Optimisation Based Clearance of Nonlinear Flight Control Laws
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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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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
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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?
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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 • SecttJ 10));(max(
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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
lΔ
• Problem becomes extremely computationally expensive
• Need efficient search methods to find “worst - case” uncertain parameter combinations
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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
ADMIRE model
• Dynamics …(1)
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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
ADMIRE model
•Control algorithm …(2)
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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
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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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
General optimisation framework
The philosophy
JMax
Reference inputs Uncertain parameters
Mach AltitudeLevel Trim
Finite time history Optimisation
Algorithm
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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)
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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 operatorsRoulette selection 0.6Single point crossover
0.9
Binary uniform mutation
0.005
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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
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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]
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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
Global Optimisation SchemeDifferential Evolution
•Random initialisation
•Mutation
•Crossover
•Evaluation and selection•Termination criteria same as that of GA
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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]
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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
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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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
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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)
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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]
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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
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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]
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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
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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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 offlight envelope:
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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
Optimisation Performance
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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
Clearance Results
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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
Clearance Results
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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
Clearance Results
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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
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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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
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Department of Engineering, Control & Instrumentation Research Group
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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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
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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Department of Engineering, Control & Instrumentation Research Group
22 – Mar – 2006
Computation of worst-case pilot inputs
Rudder
Input of Rate LimiterOutput of Rate Limiter
•What’s the problem?
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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