SU\G(𝔸)/K·U
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All published works (125)

The effect of COVID certificates on vaccine uptake, health outcomes, and the economy

2022-07-08
Miquel Oliu‐Barton , Bary Pradelski , Nicolas Woloszko +6 more
In the COVID-19 pandemic many countries required COVID certificates, proving vaccination, recovery, or a recent negative test, to access public and private venues. We estimate their effect on vaccine uptake … In the COVID-19 pandemic many countries required COVID certificates, proving vaccination, recovery, or a recent negative test, to access public and private venues. We estimate their effect on vaccine uptake for France, Germany, and Italy using counterfactuals constructed via innovation diffusion theory. The announcement of COVID certificates during summer 2021 were associated - although causality cannot be directly inferred - with increased vaccine uptake in France of 13.0 (95% CI 9.7-14.9) percentage points (p.p.) of the total population until the end of the year, in Germany 6.2 (2.6-6.9) p.p., and in Italy 9.7 (5.4-12.3) p.p. Based on these estimates, an additional 3979 (3453-4298) deaths in France, 1133 (-312-1358) in Germany, and 1331 (502-1794) in Italy were averted; and gross domestic product (GDP) losses of €6.0 (5.9-6.1) billion in France, €1.4 (1.3-1.5) billion in Germany, and €2.1 (2.0-2.2) billion in Italy were prevented. Notably, in France, the application of COVID certificates averted high intensive care unit occupancy levels where prior lockdowns were instated.
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Error analysis of a demodulation procedure for multicarrier signals with slowly-varying carriers

2021-08-23
Dilshad Surroop , Pascal Combes , Philippe Martin
We propose a procedure to demodulate analog signals encoded by a multicarrier modulator, with slowly-varying carrier shapes. We prove that the asymptotic demodulation error can be made arbitrarily small. The … We propose a procedure to demodulate analog signals encoded by a multicarrier modulator, with slowly-varying carrier shapes. We prove that the asymptotic demodulation error can be made arbitrarily small. The intended application is the "sensorless" control of AC electric motors at or near standstill, through the decoding of the PWM-induced current ripple.
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Error Estimates in Second-Order Continuous-Time Sigma-Delta Modulators

2021-05-13
Dilshad Surroop , Pascal Combes , Philippe Martin
Continuous-time Sigma-Delta (CT-ΔΣ) modulators are oversampling Analog-to-Digital converters that may provide higher sampling rates and lower power consumption than their discrete counterpart. Whereas approximation errors are established for high-order discrete … Continuous-time Sigma-Delta (CT-ΔΣ) modulators are oversampling Analog-to-Digital converters that may provide higher sampling rates and lower power consumption than their discrete counterpart. Whereas approximation errors are established for high-order discrete time ΔΣ modulators, theoretical analysis of the error between the filtered output and the input remain scarce. This paper presents a general framework to study this error: under regularity assumptions on the input and the filtering kernel, we prove for a second-order CT-ΔΣ that the error estimate may be in o(1/N)2, where N is the oversampling ratio. The whole theory is validated by numerical experiments.
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Error estimates in Second-order Continuous-Time Sigma-Delta modulators

2020-11-25
Dilshad Surroop , Pascal Combes , Philippe Martin
Continuous-time Sigma-Delta (CT-$\Sigma\Delta$) modulators are oversampling Analog-to-Digital converters that may provide higher sampling rates and lower power consumption than their discrete counterpart. Whereas approximation errors are established for high-order discrete … Continuous-time Sigma-Delta (CT-$\Sigma\Delta$) modulators are oversampling Analog-to-Digital converters that may provide higher sampling rates and lower power consumption than their discrete counterpart. Whereas approximation errors are established for high-order discrete time $\Sigma\Delta$ modulators, theoretical analysis of the error between the filtered output and the input remain scarce. This paper presents a general framework to study this error: under regularity assumptions on the input and the filtering kernel, we prove for a second-order CT-$\Sigma\Delta$ that the error estimate may be in $o(1/N^2)$, where $N$ is the oversampling ratio. The whole theory is validated by numerical experiments.

A demodulation procedure for multicarrier signals with slowly-varying carriers

2020-11-25
Dilshad Surroop , Pascal Combes , Philippe Martin
We propose a causal and implementable procedure to demodulate signals encoded by a multicarrier modulator, with slowly-varying carrier shapes. The intended application is the sensorless control of AC motors at … We propose a causal and implementable procedure to demodulate signals encoded by a multicarrier modulator, with slowly-varying carrier shapes. The intended application is the sensorless control of AC motors at low velocity by decoding the PWM-induced current ripple.
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Sensorless rotor position estimation by PWM-induced signal injection

2020-10-18
Dilshad Surroop , Pascal Combes , Philippe Martin +1 more
We demonstrate how the rotor position of a PWM-controlled PMSM can be recovered from the measured currents, by suitably using the excitation provided by the PWM itself. This provides the … We demonstrate how the rotor position of a PWM-controlled PMSM can be recovered from the measured currents, by suitably using the excitation provided by the PWM itself. This provides the benefits of signal injection, in particular the ability to operate even at low velocity, without the drawbacks of an external probing signal. We illustrate the relevance of the approach by simulations and experimental results.
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Sensorless rotor position estimation by PWM-induced signal injection

2020-09-10
Dilshad Surroop , Pascal Combes , Philippe Martin +1 more
We demonstrate how the rotor position of a PWM-controlled PMSM can be recovered from the measured currents, by suitably using the excitation provided by the PWM itself. This provides the … We demonstrate how the rotor position of a PWM-controlled PMSM can be recovered from the measured currents, by suitably using the excitation provided by the PWM itself. This provides the benefits of signal injection, in particular the ability to operate even at low velocity, without the drawbacks of an external probing signal. We illustrate the relevance of the approach by simulations and experimental results.

Adding virtual measurements by PWM-induced signal injection

2020-07-01
Dilshad Surroop , Pascal Combes , Philippe Martin +1 more
We show that for PWM-operated devices, it is possible to benefit from signal injection without an external probing signal, by suitably using the excitation provided by the PWM itself. As … We show that for PWM-operated devices, it is possible to benefit from signal injection without an external probing signal, by suitably using the excitation provided by the PWM itself. As in the usual signal injection framework conceptualized in [1], an extra "virtual measurement" can be made available for use in a control law, but without the practical drawbacks caused by an external signal.
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Error estimates in Second-order Continuous-Time Sigma-Delta modulators

2020-01-01
Dilshad Surroop , Pascal Combes , Philippe Martin
Continuous-time Sigma-Delta (CT-$\Sigma\Delta$) modulators are oversampling Analog-to-Digital converters that may provide higher sampling rates and lower power consumption than their discrete counterpart. Whereas approximation errors are established for high-order discrete … Continuous-time Sigma-Delta (CT-$\Sigma\Delta$) modulators are oversampling Analog-to-Digital converters that may provide higher sampling rates and lower power consumption than their discrete counterpart. Whereas approximation errors are established for high-order discrete time $\Sigma\Delta$ modulators, theoretical analysis of the error between the filtered output and the input remain scarce. This paper presents a general framework to study this error: under regularity assumptions on the input and the filtering kernel, we prove for a second-order CT-$\Sigma\Delta$ that the error estimate may be in $o(1/N^2)$, where $N$ is the oversampling ratio. The whole theory is validated by numerical experiments.
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Error analysis of a demodulation procedure for multicarrier signals with slowly-varying carriers

2020-01-01
Dilshad Surroop , Pascal Combes , Philippe Martin
We propose a procedure to demodulate analog signals encoded by a multicarrier modulator, with slowly-varying carrier shapes. We prove that the asymptotic demodulation error can be made arbitrarily small. The … We propose a procedure to demodulate analog signals encoded by a multicarrier modulator, with slowly-varying carrier shapes. We prove that the asymptotic demodulation error can be made arbitrarily small. The intended application is the "sensorless" control of AC electric motors at or near standstill, through the decoding of the PWM-induced current ripple.
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Sensorless rotor position estimation by PWM-induced signal injection

2020-01-01
Dilshad Surroop , Pascal Combes , Philippe Martin +1 more
We demonstrate how the rotor position of a PWM-controlled PMSM can be recovered from the measured currents, by suitably using the excitation provided by the PWM itself. This provides the … We demonstrate how the rotor position of a PWM-controlled PMSM can be recovered from the measured currents, by suitably using the excitation provided by the PWM itself. This provides the benefits of signal injection, in particular the ability to operate even at low velocity, without the drawbacks of an external probing signal. We illustrate the relevance of the approach by simulations and experimental results.
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Adding virtual measurements by PWM-induced signal injection

2020-01-01
Dilshad Surroop , Pascal Combes , Philippe Martin +1 more
We show that for PWM-operated devices, it is possible to benefit from signal injection \emph{without an external probing signal}, by suitably using the excitation provided by the PWM itself. As … We show that for PWM-operated devices, it is possible to benefit from signal injection \emph{without an external probing signal}, by suitably using the excitation provided by the PWM itself. As in the usual signal injection framework conceptualized in [1], an extra "virtual measurement" can be made available for use in a control law, but without the practical drawbacks caused by an external signal.
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Exact Controllability of a Linear Korteweg--de Vries Equation by the Flatness Approach

2019-01-01
Philippe Martin , Ivonne Rivas , Lionel Rosier +1 more
We consider a linear Korteweg--de Vries equation on a bounded domain with a left Dirichlet boundary control. The controllability to the trajectories of such a system was proved in the … We consider a linear Korteweg--de Vries equation on a bounded domain with a left Dirichlet boundary control. The controllability to the trajectories of such a system was proved in the last decade by using Carleman estimates. Here, we go a step further by establishing the exact controllability in a space of analytic functions with the aid of the flatness approach.
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Controllability of the 1D Schrödinger equation using flatness

2018-03-20
Philippe Martin , Lionel Rosier , Pierre Rouchon
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Delay-Robust Control Design for Two Heterodirectional Linear Coupled Hyperbolic PDEs

2018-01-26
Jean Auriol , Ulf Jakob F. Aarsnes , Philippe Martin +1 more
We detail in this paper the importance of a change of strategy for the delay robust control of systems composed of two linear first-order hyperbolic equations. One must go back … We detail in this paper the importance of a change of strategy for the delay robust control of systems composed of two linear first-order hyperbolic equations. One must go back to the classical tradeoff between convergence rate and delay robustness. More precisely, we prove that, for systems with strong reflections, canceling the reflection at the actuated boundary will yield zero delay robustness. Indeed, for such systems, using a backstepping controller, the corresponding target system should preserve a small amount of this reflection to ensure robustness to a small delay in the loop. This implies, in some cases, giving up finite time convergence.
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Exact controllability of a linear Korteweg-de Vries equation by the flatness approach

2018-01-01
Ivonne Rivas , Philippe Martin , Lionel Rosier +1 more
We consider a linear Korteweg-de Vries equation on a bounded domain with a left Dirichlet boundary control.The controllability to the trajectories of such a system was proved in the last … We consider a linear Korteweg-de Vries equation on a bounded domain with a left Dirichlet boundary control.The controllability to the trajectories of such a system was proved in the last decade by using Carleman estimates.Here, we go a step further by establishing the exact controllability in a space of analytic functions with the aid of the flatness approach.
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Obtaining the current-flux relations of the saturated PMSM by signal injection

2017-10-01
Pascal Combes , François Malrait , Philippe Martin +1 more
This paper proposes a method based on signal injection to obtain the saturated current-flux relations of a PMSM from locked-rotor experiments. With respect to the classical method based on time … This paper proposes a method based on signal injection to obtain the saturated current-flux relations of a PMSM from locked-rotor experiments. With respect to the classical method based on time integration, it has the main advantage of being completely independent of the stator resistance; moreover, it is less sensitive to voltage biases due to the power inverter, as the injected signal may be fairly large.
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A global observer for attitude and gyro biases from vector measurements

2017-07-01
Philippe Martin , Ioannis Sarras
We consider the classical problem of estimating the attitude and gyro biases of a rigid body from vector measurements and a triaxial rate gyro. We propose a simple "geometry-free" nonlinear … We consider the classical problem of estimating the attitude and gyro biases of a rigid body from vector measurements and a triaxial rate gyro. We propose a simple "geometry-free" nonlinear observer with guaranteed uniform global asymptotic convergence and local exponential convergence; the stability analysis, which relies on a strict Lyapunov function, is rather simple. The excellent behavior of the observer is illustrated through a detailed numerical simulation.
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Obtaining the Current-Flux Relations of the Saturated PMSM by Signal Injection

2017-05-12
Pascal Combes , François Malrait , Philippe Martin +1 more
This paper proposes a method based on signal injection to obtain the saturated current-flux relations of a PMSM from locked-rotor experiments. With respect to the classical method based on time … This paper proposes a method based on signal injection to obtain the saturated current-flux relations of a PMSM from locked-rotor experiments. With respect to the classical method based on time integration, it has the main advantage of being completely independent of the stator resistance; moreover, it is less sensitive to voltage biases due to the power inverter, as the injected signal may be fairly large.
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Obtaining the Current-Flux Relations of the Saturated PMSM by Signal Injection

2017-01-01
Pascal Combes , François Malrait , Philippe Martin +1 more
This paper proposes a method based on signal injection to obtain the saturated current-flux relations of a PMSM from locked-rotor experiments. With respect to the classical method based on time … This paper proposes a method based on signal injection to obtain the saturated current-flux relations of a PMSM from locked-rotor experiments. With respect to the classical method based on time integration, it has the main advantage of being completely independent of the stator resistance; moreover, it is less sensitive to voltage biases due to the power inverter, as the injected signal may be fairly large.
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Flatness and null controllability of 1‐D parabolic equations

2016-10-01
Philippe Martin , Lionel Rosier , Pierre Rouchon
Abstract We present a recent result on null controllability of one‐dimensional linear parabolic equations with boundary control. The space‐varying coefficients in the equation can be fairly irregular, in particular they … Abstract We present a recent result on null controllability of one‐dimensional linear parabolic equations with boundary control. The space‐varying coefficients in the equation can be fairly irregular, in particular they can present discontinuities, degeneracies or singularities at some isolated points; the boundary conditions at both ends are of generalized Robin‐Neumann type. Given any (fairly irregular) initial condition θ 0 and any final time T , we explicitly construct an open‐loop control which steers the system from θ 0 at time 0 to the final state 0 at time T . This control is very regular (namely Gevrey of order s with 1 < s < 2); it is simply zero till some (arbitrary) intermediate time τ, so as to take advantage of the smoothing effect due to diffusion, and then given by a series from τ to the final time T . We illustrate the effectiveness of the approach on a nontrivial numerical example, namely a degenerate heat equation with control at the degenerate side. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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Energy-based Modeling of AC Motors

2016-09-26
Pascal Combes , Al Kassem Jebai , François Malrait +2 more
We propose an approach to modeling of AC motors entirely based on analytical mechanics. Symmetry and connection constraints are moreover incorporated in the energy function from which the models are … We propose an approach to modeling of AC motors entirely based on analytical mechanics. Symmetry and connection constraints are moreover incorporated in the energy function from which the models are derived. The approach is especially suited to handle magnetic saturation, but also directly recovers the standard unsaturated models of the literature. The theory is illustrated by some experimental data.
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Adding virtual measurements by signal injection

2016-07-01
Pascal Combes , Al Kassem Jebai , François Malrait +2 more
We propose a method to "create" a new measurement output by exciting the system with a high-frequency oscillation. This new "virtual" measurement may be useful to facilitate the design of … We propose a method to "create" a new measurement output by exciting the system with a high-frequency oscillation. This new "virtual" measurement may be useful to facilitate the design of a suitable control law. The approach is especially interesting when the observability from the actual output degenerates at a steady-state regime of interest. The proposed method is based on second-order averaging and is illustrated by simulations on a simple third-order system.
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An analysis of the benefits of signal injection for low-speed sensorless control of induction motors

2016-06-01
Pascal Combes , François Malrait , Philippe Martin +1 more
We analyze why low-speed sensorless control of the IM is intrinsically difficult, and what is gained by signal injection. The explanation relies on the control-theoretic concept of observability applied to … We analyze why low-speed sensorless control of the IM is intrinsically difficult, and what is gained by signal injection. The explanation relies on the control-theoretic concept of observability applied to a general model of the saturated IM. We show that the IM is not observable when the stator speed is zero in the absence of signal injection, but that observability is restored thanks to signal injection and magnetic saturation. The analysis also reveals that existing sensorless algorithms based on signal injection may perform poorly for some IMs under particular operating conditions. The approach is illustrated by simulations and experimental data.
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A simple global observer for attitude and gyro biases

2016-04-13
Philippe Martin , Ioannis Sarras
We consider the classical problem of estimating the attitude and gyro biases of a rigid body from vector measurements and a triaxial rate gyro. We propose a simple geometry-free nonlinear … We consider the classical problem of estimating the attitude and gyro biases of a rigid body from vector measurements and a triaxial rate gyro. We propose a simple geometry-free nonlinear observer with guaranteed global asymptotic convergence and a straightforward stability analysis. The excellent behavior of the observer is illustrated on a detailed numerical simulation.
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A global observer for attitude and gyro biases from vector measurements

2016-04-13
Philippe Martin , Ioannis Sarras
We consider the classical problem of estimating the attitude and gyro biases of a rigid body from vector measurements and a triaxial rate gyro. We propose a simple geometry-free nonlinear … We consider the classical problem of estimating the attitude and gyro biases of a rigid body from vector measurements and a triaxial rate gyro. We propose a simple geometry-free nonlinear observer with guaranteed uniform global asymptotic convergence and local exponential convergence; the stability analysis, which relies on a strict Lyapunov function, is rather simple. The excellent behavior of the observer is illustrated through a detailed numerical simulation.

On the Reachable States for the Boundary Control of the Heat Equation

2016-01-28
Philippe Martin , Lionel Rosier , Pierre Rouchon
We are interested in the determination of the reachable states for the boundary control of the 1D heat equation. We consider either one or two boundary controls. We show that … We are interested in the determination of the reachable states for the boundary control of the 1D heat equation. We consider either one or two boundary controls. We show that reachable states associated with square integrable controls can be extended to analytic functions on some square of | ${\mathbb C}$ |⁠, and conversely, that analytic functions defined on a certain disk can be reached by using boundary controls that are Gevrey functions of order 2. The method of proof combines the flatness approach with some new Borel interpolation theorem in some Gevrey class with a specified value of the loss in the uniform estimates of the successive derivatives of the interpolating function.
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A global exponential observer for velocity-aided attitude estimation

2016-01-01
Philippe Martin , Ioannis Sarras , Minh‐Duc Hua +1 more
We propose a simple nonlinear observer for estimating the attitude and velocity of a rigid body from the measurements of specific acceleration, angular velocity, magnetic field (in body axes), and … We propose a simple nonlinear observer for estimating the attitude and velocity of a rigid body from the measurements of specific acceleration, angular velocity, magnetic field (in body axes), and linear velocity (in body axes). It is uniformly globally exponentially convergent, and also enjoys other nice properties: global decoupling of pitch and roll estimation from magnetic measurements, good local behavior, and easy tuning. The observer is "geometry-free", in the sense that it respects only asymptotically the rotational geometry. The good behavior of the observer, even when the measurements are noisy and biased is illustrated in simulation.
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Null Controllability of One-dimensional Parabolic Equations by the Flatness Approach

2016-01-01
Philippe Martin , Lionel Rosier , Pierre Rouchon
We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular. Considering generalized Robin--Neumann boundary conditions at both extremities, we … We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular. Considering generalized Robin--Neumann boundary conditions at both extremities, we prove the null controllability with one boundary control by following the flatness approach, which provides explicitly the control and the associated trajectory as series. Both the control and the trajectory have a Gevrey regularity in time related to the $L^p$ class of the coefficient in front of $u_t$. The approach applies in particular to the (possibly degenerate or singular) heat equation $(a(x)u_x)_x-u_t=0$ with $a(x)>0$ for a.e. $x\in (0,1)$ and $a+1/a \in L^1(0,1)$, or to the heat equation with inverse square potential $u_{xx}+(\mu/|x|^2)u-u_t=0$ with $\mu\ge 1/4$.
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A global observer for attitude and gyro biases from vector measurements

2016-01-01
Philippe Martin , Ioannis Sarras
We consider the classical problem of estimating the attitude and gyro biases of a rigid body from vector measurements and a triaxial rate gyro. We propose a simple "geometry-free" nonlinear … We consider the classical problem of estimating the attitude and gyro biases of a rigid body from vector measurements and a triaxial rate gyro. We propose a simple "geometry-free" nonlinear observer with guaranteed uniform global asymptotic convergence and local exponential convergence; the stability analysis, which relies on a strict Lyapunov function, is rather simple. The excellent behavior of the observer is illustrated through a detailed numerical simulation.
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Energy-based Modeling of AC Motors

2016-01-01
Pascal Combes , Al Kassem Jebai , François Malrait +2 more
We propose an approach to modeling of AC motors entirely based on analytical mechanics. Symmetry and connection constraints are moreover incorporated in the energy function from which the models are … We propose an approach to modeling of AC motors entirely based on analytical mechanics. Symmetry and connection constraints are moreover incorporated in the energy function from which the models are derived. The approach is especially suited to handle magnetic saturation, but also directly recovers the standard unsaturated models of the literature. The theory is illustrated by some experimental data.
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An analysis of the benefits of signal injection for low-speed sensorless control of induction motors

2015-11-30
Pascal Combes , François Malrait , Philippe Martin +1 more
We analyze why low-speed sensorless control of the IM is intrinsically difficult, and what is gained by signal injection. The explanation relies on the control-theoretic concept of observability applied to … We analyze why low-speed sensorless control of the IM is intrinsically difficult, and what is gained by signal injection. The explanation relies on the control-theoretic concept of observability applied to a general model of the saturated IM. We show that the IM is not observable when the stator speed is zero in the absence of signal injection, but that observability is restored thanks to signal injection and magnetic saturation. The analysis also reveals that existing sensorless algorithms based on signal injection may perform poorly for some IMs under particular operating conditions. The approach is illustrated by simulations and experimental data.
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A simple model-based state estimator for the quadrotor using only inertial measurements

2015-10-12
Philippe Martin , Ioannis Sarras
We propose a nonlinear observer to estimate the state (orientation and in-plane velocity vector) of the quadrotor, based on a drag-force-enhanced model. It is an alternative to recent works using … We propose a nonlinear observer to estimate the state (orientation and in-plane velocity vector) of the quadrotor, based on a drag-force-enhanced model. It is an alternative to recent works using a similar model together with an Extended Kalman Filter (EKF). But while enjoying the benefits of an enhanced model, it does not have the usual drawbacks of an EKF: indeed, the computational cost is much lower, the tuning is easier, and above all the guaranteed domain of convergence is very large.
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On the reachable states for the boundary control of the heat equation

2015-09-30
Philippe Martin , Lionel Rosier , Pierre Rouchon
We are interested in the determination of the reachable states for the boundary control of the one-dimensional heat equation. We consider either one or two boundary controls. We show that … We are interested in the determination of the reachable states for the boundary control of the one-dimensional heat equation. We consider either one or two boundary controls. We show that reachable states associated with square integrable controls can be extended to analytic functions onsome square of C, and conversely, that analytic functions defined on a certain disk can be reached by using boundary controlsthat are Gevrey functions of order 2. The method of proof combines the flatness approach with some new Borel interpolation theorem in some Gevrey class witha specified value of the loss in the uniform estimates of the successive derivatives of the interpolating function.
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Null controllability using flatness: A case study of a 1-D heat equation with discontinuous coefficients

2015-07-01
Philippe Martin , Lionel Rosier , Pierre Rouchon
The approach recently proposed by the authors for the null controllability of 1-D parabolic equations is applied to the nontrivial case of a heat equation with discontinuous coefficients and subjected … The approach recently proposed by the authors for the null controllability of 1-D parabolic equations is applied to the nontrivial case of a heat equation with discontinuous coefficients and subjected to Robin boundary conditions. The control steering the system to zero from a discontinuous initial state is comprehensively derived, together with the corresponding trajectory. Numerical experiments illustrate several features of the theory and demonstrate its effectiveness.

A semi-global model-based state estimator for the quadrotor using only inertial measurements

2015-01-01
Philippe Martin , Ioannis Sarras
We propose a nonlinear observer to estimate the state (orientation and in-plane velocity vector) of the quadrotor, based on a drag-force-enhanced model. It is an alternative to recent works using … We propose a nonlinear observer to estimate the state (orientation and in-plane velocity vector) of the quadrotor, based on a drag-force-enhanced model. It is an alternative to recent works using a similar model together with an Extended Kalman Filter (EKF). But while enjoying the benefits of an enhanced model, it does not have the usual drawbacks of an EKF: indeed, the computational cost is much lower, the tuning is easier, and above all the guaranteed domain of convergence is very large.
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On the reachable states for the boundary control of the heat equation

2015-01-01
Philippe Martin , Lionel Rosier , Pierre Rouchon
We are interested in the determination of the reachable states for the boundary control of the one-dimensional heat equation. We consider either one or two boundary controls. We show that … We are interested in the determination of the reachable states for the boundary control of the one-dimensional heat equation. We consider either one or two boundary controls. We show that reachable states associated with square integrable controls can be extended to analytic functions onsome square of C, and conversely, that analytic functions defined on a certain disk can be reached by using boundary controlsthat are Gevrey functions of order 2. The method of proof combines the flatness approach with some new Borel interpolation theorem in some Gevrey class witha specified value of the loss in the uniform estimates of the successive derivatives of the interpolating function.
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An analysis of the benefits of signal injection for low-speed sensorless control of induction motors

2015-01-01
Pascal Combes , François Malrait , Philippe Martin +1 more
We analyze why low-speed sensorless control of the IM is intrinsically difficult, and what is gained by signal injection. The explanation relies on the control-theoretic concept of observability applied to … We analyze why low-speed sensorless control of the IM is intrinsically difficult, and what is gained by signal injection. The explanation relies on the control-theoretic concept of observability applied to a general model of the saturated IM. We show that the IM is not observable when the stator speed is zero in the absence of signal injection, but that observability is restored thanks to signal injection and magnetic saturation. The analysis also reveals that existing sensorless algorithms based on signal injection may perform poorly for some IMs under particular operating conditions. The approach is illustrated by simulations and experimental data.
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Energy-based modeling of electric motors

2014-12-01
Al Kassem Jebai , Pascal Combes , François Malrait +2 more
We propose a new approach to modeling electrical machines based on energy considerations and construction symmetries of the motor. We detail the approach on the Permanent-Magnet Synchronous Motor and show … We propose a new approach to modeling electrical machines based on energy considerations and construction symmetries of the motor. We detail the approach on the Permanent-Magnet Synchronous Motor and show that it can be extended to Synchronous Reluctance Motor and Induction Motor. Thanks to this approach we recover the usual models without any tedious computation. We also consider effects due to non-sinusoidal windings or saturation and provide experimental data.
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Velocity-aided attitude estimation for accelerated rigid bodies

2014-12-01
Minh‐Duc Hua , Philippe Martin , Tarek Hamel
Two nonlinear observers for velocity-aided attitude estimation, relying on gyrometers, accelerometers, magnetometers, and velocity measured in the body-fixed frame, are proposed. As opposed to state-of-the-art body-fixed velocity-aided attitude observers endowed … Two nonlinear observers for velocity-aided attitude estimation, relying on gyrometers, accelerometers, magnetometers, and velocity measured in the body-fixed frame, are proposed. As opposed to state-of-the-art body-fixed velocity-aided attitude observers endowed with local properties, both observers are (almost) globally asymptotically stable, with very simple and flexible tuning. Moreover, the roll and pitch estimates are globally decoupled from magnetometer measurements.
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Null controllability of the heat equation using flatness

2014-10-25
Philippe Martin , Lionel Rosier , Pierre Rouchon
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Null controllability of one-dimensional parabolic equations

2014-10-09
Philippe Martin , Pierre Rouchon , Lionel Rosier
We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular. Considering generalized Robin-Neumann boundary conditions at both extremities, we … We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular. Considering generalized Robin-Neumann boundary conditions at both extremities, we prove the null controllability with one boundary control by following the flatness approach, which provides explicitly the control and the associated trajectory as series. Both the control and the trajectory have a Gevrey regularity in time related to the $L^p$ class of the coefficient in front of $u_t$. The approach applies in particular to the (possibly degenerate or singular) heat equation $(a(x)u_x)_x-u_t=0$ with $a(x)>0$ for a.e. $x\in (0,1)$ and $a+1/a \in L^1(0,1)$, or to the heat equation with inverse square potential $u_{xx}+(\mu / |x|^2)u-u_t=0$ with $\mu\ge 1/4$.
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Null controllability of one-dimensional parabolic equations by the flatness approach

2014-10-09
Philippe Martin , Lionel Rosier , Pierre Rouchon
We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular.Considering generalized Robin-Neumann boundary conditions at both extremities, we prove … We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular.Considering generalized Robin-Neumann boundary conditions at both extremities, we prove the null controllability with one boundary control by following the flatness approach, which providesexplicitly the control and the associated trajectory as series. Both the control and the trajectory have a Gevrey regularity in time related to the $L^p$ class of the coefficient in front of $u\_t$.The approach applies in particular to the (possibly degenerate or singular) heat equation $(a(x)u\_x)\_x-u\_t=0$ with $a(x)\textgreater{}0$ for a.e. $x\in (0,1)$ and $a+1/a \in L^1(0,1)$, or to the heat equation with inverse square potential $u\_{xx}+(\mu / |x|^2)u-u\_t=0$with $\mu\ge 1/4$.
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Controllability of the 1D Schrodinger equation by the flatness approach

2014-04-03
Philippe Martin , Lionel Rosier , Pierre Rouchon
We derive in a straightforward way the exact controllability of the 1-D Schrodinger equation with a Dirichlet boundary control. We use the so-called flatness approach, which consists in parameterizing the … We derive in a straightforward way the exact controllability of the 1-D Schrodinger equation with a Dirichlet boundary control. We use the so-called flatness approach, which consists in parameterizing the solution and the control by the derivatives of a flat output. This provides an explicit control input achieving the exact controllability in the energy space. As an application, we derive an explicit pair of control inputs achieving the exact steering to zero for a simply-supported beam.
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Energy-based modeling of electric motors

2014-03-26
Al Kassem Jebai , Pascal Combes , François Malrait +2 more
We propose a new approach to model electrical machines based on energy considerations and construction symmetries of the motor. We detail the approach on the Permanent-Magnet Synchronous Motor and show … We propose a new approach to model electrical machines based on energy considerations and construction symmetries of the motor. We detail the approach on the Permanent-Magnet Synchronous Motor and show that it can be extended to Synchronous Reluctance Motor and Induction Motor. Thanks to this approach we recover the usual models without any tedious computation. We also consider effects due to non-sinusoidal windings or saturation and provide experimental data.
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Velocity-aided Attitude Estimation for Accelerated Rigid Bodies

2014-01-01
Minh‐Duc Hua , Philippe Martin , Tarek Hamel
Two nonlinear observers for velocity-aided attitude estimation, relying on gyrometers, accelerometers, magnetometers, and velocity measured in the body-fixed frame, are proposed. As opposed to state-of-the-art body-fixed velocity-aided attitude observers endowed … Two nonlinear observers for velocity-aided attitude estimation, relying on gyrometers, accelerometers, magnetometers, and velocity measured in the body-fixed frame, are proposed. As opposed to state-of-the-art body-fixed velocity-aided attitude observers endowed with local properties, both observers are (almost) globally asymptotically stable, with very simple and flexible tuning. Moreover, the roll and pitch estimates are globally decoupled from magnetometer measurements.
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Controllability of the 1D Schrödinger equation by the flatness approach

2014-01-01
Philippe Martin , Lionel Rosier , Pierre Rouchon
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Controllability of the 1D Schrodinger equation by the flatness approach

2014-01-01
Philippe Martin , Lionel Rosier , Pierre Rouchon
We derive in a straightforward way the exact controllability of the 1-D Schrodinger equation with a Dirichlet boundary control. We use the so-called flatness approach, which consists in parameterizing the … We derive in a straightforward way the exact controllability of the 1-D Schrodinger equation with a Dirichlet boundary control. We use the so-called flatness approach, which consists in parameterizing the solution and the control by the derivatives of a "flat output". This provides an explicit control input achieving the exact controllability in the energy space. As an application, we derive an explicit pair of control inputs achieving the exact steering to zero for a simply-supported beam.
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Null controllability of one-dimensional parabolic equations by the flatness approach

2014-01-01
Philippe Martin , Lionel Rosier , Pierre Rouchon
We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular.Considering generalized Robin-Neumann boundary conditions at both extremities, we prove … We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular.Considering generalized Robin-Neumann boundary conditions at both extremities, we prove the null controllability with one boundary control by following the flatness approach, which providesexplicitly the control and the associated trajectory as series. Both the control and the trajectory have a Gevrey regularity in time related to the $L^p$ class of the coefficient in front of $u\_t$.The approach applies in particular to the (possibly degenerate or singular) heat equation $(a(x)u\_x)\_x-u\_t=0$ with $a(x)\textgreater{}0$ for a.e. $x\in (0,1)$ and $a+1/a \in L^1(0,1)$, or to the heat equation with inverse square potential $u\_{xx}+(\mu / |x|^2)u-u\_t=0$with $\mu\ge 1/4$.
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Energy-based modeling of electric motors

2014-01-01
Al Kassem Jebai , Pascal Combes , François Malrait +2 more
We propose a new approach to model electrical machines based on energy considerations and construction symmetries of the motor. We detail the approach on the Permanent-Magnet Synchronous Motor and show … We propose a new approach to model electrical machines based on energy considerations and construction symmetries of the motor. We detail the approach on the Permanent-Magnet Synchronous Motor and show that it can be extended to Synchronous Reluctance Motor and Induction Motor. Thanks to this approach we recover the usual models without any tedious computation. We also consider effects due to non-sinusoidal windings or saturation and provide experimental data.
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Common Coauthors

Coauthor Papers Together
Pierre Rouchon 73
Lionel Rosier 24
Pascal Combes 23
François Malrait 19
Dilshad Surroop 11
Al Kassem Jebai 11
Silvère Bonnabel 8
Ioannis Sarras 7
Jean Lévine 6
Michel Fliess 4
Pascal R. Buenzli 4
Michel Fliesś 3
Minh‐Duc Hua 3
C. Gruber 3
Erwan Salaün 3
J. Piasecki 3
Tarek Hamel 3
Béatrice Laroche 3
B. Misra 2
F. Cornu 2
AlKassem Jebai 2
Jean-Paul Marchand 2
V. Ballenegger 2
Ivonne Rivas 2
Joel L. Lebowitz 2
L. Frachebourg 2
Massimiliano Sassoli de Bianchi 2
François Ollivier 2
Richard M. Murray 2
Nicolas Petit 2
William B. Dunbar 2
A. Alastuey 2
David Brydges 2
Patrick Artus 1
B. Jancovici 1
Franois Malrait 1
J. Bonvin 1
X. Zotos 1
M. Bouziane 1
Guntram B. Wolff 1
Tarik Yalcin 1
Philippe Aghion 1
Jean Lévine 1
Nicolas Woloszko 1
G. Nenciu 1
Florent Di Meglio 1
Ch. Oguey 1
L. Blum 1
Jean Auriol 1
Lionel Guetta-Jeanrenaud 1

Commonly Cited References

Motion planning for the heat equation

2000-01-01
Béatrice Laroche , Philippe Martin , Pierre Rouchon
The paper gives an explicit open-loop control, able to approximately steer the one-dimensional heat equation with control on the boundary from any state to any other state. The control is … The paper gives an explicit open-loop control, able to approximately steer the one-dimensional heat equation with control on the boundary from any state to any other state. The control is obtained thanks to a parametrization of the solutions of the heat equation by a series involving infinitely many derivatives of the system 'flat output'. Copyright © 2000 John Wiley & Sons, Ltd.

Real and Complex Analysis.

1987-01-01
G. A. Garreau , Walter Rudin
Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive … Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Null controllability of the heat equation using flatness

2014-10-25
Philippe Martin , Lionel Rosier , Pierre Rouchon
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Adding virtual measurements by signal injection

2016-07-01
Pascal Combes , Al Kassem Jebai , François Malrait +2 more
We propose a method to "create" a new measurement output by exciting the system with a high-frequency oscillation. This new "virtual" measurement may be useful to facilitate the design of … We propose a method to "create" a new measurement output by exciting the system with a high-frequency oscillation. This new "virtual" measurement may be useful to facilitate the design of a suitable control law. The approach is especially interesting when the observability from the actual output degenerates at a steady-state regime of interest. The proposed method is based on second-order averaging and is illustrated by simulations on a simple third-order system.
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Symmetry-Preserving Observers

2008-12-01
Silvère Bonnabel , Philippe Martin , Pierre Rouchon
This paper presents three non-linear observers on three examples of engineering interest: a chemical reactor, a non-holonomic car, and an inertial navigation system. For each example, the design is based … This paper presents three non-linear observers on three examples of engineering interest: a chemical reactor, a non-holonomic car, and an inertial navigation system. For each example, the design is based on physical symmetries. This motivates the theoretical development of invariant observers, i.e, symmetry-preserving observers. We consider an observer to consist in a copy of the system equation and a correction term, and we give a constructive method (based on the Cartan moving-frame method) to find all the symmetry-preserving correction terms. They rely on an invariant frame (a classical notion) and on an invariant output-error, a less standard notion precisely defined here. For each example, the convergence analysis relies also on symmetries consideration with a key use of invariant state-errors. For the non-holonomic car and the inertial navigation system, the invariant state-errors are shown to obey an autonomous differential equation independent of the system trajectory. This allows us to prove convergence, with almost global stability for the non-holonomic car and with semi-global stability for the inertial navigation system. Simulations including noise and bias show the practical interest of such invariant asymptotic observers for the inertial navigation system.
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Coulomb systems at low density

1999-01-01
David Brydges , Philippe Martin
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Null boundary controllability for semilinear heat equations

1995-11-01
Yung-Jen Lin Guo , Walter Littman
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Adding virtual measurements by PWM-induced signal injection

2020-07-01
Dilshad Surroop , Pascal Combes , Philippe Martin +1 more
We show that for PWM-operated devices, it is possible to benefit from signal injection without an external probing signal, by suitably using the excitation provided by the PWM itself. As … We show that for PWM-operated devices, it is possible to benefit from signal injection without an external probing signal, by suitably using the excitation provided by the PWM itself. As in the usual signal injection framework conceptualized in [1], an extra "virtual measurement" can be made available for use in a control law, but without the practical drawbacks caused by an external signal.
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A differential geometric setting for dynamic equivalence and dynamic linearization

1995-01-01
Jean‐Baptiste Pomet
This paper presents an (infinite-dimensional) geometric framework for control systems, based on infinite jet bundles, where a system is represented by a single vector field and dynamic equivalence (to be … This paper presents an (infinite-dimensional) geometric framework for control systems, based on infinite jet bundles, where a system is represented by a single vector field and dynamic equivalence (to be precise: equivalence by endogenous dynamic feedback
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Contróle Exact De Léquation De La Chaleur

1995-01-01
Gilles Lebeau , Luc Robbiano
(1995). Controle Exact De Lequation De La Chaleur. Communications in Partial Differential Equations: Vol. 20, No. 1-2, pp. 335-356. (1995). Controle Exact De Lequation De La Chaleur. Communications in Partial Differential Equations: Vol. 20, No. 1-2, pp. 335-356.
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Controllability of Evolution equations

1996-01-01
A. V. Fursikov , Oleg Imanuvilov

A Lie-Backlund approach to equivalence and flatness of nonlinear systems

1999-05-01
Michel Fliess , Jean Lévine , Philippe Martin +1 more
A new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent … A new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives. This is a Lie-Backlund isomorphism. The authors prove that, although the state dimension is not preserved, the number of input channels is kept fixed. They also prove that a Lie-Backlund isomorphism can be realized by an endogenous feedback. The differentially flat nonlinear systems introduced by the authors (1992) via differential algebraic techniques, are generalized and the new notion of orbitally flat systems is defined. They correspond to systems which are equivalent to a trivial one, with time preservation or not. The endogenous linearizing feedback is explicitly computed in the case of the VTOL aircraft to track given reference trajectories with stability.
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Nonlinearizable single-input control systems do not admit stationary symmetries

2002-05-01
Witold Respondek , Issa Amadou Tall

Null Controllability of One-dimensional Parabolic Equations by the Flatness Approach

2016-01-01
Philippe Martin , Lionel Rosier , Pierre Rouchon
We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular. Considering generalized Robin--Neumann boundary conditions at both extremities, we … We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular. Considering generalized Robin--Neumann boundary conditions at both extremities, we prove the null controllability with one boundary control by following the flatness approach, which provides explicitly the control and the associated trajectory as series. Both the control and the trajectory have a Gevrey regularity in time related to the $L^p$ class of the coefficient in front of $u_t$. The approach applies in particular to the (possibly degenerate or singular) heat equation $(a(x)u_x)_x-u_t=0$ with $a(x)>0$ for a.e. $x\in (0,1)$ and $a+1/a \in L^1(0,1)$, or to the heat equation with inverse square potential $u_{xx}+(\mu/|x|^2)u-u_t=0$ with $\mu\ge 1/4$.
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Differential flatness and absolute equivalence

2002-12-17
Michiel Van Nieuwstadt , Muruhan Rathinam , Richard M. Murray
In this paper we give a formulation of differential flatness-a concept originally introduced by Fliess, Levine, Martin, and Rouchon (1992)-in terms of absolute equivalence between exterior differential systems. Systems which … In this paper we give a formulation of differential flatness-a concept originally introduced by Fliess, Levine, Martin, and Rouchon (1992)-in terms of absolute equivalence between exterior differential systems. Systems which are differentially flat have several useful properties which can be exploited to generate effective control strategies for nonlinear systems. The original definition of flatness was given in the context of differential algebra, and required that all mappings be meromorphic functions. Our formulation of flatness does not require any algebraic structure and allows one to use tools from exterior differential systems to help characterize differentially flat systems. In particular, we show that in the case of single input control systems (i.e., codimension 2 Pfaffian systems), a system is differentially flat if and only if it is feedback linearizable via static state feedback. However, in higher codimensions feedback linearizability and flatness are not equivalent: one must be careful with the role of time as well the use of prolongations which may not be realizable as dynamic feedbacks in a control setting. Applications of differential flatness to nonlinear control systems and open questions are also discussed.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>
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On dynamic feedback linearization

1989-08-01
B. Charlet , Jean Lévine , R. Marino

Rank Invariants of Nonlinear Systems

1989-05-01
Maria Domenica Di Benedetto , Jessy W. Grizzle , Claude H. Moog
A linear algebraic framework for the analysis of rank properties of nonlinear systems is introduced. This framework gives a high-level interpretation of several existing algorithms built around the recursive computation … A linear algebraic framework for the analysis of rank properties of nonlinear systems is introduced. This framework gives a high-level interpretation of several existing algorithms built around the recursive computation of certain algebraic ranks associated with right-invertibility, left-invertibility, and dynamic decoupling. Furthermore, it can be used to establish links between these algorithms and the differential algebraic approach, as well as to solve some static and dynamic noninteracting control problems.

Null-controllability of one-dimensional parabolic equations

2007-10-12
Giovanni Alessandrini , Luis Escauriaza
We prove the interior null-controllability of one-dimensional parabolic equations with time independent measurable coefficients. We prove the interior null-controllability of one-dimensional parabolic equations with time independent measurable coefficients.
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Estimation of saturation of Permanent-Magnet Synchronous Motors through an energy-based model

2011-05-01
Al Kassem Jebai , François Malrait , Philippe Martin +1 more
We propose a parametric model of the saturated Permanent-Magnet Synchronous Motor (PMSM) together with an estimation method of the magnetic parameters. The model is based on an energy function which … We propose a parametric model of the saturated Permanent-Magnet Synchronous Motor (PMSM) together with an estimation method of the magnetic parameters. The model is based on an energy function which simply encompasses the saturation effects. Injection of fast-varying pulsating voltages and measurements of the resulting current ripples then permit to identify the magnetic parameters by linear least squares. Experimental results on a surface-mounted PMSM and an interior magnet PMSM illustrate the relevance of the approach.
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Lectures on entire functions

1996-07-23
B. Yu. Levin
Part I. Entire Functions of Finite Order: Growth of entire functions Main integral formulas for functions analytic in a disk Some applications of the Jensen formula Factorization of entire functions … Part I. Entire Functions of Finite Order: Growth of entire functions Main integral formulas for functions analytic in a disk Some applications of the Jensen formula Factorization of entire functions of finite order The connection between the growth of an entire function and the distribution of its zeros Theorems of Phragmen and Lindelof Subharmonic functions The indicator function The Polya Theorem Applications of the Polya Theorem Lower bounds for analytic and subharmonic functions Entire functions with zeros on a ray Entire functions with zeros on a ray (continuation) Part II. Entire Functions of Exponential Type: Integral representation of functions analytic in the half-plane The Hayman Theorem Functions of class $C$ and their applications Zeros of functions of class $C$ Completeness and minimality of system of exponential functions in $L^2(0,a)$ Interpolation by entire functions of exponential type Interpolation by entire functions of the spaces $L_\pi$ and $B_\pi$ Sin-type functions Riesz bases formed by exponential functions in $L^2(-\pi,\pi)$ Completeness of the eigenfunction system of a quadratic operator pencil Part III. Some Additional Problems of the Theory of Entire Functions: Carleman's and R. Nevanlinna's formulas and their applications Uniqueness problems for Fourier transforms and for infinitely-differentiable functions The Matsaev Theorem on the growth of entire functions admitting a lower bound Entire functions of the class $P$ S. N. Bernstein's inequality for entire functions of exponential type and its generalizations Bibliography Author index Subject index.

Automatique et corps différentiels

1989-01-01
Michel Fliess

A new higher order chain rule and Gevrey class

1989-01-01
Takesi Yamanaka

Numerical approximation of null controls for the heat equation: Ill-posedness and remedies

2010-07-09
Arnaud Münch , Enrique Zuazua
The numerical approximation of exact or trajectory controls for the wave equation is known to be a delicate issue, since the pioneering work of Glowinski–Lions in the nineties, because of … The numerical approximation of exact or trajectory controls for the wave equation is known to be a delicate issue, since the pioneering work of Glowinski–Lions in the nineties, because of the anomalous behavior of the high-frequency spurious numerical waves. Various efficient remedies have been developed and analyzed in the last decade to filter out these high-frequency components: Fourier filtering, Tychonoff's regularization, mixed finite-element methods, multi-grid strategies, etc. Recently convergence rate results have also been obtained. This work is devoted to analyzing this issue for the heat equation, which is the opposite paradigm because of its strong dissipativity and smoothing properties. The existing analytical results guarantee that, at least in some simple situations, as in the finite-difference scheme in 1 − d, the null or trajectory controls for numerical approximation schemes converge. This is due to the intrinsic high-frequency damping of the heat equation that is inherited by its numerical approximation schemes. But when developing numerical simulations the topic appears to be much more subtle and difficult. In fact, efficiently computing the null control for a numerical approximation scheme of the heat equation is a difficult problem in itself. The difficulty is strongly related to the regularizing effect of the heat kernel. The controls of minimal L2-norm are characterized as minima of quadratic functionals on the solutions of the adjoint heat equation, or its numerical versions. These functionals are shown to be coercive in very large spaces of solutions, sufficient to guarantee the L2 character of controls, but very far from being identifiable as energy spaces for the adjoint system. The very weak coercivity of the functionals under consideration makes the approximation problem exponentially ill-posed and the functional framework far from being well adapted to standard techniques in numerical analysis. In practice, the controls of the minimal L2-norm exhibit a singular highly oscillatory behavior near the final controllability time, which cannot be captured numerically. Standard techniques, such as Tychonoff's regularization or quasi-reversibility methods, allow a slight smoothing of the singularities but significantly reduce the quality of the approximation. In this paper we develop some more involved and less-standard approaches which turn out to be more efficient. We first discuss the advantages of using controls with compact support with respect to the time variable or the effect of adding numerical dissipative singular terms. We also develop the numerical version of the so-called transmutation method that allows writing the control of a heat process in terms of the corresponding control of the associated wave process, by means of a 'time convolution' with a one-dimensional controlled fundamental heat solution. This method, although it can be proved to converge, is also subtle in its computational implementation. Indeed, it requires using convergent numerical schemes for the control of the wave equation, a problem that, as mentioned above, is delicate in itself. But we also need to compute an accurate approximation of a controlled fundamental heat solution, an issue that requires its own analysis and significant numerical and computational new developments. These methods are thoroughly illustrated and discussed in the paper, accompanied by some numerical experiments in one space dimension that show the subtlety of the issue. These experiments allow one to compare the efficiency of the various methods. This is done in the case where the control is distributed in some subdomain of the domain where the heat process evolves but similar results and numerical experiments could be derived for other cases, such as the one in which the control acts on the boundary. The techniques we employ here can also be adapted to the multi-dimensional case.
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On the Dirichlet boundary controllability of the one-dimensional heat equation: semi-analytical calculations and ill-posedness degree

2011-04-18
Faker Ben Belgacem , Sidi Mahmoud Kaber
The ill-posedness degree for the controllability of the one-dimensional heat equation by a Dirichlet boundary control is the purpose of this work. This problem is severely (or exponentially) ill-posed. We … The ill-posedness degree for the controllability of the one-dimensional heat equation by a Dirichlet boundary control is the purpose of this work. This problem is severely (or exponentially) ill-posed. We intend to shed more light on this assertion and the underlying mathematics. We start by discussing the framework liable to fit an efficient numerical implementation without introducing further complications into the theoretical analysis. Afterward we expose the Fourier calculations that transform the ill-posedness issue to the one related to linear algebra. This consists of investigating the singular values of some infinite structured matrices that are obtained as sums of Cauchy matrices. Utilising the Gershgorin–Hadamard theorem and the Collatz–Wielandt formula, we are able to provide the lower and upper bounds for the largest singular value of these matrices. After checking whether they are also solutions of some symmetric Lyapunov (or Sylvester) equations with a very low displacement rank, we use an estimate that improves Penzl's former result to bound the ratio's smaller/largest singular values of these matrices. Accordingly, the controllability problem is confirmed to be severely ill-posed. The bounds proved here will be supported by computations made using Matlab procedures. At last, the well-known explicit inverse of Cauchy-type matrices allows us to provide a formal exponential series representation of the Dirichlet control in a long horizon controllability. That series has to be checked afterward to discover whether it is convergent or not and to find out if the desired state can be reached. Here again, some examples ran within Matlab will be discussed and commented upon.

Applications of lie groups to differential equations

1990-09-01
Peter J. Olver
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Nonlinear Control Systems

1989-01-01
Alberto Isidori

Sur l'équivalence absolue de certains systèmes d'équations différentielles et sur certaines familles de courbes

1914-01-01
Élie Cartan
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Uniform controllability properties for space/time-discretized parabolic equations

2011-03-09
Franck Boyer , Florence Hubert , Jérôme Le Rousseau
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Sur la nature analytique des solutions des équations aux dérivées partielles. Premier mémoire

1918-01-01
Maurice Gevrey
ou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques ou impression de ce fichier doit contenir la présente mention de copyright.Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
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Exterior Differential Systems

1991-01-01
Robert L. Bryant , Shiing-Shen Chern , Robert B. Gardner +2 more

Nilpotent bases for a class of nonintegrable distributions with applications to trajectory generation for nonholonomic systems

1994-03-01
Richard M. Murray
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Energy-based modeling of electric motors

2014-12-01
Al Kassem Jebai , Pascal Combes , François Malrait +2 more
We propose a new approach to modeling electrical machines based on energy considerations and construction symmetries of the motor. We detail the approach on the Permanent-Magnet Synchronous Motor and show … We propose a new approach to modeling electrical machines based on energy considerations and construction symmetries of the motor. We detail the approach on the Permanent-Magnet Synchronous Motor and show that it can be extended to Synchronous Reluctance Motor and Induction Motor. Thanks to this approach we recover the usual models without any tedious computation. We also consider effects due to non-sinusoidal windings or saturation and provide experimental data.
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Global controllability and stabilization for the nonlinear Schrödinger equation on an interval

2009-02-09
Camille Laurent
We prove global internal controllability in large time for the nonlinear Schrodinger equation on a bounded interval with periodic, Dirichlet or Neumann conditions. Our strategy combines stabilization and local controllability … We prove global internal controllability in large time for the nonlinear Schrodinger equation on a bounded interval with periodic, Dirichlet or Neumann conditions. Our strategy combines stabilization and local controllability near 0. We use Bourgain spaces to prove this result on L2. We also get a regularity result about the control if the data are assumed smoother.
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A linear algebraic framework for dynamic feedback linearization

1995-01-01
E. Aranda-Bricaire , Claude H. Moog , Jean‐Baptiste Pomet
To any accessible nonlinear system we associate a so-called infinitesimal Brunovsky form. This gives an algebraic criterion for strong accessibility as well as a generalization of Kronecker controllability indices. An … To any accessible nonlinear system we associate a so-called infinitesimal Brunovsky form. This gives an algebraic criterion for strong accessibility as well as a generalization of Kronecker controllability indices. An output function which defines a right-invertible system without zero dynamics is shown to exist if and only if the basis of the Brunovsky form can be transformed into a system of exact differential forms. This is equivalent to the system being differentially flat and hence constitutes a necessary and sufficient condition for dynamic feedback linearizability.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

High-Index Differential Algebraic Equations

1995-01-01
Stephen L. Campbell
ABSTRACT In the last few years there has been considerable research on differential algebraic equations (DAEs) f(x1, x, t) = 0, where fx 1 is identically singular. The index provides … ABSTRACT In the last few years there has been considerable research on differential algebraic equations (DAEs) f(x1, x, t) = 0, where fx 1 is identically singular. The index provides one measure of the singularity of a DAE. Most of the numerical analysis literature on DAEs to date has dealt with DAEs with indices no larger than three, because of technical difficulties and because many basic applications including constrained mechanical systems have this index. This paper discusses several situations where DAEs of index higher than three occur naturally. It will also discuss the relationship between certain concepts in nonlinear control theory such as relative degree and zero dynamics, the index, and constrained mechanical systems.

Handbook of Mathematical Functions

1966-02-01
Donald A. McQuarrie
First Page First Page

Über den Begriff der Klasse von Differentialgleichungen

1935-01-01
David Hilbert

Null controllability of the 1D heat equation using flatness

2013-01-01
Philippe Martin , Lionel Rosier , Pierre Rouchon
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Measurement of the Casimir Force between Parallel Metallic Surfaces

2002-01-15
G. Bressi , G. Carugno , Roberto Onofrio +1 more
We report on the measurement of the Casimir force between conducting surfaces in a parallel configuration. The force is exerted between a silicon cantilever coated with chromium and a similar … We report on the measurement of the Casimir force between conducting surfaces in a parallel configuration. The force is exerted between a silicon cantilever coated with chromium and a similar rigid surface and is detected by looking at the shifts induced in the cantilever frequency when the latter is approached. The scaling of the force with the distance between the surfaces was tested in the 0.5-3.0 microm range, and the related force coefficient was determined at the 15% precision level.
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Practical stabilization of driftless systems on lie groups: The transverse function approach

2003-09-01
Pascal Morin , Claude Samson
A general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented. The approach is based on the concept of bounded transverse … A general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented. The approach is based on the concept of bounded transverse functions, the existence of which is equivalent to the system's controllability. Its outcome is the practical stabilization of any trajectory, i.e., not necessarily a solution of the control system, in the state-space. The possibility of applying the approach to an arbitrary controllable smooth driftless system follows in turn from the fact that any controllable homogeneous approximation of this system can be lifted (via a dynamic extension) to a system on a Lie group. Illustrative examples are given.
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Non-Linear Symmetry-Preserving Observers on Lie Groups

2009-07-01
Silvère Bonnabel , Philippe Martin , Pierre Rouchon
In this paper we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between … In this paper we give a geometrical framework for the design of observers on finite-dimensional Lie groups for systems which possess some specific symmetries. The design and the error (between true and estimated state) equation are explicit and intrinsic. We consider also a particular case: left-invariant systems on Lie groups with right equivariant output. The theory yields a class of observers such that error equation is autonomous. The observers converge locally around any trajectory, and the global behavior is independent from the trajectory, which reminds of the linear stationary case.
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Exact Results for the Two-Dimensional One-Component Plasma

1981-02-09
B. Jancovici
At some special temperature ${T}_{0}$, the distribution functions of a two-dimensional one-component plasma are explicitly computed up to the four-body one. The correlations have a Gaussian falloff. The distribution functions … At some special temperature ${T}_{0}$, the distribution functions of a two-dimensional one-component plasma are explicitly computed up to the four-body one. The correlations have a Gaussian falloff. The distribution functions at ${T}_{0}$ are used for building a temperature expansion around ${T}_{0}$.

Exact Controllability for the Schrödinger Equation

1994-01-01
E. Machtyngier
The exact controllability of Schrödinger equation in bounded domains with Dirichlet boundary condition is studied. Both the boundary controllability and the internal controllability problems are considered. Concerning the boundary controllability, … The exact controllability of Schrödinger equation in bounded domains with Dirichlet boundary condition is studied. Both the boundary controllability and the internal controllability problems are considered. Concerning the boundary controllability, the paper proves the exact controllability in $H^{ - 1} (\Omega )$ with $L^2 $-boundary control. On the other hand, the exact controllability in $L^2 (\Omega )$ is proved with $L^2 $-controls supported in a neigh-borhood of the boundary. Both results hold for an arbitrarily small time. The method of proof combines both the HUM (Hilbert Uniqueness Method) and multiplier techniques.

Locally Distributed Control and Damping for the Conservative Systems

1997-09-01
Kangsheng Liu
In this paper we note the equivalence between exact controllability and exponential stabilizability for an abstract conservative system with bounded control. This enables us to establish a frequency domain characterization … In this paper we note the equivalence between exact controllability and exponential stabilizability for an abstract conservative system with bounded control. This enables us to establish a frequency domain characterization for the exact controllability/uniform exponential decay property of second-order elastic systems, such as the wave equation and the Petrovsky equation, with (locally) distributed control/damping. A piecewise multiplier method for frequency domain is introduced. For several classes of PDEs on regions which are not necessarily smooth, we obtain a sufficient condition for the subregion on which the application of control/damping will yield the exact controllability/uniform exponential decay property. This result provides useful information for designing the location of controllers/dampers for distributed systems with a law of conservation.

On exact and approximate boundary controllabilities for the heat equation: A numerical approach

1994-09-01
C. Carthel , Roland Glowinski , J. L. Lions

Exact boundary controllability of the nonlinear Schrödinger equation

2008-12-12
Lionel Rosier , Bing‐Yu Zhang

The Casimir force at high temperature

2005-08-30
Pascal R. Buenzli , Philippe Martin
The standard expression of the high-temperature Casimir force between perfect conductors is obtained by imposing macroscopic boundary conditions on the electromagnetic field at metallic interfaces. This force is twice larger … The standard expression of the high-temperature Casimir force between perfect conductors is obtained by imposing macroscopic boundary conditions on the electromagnetic field at metallic interfaces. This force is twice larger than that computed in microscopic classical models allowing for charge fluctuations inside the conductors. We present a direct computation of the force between two quantum plasma slabs in the framework of non-relativistic quantum electrodynamics including quantum and thermal fluctuations of both matter and field. In the semi-classical regime, the asymptotic force at large slab separation is identical to that found in the above purely classical models, which is therefore the right result. We conclude that when calculating the Casimir force at non-zero temperature, fluctuations inside the conductors cannot be ignored. Aspects of this subject are treated in a companion letter by Jancovici B. and Šamaj L., Casimir force between two ideal-conductor walls revisited (Europhys. Lett., 72 (2005) 35).
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Universality in some classical Coulomb systems of restricted dimension

1996-08-01
Peter J. Forrester , B. Jancovici , Gabriel Téllez
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Control and stabilization of the Korteweg-de Vries equation: recent progresses

2009-11-02
Lionel Rosier , Bing‐Yu Zhang

Equilibrium correlations in charged fluids coupled to the radiation field

2006-03-09
Sami El Boustani , Pascal R. Buenzli , Philippe Martin
We provide an exact microscopic statistical treatment of particle and field correlations in a system of quantum charges in equilibrium with a classical radiation field. Using the Feynman-Kac-Ito representation of … We provide an exact microscopic statistical treatment of particle and field correlations in a system of quantum charges in equilibrium with a classical radiation field. Using the Feynman-Kac-Ito representation of the Gibbs weight, the system of particles is mapped onto a collection of random charged wires. The field degrees of freedom can be integrated out, providing an effective pairwise magnetic potential. We then calculate the contribution of the transverse field coupling to the large-distance particle correlations. The asymptotics of the field correlations in the plasma are also exactly determined.
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