Transcript Dr. Vladimir Kucera

The H2 Control Problem
A Detailed Comparison of State-Space
and Transfer-Function Solutions
Vladimír Kučera
Czech Technical University in Prague
SWAN 2006
The University of Texas at Arlington
Motivation
For analytical design of control systems,
it is often convenient to measure system performance
in terms of norm of the closed-loop system transmittance
from the exogenous signals to the regulated variables
One common measure of performance for a linear system
is the H2 norm of its transfer function
The H2 norm is relevant when minimizing
 the variance of stochastic signals
 the peak amplitude of deterministic signals
Czech Technical University in Prague
1
Problem Formulation
Given a linear, finite dimensional, time invariant plant P,
find a controller C that stabilizes the control system
and minimizes the norm
of its transfer function T
v
from v to z
P
1 
T
T 
 Tr T ( j) T ( j) d 
2  
z
y
u
C
in RH2
(the space of strictly proper stable rational matrices).
Czech Technical University in Prague
2
Methods of Solution
 Systems
in state space form
An optimal controller is obtained in observer form
by solving two algebraic Riccati equations
 Systems
described by transfer functions
The optimal controller transfer function is obtained
via two inner-outer factorizations
and two proper-stable projections
Czech Technical University in Prague
3
Comparison
It is well understood that the inner-outer factorization
corresponds to solving an algebraic Riccati equation.
However, why are the proper-stable projections
not needed in the state-space approach?
Efficient numerical algorithms are now available
to handle operations on and among polynomial matrices.
However, why is the state space algorithm still preferred?
Czech Technical University in Prague
4
Standard Assumptions
Given a state-space realization of the plant
P
P   11
 P21
F G1 G2 

P12 
:   H 1 0 J 12 

P22  
 H 2 J 21 0 
It is assumed that ( F, G2) is stabilizable, ( F, H2) is detectable,
the matrices
 F  j I G 2  ,  F  j I G 1 
 H1
J12   H 2
J 21 
have full column and row rank, respectively, for all finite ω
and
T
T
J 12
J 12  I , J 21J 21
 I.
Czech Technical University in Prague
5
Fractional Representations
Firstly, P is represented
in the form of doubly coprime factorizations over RH
(the space of proper stable rational matrices)
P  D 1 N  N D 1
where
and
 I D12 
 N 11 N 12 
D
, N 


0
D
N
N

 21
22 
22 
I
0 
 N 11 N 12 

N 
, D

 D21 D22 
 N 21 N 22 
Note the block-triangular structure of D and D .
Czech Technical University in Prague
6
Stabilizing Controllers
Then all controllers that stabilize P are parametrized as
CS (W )  ( X  WN 22 )1 (Y  WD22 )
 (Y  D22W )( X  N 22W )1
where X, Y and X , Y are proper stable rational matrices
that satisfy the Bézout identity
 D22  N 22   X N 22   I 0 



 Y


X
0
I
Y
D



22  
and where W is a free parameter that ranges over RH .
Czech Technical University in Prague
7
Transfer Function Solution
The closed loop transfer function is
T  P11  P12 CS ( I  P22 CS ) 1 P21
where CS is any stabilizing controller.
The strategy to minimize T is to
express T as a function of the parameter W
using any but fixed doubly coprime factorization of P,
then manipulate the expression
so that T 2 as a function of W has no linear term.
Two inner-outer factorizations
and two proper-stable projections are used in the process.
Czech Technical University in Prague
8
Norm Minimization (1)
Using doubly coprime factors,
T  N 11  [ D12 ( X  N 22W )  N 12 (Y  D22W )] N 21
 
~
V
U N 21
~ T is inner ( ~ ~ 
where U T is outer, N
N 21 N 21  I ),
21
and
~*
~*
TN 21
 N 11 N 21
 VU
~*
 ( N 11 N 21  S )  (VU  S )



 
T1  RH 2 V1  RH 2
*
where S is the proper stable part of N 11 N~ 21
.
Then
2
2
2
T  T1  V1 .
Czech Technical University in Prague
9
Norm Minimization (2)
Now V1 is a function of W ,
V1  [( D12 X  N 12Y )U  S ]  N 12 (WU )


~
N11K
N 12U
~ is inner ( ~  ~
where U is outer, N
N 12 N 12  I ),
12
~*
~*
and
N 12V1  N 12 N 11 K  U WU
~*
 ( N 12
N 11 K  S )  (U WU  S )


 
V  RH 2
T2  RH 2
~*
where S is the proper stable part of N 12
N 11 K .
Then
2
2
2
V1  T2  V .
Czech Technical University in Prague
10
Optimal Controller
To summarize,
T
2
2
2
 T1  T2  V
2
where only the final term
depends on W.
V  U WU  S
Thus the optimal controller C0 corresponds to V  0,
hence
C0  CS (U 1 S U 1 )
and
T 2  T1 2  T2 2 .
Czech Technical University in Prague
11
State Space Solution
The strategy to minimize T is to
express T as a function of the parameter W
using special doubly coprime factorizations of P
(and the corresponding solutions to the Bézout identity),
then manipulate the expression
so that T 2 as a function of W has no linear term.
Solution of two algebraic Riccati equations
is used in the process.
Czech Technical University in Prague
12
Operations with Systems
Denote a system transfer function
F G
S  H ( sI  F )1 G  J :  
 H J 
Then
0
G2 
 F2
S1 S 2  G1 H 2 F1 G1 J 2 


 J1 H 2 H1 J1J 2 
 F
S : S (  s )  
T

G

T

T
H 

JT 
T
G1 
 F1 0
S1  S 2   0 F2
G2 


 H1 H 2 J1  J 2 
1
1


F

GJ
H

GJ
1
S 
1
1 
J
H
J


Czech Technical University in Prague
13
Plant Matrix Fractions (1)
Construction of right coprime, proper stable factors
w
u
x
G2
H2
y
F
L
1
u  [ I  L ( sI  F  G 2 L) G 2 ]w ,

 F G 2 L G 2 
D22  
I 
 L
y  H 2 ( sI  F  G 2 L)1 G 2 w



 F G 2 L G 2 
N 22  
0 
 H2
1
y  N 22 D22
u  P22 u
where L is a matrix such that F + G2L is stable.
Czech Technical University in Prague
14
Plant Matrix Fractions (2)
Construction of left coprime, proper stable factors
K
u
x̂
G2
e
y
–
H2
F
e  H 2 ( sI  F  KH 2 )1 G2 u  [ I  H 2 ( sI  F  KH 2 )1 K ] y



F  KH 2 G2 
N 22  
 H 2
0 
0  N 22 u  D22 y
F  KH 2 K 
D22  
 H 2
I 
1
y  D22
N 22 u  P22 u
where K is a matrix such that F + KH2 is stable.
Czech Technical University in Prague
15
Bézout Equations
Solution pairs to the Bézout equations
D22 X  N 22Y  I
D22 X  N 22Y  I
can be explicitly constructed as
F  G2 L K 

X
  H 2
I 
Y 

F  G2 L K 
L
0 
F  KH 2 G2 

X
  L
I 
Y 

F  KH 2 K 
L
0 
Proof by direct verification,
using the rules for operations with systems.
Czech Technical University in Prague
16
Inner Functions
Select L so as to make N 12 inner
 FLT
N N 12  
T
  G2

12
H 1TL   FL

J 12T   H 1 L
 FL
  H 1TL H 1 L
 T
 J 12 H 1 L
0
 FLT
 G2T
G2  with the notation FL : F  G2 L
J 12 
H 1 L : H 1  J 12 L
G2 
H 1TL J 12 

T
J 12 J 12 
FL

  FLT X L  X L FL  H 1TL H 1 L

 L  (G2T X L  J 12T H 1 )
similarity transformation
0
 I
  X L I 
0
 FLT
 G2T
G2

LT  (G2T X L  J 12T H 1 )T   I

I

on putting
FLT X L  X L FL  H1TL H1 L  0, L   (G2T X L  J 12T H1 )
Czech Technical University in Prague
17
Strictly Proper Stable Rational Functions

N 11 belongs to RH 2
For such a gain L, N 12
T


F

N 12
N 11   LT
  G2
H 1TL   FL
T 
J 12
  H1L
 FL
  H 1TL H 1 L
 T
 J 12 H 1 L
 FL
 0

 0
0
 FLT
 G2T
0
 FLT
 G2T
G1 
0 
G1 
0

0 
G1 
 FLT

X LG1  
  G T
2
0 
with the notation FL :  F  G2 L
H 1 L : H 1  J 12 L
similarity transformation
0
 I
  X L I 
X LG1 


RH
2

0 
By duality, a gain K can be selected so as
to make N 21T inner and N 11K N 21 belong to RH 2 .
Czech Technical University in Prague
18
Optimal Controller


Now TN 21
 N 11 N 21
 V1





 RH 2  RH 2


N 12
V1  N 12
N 11 K  
W


 RH 2  RH 2
and
yield
T
2
2
 N 11  N 11K
2
 W
so that minimum is achieved for W = 0.
The optimal controller results in the observer-based form
C S  X 1Y  Y X 1  

F  G2 L  KH 2 K 
L
0 
applying the rules for operations with systems.
Czech Technical University in Prague
19
Comparison (1)

The transfer-function approach
takes any but fixed doubly coprime factors of P,
while the state-space approach
parametrizes all such factors using stabilizing gains K a L.

The transfer-function approach
extracts the inner factors from N 21 and N 12 ,
while the state-space approach
shapes these matrices to make them inner by selecting K and L.



This selection makes N 11 N 21
and N 12
N 11 K belong to RH 2,
hence no need to extract their proper stable parts S and S .
Czech Technical University in Prague
20
Comparison (2)

Thus, the difference between the two approaches derives from
a different construction and use of doubly coprime factors.

No need to solve the Bézout equations
in the state-space approach;
a particular solution can be explicitly constructed.

The observer-based form of the optimal controller is a result of
taking that particular solution to the Bézout equations.

In addition, the design parameters K and L
directly define the optimal controller R0.
Hence, the doubly coprime factors need not be calculated.
Czech Technical University in Prague
21
Computational Aspects
A wind gust disturbance rejection controller for an F-8 aircraft
is designed to compare the two approaches.
The linearized, longitudinal state equations have
5 states,
2 control inputs, 3 external inputs,
2 measurement outputs, 4 performance outputs.
The function h2syn of the MATLAB Robust Control Toolbox
is compared with the upcoming Version 3
of the Polynomial Toolbox for MATLAB.
Czech Technical University in Prague
22
Comparison (3)

Robust Control Toolbox contains a dedicated function h2syn
while Polynomial Toolbox offers general purpose functions:
ldf and rdf to create polynomial matrix fractions,
spf and spcof to calculate spectral factors,
axybc to extract proper stable parts,
fact and xab and axb to calculate the optimal controller.

The toolboxes return different optimal controllers,
in the observer-based form and in the matrix-fraction form.
 A good
match is observed: the minimum H2 norm achieved
0.2778 9787 for the state-space controller
0.2778 9788 for the transfer-function controller.
Czech Technical University in Prague
23
Comparison (4)
Although yielding almost identical results,
the two synthesis procedures are not equivalent.
In fact, the state-space algorithm
is more efficient than the transfer-function one.

The critical part of the transfer-function algorithm
is the final substitution of the optimal W into CS to obtain C0 .
This substitution generates common factors
that must be cancelled in order to obtain
the optimal controller in reduced form.

The state-space algorithm fixes the order of the controller
to equal that of the plant.
Czech Technical University in Prague
24
Comparison (5)
The computational complexity of the state-space design
depends critically on the size of the state vector x,
while the transfer-function algorithm depends largely on
the sizes of the control inputs u and the measurement outputs y.
That is why the transfer-function algorithm
is most efficient in the single-input single-output case.
Why in general is the state space design simpler?
Because the state model carries more information,
which makes it possible to parametrize the plant matrix fractions
through the stabilizing gains K and L,
and this information is fully exploited in the process.
Czech Technical University in Prague
25
References
V. Kučera,
„The H2 control problem: state-apace and transfer-function solutions,“
in Proc. IEEE Mediterranean Conference on Control and Automation,
Ancona 2006.
V. Kučera,
„The H2 control problem: a general transfer-function solution ,“
International Journal of Control,
to be published.
Czech Technical University in Prague
26