International Journal of Automation and Computing
9(4), August 2012, 358-368
DOI: 10.1007/s11633-012-0656-y
Exponential Nonlinear Observer Based on the Differential
State-dependent Riccati Equation
Hossein Beikzadeh1
1
2
Hamid D. Taghirad2
Department of Electrical and Computer Engineering, University of Alberta, Edmonton T6G 2V4, Canada
Faculty of Electrical and Computer Engineering, K. N. Toosi University of Technology, Tehran 16315-1355, Iran
Abstract: This paper presents a novel nonlinear continuous-time observer based on the differential state-dependent Riccati equation
(SDRE) filter with guaranteed exponential stability. Although impressive results have rapidly emerged from the use of SDRE designs for
observers and filters, the underlying theory is yet scant and there remain many unanswered questions such as stability and convergence.
In this paper, Lyapunov stability analysis is utilized in order to obtain the required conditions for exponential stability of the estimation
error dynamics. We prove that under specific conditions, the proposed observer is at least locally exponentially stable. Moreover, a
new definition of a detectable state-dependent factorization is introduced, and a close relation between the uniform detectability of
the nonlinear system and the boundedness property of the state-dependent differential Riccati equation is established. Furthermore,
through a simulation study of a second order nonlinear model, which satisfies the stability conditions, the promising performance of
the proposed observer is demonstrated. Finally, in order to examine the effectiveness of the proposed method, it is applied to the
highly nonlinear flux and angular velocity estimation problem for induction machines. The simulation results verify how effectively
this modification can increase the region of attraction and the observer error decay rate.
Keywords: Detectability, direct method of Lyapunov, exponential stability, nonlinear observer, region of attraction, state-dependent
Riccati equation (SDRE) technique.
1
Introduction
The problem of estimating the state of a nonlinear dynamical system has attracted considerable attention since
the development of linear observers[1, 2] . Numerous design
methods exist for various classes of systems: feedback linearization, variable structure techniques, extended Kalman
filter, high gain observers, Lyapunov-based observer design,
state-dependent Riccati equation (SDRE) technique, etc,
among others (see [3 – 6] and the references cited therein).
The most popular reported method is the extended Kalman
filter (EKF)[7, 8] . In spite of its satisfactory results, there is
documented evidence of erratic filter behavior such as the
premature collapse of the error covariance[9] and the loss of
system observability[10] .
The state-dependent Riccati equation (SDRE) techniques are rapidly emerging as general design and synthesis
methods of nonlinear feedback controllers and estimators
for a broad class of nonlinear problems[11] . 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[10] . The resulting 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 filter is based on parameterization that brings
the nonlinear system to a linear-like structure with statedependent coefficients (SDC).
As shown in [12], in the multivariable case, the SDC parameterization 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 observability in a traditional estimation
Manuscript received September 13, 2010; revised November 2, 2011
method[10] .
Indeed, an algebraic Riccati equation is solved at every
time step to obtain the SDRE observer gain. Hence in
[13 – 15], this method is called the state-dependent algebraic
Riccati equation (SDARE) observer. One drawback of the
SDARE is its high computational load for large scale systems. The other is that if loss of observability occurs during
certain time intervals, then the algebraic Riccati equation
may not have a solution and the SDARE cannot be used
during these time-intervals.
Although the SDRE observer has been empirically implemented in a number of applications[13,16−21] , its practical usefulness is accompanied with a heuristic theoretical
derivation. In fact, there is no mathematically rigorous stability analysis regarding this estimating method except, the
theoretical results which are provided in a recent paper[21] .
In [21, 22], by splitting the state-dependent matrices of the
SDC form into a constant element and a state-dependent
incremental element, it is shown that the estimator is locally asymptotically stable under some observability and
Lipschitzian conditions. However, the provided results are
essentially local and require certain simplicity and boundedness assumptions on the incremental matrices in a neighborhood of the origin, which may be violated for many practical
systems.
Recently an alternative form of this estimator, namely
the state-dependent differential Riccati equation (SDDRE)
observer, which has the same structure as the linear Kalman
filter has been offered[15, 23−25] . This alternatively addresses
the issues of high computational load and the potentially
overly restrictive observability requirement in the SDARE.
The key idea underlying this new method is to remove
the infinite-time horizon assumption and to use differential rather than algebraic Riccati equation. In [23, 25], a
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discrete-time form of the SDDRE observer is considered and
two distinct sufficient conditions set for its asymptotic stability are obtained. However, these conditions can be verified only based on the simulation results and are confined to
unforced systems. The stochastic stability and robust synthesis of the SDDRE state estimator for nonlinear systems
exposed to disturbance inputs are addressed in [26, 27], respectively.
In this paper, motivated by the exponential stability results provided in [28, 29] regarding the widely used EKF,
the continuous-time SDDRE formulation is modified in order to have an exponentially stable observer, i.e., the estimation error goes to zero exponentially[30] . The suggested
methodology is based on the regulation theory with prescribed degree of stability[31] and exponential data weighting for linear stochastic systems[32] . Using the Lyapunov
stability analysis, a set of sufficient conditions is obtained
which ensures the local exponential stability of the proposed
observer. These conditions do not implicate any particular
splitting or simplicity limitations on the SDC factorization
like those imposed in [21], and can be satisfied easier than
those conditions in [23, 25] which are completely simulationbased. Furthermore, a new definition of a detectable SDC
parameterization is introduced, which is derived from interesting results presented in [33], and is closely related to the
existence of bounded and positive definite solutions for the
state-dependent differential Riccati equations.
By this means, a new SDRE-based observer with guaranteed exponential stability is obtained, which inherits the
elaborated properties of the SDRE technique. Moreover,
this modification allows an effective treatment of the nonlinearities and under a specific condition grants in advance assignment of the degree of stability[31] . Comparative studies
reveal the preferable performance and an increased region of
attraction compared to the common SDRE-based observers.
This paper is organized as follows. In Section 2, we state
the necessary preliminaries and introduce the proposed observer. Then in Section 3, by choosing an appropriate Lyapunov function, we show that the proposed observer is an
exponential observer. The role of uniform detectability in
this context is discussed in Section 4. Section 5 which contains two simulation examples, states the estimation of a
simple nonlinear system and an induction machine respectively, illustrates the recent SDRE-based observer in a deterministic setting. We also compare the results of the proposed observer to that of the SDARE and SDDRE observers
given in the literature. Finally, the conclusions are given in
Section 6. Throughout this paper, k·k denotes the Euclidian
norm of real vectors or the induced norm of real matrices.
Moreover, Rq is the real q-dimensional vector space and C 1
is the space of continuously differentiable functions.
2
Proposed SDRE observer and some
preliminaries
Consider a nonlinear continuous-time system affine in the
input that is represented by
ẋ(t) = f (x(t)) + g(x(t))u(t)
(1)
y(t) = h(x(t))
(2)
where x(t) ∈ Rn is the state, u(t) ∈ Rp the input, and
y(t) ∈ Rm the output. Assume that by direct parameterization, the nonlinear dynamics can be rewritten in the
following state-dependent coefficient (SDC) form:
ẋ(t) = A(x(t))x(t) + B(x(t))u(t)
(3)
y(t) = C(x(t))x(t)
(4)
where
f (x(t)) = A(x(t))x(t)
g(x(t)) = B(x(t)),
h(x(t)) = C(x(t))x(t).
(5)
The former parameterization for a continuous A(x) is
possible if f (0) = 0 and f (x) ∈ C 1[34, 35] . However, as discussed in [17], even if f is only continuous, i.e., f (x) ∈ C 0 ,
finding a continuous factorization is still possible (but not
guaranteed). Also see [11] for effective handling of some
situations which may prevent a straightforward parameterization. It is also shown in [12] that in multivariable case,
the SDC parameterization is not unique.
We first make the following assumptions on the chosen
SDC form, the control input u(t), and the signal to be estimated x(t).
Assumption 1. The SDC parameterization is chosen
such that A(x), B(x) and C(x) are at least locally Lipschitz
(see [21, 27]). i.e., there exist constants kA , kB , kC > 0 such
that
kA(x1 ) − A(x2 )k 6 kA kx1 − x2 k
(6)
kB(x1 ) − B(x2 )k 6 kB kx1 − x2 k
(7)
kC(x1 ) − C(x2 )k 6 kC kx1 − x2 k
(8)
for x1 , x2 ∈ Rn with kx1 − x2 k 6 εA , kx1 − x2 k 6 εB , and
kx1 − x2 k 6 εC , respectively.
It should be mentioned that if the SDC form fulfills the
Lipschitz condition globally in Rn , then all the results in
this and the ensuing sections will be valid globally.
Assumption 2. The time varying state-dependent matrix C(x(t)) is bounded by
kC(x(t))k 6 c̄
(9)
where c̄ > 0 is a real number.
Assumption 3. Assume that there exist σ, ρ > 0 such
that for all t > 0
kx(t)k 6 σ,
ku(t)k 6 ρ.
(10)
These assumptions will be used in our stability analysis in
the next section.
Remark 1. If A1 (x) and A2 (x) are two distinct parameterization of f (x), then
Ã(x) = M (x)A1 (x) + (I − M (x))A2 (x)
(11)
is also a parameterization of f (x) for all matrix valued functions M (x) ∈ Rn×n (see [10, 12, 25]). This is also valid for
the output matrix C(x). These additional degrees of freedom provided by nonuniqueness of the SDC parameterization can be used to either enhance the observer performance
and avoid loss of observability, or particularly satisfy (6) –
(8).
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International Journal of Automation and Computing 9(4), August 2012
Remark 2. The Lipschitz conditions (6) – (8) are also
considered in [21, 25]. Consequently, compared to the previous studies on the SDRE observers, we have not imposed
any new restrictive assumptions on the chosen SDC form.
Remark 3. Inequalities (9) – (10) are mild conditions. In
particular, for many applications, the state variables, which
often represent physical quantities, are bounded. Boundedness of the control input seems also to be a trivial hypothesis. Thus, (10) are satisfied easily. Besides, if C(x) fulfills
(9) for every reasonable value of the state vector x(t), we
may suppose without loss of generality that (9) is also satisfied.
Now, the proposed observer is developed. For the dynamical system given by (3) – (4), let us introduce an observer
as follows
Adding and subtracting A(x̂(t))x(t) to the whole equation
together with adding and subtracting C(x̂(t))x(t) into the
bracket lead to
ė(t) =A(x̂(t))x(t) − A(x̂(t))x̂(t) + A(x(t))x(t)−
A(x̂(t))x(t) + [B(x(t)) − B(x̂(t))]−
K(t)[C(x̂(t))x(t) − C(x̂(t))x̂(t)+
C(x(t))x(t) − C(x̂(t))x(t)].
So the error dynamics are given by
ė(t) = [A(x̂(t)) − K(t)C(x̂(t)] e(t)+
φ(x(t), x̂(t), u(t)) − K(t)χ(x(t), x̂(t))
φ(x(t), x̂(t), u(t)) = [A(x(t)) − A(x̂(t))] x(t)+
(13)
where P (x(t)) ∈ Rn×n is symmetric and computed through
the following state-dependent differential Riccati equation:
Ṗ (x(t)) = (A(x̂(t)) + αI) P (x(t)) + P (x(t))(AT (x̂(t))+
αI) − P (x(t))C T (x̂(t))R−1 C(x̂(t))P (x(t)) + Q
(14)
with a positive real number α > 0, and symmetric positive
definite matrices Q ∈ Rn×n and R ∈ Rm×m . It should be
noted that (14) renders to the Riccati equation of the usual
SDDRE[15] for α = 0. Moreover, the observer gain K and
the solution P of (14) are state-dependent. However, their
state dependency is omitted thereafter just for notational
convenience.
Remark 4. Note that (14) is indeed a modified version
of the Riccati equation with an additional term 2αP (x(t)).
The system dynamics is still incorporated in obtaining
P (x(t)) and our stability analysis in the next section.
Remark 5. The scalar α is a design parameter, which
indirectly indicates the error decay rate of the proposed
exponential observer. This verity as well as the feasibility
of α will be clarified in the next section.
Remark 6. In a stochastic framework, a common choice
for the matrices Q and R are the covariances of the corrupting noise signals. However, this is not the only possibility.
Especially, for a deterministic estimation problem that is
tackled in this paper, any other positive definite matrices
can be chosen as well.
Define the estimation error by
e(t) = x(t) − x̂(t).
(15)
By subtracting (12) from (3), the error dynamics is obtained
ė(t) =A(x(t))x(t) + B(x(t))u(t) − A(x̂(t))x̂(t)−
B(x̂(t))u(t) + K(t) [y(t) − C(x̂(t))x̂(t)] .
[B(x(t)) − B(x̂(t))]u(t)
(19)
χ(x(t), x̂(t)) = [C(x(t)) − C(x̂(t))] x(t).
(20)
(12)
where x̂(t) denotes the state estimate and the observer gain
K(t) is a time varying n×m matrix. We define the observer
gain by
K(x(t)) = P (x(t))C T (x̂(t))R−1
(18)
where
˙
x̂(t)
=A(x̂(t))x̂(t) + B(x̂(t))u(t)+
K(x(t)) [y(t) − C(x̂(t))x̂(t)]
(17)
(16)
In order to analyze the error dynamics, we make use of
the following two definitions.
Definition 1. The equilibrium point e(t) = 0 of (18) is
locally exponentially stable, if there exist constants ε, η, θ >
0 such that
t
ke(t)k 6 η ke(0)k exp(− )
(21)
θ
holds for every t > 0 and for every solution e(·) of equation (18), originating from an initial state inside Bε =
{e ∈ Rn | kek < ε}[36] .
Definition 2. The observer given by (12) – (14) is an exponential observer, if the differential (18) of the estimation
error has a locally exponentially stable equilibrium point at
e(t) = 0[30] .
3
Stability analysis
In this section, we determine sufficient conditions that
guarantee exponential stability of the proposed observer.
Theorem 1 states the main result of this paper. Note that
the matrix inequalities Ω 6 ∆ and further Ω − ∆ 6 0 mean
that the matrix Ω − ∆ is negative semidefinite.
Theorem 1. Consider the nonlinear continuous-time
system (1) – (2) put into the SDC form (3) – (4), along with
the proposed SDRE-based observer (12) – (14). Let Assumptions 1 – 3 hold and the solution P (t) of differential
Riccati equation (14) is bounded via
pI 6 P (t) 6 p̄I
(22)
for some positive real numbers p, p̄ > 0. Then the proposed
observer is an exponential observer in the sense of Definition
2, provided that the design parameter α satisfies
α>
qp
c̄kC σp
+ kA σ + kB ρ − 2
r
2p̄
(23)
where q = λmin (Q) and r = λmin (R).
Remark 7. Inequality (22) which seems the key condition in our stability analysis, is closely related to observability and detectability properties of the system to be observed. These relations will be discussed in Section 4.
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Remark 8. The Lipschitz constants kA , kB , and kC are
derived analytically from the system dynamics. The bounds
r and q can also be obtained from the weighting matrices
Q and R, respectively. Likewise, ρ can be determined by
considering the actuators saturation limit, although conservatively, and values of σ, c̄ are calculated analytically. The
bounds p and p̄ for the matrix P (t) are related to the detectability of the nonlinear system (see the next section).
Accordingly, (23) can be verified in advance. Also, see the
following two remarks.
Remark 9. Inequality (23) roughly means that α should
be chosen sufficiently large. Surprisingly, this is in accordance with the purpose of performance improvement which
calls for a smaller time constant (see the proof of Theorem
1 and also Remark 12).
Remark 10. It can be shown that, (23) is obviated while
the estimation error remains still exponentially stable, provided that (6) – (8) are replaced by more restricted Lipschitz
conditions with exponent two, e.g., kA(x1 ) − A(x2 )k 6
kA kx1 − x2 k2 (see [26]). The proof of this theorem can
be modified easily for this case.
To prove Theorem 1, we state the following preparatory
lemma.
Lemma 1. Consider a positive-definite m × m matrix R
with R > rI for some r > 0. Assume that the matrix K(t)
and the nonlinearities φ(x(t), x̂(t), u(t)) and χ(x(t), x̂(t))
are given by (13), (19), and (20), respectively. Then under the assumptions of Theorem 1, there exist real numbers
ε, κ > 0 such that Π(t) = P −1 (t) satisfies the inequality
T
T
T
K
1
1
ke(t)k2 6 V (e(t), t) 6 ke(t)k2
p̄
p
(30)
which imply that V (e(·), ·) is positive definite and decrescent, thus this function is an appropriate Lyapunov function candidate. Taking time derivative of the Lyapunov
function, we get
V̇ (e(t), t) =ėT (t)Π(t)e(t) + eT (t)Π̇(t)e(t)+
eT (t)Π(t)ė(t).
(31)
Inserting ė(t) according to differential equation (18) yields
with a few rearrangements
V̇ (e(t), t) =eT (t)Π̇(t)e(t)+
eT (t) [A(x̂(t)) − K(t)C(x̂(t))]T e(t)+
eT (t)Π(t) [A(x̂(t)) − K(t)C(x̂(t))] e(t)+
2eT (t)Π(t)[φ(x(t), x̂(t), u(t))−
K(t)χ(x(t), x̂(t))].
(32)
Applying Lemma 1 and considering (13) yields
By the hypothesis that A(x), B(x), and C(x) are locally
Lipschitz and the use of (10), we have
k[B(x) − B(x̂)] uk 6 (kA σ + kB ρ) kx − x̂k .
(26)
1
,
r
we
k(x − x̂)T Πφ(x, x̂, u) − (x − x̂)T ΠKχ(x, x̂)k 6
(kA σ + kB ρ)
c̄kC σ
kx − x̂k + kx − x̂k
kx − x̂k
p
r
(27)
for kx − x̂k 6 ε with ε = min(εA , εB , εC ). Thus (24) follows
immediately with
(28)
(33)
for ke(t)k 6 ε, where κ is given by (28) and ε =
min(εA , εB , εC ). Considering
Π̇(t) = −Π(t)Ṗ (t)Π(t)
(34)
and the differential Riccati equation (14) leads to
V̇ (e(t), t) 6 −2αeT (t)Π(t)e(t) − eT (t) [Π(t) ×
i
QΠ(t) + C T (x̂(t))R−1 C(x̂(t)) e(t) + 2κ ke(t)k2 .
kφ(x, x̂, u)k 6 k[A(x) − A(x̂)] xk +
kχ(x, x̂)k = k[C(x) − C(x̂)] xk 6 kC σ kx − x̂k .
°
°
Considering (26), kΠk 6 p1 , kCk 6 c̄, and °R−1 ° 6
obtain
AT (x̂(t))Π(t) − 2C T (x̂(t))R−1 C(x̂(t))]e(t)+
2κ ke(t)k2
k(x − x̂)T Πφ(x, x̂, u)k + k(x − x̂)T C(x̂)T R−1 χ(x, x̂)k.
(25)
(kA σ + kB ρ)
c̄kC σ
+
.
p
r
with Π(t) = P −1 (t). Because of (22), we have the following
bounds for the Lyapunov function:
=
T
κ=
(29)
V̇ (e(t), t) 6 eT (t)[Π̇(t) + Π(t)A(x̂(t))+
k(x − x̂) Πφ(x, x̂, u) − (x − x̂) ΠKχ(x, x̂)k 6
kx − x̂k
V (e(t), t) = eT (t)Π(t)e(t)
2
(x − x̂) Πφ(x, x̂, u) − (x − x̂) ΠKχ(x, x̂) 6 κ kx − x̂k
(24)
for every kx − x̂k 6 ε.
Proof. Applying the triangular inequality,
P C T R−1 and ΠP = I yield to
¤
Proof of Theorem 1. We consider differential equation (18) of the estimation error and prove its exponential
stability by choosing the Lyapunov function
(35)
Denote the smallest eigenvalue of the positive-definite matrix Q by q, then we have qI < Q. Together with bounds
(22) for P (t) we obtain
¶
µ
q
−
2κ
ke(t)k2 . (36)
V̇ (e(t), t) 6 −2αV (e(t), t) −
p̄2
According to (30), we get
p
− ke(t)k2 6 −pV (e(t), t) 6 − ke(t)k2
p̄
(37)
µ
¶
qp
V̇ (e(t), t) 6 − 2α + 2 − 2κp V (e(t), t)
p̄
(38)
hence
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International Journal of Automation and Computing 9(4), August 2012
±
for ke(t)k 6 ε. Therefore, if 2α + qp p̄2 − 2κp > 0, it
then follows that V̇ (e(t), t) is locally negative definite. By
applying standard results concerning the direct method of
Lyapunov (see [36]), we conclude that the differential equation (18) has a uniformly asymptotic stable equilibrium at
0. Moreover, by separating variables and integration we
have
µ ·
¸ ¶
qp
V (e(t), t) 6 V (e(0), 0) exp − 2α + 2 − 2κp t . (39)
p̄
Together with (30) lead to
s
µ ·
¸ ¶
qp
p̄
ke(0)k exp − α + 2 − κp t
ke(t)k 6
p
2p̄
(40)
q
±
i.e., (21) is valid with η= p̄/p and θ−1=α + qp 2p̄2−κp. ¤
Remark 11. For α = 0, we have the standard differential
SDRE observer[15] which, according to this theorem, is an
exponential observer if
qp
− κp > 0.
2p̄2
(41)
It is clear that, satisfaction of (23) for the proposed observer
is much easier than (41) for the SDDRE. In fact, the free
parameter α can be chosen such that (23) holds. On the
other hand, (41) depends on the parameters which are not
under our control, and
± thus it may be violated.
Remark 12. If qp 2p̄2 − κp > 0, then the time constant
θ for the exponential error decay in (21) satisfies θ < α−1 .
In this situation, selecting an appropriate α > 0 and by
using θ < α−1 , it can be seen that the constant θ in (21)
can be assigned in advance, i.e., we obtain an observer with
a prescribed degree of stability (see [28, 31]).
4
Relation to detectable SDC parameterization
According to (22), bounds are required for the solution
P (t) of the differential Riccati equation (14) to prove the
convergence of the estimation error. Interesting results between the observability of the nonlinear system and existence of positive definite solutions for the differential Riccati equations are given in [33]. Lower and upper bounds for
the error covariances, P (t), have been also obtained using
dual optimal control problems.
In [25], a discrete-time SDRE-based observer is considered and using the results presented in [37 – 38], a priori
bounds on the Riccati equation are obtained. These bounds
follow from a uniform observability condition. However, the
nonlinear system is treated like a frozen-in-time linear system, and the state-dependency fact is ignored.
In this section, based on the results given in [33], we
discuss how the following uniform detectability notation
is related to the boundedness of the solution P (t) of the
continuous-time state-dependent differential Riccati equation (14).
Definition 3. The pair {C(x), A(x)} is called a uniformly detectable SDC parameterization of the nonlinear
system (1) – (2), if there exist a bounded matrix valued
function Λ(x) and a real number γ > 0 such that
ω T [A(x) + Λ(x)C(x)] ω 6 −γ kωk2
(42)
n
holds for all x ∈ R .
Another detectability definition associated with the SDC
form, has been proposed in the literature (see [21, 22, 34])
where A(x) is called as a detectable (observable) parameterization if the pair {C(x), A(x)} is pointwise detectable
(observable) in the linear sense for all x in the domain
of interest. This definition is clearly equivalent to Definition 3. The reason is that, this pointwise detectability
implies that there exists a matrix Λ(x) such that the matrix A(x) + Λ(x)C(x) has all eigenvalues in the open left
half plane and therefore, inequality (42) will be satisfied.
Remark 13. Note that the pointwise detectability condition is not necessarily equivalent to nonlinear detectability. Due to the nonuniqueness of the SDC form, different choices may yield different state-dependent observability matrices[21] and thus different pointwise observability
characteristics. The uniform detectability condition provided in Definition 3 will inherit this property as well (see
[39] for rigorous establishment of connections between the
pointwise controllability and true nonlinear controllability).
Now it is possible to state the following lemma.
Lemma 2. Consider the nonlinear system (3) – (4), the
solution P (t) of the differential Riccati equation (14) and
assume that the following conditions hold:
1) The system matrix A(x) is norm bounded, i.e., kAk <
∞;
2) The SDC parameterization is chosen such that the
pair {C(x), A(x) + αI} is uniformly detectable according
to Definition 3;
3) The initial condition P (0) of the differential Riccati
equation (14) is positive definite.
Then P (t) satisfies (22) in Theorem 1.
The proof of Lemma 2 is in the Appendix.
This lemma indicates that (23) can be replaced by the
meaningful uniform detectability condition, which incorporates an important notion into picture.
Remark 14. If the nonlinear system given by (3) – (4)
is uniformly detectable in the sense of (42), then the pair
{C(x), A(x) + αI} will be also uniformly detectable for the
same bounded matrix value function provided that α < γ.
Clearly, the reverse is always true.
5
Simulation examples
5.1
Second order nonlinear model
Consider the following continuous-time unforced nonlinear system with state x = [x1 (t) x2 (t)]T
(
ẋ1 (t) = 0.01x1 (t) − x2 (t)
(43)
ẋ2 (t) = x1 (t) − 0.003x22 (t)
y(t) = x1 (t).
(44)
This model can be parameterized as follows:
ẋ(t) = A(x)x
y(t) = C(x)x.
In which
(45)
H. Beikzadeh and H. D. Taghirad / Exponential Nonlinear Observer Based on the Differential · · ·
"
A(x) =
363
#
0.01
1
−1
−0.003x2
(46)
and C(x) = [1 0] is a constant matrix. The statedependent observability matrix is
"
# "
#
C(x)
1
0
O(x) =
=
.
(47)
C(x)A(x)
0.01 −1
function (29), when the proposed SDRE-based estimator is
used. We see that V (e(t), t) is positive definite and decreasing for all t > 0. This verifies the convergence and stability
of the proposed observer.
Since O(x) has full rank throughout R2 , the system is observable. It can be verified through some algebraic calculations that the SDC form fulfills the uniform detectability
condition of Definition 3 with
"
#
−(0.01 + a) + 0.003x2
Λ(x) =
(48)
(b − 1) + 0.003(−a + 0.003x2 )
for some positive real numbers a and b.
The pair
{C(x), A(x) + αI} is also uniformly detectable for the same
matrix value function given in (48), if the following inequality holds
α2 + b
.
(49)
α
√
Note that it is necessary that α < b. For a certain parameter α > 0, it is always possible to choose a, b > 0 such
that the foregoing inequality is met. Therefore, according
to Lemma 2 the differential Riccati equation (14) has a
bounded and positive definite solution.
Obviously, the output matrix C is a Lipschitz matrix and
satisfies (8) with any positive real number as kC . It follows
from (46) that for all x, x̃ ∈ R2
"
#
0
0
A(x) − A(x̃) =
.
(50)
0 −0.003(x2 − x̃2 )
2α < a <
Fig. 1 The actual state and the estimated state provided by the
SDARE, SDDRE, and the proposed SDDRE observer
Hence, (50) implies that
kA(x) − A(x̃)k 6 kA kx − x̃k
(51)
where kA = 0.003. The system (43) is simulated from the
initial condition x0 = [−0.1 0.1]T . The proposed SDRE
observer is simulated using (12) – (14) with initial condition x̂0 = [0.5 −0.5]. We choose α = 10, P (0) = 10I2 ,
Q = 10I2 and R = 1.
The values of c̄, p, p̄, r, q and σ are evaluated after the entire simulation has been performed, and are listed in Table
1. Hence, it can be seen from Table 1 that (23) is satisfied. Hence, this implies that e(t) → 0 exponentially as
t → ∞. Note that in this example kB ρ = 0. However, it
can be checked that for this example, the situations which
are stated in Remark 10 are encountered and hence, (23) is
obviated automatically.
For the sake of comparison, we also simulated the
SDARE and the standard SDDRE from the same initial
condition and similar values for Q and R. Fig. 1 shows
the actual state and the estimated one obtained from these
three observers. Fig. 2 shows the norm of the error in the estimates. As it can be easily observed from these figures, the
proposed observer has performed much better in comparison with two other observers. Fig. 3 visualizes the Lyapunov
Fig. 2 The norm of the estimation error for three different
SDRE-based observers
Fig. 3 The Lyapunov function V (e(t), t) for the proposed observer
364
International Journal of Automation and Computing 9(4), August 2012
Table 1
Values of various bounds used in Theorem 1
Bound
c̄
p
p
r
q
δ
kA
kC
α
Value
1
10
84.59
1
10
0.149
0.003
0.1
10
5.2
Induction motor
ẋ1 (t) = k1 x1 (t) + u1 (t)x2 (t) + k2 x3 (t) + u2 (t)
ẋ2 (t) = −u1 (t)x1 (t) + k1 x2 (t) + k2 x4 (t)
ẋ3 (t) = k3 x1 (t) + k4 x3 (t) + (u1 (t) − x5 (t)) x4 (t)
ẋ4 (t) = k3 x2 (t) − (u1 (t) − x5 (t)) x3 (t) + k4 x4 (t)
(52)
In the above, x1 , x2 and x3 , x4 are the components of the
stator and the rotor flux, respectively, in the plane perpendicular to the rotation axis. x5 is the angular velocity. The
inputs are denoted by u1 as the frequency and u2 the amplitude of the stator voltage, respectively, and u3 the load
torque. k1 , · · · , k 6 are several parameters depending on the
machine structure and the considered drive system.
The output equations are given by
y1 (t) = k7 x1 (t) + k8 x3 (t)
y2 (t) = k7 x2 (t) + k8 x4 (t)
(53)
in which k7 and k8 are user defined parameters, and hence,
y1 (t) and y2 (t) are the normalized stator currents. Consider the following SDC parameterization of (52) and (53)
(parameter t is omitted):


k1
0
k2
0
0
 0
k1
0
k2
0 




A(x) =  k3
(54)
0
k4 −x5 0 


 0
k3
0
k4
x3 
k5 x4 −k5 x3 0
0
0


x2
1 0
 −x
0 0 
1




B(x) =  x4
(55)
0 0 


 −x3 0 0 
0
0 k6
"
#
k7 0 k8 0 0
C(x) =
(56)
0 k7 0 k8 0
inequality (8) is evidently satisfied by the above output matrix (56). But, confirming the Lipschitz condition for the
matrices (54) and (55) will be much more involved and necessitate some calculations. For the matrix B(x), we have
kB(x) − B(x̂)k =
p
(x1 − x̂1 )2 + (x2 − x̂2 )2 + (x3 − x̂3 )2 + (x4 − x̂4 )2 6
kB kx − x̂k
ckC σp
r
− kA σ +
qp
2p2
9.86
matrix A(x), we obtain
To demonstrate the effectiveness of the proposed method,
it is applied to estimate the flux and angular velocity for
induction machines[40] . The normalized state equations of
a symmetrical three phase induction machine are expressed
as
ẋ5 (t) = k5 (x1 (t)x4 (t) − x2 (t)x3 (t)) + k6 u3 (t).
α−
(57)
with x, x̂ ∈ R5 . Therefore, (7) is met and any positive real
number can be assigned as kB . Likewise, for the system
kA(x) − A(x̂)k =
³
´
p
max |x3 − x̂3 | , |x5 − x̂5 | , k5 (x3 − x̂3 )2 + (x4 − x̂4 )2 .
(58)
The quantity kA(x) − A(x̂)k takes one of the terms within
the parentheses, depending on the relative position of the
vectors x and x̂ and also the value of k5 . Nevertheless, it
can be said that (6) is also verified with kA = max (1, k5 ).
We now proceed to implement the proposed SDRE-based
observer.
For the simulations, we set k1 = −0.186, k2 = 0.178,
k3 = 0.225, k4 = −0.234, k5 = −0.081, k6 = −0.018, k7 =
4.643, k8 = −4.448 and the input vector u(t) = [1 1 0]T .
First, we consider a small initial estimation error by choosing x(0) = [0.2 −0.6 −0.4 0.1 0.3] for the system to
be observed and x̂(0) = [0.3 −0.3 0.2 0 0.7] for the
observer. We choose α = 2, Q = I5 , R = I2 , and the initial
condition of the differential Riccati equation (14) is set to
P (0) = 10I2 . For the sake of comparison, we also simulated
the SDARE observer by choosing Q = I5 , R = 10I2 and the
SDDRE observer by setting Q = I5 , R = I2 and P (0) = I2
for the same initial condition x̂(0).
The actual and the estimated values for the first state
variable x1 (t) (one of stator flux components) and the 5th
state variable x5 (t) (angular velocity) obtained from these
three observers, are shown in Figs. 4 and 5, respectively.
These figures point out that for the proposed observer,
the state estimates do converge to the actual states more
rapidly, compared with the usual SDRE observers.
Fig. 4 The actual state x1 (t) and the estimated values obtained
from the SDARE, SDDRE, and the proposed observer with small
initial error
Now let us consider a large initial estimation error by
choosing the previous initial condition for the system and
x̂(0) = [0.5 0.1 0.3 −0.2 4] for the observer. Similarly, the proposed observer is simulated for α = 1, Q = I5 ,
R = I2 and P (0) = 100I2 . The usual SDDRE observer is
H. Beikzadeh and H. D. Taghirad / Exponential Nonlinear Observer Based on the Differential · · ·
365
also simulated for Q = 10I5 , R = I2 and P (0) = 10I2 with
the same initial condition. Figs. 6 – 8 show the simulation
results.
Fig. 7 The norm of the estimation error for the proposed SDDRE observer with large initial error. The error does converge
to asymptotically
Fig. 5 The actual and the estimated angular velocity obtained
using three different observers with small initial error
As shown in Fig. 6, the estimation error for the proposed
observer approaches zero, while that is not true for the SDDRE observer. Further simulation results show that the
proposed observer has usually a larger region of attraction
than that in the standard SDDRE observer, i.e., the initial estimation error may be larger. In fact, if the initial
estimation error increases, then the SDDRE observer will
diverge. On the contrary, for the proposed observer the
parameter α can be chosen such that (23) is satisfied and
hence, the estimation error converges exponentially to zero
(see Remark 11). Fig. 7 shows the norm of error between
the state estimates and the actual state for the proposed
SDRE-based observer, and the Lyapunov function (29) is
depicted in Fig. 8. We see that the Lyapunov function is
increasing over a small period of time. This means that
(38) is sufficient but not necessary for the error decay.
Fig. 8 Lyapunov function V (e(t), t) for the proposed SDDRE
observer with large initial error
Remark 15. The result in this paper provides sufficient
conditions to guarantee exponential convergence of the estimation error to zero. As seen in the simulation example
for induction machine, these conditions are not necessary
and may be conservative as well. Furthermore, since the
bounds obtained by using the norm operators are conservative, in some applications the sufficient conditions cannot
be satisfied easily.
Remark 16. Adjustment of the free parameter α and
the initial condition P (0) of the differential Riccati equation
(14) has an important effect on the performance of the proposed observer. Especially, when the sufficient conditions
in Theorem 1 are not satisfied or their correctness is difficult to be verified. In this situation, appropriate selection
of these parameters is advised in order to avoid undesirable
oscillations in the output of the proposed observer.
6
Fig. 6 Angular velocity x5 (t) for original system, usual SDDRE
and the proposed observer with large initial error
Conclusions
The purpose of this paper is to develop an alternative
SDRE-based observer for nonlinear continuous-time systems, in which the estimation error converges to zero exponentially. In addition to present a rigorous stability analysis, we claimed that the suggested methodology can be used
366
International Journal of Automation and Computing 9(4), August 2012
to enhance the observer performance significantly and to
achieve a prescribed degree of stability as well. Employing
an appropriate Lyapunov function, we provided sufficient
conditions which guarantee local exponential stability of the
proposed observer. The acquired results are not restricted
to unforced systems and do not implicate any simplicity
limitations on the SDC factorization, like those imposed
in Banks et al.[21] and Lewis[22] . A key stability condition included is that the solution of the state-dependent
differential Riccati equation remains positive definite and
bounded. Introducing an interesting novel definition of a
detectable SDC parameterization, this condition is reduced
to a uniform detectability condition which can be checked
in advance.
In order to evaluate the effectiveness of the proposed observer, it is first applied to a simple nonlinear model which
satisfies the stability conditions. The simulation results verify the effectiveness of the proposed observer in comparison
with the common SDARE and SDDRE observers. Moreover, by applying the proposed observer to an induction
motor for estimation of the rotor flux and angular velocity,
we showed that the conditions which guarantee the asymptotic convergence are not necessary. Furthermore, numerical simulations of this estimation example show a superior
performance and an increased domain of attraction compared to that of the usual SDDRE observer. Hence, we can
tolerate larger initial estimation errors using such structure
in the estimation which is a considerable outcome.
Appendix
From (A3), we have
Z
kω(0)k2 − 2
T
¡
¢
ω T (t) ĀT (x) + H T (x)ΛT (x) ω(t)dt = khk2
0
(A6)
and from (A1) we conclude that kω(0)k2 6 khk2 and
Z
T
kω(t)k2 dt 6
0
This lemma follows directly from [33, Theorems 4 and
1
7]. Write©H(x) = R−ª2 C(x) and Ā(x) = A(x) + αI. Since
the pair C(x), Ā(x) is uniformly detectable, there exists
a bounded matrix valued function Λ(x) such that
£
¤
ω T Ā(x) + Λ(x)H(x) ω 6 −γ kωk2 , γ > 0
(A1)
for all x ∈ Rn . Consider the dual control problem
−ω̇ = ĀT (x)ω + H T (x)v,
ω(T ) = h
(A2)
where T is the finite time horizon, h is given and v is the
control signal. Applying a feedback control v(t) = ΛT (x)ω
in (A2) yields
³
´
−ω̇ = ĀT (x) + H T (x)ΛT (x) ω, ω(T ) = h
(A3)
(A7)
Thus, from (A4) it follows that
Ã
hT P (T )h 6 hT
kP (0)k +
kQk + kΛ(x)k2
2γ
!
h
(A8)
which indicates that for every T > 0, the matrix P (T ) fulfills
kP (T )k 6 kP (0)k +
kQk + kΛk2
= p̄
2γ
(A9)
with kΛk = supx∈Rn kΛ(x)k, and thus P°(t) is °bounded.
Similarly,o since Q is full rank and °Ā(x)° < ∞, the pair
n
1
Ā(x), Q 2 is uniformly controllable. Hence, there exists a
bounded matrix valued function Γ(x) such that
³
1
´
λT A(x) + Q 2 Γ(x) λ > δ kλk2 ,
δ>0
(A10)
for all x ∈ Rn . Using an optimal control problem similar to (A2)
along with (A10), it can be concluded that Π(t), i.e., the inverse
of P (t), satisfies
kΠ(T )k 6 kΠ(0)k +
Proof of Lemma 2
khk2
.
2γ
kHk2 + kΓk2
=p
2δ
(A11)
with kΓk = supx∈Rn kΓ(x)k. This implies the positive definiteness of the solution P (t).
¤
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Hossein Beikzadeh
received his B. Sc.
and M. Sc. degrees in electrical engineering
from K. N. Toosi University of Technology,
Iran in 2006 and 2009, respectively. He is
currently a Ph. D. candidate in the Department of Electrical and Computer Engineering University of Alberta, Canada.
His research interests include nonlinear
systems analysis, nonlinear observer design,
robust control, and sampled-data control.
E-mail: [email protected] (Corresponding author)
Hamid D. Taghirad received his B. Sc.
degree in mechanical engineering from
Sharif University of Technology, Iran in
1989, his master degree in mechanical engineering in 1993, and his Ph. D. degree in
electrical engineering in 1997, both from
McGill University, Canada. He is currently
an associate professor with the Department
of Electrical Engineering, and the Director
of the Advanced Robotics and Automated
System, ARAS Research Center at K. N. Toosi University of
Technology, Iran. He was appointed as the director of the office
of international scientific cooperation of the university in 2007.
He is a senior member of the IEEE, and his publications include
2 books and more than 90 papers in international journals and
conference proceedings.
His research interests include robust and nonlinear control applied on robotic systems.
E-mail: [email protected]