Modern Control for quantum optical systems ISC-Meeting Hannover 29.08.2011 Maximilian Wimmer Quantum Controls Group

Outline Traditional and Modern Control Control Theory Nomenclature LQG Method Our Plans

Control History First mentioned control: water clocks (1500 ac) Serial production of governor controllers for steam engines started 1769 Begin of mathematical description of control problems by Maxwell (late 19th Century) Development of Bode- and Nyquist-plots and root locus method (beginning of 20th Century) 1967 Kalman Filter, first used for Apollo missions in 70s Since 80s Digital Signal Processing, with new numerical methods

Traditional and Modern Control - Ideal for systems with few inputs/outputs - Bode plots, Nyquist stability criteria Modern - Ideal for MiMo-systems with nested loops - Robust control - Optimal control (LQG-Method)

Control Theory Nomenclature (I) State Space model with disturbance vector w(t) observation vector z(t) with measurement noise v(t)

Control Theory Nomenclature (II)

Control Of A Linear Cavity Using: - Linear Quadratic Gaussian controller - Kalman filtering - subspace filtering and model reduction Frequency locking of an optical cavity using linear–quadratic Gaussian integral control S Z Sayed Hassen1, M Heurs1, E H Huntington1, I R Petersen1 and M R James2 Example to show methology and techniques of modern control

LQG Design (I) Get the state space model - by theoretical models (eg. linear cavity, homodyne detection, laser white noise…) - via frequency response measurements - fit model with sub space system identification

LQG Design (II) Choose general performace criteria Cost functional With design parameters Q and R Which correspond to small detuning and control effort

LQG Design (III) Minimize cost functional J includes the laser noise z(t) in an integral Form L(z) with a new design parameter Q Design a Kalman-Filter for good estimation of system variables including quantum ones

LQG Design (IV) Set design parameters Discretise (eg. sampling) Full (15th) and reduced-order (6th) continuous LQG controller Bode plots phase (°) magnitude (dB) frequency (Hz) Set design parameters Discretise (eg. sampling) Solve equations (MatLab) Finally you get the controller

Results Closed loop transfer function (disturbance angle (°) gain (dB) freq (Hz) Closed loop transfer function (disturbance supression function)

Plans Build a squeezer (MIMO-System) Design a digital controller (LQG, dSPACE) Show the advantages of modern approach But first get rid of problems getting started with new lab

Our Group Imagine Photo here Michéle Heurs Timo Denker Dirk Schütte Maximilian Wimmer Imagine Photo here

Thank you for your attention

Anhang Cost functional is minimized if P is determined by solution of Riccati equation