For engineers, a wide variety of information cannot be directly obtained through measurements. Some parameters (constants of an electrical actuator, delay in a transmission, etc.) or internal variables (robot's posture, torques applied to a robot, localization of a mobile robot, etc.) are unknown or unmeasured. In addition, usually the signals from sensors are distorted and tainted by measurement noises. In order to simulate, to control or to supervise processes, and to extract information conveyed by the signals, one has to estimate parameters or variables.

Estimation techniques are, under various guises, present in many parts of control, signal processing and applied mathematics. Such an important area gave rise to a huge international literature. From a general point of view, the performance of an estimation algorithm can be characterized by three indicators:

The computation time (the time needed to obtain the estimation). Obviously, the estimation algorithms should have as small as possible computation time in order to provide fast, real-time, on-line estimations for processes with fast dynamics.

The algorithm complexity (the easiness of design and implementation). Estimation algorithms should have as low as possible algorithm complexity, in order to allow an embedded real-time estimation (for example, in Internet of things, the embedded computation power is limited and can be even more limited for small sensors/actuators devices). Another question about complexity is: can an engineer appropriate and apply the algorithms? For instance, an algorithm application is easier if the parameters have a physical meaning w.r.t. the process under study.

The robustness. The estimation algorithms should exhibit as much as possible robustness with respect to a large class of measurement noises, parameter uncertainties, discretization steps and other issues of numerical implementation. A complementary point of view on robustness is to manage a compromise between existence of theoretical proofs versus universalism of the algorithm. In the first case, the performance is guaranteed in a particular case (a particular control designed for a particular model). In the second case, an algorithm can be directly applied in “most of the cases", but it may fail in few situations.

Within the very wide area of estimation, *Non-A POST* addresses 3 particular theoretical challenges:

1) Development of theory of dynamical homogeneous systems;

2) Estimate on-line the derivatives of a signal;

3) Design of control and estimation algorithms converging in finite and in fixed time.

All of them are connected with the central idea of designing or exploiting algorithms with the finite-time convergence property. In
particular, the *non-asymptotic* estimation techniques (numerical differentiation, finite-time differentiators or observers) constitute a central objective of the project, explaining the name *Non-Asymptotic estimation for on-line systems*. Below, these 3 challenges will be shortly described in relation to the above indicators.

The researches developed by *Non-A POST* are within the continuity of the project-teams *Non-A* and *ALIEN* in what concerns the *algebraic tools* that are developed for finite-time estimation purposes. However, *Non-A POST* also aims at developing complementary estimation techniques, still aiming at the finite-time performance but based on the so-called *higher-order sliding mode* algorithms, interval estimation techniques and, as well as, fixed-time algorithms.

*Non-A POST* also wants to confront these theoretical challenges with some application fields: Networked robots, Nano/macro machining, quadrotors, active flow control, *etc*. Today, most of our effort (*i.e.*, engineering staff) is devoted to the first item, according to the theme "Internet of Things" (IoT) promoted by Inria in its Strategic Plan for the Lille North-Europe research center.

Homogeneity is a property of mathematical objects, such as functions
or vector fields, to be scaled in a consistent manner with respect
to a scaling operation (called a dilation) applied to their argument
(a kind of symmetry). The first rise of homogeneity deals with homogeneous
polynomials investigated by L. Euler in 18^{th} century. In 50s and 60s more generic
notions of homogeneity (weighted and coordinate-free or geometric)
have been introduced by V.I. Zubov and his group.
For example, a function

for some *e.g*. for

denote a solution corresponding to the initial condition

So homogeneous systems possess several important and useful properties: their local behavior
is the same as global one, the rate of convergence to the origin can
be identified by degree of homogeneity, the stability is robust to various perturbations.
There are also plenty of researches performed in the last 30 years
and the members of *Non-A POST* team extended these notions of
homogeneity to discontinuous systems (in a geometric framework), time-delay systems, partial differential equations,
time-varying systems, and recently to discrete-time models (together with the concept of local homogeneity). They also proposed plenty of control and estimation algorithms based on homogeneity.

Advantages of homogeneous algorithms taking into account the above mentioned criteria:

**A1)** The rate of convergence in homogeneous systems can be qualified by its degree (finite-time and fixed-time for negative and positive degrees, respectively).

**A2)** Due to symmetry of these systems they admit special discretization tools (also developed by the members of *Non-A POST* team), which make simpler they realization for on-line scenarios.

**A3)** The internal symmetry of these dynamics makes them inherently robust with respect to external perturbations, measurement noises and delays, which is especially important in networked systems.

Estimating the derivative of a (noisy) signal with a sufficient accuracy can be seen as a key problem in domains of control and diagnosis, as well as signal and image processing. At the present stage of our research, the estimation of the *One of the open questions is about the robustness issues (Indicator 3) with respect to the the parameters and the numerical implementation choices.*

Two classes of techniques are considered here (**Model-based** and **Model-free**), both of them aiming at non-asymptotic estimation.

In what we call *model-based techniques*, the derivative estimation is regarded as an observation problem, which means the software-based reconstruction of unmeasured variables and, more generally, a left inversion problem

*Model-free techniques* concern the works initiated by *ALIEN* and *Non-A* teams, which rely on the only information contained in the output signal and its derivatives.

To design an estimation or control algorithm we have to select a performance
criterion to be optimized. Stability is one of the main performance
indexes, which has to be established during analysis or design of
a dynamical system. Stability is usually investigated
with respect to an invariant mode (*e.g*., an equilibrium, desired
trajectory or a limit cycle), then another important characteristics
is the time of convergence of the system trajectories to this mode,
which can be *asymptotic* (in conventional approaches) or *finite-time*
(being the focus of *Non-A POST* team). In the latter case the limit
mode has to be exactly established in a finite time dependent on initial
deviations (if such a time is independent on initial conditions, then
this type of convergence is called *fixed-time*). If the rate
of convergence is just faster than any exponential of time, then such
a convergence is called *hyperexponential*. The notion of finite-time
stability has been proposed in 60s by E. Roxin and it has been developed in many works later,
where a particular attention is paid to the time of convergence for
trajectories to a steady state (it is worth to note that there exists
another notion having the same name, *i.e*. finite-time or short-time
stability, which is focused on analysis of a dynamical system behavior on bounded intervals of time,
and it is completely different and not considered here). For example,
the following simple scalar dynamics:

has the solution

which possesses a finite-time convergence with the settling time *Non-A POST* team obtained many results on analysis and design of control
and estimation algorithms in this context. A useful and simple method to deal with these three types of convergence
(finite-time, fixed-time or hyperexponential) is based on the theory of *homogeneous* systems.

The application of the developed
control and estimation algorithms for different scenarios in IoT is a priority for *Non-A POST* team. Participation in different potential applications allows the team to better understand the features of IoT and their required
performances. A list of possible applications, partially already addressed
in the team, is as follows:

smart bivalve-based biosensor for water quality monitoring (ANR project WaQMoS): presence of persistent external perturbations, which are hard to measure, and important model uncertainty make application of conventional techniques complicated; another issue is consensus seeking between animals for a contamination detection;

control and estimation for flying vehicles, *e.g*. quadrotors
or blimps (1 PhD ONERA, 2 PhDs EC Lille): nonlinearity of the model
and its uncertainty coupled with important aerodynamic perturbations
have to be compensated by fast (finite- or fixed-time) and robust
control and estimation algorithms;

human behavior modeling and estimation with posterior design of algorithms for human-computer interaction (ANR project TurboTouch): robust finite-time differentiators demonstrate good estimation capabilities needed for prediction in this application;

human physiological characteristics estimation (like emotion detection, galvanic skin response filtering, fatigue evaluation in collaborations with Neotrope and Ellcie Healthy): intelligent robust filtering and finite-time distributed estimation are key features in this scenario;

path planning for autonomous vehicles taking into account behavior of humans (PhD CIFRE with SEQUEL team and Renault): application of interval estimation and prediction techniques to treat the uncertainty of the environment by reducing computational complexity of reinforcement learning;

flow control (in the framework of ContrATech subprogram of CPER ELSAT): the case of control and
estimation of a distributed-parameter system with very fast and uncertain
dynamics, where finite-time solutions developed by *Non-A POST* team are necessary

Involvement in various real-world scenarios will allow *Non-A POST* to develop
demonstrators of disposed technologies with application to IoT.

Gabriele Perozzi (a PhD student of the team) get the creativity prize of FR CNRS TTM (La Fédération de Recherche Transports Terrestres & Mobilité);

Hafiz Ahmed (a former PhD student of the team) is a winner of Annual European PhD Award on Control for Complex and Heterogeneous Systems.

*Adaptive Homogeneous Filtering*

Keywords: Automatic differentiation - Filtering

Functional Description: allows to reconstruct a signal based on derivatives estimation and to filter high amplitude and wide frequencies spectrum perturbations.

Contact: Denis Efimov

The problem of the synthesis of a homogeneous Lyapunov function for an asymptotically stable homogeneous system is studied in . First, for systems with nonnegative degree of homogeneity, several expressions of homogeneous Lyapunov functions are derived, which depend explicitly on the supremum or the integral (over finite or infinite intervals of time) of the system solutions. Second, a numeric procedure is proposed, which ensures the construction of a homogeneous Lyapunov function.

A transfer contract with Ellcie Healthy on intelligent filtering of measurements in smart eyeglasses.

*Non-A POST* team hosts CPER Data ControlHub (an on-line laboratory for control system experimentation) and participates at ContrATech subprogram of CPER ELSAT.

Inria Project Lab (IPL) IPL COSY.

ANR project Finite4SoS (Finite time control and estimation for Systems of Systems), coordinator: W. Perruquetti, 2015-2020.

ANR project WaQMoS (Coastal waters quality surveillance using bivalve mollusk-based sensors), coordinator: D. Efimov, 2015-2020.

ANR project TurboTouch (High-performance touch interactions), coordinator: G. Casiez (MJOLNIR team, Inria), 2014-2019.

ANR project DIGITSLID (DIGITal set-valued and homogeneous SLIding mode control and Differentiators: the implicit approach), coordinator: Bernard Brogliato (TriPOP team, Inria), 2018−2022.

ANR project ROCC-SYS (Robust Control of Cyber-Physical Systems), coordinator: L. Hetel (CNRS, EC de Lille), 2013-2018.

We are also involved in several technical groups of the GDR MACS (CNRS, "Modélisation, Analyse de Conduite des Systèmes dynamiques", see http://

UCoCoS: the objectives of the project are to create a control-oriented framework for complex systems, and to define a common language, common methods, tools and software for the complexity scientist. The principal investigators are: W. Michiels, J.-P. Richard and H. Nijmeijer.

Title: Homogeneity Tools for Sliding Mode Control and Estimation

International Partner (Institution - Laboratory - Researcher):

UNAM (Mexico)

Prof. Leonid Fridman

2016–2018

The team *Non-A POST* is developing an estimation theory, built around differential algebra and operational calculation on the one hand, and high gain algorithms (such as sliding mode) on the other hand. The Mexican partner team comes from "Sliding Mode Control" laboratory of UNAM. There exists a strong intersection of interests of both teams (application of homogeneity for design of sliding mode control and estimation algorithms, and analysis of finite-time stability). That is why there exists a long history of collaboration between these two teams. The goal of the project is development of control and estimation algorithms converging in fixed or in finite time by applying the last generation sliding mode techniques and the homogeneity theory. The project realization is planned in the form of short-time visits of permanent staff and visits of PhD students for a long period of stay. Such visits are very important for young scientists, and also help Non-A team to prepare and find good PhDs/post-docs for future.

RECoT, "Robust Estimation and Control with Time Constraints", 2018–2020

International Partner: IBM Research, Dublin (Dr. Sergiy Zhuk)

Non-A Post team of Inria deals with control and estimation of on-line (dynamical) systems with applications to robotics, biological systems, human-machine interfaces and active ow control. The key feature of the developed algorithms is a robustness and a non-asymptotic convergence allowing to fulfill some time constraints. The main methodology is a homogeneity (dilation symmetry) approach. IBM Research team develops minimax algorithms for state estimation and identification of dynamical systems with applications to computational fluid dynamics and image assimilation problems. The key feature of the resulting algorithms is the exact or approximate description of the reachability set of the underlying dynamical system in finite or infinite dimensions. The methodology is relies upon duality and Lyapunov exponents. The objectives of the collaboration are an exchange of the scientific knowledge and the joint research of the following problems: homogeneous observers design using minimax approach; development of fast and consistent computational algorithms for digital implementation of homogeneous controllers and observers; extension of sliding mode control methodology to infinite-dimensional models using minimax approach; the minimax observer-based control design for turbulent flows.

ITMO University, Saint-Petersburg, Russia

Tel-Aviv University, Tel-Aviv, Israel

CINVESTAV-IPN, Mexico, Mexico

Hangzhou Dianzi University, Hangzhou, China

Brandenburg University of Technology, Cottbus, Germany

Richard J.-P., EUCA-IEEE ECC, Limassol, Cyprus

Richard J.-P., IFAC TDS, Budapest, Hungary

Richard J.-P., IARA VEHICULAR, Venice, Italy

Efimov D., IFAC CHAOS, Eindhoven, Netherlands

Efimov D., IFAC MICNON, Guadalajara, Mexico

The members of the team serve as reviewers to all major conferences in the field: IEEE CDC, ECC, ACC *etc*.

Polyakov A., International Journal of Robust and Nonlinear Control

Polyakov A., Journal of Optimization Theory and Applications

Efimov D., IFAC Journal on Nonlinear Analysis: Hybrid Systems

Efimov D., Asian Journal of Control

Efimov D., IEEE Transactions on Automatic Control

The members of the team serve as reviewers to all major journals in the field: IEEE Trans. Automatic Control, Automatica, Systems & Control Letters, SIAM Journal on Optimization and Control, Int. Journal of Robust and Nonlinear Control *etc*.

Richard J.-P., Investigator for the CNRS FR TTM

Efimov D., Chair of EECI PhD Award