ROBUST KALMAN FILTERING FOR LINEAR DISCRETE TIME UNCERTAIN SYSTEMS

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1 International Journal o Advances in Engineering & echnology, Sept 202. IJAE ISSN: ROBUS KALMAN FILERING FOR LINEAR DISCREE IME UNCERAIN SYSEMS Munmun Dutta, Balram imande 2 and Raesh Mandal 3 Department o Electrical & Electronic Engineering, DIMA, Raipur, India 2 Department o Electronic & elecommunication Engineering, DIMA, Raipur, India 3 Department o Electronic & elecommunication Engineering, SSCE, Bhilai, India ABSRAC In this paper, a robust inite-horizon Kalman ilter is designed or discrete time-varying uncertain systems, basically the problem o inite horizon robust Kalman iltering or uncertain discrete-time systems is studied. he system under consideration is subject to time-varying norm-bounded parameter uncertainty in both the state and output matrices. he problem addressed is the design o linear (lters having an error variance with an optimized guaranteed upper bound [] or any allowed uncertainty. he Kalman Filter (KF) is a set o equations that provides an eicient recursive computational solution o the linear estimation problem.basicaly it is a numerical method used to trac a time-varying signal in the presence o noise. In this paper we study the robust Kalman iltering problem or linear uncertain discrete time-delay systems. Developing KF algorithms or such systems is an important problem to obtain optimal state estimates by utilizing the inormation on uncertainties and time-delays. First, designing o KF or nominal discrete-time systems is studied. Considering the covariance o the error in the estimation the KF algorithm is derived which is urther tested on the system under consideration is subject to time-varying norm-bounded parameter uncertainty in both the state and output matrices[6]. KEYWORDS- Uncertain discrete-time systems, Linear discrete time system, Robust Kalman iltering (RKF), Kalman iltering (KF), Kalman gain. I. INRODUCION Filtering is desirable in many situations in engineering and embedded systems. A good iltering algorithm can remove the noise rom electromagnetic signals while retaining the useul inormation. One o the undamental problems in control and signal processing is the estimation o the state variables o a dynamic system through available noisy measurements. Basically the KF is an estimator which estimates the uture state o a linear dynamic system rom series o noisy measurement. It is the problem o estimating the instantaneous state o a linear system rom a measurement o outputs that are linear combinations o the states but corrupted with Gaussian white noise []. It has come to be well recognized that the popular Kalman Filtering theory is very sensitive to system data and has poor perormance robustness when a good system model is hard to obtain or the system drits. hus, the standard Kalman Filter may not be robust against modeling uncertainty and disturbances. his has motivated many studies on robust Kalman Filter design, which may probably yield a suboptimal solution with respect to the nominal system. Our interest in this paper is to tacle the iltering problem or a class o discrete time-delay uncertain systems. Necessary and suicient conditions or the design o such a robust ilter with an optimized upper bound or the error variance were given[3]. 276 Vol. 4, Issue 2, pp

2 International Journal o Advances in Engineering & echnology, Sept 202. IJAE ISSN: II. PROBLEM FORMULAION In this paper irstly we use Kalman Filter to estimate the state o a linear discrete system deined by equation-(3.) & (3.2) and it is observed that KF wors well or linear systems without considering the time-delay[2]. One o the problems with the Kalman ilter is that they may not robust against modeling uncertainties. he Kalman ilter algorithm is the optimal ilter or a system without uncertainties. he perormance o a KF may be signiicantly degraded i the actual system model does not match the model on which the KF was based. Hence a new ilter is proposed which addresses the uncertainties in process and measurement noise covariance deined by equations(4.) & (4.2) and gives better results than the standard Kalman ilter. So in coming sections it has been shown that the KF is the most widely used ilter or its better perormance. It is a very simple and intuitive concept with good computational eiciency. But when uncertainty and time delay is included then the perormance o KF may degraded, so RKF is considered which is robust against large parameter uncertainty and time delay. In coming sections a perormance comparison between RKF and KF is obtained or linear discrete-time uncertain system [5]. III. KALMAN FILERING FOR LINEAR DISCREE IME SYSEM he Kalman ilter (KF) is a tool that can estimate the variables o a wide range o processes. In mathematical terms we would say that a Kalman ilter estimates the states o a linear system. he Kalman ilter not only wors well in practice, but it is theoretically attractive because it can be shown that o all possible ilters, it is the one that minimizes the variance o the estimation error. he KF is an extremely eective and versatile procedure or combining noisy sensor outputs to estimate the state o the system with uncertain dynamics. When applied to a physical system, the observer or ilter will be under the inluence o two noise sources: (i) Process noise, (ii) Measurement noise [3]. In order to use a Kalman ilter to remove noise rom a signal, the process that we are measuring must be able to be described by a linear system. n We use a KF to estimate the state x R o a discrete time controlled system. A linear system is described by the ollowing two equations: x = Ax + + Bw (3.) y = Cx + v (3.2) n m q Where, x R is the system state, y R is the measured output, w R is the process noise, p v R is the measurement noise[5]. n n he matrix A R is the dynamics matrix which relates the sate at time step to the sate at time n m m step +. he matrix B R called noise matrix. he matrix C R relates the state measurement y. 3. he Kalman Filter Design Suppose we have a linear discrete-time system given as equation (3.) and (3.2), our objective is to design KF in the orm o an equation (3.3), and determine a gain matrix which minimizes the mean square o the error e. = ˆ A x + + y (3.3) Note that the matrix A & K both are time varying matrices to be determined in order that the estimation error e = x is guaranteed to be smaller than a certain bound or all uncertainty matrices, i.e., the estimation error dynamics satisies equation (2.4) [2] (( )( ) ) Ε x x S. he matrix l l S R in the Equation (3.4) is the covariance o estimation error. (3.4) 277 Vol. 4, Issue 2, pp

3 International Journal o Advances in Engineering & echnology, Sept 202. IJAE ISSN: Now, we ind a solution to the Kalman iltering problem over inite horizon [ 0, N ] will be given using a Discrete Riccati Equation (DRE) approach. For the purpose, consider the augmented vector [3], x e 2n ξ =, = (3.5) ξ + x + + e + =. = + + (3.6) Using Equation (3.5)-(3.6) we get, e ˆ + = Ax + Bw A x + K Cx + K v (3.7) Where q p w R is the process noise, v R is the measurement noise. Further on solving equation (3.7) we get- ( ) ( ) ( ) e = ˆ ˆ ˆ A K C x A K C x + A K C x + Bw + A x K v (3.8) ( ) + = A + K C + K Ce + K v (3.9) Using equation (3.8) and (3.9), we get the state space equation or the estimation error e as ollows[8], ξ+ = Ac ξ + Gη, (3.0) e w A K C A A K C ξ =,, A c η = v = K C A K C Where, B K G =. 0 K he covariance matrix o ξ o the error system satisying the bound [4], [ ] ξ ξ ξ Σ, K 0 N (3.) ater some algebraic manipulation we get cσ c + Σ A A GW G + where W diag { W, V} =, equation (3.2) is called matrix inequality equation. (3.2) Note that the optimal solution o Equation (3.2) should be o the ollowing partitioned rom [2]. ( ) ( ) ( xe ˆ ) ( ˆ ˆ x x ) Ε ee ˆ Ε e x Σ, Σ2, S 0 Σ = = = 2, 22, 0 P S Ε Ε Σ Σ (3.3) Since our main aim is to reduce covariance o estimation error Σ,, using equation (3.) & (3.3) we get, 278 Vol. 4, Issue 2, pp

4 International Journal o Advances in Engineering & echnology, Sept 202. IJAE ISSN: ( e e ), 0, N ] Ε Σ (3.4) he A and Kalman gain K can be denoted as - A = A C ( ) K = AQ C R ( ) ( ) Ultimately we have (3.5) (3.6) S = AQ A + AQ C R AQ C + BWB (3.7) According to all these result we can say that, the ilter (3.3) is a quadratic estimator with an upper bound o error covariances.[4] IV. ROBUS KALMAN FILERING FOR LINEAR DISCREE-IME UNCERAIN SYSEMS A Robust Kalman Filter (RKF) is a robust version o the KF but with necessary modiication to account or the parameter uncertainty. he RKF usually yields a suboptimal solution with respect to the nominal system but it guarantees an upper bound to the iltering error covariance in spite o large parameter uncertainties. he standard KF approach does not address the issue o robustness against large parameter uncertainty in the linear process model. he perormance o a KF may be signiicantly degraded i the system having uncertainty. In this chapter, we consider the problem o Robust Kalman Filtering or discrete-time systems with norm-bounded parameter uncertainty in both the state and output matrices [2]. 4. Facts Σ (4..)For any real matrices, Σ2 Σ and 3 with appropriate dimension and Σ3 Σ3 ΣΣ2Σ 3 + Σ 2 Σ 3 Σ α ΣΣ + Σ2 Σ 2.[] I, it ollows that, (4..2) Let Σ, Σ 2 and Σ 3 and 0 H ( t ) be a real matrix unction satisying ( ) ( ) µσ Σ < R, the ollowing matrix inequality holds, 2 2 ( ( ) ) ( ) < R = R be real constant matrices o compatible dimensions and ( ) µ ( µ ) Σ 3 +Σ H t Σ2 R Σ 3 +Σ2 H t Σ ΣΣ +Σ3 R Σ2 Σ2 Σ 3.[] H t H t I. hen or any µ > 0 satisying 4.2 he Robust Kalman Filter Design or discrete - time uncertain system n We use a RKF to estimate the state x R o a discrete time uncertain controlled system. he system is described by a linear stochastic dierence equation as[3], x + = ( A + A ) x + Bw (4.) y = ( C + C ) x + v (4.2) In the ollowing v and w will be regarded as zero mean, uncorrelated white noise sequence with covariance R andq. ( 0, ) v = N R (4.3) 279 Vol. 4, Issue 2, pp

5 International Journal o Advances in Engineering & echnology, Sept 202. IJAE ISSN: ( 0, ) w = N Q (4.4) In order to obtain good results rom RKF we are assumed the uncertainty matrix in the ollowing structure [2]. A H F E C = H 2 (4.5) I J where, F K R is an unnown real time varying matrix and H, H2 and E are nown real constant matrices o appropriate dimensions that speciy how the elements o A and C are aected by uncertainty in F. Our objective is to design KF in the orm o an Equation (4.6), and determine a gain matrix which minimize the mean square o the error e, [0] ( ) + = A + K y C + Ce (4.6) ( ε ) ( ε ) A = ε AS E I ES E E e C = ε CS E I ES E E e ( )( ε ) 2 K = AQ C + H H CQ C + R ε the estimation error e = x dynamics must satisies equation (4.0), (( ˆ )( ˆ ) ) Ε x x x x S he matrix (4.7) (4.8) (4.9) (4.0) l l S R in the equation (4.0) is the covariance o estimation error. On urther simpliying we get, e ( ) { ˆ + = A+ A x + Bw A x + K ( C+ C ) x + v } (4.) ( ) + = K Ce + A K C + K Ce + K Cx ˆ + K v (4.2) using equation (4.) & (4.2) we get the state space equation or the estimation error e as, where, e w A K C A A K C ξ =,, A c η = v = K C A K C H K H2 B K Hc =, G =, Ec = [ E E] K H2 0 K he covariance matrix o ξ o the error system satisying the bound, ξ ξ ξ + + Σ+ OR {( ) } ( ) { } Ε Ac + Hc F Ec ξ + Gη Ac + Hc F Ec ξ + Gη Σ + (4.3) Our main aim is to reduce covariance o estimation error i.e S Ε( e e ) Σ, 0, N ] (4.4) 280 Vol. 4, Issue 2, pp

6 International Journal o Advances in Engineering & echnology, Sept 202. IJAE ISSN: Now we are going to select the optimal value o ( ) ( % %, ) ( % ) A = A KC I+ PEM E Σ EM E I+ PEM E ( )( ε ) 2 K = AQ C + H H CQ C + R ε Using above equations we get, Σ = AQA + ε H H + BWB AQC + ε H H, + ( 2 ) ( Rε CQC ) ( AQC ε HH 2 ) + + A and K to minimize (, ) + = + ε + ( + ε 2 ) ( Rε CQC ) ( AQC ε HH 2 ) S AQA H H BWB AQC H H tr Σ, [ 0, N ], (4.5) (4.6) (4.7) + + (4.8) According to all these result we can say that, the ilter (4.6) is a quadratic estimator with an upper S.[5] bound o error covariance V. EXPERIMENAL RESULS & SIMULAIONS 5. Simulation Result o Kalman Filter (KF) Consider the ollowing uncertain discrete time system[3], x = x w δ y = x + v + [ 00 0] Where δ is an uncertain parameter satisying δ 0.3. Note that the above system is o the orm o system (4.)-(4.2) with[3], 0 H =, 2 0, 0 H = E = P = S =, W = V = 0 [ ] Now applying Kalman Filter or the above system,the simulation results are as ollows: 28 Vol. 4, Issue 2, pp

7 International Journal o Advances in Engineering & echnology, Sept 202. IJAE ISSN: FIGURE : No. o Iteration vs. Eigen Values o Covariance Matrix with KF 5.2 Simulation Result o Robust Kalman Filter (RKF) Applying Robust Kalman Filter or the same uncertain discrete time system as described in section 5., the simulation [2] results are as ollows: VI. FIGURE 2: No. o Iteration vs. Eigen Values o Covariance Matrix with RKF CONCLUSION In this paper, a ormulation has been presented or the eective design o RKF or linear discrete-time uncertain system. Necessary and suicient conditions or the design o robust quadratic ilters are obtained. We have also analyzed that when the system has uncertainty, the RKF gives better perormance as compare to KF. Results are illustrated by a numerical example. Fig. shows the graph between no. o iteration and Eigen values o covariance matrix, here when δ =0, the KF perormance is better but, when δ =0.3, the KF perormance is degraded. By increasing the uncertain parameter some spies are included in system and it is diicult to KF to estimate the states. Fig. 2 shows the perormance o RKF it illustrate the graph between no. o iteration and Eigen values o covariance matrix, when δ =0.3. From the above said igures we can see that estimating the states with RKF is better than KF because the Eigen value o covariance matrix settles at 200 or δ =0 and gives poor perormance in case o δ =0.3 but when RKF is considered or a system with δ =0.3 it gives better result than KF, as its eigen values settles at much lower level than o KF. In this paper, we have given a solution to the problem o inite horizon robust Kalman iltering or uncertain discrete-time systems. We have also analyzed the easibility and convergence properties o such robust ilters. Results are illustrated by a numerical example. 282 Vol. 4, Issue 2, pp

8 International Journal o Advances in Engineering & echnology, Sept 202. IJAE ISSN: REFERENCES [] X. Zhu, Y. C. Soh, and L. Xie, Robust Kalman ilters design or discrete time-delay systems, Circuits, Syst., Signal Process., vol. 2, no. 3, pp , [2] Liao, B and Zhang,Z.G,;Chan,S.C A New Robust Kalman Filter-Based Subspace racing Algorithm in an Impulsive Noise Environment, Volume: 57, Issue: 9,IEEE ransaction (200). [3] Mohamed,S.M.K and Nahavandi.S Robust Finite-Horizon Kalman Filtering or Uncertain Discrete-ime Systems,IEEE Proceeding on Automatic control, Volume: 57, Issue: 6 Page(s): (202). [4] Chen,B. and Yu,L;Zhang, Robust Kalman iltering or uncertain state delay systems with random observation delays and missing measurements, IEEE Proceeding on Control heory Appllation, Volume: 5, Issue: 7,Page(s): (20). [5] X. Zhu, Y. C. Soh, L. Xie, Design and Analysis o Discrete-ime Robust Kalman ilter, Automatica, Volume 38, pp , (2002). [6] M. S. Mahmoud, L. Xie, and Y. C. Soh, Robust Kalman Filtering or Discrete State-Delay Systems, IEEE Proceeding on Control heory Appllation, Volume 47, pp-63 68, (2000). [7] Rodrigo Fontes Souto and Joao Yoshiyui Ishihara, Robust Kalman Filter or Discrete-ime Systems With Correlated Noise, Ajaccio, France, [8] Z.Wang, J.Lam and X.Liu, Robust Kalman iltering or Discrete-imr Marovian Jump Delay Sustem, IEEE Signal Processing Letters, Volume, pp (2004). [9] X.Lu, H.Zhang and W.Wang, Kalman Filtering or Multiple ime Delay System, Automatica, Volume 4, pp (2005). [0] I. R. Petersen, A. V. Savin, Robust Kalman Filtering or Signaling and System with Large Uuncertanities, Birhauser Boston, New Yor, (999). [] D.E.Catlin, Estimation Control and the Discrete Kalman Filter, Springer, New Yor, (989) [2] MALAB 6., he MathWors Inc., Natic, MA, 2000). AUHORS BIOGRAPHIES Munmun Dutta, born in Rajasthan, India, in 985. He received the Bachelor degree in Electrical Engineering rom the NI Raipur, India, in 2007 and pursuing Master degree in Power System Engineering with the Department o Electrical & Electronic Engineering rom the DIMA,Raipur, India. Her main area o interest are Control system engineering, Power system engineering, esting & commissioning o electrical installations. Balram imande, Born in 970, currently woring as Reader in department o E&C at DIMA Raipur. He is having 9.5 years Industrial experience and 7.5 years teaching experience. He obtained B.E. (Electronics) rom Nagpur University (INDIA) in 993 and M. ech. orm B.I.. Durg (ailiated to C.S.V..U. Bhilai) in He published 02 papers in National Journal LAB EXPERIMENS, (LE) ISSN no KARENG/200/8386 and 02 Papers in International Journal IJECSE ISSN His area o research/interest is embedded system design and Digital Image Processing. Raesh Mandal, born in Raipur, India, in 985. He received the Bachelor degree in Electronics & elecommunications Engineering rom the MPCCE& Bhilai, India, in 2007 and the Master degree in Electrical Engineering rom NI,Rourela, India. He is currently woring in Shri Shanaracharya echnical Campus As an Assistant Proessor, C.G, India His main area o interest are Control system engineering, Signal processing research. 283 Vol. 4, Issue 2, pp