Autonomous University of Kuevo Leon, Mexico. Leonid Fridman. National Autonomous University of Mexico. Mikhail Skliar. Department of Chemical and Fuels ...
Proceedings of the 41st IEEE Conference on Decision and Control Las Vegas, Nevada USA, December 2002
ThA07-4
OPTIMAL AND ROBUST INTEGRAL SLIDING MODE FILTER DESIGN FOR SYSTEMS W I T H CONTINUOUS AND DELAYED MEASUREMENTS Michael 1’. Basin Department of Phy-sical and Mahhematical Sciences Autonomous University of Kuevo Leon, Mexico Leonid Fridman National Autonomous University of Mexico Mikhail Skliar Department of Chemical and Fuels Engineering University of Utah, US;\
A b s t r a c t . In this paper, the optimal filtering problem for a linear system over observations with a fixed delay is treated, proceeding from the general expression for the stochastic It0 differential of the optimal estimate and its variance. As a result:the optimal filtering equations similar to the traditional Kalman-Bucy filter are ohtained in the form dual to the Smith predictor, commonlg used control structure for model-based time delay compensation. The paper then presents a robustification algorithm for the obtained optimal filter based on sliding mode compensat,ion of disturbances. As a result, the sliding mode control of observations leading to suppression of the disturbances in a finite time is designed. This control algorithm also guarantees finite-time convergence of the estimate based on the corrupted observations t o the optimal estimate satisf)-ing the obtained optimal filtering equations over delayed observations.
1. Introduction
The optimal filteriiig and coutrol prohlenis for linear systems \\-it11 nieasurenient delaj-s and its dual optimal coutrol problem rerirain t,lieoretically unsolved in t,heir most general fornrulat,iouwith niultiple and time-varying delays, although the importance of t,he optimal filtering problem for h e a r dynamic systeins with observation delays was recognized a long time ago. The duality of the control and filtering problems in linear systems implies that the optimal state estimation for the system with measurement delays is closely related to the optimal quadratic regulator problem \vit.h delays in inputs, which was extensively studied ([1],[2]and references therein). A significantly smaller number of publications consider the problem of optimal filtering for systems with measurement delays. As a representative example, a vast major-
0-7803-7516-5/02/$17.0002002 IEEE
ity of papers presented during the recent IFAC Workshop on Time Delay Systems [3]deal with the controller design for systems with delays. Most of the relevant prior work on optimal filtering considers a single delayed measurement. Both discrete and continuous cases were studied. However, the optimal filter for fusing multiple measurements with different (and time-varying) delays, including the measurements with no time delays, is not known. Further details on this subject can be found in [4]. additional results were obtained in [5] and [SI for the systems with identical delays in state and measurement equations. The Kalman filters for discrete linear stochastic systems with measurements delayed by a fixed fraction of the sample time can be handled optimally after modifying the observation equation, as shown in [7]. It was suggested in [8] and [9] that the problem with multiple time delays can be converted t o the known optimal filtering problem by introducing new fictitious states that correspond to the delayed states of the original system. Clearly, such an approach will lead to a substantial increase of the order of the state model for a large number of delayed measurements and cannot be used for the systems with time-wrying and continuum delays. In discrete systems, a practical filter incorporating the delayed measurements was ohtained in [lo], which requires the calculat.ion of a correction term that is added to the state estimation when the delayed measurement becomes available. Such a filter is optimal nndcr certain condit,ions described in [lo]. .4n alt,ernat,ivefiltering mct,hod, based on the extrapolation of the dcla.yed observations, is dcvclopcd in [ll]and allows for the incorporation of a considerable nnmbcr of delayed and non-delayed measurements. For systems with deterministic uncertainties, the robust H , filtering problem was treated in [I21 for discrete systems and in [13, 141 for continuous-time systems. Comprehensive reviews of theory and algorithms tems are given in [IS, IS].
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Based on the analysis of current state-of-the-art, it can be concluded that the general optimal filter design problem for linear systems with delays remains unsolved. Existing solutions are typically limited in a number of allowed delayed measurements, the requirements that delays are time invariant, and other method-specific assumptions [4, 101. In this paper, the optimal filtering problem for linear systems with observations subject t o k e d delay is rigorously treated proceeding from the general expression for the stochastic Ito differential of the optimal estimate and its variance [17]. As a result, the optimal filtering equations similar to the traditional Kalman-Bucy filter are obtained. However, unlike the traditional filter, the obtained result contains specific delay-dependent adjustments t o the filter gain matrix and the quadratic term of the variance equation. The structure of the developed filter is dual to the Smith predictor [18], a commonly used controller for model-based delay compensation. It is well known that the optimal filters (as well as regulators) are not robustly stable. Indeed, the presence of non-Gaussian disturbances in the observation equation, that cannot be processed through the Kalman-Bucy filter, leads t,o sharp deterioration of the quality of state estimat.ion. For this reason, the problem of robustness of optimal filtering must be addressed. It was previously shown that sliding mode technique can be used t o robustify the optimal algorithms for systems without. delay [19]. At the same time, it is not possible to use sliding modes for systems with delayed control 1191 and the robustness of sliding mode control in the space of predictor variables typically imposes many restrictions on uncertainties ((20, 21, 221). The application of the sliding mode technique to robustification of the optimal filters for stochastic systems was initiated by Drakunov [23]. In this paper, a rohustification algorithm for the obt,ained optimal filter based on int,egd sliding mode compensat,ion of disturbances [24] is proposed, and t,he sliding mode control of observations leading to the supprcssion of t,he disturbances in a finite time is designed. This developed algorithm guarantees finite-time convergence of the estimate based on the corrupted observations t o the optimal estimate satisfying the obtained optimal filtering equations with delayed observations.
hy an ordinary differential equation governing the dynamics of the state space system
+
+
to) = 20, (1) and a delay-differential equation for the observation prccess: d z ( t ) = (ao(t) a(t)z(t))dt b ( t ) d W i ( t ) ,
d y ( t ) = (&(t)
+ A ( t ) z ( t- h))dt+ B(t)dWZ(t),
(2)
where z ( t ) E R" is the state vector, y ( t ) E R" is the observation process, the initial condition xo E R" is a Gaussian vector such that 20, W'l(t), W z ( t ) are independent. The observation process y ( t ) depends on the delayed state x ( t - h ) , where h is a fixed delay. The vector-valued function ao(s) describes the effect of system inputs (controls and disturbances). All coefficients in equations are deterministic continuous functions of bounded variation. It is also assumed that 4(t:s) is a nonzero matrix and B ( t , s ) B T ( t , s )is a positive definite matrix. All coefficients in (1)-(2) are deterministic functions of appropriate dimensions. The estimation problem is t o find the estimate of the system state z ( t ) based on the observation process Y ( t )= {y(s),O 5 s 5 t ] , which minimizes the Euclidean 2-norm
J = E[(s(t) 2 ( t ) ) T ( s (t )i ( t ) ) ] a t each time t. hi other words, our objective is to find the conditional expectation
m(t)= 2 ( t ) = E ( s ( t )1 Ffy). As usual, the matrix function P ( t ) = E[(z(t)- m ( t ) ) ( z ( t-) r n ( t ) ) T I Ffy] is the esaimate variance. The proposed solution t o this optimal filtering problem is based on the formulas for the Ita differential of t.he conditional expectation E ( x ( t ) 1 F Z ) and its variance P ( t ) [17] and will serve as the basis for constructing a robust filter using sliding mode design.
3. Optimal Filter w i t h Delayed Observations 2. Filtering w i t h Delayed Observations
Let (0,F, P ) be a complete probability space with an increasing right-continuous family of o-algebras FA, t 2 0, and let (Wl(t),Ft,t 0) and (IVz(t);Ft,t 0) be independent Wiener processes. The partially ohseryed Ft-measurable random process ( ~ ( ty)( ,t ) ) is described
>
>
In the simplified situation of one fixed delay, the optimal filtering equations could be obtained directly from the formula for the Ita differential of the conditional expectation m ( t ) = E ( z ( t )I F,") [17]
dm(t) =E($+)
I F:)+E(z[Vl-E(lpl(s) I F31T I
x(B(t:s)BT(t,s))-l(dY(t) - E(ipl(4 I Ffy)dt)>
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+
+
where ~ ( z = ) ao(t) a ( t ) z ( t ) and i p l ( z ) = .4o(t) .-I(t)z(t - h). After obvious manipulations and noting that E ( z ( t - h) I F z ) = E ( z ( t - h) I F z h ) = m(t - h ) , the estimate equation takes the following form:
dm(t) = ao(t) + a(t)rn(t) +E(Z(t)[.d(t)(Z(t - h ) - m(t - h))]'
IFz)
+
x (B(t)B'(t))-l(dy(t) - (Ao,(t) A(t)m(t- h))dt) = ao(t)
+a(t)m(t)
I FZ)A'(t) x(B(t,s)BT(t,s))-l(d2/(t) - (ilo(t)+ A ( t ) m ( t- h ) ) d t ) . + E ( z ( t ) ( z (t h ) - m(t - h))'
The obtained optimal state estimate equation is similar to the tradit.iona1 Kalman filter, with an exception of the term E ( z ( t ) ( z ( t- h ) - m ( t - h))' I F:) instead of the familiar P ( t )= E ( ( z ( t )- m ( t ) ) ( z ( t )-m(t))' I F:). However, the former term can he expressed as a function of the variance, using the Cauchy formula for z ( t ) as the solution of the linear equation (1) and its conditional expectation, m(t).Indeed,
z ( t )- m.(t) = @ ( t , t h ) ( z ( t- h ) - m.(t - h))+ (3)
l-h I
@(kr ) b ( r ) d W( r ) ,
where @ ( t , r )is the state transition matrix of the h e niogeneous equation (1). @(t,r) is the solution of the matrix equation
where I is the identity matrix, and can be written as t the matrix exponeut,ial: @ ( t , r )= exp(J,-, a(s)ds). Thus, the delay-dependent term in the estimate equation is equal to E ( z ( t ) ( z ( t- h) - m(t - h))' l I;y) = E ( z ( t ) ( z ( t )- m(t))' I Fy)exp(-JLhaT(s)cis) = P ( t )esp (a T ( s ) d s ) ,and the entire equation takes the form dm(t) = ao(t) a(t)rn(t)
At-,
+
(4)
x exp (-
x (B(t)BT(t))-'(dy(t)- ( & ( t ) + d ( t ) m ( t- / t ) ) d t ) . The gain matrix of thc optimal filter is now cqual P ( t ) e a p ( - S ~ - h a T ( s ) d s ) ( B ( t ) B T ( t ) )which - l , is asimilar expression to the traditional Kalman filter gain with an exception of the delay-dependent adjustment exp (- .f& a'(s)ds). To obtain the optimal filter in the closed form, we now need to find the equation for P ( t ) . The starting point is
Lh
aT(s)ds)A T ( t )( E (t , 8 ) B T (t , s))
-'
For linear systems with Gaussian noises and initial conditions, z ( t ) itself is conditionally Gaussian. Since conditional third central moment E ( ( z ( t )- m ( t ) ) ( z ( t )rn(t))(z(t) - m(t) I F:) is equal t o zero for conditionally Gaussian z ( t ) ,the last equation yields that
dP(t) = P(t)a'(t)
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+ a(t)P(t)+ b(s)bT(s)
(5)
-P(t) exp (-
/'
a T ( s ) d s ) A T ( t ) ( B ( ts)BT(t,s))-' ,
t-h
xA(t)exp(-
1;
a(s)d.s)P(t)
h
Tlie system of equations (4) and (5) constitutes the closed-form optimal filter for the posed problem. The initial conditions for the filter equations (4) and (5) are as follows: m(to) = E[z(to)IF:] and
P(t0) = E[(z(to)- m(to)(z(to) -m w =IF 3 . The obtained filter is very similar to the conveidional Kalman-Bucy filter, except for the delay-dependent factors in the estimate and variance equations. In its structure, the developed filter resemhles the Smith predictor [lS], a commonly used controller parameterization for model-based time delay compensation. The obtained 81ter is optimal under the described assumptions. The optimality is evident, since the result is directly derived from the exact Ito differential for the conditional expectation and variance. In the case of a constant matrix a in the state equation, the optimal filter takes the especially simple form (exp (- J f - h aTds) = exp (-aTh))
dm(t)= ao(t) + a m ( t )+ p(t)eXp(-QTh)4T(t)
x(B(t)BT(t))-'(dy(t)-(AO(t)+A(t)m(t-h))dt), (?a)
+
d P ( t ) = P(t)aT + a P ( t ) b(s)bT(s)
(gi(z(t),t)+ g z ( z ( t - h ) ; t ) ) d t , (6) wheregl(z(t),t) and gz(z(t-h),t) arenon-Gaussianand, possibly, deterministic disturbances not hearing any useful information, and possibly depending on the current and delayed states. An example of the state-dependent disturbances is encountered during electromagnetic well logging [25],when a parasitic sideways impulse from t,he transmitter affects the receiver direchly, and after reflection from the well border? leading to the same time delay as a prospecting wave. Such dist.urhances obviously deteriorate t.he quality of estimation and should he eliminated. We further assume that the disturbances satisfy the following matching condit.ious: gl(z(t),t)= Kn(z(t),t),
gz(z(t - h ) , t ) ~ K-Yz(z(t = -h),t), I171(~(t),t)ll5 q1Ilz(t)ll+PI, YI,PI 1/72(z(t- h),t)l/ 5 qzIlz(t - h)ll: QZ
7 1 ( z ( t o ) , t o ) = 72(z(to - h),to) = 0,
providing reasonable restrictions on their growth and assuming no disturbances at the initial moment. The last condition, n ( z ( t o )to) : = h ( z ( t 0- h ) ,t o ) = 0, implies that the measurement model is initially unbiased. The observation process ( 6 ) can be separated into the useful and parasitic parts, y ( t ) = y o ( t ) yJt), where dyo(t) = (.4o(t) A ( t ) z ( t - h ) ) d t B(t)dll'2(t) and d y p ( t ) = ( g I ( z ( t ) , t )+gz(z(t - h),t))dt. If only the useful signal yo(t) is present, the desired optimal estimate mo(t) is given by the equation ( 4 4 :
+
- ~ ( texp ) ( - a ' h ) ~ ~ ( (t )~ ( t ) ~ ' ( t ) ) - ' xA(t)exp (-ah)P(t). (54 Next, we propose the robust implementation of the designed optimal filter to deal with the case of observation signals corrupted with non-Gaussian and, possibly, deterministic, unknown disturbances, which do not bear any useful information and should be suppressed to guarantee the quality of estimation. The robust filter is designed using the sliding mode assignment of t.he observation manifold and subsequent movement to and along this manifold using relay control.
4. Robust Sliding Mode Filter with Delayed
Observations
Assume that the observation proccss (2) is rorruptcd by unknown disturbanccs:
d y ( t ) = (.4o(t)
+ A(t)z(t- h))dt + B ( t ) d W z ( t ) +
> 0: > 0,
+
+
d m o ( t ) = ao(t) + 4 t ) m o ( t ) +P(t)exp(-
6,,+
aT(s)ds)ilT(t)
x ( B ( t ) B T ( t ) ) - l ( d y o ( t )- (.4o(t)
.4(t)mo(t - h ) ) d t ) .
Tbe idea is to use tlie previous equation as the definition of tlie sliding mode manifold and to desigii an observation control such that the estimate m ( t ) ,based on the corrupted observations y ( t ) , is driven to tlie sliding manifold. !&'eseek an observation cont.rol as an additive term K y l ( t ) in the measurement model ( G ) , such that ifm(t) is based on t,he corrected measurements y ( t )+Kyl(t), convergence of m ( t ) to the optimal estimate mo(t)is guaranteed. The following sliding mode t,echnique solves this problem. Let s ( t ) = 0 be anot,her sliding manifold that would be defined in such a way that convergence t,o this sliding manifold also implies convergence of m ( t ) to the optimal estimate mo(t).
2597
+
w h e r e g i ( d t ) , t ) = K - r l ( d t ) , t ) , Ilri(d%t)lI5 qi11dt)ll p l , ql,pl > 0, and n ( z ( t o ) , t a= ) 0, the observation process (9) can be separated in the useful and parasitic parts, y(t) = y o ( t ) y,(t), where dyo(t) = (.4o(t) A ( t ) z ( t ) ) d t + B ( t ) d ~ , ~and z ( td) y p ( t ) = gl(z(t),t)dt. The d s ( t ) = dz(t)+G(t)[(Ao(t)+A(t)z(t-h))dt+B(t)dCI’z(t) optimal estimate mo(t) based on yo(t) is given by the Kalman-Bucy filtering equations + K ( n ( z ( t ) , t ) r l t + n ( z ( t - h ) , t ) d t ) + K ~ ~ , q= ( t 0, ) l (7) dVlo(t) = @(t) a(t)mo(t) P ( t ) A T ( t ) where G ( t )= dso(y(t))/dy, and Ky,,,(t) stands for the Define s ( t ) as s ( t ) = z ( t ) so(t), where so(t) = nLo(t) is the ideal sliding manifold and z ( t ) is an auxiliary variable to he assigned. During the motion along the sliding manifold, d s ( t ) / d t = 0, which yields:
+
+
+
+
values of the observation control K y l ( t ) on the sliding manifold. Clearly, to compensate for unknown observation disturbances on the sliding manifold, we need to have yieq(t)
= -(7i(z(t):t) + YZ(Z(~ - h),t)).
x(B(t)B’(t))-l(dyo(t)
- (4o(t)
+ A(t)mo(t))W
The auxiliary variable z(t) satisfies the equation d z ( t ) = -G(t)[(Ao(t)+A(t)z(t))dt+B(t)dWz(t)], t 2h
(8)
ensuring that G[$(t) - do(t)] = 0, m(t)= ma(t),and the desired sliding mode manifold s ( t ) = 0 is found from the solution of the following equation:
+
In doing so; and noting that ds(t)/dt = d z ( t ) / d t dso(t)/dt, the auxiliary variable z ( t ) is defined by equation dz(t) =
+
s(t) = G ( t ) [ K ( n ( z ( t ) , t ) + If~ieq(t)l
-G(t)[(Ao(t)+A(t)z(t-h))dt+B(t)dlVz(t)], t 2h
which ensures that G [ o ( t )-uo(t)]= 0 and m(t)= mo(t) as long as m(to) = mo(to) = E[z(to)I Fi]. With the proposed selection of Kyleq(t), the effect of unknown observation disturbances is reduced on the sliding manifold. In view of the equations (8) and (7), s ( t ) satisfies the following equation:
Finally, the observation control Kyl(t) that enables us
to reach the sliding manifold s ( t ) = 0 in a finite time is assigned using the relay control design as Y l W = -hf(z(t),z(t - h),t))sign[s(t)l,
whcreAl =q(liz(t)ll)+p: q > q l ; p > p i . Onthesliding manifold s ( t ) = 0,
S ( t ) = G ( t ) [ l ( ( y t ( z ( t ) , t ) + ~ z ( z ( t - h ) , t ) d+Kyleq(t)l, t)
Yl(t) = Yl&) assuming t.hat sliding mode must be achieved on the manifold s ( t ) = 0. The only thing left to do is to design K y l ( t ) that enables us t o reach the sliding manifold s ( t ) = 0 in a finite time, where Kyl ( t ) = Kyl,,(t). This can he done by using conventional relay control design in the form yl(t) = - A M t ) , Z ( t - h),t))sign[s(t)l, There .&I = s(ll4t)ll + I l d t - h)ll) + P , 4 P > Pl.
> 41, 42
and
4.1. Robust Kalrnan-Bucy filter An important particular case corresponds to disturbed measurement, hut without time delays, h = 0. The described sliding mode technique for suppressing observational disturbances yields a robust version of the classical Kalman-Bucy filter [26] in the presence of non-Gaussian or deterministic noises. Indeed, if the observation equation is in the form
+
+
dy(t) = ( A o ( t ) A ( t ) z ( t ) ) d t B(t)dlVz(t) +91(z(t),t)dt,
(9)
= -(n(dt):t),
thus compensating for unknown disturbances in ohservations and providing the estimate m o ( t ) , which coincides with the hest Kalman-Bucy estimate, starting with the moment when the sliding manifold s(t) = 0 is reached. References [l] R. L. -4lford and E. B. Lee, “Sampled data hereditary systems: linear quadratic theory,” IEEE Punsactions on Automatic Control, pp. 60-65, 1986.
[Z] E. D. Lee and A. Olbrot, “Observability and related structural results for linear hereditary syst,ems,” Int. J . Control, vol. 34, pp. 1061-1078,1970.
C. T. Abdallab and K. G. (Eds.), Pmc. 3rd IFAC Workshop on Time Delay Systems. Madison: OhlIiIPRESS, 2001. [3]
141 S. C. A. Thomopoulos, “Decentralized filtering in the presence of delays: Discrete-time and continuoustime cases,’. Information Sciences, 1994. (51 G. N. hlil’shtein and S. A. P’yanzin, “A discretization method in designing an optimal filter for systems with delay,” Automation and Remote Control, 1990.
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[6] S. J. Lee, S. M. Hong, and G. G. Goodwin, “Loop transfer recovery for linear systems with delays in the state and output,” International Journal of Control, 1994.
171 G. T. Andersen, A. C. Cristensen, and 0. Ravn, “Augmented models for improving vision control of a mobile robot,“ in Proc. 3rd IEEE Conf. on Control Applicntions, 1994. [SI F. H. Hsiao and S. T. Pan, “Robust Kahnan filter synthesis for uncertain multiple timedelay stochastic systems,” ASME Tmnsactiom. J. of Dynamic Systems, Measurement, and Control, 1996. [9] E. Kaszkurewicz and .4. Bhaya, “Discrete-time state estimat.ion with two counters and measurement delay,” in Pmc. 35th IEEE Cunf. on Decision and Control, 1996. [lo] H. L. .Ilexander, “State estimation for distributed systems with sensing delay;“ SPIE. Data Structures and Target Classification, vol. 1470, 1991. [11] T. D. Larsen, N. A. Andersen, 0. Ravn, and N. K. Poulsen, “Incorporation of the time-delayed measure ments in a discretetime Kalman filter,” in Proc. 37th Conf. on Decision and Control, pp. 3972-3977, 1998.
[U] C. E. de Souza, R. h4. Palhares, and P. L. D. Peres, “Robust H , filter design for uncertain discretetime statedelayed systems: An LMI approach,” in Pmc. 38th IEEE Conf. on Decision and Control, pp. 23472352, 1999. (131 C. E. de Souza, R. M. Palhares, and P. L. D. Peres, “Robust H , filtering for uncertain linear systems with multiple time-varying state delays: An LMI approach,” in Pmc. 38th IEEE Conf on Decision and Control, pp. 2023-2028, 1999. [14] A. Fattouh, 0. Sename, and 3. M. Dion, ‘‘Hobserver design for timodelay systems,” in Proc. 37th IEEE Conf. on Decision and Control, pp. 4545-4546, 1998. [lj] 1’.B. Kolmanovskii and L. E. Shaikhet, Control of Systems with Afterefect. Providence: American Mathematical Society, 1996.
[16] V. B. Kolmanovskii and A. D. Myshkis, Intmduction to the Theoy and Applications of Functional Differential Equations. New York Kluwer, 1999. 1171 V. S . Pugachev ~ and I. N. Sinitsyn, Stochastic differential systems; analysis and filtering. Chichester: John Wiley & Sons, 1987. [I81 0. J. hl. Smith, Feedback Contml Systems. New York McGraw Hill, 1958.
disturbances,” ASME Bansactiom. J. of Dynamic Systems, Measurement, and Control, vol. 122, pp. 732-737, 2000.
[20] Y.-H. Fbh and J.-H. Oh, “Robust stabilization of uncertain input delay systems by sliding mode control with delay compensation,” Automatica: vol. 35, pp. 1861-1865,1999. I211 S. K . Nguang, “Comments on robust stabilization of uncertain input delay systems by sliding mode control with delay compensation,” Automatica, vol. 37, p. 1677, 2001. [22] L. Ridman, A. Polyakov, and P. Acosta, “Robust eigenvalue assignment for uncertain delay control systems,” in Proc. 3rd IFAC Workshop on Time Delay Systems, pp. 239-244, Madison: OMKIPRESS, 2001. [23] S. V. Drakunov, “An adaptive quasioptimal filter with discont,inuousparameters,” Automation and Remote Control, vol. 44, pp. 1167-1175, 1983. [24] V. Utkin, J. Guldner, and J. Shi, Sliding modes control of Electromechanical Systems. Taylor and Francis, 1999. [25] M. Wilt, C. Schenkel, B. Spies, C. Torres-Verdin, and D. Alumbaugh, “Measurements of surface and borehole electromagnetic fields in 2-D and 3-D geology:“ in Three-dimensional electromagnetics, Geohysical derelopment series, pp. 545-564, Tulsa: Society of Exploration Geophysicists, 1999. [26] R. E. Kalman and R. S. Bucy, “New results in linear filtering and prediction theory,“ ASME J o u r n u l of Basic Engineering, Ser: D, vol. 83: pp. 95-108, 1961. [27] A. Fattouh, 0. Sename, and J. h l . Dion, “An unknown input observer design for linear timodelay systems,” in Pmc. 38th IEEE Conf. on Decision and Control, pp. 4222-4227, 1999. [28] L. Fridman, E. Ridman, and E. Shust,in, “Stcady modes and sliding modes in relay control systrms n-it,h delay,” in Sliding mode control in engineering, Cont,rol Engineering, pp. 261-294, Yew York Marcel Dekker, 2002. [29] R. Murray and S. Sastry, “Nonbolonomic motion planning: steering using sinusoids,” IEEE Trans. Automat. Contr., vol. 38, pp. 700-716, 1993. [30] A. Poznyak, R. Martinez-Guerra, and -4.OsorioCordero, “Robust high gain observer for nonlinear closed loop stochastic systems,” Mathematical Methods in Engineering, vol. 6, pp. 31-60, 2000.
[IQ] E. Fridman, L. Fridman, and E. Shustin, “Steady modes in relay control systems with delay and periodic
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