It is well known that the spectral abscissa of a given dynamical is nothing but the corresponding solutions' exponential decay. The analytical characterization of the spectral abscissa for infinite dimensional dynamical systems is an old problem which is still nowadays a question of ongoing interest due to its links with stability problems. We produced several works in this topic dealing with reduced order retarded Time-delay systems and emphasized a property that we call multiplicity induced-dominancy. In the paper , the interest of using time-delay in the controller design as a control parameter is underlined and the way to assign a dominant spectral value is demonstrated. As a matter of fact, it is shown that the multiplicity of given spectral value may reach the degree of the corresponding quasipolynomial. Furthermore, when this holds, then using a particular factorization of the quasipolynomial, such a multiple spectral value is shown to be the corresponding spectral abscissa. A generalization of such a result to generic second order retarded equation with a single delay is established in . More precisely, a parametric characterization of the spectral abscissa is established using the principle argument theorem. Furthermore, in the work , the potential applicability of such a parametric characterization in controller design in concrete application is demonstrated. As a matter of fact, a third order retarded system modeling the dynamics of Mach number in a wind tunnel is considered and a delayed controller design based on the spectral abscissa assignment is proposed.

It is well known in dynamical system theory that real spectral values correspond to non oscillating solutions. In the paper we made a connexion between the degree of a given quasipolynomial and the admissible number of non oscillating modes for the corresponding Time-delay system. More precisely, we have shown that the assignment of at most $n$ real spectral values is possible for generic quasipolynomial function of degree $n$. Namely, explicit formulas on the quasipolynomial's coefficients guaranteeing the coexistence of $n$ negative spectral values are obtained. Furthermore, a new quasipolynomial factorization technique, analogous to the one we developed for multiple spectral values for the proof of the dominancy of $n$ distinct negative spectral values is obtained.

In the paper a robust alternative of the delayed resonator is proposed by spectral approach where a double root assignment at the excitation frequency is proposed. Such an excitation frequency is projected to widening the stop-band in the active absorber frequency response. It is shown that the performance sensitivity to the mismatch between the design and true excitation frequency is considerably decreased. Additionally, the overall scheme is supplemented by a control loop which improves the stability margin.

The behavior of characteristic roots of time-delay systems, when the delay is subject to small variations is investigated in . We performed an analysis by means of the Weierstrass polynomial which are employed to study the stability behavior of the characteristic roots with respect to small variations on parameters. Analytic description and splitting properties of the Puiseux series expansions of critical roots are characterized by allowing a full description covering all the cases that can be encountered.

In the paper the migration of double imaginary roots of the systems characteristic equation when two parameters are subjected to small deviations is geometrically investigated. Under the least degeneracy assumptions, the local stability crossing curve is shown to have a cusp at the point that corresponds to the double root, which divides the neighborhood of this point into two sectors (called S-sector and a G-sector). We have shown that when the parameters move into the G-sector, one of the roots moves to the right half-plane, and the other moves to the left half-plane. However, when the parameters move into the S-sector, both roots move either to the left half-plane or the right half-plane depending on the sign of a quantity that depends on the characteristic function and its derivatives up to the third order.

Retarded dynamical systems with delay dependent coefficients is a class of systems which is frequently encountered in various scientific and engineering applications. The paper provides an overview of the stability analysis of such systems which generalizes those on systems with delay-independent coefficients. Methods of analysis for systems with a single delay and commensurate delays are presented, their application to output feedback control and a geometric perspective that establishes a link between systems with and without delay-dependent coefficients.

The paper presents a systematic method to analyse the stability of systems with single delay in which the coefficient polynomials of the characteristic equation depend on the delay. With respect to the literature on the topic, a less restrictive method to analyse stability is presented. It is found that a much richer behavior is possible when the restrictive assumptions are removed. The interval of interest for the delay is partitioned into subintervals so that the magnitude condition generates a fixed number of frequencies as functions of the delay within each subinterval. The crossing conditions are expressed in a general form, and a simplified derivation for the first-order derivative criterion is obtained.

Through an LMI approach it was possible to determine envelopes and use them not only to study stability but to design robust controllers for retarded and neutral time-delay systems. The controller designed is robust to parametric uncertainties and can guarantee delay independant stability or delay-dependant $\alpha -stability$ .

The Walton and Marshall method allows to determine stability windows of delay systems of the retarded and neutral type. We noticed that some delay systems of the neutral type do not behave as claimed in and analyzed carefully the position of the poles of such systems in the right half-plane.

An important aspect of nonlinear systems is the fact that stability might only be a local property. This means that associated to a stable equilibrium point or periodic trajectory, there is a region of attraction. Such a region is formed by points of trajectories converging to the stable sets. An important task of practical interest is then to estimate these regions via numerical methods that rely on the model of the system. As an illustration, it might be of interest to know the region of safe operation of an electric motor in order to preserve its integrity or, in the case of an autonomous vehicle, limit the operating condition for safety purposes.

For the particular class of Lurie systems, namely systems defined by the interconnection of a linear system and a static nonlinearity, it is possible to compute estimates based on sector inequalities characterizing the nonlinearities in the system. If further information, such as the slope of the nonlinearity is available, one can better characterize local properties such as regions of stability, and input-output relations such as reachable sets and local nonlinear gains.

To obtain these characterizations we rely on numerical methods based on convex optimization. These methods are based on the solution of Lyapunov inequalities yielding Lyapunov functions that are quadratic on both the states and the nonlinearity and has an integral term on the nonlinearity .

Moreover, whenever a more precise characterization of the nonlinearity is at hand as for instance nonlinearities having rational Jacobian, one can generalize the local analysis methods using polynomial optimization. This includes the case of standard Lurie systems by considering the interconnection of a polynomial system with static sector nonlinearities that have rational Jacobian. In this setting we have proposed conditions that relax the requirement on the candidate Lyapunov function , which serve as stability certificates, from being sum-of-squares of polynomial with respect to the nonlinearities and the Lurie-Postnikov terms from being non-negative.

Further to the stability analysis we were interested in another important phenomenon and its analysis through numerical methods : the existence of limit cycles on nonlinear systems. Such a phenomenon is relevant since it can be used as a method to design stable oscillators with known amplitude and frequency but also to evaluate and suppress undesirable oscillations in engineered systems. In order to proceed with this analysis we have limited our attention to a particular class of systems defined by a Liénard systems and formulate sufficient conditions for existence and uniqueness of limit cycles for systems with a non-differentiable vector field. As an application we consider the example of a linear system with saturation . Moreover, for planar saturating systems we present sufficient conditions for the existence of periodic orbits and we characterize inner and outer sets bounding the periodic orbits. A method to build these bounds, based on the solution to a convex optimization problem is proposed and numerical examples optimizing the region bounding the limit cycle illustrate the technique .

We worked on the problem of improving a fundamental control design technique for nonlinear systems called backstepping by using a fundamentally new approach which consists in introducing in the control artificial delays or using dynamic extensions.

In , we provided backstepping results for a large class of partially linear systems with an arbitrarily large number of integrators. We proposed control laws whose size respects some constaints given a priori. The key tool is a dynamic extension that contains only one artificial delay, which is in sharp contrast with our prior contributions. We also showed that the closed-loop system is robust, in the input-to-state stability sense, with respect to a large class of model uncertainties, and robust with respect to delays in the measurements. We illustrated the result using an example that is beyond the scope of classical backstepping.

The paper also provides a crucial backstepping result. We explained how globally asymptotically stabilizing output feedbacks can be constructed for a family of nonlinear systems using only a dynamic extension and a "Converging Input-Converging State" assumption and no additional delays. The technique presents several advantages. It provides control laws whose expressions are simple. It makes it possible to stabilize systems in the presence of uncertain terms, which are not necessarily of class $C^1$ and which prevent the use of the classical backstepping technique. It applies in cases where only part of the state variables can be measured.

The family of the switched systems is frequently encountered in practice. It can be used to approximate time-varying systems to ease their stability analysis or control.

In we provided theoretical results for the stability and robustness analysis of nonlinear switched time-varying systems with uncertainties and time-varying delays. The delays are allowed to be discontinuous and arbitrarily long with known upper bounds. We established the results via an adaptation of Halanay's inequality and a trajectory based technique. Also, we used the results for designing switched controllers that stabilize linear time-varying systems with time-varying delays.

The contribution proposed a new technique of construction of observers making possible to stabilize by output feedback a class of continuous-time switched linear systems with a time-varying delay in the output. The motivation of this paper is strong: frequently measurements are affected by pointwise time-varying delays. For stability analysis, we developped an extension of the trajectory based approach. A stability condition is given in terms of the upper bound on the time-varying delay to ensure global exponential stability of the switched feedback systems. It is worth observing that the main result applies in cases where some of the subsystems of the switched system are not stabilizable and not detectable.

The paper is also devoted to classes of nonlinear time-varying continuous-time systems with outputs. For a first family of systems, we builded an observer in the case where a state dependent disturbance affects the linear approximation. A fundamental feature of our observer is the fact that it converges after a predetermined finite time. When the disturbances are the zero functions, it provides exact values of the state and it provides an approximate estimate when there are nonzero disturbances. We used this construction to design a globally exponentially stabilizing dynamic output feedback for a second family of nonlinear systems whose outputs are only available on some finite time intervals. Our technique consists in switching between control laws. We applied the control design to the controlled Mathieu equation, which arises in the study of vibrations of an elliptic membrane.

The paper is devoted to a stability analysis for a class of nonlinear systems with a time-varying delay taking both large and small values in an alternating manner, precluding the application of most of the classical control design techniques. The type of assumption we imposed is the following: we imposed on the delay to be "small" on "long" time intervals and possibly "large" on "small" time-intervals. Bearing in mind this key property, we first introduced the concept of delay-hybrid-dependent stability, which grasps the features of the delays described above and represented the studied system as a system with a switched delay. Then by using switching techniques and Lyapunov-Krasovskii functionals (LKFs), we provided a new stability criterion.

We produced several works which pertain to the case where only a part of the state variables can be measured.

In the paper , we adopted a technique based on the indroduction of several observers in cascade (such a cascade is called 'sequential observer') for a class of time-varying linear systems in which the inputs and outputs containing sampling and arbitrarily long delays. The observers are of a continuous-discrete type. We used the observers to design controllers that ensure a strong robustness property with respect to uncertainties in the system and the output, under delays and sampling. A fundamental aspect of the approach is that it produces the observers and controllers without distributed terms. We have assessed the performance of the control laws through two examples, which inlcude a DC motor model that illustrates the utility of the work in engineering applications.

In two papers, we developped the theory of the finite time observers. In , we study a class of linear continuous-time time-varying systems with piecewise continuous disturbances and piecewise constant outputs. Under a classical assumption of observability, we designed a new type of observers to estimate the solutions of the system in a predetermined finite time. In contrast to the well-established finite time observer design techniques which estimate the system state using a continuous output, our proposed observer applies when only piecewise constant measurements are available. In , we construct finite-time reduced order observers for a broad family of nonlinear time-varying continuous-time systems. The motivation for this is the fact that in practice the time-varying aspect of a system may be an obstacle to the design of full-order finite-time observers, but not for the design of reduced order ones. We illustrated our results using a tracking problem for nonholonomic systems in chained form.

Two of our works present construction of asymptotic observers without delay. The paper solves an $H_{\infty }$ observer design problem for a class of descriptor nonlinear systems. The method we established is theoretical and can be applied to many automatic control design problems such as unknown input estimation problem, which plays an important role in control systems, namely for diagnosis and fault tolerant control. The design relies on the Linear Matrix Inequality condition (LMI) technique. We applied our resut to a model of a flexible joint robot system.

The work is dedicated to the design of a smooth six-degree-of-freedom observer to estimate the incorporating linear and angular velocity, called dual angular velocity, for a rigid body. The approach is based on the dual-quaternion description and we proved that the estimation errors exhibit asymptotic convergence. Furthermore, to achieve tracking control objective, we combined the proposed observer with an independently designed proportional-derivative-like feedback control law (using full-state feedback), and a special Lyapunov "strictification" process is employed to ensure a separation property between the observer and the controller. We performed numerical simulations for a prototypical spacecraft hovering mission application.

The presence of delays to big for being neglected is an obstacle to the design of stabilizing controllers in many cases. We have made efforts to overcome this challenge by developping several techniques.

In the paper , we investigated the design of a prediction-based controller for a linear system subject to a problematic time-varying input delay: the delay we considered is not necessarily "First-In/First-Out". The feedback law we proposed uses the current delay value in the prediction. It does not exactly compensate the delay in the closed-loop dynamics but does not require to predict future delay values, contrary to classical prediction techniques. Modeling the input delay as a transport Partial Differential Equation, we proved asymptotic stabilization of the system state, provided that the average $L_2$-norm of the first derivative of the delay over some time-window is sufficiently small and that the average time between two discontinuities (average dwell time) is sufficiently large.

In the paper , we adopted another type of strategy: we used a new sequential predictors approach to build uniformly globally exponentially stabilizing feedback controls for a large class of linear time-varying systems that contain an arbitrary number of different delays. This allows different delays in different components of the input. We illustrated our work in an example from identification theory, and in an Euler-Lagrange system arising from two-link manipulator systems.

The paper continues our works on the chemostat model with an arbitrary number of competing species, one substrate, and constant dilution rates. We allowed delays in the growth rates and additive uncertainties. Using constant inputs of certain species, we derived bounds on the sizes of the delays that ensure asymptotic stability of an equilibrium when the uncertainties are zero, which can allow persistence of multiple species. In the presence of delays and uncertainties, we provided bounds on the delays and on the uncertainties that ensure, with respect to uncertainties, the robustness property called "input-to-state stability".

Explicit Model Predictive Control often has a complex solution in terms of the number of regions required to define the solution and the corresponding memory requirement to represent the solution in the online implementation. An alternative approach to constrained control is based on the use of controlled contractive sets. However, polytopic controlled contractive sets may themselves be relatively complex, leading to a complex explicit solution, and the polytopic structure can limit the size of the controlled contractive set. Our recent results develop a method to obtain a larger controlled contractive set by allowing higher order functions in the definition of the contractive set, and explores the use of such higer-order contractive sets in controller design leading to a low complexity explicit control formulation.

The work pursues a framework of set characterization of dynamical systems with state-dependent disturbances. It aims to propose a new approach to analysis and design of nonlinear systems involving non-differentiability and asymmetric components which hamper application and effectiveness of local linearization methods. Several characterizations of ultimate bounds are developed. The utility of shifting the fix point is formulated as a parametrization of the ultimate bounds.

Several optimization-based control design techniques can be cast in the form of parametric optimization problems. The multi-parametric quadratic programming (mpQP) represents a popular class often related to the control of constrained linear systems. The complete solution to mpQP takes the form of explicit feedback functions with a piecewise affine structure, valid in polyhedral partitions of the feasible parameter space known as critical regions. The recently proposed combinatorial approach for solving mpQP has shown better efficiency than geometric approaches in finding the complete solution to problems with high dimensions of the parameter vectors. The drawback of this method, on the other hand, is that it tends to become very slow as the number of constraints increases in the problem. This work presents an alternative method for enumerating all optimal active sets in a mpQP based on theoretical properties of adjacent critical regions and their corresponding optimal active sets. Consequently, it results in excluding a noticeable number of feasible but not optimal candidate active sets from investigation. Therefore, the number of linear programs that should be solved decreases noticeably and the algorithm becomes faster. Simulation results confirm the reliability of the suggested method in finding the complete solution to the mpQPs while decreasing the computational time compared favourably with the best alternative approaches.

In the work , an aluminium-based flexible structure embedded in a mobile support subjected to an acceleration is considered. Such a flexible beam is equipped with two piezoelectric patches. One of them is used as an actuator and the second acts as a sensor. These patches are supposed to be rigidly bounded on the beam, one on each side, located at the clamped edge. The whole device is called a piezo-actuated beam which is generally modeled by Euler-Bernoulli equations. Finite element modeling is then applied to reduce the PDE system to a linear finite-dimensional system. Then, the peak of resonance of the first bending mode is damped by using a delayed output-feedback controller, without affecting the neglected vibrating modes in the reduced order model. The proposed controller design is based on the spectral abscissa characterization using the multiplicity property.

In this work and the following, we applied the technique of to problems arising from applications. The paper is devoted to the problem of stabilizing a nonlinear system approximated in a neighborhood of the origin by a saturated chain of integrators when the variables are not accurately measured. We used our control design to solve a control problem that arises in the context of vision based landing of a civil aircraft. In , we solved the problem of stabilizing a nonlinear system when the variables are not accurately measured and cannot be differentiated. The proposed method was first motivated and thus finally applied to the vision based control problem of a landing airliner.

The contribution is distinct from the papers mentioned above because it uses more traditional backstepping tools. It is devoted to power electronics devices. We proposed a nonlinear control law for a DC/DC boost converter dedicated to extract the maximum power from a photovoltaic (PV) array, taking into account the constraints of the control action. We performed simulations on SimPowerSystems to validate how the developed control strategy is able to properly control the converter.

A novel distributed approach to treat the wind farm (WF) power maximization problem accounting for the wake interaction among the wind turbines (WTs) is presented. Power constraints are also considered within the optimization problem. These are either the WTs nominal power or a maximum allowed power injection, typically imposed by the grid operator. The approach is model-based. Coupled with a distributed architecture it allows fast convergence to a solution, which makes it exploitable for real-time operations. The WF optimization problem is solved in a cooperative way among the WTs by introducing a new distributed particle swarm optimization algorithm, based on cooperative co-evolution techniques. The algorithm is first analyzed for the unconstrained case, where we show how the WF problem can be distributed by exploiting the knowledge of the aerodynamic couplings among the WTs. The algorithm is extended to the constrained case employing Deb's rule. Simulations are carried out on different WFs and wind conditions, showing good power gains and fast convergence of the algorithm. To appear in Rnewable Energy, 2019.

The inertial stabilisation of the line of sight of an imager fixed on a mobile carrier is considered in order to acquire good quality images despite the disturbances generated by the carrier.

A double stage mechanical stabilisation architecture is proposed, where a second stabilisation stage, based on a piezoelectric actuator, is added to the usual structure. The piezoelectric actuator transfer function and hysteresis are characterized through experiments.

In order to design the controllers of both stages, a high-level image quality criterion (the Modulation Transfer Function MTF) is considered, together with design constraints on the main variables of interest. The criterion and the constraints are evaluated by realistic simulations based on some input and noise profiles measured on a real-life system. The MTF evaluation being time-consuming, a Bayesian optimisation method specially dedicated to expensive-to-evaluate functions is used to obtain the parameters of the controllers. The obtain experimental results are displayed and their performances discussed. To appear in the International Journal of Systems and Sciences in 2019.

Demand-Side Management (DSM) strategies, such as load shifting and nighttime set-back, exploit the thermal inertia of buildings to make the operation of District Heating Systems (DHSs) more efficient. The control strategy requires a building model to assess the flexibility of buildings in handling demand modulation, without jeopardizing the thermal comfort. Reduced Order Models (ROMs) with few parameters are often used for this end; in many previous works their parameters have been identified using time-series data including indoor temperature measurements. However, at a city scale and due to privacy rights, such internal signals are usually unavailable. Thereby, identifying the ROM shall rely solely on measurements available at the substation level.

In our work, we develop and demonstrate a method respecting this practical constraint to identify a first and a second order building model. In literature, a rather simplified approach had been proposed to derive a first order building model from substation measurements. We compare the performance of our methodology with respect to the latter, using the same model structure. As for the second order model, its structure is more relevant to account for different dynamics in buildings equipped with hydronic heating systems or featuring important internal thermal inertia. Data used for the identification is restricted to the heat flux delivered from the DHS, both supply and return water temperatures, mass flowrate across the substation’s heat-exchangers and the outdoor temperature. Validation of the proposed approach is carried out using a representative white-box model of a building and its substation written in the Modelica language. Implementation of advanced control strategies for DHSs based on this model identification is in prospect.

Our project is about the modeling and analysis of healthy and unhealthy cell population dynamics, with a particular focus on hematopoiesis, which is the process of blood cell production and continuous replenishment. We point out that medical research is now looking for new combined targeted therapies able to overcome the challenge of cancer cells (e.g. to stop overproliferation, to restore normal apoptosis rates and differentiation of immature cells, and to avoid the high toxicity effects that characterize heavy non-selective chemotherapy). In that quest, the ultimate goal behind mathematical studies is to provide some inputs that should help biologists to suggest and test new treatment, and to contribute within multi-disciplinary groups in the opening of new perspectives against cancer. Thus, our research project is imbued within a similar spirit and fits the expectations of a better understanding of the behavior of healthy and unhealthy blood cell dynamics. It involve intensive collaboration with hematologists from Saint Antoine hospital in Paris, and aims to analyze the cell fate evolution in treated or untreated leukemia, allowing for the suggestion of new anti-leukemic combined chemotherapy.

Cells have amazing features that allow them to guide their development paths and determine their individual and collective fates. Dedifferentiation and transdifferentiation (cell plasticity) are little understood phenomena that allow cells to regress from an advanced differentiated state to a less differentiated one, including the case where cells lose their specific function and become stem cells.

We have introduced cell plasticity into a class of mathematical models we are interested in. We explored a new model involving a dedifferentiation function in the case of two cell maturity stages (stem cells and progeny). We have highlighted the role that dedifferentiation may have in the survival of cancer cells during therapy. The latter hypothesis appears to be in line with some medical observations .

We have also developed and analyzed a model taking into the account the fact that few cells of the proliferating compartment may be arrested during an unlimited time .

Migratory fluxes of humains and of insects of various species have favoured the spreading of diseases world-wide. In particular the Ae Aegypti and Ae Albopictus mosquitoes of the Aedes family are vectors able to transmit and spead among humans a variety of diseases: Dengue, Zika, Chikungunya, Yellow fever and the newly discovered Mayaro.

We have continued to analyze SIR models with time-varying parameters to predict dengue epidemics and compared numerical simulations with real data from Dengue epidemics in Rio de Janeiro in order to estimate the infectivity rate and predict what are the periods more at risk of infection , .