Mae 331 Lecture 13
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Transcript of Mae 331 Lecture 13
Time ResponseRobert Stengel, Aircraft Flight Dynamics
MAE 331, 2010
• Time-domain analysis
– Transient response to initial conditions and inputs
– Steady-state (equilibrium) response
– Continuous- and discrete-time models
– Phase-plane plots
– Response to sinusoidal input
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 (LTI)Longitudinal Model
! !V (t)
! !" (t)
! !q(t)
! !#(t)
$
%
&&&&&
'
(
)))))
=
*DV
*g *Dq
*D#
LV
VN
0Lq
VN
L#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)
$
%
&&&
'
(
)))
• Steady, level flight
• Simplified control effects
• Neglect disturbance effects
• What can we do with it?– Integrate equations to obtain time histories of initial condition, control, and
disturbance effects
– Determine modes of motion
– Examine steady-state conditions
– Identify effects of parameter variations
– Define frequency response
Gain insights aboutsystem dynamics
Linear, Time-Invariant
System Model
• General model contains– Dynamic equation (ordinary differential equation)
– Output equation (algebraic transformation)
!˙ x (t) = F!x(t) + G!u(t) + L!w(t), !x(to) given
!y(t) = Hx!x(t) + Hu!u(t) + Hw!w(t)
• State and output dimensions need not be the same
dim !x(t)[ ] = n "1( )
dim !y(t)[ ] = r "1( )
System Response to Inputsand Initial Conditions
• Solution of the linear, time-invariant (LTI) dynamic model
!!x(t) = F!x(t) +G!u(t) + L!w(t), !x(to ) given
!x(t) = !x(to ) + F!x(" ) +G!u(" ) + L!w(" )[ ]to
t
# d"
• ... has two parts
– Unforced (homogeneous) response to initial conditions
– Forced response to control and disturbance inputs
Response toInitial Conditions
Unforced Response
to Initial Conditions
• The state transition matrix, !, propagates thestate from to to t by a single multiplication
!x(t) = !x(to) + F!x(" )[ ]
to
t
# d" = eF t$ to( )!x(t
o) = % t $ t
o( )!x(to )
eF t! to( )
= Matrix Exponential
= I + F t ! to( ) +1
2!F t ! to( )"# $%
2
+1
3!F t ! to( )"# $%
3
+ ...
= & t ! to( ) = State Transition Matrix
• Neglecting forcing functions
Initial-Condition Response
via State Transition
! = I + F "t( ) +1
2!F "t( )#$ %&
2
+1
3!F "t( )#$ %&
3
+ ...
!x(t1) = " t
1# t
o( )!x(to )
!x(t2) = " t
2# t
1( )!x(t1)
!x(t3) = " t
3# t
2( )!x(t2 )
…
• If (tk+1 – tk) = !t = constant,state transition matrix isconstant
!x(t1) = " #t( )!x(t
o) = "!x(t
o)
!x(t2) = "!x(t
1) = "
2!x(t
o)
!x(t3) = "!x(t
2) = "
3!x(t
o)
…
• Propagation of "x
Discrete-Time Dynamic Model
!x(tk+1) = !x(t
k) + F!x(" ) +G!u(" ) + L!w(" )[ ]
tk
tk+1
# d"
!x(tk+1) = " #t( )!x(t
k) +" #t( ) e
$F % $ tk( )&'
()
tk
tk+1
* d% G!u(tk) + L!w(t
k)[ ]
= "!x(tk) + +!u(t
k) + ,!w(t
k)
• Response to continuous controls and disturbances
• Response to piecewise-constant controls and disturbances
! = eF" t
# = eF" t
$ I( )F$1G
% = eF" t
$ I( )F$1L
Ordinary Difference Equation
Control- and Disturbance-Effect
Matrices
! = eF" t # I( )F#1
G
= I #1
2!F"t +
1
3!F2"t 2 #
1
4!F3"t 3 + ...$
%&'()G"t
* = eF" t # I( )F#1
L
= I #1
2!F"t +
1
3!F2"t 2 #
1
4!F3"t 3 + ...$
%&'()L"t
!x(tk) = "!x(t
k#1) + $!u(t
k#1) + %!w(t
k#1)
• As !t becomes very small
!" t#0
$ #$$ I + F"t( )
%" t#0
$ #$$ G"t
&" t#0
$ #$$ L"t
Discrete-Time Response to Inputs
!x(t1) = "!x(t
o) + #!u(t
o) + $!w(t
o)
!x(t2) = "!x(t
1) + #!u(t
1) + $!w(t
1)
!x(t3) = "!x(t
2) + #!u(t
2) + $!w(t
2)
!
• Propagation of "x, with constant !, #, and $
!t = tk+1
" tk
Continuous- and Discrete-Time
Short-Period System Matrices
• !t = 0.1 s
• !t = 0.5 s
F =!1.2794 !7.9856
1 !1.2709
"
#$
%
&'
G =!9.069
0
"
#$
%
&'
L =!7.9856
!1.2709
"
#$
%
&'
! =0.845 "0.694
0.0869 0.846
#
$%
&
'(
) ="0.84
"0.0414
#
$%
&
'(
* ="0.694
"0.154
#
$%
&
'(
! =0.0823 "1.475
0.185 0.0839
#
$%
&
'(
) ="2.492
"0.643
#
$%
&
'(
* ="1.475
"0.916
#
$%
&
'(
• Continuous-time(“analog”) system
• Discrete-time (“digital”) system
!t has a large effecton the “digital” model
!t = tk+1
" tk
! =0.987 "0.079
0.01 0.987
#
$%
&
'(
) ="0.09
"0.0004
#
$%
&
'(
* ="0.079
"0.013
#
$%
&
'(
• !t = 0.01 s
Example: Aerodynamic
Angle, Linear Velocity, and
Angular Rate Perturbations
Learjet 23
MN = 0.3, hN = 3,050 m
VN = 98.4 m/s
!" ! !wVN
; !" = 1°# !w = 0.01745 $ 98.4 = 1.7m s
!% ! !vVN
; !% = 1°# !v = 0.01745 $ 98.4 = 1.7m s
!p = 1° / s; !wwingtip = !p b2
"#
$% = 0.01745 & 5.25 = 0.09m s
!q = 1° / s; !wnose = !q xnose ' xcm[ ] = 0.01745 & 6.4 = 0.11m s
!r = 1° / s; !vnose = !r xnose ' xcm[ ] = 0.01745 & 6.4 = 0.11m s
• Aerodynamic angle and linear velocity perturbations
• Angular rate and linear velocity perturbations
Example: Continuous- andDiscrete-Time Models
! !q
! !"
#
$%%
&
'((=
)1.3 )8
1 )1.3
#
$%
&
'(
!q
!"
#
$%%
&
'((+
)9.1
0
#
$%
&
'(!*E
!!p
! !"
#
$%%
&
'(() *1.2 0
1 0
#
$%
&
'(
!p
!"
#
$%%
&
'((+
2.3
0
#
$%
&
'(!+A
!!r
! !"
#
$%%
&
'(() *0.11 1.9
*1 *0.16
#
$%
&
'(
!r!"
#
$%%
&
'((+
*1.10
#
$%
&
'(!+R
• Note individual acceleration and difference sensitivities to state and control perturbations
Short Period
Roll-Spiral
DutchRoll
!qk+1
!" k+1
#
$
%%
&
'
((=
0.85 )0.7
0.09 0.85
#
$%
&
'(
!qk
!" k
#
$
%%
&
'
((+
)0.84
)0.04
#
$%
&
'(!*Ek
!pk+1!"k+1
#
$%%
&
'(() 0.89 0
0.09 1
#
$%
&
'(
!pk!"k
#
$%%
&
'((+
0.24
*0.01
#
$%
&
'(!+Ak
!rk+1
!"k+1
#
$%%
&
'(() 0.98 0.19
*0.1 0.97
#
$%
&
'(
!rk
!"k
#
$%%
&
'((+
*0.110.01
#
$%
&
'(!+Rk
Differential Equations Produce
State Rates of Change
Difference Equations
Produce State Increments
Learjet 23
MN = 0.3, hN = 3,050 m
VN = 98.4 m/s !t = 0.1sec
Initial-Condition Response
• Doubling the initial condition doubles the output
!!x1
!!x2
"
#
$$
%
&
''=
(1.2794 (7.9856
1 (1.2709
"
#$
%
&'
!x1
!x2
"
#
$$
%
&
''+
(9.069
0
"
#$
%
&'!)E
!y1
!y2
"
#
$$
%
&
''=
1 0
0 1
"
#$
%
&'
!x1
!x2
"
#
$$
%
&
''+
0
0
"
#$
%
&'!)E
% Short-Period Linear Model - Initial Condition
F = [-1.2794 -7.9856;1. -1.2709];
G = [-9.069;0];
Hx = [1 0;0 1];
sys = ss(F, G, Hx,0);
xo = [1;0];
[y1,t1,x1] = initial(sys, xo);
xo = [2;0];
[y2,t2,x2] = initial(sys, xo);
plot(t1,y1,t2,y2), grid
figure
xo = [0;1];
initial(sys, xo), grid
Angle of AttackInitial Condition
Pitch RateInitial Condition
Phase Plane Plots
State (“Phase”)-Plane Plots
• Cross-plot of one component againstanother
• Time or frequency not shown explicitly
% 2nd-Order Model - Initial Condition Response
clear
z = 0.1; % Damping ratio
wn = 6.28; % Natural frequency, rad/s
F = [0 1;-wn^2 -2*z*wn];
G = [1 -1;0 2];
Hx = [1 0;0 1];
sys = ss(F, G, Hx,0);
t = [0:0.01:10];
xo = [1;0];
[y1,t1,x1] = initial(sys, xo, t);
plot(t1,y1)
grid on
figure
plot(y1(:,1),y1(:,2))
grid on
!!x1
!!x2
"
#$$
%
&''(
0 1
)*n
2 )2+*n
"
#$$
%
&''
!x1
!x2
"
#$$
%
&''+
1 )10 2
"
#$
%
&'
!u1
!u2
"
#$$
%
&''
Dynamic Stability Changesthe State-Plane Spiral
• Damping ratio = 0.1 • Damping ratio = 0.3 • Damping ratio = –0.1
Superposition ofLinear Responses
Step Response
• Stability, speed of response,and damping areindependent of the initialcondition or input
• Doubling the inputdoubles the output
!!x1
!!x2
"
#
$$
%
&
''=
(1.2794 (7.9856
1 (1.2709
"
#$
%
&'
!x1
!x2
"
#
$$
%
&
''+
(9.069
0
"
#$
%
&'!)E
!y1
!y2
"
#
$$
%
&
''=
1 0
0 1
"
#$
%
&'
!x1
!x2
"
#
$$
%
&
''+
0
0
"
#$
%
&'!)E
% Short-Period Linear Model - Step
F = [-1.2794 -7.9856;1. -1.2709];
G = [-9.069;0];
Hx = [1 0;0 1];
sys = ss(F, -G, Hx,0); % (-1)*Step
sys2 = ss(F, -2*G, Hx,0); % (-1)*Step
% Step response
step(sys, sys2), grid
!"E t( ) =0, t < 0
#1, t $ 0
%&'
('
Superposition of Linear Responses
• Stability, speed of response, and damping areindependent of the initial condition or input
% Short-Period Linear Model - Superposition
F = [-1.2794 -7.9856;1. -1.2709];
G = [-9.069;0];
Hx = [1 0;0 1];
sys = ss(F, -G, Hx,0); % (-1)*Step
xo = [1; 0];
t = [0:0.2:20];
u = ones(1,length(t));
[y1,t1,x1] = lsim(sys,u,t,xo);
[y2,t2,x2] = lsim(sys,u,t);
u = zeros(1,length(t));
[y3,t3,x3] = lsim(sys,u,t,xo);
plot(t1,y1,t2,y2,t3,y3), grid
!!x1
!!x2
"
#
$$
%
&
''=
(1.2794 (7.9856
1 (1.2709
"
#$
%
&'
!x1
!x2
"
#
$$
%
&
''+
(9.069
0
"
#$
%
&'!)E
!y1
!y2
"
#
$$
%
&
''=
1 0
0 1
"
#$
%
&'
!x1
!x2
"
#
$$
%
&
''+
0
0
"
#$
%
&'!)E
Example: Continuous- andDiscrete-Time LTILongitudinal Models
Short Period
Phugoid
! !V! !"
#
$%%
&
'(() *0.02 *9.8
0.02 0
#
$%
&
'(
!V!"
#
$%%
&
'((+
4.7
0
#
$%
&
'(!+T
! !q
! !"
#
$%%
&
'((=
)1.3 )8
1 )1.3
#
$%
&
'(
!q
!"
#
$%%
&
'((+
)9.1
0
#
$%
&
'(!*E
!qk+1
!" k+1
#
$
%%
&
'
((=
0.85 )0.7
0.09 0.85
#
$%
&
'(
!qk
!" k
#
$
%%
&
'
((+
)0.84
)0.04
#
$%
&
'(!*Ek
!Vk+1
!"k+1
#
$%%
&
'((=
1 )0.980.002 1
#
$%
&
'(
!Vk
!"k
#
$%%
&
'((+
0.47
0.0005
#
$%
&
'(!*Tk
Learjet 23MN = 0.3, hN = 3,050 mVN = 98.4 m/s
Differential Equations Produce
State Rates of Change
Difference Equations
Produce State Increments
!t = 0.1sec
Example: Superposition ofContinuous- and Discrete-Time
Longitudinal Models
Phugoid and Short Period
! !V! !"
! !q
! !#
$
%
&&&&&
'
(
)))))
=
*0.02 *9.8 0 0
0.02 0 0 1.3
0 0 *1.3 *8*0.002 0 1 *1.3
$
%
&&&&
'
(
))))
!V!"
!q
!#
$
%
&&&&&
'
(
)))))
+
4.7 0
0 0
0 *9.10 0
$
%
&&&&
'
(
))))
!+T!+E
$
%&
'
()
!Vk+1!" k+1
!qk+1!# k+1
$
%
&&&&&
'
(
)))))
=
1 *0.98 *0.002 *0.060.002 1 0.006 0.12
0.0001 0 0.84 *0.69*0.0002 0.0001 0.09 0.84
$
%
&&&&
'
(
))))
!Vk!" k
!qk!# k
$
%
&&&&&
'
(
)))))
+
0.47 0.0005
0.0005 *0.0020 *0.840 *0.04
$
%
&&&&
'
(
))))
!+Tk!+Ek
$
%&&
'
())
Learjet 23MN = 0.3, hN = 3,050 mVN = 98.4 m/s
Differential EquationsProduce State Rates ofChange
DifferenceEquations ProduceState Increments
!t = 0.1sec
Equilibrium Response
Equilibrium Response
!!x(t) = F!x(t) +G!u(t) + L!w(t)
0 = F!x(t) +G!u(t) + L!w(t)
!x* = "F"1G!u *+L!w *( )
• Dynamic equation
• At equilibrium, the state is unchanging
• Denoting constant values by (.)*
Steady-State Condition• If the system is also stable, an equilibrium point
is a steady-state point, i.e.,– Small disturbances decay to the equilibrium condition
F =f11
f12
f21
f22
!
"
##
$
%
&&; G =
g1
g2
!
"
##
$
%
&&; L =
l1
l2
!
"
##
$
%
&&
!x1*
!x2*
"
#$$
%
&''= (
f22
( f12
( f21
f11
"
#$$
%
&''
f11f22( f
12f21( )
g1
g2
)
*++
,
-..!u *+
l1
l2
)
*++
,
-..!w *
"
#$$
%
&''
• 2nd-order example
sI ! F = " s( ) = s2 + f11+ f
22( )s + f11f22! f
12f21( )
= s ! #1( ) s ! #
2( ) = 0
Re #i( ) < 0
System Matrices
Equilibrium Response
Requirement forStability
Equilibrium Response of
Approximate Phugoid Model
!xP* = "F
P
"1G
P!u
P*+L
P!w
P*( )
!V *
!" *#
$%%
&
'((= )
0VN
LV
)1g
VNDV
gLV
#
$
%%%%%
&
'
(((((
T*T
L*T
VN
#
$
%%%
&
'
(((!*T *
+
DV
)LV
VN
#
$
%%%
&
'
(((!V
W
*
+
,--
.--
/
0--
1--
• Equilibrium state with constant thrust and wind perturbations
Steady-State Response of
Approximate Phugoid Model
!V *= "
L#T
LV
!#T *+ !V
W
*
!$ * =1
gT#T + L#T
DV
LV
%
&'(
)*!#T *
• With L!T ~ 0, steady-state velocity depends only on thehorizontal wind
• Constant thrust produces steady climb rate
• Corresponding step response, with L!T = 0
Equilibrium Response ofApproximate Short-Period Model
!xSP* = "F
SP
"1G
SP!u
SP*+L
SP!w
SP*( )
!q*
!" *
#
$%%
&
'((= )
L"
VN
M"
1 )Mq
#
$
%%%
&
'
(((
L"
VN
Mq+ M"
*+,
-./
M0E
)L0E
VN
#
$
%%%
&
'
(((!0E* )
M"
)L"
VN
#
$
%%%
&
'
(((!"
W
*
1
233
433
5
633
733
• Equilibrium state with constant elevator and wind perturbations
Steady-State Response ofApproximate Short-Period Model
• Steady pitch rate and angle of attack are not zero
• Vertical wind reorients angle of attack
!q* = "
L#
VN
M$E
%&'
()*
L#
VN
Mq+ M#
%&'
()*
!$E*
!# *= "
M$E( )L#
VN
Mq+ M#
%&'
()*
!$E + !#W
*
with L!E = 0
Scalar Frequency Response
Speed Control ofDirect-Current Motor
u(t) = Ce(t)
where
e(t) = yc (t) ! y(t)
• Control Law (C = Control Gain)
Angular Rate
Characteristics
of the Motor
• Simplified Dynamic Model
– Rotary inertia, J, is the sum of motor and load
inertias
– Internal damping neglected
– Output speed, y(t), rad/s, is an integral of thecontrol input, u(t)
– Motor control torque is proportional to u(t)
– Desired speed, yc(t), rad/s, is constant
Model of Dynamics
and Speed Control
• Dynamic equation
y(t) =1
Ju(t)dt
0
t
! =C
Je(t)dt
0
t
! =C
Jyc (t) " y(t)[ ]dt
0
t
!
dy(t)
dt=u(t)
J=Ce(t)
J=C
Jyc (t) ! y(t)[ ], y 0( ) given
• Integral of the equation, with y(0) = 0
• Direct integration of yc(t)
• Negative feedback of y(t)
Step Response of
Speed Controller
y(t) = yc 1! e!
C
J
"#$
%&'t(
)**
+
,--= yc 1! e
.t() +, = yc 1! e! t /(
)*+,-
• where– " = –C/J = eigenvalue or
root of the system (rad/s)
– # = J/C = time constant ofthe response (sec)
Step input :
yC (t) =0, t < 0
1, t ! 0
"#$
%$
• Solution of the integral,
with step commandyct( ) =
0, t < 0
1, t ! 0
"#$
%$
Angle Control of a DC Motor
• Closed-loop dynamic equation, with y(t) = I2 x(t)
u(t) = c1yc (t) ! y1(t)[ ]! c2y2(t)
!x1(t)
!x2(t)
!
"
##
$
%
&&=
0 1
'c1/ J 'c
2/ J
!
"##
$
%&&
x1(t)
x2(t)
!
"
##
$
%
&&+
0
c1/ J
!
"##
$
%&&yc
• Control law with angle and angular rate feedback
!n= c
1J ; " = c
2J( ) 2!n
c1 /J = 1
c2 /J = 0, 1.414, 2.828% Step Response of Damped
Angle Control
F1 = [0 1;-1 0];
G1 = [0;1];
F1a = [0 1;-1 -1.414]; F1b = [0 1;-1 -2.828];
Hx = [1 0;0 1];
Sys1 = ss(F1,G1,Hx,0); Sys2 = ss(F1a,G1,Hx,0);
Sys3 = ss(F1b,G1,Hx,0);
step(Sys1,Sys2,Sys3)
Step Response of Angle Controller,
with Angle and Rate Feedback
• Single natural frequency, three damping ratios
Angle Response to a Sinusoidal
Angle Command
Amplitude Ratio (AR) =ypeak
yC peak
Phase Angle = !360"tpeak
Period, deg
• Output wavelags behind theinput wave
• Input and outputamplitudesdifferent
Effect of Input Frequency on Output
Amplitude and Phase Angle
• With low inputfrequency, inputand outputamplitudes areabout the same
• Lag of angleoutput oscillation,compared to input,is small
• Rate oscillation“leads” angleoscillation by ~90deg
yc(t) = sin t / 6.28( ), deg !
n= 1 rad / s
" = 0.707
At Higher Input Frequency, Phase
Angle Lag Increasesyc(t) = sin t( ), deg
At Even Higher Frequency,
Amplitude Ratio Decreases and
Phase Lag Increasesyc(t) = sin 6.28t( ), deg
Angle and RateResponse of a
DC Motor overWide Input-FrequencyRange
! Long-term responseof a dynamic systemto sinusoidal inputsover a range offrequencies
! Determineexperimentally fromtime response or
! Compute the Bodeplot of the system!stransfer functions(TBD)
Very low damping
Moderate damping
High damping
Next Time:Root Locus Analysis