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Recursive Nonlinear Set–Theoretic Estimation

Based on Pseudo Ellipsoids

Uwe D. Hanebeck

Institute of Automatic Control Engineering

Technische Universit¨at M¨unchen

80290 M¨unchen, Germany

Uwe.Hanebeck@ieee.org

Abstract

In this paper, the problem of estimating a vector

x

of unknown quantities based on a set of measure-

ments depending nonlinearly on x

is considered. The

measurements are assumed to be taken sequentially

and are corrupted by unknown but bounded uncer-

tainties. For this uncertainty model, a systematic de-

sign approach is introduced, which yields closed–form

expressions for the desired nonlinear estimates. The

estimates are recursively calculated and provide solu-

tion sets X containing the feasible sets, i.e., the sets

of all x

consistent with all the measurements available

and their associated bounds. The sets X are tight

upper bounds for the exact feasible sets and are in

general not convex and not connected. The proposed

design approach is versatile and the resulting nonlin-

ear ﬁlter algorithms are both easy to implement and

eﬃcient.

1 Introduction

Estimating the state of a dynamic system based on

a sequence of uncertain measurements is a standard

problem in many applications. Usually, this problem

is approached in a stochastic setting. Alternatively,

set–theoretic methods can be used by assuming a pri-

ori bounds on the uncertainties. Estimation then con-

sists of constructing sets of possible states, which are

consistent with the a priori bounds and the measure-

ments. Good overviews about this topic can be found

in [1, 3, 19].

Most work has been done in set–theoretic state es-

timation for linear systems [17, 18, 21, 22]. Appli-

cations in the ﬁeld of speech processing are found in

[4, 5]. Robotic applications are discussed in [7, 16]. A

comparison of stochastic and set–theoretic estimation

is given in [6].

In the case of nonlinear systems, the complex sets

resulting from the estimation procedure are either

approximated by simple–shaped sets, e.g. ellipsoids,

boxes [15, 12], polytopes [20], or by the union of simple

sets [11, 12].

In [20], an approach similar to the Extended

Kalman Filter (EKF) is pursued. As in the EKF,

the nonlinear state equations are linearized about the

current state estimate. Unlike the EKF, linearization

errors are not neglected, but rather considered as ad-

ditional exogeneous disturbances. Estimation is per-

formed recursively and provides polytopes as approx-

imation of the posterior feasible set.

The procedure in [15] works without linearization

of the nonlinear state equations and provides (recur-

sively) the smallest axis–aligned box enclosing the

posterior feasible set.

In [11], the posterior feasible set is characterized by

enclosing it between internal and external unions of

boxes on the basis of interval analysis. Recursive es-

timation has not been addressed. A more advanced

version of this approach with lower computational

complexity has been introduced in [12].

In this paper, a new nonlinear ﬁltering algorithm

for nonlinear systems is proposed, that does not rely

in any way on linearization. In addition, the new ap-

proach is not based on a grid or on propagating par-

ticles, but provides a ﬁnite–dimensional closed–form

representation of the resulting complex–shaped sets.

This includes nonconvex sets or sets that are not even

connected. When applying the new ﬁlter recursively

to a sequential stream of measurements, the size of the

analytical representation of the resulting sets does not

grow with the number of measurements.

The key idea of the proposed ﬁlter is to trans-

form the original N –dimensional space S to an L–

dimensional hyperspace S

∗

with L>N.Thisre-

sults in an N–dimensional manifold U

∗

, called the

universal manifold, in the L–dimensional transformed

space S

∗

. Complex–shaped subsets of the original

N–dimensional space are then represented by N–

dimensional submanifolds of U

∗

in the space S

∗

.

These submanifolds are deﬁned by the intersection

of L–dimensional simple–shaped sets, e.g. ellipsoids,

with the universal manifold U

∗

. Furthermore, the

nonlinear measurement equation is transformed to a

linear one in the hyperspace S

∗

. Hence, nonlinear ﬁl-

tering can be performed by a linear ﬁlter operating in

the transformed space S

∗

.

Section 2 formulates the problem of nonlinear set–

theoretic estimation. In Sec. 3 the concept of modeling

complex–shaped sets is introduced. The nonlinear ﬁl-

tering algorithm is given in Sec. 4. The performance of

the new nonlinear ﬁltering algorithm is demonstrated

by estimating the parameters of a SCARA–type robot

manipulator in Sec. 5.

2 Problem Formulation

Consider a nonlinear discrete–time dynamic system

with system state x

k

(not directly observable) at time

step k.Measurementsˆz

k

of the system output are

taken at time instants k =1, 2,... according to the

nonlinear measurement equation

ˆz

k

= h

k

(x

k

)+v

k

(1)

with measurement uncertainty v

k

, which represents

exogenous noise sources or model parameter uncer-

tainties.

The uncertainties v

k

, k =1, 2,..., are assumed to

be bounded by a known set V

k

according to v

k

∈V

k

.

The set can be of complicated shape, i.e., can be

nonconvex or not connected.

The goal is to estimate at each time instant k the

state x

k

based on all available measurements ˆz

l

for

l =1, 2,... ,k. Of course, a recursive estimation pro-

cedure is preferred, which calculates a state estimate

based on the estimate at the previous time step and

the current measurement. Therefore, it is not required

to store and reprocess all measurements. Further-

more, instead of trying to construct point estimates,

we prefer to calculate at each time instant k all states

that are compatible with the measurements and their

corresponding uncertainties.

On a theoretical level, the problem can easily be

solved: Let X

s

k−1

denote the set of all states com-

patible with all the measurements up to time step

k − 1 and their respective uncertainties. Furthermore,

X

m

k

denotes the set of states deﬁned solely by the

measurement at time k according to

X

m

k

= {x

k

:ˆz

k

− h

k

(x

k

) ∈V

k

} .

Then the estimate X

s

k

is given by the intersection

X

s

k

= X

s

k−1

∩X

m

k

.

However, representing these sets in practical applica-

tions at least approximately by a ﬁnite set of parame-

ters is not a trivial task. On one hand, the parameter

set should not be too large, even more, the approx-

imation should degrade gracefully with a decreasing

number of parameters. On the other hand, the num-

ber of parameters should not be permanently growing

with an increasing number of incoming measurements.

Hence, the remainder of this paper is concerned with

a new parametric representation of complex–shaped

sets and an eﬃcient procedure for calculating the cor-

responding parameters.

3 Pseudo Ellipsoids

The key idea of this paper is to represent an un-

certainty X

k

with a complicated shape in the N –

dimensional original space S by a simpler shaped un-

certainty X

∗

k

in an L–dimensional hyperspace S

∗

with

L>N.Pointsx

k

in S are related to points x

∗

k

in S

∗

via a nonlinear transformation T (.) according to

x

∗

k

= T (x

k

)=

T

1

(x),T

2

(x), ... , T

L

(x)

T

.

Hence, T

(.) deﬁnes an N –dimensional manifold U

∗

in

an L–dimensional space.

In addition, L–dimensional sets X

∗

k

of simple shape

are deﬁned in the transformed space S

∗

. Here, ellip-

soidal sets according to

X

∗

k

=

x

∗

k

:(x

∗

k

− ˆx

∗

k

)

T

(X

∗

k

)

−1

(x

∗

k

− ˆx

∗

k

) ≤ 1

are used, where ˆx

∗

k

is the ellipsoid midpoint and X

∗

k

is

a symmetric positive deﬁnite matrix.

The intersection of an ellipsoid X

∗

k

with the mani-

fold U

∗

deﬁnes a submanifold of U

∗

, which, in turn,

deﬁnes a complicated set in the original space S.

Remark 3.1 A complex–shaped set in the original

space S is deﬁned by both the transformation T

(.) and

the pseudo ellipsoid X

∗

k

.

In many cases, the type of transformation T

(.)re-

sults directly from the nonlinearities considered. For

example, when considering polynomial nonlinearities,

a polynomial transformation is used. For trigonomet-

ric nonlinearities, a trigonometric transformation can

be used.

However, to simplify application of the new ﬁltering

approach, a generic transformation is helpful. For that

purpose, Bernstein polynomials are used, since their

approximation capabilities are suﬃcient for a large

class of nonlinearities. In addition, they lead to better

conditioned calculations than standard polynomials.

Multidimensional Bernstein polynomials are deﬁned

on the basis of one–dimensional Bernstein polynomi-

als, which on the interval [l, r]aregivenby

H

n

i

(x)=

n

i

l − x

l − r

i

r − x

r − l

n−i

for i =0,... ,n.With

x

k

=

x

k,1

x

k,2

... x

k,N

T

,

the above transformation is deﬁned by

T

i

(x

k

)=

N

j=1

H

L

j

−1

i

j

(x

k,j

) ,

for i

j

=0,... ,L

j

− 1, j =1,... ,N, L =

N

j=1

L

j

,

and i =

N

j=1

i

j

.

4 Filtering

Based on the concept of pseudo ellipsoids, which

represent complex–shaped sets in the original space S

by pseudo ellipsoids in the hyperspace S

∗

, the nonlin-

ear ﬁlter step can now be performed by a linear ﬁlter

in the hyperspace S

∗

. For that purpose, a pseudo–

linear expansion of the nonlinear measurement equa-

tion h

k

(.) is performed according to

h

k

(x

k

)=H

∗

k

x

∗

k

+ e

h

k

≈ H

∗

k

x

∗

k

,

where e

h

k

represents the approximation error deﬁned

by e

h

k

= h

k

(x

k

) − H

∗

k

x

∗

k

.

In general, the expansion can be enhanced by an

additional nonlinear transformation g

(.)ofthemea-

surements according to

g

(ˆz

k

− v

k

)=g(h

k

(x

k

)) .

The left hand side can be approximated by

g

(ˆz

k

− v

k

)=ˆz

∗

k

− G

∗

k

v

∗

k

− e

v,∗

k

≈ ˆz

∗

k

− G

∗

k

v

∗

k

,

where ˆz

∗

k

and G

∗

k

are nonlinear functions of ˆz

k

and

v

∗

k

is a nonlinear function of v

k

. e

v,∗

k

accounts for the

approximation error.

The right hand side is again approximated accord-

ing to

g

(h

k

(x

k

)) = H

∗

k

x

∗

k

+ e

h,∗

k

≈ H

∗

k

x

∗

k

with approximation error e

h,∗

k

.Asaresult,themea-

surement equation in the hyperspace is obtained ac-

cording to

z

∗

k

= H

∗

k

x

∗

k

+ e

h,∗

k

+ G

∗

k

v

∗

k

+ e

v,∗

k

w

∗

k

with w

∗

k

representing the total uncertainty.

Let the set of all predicted states be given by the

set X

p

k

, which is deﬁned in the transformed space S

∗

by

X

p,∗

k

=

x

∗

k

:(x

∗

k

− ˆx

p,∗

k

)

T

(E

p,∗

k

)

−1

(x

∗

k

− ˆx

p,∗

k

) ≤ 1

.

Furthermore, let w

∗

k

be bounded by the set

W

∗

k

=

w

∗

k

:(w

∗

k

)

T

(W

∗

k

)

−1

w

∗

k

≤ 1

.

Then, the set deﬁned by the measurement is given by

X

m,∗

k

=

x

∗

k

:(ˆz

∗

k

− H

∗

k

x

∗

k

)

T

(W

∗

k

)

−1

(ˆz

∗

k

− H

∗

k

x

∗

k

) ≤ 1

.

The fusion result is given by a set X

s,∗

k

(again an el-

lipsoid in the transformed space, but a set of compli-

cated shape in the original space!) that contains the

intersection of the ellipsoids X

p,∗

k

and X

m,∗

k

. Hence,

X

s,∗

k

is obtained by a linear set–theoretic ﬁlter in the

hyperspace S

∗

[19]

X

s,∗

k

=

x

∗

k

:(x

∗

k

− ˆx

s,∗

k

)

T

(E

s,∗

k

)

−1

(x

∗

k

− ˆx

s,∗

k

) ≤ 1

with

ˆx

s,∗

k

=ˆx

p,∗

k

+ λ

∗

k

E

p,∗

k

(H

∗

k

)

T

W

∗

k

+ λ

∗

k

H

∗

k

E

p,∗

k

(H

∗

k

)

T

−1

ˆz

∗

k

− H

∗

k

ˆx

p,∗

k

,

and

E

s,∗

k

= d

∗

k

P

s,∗

k

P

s,∗

k

= E

p,∗

k

− λ

∗

k

E

p,∗

k

(H

∗

k

)

T

W

∗

k

+ λ

∗

k

H

∗

k

E

p,∗

k

(H

∗

k

)

T

−1

H

∗

k

E

p,∗

k

,

where

d

∗

k

=1 + λ

∗

k

− λ

∗

k

ˆz

∗

k

− H

∗

k

ˆx

p,∗

k

T

W

∗

k

+ λ

∗

k

H

∗

k

E

p,∗

k

(H

∗

k

)

T

−1

ˆz

∗

k

− H

∗

k

ˆx

p,∗

k

.

Using this form of bounding ellipsoid for the exact

intersection of X

p,∗

k

, X

m,∗

k

in the transformed space S

∗

oﬀers the advantage that the resulting set X

s

k

(λ

k

) in

the original space S possesses the following property,

which is desirable in applications in the sense, that no

new uncertainty is introduced:

l

1

l

2

x

EE

x

Figure 1: Schematical top view of the considered

type of 2D SCARA robot manipulator with segment

lengths l

1

, l

2

and joint angles φ

1

, φ

2

.

Lemma 4.1 If the two sets X

p

k

und X

m

k

overlap, the

ﬁltering result X

s

k

(λ

k

) contains the exact intersection

X

p

k

∩X

m

k

and is itself contained in their union X

p

k

∪

X

m

k

. Hence, it holds

(X

p

k

∩X

m

k

) ⊂X

s

k

(λ

k

) ⊂ (X

p

k

∪X

m

k

)

for all λ

k

∈ [0, ∞].

The fusion parameter λ

k

is selected in such a way,

that a certain measure of the size of the set X

s

k

is min-

imized. (How to obtain the minimum volume ellipsoid

in a linear setting is discussed in [2].)

5 Simulation Example

We consider a SCARA–type robot manipulator

with two degrees of freedom according to Fig. 1. The

segment lengths l

1

, l

2

are assumed to be unknown and

are not amenable to direct measurements. Only uncer-

tain measurements of the distance r of the end–eﬀector

from the origin are available for diﬀerent angles φ

1

, φ

2

.

The position of the end–eﬀector with respect to the

origin is given by

x

EE

=

l

1

cos(φ

1

)+l

2

cos(φ

1

+ φ

2

)

l

1

sin(φ

1

)+l

2

sin(φ

1

+ φ

2

)

.

Hence, the distance r of the end–eﬀector from the

origin is given by

r =

l

2

1

+ l

2

2

+2l

1

l

2

cos(φ

2

) .

Given this nonlinear relation, the segment lengths l

1

,

l

2

are estimated (N = 2) based on measured distances

ˆr

k

, k =1, 2,... for diﬀerent angles φ

2,k

.Themea-

surement uncertainties are assumed to be bounded

according to

|r

k

− ˆr

k

|≤R,

which gives

ˆr

k

=

l

2

1

+ l

2

2

+2l

1

l

2

cos(φ

2,k

)+v

k

with v

2

k

≤ R

2

.

The parameters used during the simulation are

l

1

= 500 mm ,

l

2

= 350 mm ,

R =40mm .

The ﬁltering procedure starts with an axis–aligned

uncertainty box

X

p

1

=[0, 1000] × [0, 1000] mm

2

.

Sequentially, measurements ˆr

k

, k =1,... ,4, for

φ

2,1

=90

o

, φ

2,2

= 120

o

, φ

2,3

= 160

o

, φ

2,4

= 170

o

are used to update the initial estimate.

The function g

(.) has been selected to g(x)=

x

2

x

4

x

6

T

, T

i

(x), i =1,... ,L, are chosen as

multidimensional Bernstein polynomials. The results

are visualized

1

for n = 6 in Fig. 2, where n is the

order of the one–dimensional Bernstein polynomials,

which gives L = 49. Obviously, the result of the pro-

posed new ﬁlter is a tight approximation of the exact

estimation result at every time step.

6 Conclusions

In this paper, an eﬃcient algorithm for the recur-

sive calculation of tight closed–form approximations of

the feasible sets in nonlinear set–theoretic estimation

problems has been presented. The resulting sets are

a much better approximation compared to simple sets

like hyper–rectangles or ellipsoids and are in general

of complex shape, i.e., nonconvex and not connected.

The same methodology has been applied to stochas-

tic nonlinear systems [10]. For the case of mixed

stochastic and set–theoretic uncertainties, the ﬁlter-

ing algorithms presented in [8, 9] will be generalized

to nonlinear systems.

References

[1] F. L. Chernousko, State Estimation for Dynamic

Systems, CRC Press, 1994.

[2] M.-F. Cheung, S. Yurkovich, K. M. Passino, “An

Optimal Volume Ellipsoid Algorithm for Parame-

ter Set Identiﬁcation”, IEEE Tr ansactions on Au-

tomatic Control, Vol. 38, No. 8, pp. 1292–1296,

1993.

1

The result of updating the estimate based on measurement

k is denoted X

s

k

. However, in the next time step, the same

estimate is denoted X

p

k+1

to be consistent with the notation

introduced above.

Figure 2: Results of sequential estimation of the lengths l

1

, l

2

of the 2D SCARA robot manipulator.

[3] P. L. Combettes, “The Foundations of Set The-

oretic Estimation”, Proceedings of the IEEE,

Vol. 81, No. 2, pp. 182–208, 1993.

[4] J. R. Deller, “Set Membership Identiﬁcation in

Digital Signal Processing”, IEEE ASSP Maga-

zine, Vol. 6, pp. 4–20, 1989.

[5] J. R. Deller, M. Nayeri, S. F. Odeh, “Least–

Squares Identiﬁcation with Error Bounds for

Real–Time Signal Processing and Control”, Pro-

ceedings of the IEEE, Vol. 81, No. 6, pp. 815–849,

1993.

[6] G.D.Hager,S.P.Engelson,S.Atiya,“OnCom-

paring Statistical and Set–Based Methods in Sen-

sor Data Fusion”, Proceedings of the 1993 IEEE

International Conference on Robotics and Au-

tomation, Atlanta, Georgia, pp. 352–358, 1993.

[7] U. D. Hanebeck, G. Schmidt, “Set–theoretic Lo-

calization of Fast Mobile Robots Using an An-

gle Measurement Technique”, Proceedings of the

1996 IEEE International Conference on Robotics

and Automation (ICRA’96), Minneapolis, Min-

nesota, pp. 1387–1394, 1996.

[8] U. D. Hanebeck, J. Horn, and G. Schmidt, “On

Combining Statistical and Set–Theoretic Estima-

tion”, Automatica, Vol. 35, No. 6, pp. 1101–1109,

1999.

[9] U. D. Hanebeck and J. Horn, “Fusing Information

Simultaneously Corrupted by Uncertainties with

Known Bounds and Random Noise with Known

Distribution”, Information Fusion, Vol. 1, No. 1,

pp. 55–63, 2000.

[10] U. D. Hanebeck, “Optimal Filtering for Polyno-

mial Measurement Nonlinearities with Additive

Non–Gaussian Noise”, Proceedings of the Amer-

ican Control Conference (ACC’2001), Arlington,

Virginia, 2001.

[11] L. Jaulin and E. Walter, “Set Inversion via In-

terval Analysis for Nonlinear Bounded–error Es-

timation”, Automatica, Vol. 29, No. 4, pp. 1053–

1064, 1993.

[12] L. Jaulin, “Interval constraint propagation with

application to bounded–error estimation”, Auto-

matica, Vol. 36, No. 10, pp. 1547–1552, 2000.

[13] T. H. Kerr, “Real–Time Failure Detection: A

Static Non–Linear Optimization Problem That

Yields a Two Elliposid Overlap Test”, Jour-

nal on Optimization Theory Applications,Vol.2,

pp. 509–536, 1977.

[14] M. Milanese, G. Belaforte, “Estimation Theory

and Uncertainty Intervals Evaluation in the Pres-

ence of Unknown but Bounded Errors: Linear

Families of Models and Estimates”, IEEE Trans-

actions on Automatic Control, Vol. 27, pp. 408–

414, 1982.

[15] M. Milanese, A. Vicino, “Estimation Theory for

Nonlinear Models and Set Membership Uncer-

tainty”, Automatica, Vol. 27, No. 2, pp. 403–408,

1991.

[16] A. Preciado, D. Meizel, A. Segovia, M. Rombaut,

“Fusion of Multi–Sensor Data: A Geometric Ap-

proach”, Proceedings of the 1991 IEEE Interna-

tional Conference on Robotics and Automation,

Sacramento, California, pp. 2806–2811, 1991.

[17] A. K. Rao, Y.–F. Huang, S. Dasgupta, “ARMA

Parameter Estimation Using a Novel Recursive

Estimation Algorithm with Selective Updating”,

IEEE Transactions on Signal Processing, Vol. 38,

No. 3, pp. 447–457, 1990.

[18] A. Sabater, F. Thomas, “Set Membership Ap-

proach to the Propagation of Uncertain Geomet-

ric Information”, Proceedings of the 1991 IEEE

International Conference on Robotics and Au-

tomation, Sacramento, California, pp. 2718–2723,

1991.

[19]

F. C. Schweppe, Uncertain Dynamic Systems.

Prentice–Hall, 1973.

[20] J. S. Shamma, K.–Y. Tu, “Approximate Set–

Valued Observers for Nonlinear Systems”, IEEE

Transactions on Automatic Contr ol, Vol. 42,

No. 5, pp. 648–658, 1997.

[21] A. Vicino, G. Zappa, “Sequential Approxima-

tion of Feasible Parameter Sets for Identiﬁcation

with Set Membership Uncertainty”, IEEE Trans-

actions on Automatic Control, Vol. 41, No. 6,

pp. 774–785, 1995.

[22] E. Walter, H. Piet-Lahanier, “Exact Recursive

Polyhedral Description of the Feasible Parame-

ter Set for Bounded Error Models”, IEEE Trans-

actions on Automatic Control, Vol. 34, No. 8,

pp. 911–915, 1989.