The document discusses parameter estimation of an aerodynamic model from experimental bifurcation diagrams. It summarizes an experiment that captured dynamic characteristics of a scale aircraft model, observing a post-stall pitch oscillation. The author proposes a novel method of parameter estimation using bifurcation analysis rather than time domain analysis to estimate dynamic parameters governing the limit cycle oscillation. However, problems were encountered in that the equations of motion were incomplete and inaccuracies in a numerical simulation prevented successful implementation of the bifurcation-based estimation method. Further work is needed to fully understand and model the system to allow the proposed approach.

1. Parameter Estimation of an Aerodynamic Model from Experimental
Bifurcation Diagrams
Andrew Salmon
Background
Parameter Estimation has longbeen used as a way to identify the static and dynamic parameters that govern
an aircraft’s motion,specifically when empirical data of the aircraft’s responseexists.The University of Bristol
has developed a free oscillation wind tunnel rigtunnel to allowthe capture of the dynamic characteristicsof
scalemodels,with which J. Pattinson in [1] conducted a range of experiments that produced time history data
of many different manoeuvres and control surfaceramps,and provided the means for parameter estimation
to be implemented on the model (a scaleBAE Hawk) [2]. In these experiments, a particularpost-stall pitch
oscillation was observed in 1 Degree of Freedom (1DOF) pitch constraint,the underlyingmechanismof which
is thought to stem from scalemodel inaccuracies,downwash lagbetween the tail and the wing, and the
dynamic stall phenomena. The dynamics of this example emulates a limitcycle,a bounded oscillation with
constantamplitude and phase, the frequency of which is around 1.8Hz. This presents a problem: for classical
parameter estimation to be applied to identify the dynamic parameters present in the equations of motion,
the experimental data and the numerical simulation of the limitcyclemust be superimposed and the error of
their differences reduced by means of an optimiser.For steady state responses with small amplitude
disturbances,the Equation Error Method (EEM) suffices,as the problem reduces to linear least-squares.
However, the dynamic parameters which govern the Limit CycleOscillation (LCO) arehighly non-linear.This
alonewould not typically flagany issues when implementing parameter estimation,however the frequency of
this specific phenomena is high,with process noise(turbulence) very much engrained into the empirical data.
The more sophisticated Output Error Method (OEM) of parameter estimation,which can handle process noise
and has a stochastic grounding,cannot fitthe two curves as they will alwaysbetoo dissimilar dueto the high
frequency. A successful implementation of this method would require the parameter values to be guessed
extremely closely with a numerical model that accounted for a very high level of experimental noise.The
problem is threefold:
 The inaccuracies in theexperimental data due to rigfriction and model vibration causethe LCO
frequency to be slightly differentto what would be predicted and variablethroughout the
experiment: the factor would be small butsignificantgiven the number of oscillations experienced
over one experimental run.
 The phasealignment is crucial to the matching process.Assuminga correct frequency estimate, the
onset of the limitcyclewould have to be simulated atexactly the same time point as the
experimental data to a degree of accuracy notpossiblewithouta highly sophisticated model.
 Initial guesses for the dynamic parameters would need to be nearly exact to begin with to consider
the above at all –these guesses areunobtainableusingthe linear region as the dynamic
characteristicsarenotexcited.
The OEM works on the basis of maximumlikelihood estimation,theupshot being with such a high frequency
LCO, the solver will regard the cost of the oscillation variablean error around a mean, and try and flatten the
curve, and will get less and less successful as each oscillation occurs.Thecost of an inaccuratephaseis large
that a flatlineof the average frequency, therefore the method is flawed for this instance. The processingtime
for such a method is also a limitation –a successful matchingwould have to analysea significantlength of
curve in this instance,makingthe method impractically timeexpensive
The need to characteriseoscillatory systems is notlimited to the constraints of aerospace:the applicationsare
numerous and spread many fields.Attempts have been made in the micro-biology field to characterisethe
oscillatory behaviour circadian rhythms of cells [3],in which the approach taken was to use a DiscreteFourier
Transform(DFT) on the data for the frequency analysis,and PhaseResponseCurve (PRC) tool which used
experimental data of phase change as a separateinput in the parameter estimati on. Whilstthis method holds
significantpotential,the framework around which such a method would work is not yet developed and would

2. not be practical for this project.The described limits would still apply,and phaseresponseexperiments would
need to be undertaken.
Project Description
Bifurcation methods areused in the analysisof non-linear models as a framework for solvingsystems in which
the states evolve over time [4] – given initial conditions and a vector of variableparameters,the steady state
behaviour of the system can be found under the progression of one or more activeparameters. This comes
under the general scope of continuation techniques.Such methods are extremely useful, as the discrete
nature of stability/instability means thatthe transitions observed can be focused on a singlepointand then
described with simpleproperties.Steady states includeboth equilibria(described as a curveagainst
continuation parameter) and LCO’s (described as a curve of maxima and minima with frequency properties),
and can be easily verified by experiment. Unsteady branches where equilibriumis held in an unstablemanner
are harder to realise, butpossiblewith the inclusion of feedback. The onset and development of the post-stall
oscillation can besuccessfully simulated usingthese techniques, and this provides a useful resource for
developing alternativeparameter estimation methods.
A comparison of the experimental bifurcation diagramand the computed one is shown above (figure1). The
activeparameter is the elevator angleof the Hawk model, and the observed variableis themodel pitch angle
(constrained to only allowmotion in this degree of freedom). The continuation starts in a stablecondition of
10° of elevator causing~-20° of pitch deflection, then decreases the elevator to -17°, and envelope which
includes the onset and deterioration of the LCO. A Hopf point is a local bifurcation (pointof stability transition)
in which the system loses stability as a complex conjugatepair of its eigenvalues has its real parts turn negative
to positive,out of which the system develops a limitcycle.A ‘supercritical’Hopf bifurcation occursatdirectly
over the point of LCO onset at -5° elevator deflection in the experiment and the complex conjugate forks
followits maxima and minima as the continuations progresses.A‘fold’bifurcation occurs as thelimitcycle
begins to decrease in size,from which an unstablelimitcyclecurvetracks the system from the fold to a second
Hopf point, located where the limitcyclevanished.These dynamics areobserved with no account of rig
bearingfriction in the numerical model: the inclusion of such factors causes theHopf points to shiftto an
unexpected deterioration of the LCO halfway alongthe continuation – this a phenomena to be explored at a
later date.
The above description of the continuation discusses theobservation of stableequilibriaand limitcycles thatfit
the experimental data. This is made possibleby usingthe EEM to firstestimate the static parameters usingthe
time history form and numerical simulation of the experiment, then guess the correct values of the dynamic
parameters until a reasonablefitof the limitcycleis achieved.Herein lies the foundation of the problem. A
novel method of parameter estimation can be implemented usingthe bifurcation domain,as opposed to the
time domain,to formally estimatethe dynamic parameters.This method will havethe benefit of bypassingthe
directfollowingof the limitcycle,instead usingthe maxima and minima of the continuation LCO and individual
Figure 1 –incorrect yet complete continuation ofthe limit cycle experiment

3. values of frequency and phaseto generate a curve comparableto the maximum and minimum data points
observed in experiment. To configurethe estimation usingextrema absolutevalues would be simple:the
writingof a costfunction to minimisethe difference of two curves would allowthe OEM to work as designed
and factor in elements of experimental noise. The challengingaspectof the method is developinga cost
function that will accountfor a difference in phase, and try to minimiseitdepending on only the experimental
values.A potential solution is to view the problem as an optimisation perpendicular to the maximum and
minima.If one could label the bifurcation points and constrain themto individual experimental points along
the stablelimitcyclecurve, then minimisethe errors ‘laterally’(translatinga changein parameter valueinto a
change in phaseand maxima position),the parameters could be fitted successfully,and Hopf points verified.
Given the experimental anomaly of the reduction in sizeof the limitcycleat16° pitch (figure2), this analysis
would have to be done in the final third of the LCO, particularly dueto the constantamplitudeof the
oscillation.If this method was successful,itcould be implemented over the entire limitcycleto include the
anomaly to investigatewhether or not the reduction could be expected under certain values of the
parameters, or whether the rigwas to blamefor this.Current work suggests the 1DOF pitch constraintis
absorbingsomesurplus energy here, and in fact the limitcyclehas elements in the roll degree of freedom.
Knowledge of the values of the Hawk model’s dynamic parameters would be extremely useful when predicting
dynamics in multipledegrees of freedom, and in theory the novel PE method should be ableto be applied to
other oscillationswhere the bifurcation curvecan be computed.
Problems encountered
Unfortunately, this projectdid not span out in the way intended. An essential requirement of the proposed
method described above is to have a complete and accurateset of differential equations of motion that a) can
be integrated to form the numerical time history simulation of the experiment and b) are in a form that a
continuation programme can accept. What was initially given was a set of equations in partial form,without
the detail required to generate the post-stall behaviour.Themissingpartspecifically was theimplementation
of downwash delay into the equations,a time dependant phenomena that required the angleof attack on the
main wing a time period before to be applied to the tail (equation 1). Upon realisingthis error, the equations
were altered accordingly and a limitcycleof the correctshape was produced.
𝛼 𝑑𝑤 = 𝜀 𝐶 𝐿 𝐶 𝐿 𝑤( 𝒕 − 𝝉 𝑬) (1)
50 100 150 200 250 300 350 400 450 500 550
10
15
20
25
30
Pitchangle
(deg.)
090622_20ms_1dof_ramp_part2_OEM Results Exp Sim
Figure 2 –experimental data ofthe limit cycle seen in [1]
Figure 3 –correct limit cycle time history
121 122 123 124 125 126 127 128 129 130
13
14
15
16
17
18
Pitchangle
(deg.)
090622_20ms_1dof_ramp_part2_OEM Results

4. However, this limitcyclewas deceivingly similar to what was expected, yet different in some important ways:
the onset and decay occurred at different times to what was expected, the maximum amplitudewas larger
than the experimental by approximately 10%, and the frequency of oscillation was doublewhatwas seen in
the experiments (figure 3 and 4)
Upon receivingthe original scripts thatwere used to calculate this simulation,theequations were further
analysed to discover the sourceof the problem. J.Pattinson for [2] adapted a series of MATLAB files from[5] to
develop a framework around which Numerical Simulation,Parameter Estimation and Continuations could be
sequentially,thus establishinga common format from which each could be designed. Pattinson was clearly
successful in hisattempts to run this code, however, the state in which itwas left was inoperableand in many
areas flawed,suggesting some editing had been done after the publishingof his thesis.
The equations of motion for this experiment aremore complex than their description gives away in [1].
Amongst several adaptions to make inputs more realistic,a key difference was a setup for includingprevious
values of the model state in the current solution cyclefor the ode solver,specifically 𝜃̈ in thecalculation of
model pitch. An attempt is firstmadeby the solver for a singletime integration, evaluated, and then fed back
into italongwith the unintegrated state values to perform the next solution.The output is favourableand
creates a limitcyclewhich mirrors the experimental values very well, however this method of achievingthe
correct values is setup around a specific ODEsolver,and does not output the correct results without it(i.e.
manually runningthe integration time step by time step). The use of a variablestep solver seems essential.
Due to the state being reused in each iteration,an unsuccessful solution statefrom the solver will befed back
into the equations,repeating until a solution is found. Similarly,inputs such as recorded airspeed are used and
changed in every time step to increasethe accuracy of the response, which requires an additional time
dependant input into the equations.
-18 -16 -14 -12 -10 -8 -6 -4 -2
5
10
15
20
25
30
35
40
Hawk elevator (deg.)
Pitchangle(deg.)
Figure 5 - continuation curve overlaid on experimental data
600 650 700 750 800 850 900 950 1000 1050 1100
-15
-10
Time (s)
720 721 722 723 724 725 726 727 728 729 730
14
16
18
Time (s)
alpha
alpha
Figure 4 –incorrect limit cycle time history

5. The upshot of this is that the equations must allowthe storage and reuse of the evaluated states, and call the
correct valueof 𝜃̈ each time an iteration is made by the continuer. Unless the continuer works with exactly
the same trail steps as the ODE solver,the state values returninginto the equations will notbe correctfor the
solution,and an incorrect continuation will beseen. AUTO, the chosen continuation programme would need
to be modified in its original FORTRAN to incorporatesuch a delay function, rulingoutits use in this project. An
alternate solution would be to compute every valueof every evaluated state and feed this in as a singleinput
to the non-delayed equations,but this too would require editingof lower level scripts in AUTO. Pattinsons
chosen continuation programme MATCONT was successfully modified to acceptthese delays,but the
bifurcation curveis inaccuratewhen reproduced here (figure 5). The reasons why this occurred were
postulated and attempts to correctthe equations were made but with no success.The fact remains that the
equations will output an accuratetime history,yet they will notreplicatea good bifurcation. Further work is
needed to understand why the equations will notproduce an accuratebifurcation.Oncethis is complete, the
maxima and minima of the stablelimitcyclecan be used in the ways mentioned in the discussion,and the
novel parameter estimation method explored.
References
[1] Pattinson, J., Lowenberg, M. H., & Goman, M. G. (2013). Investigationof Poststall Pitch Oscillations of an Aircraft Wind-
Tunnel Model. Journal of Aircraft, 50(6), 1843–1855. doi:10.2514/1.C032184
[2] Pattinson, J. (2010). Development andEvaluationof a WindTunnelManoeuvre Rig, (November).
[3] Nabavi, S., & Williams, C. M. (2012). A novel cost function to estimate parameters of oscillatorybiochemical systems.
EURASIPJournal onBioinformatics andSystems Biology, 2012(1), 3. doi:10.1186/1687-4153-2012-3
[4] Thompson, J. M. ., & Stewart, H. . (2002). Nonlinear Dynamics andChaos (Second., pp. 106–131).
[5] Jategaonkar, R. V. (2006). Flight Vehicle System Identification (1st ed.). AmericanInstitute of Aeronautics and
Astronautics.