Robust sdre filter design for nonlinear uncertain systems with an h performance criterion

ISA Transactions 51 (2012) 146–152
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ISA Transactions
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Robust SDRE filter design for nonlinear uncertain systems with an H∞
performance criterion
Hossein Beikzadeh, Hamid D. Taghirad∗
Advanced Robotics and Automated Systems (ARAS), Department of Systems and Control, Faculty of Electrical and Computer Engineering, K. N. Toosi University of Technology,
P.O. Box 16315-1355, Tehran, 16314, Iran
a r t i c l e i n f o
Article history:
Received 13 April 2011
Received in revised form
8 June 2011
Accepted 10 September 2011
Available online 19 October 2011
Keywords:
SDRE filter
Robust H∞ filter
Nonlinear systems
Filter design
Modeling uncertainty
Measurement noise
Input disturbance
a b s t r a c t
In order to remedy the effects of modeling uncertainty, measurement noise and input disturbance on
the performance of the standard state-dependent Riccati equation (SDRE) filter, a new robust H∞ SDRE
filter design is developed in this paper. Based on the infinity-norm minimization criterion, the proposed
filter effectively estimates the states of nonlinear uncertain system exposed to unknown disturbance
inputs. Moreover, by assuming a mild Lipschitz condition on the chosen state-dependent coefficient form,
fulfillment of a modified H∞ performance index is guaranteed in the proposed filter. The effectiveness of
the robust SDRE filter is demonstrated through numerical simulations where it brilliantly outperforms
the conventional SDRE filter in presence of model uncertainties, disturbance and measurement noise, in
terms of estimation error and region of convergence.
© 2011 ISA. Published by Elsevier Ltd. All rights reserved.
1. Introduction
The state-dependent Riccati equation (SDRE) techniques are
rapidly emerging as general design and synthesis methods of non-
linear feedback controllers and estimators for a broad class of non-
linear problems [1]. Essentially, the SDRE filter, developed over the
past several years, is formulated by constructing the dual problem
of the SDRE-based nonlinear regulator design technique [2]. The re-
sulting observer has the same structure as the continuous steady
state linear Kalman filter. In contrast to the EKF which uses the
Jacobian of the nonlinearity in the system dynamics, the SDRE fil-
ter is based on parameterization that brings the nonlinear sys-
tem to a linear-like structure with state-dependent coefficients
(SDCs). As it is shown in [3], in the multivariable case, the SDC pa-
rameterization is not unique. Consequently, this method creates
additional degrees of freedom that can be used to overcome the
limitations such as low performance, singularities and loss of ob-
servability in a traditional estimation method [2]. Furthermore,
such representation can fully capture the nonlinearities of the sys-
tem, and therefore, this technique has been extensively used for
nonlinear state/parameter estimation within aerospace [4,5] and
power electronics applications [6,7].
∗ Corresponding author. Tel.: +98 21 8406 2321; fax: +98 21 8864 2066.
E-mail addresses: taghirad@kntu.ac.ir, hamid@cim.mcgill.ca (H.D. Taghirad).
There are two commonly used approaches for the SDRE filtering
technique. The first approach, proposed originally by Mracek et al.
in [2], is essentially constructed by considering the dual problem
of the well-known SDRE nonlinear control law. The resulting filter
has the same structure as the steady-state linear Kalman filter
and the Kalman gain is obtained by solving a state dependent
algebraic Riccati equation (SDARE) [2]. However, as reported in [8],
this solution may be computationally expensive for large scale
systems and depends significantly on the observability property
of the system. The second approach that is recently suggested in
the literature has the same structure as the linear Kalman filter
[9,10]. Indeed, it removes the infinite time horizon assumption and
requires the integration of a state-dependent differential Riccati
equation (SDDRE) [9]. This alternative approach addresses the
issues of high computational load and the restrictive observability
requirement in the algebraic form of the estimator.
Although the effectiveness of the SDRE filter has been demon-
strated through impressive simulation results, only few rigorous
mathematical investigations on the filter have been considered
in the literature [11,12]. Assuming certain observability and
Lipschitz conditions on the SDC factorization and considering an
incremental splitting of the state-dependent matrices, the local
convergence of the continuous-time algebraic SDRE observer is
proven in [11]. It is also shown in [1], how this observer asymp-
totically converges to the first-order minimum variance estimate
given by the EKF. The analysis is based on stable manifold the-
ory and Hamilton–Jacobi–Bellman (HJB) equations. Moreover, the
0019-0578/$ – see front matter © 2011 ISA. Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.isatra.2011.09.003

H. Beikzadeh, H.D. Taghirad / ISA Transactions 51 (2012) 146–152 147
analogous discrete-time difference observer is treated in [8,10],
where two distinct sufficient conditions sets for its asymptotic sta-
bility are provided.
All the theoretical results cited above are confined to the
nonlinear deterministic processes and they assume that the
system model is perfectly known. Applying the standard SDRE
filters to general stochastic systems, that is inevitable in practical
purposes, requires accurate specification of the noise statistics
as well. However, model uncertainty and incomplete statistical
information are often encountered in real applications which may
potentially give rise to excessive estimation errors. To tackle such
difficulties, in this paper a robust SDRE filter is proposed with
guaranteed H∞ performance criterion. The motivation of this
paper stems from the fact that in contrast to successful derivation
of an H∞ formulation concerning the SDRE control, accomplished
by Cloutier et al. in [13], there is no documented similar attempt
concerning its filtering counterpart.
Since the pioneer works of linear H∞ filtering designs [14,15],
the nonlinear H∞ filtering problem has been studied by a number
of authors (see [16,17] for a basic study, [18] for a general
stochastic investigation, and [19,20] for approximate solutions).
In this paper, a general continuous-time nonlinear uncertain
model is considered as represented by Nguang and Fu, [16], to
develop a robust H∞ SDRE-based filter for nonlinear uncertain
systems exposed to additive disturbance inputs. The proposed
filter does not involve solving the Hamilton–Jacobi inequalities
(HJIs) in [17,18], which is a time-consuming task. In addition, it
obviates the need for linearization procedure of the extended H∞
techniques [19], and the Riccati-based filtering design [20], and
exhibits robustness against both unknown disturbances and model
uncertainties. In fact in this paper the standard differential SDRE
filter is reformulated into a robust filter such that the estimation
error infinity norm is bounded and the filter achieves a prescribed
level of disturbance attenuation for all admissible uncertainties.
The key assumption made in this paper is that the SDC
parameterization is chosen such that the state-dependent matrices
are at least locally Lipschitz. Note that this result is substantially
different from the well-established methods associated with
the Lipschitz nonlinear systems, which decompose the entire
model into a linear unforced part and a Lipschitz nonlinear
uncertain part [21]. In comparison to the algorithms in [16–18],
another advantage of the proposed method is its simplicity, as no
complicated computation procedures are required to implement
the estimator. It can be implemented systematically and inherits
the elaborated capabilities of the SDRE-based filters [22], as well.
The rest of the paper is organized as follows. Section 2 provides
the necessary backgrounds and formulations of the uncertain SDC
description along with the robust SDRE filter design. In Section 3,
by employing an appropriate Lyapunov function the performance
index for the proposed filter is derived, which can be regarded
as a modification of the conventional H∞ performance criterion.
Section 4 provides simulation examples to illustrate some def-
inite superiority of the proposed filter over the corresponding
usual SDRE-based filters. Finally, some conclusions are drawn in
Section 5.
2. Robust SDRE filter and preliminaries
Consider a smooth nonlinear uncertain system described by
continuous-time equations of the following form:
˙x(t) = f (x) + f (x) + G(t)w0(t) (1)
y(t) = h(x) + h(x) + D(t)v0(t) (2)
where x(t) ∈ Rn
is the state, y(t) ∈ Rm
is the measured output, and
w0(t) ∈ Rp
and v0(t) ∈ Rq
are process and measurement noises
with unknown statistical properties, which stand for exogenous
disturbance inputs. For the sake of simplicity, we restrict ourselves
to unforced noise-driven systems, a slightly more general repre-
sentation than that of [16]. Some remarks on the forced case and
affine in control input are given in Section 3. The nonlinear system
dynamic f (x) and the observation model h(x) are assumed to be
known as C1
-functions. G(t) and D(t) are time varying known ma-
trices of size n × p and m × q, respectively. Also, f (x) and h(x)
represent the system model uncertainties.
Assumption 1. Let the model uncertainties satisfy
[
f (x)
h(x)
]
=
[
E1(t)∆1(t)
E2(t)∆2(t)
]
N(x) (3)
in which N(x) ∈ C1
, E1(t) and E2(t) are known matrix functions
with appropriate dimensions that characterize the structure of the
uncertainties. Also, ∆1(t) and ∆2(t) are norm-bounded unknown
matrices. By performing direct parameterization, the nonlinear
dynamics (1) and (2) accompanied by Assumption 1 can be put into
the following uncertain state-dependent coefficient (SDC) form
˙x(t) = A(x)x + E1(t)∆1(t)N(x) + G(t)w0(t) (4)
y(t) = C(x)x + E2(t)∆2(t)N(x) + D(t)v0(t) (5)
where, f (x) = A(x)x, and h(x) = C(x)x. Note that the SDC
parameterization is unique only if x is scalar [12] (also see the
Remark 1 given below). Besides, the smoothness of the vector
functions f (x) and h(x) with f (0) = h(0) = 0 makes it feasible
[2,3] (see also [22] for effective handling of situations which
prevent a straightforward parameterization).
Remark 1. If A1(x) and A2(x) are two distinct factorization of f (x),
then
A3(x) = M(x)A1(x) + (I − M(x))A2(x)
is also a parameterization of f (x) for each matrix-valued function
M(x) ∈ Rn×n
. This is an exclusive characteristic of all SDRE-based
design techniques, which has been successfully used not only
to avoid singularity or loss of observability, but also to enhance
performance (cf. [1,2,10]). Moreover, it may be used to satisfy the
Lipschitz condition in our filtering design (see Remark 6).
Let us define the signal to be estimated as follows
z(x(t)) = L(t)x(t) (6)
where z(x) ∈ Rs
can be viewed as the filter output, and L(t) is a
known s × n matrix bounded via
lI ≤ LT
(t)L(t) ≤ ¯lI (7)
for every t ≥ 0 with some positive real numbers l, ¯l.
We seek to propose a dynamic filter for the uncertain SDC
model, given by (4) and (5), which robustly estimates the
quantity z(x) from the observed data y(t) with a guaranteed H∞
performance criterion. In other words, it is desired to ensure a
bounded energy gain from the input noises (w0(t), v0(t)) to the
estimation error in terms of the H∞ norm. The proposed filter has
an SDRE-like structure and is prescribed to be
˙ˆx = A(ˆx)ˆx + K(t)[y(t) − C(ˆx)ˆx] + µ−2
P(t)∇xN(ˆx)T
N(ˆx). (8)
In the above, µ > 0 is a free design parameter and ∇x denotes
the gradient with respect to x. Furthermore, ˆx(t) represents the
estimated state vector, and the filter gain matrix, K(t) ∈ Rn×m
, is
defined as
K(t) = P(t)CT
(ˆx)R−1
(9)
in the same way as for the usual SDRE filter. The positive definite
matrix P(t) is updated through the following state-dependent
differential Riccati equation (SDDRE)

148 H. Beikzadeh, H.D. Taghirad / ISA Transactions 51 (2012) 146–152
˙P(t) = A(ˆx)P(t) + P(t)AT
(ˆx) + Γ (t)Q Γ T
(t)
− P(t)[CT
(ˆx)R−1
C(ˆx) − µ−2
∇xN(ˆx)T
∇xN(ˆx)
− λ−2
∇xz(ˆx)T
∇xz(ˆx)]P(t) (10)
with
Γ (t) = [µE1(t) G(t)], (11)
positive definite matrix Q ∈ Rn×n
, symmetric positive definite
matrix R ∈ Rm×m
, and a given positive real value λ > 0 that
indirectly indicates the level of disturbance attenuation in our
robust filter design. This variety as well as the exact role of the free
parameter µ will be clarified in the next section.
Remark 2. It can be easily seen that with λ, µ → ∞, the proposed
filter reverts to the standard differential SDRE filter [9]. Meanwhile,
setting µ = ∞ together with replacing A(ˆx) and C(ˆx) by the
Jacobian of f (x) and h(x), respectively, in (9) and (10) render the
structure of the extended H∞ filter [19,23].
Before analyzing the performance of the robust SDRE filter, we
recall two preparatory definitions within the H∞ filtering theory.
Definition 1 (Extended L2-Space). The set L2[0, T] consists of all
Lebesgue measurable functions g(t) ∈ R+
→ Rr
such that
∫ T
0
‖g(t)‖2
dt < ∞ (12)
for every T ≥ 0 with ‖g(t)‖ as the Euclidean norm of the vector
g(t) (see, e.g., [17,21]).
Definition 2 (Robust H∞ SDRE Filtering). Given any real scalar γ >
0, the dynamic SDRE filter (8)–(10) associated with the dynamics
(4)–(6) is said to satisfy the H∞ performance criterion if
∫ T
0
‖z(t) − ˆz(t)‖2
dt ≤ γ 2
∫ T
0
(‖w0(t)‖2
W + ‖v0(t)‖2
V )dt (13)
holds for all T ≥ 0, all v0(t), w0(t) ∈ L2[0, T], and all admissible
uncertainties. Where, ‖w0(t)‖W and ‖v0(t)‖V are taken to be
Euclidean norms scaled by some positive matrices W and V,
respectively.
Remark 3. Inequality (13) implies that the L2-gain from the exoge-
nous inputs (w0(t), v0(t)) to z(t) − ˆz(t), called the generalized es-
timation error, is less than or equal to some minimum value γ 2
. It
only necessitates that the disturbances have finite energy which is
a familiar mild assumption.
Remark 4. Definition 2 is derived from what is declared by Nguang
and Fu in [16]. The difference is that we consider two distinct noise
sources with scaled Euclidean norms. These scalings, which are
similar to those introduced in [19], may be interpreted as simple
weights because H∞ filtering does not rely on the availability of
statistical information.
3. H∞ performance analysis
In this section, we analyze the estimation error dynamics to
derive an interesting feature of the proposed robust filter, which
will be properly called modified H∞ performance index. This
criterion reveals the ability of the filter to minimize the effects of
disturbances and uncertainties on the estimation error.
In order to facilitate our analysis, we adopt the following
notation
w(t) = [µ−1
∆1(t)N(x) w0(t)]T
v(t) = [E2(t)∆2(t)N(x) v0(t)]T
(14)
in which, the uncertainties are treated as bounded noise signals.
The estimation error is defined by
e(t) = x(t) − ˆx(t). (15)
Subtracting (8) from (4) and considering (13) and (11), the error
dynamics is expressed as
˙e(t) = A(x)x + Γ (t)w(t) − A(ˆx)ˆx
− K(t)[y(t) − C(ˆx)ˆx] − µ−2
P(t)∇xN(ˆx)T
N(ˆx). (16)
Adding and subtracting A(ˆx)x to the whole equation, and adding
and subtracting C(ˆx)x into the bracket lead to
˙e(t) = A(ˆx)(x − ˆx) + (A(x) − A(ˆx))x
− K(t)[C(ˆx)(x − ˆx) + (C(x) − C(ˆx))x + v(t)]
+ Γ (t)w(t) − µ−2
P(t)∇xN(ˆx)T
N(ˆx) (17)
rearranging the terms together with (15), we have
˙e(t) = [A(ˆx) − K(t)C(ˆx)]e(t) + α(x, ˆx) − K(t)β(x, ˆx)
− K(t)v(t) + Γ (t)w(t) − µ−2
P(t)∇xN(ˆx)T
N(ˆx) (18)
where the nonlinear functions α(x, ˆx) and β(x, ˆx) are given by
α(x, ˆx) = [A(x) − A(ˆx)]x (19)
β(x, ˆx) = [C(x) − C(ˆx)]x. (20)
The requirements for Theorem 1 given below, which embodies
the main result of this paper, are summarized by the following
assumptions.
Assumption 2. The state-dependent matrix C(x) and the state
vector x(t) are bounded via
‖C(x)‖ ≤ ¯c (21)
‖x(t)‖ ≤ σ (22)
for all t ≥ 0 and some positive real numbers ¯c, σ > 0.
Remark 5. Note that the assumption above is not severe. In
particular, for many applications boundedness of the state
variables, which often represent physical quantities, is natural.
Besides, if C(x) fulfill (21) for every physical reasonable value of
the state vector x(t), we may suppose without loss of generality
that (21) holds.
Assumption 3. The SDC parameterization is chosen such that A(x)
and C(x) are at least locally Lipschitz, i.e., there exist constants
kA, kC > 0 such that
‖A(x) − A(ˆx)‖ ≤ kA‖x − ˆx‖ (23)
‖C(x) − C(ˆx)‖ ≤ kC ‖x − ˆx‖ (24)
hold for all x, ˆx ∈ Rn
with ‖x − ˆx‖ ≤ εA and ‖x − ˆx‖ ≤ εC ,
respectively.
It should be mentioned that if the SDC form fulfills the Lipschitz
condition globally in Rn
, then all the results in this section will be
valid globally.
Remark 6. Inequalities (23) and (24) are the key conditions in
our performance analysis. They are similar to Lipschitz conditions
imposed in [11] and may be difficult to satisfy for some nonlinear
dynamics. Nevertheless, additional degrees of freedom provided
by nonuniqueness of the SDC parameterization can be exploited
to realize Assumption 3.
With these prerequisites we are able to state the following
theorem, which demonstrates how a modified H∞ performance
index is met by applying the proposed SDRE filter.

Theorem 1. Consider the nonlinear uncertain system of (4)–(6)
along with the robust SDRE filter described by (8)–(10) with some
λ, µ > 0 and positive definite matrices Q and R. Under Assumptions 2
and 3 the generalized estimation error z(t) − ˆz(t) fulfills a modified
type of the H∞ performance criterion introduced in Definition 2,
provided that the SDDRE (10) has a positive definite solution for all
t ≥ 0 and λ is chosen such that
λ−2
l > 2κ (25)
where,
κ =

kA
p
+
¯ckC
r

σ (26)
and r, p > 0 denote the smallest eigenvalue of the positive definite
matrices R and P(t), respectively. Furthermore, the disturbance
attenuation level, γ in (13), is given by
γ 2
=
¯l
λ−2l − 2κ
(27)
with ¯l, l in (7).
Remark 7. For usual differential SDRE filter, the solution of the
standard SDDRE is positive definite and has an upper bound if
the SDC form satisfies a certain uniform detectability condition as
stated in [11]. Unfortunately, this condition cannot be applied to
get similar results for the H∞-filtering-like SDDRE (10). However,
it is a well-known problem arising in H∞ control as well as H∞
filtering that the solutions of the related Riccati equations suffered
from lack of being positive definite (cf. [20]).
Remark 8. The existence of a positive definite solution P(·) for the
SDDRE (10) depends mainly on an appropriate choice of λ and
µ. To find suitable values for λ and µ one can employ a binary
search algorithm, which is widely used to solve H∞ control and
H∞ filtering problems (see, e.g., [20,21]).
Remark 9. Clearly, the filter attenuation constant γ is indirectly
specified by the design parameter λ while it is independent of
µ. The extra design parameter µ has turned out to be very
useful for ensuring solvability of (10) with the desired positive
definiteness property. Furthermore, it scales the uncertainty norm
in the proposed performance index (see the proof of Theorem 1).
Remark 10. Inequality (25) roughly means that λ must chosen
sufficiently small. This is in accordance with the purpose of
performance improvement which calls for a small value of γ in
(27). Furthermore, it can be shown that (25) is obviated while the
estimation error still assures the same performance index with
different attenuation constant γ 2
= λ2
(2¯l/l), if inequalities (23)
and (24) are replaced by more restricted Lipschitz conditions of
order two, e.g., ‖A(x) − A(ˆx)‖ ≤ kA‖x − ˆx‖2
. The proof of theorem
can be modified easily for this case.
To prove Theorem 1 the following lemma is required.
Lemma 1. Let inequalities (21)–(24) are satisfied, then for an
estimation error ‖e‖ ≤ ε, Π(t) = P(t)−1
satisfies the following
inequality
(x − ˆx)T
Π(t)[α(x, ˆx) − K(t)β(x, ˆx)] ≤ κ‖x − ˆx‖2
(28)
where ε = min(εA, εC ). The positive real scalar κ, the matrix
K(t), and the nonlinearities α, β are given by (26), (9), (19),
and (20) respectively.
Proof. Applying the triangle inequality, K = PCT
(ˆx)R−1
and ΠP =
I leads to
‖(x − ˆx)T
Πα(x, ˆx) − (x − ˆx)T
ΠKβ(x, ˆx)‖
≤ ‖(x − ˆx)T
Πα(x, ˆx)‖ + ‖(x − ˆx)T
CT
(ˆx)R−1
χ(x, ˆx)‖. (29)
In view of the Lipschitz conditions on the SDC form and inequality
(22), the nonlinear functions α, β are bounded via
‖α(x, ˆx)‖ = ‖[A(x) − A(ˆx)]x‖ ≤ kAσ‖x − ˆx‖ (30)
‖β(x, ˆx)‖ = ‖[C(x) − C(ˆx)]x‖ ≤ kC σ‖x − ˆx‖ (31)
with ‖x − ˆx‖ ≤ εA and ‖x − ˆx‖ ≤ εC , respectively. Choosing
ε = min(εA, εC ) and employing (30), (31), (21), ‖Π‖ ≤ 1/p, and
‖R−1
‖ ≤ 1/r in (29), we obtain
‖(x − ˆx)T
Πφ(x, ˆx, u) − (x − ˆx)T
ΠKχ(x, ˆx)‖
≤ ‖x − ˆx‖
kAσ
p
‖x − ˆx‖ + ‖x − ˆx‖
¯ckC σ
r
‖x − ˆx‖ (32)
therefore, (28) follows immediately with κ given in (26).
Proof of Theorem 1. Choose a Lyapunov function as
V(e(t)) = eT
(t)Π(t)e(t) (33)
with Π(t) = P(t)−1
, which definitely is positive definite sinceP(t)
in (10) is positive definite. Take time derivative of V(e)
˙V(e(t)) = ˙eT
(t)Π(t)e(t) + eT
(t) ˙Π(t)e(t) + eT
(t)Π(t)˙e(t). (34)
Insert (18) and (10) in (34) along with considering ˙Π(t) = −Π(t)
˙P(t)Π(t), yield with a few rearrangement to
˙V(e(t)) = eT
[−λ−2
LT
L]e + 2eT
Π[α − Kβ]
+ wT
Γ T
Πe + eT
ΠΓ w − vT
R−1
C(ˆx)e
− eT
CT
(ˆx)R−1
v − eT
CT
(ˆx)R−1
C(ˆx)e
+ µ−2
{−eT
∇xN(ˆx)T
N(ˆx)
− eT
∇xN(ˆx)T
N(ˆx) − (∇xN(ˆx)T
N(ˆx))T
e}. (35)
Let us set s = Q −(1/2)
w − (HQ (1/2)
)T
Πe and η = v + C(ˆx)e, then
(35) can be rewritten as
˙V(e(t)) = eT
[−λ−2
LT
L]e + 2eT
Π[α − Kβ]
+ wT
Q −1
w − sT
s + vT
R−1
v − ηT
R−1
η
+ µ−2
{−eT
[∇xN(ˆx)T
∇xN(ˆx)]e
− eT
∇xN(ˆx)T
N(ˆx) − (∇xN(ˆx)T
N(ˆx))T
e} (36)
where Q = Q (1/2)
Q (1/2)T
, R = R(1/2)
R(1/2)T
. Completing the square
in the accolade of (36) and use triangular inequality, to obtain by
virtue of Lemma 1.
˙V(e(t)) ≤ eT
[−λ−2
LT
L]e + 2κ‖e‖2
+ wT
Q −1
w + vT
R−1
v + µ−2
NT
(ˆx)N(ˆx). (37)
This holds, provided that the estimation errors satisfy ‖e‖ ≤ ε, in
which, ε = min(εA, εC ).
The use of −¯lI ≤ −LT
L ≤ lI leads to
˙V(e(t)) ≤ −
λ−2
l − 2κ
¯l
eT
[LT
L]e
+ wT
Q −1
w + vT
R−1
v + µ−2
NT
(ˆx)N(ˆx). (38)
By integrating both sides of (38) over the time interval [0, T], the
H∞ performance index of the proposed filter is derived as
∫ T
0
‖L(t)e(t)‖2
dt ≤ γ 2
∫ T
0
(‖w(t)‖2
Q −1 + ‖v(t)‖2
R−1
+ ‖N(ˆx)‖2
µ−2 + eT
(0)Π(0)e(0))dt (39)

where γ 2
= ¯l/(λ−2
l − 2κ) is a positive real number if λ−2
l > 2κ,
and indicates the filter attenuation constant. Clearly, (39) can be
viewed as a modification of (13) in the sense that it incorporates
the effects of model uncertainties and initial estimation errors. This
concludes the proof of theorem.
Remark 11. Note that, γ is not only an index to indicate the dis-
turbance attenuation level, but also it is an important parame-
ter describing filter estimation ability in the worst case. In other
words, decreasing γ will enhance the robustness of the filter. The
SDRE-based H∞ control offered in [13], is based on a game theo-
retic approach and exhibits robustness only against disturbances.
However, the significance of (39) is that it is derived from
Lyapunov-based approach and guarantees robustness against the
system model uncertainty as well as the process and measurement
noises.
We now endeavor to extend our results to a class of uncertain
forced systems. Suppose the state equation (4) is controlled by the
input u(t) ∈ Rl
as follows
˙x(t) = A(x)x + B(x)u + E1(t)∆1(t)N(x) + G(t)w0(t) (40)
where B(x) ∈ Rn×l
is a known matrix function. Eq. (40) together
with (5) represents an uncertain form of the nonlinear affine sys-
tem used in the SDRE control technique. We claim that, under cer-
tain conditions, the proposed filter will successfully work for the
given forced system, as well. The following corollary evolves this
fact.
Corollary 1. Let the control input u(t) is norm-bounded, i.e., ‖u(t)‖
≤ ρ. Furthermore consider some ρ > 0, and a locally Lipschitz
control matrix B(x), for kB > 0 and ‖x − ˆx‖ ≤ εB:, i.e. ‖B(x) −
B(ˆx)‖ ≤ kB‖x − ˆx‖. Then under the conditions of Theorem 1,
applying (8)–(10), with an additive term of B(ˆx)u in (8), to the given
system (40) and (5) achieves the same performance index as (40). The
only difference to previous result is that in this case, κ = (kAσ +
kBρ)/p + ¯ckC σ/r and ε = min(εA, εB, εC ).
Proof. The proof is in complete analogy to that of Theorem 1.
4. Illustrative examples
4.1. Example 1
To illustrate the performance improvement of the proposed
SDRE filter over the usual algebraic and differential SDRE filters,
we consider a second-order nonlinear uncertain system expressed
as
˙x(t) =
[
x2
1 − 2x1x2 + (−1 + δ1(t))x2
x1x2 + x2 sin x2 + (1 + δ2(t))x1
]
+
[
1
1
]
w0(t) (41)
y(t) = x1 + v0(t) (42)
where, x = [x1 x2]T
and δ1(t), δ2(t) are unknown time varying
functions satisfying the following condition




[
δ1(t)
δ2(t)
]


 ≤ 1. (43)
The disturbing noise signals w0(t) and v0(t) are determined from
two different distributions with unknown statistics, in which w0(t)
is a white Gaussian process noise, while v0(t) is a uniformly
distributed measurement noise.
Obviously, (41) and (42) have the general form of (1) and (2)
and can be rewritten to the uncertain SDC form (4) and (5) by any
suitable parameterization. Among several possible choices, let us
set
A(x) =
[
x1 − 2x2 −1
−1 x1 + sin x2
]
(44)
C(x) = [1 0]. (45)
In addition, in this example there is no measurement uncertainty
(∆2 ≡ 0) while the state equation uncertainty is described by
∆1(t) =
[
0 δ1(t)
δ2(t) 0
]
, N(x) = x (46)
with E1(t) = I2.
Note that (45) is a trivial choice, and one can choose other forms
such as C(x) = [1 + x2 x1]. The reason of our choices, (44) and
(45), is mainly related to the compliance of inequalities (21), (23),
and (24). Firstly, it follows from (44) that for all x, ˆx ∈ R2
,
A(x) − A(ˆx)
=
[
(x1 − ˆx1) − 2(x2 − ˆx2) 0
0 (x1 − ˆx1) + (sin x2 − sin ˆx2)
]
. (47)
Since
|(x1 − ˆx1) − 2(x2 − ˆx2)| ≤
√
5‖x − ˆx‖
and
|(x1 − ˆx1) + (sin x2 − sin ˆx2)| ≤
√
2‖x − ˆx‖,
it can be deduced that (23) is globally satisfied with kA =
√
5.
Secondly, the selected output matrix (45) fulfills (21) with ¯c =
1 and (24) with any positive real Lipschitz constant such as
kC = 0.001. Considering these facts in addition to the Lyapunov
stability of (41), we may conclude that Assumptions 2 and 3 hold.
Next, implement the robust SDRE filter according to (8)–(10) to
obtain the desired performance for the system (41) and (42). The
differential equations are solved numerically by the Runge–Kutta
method, choosing the initial conditions x(0) = [−0.5 0.5]T
for
the system to be observed, ˆx(0) = [0.5 −0.5]T
for the filter and
P(0) = 10I2 for the SDDRE (10). The design details are summarized
as follows.
The filter output, z(x) in (6), is assumed to be x(t) itself.
Therefore, in this case, L(t) is the identity matrix and ¯l, l may be
considered as one. Also choose the weighting matrices Q = I2 and
R = 0.1. The appropriate values for λ and µ are obtained using
a binary search algorithm similar to that proposed in [12]. By this
means, it turns out that λ = 0.5 and µ = 0.004 are sufficient
for P(t) in (10) to be always positive definite. Besides, this value of
λ will satisfy (25). This fact can be easily verified by inserting the
values of kA, kC , and ¯c, which are analytically determined above,
along with r = 0.1, σ = 0.707, and p = 10 into (25) which yields
to κ = 0.165.
So far, all the sufficient conditions in Theorem 1 have been
ensured, and hence, it is expected to reach to the modified H∞
performance index obtained in (39). This is verified through the
simulation results depicted in Fig. 1. The figure shows the true
state of the system together with the estimated value obtained
from the robust SDRE filter. It is clear that the filter performs
as expected, and the estimated signals converge quickly to the
corresponding actual ones in spite of the considered disturbances
and modeling uncertainties. Note that according to (27), the given
λ = 0.5 guarantees an attenuation level of γ 2
= 0.27. This means
the energy gain from the disturbances to the estimation errors is
bounded by 0.27.
For the sake of comparison, the standard algebraic and differ-
ential SDRE filters, namely SDARE filter and SDDRE filter, were also
simulated with the same weighting matrices Q , R and the same ini-
tial conditions ˆx(0). The results of this simulation are illustrated in
Fig. 2. It is observed that in these two cases, the estimated signals
do not track the true ones and exhibit divergence.

Fig. 1. The actual and the estimated states by the robust SDRE filter.
Fig. 2. The actual and the estimated states by the algebraic and differential SDRE
filters.
4.2. Example 2 (induction motor).
In order to show the effectiveness of the proposed filtering
scheme, let us implement it for the estimation of flux and angular
velocity of an induction motor. The normalized state equation of
such system may be written as follows [24,25]. Note that in this
example a forced system is considered for the robust SDRE filter
implementation.
˙x1(t) = k1x1(t) + u1(t)x2(t) + k2x3(t) + u2(t)
˙x2(t) = −u1(t)x1(t) + k1x2(t) + k2x4(t) (48)
˙x3(t) = k3x1(t) + k4x3(t) + (u1(t) − x5(t))x4(t)
˙x4(t) = k3x2(t) − (u1(t) − x5(t))x3(t) + k4x4(t)
˙x5(t) = k5(x1(t)x4(t) − x2(t)x3(t)) + k6u3(t).
In the above, x1, x2 and x3, x4 are the components of the stator
and the rotor flux, respectively, and x5 is the angular velocity. The
inputs are denoted by u1 as the frequency u2 as the amplitude of the
stator voltage, and u3 as the load torque. Furthermore, k1, . . . , k6
are some constants determined from the structure of the motor
and its drive system [24]. The output equations are given as
y1(t) = k7x1(t) + k8x3(t)
y2(t) = k7x2(t) + k8x4(t)
(49)
in which, k7 and 8 k8 are user defined parameters, to generate the
normalized stator current denoted by y1(t) and y2 (t). Consider the
following SDC parameterization of (48) and (49)
A(x) =





k1 0 k2 0 0
0 k1 0 k2 0
k3 0 k4 −x5 0
0 k3 0 k4 x3
k5x4 −k5x3 0 0 0





(50)
B(x) =





x2 1 0
−x1 0 0
x4 0 0
−x3 0 0
0 0 k6





(51)
C(x) =
[
k7 0 k8 0 0
0 k7 0 k8 0
]
. (52)
Inequality (24) is evidently satisfied by the above output matrix
(52). But, confirming the Lipschitz condition for the matrices (50)
and (51) necessitate some calculations. For matrix B(x) we have
‖B(x) − B(ˆx)‖
=

(x1 − ˆx1)2 + (x2 − ˆx2)2 + (x3 − ˆx3)2 + (x4 − ˆx4)2
≤ kB‖x − ˆx‖ (53)
with x, ˆx ∈ R5
. Therefore, Lipschitz condition for the matrices (51)
is met for any positive real number kB. Likewise, for the system
matrix A(x) we may derive:
‖A(x) − A(ˆx)‖
= max(|x3 − ˆx3|, |x5 − ˆx5|, k5

(x3 − ˆx3)2 + (x4 − ˆx4)2). (54)
Hence, it can be concluded that Lipschitz condition for the matrices
(50) is also satisfied for kA = max(1, k5).
We now proceed to implement the proposed robust SDRE
filter. The following parameters are set in the simulations. k1 =
−0.186, k2 = 0.176, k3 = 0.225, k4 = −0.234, k5 = −0.1081,
k6 = −0.018, k7 = 4.643, k8 = −4.448. The control input is
considered as u(t) = [1 1 0]T
, and the initial state conditions
are set to: x(0) = [0.2 − 0.6 − 0.4 0.1 0.3]T
.
Let us consider a relatively large initial estimation error by
choosing ˆx(0) = [0.5 0.1 0.3 −0.2 4]T
, and design the
proposed robust SDRE filter with the following design parameters:
Q = 0.04I5, R = 0.06I2, λ2
= 0.7, µ2
= 0.008.
For sake of comparison, the standard SDDRE filter, is also
simulated with the same weighting matrices Q , R and the
same initial conditions ˆx(0). The results of these simulations are
illustrated in Fig. 3. As it is observed in this figure the estimation
error for the angular velocity estimates for the robust SDRE filter
converges to zero, while the estimation error diverges in the case
of SDDRE filter. This indicates that a larger region of convergence
is attainable for robust SDRE filter in this case.
Furthermore, in order to illustrate that the conditions given in
Theorem 1, are just sufficient conditions, the Lyapunov function
(33) is evaluated and plotted in Fig. 4. It can be seen that
although the Lyapunov function is not monotonically decreasing,
the estimation error converges to zero. This reveals the fact that
the obtained results are not necessary conditions for error decay.

Fig. 3. The actual angular speed and its estimates by SDDRE filter and the proposed
robust SDRE filter for large initial error.
Fig. 4. The values of the Lyapunov function (33) of the proposed robust SDRE filter
for large initial error.
5. Conclusion
To overcome the destructive effects of uncertain dynamics
and unknown disturbance inputs on the performance of the
usual SDRE filters a new robust H∞ filter design is developed in
this paper. The proposed filter can be systematically applied to
nonlinear continuous-time systems with an uncertain SDC form.
It is proved that under specific conditions the proposed filter
guarantees the modified H∞ performance criterion by choosing
an appropriate Lyapunov function. This criterion is modified in
the sense that it incorporates both the effects of disturbances
and model uncertainties in the H∞ norm minimization. Numerical
simulations show the promising performance of the robust SDRE
filter in comparison to the standard SDRE filters, in terms of
estimation error and region of convergence. The obtained results
nominate the proposed filter as a viable H∞ filtering method for
practical applications.
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