Linear dynamic filtering with noisy input and output 1

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1 Katholieke Universiteit Leuven Departement Elektrotechniek ESAT-SISTA/TR Linear dynamic filtering with noisy input and output 1 Ivan Markovsky and Bart De Moor 2 2 November 22 Published in the proc of 13th IFAC Symposium on System Identification (SYSID-23) pages This report is available by anonymous ftp from ftpesatkuleuvenacbe in the directory pub/sista/markovsky/reports/2-191psgz 2 KULeuven, Dept of Electrical Engineering (ESAT), Research group SCD (SISTA), Kasteelpark Arenberg 1, 31 Leuven-Heverlee, Belgium, Tel 32/16/ , Fax 32/16/ , WWW: ivanmarkovsky@esatkuleuvenacbe Research Council KUL: GOA Mefisto 666, several PhD/postdoc & fellow grants; Flemish Government: - FWO: PhD/postdoc grants, projects, G2499 (multilinear algebra), G472 (support vector machines), G1972 (power islands), G1413 (Identification and cryptography), G4913 (control for intensive care glycemia), G123 (QIT), research communities (ICCoS, ANMMM); - AWI: Bil Int Collaboration Hungary/Poland; - IWT: PhD Grants, Soft4s (softsensors), Belgian Federal Government: DWTC (IUAP IV 2 ( ) and IUAP V 22 (22 26), PODO II (CP/4: TMS and Sustainibility); EU: CAGE; ERNSI; Eureka 263 IMPACT; Eureka 2419 FliTE; Contract Research/agreements: Data4s, Electrabel, Elia, LMS, IPCOS, VIB;

2 Abstract We establish the equivalence between the optimal least-squares state estimator for a linear time-invariant dynamic system with noise corrupted input and output, and an appropriately modified Kalman filter The approach used is algebraic and the result shows that the noisy input/output filtering problem is not fundamentally different from the classical Kalman filtering problem The result is illustrated with a simulation example

3 LINEAR DYNAMIC FILTERING WITH NOISY INPUT AND OUTPUT Ivan Markovsky and Bart De Moor ESAT, SCD-SISTA, KULeuven, Kasteelpark Arenberg 1, B-31 Leuven-Heverlee, Belgium Tel: , Fax: Abstract: We establish the equivalence between the optimal least-squares state estimator for a linear time-invariant dynamic system with noise corrupted input and output, and an appropriately modified Kalman filter The approach used is algebraic and the result shows that the noisy input/output filtering problem is not fundamentally different from the classical Kalman filtering problem The result is illustrated with a simulation example Keywords: dynamic errors-in-variables model, Kalman filtering, optimal smoothing, total least squares 1 INTRODUCTION Optimal least-squares state estimation for linear dynamical systems is a well developed topic with many practical applications A central result is the Kalman filter In the discrete-time case, one considers the model x(t + 1) = Ax(t)+Bu(t)+w(t), y(t) = Cx(t)+Du(t)+v(t), (1) with x() = x and t =,1, Here u(t) R m, y(t) R l, and x(t) R n are the input, output, and state vectors at time instant t When the model matrices A, B, C, and D are known, the state x depends linearly on the input u and the output y, so that it can be estimated from the input/output (I/O) data via the leastsquares method The unknown process noise w and the measurement noise v play the role of equation errors in (1) Even though v and w are unknown, we assume that they are zero mean, white, Gaussian noises, with covariance matrix E w(t) v(t) ] w T (t + τ) v T (t + τ)] = Q(t) S(t) S T δ(τ) (t) R(t) The importance of the Kalman filter is that it solves the least-squares problem recursively and in real time Model (1), however, is in a certain sense asymmetric The process noise w acts as an unobserved input while the measurement noise v represents measurement error on y However, u is assumed to be noiseless! In the paper, we pose and answer the question: How should we modify the Kalman filter when both the input and the output of the system are noisy? The paradigm of treating the input and the output on an equal footing leads to the behavioral approach (Polderman and Willems, 1998) A derivation of the Kalman filter in the behavioral context is given in (Fagnani and Willems, 1997) In this paper, we consider the deterministic discretetime LTI state-space system x(t + 1) = Ax(t)+Bu(t), x() = x, y(t) = Cx(t)+Du(t), t =,1, together with the measurement errors model (2) u d (t) = u(t)+ũ(t), = y(t)+ỹ(t) (3) and refer to the model (2) together with the measurement errors model (3) as the noisy I/O model (see Figure 1) We note that (2 3) is an errors-in-variables (EIV) model in the sense of, eg, (Söderström et

4 al, 22; Zheng, 22), when the problem would be to estimate A, B, C, and D But since the purpose of this paper is to estimate x, knowing A, B, C, and D, we call it a noisy I/O model Kalman filter Section 6 further establishes the optimal estimates of the true input/output signals In Section 7, we confirm the results on an example and in Section 8, give conclusions x u ũ u d (A,B,C,D) Fig 1 Block scheme of the noisy I/O model A full analysis of a static EIV estimation problem, also known as the total least squares (TLS) problem, is given in the monograph (Van Huffel and Vandewalle, 1991) The dynamic equivalent of the TLS problem, see (Aoki and Yue, 197; De Moor and Roorda, 1994), is a system identification problem: given noisy I/O measurements, find the model (2) The dynamic TLS problem can be expressed as a static TLS problem with Toeplitz (or Hankel) structured data matrices The structured total least squares problem is treated in (De Moor, 1993; Lemmerling, 1999) Different type of estimation problem, in the EIV context, occurs when the model is exactly known and the state vector has to be estimated from the noisy I/O measurements We refer to this latter problem as the noisy I/O state estimation problem In (Guidorzi et al, 23; Diversi et al, 23), the noisy I/O state estimation problem has been considered, using the language of transfer functions The authors there claim that it is fundamentally different problem from the Kalman filtering problem and describe new recursive algorithms for its solution In this paper, we prove that the noisy I/O state estimation problem is not fundamentally different from the Kalman filtering problem Its solution boils down to the solution of a sequence of linear least-squares problems with a special structure coming from the state space equation (1) Our approach is linear algebraic We represent the model over a finite time horizon as a set of linear equations and apply standard linear algebra techniques for its analysis The optimal solution is shown to be a Kalman filter with correlated process and measurement noises The continuous-time version of the noisy I/O state estimation problem is treated in (Markovsky et al, 22), where a completion of squares approach is used and the solution is also shown to be a Kalman filter-type system In Section 2, we define the smoothing and filtering noisy I/O state estimation problems Two explicit block solutions of the smoothing problem are derived in Section 3 They are weighted least-squares problems In Section 4, we transform the noisy I/O model in the form (1) and define the Kalman filter for the resulting system as the modified Kalman filter In Section 5, we prove the deterministic equivalence of the estimates of the noisy I/O filter and the modified y d y ỹ 2 PROBLEM FORMULATION An on-line, recursive estimation procedure can be realized by a causal dynamical system called a filter Such a filter operates on previous and current measurements and produces an estimate of the to-beestimated signal for the current moment of time The problem of a filter synthesis is referred to as a filtering problem A smoothing problem is an estimation problem, in which on the basis of the available measurements, an estimate for the to-be-estimated signal is produced for the whole (past) period of observation We introduce some notation used in the rest of the paper A signal variable, without time index, denotes the vector obtained by stacking one over another the signal samples for the consecutive time instances For example, over the time horizon,1,,t f 1, the vector of the consecutive input samples u is defined as u := u T () u T (t f 1)] T and the vector of the consecutive state samples is x := x T () x T (t f 1) x T (t f )] T For a time indexed matrix sequence {V(t)} t f 1 t=, we denote by V, without time argument, the block matrix V := blk diag ( V(),,V(t f 1) ) Definition 1 (Optimal noisy I/O smoothing problem) Assume that the measurement errors ũ and ỹ are random, centered, normal, uncorrelated, and white with known covariance matrices cov ( ũ(t) ) =: Vũ(t), cov ( ỹ(t) ) =: Vỹ(t), (4) and that the initial condition x is unknown Then, given the matrices A, B, C, D, the optimal noisy I/O smoothing problem is defined as 1/2 û ] min Vũ ud 2 û,ŷ, ˆx Vỹ ŷ y d (5) ˆx(t + 1) = A ˆx(t)+Bû(t) st ŷ(t) = C ˆx(t)+Dû(t) for t =,1,,t f 1 The optimal smoothed state estimate ˆx(,t f ) is the solution of (5) Under the normality assumption for the noises, ˆx(,t f ) is the minimum variance estimate, the maximum likelihood estimate, and the conditional expectation estimate of the state x The equivalence is well known in the Kalman filter case (Willems, 22; Bryson and Ho, 1975; Anderson and Moore, 1979), and is shown in the noisy I/O case in (Guidorzi et al, 23)

5 Definition 2 (Optimal noisy I/O filtering problem) Given the model (2 3), satisfying the assumptions of Definition 1, the optimal noisy I/O filtering problem is to find a dynamical system, z(t + 1) = A f (t)z(t)+b f (t), (6) ˆx(t) = C f (t)z(t)+d f (t), such that ˆx(t) = ˆx(t,t + 1), where ˆx( ) is the solution of (6), ie, the optimal filtered state estimate, and ˆx(,t + 1) is the optimal smoothed state estimate with a time horizon t SMOOTHING BY BLOCK PROCESSING In this section, we write the optimal noisy I/O smoothing problem (5) as a weighted least-squares (WLS) problem This representation is used as a conceptual tool for the analysis and not as a means to carry out the actual computations needed for the estimation We represent the I/O dynamics of the system (2), over the time horizon,,t f 1, explicitly as where Γ := C CA CA t f 2 y = Γx + Tu, (7) H() H(1) H(), and T := H(t f 1) H(t f 2) H() The matrix Γ is an extended observability matrix and T is a Toeplitz matrix formed from the Markov parameters H() = D, H(t) = CA t 1 B, t = 1,,t f 1 Using (7), we see that the optimal noisy I/O smoothing problem (5) is a weighted least-squares problem 1 ( ) min Vũ 2 ud I 2 ˆx,û (8) Vỹ y d Γ T]ˆx û Alternatively, we represent the input/state/output dynamics of the system, over the time horizon,,t f 1, as y() y(1) y(t f 1) C A I x() C x(1) = A I x(2) + C x(t f ) A I } {{ } A D B D u() + B u(1) D u(t f 1) B } {{ } B Substituting y d ỹ for y and u d ũ for u (see (3)), we have y d () y d (1) u d () u d (1) +B = y d (t f 1) u d (t f 1) }{{} ȳ d I ũ() ũ(1) I ỹ() = A x+b + ỹ(1), ũ(t f 1) I ỹ(t f 1) } {{ } C or with the definition of the new variables ȳ d +Bu d = A x+bũ+cỹ (9) Using (9), the optimal noisy I/O smoothing problem is equivalent to the following problem 1/2 min Vũ u 2 ˆx, u, y Vỹ y (1) u st ȳ d +Bu d = A ˆx+B C], y which solution is given in Section 5 The solutions (8) and (1) of the noisy I/O smoothing problem are not recursive and thus not practical for large data sets See (Markovsky et al, 22) for recursive solution of an equivalent continuous-time problem In the discrete-time case, the recursive solution is given by two time-varying filters; one running backward in time and one running forward in time The forward recursion is defined by a time-varying Riccati equation The backward filter produces the optimal smoothed state estimate 4 THE MODIFIED KALMAN FILTER We convert the noisy I/O model (2 3) in the form (1) by substituting u d (t) ũ for u(t) and ỹ for y(t) (see (3)) in (2) x(t + 1) = Ax(t)+Bu d (t) Bũ(t) = Cx(t)+Du d (t) Dũ(t)+ỹ(t) and define (fake) process noise w and measurement noise v by w := Bũ and v := Dũ+ỹ The resulting system x(t + 1) = Ax(t)+Bu d (t)+w(t) = Cx(t)+Du d (t)+v(t) is in the form (1), where Q(t) S(t) B Vu (t) S T = (t) R(t) D I (11) ] T B V y (t) D I

6 We call the Kalman filter of the modified system (11), ie, the system z(t + 1) = A KF (t)z(t)+b KF (t), (12) ˆx(t) = C KF (t)z(t)+d KF (t), where A KF (t) = ( A K(t)C ), B KF (t) = B K(t)D, K(t) ], and C KF (t) = I P(t)C T( CP(t)C T + R(t) ) 1 C, D KF (t) = P(t)C T( CP(t)C T + R(t) ) 1 D I ], K(t) = ( AP(t)C T + S(t) )( CP(t)C T + R(t) ) 1, P(t + 1) = AP(t)A T ( AP(t)C T + S(t) ) ( CP(t)C T + R(t) ) 1( AP(t)C T + S(t) ) T + Q(t), the modified Kalman filter It recursively solves (9) for the last block entry of the unknown x The solution is in the sense of the weighted least-squares problem 1/2 min Ve ê 2 st ȳ d +Bu d = A ˆx+ê (13) ˆx,ê The variable ê accounts for the cumulative noise ũ ] e := B C] ỹ added to the equation and the covariance matrix of e is V e = BVũB T +CVỹC T When the measurement noise covariances Vũ(t) and Vũ(t) does not depend on t, one can replace the timevarying Kalman filter with the (suboptimal) timeinvariant filter, obtained by replacing P(t) in (12) with the steady-state solution P of the algebraic Riccati equation P = APA T ( A PC T +S )( C PC T +R ) 1 ( A PC T +S ) T +Q In the following section, we investigate the relation between the modified Kalman filter (12) state estimate and the noisy I/O filter state estimate 5 EQUIVALENCE OF THE MODIFIED KALMAN FILTER AND THE NOISY I/O FILTER Consider the linear system of equations (9) and the two solution methods (1) and (13) Denote ũ ] Vũ z := ȳ d +Bu d, δ :=, V ỹ δ :=, Vỹ u ˆδ :=, and D := B C] y We want to find a relation between the solutions of the following problems 1/2 min Ve ê 2 2 st z = A ˆx+ê, (14) ˆx,ê where V e = DV δ D T and min ˆx, ˆδ V 1/2 δ ˆδ 2 2 st z = A ˆx+D ˆδ (15) The first problem is a weighted least-squares problem and its solution is ˆx KF = (A T Ve 1 A ) 1 A T Ve 1 z (16) The noisy I/O estimation problem (15) is a minimumnorm type problem and its solution is (see Lemma 3 in the Appendix) ˆx = I]V e A A ] 1 z, (17) where A is a matrix which columns form a basis for the orthogonal complement of the range space of A We transform (16) and (17) by the change of variables Then and A := V 1/2 e A and z := V 1/2 e z ˆx KF = ( A T A ) 1 A T z, (18) ˆx = I] A A ] 1 z (19) ((19) follows from the identity A = Ve 1/2 A ) Now the question of the solutions equivalence is answered by Theorem 4, see the Appendix, which states that ˆx KF = ˆx Thus the two solutions are deterministically equal and the noisy I/O filtering problem is solved by the modified Kalman filter (12), ie A f = A KF, B f = B KF, C f = C KF, and D f = D KF 6 OPTIMAL ESTIMATION OF THE TRUE INPUT/OUTPUT SIGNALS Up to now we were interested in the optimal filtering in the sense of state estimation In this section, we show how the optimal estimates of the input and the output can be derived from the modified Kalman filter The solutions ê and ˆδ of (14) and (15), respectively, satisfy the following relation ê = V e A V e A A ] 1 z = DV δ D T A V e A A ] 1 z = D ˆδ This implies that the state estimate ˆx, the one-stepahead prediction z(t + 1), and the optimal input estimate û satisfy the equation z(t + 1) = A ˆx(t)+Bû(t) (2) Then we can find û exactly from ˆx and z(t + 1), obtained from the modified Kalman filter (12) In fact, (2) and the Kalman filter equations imply that û(t) = E(t)z(t)+F(t), where E(t) := VũD T ( CP(t)C T + R(t) ) 1 C and F(t) := I VũD T ( CP(t)C T + R(t) ) 1 D, VũD T( CP(t)C T + R(t) ) ] 1 The optimal output estimate is ŷ(t) = C ˆx(t) + Dû(t)

7 7 NUMERICAL EXAMPLE In this section, we verify numerically the equivalence of the solutions established in Section 5 The particular system, we use, is A = ], B = 1 ], C = ], and D = 5381 The time horizon is t f = 1, the initial state is x =, and the input signal is a normal white noise sequence with unit variance The input and the output of the system are corrupted by independent, centered, normal, white noises with variances var(ũ(t)) = 4 and var(ỹ(t)) = 4 for all t The estimate of the noisy I/O filter is computed directly from the definition, ie, we solve a sequence of optimal smoothing problems with increasing timehorizon Every smoothing problem is a weighted leastsquares problem that is solved explicitly according to (8) The last block entries of the obtained sequence of solutions form the noisy I/O filter state estimate We compare the noisy I/O filter estimate with the estimate of the modified Kalman filter (12) The experiment is carried out in MATLAB The state estimate ˆx KF obtained by the modified Kalman filter is up to the numerical errors equal to the state estimate ˆx f obtained by the noisy I/O filter, ˆx KF ˆx f = 57723e 15 This is the desired numerical verification of the theoretical result of the paper The absolute errors of estimation ˆx x 2, û u 2, ŷ y 2 for all estimation methods, discussed in the paper is given in Table 1 Table 1 Comparison of the absolute errors of the state, input, and output estimates for all methods and the noisy data (MKF modified Kalman filter) Method ˆx x 2 û u 2 ŷ y 2 optimal smoothing optimal filtering time-varying MKF time-invariant MKF noisy data CONCLUSIONS We considered optimal noisy I/O estimation problems for discrete-time LTI systems The filtering problem is solved via a modified Kalman filter The equivalence between the optimal noisy I/O filter and the modified Kalman filter is proven algebraically using explicit state-space representation of the system ACKNOWLEDGEMENTS Ivan Markovsky is a research assistant and Dr Bart De Moor is a full professor at the Katholieke Universiteit Leuven, Belgium Our research is supported by Research Council KUL: GOA Mefisto 666, several PhD/postdoc & fellow grants; Flemish Government: FWO: PhD/postdoc grants, projects, G2499 (multilinear algebra), G472 (support vector machines), G1972 (power islands), G1413 (Identification and cryptography), G4913 (control for intensive care glycemia), G123 (QIT), research communities (ICCoS, ANMMM); AWI: Bil Int Collaboration Hungary/Poland; IWT: PhD Grants, Soft4s (softsensors), Belgian Federal Government: DWTC (IUAP IV 2 ( ) and IUAP V 22 (22 26), PODO II (CP/4: TMS and Sustainibility); EU: CAGE; ERNSI; Eureka 263 IMPACT; Eureka 2419 FliTE; Contract Research/agreements: Data4s, Electrabel, Elia, LMS, IPCOS, VIB 9 REFERENCES Anderson, B D O and J B Moore (1979) Optimal Filtering Prentice Hall Aoki, M and P C Yue (197) On certain convergence questions in system identification SIAM J Control 8(2), Bryson, A and Y Ho (1975) Applied Optimal Control Hemisphere, Washington, DC De Moor, B (1993) Structured total least squares and L 2 approximation problems Lin Alg and Its Appl , De Moor, B and B Roorda (1994) L 2 -optimal linear system identification structured total least squares for SISO systems In: In the proceedings of the CDC pp Diversi, R, R Guidorzi and U Soverini (23) Algorithms for optimal errors-in-variables filtering Systems & Control Letters 48, 1 13 Fagnani, F and J C Willems (1997) Deterministic Kalman filtering in a behavioral framework Systems & Control Letters 32, Guidorzi, R, R Diversi and U Soverini (23) Optimal errors-in-variables filtering Automatica 39, Lemmerling, P (1999) Structured total least squares: analysis, algorithms and applications PhD thesis ESAT/SISTA, KU Leuven Markovsky, I, J C Willems and B De Moor (22) Continuous-time errors-in-variables filtering In: Proc of the Conference on Decision and Control pp Meyer, C D (2) Matrix Analysis and Applied Linear Algebra SIAM Polderman, J W and J C Willems (1998) Introduction to mathematical systems theory Springer- Verlag Söderström, T, U Soverini and K Mahata (22) Perspectives on errors-in-variables estimation for dynamic systems Signal Processing 82, Van Huffel, S and J Vandewalle (1991) The total least squares problem: Computational aspects and analysis SIAM, Philadelphia Willems, J C (22) Deterministic Kalman filtering J of Econometrics, to appear

8 Zheng, W X (22) A bias correction method for identification of linear dynamic errors-invariables models IEEE Trans on Aut Control 47(7), Appendix A SOLUTION OF THE OPTIMIZATION PROBLEM (15) AND PROOF OF THE STATE ESTIMATES EQUIVALENCE Lemma 3 Assuming that A is full rank and V δ is positive definite The solution of the minimum-norm type problem (15) is ˆδ Vδ D = T A ] V ˆx I e A A ] 1 z PROOF The Lagrangian of (15) is L( ˆx, ˆδ,λ) = ˆδ T V 1 δ ˆδ + λ T (A ˆx+Dδ z) The first order optimality condition L 1 = 2V ˆδ +D T λ = δ δ ˆδ = 1 2 V δd T λ, (A1) L ˆx = A T λ = λ = 2A λ, (A2) L λ = A ˆx+D ˆδ z = z = A ˆx+D ˆδ (A3) is a necessary and sufficient condition for a global minimum The matrix A is any matrix which columns form a basis for the range space of A T Substituting (A1) and (A2) in (A3), we have ] λ DV δ D T A A ] = z ˆx Using V e = DV δ D T and the assumption that A is full rank, the result follows Theorem 4 For a full rank matrix A R m n, with m greater then n, 1, (A T A) 1 A T = n (m n) I n ] A A] (A4) and the right-hand side is 1 I n ] A A = In ] U 2 U 1 Σ 1 V T] 1 U12 U 11 Σ 1 V T 1 = I n ] U 21 Σ 1 V T (A5) U 22 To find explicitly the inverse matrix in (A5), we use the formula for inverse of a block matrix (Meyer, 2, p123) B C D E where ] 1 = B 1 + B 1 CS 1 DB 1 B 1 CS 1 S 1 DB 1 S = E DB 1 C S 1 ], is the Schur complement of B in the matrix B C D E] For the block matrix in (A5), the Schur complement of U 12 is Then I n ] S = U 21 Σ 1 V T U 22 U 1 12 U 11Σ 1 V T = (U 21 U 22 U 1 12 U 11)Σ 1 V T A A] 1 = VΣ 1 1 (U 21 U 22 U 1 12 U 11) 1 U 22 U 1 12 I] (A6) Because of the orthogonality of U, U T 1 U 2 = U T 11U 12 +U T 21U 22 = Then U 22 U 1 12 = U T 21 U T 11 (A7) (U 21 U 22 U 1 12 U 11) 1 = (U 21 +U T 21 U T 11 U 11) 1 = ( U21 T (U 21 T U 21 +U11 T U 11) ) 1 = U T }{{} 21 (A8) U1 T U 1=I Substitution of the expressions of (A7) and (A8) into (A6) establishes the identity where A is a matrix which columns form a basis for the orthogonal complement of the range space of A, ie, Range(A ) = Null(A T ), and rank(a ) = m n PROOF In the proof, we use the SVD of the matrix A A = UΣV T Σ1 = U 1 U 2 ] V T = U 1 Σ 1 V T Partition U as follows n m n U = U 1 U 2 ] = U11 U 12 U 21 U 22 m n n The matrix U 2 satisfies Range(U 2 ) = Null(A T ) and rank(u 2 ) = m n, so it serves as a particular A The left-hand side of the desired identity (A4) is (A T A) 1 A T = VΣ 1 1 U 1 T = VΣ 1 1 U T 11 U21 T ],

Structured weighted low rank approximation 1

Departement Elektrotechniek ESAT-SISTA/TR 03-04 Structured weighted low rank approximation 1 Mieke Schuermans, Philippe Lemmerling and Sabine Van Huffel 2 January 2003 Accepted for publication in Numerical