Optimisation Based Clearance of Nonlinear Flight Control Laws

Department of Engineering, Control & Instrumentation Research Group

22 – Mar – 2006

Optimisation Based Clearance of

Nonlinear Flight Control Laws

Prathyush P. MenonJongrae Kim

Declan G. BatesIan Postlethwaite

Control & Instrumentation Research Group,Department of Engineering,

University of Leicester,Leicester LE1 7RH, UK.


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•Nonlinear flight clearance

•A general optimisation framework

•Worst case uncertainty evaluation

•Clearance over regions of the flight envelope

•Worst case input identification

•Summary

Overview


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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?


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• Nonlinear flight clearance criterion – Based on time response, peak overshoot– AoA limit exceedance • SecttJ 10));(max(


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•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

lΔ

• Problem becomes extremely computationally expensive

• Need efficient search methods to find “worst - case” uncertain parameter combinations


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ADMIRE model

• Dynamics …(1)


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ADMIRE model

•Control algorithm …(2)


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• 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

u(t))h(x(t),y(t))w(t),u(t),f(x(t),(t)x

)Δ̂(t),yg(x(t),u(t) REF


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General optimisation framework

The philosophy

JMax

Reference inputs Uncertain parameters

Mach AltitudeLevel Trim

Finite time history Optimisation

Algorithm


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Global Optimisation Schemes

•Several algorithms evaluated:

– Genetic algorithms (GA)

– Differential evolution (DE)

– Hybrid GA / Hybrid DE

– Dividing Rectangles (DIRECT)


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Global Optimisation Scheme

Genetic algorithms

•Search space

•Accuracy 1e-6 •Chromosomes length 105 bits (5 genes)• Initial population 50

•Genetic operatorsRoulette selection 0.6Single point crossover

0.9

Binary uniform mutation

0.005


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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


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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]


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Global Optimisation SchemeDifferential Evolution

•Random initialisation

•Mutation

•Crossover

•Evaluation and selection•Termination criteria same as that of GA


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DE Results

•Better convergence to global optimum

•Reduced number of simulations (~3000)

[0.1000, 0.0750, 0.0500, 0.18309, 0.0500, 36.0908]


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Global Optimisation Scheme Global optimisation comparison statistics

Optimisation

Trials Avg.

Max. Min.

Std. Dev.

Prob. of success

GA 100 4485 7500 2400828.364 65%

DE 100 3086 4176 1152 567.57 90%

Tria

lsTr

ials

Tria

lsTr

ials


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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


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•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)


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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]


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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


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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]


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Hybrid Optimisation Scheme Hybrid optimisation comparison statistics

Optimisation

Trials Avg.

Max. Min.

Std. Dev.

Prob. of success

HGA 100 2011 4468 1357 547.42 92%HDE 100 1106 1434 477 192.42 98%

Tria

lsTr

ials

Tria

lsTr

ials


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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 offlight envelope:


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Optimisation Performance


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Clearance Results


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Clearance Results


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Clearance Results


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Clearance ResultsWorst 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


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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


22 – Mar – 2006

Computation of worst-case pilot inputs•Klonk inputs:

deg16.0038αmax

]t[tt,(t)y f0REF

Global Optimisation 12Xx(t)

Δ(t),yREF

FULL NONLINEAR AIRCRAFT SIMULATION MODEL

u(t))h(x(t),y(t))w(t),u(t),f(x(t),(t)x

)Δ̂(t),yg(x(t),u(t) REF

Mach AltitudeLevel Trim


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Computation of worst-case pilot inputs•Worst-case inputs: deg66.4316αmax

0.0611 0.0648 -0.0020 -0.0022 0.0418 66.4316cgx

massemC

alC

mC

max

Time: 3hrs. 5mins.

deg58.0721αIIAnalysis max

deg16.0038αKlonk max

deg27.066α:IAnalysis max

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.


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Computation of worst-case pilot inputs

Rudder

Input of Rate LimiterOutput of Rate Limiter

•What’s the problem?


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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

Optimisation Based Clearance of Nonlinear Flight Control Laws

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