fvsysid shortcourse 4 methods

AIAA Short Course: Flight Vehicle System Identification in Time Domain, Aug. 2006 Methods/1Dr. Ravindra Jategaonkar

Methods for Flight Vehicle System Identification

ParameterAdjustments

Model Response

ResponseError

ActualResponseInput

Maneuver

ModelValidation

ComplementaryFlight Data

Identification Phase

Validation Phase

OptimizedInput

Flight Vehicle

IdentificationCriteria

EstimationAlgorithm /Optimization

MathematicalModel /

Simulation

Data Collection& Compatibility

easurementsM

ethodsM

odelsMModel Structure

A Priori Values,lower/upperbounds

-+


This page is left intentionally blank.


Model Equations

State equations:x state vectoru input vectorw additive process noise, N(0,I)F process noise distribution matrix

Observation equations:y output variables

Discrete measurements:v measurement noise, N(0,I)

Unknown parameters:βx system parameters of the state eqs.βy system parameters of the observation eqs.λ elements of noise distribution matrixx0 initial conditions

00 )();()(]),(),([)( xtxtwFtutxftx x =+= λβ&

],),(),([)( yxtutxgty ββ=

NktvGtytz kkk ,...,1),()()( =+=

);;;( 0TTT

yTx

T xλββ=Θ


Model Extensions (I)

• Multi run evaluation (concatenating several flight maneuvers)• Bias errors in measurements of input and output variables

Nonlinear systems

nz number of time slices /experiments analyzed simultaneouslyN total number of data points analyzedm number of output variablesn number of state variablesp number of input variables

Unknown parameters:

n initial conditions and m + p bias parameters.Not all of them can be estimated (linear dependence, high correlation).

NkktvGktyktznzllybzyxlubututxgty

lxbxltxxlubututxftx

,...,1)()()(,...,1,),(],),,()(),([)(

),(0),0(],),,()(),([)(

=+=

=Δ+Δ−==Δ−=

βββ&

{ } { } b ,. . ,b ; b ,. . ,b ; b ,. . ,b ; ; = Tnzu,

Tu,1

Tnzy,

Ty,1

Tnzx,

Tx,1

Ty

Tx

T ββΘ


Model Extensions (II)

Linear Systems:

- State variable transformation x:= x-x0 leads to equivalent model:

- bias terms bx = A x0 - B Δuby = C x0 - D Δu + Δz

– exactly n + m bias parameters

– all can be estimated

NktvGtytznzlbtuDtxCbtutxgty

txbtuBtxAbtutxftx

kkk

lylyyx

llxlxx

,,...1)()()(,,...1)()(],),(),([)(0)()()(]),(),([)(

,,

,0,,

=+==++=+=

=++=+=ββ

β&


Parameter Estimation Methods

Equation error - Linear in parameters and independent variables (one-shot)- Nonlinear least squares (iterative solution)- Accounts for process noise

Output error - Accounts for measurement noise- Nonlinear optimization problem: iterative solution

Filter error - Accounts for both process and measurement noise- state and parameter estimation

EKF/UKF - Filtering approach to parameter estimation

Generalized approach

No Measurementnoise

Filter Error Equation Error Output Error

Special caseSpecial case

No Processnoise

Complications:1) Presence of noise --> no longer possible to exactly identify the values of unknown parameters; instead , the values must be estimated by some criterion (to average out the noise effects).

2) Modeling errors --> are deterministic, but are treated simply as noise (rigorouslynot justifiable, but probably the best approach).


Regression Analysis (1)Model postulate

yi dependent variable (response variable)x1,...,xr independent variables (regressors, explanatory variables)θi1,...,θir unknown parametersei equation error

Assumptions:- Measurements of y corrupted by noise (zero mean)- Measurements of x are exact and noise free

Equation error:

LS cost function: Directly in terms of error. Not based on probability theory.

LS method applicable to linear as well as nonlinear models:- As for the cost function, it is immaterial whether the errors result from

a linear or nonlinear model.

- The minimization procedure is model type dependent.

Nktetxtxtxty kikrirkikiki ,,...1)()(...)()()( 2211 =++++= θθθ

εεεθ TN

1k

221)k(

21)(J ∑

===

θε )()()( kxkyk T−=


Linear least squares:

One shot procedure to solve Normal equation:

Nonlinear least squares problems can be solved only iteratively.

Statistical properties:

- Unbiased estimates under given assumptions (noise zero mean; independent variables exact)

- In the presence of noise and systematic (bias, scale factor) errors in x,LS estimates are biased and inefficient.

- Leads to two step procedure of aerodynamic model identification:

a) FPR to eliminate systematic instrument errors

b) estimate aerodynamic parameters using LS method

Regression Analysis (2)

YXXX TTT =θ̂)(

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

=

)(...)()(

)2(...)2()2()1(...)1()1(

21

21

21

NxNxNx

xxxxxx

X

nq

nq

nq

TNyyyY )](...)2()1([=



Total least squares (TLS)Orthogonal Distance Regression (ODR) Error-in-Variable (EIV) Modeling

TLS accounts for noise in the independent variables.

TLS solution based on SVDand rank reduction. Simple linear solution not possible.

Equivalently, : Includes a correction term

smallest singular value.

TLS estimates are unbiased, provided noise is zero mean.

No general procedure within the framework of regression techniques to account for systematic errors.

x

y

(x1, y1)

(x2, y2)(x3, y3)

δyOLS

δyTLS

y = ax

YXIXX Tn

TTLS

121 )( −

+−= σθ1+nqσ


80

5550

time s20

angle of attack

deg

0 100

5550

time s

55

0 1050

Data Partitioning

possible because LS does not rely on the temporal relation between data points

Improved information contents leads to• reduced correlation• improved parameter estimates



Measurementnoise

Dynamic System++

Measuredresponse

Mathematical Model

Integration of State Eq.Observation Equations

Parameter update By optimization ofLikelihood Function

+-

Input

Sensitivities ResponseError

z

y

(z-y)

Advantages: Computationally (relatively) simple; Readily extendable tononlinear systems, widely applied (vast expertise available)

Limitations: Accounts for Measurement noise only; biased estimates in the presence of atmospheric turbulence

Arguments: Flight tests to be performed in steady atmosphere(hypothetical assumption; rarely feasible due to tight time schedules and cost factors)

Output Error Method (1)


Assumptions1) Mathematical model: State space

2) Input sequence {u(k), k=1, 2, ... , N} is exogenous;i.e., generated independent of the system output==> special treatment for unstable aircraft with FCS

3) Control inputs {u(k)} are sufficiently and adequately (i.e., in magnitudeand frequency) varied to excite various modes of the system

4) System is corrupted by measurement noise only;i.e., state equations represent a deterministic system==> special treatment for process noise case (Stochastic system)

5) Measurement errors v(k) at different discrete time points are statisticallyindependent and Gaussian distributed with zero mean and covariancematrix R;

i.e., ll kT

kk RtvtvEtvE δ== )}()({;0)}({

Output Error Method (2)


Likelihood Function (1)

Definition of Likelihood function:Conditional probability density function of measurements z(k)for given Θ and R.

For Gaussian distribution (assumption 5), the conditional probability densityp(z(k)| Θ, R) at a particular discrete time point k is given by:

Mathematically strictly speaking, it would be more appropriate to write the likelihood function for the given data as p(z(k)| Θ, R, u).

However, the argument u is dropped without loss of generality:What is the reason?

Ref. for Eq. (1):Davenport, W. B. and Rot, W. L., “Random Signals and Noise”, McGraw-Hill NY, 1958.

[ ] [ ])}()({)}()({exp)2(),)(( 1212/1

kkT

kkm

k tytzRtytzRRtzp −−−=Θ −−π Eq. (1)



The reasons for dropping the argument u:- Identical control inputs are used to excite the systemas well as the postulated model

- Control inputs are exogenous, noise free and known a-priori

Measurement errors at various discrete points are assumed independent:=> the likelihood function can be expressed as:

From Eqs. 1 and 2, the likelihood function follows as:

m: number of output variables, i. e., the dimension of the vectors z and y.N: number of data points.

∏=

Θ=ΘN

kRkzpRNzzzp

1),)((),)(,...),2(),1(( Eq. (2)

[ ]⎥⎥⎦

⎤

⎢⎢⎣

⎡−−−=Θ ∑

=

−− N

kkk

Tkk

NmN tytzRtytzRRzp

1

1212/

)}()({)}()({exp)2(),( π Eq. (3)


is the probability of the measurements for a given parameter vector Θ and the noise covariance matrix R;

The Likelihood function p does not represent the probability distribution of the unknown parameters; but of the measurements .

The parameters does not have any probability density, since they are not random.

Maximum-Likelihood estimation means that the Θ -vector is searched which maximizes the function

Such a vector is ”the most plausible”, because it gives the highest probability to the measurements.

Equivalently, negative logarithm of the likelihood-function is minimized:

),|( Rzp Θ

),|( Rzp Θ

)2ln())ln(det()]()([)]()([),( 221

121 πmNN

kkN

k

Tkk RtytzRtytzRzLJ ++−−=Θ= ∑

=

−

Eq. (4)



)2ln())ln(det()]()([)]()([),( 221

121 πmNN

kkN

k

Tkk RtytzRtytzRzLJ ++−−=Θ= ∑

=

−

Optimization of Likelihood Function (1)

)]()([)]()([1

1kk

N

k

Tkk tytzRtytzJ −−= ∑

=

−

Cost function:

The last term is a constant: can be neglected without affecting the optimization.

Two cases are of interest:

Case 1: R Known:

Apply any optimization procedure (Newton based methods)

Case 2: R Unknown: Brute-force approach to combine Θ and R into singlevector and apply any standard optimization method:Never practiced, because there is no closed formsolution to this minimization problem. During the optimization, the estimates of Θ dependon the R and vice versa.

use relaxation strategy (two step optimization procedure)


Equating to zero gives maximum likelihood estimate of R:

For any fixed value of Θ, Eq. (6) maximizes the likelihood function w.r.t. R.

Estimation problem in this case reduces to minimization of J w.r.t. Θ;subject to R given by Eq. (5).

Substituting Eq. (5) in Eq. (4) yields:

Minimization of J1(Θ) is equivalent to minimization of:

R/J ∂∂

Tkk

N

kkk tytztytz

NR )]()([)]()([1

1−−= ∑

=Eq. (5)

)2ln())ln(det()( 2221

1 πmNN RNmJ ++=Θ

T

kkN

kkk tytztytz

NRJ ⎟

⎟⎠

⎞⎜⎜⎝

⎛−−==Θ ∑

=)]()([)]()([1det)det()(

12

Eq. (6)

Eq. (7)



Relaxation algorithm:

1) Choose suitable initial values for Θ

2) Estimate R using Eq. (5)

3) Minimize J2(Θ), Eq. (7), with respect to Θ using any suitable optimization method

4) Iterate on step 2 and check for convergence.

Proof of global convergence of relaxation procedures is difficult,but they are convenient to use and mostly work well in practice.



Iterative method of finding a zero of a nonlinear function J(Θ) of several variables.

i.e., zero of the gradient of the cost function J

Necessary condition0=Θ∂∂J

Taylor series expansion about the k-th value:

( ) ( ) ( ) 122

1 ++ ΔΘΘ∂∂+Θ∂∂=Θ∂∂ kkkk JJJ

( )kkk Θ−Θ=ΔΘ ++ 11

with

Where is the second gradient of the cost function w.r.t. Θ,(Hessian matrix) at the k-th iteration

( )kJ2Θ∇

Optimization Algorithms (1)

Newton Raphson Method


Tangent line at

kΘ1k+Θ

)(Θ∇Θ J

Slope of tangent line =

kΘ

( ))(Θ∇∇ ΘΘ J

The change in Θ on the k+1-th iteration

to make approximately zero is:( ) 1+Θ∂∂ kJ

( )kk

k JJ Θ∂∂⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ Θ∂∂−=ΔΘ

−122

( ) ( ) ( ) ==>=ΔΘΘ∂∂+Θ∂∂=Θ∂∂ ++ 0122

1 kkkk JJJ

The Newton-Raphson algorithm:



Simple Test Cases

1) Quadratic cost function28θ=JLet the cost function be:

The first gradient: ( ) θ16=Θ∂∂J

The second gradient: 1622 =Θ∂∂ J

The starting value for unknown parameter: 40 =θ

[ ] 44*1616 1 −=−=ΔΘ −kThe parameter update:

The updated parameter: 0440 =−=ΔΘ+Θ=Θ k

Which happens to be the minimum.Optimum in a single step: -- Quadratic cost function;

second gradient constant.

Minimum in a single step starting from any initial value. ExQuadFu

n.m



Simple Test Cases

2) Non quadratic cost function32 4.04 θθ +=JLet the cost function be:

The first gradient: ( ) 22.18 θθ +=Θ∂∂J

The second gradient: θ4.2822 +=Θ∂∂ J

The starting value for unknown parameter: 40 =θ

[ ] 9091.2]4*2.14*8[4*4.28 211 −=++−=ΔΘ −

The parameter update:

0909.19091.24101 =−=ΔΘ+Θ=Θ

ExNonQuadFun.m

[ ] 9564.0]0909.1*2.10909.1*8[0909.1*4.28 212 −=++−=ΔΘ −

1345.09564.00909.1212 =−=ΔΘ+Θ=Θ

Iteration 1:

……

Iteration 2:



)]()([)]()([)(1

21

kkN

k

Tkk tytzWtytzJ −−=Θ ∑

=

Recall our ML cost function defined as:

Partial differentiation of J w.r.t Θ yields:

0)]()([][1 =−Θ∂∂−=Θ∂∂ ∑ kkT

N tytzWyJ

and the partial differentiation of w.r.t Θ yields:Θ∂∂J

∑∑ −Θ∂∂+Θ∂∂Θ∂∂=Θ∂∂ ][][][][ 221122 yzWyyWyJ TN

TN

Computation of the first gradient is relatively straightforward;

computation of of the second gradient very time consuming.

Θ∂∂y22 Θ∂∂ y


)( 1−= RW


Modified Newton-Raphson (Gauss-Newton) method:contribution due to goes to zero as the process converges.

Residuals z(k)-y(k) summed over sufficiently large N are zero mean.

)]()([)]()([)(1

21

kkN

k

Tkk tytzWtytzJ −−=Θ ∑

=

Quasi-Linearization method:Recall our cost function defined as:

Gradient of J is given by:

0)]()([][ =−Θ∂∂−=Θ∂∂ ∑ kkT tytzWyJ

∑ Θ∂∂Θ∂∂=Θ∂∂ ][][122 yWyJ TN

22y Θ∂∂

( )kk

k JJ Θ∂∂⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ Θ∂∂−=ΔΘ

−122



Substituting linearized y(Q) in the previous equation leads to:

0}])({[][ =ΔΘΘ∂∂−−Θ∂∂−=Θ∂∂ ∑ yyzWyJ T

0][][][][ =ΔΘΘ∂∂Θ∂∂+−Θ∂∂∑ ∑ yWyyzWy TT

∑∑ −Θ∂∂=ΔΘΘ∂∂Θ∂∂ ][][][][ yzWyyWy TT

( ) kyyy ΔΘΘ∂∂+Θ≈Θ )()( 0

Quasi-Linearization of y(Θ) about Θ0 yields:

Gauss-Newton and Quasi-linearination methods are equivalent.



Unconstrained Gauss-Newton Method (summary)

Iterative parameter update starting from initial guess

Information Matrix:

Gradient vector:

Iterative update requires:

• computation of observation variables y

• computation of response gradients

( ) ( )∑ ⎥⎦

⎤⎢⎣⎡

Θ∂∂

⎥⎦⎤

⎢⎣⎡

Θ∂∂

≈Θ∂

∂=

=

−N

kk

Tk tyRtyJF

1

12

2

( ) [ ]∑ −⎥⎦⎤

⎢⎣⎡

Θ∂∂

=Θ∂

∂=

=

−N

kkk

Tk tytzRtyJG

1

1 )()(

Θ∂∂ /y


ΔΘ+Θ=Θ + ii 1 GF 1−−=ΔΘwith


Sensitivity matrix

Partial differentiation of the observation equations y=g(x,u,Θ) w.r.t. Θ yields:

Thus, to compute the response gradients, we need gradients of the states.

Partial differentiation of the state equations yields:

Analytical expressions

Or

Numerical approximations

Θ∂∂

+Θ∂

∂∂∂

=Θ∂

∂ gxxgy

),u,x(fx Θ=&

Θ∂∂

+Θ∂

∂∂∂

=⎟⎠⎞

⎜⎝⎛

Θ∂∂ fx

xfx&



Response gradient approximated by finite difference approximation:

Forward difference:

Central difference:

perturbed response variables

Numerical approximation offers flexibility to handle conveniently different model structures without software changes

jkt

jejuxiyktjejuxpiy

ijkty

ΘΘ−Θ−Θ+Θ

≈Θ∂

∂⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

δδδ

2),,,(),,,(

)(

)( jpy Θ

jktiyktpiy

ijkty

Θ−

≈Θ∂

∂⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

δ)()(

)(



Numerical Aspects:

- Gauss-Newton method results in a full step within parameter space- J(Θ) not a simple function of Θ- Approximation in computation of Hessian- Numerical approximation of gradients- Poor performance for starting values far from optimum- intermediate local divergence or stalling

Solutions:

- Heuristic approachon divergence, halving of parameter update ΔΘ

- Line Search (λ)

Damping strategy widens the convergence region and overcomesnumerical problems

ΔΘ+Θ=+Θ λii 1 with GF 1−−=ΔΘ



Bounded-Variable Gauss-Newton Method

Parameters often physically constrained to lie in a certain range,e.g.,

- non negative time delays,

- Oswald factor (increase in drag due to nonelliptical lift distribution) < 1

Linearly-constrained optimization problem

Classical approaches:Barrier function,

Langrangian,

Active set strategy

)(min ΘJ subject to maxmin Θ≤Θ≤Θ



BVGN Method (continued): Active set strategy

• Simple, direct and efficient approach (retains the advantages of GN method)

• update of the free parameters via

Requires sorting of gradient vector and information matrix to compute ΔΘ

• freeze parameters that hit the bounds

• re-activate parameter if the gradient points no longer outside the feasible region (Kuhn-Tucker optimality condition)

Optimality conditions guarantee that the gradients for the variables hitting the bounds are such that they point outwards of the feasible region, implying that any further minimization of the cost function would be possible only when the particular parameters are not constrained within the specified limits.

0.02

0.01

0.0

-0.005

1/radUnconstrainedGauss-Newton

Bounded-variableGauss-Newton

Clδr

0 2 4 6Number of iterations

Upperbound

freefreefree GF 1−=ΔΘ

;max_0 iii forG Θ=Θ< min_0 iii forG Θ=Θ>



Levenberg-Marquardt Method

λ Levenberg-Marquardt parameter

λ controls whether the update direction is more like:

steepest descent (λ → ∞) or

Gauss-Newton direction (λ → 0)

Selection of λ:

- requires solution for two values of LM-parameter

- LM-overhead minor part, major part spent oncomputation of sensitivity matrix

- adaptation of λ guarantees convergence to the optimum

ΔΘ+Θ=Θ + ii 1 with GIF 1)( −+−=ΔΘ λ



Direct Search Methods

• Simplex method of Nelder and Mead• Subspace searching method of Rowan• Powell's conjugate direction minimum search• Jacob's heuristic minimum search

Simplex method:- reflection, expansion, contraction and shrinkage of q-dimensional

convex hull (a geometrical figure with q+1 vertices - Simplex)

- Robustness (w.r.t. initial values and discrete nonlinearities)

- Inefficient for large number of parameters

Subplex method:- Generalization of Simplex method

- Decomposes higher dimensional problem into smaller-dimensionalsubspaces in which Simplex method search is more efficient.



Integration Methods

• Euler integration• Runge-Kutta 2nd order• Runge-Kutta 3rd order• Runge-Kutta 4th order• Runge-Kutta-Fehlberg 4th and 5th order with step size control

- Computational load and accuracy increases with the order- Step size control is useful for high bandwidth models

• Gear's backward differentiation formula for stiff systems- Special integration method for systems with large and small eigenvalues- inefficient for non-stiff systems- Not possible for retarded systems


Solution of Linear Algebraic Equation

Calculation of the inverse of the information matrix via

• Cholesky factorization

• Singular value decomposition

Standard LAPACK subroutines for SVD.Lower bound for singular values ε


Hybridization of Optimization and Integration Methods

• To generate better starting values using more robust (but slower) direct search methods before applying the Gauss-Newton method

• To overcome numerical problems near optimum due to finite-difference approximation used in the Gauss-Newton method

• To reduce computational time by using a simpler integration method for the first iterations


A-priori Information

Cost function modified by additional term that penalizes deviations from the a-priori values

Θ∗ a-priori values of the derivativesR2 error covariance matrix of the a-priori valuesW weighting factor

W → 0: suppression of the a-priori valuesW → ∞: suppression of the measurements

Modified system of equations for parameter increment

*)(*)()]()([)]()([)( 12

1

1 Θ−ΘΘ−Θ+−∑ −=Θ −

=

− RWtytzRtytzJ Tkk

N

k

Tkk

*)(12)]()([1)(1

2)(1)(

Θ−Θ−−⎥⎦

⎤⎢⎣

⎡−−

Θ∂

∂=ΔΘ

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

−⎥⎦

⎤⎢⎣

⎡

Θ∂

∂−⎥⎦

⎤⎢⎣

⎡

Θ∂

∂∑∑ RWk

ktyktzRkty

kRWkty

RT

kty


Statistical Accuracy of Parameter Estimates

Measure of accuracyClues into the effectiveness, or lack thereof, of model parameters

Fischer information matrix provides a good approximation to the parameter error covariance matrix P:

Standard deviations (Cramer-Rao bounds):

Correlation coefficients:

Model fit and parameter accuracy: get back to this later (Model Validation).

( ) ( )1

1

1−

=

−

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

∑ ⎥⎦⎤

⎢⎣⎡

Θ∂∂

⎥⎦⎤

⎢⎣⎡

Θ∂∂

≈N

kk

Tk tyRtyP

iii p=Θσ

ji

ijji pp

p=ΘΘρ


Implementation Aspects for Large Scale Systems (1)

Gauss-Newton, Bounded-Variable GN or LM methods require solving the linear algebraic equation once per iteration.

Computational time:Information matrix Matrix Inversion

240 sec 0.1 secexample: 10 states, 19 observations, 3772 data points, 59 parameters

Sensitivity matrix for each time point involves (m*q*N) entries.m=60; q=1000; N=80000

Avoid storing huge matrix:- simulations for different parameter variations are run in parallel- sensitivity matrix is determined for the current data point- summands of the information matrix and gradients are immediately

calculated and summed up.

Θ∂∂ /y


Implementation Aspects for Large Scale Systems (2)

Computation of sensitivity matrix

+ Symmetric squarematrix (compute only lower triangular part)

For multiple experimentevaluation, Sparse matrixstructure

1 2 3Time segmentBlock for bias parameters of observation equation

1 2 3Time segmentBlock for initial conditions

Block forsystem parameters

0000

0000

0000

0000

0000

0000


Different Approaches to Account for Turbulence (1)

Two possible approaches:

1) to measure the wind components, more appropriately said, derive them from other measured variables such as true airspeed inertial speed, attitude angles, and flow angles

2) to model (generically or explicitly) the turbulence mathematicallyand estimate the corresponding parameters.

Complications:1) Presence of noise --> parameters must be estimated by some statistical criterion.

2) Modeling errors --> are deterministic, but are treated simply as noise:(rigorously not justifiable, but probably the best approach).

3) Process noise --> Stochastic system: State estimator is necessary



The first approach:A data pre-processing step:yields wind components along the three body-fixed coordinates, those can be treated as known inputs and accounted for in the estimation through minor modifications of the postulated models.

Advantage:Fairly simple output error method can be applied directly.

Disadvantages:Requires precise measurements of the said variables. Any inaccuracies in the measurements, for example those resultingfrom calibration errors or time delays in the recorded flow angles, will affect the derived wind, and consequently the estimates of theaerodynamic derivatives.


The second approach: Simultaneous modeling of turbulence

Two possible approaches to account for the atmospheric turbulence in the estimation of parameters:

1) Explicit modeling of gust spectrum (Dryden)

2) Generic model for process noise

Differences in the two options:- First option based on physical characteristics of gust- State vector augmentation- Leads to additional state equations

- Second option is more generic- Treat turbulence as white noise- No additional states



Consider the longitudinal motion:

Approach 1: Explicit modeling of gust spectrum (1)

x

z

αVγθ

u

w

Steady atmospheric conditions.

ee

e

e

q

u

u

MUZ

X

q

u

MMUZUZ

gXX

q

u

δ

θ

α

θ

α

δ

δ

δ

α

α

α

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ −

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0

/

01000001//

0

000

&

&

&

&


State vector: Tqux ],,,[ θα=

Input vector: ][ eu δ=

],,,,,[ θα quaay zx=Observation vector:

(.)(.)(.) ,, MZX LUnknown parameters:

To account for the turbulence in the longitudinal and vertical directions, we have to model the stochastic disturbances.

Assume stationary and homogenous gusts.



Exact determination of the Autocorrelation function and power spectral density of turbulence is not possible.

The PSD, S(Ω), at low frequencies tends to have a constant value:

At higher frequencies the PSD, and in turn the energy, falls off:

The different types of spectrum result depending upon the value of m.

Kolmogoroff, von Weizsaecker and Onsager determined a value of:

Von Karman spectrum:

.const)(Slim0

=Ω→Ω

m~)(Slim −

∞→ΩΩΩ

3/5m =

[ ] 6/52w

w2w

)L339,1(1

L2)(SΩ+

σ=Ω



22w

w2w

L1L2)(S

Ω+σ=Ω

Dryden

von Karman

Measured power spectral density*

Heisenberg: m = 7

Pritchard: 1.2 < m < 2,2

Dryden: m = 2 (Pragmatic value)

Both von Karman and Dryden spectrum define PSD of turbulence through just two parameters;

- the variance - characteristics length

Advantage of Dryden spectrum: - Simple structure and simple realization- Amenable to state space representation through simple filter(starting from white noise process).



In order to incorporate such an explicit model in the parameter estimation, we have to arrive at a model form compatible with the state space model of our system equations


Consider a first order Gauss-Markov process of the form:

ξbxax +−=&

),0(~)( xqt Nξis an arbitrary variable (either longitudinal or vertical gust component)

is the white noise (zero mean and variance ).

xxq

For a stationary Gauss-Markov process, the power spectral density is given by

22

2

)/(11)(

aa

qbS x

xω

ω+

=LUa E /0ω=

LUb E πω /20=

L Characteristic length (meter); ωΕ Corner frequency in rad/s;qx mean square intensity of the gust


For longitudinal and vertical gusts, typically,

ωE = 1 and 2.4 rad/s

they are characterized through the mean square intensities and scale (characteristic) lengths:

Combining state models for longitudinal and vertical gusts(two additional states ug and wg)with model for the longitudinal motion leads to an extended model representation of the form:

),( 2ug Lu ),( 2

wg Lw

The unknown parameters are the intensities and scale lengthsof the gusts:

),( 2ug Lu ),( 2

wg Lw



⎥⎦

⎤⎢⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−

−

−−−−−−

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

wg

ug

wE

u

ee

e

e

g

g

w

Eu

q

uu

uu

g

g

LULU

MUZ

X

wu

q

u

LU

LU

UMMMUZUZUZUZUXXgXX

wu

q

u

ξξ

πωπ

δ

θ

α

ω

θ

α

δ

δ

δ

αα

αα

αα

/200/200000000

000

/

00000

00000000100

/000//01///0

0

0

0

0

0

0

20000

0

&

&

&

&

&

&

ugξ wgξThe inputs and are the white noise processes, which are generated using the random number generator



Parameter estimation based on such an extended model is by no means simpler than the generic approach (Approach 2) discussed next:

- still have to apply the complex filter error methodincorporating a suitable state estimator.

- Model dependent (longitudinal mode)additional states for lateral-directional motion

- estimation of the scale lengths and gust intensities posed convergence problems:inconsistent estimates compared to the expected values based on physical understanding of the atmosphere.



State equations:

x state vectoru input vectorw additive process noise, N(0,I)F process noise distribution matrix

Observation equations:

Discrete measurements:

v measurement noise, N(0,I)

Unknown parameters:

Filter Error method

00 )()()(]),(),([)( xtxtwFtutxftx =+= λβ&

]),(),([)( xtutxgty β=

Nktvtytz kkk ,...,1),()()( =+=

);;( 0TTT

xT xλβ=Θ

Approach 2: Atmospheric Turbulence as white noise


Filter Error Method (1)

Cost function: Likelihood function

y~

⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧

⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ∑

=−−−−

−=Θ

N

k ktyktzRT

ktyktzNRmRzp1

)(~)(1)(~)(21exp2/det)2(),|( π

Measurementnoise

Dynamic System ++

Measuredresponse

Mathematical Model

State EstimatorObservation Equations

Parameter update By optimization ofLikelihood Function

+-

Input

Sensitivities ResponseError

z

z-

Processnoise

y~


xkkk btuBtxtx Ψ+Ψ+Φ=+ )()(ˆ)(~1

ykkk btuDtxCty ++= )()(~)(~

)](~)([)(~)(ˆ kkkk tytzKtxtx −+=Nk

tvtytz

uDxCy

wFuBxAx

kkk,...,2,1

);()()(=+=

+=

++=&

....!3)(

!2)(

....!2)(

32

2

0

22

+Δ

+Δ

+Δ==Γ

+Δ

+Δ+==Φ

∫Δ

Δ

tAtAtIde

tAtAIe

tA

tA

ττ

1−= RCPK T


Linear system State estimator

where the state transition matrix and its integral is given by:

Kalman Gain:


Covariance matrix of the state prediction error P

Riccati Equation (First-order approximation):

Solve Riccati equation by Potter’s method of Eigenvetor decomposition:

Define the Hamiltonian matrix as:

Compute the eigenvalues and eigenvectors.

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

−−Δ

−

−TACRTCt

TFFA11

01 1 =+Δ

−+ − TTT FFCPRPCt

PAAP



Where the eigenvectors corresponding to eigenvalues with positive real parts are in the left partition.

For controllable and observable system it turns out that exactlyone-half of the eigenvalues will have +ve real parts.

Solution is then given by: 12111−−= XXP

Partition the eigenvectors into four equal size matrices:

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

2221

1211XX

XX



Optimization of cost function:

Relaxation algorithm

1st step: Estimate covariance matrix R

2nd step: Apply Gauss-Newton method

,1 ΔΘ+Θ=Θ + ii GF −=ΔΘ

⎥⎦

⎤⎢⎣

⎡Θ∂

∂⎥⎦

⎤⎢⎣

⎡Θ∂

∂= −

=∑

)(~)(~1

1

kTN

k

k tyR

tyF

)](~)([)(~

1

1kk

TN

k

k tytzRty

−⎥⎦

⎤⎢⎣

⎡Θ∂

∂−= −

=∑G

Tkk

N

kkk tytztytz

NR )](~)([)](~)([1

1−−= ∑

=



Two-Step procedure: (for both Output error and Filter error methods)

Step 1: Estimate R for specified/updated parameters

Step 2: for known R, estimate parameters

Iterate on step 1 and 2 till convergence

For filter error method:

For physical meaningful results, the indirectly obtained GGT must be

positive semidefinite.

Results from constraining the diagonal elements of KC < 1.

Optimization subject to nonlinear inequality constraints;

solved by quadratic programming method.

TT CPCGGR +=



xx

kkkkk b

btuBtuBtx

txtxΘ∂Ψ∂

+Θ∂

∂Ψ+

Θ∂Ψ∂

+Θ∂

∂Ψ+

Θ∂Φ∂

+Θ∂

∂Φ=

Θ∂∂ + )()()(ˆ)(ˆ)(~

1

0)1(~,)()(

)(ˆ)(~1 =

Θ∂∂

Θ∂∂

Ψ+Θ∂

∂Ψ+

Θ∂∂

Ψ+Θ∂

∂Φ≈

Θ∂∂ + xb

tuBtxAtxtx xkk

kk

Θ∂

∂+

Θ∂∂

+Θ∂

∂+

Θ∂∂

=Θ∂

∂ ykk

kk btuDtxCtx

Cty

)()(~)(~)(~

)](~)([)(~)(~)(ˆ

kkkkk tytzKty

Ktxtx

−Θ∂

∂+

Θ∂∂

−Θ∂

∂=

Θ∂∂

Gradients



Solve Lyapunov equations to get

1−= RCPK T

Needs gradient of K (Kalman gain matrix):

11 −−

Θ∂∂

+Θ∂

∂=

Θ∂∂ RCPRCPK T

T

Partial differentiation of the Riccati equation,

011

11

11

11

=Θ∂

∂+

Θ∂∂

+Θ∂

∂Δ

−Θ∂

∂Δ

−

Θ∂∂

Δ−

Θ∂∂

Δ−

Θ∂∂

+Θ∂

∂+

Θ∂∂

+Θ∂

∂

−−

−−

TT

TT

TTT

T

FFFFPCRPCt

PCRPCt

CPRCPt

CPRCPt

APAPPAPA

Θ∂∂P



Nonlinear Systems:

State Estimator:

Prediction step

Correction step:

Steady state gain Matrix:

Riccati equation for covariance matrix of state prediction error:

∫−

+−=kt

ktdtktutxfktxktx

1]),(),([)1(ˆ)(~ β

]),(),(~[)( βktuktxgkty =

)]()([)(~)(ˆ ktyktzKktxktx −+=

0

]),(),([;1ttx

tutxgCRTCPK=∂

∂=−=⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ β

0

]),(),([;011ttx

tutxfAwhereTFFCPRTPCtTPAAP

=∂∂==+−

Δ−+

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ β



Response gradient approximated by finite difference approximation:Forward difference or Central difference

Perturbed response variables

∫−

++−=kt

ktdtktutpxfktpxktpx

1]),(),([)1(ˆ)(~ δββ

]),(),(~[)( δββ += ktuktpxgktpy

)]()([)(~)(ˆ ktpyktzpKktpxktpx −+=

)( jpy Θ

)( jpy ΘComputation of by numerical integration of straightforward.

Perturbed gain matrix:

(solution of Riccati equation with perturbed system matrices ).

)( jpy Θ

1−= RTpCpPpK

pCandpA



Comparative performance of Output Error and Filter Error methods

Time history plots of measured and estimated responses

Power spectral densities of residuals

comparison of parameter estimates

Example 1:Estimation of lift, drag and pitching moment

coefficients

Example 2:Estimation of lateral-directional derivatives


Comparative performance (2) Time histories

Output error method Filter error method


Output error method Filter error method

Comparative performance (3) PSD of Residuals


Filter Error Method0.22

0.20

0.18

0.16

Output Error Method0.22

0.20

0.18

0.16

FlightNo.

209112214219221216222223

Cnβ

Cnβ

15-3 3 9Angle of Attack, deg

flight inflight inturbulence

flight inC-160

Output Error Method

Mach Number

-0.18

-0.24

-0.30

-0.36

Cnζ

-0.18

-0.24

-0.30

-0.36

0.15 0.25 0.35 0.45 0.55

Cnζ

Filter Error Method

Weathercock stability (yaw stiffness) Rudder effectiveness

Comparative performance (4) Practical Utility


Pros and Cons of Filter Error method:

- Computationally complex

- Extension to nonlinear systems possible (Prediction step on nonlinear model; Correction step based onfirst order approximation found adequate)

- Difficult for multiple experiment analysis

- Clues regarding modeling errors obliterated

- Capability to analyze data in turbulent atmospheric conditions

- Also, in seemingly steady atmospheric conditions yields better results

- Marked differences in derivatives w.r.t. states

- Comparable estimates of control derivatives(because they are identified from the deterministic inputs, noiselevel in control inputs is low).

- Improved convergence

Comparative performance (5) Practical Utility


This page is left intentionally blank


Identification of Unstable Systems (1)

Difficulties:

1) Open loop plant identification:the basic / uncontrolled aircraft is unstable(due to the aerodynamic design)

2) Aircraft states and controls are highly correlated(due to the design of flight control laws)

3) Aircraft may be excited by process noise(e.g., induced by forebody vortices)

Unstable Aircraft

MotionFlight Control

SurfaceDeflections

PilotCommands


Difficulties Consequences Solution

Instability Integration of Eqs. of Filter error methodmotion leads to divergence EKF(overflow problems) Stabilized output error

Equation DecouplingRegressionFrequency domain methods

Correlated Correlated estimates, Reduced aerodynamic modelstates & controls convergence problems - fix some derivatives

- combined derivatives- mixed estimation

Avoid correlation- separate surface excitation- Optimized input

Process noise makes estimation more Use appropriate methoddifficult - Filter error

- Regression- EKF



In the case of highly unstable systems, special techniques andmodifications are necessary to prevent the growth of errors introduced by poor initial guess values, round off or discretizationerrors, and propagated by inherent instabilities of the system equations. Such approaches are based either on limiting the integration interval or making more efficient use of observed data.

-Least squares (LS) - Total least squares (TLS)

- Filter error method (FEM) - Output error method (OEM)

(LS, TLS, FEM, OEM already covered)

- Equation decoupling

- Eigen value transformation

- Extended Kalman filter / Unscented Kalman filter

- Stabilized output error



Output error method with artificial stabilization

S stabilization matrix 0 < S < Istabilized state vector

S independent of system parameters Θ

Stabilization matrix S corresponds to Kalman Filter gain matrix for the filter error method

S = 0 => output error method

S = I => equation error method

Unstable systems: numerical integration without stabilization diverges

)]()([)()(ˆ kkkk tytzStxtx −+=

x̂

Artificial Stabilization



Equation Decoupling

Reformulate state equations using measured states in such a way that each differential equation can be integrated independently.

AD - Diagonal matrix containing diagonal elements of AAOD - Matrix containing off-diagonal elements of A

Decoupled system, implied by diagonal AD, mostly stable- hence suitable for unstable systems.

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡+= )(

)()]();([)()()( tmxtu

ODABtxDAtx βββ&



Eigen value transformation:

Transformation though a real positive σ larger than the largest real part of the most unstable eigenvalue

Leads to transformed stable system:

Apply classical output error method

)()(~,)()(~,)()(~ tuetutyetytxetx ttt TTT σσσ −−− ===

)()(~,)()(~,)()(~ tvetvtwetwtzetz ttt TTT σσσ −−− ===

)(~)(~)(~)()(~ twFtuBtxIAtx T ++−= σ&

)(~)(~)(~ tuDtxCty +=

)(~)(~)(~kkk tvGtytz +=

−σ

jω

x x

xx

Original Im axis

TransformedIm axis

σΤ

σ0



EKF: Parameter Estimation by State Augmentation

Augmentation of state vector by unknown parameters:

0=ΘΘ= ⎥⎦

⎤⎢⎣

⎡ &withxax

Extended model of the dynamic system:

)()](),([)( tawaFtutaxaftax +=&

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+=

0)(

000

0]),(),([ twFtutxf β

)](),([)( tutaxagtay =

)()()( ktvGktyktz +=

State estimation of augmented system by extended Kalman filter.

FFT and GGT to be specified a priori ==> Filter tuning.

-0.6

-0.4

-0.2

CNBET[/rad]

0.05

0.15

0.25

CLP*[/rad]

-1

00.5

CLP*

CNR*[/rad]

-0.05

-0.03

-0.01

0 5 10 15 20time

CLDA[/rad]

sec

Convergence plots of estimates



Prediction step: ∫+=−

−kt

ktdttutxftxktax kaaka

1

)](),([)(ˆ)(~1

)](),(~[)(~kkaak tutxgty =

TaakTakakaka FFtttPttP Δ+ΦΦ= − )()(ˆ)()(~

1


where the transition matrix is given by: tAa ae Δ=Φ

and the linearized state matrix by:)1(ˆ

)(

−=∂

∂=

kaaaa

katxxx

ftA

obtained by numerical difference approximation.



Correction step (Update):

)](~)([)()(~)(ˆ kkkakka tytztKtaxtx −+=

)()()]()([)(~)]()([)(~)]()([)(ˆ

kTaT

kaTkakakakaka

kakakaka

tKGGtKtCtKItPtCtKItPtCtKItP

+−−=

−=


1])()(~)([)()(~)( −+= TGGtCtPtCtCtPtK kTakakak

Takaka

The linearized observation matrix is given by:)1(~

)(

−=∂

∂=

kaaaa

katxxx

gtC

The long form better conditioned for numerical computations andEnsures that the covariance matrix P is positive definite.



eeqw

eeqw

MqMwMq

ZqZUwZw

δ

δ

δ

δ

++=

+++=

&

& )( 0

qqww

ZqZwZa eeqwz

==

++= δδ

⎥⎦

⎤⎢⎣

⎡ +=

qw

qwMM

ZUZA 0

wKpe += δδ

Observation equations:

The static stability parameter, Mw, is so adjusted as to result in an unstable system with time-to-double of 1 s.

State equations:

A feedback proportional to the vertical velocity

System matrix

U0 of 44.57 m/s

Eigenvalues: (0.6934, -5.8250)

Time to double: roughly 1 sgiven by

TT

σ2ln

2 =

Example: Simulated data



Output error method

Example: Simulated data -- Estimated responses for starting parameter values

Stabilized output error method



Example: Simulated data -- Estimated responses

Output error method Stabilized output error method



666666663; 33; 3NparSys

2; 3; 12; 3; 12; 3; 12; 3; 12; 3; 12; 3; 12; 3; 32; 3; 30; 1; 30; 1; 3Nx; Ny; Nu

xdot_TC08 _uACobs_TC08_uAC

xdot_TC08_uACobs_TC08_uAC




xdot_TC10_uAC_EigTobs_TC08_ uAC

xdot_TC07_uAC_EqDecoupobs_TC08_uAC

xdot_TC06_uAC_RegStobs_TC06_ uAC_ RegSt

----Function name for state and observation equations

988881076----test case

ml_oemml_oemmainRPEmainRPEml_femml_oemml_oemml_oemuAC_regTLSuAC_regLSProgram name

0.6925-5.8395

-0.0873-3.1490

0.6898-5.8493

0.6913-5.8538

0.6930-5.8405

0.6936-5.8495

0.6900-5.8262

0.6919-5.8290

0.6887-5.8043

0.6887-5.8042

0.6934-5.8250

Eigen-values

-12.815(0.01)

-7.7214(1.87)

-12.8122(0.02)

-12.8247(0.02)

-12.8152(0.01)

-12.8262(0.01)

-12.773(0.01)

-12.7687-12.7059-12.7056-12.784

-3.7201(0.01)

-1.9277(4.04)

-3.7343(0.03)

-3.7373(0.03)

-3.71967(0.0)

-3.72645(0.01)

-3.7086(0.0)

-3.71215-3.6907-3.6906-3.7067

0.21709(0.01)

0.05216(5.85)

0.21713(0.04)

0.2175(0.04)

0.21717(0.01)

0.21776(0.01)

0.21616(0.01)

0.216340.21480.21480.2163

-6.1711(0.02)

-9.9477(8.72)

-6.2653(0.01)

-6.2622(0.01)

-6.16658(0.01)

-6.12825(0.01)

-6.1681(0.01)

-6.2632-6.2632-6.2632-6.2632

-1.4768*-1.4768*-1.4768*-1.4768*-1.4768*-1.4768*-1.47680*-1.47680*-1.4768-1.4768-1.4768

-1.4268(0.01)

-1.3086(2.48)

-1.4252(0.0)

-1.4252(0.0)

-1.42780(0.01)

-1.4294(0.01)

-1.4277(0.01)†

-1.4249-1.4249-1.4249-1.4249

SOEMOEMUKFEKFFEMEigTransfEqDecplLS/OEMTLSLSNominal value

Example: Simulated data -- Estimated parameters (Summary)

wZ

qZ

eZδ

wM

qM

eM δ



This page is left intentionally blank


Recursive Parameter Estimation (1)

Offline methods:

• All data points are processed together at a time, yielding parameters representing average system behavior.

• Assume availability of data set over a fixed interval of time.

• Implicitly assume system parameters constant over the period of observation.

Recursive methods:

• Utilize the data point-by-point as they become available.

• They are approximations of the more elaborative non-recursive methods.

• By nature, they cater to systems with time-varying parameters.

• Computer memory requirements are small, because storage of past data is not required.

Ability to track time-varying parameters helps indirectly in aerodynamic modeling

leads to accounting for nonlinearities in system model, at least to some extent, although the postulated model may be linear.

Likewise, it also helps to account for the changes in the flight conditions


Limitations associated with recursive estimation methods:

Convergence of standard RPE methods is slow and may not be adequate for real-time fault detection or to detect sudden changes in the dynamics.

Convergence can be improved by incorporating a forgetting factor to discard older data and thereby relying more on recent data.

Leads to increased noise sensitivity. Shorter records are necessary for faster adaptation, whereas longer records are necessary to distinguish noise.

A compromise between the rate of tracking parameter changes and noise sensitivity is necessary.

Wrong choice of forgetting factor can result in estimates oscillating around the true values.

Uninterruptedly running recursive estimation may face numerical problems: lack of or limited information content pertaining to dynamic motion. During parts of the flight phases, like steady level flight, the control and motion variables could be below noise level.

Verification of data collinearity is not a part of RPE methods. Nearly correlated states and controls affect estimation of stability and control derivatives.



Basically, any iterative off-line estimation method can be transformed into recursive form.

Two types of recursive methods:

Least Squares Based Recursive Methods

Recursive Least Squares (RLS)

Fourier Transform Regression (FTR)

Filtering approach

nonlinear state estimation by augmenting the basic system stateswith system parameters:

Extended Kalman Filter (EKF)

Unscented Kalman filter (UKF)

Extended forgetting factor recursive least squares (EFRLS)

Examples

Short period motion;

Longitudinal motion; Flight test Data (with turbulence)



)](ˆ)1()1([)1()(ˆ)1(ˆ kkxkykKkk T θθθ +−+++=+

)1()()1()1()()1(

+++

+=+

kxkPkxkxkPkK Tλ

{ })()1()1()(1)1( kPkxkKkPkP T ++−=+λ

Nktetxtxtxty kikrirkikiki ,,...1)()(...)()()( 2211 =++++= θθθ

YX)XX(ˆ T1T −=θ

Recall, from least squares estimation:

Recursive least squares (RLS)



i.e., samples older than T0 = 1/(1-λ) with reduced weighting.

∑

∑

=−

=

=

=N sk

N

s

k

1s2

N

1k2

N

)(J:functioncost Modified

)(J :RLS Standard:functionCost

ελ

εRLS: Recursive Least Squares -- forgetting factor

0

1

0.5

0.999

0 1000500

0.99

0.98 0.97

Weighting curves forDifferent forgetting factors



Standard RLS: Infinite memory, (i.e., entire data is given equal weighting)

RLS with Forgetting factor:

T0: Memory index (Typical choice: 0.98 ~ 0.995)- Improved convergence

- at the cost of increased sensitivity to noise (λ << 1 -> oscillations)- time varying forgetting factor or Kalman filter

- Thumb rule: balancing between process and measurement noise- larger process noise variance -> λ small

- Process noise variance relatively smaller than measurement noisevariance - > λ large (i.e., use more data for averaging).

RLS: Recursive Least Squares – forgetting factor



Methodology

The fourier transform of a signal x(t) is given by:

Simple Euler approximation to the finite Fourier transform

Apply Fourier transform to state space eq.

Equation error formulation for the k-th state equation:

and are the k-th row of matrices A and B, and is the k-thelement of vector for frequency ; m the frequencies of interest.

tXeixtxdtetxx Ni

T itjtj Δ=Δ==>= ∑∫−

=−

)()(~;)()(~10 ωωω

ωω

FTR: Fourier Transform Regression

)(~)(~)()(~)(~)(~

ωωωωωωω

uDxCyuBxAxj

+=+=

∑ −−= =mn kkkn nuBnxAnxjkJ 1

2|)(~)(~)(~|21 ω

)(~ nxkx~ nω

kA kB



Denoting vector of unknown parameter as , formulate problem as LS Regression with complex data:

ε represents the complex equation error in the frequency domain.Minimization yields:

On-Line VersionFor a given frequency , the discrete Fourier transform at the i-th time stepis related to that at the preceding step by:

[ ] ( )YXXX TT Re)Re(ˆ 1−=θ

θ

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

)(~.

)2(~)1(~

2

1

mxj

xjxj

Y

km

k

k

ω

ωω

where

⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

(m)u~ )(~.

)2(u~ )2(~(1)u~ )1(~

T

T

mx

x

x

X

Tk

Tk

Tk

andεθ += XY

nω

tijtjtijtijiii eeeexXX n Δ−Δ−Δ−Δ−

− −=+= )1(1 ;)()( ωωωωωω




Low computational effort:at each time point, discrete Fourier transform can be calculated using one

addition and two multiplications.

The m frequencies over which the cost function is evaluated can be selected as evenly spaced between and

Excluding the zero frequency removes trim values and measurement biases.

The most computational effort is the inversion of the matrixperformed by SVD.

- No tuning of any parameter- Good convergence for low to moderate noise.- Applicable to linear models- Aerodynamic zero terms can not be estimated

it is necessary to remove the trim values from the measurements of the motion variables as well as controls before using them in the algorithm.

Failure to do so leads to erratic behavior of the FTR algorithm, resulting from disproportionately large component at zero frequency.

minω maxω




Augmentation of state vector by unknown parameters:

0=ΘΘ= ⎥⎦

⎤⎢⎣

⎡ &withxax

Extended model of the dynamic system:

)()](),([)( tawaFtutaxaftax +=&

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡+=

0)(

000

0]),(),([ twFtutxf β

)](),([)( tutaxagtay =

)()()( ktvGktyktz +=

State estimation of augmented system by extended Kalman filter.

FFT and GGT to be specified a priori ==> Filter tuning.

-0.6

-0.4

-0.2

CNBET[/rad]

0.05

0.15

0.25

CLP*[/rad]

-1

00.5

CLP*

CNR*[/rad]

-0.05

-0.03

-0.01

0 5 10 15 20time

CLDA[/rad]

sec

Convergence plots of estimates

EKF: Extended Kalman Filter



Tuning of covariance matrices Q and R

Best when noise characteristics are known

11111 Update,Covariance .61ˆ1111

ˆ1

ˆ Update,State .5

11111111 n,computatioGain Kalman .4

111 ion,extrapolat Covariance .3

ˆ11

ˆ ion,extrapolat State 2.

and , matrices, Covariance and vector,(state)parameter Initialise 1.

+++−+=+

++−++++=+

−+++++++=+

+++=+

+=+

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

kPkHkKkPkPkxkHkzkKkk

kRTkHkPkHT

kHkPkK

kQTkkPkkP

kkk

kRkQkPk

θθ

φφ

θφθ

θ

EKF: Extended Kalman Filter



Unscented Kalman Filter UKF

The EKF performance is sensitive to nonlinearities in the system model.Strong system nonlinearities, or in other words the higher-order terms neglected in the propagation of states and error covariances, and wrong values of noisestatistics may result in biased estimates and in a worst case lead to divergence

Fundamental problem of not accounting for nonlinear transformation the random variables undergo:

Sigma point filters (SPF) / Unscented Kalman filter (UKF)- retains the standard Kalman filter form, - involves no local iterations, and has a better performance

UKF: - Propagates a finite set of (sigma) points through nonlinear dynamics- Approximates the distribution (mean and covariance) through a

weighted sum and outer (cross) product of the propagated points.

In contrast to the first order approximation used in the EKF for covariance propagation, in the UKF nonlinear dynamics are used without approximations.



Unscented Kalman Filter UKF


sigmapoints

Sigma-Point

( )χϒ= f

transformedsigma points

S-Pcovariance

S-Pmean

iiY = f(X)

mean

covariance

Actual (sampling)

( )=yfx

truemean

y=f(x)

truecovariance

Linearized (EKF)

y f x= =( )y = f(x),

EKFcovariance

EKFmean

truemean, covariance

linear propagation

TPP ΦΦ= ˆ~

Algorithmic steps: UKF Much more complex than EKF (See Ref. 1)


Knowledge of noise covariances not required, but needs tuning of forgetting factor

Applicable to linear systems, with only states as observations

EFRLS: Extended Forgetting factor Recursive Least Squares


An alternative approach based on an extension of the RLS with forgetting factor incorporating dynamic system matrix.

)()()()1( kwkxkkx +Φ=+

)()()( kxkCky = )()()( kvkykz +=

)]()()1()1([)1()()()()1( kxkkCkzkLkkxkkx Φ+−++Φ+Φ=+

1)]1()()()()1(

[)1()()()1(−+ΦΦ+

++Φ=+

kCkkPkkC

IkCkkPkLTT

TT λ

)()()]()1()1([)(1)1( kkPkkCkLIkkP TΦΦ++−Φ=+λ

Algorithm


X = TMoMZoZq ...] ... [ ααα L

u = ][ eδ

y= Tq ] [α

eeMqqMMoMq

eeZqqZZoZ

δδαα

δδααα

+++=

+++=

&

& '

Example 1: Short period dynamics:

Mq: total pitch damping;

)1(' qZqZ +=


Flight test data: elevator multistep inputs


-7.217(0.12)

-7.074(0.35)

-7.072(0.37)

-7.254(2.11)

-7.129-7.208(0.35)

-2.012(0.20)

-1.967(0.54)

-1.965(0.57)

-2.031(2.96)

-1.972-2.006(0.57)

-4.914(0.11)

-4.873(0.25)

-4.871(0.26)

-4.929(1.52)

-4.846-4.923(0.33)

0.474(0.10)

0.468(0.26)

0.467(0.28)

--0.4650.475(0.3)

0.679(0.67)

0.652(3.49)

0.648(3.57)

0.665(5.47)

0.6690.676(1.76)

0.108(1.88)

0.103(9.34)

0.102(9.61)

0.105†

(1.29)0.1090.104

(5.28)

-0.484(0.50)

-0.496(2.24)

-0.496(2.28)

-0.477(3.74)

-0.500-0.483(1.36)

-0.0090(2.78)

-0.0077(14.6)

-0.0075(15.2)

---0.0075-0.009(7.25)*

UKFaugUKFEKFFTREFRLS

EstimatesReference values (OEM)

Parameter

0Z

αZ

qZ

eZδ

0M

αM

qM

eMδ




Example 2: Non-dimensional lift, drag, pitching moment derivatives:


)cossin(

)sin()cos(

)cos()sin(

TtzTtxy

em

y

Te

L

Te

D

IF

CI

cSqq

qmVF

VgqC

mVSq

mF

gCmSqV

σσ

θ

σαθαα

σαθα

ll&

&

&

&

++=

=

+−−++−=

++−+−=

eemo

mqmmVmm

LLVLL

DDVDD

CVcqCC

VVCCC

CVVCCC

CVVCCC

δα

α

α

δα

α

α

++++=

++=

++=

200

00

00

zx aaqqV ,,,,,, &θα

TemmqmmVmLLVLDDVD CCCCCCCCCCC ][ 000 δααα=Θ

State equations

Observation variables:

Aerodynamic model:

Unknown paramaters:




-1.552(1.31)

-1.539(1.65)

-1.533(1.65)

-1.529(1.27)

-35.098(2.27)

-35.363(2.82)

-34.937(2.85)

-34.710(2.27)

-0.983(1.24)

-0.970(1.54)

-0.971(1.54)

-0.968(1.12)

0.0022(154)

0.0045(92.2)

0.0046(90.5)

0.0039(82.1)

0.115(3.40)

0.112(4.29)

0.112(4.28)

0.112(3.27)

4.303(1.13)

4.289(1.14)

4.303(1.14)

4.328(1.08)

0.157(10.6)

0.147(11.4)

0.144(11.7)

0.149(11.1)

-0.099(20.0)

-0.087(22.9)

-0.0853(23.5)

-0.0929(21.1)

UKFaugUKFEKF

RPE methodsFilter error method (FEM)

Parameter

0LC

LVC

αLC

0mC

mVC

αmC

mqC

emC δ


Sensor calibration model

αααα dpnbdynpKmdp Δ+=

αΔαταταα dp)t(nb)qt(dynpK)t(mdp +−−=

Scale factor and bias

Time delay

Estimation of Time Delays (1)

Typical examples

1: Recorded data

2. Multi-point aerodynamic model- Downwash lag /

transit time effect Transit time

)t(M)/r(CC DLCHLmDLC τ−δ=Δ τμεε l

where CLε stabilizer lift coefficient,ε stabilizer anglerh distance from wing NP to tail NPδDLC(t-t): time delayed DLC deflectionMτ unknown downwash parameter


Estimation of Time Delays (2)

Time delay in δe

-0.6 0 0.4sec

Cmw

0

-2.5-0.6 0 0.4sec

Costfunction

Time delay in δe

Time delays affect the estimates

Example: Estimation of pitching moment derivatives


Estimation of Time Delays (3)Different Approaches: Advantages and Limitations

Data pre-processing:- Time shifting through a specified fixed value- Simple, commonly used procedure- Limited to measured parameters- Only part to read flight data needs to be modified

Approximation by a first order lag- Simple procedure, requiring no modifications to estimation program- Additional first order differential eq. for each variable to be time shifted- Some dynamic effects due to approximation, albeit small.

Delay matrix- Accurate representation of time shift- Possible for both positive and negative values- Flexible: Can be applied to any parameter (directly measured as well asthose computed in the model)

- Requires estimation program for Nonlinear systems and a specialsubroutine for delay matrix.


Neural Networks for Aerodynamic ModelingNeural Networks - Pros and Cons ATTAS Lateral-Directional Motion

Flight data Neural network output

Predictive Capability

-0.1

0.0

0.15

-0.025

0

0.025

times

Side force coefficient

Yawing moment coeff.

10 20 30 400

Training Phase

-0.15

0

0.15

-0.05

0

0.05

time0 20 40 60s

Side force coefficient

Yawing moment coefficient

Nonlinear Input-Output subspace mapping

x c Rn y c Rmf

Less attractive for classical SysId applications

Incremental update or fine tuning ofsub models difficult

Network properties:- Number of hidden layers and nodes- Weights estimated by Back-Propagation

Black-box approach:- No physical significance to the structureor to the weights

Application spectrum:- Sensor failure detection- Highly NL phenomenon: stall, separated flows- Unstable systems


References (1)Jategaonkar, R. V., Flight Vehicle System Identification: A Time Domain Methodology,Volume 216, AIAA Progress in Astronautics and Aeronautics SeriesPublished by AIAA Reston, VA, Aug. 2006, ISBN: 1-56347-836-6http://www.aiaa.org/content.cfm?pageid=360&id=1447

Chapman, G. T. and Yates, L. A., “Nonlinear Aerodynamic Parameter Estimation and Model StructureIdentification”, AIAA Paper 92-4502, Aug. 1992.

Chen, W., and Valasek, J., “Observer/Kalman Filter Identification for On-Line System Identification ofAircraft”, AIAA Paper 99-4173, Aug. 1999.

Hamel, P. G. and Jategaonkar, R. V., “Evolution of Flight Vehicle System Identification”, Journal of Aircraft, Vol. 33, No. 1, Jan.-Feb. 1996, pp. 9-28.

Jategaonkar, R. V. and Plaetschke, E., “Algorithms for Aircraft Parameter Estimation Accounting for Process and Measurement Noise”, Journal of Aircraft, Vol. 26, No. 4, 1989, pp. 360-372.

Jategaonkar, R. V. and Thielecke, F., “Evaluation of Parameter Estimation Methods for Unstable Aircraft”, Journal of Aircraft, Vol. 31, No. 2, May-June 1994, pp. 510-519.

Jategaonkar, R. V., “Bounded-Variable Gauss-Newton Algorithm for Aircraft Parameter Estimation”,Journal of Aircraft, Vol. 37, No. 4, July-Aug. 2000, pp.742-744.

Julier, S., Uhlmann, J., and Durrant-Whyte, H. F., “A New Method for the Nonlinear Transformation of Means and Covariances in Filters and Estimators,” IEEE Transactions on Automatic Control, Vol. 45, No. 3, 2000, pp. 477-482.

Klein, V., “Estimation of Aircraft Aerodynamic Parameters from Flight Data”, Progress in Aerospace Sciences, Vol. 26, Pergamon, Oxford, UK, 1989, pp. 1-77.


References (2)

Maine, R. E. and Iliff, K. W., “Identification of Dynamic Systems - Applications to Aircraft. Part 1: The Output Error Approach”, AGARD AG-300, Vol. 3, Pt. 1, Dec. 1986.

Morelli, E. A., “Real-Time Parameter Estimation in the Frequency Domain”, AIAA Paper 99-4043, Aug. 1999.

Murray-Smith, D. J., “Methods for the External Validation of Continuous System Simulation Models: A Review”, Mathematical and Computer Modelling of Dynamical Systems, Vol. 4, No. 1, 1998, pp. 5-31.

Plaetschke, E., “Ein FORTRAN-Program zur Maximum-Likelihood-Parameterschätzung in nichtlinearenSystemen der Flugmechanik – Benutzeranleitung”, DFVLR-Mitt. 86-08, Feb. 1986.

Seanor, B., Song, Y., Napolitano, M. R., and Campa, G., “Comparison of On-Line and Off-Line ParameterEstimation Techniques Using the NASA F/A-18 HARV Flight Data”, AIAA paper 2001-4261, Aug. 2001.

Wan., E. A., and van der Merwe, R., “The Unscented Kalman Filter for Nonlinear Estimation,” Proceedings of the IEEE Symposium 2000 on Adaptive Systems for Signal Processing, Communication and Control AS-SPCC, Lake Louise, Alberta, Canada, 1-4 Oct. 2000, pp. 153-158.

fvsysid shortcourse 4 methods

Documents