Equivalence of recursive three‐step filter and infinity augmented Kalman filter for linear discrete‐time stochastic systems with direct feedthrough

IET Control Theory & ApplicationsVolume 14, Issue 17 p. 2696-2702 Brief PaperFree Access Equivalence of recursive three-step filter and infinity augmented Kalman filter for linear discrete-time stochastic systems with direct feedthrough Bo Ding, Corresponding Author Bo Ding dingbo@yzu.edu.cn orcid.org/0000-0002-7736-1758 Department of Automation, College of Information Engineering, Yangzhou University, Yangzhou, 225127 People's Republic of ChinaSearch for more papers by this authorTianping Zhang, Tianping Zhang Department of Automation, College of Information Engineering, Yangzhou University, Yangzhou, 225127 People's Republic of ChinaSearch for more papers by this author Bo Ding, Corresponding Author Bo Ding dingbo@yzu.edu.cn orcid.org/0000-0002-7736-1758 Department of Automation, College of Information Engineering, Yangzhou University, Yangzhou, 225127 People's Republic of ChinaSearch for more papers by this authorTianping Zhang, Tianping Zhang Department of Automation, College of Information Engineering, Yangzhou University, Yangzhou, 225127 People's Republic of ChinaSearch for more papers by this author First published: 01 October 2020 https://doi.org/10.1049/iet-cta.2019.1387AboutSectionsPDF 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 In this study, the recursive three-step filter method introduced by Gillijns and Moor (2007) is proved strictly to be equivalent to the augment Kalman filter (AKF) when the covariance of process noise about the input approaches infinity. This result indicates that if the information about the exogenous input is unknown, the AKF can also be used to simultaneously obtain the state and exogenous input estimation. Numerical examples illustrate the authors' main results. 1 Introduction The problems of optimal estimation for linear discrete-time stochastic systems subjected to unknown input have drawn increasing attention in the past several decades because of its many useful applications such as fault reconstruction and diagnosis [1, 2], fault tolerant control [3], real-time monitoring [4], multi-sensor fusion [5] and so on. If the information about exogenous input cannot be obtained, a general approach to estimate the state is minimum variance-unbiased (MVU) filter, which was first presented by Kitanidis [6]. Later, this approach was extended to a more common parameterised solution by Darouach and Zasadzinski [7] and its stability condition was obtained by Keller et al. [8]. To both estimate the state and unknown input, a robust two-stage Kalman filter was introduced by Hsieh [9]. A recursive MVU filter and its global optimality were established by Gillijns and Moor [10]. Later, the stability properties of the recursive MVU filter were achieved by Fang and Callafon [11]. It should be noted that the above-mentioned method is only valid in the system where the input only affects the state. In the direct feedthrough stochastic systems, where the exogenous unknown input affects both state and measurement equations, the problem becomes more complicated because the estimate error is correlated with the system noise. To simultaneously estimate the state and exogenous input in this case, many useful results have been achieved in the literature. A recursive three-step filter (RTSF) to both estimate the state and input was established by Gillijns and Moor [12]. By using singular value decomposition and input transformations, Cheng et al. [13] proposed a recursive state estimator with a rank deficient direct feedthrough matrix. Hsieh [14, 15] proposed a novel globally optimal state estimators without transformation matrices. Based on the expectation–maximum criterion, Lan et al. [16] introduced a joint estimation and identification filter via iterative optimisation. Recently, Wang et al. [17] and Yong et al. [18] both established a unified filter for simultaneously estimating the state and unknown input. On the other hand, if the information of exogenous input is assumed to be known, the augment Kalman filter (AKF) can be used to deal with the estimation problem. For the AKF method, the model of exogenous input can be expressed as a random walks or stochastic process with known covariance [the dynamic formula can be seen in (15)]. Then the input can be augmented into the state of systems and based on the Kalman filter, the exogenous input and state can be simultaneously obtained. It is known that (15) involves the prior information about the input. In particular, the uncertainty of the information about exogenous input can be reflected in its covariance . If is large, the input at time steps k and will be less related with each other, which means that the prior information about the input is more uncertain. Moreover, as tends to infinity, the information about input will become totally unknown, which is just the case of the RTSF method. Intuitively, when tends to be infinity, the result of AKF will be equal to that of the RTSF method. More recently, the equivalence between AKF in the infinity condition and MVU filter has been proved, respectively, in [19, 20] for the system that unknown input only affects the state. However, in the direct feedthrough systems, whether or not the result of corresponding AKF in the infinity condition will be equal to that of RTSF is still unknown. For the sake of simplicity, when approaches infinity, the AKF is called the infinity AKF (IAKF) for short. In this work, by using matrix limit and mathematical induction, a strict proof is given to show that RTSF is equivalent to IAKF. The contribution of this paper is twofold. First, the property of IAKF is discussed for the linear discrete-time systems with direct feedthrough. Second, a strict proof of the equivalence between RTSF and IAKF is provided. To be specific, the result of AKF will be equal to that of the RTSF method when tends to infinity, which can be visually shown as follows: This result indicates that when we choose a large , the result of AKF method will be close to that of RTSF. Therefore, if the information about the exogenous input is totally unknown, the AKF method can also be used to obtain the state and unknown input estimation by only choosing a proper large , which is easy to implement in some practice. This work can be regarded as a new contribution in the area of simultaneously estimating the state and input for linear discrete-time stochastic systems with direct feedthrough. The rest of this paper is organised as follows. A brief review about RTSF and AKF is introduced in Section 2. The main results are demonstrated in Section 3, where the RTSF is equivalent to AKF under the infinity condition is proved. Two numerical examples are given in Section 4, and a brief conclusion is provided in Section 5. Notation : The notation used throughout the paper is fairly standard. denotes the n -dimensional Euclidean space. represents the set of all real matrices. For a random vector , its expectation is denoted by . For a matrix , its transpose, rank, determinant, inverse are given by , , , . For a symmetric matrix S, indicates S is positive (semi-positive) definite. represents the identity matrix with order n. stands for a matrix with elements on the diagonal. 2 Preliminaries Consider the linear discrete-time stochastic systems with direct feedthrough as follows: (1) where is the state vector, is the unknown exogenous input vector and is the measurement vector. The system noise and the measurement noise are mutually uncorrelated white-noise sequences, with known covariance matrices and , respectively. , , and are known matrices. It is assumed that . The initial state is independent of and for all time step k. 2.1 RTSF method If no prior information about the dynamical evolution of the unknown input is obtained, the state and unknown input can be estimated by the RTSF method [12]. In the following, a brief result about the RTSF method will be given. Consider the recursive filter with the following form: (2) (3) (4) where and are the state estimation and unknown input estimation, respectively, and is the intermediate value. and are the gain matrices. The calculation of and can be expressed as Lemmas 1 and 2, respectively. Lemma 1.([12], Theorem 2) If is unbiased, and let be given by (5) where with (6) and Then (3) is the RTSF estimator of given the innovation . The variance of the corresponding input estimate can be calculated by . Lemma 2.([12], Theorem 3) If satisfies (5), then the optimal gain matrix can be given by (7) where .The minimum variance can be obtained by (8) and are calculated by (9) Notice that filter (2)–(4) can be rewritten as (10) The following lemma and theorems show the property of gain matrices and . Lemma 3.([12], Theorem1 and Equation (10)) The gain matrix presented in (4) and presented in (7) satisfy the following equations: (11) (12) Theorem 1.The matrix presented in (7) can be rewritten as (13) Proof.See the Appendix. □ Theorem 2.The gain matrix presented in (5) can be rewritten as (14) Proof.See the Appendix. □ 2.2 AKF method If the model of exogenous input is expressed as a stochastic process as follows: Assumption 1.(15) where is the white noise and its covariance matrix is , which is assumed to be known. The initial estimation of is and its covariance is . Then a general method to obtain the state estimation and exogenous input estimation is the AKF method, which is briefly demonstrated as follows. Combined with (1) and (15), and let (16) (17) then the dynamics of augmented system can be expressed as (18) and the initial estimation is the initial covariance is Now based on the principle of Kalman filter [21], the augmented state estimation can be obtained as the following equations: (19) (20) (21) (22) (23) where represents a priori estimation of and represents a posteriori one. and denote the covariance of the estimation error, respectively. is the gain of Kalman filter. Based on formulas (19)–(22) and the definition of , the state and input can be simultaneously estimated. Let (24) Then in light of (21), (20) and (23), we have (25)–(27) (25) (26) (27) where (28) It is known that can also be represented by , then it follows that (29) Note that (15) shows the information about input , and the uncertainty of its information can be reflected in the covariance . If is small, then it indicates that the inputs at time step k and time step are closely related. Conversely, if is large, then it means the inputs at time step k and time step are little to do with each other, which reflects the uncertainty of the information that input is huge. Moreover, if approaches infinity, it indicates that the information about the input is unavailable. In Section 3, we will show that when is infinity, the result of the AKF method is equivalent to that of the RTSF method. 3 Equivalence between IAKF and RTSF Before giving the proof of the equivalence between IAKF and RTSF, we introduce the definition and property of matrix limit. 3.1 Matrix limit Definition 1.([19], Definition 3.2) Let be a matrix sequence. If and each diagonal elements of approach to infinity, i.e. , as , we say matrix tend to infinity and simply write . Some important properties about the matrix limit are given in the following lemma. Lemma 4.([19], Lemma 3.4) Let , , , , , and . If , then (30) (31) (32) By using Lemma 4 and Matrix Inversion lemma, we have the following lemma, whose proof is omitted. Lemma 5.Let , , , , , , and . If , then (33) (34) (35) where . 3.2 Evidence when : To demonstrate the equivalence between IAKF and RTSF, the AKF method is rewritten at first. Substituting (19) into (22) and based on (16), we have (36), which is shown at the bottom of this page (36) Notice (11) and (12), if and , it can be obtained that and , then (36) will be equal to (10). That is if and , then the result of AKF method will be equal to that of the RTSF method. The following two theorems illustrate that and as . Then according to the above discussions, the IAKF will be equal to the RTSF. Theorem 3.Consider the AKF method (18)–(23) if then the covariance of estimation error (27) will equal to that of the RTSF method, i.e. the following equation is valid: (37) Proof.See the Appendix. □ Theorem 4.Consider the RTSF method (2)–(5) and AKF method (18)–(23). If , then and , where is shown in (10), and are shown in (24). Proof.According to (37), it can be shown that and . Then compared with (13) and (29), it can be concluded that (38) In light of (14) and (29), it can be obtained that (39) Thus the proof is completed. □ 4 Numerical simulations In this section, to verify the main result, two numerical simulations are conducted. Example 1.Consider one-dimensional model as follows: where the covariance of the process noise is and the covariance of the measurement noise is . The initial state is and its covariance is .By using the RTSF method, the steady-state gains given by (5) and (7) are and , respectively. The covariances of the estimation error are and .Then consider the AKF method with different covariance of the input. These results can be shown in Table 1. It can be seen that the results of AKF are tending to that of RTSF as the value of becomes larger, which illustrates the validness of Theorems 3 and 4. Example 2.Consider the following linear discrete-time stochastic system: where the initial state is and its covariance is .According to (2)–(9), the result of RTSF filter can be calculated as Now let and then use the AKF method to deal with the same estimation problem. The corresponding steady-state values are as follows: It is shown that when the is large enough, the result of AKF is very close to that of RTSF. Therefore, if the prior information of the input is unavailable, the AKF can also be selected to estimate the state and input by only choosing a proper large . It is known that AKF has the same program code as KF. Compared with RTSF, the AKF is easy to implement in some cases since the code of KF is not required to change. Table 1. Result of the AKF filter with different 0.12180027 0.81222824 0.00809153 0.01499581 1 0.01944353 0.97112561 0.010504825 0.02002164 0.00208305 0.99692309 0.01094640 0.02089480 0.00020982 0.99969023 0.01099459 0.02098939 0.00002099 0.99996900 0.01099945 0.02099893 0.00000209 0.99999690 0.01099994 0.02099989 5 Conclusions In this work, the RTSF method is proved to be equivalent to the AKF when the covariance of process noise about the exogenous input towards infinity. This result is meaningful in theory. It indicates that even if we do not know the prior information about the input, the state and unknown input estimation can also be obtained by the AKF method with a proper large . This work can be shown as a new contribution in the area of simultaneously estimating the state and input for linear discrete-time stochastic systems with direct feedthrough. Future results will be anticipated by using the proposed thought to more complex systems, such as descriptor system [22], non-linear system [23] or system with multiplicative noise [24]. 6 Acknowledgments This work was supported by the National Natural Science Foundation of China (NSFC) under grant No. 61803330. Appendix 8 8.1 Proof of Theorem 1 Notice , and , then (8) can be expressed as (40) Note that , then we have (41) (42) Then we have (43) It is known that , then it can be obtained that (13) is valid. Thus, Theorem 1 holds. 8.2 Proof of Theorem 2 Note that and , then it follows that (44) Note that is invertible, then it follows that (14) is valid. Thus, Theorem 2 holds. 8.3 Proof of Theorem 3 This proof is conducted by mathematical induction. First, when , then it is obvious that (37) is valid at time step 0. Second, assume that (37) holds at time step k, in the following we will show this equality also holds at time step . Based on (26) and (6), can be rewritten as (45) can be expressed as (46) Note that and , it follows that . According to Lemma 5 and (46), we have (47) where . Denote , then it follows that and . Denote and , then based on Lemma 5, it follows that (48) (49) It is obvious that as . Now according to (25), (27) and the above formulas, , , and can be calculated by (50)–(53): (50) (51) (52) (53) Based on (50)–(53), it is obvious that (37) also holds at time step . Thus the proof is completed. 7 References 1Liu Y. Wang Z. Zhou D.: 'State estimation and fault reconstruction with integral measurements under partially decoupled disturbances ', IET Control Theory Appl., 2018, 12, (10), pp. 1520– 1526 2Hassanabadi A. Shafiee M. Puig V.: 'Actuator fault diagnosis of singular delayed LPV systems with inexact measured parameters via PI unknown input observer ', IET Control Theory Appl., 2017, 11, (12), pp. 1894– 1903 3Hashemi M. Egoli A. Naraghi M et al.: 'Saturated fault tolerant control based on partially decoupled unknown-input observer: a new integrated design strategy ', IET Control Theory Appl., 2019, 13, (13), pp. 2104– 2113 4Torres Zúñiga I. Vargas A. Moreno J.: 'On the practical estimation of unknown inputs for polytopic LTI systems ', IET Control Theory Appl., 2017, 12, (4), pp. 466– 476 5Geng H. Liang Y. Liu Y. et al.: 'Bias estimation for asynchronous multi-rate multi-sensor fusion with unknown inputs ', Inf. Fusion, 2018, 39, pp. 139– 153 6Kitanidis P.: 'Unbiased minimum-variance linear state estimation ', Automatica, 1987, 23, (6), pp. 775– 778 7Darouach M. Zasadzinski M.: 'Unbiased minimum variance estimation for systems with unknown exogenous inputs ', Automatica, 1997, 33, (4), pp. 717– 719 8Keller J. Darouach M. Caramelle L.: 'Kalman filter with unknown inputs and robust two-stage filter ', Int. J. Syst. Sci., 1998, 29, (1), pp. 41– 47 9Hsieh C.: 'Robust two-stage Kalman filters for systems with unknown inputs ', IEEE Trans. Autom. Control, 2000, 45, (12), pp. 2374– 2378 10Gillijns S. Moor B.: 'Unbiased minimum-variance input and state estimation for linear discrete-time systems ', Automatica, 2007, 43, (1), pp. 111– 116 11Fang H. Callafon R.: 'On the asymptotic stability of minimum-variance unbiased input and state estimation ', Automatica, 2012, 48, (12), pp. 3183– 3186 12Gillijns S. Moor B.: 'Unbiased minimum-variance input and state estimation for linear discrete-time systems with direct feedthrough ', Automatica, 2007, 43, (5), pp. 934– 937 13Cheng Y. Ye H. Wang Y. et al.: 'Unbiased minimum-variance state estimation for linear systems with unknown input ', Automatica, 2009, 45, (2), pp. 485– 491 14Hsieh C.: 'On the global optimality of unbiased minimum-variance state estimation for systems with unknown inputs ', Automatica, 2010, 46, (4), pp. 708– 715 15Hsieh C.: 'A framework for globally optimal state estimation for systems with unknown inputs (II): untrammeled approach ', Asian J. Control, 2012, 14, (4), pp. 1002– 1011 16Lan H. Liang Y. Yang F. et al.: 'Joint estimation and identification for stochastic systems with unknown inputs ', IET Control Theory Appl., 2013, 7, (10), pp. 1377– 1386 17Wang H. Zhao J. Xu Z. et al.: 'Input and state estimation for linear systems with a rank-deficient direct feedthrough matrix ', ISA Trans., 2015, 57, pp. 57– 62 18Yong S. Zhu M. Frazzoli E.: 'A unified filter for simultaneous input and state estimation of linear discrete-time stochastic systems ', Automatica, 2016, 63, pp. 321– 329 19Ding B. Zhang T. Fang H.: 'On the equivalence between the unbiased minimum-variance estimation and the infinity augmented kalman filter ', Int. J. Control, 2019, doi: 10.1080/00207179.2019.1574984 20Bitmead R. Hovd M. Abooshahab M.: 'A Kalman-filtering derivation of simultaneous input and state estimation ', Automatica, 2019, 108, (108478), pp. 1– 5 21Simon D.: ' Optimal state estimation: Kalman, H infinity, and nonlinear approaches ' ( John Wiley & Sons, New York, 2006) 22Wang Z.: 'Parametric circular eigenvalue assignment for descriptor systems via state feedback ', IMA J. Math. Control Inf., 2000, 17, (1), pp. 57– 66 23Liu S. Wei G. Song Y. et al.: 'Extended Kalman filtering for stochastic nonlinear systems with randomly occurring cyber attacks ', Neurocomputing, 2016, 207, pp. 708– 716 24Geng H. Wang Z. Liang Y. et al.: 'Tobit Kalman filter with time-correlated multiplicative sensor noises under redundant channel transmission ', IEEE Sensors J., 2017, 17, (24), pp. 8367– 8377 Volume14, Issue17November 2020Pages 2696-2702 ReferencesRelatedInformation