Adaptive Control Loops for Advanced LIGO - PowerPoint Presentation

1. Adaptive Control Loops for Advanced LIGO Brett Shapiro25 February 20111G1100161

2. Control Loops Keep LIGO RunningEvolving seismic noise from: weather people… adaptive control also makes a very good thesis topic…2

3. How are Adaptive Loops Useful?Simple example: Push on the pendulum to maintain a constant L.Control LawL3G1100161Applied ForceApplied Force

4. Adaptive Control for Isolation SystemsSuspensions: Angular mirror control Damping (modal or classical) Length control?Seismic Control Isolation loops sensor blending?Can be used to optimize in real time: aLIGO noise budget amplification (from sensor noise, barkhausen noise, etc) actuator forces RMS error signals just about anything else4G1100161

5. LASTI Quad-Triple CavityQuad PendulumTriple Pendulum16 meters5G1100161

6. Adaptive Control MEDM Screen

7. Questions to AnswerWhat is being controlled in this experiment?What is the adaptation optimizing?What parameters are being adapted?How are they being adapted?G11001617

8. TripleLASTI Experimental Adaptive SetupControl LawQuadOptimization Goals:Cavity Length RMSAvoid saturating the actuatorsPenultimate massTest massSolution: Least squares adaptationTest mass (TM)Penultimate mass (PUM)Top mass (Top)Length measurement:AKA ‘Error signal’8

9. Control Block Diagram Diagram3 primary components to adaptive control:1. ControlTriple+-ControlCavity NoiseError signal9G1100161

10. Adaptive Control Architecture3 primary components to adaptive control:1. Control – parameterized in terms of adapting parameters2. Adaptation algorithm – updates control parametersTriple+-ControlAdaptCavity NoiseError signal10G1100161

11. Adaptive Control Architecture3 primary components to adaptive control:1. Control – parameterized in terms of adapting parameters2. Adaptation algorithm – updates control parameters3. Costs – variables that are optimized with adaptationTriple+-ControlCostAdaptCavity NoiseError signal11G1100161

12. Adaptive Control Architecture3 primary components to adaptive control:1. Control – parameterized in terms of adapting parameters2. Adaptation algorithm – updates control parameters - uses a least squares optimization routine3. Costs – variables that are optimized with adaptationTriple+-ControlCostAdaptCavity NoiseError signal12G1100161

13. Cost Box: Summation of CostsTriple+-ControlCostAdaptCavity NoiseError signalThe cost box measures the performance values we care about and scales them to get the costs: 1. Error RMS 2. PUM force RMS 3. Test mass force RMS

14. Control BoxTriple+-ControlCostAdaptCavity NoiseError signalControl filters are parameterized in terms of: PUM crossover frequencies Test mass crossover frequency

15. Adaptation BlockTriple+-ControlCostAdaptCavity NoiseError signal

16. Adaptation Algorithm: Least Squares Minimization Approach Total system cost Cost gradient System Jacobian matrixDefinitions for optimizationv = total costc = list of costs we want to optimizeθ = list of adjustable control parametersJ = System Jacobian matrix Variable list JT yields gradient descent (1st order). J-1 yields Gauss-Newton (2nd order).Optimization routine

17. Adaptation Algorithm: quadratic minimization approachv = total costc = list of costs we want to optimizeθ = list of adjustable control parametersJ = system Jacobian matrixα = user defined scalar step sizeVariable listAdaptation Problem Question: What is J? Answer: We do not know. It depends on θ and c, and other unknown or unmodeled parameters. Good news: we can estimate it in real-time. A recursive least squares algorithm (RLS) is used.

18. Results

19. Simulated Results19G1100161

20. Results20G1100161

21. Results21G1100161

22. ConclusionsAdaptive control is a powerful real-time self-tuning method for many aLIGO loops.Can target arbitrary performance requirements:Avoiding actuator saturationsMinimizing noise amplificationReal-time RLS estimation of system response compensates for unknowns adequately.Adaptation speed is limited by RMS averagingComplexities beyond this talk existAchieving good estimates with the JacobianSetting good stopping and starting conditions22G1100161

23. Backups23G1100161

24. Feedback Filter Box24G1100161

25. Transfer Function Canonical formsLaplace Transfer functionState space transfer function in observer canonical form25G1100161

26. Parametric Transfer FunctionsTest mass feedback filter: Laplace formTest mass feedback filter: State space formωugf = unity gain freq.p -> gives phase margin around ugff -> f*ωugf is the freq. of a low pass poleK is a constant scaling factor which compensates for the plant gain26G1100161

27. Parametric Filter in LIGO’s RCG Simulink EnvironmentG110016127

28. Parametric Controller TFs28G1100161

29. Parametric Controller TFs29G1100161

30. Cost Box DetailsError RMS scalingActuator RMS scalingThe error RMS is scaled relative to a target value (such as the design value).The actuator RMS is scaled relative to its saturation limit. In this case, the scaled value is set to ‘explode’ when the RMS reaches the saturation limit.30G1100161

31. Adaptation Algorithm: quadratic minimization approach1st order algorithm: gradient descent methodv = total costc = list of costs we want to optimizeθ = list of adjustable control parametersJ = system Jacobian matrixα = user defined scalar step sizet = current time stepVariable list2nd order algorithm: Gauss-Newton methodFor noninvertible J(+ -> pseudoinverse)For invertible JCost Reduction Choices

32. Adaptation Algorithm: quadratic minimization approachv = total costc = list of costs we want to optimizeθ = list of adjustable control parametersJ = system Jacobian matrixα = user defined scalar step sizet = current time stepVariable listJacobian definitionAdaptation ProblemK = Jacobian update gainApproximate errorRLS updating

33. RLS for Jacobian EstimationG110016133Definitions:i = row index of the Jacobian matrixλ = exponential forgetting factor.0 < λ ≤ 1Cost function optimized by RLSIterative RLS algorithm1.2.3.0. Initialize J and P4. Go back to 1.

34. Convergence of JacobianG110016134A necessary condition for the convergence of RLS algorithm to the true J is that this matrix is nonsingular.A real-time estimate of the invertiblility of this matrix is to calculate this matrix over a finite number of time steps and then calculate its condition number. The condition number is the smallest eigenvalue divided by the largest.Condition number R = (max eigenvalue of Θ)/(min eigenvalue Θ)

35. Results35G1100161

36. ResultsG110016136

37. Thesis ContributionsThe literature has very little on generating real-time estimates for how controller parameters influence the statistics of linear system performance. Similarly there is little in the literature on using real-time function minimization techniques to optimize linear system performance.Optimization of Jacobian estimation accuracy.Optimizing adaptation rate using the measured statistics of the stochastic system performance.Use of singular value decomposition to quantify behavior of control adaption and system Jacobian.G110016137