Optimisation Based Clearance of Nonlinear Flight Control Laws

1. 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. Overview • Nonlinear flight clearance • A general optimisation framework • Worst case uncertainty evaluation • Clearance over regions of the flight envelope • Worst case input identification • Summary

3. 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. Nonlinear flight clearance • Nonlinear flight clearance criterion • Based on time response, peak overshoot • AoA limit exceedance

5. 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. ADMIRE model • Dynamics …(1)

7. ADMIRE model • Control algorithm …(2)

8. AIRCRAFT MATHEMATICAL MODEL ADMIRE model • 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

9. The philosophy Reference inputs Uncertain parameters Mach Altitude Level Trim Finite time history Optimisation Algorithm General optimisation framework

10. Global Optimisation Schemes • Several algorithms evaluated: • Genetic algorithms (GA) • Differential evolution (DE) • Hybrid GA / Hybrid DE • Dividing Rectangles (DIRECT)

11. Global Optimisation Scheme Genetic algorithms • Search space • Accuracy 1e-6 • Chromosomes length 105 bits (5 genes) • Initial population 50 • Genetic operators

12. 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. [0.1000, 0.0750, 0.0500, 0.18309, 0.0500, 36.0908] 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)

14. Global Optimisation Scheme Differential Evolution • Random initialisation • Mutation • Crossover • Evaluation and selection • Termination criteria same as that of GA

15. [0.1000, 0.0750, 0.0500, 0.18309, 0.0500, 36.0908] Global Optimisation Scheme DE Results • Better convergence to global optimum • Reduced number of simulations (~3000)

16. Global Optimisation Scheme Global optimisation comparison statistics Trials Trials Trials Trials

17. 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. Hybrid Optimisation Scheme Hybrid genetic algorithm (HGA) • 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

19. [0.1000, 0.0750, 0.0500, 0.18309, 0.0500, 36.0908] Hybrid Optimisation Scheme HGA Results • Faster convergence to global optimum • Smaller No. of simulations (~2000) • Good reliability (92%)

20. 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. [0.1000, 0.0750, 0.0500, 0.18309, 0.0500, 36.0908] Hybrid Optimisation Scheme HDE Results • Faster convergence to global optimum • Significantly fewer No. of simulations (~1000) • Excellent reliability (98%)

22. Hybrid Optimisation Scheme Hybrid optimisation comparison statistics Trials Trials Trials Trials

23. Flight envelope clearance Optimisation based clearance over a continuous region of flight envelope: 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

24. Optimisation Performance

25. Clearance Results

26. Clearance Results

27. Clearance Results

28. 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. 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. Computation of worst-case pilot inputs • Klonk inputs: FULL NONLINEAR AIRCRAFT SIMULATION MODEL Mach Altitude Level Trim Global Optimisation

31. 0.0611 0.0648 -0.0020 -0.0022 0.0418 66.4316 Computation of worst-case pilot inputs • Worst-case inputs: 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. Computation of worst-case pilot inputs • What’s the problem? Input of Rate Limiter Output of Rate Limiter Rudder

33. 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