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Social distancing has been enacted in order to mitigate the spread of COVID-19. Like many authors, we adopt the classic epidemic SIR model, where the infection rate is the control variable. Its differential flatness property yields elementary closed-form formulae for open-loop social distancing scenarios, where, for instance, the increase of the number of uninfected people may be taken into account. Those formulae might therefore be useful to decision makers. A feedback loop stemming from model-free control leads to a remarkable robustness with respect to severe uncertainties and mismatches. Although an identification procedure is presented, a good knowledge of the recovery rate is not necessary for our control strategy.

In order to increase the number of situations in which an intelligent vehicle can operate without human intervention, lateral control is required to accurately guide it in a reference trajectory regardless of the shape of the road or the longitudinal speed. Some studies address this problem by tuning a controller for low and high speeds and including an output adaptation law. In this paper, a strategy framed in the Model-Free Control paradigm is presented to laterally control the vehicle over a wide speed range. Tracking quality, system stability and passenger comfort are thoroughly analyzed and compared to similar control structures. The results obtained both in simulation and with a real vehicle show that the developed strategy tracks a large number of trajectories with high degree of accuracy, safety and comfort.

We present some “in silico” experiments to design combined chemo- and immunotherapy treatment schedules. We introduce a new framework by combining flatness-based control, which is a model-based setting, along with model-free control. The flatness property of the used mathematical model yields straightforward reference trajectories. They provide us with the nominal open-loop control inputs. Closing the loop via model-free control allows to deal with the uncertainties on the injected drug doses. Several numerical simulations illustrating different case studies are displayed. We show in particular that the considered health indicators are driven to the safe region, even for critical initial conditions. Furthermore, in some specific cases there is no need to inject chemotherapeutic agents.

Due to severe mathematical modeling and calibration difficulties open-loop feedforward control is mainly employed today for wastewater denitrification, which is a key ecological issue. In order to improve the resulting poor performances a new model-free control setting and its corresponding “intelligent” controller are introduced. The pitfall of regulating two output variables via a single input variable is overcome by introducing also an open-loop knowledge-based control deduced from the plant behavior. Several convincing computer simulations are presented and discussed.

This communication presents a longitudinal model-free control approach for computing the wheel torque command to be applied on a vehicle. This setting enables us to overcome the problem of unknown vehicle parameters for generating a suitable control law. An important parameter in this control setting is made time-varying for ensuring finite-time stability. Several convincing computer simulations are displayed and discussed. Overshoots become therefore smaller. The driving comfort is increased and the robustness to time-delays is improved.

Recent advances in time series, where deterministic and stochastic modelings as well as the storage and analysis of big data are useless, permit a new approach to short-term traffic flow forecasting and to its reliability, i.e., to the traffic volatility. Several convincing computer simulations, which utilize concrete data, are presented and discussed.

Electricity production via solar energy is tackled via short-term forecasts and risk management. Our main tool is a new setting on time series. It allows the definition of "confidence bands" where the Gaussian assumption, which is not satisfied by our concrete data, may be abandoned. Those bands are quite convenient and easily implementable. Numerous computer simulations are presented.

Recent investigations on the longitudinal and lateral control of wheeled autonomous vehicles are reported. Flatness-based techniques are first introduced via a simplified model. It depends on some physical parameters, like cornering stiffness coefficients of the tires, friction coefficient of the road, ..., which are notoriously difficult to identify. Then a model-free control strategy, which exploits the flat outputs, is proposed. Those outputs also depend on physical parameters which are poorly known, i.e.,the vehicle mass and inertia and the position of the center of gravity. A totally model-free control law is therefore adopted. It employs natural output variables, namely the longitudinal velocity and the lateral deviation of the vehicle. This last method, which is easily understandable and implement able, ensures a robust trajectory tracking problem in both longitudinal and lateral dynamics. Several convincing computer simulations are displayed.

This communication is devoted to a “practical” comparison between high-order sliding modes and the recently introduced model-free control. The perfect knowledge of the relative degree of the output variable, which is a standard assumption for sliding modes, is assumed here. Our comparisons are based on two concrete case-studies and on numerous computer simulations. The smoothness of the input variables, the robustness with respect to noises and the straightforward extendibility of the model-free controllers to MIMO systems are highlighted.

This paper simplifies several aspects of the practical implementation of the newly introduced model-freecontrol and of the corresponding intelligentPID controllers (M. Fliess, C. Join, “Model-free control and intelligent PID controllers: towards a possible trivialization of nonlinear control?,” 15thIFAC Symp. System Identif, Saint-Malo, 2009). Four examples with their computer simulations permit to test our techniques.

Non-linear state estimation and some related topics like parametric estimation, fault diagnosis and perturbation attenuation are tackled here via a new methodology in numerical differentiation. The corresponding basic system theoretic definitions and properties are presented within the framework of differential algebra, which permits to handle system variables and their derivatives of any order. Several academic examples and their computer simulations, with online estimations, illustrate our viewpoint.

This paper deals with real time and possibly closed loop identification of delay systems. It is based on non-asymptotic algebraic estimation techniques initiated in (Fliess M., 2003). Numerical simulations with noisy data are provided.

A closed loop parametrical identification procedure for continuous-time constant linear systems is introduced. This approach which exhibits good robustness properties with respect to a large variety of additive perturbations is based on the following mathematical tools: module theory; differential algebra; operational calculus. Several concrete case-studies with computer simulations demonstrate the efficiency of our on-line identification scheme.

A closed loop parametrical identification procedure for continuous-time constant linear systems is introduced. This approach which exhibits good robustness properties with respect to a large variety of additive perturbations is based on the following mathematical tools: (1) module theory; (2) differential algebra; (3) operational calculus. Several concrete case-studies with computer simulations demonstrate the efficiency of our on-line identification scheme.

For constant linear systems we are introducing integral reconstructorsand generalized proportional-integral controllers, which permit to bypass the derivative term in the classic PID controllers and more generally the usual asymptotic observers. Our approach, which is mainly of algebraic flavour, is based on the module-theoretic framework for linear systems and on operational calculus in Mikusiński's setting. Several examples are discussed.

We are extending to linear recurrent codes, i.e., to time-varying convolutional codes, most of the classic structural properties of fixed convolutional codes. We are also proposing a new connection between fixed convolutional codes and linear block codes. These results are obtained thanks to a module-theoretic framework which has been previously developed for linear control.

For constant linear systems we are introducing integral reconstructors and generalized proportional-integral controllers, which permit to bypass the derivative term in the classic PID controllers and more generally the usual asymptotic observers. Our approach, which is mainly of algebraic flavour, is based on the module-theoretic framework for linear systems and on operational calculus in Mikusinski's setting. Several examples are discussed.

For constant linear systems we are introducing integral reconstructors and generalized proportional-integral controllers, which permit to bypass the derivative term in the classic PID controllers and more generally the usual asymptotic observers. Our approach, which is mainly of algebraic flavour, is based on the module-theoretic framework for linear systems and on operational calculus in Mikusiński's setting. Several examples are discussed.

We derive PID-like regulators utilizing flatness-based predictive control (M. Fliess and R. Marquez, 2000, Continuous-time linear predictive control and flatness: a module-theoretic setting with examples, vol. 73, 606-623), generalized PI controllers (M. Fliess, R. Marquez, E. Delaleau, and H. Sira-Ramirez, 2001, Correcteurs proportionnels-intégraux généralisés, ESAIM: Control, Optimisation, and Calculus of Variations) and a kind of predictors which are reminiscent of Smith predictors. Our formalism is analogous to the two-sided Laplace transform.

Two important control schemes, PID and Model Predictive Control, are analyzed. Their respective characteristics are combined to propose a PID strategy based on predicted system trajectories. It is shown that predicted input and output trajectories can be easily generated using the flat outputs of the system. Our approach is illustrated by two examples, a simple DC motor and an identified cement mill model.

In this paper we use results and techniques from the theory of rational power series to show that the complement of a one-letter stochastic language is stochastic, but that the family of stochastic languages is closed neither under union and intersection nor under product and homomorphism. We also give a condition on the poles of a rational one-variable power seriesr to ensure that the stochastic language defined byr and any cut-point is rational.

R.E. Kalman introduced the concept of controllability for finite-dimensional linear systems at the beginning of the sixties. It has played since then a key role. A vast literature has been devoted to various extensions to nonlinear and infinite-dimensional systems. We first interpret Kalman's controllability in a new module-theoretic language, which is independent of any distinction between the system variables. Its nonlinear extension leads to the notion of differentially flat systems, which were introduced in 1992 by J. Lévine, P. Martin, P. Rouchon, and the author. They permit a straightforward trajectory tracking, which has been illustrated by numerous applications. In 1995 H. Mounier worked out the case of linear delay systems and discovered a new type of controllability which is verified by most realistic case-studies. We will conclude with some examples of systems governed by partial differential equations, which were recently studied by H. Mounier, P. Rouchon, J. Rudolph, and the author.

By employing differential vector spaces, we obtain results for linear systems with constant or varying coefficients and with lumped or distributed parameters which were only known for constant linear systems described by rational transfer matrices. Questions such as the characterization of input and output, input-output equivalence, invertibility, feedback decoupling and realization are investigated.

The motion planning and the synthesis of a tracking controller of a flexible robot arm is studied in an algebraic framework using Mikusiński's operational calculus. Experimental results are reported.

Differential algebra permits the generalization of several features of symplectic formalism to mechanics with perfect nonholonomic constraints.

The stability of an elementary type of stochastic differential equations driven by multidimensional noises is explained by using non-commutative power series.

We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.

Recent theoretical advances in nonlinear systems theory insist on the relevance of dynamics where control derivatives appear. We discuss here a simplified model of a crane which exhibits such a description.

A module-theoretic approach to linear controllability is shown to be equivalent to Willems' trajectory characterization. We also sketch a computational-free derivation of Brunovsky's canonical form and show how to relate the notion of numerical state-space to our algebraic standpoint.

The controllability of linear delay differential systems is attacked via module theory. Many of the notions which have been proposed in the literature are interpreted and compared via a change of the ground ring. New definitions are also suggested which, as demonstrated by several case studies, turn out to be often encountered in practice.

In engineering, as well as in physics, one often encounters implicit systems of ordinary differential equations of the form Fi(ẏ1…ẏm, y1…ym) = 0, i = 1,…, m, in the unknowns y1…ym, where the Jacobian matrix (∂Fi/∂ẏj)i,jis identically singular. We first state a condition under which the above equation may be solved. Then, we give a decomposition, which provides as a byproduct a clear-cut definition of the index. Since the Fi's must be differentiated an arbitrary number of times, we employ tools stemming from the differential geometry of infinite jets and prolongations like Lie-Bäcklund morphisms.

Several kinds of finite poles and zeros are defined and their relationships are studied in an intrinsic manner. Some of these definitions are more precise than the usual ones (all kinds of multiplicities of an input-output decoupling zero are defined here). In addition, we believe that our approach clarifies those notions, and it makes it possible to very concisely establish fundamental relationships between various poles and zeros. Some new relationships are derived. The notions of inputoutput stability and transfer stability are also introduced. Several examples illustrate the theory.

Les notions de dérivations issues de l’analyse fonctionnelle classique ne conviennent pas toujours à une évolution dynamique. Nous montrons qu’une dérivation, introduite par R. Ree à propos des intégrales itérées de K.-T. Chen, possède une interprétation infinitésimale qui prend naturellement en compte la causalité. Pour cette dérivation, le développement en série génératrice non commutative apparaît comme un développement taylorien.

Tensoring the module of a linear system with the quotient field of the ring of linear differential operators yields an algebraic framework where transfer matrices most naturally appear and which easily applies to the time-varying case. Several basic structural properties are examined : transfer algebra, matrix fraction decompositions and their connections with controllability and observability, realization, input-output inversion, model-matching.