1. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Optimisation Based Clearance of Nonlinear Flight Control Laws Prathyush P. Menon Jongrae Kim Declan G. Bates Ian Postlethwaite Control & Instrumentation Research Group, Department of Engineering, University of Leicester, Leicester LE1 7RH, UK.

2. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Nonlinear flight clearance A general optimisation framework Worst case uncertainty evaluation Clearance over regions of the flight envelope Worst case input identification Summary Overview

3. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Nonlinear flight clearance Control algorithms usually designed based on linear models Robust performance over the whole flight envelope Controller gains are scheduled for the whole envelope How can we effectively “clear” the controller over the whole envelope?

4. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Nonlinear flight clearance Nonlinear flight clearance criterion –Based on time response, peak overshoot –AoA limit exceedance

5. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Nonlinear flight clearance The uncertain parameters define a multidimensional (dimension ‘l’) hyper box The worst case need not be at the vertices (max or min values) Industry needs efficient, reliable and easily portable methods Problem becomes extremely computationally expensive Need efficient search methods to find “worst - case” uncertain parameter combinations

6. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 ADMIRE model Dynamics …(1)

7. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 ADMIRE model Control algorithm …(2)

8. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 ADMIRE –Simulink model –Long. controller scheduled over the flight envelope –SAAB phase compensation rate limiter active –Nonlinear stick shaping elements present –Reference inputs limited to ±40 N (for this study) – Uncertain parameters are bounded ADMIRE model AIRCRAFT MATHEMATICAL MODEL

9. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 General optimisation framework The philosophy Reference inputs Uncertain parameters Mach Altitude Level Trim Finite time history Optimisation Algorithm

10. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Global Optimisation Schemes Several algorithms evaluated: –Genetic algorithms (GA) –Differential evolution (DE) –Hybrid GA / Hybrid DE –Dividing Rectangles (DIRECT)

11. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Global Optimisation Scheme Genetic algorithms Search space Accuracy 1e-6 Chromosomes length 105 bits (5 genes) Initial population 50 Genetic operators Roulette selection0.6 Single point crossover0.9 Binary uniform mutation 0.00 5

12. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Global Optimisation Scheme Genetic algorithms (cont.) Termination criteria – improvement on the solution accuracy ≤ 1e-6 – for a defined number of generations, fixed at 15 – stop iteration Each trial gives different total number of simulations

13. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Global Optimisation Scheme GA Results Slow convergence to global optimum No. of simulations very high (~5000) Computationally prohibitive – slow (~ 3-4 hours for each test point) [0.1000, 0.0750, 0.0500, 0.18309, 0.0500, 36.0908]

14. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Global Optimisation Scheme Differential Evolution Random initialisation Mutation Crossover Evaluation and selection Termination criteria same as that of GA

15. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Global Optimisation Scheme DE Results Better convergence to global optimum Reduced number of simulations (~3000) [0.1000, 0.0750, 0.0500, 0.18309, 0.0500, 36.0908]

16. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Global Optimisation Scheme Global optimisation comparison statistics OptimisationTrialsAvg.Max.Min.Std. Dev. Prob. of success GA100448575002400828.36465% DE100308641761152567.5790% Trials

17. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Hybrid Optimisation Scheme Hybrid global and local optimisation schemes Exploit the advantages of both schemes Question: When to switch between the schemes? Standard approach: run global algorithm, then run local algorithm We use a more sophisticated decision making scheme based on one proposed by Lobo and Goldberg, 1996

18. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Hybrid Optimisation Scheme Probabilistic switching scheme Weighted reward for each algorithm – Probability of algorithm being selected depends on improvement in cost function Initial probabilities selected to favour use of GA at beginning “fmincon” is the local algorithm (SQP) Termination criteria same as previous cases Hybrid genetic algorithm (HGA)

19. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Hybrid Optimisation Scheme HGA Results Faster convergence to global optimum Smaller No. of simulations (~2000) Good reliability (92%) [0.1000, 0.0750, 0.0500, 0.18309, 0.0500, 36.0908]

20. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Hybrid Optimisation Scheme Hybrid differential evolution Global optimisation used is DE Local optimisation is “fmincon” (SQP) Switching scheme –Simple method; Starts with DE –When there is no improvement from successive iterations: –choose a random initial solution from the current iteration set –apply local optimisation –replace solution from local if improvement occurs Termination criteria: same as previous cases

21. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Hybrid Optimisation Scheme HDE Results Faster convergence to global optimum Significantly fewer No. of simulations (~1000) Excellent reliability (98%) [0.1000, 0.0750, 0.0500, 0.18309, 0.0500, 36.0908]

22. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Hybrid Optimisation Scheme Hybrid optimisation comparison statistics OptimisationTrialsAvg.Max.Min.Std. Dev. Prob. of success HGA100201144681357547.4292% HDE10011061434477192.4298% Trials

23. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Flight envelope clearance Mach [ 0.4 - 0.5 ] Altitude [ 1000 - 4000 ] Uncertainties same as discussed earlier Stick input now to 80N. We apply Hybrid DE scheme over the region of flight envelope Optimisation based clearance over a continuous region of flight envelope:

24. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Optimisation Performance

25. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Clearance Results

26. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Clearance Results

27. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Clearance Results

28. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Clearance Results Worst case Flight condition P. P. Menon, J. Kim, D.G. Bates and I. Postlethwaite, ``Clearance of nonlinear flight control laws using hybrid evolutionary optimisation”, to appear in IEEE Transactions on Evolutionary Computation 2006

29. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Deterministic global optimisation Disadvantages of stochastic optimisation for flight clearance: No guaranteed proof of convergence Require statistical analysis of performance Non-repeatability of results DIviding RECTangles (DIRECT) is a deterministic global optimisation algorithm with a proof of convergence Initial results of application of this method for flight clearance are very promising: P. P. Menon, D.G. Bates and I. Postlethwaite, ``A Hybrid Deterministic Optimisation Algorithm for Nonlinear Flight Clearance”, to appear in the proceedings of the American Control Conference, Boston, 2006

30. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Computation of worst-case pilot inputs Klonk inputs: Global Optimisation FULL NONLINEAR AIRCRAFT SIMULATION MODEL Mach Altitude Level Trim

31. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Computation of worst-case pilot inputs Worst-case inputs: 0.0611 0.0648-0.0020-0.00220.041866.4316 Time: 3hrs. 5mins. P. P. Menon, D. G. Bates and I. Postlethwaite, ``Computation of Worst-Case Pilot Inputs for Nonlinear Flight Control System Analysis'', AIAA Journal of Guidance, Control and Dynamics, 29(1), 2006.

32. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Computation of worst-case pilot inputs Rudder Input of Rate Limiter Output of Rate Limiter What’s the problem?

33. Department of Engineering, Control & Instrumentation Research Group 22 – Mar – 2006 Conclusions Results demonstrate that the uncertain parameter combination resulting in worst behaviour need not be at extremum bounds Hybrid optimisations schemes successfully applied to a nonlinear flight clearance problem over a continuous region of the flight envelope Flexibility of the framework also allows robust computation of worst case pilot inputs Improved accuracy and faster convergence due to hybridisation could allow the use of such methods in the industrial flight clearance process