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De Demer geregeld met MPC

Public Doctoral Defense. De Demer geregeld met MPC. Jury:. A. Haegemans, chair B. De Moor, promotor J. Berlamont, co-promotor J. Suykens P. Willems B. De Schutter (TU Delft) R. Negenborn (TU Delft). Toni Barjas Blanco. SCD Research Division ESAT – K. U. Leuven September 8 th , 2010.

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De Demer geregeld met MPC

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  1. Public Doctoral Defense De Demer geregeld met MPC Jury: A. Haegemans, chair B. De Moor, promotor J. Berlamont, co-promotor J. Suykens P. Willems B. De Schutter (TU Delft) R. Negenborn (TU Delft) Toni Barjas Blanco SCD Research Division ESAT – K. U. Leuven September 8th, 2010 Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  2. outline • Introduction • River Modeling • Nonlinear Model Predictive Controller • Nonlinear Moving Horizon Estimator • Set Invariance • Conclusions and Future research Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  3. Introduction • Floodings in the Demer basin The damage caused in the Demer basin by the most recent floodings. Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  4. Introduction • In this research: “We implement a nonlinear model predictive controller for flood regulation.” • Current: three-position controller • not based on rainfall predictions • no optimization Goal: reduction of floods Proposed control scheme Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  5. River modeling • Modeling techniques: • Finite-difference models: very accurate, too complex • Integrator-delay models: fast, linear • System identification: not based on conservation laws • Reservoir model: • Fast • Nonlinear • Accurate • Conservation laws Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  6. River modeling State variables : Inputs : • Discharges (q) • Water levels (h) • Volumes (v) • Gates • Rainfall-runoff (disturbances) Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  7. River modeling Conceptual model • Volume balance • Nonlinear H-V relation Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  8. River modeling • Downstream reach • Nonlinear gate equations (Infoworks) Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  9. River modeling • Nonlinear gate equations (Infoworks) independent of the gate level  uncontrollability (see later) Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  10. River modeling • Calibration and validation Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  11. River modeling • Calibration and validation Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  12. River modeling • Calibration and validation Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  13. Control scheme • State of the art • Classical feedback and feedforward • Optimal control • Heuristic control • Three-position control • Model predictive control • Why model predictive control ? • River dynamics are slow • Constraint handling • Rainfall predictions (model based) • MIMO Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  14. Control scheme • Model predictive control x u t Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  15. Control scheme • Practical MPC for setpoint regulation • Flood regulation • Nonlinear dynamics • Nonlinear relation discharge/gate position Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  16. Control scheme • Nonlinear model predictive control scheme (NLP) subject to the following constraints for : Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  17. Control scheme u • Simulation • Linearization with central difference scheme •  LTV system with x t k k+1 k+2 Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  18. Control scheme • SQP algorithm (ii). Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  19. Control scheme • Constraints: • Hard constraints : input • Soft constraints : water levels • Constraint strategy: • Heavy rainfall  flooding unavoidable • Constraint prioritization: remove less important constraints and resolve NLP • Cost function strategy: • Adjusting weights in order to minimize constraint violation of removed constraints Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  20. Control scheme • Uncontrollability • Equations: • Reference levels and corresponding weights in cost function Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  21. Simulations • Regulation and flood cost: • with • No uncertainty outperformed by MPC Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  22. Simulations • Gaussian uncertainty (10 % unc, increase of 0.2 %, overestimation) ±equal outperformed by MPC Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  23. State estimation • At each sampling time  estimation current state based on past measurements of a subset of the states. Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  24. State estimation • State of the art in river control: • Sensor measurements • Kalman filtering Moving horizon estimation (MHE) • MHE: • Dual of MPC • Online constrained optimization problem • Finite window in the past  computational tractability • Solves following problem: “Given the measurements of a subset of states within the past time window, find all the states in that window that match the measurements as close as possible, given the underlying system model .” Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  25. Moving horizon estimator • Nonlinear MHE scheme (NLP) Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  26. Moving horizon estimator • Linearization of nonlinear system around previous estimated state trajectory. • Linearized model: • Central difference scheme: x k-5 k-4 k-3 k-2 k-1 k t with Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  27. Moving horizon estimation • SQP • Linearize system around state trajectory obtained at the previous time step or iteration: • Solve QP and obtain a new estimated state trajectory. • Perform line-search between previous and new state trajectory. • Check convergence: • Converged  stop SQP iterations • Not converged  go to step 1 Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  28. Simulations • Gaussian uncertainty on rainfall-runoff • Measurement noise • MHE parameters • State estimates Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  29. Simulations • Comparison performance MPC with three-position controller Slightly worsened Significant improvement Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  30. Set invariance • LTV system: • Constraints: • Set invariance: Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  31. Set invariance • MPC stability (dual mode MPC): • Polytopic • Ellipsoidal Convex program Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  32. Set invariance • Low-complexity polytopes: • Vertices: • Existing algorithms : • Conservative • Fixed feedback law K • Scale badly with state dimension (vertex based  2n vertices) New algorithm with better properties Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  33. Set invariance • New algorithm : • Initial invariant and feasible set • Sequence of convex programs increasing the volume of the set while keeping it invariant and feasible until convergence • Initialization : • Convex LMI : convex Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  34. Set invariance • Volume maximization : • New invariance conditions : • Introduction of transformed variables : • New parametrization of unknown variable P: with X a symmetric inverse positive matrix Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  35. Set invariance • Algorithm : Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  36. Example • Control of temperature profile of a one-dimensional bar [Agudelo,2006]: New algorithm outperforms existing ones w.r.t. volume of set as well as computation time Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  37. Setpoint regulation • Regulation of the upstream part of the Demer • Steady state Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  38. Setpoint regulation • LQR • Linearize nonlinear model around steady state • Determine state feedback K with LQR theory • Robust state feedback • Determining a LTV system  simulation: • Invariant set + feedback K Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  39. Setpoint regulation • Simulation 1: step disturbance Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  40. Setpoint regulation • Simulation 2: • new K  LTV based on 6 linear models • 2 different step disturbances and no disturbance at the end Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  41. Setpoint regulation • Simulation 3: simulation first 200 hours of 1998 Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  42. Conclusions and future research Concluding remarks • A nonlinear model was determined accurate and fast enough for real-time control purposes. • A nonlinear MPC and MHE scheme was developed that outperformed the current three-position controller. Moreover, the scheme was robust against uncertainties. • A new algorithm was developed for the efficient calculation of low-complexity polytopes. The algorithm was used for improved setpoint regulation of the upstream part of the Demer. Future research • Coupling control scheme with finite-difference model • Extending model with flood map • Distributed MPC • Extend results to invariant low-complexity polytopes with a more general shape Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

  43. THANK YOU FOR LISTENING Toni Barjas Blanco - Public Doctoral Defense - September 8th, 2010

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