Nonlinear observers for autonomous Lipschitz continuous ...webpages.lss.supelec.fr/perso/ortega/papers/Kreisselmeier-Engel.pdf · Nonlinear Observers for Autonomous Lipschitz Continuous

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 3, MARCH 2003 451

Nonlinear Observers for AutonomousLipschitz Continuous Systems

Gerhard Kreisselmeier and Robert Engel

Abstract—This paper considers the state observation problemfor autonomous nonlinear systems. An observation mapping is in-troduced, which is defined by applying a linear integral operator(rather than a differential operator) to the output of the system. Itis shown that this observation mapping is well suited to capture theobservability nature of smooth as well as nonsmooth systems, andto construct observers of a remarkably simple structure: A linearstate variable filter followed by a nonlinearity. The observer is es-tablished in Sections III–V by showing that observability and finitecomplexity of the system are sufficient conditions for the observerto exist, and by giving an explicit expression for its nonlinearity.It is demonstrated that the existence conditions are satisfied, andhence our results include a new observer which is not high-gain,for the wide class of smooth systems considered recently in a pre-vious paper by Gauthier, et al. In Section VI, it is shown that theobserver can as well be designed to realize an arbitrary, finite ac-curacy rather than ultimate exactness. On a compact region of thestate space, this requires only observability of the system. A cor-responding numerical design procedure is described, which is easyto implement and computationally feasible for low order systems.

Index Terms—Nonlinear state observer, nonlinear systems, non-smooth systems.

I. INTRODUCTION

T HE leading results in nonlinear observer theory are forclasses of systems, which have a certain degree of smooth-

ness, i.e., are differentiable a sufficient number of times. Non-linear coordinate changes, normal forms, output injection, em-bedding, and high gain techniques are some of the most notableconcepts. See, e.g., [1]–[7] for discussions of the various ap-proaches and further references. The importance of smoothnessthroughout all this work is not surprising, because many of theideas, which are involved, have a linear systems background,linear systems are infinitely smooth, and smooth analysis is amost powerful mathematical tool.

In comparison, very little is known about observers for non-smooth systems (i.e., systems, which are Lipschitz continuous,but not differentiable everywhere), although nonsmooth sys-tems also occur frequently in practice. A major reason is thatthe observation problem is in a significant way different.

For example, a linear oscillator with a one-sided sensor

if positiveotherwise

Manuscript received January 18, 2002; revised June 25, 2002. Recommendedby Associate Editor Z. Lin.

The authors are with the Department of Electrical Engineering, the Universityof Kassel, D-34109 Kassel, Germany (e-mail: [email protected]).

Digital Object Identifier 10.1109/TAC.2002.808468

Fig. 1. Output signaly .

is a nonsmooth system, which produces output measurementsof the form in Fig. 1.

In order to reconstruct the state vector from measurements itwould, for example, not be sufficient to know the output on aninterval, where the output is identically zero. However, it wouldbe sufficient to know the output on a sufficiently large interval,e.g., for any , because this always includesa subinterval, on which is strictly positive and, hence, thesystem is linear and observable.

The point in this example is a phenomenon, that appears tooccur more generally in nonsmooth systems: The state informa-tion is contained in the output signal, but is in a significant wayunequally distributed in time. It would appear unlikely, that localmethods are useful in such cases. Looking for possible nonlocalconcepts, full state information becomes a main point of con-cern, and this leads very naturally toward thinking about a moreextensive use of the (complete) output signal history.

The moving horizon approach, considered, e.g., in [8]–[10]goes in this direction. The idea is to store measurements from the(sliding) interval , and to generate a state estimate so asto asymptotically match the predicted output with the measuredone on the whole interval. Thereby the observation problem isconverted into the problem of (asymptotic) online minimizationof an error criterion [8], [9], or solving a set of nonlinear equa-tions [10], respectively. Algorithms to do this are the main issueof the approach. Under nonsmoothness and/or a lack of globalconvexity this is again a tough problem. Recent suggestions in[9] and [10] are gradient descent based and give local conver-gence, assuming some smoothness of the system to be observed.We note that the structure of this kind of observer comprisesa data storage part, which is distributed parameter system typefor continuous time measurements (this can be avoided by usingsampled measurements only at the expense that a major assump-tion, similar to the finite complexity property in this paper, has tobe imposed in addition to observability on the continuous timesystem), and a nonlinear dynamics part, which is created by the

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452 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 48, NO. 3, MARCH 2003

asymptotic nature of the minimization/equation solving algo-rithm.

This paper takes a different, more system theoretically ori-ented perspective of the subject. We pursue the idea of an ob-servation mapping and its analysis as being central to the ob-servation problem and the starting point for constructing ob-servers. To include nonsmooth systems, a special observationmapping is introduced, which uses the complete output historyof the system, is detectable from measurements and, under con-ditions explored in detail, captures the observability nature ofthe system in a mapping of finite dimension.

Specifically, instead of using successive derivativesof the output, we use successive integrals

of the output. The integral oper-ations are taken up to a sufficient order, are taken over thecomplete output history, and are equipped with stable kernelsfor well definedness. This results in a nonlinear observer ofa remarkably simple structure: A linear state variable filterfollowed by a nonlinearity, the nonlinearity being an extendedinverse of the observation mapping. The inverse is given in anexplicit form.

In what follows, the basic concept is introduced, exploredtheoretically in some detail, related to known results for smoothsystems, and is then extended to include finite time convergentand arbitrarily accurate observers as well as an easy numericaldesign.

II. THE OBSERVATION CONCEPT

A. Problem Statement

We consider nonlinear systems of the form

(1)

(2)

with state and output . Throughout this paper,the basic assumption is that and areglobally Lipschitz continuous.

By the Lipschitz property, has a uniquely defined state-trajectory , which passes through the point

for each and . Moreover, the state-trajectory and the associated output are exponentiallybounded.

The problem is to asymptotically observe the unknown stateof the system using only present and past output measurements.The focus will be on some closed, but not necessarily bounded,set , which is invariant under , i.e., trajectories whichstart in remain in for all future times.

B. Basic Idea

Let us define an observation mapping

(3)

where is a controllable pair, ,( th-order for short), and sufficiently stable such that the in-tegral is well defined.

The mapping assigns to each state ,via the corresponding output history , of the

Fig. 2. Image ofq(x).

system, a point . Since is by definition an integralof past outputs, it is well suited for computation from measure-ments. In particular, along each system trajectorywe have that satisfies the equation

. Therefore, the current value ofcan be generated asymp-totically from a model of this equation. This suggests the fol-lowing observer structure:

(4)

(5)

Its inherent feature is that converges toexponentially as . The state estimate is thenformed from using a nonlinear mapping ,which in an ideal case satisfies and is a so-calledextended inverse of .

The idea is best illustrated by an example.Example 11 :

whenotherwise

This system is first order, not differentiable at , and

we have . Taking , the

observation mapping becomes

Its image is shown in Fig. 2.Since is injective, an extended inverse mappingexists, and can be obtained by taking the intersection of the

line , , with the image of ,and thereby to find . It is not hard to verify that the resultingobserver

is globally exponentially stable. The extended inverseis evenlinear in this case, by surprise.

The above idea gives an observer more generally, accordingto the following theorem.

1A. J. Krener, Presentation at a control theory meeting, sponsored by theMathematical Research Institute, Oberwolfach, Germany, 1999.

KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS 453

Theorem 1: Suppose is sufficiently stable, so thatis welldefined.

If

i) is injective;ii) satisfies

a) , ;b) 2 , ;

then (4) and (5) represent an observer forin .Proof: Consider any pair of initial conditions

From the definitions of and , it is evident thatsatisfies

(6)

If , then and, hence, . If ,then as , andfollows.

It remains to satisfy the assumptions of Theorem 1. This isdone in two steps. The first step (Section III) shows thatcanbe made uniformly injective, provided is observable and hasa further property called finite complexity. The second step(Section IV) shows that a uniformly injectiveis also sufficientfor the existence and an explicit construction of.

C. Notations

The following notations are used.(Euclidean) norm of a vector, and induced normof a matrix,respectively.Space of square integrable functions on the in-terval .

-norm on the interval , short notation. For a vector of functions

.

Inner product of functions on , shortnotation . For vectors offunctions, denotes a matrix with entries

.Set of continuous, strictly monotone increasingfunctions , satisfying .We write if is in class .Subset of , where the state is to be observed.

neighborhood of the point ., .

D. Observability

Suppose is the present state of the system. Then, the systemoutput, seen backward in time, is , .Throughout this paper, we consider the exponentially weightedversion of this output

and its variations

2This notation means that any sequence(z ; x ) 2 �G which satisfiesz � q(x ) ! 0, impliesQ(z )� x ! 0 ask ! 1.

where is chosen such that3

i) ;ii) ;

for all . Occasionally, we also write and, respectively, to make the dependence on time

more explicit.An observable system is basically unterstood here as a system

whose output time history uniquely determines its presentstate. The formal definition of observability is as follows.

Definition: is said to be observable in, if there is asuch that

(7)

for all .Observability thus allows to conclude smallness of

from smallness of uniformly in . If is compact,then observability in this sense only requires forall .4

Observability in finite time is defined accordingly, withreplaced by .

E. Choice of

Recall that the observation mapping is given by

(8)

where the observer filter pair is to be chosen. For conve-nience, we preselect a set of pairs of different dimensionas follows.

Let be chosen as a sequence of real or complex num-bers (with complex numbers appearing in complex conjugatepairs) such that

i) , ;ii) ;

and let denote the set of integers

the set is complex conjugate (9)

Then, by a result from [11], there is a uniquely defined se-quence of functions , which is an orthonormal basisin and has the following property. For eachthere is exactly one real, th-order pair with spectrum

such that

... (10)

3Any value of� greater than the Lipschitz constant off(x) is appropriate.From design considerations, smaller values would be preferred if possible.

4Note thaty(t; x) is continuous inx. A suitable' 2 K would be'(s) :=(s=s) �min ky(x)� y(x )k wheres := max jx � x j.


An explicit formula to obtain from is givenin [11]. The associated th-order observation mapping can thenbe rewritten in the form

(11)

As a result of this preselection, only the choice of dimensionremains to be discussed in the later sections.

It is worth pointing out that is in and, there-fore, has an orthonormal series representation

(12)

where the coefficients are . Hence, the ob-servation mapping of order represents the first coef-ficients of the orthonormal expansion of .

Finally, we remark that orthonormal coordinates are used hereonly to simplify the subsequent analysis. To actually build theobserver, it is sufficient to set up the observer filter pair con-trollable, of suitable dimension and with spectrum

, because this only amounts to a similarity trans-formation of and the corresponding linear transformationof .

III. U NIFORMLY INJECTIVEOBSERVATION MAPPINGS

It is shown in this section that the observation mapping canbe made uniformly injective, just by taking its dimension suf-ficiently large, provided the system is observable and of finitecomplexity. The property of finite complexity is introduced andexplored by giving suitable conditions.

Definition: The observation mapping is said to be injectiveof uniform measure in (uniformly injective, for short), if thereis a such that

(13)

for all .Note that if is compact then injectivity implies uniform

injectivity, because is continuous.

A. Finite Complexity

While observability characterizes the variations with re-spect to the distance of states , the property of finitecomplexity characterizes them as functions of time.

Definition: is said to be offinite complexityin , ifthere exists a finite number of piecewise continuous func-tions , combined to a vector

, such that for some

(14)

for all .Example 1 (Continued):The first-order system of example

1 has weighted output

otherwise

Hence, is a linear combination of and, which are in and linearly indepen-

dent. Therefore, and

which implies finite complexity.Example 2: A linear system , is

typically considered in . It has output variation

Condensing the linearly independent entries ofinto a vector , this can be rewritten in the form

. Finite complexity then follows bythe same argument as in the previous example. As a result, alinear system is always of finite complexity in , whether ornot the pair is observable.

The finite complexity property enables the following basicresult.

Theorem 2: If is observable and of finite complexity in, then there is an integer such that is uniformly

injective in .A proof is given in Appendix A.Note that if is uniformly injective for some dimension

, then it is so for all , . In other words,a candidate is appropriate if it is large enough.

B. Conditions for Finite Complexity

Useful conditions for finite complexity are given in twoLemmata. They indicate that this property is not overly re-strictive. On the contrary, it appears to be essential for wellconditionedness of the observation problem.

Lemma 1: For to be of finite complexity in , it is neces-sary that there are constants, , such that

i) ;

ii) ;

for all , where denotes the Fouriertransform of .

A proof is given in Appendix B.Finite complexity of a system thus implies that its output vari-

ations have a low frequency portion and a finite time in-terval portion, each of which is of the same order of magnitudeas the variation itself. Thereby, the variations are sufficientlywell conditioned to be captured from real world measurements.This is further reflected in the following necessary and sufficientcondition.

Lemma 2: is of finite complexity in , if and only if thereexist , , such that for each therelation

(15)

holds for at least one .A proof is given in Appendix C.


Lemma 2 combines the low frequency and finite time intervalaspects. The quantity of interest is ,which is obtained by passing through a first order low passfilter with transfer function (or, equivalently, bypassing through a filter to obtain

and then . As a result, the conditonof Lemma 2 can be interpreted in the sense thatis a simulta-neous low frequency/finite time portion from . The relativesize of this portion, the bandwidth and the time horizon can takeany nontrivial values.

To give an idea of the nonsmoothness, which can be dealtwith, we give a simple example of a nonlinear system, which isobservable and of finite complexity and, therefore, has an ob-server of the proposed kind.

Example 3: Consider a linear oscillator with a nonlinearoutput


where denotes the first state variable and . Themeasured output is simply the positive branch of the sinusoid

. Fig. 1, which has beenpresented in the introduction section, illustrates this kind ofnonsmoothness.

This system is obviously observable. Moreover, it has a fi-nite complexity, because it satisfies the condition of Lemma2. The main point in showing this (details are omitted for thesake of brevity) is the following. There is always an interval

on which does not changesign and is a sinusoid of amplitude in case of

and of amplitude in caseof , respectively, where denotesthe angle between and . This results in

for at least one . On the other hand, because .

Combining the two, finite complexity follows.To construct the observation mapping, we can use the fact

that describes a circle in , and thereforemoves on a closed path in , which is the periodic solutionof . The calculation of is completewith one such solution and the fact that due to

for .Based on ana priori choice of an eigenvalue sequence

, a natural initial guess of an observerdimension is (which is a lower bound for

to be injective). The pair can then taken to be, . Computing

as described above, and plotting its entries versus(which is not illustrated here) reveals that is not

injective, because the origin is not in the interior of this closedcontour (which is due to the mean value of ). A naturalsecond guess to try is . The pair is nowtaken to be

Fig. 3. Observation mappingq(x), jxj = 1 in the(q ; q ) plane.

(a)

(b)

Fig. 4. (a) Measured outputy and observed statex . (b) Observation errors(x � x ) , i = 1; 2.

i.e., with eigenvalues according to the a priori choice, and inconvenient coordinates such that .This gives zero mean value for and . Theplot of versus , which is illustrated in Fig. 3, givesevidence that this is injective. The observers (4) and (5)are finally implemented with from above, and withrealized by the extended inverse formula (28), which is given inSection VI-B, using in (b1) and in (28).

Fig. 4(a) and (b) illustrate the observation process for initialconditions , . The convergence is roughly

corresponding to the slowest eigenvalue of.

C. Conditions for Finite Complexity: Mildly Smooth Systems

This subsection gives extended conditions for finite com-plexity, taking advantage of some mild smoothness. Thestanding assumption is again that is globally Lipschitz.


Assumptions regarding smoothness will be stated individuallyin each result.

We start with a general condition on finite complexity, whichis formulated in terms of the weighted outputand the weightedoutput derivative . The notations are ,

and , respectively.Lemma 3: If exists and is continuous, and if there

are , , such that

i) ;

ii) ;for all , then is of finite complexity in .

The proof is given in Appendix D.For systems which are first order observable (i.e., systems

which satisfy ,for some ), the condition for finite complexity can besimplified drastically. The subsequent theorem reduces it to aLipschitz condition on in the first assumption,while the second assumption states first order observability infinite time.

Theorem 3: If there are , such that

i) is Lipschitz in the set;

ii) , ;then is observable and of finite complexity in.

A proof is given in Appendix E.For checking the second hypothesis of Theorem 3 in applica-

tions, the subsequent Corollary can be useful. It points out that(under some mild extra assumptions) a system is first-order ob-servable in finite time, if on some interval its lineariza-tions are observable.

Corollary 1: If

i) is compact;ii) have continuous first order partial derivatives;iii) is observable on in ;iv) for each the linearized system ,

, ,is observable on ;

for some , then there is a such that

A proof is given in Appendix F.The result, as given by Theorem 3 and Corollary 1, has a

fairly wide span of applicability. Linear systems, considered in, are clearly included in Theorem 3, because their ob-

servability is always first order. However, Theorem 3 also in-cludes the following situation from the literature as a specialcase.

Example 4: For systems which are of the form (or can beembedded in this form)

......

where is globally Lipschitz, a so-called high gain ob-server was established in [4].

It turns out that Theorem 3 applies to this class of systems onthe set . It is important here to cover all of , becausethis class includes systems which have unbounded solutions. Bythe Lipschitz assumption on , the first hypothesis of The-orem 3 is obviously met. Moreover it is proved in Appendix Gthat this class of systems is first order observable in ,and hence also satisfies the second hypothesis.

In cases where is continuously differentiable and iscompact, first-order observability may be concluded more easilyby using Corollary 1. Here, the main point is to realize that

has a special structure with , andthe fact that this structure carries over to the linearized system.The linearized system is therefore observable on every non-trivial interval . This is in fact stronger thanrequired by Corollary 1 (which would allow unobservability onsome subinterval) to conclude first-order observability of thenonlinear system in .

As a result, Theorem 3 applies, i.e., each system in the givenclass is observable and of finite complexity in, and thereforealso has an observer of the form proposed here, which is nothigh gain.

IV. I NVERSEMAPPING

A mapping , which satisfies Assumption ii) of Theorem 1,is referred to as an extended inverse of the observation mapping.This section shows that, based on uniform injectivity, which waspreviously established, such an inverse exists and can be givenin an explicit form.

Lemma 4: If is uniformly injective in , then an extendedinverse exists.

The proof is given in Appendix H.To be constructive about an extended inverse, we introduce

the explicit formula5

(16)

where is a weight, and is the volume increment in. This formula defines as a weighted average of all .

A suitable weight is

(17)

For a discrete (approximating) version of this formula, seeSection VI-B.

Theorem 4: If is uniformly injective in , and is notlocally thin6 , then , as defined by

a) (16) and (17), in case is bounded;b) (29)–(31), in case is not bounded;

is continuous and an extended inverse of.

5Forz 2 q(G) the right-hand side is to be unterstood in the sense of the limit,obtained when replacingw(z; x) byw (z; x) := 1=["+ jz � q(x)j] andletting " ! 0.

6We say thatG is not locally thin, if dX � c � dX for each

� 2 G and" 2 (0; " ], whereG := U (�) \ G andc; " are positiveconstants.


Fig. 5. Finite-time observer structure.

A proof is given in Appendix I.The idea behind this construction ofis as follows. The state

observation error of the nonlinear observer becomesand can be written in the form

(18)

From (6), we have as . Thereby, thenormalized weight , seen as a functionof , tends to a -distribution at . Convergence

then follows from (18). The proof of Theorem 4 makesthis intuitive idea rigorous.

V. FINITE-TIME OBSERVERS

Let be an observation mapping, which is defined usinga finite time interval (rather than )

(19)

Comparing with , it is seen that along any trajectoryof the system

(20)

This motivates an observer of the form

(21)

(22)

(23)

defined for and with initial conditions ,. The structure is again open loop and includes

now a delay, as is illustrated in Fig. 5.Theorem 5:

i) is injective;ii) satisfies for all ;

then (21)–(23) are an observer for, whose state es-timate converges to the true state in finite time, i.e.,

for .Proof: As before, along any trajectory of the system

we have

Rewriting as

(24)

it follows that and, thereby,for all , i.e., finite-time convergence regardless of theinitial conditions.

(a)

(b)

Fig. 6. (a) Measured outputy and observed statex . (b) Observation errors(x � x ) , i = 1, 2.

For initial conditions ,the same argument gives for all .

Note that the assumptions made in Theorem 5 are in factweaker than those in Theorem 1. In particular, any kind of anextended inverse is now suitable, because convergence occursin finite time. The latter is a structural property of this observer.

We also note that injectivity of and are the sameproblems on different horizons. Therefore, with observabilityand finite complexity redefined on the interval and with

replaced by accordingly, the results of Section IIIcarry over to the finite-time case. In particular, observability andfinite complexity together guarantee that is uniformly in-jective, for some . The extended inverse of Section IVmay then be used to complete the observer.

The finite time observer is illustrated using Example 3 again.Example 3 (Continued):It was shown previously that is

observable and of finite complexity on the interval .Since has periodic solutions with period , it is also ob-servable and of finite complexity on the interval . Usingthe same observer dynamics as before and the fact thatis periodic, we find that

Hence, is also injective, and its extended inverse can beobtained as from the extendedinverse of . The resulting observer is simulated with

as above, and . Fig. 6 illustrates the finite timeconvergence, which is in contrast to the asymptotic convergencein Fig. 4.


VI. A RBITRARILY ACCURATE OBSERVERS

Under real world conditions, an observer can only give an ap-proximation of what it is wanted to give theoretically. Therefore,an observer may as well be designed for an arbitrary finite ac-curacy, rather than for ultimate exactness. We develop this con-cept for practical situations, where the trajectories of the systemare bounded, so that can be taken compact. The result areobservers of any desired accuracy, which require only observ-ability of the system and are, therefore, of widest applicability.

A. Arbitrary Finite Accuracy

Recall from Section II-E, that our observation mapping rep-resents the first coefficients of an orthonormal expansion of

. Therefore, the information for a prescribed accuracy isnaturally contained in this representation, if we takesuffi-ciently large. Together with observability, this gives rise to thefollowing Lemma.

Lemma 5: If is observable in , and is compact, thenfor each desired accuracy there exist ,such that

(25)

where .A proof is given in Appendix J.With an accuracy in the sense of (25), the set of all points ingiving the same value of, is located strictly in the interior

of a ball of radius in . Therefore, can be recovered froma given , up to an error of size .

As a suitable mapping, which does such an inversion approx-imately, we consider the expression

(26)

with

(27)

where is a positive constant, to be chosen in combination with(respectively, ) so as to achieve a joint accuracyof the

resulting observer.Theorem 6: Let be observable in , and let be compact.

Then for each there exist and , such that(4), (5), (26), (27)7 are an observer of accuracyin , i.e.,

i) as for all;

ii) , for all.

A proof is given in Appendix K.The result of Theorem 6 is very general and widely appli-

cable. It says that for an observable system on a compact set,there is always an arbitrarily accurate observer, and the proposedobserver structure is an easy realization, where one needs onlytake the dimension sufficiently large and the constantsuf-ficiently small.

7In caseG is locally thin,G is to be replaced by�U (G) in (26) with a constant� > 0, which is tolerable from the desired accuracy (see the proof of details).

Note that the results of this section and the previous one, re-spectively, combine nicely to an observer, which attains any de-sired accuracy in finite time.

B. Numerical Design

Consider any Lipschitz continuous system(not necessariliyobservable or of finite complexity), with a compact design re-gion . Based on an a priori choice of a sequenceas described in Section II-E, the following steps toward an ob-server can always be taken.

a) The choice of an integer gives (see Section II-E)an th-order pair , which defines the observer dy-namics .

b) The choice of some (uniformly continuous) nonlinearitydefines the observed state .

The result of a) and b) is an observer of accuracy, i.e., an ob-server which satisfies properties i) and ii) of Theorem 6, where8

An observer design, thus, becomes a selection ofand , soas to make as small as desired.

By Theorem 6 we know that if is observable in , then sucha selection is always possible and, in particular,as defined by(26) and (27) is an appropriate nonlinearity. Replacing in (26)the ratio of integrals by a corresponding (approximating) ratioof sums, is then also an appropriate nonlinearity. It can be usedas a choice of in step b), and can be designed numerically asfollows.

b1) Select a set of points sufficiently dense andproperly distributed in (e.g., ,where is the discretization density).b2) Compute via integrating (1) backward in timeand evaluating (3).b3) Define

(28)

Based on these, an observer design can now proceed iteratively(and without the need to have checked observability ofbeforehand) as follows. Each iteration involves going throughsteps a) and b1)–b3), and delivers an observer with accuracy

as a candidate solution. A convenientestimate of this accuracy can be obtained from the computeddata as .

In each iteration the observer dimension is increased(starting from some ) to further improve the accuracy,while keeping and sufficiently small, until the desiredaccuracy is attained. By Theorem 6, this will succeed withina finite number of iterations, if is observable. In case thedesired accuracy is not attained for reasonably largeandsmall this indicates that may be not observable.

Note that the main computations are in step b2), and that theseareoffline computations. The nonlinearity (28) is to be imple-

8Recall from Section II-B thatz(t) ! q(x (t)) ast ! 1, which impliesthat jx (t) � x (t)j = jQ(z(t))� x (t)j ! [0; ].


mented as part of the final observer. Its evaluation to determinethe observer output areonlinecomputations.9

VII. CONCLUDING REMARKS

An integral operator based observation mapping isintroduced, which captures the observability nature ofautonomous nonlinear systems, whether or not they aresmooth. What makes this mapping constructive and wellsuited for theoretical analyzes is that anth-order observationmapping represents anth-order orthonormal series expansionof the complete output signal history.

The observer has a particularly simple open loop structure,which divides the observation process into two subsequentstages. In the first stage, linear observer dynamics generatea convergent estimate of the observation mapping value thatcorresponds to the current output signal history. That is,observation takes place here completely in the output functionspace, the latter being respresented implicitly by its finite-dimensional vector space equivalent . Accordingly, thisstage is coordinate free. The subsequent second stage merelymaps the estimate into the state coordinates of the system, viaan extended inverse of the observation mapping. An explicitinversion formula is presented.

In the given observer structure, it only remains to choose theobserver dynamics with (almost) arbitrary eigenvalues and asufficiently large dimension to complete the observer. By con-struction, its convergence is global.

The main assumptions to be made are observability and finitecomplexity. Finite complexity is introduced and characterizedby a necessary and sufficient condition. Roughly, this propertyis well conditionedness of the output variations in a finite time/finite bandwidth sense. While these conditions are difficult tocheck in general, it is demonstrated that they are satisfied and,hence, our results include a new observer which is not highgain, for the wide class of smooth systems considered recentlyin [4].

The proposed observer structure also works nicely incases where the system is only observable. Applied with anydimension , it results in a finite accuracy observer, which isthe more accurate the larger is. A computerized numericaldesign of such observers is presented, which is easy to im-plement, and which is computationally feasible for low ordersystems.

Finally, we point out that our single-output results readilycarry over to the multiple-output case with an observation map-ping, which comprises the individual observation mappings ofeach output. Extensions to nonautonomous systems (i.e., sys-tems with inputs) are nontrivial. Results in this direction havebeen obtained in [12] and are currently under preparation forpublication.

9Any alternate form ofQ (e.g., based on neural nets, fuzzy logic, or interpo-lation/approximation techniques) may be used to optimize its implementationsand/or evaluations.

APPENDIX

A. Proof of Theorem 2

Let be of finite complexity in , i.e., (14) holds for some, . Since is an orthonormal basis in

(see Section II-E), and , we have

where is square summable. Therefore, aninteger exists such that

Letting and ,we can rewrite

where . This can be substituted in (14) to get

and, thus

Using observability, it finally follows that

i.e., is uniformly injective in .

B. Proof of Lemma 1

Let be of finite complexity in , i.e., (14) holds for some, .

Then

and, using (14)

Taking sufficiently large implies part i) of the Lemma.There is no loss of generality assuming that .

To see this, we can obtain in the same way as before

Hence, to establish finite complexity,can be restricted to benonzero only on a (sufficiently large) finite interval. On thelatter, we have , because is piecewisecontinuous by assumption.


With , the Fourier transforms of and(taken for the signals as defined on and continued iden-tically zero on , respectively, and by Parseval’s Theorem

This gives, using (14)

Taking sufficiently large, this implies part ii) of the Lemma.

C. Proof of Lemma 2

a) Sufficiency: Let

otherwise

and let denote its Fourier transform. Then, for

The order of integrals can be switched due to Fubini’s Theorem[13]. Then, for

where is a constant resulting from the integral, which isbounded.

Suppose now that the condition of the Lemma holds, i.e., forsome we have . Then

and with it follows that

Let denote the integer such that . Then,we have

Further define a vector with components

whenotherwise

for where is the smallest integer greater thanThen, it follows that

and finite complexity is immediate from its definition. Thisproves sufficiency.

b) Necessity: Suppose is of finite complexity, i.e., (14)holds for some , . Since , there is asuch that . Thereby

Let and define where

otherwise

Further rewrite as , where is chosenso that . Since is piecewise continuous, canbe taken sufficiently large, so that Then, itfollows that

and

As a consequence, there is an integer (which de-pends on ) such that, with the notation , wehave


i.e., the condition of the Lemma is implied. This provesnecessity.

D. Proof of Lemma 3

Since and are related by , assump-tion ii) implies

where is an appropriate constant. For all ,this gives

By assumption i), there is a such that

hence

Let be chosen such that andThen does not change sign for

, and there is an interval of lengthsuch that . Integration

thus gives

and finite complexity follows from Lemma 2.

E. Proof of Theorem 3

is observable by assumption ii). We prove finite com-plexity by showing that under the assumptions of the Theorem,Lemma 3 applies.

Since is Lipschitz, we havefor some . Combining this with ii) it follows that the firstcondition of Lemma 3 is satisfied.

Let and let denote its Lipschitzconstant. Then

where the right-hand side is bounded proportional to on, because is Lipschitz. Taking norms and using ii) it

follows, with an appropriate constant , that

i.e., the second condition of Lemma 3 is satisfied as well. Thiscompletes the proof.

F. Proof of Corollary 1

For any , we can write

where denotes the state transition matrix which is asso-ciated with the linearization of along the trajectory passingthrough the point , and is a remainder term, which is de-fined by the above relation in case , and byfor all .

By assumptions i) and iv), there is a , such that for all

Moreover, is continuous in its arguments on the com-pact set , and satisfies for all

. Therefore, a constant exists such thatimplies . Hence

By assumption iii), we also haveCombining the two gives

where is the maximum distance of two points inand. This completes the proof.

G. Proof of First Order Observability of Example 4

Let , and consider any . Since istimes continuously differentiable with respect to time due to

the special structure of, application of Taylor’s formula gives

where

and

for some . Due to the special structure of,we have

With notations

we can write


For any choice of , this gives

for some , where the last inequality follows because thesmallest eigenvalue of

...

......

. . ....

. . .

is bounded below proportional to .Also, due to the special structure of, we have

where and denotes the Lipschitz constantof .

For any choice of this can be used, in the definitionof , to conclude that

for some constant .Combining the two arguments, it follows that

Taking sufficiently small, this gives a (which is in-dependent of ) such that . This provesfirst-order observability of in .

H. Proof of Lemma 4

We note that is closed. To see this, let be asequence in such that as . By uniforminjectivity we have

from which we can conclude that is a Cauchy sequence.Hence, converges to some. Since is closed, is in .By continuity follows.

Define, for example

ifif

where denotes the projection of on (i.e., issome point in which minimizes the distance ), whichis made unique by choice where necessary.

Then , holds by definition. Letbe varied such that Then, by the

projection property, . Combining the two gives. Due to uniform injectivity this implies

. Since is either or , it follows that .

I. Proof of Theorem 4

a) Consider the case whereis bounded. Sinceis uniformlyinjective in and is Lipschitz, there is a and a constant

such that

for all in .Let , and let denote the projection of on

as in the proof of Lemma 4. Further let and defineThen

This is so in case of by the definition ofprojection, and in case of because

Defining , we thus obtain

where is an appropriate constant, due to the boundedness of.On the other hand

and for


where , because is not locally thin. Together, this gives

where is the maximum distance of two points in, and thelast inequality follows because, are arbitrary. Letting

on both sides, it is found that all limits exist and

This proves for all , because the projec-tion of on equals , hence .

Letting and , we also have, which combine to . As a consequence,

and , which proves that .Finally, is continuous, because this is obviously so in

and ; and also in the transition as just shown.b) In case is not bounded, the extended inverse formula is

modified to

(29)

(30)


(31)

where are appropriate constants. The effect is thatthe weight is nonzero only on a bounded subset of, where

is sufficiently small. This is relevant to keep the integrals,which are taken over the set in the same way bounded.The proof is then as in case a).

J. Proof of Lemma 5

Let . Using the orthonormal series representation of(see Section II-E), and letting and , we have

(recall that , ; and

Hence, for each there is a such that. This extends to for , where

can be taken independent of, because andare continuous and is compact. Since has a finite cover ofsuch neighborhoods , the largest of the correspondingcan be taken as an , which gives for all

.Using observability, this gives

hence

Taking proves the Lemma.

K. Proof of Theorem 6

a) Suppose that is not locally thin. Let the observation map-ping be of accuracy . Then

for all .For an arbitrary , we obtain

where is an appropriate constant, and the fact thatis notlocally thin is relevant.

Further

Therefore

where is the maximum distance of two points in.To satisfy the theorem, we choose , such that

. This gives , which proves part ii) of thetheorem.

Since is uniformly continuous on any bounded subsetof , which contains in its interior, it follows that

implies and, hence,, which proves part i) of the theorem.

a) Suppose is locally thin.


Let be of accuracy as before, and define

the closure of

where is chosen such that

Such a constant exists, because and are continuousand , are compact. In addition, is not locally thin, andcase a) applies with replaced by .

REFERENCES

[1] H. Nijmeijer and T. I. Fossen, Eds.,New Directions in Nonlinear Ob-server Design. New York: Springer-Verlag, 1999.

[2] N. Kazantzis and C. Kravaris, “Nonlinear observer design using Lya-punov’s auxiliary theorem,”Syst. Control Lett., vol. 34, pp. 241–247,1998.

[3] G. Ciccarella, M. D. Mora, and A. Germani, “A luenberger-like observerfor nonlinear systems,”Int. J. Control, vol. 57, no. 3, pp. 537–556, 1993.

[4] J. P. Gauthier, H. Hammouri, and S. Othman, “A simple observer fornonlinear systems, application to bioreactors,”IEEE Trans. Automat.Contr., vol. 37, pp. 875–880, June 1992.

[5] M. Zeitz, “The extended luenberger observer for nonlinear systems,”Syst. Control Lett., vol. 9, pp. 149–156, 1987.

[6] D. Bestle and M. Zeitz, “Canonical form observer design for nonlineartime-variable systems,”Int. J. Control, vol. 38, no. 2, pp. 419–431, 1983.

[7] A. J. Krener and A. Isidori, “Linearization by output injection and non-linear observers,”Syst. Control Lett., vol. 3, pp. 47–52, 1983.

[8] H. Michalska and D. Q. Mayne, “Moving horizon observers andobserver-based control,”IEEE Trans. Automat. Contr., vol. 40, pp.995–1006, June 1995.

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[10] P. E. Moraal and J. W. Grizzle, “Observer design for nonlinear systemswith discrete-time measurements,”IEEE Trans. Automat. Contr., vol.40, pp. 395–404, Mar. 1995.

[11] A. Linnemann, “Convergent Ritz approximations of the set of stabilizingcontrollers,”Syst. Control Lett., vol. 36, pp. 151–156, 1999.

[12] R. Engel, “Observers for nonlinear systems,” Ph.D. dissertation (inGerman), Univ. of Kassel, Kassel, Germany, 2002.

[13] E. Hewitt and K. Stromberg,Real and Abstract Analysis. New York:Springer-Verlag, 1965, pp. 386–386.

Gerhard Kreisselmeierwas born in Hamburg, Ger-many, in 1943. He received the Dipl. Ing. degree fromthe Technical University of Hannover, Germany,and the Dr. Ing. degree from the Ruhr-University,Bochum, Germany, both in electrical engineering, in1968 and 1972, respectively.

From 1968 to 1970, he was with SiemensCompany, Erlangen, Germany, in the field ofprocess control, and from 1970 to 1985, with theDFVLR-Jnstitute fcir Flight Systems Dynamics,Oberpfaffenhofen, Germany, where he did research

in control with aerospace applications. Since 1985, he has been a Professor ofControl and Systems Theory in the Department of Electrical Engineering, theUniversity of Kassel, Kassel, Germany.

Robert Engel was born in Friedrichshafen, Ger-many, in 1972. He received the Dipl. Ing. degreefrom the University of Ulm, Ulm, Germany, andthe Dr. Ing. degree from the University of Kassel,Kassel, Germany, both in electrical engineering, in1996 and 2002, respectively.

Since then, he has been a Research Associate in theDepartment of Electrical Engineering, the Universityof Kassel. His current research interests include con-trol and state observation of nonlinear systems.

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