Pierre Rouchon

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

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.

Flatness of Heavy Chain Systems

In this paper the flatness [M. Fliess, J. Levine, P. Martin, and P. Rouchon, Internat. J. Control, 61 (1995), pp. 1327--1361, M. Fliess, J. Levine, P. Martin, and P. Rouchon, IEEE Trans. Automat. Control, 44 (1999), pp. 922--937] of heavy chain systems, i.e., trolleys carrying a fixed length heavy chain that may carry a load, is addressed in the partial derivatives equations framework. We parameterize the system trajectories by the trajectories of its free end and solve the motion planning problem, namely, steering from one state to another state. When considered as a finite set of small pendulums, these systems were shown to be flat [R. M. Murray, in Proceedings of the IFAC World Congress, San Francisco, CA, 1996, pp. 395--400]. Our study is an extension to the infinite dimensional case. nUnder small angle approximations, these heavy chain systems are described by a one-dimensional (1D) partial differential wave equation. Dealing with this infinite dimensional description, we show how to get the explicit parameterization of the chain trajectory using (distributed and punctual) advances and delays of its free end. nThis parameterization results from symbolic computations. Replacing the time derivative by the Laplace variable

Dynamics and solutions to some control problems for water-tank systems

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A distributed parameter approach to the control of a tubular reactor: a multivariable case

yields a second order differential equation in the spatial variable where s is a parameter. Its fundamental solution is, for each point considered along the chain, an entire function of s of exponential type. Moreover, for each, we show that, thanks to the Liouville transformation, this solution satisfies, modulo explicitly computable exponentials of s, the assumptions of the Paley--Wiener theorem. This solution is, in fact, the transfer function from the flat output (the position of the free end of the system) to the whole state of the system. Using an inverse Laplace transform, we end up with an explicit motion planning formula involving both distributed and punctual advances and delays operators.

Periodic input estimation for linear periodic systems: Automotive engine applications

We consider a tank containing a fluid. The tank is subjected to directly controlled translations and rotations. The fluid motion is described by linearized wave equations under shallow water approximations. For irrotational flows, a new variational formulation of Saint-Venant equations is proposed. This provides a simple method to establish the equations when the tank is moving. Several control configurations are studied: one and two horizontal dimensions; tank geometries (straight and nonstraight bottom, rectangular and circular shapes), tank motions (horizontal translations with and without rotations). For each configuration, we prove that the linear approximation is steady-state controllable and provide a simple and flatness-based algorithm for computing the steering open-loop control. These algorithms rely on operational calculus. They lead to second order equations in space variables whose fundamental solutions define delay operators corresponding to convolutions with compact support kernels. For each configuration, several controllability open-problems are proposed and motivated.

Observing a quantum Maxwell demon at work

The start-up and the shutdown of tubular fixed-bed reactors can be improved by an explicit trajectory design and active control. By using the inflow concentration as a control the outflow concentration can be changed in arbitrary finite time. Explicit series expansions of the control inputs are calculated using the distributed parameter system model. It is shown that these promising techniques are not restricted to the single-input case, but can easily be extended to certain multivariable models. The control laws derived are new in the scalar case.

Real-time combustion torque estimation on a diesel engine test bench using time-varying Kalman filtering

In this paper, we consider periodic linear systems driven by T0-periodic signals that we desire to reconstruct. The systems under consideration are of the form [emailxa0protected]?=A(t)x+A0(t)w(t), y=C(t)x, [emailxa0protected]?R^n, [emailxa0protected]?R^m, [emailxa0protected]?R^p, (m=

NULL CONTROLLABILITY OF THE STRUCTURALLY DAMPED WAVE EQUATION WITH MOVING CONTROL

Significance Maxwell’s demon plays a central role in thermodynamics of quantum information, yet a full experimental characterization is still missing in the quantum regime. Here we use superconducting circuits to realize a quantum Maxwell demon in which all thermodynamic quantities can be controlled and measured. Using power detection resolved at the single microwave photon level and unprecedented tomography techniques, we directly measure the extracted work while tracking the qubit and cavity entropies and energies. We are thus able to fully characterize the demon’s memory after the work extraction and show that it takes full part in the thermodynamic process. The experiment establishes superconducting circuits as a testbed well suited to perform quantum thermodynamics experiments. In apparent contradiction to the laws of thermodynamics, Maxwell’s demon is able to cyclically extract work from a system in contact with a thermal bath, exploiting the information about its microstate. The resolution of this paradox required the insight that an intimate relationship exists between information and thermodynamics. Here, we realize a Maxwell demon experiment that tracks the state of each constituent in both the classical and quantum regimes. The demon is a microwave cavity that encodes quantum information about a superconducting qubit and converts information into work by powering up a propagating microwave pulse by stimulated emission. Thanks to the high level of control of superconducting circuits, we directly measure the extracted work and quantify the entropy remaining in the demon’s memory. This experiment provides an enlightening illustration of the interplay of thermodynamics with quantum information.

Active signal restoration for the telegraph equation

We propose an estimator of the combustion torque on a diesel engine using as only sensor the easily available instantaneous crankshaft angle speed. The observer consists in a Kalman filter designed on a physics-based time-varying model for the engine dynamics. Convergence is proven, using results from the literature by establishing the uniform controllability and observability properties of this periodic system. A test bench and development environment is presented. Performance is studied through simulations and real test bench experiments.

Experimental motion planning in airpath control for HCCI engine

We investigate the internal controllability of the wave equation with structural damping on the one-dimensional torus. We assume that the control is acting on a moving point or on a moving small interval with a constant velocity. We prove that the null controllability holds in some suitable Sobolev space and after a fixed positive time independent of the initial conditions.