5/13/2003
Robust Control 8
Robust Stability
Harry G. Kwatny
Department of Mechanical Engineering &
Mechanics
Drexel University
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Robust Stability and Performance - Outline | MEM 633, Study notes of Mechanical Engineering
Material Type: Notes; Class: Robust Control Systems I; Subject: Mechanical Engr & Mechanics; University: Drexel University; Term: Spring 2003;
Typology: Study notes
Pre 2010
1 / 46
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Robust Stability and Performance - Outline | MEM 633 and more Study notes Mechanical Engineering in PDF only on Docsity!
5/13/
Robust Control 8 Robust Stability
Harry G. Kwatny Department of Mechanical Engineering &Mechanics Drexel University
Outline •^ Modeling Uncertainty •^ Robust Stability/Robust Performance •^ Robust Stability of the
M
-^ ∆^
Structure
•^ Uncertainty using Coprime Factors •^ The Structured Singular Value
Example
( )^
(^
2 2 2
A spacecraft attitude control model that includes the rigid body dynamicsplus one flexible mode is given by the transfer function
2
2 1
A nominal model that includes only the rigid body d
p
s^
s
G^
s^
s^ s +^ +^ s =^
+^ +
( ) ( )^
( )^
(^
( )^
(^
)^ ( )
(^
2
2
2
2
2 2
(^22)
ynamics is
2 Additive uncertainty:
2
2
2
1 1
1
Multiplicative uncertainty:
1
2
1
p
p
G s p
s
s^
s
G^
s^
G s
s^
s^
s
s^ s
s G^ G
s
G^
s^
G s
G^
s^
s
=
+^
+^
−
=^
- ∆^ ⇒
∆ =
−^
=^
+^ +
+^ + −^
−
=^ + ∆
⇒^ ∆ =
=^
+^ +
M-
∆^
Structure
y ∆
u ∆
y ∆
u ∆
(^
(^1) )
11
12
22
21
N^
F^
P^
P^
I^ P K
− P
=^
+^
−
y ∆
u ∆ 11 M^
N =
Singular Values^ Recall a singular value and the corrspondin
g^ sin
gular vectors
of a rectangular matrix
are, respectivel
y, the scalar
and
the two vectors
,^
that satisfy
The singular value decomposition of
T
A
u v
Av^
u A u
v
σ
σ= σ=
is
T
A
A^
U^
V
=^
Robust Stability/Robust Performance
(^
22
21
11
12
22
The perturbed system closed loop transfer function
is
Nominal Stability (NS)
is stable
Nominal Performance (NP)
1; & NS
Robust Stability (RS)
w^
z
F^
N^
N^
I^
N^
N N
N
−
∆
∞
=^
+^
∆^
−^
⇔^
is stable
,^
1; & NS
Robust Performance (RP)
1,^
,^
1; & NS
F^ ∆^ F
∞
∆^ ∞^
∞ ∀∆
⇔^
<^
Determinant Stability Condition
( )^
Assume
,^
are stable and
belon
gs to
a convex set of perturbations, such that if
' is a member
then so is
' for any real scalar
with
- Then the
system is stable for all admissib Theorem:
M^
s^
s
c^
c^
c
M
∆^
∆^
(^
(^
(^
(^
(^
)^
(^
le perturbations if
and only ifThe Nyquist plot of det
does not encircle
the origin for alldet
0,^
1,^
,^
,^
1,^
, dim
i
I^
M
I^
M^
j
M^
j^
i^
M
−^
⇔^
−^
∆^
≠^
∀^
⇔^
∆^
≠^
∀^
Spectral Radius Stability Condition
( )^
( )
Assume
,^
are stable and
belongs to
a set of perturbations, such that if
' is a member then so is
' for any complex scalar
with
- Then the
The system is stabl
e for all admissibl
ore
e
m:^
M^
s^
s
c^
c^
c^
M
∆^
∆^
≤^
(^
) (^
) (^
) (^
)
perturbations if and only if
1,^
max
M^
j M^
j
∆
∆^
<^
∀^
∆^
<^
Coprime Factorization
1
1
A transfer matrix can be written in (left or ri
ght) coprime factored form
where
,^ are stable, i.e.,
contains all of the RHP-zeros of
and
contains all of the RHP-poles of
r^ r
G^
D^ N
N D N D
N^
G^
D
−^
−
=^
=
A^ A
as RHP-zeros.
,^ are coprime, i.e., they have no common RHP-zeros which resultin cancelation. Formally, they satisf
y^ the Bezout identit
y: there exist
stable
,^
that satisfy
or
l^
l
G
N D
U V N U
DV
I
+^
= *^
*^
*^
a coprime factorization is called normalized if
or
r^
r
l^ l^
l^ l^
r^ r^
r^ r
UN^
VD^
I
N N
D D
I
N N
D D
I +^
=
+^
=^
+^
=
Example
( )^
(^
)(^
(^
)(^
( )^
( )^
(^
)(^
)^
( )^
(^
)(^
)^
1 0
2
2
1
0
1
0
An obvious factorization is
Another one is
,^
,^ ,
s^
s
G s
s^
s s^
s
N^ s
D s s^
s
s^
s^
s^
s
N^ s
D s
a^ a
s^
a s^
a^
s^
a s^
a
−^
=^
−^
−^
=^
+^
−^
+^
−^
=^
=^
+^
+^
+^
Uncertainty Using Coprime Factors
(^
)^ (
)
1
(^11)
Recall that a transfer matrix can be written incoprime factored formSuppose we allow separate perturbations in the numeratorand denominator so that the actual plant is
,
r^ r
p^
D^
N
G^
D^
N^
N D
G^
D^
N
−^
− −
=^
= =^
∆^
∆^
∆
A^
A A
A^
[^
]
[^
]^
(^
)
[^
]
1
1
This is equivalent to a system in
structure with
=^
,
RS^
1/ N^
D
N^
D N^
D
M K M^
I^ GK
D
I
M
ε
ε
∞ − − ε ∞
∞
∆^
≤ − ∆ ^
∆^
∆^
∆^
= −^
^ ^
⇒^
∀^
∆^
∆^
≤^
⇔^
<
A
Uncertainty Using Coprime Factors, Contunued
y ∆
u ∆
(^
(^1) ) 1
K^ I
GK
D I
−^ − ^ −
^ ^ ^
A [^
] N^
D ∆^
∆
∆ D
∆ N N^ l
(^1) − D l K −
Controller Design with Coprime Uncertainty
(^
)^ (
) [^
]
{^
}
(^
1 1 )
1
Consider the family of perturbed plantswith some stability margin
The system is robustly stabilized by the controller
if
and only if
1
Robust Stabili
p^
D^
N^
N^
D
G^
D^
N
u^
Ky
K^
I^ GK
D I
ε
ε
γ
ε
−
∞
−^
− ∞
=^
∆^
∆^
∆^
∆^
<
=
^
−
≤
^ ^
A^
A A
Find the lowest achievable
and the corresponding maximum stability mar
gin^
and the
corresp
zer Design onding cont
Problem: roller
. K
γ
ε
Solution to RSDP
(^
(^
(^
)^
(^
(^
)^
(^
( )^
(^
1/ 2
1 min^
max 1
1
1
1
1
1
1
1
1
where
,^
are the unique positive definite sol'ns of the ARE's
0 0
and
minimal realization
,^ ,^
, T
T^
T^
T^
T
T T^
T^
T^
T
XZ
X Z A^
BS^
D C Z
Z
A^
BS^
D C
ZC R
CZ
BS
B
A^
BS^
D C
X
X
A BS
D C
XBS
B X
C R
C
G s
A B C D
R^
I γ^
ε
ρ − −
−^
−^
−
−^
−^
−^
−
=^
=^
−^
+^
−^
−^
+^
=
−^
+^
−^
−^
+^
=
⇔ =^
+^
T^ ,
T
DD
S^
I^ D D =^