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In [1] we have shown that almost all dynamical systems are observable with respect to an almost arbitrary sample program consisting of 2n + 1 samples (n is the dimension of the differentiable manifold supporting the dynamical system). In this paper we construct a dynamical system which is unobservable with respect to any sample program consisting of 1n samples. Small perturbations of the dynamics do not destroy the non-observability. This shows that the results obtained in [1] are the best ones possible in general.

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... Der Beobachterfehler klingt also deutlich schneller ab als das geregelte System. 3 das System exponentiell und nicht nur asymptotisch stabilisiert. ...

... Der Einfluß der Ausgangsfunktion in ENBNF zeigt sich bereits bei einem Vergleich der Transformationen von ANDF auf ENBNF. Im Fall einer linearen Ausgangsfunktion h * weist diese Transformation (4.57) nur nichtlineare Anteile auf, während die Transformation (4.62), die aus der Wahl der nichtlinearen Funktion h * (x * n (k)) = x * n (k) 3 der Zustände in ENBNF für k > 0 und beliebige Anfangswerte x * (0) liefert ...

... führen auf e ✷ 1 (2) = 2.25 e ✷ 1 (0), (6.9) was mehr als eine Verdopplung des Anfangsfehlers darstellt. 3 Vorausgesetzt, alle Eigenwerte liegen innerhalb des Einheitskreises. für das Einschwingverhalten der ENBNFund NABKNF-Beobachter in Abhängigkeit der Beobachteranfangswertê ...

Modern control design methods are based on the knowledge of all state variables of the considered system. Since a measurement of all states is in most cases not possible or too expensive, the use of observers is of great importance. Up to now, nonlinear observers have mainly been studied for continuous-time systems, however, discrete-time representations are of increasing interest. For a relatively small class of systems an observer design with linearizable error dynamics based on canonical forms is possible. This work gives an extension of the so called "Two-Step-Transformation" to nonlinear observer canonical form. This extension allows to enlarge the class of transformable systems considerably. Considering past measurements of the systems in- and output variables leads to the so called extended nonlinear observer canonical form which also allows to design an observer with linearizable error dynamics. The transformation into extended observer form includes several degrees of freedom which help to select the structure and the characteristics of the resulting observers. The extended observer form exists for every strongly locally observable system with one output. The transformation of a system with several outputs is subject to further conditions. Compared to the transformation into classical observer form, these conditions are noticeably less restrictive. The observers via extended observer form are compared to another design procedure, which can be found in the literature. For the latter, an alternative structure and an extension to systems with several outputs is presented in this work. The comparison of all considered observers includes the transient behaviour, robustness to noisy measurements, parameter sensitivity and the feasibility of the design procedure. One of the main tasks to use observers is the state feedback of dynamical systems. Since the separation principle which holds for all linear, timeinvariant systems does not hold in the nonlinear case, this work also focuses on the problem of nonlinear discrete-time observers for nonlinear state feedback. An experimental investigation of the closed loop dynamics was carried out for the stabilization of an inverted pendulum. The results show the general applicability to technical systems of all considered observers and furthermore significant differences between some observers in the closed loop were emphasized.

... Nous étudions dans ce chapitre le problème de la généricité de l'observabilité. Plusieurs rêsultats ont été montres dans le cas continu-discret par Dirk Aeyels dans [2] et [3] et dans le cas continu par J.P. Gauthier et I.A. Kupka dans [22]. Nous commençons par les rappeler au début de ce chapitre, ensuite nous présentons notre travail dans lequel nous nous intéressons à l'étude de la généricité de I'observabilité pour les systèmes discrets avec contrôles. ...

... Arrant de présenter le théorème, nous donnons les deux lemmes suivantes. Dirk Aeyels a construit dans [3] un exemple de système dynamique non observable pour tout programme d'échantionnage constitué seulemenl de 2n points. Des petites perturbations de ce système ne modifient pas la non observabilité du système. ...

... Notons que Dirk Aeyels a montré dans [2] et [3] quelques résultats concernant la généricité de I'observabilité pour les systèmes continus-discrets et que J.P. Gauthier La topologie utilisée sur I'espace C*(X,Y) est la topologie de Whitney, voir [23]. Nous énonçons ci-dessous quelques théorèmes qui jouent un rôle important dans la démonstration de nos résultats. ...

Cette thèse est consacrée au problème de l'observabilité pour les systèmes non linéaires discrets. Nous étudions au début le problème de la conservation de l'observabilité après discrétisation. Ce problème est motivé par le fait que pour un système commandé par ordinateur, une commande constante (par morceaux) est appliquée aux instants 0,[delta], 2[delta],..., avec une mesure (complète ou partielle) de l'état effectuée aux mêmes instants. Nous montrons que si l'état et l'entrée évoluent dans des espaces compacts, si le système continu est observable pour toute entrée et uniformément infinitésimalement observable, alors le système discrétisé est aussi observable pour toute entrée constante par morceaux et M bornée, pourvu que le pas de discrétisation soit assez petit. Nous donnons des contre-exemples montrant que chacune des hypothèses est utile. Nous montrons aussi la conversation presque partout de l'observabilité après discrétisation, lorsque le système est analytique et que l'observabilité infinitésimale n'est plus assurée. Dans la deuxième partie, nous étudions le problème de la généricité de l'observabilité pour les systèmes discrets (avec entrée), l'état et l'entrée évoluant dans des variétés compactes et connexes. Nous montrons la densité (pour la topologie de Whitney) de l'ensemble des systèmes fortement observables lorsque la dimension de l'espace des sorties est strictement supérieure à celle de l'espace des entrées, et que l'on observe 2n+1 valeurs successives de la sortie, où n est dimension de l'espace des états

... Often, like in 15] a drive/response, or transmitter/receiver, viewpoint is assumed. In a discrete-time context, this typically allows for a description of the transmitter as a n-dimensional dynamical system x 1 (k+1) = f 1 (x 1 (k); x 2 (k)) (1) x 2 (k+1) = f 2 (x 1 (k); x 2 (k)) (2) This paper has been submitted to the European Control Conference 1999, Karlsruhe, Germany where x 1 ( ) and x 2 ( ) are vectors of dimension m and l, with m + l = n and x(k) = (x 1 (k); x 2 (k)). Given x 1 ( ) as the drive signal, the receiver dynamics are taken as a copy of (2)x 2 (k+1) = f 2 (x 1 (k);x 2 (k)): ...

... (3) Synchronization of transmitter and receiver now corresponds to the asymptotic matching of (2) and (3), that is lim k!1 kx 2 (k) ?x 2 (k)k = 0: ...

... (5) such that (4) holds, whatever initial conditions (1), (2) and (5) have. Although (5) enlarges the idea of using the copy (3) for (2), there are many systems for which (4) will not be met, no matter howf 2 in (5) is chosen. ...

A method is described for the synchronization of nonlinear discrete time dynamics. The methodology consists of constructing observer-receiver dynamics that exploit at each time instant the drive signal and buffered past values of the drive signal. In this way, the method can be viewed as a dynamic reconstruction mechanism, in contrast to existing static inversion methods from the theory of dynamical systems. Keywords: Synchronization, observers, nonlinear discrete time systems 1 Introduction Following Pecora and Carroll [15] a huge interest in the synchronization of two coupled systems has arisen. This research is partly motivated by its possible use in secure communications, cf. [6]. Often, like in [15] a drive/response, or transmitter/receiver, viewpoint is assumed. In a discrete-time context, this typically allows for a description of the transmitter as a n-dimensional dynamical system x 1 (k+1) = f 1 (x 1 (k); x 2 (k)) (1) x 2 (k+1) = f 2 (x 1 (k); x 2 (k)) (2) This paper has...

... • The second setting conforms to what one can actually do in a current clamp experiment, namely observe only the membrane voltage V(t) given the stimulating current I stim (t). This requires us to add to the basic DDF formulation the idea of constructing enlarged state spaces from the observed variables and their time delays [37,3,4,1,20]. This method is familiar and essential in the study of nonlinear dynamics and will be explained in the present context. ...

... What we observe is the operation of the full dynamics projected down to the single dimension V(t). To proceed we must effectively 'unproject' the dynamics back to a 'proxy space', comprised of the voltage and its time delays [37,3,4,1,20], which is equivalent to the original state space of V(t) and the gating variables for the ion channels. This is accomplished as follows: If we have observed V(t), we can define D E -dimensional ('unprojected') proxy space vectors S(t n ) via time delays of 13 . ...

... One can expect to achieve computational efficiency and allow the exploration of larger biological networks when using the DDF construction to capture the neuron biophysics. 4. Select a radial basis function (RBF) ψ([u − u c (q)] 2 , σ). ...

Using methods from nonlinear dynamics and interpolation techniques from applied mathematics, we show how to use data alone to construct discrete time dynamical rules that forecast observed neuron properties. These data may come from from simulations of a Hodgkin-Huxley (HH) neuron model or from laboratory current clamp experiments. In each case the reduced dimension data driven forecasting (DDF) models are shown to predict accurately for times after the training period.
When the available observations for neuron preparations are, for example, membrane voltage V(t) only, we use the technique of time delay embedding from nonlinear dynamics to generate an appropriate space in which the full dynamics can be realized.
The DDF constructions are reduced dimension models relative to HH models as they are built on and forecast only observables such as V(t). They do not require detailed specification of ion channels, their gating variables, and the many parameters that accompany an HH model for laboratory measurements, yet all of this important information is encoded in the DDF model.
As the DDF models use only voltage data and forecast only voltage data they can be used in building networks with biophysical connections. Both gap junction connections and ligand gated synaptic connections among neurons involve presynaptic voltages and induce postsynaptic voltage response. Biophysically based DDF neuron models can replace other reduced dimension neuron models, say of the integrate-and-fire type, in developing and analyzing large networks of neurons.
When one does have detailed HH model neurons for network components, a reduced dimension DDF realization of the HH voltage dynamics may be used in network computations to achieve computational efficiency and the exploration of larger biological networks.

... That a solution to the above synchronization problem, or observer problem, may be feasible under certain conditions may be deduced from the Takens embedding theorem [17], which is closely related to the observability property for nonlinear dynamical systems [18], [19]. In essence the ob-servability property states that the history of the transmitted signal contains all the information required to reconstruct a state variable for the master dynamics. ...

... A generalization of the above example is the class of systems of Lur'e type, considered in, e.g., [38], [31], otherwise known as the output injection case (19) Here are constant matrices of appropriate dimensions. Suppose that the solutions of (19) are well defined on [ ). ...

... A generalization of the above example is the class of systems of Lur'e type, considered in, e.g., [38], [31], otherwise known as the output injection case (19) Here are constant matrices of appropriate dimensions. Suppose that the solutions of (19) are well defined on [ ). Assuming that the matrix pair is detectable, a full observer system takes the form: (20) It suffices to choose such that is asymptotically stable. ...

In the literature on dynamical systems analysis and the control of
systems with complex behavior, the topic of synchronization of the
response of systems has received considerable attention. This concept is
revisited in the light of the classical notion of observers from
(non)linear control theory,

... Often, like in 14] a drive/response, or transmitter/receiver, viewpoint is assumed. In a discrete-time context, this typically allows for a description of the transmitter as a n-dimensional dynamical system x 1 (k+1) = f 1 (x 1 (k); x 2 (k)) (1) x 2 (k+1) = f 2 (x 1 (k); x 2 (k)) (2) where x 1 ( ) and x 2 ( ) are vectors of dimension m and l, with m+l = n and x(k) = (x 1 (k); x 2 (k)). Given x 1 ( ) Corresponding author. ...

... Synchronization of transmitter and receiver now corresponds to the asymptotic matching of (2) and (3), that is lim k!1 kx 2 (k) ?x 2 (k)k = 0: ...

... such that (4) holds, whatever initial conditions (1), (2) and (5) have. Although (5) enlarges the idea of using the copy (3) for (2), there are many systems for which (4) will not be met, no matter howf 2 in (5) is chosen. ...

A method, based on ideas from control theory, is described for the synchronization of discrete time transmitter /receiver dynamics. Conceptually, the methodology consists of constructing observer-receiver dynamics that exploit at each time instant the drive signal and past values of the drive signal. In this way, the method can be viewed as a dynamic reconstruction mechanism. PACS numbers: 02.10.Jf 02.90.+p 05.45.+b 47.52.+j 89.90.+n 1 Introduction Following Pecora and Caroll [14] a huge interest in the synchronization of two coupled systems has arisen. This research is partly motivated by its possible use in secure communications, cf. [6]. Often, like in [14] a drive/response, or transmitter/receiver, viewpoint is assumed. In a discrete-time context, this typically allows for a description of the transmitter as a n-dimensional dynamical system x 1 (k+1) = f 1 (x 1 (k); x 2 (k)) (1) x 2 (k+1) = f 2 (x 1 (k); x 2 (k)) (2) where x 1 (Delta) and x 2 (Delta) are vectors of dimension m ...

... ,ˆˆ, which can be understood as a set of estimated states that are a type of inversion of the dynamic system associated to the system's past history. This set will only have one element if, and only if, the system is observable [4], which can be difficult to ascertain when considering a generalized non-linear system. It is therefore assumed that the model represented by (1) is perfectly known and observable. ...

... The observability of the system (1) is restricted by the existence and uniqueness of the solution of problem (5), i.e. this solution will have a unique element if, and only if, the system is observable. For further details, see [4], [9], [12], [14]. ...

Este artículo propone una novedosa estrategia aplicada al problema de estimación no lineal de estados en un motor de inducción (MI). Este método permite, en general, la estimación de variables de difícil acceso o que simplemente no se pueden medir. Lo cual es posible a través de mediciones indirectas, considerando un modelo dinámico del proceso y un algoritmo de estimación basado en optimización no lineal. El principal atractivo de esta estrategia de estimación de estados denominada MHSE (Moving Horizon State Estimation) en un MI, que permite conocer la magnitud del flujo, la velocidad o posición del rotor, es su simplicidad de implementación, sus buenas características de convergencia, su independencia de estructuras preestablecidas de modelos y su fácil sintonía. Resultados en simulación muestran la efectividad del método propuesto efectuando la estimación de la velocidad de un MI bajo diferentes puntos de operación This article proposes an innovative strategy to the problem of non-linear estimation of states in an induction motor, (IM). This method allows the estimation of variables that are difficult to access or that are simply impossible to measure. The estimation is made possible by using indirect measures, through the consideration of a dynamic model of the process and an estimation algorithm based on non-linear optimization. This state estimation strategy (a.k.a. Moving Horizon State Estimation or MHSE) in an IM allows the determination of the flux magnitude and the velocity or position of the rotor. Its principal advantages are the simplicity of its implementation, good convergence characteristics, independence from pre-established model structures, and easy tuning. Simulated results corroborate the effectiveness of the proposed method through estimates of the velocity of an IM under different operational situations

... 208 and 308, as well as the recent paper [19] which describes other variations which are biologically more accurate. In (1), the state z is the vector (M, E). We assume first that there is no true external input to the system, so experiments consist of simply letting the system evolve from its initial state up to certain time T , and measuring M (T ) at the end of the interval [0, T ]. (That is, the measured quantity is the amount of RNA; currently gene arrays are used for that purpose.) ...

... The material in Section 2.3 on genericity is motivated by, and shares many of the techniques with, the theory of manifold embeddings (see also Section 6.6 below, as well as the remark in the proof about one to one maps). Closely related is also the work of Takens [28], which shows that generically, a smooth dynamical system on an r-dimensional manifold can be embedded in R 2r+1 , as well as the control-theory work of Aeyels on generic observability, which shows in [2] that for generic vector fields and observation maps on an r-dimensional manifold, 2r+1 observations at randomly chosen times are enough for observability, and in [1] that this bound is best possible. Aeyels proofs, in particular, are based on transversality arguments of the general type that we use. ...

Given a set of differential equations whose description involves unknown parameters, such as reaction constants in chemical kinetics, and supposing that one may at any time measure the values of some of the variables and possibly apply external inputs to help excite the system, how many experiments are sufficient in order to obtain all the information that is potentially available about the parameters? This paper shows that the best possible answer (assuming exact measurements) is 2r+1 experiments, where r is the number of parameters.

... The idea is that information resides in the temporal derivatives of y l (t) in addition to that contained in the measurement itself at each t n . As we do not have that derivative information directly, we may approximate it with known data as finite differences such as [y l (t n + τ ) − y l (t n )]/τ , representing dy l (t)/dt with τ some multiple of the time differences between measurements [Aeyels, 1981a, Aeyels, 1981b, Mañé, 1981, Sauer et al, 1991, Takens, 1981, Abarbanel, 1996, Kantz & Schreiber, 2004. As Takens noted in the context of nonlinear dynamical systems, the new information beyond y l (t n ) lies in y l (t n + τ ), so that we can establish an extended state space by creating L data vectors of dimension D M from y l (t n ) and its time delayed versions: Y k;l (t) = y l (t n ), y l (t n + τ ), ..., y l (t n + (k − 1)τ ) ; k = 1, 2, ..., D M . ...

Utilizing the information in observations of a complex system to make
accurate predictions through a quantitative model when observations are
completed at time $T$, requires an accurate estimate of the full state of the
model at time $T$.
When the number of measurements $L$ at each observation time within the
observation window is larger than a sufficient minimum value $L_s$, the
impediments in the estimation procedure are removed. As the number of available
observations is typically such that $L \ll L_s$, additional information from
the observations must be presented to the model.
We show how, using the time delays of the measurements at each observation
time, one can augment the information transferred from the data to the model,
removing the impediments to accurate estimation and permitting dependable
prediction. We do this in a core geophysical fluid dynamics model, the shallow
water equations, at the heart of numerical weather prediction. The method is
quite general, however, and can be utilized in the analysis of a broad spectrum
of complex systems where measurements are sparse. When the model of the complex
system has errors, the method still enables accurate estimation of the state of
the model and thus evaluation of the model errors in a manner separated from
uncertainties in the data assimilation procedure.

... The generated current profile is in-turn fed to the battery pack model and the resulting output voltage and the trajectories of the internal states and parameters are recorded similarly to [25]. Taking into account the minimum number of samples required to estimate n parameters from data, 2n + 1, as suggested in [26], the information matrix is computed along the trajectory of the states and the associated significance of each element of aSPs is computed as described in Alg. 2. The significance metrics, computed at each instance based on a receding history, are then averaged to compute the significance metric over the entire drive-cycle. evaluated over a rolling data-set obtained from driving the heavy-duty vehicle model to follow the Urban Assault Cycle (UAC) [24]. ...

Enforcing constraints on the maximum deliverable power is essential to protect lithium-ion batteries and to maximize resource utilization. This paper describes an algorithm to address the estimation of power capability of battery systems accounting for thermal and electrical constraints. The algorithm is based on model inversion to compute the limiting currents and, hence, power capability. The adequacy of model inversion significantly depends on the accuracy of model states and parameters. Herein, these are estimated by designing cascading estimators whose structure is determined by quantifying the relative estimability of states and parameters. The parameterized battery model and the estimation algorithms are integrated with a power management system in a model of a series hybrid electric vehicle to demonstrate their effectiveness.

... This is a reasonable point of view, but faces two important difficulties: (i) from the moment that the continuous-time system description is abandoned and is substituted by a discrete-time description, the inter-sample dynamic behavior is lost (ii) any errors in the sampling schedule, get transferred into errors in the discrete-time description As a consequence, available design methods (i) do not provide an explicit estimate of the error in between two consecutive sampling times and (ii) do not account for perturbations of the sampling schedule. Moreover, due to observability issues, the magnitude of the sampling period cannot be arbitrary (see [1,26]). ...

In this work, a sampled-data nonlinear observer is designed using a continuous-time design coupled with an inter-sample output predictor. The proposed sampled-data observer is a hybrid system. It is shown that under certain conditions, the robustness properties of the continuous-time design are inherited by the sampled-data design, as long as the sampling period is not too large. The approach is applied to linear systems and to triangular globally Lipschitz systems.

... Dans notre cas, Si n est la dimension du vecteur d'état, un nombre de points h sur l'horizon égal à 2 1 n ⋅ + est suffisant pour recouvrir l'état [Aeyels, 1981]. De plus, une longueur de l'horizon lh plus grande que n donnera une plus faible sensibilité au bruit de mesure [Boillereaux, 1996]. ...

This thesis proposes an original method to estimate states in non-linear discrete-time systems with global convergence properties. The method is based on an Interval Moving Horizon State Estimation Method (IMHSE), which is coupled to a technique of global optimisation of nonlinear functions that uses interval arithmetic. In other words, the principal idea is to transform the problem of state estimation from a dynamic system into a static problem of global nonlinear optimisation over a considered time horizon by interval analysis. Offline measures or delayed measures can be easily used in this interval observer to reconstruct the state variables that are described using a representation by interval numbers. The work also considers model fault detection by a multimodel IMHSE (as an extra property given to our observer). The goal of this multimodel observer approach is to detect dynamic variations of the involved model parameters in time. These variations are taken into account using several different models that are commuted and used by our interval observer to reconstruct the states of the system. Put simply, this approach consists of using a model for the nominal dynamic state(s) and other models to describe situations of anomalous working (perturbed parameters). The algorithm allows us to know on line which model best describes the behaviour of the system. The proposed technique is applied to biotechnological complex process models such as solid substrate fermentation, and to bioprocesses described by a hybrid model. The results obtained through experimental and computer simulation demonstrate that this kind of estimator has advantages over other observers and filters, and that it can be easily implemented in an industrial context.

... Dans le cas linéaire, cette condition est indépendante de l'entrée (la condition du rang dépend que de et ), et elle est également suffisante pour garantir l'existence d'un observateur à vitesse de convergence exponentielle et arbitrairement rapide [Luenberger, 1971;Aeyels, 1981]. ...

This thesis proposes a general methodology for identifying and reconstructing sensor faults on dynamical processes. This identification theory provides a general framework for the problem of "observability with unknown inputs". Next, a framework for fault detection and isolation of sensors and actuators is proposed. The FDI sheme is based on bank of high-gain observers. A simulation study of a waste water treatment plant shows the effectiveness of the proposed approach.The second point evoked in the thesis is the observability of nonlinear dynamic systems and state estimation. The Extended Kalman Filter (EKF) is a widely used observer for such nonlinear systems. However, it suffers from the lack of theoretical justifications. The EKF, when applied to a system put in a normal form of observability, it acquires the property of global exponential convergence. Unfortunately, this latter observer (HG-EKF) is very sensitive to measurement noise. In order to combine the behaviors of the EKF (efficiency with respect to noise smoothing) and of the HG-EKF (reactivity to large estimation errors), (Boizot et al, 2010) proposed an adaptive high gain observer. This observer is applied to a MIMO nonlinear system of an Activated Sludge Process. A comparison study of the performances of the three observers under consideration is carried out. Results show a clearly better state estimation for the adaptive observer.

... In particular, it is a generic property for polynomial systems of bounded degrees, that the state can be polynomially expressed in terms of the first N + 1 derivatives y, · · · , y (N ) of the output, provided N ≥ 2n. Similar results have been shown by F. Takens and D. Aeyels for smooth systems, and by Gauthier and Kupka for real analytic systems; see [1,2,10,18]. There is also a very nice recent survey paper by E. Sontag [17] with potential applications to biology in mind. ...

E. Sontag has introduced the concept of algebraic observability for n-dimensional polynomial systems. It is a stronger notion than the usual concept of observability and implies the existence of a polynomial expression of the state variables in terms of a nite number of derivatives of the output function. We prove that algebraic observability is a generic property for polynomial systems of bounded degrees. Explicit geometric characterizations of algebraic observability via polynomial embeddings are derived and it is shown that the state variables of an algebraically observable system can be expressed as a polynomial in the rst 2n + 1 derivatives of the output.

... Notice that in the IMHSE method, the only parameter to adjust is the length of the horizon (lh). For example, in [6,7] we can find theoretical and experimental relations between the length of the horizon, the time constant of the system, the number of points on the horizon sufficient to distinguish the states [31] and the numeric observability involved. ...

This work proposes an original method to estimate states in non-linear discrete-time systems with global convergence properties. The approach is based on the minimisation of a criterion (non-linear function, differentiable or not) that is the Euclidean norm of the difference between the estimated output and the measured output of the system over a considered time horizon. This method is based on an interval moving horizon state estimation method, called IMHSE, which is coupled to a technique of global optimisation of non-linear functions that uses interval arithmetic. The system states are described using a representation by interval numbers. The proposed technique is applied to biotechnological complex process models (solid substrate fermentation), and the results obtained through experimental and computer simulation demonstrate that this kind of estimator offers advantages over other observers and filters and can be easily implemented in an industrial context.

... (i) from the moment that the continuous-time system description is abandoned and is substituted by a discrete-time description, the inter-sample dynamic behavior is lost (ii) any errors in the sampling schedule, get transferred into errors in the discrete-time description As a consequence, available design methods (i) do not provide an explicit estimate of the error in between two consecutive sampling times and (ii) do not account for perturbations of the sampling schedule. Moreover, due to observability issues, the magnitude of the sampling period cannot be arbitrary (see [1,26]). Finally, optimization-based approaches for nonlinear observer design [2,13,23,24,29] are also based on a discretetime description of the dynamics and therefore share all the above difficulties, but, because they utilize a large number of measurements, offer the advantage of reduced sensitivity to measurement errors at the expense of higher memory requirements and computational cost . ...

In this work, a sampled-data nonlinear observer is designed using a continuous-time design coupled with an inter-sample output predictor. The proposed sampled-data observer is a hybrid system. It is shown that under certain conditions, the robustness properties of the continuous-time design are inherited by the sampled-data design, as long as the sampling period is not too large. The approach is applied to triangular globally Lipschitz systems.

... If the observed time series s(n) comes from projecting onto the s-axis, then points that appear to be nearby in time t n = t 0 + n t may be neighbors due to the projection rather than due to the dynamics that moves the actual system of interest forward in time in a higher-dimensional space. Nonlinear time series methods for unfolding the scalar time series (Aeyels, 1981a(Aeyels, , 1981bTakens, 1981) use the data s(t n ) = s(n), along with the time delays of the data at time points t n + qτ t = t 0 + (n + qτ ) t : s(n + qτ ). τ and q are integers. ...

Tasking machine learning to predict segments of a time series requires estimating the parameters of a ML model with input/output pairs from the time series. We borrow two techniques used in statistical data assimilation in order to accomplish this task: time-delay embedding to prepare our input data and precision annealing as a training method. The precision annealing approach identifies the global minimum of the action ([Formula: see text]). In this way, we are able to identify the number of training pairs required to produce good generalizations (predictions) for the time series. We proceed from a scalar time series [Formula: see text] and, using methods of nonlinear time series analysis, show how to produce a [Formula: see text]-dimensional time-delay embedding space in which the time series has no false neighbors as does the observed [Formula: see text] time series. In that [Formula: see text]-dimensional space, we explore the use of feedforward multilayer perceptrons as network models operating on [Formula: see text]-dimensional input and producing [Formula: see text]-dimensional outputs.

... If the observed time series s(n) comes from projecting onto the s-axis, then points which appear to be nearby in time t n = t 0 + n∆t may be neighbors due to the projection rather than due to the dynamics that moves the actual system of interest forward in time in a higher dimensional space. Nonlinear time series methods for unfolding the scalar time series (Aeyels, 1981a(Aeyels, , 1981bTakens, 1981) use the data s(t n ) = s(n) along with the time delays of the data at time points t n + qτ ∆t = t 0 + (n + qτ )∆t : s(n + qτ ). τ and q are integers. ...

Tasking machine learning to predict segments of a time series requires estimating the parameters of a ML model with input/output pairs from the time series. Using the equivalence between statistical data assimilation and supervised machine learning, we revisit this task. The training method for the machine utilizes a precision annealing approach to identifying the global minimum of the action (-log[P]). In this way we are able to identify the number of training pairs required to produce good generalizations (predictions) for the time series. We proceed from a scalar time series $s(t_n); t_n = t_0 + n \Delta t$ and using methods of nonlinear time series analysis show how to produce a $D_E > 1$ dimensional time delay embedding space in which the time series has no false neighbors as does the observed $s(t_n)$ time series. In that $D_E$-dimensional space we explore the use of feed forward multi-layer perceptrons as network models operating on $D_E$-dimensional input and producing $D_E$-dimensional outputs.

... tel-00198362, version 1 -17 Dec 2007 Si n est la dimension du vecteur d'état, un nombre de points h sur l'horizon égal à 2 1 n ⋅ + est suffisant pour recouvrir l'état [Aeyels, 1981]. De plus, une longueur de l'horizon lh plus grande que n donnera une plus faible sensibilité au bruit de mesure [Boillereaux, 1996]. ...

Cette thèse propose une méthode originale d'estimation ensembliste d'états de procédés nonlinéaires discrets, qui est globalement convergente. La méthode est basée sur une technique d'estimation à horizon glissant par intervalles (IMHSE), couplé à une technique d'optimisation globale de fonctions non-linéaires qui utilise l'arithmétique par intervalles. En d'autres termes, la méthode IMHSE résout le problème d'estimation d'état d'un système dynamique par un problème statique d'optimisation globale non-linéaire par intervalles, sur un horizon de temps prédéfini. Les mesures faites hors ligne dans un procédé peuvent être utilisées facilement dans cet observateur ensembliste pour reconstruire les variables de l'état qui sont représentés par intervalles. Ce travail considère aussi la détection de dysfonctionnement d'un modèle en utilisant un observateur IMHSE multi-modèles (une propriété de plus donnée à notre observateur). L'objectif de cette approche multi-modèles est de détecter les variations dynamiques des paramètres du modèle dans le temps. Ces variations sont prises en considération en utilisant plusieurs modèles différents. Ces modèles seront commutés par notre observateur ensembliste pour reconstruire les états du système. Mis d'une façon simple, cette approche consiste à utiliser un modèle nominal pour l'état et d'autres modèles pour décrire les situations possibles de fonctionnement anormal (paramètres perturbés). L'algorithme nous permet de connaître en ligne quel est le meilleur modèle qui décrit le comportement réel du système. La technique proposée a été appliquée sur des modèles de procédés complexes biotechnologiques tel que la fermentation sur substrat solide, et à des bioprocédés décrit par des modèles hybrides. Les résultats obtenus par simulation montrent que ce type d'observateur a des avantages sur les autres observateurs et filtres, et qu'il peut être facilement appliqué dans un contexte industriel.

We review the major ideas involved in the control of chaos. We present the Ott-Grebogi-Yorke (OGY) method of controlling chaos, which is a particular case of the pole placement technique, but which is the one leading to the shortest time to achieve the control of chaotic systems. Implementation using only measured time series in experimental settings is also described.

We study the Newton observer design, developed by Moraal and Grizzle, when the exact discrete-time model of the sampled-data plant is not known analytically. We eliminate the dependence on this exact model with a “hybrid” reconstruction that makes use of continuous-time filters to produce the numerical value of the exact model. We then implement the Newton method with finite-difference and secant approximations for the Jacobian. Despite the continuous-time filters, the proposed hybrid redesign preserves the sampled-data characteristic of the Newton observer because it only employs discrete-time measurements of the output. It also offers flexibility to be implemented with nonuniform, or event-driven, sampling. We finally study how a line search scheme can be incorporated in this hybrid Newton observer to enlarge the region of convergence.

This paper focuses on the development of asymptotic observers for
nonlinear discrete-time systems. It is argued that instead of trying to
imitate the linear observer theory, the problem of constructing a
nonlinear observer can be more fruitfully studied in the context of
solving simultaneous nonlinear equations. In particular, it is shown
that the discrete Newton method, properly interpreted, yields an
asymptotic observer for a large class of discrete-time systems, while
the continuous Newton method may be employed to obtain a global
observer. Furthermore, it is analyzed how the use of Broyden's method in
the observer structure affects the observer's performance and its
computational complexity. An example illustrates some aspects of the
proposed methods; moreover, it serves to show that these methods apply
equally well to discrete-time systems and to continuous-time systems
with sampled outputs

We discuss the problem of determining unknown fixed parameters and unobserved state variables in nonlinear models of a dynamical system using observed time series data from that system. In dynamical terms this requires synchronization of the experimental data with time series output from a model. If the model and the experimental system are chaotic, the synchronization manifold, where the data time series is equal to the model time series, may be unstable. If this occurs, then small perturbations in parameters or state variables can lead to large excursions near the synchronization manifold and produce a very complex surface in any estimation metric for those quantities. Coupling the experimental information to the model dynamics can lead to a stabilization of this manifold by reducing a positive conditional Lyapunov exponent (CLE) to a negative value. An approach called dynamical parameter estimation (DPE) addresses these instabilities and regularizes them, allowing for smooth surfaces in the space of parameters and initial conditions. DPE acts as an observer in the control systems sense, and because the control is systematically removed through an optimization process, it acts as an estimator of the unknown model parameters for the desired physical model without external control. Examples are given from several systems including an electronic oscillator, a neuron model, and a very simple geophysical model. In networks and larger dynamical models one may encounter many positive CLEs, and we investigate a general approach for estimating fixed model parameters and unobserved system states in this situation.

In this paper, it is proven that, generically, a nonlinear discrete system given on a compact manifold X is observable.

The genericity of a nonlinear discrete system given on a compact manifold 'X', which is observable, is described. As the genericity of the observability for uncontrolled continuous-time systems is proved, similarly the same result is established for controlled continuous-time systems. An uncontrolled system given on a compact manifold 'X' is considered and the discretization of the vector field 'f' is used. It is observed that the tools used to prove the results are also the tools of transversality theory. The results obtained are more general as a diffeomorphism 'f' is used, which does not necessarily derive from the flow of a vector field.

This paper deals with the problem of synchronization, or observer
design, of chaotic dynamical systems. It is argued that the complex
nature of the transmitter dynamics may provide additional tools for
finding a suitable observer. A number of characteristic examples
illustrate the idea, and reveal some challenging open problems in this
context

We propose an approach to designing what we call a model-based
indirect sensor. This methodology allows estimation of non-measurable
variables from indirect measurements, by coupling a dynamical model of
the process and an estimation algorithm. This method is especially
developed with the aim of allowing a nonspecialist in process control to
perform, with limited mathematical development, the state estimation of
the processes he has to monitor or control. The main interest of this
algorithm is its great simplicity of implementation. Moreover, we
present a concept to tune this algorithm based on what we call numerical
observability. The performance of this method is illustrated on a real
bioprocess

Development of exact asymptotic observers for nonlinear
discrete-time systems is addressed. It is argued that instead of trying
to imitate the linear observer theory, the problem of constructing a
nonlinear observer can be more fruitfully studied in the context of
solving simultaneous nonlinear equations. In particular, it is shown
that Newton's algorithm, properly interpreted, yields an asymptotic
observer for a large class of discrete-time systems. The utility of the
observer for closed-loop, observer-based, feedback control is also
established. Some non-local aspects of the results are also discussed

Optimisation based algorithms known as Moving Horizon Estimator (MHE) have been developed through the years. In this work, we propose two solutions to decrease the computational cost of MHE, limiting its applicability in real-time applications. The proposed solutions rely on output filtering and adaptive sampling. The use of filters reduces the total amount of data by shortening the length of the moving window (buffer) and consequently decreasing the time consumption for plant dynamics integration. The proposed adaptive sampling policy allows for discarding data that do not yield significant improvements in the estimation error. Simulations on several cases are provided to corroborate the effectiveness of the proposed strategies.

Given a set of differential equations whose description involves unknown parameters, such as reaction constants in chemical
kinetics, and supposing that one may at any time measure the values of some of the variables and possibly choose external
inputs to help excite the system, how many experiments are sufficient in order to obtain all the information that is potentially
available about the parameters? This paper shows that the best possible answer (assuming exact measurements) is 2r+1 experiments, where r is the number of parameters. Moreover, a generic set of such experiments suffices.

SUMMARYA set membership method for right inversion of nonlinear systems from data is proposed in the paper. Both the cases where the system to invert is known or unknown and therefore identified from data are addressed. The method does not require the invertibility of the regression function describing the system and ensures tight bounds on the inversion error. In the case of unknown system, the method allows the derivation of a robust right-inverse, guaranteeing the inversion error bound for all the systems belonging to the uncertainty set which can be defined from the available prior and experimental information. Based on such a set membership inversion, two methods for robust control of nonlinear systems from data are introduced: nonlinear feed-forward control (NFFC) and nonlinear internal model control (NIMC). Both the design methods ensure robust stability and bounded tracking errors for all the systems belonging to the involved uncertainty set. Two applicative examples of robust control from data are presented: NFFC control of semi-active suspension systems and NIMC control of vehicle lateral dynamics.Copyright © 2013 John Wiley & Sons, Ltd.

Optimisation-based algorithms known as Moving Horizon Estimator (MHE) have been developed through the years. This paper illustrates the implementation of the policy introduced in the companion paper submitted to the 18th IFAC Workshop on Control Applications of Optimization [Oliva and Carnevale, 2022], in which we propose two techniques to reduce the computational cost of MHEs. These solutions mainly rely on output filtering and adaptive sampling. The use of filters reduces the total amount of data used by MHE, shortening the length of the moving window (buffer) and consequently decreasing the time consumption for plant dynamics integration. Meanwhile, the proposed adaptive sampling policy discards those sampled data that do not allow a sensible improvement of the estimation error. Algorithms and numerical simulations are provided to show the effectiveness of the proposed strategies.

Les travaux présentés dans cette thèse portent sur l'observabilité et l'observateur des systèmes dynamiques non linéaires. Elle comporte deux parties relativement indépendantes. La première partie traite le problème de la transformation d'un système d'Euler-Lagrange afin de résoudre un certain problème intéressant tel que la conception de l'observateur, la stabilisation par retour de sortie. Dans un premier temps, l'étude est consacrée au développement de l'équivalence affine des systèmes d'Euler-Lagrange. Nous caractérisons d'une manière rigoureuse une classe de ces systèmes. Nous aboutissons, d'une manière plus simple, aux résultats prouvés par Spong et Bedrossian en 1992. Dans un deuxième temps, nous apportons une contribution à la construction d'observateurs et la stabilisation par retour de sortie pour une classe de systèmes d'Euler-Lagrange en transformant un tel système en une forme triangulaire. Dans la deuxième partie, nous étudions le problème de la généricité de l'observabilité différentielle pour les systèmes discrets avec entrée, sachant que l'état et l'entrée évoluent dans des variétés compactes et connexes. En conséquence, nous montrons la densité (pour la topologie de Whitney) de l'ensemble des systèmes fortement différentiellement observables lorsque la dimension de l'espace des sorties est strictement supérieure a celle de l'espace des entrées, et que l'on observe (2n + I ) valeurs successives de la sortie, 02 n est la dimension de l'espace des états

The problem of forecasting the behavior of a complex dynamical system through analysis of observational time-series data becomes difficult when the system expresses chaotic behavior and the measurements are sparse, in both space and/or time. Despite the fact that this situation is quite typical across many fields, including numerical weather prediction, the issue of whether the available observations are "sufficient" for generating successful forecasts is still not well understood. An analysis by Whartenby et al. (2013) found that in the context of the nonlinear shallow water equations on a β plane, standard nudging techniques require observing approximately 70 % of the full set of state variables. Here we examine the same system using a method introduced by Rey et al. (2014a), which generalizes standard nudging methods to utilize time delayed measurements. We show that in certain circumstances, it provides a sizable reduction in the number of observations required to construct accurate estimates and high-quality predictions. In particular, we find that this estimate of 70 % can be reduced to about 33 % using time delays, and even further if Lagrangian drifter locations are also used as measurements.

The data assimilation process, in which observational data is used to estimate the states and parameters of a dynamical model, becomes seriously impeded when the model expresses chaotic behavior and the number of measurements is below a critical threshold, Ls. Since this problem of insufficient measurements is typical across many fields, including numerical weather prediction, we analyze a method introduced in Rey et al. (2014a, b) to remedy this matter, in the context of the nonlinear shallow water equations on a β-plane. This approach generalizes standard nudging methods by utilizing time delayed measurements to augment the transfer of information from the data to the model. We will show it provides a sizable reduction in the number of observations required to construct accurate estimates and high-quality predictions. For instance, in Whartenby et al. (2013) we found that to achieve this goal, standard nudging requires observing approximately 70 % of the full set of state variables. Using time delays, this number can be reduced to about 33 %, and even further if Lagrangian drifter information is also incorporated.

The goal of this paper is to apply some concepts and techniques from algebraic geometry to study the observability of nonlinear continuous-time polynomial systems. After deriving some new results for observability and embeddings, it is shown how to use such concepts to easily design high-gain observers. The proposed technique is illustrated by an application to the well known Rossler oscillator.

This paper focuses on the design of nonlinear observers for discrete-time systems by means of a so called extended nonlinear observer canonical form which is computed via a nonlinear observability canonical form. In contrast to other approaches by Brodmann 1994 and Lin and Byrnes 1995 using a two-step-transformation, past measurements of the system output are used. This allows to extend the class of systems for which an observer can be designed and leads to several observers with different characteristics. An application to a pendulum on a cart shows the efficiency of the design method.

Information in measurements of a nonlinear dynamical system can be transferred to a quantitative model of the observed system to establish its fixed parameters and unobserved state variables. After this learning period is complete, one may predict the model response to new forces and, when successful, these predictions will match additional observations. This adjustment process encounters problems when the model is nonlinear and chaotic because dynamical instability impedes the transfer of information from the data to the model when the number of measurements at each observation time is insufficient. We discuss the use of information in the waveform of the data, realized through a time delayed collection of measurements, to provide additional stability and accuracy to this search procedure. Several examples are explored, including a few familiar nonlinear dynamical systems and small networks of Colpitts oscillators.

Transferring information from observations to models of complex systems may meet impediments when the number of observations at any observation time is not sufficient. This is especially so when chaotic behavior is expressed. We show how to use time-delay embedding, familiar from nonlinear dynamics, to provide the information required to obtain accurate state and parameter estimates. Good estimates of parameters and unobserved states are necessary for good predictions of the future state of a model system. This method may be critical in allowing the understanding of prediction in complex systems as varied as nervous systems and weather prediction where insufficient measurements are typical.

A notion of local observability, which is natural in the context of nonlinear input/output regulation, is introduced. A simple characterization is provided, a comparison is made with other local nonlinear observability definitions, and its behavior under constant-rate sampling is analyzed.

A method is described for the synchronization of nonlinear discrete-time dynamics. The methodology consists of constructing observer–receiver dynamics that exploit at each time instant the drive signal and buffered past values of the drive signal. In this way, the method can be viewed as a dynamic reconstruction mechanism, in contrast to existing static inversion methods from the theory of dynamical systems. The method is illustrated on a few simulation examples consisting of coupled chaotic logistic equations. Also, a discrete-time message reconstruction scheme is simulated using the extended observer mechanism.

We study the Newton observer design, developed by Moraal and Grizzle (1995), when the exact discrete-time model of the sampled-data system is not available analytically. The hybrid reconstruction of the Newton observer in (Arcak, 2006) eliminates the dependence on this exact discrete-time model by numerical integration and finite difference Jacobian approximations. This paper reduces the computational cost of (Arcak, 2006) using an inexact Newton method and the generalized minimum residuals (GM-RES) algorithm. It then studies how a line search scheme can be incorporated in this modified Newton observer to enlarge the region of convergence

The authors address the problem of observability of polynomial discrete-time systems. The ideal theoretic definition is translated to effective computations in terms of Grobner bases. Linear system observability is a special case, and for general polynomial systems n samples are needed to determine observability, where n is the state dimension. The formulation yields a decision criterion as well as an implicit form of an observer

Using methods from nonlinear dynamics and interpolation techniques from applied mathematics, we show how to use data alone to construct discrete time dynamical rules that forecast observed neuron properties. These data may come from simulations of a Hodgkin-Huxley (HH) neuron model or from laboratory current clamp experiments. In each case, the reduced-dimension, data-driven forecasting (DDF) models are shown to predict accurately for times after the training period.
When the available observations for neuron preparations are, for example, membrane voltage V(t) only, we use the technique of time delay embedding from nonlinear dynamics to generate an appropriate space in which the full dynamics can be realized.
The DDF constructions are reduced-dimension models relative to HH models as they are built on and forecast only observables such as V(t). They do not require detailed specification of ion channels, their gating variables, and the many parameters that accompany an HH model for laboratory measurements, yet all of this important information is encoded in the DDF model. As the DDF models use and forecast only voltage data, they can be used in building networks with biophysical connections. Both gap junction connections and ligand gated synaptic connections among neurons involve presynaptic voltages and induce postsynaptic voltage response. Biophysically based DDF neuron models can replace other reduced-dimension neuron models, say, of the integrate-and-fire type, in developing and analyzing large networks of neurons.
When one does have detailed HH model neurons for network components, a reduced-dimension DDF realization of the HH voltage dynamics may be used in network computations to achieve computational efficiency and the exploration of larger biological networks.

The dynamic behavior of many processes is characterized by time delays due to measurement delays, which put strict limitations on the performance of the control system. In this paper a time-delay factorization strategy for the nonlinear model predictive control (NMPC) and state estimation of multiple-input multiple-output (MIMO), nonlinear, open-loop unstable processes having output-measurement delays, and subject to unmeasured disturbances is proposed.At first, the NMPC algorithm based on a nonlinear programming approach is presented. Then, on-line parameter-identification and state-estimation schemes are combined with the NMPC algorithm to maintain the process at a steady-state which is unstable for the open-loop system. Finally, the effectiveness of the proposed method is demonstrated via simulation on the control of a catalytic continuous stirred tank reactor (CSTR).

In variational formulations of data assimilation, the estimation of parameters or initial state values by a search for a minimum of a cost function can be hindered by the numerous local minima in the dependence of the cost function on those quantities. We argue that this is a result of instability on the synchronization manifold where the observations are required to match the model outputs in the situation where the data and the model are chaotic. The solution to this impediment to estimation is given as controls moving the positive conditional Lyapunov exponents on the synchronization manifold to negative values and adding to the cost function a penalty that drives those controls to zero as a result of the optimization process implementing the assimilation. This is seen as the solution to the proper size of ‘nudging’ terms: they are zero once the estimation has been completed, leaving only the physics of the problem to govern forecasts after the assimilation window.
We show how this procedure, called Dynamical State and Parameter Estimation (DSPE), works in the case of the Lorenz96 model with nine dynamical variables. Using DSPE, we are able to accurately estimate the fixed parameter of this model and all of the state variables, observed and unobserved, over an assimilation time interval [0, T]. Using the state variables at T and the estimated fixed parameter, we are able to accurately forecast the state of the model for t > T to those times where the chaotic behaviour of the system interferes with forecast accuracy. Copyright

In this article, we propose a Moving-Horizon State-Estimation method, applied to a neural dynamical process model. Firstly, the approach chosen to represent a nonlinear dynamical system by a neural network is explained. After that, the MHSE method, used to perform the state estimation, is presented. The algorithm performances are showed on a biotechnological process. The combination of the MHSE method and the neural network permits a particularly efficient estimation of the state of the process. with a nonlinear model easy to build thanks to the neural network, and with an easy tuning due to the choice of the MHSE method.

Stable M~l'tm.~: u~l h*,h" o eke

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Genetic observability of differentiable systems

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