Mean-square H∞ filter design: Application to a 2DOF helicopter

Abstract

This paper designs the central finite-dimensional H_∞ filter for linear stochastic systems with integral-quadratically bounded deterministic disturbances, that is sub-optimal for a given threshold y with respect to a modified Bolza-Meyer quadratic criterion including the attenuation control term with the opposite sign. The original H_∞ filtering problem for a linear stochastic system is reduced to the corresponding mean-square H₂ filtering problem, using the technique proposed in [1]. In the example, the designed filter is applied to estimation of the pitch and yaw angles of a two degrees of freedom (2DOF) helicopter.

Full text

Page 1

Mean-Square H∞Filter Design: Application to a 2DOF Helicopter
Michael Basin Santiago Elvira-CejaEdgar N. Sanchez
Abstract—This paper designs the central finite-dimensional
H∞
filterfor linearstochastic
quadratically bounded deterministic disturbances, that is sub-
optimal for a given threshold γ with respect to a modified Bolza-
Meyer quadratic criterion including the attenuation control
term with the opposite sign. The original H∞filtering problem
for a linear stochastic system is reduced to the corresponding
mean-square H2filtering problem, using the technique proposed
in [1]. In the example, the designed filter is applied to estimation
of the pitch and yaw angles of a two degrees of freedom (2DOF)
helicopter.
systemswithintegral-
I. INTRODUCTION
Over the past two decades, considerable attention has been
paid to the H∞ estimation problem for deterministic and
stochastic systems. The seminal papers on H∞control ([1])
and estimation ([2], [3], [4]) established a background for
consistent treatment of controller/filtering problems in the
H∞framework. The H∞filter design implies that the resulting
closed-loop filtering system is robustly stable and achieves a
prescribed level of attenuation from the disturbance input to
the output estimation error in L2/l2-norm. A large number of
results on this subject have been reported for systems in the
general situation (see, for example, [5]–[23] and references
therein). Sufficient conditions for existence of an H∞ filter,
where the filter gain matrices satisfy Riccati equations, were
obtained for linear deterministic systems in [4] and linear
systems with state delay in [24] or with measurement delay
in [25]. However, the criteria of existence and suboptimality
of solution for the central H∞ filtering problems based on
the reduction of the original H∞problem to the induced H2
one, similar to those obtained in [1], [4] for linear systems,
remain yet undeveloped for linear stochastic systems with
integral-quadratically bounded deterministic disturbances.
This paper presents the central (see [1] for definition)
finite-dimensional mean-square H∞filter for linear stochastic
systems, that is suboptimal for a given threshold γ with
respect to a modified Bolza-Meyer quadratic criterion includ-
ing the attenuation control term with the opposite sign. In
contrast to results previously obtained for linear systems [4],
[24], [25], this paper reduces the original H∞filtering prob-
lem to the corresponding mean-square H2filtering problem,
using the technique proposed in [1].
The authors thank the Mexican National Science and Technology Coun-
cil (CONACyT) for financial support under Grant 55584 and Sabbatical
Fellowship for the first author.
M. Basinis withDepartment
Sciences, AutonomousUniversity
mbasin@fcfm.uanl.mx, mbasin2007@gmail.com
Theother authorsare with
GraduateStudies(CINVESTAV),
{jelvira,sanchez}@gdl.cinvestav.mx
ofPhysical
Nuevo
andMathematical
Leon,ofMexico
Center
Campus
for
Guadalajara,
Research and
Mexico
Designing the central suboptimal mean-square H∞ filter
for linear stochastic systems presents a significant advantage
in filtering theory and practice, since (1) it enables one
to address filtering problems for linear stochastic time-
varying systems, where the linear matrix inequality tech-
nique is hardly applicable and the Hamilton-Jacobi-Bellman
equation-based methods fail to provide a closed-form solu-
tion, (2) the obtained mean-square H∞ filter is suboptimal,
that is, optimal for any fixed γ with respect to the H∞noise
attenuation criterion, and (3) the obtained mean-square H∞
filter is finite-dimensional.
It should be commented that the proposed design of
the central suboptimal mean-square H∞ filters for linear
stochastic systems with integral-quadratically bounded dis-
turbances naturally carries over from the design of the opti-
mal mean-square H2filters for linear stochastic systems with
unbounded disturbances (white noises). The entire design
approach creates a complete filtering algorithm for handling
the linear stochastic time-varying systems with unbounded or
integral-quadratically bounded disturbances optimally for all
thresholds γ uniformlyor for any fixed γ separately. A similar
algorithm for linear deterministic systems was developed in
[4].
The designed filter is applied to estimation of the pitch and
yaw angles of a two degrees of freedom (2DOF) helicopter.
The simulation results show a reliable performance of the
filter, in particular, the obtained attenuation level is five times
less than a given threshold.
The paper is organized as follows. Section 2 presents the
mean-square H∞filter problem statement for linear stochastic
time-varying systems. The central suboptimal mean-square
H∞filter is designed in Section 3. In Section 4, the designed
filter is applied to estimation of the pitch and yaw angles of
a two degrees of freedom (2DOF) helicopter. Conclusions
are given in Section 5.
II. MEAN-SQUARE H∞FILTERING PROBLEM STATEMENT
Let (Ω,F,P) be a complete probability space with an
increasing right-continuous family of σ-algebras Ft,t ≥ t0,
and let (W1(t),Ft,t ≥t0) and (W2(t),Ft,t ≥t0) be independent
Wiener processes. Consider the following linear stochastic
time-varying system S1:
dx(t)
x(t0)
dy1(t)
y2(t)
z(t)
=
=
(A(t)x(t)+B(t)u(t)+G(t)ω(t))dt+b(t)dW1(t),
x0,
C1(t)x(t)dt +h(t)dW2(t),
C2(t)x(t)+H(t)ω(t),
L(t)x(t),
(1)
=
=
(2)
(3)
=
(4)
2011 American Control Conference
on O'Farrell Street, San Francisco, CA, USA
June 29 - July 01, 2011
978-1-4577-0079-8/11/$26.00 ©2011 AACC66

Page 2

where x(t) ∈ Rnis the unmeasured state, u(t) ∈ Rlis
is a known input signal,y1(t) ∈ Rm1and y2(t) ∈ Rm2are
the measured observations, z(t) ∈ Rqis the output to be
estimated, ω(t) ∈ Ls
2[0,∞) is the deterministic disturbance
input, A(t), B(t),G(t), b(t), C1(t), h(t), C2(t), H(t), and L(t)
are known deterministic continuous time-varying functions
of appropriate dimension. The initial condition x0∈ Rnis
a Gaussian random variable such that x0, W1(t) ∈ Rp1, and
W2(t) ∈ Rp2are independent. It is assumed that h(t)hT(t) is
a positive definite matrix.
For the system given by (1)–(4), the following assumptions
are made over the time interval [t0,t1]:
• (A(t),b(t)) is stabilizable and (C1(t),A(t)) is detectable;
(C1)
• (A(t),G(t)) is stabilizable and (C2(t),A(t)) is de-
tectable;
(C2)
• (A(t),B(t)) is stabilizable and (L(t),A(t)) is detectable,
and
(C3)
• H(t)GT(t) = 0 and H(t)HT(t) is a positive definite
matrix.
(C4)
As usual, the first two assumptions ensure that the esti-
mation error, provided by the designed filter, converge to
zero [26]. The noise orthogonality condition H(t)GT(t) = 0
is technical and represents the independence between the
state and measurement deterministic disturbances. Extensive
comments on the assumption (C4) can be found in [1].
The filtering problem to be addressed is as follows:
develop a central suboptimal mean-square H∞filter for the
linear stochastic system (S1) as a linear filter based on the
observations {y1(s),t0≤ s ≤ t} and {y2(s),t0≤ s ≤ t} such
that the following three requirements are satisfied.
1) The resulting dynamics of the estimation error
E(x(t))−m(t), where x(t) is the state of (S1) and
m(t) is the mean-square H∞estimate produced by the
designed filter, is asymptotically stable in the absence
of disturbances, ω(t) ≡ 0. Here, E(x(t)) denotes the
expectation of stochastic process x(t).
2) The variance of the mean-square H∞estimate m(t) of
the system state x(t), based on the observation process
Y(t) = {y1(s),0 ≤ s ≤ t}, is equal to the minimum
estimation error variance ([27])
E[(x(t)−E(x(t)|FY
t))(x(t)−E(x(t) |FY
t))T|FY
t] (5)
at every time moment t. Here, E[ξ(t) | FY
conditional expectation of a stochastic matrix process
ξ(t) = (x(t)−E(x(t) | FY
respect to the σ - algebra FY
vation process Y(t) in the interval [t0,t].
3) Given a noise attenuation level γ, the H∞ noise at-
tenuation condition (6) is ensured. More specifically,
for any nonzero disturbance input ω(t)∈Ls
inequality
t] means the
t))(x(t)−E(x(t) | FY
generated by the obser-
t))Twith
t
2[0,∞), the
?z(t)−L(t)m(t)?2
γ2?
2<
?ω(t)?2
2+E(xT(t0))RE(x(t0))
?
(6)
holds, where ?f(t)?2
lected filter horizon, R is a symmetric positive definite
matrix, and γ is a given real positive scalar.
2:=?t1
t0fT(t)f(t)dt, t1 is the se-
III. CENTRAL SUBOPTIMAL MEAN-SQUARE H∞FILTER
DESIGN
The proposed design of the suboptimal mean-square H∞
filter for linear stochastic systems is based on the general
result (see Theorem 3 in [1]) reducing the H∞ controller
problem to the corresponding optimal H2controller problem.
In this paper, only the filtering part of this result, valid for the
entire controller problem, is used. Then, the optimal mean-
square Kalman-Bucy filter for linear stochastic systems [28]
and the H∞filter for linear systems (Theorem 4 in [4]) are
employed to obtain the desired result, which is given by the
following theorem.
Theorem 1. The central suboptimal mean-square H∞
filter for the linear stochastic system (1)–(4), ensuring the
minimum of the mean-square criterion (5) and the H∞noise
attenuation condition (6), is given by the equation for the
mean-square H∞estimate m(t)
dm(t) = (A(t)m(t)+B(t)u(t))dt+
(7)
P(t)CT
1(t)(h(t)hT(t))−1[dy1(t)−C1(t)m(t)dt]+
S(t)CT
2(t)(H(t)HT(t))−1[y2(t)−C2(t)m(t)]dt,
with initial condition m(t0)=E(x(t0)|FY
function P(t) (minimum estimation error variance) is the
solution of the differential Riccati equation
t0), where the matrix
˙P(t) = A(t)P(t)+P(t)AT(t)+
(8)
b(t)bT(t)−P(t)CT
1(t)(h(t)hT(t))−1C1(t)P(t),
with initial condition P(t0) = E[(x(t0) − m(t0))(x(t0) −
m(t0))T| FY
t0], and the symmetric matrix function S(t) is the
solution of the differential Riccati equation
˙S(t) = A(t)S(t)+S(t)AT(t)+G(t)GT(t)−
(9)
S(t)[CT
2(t)(H(t)HT(t))−1C2(t)−γ−2LT(t)L(t)]S(t),
with initial condition S(t0) = R−1.
Proof. First, let us design the estimate ¯ x(t) satisfying the
minimum variance condition (5) of Section 2. As known
[27], this mean-square estimate is given by the conditional
expectation ¯ x(t) = E(x(t) | FY
t) of the system state x(t) with
respect to the σ - algebra FY
t, generated by the observations
(2) in the interval [t0,t], and is produced by the Kalman-
Bucy filter [28] applied to the linear stochastic system (1)
over the linear observations (2) in the presence of Gaussian
disturbances (Wiener processes) W1(t) and W2(t). The cor-
responding filtering equations for the estimate ¯ x(t) and the
estimation error variance¯P(t) take the form
d¯ x(t) = (A(t)¯ x(t)+B(t)u(t)+G(t)ω(t))dt+
(10)
¯P(t)CT
1(t)(h(t)hT(t))−1[dy1(t)−C1(t)¯ x(t)dt],
with the initial condition ¯ x(t0) = E(x(t0) | FY
t0), and
˙¯P(t) = A(t)¯P(t)+¯P(t)AT(t)+b(t)bT(t)−
(11)
67

Page 3

¯P(t)CT
1(t)(h(t)hT(t))−1C1(t)¯P(t),
with the initial condition
¯P(t0) = E((x(t0)− ¯ x(t0))(x(t0)− ¯ x(t0))T| FY
t0).
Note that the latter equation coincides with (8). Now, apply-
ing the central suboptimal H∞filter for linear systems [4] to
the estimate ¯ x(t) governed by equations (10), (11) yields the
central suboptimal mean-square H∞ estimate equation (7),
where the matrix function ¯P(t) satisfies equation (8), and
the matrix S(t) in (7) satisfies equation (9).
Note that filter (7)–(9) yields, in view of Theorems 3 and
4 in [1], the asymptotic stability of the mean value E(¯ x(t))
of the estimate (10) in the absence of disturbances and the
prescribed attenuation level γ for this variable: ?E(¯ x(t))?2
γ2?ω(t)?2
and the variances of the estimated errors produced by the
estimates ¯ x(t) and m(t) are equal, the conditions 1–3 of
Section 2 hold. The theorem is proved. ?
Remark 1. The convergence of the designed mean-square
H∞state estimate m(t) to the real state value x(t) is assured
by the conditions (C1) and (C2) in view of the results of
Theorem 7.4 and Section 7.7 in [26]. Note that boundedness
of the noise-output H∞norm for the system (S1), controlled
by filter (7)–(9), i.e., admissibility of the mean-square H∞fil-
ter (7)–(9), is determined by the conditions I–III of Theorem
3 in [1].
Remark 2. According to the comments in Subsection
V.G in [1], the obtained central mean-square H∞filter (7)–
(9) presents a natural choice for H∞ filter design among
all admissible H∞ filters satisfying the inequality (6) for a
given threshold γ, since it does not involve any additional
actuator loop (i.e., any additional external state variable) in
constructing the filter gain matrix. Moreover, the obtained
central mean-square H∞filter has the suboptimality property,
i.e., it minimizes the criterion
2<
2+ E(xT(t0))RE(x(t0)). Since E(¯ x(t)) = E(x(t))
J = ?z(t)−L(t)m(t)?2
2−γ2?
?ω(t)?2
2+E(xT
0)RE(x0)
?
Remark 3. Following the discussion in Subsection V.G in
[1], note that the complementarity condition always holds for
the obtained filter (7)–(9), since the positive definiteness of
the initial condition matrix R implies the positive definiteness
of the filter gain matrix S(t) as the solution of (9).
IV. EXAMPLE
This section presents the design of the central suboptimal
mean-square H∞filter to estimate the pitch and yaw angles
for a 2DOF helicopter, ensuring the minimum of the mean-
square criterion (5) and the H∞noise attenuation condition
(6) holds for γ = 1.1.
Let the 2DOF helicopter system with the state space
representation
˙ x(t) = Ax(t)+B(t)u(t)+Gω(t)+bψ1(t),x(t0) = x0
(12)
y(t) =C1x(t)+hψ2(t)
(13)
y2(t) =C2x(t)+Hω(t)
(14)
z(t) = Lx(t),
(15)
where the state vector is: x = [Θ,Ψ,˙Θ,˙Ψ]T, in which Θ and
Ψ are pitch and yaw angles respectively,˙Θ and˙Ψ are pitch
and yaw rates respectively. The matrices are:
A =



0
0
0
0
0
0
0
0
1
0
0
1
0
−9.2751
0
−3.4955



B =



0
0
0
0
2.3667
0.2410
0.0790
0.7913



b =



0
0
0
0
0.9024
0.0919
0.0876
0.8772



G =



0
0
0
0
0
0
0
0
0
0
0
0
0.9024
0.0919
0.0876
0.8772



C1=C2= L =
?
1
0
0
1
0
0
0
0
?
h =
?
1
0
0
1
?
H =
?
0
0
0
0
1
0
0
1
?
.
Here, u(t) is the motor voltage input, ω(t) is an L2
disturbance input, ψ1(t) and ψ2(t) are Gaussian white noises,
which are the weak mean square derivatives of standard
Wiener processes W1(t) and W2(t) (see [27]), respectively.
The Wiener processes are considered independent of each
other and of a Gaussian random variable x0serving as the
initial condition in (12). Equations (12) and (13) present the
conventional form for equations (1) and (2), which is actually
used in practice [29]. It can be easily verified that the noise
orthogonality condition holds for the system (12)–(15).
The filtering problem to be addressed is the same as
described at Section 2. The filtering horizon is set fromt0= 0
to t1= 80 s.
The central suboptimal mean-square H∞ filter takes the
following form for the system (12)–(15)
2
˙ m(t) =



0
0
0
0
0
0
0
0
1
0
0
1
0
−9.2751
0
−3.4955


m(t)
(16)
+P(t)



1
0
0
0
0
1
0
0



?
y(t)−
?
1
0
0
1
0
0
0
0
?
m(t)
?
+S(t)



1
0
0
0
0
1
0
0



?
y2(t)−
?
1
0
0
1
0
0
0
0
?
m(t)
?
,
68

Page 4

with m(0) = m0= E(x0| FY
solutions to the differential Riccati equations
0), where S(t) and P(t) are the
˙S(t) =



0
0
0
0
0
0
0
0
1
0
0
1
0
−9.2751
0
−3.4955


S(t)
(17)
+S(t)





0
0
1
0
0
0
0
1
0
0
0
0
0
−9.2751
0
−3.4955



−S(t)

0.1736
0
0
0
00
0
0
0
0
0
0
0
0.1736
0
0


S(t)
+



0
0
0
0
0
0
0
0
1
0
0
1
0.8220
0.1597
0.1597
0.7779


,
S(0) = S0= R−1,
and
˙P(t) =



0
0
0
0
0
0
0
0
1
0
0
1
0
−9.2751
0
−3.4955


P(t)
(18)
+P(t)



0
0
0
0
0
0
1
0
0
0
0
1
0
0
0
0
0
−9.2751
0
−3.4955



−P(t)



1
0
0
0
0
1
0
0
0
0
0
0


P(t)+



0
0
0
0
0
0
0
0
1
0
0
1
0.8220
0.1597
0.1597
0.7779


,
P(0) = E[(x0−m0)(x0−m0)T| FY
0] = P0,
respectively.
Numerical
solving
equations (16)-(18), with the following initial values:
x0
=[−0.7069,0,0,0]T,
R
=[0.8,0.5,0.2,0.1;0.5,0.8,0.7,0.2;0.2,0.7,0.9,0.4;
0.1,0.2,0.4,0.9], and m0=[0.1745,−0.5236,−0.15,−0.4]T.
Themotorvoltage
u(t) = [12.495,−3.835]T,
L2
disturbance
ω(t) = [ω1(t),ω2(t),ω3(t),ω4(t)]T
realized as ω1(t) = 1/(1 + t), ω2(t) = 0.1(1 − e−0.3t),
ω3(t) = 0.1745/(1+t), and ω4(t) = 0.0873(1−e−0.3t). The
attenuation level value is set to γ = 1.1. The disturbances
ψ1(t) and ψ2(t) in (12),(13) are realized using the built-in
MatLab white noise function.
As a result of the numerical simulation, the following
graphs are presented: graph of the noise-output H∞ norm
(Figure 1); graphs of the pitch angle and the corresponding
estimation error (Figure 2); graph of the yaw angle and the
corresponding estimation error (Figure 3).
simulations
system
resultsareobtained
filteringthe(12)–(15),andthe
P0
=
diag(10,10,15,5),
the
is
Note that the maximum value of the noise-output H∞norm
T = ?z(t)−L(t)m(t)?/(?ω(t)?2
in the considered simulation interval, which is five times less
than the given H∞attenuation level, γ = 1.1. In addition, the
estimation errors converge to zero.
2+E(x0)RE(x0))1/2is 0.208
V. CONCLUSIONS
This paper designs the central finite-dimensional H∞ fil-
ter for linear stochastic systems with integral-quadratically
bounded deterministic disturbances, that is suboptimal for
a given threshold γ with respect to a modified Bolza-
Meyer quadratic criterion including the attenuation control
term with opposite sign. The designed filter is applied to
estimation of the pitch and yaw angles of a two degrees of
freedom (2DOF) helicopter. The simulation results show a
reliable performance of the filter, in particular, the obtained
attenuation level is five times less than a given threshold.
This significant improvement is obtained due to the more
reasonable selection of the filter gain matrix in the designed
filter. Although this conclusion follows from the developed
theory, the numerical simulation serves as a convincing
illustration. The presented approach would be applied in the
future to obtain the central suboptimal mean-square H∞filters
for nonlinear polynomial stochastic systems.
REFERENCES
[1] Doyle JC, Glover K, Khargonekar PP, Francis BA. State-space solu-
tions to standard H2and H∞control problems, IEEE Trans. Automat.
Contr., 1989; 34: 831–847.
[2] Yaesh I, Shaked U. Game theory approach to optimal linear estimation
in the minimum H∞ norm sense, Proc. 28th IEEE Conf. Decision
Control, Tampa, FL, 1989, pp. 421-425.
[3] Shaked U. H∞ minimum error state estimation of linear stationary
processes, IEEE Trans. Automat. Contr., 1990; 35: 554-558.
[4] Nagpal KM, Khargonekar PP. Filtering and smoothing in an H∞
setting, IEEE Trans. Automat. Contr., 1991; 36: 152–166.
[5] Nguang SK, Fu MY. Robust nonlinear H∞filtering, Automatica, 1996;
32: 1195–1199.
[6] Xie LH, De Souza CE, Wang YY. Robust fitering for a class of
discrete-time uncertain nonlinear systems, International Journal of
Robust and Nonlinear Control, 1996; 6: 297–312.
[7] Fridman E, Shaked U. On regional nonlinear H∞filtering, Systems and
Control Letters, 1997; 29: 233–240.
[8] McEneaney WM. Robust H∞filtering for nonlinear systems, Systems
and Control Letters, 1998; 33: 315–325.
[9] Xu SY, Dooren PV. Robust H∞filtering for a class of nonlinear systems
with state delay and parameter uncertainty, Int. J. Control, 2002; 75:
766–774.
[10] Gao H, Wang C. Delay-dependent robust H∞and L2-L∞filtering for a
class of uncertain nonlinear time-delay systems, IEEE Trans. Automat.
Control, 2003; 48: 1661–1666.
[11] Wang Z, Lam J, Liu X. Nonlinear filtering for state delayed systems
with Markovian switching, IEEE Trans. Signal Processing, 2003; 51:
2321–2328.
[12] Xu SY, Chen TW. Robust H∞filtering for uncertain impulsive stochas-
tic systems under sampled measurements, Automatica, 2003; 39: 509–
516.
[13] Zhang WH, Chen BS, Tseng CS. Robust H∞ filtering for nonlinear
stochastic systems, IEEE Trans. Signal Processing, 2005; 53: 589–
598.
[14] Gao H, Wang C. A delay-dependent approach to robust H∞ filtering
for uncertain discrete-time state-delayed systems, IEEE Trans. Signal
Processing, 2004; 52: 1631–1640.
[15] Gao H, Lam J, Xie L, Wang C. New approach to mixed H2/H∞
filltering for polytopic discrete-time systems, IEEE Trans. Signal
Processing, 2005; 53: 3183–3192.
69

Page 5

[16] Duan ZS, Zhang JX, Zhang CS, Mosca E. A simple design method
of reduced-order filters and its applications to multirate filter bank
design, Signal Processing 2006; 86: 1061–1075.
[17] Duan ZS, Zhang JX, Zhang CS, Mosca E. Robust H2and H∞filtering
for uncertain linear systems, Automatica 2006; 42: 1919–1926.
[18] Gao H, Chen TW. H∞ estimation for uncertain systems with limited
communication capacity. IEEE Trans. Automat. Control 2007; 52:
2070–2084.
[19] Wang Z, Liu Y, Liu X. H∞ filtering for uncertain stochastic time-
delay systems with sector-bounded nonlinearities, Automatica 2008;
44: 1268–1277.
[20] Wang Z, Ho D.W.C, Liu Y, Liu X. Robust H∞ control for a class
of nonlinear discrete time-delay stochastic systems with missing
measurements, Automatica 2009; 45: 684–691.
[21] Wei G, Wang Z, Shu H. Robust filtering with stochastic nonlinearities
and multiple missing measurements, Automatica 2009; 45: 836–841.
[22] Gao H, Liu X, Lam J. Stability analysis and stabilization for discrete
fuzzy systems with time-varying delay, IEEE Trans. Systems, Man and
Cybernetics, Part B, 2009; 39: 306–317.
[23] Shen B, Wang Z, Shu H, Wei G. H∞filtering for nonlinear discrete-
time stochastic systems with randomly varying sensor delays, Auto-
matica 2009; 45: 1032–1037.
[24] Fattou A, Sename O, Dion J. H∞ observer design for time-delay
systems, Proc. 37th IEEE Conf. Decision Control, Tampa, FL, 1998,
pp. 4545–4546.
[25] Pila A, Shaked U, de Souza CE. H∞ filtering for continuous-time
linear systems with delay, IEEE Trans. Automat. Control, 1999; 44:
1412–1417.
[26] Jazwinski AH. Stochastic Processes and Filtering Theory. Academic
Press: New York, 1970.
[27] Pugachev VS, Sinitsyn IN. Stochastic Systems: Theory and Applica-
tions. World Scientific: Singapore, 2001.
[28] Kalman RE, Bucy RS. New results in linear filtering and prediction
theory, ASME Trans., Part D (J. of Basic Engineering) 1961; 83: 95–
108.
[29] K. J. ˚ Astr¨ om, Introduction to Stochastic Control Theory, Academic
Press, New York, 1970.
70