## Descriptif

**Automatic Control with applications in Robotics and in Quantum Engineering****Responsible for the course**: Ugo Boscain - Mazyar Mirrahimi

Automatic Control deals with physical systems that are \controlled by external agents" as an aircraft controlled by rotors or an electron controlled via an external electric eld. The purpose of this course is to present the fundamental principles of control theory (non-commutativity of dynamics, controllability, stability, feedback, observability, robustness to uncertainty, optimal control).

Such notions are presented from a mathematical viewpoint giving the proofs of the most fundamental theorems. We will try to develop as much as possible the concept of dynamics evolving at dierent scales (averaging).

Nevertheless, more than half of the course is dedicated to applications in engineering based on physical equations coming from mechanics, quantum physics, hydrodynamics etc. Special focus will be given to applications in robotics and quantum engineering (control of Qubits for quantum computers, control of spin 1/2 particles in Nuclear Magnetic Resonance, controls of molecules via LASERS).

Certain applications in image processing and in nance will be also studied. These applications will be developed in the TDs and in the TPs in which the students learn how to program the control laws in Scilab or Python.

The course is organized in 9 lectures followed by 9 TDs. Two additional TPs in the middle of the course should permits to the students to become more familiar with the concept presented in the lectures. Such additional TPs are facultative, however an active participation to them will permits to increase the final grade.

**Week 1 :** **Free dynamical systems.** Flows. Qualitative analysis. Phase portrait. Stability. Lyapunov functions. LaSalle principle. Limit cycles. Slow and fast varying systems (averaging).**TD 1 :**

Evolution of a particle of spin 1/2 in a magnetic eld (NMR).

Dynamics of an aircraft submitted to aerodynamics forces.

Dynamics of populations.

Stability of a geostationary orbit for a satellite.

**Week 2 :** **Controllability, stability and observability of linear systems**. Kalman condition. Flat systems. Pole placement. Stabilization by static feedback. Observer of Luenberger and reconstruction of

the state. Motion planning.**TD 2 :**

Observer for the state of charge of a battery.

Stabilization of a satellite on a geostationary orbit.

Control of an articial muscle.

Stabilization of load's position of a crane.

Modeling a condenser microphone as a control system.**Week 3 :** **Controllability and stabilizability of non-linear systems 1**. Lie brackets. Accessibility, Controllability. The Krener and the Chow theorem. Systems subject to a periodic drift. Poisson stability.

Weak Hormander condition.**TD 3 :**

Controllability of a drone moving on the plane (with controls on the acceleration).

Controllability of a UAV veichule.

Controllability of a ball rolling on the plane without slipping or twisting.

**Week 4 : Controllability and stabilizability of non-linear systems 2**. The Brockett condition for the stabilizability of a nonlinear system. Regularity of a stabilizing feedback. Control Lyapunov func-

tions. Stabilization via backstepping. Brownian motion in place of the controls: the support theorem by

Stroock and Varadhan.**TD 4 :**

A model of volatility in Finance.

Stabilization of an inverted pendulum.

Stabilization of a multi-pendulum for the detection of gravitational waves.

**Week 5 : ** **Models for finite dimensional quantum systems**. Isolated systems and their description via the Schroedinger equation. Systems in interaction with the environment and their description via the Lindblad equation. The adiabatic theory. Conical intersections in the spectrum. The rotating wave approximation (RWA).**TD 5:**

Controlling a Qubit via two controls.

Controlling a Qubit via one control.

Controlling a Qubit subject to dissipation

Controlling using a chirp pulse.

The STIRAP process for a 3 level quantum systems.

Facultative TP 1 (on computer).

Implementation on computer of population transfer problems in quantum mechanics. STRAP and chirp pulses.

**Week 6 :** **Optimal Control for linear systems.** The Linear Quadratic problem (LQ). The Riccati equation. Motion planning. Deterministic Kalman lter. Stochastic Kalman lter. Estimation of parameters.**TD 6 :**

Optimal stabilization of a missile.

Tracking of objects in computer vision.

Optimal control in cancer chemotherapy.

Tracking load's position of a crane.**Week 7: Optimal control in the non-linear case**. The Pontryagin Maximum Principle. Normal and abnormal extremals. Chattering controls.**TD 7 :**

Minimizing the consumption of energy of a drone.

Reconstruction of interrupted curves in image processing.

Automatic optimal parking of a track with and without trailers.

**Week 8. The minimum time problem**. Switching functions. Singular trajectories. Minimum time stabilizing feedback.**TD 8 :**

Construction of the time optimal synthesis for an unmanned aerial vehicle (UAV) ying at a constant

altitude (HALE type) to provide a target supervision.

Facultative TP 2 (on computer).

Realization in Scilab or Python of the time optimal synthesis for the UAV drone of TD 7.

**Week 9 : Robustness to uncertainty. Criterium of Nyquist, Transfer function. Stability margins.**

Resonance. PID. Robustness of the LQ control. Adaptive control. Stability of switching systems under random switchings.**TD 9 :**

Control of a material exhibiting magnetic hysteresis.

Earthquake resistant structures.

Robustness of adiabatic control in quantum mechanics.

Robustness of Chirp and STIRAP for quantum control.

Language of the course: English

Numerus Clausus : 80 students

**http://www.cmapx.polytechnique.fr/~boscain/AUTOMATICS/**

Credits ECTS : 4