Optimal Control and Estimation:
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1 Optimal Control and Estimation: An ICFD Retrospective N.K. Nichols University of Reading ICFD 25th Anniversary
2 With thanks to: CEGB / National Power SSEB BP Exploration British Gas Institute of Hydrology (CEH) HR Wallingford DRA RAE Bedford Met Office for support for 26 PhD students and 7 post-docs to work on this research
3 Outline 1. Introduction 2. Tidal Energy 3. Flight Control 4. Numerical Weather Prediction 5. Conclusions / Future
4 1. Introduction
5
6
7 Control problem: Find the input or control function that produces a given response from the system State estimation problem: Given measured output from system, find the state of the system at a specified time.
8 2. Tidal Energy
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10 Control is discontinuous!
11 Results No control Energy = With control Energy =
12 Severn Barrage Model
13 Fluid conservation Momentum conservation Boundary conditions Periodic conditions Control constraints
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16 3. Automatic Flight Control Systems
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22 2
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24 Bifurcation Diagrams Jet Fighter No control Robust decoupling control
25 4. Numerical Weather Prediction Met Office Website
26 Significant Properties: Very large number of unknowns ( ) Few observations ( ) System nonlinear unstable/chaotic Multi-scale dynamics Stochastic errors in model and observations
27 State Estimation Aim: Find the best estimate (analysis) of the true state of a system, consistent with both observations and the system dynamics given: Numerical prediction model Observations of the system (over time) Background state (prior) Estimates of the errors
28 State Estimation Aim: Find the best estimate (analysis) of the true state of a system, consistent with both observations and the system dynamics: State Numerical prediction model Observations of the system (over time) Background state (prior) Estimates of the errors
![Best Unbiased Estimate min J ( x 0 ) = min 1 ( x 0 x b )T B 1 ( x 0 x b ) 2 n + ( H i [x i ] y io )T R i 1 ( H i [x i ] y io ) i= 0 subject to where x b -](/docs-images/76/74437920/images/29-0.jpg "Background state (prior estimate) y i - Observations H i - Observation operator B - Background error covariance matrix R i - Observation error covariance")
29 Best Unbiased Estimate min J ( x 0 ) = min 1 ( x 0 x b )T B 1 ( x 0 x b ) 2 n + ( H i [x i ] y io )T R i 1 ( H i [x i ] y io ) i= 0 subject to where x b - Background state (prior estimate) y i - Observations H i - Observation operator B - Background error covariance matrix R i - Observation error covariance matrix
30 Algorithm Uses Approximate Gauss-Newton method: Solves sequence of linearized least squares problems by inner gradient iteration procedure Finds gradients by adjoint integration Truncates inner loop iterations Uses approximate linear system models Theoretical convergence results obtained by reference to GaussNewton method (SIOPT).
31 New Research Find approximate linear system models using optimal reduced order modeling techniques from control theory to improve computational efficiency Test feasibility of approach: compare solutions using Low resolution linear model Optimal reduced order model for a simple shallow water flow model.
32 Norm of analysis error for 1-D SWE model Log Error Low Res Model of order = 200 Norm = vs Reduced Model of order = 200 Norm = Low Res Model of order = 200 Norm = vs Reduced Model of order = 80 Norm = Component of state Red (dotted) = Low Resolution Model Green (dashed) = Reduced Order Model
33 5. Conclusions / Future Many exciting open problems exist with valuable applications New research areas include: - PDE constrained optimization - Stochastic PDE control and estimation (data assimilation) - Model Order Reduction - Uncertainty in prediction
34 References A.S. Lawless, N.K. Nichols, C. Boess and A. Bunse-Gerstner, Using model reduction methods within incremental four-dimensional variational data assimilation, Monthly Weather Review, 136, 2008, S. Gratton, A.S. Lawless and N.K. Nichols, Approximate GaussNewton methods for nonlinear least squares problems, SIAM Journal on Optimization, 18, 2007, D.M. Littleboy and N.K. Nichols, Modal Coupling in Linear Control Systems Using Robust Eigenstructure Assignment, Linear Algebra and Applications, , 1998, L.P. Gibson, N.K. Nichols and D.M. Littleboy, Bifurcation analysis of eigenstructure assignment control in a simple nonlinear aircraft model, J. of Guidance, Control and Dynamics, Vol. 21, 1998, J. Kautsky, N.K. Nichols and P. Van Dooren, Robust pole assignment in linear state feedback, International J. of Control, 41, 1985, N.R.C. Birkett, N.K. Nichols and D.A.C. Nicol, Dynamic models for optimal control of tidal power schemes, ICE Tidal Power Symposium, Thomas Telford, London, 1987, N.R.C. Birkett and N.K. Nichols, Optimal control problems in tidal power generation, Industrial Numerical Analysis - Case Histories, (eds. C. Elliott and S. McKee), Clarendon Press, Oxford, 1986, N.R.C. Birkett, B.M. Count, N.K. Nichols and D.A.C. Nicol, Optimal control problems in tidal power generation - full estuary model, Applied Optimization Techniques in Energy Problems, (ed. Hj. Wacker), B.G. Teubner, Stuttgart, 1985, N.R.C. Birkett, B.M. Count and N.K. Nichols, Optimal control problems in tidal power, Water Power and Dam Construction, Jan. 1984,
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