Optimal Estimation for Continuous-Time Systems With Delayed Measurements

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1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 5, MAY satisfies Lipschitz conition. Base on the result on the existence an uniqueness of solution to CSSP, necessary an sufficient conitions for generalize quaratic stability is obtaine by using S-proceure approach an matrix inequality technique. The propose convex optimization approachguarantees global exponential stability an simultaneously maximizes the tolerable perturbation boun. The approach presente in this note has improve an generalize the results an techniques in the literature. The assumptions of inex one an single equilibrium systems will be relaxe an investigate in our future work. REFERENCES [1] S. P. Boy, L. E. Ghaoui, E. Feron, an V. Balakrishnam, Linear Matrix Inequalities in System an Control Theory. Philaelphia, PA: SIAM, 1994, vol. 15. [2] H. G. Chen an K. W. Han, Improve quantitative measures of robustness for multivariable systems, IEEE Trans. Autom. Control, vol. 39, no. 4, pp , Apr [3] J. H. Chou an W. H. Liao, Stability robustness of continuous-time perturbe escriptor systems, IEEE Trans. Circuits Syst. I, vol. 46, no. 9, pp , Sep [4] L. Dai, Singular Control Systems. Berlin, Germany: Springer-Verlag, [5] C. H. Fang an F. R. Chang, Analysis of stability robustness for generalize state-space systems withstructure perturbations, Syst. Control Lett., vol. 21, pp , [6] J. H. Kim, Comments on Improve quantitative measures of robustness for multivariable systems, IEEE Trans. Autom. Control, vol. 40, no. 9, pp , Sep [7] J. L. Lin an S. J. Chen, Robust analysis of uncertain linear singular systems withoutput feeback control, IEEE Trans. Autom. Control, vol. 44, no. 10, pp , Oct [8] D. W. C. Ho an G. Lu, Robust stabilization for a class of iscrete-time nonlinear systems via output feeback: The unifie LMI approach, Int. J. Control, vol. 76, pp , [9] I. Masubuchi, Y. Kamitane, A. Ohara, an N. Sua, H control for escriptor systems: A matrix inequalities approach, Automatica, vol. 33, pp , [10] Y. Nesterov an A. Nemirovsky, Interior Point Polynomial Methos in Convex Programming. Philaelphia, PA: SIAM, 1994, vol. 13. [11] D. M. Stipanovic an D. D. Siljak, Robust stability an stabilization of iscrete-time nonlinear systems: The LMI approach, Int. J. Control, vol. 74, pp , [12] S. Y. Xu, P. V. Dooren, R. Stefan, an J. Lam, Robust stability an stabilization for singular systems withstate elay an parameter uncertainty, IEEE Trans. Autom. Control, vol. 47, no. 7, pp , Jul [13] V. A. Yakubovich, The S-proceure in nonlinear control theory, VestnikLeningra Univ. Math., vol. 4, pp , Englishtranslation. [14] W. Y. Yan an J. Lam, On quaratic stability of systems withstructure uncertainty, IEEE Trans. Autom. Control, vol. 46, no. 11, pp , Nov [15] A. Rehm an F. Allgower, H control of escriptor systems with norm-boune uncertainties in the system matrices, in Proc. Amer. Control Conf., Chicago, IL, Jun , 2000, pp [16], Self-scheule H output feeback control of escriptor systems, Comput. Chem. Eng., vol. 24, pp , Optimal Estimation for Continuous-Time Systems With Delaye Measurements Huanshui Zhang, Xiao Lu, an Daizhan Cheng Abstract This note focuses on the traitional problem of the Kalman filtering for linear continuous-time systems. Although the problem has been stuie wiely in the past ecaes, little work has been one for the time-elaye systems an some funamental problems remain to be solve. This note proposes a new tool, namely, reorganization innovation analysis approach, to investigate the filtering problem for systems with elaye measurements. The Kalman filter is given in terms of the solution of stanar Riccati equations. The performance is clearly emonstrate through analytical results an simulation. The solve problem in this note is relate with some more complicate problems such as fixe-lag smoothing, control with preview an control with input elays. Inex Terms Continuous-time system, Kalman filter, reorganize innovation analysis, Riccati equations, time-elaye systems. I. INTRODUCTION Linear estimation has important applications in many fiels, such as communication, control, econometrics an signal processing, etc. The problem has attracte significant attention in the past 50 years [4], [6], [7]. There are two main approaches to the linear estimation. One is the minimum variance estimation which is terme as H 2 [5], [10], [16]. The other is the H1 estimation which has emerge as an alternative since the 1980s [6], [11]. For the systems without elay, most of the estimation or control problems uner the two performances have been well stuie. In the case of time-elay, however, the estimation or control problem is muchmore complicate an some problems remain to be investigate. The Kalman filter (H 2 estimation), which aresses the minimization of filtering error covariance, has been a classical tool in signal processing, communication an control applications. It has been wiely stuie via Riccati equation approach[7]. However, the Kalman filtering formulation is only applicable to the stanar systems without elays. In the time elays context, the optimal estimation has been stuie via partial ifferential equation (PDE) [10] for continuous-time systems or state augmentation metho for iscrete-time systems. Note that the PDE is very ifficult to be solve in general (in fat it is impossible to have a analytical solution) an the state augmentation leas to very expensive cost. Very recently, [1] an [2] stuie the estimation an control problem for observation-elay systems via ifferential Riccati-type equations. In this note, we consier the minimum mean square error (MMSE) estimation problem for linear continuous time varying systems with current an time-elay measurements. Suchproblem has obvious applications to many engineering problems. Furthermore, the problem has been shown to be relate with some complicate problems such as H1 fixe-lag smoothing [14], [15], preview control [12] an H1 control withcontrol input signal elays [8], [9]. Our aim is to present the /$ IEEE Manuscript receive September 19, 2004; revise January 8, Recommene by Associate Eitor R. Srikant. This work was supporte by the National Nature Science Founation of China uner Grant The work of H. Zhang was supporte by the Taishan Scholarship Construction Engineering. H. Zhang is with School of Control Science an Engineering, Shanong University, Jinan , China (huanshuizhang@hotmail.com). X. Lu is with Dalian University of Technology, Dalian , China. D. Cheng is with Chinese Acaemy of Sciences, Beijing , China. Digital Object Ientifier /TAC

2 824 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 5, MAY 2006 Kalman filtering formulation for suchproblem in terms of the ifferential Riccati equation. It will be shown that, ifferent from the stanar elay-free system, the Kalman filtering formulation for the time elay systems consists of two stanar Kalman filters withthe same imension as the original systems. The rest of the note is organize as follows. The problem formulation is given in Section II. The main results for the H 2 filter is presente in Section III. In Section IV, a numerical example is constructe to illustrate the main result. Section V contains some concluing remarks. II. PROBLEM FORMULATION We consier the following linear system for H 2 estimation problem: _x(t) =8(t)x(t) +0(t)u(t) (1) x(t) 2 R n an u(t) 2 R r, represent the state an input noise, respectively, 8(t) an 0(t) are boune time-varying matrices withappropriate imensions. Assume that the state x is observe by ifferent systems withelays whichare escribe by y(t) =H(t)x(t)+v(t) (2) y 1(t) =H 1(t)x(t 0 ) +v 1(t); > 0 (3) y(t) 2 R p an y 1(t) 2 R p are respectively the current an the elaye output measurement, v(t) 2 R p an v 1(t) 2 R p are the relate the measurement noises. It is assume that the input u an measurement noise v(t) are from L 2[0;T], an the elaye measurement noise v 1 (t) is from L 2 [; T ], while T > 0 is the time-horizon of filtering. Let y (t) enote the observation of the system (2), (3) at time t an v (t) the relate observation noise at time t, then we have y (t) = y(t); 0 t< col fy(t); y 1(t)g ; t v(t); 0 t< v (t) = (5) col fv(t); v 1 (t)g ; t : Now, we make the following stanar assumptions for the systems (1) (3) Assumption 2.1: The initial state x(0) an the noises u(t), v(t), v 1(t) are mutually uncorrelate white noises with zero means an known covariance matrices as (4) hx(0); x(0)i =5 0 (6) hu(t); u( )i = Q u(t)(t 0 ) (7) hv(t); v( )i = Q v (t)(t 0 ) (8) hv 1(t); v 1( )i = Q v (t)(t 0 ): (9) Problem P: Given the observation ffy ( )gj 0t g, fin a linear least mean square error estimator ^x(tjt) of x(t). In the following, for the convenience of iscussions we will enote that t 1 1 = t 0 : III. MAIN RESULTS The basic iea to eal with the time elay in this note is to reorganize the observations from ifferent channel as new elay-free observations, an introuce the innovation associate with the observations. The optimal filter is then erive by using the reorganization innovation an the projection formula. For the simplicity of iscussions we first suppose that the time t>. The case of 0 t will be consiere later. A. Reorganization Innovation We first efine the reorganization observation in the following lemma. Lemma 3.1: The linear space of Lfy ( ); 0 tg is equivalent to LfY 2( )j 0t Y 1( )j t <t g (10) Y 2 ( ) an Y 1 ( ) are the reorganize new observations as satisfy that with Y 2 ( ) 1 = y( ) y 1 ( + ) (11) Y 1 ( ) 1 = y( ) (12) Y i ( )=H i ( )x( )+V i ( ); i =1; 2 (13) H 2 ( ) 1 = H( ) H 1 ( + ) H 1 ( ) 1 = H( ): (14) V 2 ( ) = 1 v( ) V 1 ( ) = 1 v( ): (15) v 1( + ) Moreover, V i ( ) is white noise with zero mean an covariance matrix as Q V ( )= Qv( ) 0 Q V ( )=Q v ( ): (16) 0 Q v ( + ) Proof: The proof is straightforwar an omitte. In the above, Y 2( ) an Y 1( ) are the reorganization observations. Now we introuce the innovation associate with the reorganization observation. Definition 3.1: Consier the linear space of (10), for any s>t 1 = t 0 enote w 1 (s) 1 = Y 1 (s) 0 ^Y 1 (s) (17) ^Y 1(s) is the projection of Y 1(s) onto linear space LfY 2 ( )j 0t Y 1 ( )j t <<s g : (18) For 0 s t 1, enote w 1 2(s) = Y 2(s) 0 ^Y 2(s) (19) ^Y 2 (s) is the projection of Y 2 (s) onto linear space LfY 2( )j 0<s g : (20) In the above, w 1 (s) an w 2 (s) are the preiction error of the reorganization observation. It is easy to observe that w 1(s) an w 2(s) have the following relationships as: w 2(s) =H 2(s)~x(s; 2) + V 2(s); 0 s t 1 (21) w 1 (s) =H 1 (s)~x(s; 1) + V 1 (s); s > t 1 (22) ~x(s; 2) = x(s) 0 ^x(s; 2); 0 s t 1 (23) ~x(s; 1) = x(s) 0 ^x(s; 1); s > t 1 (24) while ^x(s; 2) is the projection of x(s) onto linear space of (20) an ^x(s; 1) is the projection of x(s) onto linear space of (18). Lemma 3.2: The stochastic process w efine in Definition 3.1 is mutually uncorrelate an fw 2 ( )j 0t w 1 ( )j t <t g (25) spans the same linear space as Lfy ( ); 0 tg. Proof: It is reaily seen from (19) that w 2 (s) for s t 1 (or w 1(s), s > t 1) is a linear combination of the observations

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 5, MAY Fig. 1. True state ^ ( ) an its estimation value withelay. fy 2 ( )j 0s g (or fy 2 ( )j 0t ; Y 1 ( )j t <s g). Conversely, Y 2 (s), s t 1, (or Y 1 (s), s > t 1 ) can be given in terms of a linear combination of w 2( )j 0<s (or fw 2( )j 0t ; w 1( )j t <<sg). Thus, fw 2 ( )j 0t ; w 1 ( )j t <t g spans the same linear space as LfY 2 ( )j 0t ; Y 1 ( )j t <t g or equivalently Lfy ( )j 0t g. Next, we show that fw i(1);i = 1; 2g is an mutually uncorrelate sequence. In fact, for any s > t 1 an t 1 t = t 0 1,it follows from (17) (19) that E w 1 (s)w 2 ( ) 0 = H 1 (s)e ~x(s; 1)w 2 ( ) 0 + E V 1 (s)w 2 () 0 : (26) Note that E[V 1 (s)w 2 ( ) 0 ] = 0. Since ~x(s; 1) is the state preiction error, it follows that E[~x(s; 1)w 2 ( ) 0 ] = 0, an thus E[w 1(s)w 2( ) 0 ]=0, which implies that w 2( ), (0 t 1) is uncorrelate with w 1 (s), (s >t 1 ). Similarly, it can be verifie that w 2 (s) is uncorrelate with w 2 ( ) for s 6= an w 1 (s 0 ) is uncorrelate with w 1( 0) for s 0 6= 0. Hence, fw 2( )j 0t ; w 1( )j t <t g is an innovation sequence. This completes the proof of the lemma. w i ( ), i =1; 2, is terme as reorganize innovation, which plays important role for eriving the optimal estimator. B. Riccati Equation Given time instant t an t 1 = t 0, enote P 2 ( ) 1 = E ~x(; 2)~x(; 2) 0 ; 0 t 1 (27) P 1( ) 1 = E ~x(; 1)~x(; 1) 0 ; > t 1 (28) P 2( ) an P 1( ) are the covariance matrices of preiction error of system state. We shall show that P 2 ( ) an P 1 ( ) satisfy Riccati equation. Theorem 3.1: 1) P 2( ) is the solution of the following Riccati equation: P 2 ( ) =8( )P 2 ()+P 2 ()8 0 ( )0K 2 ( )Q V ( )[K 2 ( )] 0 +0( )Q u ( )0 0 ( ) P 2 (0) = 5 0 (29) K 2 ( )=P 2 ( )H2( 0 )Q 01 V ( ): (30) 2) P 1 ( ) for > t 1 is the solution of the following Riccati equation: P 1 ( ) =8( )P 1()+P 1()8 0 ( )0K 1( )Q V ( )[K 1( )] 0 20( )Q u ( )0 0 ( ) P 1 (t 1 )=P 2 (t 1 ) (31) K 1( )=P 1( )H1( 0 )Q 01 V ( ): (32) Proof: First, it is obvious that P 2 is the solution of the stanar Riccati (29) which is associate with the Kalman filtering of the system (1) an (13) with i =2. Secon, note that for > t 1, ^x(; 1) is the projection of x( ) into the linear space Lfw 2(s)j 0st ; w 1(s)j 0<s<g an can be obtaine by applying the projection formula as ^x(; 1) = 0 t hx( ); w 2 (s)i Q 01 V (s)w 2(s)s + t hx( ); w 1 (s)i Q 01 V (s)w 1(s)s: (33) By ifferentiating bothsies of (33) withrespect to,wehave ^x(; 1) =8( )^x(; 1) + hx( ); w 1 ( )i Q 01 V ( )w 1( ) =8( )^x(; 1) + K 1( )w 1( ) (34) K 1 ( )=P) 1 ()H 0 1( )Q 01 V ( ) withinitial conition ^x(t 1; 2). Therefore, 6 1( ) 1 = h^x(; 1); ^x(; 1)i satisfies the equation 6 1( ) =8( )6 1 ( )+6 1 ( )8 0 ( )+K( ) 1 Q V ( )[K 1 ( )] 0 : (35)

4 826 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 5, MAY 2006 Fig. 2. True state ^ ( ) an its estimation value withelay. It is also clear that 5( ) = 1 hx( ); x( )i satisfies the linear ifferential equation 5( ) =8( )5( )+5()8 0 ( )+0()Q u ( )0 0 ( ): (36) In view of the ecomposition x( )=^x(; 1) + ~x(; 1), we have that 5( ) = 6 1 ( )+P 1 ( ). Then, (31) follows, which completes the proof. C. Optimal Estimate ^x(tjt) In this subsection, we will calculate the optimal estimate x(tjt) base on the Riccati equation given in last subsection. Theorem 3.2: Consier (1) (3) an given time t>, the optimal estimate ^x(tjt) is given by ^x(tjt) =^x(t; 1) (37) ^x(t; 1) is compute through the following steps. Step 1) Calculating ^x(; 2) for = t 1 ^x(; 2) =8( )^x(; 2) + K 2( )[Y 2( ) 0H 2( )^x(; 2)] ^x(0; 2) = 0 (38) K 2( ) = P 2( )H2( 0 )Q 01 V ( ) an P2( ) is calculate by (29), i.e., P 2 ( ) =8( )P 2 ()+P 2 ()8 0 ( )0K 2 ( )Q V ( )[K 2 ( )] 0 +0( )Q u ( )0 0 ( ) P 2 (0) = 5 0 : (39) Step 2) Calculating ^x(; 1) for t 1 < t ^x(; 1) =8( )^x(; 1) + K 1 ( )[Y 1 ( ) 0H 1 ( )^x(; 1)] ^x(t 1 ; 1) = ^x(t 1 ; 2) (40) K 1( ) = P 1( )H1( 0 )Q 01 V ( ), an P1( ) is calculate by (31), i.e., P 1 ( ) =8( )P 1 ()+P 1 ()8 0 ( )0K 1 ( )Q V ( )[K 1 ( )] 0 +0( )Q u ( )0 0 ( ) P 1 (t 1 )=P 2 (t 1 ): (41) Step 3) The estimator ^x(t; 1) is compute from Step 2) for = t. Proof: (37) is obtaine immeiately from Definition 3.1. (39) an (41) follow irectly from (29) an (31) respectively. From the stanar Riccati equation, we can conclue (38). From (17) an (34), we can obtain (40). Remark3.1: When 0 t, the optimal estimator ^x(tjt) is the projection of state x(t) onto the linear space generate by the observation of fy( ); 0 tg. Note the observation y( ) is from (2) an elay free, thus ^x(tjt) is stanar Kalman filter associate withsystem (1), (2) which can be compute by ^x(; 1) =8( )^x(; 1) + K 1 ( )[y1() 0H 1 ( )^x(; 1)] ^x(0; 1) = 0 (42) K 1( )=P 1( )H1( 0 )Q 01 ( ), an P1( ) is calculate by V P 1 ( ) =8( )P 1()+P 1()8 0 ( )0K 1( )Q V ( )[K 1( )] 0 +0( )Q u ( )0 0 ( ) P 1 (0) = 5 0 : (43) When t>, a flow chart for calculating ^x(tjt) is rawn in Fig. 4. IV. NUMERICAL EXAMPLE In the section, we illustrate the results obtaine in previous section witha simple example. Consier the continuous-time moel (1) (3) with then 8(t) = (t) = H(t) =[1 1] H 1 (t) =[2 1] (44) H 1 (t) =[1 1] H 2 (t) = : (45) The initial state x(0) = 1 0:5, ^x(0) = 0 0 an P 0 = In the simulation, u(t), v0(t) an v1(t) are assume to be uncorrelate Gaussian noises withzero means an known covariance matrices Q u(t) =1, Q v (t) =Q v (t) =1, let sampling perio Ts =0:02 s, = 0:4 s. The simulation results are rawn in Figs It can be observe from the simulation results that the propose metho prouces very goo performance, so the technique propose in this note is efficient.

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 5, MAY Fig. 3. Sum of the variance with elay an without elay. [1] an [2], the more general cases for the systems with both measurement elays an state elays were consiere. REFERENCES Fig. 4. Flow chart for calculating ^x( ) with. V. CONCLUSION The note stuie the Kalman filtering for continuous-time systems withelaye measurements. The optimal estimate is erive by eveloping a new tool calle reorganize innovation approach. It consists of two ifferent Kalman filters that have the same imension as the original system. Comparing withthe conventional approachvia solving partial ifferential equation (PDE), a significant avantage of new approachis that it provies a close-form solution. Simulation results showe that the new approach is efficient. Moreover, we believe that the presente results give an useful benchmark for ealing with time elay problems suchas controller an filter esign in H 2 or H1 performance [8], [9], [13]. It shoul be pointe out that the presente results in this note are limite to the systems with only current an one channel elaye measurements. This is in comparison with some recent publications as in [1] M. V. Basin an R. Martínez Zúñiga, Optimal linear filtering over observations withmultiple elays, Int. J. Robust Nonlinear Control, vol. 14, no. 8, pp , [2] M. V. Basin, M. A. Alcorta Garcia, an J. G. Roríguez González, Optimal filtering for linear systems withstate an observation elays, Int. J. Robust Nonlinear Control, vol. 15, no. 17, pp , [3] N. Wiener, Extrapolation Interpolation, an Smoothing of Stationary Time Series. New York: Wiley, [4] B. D. O. Anerson an J. B. Moore, Optimal Filtering. Englewoo Cliffs, N.J.: Prentice-Hall, [5] N. Briggs an R. Vinter, Linear filtering for time-elay systems, IMA J. Math. Control Inf., vol. 6, pp , [6] K. M. Nagpal an P. P. Khargonekar, Filtering an smoothing in an setting, IEEE Trans. Autom. Control, vol. 36, no. 2, pp , Feb [7] T. Kailath, A. H. Saye, an B. Hassibi, Linear Estimation. Englewoo Cliffs, NJ: Prentice-Hall, [8] G. Tamor, The stanar problem in systems witha single input elay, IEEE Trans. Autom. Control, vol. 45, no. 3, pp , Mar [9] K. Uchia, K. Ikea, T. Azuma, an A. Kojima, Finite-imensional characterizations of control for linear systems withelays in input an output, Int. J. Robust Nonlinear Control, Jul [10] H. Kwakernaak, Optimal filtering in linear systems withtime elays, IEEE Trans. Autom. Control, vol. AC-12, no. 2, pp , Apr [11] U. Shake an Y. Theoor, optimal estimation. A tutorial, in Proc. 31st Conf. Decision an Control, Tucson, AZ, 1992, pp [12] A. Kojima an S. Ishijima, control for preview an elaye strategies, in Proc. 40th IEEE Conf. Decision an Control, Orlano, FL, 2001, pp [13] H. Zhang, L. Xie, an Y. C. Soh, A unifie approach to linear estimation for iscrete-time systems-part I: estimation, in Proc. 40th IEEE Conf. Decision an Control, Dec. 2001, pp [14] H. Zhang, D. Zhang, an L. Xie, Fixe-lag smoothing an preiction for linear continuous-time systems, in Proc. Amer. Control Conf., Denver, CO, 2003, pp [15] H. S. Zhang, L. Xie, W. Wang, an X. Lu, An innovation approach to H infinity fixe-lag smoothing for continuous time-varying systems, IEEE Trans. Autom. Control, vol. 49, no. 12, pp , Dec [16] X. Lu, H. S. Zhang, W. Wang, an K. L. Teo, Kalman filtering for multiple time-elay systems, Automatica, vol. 41, no. 8, pp , 2005.