the corresponding diagonal elements of the error covariance for the UBB-REKF and the AEKF in the presence of modelling uncertainty are shown in Fig. 3. Fig. 3Open in figure viewerPowerPoint RMSE and the calculated upper bound in the presence of uncertainty Obviously, the RMSE of the UBB-REKF, unlike the AEKF, is always below its calculated upper bound. Hence, the proposed algorithm is appropriate for handling the uncertainties. The standard AEKF, on the contrary, does not guarantee the containment of the RMSE and is not robust against the deterministic uncertainties. Furthermore, Fig. 4 demonstrates that the UBB-REKF can instantly estimate the unknown inputs whereas the uncertainties cause the divergence of the AEKF. Fig. 4Open in figure viewerPowerPoint Comparison of the estimated inputs in the presence of uncertainty Next, the proposed robust algorithm is compared with the REKF, which no unknown input is considered in its design. For a fair comparison, we consider the acceleration input as a known input in the REKF. In this case, a total of 200 Monte–Carlo simulation runs are employed, and the average RMSE is computed for each of the time steps. Fig. 5 shows that the average RMSE of the UBB-REKF is always smaller than that of the REKF. Moreover, Table 2 compares the average RMSE computed by Monte–Carlo runs at . Thus, the simulation results reveal that the UBB-REKF estimates the states more accurately, and it is superior to the REKF for the considered system. Fig. 5Open in figure viewerPowerPoint Comparison of the average RMSE computed by 200 Monte–Carlo runs for the UBB-REKF and the REKF Table 2. Monte–Carlo simulation results at t = 200 Average RMSE in UBB-REKF REKF position X (m) 2.95 9.78 position Y (m) 2.97 9.13 velocity Vx (m/s) 7.18 58.63 velocity Vy (m/s) 6.84 49.89 Finally, the performance of the UBB-REKF is evaluated when it is initialised using different values of the estimate vector. To that end, vectors of uniformly distributed random numbers are used in the filter initialisation, and the average RMSE and the corresponding upper bound obtained from 50 Monte–Carlo runs are plotted in Fig. 6. It is apparent that the average RMSE is bounded by the optimal upper bound immediately after the initial steps. Fig. 6Open in figure viewerPowerPoint Monte–Carlo simulation results for the average RMSE of the UBB-REKF initialised by the uniformly distributed random numbers 6 Conclusion In this study, we have put forward a new approach to non-linear robust estimation in the presence of unknown inputs and parametric uncertainties based on a UBB model approach. The proposed UBB-REKF gives a minimised upper bound on the RMSE. The simulation results indicate that the suggested filter has flawless robustness against the uncertainties, and the RMSE is confined by the calculated optimal bound. The great accuracy, despite unknown inputs, and the familiar structure are the distinct advantages of the UBB-REKF. Moreover, future work will concentrate on state-space models with UBB state-dependent noises on account of their very existence in many actual systems. 8 Appendix 8.1 Appendix 1: analysis of linear systems Consider the linear dynamic system (65) where the initial state and process noise belong to ellipsoidal sets. According to the set theory, for a closed convex set , the support function is defined as (66) When is an ellipsoid, it is defined by (67) where the size and shape of the ellipsoid is determined by , and its support function is expressed as (68) The vector sum (Minkowski sum) of two closed convex sets is also a closed convex set with the support function (69) Thus, from (68) and (69), the vector sum of two ellipsoids is (70) It is obvious that (70) is not, in general, the support function of an ellipsoid. Therefore, a bounding ellipsoid that contains is developed. According to a special case of Holder's inequality, we have (71) From (70) and (71), the support function of the bounding ellipsoid is (72) For the linear model (65), let denote the set of all possible . Using (69), we have (73) Since and are considered as ellipsoids, (73) can be rewritten as (74) Similar to (70), (74) shows that the resulting is not necessarily an ellipsoid. To solve this problem, we consider two stages. First, we assume a bounding ellipsoid is given. Second, we are supposed to find the equation determining assuming that . This logic results in a difference equation that gives a bounding ellipsoid for as follows: (75) where the difference equation of the shaping matrix is (76) Obviously, the structure of the difference equation (76) is similar to that of the Bayesian model. 8.2 Appendix 2: Proof of Theorem 1 Proof.Let . Since , from (41) and (51), we have (77) Combining (47) and (52) with (77), we conclude that (78) Let and assume that . According to (41) and (51), there is (79) From (47), (52) and (79), it is clear that (80) By virtue of (78) and (80), we conclude that (56) is satisfied.Furthermore, in order to minimise the upper bound , we have (81) From (81), the optimal filtering gain is obtained by (57). Moreover, the proof of (49) and (50) can be found in [19]. □ 7 References 1Alsofyani I.M. 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