Abstracts

Jean-Pierre Barbot (ENSEA, Université Cergy-Pontoise)

Title: "Left inversion of  nonlinear  dynamic systems"

Abstract: This presentation is devoted to the left inversion of nonlinear dynamic systems, i.e. the recovering for unknown inputs from measured outputs. First, a brief review of left inversion in control theory and signal processing will be given. Then, applications will be mentioned.

Ugo Boscain (CNRS & Sorbonne Université, France)

Title: "Geometric confinement of the curvature laplacian on almost Riemannian manifolds"

Abstract: Two-dimensional almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step assumption the singular set Z, where the structure is not Riemannian, is a 1D embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Beltrami operator $\Delta$ cannot. This is due to the fact that (under a natural compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the manifold without the singular set. In the literature such counterintuitive phenomenon is called geometric confinement. For the heat equation an intuitive explanation of this fact can be given in terms of random walks. For the Schroredinger equation  an intuitive explanation is more subtle since the evolution of a quantum particle on a manifold can be done in several non-equivalent ways.
  In this talk I will describe the evolution (and the confinement) of a quantum particle described by the curvature Laplacian $-\Delta+cK$ (here $K$ is the Gaussian curvature and c>0 a constant) which originates in coordinate free quantization procedures (as for instance in path-integral or covariant Weyl quantization). Joint work with Eugenio Pozzoli and Ivan Beschastnyi. 

Bronisław Jakubczyk (Institute of Mathematics, Polish Academy of Sciences, Poland)

Title: "Gauge equivalence of control systems and conjugate points"

Abstract: Click here

Frédéric Jean (ENSTA Paris, IP Paris, France)

Title: " Inverse optimal control problem, general methodology and non-autonomous LQ case "

Abstract: Given a control system and a set of optimal trajectories, is it possible to recover the cost for which the trajectories are minimized? In this talk, we present a general approach to answering this question in the widely used class of optimal control problems defined by control-affine systems and quadratic costs. The basic notion is that of cost equivalence – a generalization of affine equivalence of metrics in the Riemannian and sub-Riemannian case - and the key element of our approach is the use of ample geodesics. We will also explore the non-autonomous setting with a comprehensive study of the linear-quadratic case based on a new characterization of optimal syntheses.

Philippe Jouan (Université de Rouen, France) 

Title: "Generalized Solutions to Degenerate Dynamical Systems"

Abstract: Click here

Arthur J. Krener (Naval Postgraduate School, USA) 

Title: "Nonlinear Observers with Linear Error Dynamics"

Abstract: It is my pleasure to participate in this workshop honoring the outstanding scientific career of Witold Respondek. Vitek is the ”Great Linearizer”. With Jacubcyk he was the first to give necessary and sufficient conditions for existence of a nonlinear change of state coordinates and a nonlinear state feedback that transforms a non- linear controlled dynamics into a linear controlled dynamics. With Krener he gave necessary and sufficient conditions for existence of nonlinear change of state coordinates, a nonlinear change of output coordinates and a nonlinear input-output injection that transforms a nonlinear controlled and observed dynamics into a form which ad- mits an observer with linear, exponentialy stable error dynamics in the transformed variables. This work inspired a large number of susequent studies that we will discuss. Perhaps the most interesting is the work of Kazantzis and Kravaris. For nonlinear observed (but uncontrolled) dynamics they showed how to construct an observer with linear, exponentialy stable error dynamics in the transformed variables. Such observers are now known as KKL observers. Krener and Xiao showed that any observer with linear, exponentialy stable error dynamics in transformed variables is an extension of a KKL observer. This extension of a KKL observer by Krener and Xiao in- troduces free parameters which can be tuned by machine learning. Kreisselmeier and Engel introduced a different extension of a KKL observer with linear error dynamics and their observers have more free parameters which might also be tuned by machine learning.

Carole Le Guyader (INSA Rouen Normandie)

Title: "Mathematical foundations of a level-set-based shape convex hull identification algorithm"

Abstract: In this work, we introduce a proper mathematical framework to analyse a novel object convex hull extraction algorithm phrased in a level-set framework. The convex shape to be identified is modelled as the zero level-line of an unknown signed-distance function viewed as a minimiser of a suitably designed functional. This latter is stated in a subspace of the Special Bounded Hessian function space, whilst a non-negativity criterion satisfied by the absolutely continuous part of the Laplacian of the unknown on the whole domain expresses the convexity constraint. A result of existence of minimisers is established for this preliminary problem. Then, to implement this optimisation problem, we provide a counterpart elliptic approximation, complemented by a Γ-convergence result and a splitting-strategy-based algorithm involving subproblems with closed-form solutions. Numerical experiments showing that the proposed approach is an appropriate compromise between mathematical thoroughness and algorithmic relevancy conclude the presentation. 

Andrzej J. Maciejewski (University of Zielona Gora, Poland)

Title: "Non-integrability of the $n$-body problem"

Abstract: We prove that the classical planar $n$-body problem when restricted to a common level of the energy and the angular momentum is not integrable except in the case when both values of these integrals are zero. This is a joint work with Maria Przybylska and Thierry Combot

Wiktor Malesza (Warsaw University of Technology, Institute of Control & Industrial Electronics, Poland) 

Title: "On region-dependent control systems"

Abstract: This presentation is about control systems whose dynamics exhibit specific relationships with regions in the state-space will be presented. In the case of region-invariant control systems the well known design using gain-scheduling technique based on the concept of parameterized linearization families will be applied to positive systems, that is, systems whose all trajectories starting in the non-negative orthant remain there for all non-negative controls. Depending on a set of parameters, the linearization of the non-linear controlled system is done for the relevant operating points. The considerations presented will be illustrated with a computational example.

Claude Moog (LS2N, Nantes, France)

Title: "Classification and decomposition of mechanical control systems"

Abstract: Motivated by the control of biped walking robots, this talk focuses

1. on characteristic properties of elementary mechanical systems such as the Acrobot or the Pendubot and 

2. on the decomposition of more complex higher order mechanical systems.

Whenever a higher order mechanical system can be decomposed into the cascade of lower order systems, then its control is also decomposed into simplified control problems. The model of a biped robot may be decomposed – up to some approximation – into a cascade which includes the Acrobot model. The latter stands for the hips and legs.
The classification of elementary mechanical systems is based on the involutivity of some distributions. The decomposition of a nonlinear system into the cascade of a given system with an other system is characterized as well thanks to geometric tools. It can be viewed as an extension of results on partial linearization.

References

1. S.J. Li, C.H. Moog and W. Respondek, Maximal feedback linearization  and its internal dynamics with applications to mechanical systems on R4, Int. Journal of Robust and Nonlinear Control, 2019, 29, 9, pp. 2639-2659.
   
2. D. Maalouf, C.H. Moog, Y. Aoustin and S.J. Li, Classification of two-degree-of-freedom underactuated mechanical systems, IET Control Theory & Applications, 25 June 2015, Vol. 9, Iss. 10, pp. 1501–1510.
3. E. Aranda-Bricaire, C. Califano, C.H. Moog, “Immersion of Nonlinear Systems into Higher Order Systems”, 20^th IFAC WC, Toulouse, July 2017, IFAC-PapersOnLine, 2017, 50, (1), pp. 9480-9484.

Florentina Nicolau (ENSEA, Université Cergy-Pontoise, France)

Title: "Constructing flat inputs for two-output systems"

Abstract: Click here

Title: "Synchronization of multiple oscillators, the role of coupling and geometry"

Abstract: Synchronization of oscillators may appear in various ways, of which full synchronization of coupled identically oscillators is best known and dates back to the earliest work of Christiaan Huygens in the 17th century. However, also other types of synchronization have been observed, for example, anti-phase synchronization for two coupled oscillators, or wave-like synchronization of multiple oscillators has also been reported, but even other types of coordination are possible. The objective of this presentation is to seek underlying principles to understand the possible synchronization mechanisms. Both some analysis tools as well as experimental results will be discussed.

Title: "Mechanical feedback linearization of mechanical control systems"

Abstract: We present the problem of feedback linearization of mechanical control systems that preserves the mechanical structure of the system. We formulate necessary and sufficient conditions using objects on the configuration space Q only, despite the fact that the state–space of mechanical system is the tangent bundle TQ. In contrast to the linearization of general nonlinear systems, mechanical linearization can be performed for both controllable and noncontrollable systems. In our study, we utilized some geometric tools, such as Lie brackets, distributions, covariant derivatives, and the Riemann curvature tensor that have an immediate mechanical interpretation. We illustrate our results by examples of linearizable mechanical systems. The talk is based on joint research with Witold Respondek. 

Romeo Ortega (Instituto Tecnológico Autónomo de México, Mexico)

Title: "Observability is Sufficient for State Observation of State-affine Nonlinear Systems" 

Abstract: In this talk we are interested in the problem of state observation of state-affine nonlinear systems. Our main contribution is to propose a globally exponentially convergent observer that requires only observability of the system that, as is well-known, is a necessary assump- tion for the solution of the problem. This should be contrasted with existing results that require the strictly stronger assumption of uniform complete observability of the system. To the best of the authors’ knowledge this is the first time such a result is reported in the liter- ature. Instrumental for the solution of the problem is the use of the Generalized Parameter Estimation Based Observer design proposed by the author.

William Pasillas-Lépine (Centrale Supélec, Université Paris-Saclay, France)

Title: "Automatic Train Operation: Robust regulation around constrained trajectories"

Abstract: The Automatic Train Operation (ATO) is the railway subsystem in charge of driving the train. In this presentation, we consider two typical problems encountered in ATO design: the generation of trajectories that respect constraints imposed by railways scheduling and the robust regulation around those trajectories. Railway trajectories are constrained by waypoints that impose a specific speed and acceleration to the train at certain predefined positions. For the open-loop trajectory design, we describe the train dynamics using a distance-based time-scaling and take into account input saturation (jerk limitation) and state constraints (speed and acceleration bounds). Inspired by the minimal-time synthesis, we propose two different switching policies: (a) two-switch trajectories, which cover the standard case where the speeds and accelerations at the endpoints are compatible; (b) multi-switch trajectories, which correspond to specific cases where the two-switch approach cannot be applied. Then, we use the constructed profile as a basis to design a cascaded control law. Our design includes an external loop (speed error regulation) that provides a set-point to the internal loop (acceleration error regulation).
Finally, we assess the robustness of those control loops against loop delays and measurement filtering by limiting the overshoot (expressed in terms of the H-infinity gain of the associated closed-loop transfer functions).

Jean-Baptiste Pomet (INRIA Sophia Antipolis, Université Côte d’Azur, France)

Title: "Old unsolved problems in differential flatness or dynamic linearization"

Abstract: This talk will give a partial account of results and questions on characterization of differential flatness of continuous-time control systems. This property greatly simplifies many control tasks when it occurs, hence many "flatness based" control methods and an important literature on how to chose flat outputs when they exist. The problem of deciding when a given control system is flat is however still open, probably for a long time. There has however been a consequent amount of literature  in this area in the control community since the late 80s, (and Wittek has been a prominent contributor!).  This talk will present a few results and considerations on this problem and associated puzzles. The author is afraid that his talk will not contain new results.

Pierre Rouchon (Mines-ParisTech, PSL, France)

Title: "Gate generation for open quantum systems via a monotonic algorithm with time optimization"

Abstract: We present a monotonic numerical algorithm including time optimization for generating quantum gates  for open systems. Such  systems  are assumed to be  governed   by Lindblad master equations for the  density operators on a large  Hilbert-space whereas the quantum gates are relative  to a sub-space of small  dimension.  Starting from an initial seed of the control input,  this algorithm consists in the repetition  of  the following two steps producing a new control input: (A) backwards integration of adjoint Lindblad master equations (in the Heisenberg-picture) from a set of final conditions encoding the quantum gate to generate;  (B)   forward integration of    Lindblad master equations in closed-loop where  a Lyapunov based control produced the new control input. The numerical stability is ensured by the stability of both the open-loop adjoint  backward system and the  forward closed-loop system. A clock-control input  can be added to the usual control input. The obtained  monotonic algorithm  allows then to optimise  not only the shape of the control imput, but also the gate time.   Preliminary numerical implementations   indicate that this algorithm  is well suited for cat-qubit gates, where  Hilbert-space dimensions (2 for the Z-gate and 4 for the CNOT-gate)  are much smaller than the dimension  of the  physical Hilbert-space involving mainly Fock-states (typically 20 or  larger for a single cat-qubit). This monotonic algorithm,  based on Lyapunov control techniques,  is shown to have  a straightforward  interpretation in terms of optimal control: its  stationary conditions  coincides with  the first-order optimality   conditions  for a  cost depending linearly  on the final values of the quantum states. Joint work with Paulo Sergio PEREIRA DA SILVA from University of Sao Paulo.

Timothée Schmoderer (PRISME Université d'Orléans, France)

Title: "Characterising and classifying control systems : a story that goes on"

Abstract: In this talk, we will explore the extensive history and development of the characterization and classification of control systems and their relation to the characterization and classification of nonholonomic constraints (namely, equations of the tangent bundle of the form $S(x,\dot x)=0$). Beginning with the foundational contributions of Pfaff, Darboux, and Cartan, we will examine the geometric study of control-linear systems and their equivalence to the classification of Pfaffian equations. We will then review key theorems by Witold Respondek on the linearization of nonlinear control systems and give their counterparts for nonholonomic constraints.  These seminal works collectively address dynamical systems with trajectories constrained by affine nonholonomic conditions, i.e equations of the form $\Omega(x)\dot{x}+b(x)=0$.  In the final part of the talk, we will present current research on the characterization and classification of control systems constrained by quadratic nonholonomic conditions, where trajectories satisfy quadratic relations between the velocities.

Arjan van der Schaft (University of Groningen, Netherlands)

Title: "Reciprocity of nonlinear input-state-output systems"

Abstract: Linear reciprocity is simply defined as symmetry of the impulse response or transfer matrix of a linear system. In the famous 1972 paper by Jan Willems it was detailed how in the linear case reciprocity is reflected in the state space realization. Furthermore, it was shown how to combine reciprocity with passivity in order to obtain state space realizations with special physical properties, such as relaxation systems. In this talk I will discuss extensions of this theory to the nonlinear case. Emphasis will be on pseudo-gradient systems; in particular those with a Hessian pseudo-Riemannian metric. The combination of reciprocity with passivity will be elucidated from a port-Hamiltonian perspective.