An Extended Luenberger-Like Observer and its

Application to Target Tracking

Guchuan Zhu∗ and Lahcen Saydy†

This paper addresses target tracking problems from the viewpoint of nonlinear ob-servers. It is shown that the differential geometric nonlinear control system theory pro-vides a universal framework for observability analysis of target tracking systems. Thank tothe progresses in nonlinear observer theory, a better guarantee on the convergence of thetracking algorithms can be expected. A new nonlinear observer based on the approximatelinearization of estimation error dynamics is proposed and is applied to the bearings-onlytarget tracking. The convergence of the constructed tracking observers is addressed and isconfirmed by numerical simulations.

I. Introduction

Target tracking essentially amounts to extracting in real-time the signals captured by sensors (e.g. radar,sonar, etc.) which provide information on the state of moving targets. In this work, we focus our attentionon the design of algorithms allowing to trace trajectories of moving targets. The mechanism of how to keepthe targets in the field of caption of the sensors will not be considered. The tracking model can then beexpressed as a dynamical system combined with a measurement equation representing the kinetics of thetarget and the sensing mechanism, respectively.

Target tracking often, if not always, results in a nonlinear problem, in that either the target dynamicsor the observation, if not both, are nonlinear. Popular approaches for tackling this problem rely on theExtended Kalman Filter (EKF).1,2 Though the EKF has successfully been applied to a considerable numberof engineering problems, its convergence is hard to verify in many cases. Extensive numerical simulationsare thus typically used for this purpose.

Using observers to estimate the system state is a viable alternative for target tracking. This approachhas two main advantages: it applies to nonlinear models and it provides a framework for a more rigorousstudy of the convergence issue. Indeed, during the last two decades, considerable efforts have been devotedto the study of observers in the formwork of nonlinear system by employing, in particular, the differentialgeometric method. A state-of-the-art overview of nonlinear observer theory is presented in Ref. 3 (see alsoRef. 4 and the references therein).

Though deterministic systems are in most cases an idealized modelling of physical phenomenons, thisapproximation does reflect certain practical situations. In a tracking system, for example, when the targetis very close to the sensor, the geometrical acceleration dominates the dynamics of maneuvering, the latterbeing often modelled by stochastic processes. In this case, a deterministic modeling could indeed be anappropriate one. It is worth noting that this is one of the most likely situations in that the nonlinearity isthe main source leading to the divergence of tracking algorithms.

The construction of observers is in most cases unsolvable in the original coordinates. A solution forovercoming this difficulty is to transform the original nonlinear system into a suitable form, such as observercanonical form. Observability is a necessary condition for the existence of such coordinate transformations.In Ref. 5, 6 a method for constructing first-order approximate observers, so-called extended Luenbergerobserver, for systems admitting an observer canonical form has been proposed. This method has been

∗Research Fellow, Department of Electrical Engineering, Ecole Polytechnique de Montreal, CP 6079, Succursale centre-ville,Montreal, QC, Canada H3C 3A7. e-mail: guchuan.zhu@polymtl.ca.

†Professor, Department of Electrical Engineering, Ecole Polytechnique de Montreal, CP 6079, Succursale centre-ville, Mon-treal, QC, Canada H3C 3A7. e-mail: lahcen.saydy@polymtl.ca.

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extended recently to the construction of higher-order approximate observers7 (see Ref. 8 for an alternativeapproach).

This paper presents an algorithm which is different from the extended Luenberger observer in that itdoes not require for the system to admit an observer canonical form, but merely to be observable. Theobservability condition is studied using results from differential geometric system theory. It will be seen thatthis approach considerably simplifies the analysis and the design of target tracking algorithms. A nonlinearobserver based on the approximative linearization of the estimation error dynamics is proposed. This observercan be systematically constructed and is suited for many target tracking problems. The convergence of theproposed observer will also be investigated for specific tracking problems.

The rest of the paper is organized as follows. Section II makes a brief overview of the observabilityof nonlinear systems. Section III recalls the Luenberger observer and its extensions to linear time-varyingand nonlinear systems. Section IV presents an extended Luenberger-like observer that can be employed formany target tracking problems. Section V demonstrates the observability analysis and the observer designthrough 2-dimensional bearings-only target tracking systems and reports on the corresponding simulationresults. Finally, some concluding remarks are presented in Section VI.

II. Preliminaries on the Observability of Nonlinear Systems

Due to space limitation, this paper will not introduce the exact definition of various observability notionsof nonlinear systems, neither the precise description of criteria for observability assessment. The interestedreader is referred to, for example, Ref. 9, 10 for a formal presentation on this issue.

Consider the following system: x = f(x, u)y = h(x)

, (1)

where x ∈ Rn, u ∈ Rm, and y ∈ Rp are the state, input, and output (or observation) vectors, respectively.f : Rn × Rm → Rn and h : Rn → Rp are smooth maps, and f(x, ·) is a smooth vector field at Rn for anyconstant input.

A system under the form of (1) is said to be locally observable at a point x0 if all state x can beinstantaneously distinguished by a judicious choice of input u on a neighborhood U of x0. An observabilitytest criterion, also called rank condition (see e.g. Ref. 9), states that for an admissible u, System (1) islocally observable if there exist integers k1 ≥ k2 ≥ . . . ≥ kp with

∑pi=1 k1 = n and a neighborhood U of x0

such thatdim

dLj

fhi(x)|i = 1, . . . , p; j = 0, . . . , ki − 1

= n (2)

for all x ∈ U . Recall that the Lie derivative of hi(x) along the vector field f is defined as

Lfhi(x) =∂hi(x)

∂xf.

Consider now a linear time-varying system of the form

x = A(t)x + B(t)uy = C(t)x

, (3)

where A ∈ Rn × Rn × R+, B ∈ Rn × Rm × R+, and C ∈ Rp × Rn × R+ are matrices with all theirelements of bounded functions for t ∈ I ⊂ R+. The observability of System (3) can be determined by theobservability Grammian.11 The observability test for nonlinear systems given in (2) can also be adapted tolinear time-varying systems. In fact, by writing (2) in autonomous form

dt

dτ= 1

dx

dτ= A(t)x + B(t)u

y = C(t)x

, (4)

and defining the vector field associated to (4) as

f(x, t) = A(t)x∂

∂x+

∂

∂t,

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the Lie derivative of hi(x, t) = Ci(t)x along the vector field f(x, t) is then

Lfhi(x, t) =∂hi(x, t)

∂xA(t)x +

∂hi(x, t)∂t

, i = 1, . . . , p,

and the rank condition becomes:

dim

dLjfhi(x, t)|i = 1, . . . , p; j ≥ 0

= n, ∀(t, x) ∈ I × Rn. (5)

Note that the observability of linear systems is independent of the state x and input u.Let x(t) be a nominal trajectory associated with the input u(t) and starting from x(t0). Denoting

xδ = x− x, yδ = y − y,

then the tangent linearization of System (1) can be written as:

xδ = fx,uxδ

yδ = hxxδ

, (6)

wherefx,u =

∂f

∂x(x(t), u(t)), hx =

∂h

∂x(x(t))

are the Jacobian of f(x, u) and h(x) with respect to x, respectively.The tangent linearization of a nonlinear system along a trajectory is in general a linear time-varying

system. The observability of such a system can eventually be verified using the criterion (5). But it appearsmore convenient to deduce the observability directly from the one of the original nonlinear system. Thereason is that the nominal trajectory is arbitrary, therefore it is not practical to verify the observability ofthe tangent linearization of nonlinear systems using the criterion for linear time-varying systems. In fact, theobservability of the tangent linearization can be determined from the one of the original nonlinear system.More precisely, if a nonlinear system is observable at all points in a trajectory associated to a sufficientlysmooth input u in a time interval I = [t0, T ], then its tangent linearization is observable along this trajectoryfor t ∈ I. To see that, we define

u =(

u u · · · u(n−1))T

. (7)

Noting also that as the input of the system, u, is a time function, the observability codistribution associatedto the rank condition (2) needs to be slightly modified. Let

fu = f(x, u)∂

∂x+

∂

∂t

be the vector field associated to the nonlinear dynamics (1). The Lie derivative of hi(x) along the vectorfield fu is defined as

Lfuhi(x) =∂hi(x)

∂xf +

∂hi(x)∂t

.

The Lie derivatives Lkfu

dhi can then be explicitly expressed as

L0fu

dhi =∂hi

∂x,

Lfudhi =∂hi

∂x

∂f

∂x+ fT ∂

∂x

(∂hi

∂x

)T

,

L2fu

dhi =∂(Lfudhi)

∂x

∂f

∂x+ fT ∂

∂x

(∂(Lfudhi)

∂x

)T

+ ˙uT

(∂

∂u

(∂(Lfudhi)

∂x

)T)T

,

...

Lkfu

dhi =∂(Lk−1

fudhi)

∂x

∂f

∂x+ fT ∂

∂x

(∂(Lk−1

fudhi)

∂x

)T

+ ˙uT

∂

∂u

(∂(Lk−1

fudhi)

∂x

)T

T

,

i = 1, . . . , p, k = 1, 2, . . .

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Substituting A(t) by fx,u and ci(t) by hi,x, it can be seen that for all point in the trajectory x(t)

Lkfu

dhi = dLkfx,u

hi,x, i = 1, . . . , p, k ≥ 0.

This proves that the rank condition (2) implies the rank condition (5) in a convenient domain.In practice, it is much easer to find the unobservable point(s) in the domain of interest, hence it is not

necessary to solve the nonlinear differential equations in order to determine the trajectories for observabilityinvestigations.

III. The Luenberger Observer and its Extensions to Linear Time-Varying andNonlinear Systems

The problem of constructing an observer for a linear time-invariant system:

x = Ax + Bu

y = Cx(8)

has been completely solved by Luenberger.12,13 A Luenberger observer is of the form:

˙x = Ax + Bu + K(y − Cx), (9)

where x denotes the estimate of the state. Let e = x− x denote the state estimation error, then the dynamicsof e (also called error dynamics) are given by:

e = (A−KC)e. (10)

When System (8) is observable, there exists a K such that A − KC is Hurwitz. The error dynamics areasymptotically stable and the corresponding observer is called an asymptotical observer. The problem ofLuenberger observer for the system (8) is also solvable if the later is completely detectable.

This approach can not be directly applied to linear time-varying systems, because the notion of pole ofsuch a system is no longer simply related to its stability. The solution is to transform the system into theobserver canonical form, under which a Luenberger observer can be constructed.5,6, 14

For the sake of simplicity, we present first the algorithm for a single output system and omit the inputterm. If System (3) is observable in an interval I ⊂ (0,∞), it can be transformed by a coordinates changez = H(t)x into the so-called observer canonical form:

z = A(t)zy = C(t)z

(11)

where

A(t) =

0 · · · · · · 0 a1(t)

1. . .

......

0. . . . . .

......

.... . . . . . 0 an−1(t)

0 · · · 0 1 an(t)

, C(t) =(

0 · · · 0 1)

,

obtained from:

A(t) = H(t)A(t)H−1(t) + H(t)H−1(t), (12)

C(t) = C(t)H−1(t). (13)

The invertibility of the transformation is guaranteed by the observability and, furthermore, the transfor-mation can be explicitly expressed as:

H−1(t) =(L0 L · · · Ln−1

)W (t) (14)

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where W (t) is the last column of inverse observability matrix and the operators Li are defined by:

L0W (t) = W (t),

LW (t) = W (t) + A(t)W (t), (15)

LkW (t) = L (Lk−1W (t)).

The observer in the canonical coordinates is therefore of the form:

˙z = A(t)z + K(t)(y − Cz). (16)

An obvious choice of the observer gain is:

K(t) = a(t) + α,

with a(t) =(

a1(t) · · · an(t))T

and α =(

α1 · · · αn

)T

a constant vector. The error dynamics,

ε = z − ˙z, become then:ε =

(En − αC

)ε, (17)

where En is a shift matrix:

En =

0 · · · · · · · · · 0

1. . .

...

0. . . . . .

......

. . . . . . . . ....

0 · · · · · · 1 0

.

Clearly, the error dynamics are exponentially stable if sn +αnsn−1 + · · ·+α2s+α1 is a Hurwitz polynomial.This is exactly the method employed in the Luenberger observer.

In the original coordinates, the observer is of the form:

˙x = A(t)x + K(t)(y − C(t)x), (18)

and the observer gain is given by:

K(t) = H−1(t)K(t) = H−1(t)(α + a(t)) =(α1 + α2L+ · · ·+ αnLn−1 + Ln

)W (t). (19)

Note that an explicit coordinates transformation is not required in the design of observers.5

The estimation error in the original coordinates is given by:

e = H−1(t)ε.

Since ε decays with an arbitrary rate for an appropriate choice of α, the convergence of the observer inthe original coordinates will be achieved provided that reasonable regularity conditions on the coordinatestransformation are met. This means that while the observability is in general a necessary condition, it ishowever not sufficient for the existence of asymptotic observers.

The idea of designing observer in the canonical coordinates can also be applied to multi-output systems.For such a system, the observability matrix formed by dLj

fhi(x, t), i = 1, . . . , p, j ≥ 0, may have several fullrank sub-matrices, each of them could be used in the construction of observers. It has been shown that (see,e.g., Ref. 11) a coordinates transformation, H(t), can be constructed from one such sub-matrix, Qo, whichcan render the system (3) into an observer canonical form similar to the single-output one with

A(t) =

a11(t) ap

1(t)

E... O

...a1

k1(t) ap

k1(t)

.... . .

...a1

n−kp+1(t) apn−kp+1(t)

O... E

...a1

n(t) apn(t)

, C = diag(C1, . . . , Cp),

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where E is of the form

E =

0 · · · · · · 0

1. . .

...

0. . . . . .

......

. . . . . . 00 · · · 0 1

, (20)

and O is a zero matrix, with appropriate dimensions,

Ci =(

0 · · · 0 1)

1×ki

, i = 1, . . . , p,

and∑p

i=1 ki = n. The coordinates transformation H(t) can be explicitly expressed as:

H−1(t) = ((L0, . . . ,Lk1−1)W1(t), . . . , (L0, . . . ,Lkp−1)Wp(t))

where Wi(t) is the mthi (mi =

∑ij=1 kj) column of Qo, and the operator L is defined in (15).

In the canonical coordinates, the error dynamics are given by:

ε = (A(t)− K(t)G(t)C(t))ε (21)

where G(t) is an invertible output transformation, which is often required in multi-output cases. By choosingan observer gain as:

Ki = ai(t) + αi, i = 1, . . . , p, (22)

where αi =(

αi1 · · · αi

n

)T

is a constant vector, and ai(t) is the mthi column in A(t), the error dynamics

become time-invariant:

ε =

−α11 −αp

1

E... O

...−α1

k1−αp

k1...

. . ....

−α1n−kp+1 −αp

n−kp+1

O... E

...−α1

n −αpn

ε.

In order to stabilize the error dynamics, it suffices to choose

αij = 0, j < mi−1, j > mi, i = 1, . . . , p,

and (αi

mi−1, . . . , αi

mi−1

), i = 1, . . . , p,

to be the coefficients of Hurwitz polynomials.The observer in the original coordinates is of the form of (18) and the observer gain is given by:

K(t) =((α1

1 + α12L+ · · ·+ α1

k1Lk1−1 + Lk1)W1, . . . ,

(αpn−kp+1 + αp

n−kp+1L+ · · ·+ αpnLk1−1 + Lk1)Wp

)G−1(t). (23)

Note that the algorithm for observer construction has a recursive structure, thus it can be easily imple-mented by a formal calculation software, e.g. Maple.

It will be seen in the next section that the above formulation can be formally used in the design of aLuenberger-like observer for nonlinear systems.

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The general form of an asymptotic observer for the nonlinear system (1) is:

˙x = f(x, u) + k(x, u, h(x)), x(t0) = x0 (24)

where k : Rn × Rm × Rp → Rn is a smooth function, satisfying

k(x, u, h(x)) = 0, ∀x ∈ Rn, ∀u ∈ U (25)

with U the set of all admissible inputs.A convenient approximation of k(x, u, y) can be written locally as k(x, u)(y − h(x)). In this expression,

the term k(x, u) is the observer gain, being analogue to linear observers, but in general a function of thestate estimate and the input.

The error dynamics for system (1) corresponding to the observer (24) can, therefore, be expressed as

e = f(x + e, u)− f(x, u)− k(x, u, h(x)). (26)

The observer design consists, thus, in finding a function k(x, u, h(x)) such that the error dynamics (26) arestable at the origin e = 0. Obviously, if the system can be transformed into an observer canonical form by adiffeomorphism, then the error dynamics can be exactly linearized by an output injection.15,16 The designproblem will again be reduced to the construction of Luenberger observer. Unfortunately, the conditionsfor the existence of an observer canonical form, also called the integration condition, cannot be verified formost of target tracking problems. The error dynamics can be locally linearized by using Lyapunov auxiliarytheorem under some nonresonance conditions.17,18 This approach requires to solve a partial differentialequation, which is, in general, a difficult task. In addition, it is restricted to unforced systems.

In the next section, we will present a nonlinear observer design based on the approximate linearizationof the error dynamics.

IV. An Extended Luenberger-Like Observer

Let the dynamical system˙x = f(x, u) + k(x, u)(y − h(x)) (27)

be a candidate asymptotic observer for the system (1). The corresponding error dynamics is thus given by

e = f(x, u)− f(x, u)− k(x, u)(y − h(x)). (28)

By expanding (28) around the equilibrium point e = 0, we obtain:

e =(

∂f

∂x(x, u)− k(x, u)

∂h

∂x(x)

)e + O(‖e‖2). (29)

The linear part of the error dynamics is thus time-varying.Consider a variable transformation:

ε = H(x, u)e, (30)

where u is defined in (7). H(x, u) will transform (29) into

ε =(

H(x, u)∂f

∂x(x, u) + H(x, u)−H(x, u)k(x, u)

∂h

∂x(x)

)H−1(x, u)ε + O(‖ε‖2)

=(

H(x, u)∂f

∂x(x, u) +

[∂H

∂u: ˙u

]+

[∂H

∂x: f(x, u)

]+

[∂H

∂u: k(x, u)(y − h(x))

]−H(x, u)k(x, u)

∂h

∂x(x)

)

×H−1(x, u)ε + O(‖ε‖2), (31)

where[

∂H∂x : v

]denotes the matrix

∂H11

∂xv · · · ∂H1n

∂xv

......

∂Hn1

∂xv · · · ∂Hnn

∂xv

.

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Note that [∂H

∂u: k(x, u)(y − h(x))

]

e=H−1ε

∼ O(‖ε‖),

therefore (31) can be written as:

ε =(A(x, u)− k(x, u)C(x, u)

)ε + O(‖ε‖2)

with

A(x, u) =H(x, u)∂f

∂x(x, u)H−1(x, u) +

([∂H

∂u: ˙u

]+

[∂H

∂x: f(x, u)

])H−1(x, u),

C(x, u) =∂h

∂x(x)H−1(x, u),

k(x, u) =H(x, u)k(x, u).

Now, choose an arbitrary trajectory of the system (1). It is clear that the transformation that transformsthe tangent linearization of the system along this trajectory into an observer canonical form is formallythe same as the one used in the construction of the corresponding observer, by omitting the fact that theoperation points are different. We can therefore formally transform the tangent linearization into an observercanonical form at each point, and use the algorithm presented earlier to compute the observer gain.

We first present the construction of observer in a formal way. The convergence of the algorithm will beaddressed later through particular systems.

Consider an arbitrary nominal trajectory x, along which the system is observable corresponding to theinput u. Denote by

Σ :(

∂f

∂x(x, u),

∂h

∂x(x)

)

the tangent linearization along x. Similar to the case of linear time-varying systems, if the tangent lin-earization is observable, then there exist coordinate and output transformations, H(x, u) and G(x, u), whichtransform the system Σ into an observer canonical form Σ = (A(x, u), C) with

A(x, u) =

a11 ap

1

E... O

...a1

k1ap

k1...

. . ....

a1n−kp+1 ap

n−kp+1

O... E

...a1

n apn

,

where aij , i = 1, . . . , p, j = 1, . . . , n, are functions of x and u, E is defined in (20), O is a zero matrix, and

C = GC =

C1

. . .Cp

,

Ci =(

0 · · · 0 1)

1×ki

, i = 0, . . . , p,

with the sequence of observability indices k1, . . . , kp verifying∑p

i=1 ki = n. The relationship between Σ andΣ is given by

A(x, u) =H(x, u)∂f

∂x(x, u)H−1(x, u) +

([∂H

∂u: ˙u

]+

[∂H

∂x: f(x, u)

])H−1(x, u),

C(x, u) =∂h

∂x(x)H−1(x, u).

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The coordinate transformation can be explicitly determined by:

H−1(x, u) = ((L0, . . . ,Lk1−1)W1(x, u), . . . , (L0, . . . ,Lkp−1)Wp(x, u))

where Wi is the mthi column of the inverse observability matrix, and the operator L is defined by:

LWi(x, u) = −Wi(x, u) +∂f

∂x(x, u)Wi =

∂f

∂xWi − ∂Wi

∂xf − ∂Wi

∂x˙u, i = 1, . . . , p. (32)

For an unforced system, the above operator reduces to

LjWi(x, u) =∂f

∂xWi − ∂Wi

∂xf = adj

−fWi(x, u), i = 1, . . . , p, j ≥ 0. (33)

Now we can replace the nominal trajectory by the estimated one and use the above formula to find again k(x, u), which is formally given by (22), so that the error dynamics in the transformed coordinates arelinear time-invariant and stable. In the original coordinates, the observer gain can be explicitly written as:

k(x, u) =((α11 + α1

2L+ · · ·+ α1k1Lk1−1 + Lk1)W1, . . . ,

(αpn−kp+1 + αp

n−kp+1L+ · · ·+ αpnLkp−1 + Lkp)Wp)×G−1(x, u), (34)

with (α11, . . . , α

1k1

), . . . , (αpn−kp+1, . . . , α

pn) Hurwitz vectors.

Note that the existence of the coordinate transformation H(x, u) is guaranteed by the observability ofthe tangent linearization along the estimated trajectory, which is in turn guaranteed by the observability ofthe original nonlinear system.

Similar to the case of linear time-varying systems, the observability does not necessarily imply the con-vergence of the observer. However, the coordinates transformation and the observer gain have an explicitexpression, rendering the analysis of convergence possible for the specific systems. This is what is being donein the design of target tracking algorithms (see the next section).

Finally, we notice that the construction of the proposes observer consists of two steps. In the first step,the error dynamics is approximately linearized. And then, in the second step, the observer parametersare determined in the linearized coordinates. If the noisy perturbations are not negligible, they can beincorporated into the design of the linear observer, by using, for example, a standard Kalman filter, a robustobserver (or filter), or any other appropriate technique for the construction of linear observers (or filters).In this way, one can obtain a guaranteed convergence and the desired performance, even in the presence ofnoisy perturbations.

V. Case Study: 2-Dimensional Bearings-Only Target Tracking

A bearings-only configuration provides the feature for passive target tracking or for toleration of sensorfailures.1 This problem can be studied in Cartesian coordinates,19 in polar coordinates,20 or in hybridcoordinates.21 For simplicity, this paper considers 2-dimensional bearings-only tacking problems in modifiedpolar coordinates (MPC). As shown in Fig 1, the coordinates of the target in the polar coordinates arecomposed of the distance r and the azimuth angle γ. The relative acceleration between the target and thesensor carrier is denoted by u = (ux, uy) = (utx − ucx, uty − ucy), with ux and uy the x− and the y−axescomponents of u in Cartesian coordinates. The variables in MPC are defined as:

ξ1 = γ, ξ2 = γ, ξ3 =r

r, ξ4 =

1r. (35)

Denoting by ξ the state and by u the control vectors, respectively, the dynamics of the target are givenby:

ξ =

ξ2

−2ξ2ξ3 − ξ4(ux sin ξ1 − uy cos ξ1)ξ22 − ξ2

3 + ξ4(ux cos ξ1 + uy sin ξ1)−ξ3ξ4

= f(ξ, u). (36)

In a passive tracking configuration, the only available measurement is γ. The observation equation canthen be written as:

y = h(ξ) = ξ1. (37)

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Figure 1. 2-dimensional bearings-onlytracking problem.

We suppose in the following analysis that:

• the entity of sensor-carrier is a rigid body and the accel-eration on its center of gravity is known;

• the carrier is capable of performing any desired maneu-vering.

We first verify the observability of the considered systems.Based on the discussion in Section II, we only need to addressthe observability of the original nonlinear systems. The observ-able matrix related to the system (36)-(37) is given by:

Qo =

dh(ξ)dLfu

h(ξ)dL2

fuh(ξ)

dL3fu

h(ξ)

, (38)

with

dh(ξ) =(

1, 0, 0, 0)

,

dLfuh(ξ) =

(0, 1, 0, 0

),

dL2fu

h(ξ) = (−ξ4(uy sin ξ1 + ux cos ξ1),−2ξ3,−2ξ2, uy cos ξ1 − ux sin ξ1) ,

dL3fu

h(ξ) = (−ξ4(3ξ2(uy cos ξ1 − ux sin ξ1)) − 3x3(uy sin ξ1 − ux cos ξ1) + uy sin ξ1 + ux cos ξ1),

− 3ξ4(uy sin ξ1 + ux cos ξ1) + 6ξ23 − 6ξ2

2 ,

12ξ2ξ3 − 3ξ4(uy cos ξ1 + ux sin ξ1,

− 3ξ2(uy sin ξ1 + ux cos ξ1)− 3x3(uy cos ξ1 − ux sin ξ1) +uy cos ξ1 − ux sin ξ1) .

It is straightforward to verify that Qo is full rank if and only if

β(ξ, u) = 6ξ22(ux cos ξ1 + uy sin ξ1) + 6ξ2ξ3(ux sin ξ1 − uy cos ξ1) + 3ξ4(ux sin ξ1 − uy cos ξ1)2

+ 2ξ2(ux sin ξ1 − uy cos ξ1)6=0. (39)

By a direct computation it can be shown that the above observable condition is equivalent to the one deducedby repeated differentiations of pseudo-measurements and solving a set of nonlinear differential equationspresented in Ref. 20, expressed in polar coordinates as:

2γγ(3) − 3γ2 + 4γ2 6= 0.

It is clear that a typical (and comprehensive) unobservable situation is that the carrier follows the target ona straight line. Being concerned only with differential computations, the approach of differential geometricis much simpler.

When the acceleration of the carrier is zero, u = 0, the variable ξ4 can be eliminated from (36), and thedynamics of the system will be reduced to

ξ =

ξ2

−2ξ2ξ3

ξ22 − ξ2

3

= f(ξ). (40)

The observability matrix corresponding to the reduced system is:

Qo =

1 0 00 1 00 −2ξ3 −2ξ2

. (41)

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Clearly, the reduced system (40) is observable if ξ2 6= 0. Note that ξ2 = 0 implies that the carrier should befollowing the target on straight line.

To illustrate the construction of tracking observers, we consider first the unforced system (40). Theobserver is of the form (27), thus the observer design consists in determining the observer gain. The inversionof Qo defined in (41) is

Q−1o =

1 0 00 1 0

0 − ξ3

2ξ2− 1

2ξ2

.

By choosing the initial vector to be the last column of Q−1o :

W =(

0 01

−2ξ2

)T

,

we obtain by recursion:

LW =(

0 1 2ξ3

ξ2

)T

,

L2W =(

1 −6ξ3 −6ξ23

ξ2

)T

,

L3W =(−6ξ3 6ξ2

2 + 18ξ23 12

ξ33

ξ2

)T

.

Hence, the observer gain is given by:

k(ξ) =

α3 − 6ξ3

α2 − 6α3ξ3 + 6ξ22 + 18ξ2

3

− α1

2ξ2

+ 2α2ξ3

ξ2

− 6α3ξ23

ξ2

+ 12ξ33

ξ2

,

where (α1, α2, α3) are the coefficients of a Hurwitz stable 3rd order polynomial.One can verify immediately that the obtained observer is convergent in the finite horizon as long as

ξ2 6= 0. In fact, when t → ∞, ξ2 → 0, that will cause the divergence of the observer. Thus, a system onPCM without carrier maneuvering is not suited for long time tracking.

The technique of stationary Kalman filter is used to determine the observer parameters (α1, α2, α3). Itis easy to see that in the linearized coordinates, we have

A =

0 0 01 0 00 1 1

, C =

(0 0 1

).

Imposing G =(

1 0 0)T

and letting Q and R represent the disturbance at the system dynamics andthe measurement, respectively, the observer gain for the system (A,C, G) is given by

K = PCT R−1, (42)

where P is a positive-definite matrix, which is the solution of algebraic Riccati equation

AP + PAT − PCT R−1CP + GQGT = 0. (43)

For the third order system under consideration, P is given as

P =

p11 p12 p13

p21 p22 p23

p31 p32 p33

=

2Q5/6R1/6 2Q2/3R1/3 Q1/2R1/2

2Q2/3R1/3 3Q1/2R1/2 2Q1/3R2/3

Q1/2R1/2 2Q1/3R2/3 2Q1/6R5/6

.

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The observer gain in (42) can be expressed as

K =

(Q/R)1/2

2(Q/R)1/3

2(Q/R)1/6

,

and the characteristic equation of closed-loop linearized error dynamics satisfies the 3rd order Butterworthpolynomial

s3 + 2Ωs2 + 2Ω2s + Ω3 = 0,

where Ω = (Q/R)1/6. The observer parameters are therefore given by

α1 = 2Ω, α2 = 2Ω2, α3 = Ω3.

The observer gain can then be tuned based on the specification of “signal to noise ratio,” Ω.

0 5 10 15 20−2

−1

0

1

2(b) Estimation Errors

∆ξ1 (

deg)

0 5 10 15 2020

30

40

50(a) State Variaibles and Their Estimates

ξ 1 (de

g)

0 5 10 15 20−2

0

2

4

6

∆ξ2 (

deg/

s)

0 5 10 15 20−1

−0.5

0

0.5

1

Time (second)

∆ξ3 (

1/s)

0 5 10 15 20−4

−3

−2

−1

0

ξ 2 (de

g/s)

0 5 10 15 20

−0.5

0

0.5

Time (second)

ξ 3 (1/

s)with noisewithout noise

without noise with noise

without noise

with noise

Figure 2. 2-dimensional bearings-only tracking without carrier maneu-vering: (a) state variables (solid lines) and their estimates with noisymeasurement (dashed lines) and noise free measurement (discontinuedlines), (b) estimation errors.

The convergence of the ob-server is confirmed by simulation.In the simulation, the parametersof the target are: x0 = 10000(m),y0 = 10000(m), vx = −400(m/s),and vy = −300(m/s), and the ob-server parameter is Ω = 0.75. Thesensor carrier is supposed to stayin the origin all the time. Thesimulation results are presented inFig. 2. In the simulation, the dis-turbance at the measurement issupposed to be a zero-mean whiteGaussian noise with a deviation of0.2. For the purpose of compar-ison, the simulation results cor-responding to noise free measure-ment are also shown.

Similarly, we can construct anobserver for the forced system(36). In the design, the nonlin-ear observer is constructed usingthe proposed algorithm and theobserver gain in the linearized co-ordinates is determined from a stationary Kalman filter. More precisely, the observer gain is of the form:

k(ξ) =(α1 + α2L+ α3L2 + α4L3 + L4

)W,

where (α1, α2, α3, α4) is a Hurwitz vector and

W =

(0 0

ux sin ξ1 − uy cos ξ1

β(ξ, u)

−2ξ2

β(ξ, u)

)T

,

is the last column of the inverse observable matrix with β(ξ, u) defined in (39). It can be shown that by

choosing G =(

1 0 0 0)T

and solving (43), the characteristic equation of closed-loop linearized error

dynamics corresponding to the observer gain of the form (42) satisfies the 4th order Butterworth polynomial

s4 +√

2√

2 +√

2Ωs3 + (2 +√

2)Ω2s2 +√

2√

2 +√

2Ω3s + Ω4 = 0,

where Ω = (Q/R)1/8. The observer parameters are therefore given by

α1 =√

2√

2 +√

2Ω, α2 = (2 +√

2)Ω2, α3 =√

2√

2 +√

2Ω3, α4 = Ω4.

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0 5 10 15 20−2

−1

0

1

∆ξ2 (

deg/

s)

0 5 10 15 20−10

−8

−6

−4

−2

0

2x 10

−4

Time (second)

∆ξ4 (

1/m

)

0 5 10 15 20−1

−0.5

0

0.5

1

Time (second)

∆ξ3 (

1/s)

0 5 10 15 20−0.5

0

0.5

∆ξ1 (

deg)

with noise

without noise

without noise

with noise

without noise

with noise

without noise

with noise

Figure 3. State estimation errors for 2-dimensional bearings-only trackingwith carrier maneuvering.

The simulated tracking sce-nario is the same as the previousone, except that the sensor car-rier starts to move from the ori-gin with a initial speed of zeroand a constant acceleration, ux =5(m/s2), uy = 0, is applied on thecarrier. The state estimation er-rors are shown in Fig. 3 for bothdisturbed and noise free measure-ment. The simulation resultsshow that the observer is moresensitive to measurement noisethan in the previous case. In thisexample, the coordinate transfor-mation is bounded for the speci-fied input as long as β(ξ, u) 6= 0(the set of observable points). Due to the energy limitation, the carrier cannot keep accelerating for a longtime with the presented fashion. Consequently, the divergence will happen as soon as the carrier stopsmaneuvering. In order to ensure a long term convergence, the carrier must perform a suitable maneuveringas long as possible.

Finally we note that in both forced and unforced cases, when the speed of the bearings angle is verylow, the tracking systems will operate near the unobservable point. In this case, the observers will be verysensitive to the measurement noise.

VI. Conclusions

This paper proposed to use the techniques of nonlinear observer to design target tracking algorithms,in order to take the advantage of the progresses in nonlinear control system theory in recent years. Thisapproach, combined with robust linear observer and filtering techniques would provide a valuable alternativefor target tracking, allowing to better guarantee the convergence of the tracking algorithms while obtainingdesired performance. It has been shown that differential geometric system theory provides a powerful anduniversal framework for the analysis of the observability for variant different tracing problems. A newnonlinear observer is also proposed. The construction of a such an observer is closely related to the propertyof the observability of the system. The convergence of the observer has been addressed through 2-dimensionalbearings-only tracking problems. The proposed algorithm admits a systematic design procedure, thus it iswell-suited for an implementation with formal calculation software.

Acknowledgments

This work was supported in part by Ecole Polytechnique de Montreal under a program of start-up funds.

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