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AERO 632: Design of Advance Flight Control SystemPreliminaries
Raktim Bhattacharya
Laboratory For Uncertainty QuantificationAerospace Engineering, Texas A&M University.
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
PreliminariesSignals & SystemsLaplace transformsTransfer functions – from ordinary linear differential equationsSystem interconnectionsBlock diagram algebra – simplification of interconnectionsGeneral feedback control system interconnection.
C P+u+r +
d
+
n
e +y ym−ym
AERO 632, Instructor: Raktim Bhattacharya 2 / 46
Signals & Systems
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Signals & Systems
Pu(t) y(t)
Actuator applies u(t)
Sensor provides y(t)
Feedback controller takes y(t) and determines u(t) to achievedesired behaviorThe controller is typically implemented as software, running ina micro controller
Imperfections exist in real world▶ sensors have noise▶ actuators have irregularities▶ plant P is not fully known
AERO 632, Instructor: Raktim Bhattacharya 4 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
System Response to u(t)
Pu(t) y(t)
Given plant P and input u(t), what is y(t)?P is defined in terms of ordinary differential equationsy(t) is the forced + initial condition response.
Linear Dynamics
mx+ cx+ kx = u(t) dynamics
y(t) = x(t) measurement
Nonlinear Dynamics
x− µ(1− x2)x+ x = u(t) dynamics
y(t) = x(t) measurement
In this class we focus on linear systems
AERO 632, Instructor: Raktim Bhattacharya 5 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Linear Systems
Pu(t) y(t)
Dynamics is defined by linear ordinary differential equationSuper position principle applies
u1(t) 7→ y1(t)u2(t) 7→ y2(t)
=⇒ (u1(t) + u2(t)) 7→ (y1(t) + y2(t))
AERO 632, Instructor: Raktim Bhattacharya 6 / 46
Laplace Transforms
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Laplace TransformsGiven signal u(t), Laplace transform is defined as
L{u(t)} :=
∫ ∞
0u(t)e−stdt
Exists when
limt→∞
|u(t)e−σt| = 0, for some σ > 0
Very useful in studying linear dynamical systems and designingcontrollers
AERO 632, Instructor: Raktim Bhattacharya 8 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Properties Laplace TransformsLinear operator
Additive
L{u1(t) + u2(t)} =
∫ ∞
0(u1(t) + u2(t)) e
−stdt
=
∫ ∞
0u1(t)e
−stdt+
∫ ∞
0u2(t)e
−stdt
= L{u1(t)}+ L{u2(t)}
Superposition
L{au(t)} = aL{u(t)} , a is a constant
AERO 632, Instructor: Raktim Bhattacharya 9 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Properties (contd.)1. U(s) := L{u(t)}2. L{au1(t) + bu2(t)} = aL{u1(t)}+ bL{u2(t)} = aU1(s) + bU2(s)
3. 1
sU(s) ⇐⇒
∫ t
0u(τ)dτ
4. U1(s)U2(s) ⇐⇒ u1(t) ∗ u2(t) Convolution
5. lims→0
sU(s) ⇐⇒ limt→∞
u(t) Final value theorem
6. lims→∞
sU(s) ⇐⇒ u(0+) Initial value theorem
7. − dU(s)
ds⇐⇒ tu(t)
8. L{du
dt
}⇐⇒ sU(s)− su(0)
9. L{u} ⇐⇒ s2U(s)− su(0)− u(0)
AERO 632, Instructor: Raktim Bhattacharya 10 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Important Signals
1. L{δ(t)} = 1 δ(t) is impulse function
2. L{1(t)} =1
s1(t) is unit step function at t = 0
3. L{t} =1
s2
4. L{sin(ωt} =ω
s2 + ω2
5. L{cos(ωt} =s
s2 + ω2
AERO 632, Instructor: Raktim Bhattacharya 11 / 46
Transfer Functions
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Spring Mass Damper System
m u(t)
Equation of Motionmx+ cx+ kx = u(t)
Take L{·} on both sidesL{mx+ cx+ kx} = L{u(t)}mL{x}+ cL{x}+ kL{x} = L{u(t)}m
(s2X(s)− sx(0)− x(0)
)+ c (sX(s)− x(0)) + kX(s) = U(s)
(ms2 + cs+ k)X(s) = U(s) x(0) and x(0) are assumed to be zero
AERO 632, Instructor: Raktim Bhattacharya 13 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Transfer Function
m u(t)
Pu(t) y(t)
(ms2 + cs+ k)X(s) = U(s) =⇒ X(s)
U(s)=
1
ms2 + cs+ k
Choose output y(t) = x(t) =⇒ Y (s) = X(s).Therefore
P (s) :=Y (s)
U(s)=
1
ms2 + cs+ kTransfer function
AERO 632, Instructor: Raktim Bhattacharya 14 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Transfer Function (contd.)In general
P (s) =N(s)
D(s)
where N(s) and D(s) are polynomials in s
Roots of N(s) are the zerosRoots of D(s) are the poles – determine stability
AERO 632, Instructor: Raktim Bhattacharya 15 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Response to u(t)Given
– input signal u(t) and transfer function P (s).
Determine– output response y(t)
1. Laplace transformU(s) := L{u(t)}
2. Determine Y (s) := P (s)U(s)
3. Laplace inverse
y(t) := L−1 {Y (s)} = L−1 {P (s)U(s)}
Pu(t) y(t)
AERO 632, Instructor: Raktim Bhattacharya 16 / 46
System Interconnection
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Block DiagramRepresentation of System Interconnections
SeriesParallelFeedbackA simple exampleA complex example
AERO 632, Instructor: Raktim Bhattacharya 18 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Series ConnectionG1 G2
u y
AERO 632, Instructor: Raktim Bhattacharya 19 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Parallel Connection
G1
G2
+u y
AERO 632, Instructor: Raktim Bhattacharya 20 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Feedback ConnectionG
r + y
−
AERO 632, Instructor: Raktim Bhattacharya 21 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Simple ExampleG1
r +
−G2 G3
G4
−
+
y
AERO 632, Instructor: Raktim Bhattacharya 22 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Complex Example
G1r +
−G2 G3
G4
−
+
y+
AERO 632, Instructor: Raktim Bhattacharya 23 / 46
Frequency Response
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Response to Sinusoidal InputP
u(t) y(t)
Let u(t) = Au sin(ωt)Vary ω from 0 to ∞
A linear system’s response to sinusoidal inputs is called thesystem’s frequency response
AERO 632, Instructor: Raktim Bhattacharya 25 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Response to Sinusoidal InputExample
Let P (s) = 1s+1 , u(t) = sin(t)
y(t) =1
2e−t − 1
2cos(t) + 1
2sin(t)
=1
2e−t︸ ︷︷ ︸
natural response
+1√2
sin(t− π
4)︸ ︷︷ ︸
forced response
Forced response has form Ay sin(ωt+ ϕ)
Ay and ϕ are functions of ω
AERO 632, Instructor: Raktim Bhattacharya 26 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Response to Sinusoidal InputGeneralization
In general
Y (s) = G(s)ω0
s2 + ω20
=α1
s− p1+ · · · αn
s− pn+
α0
s+ jω0+
α∗0
s− jω0
=⇒ y(t) = α1ep1t + · · ·+ αne
pnt︸ ︷︷ ︸natural
+Ay sin(ω0 + ϕ)︸ ︷︷ ︸forced
Forced response has same frequency, different amplitude and phase.
AERO 632, Instructor: Raktim Bhattacharya 27 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Response to Sinusoidal InputGeneralization (contd.)
For a system P (s) and input
u(t) = Au sin(ω0t),
forced response is
y(t) = AuM sin(ω0t+ ϕ),
where
M(ω0) = |P (s)|s=jω0 = |P (jω0)|,magnitude
ϕ(ω0) = P (jω0) phase
In polar formP (jω0) = Mejϕ.
AERO 632, Instructor: Raktim Bhattacharya 28 / 46
Fourier Analysis
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Fourier Series ExpansionGiven a signal y(t) with periodicity T ,
y(t) =a02
+∑
n=1,2,···an cos
(2πnt
T
)+ bn sin
(2πnt
T
)
a0 =2
T
∫ T
0y(t)dt
an =2
T
∫ T
0y(t) cos
(2πnt
T
)dt
bn =2
T
∫ T
0y(t) sin
(2πnt
T
)dt
AERO 632, Instructor: Raktim Bhattacharya 30 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Fourier Series ExpansionApproximation of step function
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2
0
0.2
0.4
0.6
0.8
1
1.2N=2
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2
0
0.2
0.4
0.6
0.8
1
1.2N=6
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2
0
0.2
0.4
0.6
0.8
1
1.2N=8
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2
0
0.2
0.4
0.6
0.8
1
1.2N=10
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2
0
0.2
0.4
0.6
0.8
1
1.2N=20
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2
0
0.2
0.4
0.6
0.8
1
1.2N=50
AERO 632, Instructor: Raktim Bhattacharya 31 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Fourier TransformStep function
Fourier transform reveals the frequency content of a signalAERO 632, Instructor: Raktim Bhattacharya 32 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Fourier TransformStep function – frequency content
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.2
0
0.2
0.4
0.6
0.8
1
1.2
t
y(t)
0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
ω
y(ω
)
AERO 632, Instructor: Raktim Bhattacharya 33 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Signals & SystemsInput Output
Pu(t) y(t)
Fourier Series Expansionsuperposition principle
P∑
i ui(t)∑
i yi(t)
Fourier Transform
PU(jω) Y (jω)
ui(t) = ai sin(ωit)
yiforced(t) = aiM sin(ωit+ ϕ)
Y (jω) = P (jω)U(jω)
Suffices to study P (jω) |P (jω)|, P (jω)
AERO 632, Instructor: Raktim Bhattacharya 34 / 46
Bode Plot
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
First Order System
−40
−35
−30
−25
−20
−15
−10
−5
0
Mag
nitu
de (
dB)
10−2
10−1
100
101
102
−90
−45
0
Pha
se (
deg)
Bode Diagram
Frequency (rad/s)
P (s) = 1/(s+ 1)
loglog scaledB = 10 log10(·)20dB = 10 log10(100/1)
0 10 20 30 40 50 60 70 80 90 100−1
0
1
y(t)
Fre q = 0.10 rad/s
0 1 2 3 4 5 6 7 8 9 10−1
0
1
y(t)
Fre q = 1.00 rad/s
0 1 2 3 4 5 6 7 8 9 10−1
0
1
y(t)
Fre q = 5.00 rad/s
0 1 2 3 4 5 6 7 8 9 10−1
0
1
y(t)
Fre q = 10.00 rad/s
u(t) = A sin(ω0t)
yforced(t) = AM sin(ω0t+ϕ)
AERO 632, Instructor: Raktim Bhattacharya 36 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Second Order System
−40
−30
−20
−10
0
10
Mag
nitu
de (
dB)
10−1
100
101
−180
−135
−90
−45
0
Pha
se (
deg)
Bode Diagram
Frequency (rad/s)
P (s) = 1/(s2 + 0.5s+ 1)
ωn = 1 rad/s
0 10 20 30 40 50 60 70 80 90 100−2
0
2
y(t)
Fre q = 0.10 rad/s
0 10 20 30 40 50 60 70 80 90 100−2
0
2
y(t)
Fre q = 1.00 rad/s
0 1 2 3 4 5 6 7 8 9 10−1
0
1
y(t)
Fre q = 5.00 rad/s
0 1 2 3 4 5 6 7 8 9 10−1
0
1
y(t)
Fre q = 10.00 rad/s
u(t) = A sin(ω0t)
yforced(t) = AM sin(ω0t+ϕ)
AERO 632, Instructor: Raktim Bhattacharya 37 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
S(jω) + T (jω) = 1
10−2
10−1
100
101
102
10−2
10−1
100
101
ω rad/s
Magnitude|S
(jω)|
10−1
100
101
102
10−3
10−2
10−1
100
101
ω rad/s
Magnitude|T
(jω)|
C P+u+r +
d
+
n
e +y ym−ym
P (s) = 1(s+1)(s/2+1)
C(s) = 10
S = Ger = 11+PC
= 11+10P
T = Gyr = PC1+PC
= 10P1+10P
10−2
10−1
100
101
102
−60
−50
−40
−30
−20
−10
0
10
Mag
nitu
de (
dB)
Bode Diagram
Frequency (rad/s)
STS+T
AERO 632, Instructor: Raktim Bhattacharya 38 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
All transfer functionsWith proportional controller
10−2
100
102
−30
−20
−10
0
10M
agni
tude
(dB
)
Ger
Frequency (rad/s)10
010
2−80
−60
−40
−20
0
Mag
nitu
de (
dB)
Ged
Frequency (rad/s)10
−210
010
2−30
−20
−10
0
10
Mag
nitu
de (
dB)
Gen
Frequency (rad/s)
100
102
−60
−40
−20
0
20
Mag
nitu
de (
dB)
Gyr
Frequency (rad/s)10
010
2−80
−60
−40
−20
0
Mag
nitu
de (
dB)
Gyd
Frequency (rad/s)10
010
2−60
−40
−20
0
20
Mag
nitu
de (
dB)
Gyn
Frequency (rad/s)
AERO 632, Instructor: Raktim Bhattacharya 39 / 46
Controller DesignConsiderations
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Design Using Bode Plot of P (jω)C(jω)Loop Shaping
Develop conditions on the Bode plot of the open loop transferfunction
Sensitivity 11+PC
Steady-state errors: slope and magnitude at limω → 0
Robust to sensor noiseDisturbance rejectionController roll off =⇒ not excite high-frequency modes ofplantRobust to plant uncertainty
Look at Bode plot of L(jω) := P (jω)C(jw)
AERO 632, Instructor: Raktim Bhattacharya 41 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Frequency Domain SpecificationsConstraints on the shape of L(jω)
Stea
dy-s
tate
erro
r bou
ndar
y
Sens
or n
oise
, pla
nt
unce
rtain
ty
!!c
1
|P(j!)C
(j!)|
slope ⇡ 1
!!c
1
|P(j!)C
(j!)|
slope ⇡ 1
Steady-state error boundary
Sensor noise, disturbance Plant uncertainty
Choose C(jω) to ensure |L(jω)| does not violate theconstraintsSlope ≈ −1 at ωc ensures PM ≈ 90◦
stable if PM > 0 =⇒ PC > −180◦
AERO 632, Instructor: Raktim Bhattacharya 42 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Plant UncertaintyP (jω) = P0(jω)(1 + ∆P (jω))
10−2
10−1
100
101
102
103
−250
−200
−150
−100
−50
0
50
Mag
nitu
de (
dB)
Bode Diagram
Frequency (rad/s)
TrueModelUnc+Unc−
Stea
dy-s
tate
erro
r bou
ndar
y
Sens
or n
oise
, pla
nt
unce
rtain
ty
!!c
1
|P(j!)C
(j!)|
slope ⇡ 1
!!c
1
|P(j!)C
(j!)|
slope ⇡ 1
Steady-state error boundary
Sensor noise, disturbance Plant uncertainty
AERO 632, Instructor: Raktim Bhattacharya 43 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Sensor CharacteristicsNoise spectrum
Stea
dy-s
tate
erro
r bou
ndar
y
Sens
or n
oise
, pla
nt
unce
rtain
ty
!!c
1
|P(j!)C
(j!)|
slope ⇡ 1
!!c
1
|P(j!)C
(j!)|
slope ⇡ 1
Steady-state error boundary
Sensor noise, disturbance Plant uncertainty
Gyn = − PC
1 + PC
AERO 632, Instructor: Raktim Bhattacharya 44 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Reference TrackingBandlimited else conflicts with noise rejection
0 5 10 15 20 25
0.01
0.02
0.03
0.04
0.05
Spectrum of r(t)
Frequency (Hz)
|X(f
)|
0 5 10 15 20 25
0.01
0.02
0.03
0.04
0.05
Spectrum of n(t)
Frequency (Hz)
|X(f
)|
Stea
dy-s
tate
erro
r bou
ndar
y
Sens
or n
oise
, pla
nt
unce
rtain
ty
!!c
1
|P(j!)C
(j!)|
slope ⇡ 1
!!c
1
|P(j!)C
(j!)|
slope ⇡ 1
Steady-state error boundary
Sensor noise, disturbance Plant uncertainty
Gyr =PC
1 + PC
Gyn = − PC1+PC
AERO 632, Instructor: Raktim Bhattacharya 45 / 46
Signals & Systems Laplace Transforms Transfer Functions System Interconnection Frequency Domain Analysis Design
Disturbance RejectonBandlimited else conflicts with noise rejection
0 5 10 15 20 25
0.01
0.02
0.03
0.04
0.05
Spectrum of n(t)
Frequency (Hz)
|X(f
)|
0 5 10 15 20 25
0.01
0.02
0.03
0.04
0.05
Spectrum of d(t)
Frequency (Hz)
|X(f
)|
Stea
dy-s
tate
erro
r bou
ndar
y
Sens
or n
oise
, pla
nt
unce
rtain
ty
!!c
1
|P(j!)C
(j!)|
slope ⇡ 1
!!c
1
|P(j!)C
(j!)|
slope ⇡ 1
Steady-state error boundary
Sensor noise, disturbance Plant uncertainty
Gyd =P
1 + PC
Gyn = − PC1+PC
AERO 632, Instructor: Raktim Bhattacharya 46 / 46
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