Nonlinear observers for autonomous Lipschitz continuous systems ...
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Nonlinear Observers for Autonomous Lipschitz Continuous Systems Gerhard Kreisselmeier and Robert Engel
Abstract—This paper considers the state observation problem for autonomous nonlinear systems. An observation mapping is introduced, which is defined by applying a linear integral operator (rather than a differential operator) to the output of the system. It is shown that this observation mapping is well suited to capture the observability nature of smooth as well as nonsmooth systems, and to construct observers of a remarkably simple structure: A linear state variable filter followed by a nonlinearity. The observer is established in Sections III–V by showing that observability and finite complexity of the system are sufficient conditions for the observer to exist, and by giving an explicit expression for its nonlinearity. It is demonstrated that the existence conditions are satisfied, and hence our results include a new observer which is not high-gain, for the wide class of smooth systems considered recently in a previous paper by Gauthier, et al. In Section VI, it is shown that the observer can as well be designed to realize an arbitrary, finite accuracy rather than ultimate exactness. On a compact region of the state space, this requires only observability of the system. A corresponding numerical design procedure is described, which is easy to implement and computationally feasible for low order systems. Index Terms—Nonlinear state observer, nonlinear systems, nonsmooth systems.
I. INTRODUCTION
T
HE leading results in nonlinear observer theory are for classes of systems, which have a certain degree of smoothness, i.e., are differentiable a sufficient number of times. Nonlinear coordinate changes, normal forms, output injection, embedding, and high gain techniques are some of the most notable concepts. See, e.g., [1]–[7] for discussions of the various approaches and further references. The importance of smoothness throughout all this work is not surprising, because many of the ideas, which are involved, have a linear systems background, linear systems are infinitely smooth, and smooth analysis is a most powerful mathematical tool. In comparison, very little is known about observers for nonsmooth systems (i.e., systems, which are Lipschitz continuous, but not differentiable everywhere), although nonsmooth systems also occur frequently in practice. A major reason is that the observation problem is in a significant way different. For example, a linear oscillator with a one-sided sensor if positive otherwise
Manuscript received January 18, 2002; revised June 25, 2002. Recommended by Associate Editor Z. Lin. The authors are with the Department of Electrical Engineering, the University of Kassel, D-34109 Kassel, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2002.808468
Fig. 1. Output signal y .
is a nonsmooth system, which produces output measurements of the form in Fig. 1. In order to reconstruct the state vector from measurements it would, for example, not be sufficient to know the output on an interval, where the output is identically zero. However, it would be sufficient to know the output on a sufficiently large interval, for any , because this always includes e.g., is strictly positive and, hence, the a subinterval, on which system is linear and observable. The point in this example is a phenomenon, that appears to occur more generally in nonsmooth systems: The state information is contained in the output signal, but is in a significant way unequally distributed in time. It would appear unlikely, that local methods are useful in such cases. Looking for possible nonlocal concepts, full state information becomes a main point of concern, and this leads very naturally toward thinking about a more extensive 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 , and to generate a state estimate so as (sliding) interval to asymptotically match the predicted output with the measured one on the whole interval. Thereby the observation problem is converted into the problem of (asymptotic) online minimization of an error criterion [8], [9], or solving a set of nonlinear equations [10], respectively. Algorithms to do this are the main issue of the approach. Under nonsmoothness and/or a lack of global convexity this is again a tough problem. Recent suggestions in [9] and [10] are gradient descent based and give local convergence, assuming some smoothness of the system to be observed. We note that the structure of this kind of observer comprises a data storage part, which is distributed parameter system type for continuous time measurements (this can be avoided by using sampled measurements only at the expense that a major assumption, similar to the finite complexity property in this paper, has to be imposed in addition to observability on the continuous time system), and a nonlinear dynamics part, which is created by the
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asymptotic nature of the minimization/equation solving algorithm. This paper takes a different, more system theoretically oriented perspective of the subject. We pursue the idea of an observation mapping and its analysis as being central to the observation problem and the starting point for constructing observers. To include nonsmooth systems, a special observation mapping is introduced, which uses the complete output history of the system, is detectable from measurements and, under conditions explored in detail, captures the observability nature of the system in a mapping of finite dimension. Specifically, instead of using successive derivatives of the output, we use successive integrals of the output. The integral operations are taken up to a sufficient order, are taken over the complete output history, and are equipped with stable kernels for well definedness. This results in a nonlinear observer of a remarkably simple structure: A linear state variable filter followed by a nonlinearity, the nonlinearity being an extended inverse of the observation mapping. The inverse is given in an explicit form. In what follows, the basic concept is introduced, explored theoretically in some detail, related to known results for smooth systems, and is then extended to include finite time convergent and arbitrarily accurate observers as well as an easy numerical design. II. THE OBSERVATION CONCEPT A. Problem Statement We consider nonlinear systems of the form (1) (2) and output . Throughout this paper, with state and are the basic assumption is that globally Lipschitz continuous. By the Lipschitz property, has a uniquely defined state, which passes through the point trajectory for each and . Moreover, the stateare exponentially trajectory and the associated output bounded. The problem is to asymptotically observe the unknown state of the system using only present and past output measurements. The focus will be on some closed, but not necessarily bounded, , which is invariant under , i.e., trajectories which set start in remain in for all future times.
Fig. 2.
Image of q (x).
system, a point . Since is by definition an integral of past outputs, it is well suited for computation from measurements. In particular, along each system trajectory we have that satisfies the equation . Therefore, the current value of can be generated asymptotically from a model of this equation. This suggests the following observer structure: (4) (5) converges to Its inherent feature is that exponentially as . The state estimate is then using a nonlinear mapping , formed from and is a so-called which in an ideal case satisfies . extended inverse of The idea is best illustrated by an example. Example 1 1 : when otherwise This system is first order, not differentiable at we have
. Taking
,
, and the
observation mapping becomes
Its image is shown in Fig. 2. Since is injective, an extended inverse mapping exists, and can be obtained by taking the intersection of the , , with the image of , line and thereby to find . It is not hard to verify that the resulting observer
B. Basic Idea Let us define an observation mapping (3) is a controllable pair, , where ( th-order for short), and sufficiently stable such that the integral is well defined. assigns to each state , The mapping , of the via the corresponding output history
is globally exponentially stable. The extended inverse is even linear in this case, by surprise. The above idea gives an observer more generally, according to the following theorem. 1A. J. Krener, Presentation at a control theory meeting, sponsored by the Mathematical Research Institute, Oberwolfach, Germany, 1999.
KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS
Theorem 1: Suppose is sufficiently stable, so that is well defined. If is injective; i) satisfies ii) , ; a) 2, ; b) then (4) and (5) represent an observer for in . Proof: Consider any pair of initial conditions From the definitions of and , it is evident that satisfies
where i) ii) for all
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is chosen such that3
; ; . Occasionally, we also write and , respectively, to make the dependence on time more explicit. An observable system is basically unterstood here as a system uniquely determines its present whose output time history state. The formal definition of observability is as follows. is said to be observable in , if there is a Definition: such that
(6)
(7)
, then and, hence, . If , If as , and then follows. It remains to satisfy the assumptions of Theorem 1. This is done in two steps. The first step (Section III) shows that can be made uniformly injective, provided is observable and has a further property called finite complexity. The second step (Section IV) shows that a uniformly injective is also sufficient for the existence and an explicit construction of .
. for all Observability thus allows to conclude smallness of from smallness of uniformly in . If is compact, for then observability in this sense only requires .4 all Observability in finite time is defined accordingly, with replaced by . E. Choice of Recall that the observation mapping is given by
C. Notations The following notations are used. (Euclidean) norm of a vector, and induced norm of 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 , short . For vectors of notation functions, denotes a matrix with entries . Set of continuous, strictly monotone increasing , satisfying . functions if is in class . We write 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 system , . output, seen backward in time, is Throughout this paper, we consider the exponentially weighted version of this output
(8) is to be chosen. For convewhere the observer filter pair of different dimension nience, we preselect a set of pairs as follows. be chosen as a sequence of real or complex numLet bers (with complex numbers appearing in complex conjugate pairs) 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, which is an orthonormal basis quence of functions and has the following property. For each in there is exactly one real, th-order pair with spectrum such that .. .
(10)
and its variations
z
notation means that any sequence (z ; x ) 2 2 G which satisfies 0 q(x ) ! 0, implies Q(z ) 0 x ! 0 as k ! 1.
2This
3Any value of greater than the Lipschitz constant of f (x) is appropriate. From design considerations, smaller values would be preferred if possible. 4Note that y (t; x) is continuous in x. A suitable ' K would be '(s) := (s=s) min y(x) y(x ) where s := max x x .
1
k
0
k
2
j 0 j
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An explicit formula to obtain from is given in [11]. The associated th-order observation mapping can then be rewritten in the form
Hence,
is a linear combination of , which are in dent. Therefore,
and and linearly indepenand
(11) As a result of this preselection, only the choice of dimension remains to be discussed in the later sections. is in and, thereIt is worth pointing out that fore, has an orthonormal series representation
which implies finite complexity. Example 2: A linear system , . It has output variation typically considered in
is
(12) . Hence, the obwhere the coefficients are of order represents the first coefservation mapping . ficients of the orthonormal expansion of Finally, we remark that orthonormal coordinates are used here only to simplify the subsequent analysis. To actually build the observer, it is sufficient to set up the observer filter pair conand with spectrum trollable, of suitable dimension , because this only amounts to a similarity transand the corresponding linear transformation formation of . of III. UNIFORMLY INJECTIVE OBSERVATION MAPPINGS It is shown in this section that the observation mapping can be made uniformly injective, just by taking its dimension sufficiently large, provided the system is observable and of finite complexity. The property of finite complexity is introduced and explored by giving suitable conditions. Definition: The observation mapping is said to be injective of uniform measure in (uniformly injective, for short), if there such that is a (13) . for all Note that if is compact then injectivity implies uniform injectivity, because is continuous. A. Finite Complexity with reWhile observability characterizes the variations , the property of finite spect to the distance of states complexity characterizes them as functions of time . is said to be of finite complexity in , if Definition: there exists a finite number of piecewise continuous func, combined to a vector tions , such that for some (14) . for all Example 1 (Continued): The first-order system of example 1 has weighted output otherwise
Condensing the linearly independent entries of into a vector , this can be rewritten in the form . Finite complexity then follows by the same argument as in the previous example. As a result, a , whether or linear system is always of finite complexity in is observable. not the pair The finite complexity property enables the following basic result. 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. is uniformly injective for some dimension Note that if , 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 two Lemmata. They indicate that this property is not overly restrictive. On the contrary, it appears to be essential for well conditionedness of the observation problem. Lemma 1: For to be of finite complexity in , it is necessuch that sary that there are constants , , ; i) ii)
;
, where denotes the Fourier for all . transform of A proof is given in Appendix B. Finite complexity of a system thus implies that its output varihave a low frequency portion and a finite time inations terval portion, each of which is of the same order of magnitude as the variation itself. Thereby, the variations are sufficiently well conditioned to be captured from real world measurements. This is further reflected in the following necessary and sufficient condition. is of finite complexity in , if and only if there Lemma 2: such that for each the exist , , relation (15) . holds for at least one A proof is given in Appendix C.
KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS
Lemma 2 combines the low frequency and finite time interval , aspects. The quantity of interest is through a first order low pass which is obtained by passing (or, equivalently, by filter with transfer function through a filter to obtain passing and then . As a result, the conditon is a simultaof Lemma 2 can be interpreted in the sense that . The relative neous low frequency/finite time portion from size of this portion, the bandwidth and the time horizon can take any nontrivial values. To give an idea of the nonsmoothness, which can be dealt with, we give a simple example of a nonlinear system, which is observable and of finite complexity and, therefore, has an observer of the proposed kind. Example 3: Consider a linear oscillator with a nonlinear output
Fig. 3. Observation mapping q (x),
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j j = 1 in the ( x
q ;q
) plane.
if positive otherwise where denotes the first state variable and . The measured output is simply the positive branch of the sinusoid . Fig. 1, which has been presented in the introduction section, illustrates this kind of nonsmoothness. This system is obviously observable. Moreover, it has a finite complexity, because it satisfies the condition of Lemma 2. The main point in showing this (details are omitted for the sake of brevity) is the following. There is always an interval on which does not change in case of sign and is a sinusoid of amplitude and of amplitude in case , respectively, where denotes of the 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 describes a circle in , and therefore that moves on a closed path in , which is the periodic solution . The calculation of is complete of due to with one such solution and the fact that for . Based on an a priori choice of an eigenvalue sequence , a natural initial guess of an observer (which is a lower bound for dimension is 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 closed ). A natural contour (which is due to the mean value of . The pair is now second guess to try is taken to be
(a)
(b) Fig. 4. (a) Measured output y and observed state x ^ x ) , i = 1; 2 .
(^ x
0
. (b) Observation errors
i.e., with eigenvalues according to the a priori choice, and in . convenient coordinates such that and . The This gives zero mean value for versus , which is illustrated in Fig. 3, gives plot of is injective. The observers (4) and (5) evidence that this from above, and with are finally implemented with realized by the extended inverse formula (28), which is given in in (b1) and in (28). Section VI-B, using Fig. 4(a) and (b) illustrate the observation process for initial , . The convergence is roughly conditions corresponding to the slowest eigenvalue of . C. Conditions for Finite Complexity: Mildly Smooth Systems This subsection gives extended conditions for finite complexity, taking advantage of some mild smoothness. The is globally Lipschitz. standing assumption is again that
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Assumptions regarding smoothness will be stated individually in each result. We start with a general condition on finite complexity, which is formulated in terms of the weighted output and the weighted , output derivative . The notations are and , respectively. exists and is continuous, and if there Lemma 3: If such that are , , ; i) ; ii) , then is of finite complexity in . for all 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 be simplified drastically. The subsequent theorem reduces it to a in the first assumption, Lipschitz condition on while the second assumption states first order observability in finite time. such that Theorem 3: If there are , is Lipschitz in the set i) ; , ; 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 applications, the subsequent Corollary can be useful. It points out that (under some mild extra assumptions) a system is first-order obits linearizaservable in finite time, if on some interval tions are observable. Corollary 1: If is compact; i) have continuous first order partial derivatives; ii) is observable on in ; iii) the linearized system , iv) for each , , is observable on ; , then there is a such that for some
where is globally Lipschitz, a so-called high gain observer was established in [4]. It turns out that Theorem 3 applies to this class of systems on . It is important here to cover all of , because the set this class includes systems which have unbounded solutions. By , the first hypothesis of Thethe Lipschitz assumption on orem 3 is obviously met. Moreover it is proved in Appendix G , that this class of systems is first order observable in and hence also satisfies the second hypothesis. is continuously differentiable and is In cases where compact, first-order observability may be concluded more easily by using Corollary 1. Here, the main point is to realize that has a special structure with , and the fact that this structure carries over to the linearized system. The linearized system is therefore observable on every non. This is in fact stronger than trivial interval required by Corollary 1 (which would allow unobservability on some subinterval) to conclude first-order observability of the nonlinear system in . As a result, Theorem 3 applies, i.e., each system in the given class is observable and of finite complexity in , and therefore also has an observer of the form proposed here, which is not high gain. IV. INVERSE MAPPING 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 was previously established, such an inverse exists and can be given in an explicit form. Lemma 4: If is uniformly injective in , then an extended inverse exists. The proof is given in Appendix H. To be constructive about an extended inverse, we introduce the explicit formula5 (16) is a weight, and is the volume increment in where . This formula defines as a weighted average of all . A suitable weight is
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 observability is always first order. However, Theorem 3 also includes the following situation from the literature as a special case. Example 4: For systems which are of the form (or can be embedded in this form) ..
.
.. .
(17) For a discrete (approximating) version of this formula, see Section VI-B. is not Theorem 4: If is uniformly injective in , and , as defined by locally thin6 , then a) (16) and (17), in case is bounded; b) (29)–(31), in case is not bounded; is continuous and an extended inverse of .
2 !
5For z q (G) the right-hand side is to be unterstood in the sense of the limit, obtained when replacing w (z; x) by w (z; x) := 1=[" + z q (x) ] and letting " 0. 6We say that G is not locally thin, if dX c dX for each
2
G and "
constants.
j 0
2
(0; " ], where G
:= U ( )
1 \ G and c; "
j
are positive
KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS
Fig. 5.
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Finite-time observer structure.
A proof is given in Appendix I. The idea behind this construction of is as follows. The state observation error of the nonlinear observer becomes and can be written in the form
(a)
(18) as . Thereby, the From (6), we have , seen as a function normalized weight . Convergence of , tends to a -distribution at then follows from (18). The proof of Theorem 4 makes this intuitive idea rigorous. V. FINITE-TIME OBSERVERS Let be an observation mapping, which is defined using (rather than ) a finite time interval (19) with , it is seen that along any trajectory of the system
Comparing
(20) This motivates an observer of the form (21) (22) (23) and with initial conditions , . The structure is again open loop and includes now a delay, as is illustrated in Fig. 5. Theorem 5: is injective; i) satisfies for all ; ii) then (21)–(23) are an observer for , whose state estimate converges to the true state in finite time , i.e., for . Proof: As before, along any trajectory of the system we have defined for
Rewriting
(b) Fig. 6. (a) Measured output y and observed state x ^ x ) , i = 1, 2.
(^ x
0
. (b) Observation errors
For initial conditions , the same argument gives for all . Note that the assumptions made in Theorem 5 are in fact weaker than those in Theorem 1. In particular, any kind of an extended inverse is now suitable, because convergence occurs in finite time. The latter is a structural property of this observer. and are the same We also note that injectivity of problems on different horizons. Therefore, with observability and with and finite complexity redefined on the interval replaced by accordingly, the results of Section III carry over to the finite-time case. In particular, observability and is uniformly infinite complexity together guarantee that . The extended inverse of Section IV jective, for some may 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 , it is also obSince has periodic solutions with period . Using servable and of finite complexity on the interval the same observer dynamics as before and the fact that is periodic, we find that
as (24)
and, thereby, it follows that for all , i.e., finite-time convergence regardless of the initial conditions.
Hence, is also injective, and its extended inverse can be from the extended obtained as . The resulting observer is simulated with inverse of as above, and . Fig. 6 illustrates the finite time convergence, which is in contrast to the asymptotic convergence in Fig. 4.
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VI. ARBITRARILY ACCURATE OBSERVERS Under real world conditions, an observer can only give an approximation of what it is wanted to give theoretically. Therefore, an observer may as well be designed for an arbitrary finite accuracy, rather than for ultimate exactness. We develop this concept for practical situations, where the trajectories of the system can be taken compact. The result are are bounded, so that observers of any desired accuracy, which require only observability of the system and are, therefore, of widest applicability. A. Arbitrary Finite Accuracy Recall from Section II-E, that our observation mapping represents the first coefficients of an orthonormal expansion of . Therefore, the information for a prescribed accuracy is suffinaturally contained in this representation, if we take ciently large. Together with observability, this gives rise to the following Lemma. Lemma 5: If is observable in , and is compact, then there exist , for each desired accuracy such that (25) . where A proof is given in Appendix J. With an accuracy in the sense of (25), the set of all points in giving the same value of , is located strictly in the interior of a ball of radius in . Therefore, can be recovered from a given , up to an error of size . As a suitable mapping, which does such an inversion approximately, we consider the expression (26) with
Note that the results of this section and the previous one, respectively, combine nicely to an observer, which attains any desired accuracy in finite time. B. Numerical Design Consider any Lipschitz continuous system (not necessariliy observable or of finite complexity), with a compact design re. Based on an a priori choice of a sequence gion as described in Section II-E, the following steps toward an observer can always be taken. gives (see Section II-E) a) The choice of an integer , which defines the observer dyan th-order pair . namics b) The choice of some (uniformly continuous) nonlinearity defines the observed state . The result of a) and b) is an observer of accuracy , i.e., an observer which satisfies properties i) and ii) of Theorem 6, where8
An observer design, thus, becomes a selection of and , so as to make as small as desired. By Theorem 6 we know that if is observable in , then such a 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) ratio of sums, is then also an appropriate nonlinearity. It can be used as a choice of in step b), and can be designed numerically as follows. sufficiently dense and b1) Select a set of points (e.g., , properly distributed in is the discretization density). where via integrating (1) backward in time b2) Compute and evaluating (3). b3) Define
(27) (28) where is a positive constant, to be chosen in combination with (respectively, ) so as to achieve a joint accuracy of the resulting observer. Theorem 6: Let be observable in , and let be compact. there exist and , such that Then for each (4), (5), (26), (27) 7 are an observer of accuracy in , i.e., as for all i) ; , for all ii) . A proof is given in Appendix K. The result of Theorem 6 is very general and widely applicable. It says that for an observable system on a compact set, there is always an arbitrarily accurate observer, and the proposed observer structure is an easy realization, where one needs only take the dimension sufficiently large and the constant sufficiently small.
Based on these, an observer design can now proceed iteratively (and without the need to have checked observability of beforehand) as follows. Each iteration involves going through steps a) and b1)–b3), and delivers an observer with accuracy as a candidate solution. A convenient estimate of this accuracy can be obtained from the computed . data as is increased In each iteration the observer dimension ) to further improve the accuracy, (starting from some sufficiently small, until the desired while keeping and accuracy is attained. By Theorem 6, this will succeed within is observable. In case the a finite number of iterations, if and desired accuracy is not attained for reasonably large this indicates that may be not observable. small Note that the main computations are in step b2), and that these are offline computations. The nonlinearity (28) is to be imple-
7In case G 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).
8Recall from Section II-B that z(t) q(x (t)) as t that x ^ (t) x (t) = Q(z(t)) x (t) [0; ].
j
0
j j
0
!
j!
! 1, which implies
KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS
mented as part of the final observer. Its evaluation to determine the observer output are online computations.9
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APPENDIX A. Proof of Theorem 2
VII. CONCLUDING REMARKS An integral operator based observation mapping is introduced, which captures the observability nature of autonomous nonlinear systems, whether or not they are smooth. What makes this mapping constructive and well suited for theoretical analyzes is that an th-order observation mapping represents an th-order orthonormal series expansion of the complete output signal history. The observer has a particularly simple open loop structure, which divides the observation process into two subsequent stages. In the first stage, linear observer dynamics generate a convergent estimate of the observation mapping value that corresponds to the current output signal history. That is, observation takes place here completely in the output function space, the latter being respresented implicitly by its finite. Accordingly, this dimensional vector space equivalent stage is coordinate free. The subsequent second stage merely maps the estimate into the state coordinates of the system, via an extended inverse of the observation mapping. An explicit inversion formula is presented. In the given observer structure, it only remains to choose the observer dynamics with (almost) arbitrary eigenvalues and a sufficiently large dimension to complete the observer. By construction, its convergence is global. The main assumptions to be made are observability and finite complexity. Finite complexity is introduced and characterized by a necessary and sufficient condition. Roughly, this property is well conditionedness of the output variations in a finite time/ finite bandwidth sense. While these conditions are difficult to check in general, it is demonstrated that they are satisfied and, hence, our results include a new observer which is not high gain, for the wide class of smooth systems considered recently in [4]. The proposed observer structure also works nicely in cases where the system is only observable. Applied with any dimension , it results in a finite accuracy observer, which is is. A computerized numerical the more accurate the larger design of such observers is presented, which is easy to implement, and which is computationally feasible for low order systems. Finally, we point out that our single-output results readily carry over to the multiple-output case with an observation mapping, which comprises the individual observation mappings of each output. Extensions to nonautonomous systems (i.e., systems with inputs) are nontrivial. Results in this direction have been obtained in [12] and are currently under preparation for publication.
Q
9Any alternate form of (e.g., based on neural nets, fuzzy logic, or interpolation/approximation techniques) may be used to optimize its implementations and/or evaluations.
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 integer
is square summable. Therefore, an exists such that
Letting we can rewrite
where
and
,
. 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 , . Then
, i.e., (14) holds for some
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 be nonzero only on a (sufficiently large) finite interval. On the , because is piecewise latter, we have continuous by assumption.
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With , the Fourier transforms of and (taken for the signals as defined on and continued iden, respectively, and by Parseval’s Theorem tically zero on
Suppose now that the condition of the Lemma holds, i.e., for some we have . Then
and with
Let we have
it follows that
denote the integer such that
Further define a vector
. Then,
with components when otherwise
This gives, using (14)
Taking
sufficiently large, this implies part ii) of the Lemma.
C. Proof of Lemma 2 a) Sufficiency: Let
and let
for
where is the smallest integer greater than Then, it follows that
and finite complexity is immediate from its definition. This proves sufficiency. is of finite complexity, i.e., (14) b) Necessity: Suppose , . Since , there is a holds for some such that . Thereby
otherwise denote its Fourier transform. Then, for Let
and define
where
otherwise
The order of integrals can be switched due to Fubini’s Theorem [13]. Then, for
Further rewrite as , where is chosen . Since is piecewise continuous, can so that Then, it be taken sufficiently large, so that follows that
and
As a consequence, there is an integer ) such that, with the notation pends on have
where is a constant resulting from the integral, which is bounded.
(which de, we
KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS
i.e., the condition of the Lemma is implied. This proves necessity.
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F. Proof of Corollary 1 For any
, we can write
D. Proof of Lemma 3 and Since tion ii) implies
are related by
, assump-
where is an appropriate constant. For all this gives
By assumption i), there is a
,
such that
where denotes the state transition matrix which is associated with the linearization of along the trajectory passing is a remainder term, which is dethrough the point , and , and by fined by the above relation in case for all . , such that for all By assumptions i) and iv), there is a
Moreover, is continuous in its arguments on the com, and satisfies for all pact set . Therefore, a constant exists such that implies . Hence
hence By assumption iii), we also have Combining the two gives Let
be chosen such that and Then does not change sign for , and there is an interval of length such that . Integration thus gives
where
is the maximum distance of two points in and . This completes the proof.
G. Proof of First Order Observability of Example 4 and finite complexity follows from Lemma 2. E. Proof of Theorem 3 is observable by assumption ii). We prove finite complexity by showing that under the assumptions of the Theorem, Lemma 3 applies. Since is Lipschitz, we have for some . Combining this with ii) it follows that the first condition of Lemma 3 is satisfied. and let denote its Lipschitz Let constant. Then
where the right-hand side is bounded proportional to on , because is Lipschitz. Taking norms and using ii) it , that follows, with an appropriate constant
i.e., the second condition of Lemma 3 is satisfied as well. This completes the proof.
, and consider any . Since is Let times continuously differentiable with respect to time due to the special structure of , application of Taylor’s formula gives
where
and
for some we have
With notations
we can write
. Due to the special structure of
,
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For any choice of
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Define, for example
, this gives
if if
for some , where the last inequality follows because the smallest eigenvalue of
..
.
where denotes the projection of on (i.e., is which minimizes the distance ), which some point in is made unique by choice where necessary. , holds by definition. Let Then be varied such that Then, by the . Combining the two gives projection property, . Due to uniform injectivity this implies . Since is either or , it follows that . I. Proof of Theorem 4
.. .
.. .
..
..
.
a) Consider the case where is bounded. Since is uniformly and a constant injective in and is Lipschitz, there is a such that
.. .
.
is bounded below proportional to . Also, due to the special structure of , we have
where and denotes the Lipschitz constant of . this can be used, in the definition For any choice of , to conclude that of
This is so in case of projection, and in case of
Defining
for some constant . Combining the two arguments, it follows that
Taking sufficiently small, this gives a ) such that dependent of first-order observability of in .
for all in . , and let denote the projection of on Let as in the proof of Lemma 4. Further let and define Then
(which is in. This proves
, we thus obtain
where is an appropriate constant, due to the boundedness of . On the other hand
H. Proof of Lemma 4 is closed. To see this, let be a We note that such that as . By uniform sequence in injectivity we have and for from which we can conclude that is a Cauchy sequence. converges to some . Since is closed, is in . Hence, follows. By continuity
by the definition of because
KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS
where
, because
is not locally thin. Together, this gives
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can be taken independent of , because and are continuous and is compact. Since has a finite cover of , the largest of the corresponding such neighborhoods can be taken as an , which gives for all . Using observability, this gives
hence
Taking
proves the Lemma.
K. Proof of Theorem 6 where is the maximum distance of two points in , and the are arbitrary. Letting last inequality follows because , on both sides, it is found that all limits exist and
a) Suppose that is not locally thin. Let the observation map. Then ping be of accuracy
for all . For an arbitrary This proves for all , because the projecon equals , hence . tion of and , we also have Letting , which combine to . As a consequence, and , which proves that . is continuous, because this is obviously so in Finally, and ; and also in the transition as just shown. b) In case is not bounded, the extended inverse formula is modified to
, we obtain
where is an appropriate constant, and the fact that locally thin is relevant. Further
is not
(29) (30) if positive otherwise
(31) Therefore
are appropriate constants. The effect is that where the weight is nonzero only on a bounded subset of , where is sufficiently small. This is relevant to keep the integrals, in the same way bounded. which are taken over the set The proof is then as in case a). J. Proof of Lemma 5 . Using the orthonormal series representation of Let (see Section II-E), and letting and , we have , ; and (recall that
Hence, for each . This extends to
there is a
such that for
, where
where is the maximum distance of two points in . , such that To satisfy the theorem, we choose . This gives , which proves part ii) of the theorem. is uniformly continuous on any bounded subset Since , which contains in its interior, it follows that of implies and, hence, , which proves part i) of the theorem. a) Suppose is locally thin.
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Let
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be of accuracy
as before, and define the closure of
where
is chosen such that
Such a constant exists, because and are continuous are compact. In addition, is not locally thin, and and , case a) applies with replaced by . REFERENCES [1] H. Nijmeijer and T. I. Fossen, Eds., New Directions in Nonlinear Observer Design. New York: Springer-Verlag, 1999. [2] N. Kazantzis and C. Kravaris, “Nonlinear observer design using Lyapunov’s auxiliary theorem,” Syst. Control Lett., vol. 34, pp. 241–247, 1998. [3] G. Ciccarella, M. D. Mora, and A. Germani, “A luenberger-like observer for 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 for nonlinear 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 nonlinear time-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 nonlinear observers,” Syst. Control Lett., vol. 3, pp. 47–52, 1983. [8] H. Michalska and D. Q. Mayne, “Moving horizon observers and observer-based control,” IEEE Trans. Automat. Contr., vol. 40, pp. 995–1006, June 1995. [9] M. Alamir, “Optimization based nonlinear observers revisited,” Int. J. Control, vol. 72, pp. 1204–1217, 1999.
[10] P. E. Moraal and J. W. Grizzle, “Observer design for nonlinear systems with discrete-time measurements,” IEEE Trans. Automat. Contr., vol. 40, pp. 395–404, Mar. 1995. [11] A. Linnemann, “Convergent Ritz approximations of the set of stabilizing controllers,” Syst. Control Lett., vol. 36, pp. 151–156, 1999. [12] R. Engel, “Observers for nonlinear systems,” Ph.D. dissertation (in German), 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 Kreisselmeier was born in Hamburg, Germany, in 1943. He received the Dipl. Ing. degree from the Technical University of Hannover, Germany, and the Dr. Ing. degree from the Ruhr-University, Bochum, Germany, both in electrical engineering, in 1968 and 1972, respectively. From 1968 to 1970, he was with Siemens Company, Erlangen, Germany, in the field of process control, and from 1970 to 1985, with the DFVLR-Jnstitute fcir Flight Systems Dynamics, Oberpfaffenhofen, Germany, where he did research in control with aerospace applications. Since 1985, he has been a Professor of Control and Systems Theory in the Department of Electrical Engineering, the University of Kassel, Kassel, Germany.
Robert Engel was born in Friedrichshafen, Germany, in 1972. He received the Dipl. Ing. degree from the University of Ulm, Ulm, Germany, and the Dr. Ing. degree from the University of Kassel, Kassel, Germany, both in electrical engineering, in 1996 and 2002, respectively. Since then, he has been a Research Associate in the Department of Electrical Engineering, the University of Kassel. His current research interests include control and state observation of nonlinear systems.
451
Nonlinear Observers for Autonomous Lipschitz Continuous Systems Gerhard Kreisselmeier and Robert Engel
Abstract—This paper considers the state observation problem for autonomous nonlinear systems. An observation mapping is introduced, which is defined by applying a linear integral operator (rather than a differential operator) to the output of the system. It is shown that this observation mapping is well suited to capture the observability nature of smooth as well as nonsmooth systems, and to construct observers of a remarkably simple structure: A linear state variable filter followed by a nonlinearity. The observer is established in Sections III–V by showing that observability and finite complexity of the system are sufficient conditions for the observer to exist, and by giving an explicit expression for its nonlinearity. It is demonstrated that the existence conditions are satisfied, and hence our results include a new observer which is not high-gain, for the wide class of smooth systems considered recently in a previous paper by Gauthier, et al. In Section VI, it is shown that the observer can as well be designed to realize an arbitrary, finite accuracy rather than ultimate exactness. On a compact region of the state space, this requires only observability of the system. A corresponding numerical design procedure is described, which is easy to implement and computationally feasible for low order systems. Index Terms—Nonlinear state observer, nonlinear systems, nonsmooth systems.
I. INTRODUCTION
T
HE leading results in nonlinear observer theory are for classes of systems, which have a certain degree of smoothness, i.e., are differentiable a sufficient number of times. Nonlinear coordinate changes, normal forms, output injection, embedding, and high gain techniques are some of the most notable concepts. See, e.g., [1]–[7] for discussions of the various approaches and further references. The importance of smoothness throughout all this work is not surprising, because many of the ideas, which are involved, have a linear systems background, linear systems are infinitely smooth, and smooth analysis is a most powerful mathematical tool. In comparison, very little is known about observers for nonsmooth systems (i.e., systems, which are Lipschitz continuous, but not differentiable everywhere), although nonsmooth systems also occur frequently in practice. A major reason is that the observation problem is in a significant way different. For example, a linear oscillator with a one-sided sensor if positive otherwise
Manuscript received January 18, 2002; revised June 25, 2002. Recommended by Associate Editor Z. Lin. The authors are with the Department of Electrical Engineering, the University of Kassel, D-34109 Kassel, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2002.808468
Fig. 1. Output signal y .
is a nonsmooth system, which produces output measurements of the form in Fig. 1. In order to reconstruct the state vector from measurements it would, for example, not be sufficient to know the output on an interval, where the output is identically zero. However, it would be sufficient to know the output on a sufficiently large interval, for any , because this always includes e.g., is strictly positive and, hence, the a subinterval, on which system is linear and observable. The point in this example is a phenomenon, that appears to occur more generally in nonsmooth systems: The state information is contained in the output signal, but is in a significant way unequally distributed in time. It would appear unlikely, that local methods are useful in such cases. Looking for possible nonlocal concepts, full state information becomes a main point of concern, and this leads very naturally toward thinking about a more extensive 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 , and to generate a state estimate so as (sliding) interval to asymptotically match the predicted output with the measured one on the whole interval. Thereby the observation problem is converted into the problem of (asymptotic) online minimization of an error criterion [8], [9], or solving a set of nonlinear equations [10], respectively. Algorithms to do this are the main issue of the approach. Under nonsmoothness and/or a lack of global convexity this is again a tough problem. Recent suggestions in [9] and [10] are gradient descent based and give local convergence, assuming some smoothness of the system to be observed. We note that the structure of this kind of observer comprises a data storage part, which is distributed parameter system type for continuous time measurements (this can be avoided by using sampled measurements only at the expense that a major assumption, similar to the finite complexity property in this paper, has to be imposed in addition to observability on the continuous time system), and a nonlinear dynamics part, which is created by the
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asymptotic nature of the minimization/equation solving algorithm. This paper takes a different, more system theoretically oriented perspective of the subject. We pursue the idea of an observation mapping and its analysis as being central to the observation problem and the starting point for constructing observers. To include nonsmooth systems, a special observation mapping is introduced, which uses the complete output history of the system, is detectable from measurements and, under conditions explored in detail, captures the observability nature of the system in a mapping of finite dimension. Specifically, instead of using successive derivatives of the output, we use successive integrals of the output. The integral operations are taken up to a sufficient order, are taken over the complete output history, and are equipped with stable kernels for well definedness. This results in a nonlinear observer of a remarkably simple structure: A linear state variable filter followed by a nonlinearity, the nonlinearity being an extended inverse of the observation mapping. The inverse is given in an explicit form. In what follows, the basic concept is introduced, explored theoretically in some detail, related to known results for smooth systems, and is then extended to include finite time convergent and arbitrarily accurate observers as well as an easy numerical design. II. THE OBSERVATION CONCEPT A. Problem Statement We consider nonlinear systems of the form (1) (2) and output . Throughout this paper, with state and are the basic assumption is that globally Lipschitz continuous. By the Lipschitz property, has a uniquely defined state, which passes through the point trajectory for each and . Moreover, the stateare exponentially trajectory and the associated output bounded. The problem is to asymptotically observe the unknown state of the system using only present and past output measurements. The focus will be on some closed, but not necessarily bounded, , which is invariant under , i.e., trajectories which set start in remain in for all future times.
Fig. 2.
Image of q (x).
system, a point . Since is by definition an integral of past outputs, it is well suited for computation from measurements. In particular, along each system trajectory we have that satisfies the equation . Therefore, the current value of can be generated asymptotically from a model of this equation. This suggests the following observer structure: (4) (5) converges to Its inherent feature is that exponentially as . The state estimate is then using a nonlinear mapping , formed from and is a so-called which in an ideal case satisfies . extended inverse of The idea is best illustrated by an example. Example 1 1 : when otherwise This system is first order, not differentiable at we have
. Taking
,
, and the
observation mapping becomes
Its image is shown in Fig. 2. Since is injective, an extended inverse mapping exists, and can be obtained by taking the intersection of the , , with the image of , line and thereby to find . It is not hard to verify that the resulting observer
B. Basic Idea Let us define an observation mapping (3) is a controllable pair, , where ( th-order for short), and sufficiently stable such that the integral is well defined. assigns to each state , The mapping , of the via the corresponding output history
is globally exponentially stable. The extended inverse is even linear in this case, by surprise. The above idea gives an observer more generally, according to the following theorem. 1A. J. Krener, Presentation at a control theory meeting, sponsored by the Mathematical Research Institute, Oberwolfach, Germany, 1999.
KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS
Theorem 1: Suppose is sufficiently stable, so that is well defined. If is injective; i) satisfies ii) , ; a) 2, ; b) then (4) and (5) represent an observer for in . Proof: Consider any pair of initial conditions From the definitions of and , it is evident that satisfies
where i) ii) for all
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is chosen such that3
; ; . Occasionally, we also write and , respectively, to make the dependence on time more explicit. An observable system is basically unterstood here as a system uniquely determines its present whose output time history state. The formal definition of observability is as follows. is said to be observable in , if there is a Definition: such that
(6)
(7)
, then and, hence, . If , If as , and then follows. It remains to satisfy the assumptions of Theorem 1. This is done in two steps. The first step (Section III) shows that can be made uniformly injective, provided is observable and has a further property called finite complexity. The second step (Section IV) shows that a uniformly injective is also sufficient for the existence and an explicit construction of .
. for all Observability thus allows to conclude smallness of from smallness of uniformly in . If is compact, for then observability in this sense only requires .4 all Observability in finite time is defined accordingly, with replaced by . E. Choice of Recall that the observation mapping is given by
C. Notations The following notations are used. (Euclidean) norm of a vector, and induced norm of 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 , short . For vectors of notation functions, denotes a matrix with entries . Set of continuous, strictly monotone increasing , satisfying . functions if is in class . We write 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 system , . output, seen backward in time, is Throughout this paper, we consider the exponentially weighted version of this output
(8) is to be chosen. For convewhere the observer filter pair of different dimension nience, we preselect a set of pairs as follows. be chosen as a sequence of real or complex numLet bers (with complex numbers appearing in complex conjugate pairs) 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, which is an orthonormal basis quence of functions and has the following property. For each in there is exactly one real, th-order pair with spectrum such that .. .
(10)
and its variations
z
notation means that any sequence (z ; x ) 2 2 G which satisfies 0 q(x ) ! 0, implies Q(z ) 0 x ! 0 as k ! 1.
2This
3Any value of greater than the Lipschitz constant of f (x) is appropriate. From design considerations, smaller values would be preferred if possible. 4Note that y (t; x) is continuous in x. A suitable ' K would be '(s) := (s=s) min y(x) y(x ) where s := max x x .
1
k
0
k
2
j 0 j
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An explicit formula to obtain from is given in [11]. The associated th-order observation mapping can then be rewritten in the form
Hence,
is a linear combination of , which are in dent. Therefore,
and and linearly indepenand
(11) As a result of this preselection, only the choice of dimension remains to be discussed in the later sections. is in and, thereIt is worth pointing out that fore, has an orthonormal series representation
which implies finite complexity. Example 2: A linear system , . It has output variation typically considered in
is
(12) . Hence, the obwhere the coefficients are of order represents the first coefservation mapping . ficients of the orthonormal expansion of Finally, we remark that orthonormal coordinates are used here only to simplify the subsequent analysis. To actually build the observer, it is sufficient to set up the observer filter pair conand with spectrum trollable, of suitable dimension , because this only amounts to a similarity transand the corresponding linear transformation formation of . of III. UNIFORMLY INJECTIVE OBSERVATION MAPPINGS It is shown in this section that the observation mapping can be made uniformly injective, just by taking its dimension sufficiently large, provided the system is observable and of finite complexity. The property of finite complexity is introduced and explored by giving suitable conditions. Definition: The observation mapping is said to be injective of uniform measure in (uniformly injective, for short), if there such that is a (13) . for all Note that if is compact then injectivity implies uniform injectivity, because is continuous. A. Finite Complexity with reWhile observability characterizes the variations , the property of finite spect to the distance of states complexity characterizes them as functions of time . is said to be of finite complexity in , if Definition: there exists a finite number of piecewise continuous func, combined to a vector tions , such that for some (14) . for all Example 1 (Continued): The first-order system of example 1 has weighted output otherwise
Condensing the linearly independent entries of into a vector , this can be rewritten in the form . Finite complexity then follows by the same argument as in the previous example. As a result, a , whether or linear system is always of finite complexity in is observable. not the pair The finite complexity property enables the following basic result. 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. is uniformly injective for some dimension Note that if , 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 two Lemmata. They indicate that this property is not overly restrictive. On the contrary, it appears to be essential for well conditionedness of the observation problem. Lemma 1: For to be of finite complexity in , it is necessuch that sary that there are constants , , ; i) ii)
;
, where denotes the Fourier for all . transform of A proof is given in Appendix B. Finite complexity of a system thus implies that its output varihave a low frequency portion and a finite time inations terval portion, each of which is of the same order of magnitude as the variation itself. Thereby, the variations are sufficiently well conditioned to be captured from real world measurements. This is further reflected in the following necessary and sufficient condition. is of finite complexity in , if and only if there Lemma 2: such that for each the exist , , relation (15) . holds for at least one A proof is given in Appendix C.
KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS
Lemma 2 combines the low frequency and finite time interval , aspects. The quantity of interest is through a first order low pass which is obtained by passing (or, equivalently, by filter with transfer function through a filter to obtain passing and then . As a result, the conditon is a simultaof Lemma 2 can be interpreted in the sense that . The relative neous low frequency/finite time portion from size of this portion, the bandwidth and the time horizon can take any nontrivial values. To give an idea of the nonsmoothness, which can be dealt with, we give a simple example of a nonlinear system, which is observable and of finite complexity and, therefore, has an observer of the proposed kind. Example 3: Consider a linear oscillator with a nonlinear output
Fig. 3. Observation mapping q (x),
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j j = 1 in the ( x
q ;q
) plane.
if positive otherwise where denotes the first state variable and . The measured output is simply the positive branch of the sinusoid . Fig. 1, which has been presented in the introduction section, illustrates this kind of nonsmoothness. This system is obviously observable. Moreover, it has a finite complexity, because it satisfies the condition of Lemma 2. The main point in showing this (details are omitted for the sake of brevity) is the following. There is always an interval on which does not change in case of sign and is a sinusoid of amplitude and of amplitude in case , respectively, where denotes of the 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 describes a circle in , and therefore that moves on a closed path in , which is the periodic solution . The calculation of is complete of due to with one such solution and the fact that for . Based on an a priori choice of an eigenvalue sequence , a natural initial guess of an observer (which is a lower bound for dimension is 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 closed ). A natural contour (which is due to the mean value of . The pair is now second guess to try is taken to be
(a)
(b) Fig. 4. (a) Measured output y and observed state x ^ x ) , i = 1; 2 .
(^ x
0
. (b) Observation errors
i.e., with eigenvalues according to the a priori choice, and in . convenient coordinates such that and . The This gives zero mean value for versus , which is illustrated in Fig. 3, gives plot of is injective. The observers (4) and (5) evidence that this from above, and with are finally implemented with realized by the extended inverse formula (28), which is given in in (b1) and in (28). Section VI-B, using Fig. 4(a) and (b) illustrate the observation process for initial , . The convergence is roughly conditions corresponding to the slowest eigenvalue of . C. Conditions for Finite Complexity: Mildly Smooth Systems This subsection gives extended conditions for finite complexity, taking advantage of some mild smoothness. The is globally Lipschitz. standing assumption is again that
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Assumptions regarding smoothness will be stated individually in each result. We start with a general condition on finite complexity, which is formulated in terms of the weighted output and the weighted , output derivative . The notations are and , respectively. exists and is continuous, and if there Lemma 3: If such that are , , ; i) ; ii) , then is of finite complexity in . for all 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 be simplified drastically. The subsequent theorem reduces it to a in the first assumption, Lipschitz condition on while the second assumption states first order observability in finite time. such that Theorem 3: If there are , is Lipschitz in the set i) ; , ; 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 applications, the subsequent Corollary can be useful. It points out that (under some mild extra assumptions) a system is first-order obits linearizaservable in finite time, if on some interval tions are observable. Corollary 1: If is compact; i) have continuous first order partial derivatives; ii) is observable on in ; iii) the linearized system , iv) for each , , is observable on ; , then there is a such that for some
where is globally Lipschitz, a so-called high gain observer was established in [4]. It turns out that Theorem 3 applies to this class of systems on . It is important here to cover all of , because the set this class includes systems which have unbounded solutions. By , the first hypothesis of Thethe Lipschitz assumption on orem 3 is obviously met. Moreover it is proved in Appendix G , that this class of systems is first order observable in and hence also satisfies the second hypothesis. is continuously differentiable and is In cases where compact, first-order observability may be concluded more easily by using Corollary 1. Here, the main point is to realize that has a special structure with , and the fact that this structure carries over to the linearized system. The linearized system is therefore observable on every non. This is in fact stronger than trivial interval required by Corollary 1 (which would allow unobservability on some subinterval) to conclude first-order observability of the nonlinear system in . As a result, Theorem 3 applies, i.e., each system in the given class is observable and of finite complexity in , and therefore also has an observer of the form proposed here, which is not high gain. IV. INVERSE MAPPING 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 was previously established, such an inverse exists and can be given in an explicit form. Lemma 4: If is uniformly injective in , then an extended inverse exists. The proof is given in Appendix H. To be constructive about an extended inverse, we introduce the explicit formula5 (16) is a weight, and is the volume increment in where . This formula defines as a weighted average of all . A suitable weight is
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 observability is always first order. However, Theorem 3 also includes the following situation from the literature as a special case. Example 4: For systems which are of the form (or can be embedded in this form) ..
.
.. .
(17) For a discrete (approximating) version of this formula, see Section VI-B. is not Theorem 4: If is uniformly injective in , and , as defined by locally thin6 , then a) (16) and (17), in case is bounded; b) (29)–(31), in case is not bounded; is continuous and an extended inverse of .
2 !
5For z q (G) the right-hand side is to be unterstood in the sense of the limit, obtained when replacing w (z; x) by w (z; x) := 1=[" + z q (x) ] and letting " 0. 6We say that G is not locally thin, if dX c dX for each
2
G and "
constants.
j 0
2
(0; " ], where G
:= U ( )
1 \ G and c; "
j
are positive
KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS
Fig. 5.
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Finite-time observer structure.
A proof is given in Appendix I. The idea behind this construction of is as follows. The state observation error of the nonlinear observer becomes and can be written in the form
(a)
(18) as . Thereby, the From (6), we have , seen as a function normalized weight . Convergence of , tends to a -distribution at then follows from (18). The proof of Theorem 4 makes this intuitive idea rigorous. V. FINITE-TIME OBSERVERS Let be an observation mapping, which is defined using (rather than ) a finite time interval (19) with , it is seen that along any trajectory of the system
Comparing
(20) This motivates an observer of the form (21) (22) (23) and with initial conditions , . The structure is again open loop and includes now a delay, as is illustrated in Fig. 5. Theorem 5: is injective; i) satisfies for all ; ii) then (21)–(23) are an observer for , whose state estimate converges to the true state in finite time , i.e., for . Proof: As before, along any trajectory of the system we have defined for
Rewriting
(b) Fig. 6. (a) Measured output y and observed state x ^ x ) , i = 1, 2.
(^ x
0
. (b) Observation errors
For initial conditions , the same argument gives for all . Note that the assumptions made in Theorem 5 are in fact weaker than those in Theorem 1. In particular, any kind of an extended inverse is now suitable, because convergence occurs in finite time. The latter is a structural property of this observer. and are the same We also note that injectivity of problems on different horizons. Therefore, with observability and with and finite complexity redefined on the interval replaced by accordingly, the results of Section III carry over to the finite-time case. In particular, observability and is uniformly infinite complexity together guarantee that . The extended inverse of Section IV jective, for some may 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 , it is also obSince has periodic solutions with period . Using servable and of finite complexity on the interval the same observer dynamics as before and the fact that is periodic, we find that
as (24)
and, thereby, it follows that for all , i.e., finite-time convergence regardless of the initial conditions.
Hence, is also injective, and its extended inverse can be from the extended obtained as . The resulting observer is simulated with inverse of as above, and . Fig. 6 illustrates the finite time convergence, which is in contrast to the asymptotic convergence in Fig. 4.
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VI. ARBITRARILY ACCURATE OBSERVERS Under real world conditions, an observer can only give an approximation of what it is wanted to give theoretically. Therefore, an observer may as well be designed for an arbitrary finite accuracy, rather than for ultimate exactness. We develop this concept for practical situations, where the trajectories of the system can be taken compact. The result are are bounded, so that observers of any desired accuracy, which require only observability of the system and are, therefore, of widest applicability. A. Arbitrary Finite Accuracy Recall from Section II-E, that our observation mapping represents the first coefficients of an orthonormal expansion of . Therefore, the information for a prescribed accuracy is suffinaturally contained in this representation, if we take ciently large. Together with observability, this gives rise to the following Lemma. Lemma 5: If is observable in , and is compact, then there exist , for each desired accuracy such that (25) . where A proof is given in Appendix J. With an accuracy in the sense of (25), the set of all points in giving the same value of , is located strictly in the interior of a ball of radius in . Therefore, can be recovered from a given , up to an error of size . As a suitable mapping, which does such an inversion approximately, we consider the expression (26) with
Note that the results of this section and the previous one, respectively, combine nicely to an observer, which attains any desired accuracy in finite time. B. Numerical Design Consider any Lipschitz continuous system (not necessariliy observable or of finite complexity), with a compact design re. Based on an a priori choice of a sequence gion as described in Section II-E, the following steps toward an observer can always be taken. gives (see Section II-E) a) The choice of an integer , which defines the observer dyan th-order pair . namics b) The choice of some (uniformly continuous) nonlinearity defines the observed state . The result of a) and b) is an observer of accuracy , i.e., an observer which satisfies properties i) and ii) of Theorem 6, where8
An observer design, thus, becomes a selection of and , so as to make as small as desired. By Theorem 6 we know that if is observable in , then such a 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) ratio of sums, is then also an appropriate nonlinearity. It can be used as a choice of in step b), and can be designed numerically as follows. sufficiently dense and b1) Select a set of points (e.g., , properly distributed in is the discretization density). where via integrating (1) backward in time b2) Compute and evaluating (3). b3) Define
(27) (28) where is a positive constant, to be chosen in combination with (respectively, ) so as to achieve a joint accuracy of the resulting observer. Theorem 6: Let be observable in , and let be compact. there exist and , such that Then for each (4), (5), (26), (27) 7 are an observer of accuracy in , i.e., as for all i) ; , for all ii) . A proof is given in Appendix K. The result of Theorem 6 is very general and widely applicable. It says that for an observable system on a compact set, there is always an arbitrarily accurate observer, and the proposed observer structure is an easy realization, where one needs only take the dimension sufficiently large and the constant sufficiently small.
Based on these, an observer design can now proceed iteratively (and without the need to have checked observability of beforehand) as follows. Each iteration involves going through steps a) and b1)–b3), and delivers an observer with accuracy as a candidate solution. A convenient estimate of this accuracy can be obtained from the computed . data as is increased In each iteration the observer dimension ) to further improve the accuracy, (starting from some sufficiently small, until the desired while keeping and accuracy is attained. By Theorem 6, this will succeed within is observable. In case the a finite number of iterations, if and desired accuracy is not attained for reasonably large this indicates that may be not observable. small Note that the main computations are in step b2), and that these are offline computations. The nonlinearity (28) is to be imple-
7In case G 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).
8Recall from Section II-B that z(t) q(x (t)) as t that x ^ (t) x (t) = Q(z(t)) x (t) [0; ].
j
0
j j
0
!
j!
! 1, which implies
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mented as part of the final observer. Its evaluation to determine the observer output are online computations.9
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APPENDIX A. Proof of Theorem 2
VII. CONCLUDING REMARKS An integral operator based observation mapping is introduced, which captures the observability nature of autonomous nonlinear systems, whether or not they are smooth. What makes this mapping constructive and well suited for theoretical analyzes is that an th-order observation mapping represents an th-order orthonormal series expansion of the complete output signal history. The observer has a particularly simple open loop structure, which divides the observation process into two subsequent stages. In the first stage, linear observer dynamics generate a convergent estimate of the observation mapping value that corresponds to the current output signal history. That is, observation takes place here completely in the output function space, the latter being respresented implicitly by its finite. Accordingly, this dimensional vector space equivalent stage is coordinate free. The subsequent second stage merely maps the estimate into the state coordinates of the system, via an extended inverse of the observation mapping. An explicit inversion formula is presented. In the given observer structure, it only remains to choose the observer dynamics with (almost) arbitrary eigenvalues and a sufficiently large dimension to complete the observer. By construction, its convergence is global. The main assumptions to be made are observability and finite complexity. Finite complexity is introduced and characterized by a necessary and sufficient condition. Roughly, this property is well conditionedness of the output variations in a finite time/ finite bandwidth sense. While these conditions are difficult to check in general, it is demonstrated that they are satisfied and, hence, our results include a new observer which is not high gain, for the wide class of smooth systems considered recently in [4]. The proposed observer structure also works nicely in cases where the system is only observable. Applied with any dimension , it results in a finite accuracy observer, which is is. A computerized numerical the more accurate the larger design of such observers is presented, which is easy to implement, and which is computationally feasible for low order systems. Finally, we point out that our single-output results readily carry over to the multiple-output case with an observation mapping, which comprises the individual observation mappings of each output. Extensions to nonautonomous systems (i.e., systems with inputs) are nontrivial. Results in this direction have been obtained in [12] and are currently under preparation for publication.
Q
9Any alternate form of (e.g., based on neural nets, fuzzy logic, or interpolation/approximation techniques) may be used to optimize its implementations and/or evaluations.
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 integer
is square summable. Therefore, an exists such that
Letting we can rewrite
where
and
,
. 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 , . Then
, i.e., (14) holds for some
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 be nonzero only on a (sufficiently large) finite interval. On the , because is piecewise latter, we have continuous by assumption.
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With , the Fourier transforms of and (taken for the signals as defined on and continued iden, respectively, and by Parseval’s Theorem tically zero on
Suppose now that the condition of the Lemma holds, i.e., for some we have . Then
and with
Let we have
it follows that
denote the integer such that
Further define a vector
. Then,
with components when otherwise
This gives, using (14)
Taking
sufficiently large, this implies part ii) of the Lemma.
C. Proof of Lemma 2 a) Sufficiency: Let
and let
for
where is the smallest integer greater than Then, it follows that
and finite complexity is immediate from its definition. This proves sufficiency. is of finite complexity, i.e., (14) b) Necessity: Suppose , . Since , there is a holds for some such that . Thereby
otherwise denote its Fourier transform. Then, for Let
and define
where
otherwise
The order of integrals can be switched due to Fubini’s Theorem [13]. Then, for
Further rewrite as , where is chosen . Since is piecewise continuous, can so that Then, it be taken sufficiently large, so that follows that
and
As a consequence, there is an integer ) such that, with the notation pends on have
where is a constant resulting from the integral, which is bounded.
(which de, we
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i.e., the condition of the Lemma is implied. This proves necessity.
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F. Proof of Corollary 1 For any
, we can write
D. Proof of Lemma 3 and Since tion ii) implies
are related by
, assump-
where is an appropriate constant. For all this gives
By assumption i), there is a
,
such that
where denotes the state transition matrix which is associated with the linearization of along the trajectory passing is a remainder term, which is dethrough the point , and , and by fined by the above relation in case for all . , such that for all By assumptions i) and iv), there is a
Moreover, is continuous in its arguments on the com, and satisfies for all pact set . Therefore, a constant exists such that implies . Hence
hence By assumption iii), we also have Combining the two gives Let
be chosen such that and Then does not change sign for , and there is an interval of length such that . Integration thus gives
where
is the maximum distance of two points in and . This completes the proof.
G. Proof of First Order Observability of Example 4 and finite complexity follows from Lemma 2. E. Proof of Theorem 3 is observable by assumption ii). We prove finite complexity by showing that under the assumptions of the Theorem, Lemma 3 applies. Since is Lipschitz, we have for some . Combining this with ii) it follows that the first condition of Lemma 3 is satisfied. and let denote its Lipschitz Let constant. Then
where the right-hand side is bounded proportional to on , because is Lipschitz. Taking norms and using ii) it , that follows, with an appropriate constant
i.e., the second condition of Lemma 3 is satisfied as well. This completes the proof.
, and consider any . Since is Let times continuously differentiable with respect to time due to the special structure of , application of Taylor’s formula gives
where
and
for some we have
With notations
we can write
. Due to the special structure of
,
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Define, for example
, this gives
if if
for some , where the last inequality follows because the smallest eigenvalue of
..
.
where denotes the projection of on (i.e., is which minimizes the distance ), which some point in is made unique by choice where necessary. , holds by definition. Let Then be varied such that Then, by the . Combining the two gives projection property, . Due to uniform injectivity this implies . Since is either or , it follows that . I. Proof of Theorem 4
.. .
.. .
..
..
.
a) Consider the case where is bounded. Since is uniformly and a constant injective in and is Lipschitz, there is a such that
.. .
.
is bounded below proportional to . Also, due to the special structure of , we have
where and denotes the Lipschitz constant of . this can be used, in the definition For any choice of , to conclude that of
This is so in case of projection, and in case of
Defining
for some constant . Combining the two arguments, it follows that
Taking sufficiently small, this gives a ) such that dependent of first-order observability of in .
for all in . , and let denote the projection of on Let as in the proof of Lemma 4. Further let and define Then
(which is in. This proves
, we thus obtain
where is an appropriate constant, due to the boundedness of . On the other hand
H. Proof of Lemma 4 is closed. To see this, let be a We note that such that as . By uniform sequence in injectivity we have and for from which we can conclude that is a Cauchy sequence. converges to some . Since is closed, is in . Hence, follows. By continuity
by the definition of because
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where
, because
is not locally thin. Together, this gives
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can be taken independent of , because and are continuous and is compact. Since has a finite cover of , the largest of the corresponding such neighborhoods can be taken as an , which gives for all . Using observability, this gives
hence
Taking
proves the Lemma.
K. Proof of Theorem 6 where is the maximum distance of two points in , and the are arbitrary. Letting last inequality follows because , on both sides, it is found that all limits exist and
a) Suppose that is not locally thin. Let the observation map. Then ping be of accuracy
for all . For an arbitrary This proves for all , because the projecon equals , hence . tion of and , we also have Letting , which combine to . As a consequence, and , which proves that . is continuous, because this is obviously so in Finally, and ; and also in the transition as just shown. b) In case is not bounded, the extended inverse formula is modified to
, we obtain
where is an appropriate constant, and the fact that locally thin is relevant. Further
is not
(29) (30) if positive otherwise
(31) Therefore
are appropriate constants. The effect is that where the weight is nonzero only on a bounded subset of , where is sufficiently small. This is relevant to keep the integrals, in the same way bounded. which are taken over the set The proof is then as in case a). J. Proof of Lemma 5 . Using the orthonormal series representation of Let (see Section II-E), and letting and , we have , ; and (recall that
Hence, for each . This extends to
there is a
such that for
, where
where is the maximum distance of two points in . , such that To satisfy the theorem, we choose . This gives , which proves part ii) of the theorem. is uniformly continuous on any bounded subset Since , which contains in its interior, it follows that of implies and, hence, , which proves part i) of the theorem. a) Suppose is locally thin.
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be of accuracy
as before, and define the closure of
where
is chosen such that
Such a constant exists, because and are continuous are compact. In addition, is not locally thin, and and , case a) applies with replaced by . REFERENCES [1] H. Nijmeijer and T. I. Fossen, Eds., New Directions in Nonlinear Observer Design. New York: Springer-Verlag, 1999. [2] N. Kazantzis and C. Kravaris, “Nonlinear observer design using Lyapunov’s auxiliary theorem,” Syst. Control Lett., vol. 34, pp. 241–247, 1998. [3] G. Ciccarella, M. D. Mora, and A. Germani, “A luenberger-like observer for 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 for nonlinear 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 nonlinear time-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 nonlinear observers,” Syst. Control Lett., vol. 3, pp. 47–52, 1983. [8] H. Michalska and D. Q. Mayne, “Moving horizon observers and observer-based control,” IEEE Trans. Automat. Contr., vol. 40, pp. 995–1006, June 1995. [9] M. Alamir, “Optimization based nonlinear observers revisited,” Int. J. Control, vol. 72, pp. 1204–1217, 1999.
[10] P. E. Moraal and J. W. Grizzle, “Observer design for nonlinear systems with discrete-time measurements,” IEEE Trans. Automat. Contr., vol. 40, pp. 395–404, Mar. 1995. [11] A. Linnemann, “Convergent Ritz approximations of the set of stabilizing controllers,” Syst. Control Lett., vol. 36, pp. 151–156, 1999. [12] R. Engel, “Observers for nonlinear systems,” Ph.D. dissertation (in German), 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 Kreisselmeier was born in Hamburg, Germany, in 1943. He received the Dipl. Ing. degree from the Technical University of Hannover, Germany, and the Dr. Ing. degree from the Ruhr-University, Bochum, Germany, both in electrical engineering, in 1968 and 1972, respectively. From 1968 to 1970, he was with Siemens Company, Erlangen, Germany, in the field of process control, and from 1970 to 1985, with the DFVLR-Jnstitute fcir Flight Systems Dynamics, Oberpfaffenhofen, Germany, where he did research in control with aerospace applications. Since 1985, he has been a Professor of Control and Systems Theory in the Department of Electrical Engineering, the University of Kassel, Kassel, Germany.
Robert Engel was born in Friedrichshafen, Germany, in 1972. He received the Dipl. Ing. degree from the University of Ulm, Ulm, Germany, and the Dr. Ing. degree from the University of Kassel, Kassel, Germany, both in electrical engineering, in 1996 and 2002, respectively. Since then, he has been a Research Associate in the Department of Electrical Engineering, the University of Kassel. His current research interests include control and state observation of nonlinear systems.
