Time Response of Linear, Time-Invariant (LTI) Systemsstengel/MAE331Lecture14.pdf2nd-Order...
Transcript of Time Response of Linear, Time-Invariant (LTI) Systemsstengel/MAE331Lecture14.pdf2nd-Order...
Time Response of Linear, Time-Invariant (LTI) Systems!
Robert Stengel, Aircraft Flight Dynamics!MAE 331, 2016
•! Methods of time-domain analysis–! Continuous- and discrete-time models–! Transient response to initial conditions and inputs–! Steady-state (equilibrium) response–! Phase-plane plots–! Response to sinusoidal input
Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html
http://www.princeton.edu/~stengel/FlightDynamics.html
Reading:!Flight Dynamics!298-313, 338-342!
Airplane Stability and Control!Sections 11.1-11.12!
1
Learning Objectives
Stability Augmentation!Chapter 20, Airplane Stability and
Control, Abzug and Larrabee!•! What are the principal subject and scope of the
chapter?!•! What technical ideas are needed to understand
the chapter?!•! During what time period did the events covered
in the chapter take place?!•! What are the three main "takeaway" points or
conclusions from the reading?!•! What are the three most surprising or
remarkable facts that you found in the reading?!2
Review Questions!!! Why can we normally reduce the 6th-order lateral-
directional equations to 4th-order without compromising accuracy?!
!! What are the lateral-directional modes of motion?!!! What variables are in the stability-axis 4th-order
equations?!!! Why are the stability-axis equations useful?!!! Why are most airplanes symmetric about the body-
axis vertical plane?!!! What is a “vortical wind”?!!! Under what circumstances can the 4th-order set be
replaced by two 2nd-order sets?!!! What benefits accrue from the separation, and what
are the limitations?! 3
Assignment #7"due: End of day, Dec 9, 2016!
•! Simulated flight test!•! Assess the aircraft's response at various
flight conditions using the AeroModel.m and DataTable.m files created in the last assignment !
•! 3-member teams!4
2nd-Order Short-Period (LTI) Model
•! What can we do with it?–! Integrate equations to obtain time histories of initial
condition, control, and disturbance effects–! Examine steady-state conditions–! Identify effects of parameter variations–! Determine modes of motion–! Define frequency response
Gain insights about system dynamics
5
! !q t( )! !" t( )
#
$%%
&
'((=
Mq M"
1) Lq VN*+,
-./ ) L"
VN
#
$
%%%
&
'
(((
!q t( )!" t( )
#
$%%
&
'((+
M0E
)L0EVN
#
$
%%%
&
'
(((!0E t( )
+M"
)L"VN
#
$
%%%
&
'
(((!"wind t( )
Linear, Time-Invariant (LTI) System Model
Dynamic equation (ordinary differential equation)
!!x(t) = F!x(t)+G!u(t)+L!w(t), !x(to ) given
State and output dimensions need not be the same
dim !x(t)[ ] = n "1( )dim !y(t)[ ] = r "1( ) 6
!y(t) = Hx!x(t)+Hu!u(t)+Hw!w(t)Output equation (algebraic transformation)
System Response to Inputs and 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
7
Response to Initial Conditions!
8
Unforced Response to Initial Conditions
The state transition matrix, !!, propagates the state from to to t by a single multiplication
!x(t) = !x(to )+ F!x(" )[ ]d"to
t
# = eF t$to( )!x(to ) = %% t $ to( )!x(to )
eF t!to( ) =Matrix Exponential
= I+ F t ! to( ) + 12!F t ! to( )"# $%
2+ 13!F t ! to( )"# $%
3+ ...
= && t ! to( ) = State TransitionMatrix
Neglecting forcing functions
9
Initial-Condition Response via State Transition
!! = I + F "t( ) + 12!F "t( )#$ %&
2+13!F "t( )#$ %&
3+ ...
!x(t1)="" t1 # to( )!x(to )!x(t2 )="" t2 # t1( )!x(t1)!x(t3)="" t3 # t2( )!x(t2 )!
If (tk+1 – tk) = !t = constant, state transition matrix is
constant
!x(t1)="" #t( )!x(to )=""!x(to )!x(t2 )=""!x(t1)=""
2!x(to )!x(t3)=""!x(t2 )=""
3!x(to )…
Incremental propagation of ""x
Propagation is exact10
Discrete-Time Dynamic Model
!x(tk+1)= !x(tk )+ F!x(" )+G!u(" )+L!w(" )[ ]d"tk
tk+1
#
!x(tk+1) = "" # t( )!x(tk )+"" # t( ) e$F % $tk( )&' ()d%tk
tk+1
* G!u(tk )+L!w(tk )[ ]= ""!x(tk )+ ++!u(tk )+ ,,!w(tk )
Response to continuous controls and disturbances
Response to piecewise-constant controls and disturbances
With piecewise-constant inputs, control and disturbance effects taken outside the integral
Discrete-time model of continuous system = Sampled-data model
11
Sampled-Data Control- and Disturbance-Effect Matrices
!! = eF" t # I( )F#1G
= I# 12!F" t + 1
3!F2" t 2 # 1
4!F3" t 3 + ...$
%&'()G" t
!x(tk ) = ""!x(tk#1)+ $$!u(tk#1)+ %%!w(tk#1)
As !!t becomes very small
!! " t#0$ #$$ I+ F" t( )%% " t#0$ #$$ G" t&& " t#0$ #$$ L" t 12
!! = eF" t # I( )F#1L
= I# 12!F" t + 1
3!F2" t 2 # 1
4!F3" t 3 + ...$
%&'() L" t
Discrete-Time Response to Inputs
!x(t1)=""!x(to )+##!u(to )+$$!w(to )!x(t2 )=""!x(t1)+##!u(t1)+$$!w(t1)!x(t3)=""!x(t2 )+##!u(t2 )+$$!w(t2 )!
Propagation of ""x, with constant !!, ##, and $$
!t = tk+1 " tk
13
Continuous- and Discrete-Time Short-Period System Matrices
•! !!t = 0.1 s
•! !!t = 0.5 s
F = !1.2794 !7.98561 !1.2709
"
#$
%
&'
G = !9.0690
"
#$
%
&'
L = !7.9856!1.2709
"
#$
%
&'
!! = 0.845 "0.6940.0869 0.846
#
$%
&
'(
)) = "0.84"0.0414
#
$%
&
'(
** = "0.694"0.154
#
$%
&
'(
!! = 0.0823 "1.4750.185 0.0839
#
$%
&
'(
)) = "2.492"0.643
#
$%
&
'(
** = "1.475"0.916
#
$%
&
'(
•! Continuous-time ( analog ) system
•! Sampled-data ( digital ) system
!!t has a large effect on the digital model
!t = tk+1 " tk
!! = 0.987 "0.0790.01 0.987
#
$%
&
'(
)) = "0.09"0.0004
#
$%
&
'(
** = "0.079"0.013
#
$%
&
'(
•! !t = 0.01 s
14
Continuous- and Discrete-Time Short-Period Models
! !q t( )! !" t( )
#
$%%
&
'((= )1.3 )8
1 )1.3#
$%
&
'(
!q t( )!" t( )
#
$%%
&
'((+ )9.1
0#
$%
&
'(!*E t( )
Note individual acceleration and difference sensitivities to state and control perturbations
!qk+1!" k+1
#
$%%
&
'((= 0.85 )0.7
0.09 0.85#
$%
&
'(
!qk!" k
#
$%%
&
'((+ )0.84
)0.04#
$%
&
'(!*Ek
Differential Equations Produce State Rates of Change
Difference Equations Produce State Increments
Learjet 23MN = 0.3, hN = 3,050 mVN = 98.4 m/s
!t = 0.1sec
15
!!x t( ) = F!x t( ) +G!u t( )!y t( ) = Hx!x t( ) +Hu!u t( )
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 Attack Initial
Condition
Pitch Rate Initial
Condition
16
Commercial Aircraft of the 1940s•! Pre-WWII designs, reciprocating engines•! Development enhanced by military transport and bomber versions
–! Douglas DC-4 (adopted as C-54)–! Boeing Stratoliner 377 (from B-29, C-97)–! Lockheed Constellation 749 (from C-69)
!!iiss""rriiccaall FFaacc""iiddss
17
Commercial Propeller-Driven Aircraft of the 1950s•! Reciprocating and turboprop engines•! Douglas DC-6, DC-7, Lockheed Starliner 1649, Vickers
Viscount, Bristol Britannia, Lockheed Electra 188
18
Bristol Brabazon“Jumbo Turboprop”
Brabazon Committee study for a post-WWII jet-powered mailplane with small passenger compartment
deHavilland Vampire, 1943
19
deHavilland Swallow, 1946
Commercial Jets of the 1950s•! Low-bypass ratio turbojet
engines•! deHavilland DH 106 Comet
(1951)–! 1st commercial jet transport–! engines buried in wings–! early takeoff accidents
•! Boeing 707 (1957)–! derived from USAF KC-135–! engines on pylons below
wings–! largest aircraft of its time
•! Sud-Aviation Caravelle (1959)–! 1st aircraft with twin aft-
mounted engines
deHavilland Comet
Boeing 707
Sud-Aviation Caravelle
20
Superposition of Linear Responses!
21
Step Response
•! Stability, speed of response, and damping are independent of the initial condition or input
Doubling the input 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 - 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
Step Input
!"E t( ) = 0, t < 0#1, t $ 0
%&'
('
22
Superposition of Linear Step Responses
Stability, speed of response, and damping are independent of the initial condition or input
!!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 - 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
23
2nd-Order Comparison: Continuous- and Discrete-Time LTI
Longitudinal Models
Short Period
Phugoid
! !V t( )! !" t( )
#
$%%
&
'(() *0.02 *9.8
0.02 0#
$%
&
'(
!V t( )!" t( )
#
$%%
&
'((+ 4.7
0#
$%
&
'(!+T t( )
! !q t( )! !" t( )
#
$%%
&
'((= )1.3 )8
1 )1.3#
$%
&
'(
!q t( )!" t( )
#
$%%
&
'((+ )9.1
0#
$%
&
'(!*E t( )
!qk+1!" k+1
#
$%%
&
'((= 0.85 )0.7
0.09 0.85#
$%
&
'(
!qk!" k
#
$%%
&
'((+ )0.84
)0.04#
$%
&
'(!*Ek
!Vk+1!" k+1
#
$%%
&
'((= 1 )0.98
0.002 1#
$%
&
'(
!Vk!" k
#
$%%
&
'((+ 0.47
0.0005#
$%
&
'(!*Tk
Differential Equations Produce State Rates of Change
Difference Equations Produce State Increments
!t = 0.1sec
24
Short Period
Phugoid
Equilibrium Response!
25
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"1 G!u *+L!w *( )
Dynamic equation
At equilibrium, the state is unchanging
Constant values denoted by (.)*
26
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 f12f21 f22
!
"##
$
%&&; G =
g1g2
!
"##
$
%&&; L =
l1l2
!
"##
$
%&&
!x1 *!x2 *
"
#$$
%
&''= (
f22 ( f12( f21 f11
"
#$$
%
&''
f11 f22 ( f12 f21( )g1g2
)
*
++
,
-
.
.!u *+l1l2
)
*
++
,
-
.
.!w*"
#
$$
%
&
''
2nd-order example
sI! F = " s( ) = s2 + f11 + f22( )s + f11 f22 ! f12 f21( )= s ! #1( ) s ! #2( ) = 0
Re #i( ) < 0
System Matrices
Equilibrium Response with Constant Inputs
Requirement for Stability
27
Equilibrium Response ofApproximate Phugoid Model
!xP* = "FP"1 GP!uP *+LP!wP *( )
!V *
!" *#
$%%
&
'((= )
0 VNLV
)1g
VNDV
gLV
#
$
%%%%%
&
'
(((((
T*TL*TVN
#
$
%%%
&
'
(((!*T * +
DV
)LVVN
#
$
%%%
&
'
(((!VW
*
+
,--
.--
/
0--
1--
Equilibrium state with constant thrust and wind perturbations
28
Equilibrium Response ofApproximate Phugoid Model
!V * = "L#TLV
!#T * +!VW*
!$ * =1gT#T + L#T
DV
LV
%
&'
(
)*!#T *
With L!!T ~ 0, steady-state velocity perturbation depends only on the horizontal wind
Constant thrust perturbation produces steady climb rateCorresponding dynamic response
to thrust step, with L!!T = 0
Steady horizontal wind affects velocity but not flight path angle
29
Equilibrium Response ofApproximate Short-Period Model
!xSP* = "FSP"1 GSP!uSP *+LSP!wSP *( )
!q*
!" *
#
$%%
&
'((= )
L"VN
M"
1 )Mq
#
$
%%%
&
'
(((
L"VN
Mq + M"
*
+,-
./
M0E
)L0EVN
#
$
%%%
&
'
(((!0E* )
M"
)L"VN
#
$
%%%
&
'
(((!"W
*
1
233
433
5
633
733
Equilibrium state with constant elevator and wind perturbations
30
Equilibrium Response ofApproximate Short-Period Model
Steady pitch rate and angle of attack response to elevator perturbation are not zero
Steady vertical wind affects steady-state angle of attack but not pitch rate
!q* = "
L#VN
M$E%
&'(
)*
L#VN
Mq + M#
%
&'(
)*
!$E*
!# * = "M$E( )
L#VN
Mq + M#
%
&'(
)*
!$E + !#W*
with L!E = 0
Dynamic response to elevator step with L!!E = 0
31
Phase Plane Plots!
32
A 2nd-Order Dynamic Model
!!x1!!x2
"
#$$
%
&''=
0 1() n
2 (2*) n
"
#$$
%
&''
!x1!x2
"
#$$
%
&''+ 1 (1
0 2"
#$
%
&'
!u1!u2
"
#$$
%
&''
33
!x1 t( ) : Displacement (or Position)!x2 t( ) : Rate of change of Position
! n : Natural frequency, rad/s" : Damping ratio, -
State ( Phase ) Plane Plots
Cross-plot of one component against another
Time is not shown explicitly
% 2nd-Order Model - Initial Condition Response
clear z = 0.1; % Damping ratio wn = 6.28; % Natural frequency, rad/s F = [0 1;-wn^2 -2*z*wn]; G = [1 -1;0 2]; Hx = [1 0;0 1]; sys = ss(F, G, Hx,0); t = [0:0.01:10]; xo = [1;0]; [y1,t1,x1] = initial(sys, xo, t); plot(t1,y1) grid on figure plot(y1(:,1),y1(:,2)) grid on
!!x1!!x2
"
#$$
%
&''(
0 1)*n
2 )2+*n
"
#$$
%
&''
!x1!x2
"
#$$
%
&''+ 1 )1
0 2"
#$
%
&'
!u1!u2
"
#$$
%
&''
34
Dynamic Stability Changes the State-Plane Spiral
Damping ratio = 0.1 Damping ratio = 0.3 Damping ratio = –0.1
35
Tactical Aircraft Maneuverability!Chapter 10, Airplane Stability and
Control, Abzug and Larrabee!•! What are the principal subject and scope of the
chapter?!•! What technical ideas are needed to understand
the chapter?!•! During what time period did the events covered
in the chapter take place?!•! What are the three main "takeaway" points or
conclusions from the reading?!•! What are the three most surprising or
remarkable facts that you found in the reading?!36
Scalar Frequency Response!
37
Speed Control of Direct-Current Motor
u(t) = Ce(t)wheree(t) = yc (t) ! y(t)
Control Law (C = Control Gain)
Angular Rate
38
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 the
control input, u(t)–! Motor control torque is proportional to u(t) –! Desired speed, yc(t), rad/s, is constant–! Control gain, C, scales command-following
error to motor input voltage39
Model of Dynamics and Speed Control
Dynamic equation
y(t) = 1J
u(t)dt0
t
! =CJ
e(t)dt0
t
! =CJ
yc (t) " y(t)[ ]dt0
t
!
dy(t)dt
=u(t)J
=Ce(t)J
=CJyc (t) ! y(t)[ ], y 0( ) given
Integral of the equation, with y(0) = 0
Direct integration of yc(t)Negative feedback of y(t)
40
Step Response of Speed Controller
y(t)= yc 1! e!CJ
"
#$
%
&'t(
)**
+
,--= yc 1! e
.t() +,= yc 1! e! t /(
)*+,-
•! where! "" = –C/J = eigenvalue or root of the system (rad/s)! ## = J/C = time constant of the response (sec)
•! Solution of the integral, with step command
yc t( ) =0, t < 01, t ! 0
"#$
%$
41
Angle Control of a DC Motor
Closed-loop dynamic equation, with y(t) = I2 x(t)
u(t) = c1 yc (t) " y1(t)[ ]" c2y2(t)
!x1(t)!x2 (t)
!
"##
$
%&&=
0 1'c1 / J 'c2 / J
!
"##
$
%&&
x1(t)x2 (t)
!
"##
$
%&&+
0c1 / J
!
"##
$
%&&yc
Control law with angle and angular rate feedback
!n = c1 J ; " = c2 J( ) 2!n42
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
!n = c1 J ; " = c2 J( ) 2!n
43
Angle Response to a Sinusoidal Angle Command
Amplitude Ratio (AR) =ypeakyCpeak
Phase Angle !( ) = "360#t peakPeriod
, deg
•! Output wave lags behind the input wave
•! Input and output amplitudes different
yC t( ) = yCpeaksin!t
44
Effect of Input Frequency on Output Amplitude and Phase Angle
•! With low input frequency, input and output amplitudes are about the same
•! Rate oscillation leads angle
oscillation by ~90°•! Lag of angle output
oscillation, compared to input, is small
yc (t) = sin t / 6.28( ), deg !n = 1 rad / s" = 0.707
45
At Higher Input Frequency, Phase Angle Lag Increases
yc (t) = sin t( ), deg
46
At Even Higher Frequency, Amplitude Ratio Decreases and
Phase Lag Increasesyc (t) = sin 6.28t( ), deg
47
Angle and Rate Response of a DC Motor over Wide Input-Frequency Range !!Long-term response
of a dynamic system to sinusoidal inputs over a range of frequencies!! Determine
experimentally from time response or
!! Compute the Bode plot of the system s transfer functions (TBD)
Very low damping
Moderate damping
High damping
48
Input Frequencies (previous slides)
Next Time:!Transfer Functions and
Frequency Response!Reading:!
Flight Dynamics!342-357!
49
•! Frequency domain view of initial condition response•! Response of dynamic systems to sinusoidal inputs•! Transfer functions•! Bode plots
Learning Objectives
Supplemental Material!
50
Example: Aerodynamic Angle, Linear Velocity, and Angular Rate Perturbations
Learjet 23MN = 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
51
Continuous- and Discrete-Time Dutch-Roll Models
!!r t( )! !" t( )
#
$
%%
&
'
(() *0.11 1.9
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Differential Equations Produce State Rates of Change
Difference Equations Produce State Increments
52
!t = 0.1sec
Continuous- and Discrete-Time Roll-Spiral Models
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Differential Equations Produce State Rates of Change
Difference Equations Produce State Increments
53
!t = 0.1sec
4th- Order Comparison: Continuous- and Discrete-Time Longitudinal Models
Phugoid and Short Period
! !V t( )! !" t( )! !q t( )! !# t( )
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Differential Equations Produce State Rates of Change
Difference Equations Produce State Increments !t = 0.1sec
54