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# BOEING PROBLEM

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1. BOEING PROBLEM Eugene Lavretsky, Boeing Heinz Engl Alistair Fitt Ian Frigaard Borislava Gutarts Philipp Kuegler Xinosheng Li Alfonso Limon Yajun Mei John Ockendon ….The cast in alphabetical order

2. PROBLEM: • Assume that we are interested in unmanned aircraft only • Use simplest 6 DoF aerodynamic model • Restrict motion to 2D • Try to determine lift, drag, thrust and pitching coefficients from (noisy) measurements of aircraft position, speed, pitch rate and pitch angle

3. EQUATIONS: T, L, D, M can be assumed to be the coefficients that we are after “known” (noisy)

4. PARADIGM • For simplicity we will concentrate almost entirely on the linear equation where x is a scalar, a and b are constants and u is the “input”, which will be exactly prescribed. Approximations to a and b will be denoted by , noisy versions of x(t) will be denoted by Note: we will not consider the optimal control problem here.

5. On- or Off-line? • Off-line:collect all data from a flight; return to base. • This is not wholly desirable as a great deal of information must be collected. • Also does not allow “live” experiments. • Adaptive control is “online” but does not generalise to nonlinear equations.Our goal will be to minimise predictive rather than tracking errors. • Note that in real life things are really much more complicated as the desired coefficients are not constant, but depend on the independent variables and time.

6. TYPICAL OFFLINE METHOD • Simple regression: collect all the data Then use Moore-Penrose generalised inverse to write

7. FINE TUNING: VARIOUS ALTERNATIVES: (i) Calculate the derivatives using eg Euler (ii) Integrate both sides and do numerical integration (iii)Take Laplace transforms Note that all except (iii) work just fine for nonlinear equations, only the dependence on the PARAMETERS must be linear.

8. OFFLINE RESULTS: • These can very easily be coded up: • Using Euler’s method to approximate derivatives is AWFUL. • Integrating both sides first and doing numerical integration is much better, • Taking Laplace transform is fine too, but these are ALL OFFLINE.

9. EXAMPLE RESULTS: • Use the problem a = -1/2, b = 2, u = sin(t) • (nb here the “noise” is 1% uniform – could use other sorts) • (i) Do it all exactly: get • (ii) Do it all exactly but with added noise: get • (iii) Use Euler’s method with noise: get

10. AN ONLINE METHOD: • For GENERAL PROCEDURE: (i) Based on observed find an algorithm to estimate a and b (ii) For an input u(t) determine PE (Persistence of Excitation) conditions so that Note that the noise can be added either to x(t) or to the differential equation: both cases are similar but we did not test the latter.

11. METHOD: FOR Suppose for a moment that we know the x’s:MINIMISE for a and b to give P = G-1B in the form Now replace the x’s with

12. FINE TUNING: • Now note that by doing some integration by parts the method becomes Finally, note that if we know something about the noise (say it has mean 0 and variance 2), then it is better to use to estimate the square of x(t)

13. THE PE QUESTION: This method is only feasible if the matrix G-1 exists, so we need det(G)  0. This is assured if and only if we satisfy the PE condition (“u(t) is rich enough”) This will ensure that as t and tend to the correct values. NB one way of checking this is to propose conditions on det(G) as t

14. RESULTS: • Numerical experiments mostly work very well if a and b are both O(1). • (There may be a few starting difficulties to get over but these can be sorted out by better numerical integration methods.) • However, there may be wild divergence if a, b and/or u are either very large or very small.

15. This procedure is fully online as the integrals can be updated by adding only one value • (NOTE: in examples we simply used the trapezium rule to do the integrals)

16. RESULTS 1: (a possibly difficult example: x is a “slave” to u) CASE u = 1 a = -1, b = 1 x = 1 – e-t Red dots show successive approximations to the solutions. The fact that x is a “slave to u” suggests that the method might not work – but it does - and Yajun can PROVE it!

17. RESULTS 2: (divergence) CASE u = sinh(t) a = -1, b = 1 x = 1 – e-t Red dots show successive approximations to the solutions. We see divergence, followed by convergence (to the wrong solution!)

18. RESULTS 3: (another possibly difficult example) CASE u = 1/(1+t) a = -1, b = 1 x = a mess of Ei’s Red dots show successive approximations to the solutions.

19. RESULTS 3 continued: This leads us to consider the relevance of this method to the ULTIMATE study group problem 0 = x(t) + bu(t) (*) The result of attacking this problem using a gradient method (as in “the book”) suggests that u must not decay too fast at infinity if the PE condition is to be satisfied.

20. MORE ABOUT THE PE CONDITION:

21. PARAMETER IDENTIFICATION (i) Traditional approach: F(b) = x where F is the parameter to solution map. We want to minimize b – b* where b* contains a priori information. This would lead to standard iterative method. (ii) “All at once” approach:  b - btrue  should be minimised under the constraint G(b,x) = 0. (differential equation as constraint) This leads to a saddle point problem, which can be solved using mixed finite elements.

22. PARAMETER IDENTIFICATION (iii) Abstract structure here: G(b,x,u) = 0 treat as constraint.  b - btrue  should be minimised as t , n , or ? or we minimise  b - btrue 2 +  u - uideal 2 where we have information on u: (a) u is known (prescribed) (b) feasibility constraints on u Methods for that could also lead to strategies for finding good u, and allow for error analysis. (iv) Analogy: row-action methods in tomography (use information as it comes).

23. FINALLY: 0 = x(t) + bu(t) (*) One proposal to solve (*) is to “regularise” by replacing the LHS by the term x’(t). Numerical experimentation suggests that there is an interesting trade-off between the size of  and the behaviour of u.

24. “THE BOOK” Applied Nonlinear Control J-J E. Slotine, Weiping Li, Prentice Hall 1991 FINIS