Introduction River modeling NMPC NMHE Set invariance Conclusions

Introduction River modeling NMPC NMHE Set invariance Conclusions De Demer geregeld met MPC Public Doctoral Defense Toni Barjas Blanco Jury: SCD Research Division ESAT – K. U. Leuven September 8 th, 2010 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 - Public Doctoral Defense - September 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance 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 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE 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 8 th, 2010 Set invariance Conclusions

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

Introduction River modeling NMPC NMHE Set invariance Conclusions River modeling § Modeling techniques: Ø Finite-difference models: very accurate, too complex Ø Integrator-delay models: fast, linear Ø System identification: not based on conservation laws § Reservoir model: Ø Fast Ø Nonlinear Ø Accurate Ø Conservation laws Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance Conclusions River modeling State variables : § Discharges (q) § Water levels (h) § Volumes (v) Inputs : § Gates § Rainfall-runoff (disturbances) Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance River modeling Conceptual model § Volume balance § Nonlinear H-V relation Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance River modeling § Downstream reach § Nonlinear gate equations (Infoworks) Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance River modeling § Nonlinear gate equations (Infoworks) independent of the gate level uncontrollability (see later) Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance River modeling § Calibration and validation Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance River modeling § Calibration and validation Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance River modeling § Calibration and validation Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Control scheme § State of the art Ø Classical feedback and feedforward Ø Optimal control Ø Heuristic control Ø Three-position control Ø Model predictive control § Why model predictive control ? Ø River dynamics are slow Ø Constraint handling Ø Rainfall predictions (model based) Ø MIMO Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Control scheme § Model predictive control x u t Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Control scheme § Practical MPC for setpoint regulation § Flood regulation Ø Nonlinear dynamics Ø Nonlinear relation discharge/gate position Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Control scheme § Nonlinear model predictive control scheme (NLP) subject to the following constraints for Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 : Conclusions

Introduction River modeling § Simulation NMPC NMHE Set invariance Conclusions Control scheme u x k k+1 t k+2 § Linearization with central difference scheme § LTV system with Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance Control scheme § SQP algorithm (ii). Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Conclusions Control scheme § Constraints: Ø Hard constraints : input Ø Soft constraints : water levels § Constraint strategy: Ø Heavy rainfall flooding unavoidable Ø Constraint prioritization: remove less important constraints and resolve NLP § Cost function strategy: Ø Adjusting weights in order to minimize constraint violation of removed constraints Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance Control scheme § Uncontrollability § Equations: § Reference levels and corresponding weights in cost function Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Simulations § Regulation and flood cost: Ø Ø with § No uncertainty outperformed by MPC Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Conclusions Simulations § Gaussian uncertainty (10 % unc, increase of 0. 2 %, overestimation) ±equal outperformed by MPC Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance 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 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Conclusions State estimation § State of the art in river control: Ø Sensor measurements Ø Kalman filtering Moving horizon estimation (MHE) § MHE: Ø Dual of MPC Ø Online constrained optimization problem Ø Finite window in the past computational tractability Ø 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 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance Moving horizon estimator § Nonlinear MHE scheme (NLP) Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Conclusions Moving horizon estimator § Linearization of nonlinear system around previous estimated state trajectory. x k-5 k-4 k-3 k-2 k-1 k § Linearized model: with § Central difference scheme: Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 t

Introduction River modeling NMPC NMHE Set invariance Conclusions Moving horizon estimation § SQP Ø Linearize system around state trajectory obtained at the previous time step or iteration: Ø Solve QP and obtain a new estimated state trajectory. Ø Perform line-search between previous and new state trajectory. Ø Check convergence: ØConverged stop SQP iterations ØNot converged go to step 1 Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010

Introduction River modeling NMPC NMHE Simulations § Gaussian uncertainty on rainfall-runoff § Measurement noise § MHE parameters § State estimates Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Set invariance Conclusions

Introduction River modeling NMPC NMHE Set invariance Conclusions Simulations § Comparison performance MPC with three-position controller Slightly worsened Significant improvement Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance § LTV system: § Constraints: § Set invariance: Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Conclusions Set invariance § MPC stability (dual mode MPC): § Polytopic § Ellipsoidal Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Convex program

Introduction River modeling NMPC NMHE Set invariance Conclusions Set invariance § Low-complexity polytopes: § Vertices: § Existing algorithms : Ø Conservative Ø Fixed feedback law K Ø Scale badly with state dimension (vertex based 2 n vertices) New algorithm with better properties Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance Conclusions Set invariance § New algorithm : Ø Initial invariant and feasible set Ø Sequence of convex programs increasing the volume of the set while keeping it invariant and feasible until convergence § Initialization : convex § Convex LMI : Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance Conclusions Set invariance § Volume maximization : Ø New invariance conditions : Ø Introduction of transformed variables : Ø New parametrization of unknown variable P: with X a symmetric inverse positive matrix Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance § Algorithm : Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Conclusions 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 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance Setpoint regulation § Regulation of the upstream part of the Demer § Steady state Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance 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 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Setpoint regulation § Simulation 1: step disturbance Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance 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 8 th, 2010 Conclusions

Introduction River modeling NMPC NMHE Set invariance Conclusions Setpoint regulation § Simulation 3: simulation first 200 hours of 1998 Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010 LQR cost Robust feedback cost 2. 6883 1. 1913

Introduction River modeling NMPC NMHE Set invariance Conclusions and future research Concluding remarks Ø A nonlinear model was determined accurate and fast enough for real-time control purposes. Ø A nonlinear MPC and MHE scheme was developed that outperformed the current threeposition 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. Ø Future research Ø Ø 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 8 th, 2010

Introduction River modeling NMPC NMHE Set invariance Conclusions THANK YOU FOR LISTENING Toni Barjas Blanco - Public Doctoral Defense - September 8 th, 2010