On an Output Feedback Finite-Time Stabilization Problem

Transcription

1 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO., FEBRUARY 35 [] E. K. Boukas, Control of systems with controlled jump Markov disturbances, Contr. Theory Adv. Technol., vol. 9, pp , 993. [3], New model for the inventory control problem, submitted for publication. [4] W. M. Wonham, Random differential equations in control theory, in Probabilistic Methods in Applied Mathematics, A. T. Bharucha-feid, Ed. New York: Academic, 97, vol.. [5] D. D. Sworder and V. G. Robinson, Feedback regulators for jump parameter systems with state and control dependent transition rates, IEEE Trans. Automat. Contr., vol. AC-8, pp , 974. [6] N. N. Krasovskii and E. A. Lidskii, Analysis design of controller in systems with random attributes Part, Automat. Remote Contr., vol., pp. 5, 96. [7] M. Mariton, Jump Linear Systems in Automatic Control. New York: Marcel Dekker, 99. [8] S. P. Sethi and Q. Zhang, Hierarchical Decision Making in Stochastic Manufacturing Systems. Berlin: Birkhauser, 994. [9] O. L. V. Costa and E. K. Boukas, Necessary and sufficient conditions for robust stability and stabilizability of continuous-time linear systems with Markovian jumps, J. Optim. Theory Applicat., vol. 99, Nov On an Output Feedback Finite-Time Stabilization Problem Yigwruang Hong, Jie Huang, and Yangsheng Xu Abstract This note studies the problem of finite-time output feedback stabilization for the double integrator system. A class of output feedback controllers that can achieve global finite-time stability for the double integrator system are constructed based on a finite-time separation principle. Furthermore, it is shown that the closed-loop system resulting from our control law can maintain its local finite-time stability regardless of some nonlinear perturbations. Thus, our control law actually applies to a large class of nonlinear second order systems. Index Terms Finite-time observer, finite-time stabilization, nonlinear systems, output feedback. I. INTRODUCTION The problem of state feedback finite-time stabilization of a dynamic system has been studied by quite a few people from different perspectives [], [4], [6], [], [4]. A most recent result was given by Bhat and Bernstein in [4], where they studied finite-time stabilization of a double integrator system by continuous, unbounded or bounded, state feedback control laws. In addition, the design of finite-time observers was discussed from the probabilistic variational considerations [9]. In this note, we will study the finite-time stabilization of the double integrator system by continuous output feedback control laws. We give a class of nonsmooth finite-time observer for the double integrator system. In conjunction with the state feedback finite-time stabilization Manuscript received March 8, 999; revised January 4, and August,. Recommended by Associate Editor W. Lin. This work was supported in part by a Grant from the Research Grants Council of the Hong Kong Special Administration Region under Project CUHK38/96E, and in part by the National Natural Science Foundation of China under Grants G697748, G598377, and G Y. Hong is with the Institute of Systems Science, Chinese Academy of Science, Beijing, 8, China ( yghong@iss3.iss.ac.cn). J. Huang and Y. Xu are with the Department of Mechanical and Automation Engineering, Chinese University of Hong Kong, Hong Kong ( jhuang@mae.cuhk.edu.hk; ysxu@mae.cuhk.edu.hk). Publisher Item Identifier S 8-986()6-9. control law, our finite-time observer naturally leads to an output feedback finite-time stabilization control law. Using homogeneous techniques as can be found in [3], [7], [3], we also show that our control law for the double integrator system results in a closed-loop system with certain robustness property with respect to a class of nonlinear perturbations. This robust property actually extends the applicability of our control law to a large class of second order nonlinear systems. The rest of this note is organized as follows. In Section II, we formulate the problem and review some preliminary results. Our major result is presented in Section III, where the output feedback finite-time stabilizing control laws for the double integrator systems are given. In Section IV, we further show that the zero solution of the closed-loop system resulting from our control law given in Section III is still locally finite-time stable in the presence of a class of nonlinear perturbations. Finally, we close this note in Section V with some remarks. II. PROBLEM FORMULATION AND PRELIMINARIES Let us begin by introducing some terminology and results given in [4]. Consider the system _x = f (x); f() = ; x R n ; x() = x () where f : D! R n is continuous on an open neighborhood D of the origin x =. The zero solution of () is finite-time convergent if there is an open neighborhood U D of the origin and a function T x: Unfg!(; ), such that 8 x U, the solution s t(; x ) of system () with x as the initial condition is defined and s t (; x ) Unfg for t [; T x(x )], and lim t!t (x ) s t(; x ) =. Here T x(x ) is called the settling time (with respect to initial state x ). The zero solution of () is finite-time stable if it is Lyapunov stable and finite-time convergent. When U = D = R n, then the zero solution is said to be globally finite-time stable. So far most research on the construction of time-invariant finitetime-stabilizing state feedback has been done for the so-called double integrator system which is described in the following stats space from _x = x ; x () = x _x = u; x () = x y = x : The problem was initially explored in the context of optimal control []. A discontinuous finite-time stabilizing controller for double integrators was given in [4]. In [], a constructive way to design finite-time stabilizing feedback was reported. Further result was given in [6] where a class of continuous time-invariant state feedback controllers was constructed for double integrators as follows: () u = jx j sgn(x ) jx j sgn(x ) (3) where < < and > =( ). Recently, Bhat and Bernstein have presented two other classes of continuous state-feedback stabilization control laws as follows: u = jx j sgn(x ) j j =() sgn( ) (4) u = sat(jx j sgn(x )) sat(j j =() sgn( )) (5) where = x +(=( ))jx j sgn(x ), < <, and sgn( ) and sat( ) are the sign function and the saturation function, respectively. It is noted that (5) is not only continuous but also bounded /$. IEEE

2 36 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO., FEBRUARY One of the major objectives of this paper is to design output feedback control law of the form u = (); _ = (; y); R such that the zero solution of the closed loop systems consisting of () and (6) is finite-time stable where ( ) and ( ; ) are continuous functions satisfying () = and (; ) =. To this end, we will start from the following result given in [4] and [6]. Lemma : Consider the nonlinear system described in (). Suppose there is a C (continuously differentiable) function V (x) defined in a neighborhood ^U R n of the origin, and there are real numbers c> and <<, such that V (x) is positive definite on ^U and _V (x)+cv (x) is negative semidefinite on ^U. Then the zero solution of system () is finite-time stable. The settling time, depending on initial state x() = x, is given by T x (x ) V (x) c( ) for all x in some open neighborhood of the origin. If ^U = R n and V (x) is also radially unbounded [i.e., V (x)! + as kxk!+], the zero solution of system () is globally finite-time stable. Our approach will be based on the properties of homogeneous systems. Therefore, we will also introduce some concepts about homogeneous systems which can be found, for example, in [3], [7], and [3]. A scalar function V (x) is homogeneous of degree R with dilation (r ; ;r n ), r i > ; i=;...;n, if for all > (6) V ( r x ;...; r x n )= V (x): (7) A vector field f (x) =[f (x); ;f n (x)] T is homogeneous of degree k R with dilation r =(r ;...;r n) if, for all >, f i( r x ;...; r x n)= k+r f i(x); i =;...;n: (8) We call () a homogeneous system if f (x) is a homogeneous vector field. The stability of the homogeneous systems has been extensively studied in [7], [3]. The following lemma is adapted from [3, Th. ]. Lemma : Suppose that system () is homogeneous of degree k with dilation (r ;...;r n ), f is continuous and x =is its asymptotically stable equilibrium. Then, for any positive integer j and any real number >j maxfr ;...;r n g, there is a C j homogeneous function V of degree with the same dilation (r ;...;r n) such that V is positive definite, radially unbounded, and _ V (x)j () < for all x 6=. III. GLOBAL FINITE-TIME OUTPUT FEEDBACK In this section, we will construct a class of output feedback finite-time stabilizing control law by considering the following output feedback control laws: u = ( ; ) (9) _ = k j yj sgn( y) _ = u k j yj sgn( y) () where >, >, k >, k >, () =, and () =. Clearly, when = =, () is nothing but the classic Luenberger observer []. Thus, () can be viewed as a generalization of the linear observer. Let e = x, e = x, and e =(e ;e ) T, then the error dynamics generated by systems () and () is given by _e = e k je j sgn(e ) _e = k je j sgn(e ) with e () = x and e () = x. With coordinates (x ;x ;e ;e ), the closed-loop system is _x = x _x = (x + e ;x + e ) _e = e k je j sgn(e ) _e = k je j sgn(e ): () () Remark : If the zero solution of the error dynamics () is globally finite-time stable with settling time T e(e ), then ( ; )=(x ;x ); t T e (e ): (3) Thus, if u = (x ;x ) is a state-feedback finite-time stabilizing control law for () with settling time T x(x ), and the zero solution of the closed-loop system is globally asymptotically stable, then the zero solution of the closed-loop system is also finite-time stable. Thus, in the following, we will show that, by properly selecting the parameters of the observer (), the zero solution of the error system () is finite-time stable. Furthermore, there exist finite-time state-feedback controllers (x) such that the zero solution of the closed-loop system () is globally asymptotically stable. Proposition : The zero solution of the error dynamics () is globally finite-time stable for any k i >, i =;, and for all < < ; =: (4) Proof: Note that the error system () is a homogeneous system of degree k =( = ) with dilation (= ; ), and condition (4) implies k<. By [3, Th. ], if the degree of a homogeneous system is negative, the zero solution of this system is finite-time stable if and only if the zero solution of this system is asymptotically stable. Therefore, we only need to show that the zero solution of the homogeneous system () is asymptotically stable. To this end, consider a Lyapunov function candidate for system (): V 3 (e) =k je j which is positive definite, and radially unbounded since k > and >. The derivative of V 3 along the trajectories of () is e _V 3(e) =k k ( + )je j + which is negative semidefinite since k >. Now note that _ V 3 (e) implies e which, in turn, implies e using (). By LaSalle s invariant set theorem, the zero solution of the homogeneous system () is asymptotically stable. Proposition : Consider the closed-loop system () where k i >, i =;, and < <, =. Assume (x ;x ) globally finite-time stabilizes the double integrator system and satisfies ( = x ;x )= = (x): (5) Then, the zero solution of the closed-loop system () is globally asymptotically stable.

3 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO., FEBRUARY 37 Proof: Under our assumption, the system _x = x ; _x = (x ;x ) (6) is homogeneous of degree k = ( = ) < with dilation (r ;r ) = (= ; ), and the zero solution of (6) is globally finite-time stable. By Proposition, the error system is also homogeneous of degree k = ( = ) < with dilation (r ;r )=(= ; ), and its zero solution globally finite-time stable. By Lemma, there are radially unbounded positive definite C functions V (x) and V (e) both of which are homogeneous of degree > with dilation (r ;r ) such that V _ ( +k)= (x) c V for some c >, and V _ ( +k)= (e) c V (e) for some c >. Define a candidate Lyapunov function V K (x; e) =V (x) +KV (e) where K > for the closed-loop system (). Clearly, V K (x; e) is homogeneous function of degree with dilation (r ;r ;r ;r ), and is positive definite and radially unbounded. We will show that, for sufficiently large K, the derivative of V K (x; e) along all the trajectories of the closed-loop system is negative definite. To this end, note that _V K (x; e) = _ V (x)+k _ V (e) h (x)kh (e)h 3(x; e) (7) where h (x) =c V (x) ( +k)= ; h 3 (x; e) [(x + e) (x)]: ( +k)= h (e) =c V (e) Clearly, h (x); h (e) and h 3(x; e) are continuous functions for any x R and e R, h and h are positive definite, and h 3 (x; ) =. It suffices to show V _ K (x; e) is positive definite when K is sufficiently large. In fact, it can be verified that V _ K (x; e) is also a homogeneous function of degree +k with dilation (r ;r ;r ;r ). Next, define a positive-definite function (x; e) =(jx j m=r + jx j m=r + je j m=r + je j m=r ) =m with m maxfr ;r g. Then, (x; e) is a homogeneous function of degree one with dilation (r ;r ;r ;r ). Thus, for any (x T ;e T ) 6=, there is a positive number =(x; e) and z = (z ;z ;z 3 ;;z 4 ) T S = f(x T ;e T ) T : (x; e) = g such that (x ;x ;e ;e ) = ( r z ; r z ; r z 3 ; r z 4 ). As a result, we only need to show that V _ K (x; e) is positive in the sphere S since, for any (x T ;e T ) 6=, there is z S, such that V _ K (x; e) = +k V _ K (z). Let U = f(x T ;e T ) T R 4 : h (x) +h 3(x; e) > g, and S x = f(x T ; T ) T R 4 ;x6= g. Then, U is open since h (x)+h 3 (x; e) is continuous, and S x U. Let S = S \U and S = S \U c. Then, S is compact, S [ S = S, and S \ S = ;. Clearly V _ K (x; e) >, 8 (x; e) S, and for all K>. Let N = min (x ;e ) S h (e); N = min h 3 (x; e): (x ;e ) S Since S and S are compact, N and N are finite. Clearly, N. Moreover, we claim N>. Otherwise, necessarily e =since h (e) is positive definite. That is to say that there exists a x 6= such that (x; ) S U c. However, by the definition of S x, it must also hold that (x; ) S x U which leads to a contradiction. Thus, h (x)+kh (e)+h 3(x; e) Kh (e)+h 3(x; e) KN + N ; (x; e) S : Letting K = (N =N ) shows that _ V K (x; e) is positive in S for all K>K, thus completing the proof. Combining [3, Th. ] as mentioned above with Proposition leads to the major result of this section. Theorem : The zero solution of the closed-loop system () where k i >, i = ;, and conditions (4) and (5) hold, is globally finite-time stable. Remark : It can be verified that the controller (4) given by Bhat and Bernstein satisfies condition (5) with = =. Remark 3: It is possible to give a construction of the Lyapunov function for () for the special case where k =, k ==. To this end, consider the following C scalar function V (e) = je j + e e + je j (+ )= (8) + + where = k ( + (= )) > and = k <. It is clear that V (e) is homogeneous C function of degree =+(= ) with dilation (= ; ). Thus, > maxf= ; g. Moreover, it is possible to show that (8) is positive definite, radially unbounded, and the derivative of (8) along the trajectories of () with k =, k = = is negative definite. To this end, let a = j =(+ ) e j, b = j =(+ ) e j, p =+, and q =(+ )=. Using Young s inequality [] gives ab ap p + bq q ; p + = (9) q j e e j = ab + je j which implies ( + ) je j (+ )= () V (e) + je j ( + je j (+ )= : ) Therefore, (8) is positive definite and radially unbounded since + < <. Next, consider the derivative of (8) along () _V (e) =( k )je j e +( + )je ke j sgn(e e ) k je j je j = sgn(e e ): () We need to show _ V (e) is negative definite. Note that _ V (e) is homogeneous with its degree equal to two. Therefore, as suggested in the proof of [4, Prop. ], we can limit our discussion on the closed set S = fz R : maxfjz j; jz jg =g. Consider the following four cases. Case : When e =, je j, from (), we have q (e ) def = _ V j e =; je j = ( ) e +( + )e je j = sgn(e ): Since <, we can differentiate q with respect to e to obtain dq de = e +( + ) je j (= ) : Since je j <, wehave dq de > +( + ) = + + =:

4 38 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO., FEBRUARY Thus, q will take its extreme value at either e =or e =. Then, q (e )jje j= = ( ) +( + )sgn(e ) sgn(e ) < yields that q < on S. Case : When je j and e =, from (), we have q (e ) def = V _ jje j; e =(e) = ( )je j +( + )je j sgn(e ) je j sgn(e ): Note that 4 ( ) > since < = <. Also we have je j je j since je j. Hence, if e,wehave q ( )je j + je j < : Similarly, if e <, wehave q ( )je j ( + )je j + je j < : So, q < on S. Case 3 (when e = ; je j ) and Case 4 (when je j and e = ) directly follow from Cases and, respectively, since _V (e) = V _ (e). Thus, we have shown that V _ < on S, and, hence, V _ is negative definite. IV. SECOND-ORDER NONLINEAR SYSTEMS In this section, we will further show that under our control law given in Section III, the zero solution of the closed-loop system composed of (9), (), and the following nonlinear control system _x = x _x = f (x ;x )+u () y = x is locally finite-time stable where f (x) is any C function defined in a neighborhood of the origin of R and satisfies f () =. To this end, we will first state the following result. Lemma 3: Consider the following system: _x = f (x) + ^f (x); x R n (3) where f (x) is n-dimensional continuous homogeneous vector field of degree k< with dilation (r ;...;r n ) satisfying f () =, and ^f is also a continuous vector field satisfying ^f () =. Assume the zero solution of _x = f (x) is asymptotically stable. Then, the zero solution of (3) is locally finite-time stable if lim! ^f i( r x ;...; r x n)= k+r =; i =;...;n (4) uniformly for any x S n = fx R n : kxk =g. Remark 4: It has been established in [7] (also, see [3, Th. 3]) that under the conditions of Lemma 3 (without the constraint k <), the zero solution of (3) is locally asymptotically stable. Note that Lemma 3 is different from the result in [7] and [3] in that it further concludes that the zero solution of (3) is locally finite-time stable. Since (3) is not homogeneous, Lemma 3 cannot be directly deduced from the result of [7] or [3]. Nevertheless, the proof of Lemma 3 can be obtained in a way similar to that of Theorem 3 in [3], together with the help of Lemma. Now, we are ready to state the major result of this section. Theorem : Consider the control system () and the control law u = ( ; ) _ = k je j sgn(e ) _ = u k je j sgn(e ) (5) where,, k, k, and (x) are defined as in Theorem. Then, the equilibrium at the origin of the closed-loop system composed of () and (5) is locally finite-time stable. Proof: In the coordinates of (x; e), the closed-loop system is given by _x = x _x = (x + e ;x + e )+f (x) _e = e k je j sgn(e ) _e = k je j sgn(e ) f (x): (6) When f, the system is exactly in the form of (), which is globally finite-time stable, directly from Theorem. Thus, by Lemma 3, the equilibrium at the origin of (6) is locally finite-time stable if the following perturbation vector ^f (x; e) = f (x) f (x) satisfies (4). However, by assumption, f is C and f () =. The mean-value inequality implies f ( = x ;x )=O(): Finally, using = < shows that f ( = x, x )=o( = )= o( k+r ), i =; 4, and thus ^f (x; e) satisfies condition (4). The proof is completed. Remark 5: Since f does not have to be exactly known, Theorem indicates that the output feedback homogeneous finite-time stabilizing control law has robustness property with respect to the nonlinear perturbation described by (4). V. CONCLUSION This note has addressed the problem of output feedback finite-time stabilization for a class of second-order systems. We note that, although only second-order systems are treated here, the tools used here may also be extended to handle some special class of higher order systems [8]. ACKNOWLEDGMENT The authors wish to thank the anonymous reviewers for many constructive comments regarding this note, especially for alerting us to the fact of finite-time escape phenomenon. The authors also wish to thank S. Bhat and D. Bernstein for providing a preprint of [5]. REFERENCES [] M. Athans and P. L. Falb, Optimal Control: An Introduction to Theory and Its Applications. New York: McGraw-Hill, 966. [] E. F. Beckenbach and R. Bellman, Inequalities, Third ed. Berlin, Germany: Springer-Verlag, 97. [3] S. Bhat and D. Bernstein, Finite-time stability of homogeneous systems, in Proc. Amer. Control Conf., 997, pp [4] S. P. Bhat and D. S. Bernstein, Continuous finite-time stabilization of the translational and rotational double integrators, IEEE Trans. Automat. Contr., vol. 43, pp , May 998.

5 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 46, NO., FEBRUARY 39 [5] S. Bhat and D. Bernstein, Finite-time stability of continuous autonomous systems, SIAM J. Control Optim., to be published. [6] V. T. Haimo, Finite time controllers, SIAM J. Control Optim., vol. 4, no. 4, pp , 986. [7] H. Hermes, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, in Differential Equations, Stability and Control, S. Elaydi, Ed. New York: Marcel Dekker, 99, pp [8] Y. Hong, Y. S. Xu, and J. Huang, Finite time control for robot manipulators, Syst. Control Lett., submitted for publication. [9] M. R. James, Finite time observer design by probabilistic-variational methods, SIAM J. Control Optim., vol. 3, pp , 99. [] T. Kailath, Linear Systems. Englewood Cliffs, NJ: Prentice-Hall, Inc., 98. [] M. Kawski, Stabilization of nonlinear systems in the plane, Syst. Control Lett., vol., no., pp , 989. [] E. R. Rang, Isochrone families for second order systems, IEEE Trans Automat. Contr., vol. AC-8, pp , 963. [3] L. Rosier, Homogeneous Lyapunov function for homogeneous continuous vector field, Syst. Control Lett., vol. 9, no. 4, pp , 99. [4] E. P. Ryan, Singular optimal controls for second-order saturating system, Int. J. Control, vol. 36, no. 4, pp , 979. Bounds for Least-Squares Estimators Babak Hassibi and Thomas Kaliath Abstract In this note, we obtain upper and lower bounds for the norm of the Kalman filter and the recursive-least-squares (RLS) algorithm, with respect to prediction and filtered errors. These bounds can be used to study the robustness properties of such estimators. One main conclusion is that, unlike -optimal estimators which do not allow for any amplification of the disturbances, the least-squares estimators do allow for such amplification. This fact can be especially pronounced in the prediction error case, whereas in the filtered error case the energy amplification is at most four. Moreover, it is shown that the norm for RLS is data dependent, whereas for least-mean-squares (LMS) algorithms and normalized LMS, the norm is simply unity. Index Terms Estimation,, least-squares, robustness. I. INTRODUCTION Since its inception in the early 96s, the Kalman filter (and the closely related recursive-least-squares (RLS) algorithm) has played a central role in estimation theory and adaptive filtering. Recently, on the other hand, there has been growing interest in (so-called) H estimation, with the belief that the resulting H -optimal estimators will be more robust with respect to disturbance variation and lack of statistical knowledge of the exogenous signals. Therefore, a natural question to ask is what the robustness properties of the Kalman filter and RLS algorithm are within the H framework. In an initial attempt to address this question, in this note we obtain upper and lower bounds on the H norm of the Kalman filter Manuscript received October 9, 997; revised June 3,. Recommended by Associate Editor H. Katayama. This work was supported in part by the U.S. Air Force through under Contract F P and in part by the Joint Service Electronics Program at Stanford University, Stanford, CA, under Contract DAAH4-94-G-58-P3. B. Hassibi is with the Department of Electrical Engineering, California Institute of Technology, Pasadena, CA 95 USA ( hassibi@caltech.edu). T. Kaliath is with the Information Systems Laboratory, Stanford University, Stanford, CA 9435 USA. Publisher Item Identifier S 8-986()-. and RLS algorithm, with respect to the prediction and filtered errors of the uncorrupted output of a linear time-variant system. These bounds are also of interest for several other reasons. First, they demonstrate that unlike the least-mean-squares (LMS) algorithm whose H norm is unity (independent of the input output data) [], the H norm of the RLS algorithm depends on the input output data, and therefore it may be more robust or less robust with respect to different data sets. Moreover, the exact calculation of the H norm for RLS (and for the Kalman filter) requires the calculation of the induced two-norm of a linear time-variant operator, which can be quite cumbersome, and, in addition, needs all the input output data, which may not be available in real-time scenarios. The H bounds we obtain only require simple a priori knowledge of the data, and may therefore be used as a simple check to verify whether RLS (or the Kalman filter) has the desired robustness with respect to a given application. A brief outline of the paper is as follows. In Section II, we give general upper and lower bounds for the H norm of the Kalman filter. The proofs of the upper bounds are given in Section III and are based on certain minimization properties of least-squares estimators. The proofs of the lower bounds are given in Section IV and are essentially based on computing the energy gains for suitably chosen disturbances. Section V specializes the general results of Section II to the adaptive filtering problem and discusses its various implications. The paper concludes with Section VI. II. A GENERAL H BOUND Consider the possibly time-variant state-space model x i+ = F i x i + G i u i ; x y i = Hix i + vi; i (.) where F i C nn ;G i C nm and H i C pn are known matrices, x ; fu ig; and fvig are unknown quantities and fyig is the measured output. Moreover, fv i g can be regarded as measurement noise and fu i g as process noise or driving disturbance. We shall be interested in estimating the uncorrupted output, s i = Hix i. It is well known that the Kalman filter for computing the predicted estimates of the states, denoted by ^x i, (i.e., ^x i is the least-squares estimate of x i,givenfy j;j<ig) is given by ^x i+ = F i ^x i + K p; i (y i H i ^x i ) (.) where K p; i = F i P i Hi 3 Re; i and R e; i = R i + H i P i Hi 3 and where P i satisfies the Riccati recursion P i+ = F ip if 3 i + G iq ig 3 i K p; ir e; ik 3 p; i; P =5 : (.3) [Note here that fq i;r ig and 5 are given positive definite weighting matrices.] There is also a filtered form of the Kalman filter recursions for computing, ^x iji, the least-squares estimate of x i,givenfy j;jig, which is given below ^x i+ji+ = F i ^x iji + K f; i+ (y i+ H i+ F i ^x iji ) (.4) where K f; i = P ihi 3 Re; i. Now using ^x i and ^x iji, the predicted and filtered estimation errors of the uncorrupted output, s i = H i x i, are defined as and e p; i = Hix i Hi ^xi = Hi ~xi e f; i = H ix i Hi ^xiji = Hi ~xiji : (.5) We should stress that these bounds are not for the problem of parameter estimation, for which causality is not an issue and for which the H and H solutions coincide /$. IEEE