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Advanced Problems of Longitudinal Dynamics Robert Stengel, Aircraft Flight Dynamics MAE 331, 2010 Fourth-order dynamics Steady-state response to control Transfer functions Frequency response Root locus analysis of parameter variations Numerical solution for trimmed flight condition Nichols chart Pilot-aircraft interactions Copyright 2010 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/MAE331.html http://www. princeton . edu/~stengel/FlightDynamics .html Linear, Time-Invariant Fourth-Order Longitudinal Model ! ! V (t ) ! ! " (t ) ! ! q(t ) ! ! # (t ) $ % & & & & & ( ) ) ) ) ) = * D V *g 0 * D # L V V N 0 0 L # V N M V 0 M q M # * L V V N 0 1 * L # V N $ % & & & & & & & ( ) ) ) ) ) ) ) !V (t ) !" (t ) !q(t ) !# (t ) $ % & & & & & ( ) ) ) ) ) + 0 T +T 0 0 0 L + F / V N M + E 0 0 0 0 * L + F / V N $ % & & & & & ( ) ) ) ) ) !+ E(t ) !+T (t ) !+ F(t ) $ % & & & ( ) ) ) Stability and control derivatives are defined at a trimmed (equilibrium) flight condition Initial-Condition and Step- Input Responses of a Business Jet Aircraft Initial pitch rate [!q(0)] = 0.1 rad/s Elevator step input [!!E(0)] = 1 deg Trimmed Solution of the Equations of Motion
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  • Advanced Problems of

    Longitudinal DynamicsRobert Stengel, Aircraft Flight Dynamics

    MAE 331, 2010

    Fourth-order dynamics

    Steady-state response to control

    Transfer functions

    Frequency response

    Root locus analysis of parametervariations

    Numerical solution for trimmedflight condition

    Nichols chart

    Pilot-aircraft interactions

    Copyright 2010 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html

    http://www.princeton.edu/~stengel/FlightDynamics.html

    Linear, Time-Invariant Fourth-Order

    Longitudinal Model

    ! !V (t)

    ! !" (t)

    ! !q(t)

    ! !#(t)

    $

    %

    &&&&&

    '

    (

    )))))

    =

    *DV

    *g 0 *D#LV

    VN

    0 0L#VN

    MV

    0 Mq

    M#

    * LVVN

    0 1 * L#VN

    $

    %

    &&&&&&&

    '

    (

    )))))))

    !V (t)

    !" (t)

    !q(t)

    !#(t)

    $

    %

    &&&&&

    '

    (

    )))))

    +

    0 T+T 0

    0 0 L+F /VN

    M+E 0 0

    0 0 *L+F /VN

    $

    %

    &&&&&

    '

    (

    )))))

    !+E(t)

    !+T (t)

    !+F(t)

    $

    %

    &&&

    '

    (

    )))

    Stability and control derivatives are defined at atrimmed (equilibrium) flight condition

    Initial-Condition and Step-

    Input Responses of aBusiness Jet Aircraft

    Initial pitch rate [!q(0)] = 0.1 rad/s Elevator step input [!!E(0)] = 1 deg

    Trimmed Solution of theEquations of Motion

  • Flight Conditions for

    Steady, Level Flight

    !V = f1=1

    mT cos ! + i( ) " D " mgsin#$% &'

    !# = f2=1

    mVT sin ! + i( ) + L " mgcos#$% &'

    !q = f3= M / Iyy

    !! = f4= !( " !# = q "

    1

    mVT sin ! + i( ) + L " mgcos#$% &'

    Nonlinear longitudinal model

    Nonlinear longitudinal model must be in equilibrium

    0 = f1=1

    mT cos ! + i( ) " D " mgsin#$% &'

    0 = f2=1

    mVT sin ! + i( ) + L " mgcos#$% &'

    0 = f3= M / Iyy

    0 = f4= !( " !# = q "

    1

    mVT sin ! + i( ) + L " mgcos#$% &'

    In level flight, q = 0, and " = #

    Numerical Solution to Estimate

    the Trimmed Condition Specify desired altitude and airspeed, hN and VN Guess starting values for the trim parameters throttle

    setting, elevator angle, and pitch angle (= angle of attack),!T0, !E0, and #0

    Calculate f1, f2, and f3 (f4 has same value when q = 0)

    Define a scalar,positive-definite trim cost function, e.g.,

    f1=1

    mT !T

    0,!E

    0,"

    0,hN ,VN( )cos " + i( ) # D !T0 ,!E0 ,"0 ,hN ,VN( )$% &' ( 0

    f2=

    1

    mVNT !T

    0,!E

    0,"

    0,hN ,VN( )sin " + i( ) + L !T0 ,!E0 ,"0 ,hN ,VN( ) # mg$% &' ( 0

    f3= M !T

    0,!E

    0,"

    0,hN ,VN( ) / Iyy ( 0

    J !Tk ,!Ek ," k( ) = a f1

    2( )k+ b f

    2

    2( )k+ c f

    3

    2( )k, k = 0,1,!,K

    Minimize the Cost Function withRespect to the Trim Parameters

    Cost is minimized at the bottom of the bowl, i.e., when

    J !Tk ,!Ek ," k( ) ! J uk( ) = a f1

    2( )k+ b f

    2

    2( )k+ c f

    3

    2( )k, k = 0,1,",K

    !J

    !uu=u*

    =!J

    !u1

    !J

    !u2

    !J

    !u3

    "

    #$$

    %

    &''u=u*

    = 0

    Cost is bowl-shaped

    Search in direction perpendicular to contours of equal cost

    Example of Search for TrimmedCondition (Fig. 3.6-9, Flight Dynamics)

    In MATLAB, use fminsearch [Nelder-Mead Downhill Simplex Method]

    u = fminsearch J,u0( )

  • Steady-State Response

    Steady-State Response of the

    Fourth-Order Longitudinal Model

    !VSS!" SS!qSS!#SS

    $

    %

    &&&&&

    '

    (

    )))))

    = *

    *DV *g 0 *D#LVVN

    0 0L#VN

    MV 0 Mq M#

    * LVVN

    0 1 * L#VN

    $

    %

    &&&&&&&

    '

    (

    )))))))

    *1

    0 T+T 0

    0 0 L+F /VN

    M+E 0 0

    0 0 *L+F /VN

    $

    %

    &&&&&

    '

    (

    )))))

    !+ESS!+TSS!+FSS

    $

    %

    &&&

    '

    (

    )))

    !xSS= "F

    "1G!u

    SS

    How do we calculate the equilibrium response to control?

    !!x(t) = F!x(t) +G!u(t)

    For the longitudinal model

    Algebraic Equation for

    Equilibrium Response

    !VSS!" SS!qSS!#SS

    $

    %

    &&&&&

    '

    (

    )))))

    =

    *gM+EL#VN

    $%&

    '()

    0 gM#L+F /VN[ ]

    DVL#VN

    * D#LVVN( )M+E

    $%&

    '()

    MVL#VN

    * M#LVVN( )T+T

    $%&

    '()

    D#MV * DVM#( )L+F /VN$% '(

    0 0 0

    *gM+ELVVN

    $%&

    '()

    0 L+F /VN[ ]

    $

    %

    &&&&&&&&&

    '

    (

    )))))))))

    g MV

    L#VN

    * M#LVVN( )

    !+ESS!+TSS!+FSS

    $

    %

    &&&

    '

    (

    )))

    !VSS!" SS!qSS!#SS

    $

    %

    &&&&&

    '

    (

    )))))

    =

    a 0 b

    c d e

    0 0 0

    f 0 g

    $

    %

    &&&&

    '

    (

    ))))

    !*ESS!*TSS!*FSS

    $

    %

    &&&

    '

    (

    )))

    Roles of stability and controlderivatives identified

    Result is a simple equation relatinginput and output

    4th-Order Steady-State ResponseMay Be Counterintuitive

    !VSS = a!"ESS + 0( )!"TSS + b!"FSS!# SS = c!"ESS + d!"TSS + e!"FSS

    !qSS = 0( )!ESS + 0( )!"TSS + 0( )!"FSS!$SS = f!"ESS + 0( )!"TSS + g!"FSS

    Observations

    Thrust command

    Elevator and flap commands

    Steady-state pitch rate is zero

    4th-order model neglects air density gradient effects

    !"SS = !# SS + !$SS = c + f( )!%ESS + d!%TSS + e + g( )!%FSS

    Pitch angle response

  • Effects of StabilityDerivative Variations on 4th-Order Longitudinal Modes

    Primary and Coupling Blocks of the

    Fourth-Order Longitudinal Model

    FLon =

    !DV !g 0 !D"

    LVVN

    0 0L"VN

    MV 0 Mq M"

    !LVVN

    0 1 !L"VN

    #

    $

    %%%%%%%

    &

    '

    (((((((

    =FPh FSP

    Ph

    FPhSP

    FSP

    #

    $

    %%

    &

    '

    ((

    Some stability derivatives appear only in primary blocks (DV, Mq, M")

    Effects are well-described by 2nd-order models

    Some stability derivatives appear only in coupling blocks (MV, D")

    Effects are ignored by 2nd-order models

    Some stability derivatives appear in both (LV, L")

    May require 4th-order modeling

    $M" Effect on Fourth-Order Roots

    !Lon (s) = s4+ DV +

    L"VN

    # Mq( )s3

    + g # D"( )LVVN

    + DVL"VN

    # Mq( ) # Mq L" VN # M"o$

    %&'

    ()s2

    + Mq D" # g( )LVVN

    # DVL"VN

    $%&

    '()+ D"MV # DVM"o{ }s

    + g MVL"VN

    # M"oLVVN( ) # !M" s

    2+ DVs + g

    LVVN( )

    * d(s) + kn(s)

    Group all terms multiplied byM" to form numerator for $M"

    Primary effect: The same as in theapproximate short-period model

    Numerator zeros The same as the approximate phugoid mode

    characteristic polynomial

    Consequently, effect of M" variation onphugoid mode is small

    sI !

    !DV !g

    LVVN

    0

    "

    #

    $$$

    %

    &

    '''

    = (Ph (s) = s2+ DVs + g

    LVVN

    Short

    Period

    Short

    Period

    Phugoid

    Phugoid

    MV Effect on Fourth-

    Order Roots

    Large positive value producesoscillatory phugoid instability

    Large negative value producesreal phugoid divergence

    !Lon(s) = s

    4+ D

    V+L"VN

    # Mq( )s3

    + g # D"( )LV

    VN

    + DV

    L"VN

    # Mq( ) # Mq L" V

    N

    # M"$

    %&'

    ()s2

    + Mq

    D" # g( )LV

    VN

    # DV

    L"VN

    $%&

    '()+ D"MV # DVM"{ }s

    gM"LV

    VN

    + MVD"s + g

    L"VN

    ( ) = 0

    D!

    = 0

    Short

    Period

    Phugoid

  • L" /VN and LV /VN Effects onFourth-Order Roots

    LV /VN: Damped natural frequency of thephugoid

    Negligible effect on the short-period

    L" /VN: Increased damping of the short-period

    Small effect on the phugoid mode

    Pitch and ThrustControl Effects

    Longitudinal Model

    Transfer Function Matrix With Hx = I

    HLon(s) = H

    xLonsI ! F

    Lon[ ]!1G

    Lon=

    1 0 0 0

    0 1 0 0

    0 0 1 0

    0 0 0 1

    "

    #

    $$$$

    %

    &

    ''''

    nV

    V(s) n(

    V(s) n

    q

    V(s) n)

    V(s)

    nV

    ((s) n(

    ((s) n

    q

    ((s) n)

    ((s)

    nV

    q(s) n(

    q(s) n

    q

    q(s) n)

    q(s)

    nV

    )(s) n(

    )(s) n

    q

    )(s) n)

    )(s)

    "

    #

    $$$$$$

    %

    &

    ''''''

    0 T*T 0

    0 0 L*F /VN

    M*E 0 0

    0 0 !L*F /VN

    "

    #

    $$$$$

    %

    &

    '''''

    s4+ a

    3s3+ a

    2s2+ a

    1s + a

    0

    HLon (s) =

    n!EV(s) n!T

    V(s) n!F

    V(s)

    n!E"(s) n!T

    "(s) n!F

    "(s)

    n!Eq(s) n!T

    q(s) n!F

    q(s)

    n!E#(s) n!T

    #(s) n!F

    #(s)

    $

    %

    &&&&&&

    '

    (

    ))))))

    s2+ 2*P+nP s ++nP

    2( ) s2 + 2*SP+nSP s ++nSP2( )

    There are 4 outputs and 3 inputs

    !V (s)

    !" (s)

    !q(s)

    !#(s)

    $

    %

    &&&&&

    '

    (

    )))))

    = HLon(s)

    !*E(s)

    !*T (s)

    !*F(s)

    $

    %

    &&&

    '

    (

    )))

    A Little More About

    Output Matrices With Hu = 0, suppose the output equals the state:

    !y = !x = Hx!x; then Hx = I4and

    !y1

    !y2

    !y3

    !y4

    "

    #

    $$$$$

    %

    &

    '''''

    =

    1 0 0 0

    0 1 0 0

    0 0 1 0

    0 0 0 1

    "

    #

    $$$$

    %

    &

    ''''

    !x1

    !x2

    !x3

    !x4

    "

    #

    $$$$$

    %

    &

    '''''

    !

    !V!(

    !q

    !)

    "

    #

    $$$$$

    %

    &

    '''''

    Suppose the only output is !V; then

    !y = !V = 1 0 0 0"# $%

    !V!&

    !q

    !'

    "

    #

    (((((

    $

    %

    )))))

    Suppose !V and !" are measured; then

    !y =!y

    1

    !y2

    "

    #$$

    %

    &''=

    !V!(

    "

    #$

    %

    &' =

    1 0 0 0

    0 0 0 1

    "

    #$

    %

    &'

    !V!)

    !q

    !(

    "

    #

    $$$$$

    %

    &

    '''''

  • A Little More About

    Output Matrices Output (measurement) of body-axis velocity and pitch

    rate and angle

    Transformation from [!V, !%, !q, !"] to [!u, !w, !q, !#]

    !u!w!q

    !"

    #

    $

    %%%%

    &

    '

    ((((

    =

    cos)N

    0 0 *VNsin)

    N

    sin)N

    0 0 VNcos)

    N

    0 0 1 0

    0 1 0 1

    #

    $

    %%%%%

    &

    '

    (((((

    !V!+

    !q

    !)

    #

    $

    %%%%%

    &

    '

    (((((

    Separate measurement of state and control perturbations

    !y =!x

    !u

    "

    #$

    %

    &' = Hx!x +Hu!u

    !y1

    !y2

    !y3

    !y4

    !y5

    !y6

    "

    #

    $$$$$$$$

    %

    &

    ''''''''

    =

    1 0 0 0

    0 1 0 0

    0 0 1 0

    0 0 0 1

    0 0 0 0

    0 0 0 0

    "

    #

    $$$$$$$

    %

    &

    '''''''

    !V!(

    !q

    !)

    "

    #

    $$$$$

    %

    &

    '''''

    +

    0 0

    0 0

    0 0

    0 0

    1 0

    0 1

    "

    #

    $$$$$$$

    %

    &

    '''''''

    !*E!*T

    "

    #$

    %

    &'

    Elevator-to-Normal-

    Velocity Numerator

    HxAdj sI ! FLon( )G = sin"N 0 0 VN cos"N#$

    %&

    nVV(s) n'

    V(s) nq

    V(s) n"

    V(s)

    nV'(s) n'

    '(s) nq

    '(s) n"

    '(s)

    nVq(s) n'

    q(s) nq

    q(s) n"

    q(s)

    nV"(s) n'

    "(s) nq

    "(s) n"

    "(s)

    #

    $

    ((((((

    %

    &

    ))))))

    0

    0

    M*E

    0

    #

    $

    ((((

    %

    &

    ))))

    = n*Ew(s)

    Transform though "N back to body axes

    n!Ew(s) = sin"

    N0 0 V

    Ncos"

    N#$

    %&

    nq

    V(s)

    nq

    '(s)

    nq

    q(s)

    nq

    "(s)

    #

    $

    ((((((

    %

    &

    ))))))

    M!E = M!E sin"N( )nqV(s) + V

    Ncos"

    N( )nq"(s)#$ %&

    Transfer function numerator

    Elevator-to-Normal-Velocity

    Transfer Function

    !w(s)

    !"E(s)=n"Ew(s)

    !Lon (s)=

    M"E s2+ 2#$ns +$n

    2( )Approx Ph

    s % z3( )

    s2+ 2#$ns +$n

    2( )Ph

    s2+ 2#$ns +$n

    2( )SP

    Normal velocity transfer function is analogous to angle ofattack transfer function ($" ! $w/VN)

    z3 often neglected due to high frequency

    Elevator-to-Normal-Velocity FrequencyResponse

    !w(s)

    !"E(s)=n"Ew(s)

    !Lon (s)#

    M"E s2+ 2$%ns +%n

    2( )Approx Ph

    s & z3( )

    s2+ 2$%ns +%n

    2( )Ph

    s2+ 2$%ns +%n

    2( )SP

    0 dB/dec+40 dB/dec

    0 dB/dec

    40 dB/dec

    20 dB/dec

    (n q) = 1

    Complex zeroalmost (but notquite) cancelsphugoid response

  • Elevator-to-Pitch-RateNumerator and Transfer Function

    HxAdj sI ! FLon( )G = 0 0 1 0"# $%

    nVV(s) n&

    V(s) nq

    V(s) n'

    V(s)

    nV&(s) n&

    &(s) nq

    &(s) n'

    &(s)

    nVq(s) n&

    q(s) nq

    q(s) n'

    q(s)

    nV'(s) nV

    '(s) nV

    '(s) nV

    '(s)

    "

    #

    ((((((

    $

    %

    ))))))

    0

    0

    M*E

    0

    "

    #

    ((((

    $

    %

    ))))

    = n*Eq(s)

    !q(s)

    !"E(s)=n"Eq(s)

    !Lon (s)#

    M"Es s $ z1( ) s $ z2( )s2+ 2%&ns +&n

    2( )Ph

    s2+ 2%&ns +&n

    2( )SP

    Free s in numerator differentiates pitch angle transfer function

    Elevator-to-Pitch-

    Rate FrequencyResponse

    +20 dB/dec

    +20 dB/dec+40 dB/dec

    0 dB/dec20 dB/dec

    !qSS = 0( )!ESS + 0( )!"TSS + 0( )!"FSS

    (n q) = 1

    Negligible low-frequency response,except at phugoidnatural frequency

    High-frequencyresponse wellpredicted by 2nd-ordermodel

    !q(s)

    !"E(s)=n"Eq(s)

    !Lon (s)#

    M"Es s $ z1( ) s $ z2( )s2+ 2%&ns +&n

    2( )Ph

    s2+ 2%&ns +&n

    2( )SP

    Transfer Functions of

    Angles from Elevator Input

    !"(s)!#E(s)

    =n#E"(s)

    !Lon(s); n#E

    "(s) = M#E s +

    1T"1

    $%&

    '()s + 1

    T"2

    $%&

    '()

    !"(s)

    !#E(s)=n#E"(s)

    !Lon (s); n#E

    "(s) = M#E s

    2+ 2$%ns +%n

    2( )Approx Ph

    !" (s)!#E(s)

    =n#E

    "(s)

    !Lon(s); n#E

    "(s) = M#E

    L$

    VN

    s + 1T" 1

    %&'

    ()*

    Elevator-to-Flight Path Angle transfer function

    Elevator-to-Angle of Attack transfer function

    Elevator-to-Pitch Angle transfer function

    Frequency Response ofAngles to Elevator Input

    Pitch anglefrequency response($# = $% + $")

    Similar to flightpath angle nearphugoid naturalfrequency

    Similar to angleof attack nearshort-periodnaturalfrequency

    !" SS = c!#ESS

    !$SS = f!#ESS

    !%SS = c & f( )!#ESS

  • Transfer Functions ofAngles from Thrust Input

    !"(s)!#T (s)

    =n#T"(s)

    !Lon(s); n#T

    "(s) = T#T s +

    1T"

    T

    $%&

    '()

    !"(s)!#T (s)

    =n#T"(s)

    !Lon(s); n#T

    "(s) = T#T s s +

    1T"

    T

    $%&

    '()

    !" (s)

    !#T (s)=n#T"(s)

    !Lon (s); n#T

    "(s) = T#T

    LV

    VNs2+ 2$%ns +%n

    2( )Approx SP

    Thrust-to-Flight Path Angle transfer function

    Thrust-to-Angle of Attack transfer function

    Thrust-to-Pitch Angle transfer function

    Frequency Response of Angles

    to Thrust Input Thrust primarily effects flight path angle and low-frequency pitch angle

    Feedback Control: Angles to Elevator

    Variations in control gain

    Principal effect is onshort-period roots

    Gain and Phase Margins:The Nichols Chart

  • Nichols Chart:

    Gain vs. Phase Angle Bode Plot

    Two plots

    Open-Loop Gain (dB) vs. log10&

    Open-Loop Phase Angle vs. log10&

    Nichols Chart

    Single crossplot; inputfrequency not shown

    Open-Loop Gain (dB) vs. Open-

    Loop Phase Angle

    Gain and Phase Margins

    Gain Margin

    At the input frequency, &, for which '(j&) = 180

    Difference between 0 dB and transfer function magnitude,

    20 log10 AR(j&)

    Phase Margin

    At the input frequency, &, for which 20 log10 AR(j&) = 0 dB

    Difference between the phase angle '(j&), and 180

    Axis intercepts on the Nichols Chart identify GM

    and PM

    Examples of Gain and Phase Margins

    Bode Plot Nichols Chart

    Hblue( j! ) =10

    j! +10( )

    "

    #$

    %

    &'

    1002

    j!( )2+ 2 0.1( ) 100( ) j!( ) +1002

    "

    #$$

    %

    &''

    Hgreen ( j! ) =10

    2

    j!( )2+ 2 0.1( ) 10( ) j!( ) +102

    "

    #$$

    %

    &''

    100

    j! +100( )

    "

    #$

    %

    &'

    Gain and Phase Margins in

    Pitch-Tracking TaskElevator-to-Pitch-Angle

    Bode Plot

    Elevator-to-Pitch-Angle

    Nichols Chart

    Amplitude Ratio vs. Phase Angle

    Gain Margin: Amplitude ratio below 0 dB

    when phase angle = 180

    Phase Margin: Phase angle above 180when amplitude ratio = 0 dB

  • Pilot-Vehicle Interactions

    Pilot Inputs to Control

    * p. 421-425, Flight Dynamics

    Effect of Pilot Dynamics on

    Pitch-Angle Control Task

    Pilot Transfer Function = KP1 /TP

    s +1 /TP= KP

    1 / 0.25

    s +1 / 0.25

    Pilot introduces neuromuscular lag while closing the control loop

    Example

    Model the lag by a 1st-order time constant, TP, of 0.25 s

    Pilot"s gain, KP, is either 1 or 2

    Open-Loop Pilot-Aircraft

    Transfer Function

    H (s) = KP

    1 /TP

    s +1 /TP( )

    !

    "#

    $

    %&

    M'E s +1T(1

    )*+

    ,-.s + 1

    T(2

    )*+

    ,-.

    s2+ 2/0

    ns +0

    n

    2( )Ph

    s2+ 2/0

    ns +0

    n

    2( )SP

    !

    "

    ####

    $

    %

    &&&&

  • Effect of Pilot

    Dynamics on

    Pitch-Angle

    Control Task

    Gain and phase

    margins become

    negative for pilot

    gain between 1

    and 2

    Effect of Pilot Dynamics on Elevator/Pitch-

    Angle Control Root Locus

    Pilot transfer function changes asymptotes of

    the root locus

    Next Time:Fourth-Order Lateral-Directional Dynamics

    Supplemental

    Material

  • Airspeed Frequency Response toElevator and Thrust Inputs

    Response is primarily through the lightly dampedphugoid mode

    Altitude Frequency Responseto Elevator and Thrust Inputs

    Altitude perturbation: Integral of the flight path angle perturbation

    !z(t) = "VN

    !# ($ )d$0

    t

    %!z(s)!"E(s)

    = #VN

    s

    $%&

    '()

    !* (s)!"E(s)

    !z(s)!"T (s)

    = #VN

    s

    $%&

    '()

    !* (s)!"T (s)

    High- and Low-Frequency Limits of

    Frequency Response Function

    Hij ( j! " 0)"kij j!( )

    q+ bq#1 j!( )

    q#1+ ...+ b

    1j!( ) + b0[ ]

    j!( )n

    + an#1 j!( )n#1

    + ...+ a1j!( ) + a0[ ]

    "

    kijb0

    a0

    , b0$ 0

    kij j(0)b1

    a0

    , b0

    = 0, b1$ 0, etc.

    %

    & ' '

    ( ' '

    Hij ( j!) = AR(!) ej" (! )

    Hij ( j! "#)"kij j!( )

    q+ bq$1 j!( )

    q$1+ ...+ b

    1j!( ) + b0[ ]

    j!( )n

    + an$1 j!( )n$1

    + ...+ a1j!( ) + a0[ ]

    "kij

    j!( )n$q