Downloaded from orbit.dtu.dk on: Jul 20, 2020 A Generalized Autocovariance Least-Squares Method for Covariance Estimation Åkesson, Bernt Magnus; Jørgensen, John Bagterp; Poulsen, Niels Kjølstad; Jørgensen, Sten Bay Published in: American Control Conference 2007 Link to article, DOI: 10.1109/ACC.2007.4282878 Publication date: 2007 Document Version Publisher's PDF, also known as Version of record Link back to DTU Orbit Citation (APA): Åkesson, B. M., Jørgensen, J. B., Poulsen, N. K., & Jørgensen, S. B. (2007). A Generalized Autocovariance Least-Squares Method for Covariance Estimation. In American Control Conference 2007 IEEE. https://doi.org/10.1109/ACC.2007.4282878 General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights.  Users may download and print one copy of any publication from the public portal for the purpose of private study or research.  You may not further distribute the material or use it for any profit-making activity or commercial gain  You may freely distribute the URL identifying the publication in the public portal If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Proceedings of the 2007 American Control Conference Marriott Marquis Hotel at Times Square New York City, USA, July 11-13, 2007 ThC07.2 A Generalized Autocovariance Least-Squares Method for Covariance Estimation Bernt M. Åkesson Sten Bay Jørgensen John Bagterp Jørgensen CAPEC Informatics and Mathematical Modelling CAPEC Department of Chemical Engineering Technical University of Denmark Department of Chemical Engineering Technical University of Denmark DK-2800 Kgs. Lyngby Technical University of Denmark DK-2800 Kgs. Lyngby Email: jbj@imm.dtu.dk DK-2800 Kgs. Lyngby Email: baa@kt.dtu.dk Email: sbj@kt.dtu.dk TABLE I D EFINITIONS OF S YMBOLS IN (8) FOR E ACH K ALMAN F ILTER F ORM Abstract— A generalization of the autocovariance leastsquares method for estimating noise covariances is presented. The method can estimate mutually correlated system and sensor noise and can be used with both the predicting and the filtering form of the Kalman filter. Index Terms— Covariance estimation, optimal estimation, state estimation. I. INTRODUCTION The Kalman filter requires knowledge about the noise statistics. In practical applications, however, the noise covariances are generally not known. The autocovariance leastsquares (ALS) method was presented by Odelson et al. [1] as a technique for estimating the system and sensor noise covariances from plant data. The technique was shown to give unbiased estimates with smaller variance than previously proposed methods, such as the correlation method by Mehra [2]. The objective of this paper is to demonstrate how the method can be extended to systems where the system noise and the sensor noise are mutually correlated. Moreover, the generalized method works with both the predicting and the filtering form of the Kalman filter. II. GENERALIZED AUTOCOVARIANCE LEAST-SQUARES ESTIMATION Consider a linear time-invariant system in discrete-time, xk+1 = Axk + Buk + Gwk yk = Cxk + vk ek Ā Ḡ yk −Cx̂k|k−1 A − K pC [G −K p ] yk −Cx̂k|k A − K pC [G −K p ] C̄ H̄ C I C −CK f C I −CK f and the Kalman filter gains are defined as K p = (APpCT + GSwv )(CPpCT + Rv )−1 T T −1 K f = PpC (CPpC + Rv ) (5) (6) and Pp is the covariance of the state prediction error, x̃k|k−1 = T ] is obtained xk − x̂k|k−1 . The covariance Pp = E[x̃k|k−1 x̃k|k−1 as the solution to the Riccati equation Pp = APp AT + GQw GT T − (APpCT + GSwv )(CPpCT + Rv )−1 (CPp AT + Swv GT ). (7) x̃k+1|k = Āx̃k|k−1 + Ḡw̄k (3) ek = C̄x̃k|k−1 + H̄vk (8) where ek and the system matrices have different definitions, as shown in Table I, depending on which form of the filter is used. The noise w̄k in (8) is defined as   w w̄k = k vk with properties      Q̄ w̄k 0 ∼N , Tw vk 0 S̄wv S̄wv Rv  (9) where or in the filtering form, 1-4244-0989-6/07/$25.00 ©2007 IEEE. Filtering Form (1) Assume that a suboptimal stationary Kalman filter is used to estimate the state. The filter is based on initial guesses of covariances Qw , Rv and Swv . The filter can be either in the one-step predicting form, x̂k|k = x̂k|k−1 + K f (yk −Cx̂k|k−1 ), Predicting Form A general state-space model of the measurement prediction/estimate error can be defined, where A ∈ Rnx ×nx , B ∈ Rnx ×nu , G ∈ Rnx ×nw and C ∈ Rny ×nx . The process noise wk and the measurement noise vk are zeromean white noise processes according to       Qw Swv 0 wk ∼N , T . (2) vk 0 Swv Rv x̂k+1|k = Ax̂k|k−1 + Buk + K p (yk −Cx̂k|k−1 ), Symbol (4) Q̄w = E[w̄k w̄Tk ] =  Qw T Swv    Swv S T , S̄wv = E[w̄k vk ] = wv . Rv Rv 3713 Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on November 18, 2009 at 09:39 from IEEE Xplore. Restrictions apply. ThC07.2 The autocovariance of the measurement prediction or estimate error is given by Re,0 = E[ek eTk ] = C̄PpC̄T + H̄Rv H̄ T Re, j = E[ek+ j eTk ] = C̄Ā j PpC̄T + C̄Ā j−1 GSwv H̄ T − C̄Ā j−1 K p Rv H̄ T , j ≥ 1 Re (L) = OPp OT + Z +Z " +Ψ L M "i=1 #i=1 + [(Ψ ⊗ Z)Unw ,ny ,L − D(In2x + Tnx ,nx )(K p ⊗ G) (11) .. . Re,0 " # L M T Swv i=1 # ZT Rv ΨT , i=1 (13) Z=Γ L M # Ψ=Γ G , i=1 " # L M (−K p ) + i=1 L M H̄ i=1 and   C̄  C̄Ā    O =  . ,  ..  C̄ĀL−1  0 C̄ C̄Ā .. .    Γ=   C̄ĀL−2 ··· 0 0 0 0 0 0 .. . ··· C̄Ā C̄  0 0  0 . ..  . 0 We apply the vec operator to (13) in order to state the problem as a linear least-squares problem. The vec operator performs stacking of the matrix columns to form a column matrix [3]. By applying the rules for the vec operator, we write the Lyapunov equation for Pp in stacked form, with the subscript s used as shorthand for the vec operator, i.e. vec (A) = As . vec(Pp ) = vec(ĀPp ĀT ) + vec(ḠQ̄w ḠT ) = (In2x − Ā ⊗ Ā)−1 (ḠQ̄w ḠT )s (14) We introduce three permutation matrices. For an m × n matrix A we define a permutation matrix Um,n,L , which is an mnL2 × mn matrix of zeros and ones satisfying ! vec L M A D = (O ⊗ O)(In2x − Ā ⊗ Ā)−1 . (18) = Um,n,L vec (A) . d Given a sequence of data {ei }i=1 , the estimate of the autocovariance can be computed by 1 Nd − j ∑ ei+ j eTi , Nd − j i=1 (19) where Nd is the length of the data sequence. The estimated autocovariance matrix R̂e (L) can be formed analogously to (12) using the estimates (19). Now (17) can be written in the form of a linear least-squares problem   2 (Qw )s (20) A (Swv )s  − (R̂e (L))s Φ = min Qw ,Swv ,Rw (Rv )s 2 where additional constraints may be necessary in order to ensure positive semidefiniteness of the covariance matrices. As noted in [1], a short data sequence or significant model error may result in covariance estimates that are not positive definite. This problem can be remedied by stating (20) as a convex semidefinite programming problem [1]. The optimal Kalman filter gain can then be computed from the estimated covariances by (5) or (6) after solving the Riccati equation (7). Note that for the case with a predicting Kalman filter and with Swv = 0, (17) takes the form presented in [1]. where " (17) R̂e, j = Qw Z T T + (Z ⊗ Ψ)Uny ,nw ,L Tny ,nw ](Swv )s N (12)   # Swv Ψ + Ψ L M (10)  RTe,L−1 RTe,L−2   and can be written as L M (Re (L))s =[(Z ⊗ Z)Unw ,L + D(G ⊗ G)](Qw )s + [(Ψ ⊗ Ψ)Uny ,L + D(K p ⊗ K p )](Rv )s in which The autocovariance matrix is defined as  ··· Re,0 RTe,1  Re,1 R e,0  Re (L) =  . ..  .. . Re,L−1 Re,L−2 · · · " Applying the vec operator to (13) yields (15) i=1 For a square matrix of size p × p we have the permutation matrix U p,L = U p,p,L . Finally, there is the vec-permutation matrix (or commutation matrix) Tm,n , such that for an m × n matrix A [3], [4],
vec AT = Tn,m vec (A) . (16) III. CONCLUSIONS AND FUTURE WORKS A. Conclusions A generalization of the autocovariance least-squares method by Odelson et al. has been presented. The generalization is applicable to systems with mutually correlated disturbances and also works with data generated by the filtering form of the Kalman filter. B. Future Works Solution methods for the constrained least-squares problem will be investigated. The estimation method will be applied to realistic examples. R EFERENCES [1] B. J. Odelson, M. R. Rajamani, and J. B. Rawlings, “A new autocovariance least-squares method for estimating noise covariances,” Automatica, vol. 42, no. 2, pp. 303–308, 2006. [2] R. K. Mehra, “On the identification of variances and adaptive kalman filtering,” IEEE Transactions on Automatic Control, vol. AC-15, no. 12, pp. 175–184, 1970. [3] J. W. Brewer, “Kronecker products and matrix calculus in system theory,” IEEE Transactions on Circuits and Systems, vol. 25, no. 9, pp. 772–781, 1978. [4] J. R. Magnus and H. Neudecker, “The commutation matrix: some properties and applications,” The Annals of Statistics, vol. 7, no. 2, pp. 381–394, 1979. 3714 Authorized licensed use limited to: Danmarks Tekniske Informationscenter. Downloaded on November 18, 2009 at 09:39 from IEEE Xplore. Restrictions apply.