Datta & Sarkissian, 2000 ).
The present paper describes a direct and iterative procedure to get a minimal order for a single **linear** **functional** observer. On the one hand the word direct means that the design method is not based on the solution of the Sylvester equation (4) . It has been underlined in Tsui ( 1998 ), that the calculus of the matrix T is not a necessary step. This point is a specific feature of the procedure we propose. On the other hand the word iterative indicates that we test an increasing sequence for the orders of the observers to obtain minimality. Moreover, the procedure points out if we face to the stable observer case or if some poles can be fixed at the outset. The paper is organised as follows. In the first place, a necessary and sufficient condition is outlined for the existence of a single **functional** observer. From these conditions, in the second section a design method for the observer is proposed. An example illustrates the procedure and points out that the minimal order depends on constraints on the poles of the observer.

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V. C ONCLUSION
A minimal single **linear** **functional** observers design for discrete-time LTI systems using the direct approach has been addressed. The observer is designed so that an asymptotic **functional** observer can be obtained with arbitrary convergence speed. A numerical example and simulation illustrated the effectiveness of the proposed approach. In this example, we have points out that we can fix the observation error at any desired rate.

From this state space representation, a **linear** **functional** observer has been designed in order to estimate the temperature in any desired point using few measurements and the knowledge of inputs. This kind of observer induces a relevant reduction in the observer order comparing to the initial system dimension. It has been demonstrated that the observer was able to accurately estimate the temperature evolution of a desired point, whatever the initial conditions. Moreover, it has been examplified that the order of the observer was linked to the relative positions of points of interest and to the dimension of the state space representation of the system.

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The main objective of this paper was to show the feasibility of our approach to design fast and efficient observers. Our future development will be to establish 3D thermal models for the observation of a complete power electronic module. However, these models will be naturally of large dimension and state space representations will be huge. Thus, it could be supposed that even the **linear** **functional** observer will not drastically reduce the order of the problem. In this case, the observer designed from experimentally identified transfers could be studied. Using small order transfers, the associated observer would have a limited dimension. This particular point will be developed in further work.

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Chapter 3 recasts the energy bound method as a method for linear functional output bounds for Poisson's equation, simultaneously extending the energy bound to more rele[r]

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Towards a Complete Design of **Linear** **Functional** Observers B. Larroque, F. Noureddine, F. Rotella
Abstract - This paper provides a procedure for the design of a reduced order observer of a state
**linear** **functional** for a **linear** lime-invariant system. The case, defined in[!], where the observer order p is given by the number m of single independenl **linear** functionals to be observed, is called in this paper the minimum case where p = m. The minimum case is revisited and numerically simpl{fied The aim of this paper is lo extend the minimum case to the case where m < p < n-l, narned minimal case. A constructive procedure is given to design the **linear** **functional** observer.

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Datta & Sarkissian, 2000 ).
The present paper describes a direct and iterative procedure to get a minimal order for a single **linear** **functional** observer. On the one hand the word direct means that the design method is not based on the solution of the Sylvester equation (4) . It has been underlined in Tsui ( 1998 ), that the calculus of the matrix T is not a necessary step. This point is a specific feature of the procedure we propose. On the other hand the word iterative indicates that we test an increasing sequence for the orders of the observers to obtain minimality. Moreover, the procedure points out if we face to the stable observer case or if some poles can be fixed at the outset. The paper is organised as follows. In the first place, a necessary and sufficient condition is outlined for the existence of a single **functional** observer. From these conditions, in the second section a design method for the observer is proposed. An example illustrates the procedure and points out that the minimal order depends on constraints on the poles of the observer.

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´
Equipe-Projet AT-SOP
Rapport de recherche n° 7214 — February 2010 — 56 pages
Abstract: Serre’s reduction aims at reducing the number of unknowns and equations of a **linear** **functional** system (e.g., system of partial differential equations, system of differential time-delay equations, system of difference equations). Finding an equivalent representation of a **linear** **functional** system containing fewer equations and fewer unknowns generally simplifies the study of its structural properties, its closed-form integration as well as of different numerical analysis issues. The purpose of this paper is to present a constructive approach to Serre’s reduction for determined and underdetermined **linear** **functional** systems.

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In particular, these conditions always hold in the case of a univariate Ore algebra over a field of coefficients i.e., ordinary differential/difference systems over the field of rational[r]

1. Introduction
Since Luenberger’s works ( Luenberger , 1963 , 1964 , 1966 ) a significant amount of research has been devoted to the problem of observing a **linear** **functional** of the state of a **linear** time-invariant system. The main developments are detailed in O’Reilly ( 1983 ), in Aldeen and Trinh ( 1999 ), Trinh and Fernando ( 2007 ) and Tsui (1985 , 1998) and, in the recent books Korovin and Fomichev ( 2009 ) and Trinh and Fernando ( 2012 ) and the reference therein. The problem at first glance can be formulated as follows. For the **linear** state-space model

(4)
with F a Hurwitz matrix, then an asymptotic estimation of the **linear** **functional** v(t) = Lx(t) can be get by a Luenberger observer (3) whose parameters are determined to ensure the existence of matrices T and F . A lot of methods exist to find, in some particular cases, a minimum q-order observer with the eighenvalues of matrix F fixed at the outset [22, 31]. But the general problem of the design of an asymptotic observer which order is as small as possible has not found a complete solution yet.

The part of the many-body ground state living in the low energy space will be dealt with using a quantitative version of the finite dimensional quantum de Finetti theorem [22, 12, 11, 28, 31]. This result roughly says that the reduced density matrices of any N -body state can be approximated by that of a convex combination of product states |u ⊗N ihu ⊗N |. The energy being a **linear** **functional** of the 2-body density matrix, it is then easy to obtain the Hartree energy as a lower bound. The approximation error due to this procedure is proportional to the dimension of the low-lying energy space and inversely proportional to the number of particles. A crucial step therefore consists in optimizing over the energy cut-off L (which governs the dimension of the low energy subspace) to minimize the error due to the use of the de Finetti theorem.

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The part of the many-body ground state living in the low energy space will be dealt with using a quantitative version of the finite dimensional quantum de Finetti theorem [22, 12, 11, 28, 31]. This result roughly says that the reduced density matrices of any N -body state can be approximated by that of a convex combination of product states |u ⊗N ihu ⊗N |. The energy being a **linear** **functional** of the 2-body density matrix, it is then easy to obtain the Hartree energy as a lower bound. The approximation error due to this procedure is proportional to the dimension of the low-lying energy space and inversely proportional to the number of particles. A crucial step therefore consists in optimizing over the energy cut-off L (which governs the dimension of the low energy subspace) to minimize the error due to the use of the de Finetti theorem.

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value equal to 10.5.
4.2 Application to the weather data
Ramsay and Silverman (2005) introduce the Canadian Temperature data set as one of their main examples of **functional** data. For 35 weather stations, the daily temperature and precipitation were averaged over a period of 30 years. The goal is to predict the complete log daily precipitation proﬁle of a weather station from information on the complete daily temperature proﬁle. To demonstrate the performance of the proposed RKHS **functional** regression method, we illustrate in Figure 4 the prediction of our RKHS estimate and LFR estimate for four weather stations. The ﬁgure shows improvements in prediction accuracy by RKHS estimate. The RRSS value of RKHS estimate is lower than LRF estimate in Montreal (1.52 → 1.37) and Ed- monton (0.38 → 0.25) station. RRSS results obtained in Prince Rupert station are all most equal to 0.9. We obtained a best RRSS value using LRF estimate than RKHS only in Resolute station. However using our method we can have more information about the shape of the true curve. Unlike **linear** **functional** re- gression estimate, our method deals with nonparamet- ric regression, it doesn’t impose a predeﬁned structure upon the data and doesn’t require a smoothing step which has the disadvantage of ignoring small changes in the shape of the true curve to be estimated (see ﬁgure 4).

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Rewriting the necessary and sufficient conditions for the existence of a stable UIFO for a **linear** system expressed in [23], a direct and iterative procedure to get a minimal order for a single stable **linear** **functional** observer has been proposed in [24] and for multi-**functional** observers in [25]. The herein presented paper extends these results towards minimality of LUIFO. The proposed procedure is simple, iterative and is not based on the solution of so-called Sylvester equation or on the use of canonical state space forms. The term iterative indicates that an increasing sequence for the order of the observer is tested to obtain a possible minimal order. The main feature of the proposed design procedure is the highlighting of some degrees of freedom to place some poles of the obtained observer.

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The first estimate in ( 22 ) states that, except around t = 0 and t = T , the discrepancies x(t) − ¯ x, p x (t) − ¯ p x and u(t) − ¯ u are bounded above by T 1 , which is small as T is large, but not exponentially
small: it is weaker than ( 11 ), and we speak of a **linear** turnpike estimate. The second estimate in ( 22 ) gives a bound (uniform with respect to T ) on the discrepancy y(t) − ¯ y(t) along the whole interval [0, T ]: it says that y(t) remains at a uniform distance of ¯ y(t) = y 0 + tg(¯ x, ¯ u) as t ∈ [0, T ]. Finally, the third estimate in ( 22 ) says that the constant p y − ¯ p y is (linearly) small as T is large.

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model that cluster individuals in two categories. The first step consist of a multivariate **functional** principal component analysis applied on the variance- covariance matrix on the **functional** data and their first derivatives. Then, the obtained scores are used as covariates in a generalized **linear** model to pre- dict the outcome.Yamamoto and Hwang (2017) propose a clustering method that combines a subspace separation technique with **functional** subspace clus- tering, named FGRC, that is less sensible to data variance than **functional** principal component k -means developed by Yamamoto (2012) and **functional** factorial k -means (Yamamoto and Terada (2014)). Finally, Jacques and Preda (2014b) present a Gaussian model-based clustering method based on a prin- cipal component analysis for multivariate **functional** data (MFPCA). One of the benefits of this method is that the dependency between **functional** vari- ables is managed thanks to the MFPCA. More recently, new methods based on a mix between dimension reduction and nonparametric approaches appear. Indeed, Traore et al. (2019) propose a clustering technique for nuclear safety experiment where one individual curve is decomposed into two new curves that are used in the decision making process. The first step consists in doing a dimension reduction technique on the first curves and applying a hierarchical clustering on those obtained values. Then, a semi-metric is build to compare the second curves, and the clusters are refining thanks to this comparison. But, even if this method is developped to deal with two curves for a same individual, at first the **functional** data are univariate.

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Ramsay and Silverman ( 2005 )’s book for a deep view on **Functional** Data Analysis (FDA
in short). For simplicity, the **functional** data considered throughout the paper correspond to the observations of independent realizations of X at the same points.
In this particular context, **functional** anomaly detection aims at detecting the curves that significantly differ from the others among the dataset available. Given the richness of spaces of functions, the major difficulty lies in the huge diversity in the nature of the observed differences, which may not only depend on the locations of the curves. Following in the footsteps of Hubert et al. ( 2015 ), one may distinguish between three types of anomalies: shift (the observed curve has the same shape as the majority of the sample except that it is shifted away), amplitude or shape anomalies. All these three types of anomalies can be isolated/transient or persistent, depending on their duration with respect to that of the observations. One may easily admit that certain types of anomalies are harder to detect than others: for instance, an isolated anomaly in shape compared to an isolated anomaly in amplitude (i.e. change point). Although FDA has been the subject of much attention in recent years, very few generic and flexible methods tailored to **functional** anomaly detection are documented in the machine-learning literature to the best of our knowledge, except for specific types of anomalies (e.g. change-points).

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1 Introduction
**Linear** algebra methods [Diaz 01, Murata 89, Reisig 82] based on state equation and invariants are a powerful tool for Petri net analysis. But to find **linear** invariants and to solve the fundamental equation of Petri net we have to solve **linear** diophantine systems in nonnegative integer numbers. All known methods of such systems solution [Colom 90, Contejean 97, Kryviy 99, Martinez 82, Schrejver 91, Toudic 82] possess exponential complexity with respect to space. It makes the analysis of large-scale models practically unfeasible and requires searching of new techniques, which provide essential speed-up of computations.

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Finally, in order to extend the sucient stability condition, a more ac- curate result can be conjectured, as a fractional version of the Hartman{ Grobman theorem, namely:
Theorem 3 (Conjecture). The local stability of the equilibrium x = 0 of the non-**linear** fractional dierential system d x = f ( x ) is governed by the global stability of the linearized system near the equilibrium d x = x , where