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SIMULATION OF A LINEAR QUADRATIC GAUSSIAN (LQG ) CONTROLER FOR LBTI

SIMULATION OF A LINEAR QUADRATIC GAUSSIAN (LQG ) CONTROLER FOR LBTI. Julien Lozi. STATE-SPACE DESCRIPTION. Measurement: . Actutator. Sensor. Command. Measure. Control Law. Simple integrator: . Sensitivity Function: Noise Transfer Function:. STATE-SPACE DESCRIPTION.

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SIMULATION OF A LINEAR QUADRATIC GAUSSIAN (LQG ) CONTROLER FOR LBTI

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  1. SIMULATION OF A LINEAR QUADRATIC GAUSSIAN(LQG) CONTROLER FOR LBTI Julien Lozi Julien Lozi, LBTI meeting4/11/2014

  2. STATE-SPACE DESCRIPTION • Measurement: Actutator Sensor Command Measure Control Law • Simple integrator: • Sensitivity Function: • Noise Transfer Function: Julien Lozi, LBTI meeting 4/11/2014

  3. STATE-SPACE DESCRIPTION • Second order auto-regressive (AR2) model: • Disturbance: • Matrix representation: Julien Lozi, LBTI meeting 4/11/2014

  4. STEPS OF THE LQG • Estimation: • Update: • Kalman gain: • Error update: • Prediction: • Command Julien Lozi, LBTI meeting 4/11/2014

  5. DATA • Unwrapped phase using the Matlab function “unwrap” • There are clearly some phase jumps visible • But it doesn’t matter for this demonstration Julien Lozi, LBTI meeting 4/11/2014

  6. IDENTIFICATION procedure • Identification using 30000 points (30s) • Identification of 20 vibrations • Main frequencies: • 167.8 • 10.3 • 283.3 • 14.0 • 115.9 • 100.6 • 17.4 • 123.2 • 12.2 • 120.7 Julien Lozi, LBTI meeting 4/11/2014

  7. Transfer function • For low frequencies, the difference between an integrator ( with a gain of 0.4) and the LQG is pretty small • But the LQG has stronger rejections at the vibration frequencies Julien Lozi, LBTI meeting 4/11/2014

  8. correction • Open loop: 10.4 μm rms (mostly due to phase jumps) • Closed loop with integrator (gain 0.4): 567 nm rms • Closed loop with LQG: 199 nm rms  Gain of almost 10 on the null ratio Julien Lozi, LBTI meeting 4/11/2014

  9. IDENTIFICATION with less points (6000) Condition at small frequencies: Closed loop: σ=219 nm rms, μ= 0.7 nm No condition at small frequencies: Closed loop: σ=346 nm rms, μ=232 nm Julien Lozi, LBTI meeting 4/11/2014

  10. USING the LQG to unwrap the phase The LQG predicts where the next command should be. If the unwrapping process includes a phase jump, the residual OPD will have an error close to the wavelength. If we use that information to correct those phase jump, the LQG can unwrap the phase without any group delay information. • On the data: • The phase is better unwrapped with the LQG than with the unwrap function alone • But there are still some errors. • A group delay information is still important for a sanity check • Experimentally: • There should be less phase jumps after the loop is closed, because we control the OPD better than the wavelength. Julien Lozi, LBTI meeting 4/11/2014

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