Aircraft Flight Dynamics Robert F. Stengel Lecture10 Linearized Equations and Modes of Motion

Linearized Equations Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012" – Nominal flight path" – Perturbations about the nominal flight path " – Longitudinal" – Lateral-directi

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Linearized Equations  

Robert Stengel, Aircraft Flight Dynamics 

MAE 331, 2012"

–   Nominal flight path"

–   Perturbations about the nominal flight path "

–   Longitudinal"

–   Lateral-directional "

http://www.princeton.edu/~stengel/MAE331.html !

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

The Mystery Airplane:

Convair XF-92A (1948)

!   Precursor to F-102, F-106, B-58, and F2Y"

!   M = 1.05 in a dive; under-powered, pre-area rule"

!   Landed at θ = 45°, V = 67 mph (108 km/h)"

!   Violent pitchup during high-speed turns, alleviated by wing fences"

!   Poor high-speed and good low-speed handling qualities "

Nominal and Actual

Flight Paths

Nominal and Actual Trajectories"

•   Nominal (or reference) trajectory and control history"

xN(t), uN(t), wN(t)

•   Actual trajectory perturbed by"

–   Small initial condition variation, Δxo(to) "

–   Small control variation, Δu(t)"

x(t), u(t), w(t)

= x { N(t) + Δx(t), uN(t) + Δu(t),wN(t) + Δw(t) }

€

x : dynamic state

u : control input

w : disturbance input

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Both Paths Satisfy the Dynamic Equations"

•   Dynamic models for the actual and the

nominal problems are the same"

x N (t) = f[x N (t), u N (t), w N (t)], x N ( ) t o given

x(t) = f[x(t),u(t),w(t)], x t ( ) o given

x(t) = xN(t) + Δx(t)

x(t) − xN(t) + Δx(t)

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in t *+o,tf

,-Δx(to) = x(to) − xN(to)

Δu(t) = u(t) − uN(t)

Δw(t) = w(t) − wN(t)

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in t *+o,tf

,-•  Differences in initial

condition and forcing " •  perturb rate of change and the state"

Approximate Neighboring Trajectory as a Linear Perturbation

to the Nominal Trajectory"

x(t) = xN(t) + Δx(t)

≈ f[xN(t), uN(t), wN(t),t] + ∂f

∂f

•   Approximate the new trajectory as the sum of the nominal path plus a linear perturbation"

xN(t) = f[xN(t), uN(t), wN(t),t]

x(t) = xN(t) + Δx(t) = f[xN(t) + Δx(t),uN(t) + Δu(t),wN(t) + Δw(t),t]

Linearized Equation Approximates

Perturbation Dynamics"

•   Solve for the nominal and perturbation trajectories

separately"

x N (t) = f[x N (t), u N (t), w N (t),t], x N ( ) t o given

Δx(t) ≈ ∂ f

∂ x=xN (t )

u=uN (t )

w=wN (t )

Δx(t)

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' (

) ) )

+ ∂ f

∂ ux=xN (t )

u=uN (t )

w=wN (t )

Δu(t)

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' (

) ) )

+ ∂ f

∂ wx=xN (t )

u=uN (t )

w=wN (t )

Δw(t)

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' (

) ) )

 F(t) Δx(t)+ G(t) Δu(t)+ L(t) Δw(t), Δx t ( )o given

dim(x) = n × 1 dim(u) = m × 1 dim(w) = s × 1

dim(Δx) = n × 1 dim(Δu) = m × 1 dim(Δw) = s × 1

Nominal Equation"

Perturbation Equation"

Sensitivity to Small Perturbations"

•   Sensitivity to state perturbations: stability matrix, F , is square"

∂ x x=xN(t )

u=uN(t )

w=wN(t )

=

∂ f 1

∂ x 1

 ∂ f 1

∂ x n

∂ f n

∂ x 1  ∂ f n

∂ x n

"

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' ' ' ' ' x=x

N(t )

u=uN(t )

w=wN(t )

∂u x = xN(t )

u= uN(t )

w = w (t )

∂w x = xN(t )

u= uN(t )

w = w (t )

•   Sensitivity to control and disturbance perturbations is similar, but matrices, G and L , may not be square"

dim(F) = n × n

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•   Linear and nonlinear, varying and

–   Numerical integration ( time domain )"

–   State transition ( time domain )"

–   Transfer functions ( frequency domain )"

How Is System

Response Calculated?"

Numerical Integration :"

MATLAB Ordinary Differential Equation

•  Explicit Runge-Kutta Algorithm"

•  Numerical Differentiation Formula"

* http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/ref/ode23.html.!

Shampine, L F and M W Reichelt, "The MATLAB ODE Suite," SIAM Journal on Scientific Computing, Vol 18, 1997, pp 1-22.!

•  Adams-Bashforth-Moulton Algorithm"

•  Modified Rosenbrock Method"

•  Trapezoidal Rule"

•  Trapezoidal Rule w/Back Differentiation"

MATLAB Simulation of Linear and

Nonlinear Dynamic Systems"

•   MATLAB Main Script"

% Nonlinear and Linear Examples"

clear"

tspan = [0 10]; "

xo = [0, 10];"

[t1,x1 = ode23('NonLin',tspan,xo); "

xo = [0, 1];"

[t2,x2] = ode23('NonLin',tspan,xo); "

xo = [0, 10];"

[t3,x3] = "ode23('Lin',tspan,xo);"

xo = [0, 1];"

[t4,x4] = ode23('Lin',tspan,xo);"

"

subplot(2,1,1)"

plot(t1,x1(:,1),'k',t2,x2(:,1),'b',t3,x3(:,1),'r',t4,x4(:,1),'g')"

ylabel('Position'), grid"

subplot(2,1,2)"

plot(t1,x1(:,2),'k',t2,x2(:,2),'b',t3,x3(:,2),'r',t4,x4(:,2),'g')"

xlabel('Time'), ylabel('Rate'), grid"

•   Linear System"

function xdot = Lin(t,x)"

% Linear Ordinary Differential Equation"

% x(1) = Position"

% x(2) = Rate"

xdot = [x(2)"

-10*x(1) - x(2)];"

•   Nonlinear System"

function xdot = NonLin(t,x)"

% Nonlinear Ordinary Differential Equation"

% x(1) = Position"

% x(2) = Rate ""

xdot = [x(2)"

€

˙

x 1(t) = x2(t)

˙

x 2(t) = −10x1(t) − x2(t)

€

˙

x 1(t) = x2(t)

˙

x 2(t) = −10x1(t) + 0.8x1(t) − x2(t)

Comparison of Damped

3

(t) − x2(t)

Linear plus Stiffening Cubic Spring"

Linear plus Weakening Cubic Spring"

3

(t) − x2(t)

Displacement"

Spring" Damper"

Displacement"

Rate of Change"

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Linear and Stiffening

Cubic Springs: Small and Large Initial Conditions"

•   Linear and nonlinear responses are indistinguishable with small initial condition"

Linear and Weakening

Cubic Springs: Small and Large Initial Conditions"

Linear, Time-Varying (LTV)

Approximation of

Perturbation Dynamics

Stiffening Linear-Cubic Spring Example"

!   Nonlinear, time-invariant (NTI) equation"

x 1 (t) = f 1 = x 2 (t)

x 2 (t) = f 2 = −10x 1 (t) − 10x 1 3 (t) − x 2 (t)

!   Integrate equations to produce nominal path"

x1N(0)

x2N(0)

!

"

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f1N

f2N

!

"

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0

t f

x2N(t)

!

"

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!   Analytical evaluation of partial derivatives"

∂ f1

∂ x1

= 0; ∂ f1

∂ x2

= 1

∂ f2

∂ x1

= −10 − 30x1

N

2(t) ; ∂ f2

∂ x2

= −1

∂ u = 0; ∂ f1∂ w = 0

∂ u = 0; ∂ f2∂ w = 0

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Nominal (NTI) and Perturbation

xN(t) = f[xN(t)], xN(0) given

x1N(t) = x2N(t)

x2

N(t) = −10x1N(t) − 10x1N

3

(t) − x2N(t)

Δx(t) = F(t)Δx(t), Δx(0) given

Δx1(t)

Δx2(t)

"

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'

' =

− 10 + 30x12N

(t)

"

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' '

Δx1(t)

Δx2(t)

"

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' '

x1

N(0)

x2

N(0)

!

"

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& =

0 9

!

"

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&

!   Nonlinear, time-invariant (NTI) nominal equation"

!   Perturbations approximated by linear, time-varying (LTV) equation"

Δx1(0)

Δx2(0)

"

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' ' =

0 1

"

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'

Example"

Comparison of Approximate and Exact Solutions"

xN(t)

Δx(t)

xN(t)+ Δx(t) x(t)

Initial Conditions

x2N(0) = 9

x2N(t)+ Δx2(t) = 10

x2(t) = 10

xN(t)

Δx(t)

xN(t) + Δx(t)

x(t)

Suppose Nominal Initial

Condition is Zero"

•   Nominal solution remains at equilibrium"

•   Perturbation equation is linear and time-invariant (LTI)"

"

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'

"

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' '

"

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' '

Separation of the Equations of Motion into Longitudinal and

Lateral-Directional Sets

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Rigid-Body Equations of Motion (Scalar Notation)"

Translational Position "

•  Rate of change of

Angular Position"

Translational Velocity "

•  Rate of change of

Angular Velocity

(Ixy = I yz = 0) "

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

!

"

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=

u v w x y z p q r

φ θ ψ

!

"

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State Vector"

u = X / m − gsinθ + rv − qw

v = Y / m + gsinφ cosθ − ru + pw

w = Z / m + g cosφ cosθ + qu − pv

x I= cosθ cosψ( )u + − cosφ sinψ + sinφ sinθ cosψ( )v + sinφ sinψ + cosφ sinθ cosψ( )w

y I= cosθ sinψ( )u + cosφ cosψ + sinφ sinθ sinψ( )v + − sinφ cosψ + cosφ sinθ sinψ( )w

z I= − sinθ( )u + sinφ cosθ( )v + cosφ cosθ( )w

φ = p + qsinφ + r cosφ ( ) tanθ

θ = qcosφ − rsinφ



ψ = q sinφ + r cosφ ( ) secθ

p = I zz L + I xz N − I xz(I yy − I xx − I zz)p + I xz

2

+ I zz(I zz − I yy)

" $% r

2

( )

q = M − I(xx − I zz)pr − I xz p2

− r2

( )

" $% ÷ I yy

r = I xz L + I xx N − I xz(I yy − I xx − I zz)r + I xz

2

+ I xx(I xx − I yy)

" $% p

2

( )

Reorder the State Vector"

€

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

!

"

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&new

xLat−Dir

!

"

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& =

u w x z q

θ

v y p r

φ ψ

!

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the state are

longitudinal variables"

"

Second six elements of the state are lateral-directional variables"

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

!

"

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=

u v w x y z p q r

φ θ ψ

!

"

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Longitudinal Equations of Motion "

xI= cosθ cosψ ( ) u + − cosφ sinψ + sinφ sinθ cosψ ( ) v + sinφ sinψ + cosφ sinθ cosψ ( ) w = x3= f3

q = M − I (xx− Izz) pr − Ixz p2

− r2

x Lon (t) = f[x Lon (t), u Lon (t), w Lon (t)]

yI= cosθ sinψ ( ) u + cosφ cosψ + sinφ sinθ sinψ ( ) v + − sinφ cosψ + cosφ sinθ sinψ ( ) w = x8= f8

p = IzzL + IxzN − Ixz( Iyy− Ixx− Izz) p + Ixz

2

+ Izz( Izz− Iyy)

2

( ) = x9= f9

r = IxzL + IxxN − Ixz( Iyy− Ixx− Izz) r + Ixz

2

+ Ixx( Ixx− Iyy)

( p

2

( ) = x10= f10



Lateral-Directional Equations of Motion "

•   Dynamics of position, velocity, angle, and angular rate out of the vertical plane "

xLD(t) = f[xLD(t), uLD(t), wLD(t)]

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Sensitivity to Small Motions "

F(t) =

∂ f1

∂ 1

∂ f1

∂ 2

∂ f1

∂ 12

∂ f2

∂ 1

∂ f2

∂ 2 ∂ f2

∂ 12

∂ f12

∂ 1

∂ f12

∂ 2

∂ f12

∂ 12

"

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' ' ' ' ' ' ' '

=

∂ u

∂ w ∂ u

∂ψ

∂ w

∂ w ∂ w

∂ψ

∂ ψ 



ψ



ψ

∂ψ

"

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' ' ' ' ' ' '

•   Four (6 x 6) blocks distinguish longitudinal and lateral-directional effects "

F = FLon FLat−Dir Lon

FLon Lat−Dir FLat−Dir

"

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' '

Effects of longitudinal perturbations

on longitudinal motion"

Effects of longitudinal perturbations

on lateral-directional motion"

Effects of lateral-directional perturbations on longitudinal motion"

Effects of lateral-directional perturbations

on lateral-directional motion"

Sensitivity to Control Inputs"

G = GLon GLat−Dir

Lon

GLon Lat−Dir

GLat−Dir

"

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' '

Effects of longitudinal controls on longitudinal motion"

Effects of longitudinal controls on lateral-directional motion"

Effects of lateral-directional controls

on longitudinal motion"

Effects of lateral-directional controls on lateral-directional motion"

u(t) =

δE(t)

δT (t) δF(t) δA(t) δR(t) δSF(t)

"

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' ' ' ' ' ' ' '

Elevator, deg or rad Throttle, % Flaps, deg or rad Ailerons, deg or rad Rudder, deg or rad Side Force Panels, deg or rad

Δu(t) =

ΔδE(t) ΔδT (t) ΔδF(t) ΔδA(t) ΔδR(t) ΔδSF(t)

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'

( ( ( ( ( ( ( (

Decoupling Approximation

for Small Perturbations

from Steady, Level Flight

Restrict the Nominal Flight Path

longitudinal equations

lateral-directional motions

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

!

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&N

= xLon

xLat−Dir

!

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N

=

uN

wN

xN

zN

qN

θN

0 0 0 0 0 0

!

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uN= X / m − g sinθN− qNwN

wN = Z / m + g cosθN+ qNuN

xI

N = cosθ ( N) uN+ sinθ ( N) wN

zI N = − sinθ ( N) uN+ cosθ ( N) wN

qN= M

Iyy

θN= qN

xLat − Dir N = 0

xLat − Dir

N = 0

Nominal State Vector"

Lateral-directional perturbations need not be zero"

ΔxLat − Dir N≠ 0

ΔxLat − Dir N≠ 0

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Restrict the Nominal Flight Path to

Steady, Level Flight "

•  Calculate conditions for trimmed (equilibrium) flight "

–  See Flight Dynamics and FLIGHTprogram for a

solution method "

0 = X / m − gsinθN− qNwN

0 = Z / m + g cosθN+ qNuN

VN = cosθ ( N) uN + sinθ ( N) wN

0 = − sinθ ( N) uN+ cosθ ( N) wN

0 = M

Iyy

0 = qN

Trimmed State Vector is constant"

•  Specify nominal airspeed (VN) and altitude (hN = –zN)

u w x z q

θ

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' ' ' ' ' ' '

Trim

=

uTrim

wTrim

VN( t − t0)

zN

0

θTrim

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' ' ' ' ' ' ' '

Small Perturbation Effects are

Level Flight "

•   Assume the airplane is symmetric and its nominal path

is steady, level flight"

–   Small longitudinal and lateral-directional perturbations are approximately uncoupled from each other "

–   (12 x 12) system is "

•   block diagonal"

•   constant, i.e., linear, time-invariant (LTI)"

•   decoupled into two separate (6 x 6) systems "

F = FLon 0

0 FLat−Dir

"

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'

0 GLat−Dir

"

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'

0 LLat−Dir

"

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' '

Δx Lon (t) = F Lon Δx Lon (t) + G Lon Δu Lon (t) + L Lon Δw Lon (t)

ΔxLon =

Δx1

Δx2

Δx3

Δx4

Δx5

Δx6

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'

Lon

=

Δu Δw Δx Δz Δq

Δθ

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' ' ' ' ' ' '

Perturbation Model"

ΔuLon=

ΔδT

Δδ E

Δδ F

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'

( ( (

ΔwLon=

Δuwind

Δwwind

Δqwind

"

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' ' '

Dynamic Equation"

State Vector"

Control "

Vector"

Disturbance"

Vector"

(6 x 6) LTI Lateral-Directional

Perturbation Model"

ΔxLat − Dir(t) = FLat − DirΔxLat − Dir(t) + GLat − DirΔuLat − Dir(t) + LLat − DirΔwLat − Dir(t)

ΔxLat − Dir=

Δx1

Δx2

Δx3

Δx4

Δx5

Δx6

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'

( ( ( ( ( ( ( (

Lat − Dir

=

Δv Δy Δp Δr

Δφ Δψ

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'

( ( ( ( ( ( ( (

ΔuLat − Dir=

Δδ A

Δδ R ΔδSF

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' (

) ) )

ΔwLon=

Δvwind

Δpwind

Δrwind

"

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' ' '

Dynamic Equation"

State Vector"

Control "

Vector"

Disturbance" Vector"

Trang 9

Frequency Domain

Description of LTI

System Dynamics

Fourier Transform of a Scalar Variable "

F Δx(t) [ ] = Δx( jω ) = Δx(t)e− jωt

−∞

∞

∫ dt, ω = frequency, rad / s

Δx(t) : real variable

Δx( j ω ) : complex variable

= a( ω )+ jb( ω )

= A( ω )e jϕ (ω )

A : amplitude

ϕ : phase angle

jω : Imaginary operator, rad/s

Fourier Transform of a

Scalar Variable "

Δx(t)

Δx( jω ) = a(ω ) + jb(ω )

Laplace Transform of

a Scalar Variable "

•   Laplace transformation from time domain to frequency domain !

0

∞

s = σ + jω

= Laplace (complex) operator, rad/s

Δx(t) : real variable Δx(s) : complex variable

= a(s)+ jb(s)

= A(s)ejϕ(s )

Trang 10

Laplace Transformation is a

Linear Operation "

L Δx [ 1 (t)+ Δx 2 (t) ] = L Δx [ 1 (t) ] + L Δx [ 2 (t) ] = Δx 1 (s)+ Δx 2 (s)

•   Sum of Laplace transforms !

•   Multiplication by a constant !

Laplace Transforms of

Vectors and Matrices "

L Δx(t) [ ] = Δx(s) =

Δx1(s)

Δx2(s)

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' ' '

L A(t) [ ] = A(s) =

a11(s) a12(s)

a21(s) a22(s)

!

"

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a Dynamic System "

Δx(t) = F Δx(t) + G Δu(t) + LΔw(t)

•   System equation !

sΔx(s) − Δx(0) = F Δx(s) + GΔ u(s) + LΔw(s)

dim(Δx) = (n × 1) dim(Δu) = (m × 1) dim(Δw) = (s × 1)

a Dynamic System "

dynamic equation!

sΔx(s) − FΔ x(s) = Δx(0) + GΔ u(s)+ LΔw(s)

sI − F

[ ] Δx(s) = Δx(0)+ GΔ u(s)+ LΔw(s)

•   F to left, I.C to right!

•   Combine terms!

•   Multiply both sides by inverse of (sI – F)!

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