2.1 Objectives

The goal of the project is to better understand and well formalize the effects of complex environments on the dynamics of the interconnections, as well as to develop new methods and techniques for the analysis and control of such systems.

It is well-known that the interconnection of dynamic systems has as consequence an increased complexity of the behavior of the total system.

In a simplified way, as the concept of dynamics is well-understood, the interconnections can be seen as associations (by connections of materials or information flows) of distinct systems to ensure a pooling of the resources with the aim of obtaining a better operation with the constraint of continuity of the service in the event of a fault. In this context, the environment can be seen as a collection of elements, structures or systems, natural or artificial constituting the neighborhood of a given system. The development of interactive games through communication networks, control from distance (e.g. remote surgical operations) or in hostile environment (e.g. robots, drones), as well as the current trend of large scale integration of distribution (and/or transport and/or decision) and open information systems with systems of production, lead to new modeling schemes in problems where the dynamics of the environment have to be taken into account.

In order to tackle the control problems arising in the above examples, the team investigates new theoretical methods, develops new algorithms and implementations dedicated to these techniques.

3.1 Analysis of interconnected systems

The major questions considered are those of the characterization of the stability (also including the problems of sensitivity compared to the variations of the parameters) and the determination of stabilizing controllers of interconnected dynamic systems. In many situations, the dynamics of the interconnections can be naturally modelled by systems with delays (constant, distributed or time-varying delays) possibly of fractional order. In other cases, partial differential equations (PDE) models can be better represented or approximated by using systems with delays. Our expertise on this subject, on both time and frequency domain methods, allows us to challenge difficult problems (e.g. systems with an infinite number of unstable poles).

• Robust stability of linear systems

  Within an interconnection context, lots of phenomena are modelled directly or after an approximation by delay systems. These systems may have constant delays, time-varying delays, distributed delays.

  For various infinite-dimensional systems, particularly delay and fractional systems, input-output and time-domain methods are jointly developed in the team to characterize stability. This research is developed at four levels: analytic approaches ($H_{\infty }$-stability, BIBO-stablity, robust stability, robustness metrics) 1, 2, 6, 7, symbolic computation approaches (SOS methods are used for determining easy-to-check conditions which guarantee that the poles of a given linear system are not in the closed right half-plane, certified CAD techniques), numerical approaches (root-loci, continuation methods) and by means of softwares developed in the team 6, 7.

• Robustness/fragility of biological systems

  Deterministic biological models describing, for instance, species interactions, are frequently composed of equations with important disturbances and poorly known parameters. To evaluate the impact of the uncertainties, we use the techniques of designing of global strict Lyapunov functions or functional developed in the team.

  However, for other biological systems, the notion of robustness may be different and this question is still in its infancy (see, e.g. 85). Unlike engineering problems where a major issue is to maintain stability in the presence of disturbances, a main issue here is to maintain the system response in the presence of disturbances. For instance, a biological network is required to keep its functioning in case of a failure of one of the nodes in the network. The team, which has a strong expertise in robustness for engineering problems, aims at contributing at the develpment of new robustness metrics in this biological context.

3.2 Stabilization of interconnected systems

• Linear systems: Analytic and algebraic approaches are considered for infinite-dimensional linear systems studied within the input-output framework.

  In the recent years, the Youla-Ku$\check{\mathrm{c}}$era parametrization (which gives the set of all stabilizing controllers of a system in terms of its coprime factorizations) has been the cornerstone of the success of the $H_{\infty }$-control since this parametrization allows one to rewrite the problem of finding the optimal stabilizing controllers for a certain norm such as $H_{\infty }$ or $H_2$ as affine, and thus, convex problem.

  A central issue studied in the team is the computation of such factorizations for a given infinite-dimensional linear system as well as establishing the links between stabilizability of a system for a certain norm and the existence of coprime factorizations for this system. These questions are fundamental for robust stabilization problems 1, 2.

  We also consider simultaneous stabilization since it plays an important role in the study of reliable stabilization, i.e. in the design of controllers which stabilize a finite family of plants describing a system during normal operating conditions and various failed modes (e.g. loss of sensors or actuators, changes in operating points). Moreover, we investigate strongly stabilizable systems, namely systems which can be stabilized by stable controllers, since they have a good ability to track reference inputs and, in practice, engineers are reluctant to use unstable controllers especially when the system is stable.

• Nonlinear systems

  In any physical systems a feedback control law has to account for limitation stemming from safety, physical or technological constraints. Therefore, any realistic control system analysis and design has to account for constraints appearing mainly from sensors and actuators nonlinearities and from the regions of safe operation in the state space. This motivates the study of linear systems with more realistic, thus complex, models of actuators accounting for saturation and quantization at the inputs of the system 12, 8, 11.

  The project aims at developing robust stabilization theory and methods for important classes of nonlinear systems that ensure good controller performance under uncertainty and time delays. The main techniques include techniques called backstepping and forwarding, contructions of strict Lyapunov functions through so-called "strictification" approaches 4 and construction of Lyapunov-Krasovskii functionals 5, 6, 7 or or Lyapunov functionals for PDE systems 10.

3.3 Synthesis of reduced complexity controllers

• PID controllers

  Even though the synthesis of control laws of a given complexity is not a new problem, it is still open, even for finite-dimensional linear systems. Our purpose is to search for good families of “simple” (e.g. low order) controllers for infinite-dimensional dynamical systems. Within our approach, PID candidates are first considered in the team 2, 90.

  For interconnected systems appearing in teleoperation applications, such as the steer-by-wire, Proportional-Derivative laws are simple control strategies allowing to reproduce the efforts in both ends of the teleoperation system. However, due to delays introduced in the communication channels these strategies may result in loss of closed loop stability or in performance degradation when compared to the system with a mechanical link (no communication channel). In this context we search for non-linear proportional and derivative gains to improve performance. This is assessed in terms of reduction of overshoot and guaranteed convergence rates.

• Delayed feedback

  Control systems often operate in the presence of delays, primarily due to the time it takes to acquire the information needed for decision-making, to create control decisions and to execute these decisions. Commonly, such a time delay induces desynchronizing and/or destabilizing effects on the dynamics. However, some recent studies have emphasized that the delay may have a stabilizing effect in the control design. In particular, the closed-loop stability may be guaranteed precisely by the existence of the delay. The interest of considering such control laws lies in the simplicity of the controller as well as in its easy practical implementation. It is intended by the team members to provide a unified approach for the design of such stabilizing control laws for finite and infinite dimensional plants 3, 9.

• Finite Time and Interval Observers for nonlinear systems

  We aim to develop techniques of construction of output feedbacks relying on the design of observers. The objectives pertain to the design of robust control laws which converge in finite time, the construction of intervals observers which ensure that the solutions belong to guaranteed intervals, continuous/discrete observers for systems with discrete measurements and observers for systems with switches.

Finally, the development of algorithms based on both symbolic computation and numerical methods, and their implementations in dedicated Scilab/Matlab/Maple toolboxes are important issues in the project.

4.1 Analysis and Control of life sciences systems

The team is involved in life sciences applications. The two main lines are the analysis of bioreactors models (microorganisms; bacteria, microalgae, yeast, etc..) and the modeling of cell dynamics in Acute Myeloblastic Leukemias (AML) in collaboration with St Antoine Hospital in Paris.

4.2 Energy Management

The team is interested in Energy management and considers control problems in energy networks.

4.3 Transportation Systems

The team is interested in control applications in transportation systems. In particular, the problem of collision avoidance of autonomous vehicels has been investigated under the framework of Time Varying systems. The goal is to obtain closed-loop control laws that guarantee the execution of a trajectory under uncertainties such as road and vehicle conditions.

5.1 New software

6.1 Spectral properties and partial pole placement for time-delay systems

Participants: Amina Benarab [DISCO], Catherine Bonnet [DISCO], Islam Boussaada [DISCO], Hugo Cavalera [DISCO], Yacine Chitour [L2S, Univ. Paris-Saclay, France], Sébastien Fueyo [DISCO], Ayrton Hammoumou [IPSA, France], Julien Huynh [IPSA, France], Guilherme Mazanti [DISCO], Titouan Millet [IPSA, France], Silviu-Iulian Niculescu [DISCO], Jayvir Raj [IPSA, France].

Linear time-invariant time-delay systems are infinite-dimensional and, as such, their spectra contains infinitely many values, which can be characterized as sets of roots of quasipolynomials. Understanding the location of such roots provides valuable information on the stability of the underlying time-delay system and is a fundamental step for the design of stabilizing feedback control laws 84, 89.

Several techniques for the stability analysis of time-delay systems through spectral methods developed on the DISCO team along the years were implemented in the Matlab toolbox YALTA 77. This toolbox was converted along the previous years into a Python toolbox, called YALTAPy, for which a friendly online graphical user interface, YALTAPy_Online was developed during the last year. The publication 45 provides a description of YALTAPy and YALTAPy_Online, presenting their main features and instructions of use.

Pole placement represents a classical method for controlling finite-dimensional linear time invariant systems. Basically, it consists of placing the poles of the closed-loop system in some predetermined loci in the complex plane. An overview of the extensions of this method to linear systems described by delay-differential equations is provided in 41. Among others, the finite spectrum assignment, the continuous pole placement and the partial pole placement approaches are presented and their limitations discussed. s

Since the seminal works 78, 79, it has been known that roots of quasipolynomials of large multiplicity are often dominant, a property which came to be known as the multiplicity-induced-dominancy (MID) property. The most general result concerning the MID property for roots with the maximal possible multiplicity of systems with a single delay was established in 17, showing that, in such a situation of maximal multiplicity, the MID property always holds true. The proof of such a result relies on the links between the spectrum of a time-delay system with a root of maximal multiplicity and Kummer confluent hypergeometric funcions. By exploiting in addition properties of Padé approximations of the exponential function, the recent work 58 has presented a simplified proof of that result.

Another challenging problem is the study of the MID property for roots with multiplicity smaller than the maximal one (IMID for Intermediate Multiplicity Induced Dominancy). By exploring in more details the ideas from 17, the work 43 provides new insights for the dominance of roots with multiplicity smaller than the maximal one, using the Green–Hille transformation to provide interesting results on the distribution of nonasymptotic zeros of linear combinations of Kummer functions. Also, the MID and IMID properties for second-order neutral time-delay differential equations with a single delay have been fully characterized in 14. The control methodology is summarized in a five-steps algorithm that can be exploited in the design of higher-order systems. It has been shown how the stabilization of the classical oscillator benefits from the obtained results.

Studying the MID property for systems with more than one delay is a challenging question, which is addressed in 71 for first-order systems with two delays. By considering the ratio between the smallest and the largest delay as a parameter, 71 provides a careful analysis of the behavior of the spectrum of the system with respect to this parameter, which is used to establish the MID property for roots of maximal multiplicity of such a class of systems. As a consequence, 71 also establishes that the inclusion of a second delay may help in stabilizing time-delay systems with constraints on their coefficients, with respect to a classical proportional-delayed controller.

Stabilization methods for time-delay systems based on the MID property, as well as on the related coexisting-real-roots-induced-dominancy (CRRID) property, are implemented in the DISCO Team software P3$\delta$, which is constantly updated with new features both to include new results available in the litterature and to improve user experience with the software. The paper 44 provides a description of the features recently implemented in the software, including in particular delaied studies of some examples of application of P3$\delta$ in the stabilization of systems stemming from applications.

6.2 PID tuning for controlling delayed dynamics

Participants: Catherine Bonnet, Islam Boussaada, Jianqui Chen [L2S, City University of Hong Kong, China], Jie Chen [L2S, City University of Hong Kong, China], Dan Ma [Northeastern University, China], Silviu-Iulian Niculescu.

Nowadays, the PID controller is the most used in controlling industrial processes. In [ma:hal-02479679], the MID property which is merely a delayed-output-feedback where the candidates delays and gains result from the manifold defining the maximal multiplicity of a real spectral value, is employed in the PID tuning for delayed plants. More precisely, the controller gains $(k_p,k_i,k_d)$ are tuned using the intentional multiplicity algebraic constraints allowing the stabilization of unstable delayed plants. The specificity of such a design is related to the analytical assignment of the closed-loop solution decay rate additionally to its robustness with respect to uncertain parameters.

6.3 Linear switched systems with deterministic or probabilistic switching

Participants: Yacine Chitour [L2S, Univ. Paris-Saclay, France], Guilherme Mazanti [DISCO], Pierre Monmarché [LJLL & LCT, Sorbonne Université, France], Mario Sigalotti [Inria & LJLL, Sorbonne Université, France].

Due to interaction with discrete phenomena, several control systems may operate under the presence of switching signals, which typically have an important impact on stability (see, e.g., 87). Among the strategies used in the literature to address the effects on stability of unknown switching signals, one can find probabilistic models, such as Markov switching, as well as robust notions of stability, such as stability under arbitrary switching. While it is clear that the latter is in general a more conservative measure of stability than the former, understanding in which situation there is a real advantage in considering the probabilistic framework is a nontrivial question.

This questions was addressed for continuous-time switched systems in 70. More precisely, that work compares deterministic and probabilistic Lyapunov exponents for linear switched systems in finite dimensions. The major difficulty in this analysis comes from understanding the limiting behavior of arbitrarily fast switching. From the probabilistic point of view, this amounts to analyzing the behavior of Markov processes with fast switching, which is done in that work by nontrivial adaptations of results from various works by C. Landim and collaborators, in particular 86, which consider instead Markov processes under slow switching. Thanks to this analysis, 70 shows that the Markovian switching framework tends to be “more stable” than the robust deterministic point of view, except in some very particular situations, which are completely characterized in dimensions 2 and 3. Partial results are also provided in higher dimensions.

6.4 Controllability of difference equations

Participants: Yacine Chitour [L2S, Univ. Paris-Saclay, France], Sébastien Fueyo [Inria & LJLL, Sorbonne Université, France], Guilherme Mazanti [DISCO], Mario Sigalotti [Inria & LJLL, Sorbonne Université, France].

Difference equations are mathematical models for systems in which the state at a given time is a function of the same state at previous times and of control inputs. They can be seen as the simplest form of neutral time-delay systems, in which no derivatives are present in the model, and they also appear naturally when studying some hyperbolic partial differential equations in one space dimension. When its delays are commensurable, the controllability of a difference equation is easy to study, and can be characterized by a Kalman-like criterion 82. However, the problem is more delicate for difference equations with uncommensurable delays. The work 69 makes use of the realization theory, and in particular a series of works by Y. Yamamoto, to provide characterizations of exact and approximate controllability in $L^p$ spaces. Approximate controllability in $L^p$ is completely characterized by a Hautus-type criterion, and the “closure” of such a criterion is shown to be a sufficient condition for exact controllability in $L^1$. A key step to establish these results is to study the range of the endpoint map, which is done thanks to the representation formula for solutions of difference equations provided in 82 and a generalization of the Cayley–Hamilton theorem to multivariate polynomials, and also yields upper bounds on the controllability time.

6.5 State transition matrix

Participants: Michael Malisoff [LSU], Frédéric Mazenc.

Determining fundamental matrices of time-varying linear systems is in general a difficult task, notably when disturbances are present. This prompted us to construct upper and lower bounds for the fundamental matrices. This problem is studied in 29, 32 and 47. In 29, for both continuous-time and discrete-time systems, bounds are provided and used to solve stabilization and observer design problems. In 32 and 47, we obtained less conservative bounds than those of 29 and we introduced the notion of "full Metzler matrix" and used it to produce bounds given by simple expressions. The results have been applied to a control problem for a model arising from a marine robotic application.

6.6 Event-triggered control

Participants: Frédéric Mazenc, Michael Malisoff [LSU], Corina Barbalata [LSU], Zhong Ping Jiang [Tandon School].

Event-triggered control has the advantage that it can reduce computational burdens of implementing feedback controls, by only changing control values when a significant enough event occurs. In order to decrease the number of needed switches of the control laws, we developed several results relying on the theory of the positive systems and comparison systems called interval observers. In several papers, 24, 25, 26, 27, 28, we addressed the case of continuous-time linear systems with delay, discrete-time systems, time-varying systems.

6.7 Halanay's inequality

Participants: Frédéric Mazenc, Michael Malisoff [LSU], Miroslav Krstic [Univ. California].

We developed in the contributions 31, 34 and 30 a stability analysis techniques to enable to prove stability in cases where traditional techniques, such as Lyapunov techniques, do not apply. It is based on variant of the celebrate Halanay's inequality, which is especially useful for ordinary differention equations in which a time-varying delay is inserted. In particular, we obtained vector versions of Halanay's inequality result.

6.8 Stability Analysis of Delay Systems with Lyapunov-Krasovskii methods

Participants: Giorgio Valmorbida, Ali Diab [Ampère - Lyon], William Pasillas-Lepine [Laboratoire des Signaux et Systèmes].

In the context of tele-operated systems, in particular for steer-by-wire systems, delays are critical for the stability of the closed loop. To design non-linear control strategies that allow to satisfy the requirements related to driving comfort, we need design methods based on a time-domain approach, in particular approaches based on Lyapunov-Krasovskii function computation. However, the existing numerical methods to compute Lyapunov-Krasovskii functions to certify stability of a delay system are limited since they do not correspond to the necessary and sufficient conditions of the analytical conditions.

We formulated a numerical test based on projections on generic function basis to compute the parameters of the Lyapunov Delay function. Our tests encompass existing approaches and provide superior results for examples in the literature in terms of delay margin enlargement. We are currently working towards the extension of these analysis results to the control design problems and to the analysis of discrete-time system.

6.9 Stability analysis of Piece-Wise Affine Systems

Participants: Giorgio Valmorbida, Joao Manoel Gomes da Silva Jr [Universidade Federal do Rio Grande do Sul], Francesco Ferrante [Università di Perugia].

To overcome these difficulties we have proposed an implicit representation for this class of systems in terms of ramp functions. The main advantage of such a representation lies on the fact that the ramp function can be exactly characterized in terms of linear inequalities and a quadratic equation, namely a linear complementarity condition. Thanks to the characterization of the ramp function and the implicit description of the PWA system the verification of Lyapunov inequalities related to piecewise quadratic functions can be cast as a pair of linear matrix inequalities. We are now applying the results to the local analysis and synthesis of PWA control laws. These results generalise the local analysis of saturating systems and avoid the complexity of PWA system analysis which is currently based on enumeration of the transition in a PWA partition.

We have also shown that the step function can be written as an ill-posed algebraic loop with two ramp functions. We were thus able to unify the analysis of continuous and discontinuous PWA functions and thus to perform the analysis with partition-based Lyapunov functions. As a first case to apply the developed methodology, we have studied the class of systems with ternary inputs. Thanks to the representation in terms of ramp functions, we are working towards an unified representation for the evaluation, analysis and design of discontinuous PWA dynamics, including switched systems.

More recently, we have shown that Convex Quadratic Programs can be written as an implicit equation involving ramp functions. For specific examples of QP-Model Predictive Control, we obtained solutions to the implicit equation with a speed up factor of 20-100 with respect to a standard QP interior-point solver or over active set methods using compiled code. We are now investigating the relations of other finite-step convergence algorithms for PWA equations and the method we proposed. Moreover, we aim to formalize the approximation schemes obtained with simple iterative methods to obtain rigorous bounds to approximate MPC strategies and assess its effect when these approximations are used in closed loop.

6.10 Advanced observers design: Certified decay and finite-time convergence

Participants: Frédéric Mazenc, Silviu Niculescu, Michael Malisoff [LSU].

In order to estimate the state of a dynamical system, we address in 60 the problem of designing an observer for linear time-invariant (LTI) dynamical systems including a time-delay. Sufficient conditions for the existence of the proposed observer are given using partial placement of the poles for the error. Namely, we exploit the multiplicity-induced-dominance property of the characteristic function corresponding to the system's error. The effectiveness of the proposed observer design is shown in both state lag and input lag respectively through the problems of Mach number control in a wind tunnel and stabilization of the inverted pendulum on a cart by using the delay

To improve asymptotic observers which give a valuable information only after a transient period which can be long, we designed observers that converge in a fixed finite time or almost in fixed finite time.

In 35, we studied continuous-time nonlinear systems with discrete measurements, model uncertainty, and sensor noise and provided an estimator of the state for which the observation error enjoys a variant of the exponential input-to-state stability property with respect to the model uncertainty and sensor noise. The main result is illustrated via a model for a pendulum whose suspension point is subjected to an unknown time-varying bounded horizontal oscillation.

In 23, we provided observers for a class of nonlinear systems which are not required to be affine in the unmeasured states. The observers ensure exponential convergence of the observation errors to zero. The rate of exponential convergence converges to infinity, as the growth rate of the nonlinear state-dependent part of the dynamics converges to zero, so we called the observers almost finite-time. A global and a local result are given in accordance of the type of the considered nonlinearities. We applied the results to a model of a pendulum with friction, and to dynamics with Lotka-Volterra nonlinearities.

In 33, we designed a reduced order observer for nonlinear dynamics. When continuous output measurements are available, we proved that our observers converge in a fixed finite time in the absence of perturbations, and we prove a robustness result under uncertainties in the output measurements and in the dynamics, which bounds the observation error in terms of bounds on the uncertainties. The observers contain a dynamic extension with only one pointwise delay, and they use the observability Gramian to eliminate an invertibility condition that was present in earlier finite time observer designs. We also provided analogs for cases where the measurements are only available at discrete times. We illustrated the advantages of our new observers using a DC motor dynamics

In 46, we provided a new class of finite-time observers for continuous-time nonlinear systems that are affine in the unmeasured state variable and that also contain unknown constant parameters. Our method provides exact values of the state variables and of the constant parameters, in a fixed finite time. We also provided analogs where there is a known delay in the measurements, and for discrete-time systems. Our example illustrates the ease with which our assumptions can be verified in practice.

6.11 Mean field games

6.12 Study of SIR models

Participants: Frédéric Mazenc, Michael Malisoff [LSU], Hiroshi Ito [Kyushu Institute].

In the paper 21, we studied the stability of a model of susceptible, infected, and recovered (or SIR) disease that includes quarantine of infected individuals and mass vaccination. We proposed a global strict Lyapunov function construction to design feedback controls to asymptotically stabilize a desired endemic equilibrium, and to prove a robustness result of 'input-to-state stability' type for the dynamics with a suitable restriction on the disturbances. Simulations have been carried out to illustrate the potential of the proposed feedback controls to reduce peak levels of infected individuals.

6.13 Attitude stabilization problem

Participants: Frédéric Mazenc, Andrew Miller [Univ. of Texas], Maruthi Akella [Univ. of Texas].

In 48, we solved the attitude stabilization problem for fractionated space systems where the onboard computers cannot run continuously. An auxiliary filter is constructed to imitate the intermittent computer or state acquisition of a small size, power, and weight constrained spacecraft such a CubeSat. The proposed controller does not require angular rate or inertia knowledge and guarantees attitude stabilization and boundedness for all closed loop signals for persistently excited gains driving the auxiliary filter. We have constructed a novel Lyapunov function to guarantee stability including auxiliary functions for strictification of the controller.

6.14 Energy applications: flexibility management

Participants: Frédéric Bonnans [DISCO], Kang Liu [DISCO], Thibault Moquet [DISCO], Laurent Pfeiffer [DISCO], Nadia Oudjane [EDF R&D], Cheng Wan [EDF R&D].

We investigate in 67 a general class of large-scale and nonconvex optimization problems, involving an aggregative term. This term can be interpreted as the sum of the contributions of $N$ agents to some common good, with $N$ large. These problems are motivated by applications in energy management, when many flexibilities are involved. They are small storage devices which can be used to facilitate the integration of volatile renewable energy sources. At a theoretical level, we proved that when $N$ is large, the problem is well approximated by a convex problem, formulated on sets of probability measures. We propose a numerical method based on the Frank-Wolfe algorithm, relying of parallelizable resolutions of subproblems associated with each agent. We demonstrate global estimates of optimality for the output of the method after a given number of iterations. Numerical results on a toy example confirm the efficiency of the method, which outperforms commercial solvers for large values of $N$.

6.15 Biomechanical application: Modelling the central nervous system action

Participants: Amina Benarab [DISCO], Islam Boussaada [DISCO], Silviu Niculescu [DISCO], Ali El Ati [IPSA], Tamas Insperger [BME], Csenge Molnar [BME], Karim Trabelsi [IPSA].

In 59, we investigate the human quiet stance regulation problem using a single-link inverted pendulum model in the sagittal plane via the ankle joint's passive/active torques' actions. The active torque consists of ankle muscle contractions that are activated by the delayed action of the Central Nervous System (neural controller). The passive torque is related to the intrinsic mechanical properties of the muscle-tendon-ligament component. The failure of the human quiet stance is then directly related to the failure of one or both types of torques. We propose to model the neural controller as a delayed Proportional-Derivative-Acceleration controller acting on the ankle joint's angular position. By using the multiplicity-induced-dominancy property, the critical time delay of the motor control and the critical ankle-joint stiffness are both investigated.

In 42 we consider the stabilization of a rolling balance board by means of the multiplicity-induceddominancy property. A two degree-of-freedom mechanical model of a human balancing on a rolling balance board is analyzed in the sagittal plane. The human body is modeled by an inverted pendulum which connects to the balance board through the ankle joint. The system is stabilized by the ankle torque managed by the central nervous system (CNS). The action of the CNS is modeled by a delayed full state feedback: a pointwise delay stands for all latencies in the neuromechanical system (reaction time, neuromechanical lag, etc.), achieving a good occurrence in terms of the decay rate, it shows the links with dominancy and with the exponential stability property of the solution.

6.16 Mechatronic application: Visual control of UAV

Participants: Amina Benarab [DISCO], Islam Boussaada [DISCO], José Castillo-Zamora [DISCO], Juan Escareno [Xlim].

We have exploited the effects of time-delays on the stability of Unmanned Aerial Vehicles (UAVs). In this regard, the main contribution is a symbolic/numeric application of the Multiplicity-Induced-Dominancy (MID) property in the control of UAVs rotorcrafts featuring time-delays. The MID property is considered to address two of the most representative aerial robotic platforms: a classical quadrotor vehicle and a quadrotor vehicle endowed with tilting-rotors. The aforementioned property leads to an effective delayed feedback control design (MID tuning criteria), allowing the system to meet prescribed behavior conditions based on the placement of the rightmost root of the corresponding closed-loop characteristic function/quasipolynomial. Lastly, the results of detailed numerical simulations, including the linear and non-linear dynamics of the vehicle, are presented and discussed to validate the proposal.

7.1 International initiatives

7.2 International research visitors

7.3 National initiatives

We are members of the ANR Dreamy

7.4 Regional initiatives

Laurent Pfeiffer chairs the PGMO (programme d'optimisation Gaspard Monge) project “Large-scale and non-convex multi-agent optimization for energy management" (other DISCO participants: Frédéric Bonnans, Kang Liu, and Thibault Moquet). Industrial partners: Nadia Oudjane (EDF R&D) and Cheng Wan (EDF R&D). Funding in 2022: 7000 euros.

8.1 Promoting scientific activities

8.2 Teaching - Supervision - Juries

8.3 Popularization

9.1 Major publications

• 1 articleC.Catherine Bonnet, A. R.André R. Fioravanti and J. R.Jonathan R. Partington. Stability of Neutral Systems with Commensurate Delays and Poles Asymptotic to the Imaginary Axis.SIAM Journal on Control and Optimization492March 2011, 498-516
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• 2 articleC.C. Bonnet and J.J.R. Partington. Stabilization of some fractional delay systems of neutral type.Automatica432007, 2047--2053
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• 3 articleI.Islam Boussaada, I.-C.Irinel-Constantin Morarescu and S.-I.Silviu-Iulian Niculescu. Inverted pendulum stabilization: characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains.Systems Control Lett.822015, 1--9
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• 4 bookM.M. Malisoff and F.F. Mazenc. Constructions of Strict Lyapunov Functions.Communications and Control Engineering SeriesSpringer-Verlag London Ltd.2009
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• 5 articleF.F. Mazenc, M.M. Malisoff and S.-I.S.-I. Niculescu. Reduction Model Approach for Linear Time-Varying Systems with Delays.IEEE Transactions on Automatic Control5982014, 2068--2014
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• 6 bookW.Wim. Michiels and S.-I.Silviu-Iulian. Niculescu. S.-I.Silviu-Iulian NiculescuW.Wim MichielsStability, Control, and Computation for Time-Delay Systems.Philadelphia, PASociety for Industrial and Applied Mathematics2014
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• 7 bookS.-I.S.-I. Niculescu. Delay Effects on Stability: a Robust Control Approach.269Lecture Notes in Control and Information SciencesSpringer2001
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• 8 articleI.Isabelle Queinnec, S.Sophie Tarbouriech, G.Giorgio Valmorbida and L.Luca Zaccarian. Design of Saturating State-Feedback with Sign-Indefinite Quadratic Forms.IEEE Transactions on Automatic Control2021
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• 9 bookM. B.Martha Belem Saldivar, I.Islam Boussaada, H.Hugues Mounier and S.-I.Silviu-Iulian Niculescu. Analysis and Control of Oilwell Drilling Vibrations.Springer International PublishingMay 2015
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• 10 articleG.G. Valmorbida, M.M. Ahmadi and A.A. Papachristodoulou. Stability Analysis for a Class of Partial Differential Equations via Semidefinite Programming.IEEE Transactions on Automatic Control616June 2016, 1649-1654
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• 11 inproceedingsG.Giorgio Valmorbida and F.Francesco Ferrante. On Quantization in Discrete-Time Control Systems: Stability Analysis of Ternary Controllers.CDC 2020 - 59th IEEE Conference on Decision and ControlIEEEJeju Island (virtual), South KoreaDecember 2020
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• 12 articleG.G. Valmorbida, S.S. Tarbouriech and G.G. Garcia. Design of Polynomial Control Laws for Polynomial Systems Subject to Actuator Saturation.IEEE Transactions on Automatic Control587July 2013, 1758-1770
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9.2 Publications of the year

9.3 Cited publications

• 76 articleY.Yves Achdou and M.Mathieu Laurière. Mean field games and applications: Numerical aspects.Mean field games2020, 249--307
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• 77 articleD.David Avanessoff, A. R.André R. Fioravanti and C.Catherine Bonnet. YALTA: a Matlab toolbox for the $H_{}$-stability analysis of classical and fractional systems with commensurate delays.IFAC Proceedings Volumes4625th IFAC Symposium on System Structure and Control2013, 839--844
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