11/21/2002
Robust Control 4
Nominal Controller Design
Harry G. Kwatny
Department of Mechanical Engineering &
Mechanics
Drexel University
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Nominal Controller Design - Robust Control Systems I | MEM 633, Assignments of Mechanical Engineering
Material Type: Assignment; Class: Robust Control Systems I; Subject: Mechanical Engr & Mechanics; University: Drexel University; Term: Fall 2002;
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11/21/
Robust Control 4 Nominal Controller Design
Harry G. Kwatny Department of Mechanical Engineering &Mechanics Drexel University
Outline •^ State Feedback – pole placement •^ Observers and the separation principle •^ The LQR/LQG Problem •^ A Generalization to LQR •^ Min-Max Control •^ Solving the Ricatti Equation •^ The Modern Paradigm: H
and H 2
∞
problems
Pole Assignment Problem
(^ )
{^
}^
{^
1
Given a self conjugate set of scalars 1
,^ ,^
, and vectors
,^
completely controllable,^ ,^
find a real
matrix
such
that the eigenvalues
, ran
and eige
Pole assignment
nvecto pro
k
blem: n^
n v
x^ Ax
Bu^
A B^
B
v^
n
m m^
K
λ^
λ…
×
=^ +
{^
rs of (^1
) are precisel
y^ the
given sets.
The system is controllable if and only if
for every self-conjugate set of scalars
,^
,^ there exists a
Theorem (Wo
real
matrix
nham, 19 such that
A^ BK^ n
m^ n^
K
λ^
- λ…
×^
{^
1 ) has^
,^ ,^
as its ei
genvalues. n
A^ BK
λ^
λ
+^
Some Definitions
[^ ]
Define the matrices
:^ [^ -^
|^ ] :^ {columns form a basis for ker[
]}^
,^
,
Note:controllability
dim ker rank^
rank
n k^
m k
S^
I^ A^ B
N
R^
S^
N^ R^
M^ R M
S^ n B^ m^
N^ m λ N N
λ
λ
λ
λ
λ λ
λ λ λ λλ
×^
×
=
^
=^
=^
∈^
∈
^ ^
⇒^
=
=^ ⇒^
=
Proof: Necessity^ (^
) (^
[^
]^
0
0
Im^
Im
i^
i^
i
i^ i^
i i^
i^
i i
i
i
i^
i
i
i^
i
A^ BK
v^
v I A v^
BKvv I^ A
B^
Kv
v^
v
S^
R^
v^
N
Kv^
Kv
λ
λ
λ
+^
= ⇒^
−^
= −^ ^
⇒^
−^
= ^
−
^
^
^
=^ ⇒
∈^
⇒^
∈
^
^
^
−^
−
^
^
^
Proof: Sufficiency, 1
{^
(^
(^
k
assume the set
,^
1,^ ,^
satisifies 1), 2), 3)
3)^ there exists
C
such that
By definition
Suppose,
can be chosen such that Then, it wo
i
i^ i i
i i^
i
i
i i^
i^
i
i i^
i^
i
i^
i
v^ i^
n z^
v^ N
z
S R
I^ A N
BM
I^ A N
z^ BM
z K^
M^ z^ λ Kv
λ^ λ λ
λ λ
λ^
λ
⇒^
∈^
=^ ⇒
−^
+^
−^
+^
(^
[^
]^
1
1
1 uld follow that
Thus, real
is to be chosen so that
i n
n
i
n I
K v^
v^
M^ z^
A^ BK^ M
v
K
z λ
λ
^
=^ −
−^ +
^
^
"^
Proof: Sufficiency 3^ [^
]
[^
]
[^
1 ]
3
3
1
1 1 1
1 1
3
1 1
1
1 1
1
3
1 1
3
1 1
3
post multiply by the nonsingular matrix1/ 2^
1/ 2^0 1/ 2^ 1/ 2^0 to obtain
n
n
n
n^ R
n
R^ I^
R^ I^
n^ R
I^
R^ I^
n
R^ I^
n^ R
I^
n
K^ w^
w K v^
v^
M^ z^
M^ z
K v^
jv^ v^
jv^ v^
v^ w
jw^
w^ jw
M^
z^
M^ z
j j I K v^
v^ v^
v^ w
w^
M^ z^
M^ z
λ
λ
λ
λ
λ
λ
^
=^ −^
−^
⇒
^
^
+^
−^
=^ +
−^
−^
−
^
−
^
^
^
^
^
=^
−^
−
^
"^ =
" "^
"
"^
" [^
]
(^
3
1
3
1 1
3
Finally, since a fixed eigenstructure uniquely defines
, it can
be proved that
is unique when rank
n^. I^
n^ R^
I^
n
M^ z^
M^ z^
v^ v^
v A^ BK
K^
B^ m
v
λ
λ
−
^
−^
−
^
^ = "^
"
Example: F-16 landing approach^ [^
0.507^ 3.861 ]
0
32.^
0
0.^ 0.^
1
0
0
0
0
1
0
0
0 0 1
0
E
u^
u
q^
q
u y^
q α
α
δ
θ
θ
α θ −^
−^
−
^ ^ ^
^ ^ ^
^ ^ ^
^ ^ ^
−^
−^
−
^ ^ ^
^ ^ ^
=^
^ ^ ^
^ ^ ^
−^
−^
−
^ ^ ^
^ ^ ^
^ ^ ^
^ ^ ^
^ ^ ^ =^
^ ^ ^
0
phugoid:^
0.00022640.000272^ 0. 0.9942870.
short period:
1.7036, 0.
j^ h^ 0.0740730.
j h
λ λ
^
^ ^
^
^ ^
^
^ ^
= −^
±^
=^
± ^
^ ^
^
^ ^
−^
− ^
^ ^
−
^
−
= −^
=^ −^
0.999508 0.014171, 0.0165070.022584
^
−
^ ^
−
−
Full-State Estimator
∫ ˆx^ ∫^
ˆy
Estimator Error Dynamics
(^
(^
(^
(^
) (^
, ˆ^ ˆ
ˆ^
ˆ^ ˆ, ˆ : One approach is to select
so as to place the poles
of^
. Notice that the following two pole
placement
problems are equivalent:
,^ , x^ Ax
Bu y
Cx x^ Ax
Bu^
L^ y^
y^ y^
Cx
e^ x^
x^ e^
Ae^ LC
e
A^
LC e e L
A^ LC A^ BK
A B =^ +
= =^ +
+^
−^
=
=^ −^
⇒^ =^
= +^
⇒
+
(^
controllable)
,^ ,^
observable T^ T^
T A^ C L
A C
Example: F-
Rynaski “robust observer”
1
2
3
"place observer poles at LHP plant zeros, remainder areplaced arbitrarily"
0,^ 0.04231,
0.5865, 0
0
0
,^
,
(^1) 0.
R^
R^
R
λ =^ −
− ^
^
^
^
^
^
− ^
^
^
^
^
^
=^
= ^
^
^
^
^
^
^
^
^
^
^
^ −
[^
]
4
,
1
T
R
L
^
^
^
^
^
^
^
^
^
^
^
^
−^
−
=^
= ^
^
^
^
^
^
^
^
^
^
^
^
=^
−^
−^
−
Example: F-16^ ( )
(^
)(^
)
(^
)(^
)(^
( )^
(^
)(^
)
(^
)(^
)(^
) 2
1.^
p c
s s^
s
G^ s^
s^
s^
s^
s
s^
s^
j
G^ s^
s s^
s^
s +^
=^
−^
+^
+^
+^
+^
±
=^
+^
+^
0.^
0.05^ 0.^
0.5^1
5 10 rad^ sec 0.01 (^504030) dB 2010 0
0.^
0.5^1
5 10 MAGNITUDE^ ê
-3^ -
-^
0 1 Re@gD
(^43210) gD -1 -2 -
10 −^10
Example: F-16^ 0.^
0.05^ 0.^
0.5^1
5 10 rad^ êsec 0.01^1050 -5 dB -10 -15 - 0.05^ 0.^
0.5^1
5 10 MAGNITUDE 0.01^ 0.
0.^
0.5^1
5 10 rad^ sec 0.01 -100 -120 deg -140 -160 -
0.^
0.1^ 0.
1
5 10 PHASE^ ê