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On the number of samples necessary to achieve observability

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Abstract

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 ...
Book
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. ...
Thesis
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. ...
Conference Paper
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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). ...
Preprint
Full-text available
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. ...
Article
Full-text available
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. ...
Article
Full-text available
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]. ...
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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. ...
Article
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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 . ...
Article
Full-text available
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]. ...
Article
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]). ...
Article
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]. ...
Article
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]. ...
Article
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. ...
Article
Full-text available
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. ...
Article
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 . ...
Conference Paper
Full-text available
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. ...
Article
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. ...
Preprint
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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]. ...
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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.
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