Nonlinear Observers for Autonomous Lipschitz Continuous Systems

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 3, MARCH 2003 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 THE leading results in nonlinear observer theory are for classes of systems, which have a certain degree of smoothness, ie, 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, eg, [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 (ie, 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 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: kreisselmeier@uni-kasselde) Digital Object Identifier 101109/TAC2002808468 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, eg, for any, because this always includes a subinterval, on which is strictly positive and, hence, the 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, eg, in [8] [10] goes in this direction The idea is to store measurements from the (sliding) interval, and to generate a state estimate so as 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 0018-9286/03$1700 2003 IEEE

452 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 3, MARCH 2003 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) with state and output Throughout this paper, the basic assumption is that and are globally Lipschitz continuous By the Lipschitz property, has a uniquely defined statetrajectory, which passes through the point for each and Moreover, the statetrajectory and the associated output are exponentially 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, set, which is invariant under, ie, trajectories which start in remain in for all future times B Basic Idea Let us define an observation mapping where is a controllable pair,, ( th-order for short), and sufficiently stable such that the integral is well defined The mapping assigns to each state, via the corresponding output history, of the (3) 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) Its inherent feature is that converges to exponentially as The state estimate is then formed from using a nonlinear mapping, which in an ideal case satisfies and is a so-called extended inverse of The idea is best illustrated by an example Example 1 1 : when 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 mapping exists, 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 resulting observer 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 1 A 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 453 Theorem 1: Suppose is sufficiently stable, so that is well defined If i) is injective; ii) satisfies a), ; b) 2, ; 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 If, then and, hence, If, then as, and 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 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 interval -norm on the interval, short notation For a vector of functions (6) Inner product of functions on, short notation For vectors of functions, denotes a matrix with entries Set of continuous, strictly monotone increasing functions, 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 system output, seen backward in time, is, Throughout this paper, we consider the exponentially weighted version of this output and its variations 2 This notation means that any sequence (z ;x ) 2 2 G which satisfies z 0 q(x )! 0, implies Q(z ) 0 x! 0 as k!1 where is chosen such that 3 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 present state The formal definition of observability is as follows Definition: is said to be observable in, if there is a such that for all Observability thus allows to conclude smallness of from smallness of uniformly in If is compact, then observability in this sense only requires for all 4 Observability in finite time is defined accordingly, with replaced by E Choice of Recall that the observation mapping is given by where the observer filter pair is to be chosen For convenience, we preselect a set of pairs of different dimension as follows Let be chosen as a sequence of real or complex numbers (with complex numbers appearing in complex conjugate pairs) such that i), ; ii) ; and let denote the set of integers (7) (8) the set is complex conjugate (9) Then, by a result from [11], there is a uniquely defined sequence of functions, which is an orthonormal basis in and has the following property For each there is exactly one real, th-order pair with spectrum such that (10) 3 Any value of greater than the Lipschitz constant of f (x) is appropriate From design considerations, smaller values would be preferred if possible 4 Note that y(t; x) is continuous in x A suitable ' 2 K would be '(s) := (s=s) 1 min ky(x) 0 y(x )k where s := max jx 0 x j

454 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 3, MARCH 2003 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 and, which are in and linearly independent Therefore, and (11) As a result of this preselection, only the choice of dimension remains to be discussed in the later sections It is worth pointing out that is in and, therefore, has an orthonormal series representation (12) which implies finite complexity Example 2: A linear system, is typically considered in It has output variation where the coefficients are Hence, the observation mapping of order represents the first coefficients 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 controllable, of suitable dimension and with spectrum, because this only amounts to a similarity transformation of and the corresponding linear transformation 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 is 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 respect to the distance of states, the property of finite complexity characterizes them as functions of time Definition: is said to be of finite complexity in,if there exists a finite number of piecewise continuous functions, combined to a vector, such that for some (14) for all Example 1 (Continued): The first-order system of example 1 has weighted output 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 linear system is always of finite complexity in, whether or not the pair is observable 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 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 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 necessary that there are constants,, such that i) ; ii) ; for all, where denotes the Fourier transform of A proof is given in Appendix B Finite complexity of a system thus implies that its output variations have a low frequency portion and a finite time interval 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 Lemma 2: is of finite complexity in, if and only if there exist,, such that for each the relation holds for at least one A proof is given in Appendix C (15)

KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS 455 Lemma 2 combines the low frequency and finite time interval aspects The quantity of interest is, which is obtained by passing through a first order low pass filter with transfer function (or, equivalently, by passing through a filter to obtain and then As a result, the conditon of Lemma 2 can be interpreted in the sense that is a simultaneous low frequency/finite time portion from The relative 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), jxj =1in the (q ;q ) plane if positive 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 sign and is a sinusoid of amplitude in case of and of amplitude in case of, respectively, where denotes 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 that describes a circle in, and therefore moves on a closed path in, which is the periodic solution of The calculation of is complete with one such solution and the fact that due to for Based on an a priori choice of an eigenvalue sequence, a natural initial guess of an observer dimension 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 closed contour (which is due to the mean value of ) A natural second guess to try is The pair is now taken to be Fig 4 (a) Measured output y and observed state ^x (b) Observation errors (^x 0 x ), i =1; 2 (a) (b) ie, with eigenvalues according to the a priori choice, and in convenient coordinates such that This gives zero mean value for and The plot of versus, which is illustrated in Fig 3, gives evidence that this is injective The observers (4) and (5) are finally implemented with from above, and with realized by the extended inverse formula (28), which is given in Section VI-B, using in (b1) and in (28) Fig 4(a) and (b) illustrate the observation process for initial conditions, 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 complexity, taking advantage of some mild smoothness The standing assumption is again that is globally Lipschitz

456 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 3, MARCH 2003 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 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 (ie, systems which satisfy, for some ), the condition for finite complexity can be simplified drastically The subsequent theorem reduces it to a Lipschitz condition on in the first assumption, while the second assumption states first order observability in finite 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 applications, the subsequent Corollary can be useful It points out that (under some mild extra assumptions) a system is first-order observable in finite time, if on some interval its linearizations 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 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) 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 the set It is important here to cover all of, because this class includes systems which have unbounded solutions By the Lipschitz assumption on, the first hypothesis of Theorem 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 In cases where is continuously differentiable and is 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 nontrivial interval This is in fact stronger than 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, ie, 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 formula 5 (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, see Section VI-B Theorem 4: If is uniformly injective in, and is not locally thin 6, 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 5 For z 2 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=[" + jz 0 q(x)j] and letting "! 0 We say that G is not locally thin, if dx c 1 dx for each 2 G and " 2 (0;" ], where G := U () \ G and c; " are positive constants

KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS 457 Fig 5 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) From (6), we have as Thereby, the normalized weight, seen as a function of, tends to a -distribution at Convergence 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 a finite time interval (rather than ) (19) Comparing with, it is seen that along any trajectory of 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 estimate converges to the true state in finite time, ie, for Proof: As before, along any trajectory of the system we have Rewriting as (24) it follows that and, thereby, for all, ie, finite-time convergence regardless of the initial conditions (b) Fig 6 (a) Measured output y and observed state ^x (b) Observation errors (^x 0 x ), i =1,2 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 We also note that injectivity of and are the same problems on different horizons Therefore, with observability and finite complexity redefined on the interval and with replaced by accordingly, the results of Section III carry over to the finite-time case In particular, observability and finite complexity together guarantee that is uniformly injective, for some The extended inverse of Section IV 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 Since has periodic solutions with period, it is also observable and of finite complexity on the interval Using the same observer dynamics as before and the fact that is periodic, we find that Hence, is also injective, and its extended inverse can be obtained as from the extended inverse of The resulting observer is simulated with as above, and Fig 6 illustrates the finite time convergence, which is in contrast to the asymptotic convergence in Fig 4

458 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 3, MARCH 2003 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 are bounded, so that can be taken compact The result are 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 naturally contained in this representation, if we take sufficiently large Together with observability, this gives rise to the following Lemma Lemma 5: If is observable in, and is compact, then for 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 in giving the same value of, is located strictly in the interior of a ball of radius in Therefore, can be recovered from agiven, up to an error of size As a suitable mapping, which does such an inversion approximately, we consider the expression with (26) (27) 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 Then for each there exist and, such that (4), (5), (26), (27) 7 are an observer of accuracy in, ie, i) as for all ; ii), for all 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 7 In 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) 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 region Based on an a priori choice of a sequence as described in Section II-E, the following steps toward an observer can always be taken a) The choice of an integer gives (see Section II-E) an th-order pair, which defines the observer dynamics b) The choice of some (uniformly continuous) nonlinearity defines the observed state The result of a) and b) is an observer of accuracy, ie, an observer which satisfies properties i) and ii) of Theorem 6, where 8 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 b1) Select a set of points sufficiently dense and properly distributed in (eg,, where is the discretization density) b2) Compute via integrating (1) backward in time and evaluating (3) b3) Define (28) 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 In each iteration the observer dimension is increased (starting from some ) to further improve the accuracy, while keeping and sufficiently small, until the desired accuracy is attained By Theorem 6, this will succeed within a finite number of iterations, if is observable In case the desired accuracy is not attained for reasonably large and small this indicates that may be not observable Note that the main computations are in step b2), and that these are offline computations The nonlinearity (28) is to be imple- 8 Recall from Section II-B that z(t)! q(x (t)) as t!1, which implies that j^x (t) 0 x (t)j = jq(z(t)) 0 x (t)j![0;]

KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS 459 mented as part of the final observer Its evaluation to determine the observer output are online computations 9 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 finitedimensional vector space equivalent Accordingly, this 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 the more accurate the larger is A computerized numerical 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 (ie, systems with inputs) are nontrivial Results in this direction have been obtained in [12] and are currently under preparation for publication APPENDIX A Proof of Theorem 2 Let be of finite complexity in, ie, (14) holds for some, Since is an orthonormal basis in (see Section II-E), and,wehave where integer is square summable Therefore, an exists such that Letting and, we can rewrite where and, thus This can be substituted in (14) to get Using observability, it finally follows that ie, is uniformly injective in B Proof of Lemma 1 Let be of finite complexity in, ie, (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 9 Any alternate form of Q (eg, based on neural nets, fuzzy logic, or interpolation/approximation techniques) may be used to optimize its implementations and/or evaluations Hence, to establish finite complexity, can be restricted to be nonzero only on a (sufficiently large) finite interval On the latter, we have, because is piecewise continuous by assumption

460 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 3, MARCH 2003 With, the Fourier transforms of and (taken for the signals as defined on and continued identically zero on, respectively, and by Parseval s Theorem Suppose now that the condition of the Lemma holds, ie, for some we have Then and with it follows that Let denote the integer such that Then, we have This gives, using (14) Further define a vector with components when for where is the smallest integer greater than Then, it follows that Taking sufficiently large, this implies part ii) of the Lemma C Proof of Lemma 2 a) Sufficiency: Let and finite complexity is immediate from its definition This proves sufficiency b) Necessity: Suppose is of finite complexity, ie, (14) holds for some, Since, there is a such that Thereby and let denote its Fourier transform Then, for Let and define where The order of integrals can be switched due to Fubini s Theorem [13] Then, for Further rewrite as, where is chosen so that Since is piecewise continuous, can be taken sufficiently large, so that Then, it follows that and As a consequence, there is an integer (which depends on ) such that, with the notation, we have where is a constant resulting from the integral, which is bounded

KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS 461 ie, the condition of the Lemma is implied This proves necessity D Proof of Lemma 3 Since and are related by, assumption ii) implies where is an appropriate constant For all, this gives F Proof of Corollary 1 For any, we can write where denotes the state transition matrix which is associated with the linearization of along the trajectory passing through the point, and is a remainder term, which is defined by the above relation in case, and by for all By assumptions i) and iv), there is a, such that for all By assumption i), there is a such that Moreover, is continuous in its arguments on the compact set, and satisfies for all 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 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 Let and let denote its Lipschitz constant Then where is the maximum distance of two points in and This completes the proof G Proof of First Order Observability of Example 4 Let, and consider any Since is times continuously differentiable with respect to time due to the special structure of, application of Taylor s formula gives where and 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 for some Due to the special structure of, we have With notations ie, the second condition of Lemma 3 is satisfied as well This completes the proof we can write

462 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 3, MARCH 2003 For any choice of, this gives Define, for example if if for some, where the last inequality follows because the smallest eigenvalue of where denotes the projection of on (ie, is some point in which minimizes the distance ), which is made unique by choice where necessary Then, holds by definition Let be 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 where is bounded Since is uniformly injective in and is Lipschitz, there is a and a constant such that is bounded below proportional to Also, due to the special structure of, wehave for all in Let, and let denote the projection of on as in the proof of Lemma 4 Further let and define Then where and denotes the Lipschitz constant of For any choice of this can be used, in the definition of, to conclude that This is so in case of projection, and in case of by the definition of because for some constant Combining the two arguments, it follows that Defining, we thus obtain Taking sufficiently small, this gives a (which is independent of ) such that This proves first-order observability of in H Proof of Lemma 4 We note that is closed To see this, let be a sequence in such that as By uniform injectivity we have from which we can conclude that is a Cauchy sequence Hence, converges to some Since is closed, is in By continuity follows where is an appropriate constant, due to the boundedness of On the other hand and for

KREISSELMEIER AND ENGEL: NONLINEAR OBSERVERS FOR AUTONOMOUS LIPSCHITZ CONTINUOUS SYSTEMS 463 where, because is not locally thin Together, this gives can be taken independent of, because and are continuous and is compact Since has a finite cover of such neighborhoods, the largest of the corresponding can be taken as an, which gives for all Using observability, this gives hence Taking proves the Lemma where is the maximum distance of two points in, and the last inequality follows because, are arbitrary Letting on both sides, it is found that all limits exist and K Proof of Theorem 6 a) Suppose that is not locally thin Let the observation mapping be of accuracy Then This proves for all, because the projection 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) for all For an arbitrary, we obtain where is an appropriate constant, and the fact that is not locally thin is relevant Further (30) if positive (31) where are appropriate constants The effect is that the 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,wehave (recall that, ; and Hence, for each there is a such that This extends to for, where 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 the theorem Since is uniformly continuous on any bounded subset of, which contains in its interior, it follows that implies and, hence,, which proves part i) of the theorem a) Suppose is locally thin

464 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 3, MARCH 2003 Let be of accuracy as before, and define the closure of where is chosen such that [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, PhD 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 Such a constant exists, because and are continuous and, are compact In addition, is not locally thin, 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 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