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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 1
METR4202 – Advanced Control & Robotics
Lecture 4
Review of Controller DesignFrequency Domain PD Controller Design
State-Space Feedback Control Design
Introduction to State-Space Observer Design
G. Hovland 2004-2006
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 2
Solutions: Tutorial week 4, Q5
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 3
Step 1a: Find Phase Margin and Frequency
o
M
p
n
aa
T w
6.587219.0
1823.1tan
412
2tan
90.33495.010.11
5912.0)1.0(ln
)1.0ln(
100
%ln
100
%ln
42
2
2222
=
=
++−
=Φ
=−
=
−
=
=
+
−=
+
−
=
ζ ζ
ζ
π ζ
π
π π
ζ
Desired Closed-Loop Frequency Response:
Desired Open-Loop Frequency Response:
22
2
2)(
)(
nn
n
sssU
sY
ω ζω
ω
++
=
sssU
sY
n
n
ζω
ω
2)(
)(2
2
+=
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 4
Step 1b: Desired Frequency Response
ss n
n
ζω
ω
22
2
+-
r(t) y(t)
= 22
2
2 nn
n
ss ω ζω
ω
++
r(t) y(t)
Bode Plots: (wc = 2.8 rad/sec) and Phase Margin = 58.6o
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 5
Tutorial Q5: Step 2a, Actual Response
Start with Kp=1 and Kd=0
Must increase gainby 32dB to get wc at2.8 rad/sec
Convert dB to gain:
32 dB = 10(32/20)
≈ 40 = Kp
+=+= 1)( s
K
K K sK K sG
p
d pd pc
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 6
Tutorial Q5: Step 2b, Actual Response
Phase Curve
Phase margin only15o. Needs to be liftedapprox. 45o.
Kp / Kd is the locationof the zero. The zerowill lift the phase curve.Approximately 45o liftwhere zero is.
Choose Kd = Kp /2.8 = 14.3,which gives a zero at2.8 rad/sec
+= 1)( s
K K K sG p
d pc
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 7
Tutorial Q5, Step 2c: Actual Response
Bode plots with Kp = 40 and Kd = 14.3
Phase margin now okThe zero has also increased the
crossover frequency wc.
Some fine-tuning required, but simply reduce gain while keeping zero.Kp = 30 and Kd = Kp / 2.8 = 10.7 gives desired result.
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 8
Tutorial Q5, Step 3: Verification
Peak time exactly1.0 seconds.
Overshoot slightly higherthan 10%.
Design is not accurate, sincethe PD controller and G(s)
do not match exactly
ss n
n
ζω
ω
22
2
+
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 9
Bode plots of (s+a) = a (s/a + 1)
Figure 10.6
a. magnitudeplot;b. phase plot.
Review from Lecture 1b. Asymptotic Bode plots
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 1 0
Gain and Phase Margin via Nyquist (10.6)
Figure 10.35Nyquist diagramshowing gainand phasemargins
Review from Lecture 1b. The Nyquist plot explains whywe look at 0dB and -180o for the open-loop transferfunction.
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 1 1
Tutorial w4, Q6
roots([1 6 11 6]))3)(2)(1(
1)(
+++=
ssssG
us
1
1
-1
s
1
1
-2
s
1
1
-3
y
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 1 2
Tutorial w4, Q6b
us
1
1
-1
s
1
1
-2
s
1
1
-3
y
z1z2z3dz1 /dtdz2 /dtdz3 /dt
[ ]
Du
z
z
z
C y Bu
z
z
z
A
z
z
z
DC B A
u z z
z z z z z z
z z z z
+
=+
=
==
=
−
−
−
=
+−=
+−=
+−=
3
2
1
3
2
1
3
2
1
33
322
211
0001
1
0
0
100
120
013
23
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 1 3
Tutorial w4, Q7
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 1 4
Tutorial w4, Q7: Step 1. Desired Polynomial
We already have the desired polynomial from Q1
90.33495.010.11
5912.0)1.0(ln
)1.0ln(
100
%ln
100
%ln
2
2222
=−
=
−
=
=
+
−=
+
−
=
π
ζ
π
π π
ζ
p
n
T w
2.156.41
2
1
2
22
++=
++
ss
ss nn ω ζω
6082.1996.44
1
)2.156.4)(40(
1232
+++=
+++ ssssss
The system is 3rd order. We have no zeros to cancel. Hence, weadd another pole at 40 (a decade above wn) where it has littleeffect on overshoot and peak time.
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 1 5
Tutorial w4, Q7: Step 2. Phase Variable-Form
u
x
x
x
x
x
x
asasasssssG
u
x
x
x
aaa x
x
x
+
−−−
=
+++=
+++=
+
−−−
=
1
0
0
6116
100
010
1
6116
1)(
1
0
0
100
010
3
2
1
3
2
1
01
2
2
323
3
2
1
2103
2
1
Ax Bx
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 1 6
Tutorial w4, Step 3: Controller Gains by Inspection
+−+−+−
=−
)6()11()6(
100
010
321 k k k
K B A x x x
6.38
2.188
602
6082.1996.44
1
3
2
1
23
=
=
=
+++
k
k
k
sss
Desired Polynomial:
Controller gains inphase-variable form
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 1 7
Tutorial w4, Q7: Step 4. Transformation Matrix
The transformation matrix P to convert from cascade tophase-variable form
==
−
−==
−
−==
−
156
013
001
*
2561
610
100
][
111
310
100
][
1
2
2
MX MZ
x x x x x MX
z z z z z MZ
C C P
B A B A BC
B A B A BC
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 1 8
Tutorial w4, Q7: Step 5. Final Controller Gains
[ ]6.388.48.384
159
013
001
]6.382.188602[
1
−=
−
−=
=−PK K x z
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 1 9
Tutorial w4, Q7: Verification
Peak time exactly 1.0 seconds
Overshoot exactly 10%.
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 2 0
Limitations of state-feedback control
In practice, all the states x are not measured andavailable for feedback.
For mechanical systems, such as robots, the jointangles are typically measured but the velocities not.
Reducing the number of sensors reduces the cost of theoverall control system.
Hence, we need to develop methods for state-spacecontroller design when the only measurement is thescalar y.
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 2 1
State-feedback design using observers
A) Unforced observer B) Forced observer
Details offorced observer
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 2 2
Unforced Observer
)ˆ(ˆ
)ˆ(ˆ
ˆˆˆˆ
xxC
xxAxx
xC
BxAx
Cx
BAxx
−=−
−=−
=
+=
=
+=
y y
yu
y
u
The plant dynamics
The observer dynamics
The error dynamics.Note that the errorapproaches zero given by the
transient dynamics of A
xxe ˆ−=
This is not a particularly good observer; we want the error dynamicsto be much faster than the system dynamics.
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 2 3
Forced Observer
)ˆ)((ˆ
)ˆ(ˆ
)ˆ()ˆ(ˆ
ˆˆ
)ˆ(ˆˆ
xxLCAxx
xxC
LxxAxx
xC
LBxAx
Cx
BAxx
−−=−
−=−
−−−=−
=
−++=
=
+=
y y
y y
y
y yu
y
u The plant dynamics
The forced observerdynamics
Forced error dynamics. Wecan now design the transientresponse of the error byplacing the poles of A-LC
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 2 4
Observer Design vs. Controller Design
In controller design we placed the poles of A-BK
The phase-variable or the controller canonical formsmade feedback gain selection easy: by inspection
In observer design we will place the poles of A-LC
The observer canonical form will turn out to be ideal forobserver feedback gain selection: also by inspection
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 2 5
3rd Order Observer: Observer Canonical Form
Signal-flow chart forthe unforced observer
Signal-flow chart forthe forced observer
01
2
2
3
12
2
3)(asasas
bsbsbsG
+++
++=
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 2 6
Observer Design Procedure
Desired Poles:
ObserverCanonicalForm
Here you see why
this form has the
advantage
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 2 7
Example 12.5: Observer Design
From Lecture Notes 3
Desired poles:
20136)4)(52(
))(2(
232
22
+++=+++=
+++
ssssss
pswsws nnζ
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 2 8
Observer Canonical Form
1ˆˆˆ
)ˆ(ˆˆ
xxC
LBxAx
==
−++=
y
y yu
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 2 9
Ex 12.5: Characteristic Polynomial
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 3 0
Ex 12.5: Desired Polynomial
20136)4)(52(
))(2(
232
22
+++=+++=
+++
ssssss
pswsws nnζ
Desired Poles from Example 12.4:
10 Times Faster wn:
500002500120)100)(50020(232
+++=+++= ssssss
This is the pole selection in the book,
alternatively cancel the zero at 4.
By inspection:
= 112,
= 2483,
= 49990
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 3 1
Observer Response
A) With observer feedbackgains L
B) With observer feedbackgains disconnected
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 3 2
Observable and Non-Observable Systems
All the states influencethe output
The state x1 does notinfluence the output
Observability by inspection:Convert to parallel form by eigenvectors
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 3 3
The Observability Matrix: Analytic Approach
An nth-order plant whose state and output
dynamics are:
is completely observable if the matrix
is of rank n.
Cx
BAxx
=
+=
yu
=
−1n
M O
CA
CA
C
Compare
withcontrollability
matrix
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 3 4
Example 12.6: Observability
Determine if the system above is observable!
32
3213
32
21
5
234
x x y
u x x x x
x x
x x
+=
+−−−=
=
=
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 3 5
Example 12.6: A and C Matrices
[ ]150
234
100
010
=
−−−
= CA
The observability matrix:
−−−
−−=
=
91312
334
150
2
CA
CA
C
M O
det(OM) = -344 ≠ 0 Hence, the system is observable
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 3 6
Example 12.7: Unobservability
This system looksas if it could beobservable.
21
212
21
45
4
215
x x y
u x x x
x x
+=
+−−=
=
]45[
4
215
10=
−−= CA
−−=
=
1620
45
CA
C
M O Rank is 1, not observable
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 3 7
Summary so far: Observers
We need observers for state-feedback control when notall the states x are measured.
First, check observability. If the system is observable,you can go on and design an observer.
Second, if the system is in observer canonical form, theobserver design is a simple gain matching by inspection.
If the system is not in observer canonical form, we needto first transform the model into this form.
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 3 8
Transformation Matrix P
Original System: Transformed System:
Cz
BAzz
=
+=
y
u
=
−1n
MZ O
CA
CA
C
CPx
BPAPxPx
=
+=−−
y
u11
P
APPCP
APPCP
CP
MZ
n
MX OO =
=
−−
−
11
1
)(
)(
Hence,
MX MZ OO1−
=P
We now have two ways
of obtaining P: either via thecontrollability matrices orthe observability matrices
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 3 9
Transformation to Observer Canonical Form
[ ]001
000
100
010
001
0
1
2
1
=
−
−
−
−
=
−
−
X
n
n
X
a
a
a
a
CA
This is the observer canonical form (system X):
MX MZ OO1−
=P
With this transformation matrixwe can transform any system Z
to the observer canonical form X
In Matlab: P = inv(obsv(Az,Cz)) * obsv(Ax,Cx)
Design gains in observer canonical form: Convert back to system ZLz = PLx
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 4 0
Ex. 12.8: Observer Design by Transformation
Design an observer for the plant
represented in cascade form.
Let the closed-loop performance of the observer begoverned by the characteristic polynomial:
)5)(2)(1(
1)(
+++=
ssssG
50000250012023
+++ sss
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 4 1
Ex 12.8: Cascade Form
)5)(2)(1(
1)(
+++=
ssssG
u z z z z z
z z z
+−=
+−=
+−=
33
322
211
2
5
[ ]001
100
120
015
=
−
−
−
= CA
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 4 2
Ex 12.8: Observability Test
−
−=
=
1725
015
001
2CA
CA
C
MZ O
Determinant equals 1Hence, observable
)5)(2)(1(
1)(
+++=
ssssG
Poles expanded:
1017823
+++ sss
Observer Canonical Form X
]001[
0010
1017
018
=
−
−
−
= X X CA
−
−=
=
1847
018
001
2
X X
X X
X
MX O
AC
AC
C
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M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 4 3
Ex 12.8: Observer Gains by Inspection
+−
+−
+−
=−
00)10(
10)17(
01)8(
3
2
1
l
l
l
X X X CLA
Characteristic Polynomial is:
)10()17()8(32
2
1
3 lslsls ++++++
50000250012023
+++ sss
Desired Polynomial is:
Observer design by coefficient matching
L1 = 112 L2 = 2483 L3 = 49990
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 4 4
Example 12.8: Final Design
Convert observer gains back to original cascade form:Lorig = PLX = OMZ
-1 * OMX * LX = [ 112 2147 47619]T
8/14/2019 Adv Control & Robotic Lec 4
http://slidepdf.com/reader/full/adv-control-robotic-lec-4 23/23
M E T R 4 2 0 2 / 7 2 0 2 – A A d v a n c e d C o n t r o
l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 4 5
Ex 12.8: Forced and unforced responses
Forced response
Response with observergains disconnected. This
response has the sametransient as the systemitself.
0 2 / 7 2 0 2 – A A d v a n
c e d C o n t r o l & R o b o t i c s ,
S e m e s t e r 2 ,
2 0 0 5 : P a g e : 4 6
Summary: Observer Design
The observer canonical form is used to find the observergains by inspection
As for state-space controller gains, the observer gainsneed to be transformed back to the original state-space
model using the transformation matrix P. For observer design, the transformation matrix P is
found from the observability matrices.
For controller design, the transformation matrix P isfound from the controllability matrices.
The observer poles should be typically designed 10times faster than the controller poles.
With the information in this lecture, you can nowproceed with the observer for the inverted pendulum
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