Upload
abdul-rahim
View
214
Download
0
Embed Size (px)
DESCRIPTION
This is a control system lecture note that was taught at University of Melbourne, Australia in 2014. This is the third part of its series.
Citation preview
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 1/29
Lecture slides for ELEN90055 prepared by Michael Cantoni (c) 2011, 2012, 2013
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 2/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
C s
E sR s Y s
Di(s)
U s
Do(s)
Dm sY m
s
C (s)
G0 s
Y (s)
Do(s) =
1
1 + G0(s)C (s)Y (s)
R(s) =
G0(s)C (s)
1 + G0(s)C (s)
wit on y o = 0wit on y r = 0
G0(s) = B0(s)
A0(s) nominal plant transfer function, C (s) =
P (s)
L(s) controller transfer function
B0(s), A0(s)} and P (s), L(s)} coprime (no common roots) polynomial pairs
Di(s), Do(s), Dm(s) plant input, plant output and measurement noise uncertainty
R(s), U (s), Y (s) reference, control, and plant output signals
Y (s) = D0(s) + G0(s)U (s) + Di(s)
U (s) = C (s)R(s)−Dm(s)− Y (s)
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
E sR s Y s
Di s
U s
Do s
Dm sY m s
80
60
40
20
0
20
M a g n i t u d e ( d B )
100
102
104
106
90
45
0
45
90
P h
a s e ( d e g )
Frequency response
Frequency (rad/sec)
K 100
0.01 s + 1
K = 1
K = 10
K = 1K = 10
1
1 + G0C
G0C
1 + G0C
G0(s) = 100
0.01 s + 1 C (s) = K
K = 1
K = 10
K = 1K = 10
K K
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 3/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
C s
E sR s Y s
Di(s)
U s
Do(s)
Dm sY m
s
G0 sH s
Y (s) = G0(s)C (s)H (s)
1 + G0(s)C (s) R(s) +
1
1 + G0(s)C (s)Do(s) +
G0(s)
1 + G0(s)C (s)Di(s)−
G0(s)C (s)
1 + G0(s)C (s)Dm(s)
C s
U (s) = C (s)H (s)
1 + G0(s)C (s)R(s)−
C (s)
1 + G0(s)C (s)Do(s)−
G0(s)C (s)
1 + G0(s)C (s)Di(s)−
C (s)
1 + G0(s)C (s)Dm(s)
H (s) is a stable transfer function (reference filter)
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
S 0 s + T 0 s = 1, S i0 s = G0 s S 0 s = T 0 s /C s , S u0 s = S 0 s C s = T 0 s /G0 s
Y (s) = T 0(s)H (s)R(s) + S 0(s)Do(s) + S i0(s)Di(s)− T 0(s)Dm(s)
U (s) = S u0
H (s)R(s)−Dm(s)−Do(s)−G0(s)Di(s)
For a nominal plant G0(s) = B0(s)A0(s)
and controller C (s) = P (s)L(s)
:
T 0(s) .=
G0(s)C (s)
1 + G0(s)C (s) =
B0(s)P (s)
A0(s)L(s) + B0(s)P (s) complementary sensitivity
S 0(s) .=
1
1 + G0(s)C (s) =
A0(s)L(s)
A0(s)L(s) + B0(s)P (s) (output) sensitivity
S i0(s) .=
G0(s)
1 + G0(s)C (s) =
B0(s)L(s)
A0(s)L(s) + B0(s)P (s) input-disturbance sensitivity
S u0(s) .=
C (s)
1 + G0(s)C (s) =
A0(s)P (s)
A0(s)L(s) + B0(s)P (s) control sensitivity
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 4/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
The problem spec. translates to |S 0 jω = 1−T 0 jω | << 1 for ω ∈ [0, 4 rad sec
Given a
nominal plant model G0(s) = 4(s+1)(s+2)2 and
output disturbance model do(t) = k + dv(t), where k is a constant and dv(t)has energy confined to the frequency band (0, 4) rad/sec,
design a feedback controller that desensitizes the plant output to the disturbance
Choosing
T 0(s) = G0(s)C (s)
1 + G0(s)C (s) =
1
(τ 1s + 1)2(τ 2s + 1)2
with τ 1 = 0.1 and τ 2 = 0.01, achieves |S 0( jω)| << 1 for ω ∈ [0, 4) and the
corresponding controller is
C (s) = T 0(s)
(1 − T 0(s)) ·
1
G0(s) =
106(s + 1)(s + 2)2
4((s + 10)2(s + 100)2 − 106)
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
10
100
10
104
0
100
M a g n i t u d e ( d B )
Bode Diagram
Frequency (rad/s)
10
100
10
104
0
100
M a g n i t u d e ( d B )
Bode Diagram
Frequency (rad/s)
10
100
10
104
0
100
M a
g n i t u d e ( d B )
Bode Diagram
Frequency (rad/s)
10
100
10
104
0
100
M a
g n i t u d e ( d B )
Bode Diagram
Frequency (rad/s)
C jω
G0( jω)
S 0( jω)S i0 jω
S u0 jω
T 0 jω
S u0(s) = T 0(s)G0(s)
= (s+1)(s+2)2
4(0.1s+1)2(0.01s+1)2 =⇒ |S u0( j3)| ≈ 20 dB!
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 5/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
|Λ0( jω)|
|T 0( jω)| S 0( jω)|
good tracking of low-
frequency reference
low sensitivity to high-frequency
measurement noise
good rejection of low-frequency
output disturbance
|Λ0 jω.= G0 jω C jω |
large low-frequency loop-gain
small high-frequency loop-gain
M a g n i t u d e ( d B )
Frequency response
Frequency (rad/sec)
NOTE: T 0( jω) = Λ0( jω)
1 + Λ0( jω); S 0( jω) =
1
1 + Λ0( jω)
loop-gain bandwidth ωb : |Λ0( jωb)| = 1
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
If |Λ0( jω)| ≈ 1, then both |S 0( jω)| and |T 0( jω)| could be large – all depends on
∠Λ0( jω) at these frequencies
Similarly, using 1− |Λ0| ≤ |1 + Λ0( jω)|, for frequencies where |Λ0( jω)| << 1,
|S 0( jω)| ≈ 1, |T 0( jω)| << 1, |S i0( jω)| ≈ |G0( jω)|, |S u0( jω)| ≈ |C ( jω)|.
Consider the inequalities |Λ0( jω)| − 1 ≤ |1 + Λ0( jω)| ≤ 1 + |Λ0( jω)|.For any frequency such that |Λ0( jω)| > 1,
1|Λ0(jω)|+1
≤ |S 0( jω)| ≤ 1|Λ0(jω)|−1
|Λ0(jω)||Λ0(jω)|+1 ≤ |T 0( jω)| ≤ |Λ0(jω)|
|Λ0(jω)|−1
1|C (jω)| ·
|Λ0(jω)||Λ0(jω)|+1 ≤ |S i0( jω)| ≤ 1
|C (jω)| · |Λ0(jω)||Λ0(jω)|−1
1
|G0(jω)| ·
|Λ0(jω)|
|Λ0(jω)|+1 ≤
|S u0( jω
)|≤ 1
|G0(jω)| ·
|Λ0(jω)|
|Λ0(jω)|−1 .
As such, for frequencies where |Λ0( jω)| >> 1,
|S 0( jω)| << 1, |T 0( jω)| ≈ 1, |S i0( jω)| ≈1
|C ( jω)|, |S u0( jω)| ≈
1
|G0( jω)|.
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 6/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
C s
E sR s Y s
Di(s)
U s
Do(s)
Dm sY m
s
G0 s
ess.
= limt→∞
e(t) = lims→0
sE (s) = lims→0
sS 0(s)R(s) = lims→0
s 1
1 + Λ0(s)R(s) = 0
ess = 0
ess = lims→0 s
1
1 + Λ0(s)
1
s =
1
1 + Λ0(0) = S 0(0)
S 0 s
Λ0(0) =∞ =⇒ S 0(0) = 0
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
r(t), di(t), do(t), dm(t)
G0(s) = 1s
, C (s) = K =⇒ T 0(s) = K s + K
, S 0(s) = ss + K
,
S i0(s) = 1
s + K , S u0(s) =
Ks
s + K
Λ0(s) = G0(s)C (s)
e t , u t , y t , ym t
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 7/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
G0(s) = s− 1
s + 3 , C (s) = 1
s− 1 =⇒ S 0(s) = s + 3
s + 4 , T 0(s) = 1
s + 4 ,
S i0(s) = s− 1
s + 4, S u0(s) =
s + 3
(s− 1)(s + 4)
A0(s)L(s) + B0(s)P (s) = (s + 3)(s− 1) + (s− 1) = (s− 1)(s + 4)
{A0, B0}, {L,P }
Given a strictly proper nominal plant model G0(s) = B0(s)A0(s)
and proper controller
C (s) = P (s)L(s) the closed-loop is internally stable if, and only if, all zeros of the
nominal characteristic polynomial
A0(s
)L
(s
) +B
0(s
)P
(s
)
have strictly negative real part.
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
S 0.
=1
1 + G0C ; T 0
.=
G0C
1 + G0C ; S i0
.=
G0
1 + G0C ; S u0
.=
C
1 + G0C
Λ0
.= G0C
G0(s)
C s
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 8/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
as2+ bs+ c has zeros at s =
−b±√ b2−4ac
2a
n = 2
an = 1
for p(s) = s2 + ks + (1 − k)
if [ p(s0) = 0 =⇒ (s0) < 0 ]
then 0 < k < 1
s3 + s2 + s+ 1
has zeros at
s = −1,± j
given a degree n > 0 polynomial p(s) = sn + an−1sn−1 + · · · + a1s + a0, if the
zeros of p(s) all have negative real part, then ai > 0 for i = 0, 1, . . . , n
follows by factorizing p(s) into product of first and second order polynomials with
positive coefficients, which is possible because all zeros have negative real part
so if any of the polynomial co-efficients is zero or negative we know there must be
a zero with non-negative real partthe condition is sufficient when n = 1, 2, but not in general
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
G0(s) = 1s(s+1.5)
and C (s) = K + s
0.1s+1
0.1s3 + 1.15s2 + (0.1K + 2.5)s + K
K
K > 0 and (0.1K + 2.5) > 0 (i.e. K > 0)
so stable if K > 0 and 3K
230 + 2.5 > 0
(i.e. for all K > 0)
s3 0.1 (0.1K + 2.5)
s2 1.15 K
s1 3K
230 + 2.5 0
s0
K
For p(s) =4
k=0
aksk, a4 > 0,
s4 a4 a2 a0s3 a3 a1 0
s2
b0.
=−
a4a1−a2a3
a3 b1.
=−
a4·0−a0a3
a3 = a0 0s1 c
.= −
a3b1−a1b0
b00
s0 −
b0·0−b1c
c = b1
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 9/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
G0(s) = 1s(s+1.5)
and C (s) = K + s
0.1s+1
0.1s3 + 1.15s2 + (0.1K + 2.5)s + K
K
w = s+ 1
w3 0.1 0.1K + 0.5
w2 0.85 0.9K − 1.45
w1 1
170(−K + 114)
w0 0.9K −
1.45
0.1(w − 1)3 + 1.15(w − 1)2 + (0.1K + 2.5)(w − 1) + K
= 0.1w3 + 0.85w2 + (0.1K + 0.5)w + (0.9K − 1.45)
so all closed-loop poles have real part less than −1 if 2918
< K < 114
so zeros lie in left-half plane if 0.9K − 1.45 > 0 and 1
170(114−K ) > 0
−1
(w) < 0 ⇔ (s) < −1
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
K
K > 0
1 + K · F s0 = 0
F (s) :
s0
K =1
|F (s0)|
1 + K · F (s) = 0 where F (s) = M (s)
D(s) =
m
k=1(s− β k)
n
k=1(s− αk)
e.g. 0.1s3 + 1.15s2 + (0.1K + 2.5)s + K = 0⇔ 1 + K 0.1s+1
s(0.1s2+1.15s+2.5) = 0
lim|ω|→∞
∠F ( jω) = (m− n)π
2
if this is not the case
(e.g. F (s) = −1/s(s + 1)) then
analysis must be modified to
handle additional π rad phase
(2l + 1)π = ∠F (s0) =m
k=1∠(s0−β k)−
n
k=1∠(s0−αk) for l = 0,±1,±2, . . .
magnitude condition: |K · F s0 | = 1
phase condition: ∠K · F (s0) = (2l + 1)π for l = 0, ±1, ±2, . . .
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 10/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
K
K
F s
F s
1 + K · F (s) = D(s) + KM (s)
D(s)
1 + K · F (s) = K ·
1
K D(s) + M (s)
D(s)
s s s
s− γ
(s− γ )
∠(s− γ ) =
−∠(s− γ )
γ ∠ s− γ = 0
γ
γ
s− γ s− γ
∠ s− γ = π
γ
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
n m
K →∞
K → 0
n−m
m− n
∞
∞
s = σ ηk
ηk
n > m
s = −σ
σ .=
n
k=1αk −
m
k=1β k
n−m and ηk
.=
(2k − 1)π
n−m for k = 1, . . . , n−m
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 11/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
K
F (s) n > m
|F s |→ 0 |s|→∞ n > m
|s|
K 1 + KF s 1 + K 1
sn−m
σ
s0
σ = −
(an−1 − bm−1)
n−m
=
n
k=1αk −
m
k=1β k
n−m
F (s) = sm + bm−1s
m−1 + · · · + b1s + b0
sn + an−1sn−1 + · · ·a1s + a0
=
m
k=1(s− β k)
n
k=1(s− αk)
= 1
sn−m + (an−1 − bm−1)sn−m−1 + · · · + d1s + d0 + cm−1sm−1+···+c1s+c0
sm+bm−1sm−1+···+b1s+b0
≈
1
(s− σ)n−m =
1
sn−m − σ(n−m)sn−m−1 + · · · + (n−m)(−σ)n−m−1s + (−σ)n−m
(n−m)∠ 1
s0= ∠− 1 = (2l + 1)π, l = 0,±1, . . . ,
∠s0 = 2k − 1 π/ n−m , k = 1, 2, . . . , n−m
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
0.1s3 + 1.15s2 + (0.1K + 2.5)s + K = 0 ⇔ 1 + K 0.1s+1
s(0.1s2+1.15s+2.5) = 0
i.e. F (s) = (s+10)s(s+8.5895)(s+2.9105)
Root Locus
Real Axis
I m a g i n a r y A x i s
>> z = [-10];
>> r = roots([0.1,1.15,2.5])
r =
-8.5895
-2.9105
>> p = [0 r(1) r(2)];
>> g = 1;
>> F = zpk(z,p,g)
Zero/pole/gain:
(s+10)
---------------------
s (s+8.589) (s+2.911)
>> rlocus(F);
as K →∞ two (n−m = 2)
roots approach ∞ along asymptotes
that intersect the real axis at
σ = 0− 8.5895− 2.9105 + 10
2
with angles η1 = π
2, η2 =
3π
2
as K →∞ one root approac es
the zero of F (s)
at s = −10
as K → 0 the locus of roots
approaches the poles of F (s)at s = 0,−2.9105,−8.5895
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 12/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
Λ0 jω = G0 jω C jω
2
Real Axis
I m a g i n a r y A x i s
Real Axis
I m a g i n a r y A x i s
s
s
→
2π
s + 1
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
0 3 4 5 6 7 8
0
3
4
5
Real Axis
I m a g i n a r y A x i s
Real Axis
I m a g i n a r y A x i s
Real Axis
I m a g i n a r y A x i s
Real Axis
I m a g i n a r y A x i s
s → (s + 1)2
two clockwise
encirclements
of the origin
when two zerosare encircled
s →
1
s + 1
one anti -clockwise
encirclement
of the origin
when one pole
is encircled
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 13/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
Real Axis
I m a g i n a r y A x i s
0
0
Real Axis
I m a g i n a r y A x i s
s →
s + 1
(s + 1.25)2
N = Z − P
N = number of clockwise encirclements of the origin made by
F (s) as s traverses a closed contour in clockwise direction
Z = number of zeros of F (s) inside the contour
P = number of poles of F (s) inside the contour
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
Λ0 jω = −1 for −∞ < ω <∞
s
s
ε
→∞
→ 0
|Λ0 jω p | = ∞
jω p
− jω p
x
x
1 + Λ0(s) = 1 + G0(s)C (s)
# encirclements of 0 made by 1 +Λ0 s = # encirclements of −1 made by Λ0 s
Nyquist stability criterion:
By Cauchy’s principle of the argument
the closed-loop has Z = N + P
unstable poles where
P = # RHP poles of Λ0(s)
N = # clockwise encirclements of
− 1 made by Λ0(s) as s traverses N ;
The closed-loop is stable if, and only if,
N = −P (i.e. Λ0(s) makes P anti-
clockwise encirclements of − 1.)
N (, ε)
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 14/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
limω→0
Λ0 jω
lim→∞
Λ0(ejθ)s
0 ≤ ω <∞
Λ0 s
ε→ 0
−nπΛ0(s)
n
2
−1 + j0
{Λ0(s) | s ∈ N (, ε) for sufficiently large and sufficiently small ε}
lim|s|→∞
Λ0(s) = 0
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
1
Nyquist Diagram
Real Axis
I m a g i n a r y A x i s
2 3
2
Nyquist Diagram
Real Axis
I m a g i n a r y A x i s
Λ0(s) = 1
(s + 1)2
Λ0(s) = 20
(s + 1)(s + 2)(s + 3)
ω = 0ω = +∞
ω = −∞
ω = 0
ω = +∞
ω = −∞
Λ0( jω) = 1 − ω
2
(1 + ω2)2
− j2ω
(1 + ω2)2
ω = 1
ω = −1
Λ0(s)
>> lambda = tf(1,[1 2 1]);
>> nyquist(lambda);
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 15/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
Nyquist Diagram
Real Axis
I m a g
i n a r y A x i s
Nyquist Diagram
Real Axis
I m a g i n a r y A
x i s
Λ0(s) = 5
(s + 1)(s− 2)(s + 3)
Λ0(s) = 1
s(s + 1)
ω = 0ω = +∞
ω = −∞
ω = −∞
ω = +∞
Λ0(s)
ω = 0+
ω = 0−
s = 0
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
Nyquist Diagram
Real Axis
I m a g i n a r y A x i s
Λ0(s) = 1.5(s + 1)
s(s− 4)
Λ0(s) = e−2s
(s + 1)2
ω = +∞
ω = −∞
ω = 0
ω = 0+
ω = 0−
Λ0(s)
Nyquist Diagram
Real Axis
I m a g i n a r y A x i s
Λ0(s)
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 16/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
1 + j0
−1 + j0
Nyquist Diagram
Real Axis
I m a g i n a r y A x i s
0 5 10 15 20 25 30 35 40 45 500
0.1
0.2
0.3
0.4
0.5
0.6
0.7Step Response
Time (sec)
A m p l i t u d e
L−1[
Λ0
1 + Λ0
· 1
s](t)
ω = 0
ω = +∞
closed-loop poles: s = −0.9393;
s = −0.0304± j1.3936
Λ0(s) = 0.825
s3 + s
2 + 2s + 1
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 17/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
Λ0(s) = G0(s)C (s)
Nyquist Diagram
Real Axis
I m a g i n a r
y A x i s
|a|
η
Λ0(s) = 30
(s + 1)(s + 2)(s + 3)
|1 + Λ0| = dist. from − 1 + j0 to Λ0
cross-over frequency
|Λ0( jω)| = 1
> 30◦
good
> 15 goo
< 4 goo
PM: M f = φ (degrees or radians)
GM: M g = 20 log10
( 1
|a|) = −20log
10 |a| (dB)
SP: 1
η =
1
minω |1 + Λ0( jω)| = max
ω
|S 0( jω)|
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
maxω
|Λ0 jω | < 1
M a g n i t u d e ( d B )
P h a s e ( d
e g )
Bode Diagram
Frequency (rad/sec)
M f
M g
Λ0(s) =
(s + 1)(s + 2)(s + 3) S 0 =1
1 + Λ0
20log10
1
minω |1 + Λ0( jω)|
2π
Λ0(s)
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 18/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
G s = G0 s + Gε s
G0 s
G0 s
G0(s)
G∆ s
Gε(s)
G s = G0 s 1 + G∆ s
Note: Gε s = G∆ s G0 s
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
G0(s) = 1
s(s2 + s + 1)(0.3s + 1)
G1(s) = 1s(s2 + s + 1)(0.2s + 1)
G2(s) = 1
s(s2 + s + 1)(0.5s + 1)
Gε2
G∆2
Gε1
G∆1
G0
10
100
101
0
10
M a g n i t u d e ( d B )
Frequency (rad/sec)
G0
G0(s) =s(0.5s + 1)
Gε2
Gε1
G∆2
G∆1
G0
G0(s) = 11
(10s + 1)(s2 + s + 1)
10
10
100
101
0
10
M a g n i t u d e ( d B )
Frequency (rad/sec)
10
100
101
102
0
10
20
M a g n i t u d e ( d B )
Frequency (rad/sec)
Gε2
G∆2
Gε1
G∆1
G2(s) = e−0
.2s
s(0.5s + 1)
G1(s) = 11
(20s + 1)(s2 + s + 1)
G2(s) =(5s + 1)(s2 + s + 1)
G1(s) = e−0
.4s
s(0.5s + 1)
Note: perturbations
Gε and G∆
are all stable for the
examples shown; may
not always be the case
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 19/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
Suppose G0(s) is the nominal transfer function model of a transfer function G(s)
Suppose C (s) is a strictly proper feedback controller transfer function that internally
stabilizes G0(s)
Suppose Λ0
.= G0C and Λ
.= GC have same number of RHP poles
e.g. G = G0 + Gε or G = G0(1 + G∆) with Gε, G∆ stable
Nyquist Diagram
Real Axis
I m a g i n a r y A x i s
|1 + Λ0( jω)|
Λ0 jωΛ jω
|GεC jω | = |G∆Λ0 jω |
|G( jω)| · |C ( jω)|
|1 + Λ0( jω)| < 1 or |G∆( jω)| ·
|Λ0( jω)|
|1 + Λ0( jω)| < 1 for all frequencies
Then C (s) also internally stabilizes G(s) if
The condition ensures the number of encirclements
of the critical point does not change
The condition is only sufficient (i.e. potentially conservative)
In practice may only have frequency dependent bound on
|G∆( jω)| with no phase information and so hard to do better
Result can be extended to unstable perturbations - just
need to account for appropriate number of encirclements
as required to satisfy Nyquist stability criterion
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
Suppose G0(s) is the nominal transfer function model of a transfer function G(s)
Let G∆(s) = G(s)−G0(s)G0(s)
be the multiplicative modelling error
Nominal and achieved performance similar if S ∆ jω ≈ 1 + j0 for all frequencies
his is roughly the case provided |T 0( jω)| is sufficiently small where |G∆( jω)| be-comes significant (often high frequency) so that |T 0( jω)G∆( jω)| << 1.
For a controller C (s) that achieves the nominal closed-loop sensitivity functions
S 0(s), T 0(s), S i0(s) and S u0(s)
we have by direct calculation the following perturbed sensitivity functions:
S (s) = S 0(s)S ∆(s)
T (s) = T 0(s)(1 + G∆(s))S ∆(s)
S i(s) = S i0(s)(1 + G∆(s))S ∆(s)
S u(s) = S u0(s)S ∆(s)
where S ∆(s) .= 1
1+T 0(s)G∆(s)
Model uncertainty typically limits loop-gain bandwidth
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 20/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
M a g n i t u d e ( d B )
2
3
P h a s e ( d e g )
Frequency (rad/sec)
G0S i0
T 0
C Λ0
Consider a feedback control loop with:
nominal plant model G0(s) = 2
(s + 1)(s + 2), unit step reference r(t) = ς (t),
input disturbance di(t) = A sin(t + φ), stable MME s.t. |G∆( jω)| ≤ ω√ ω2 + 400
and controller C (s) = 600 (s + 1)(s + 2)(s + 3)(s + 5)s(s2 + 1)(s + 100)
T 0 jω ≈ 1 over the range 0 ≤ ω ≤ 1,
by virtue of high gain of controller; as such,
reference tracking performance is good
gain of input sensitivity function is 0 at
1 rad/sec, by virtue of controller poles at ± j;
as such, input disturbance rejection is achieved
the LHP zeros of controller add phase to achieve
good phase margin; sensitivity peak also small
loop-gain bandwidth is maximized for fast response
while accounting for modelling uncertainty which
becomes of size 1√
2≈ 0.7 at 20 rad/sec
high gain of controller at high frequency could
be problem in the presence of measurement noise
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
1 + j0
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 21/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
ωb : |Λ0 jωb | = 1
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
S
0 s = 1− e
−τ s
T
0 (s) = e
−τ s1
10
100
101
102
0
10
M a g n i t u d e ( d B )
Bode Diagram
Frequency (rad/sec)
T
0 (s)
S
0 s
Given a plant model G(s) = e−τ s
G0(s)
the muliplicative modelling error is
G∆(s) = e−τ s
− 1
which has same magnitude plot as S 0(s)
1
τ =
1
0.02 = 5
e−τ s
e−τ s
R
Do
Y
−
+ ++
+
+
1
1− e−τ s
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 22/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
Given a nominal plant G0(s) = B0(s)A0(s)
and any controller C (s) = P (s)L(s) , suppose
s = z is a zero and s = p a pole of Λ0(s) = G0(s)C (s) (i.e. z is an uncancelledero and p an uncancelled pole of the plant or controller):
T 0(z) = B0(z)P (z)A0(z)L(z) + B0(z)P (z)
= 0 S 0(z) = 1 − T 0(z) = 1
S 0( p) = A0( p)L( p)
A0( p)L( p) + B0( p)P ( p) = 0 T 0( p) = 1− S 0( p) = 1
If s = p is an uncancelled pole of C (s) then
S i0( p) = B0( p)L( p)
A0( p)L( p) + B0( p)P ( p) = 0
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
−
α > 0
z p z > −α and p > −α
z and p
∞
0
h(t)e−ztdt = lim
s→z
H (s) for any z in region of convergence for H (s) .= L[h](s)
S 0 z = 1, T 0 z = 0, S 0 p = 0, T 0 p = 1, S 0 0 = 0, T 0 0 = 1
s = 0
E (s) = S 0(s)(R(s)−D(s))
For R(s) = 1
s and D(s) = 0 OR R(s) = 0 and D(s) = −
1
s
(i) ∞
0
e(t)e−zt dt = E (z) = S 0(z)1
z
= 1
z
and
(ii)
∞
0
e(t)e− pt dt = E ( p) = S 0( p)1
p = 0
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 23/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
z
p α
s = 0
s = p
ii
i
y t > r t
s = z
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
ln |S 0 s | = − ln |1 + Λ0 s |
∞
0
ln |S 0( jω)|dω = 0
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 24/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
X (s) = X (s)Γ(s)
where X (s) is an order q − 1 polynomial depending on the initial
conditions and Γ(s) = sq +
q−1
n=0
cnsn
Γ s
x = 0 with x(0) = x; x(t) = x for t ≥ 0
x = 0 with x(0) = 0, x(0) = x; x(t) = xt for t ≥ 0
x + ω2x = 0 with x(0) = x, x(0) = 0; x(t) = x cos(ωt) for t ≥ 0
x + ω2x = 0 with x(0) = 0, x(0) = x; x(t) = x sin(ωt) for t ≥ 0
and more generally dqx
dtq+
q−1
n=0
cndnx
dtn= 0 with
dnx
dtn= xn for n = 0, . . . , q − 1
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
Λ0 s = G0 s C s
Γ(s) =
q−1
n=0
(s− γ n) (with (γ n) ≥ 0)
γ n
S 0 γ n = 0 and T 0 γ n = 1
limt→∞
e(t) = 0
Y (s) = T 0(s)R(s)
E (s) = R(s)− Y (s) = (1− T 0(s))R(s) = S 0(s)R(s)
U (s) = S u0(s)R(s)
E s = S 0 s R s
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 25/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
C s
E sR s Y sU s
D(s)
Gi0 s Go0 s
Gi0(s) = 1 and Go0(s) = G0(s)
G0 = Go0Gi0
Γ(s) = s
2
+ 1
Γ(s) =
q−1
n=0
(s− γ n) (with (γ n) ≥ 0)
γ n Gi0 s
Y s
Y (s) = S 0(s)Go0(s)D(s) U (s) = −S u0(s)Go0(s)D(s) = −
T 0(s)
Gi0(s)D(s)
Gi0 s = G0 s and Go0 s = 1
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
Gi0(s) = 1
S = S 0S ∆
S 0 =1
1+G0C
T 0 γ n = 1
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 26/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
H (s) T 0(s)
T 0 s H s = 1
H s
s = γ n
Y s = T 0 s H s R s ; E s = 1− T 0 s H s R s ; U s = S u0 s H s R s
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
r t = K 1 sin t + K 2 + ra t
K 1 K 2 ra t
[0, 5]rad/sec 3 rad/sec
T 0 s 3 rad sec
T 0 s H s 5 rad sec
s = 0 s = ± j
C (s) = (s2+3s+2)(5s2+3s+2)
s(s2+1)(s+5)
T 0(s) = 10s2+6s+4(s2+3s+4)(s+1)2
G(s) = 2
s2+3s+2
H s
0 s 2 T 0 s H s
H (s) = (s2+3s+4)(s+1)2
(10s2+6s+4)(0.01s+1)2 T 0(s)H (s) = 1(0.01s+1)2
ra(t)
5 rad sec
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 27/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
10
10
100
101
10
103
0
60
M a g n i t u d e ( d B )
Bode Diagram
Frequency (rad/sec)
H (s)
T 0 s
S 0 s
T 0 s H s
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
C (s)E sR s Y sU s
D s
Gi0(s) Go0(s)
G0 = Go0Gi0
Gf s
Gf (s) = −Gi0(s)−1
Gi0
Y (s) = T 0(s)R(s) + S 0(s)Go0(s)(1 + Gi0(s)Gf (s))D(s)
U (s) = S u0(s)R(s)− (S u0(s)Go0(s)− S 0(s)Gf (s))D(s)
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 28/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
G0 = Go0Gi0 Gi0(s) = 2
s− 2 Go0(s) =
1
s + 1
C (s) = 3(s2 + 2s + 2)
s(0.1s + 1)(0.05s + 1)
10
100
101
102
103
0
10
20
30
M a g
n i t u d e ( d B )
Frequency (rad/sec)
Gi0 Go0
C s
S 0 s
0(s)
Λ0 s
Go0 s S 0 s
peaking of sensitivity is a result of limiting
loop-gain bandwidth to be not much more
than the bandwidth of the unstable pole
for robustness to MME above 10 rad/sec
correspondingly, injected disturbances with
frequencies in the range [2, 7] rad/sec
are not attenuated by much at plant output
if these components of the disturbance were
measurable, then feedforward could be used
to improve the level of attenuation ...
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
10
100
101
102
103
0
M a g n i t u d e ( d B )
Frequency (rad/sec)
Go0(s)S 0(s)
Go0(s)S 0(s)(1 + Gi0(s)Gf (s))
1 + Gi0(s)Gf (s)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50
0.5
1
1.5
2
2.5
0
0.5
1
1.5
2
2.5
Time (sec)
Gf (s) = −
s− 2
2(0.01s + 1) ≈ −Gi0(s)−1
7/21/2019 Control System Part 3
http://slidepdf.com/reader/full/control-system-part-3 29/29
ELEN90055 Control Systems: Part IIIM.Cantoni (c) 2011, 2012, 2013
−1 + j0