Robust Optimal Control for Nonlinear Dynamic Systems

Transcription

1 Robust Optimal Control for Nonlinear Dynamic Systems, Professor for Optimization in Engineering, Electrical Engineering Department (ESAT) K.U. Leuven, Belgium Joint work with Peter Kuehl, Boris Houska, Andreas Ilzhoefer INRIA-Rocquencourt, May 31, 2007

2 Overview Dynamic Optimization Example: Control of Batch Reactors How to Solve Dynamic Optimization Problems? (recalled) Two Challenging Applications: Robust Open-Loop Control of Batch Reactor Periodic and Robust Optimization for Flying Windmills

3 Overview Dynamic Optimization Example: Control of Batch Reactors How to Solve Dynamic Optimization Problems? Two Challenging Applications: Robust Open-Loop Control of Batch Reactor Periodic and Robust Optimization for Flying Windmills

4 Control of Exothermic Batch Reactors Cooperation between Heidelberg University and Warsaw University of Technology Work of Peter Kühl (H.G. Bock, Heidelberg) with A. Milewska, E. Molga (Warsaw)

5 Batch Reactor in Warsaw [Peter Kuehl, Aleksandra Milewska] Esterification of 2-Butanol (B) by propionic anhydride (A): exothermic reaction, fed-batch reactor with cooling jacket Aim: complete conversion of B, avoid explosion! Control: dosing rate of A

6 Safety Risk: Thermal Runaways accumulation - temperature rise - thermal runaway Try to avoid by requiring upper bounds on reactor temperature T R, and hypothetical adiabatic temperature S that would result if all A reacts with B

7 Differential (Algebraic) Equation Model (1) (2)

8 Dynamic Optimization Problem for Batch Reactor Constrained optimal control problem: minimize remaining B subject to dosing rate and temperature constraints Generic optimal control problem:

9 Overview Dynamic Optimization Example: Control of Batch Reactors How to Solve Dynamic Optimization Problems? Three Challenging Applications: Robust Open-Loop Control of Batch Reactor Periodic and Robust Optimization for Flying Windmills

10 Recall: Direct Multiple Shooting [Bock, Plitt 1984]

11 Solution of Peter s Batch Reactor Problem

12 Experimental Results for Batch Reactor Mettler-Toledo test reactor R1 batch time: 1 h end volume: ca. 2 l

13 Experimental Results for Batch Reactor (Red) large model plant mismatch Safety critical! How can we make Peter and Blue: Aleksandra s Simulation work safer? Red: Experiments

14 Experimental Results for Batch Reactor (Red) large model plant mismatch Safety critical! How can we make Peter s and Aleksandra s work safer?

15 Overview Dynamic Optimization Example: Control of Batch Reactors How to Solve Dynamic Optimization Problems? Two Challenging Applications: Robust Open-Loop Control of Batch Reactor Periodic and Robust Optimization for Flying Windmills

16 Robust Worst Case Formulation Make sure safety critical constraints are satisfied for all possible parameters p! Semi-infinite optimization problem, difficult to tackle...

17 Approximate Robust Formulation[Körkel, D., Bock, Kostina 04, 05] Fortunately, it is easy to show that up to first order: So we can approximate robust problem by: ~ Intelligent safety margins (influenced by controls)

18 Numerical Issues for Robust Approach for optimization, need further derivatives of treat second order derivatives by internal numerical differentiation in ODE/DAE solver implemented in MUSCOD-II Robust -Framework [C. Kirches] use homotopy: start with nominal solution, increase slowly, employ warm starts

19 Estimated Parameter Uncertainties for Test Reactor Standard deviation gamma T jacket 0.3 K 3.0 m catalyst 0.5 g (~10 %) 3.0 U A 10.0 W/(m 2 K) (~10 %) u offset kg/s (~10 % of upper bound)

20 Robust Open Loop Control Experiments

21 Robust Optimization Result and Experimental Test Safety margin Perturbed Scenarios (Simulated) Blue: Simulation Red: Experiments

22 Comparison Nominal and Robust Optimization Different solution structure. Model plant mismatch and runaway risk considerably reduced. Complete conversion.

23 Comparison Nominal and Robust Optimization Different solution structure. Model plant mismatch and runaway risk considerably reduced. Complete conversion.

24 Overview Dynamic Optimization Example: Control of Batch Reactors How to Solve Dynamic Optimization Problems? Two Challenging Applications: Robust Open-Loop Control of Batch Reactor Periodic and Robust Optimization for Flying Windmills

25 Conventional Wind Turbines Due to high speed, wing tips are most efficient part of wing High torques at wings and mast limit size and height of wind turbines But best winds are in high altitudes! Could we construct a wind turbine with only wing tips and generator?

26 Conventional Wind Turbines Due to high speed, wing tips are most efficient part of wing High torques at wings and mast limit size and height of wind turbines But best winds are in high altitudes! Could we construct a wind turbine with only wing tips and generator?

27 Crosswind Kite Power (Loyd 1980) use kite with high lift-to-drag-ratio use strong line, but no mast and basement automatic control keeps kites looping But where could a generator be driven?

28 New Power Generating Cycle New cycle consists of two phases: Power generation phase: add slow downwind motion by prolonging line (1/3 of wind speed) generator at ground produces power due to large pulling force Retraction phase: change kite s angle of attack to reduce pulling force pull back line Cycle produces same power as (hypothetical) turbine of same size!

29 New Power Generating Cycle New cycle consists of two phases: Power generation phase: add slow downwind motion by prolonging line (1/3 of wind speed) generator at ground produces power due to large pulling force Retraction phase: change kite s angle of attack to reduce pulling force pull back line Cycle produces same average power as wind turbine of same wing size, but much larger units possible (independently patented by Ockels, Ippolito/Milanese, D.)

30 Can stack kites, can use on sea

31 Periodic Optimal Control (with Boris Houska) Have to regard also cable elasticity ODE Model with 12 states and 3 controls forces at kite Control inputs: line length roll angle (as for toy kites) lift coefficient (pitch angle)

32 Some Kite Parameters e.g. 10 m x 50 m, like Boeing wing, but much lighter material standard wind velocity for nominal power of wind turbines

33 Solution of Periodic Optimization Problem Maximize mean power production: by varying line thickness, period duration, controls, subject to periodicity and other constraints:

34 Solution of Periodic Optimization Problem Maximize mean power production: by varying line thickness, period duration, controls, subject to periodicity and other constraints:

35 Visualization of Periodic Solution

36 Prototypes built by Partners in Torino and Delft New cycle consists of two phases: Power generation phase: add slow downwind motion by prolonging line (1/3 of wind speed) generator at ground produces power due to large pulling force Pull back phase: change kite s angle of attack to reduce pulling force pull back line Cycle allows same power production as wind turbine of same size!

37 Experimental Proof of Concept in Italy

38 What about dancing kites?

39 Optimization with dancing kites: 14 MW possible 2 x 500 m 2 airfoils kevlar line 1500 m, diameter 8 cm wind speed 10 m/s

40 Question: could kite also fly without feedback? Stability just by smart choice of open-loop controls?

41 Linearization of Poincare Map determines stability Monodromy matrix = linearization of Poincare Map. Stability Spectral radius smaller than one. Cons of Spectral radius: Nonsmooth criterion difficult for optimization uncertainty of parameters not taken into account

42 Periodic Lyapunov Equations and Stability Lyapunov Lemma [Kalman 1960]: Nonlinear system is stable periodic Lyapunov Equation with has bounded solution.

43 Robust stability optimization problem Allows to robustly satisfy inequality constraints!

44 Orbit optimized for stability (using periodic Lyapunov eq.) Long term simulation: Kite does not touch ground Open-loop stability only possible due to nonlinearity!

45 Alternative: NMPC Control after turn of wind direction

46 Summary: Nonlinear Dynamic Optimization Open-Loop Optimization (prone to model-plant-mismatch) Robust Open-Loop (no sensor feedback needed, simple) Model Predictive Control (feedback by fast online optimization)

47 Two events of interest this year Workshop on NMPC Software and Applications (NMPC-SOFAP), Loughborough, United Kingdom, April 19-20, (inv. speakers: Biegler, Findeisen, Kerrigan, Richalet, Schei) 13th Czech-French-German Conference on Optimization (CFG07), Heidelberg, Germany, September 17-21, (inv. speakers: Fletcher, Scherer, Trelat, Waechter,...) Traditionally strong in optimal control.

48 4 PhD Positions in Numerical Optimization: Sequential Convex Programming Algorithms for Nonlinear SDP Large Scale & PDE Constrained Real-Time Optimization Algorithms Fast Model Predictive Control Applications in Mechatronic Systems Shape Optimization of Mechanical Parts under Inertia Loading (deadline: June 21, 2007)