The H 2 Control Problem A Detailed Comparison of State-Space and Transfer-Function Solutions Vladimír Kučera Czech Technical University in Prague SWAN

The HThe H2 2 Control ProblemControl ProblemA Detailed Comparison of State-A Detailed Comparison of State-SSpace pace

and Transfer-and Transfer-FFunction Solutionsunction Solutions

Vladimír KučeraVladimír Kučera

Czech Technical University in PragueCzech Technical University in Prague

SWAN 2006SWAN 2006

The University of Texas The University of Texas atat Arlington Arlington

2

Czech Technical University in Prague

MotivationMotivation

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

3


Problem FormulationProblem 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

from v to z

in RH2

(the space of strictly proper stable rational matrices).

P

C

v

u y

z

dTr

21

)()( jTjTT T

4


Methods of SolutionMethods 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

5


ComparisonComparison

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?

6


Standard AssumptionsStandard Assumptions

Given a state-space realization of the plant

00:

212

121

21

2221

1211

JHJHGGF

PPPP

P

It is assumed that ( F, G2) is stabilizable, ( F, H2) is detectable,

the matrices

have full column and row rank, respectively, for all finite ω

and

212

1

121

2 ,JHGIjF

JHGIjF

., T212112

T12 IJJIJJ

7


where

and

Note the block-triangular structure of D and .

Fractional RepresentationsFractional Representations

Firstly, P is represented

in the form of doubly coprime factorizations over RH

(the space of proper stable rational matrices)

2221

1211

22

12 ,0 NN

NNN

D

DID

11 DNNDP

22212221

1211 0,

DDI

DNNNN

N

D

8


Stabilizing ControllersStabilizing Controllers

Then all controllers that stabilize P are parametrized as

where X, Y and are proper stable rational matrices

that satisfy the Bézout identity

and where W is a free parameter that ranges over RH .

I

I

DY

NX

XY

ND

0

0

22

222222

YX ,

12222

221

22S

))((

)()()(

WNXWDY

WDYWNXWC

9


Transfer Function SolutionTransfer Function Solution

The closed loop transfer function is

where CS is any stabilizing controller.

The strategy to minimize 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 as a function of W has no linear term.

Two inner-outer factorizations

and two proper-stable projections are used in the process.

211

S22S1211 PCPICPPT )(

T

2T

10


Using doubly coprime factors,

where is outer, is inner ( ),

and

where S is the proper stable part of .Then

21

212212221211

~)]()([

NU

N

V

WDYNWNXDNT

Norm Minimization (1)Norm Minimization (1)

INN 2121

~~

.21

21

2 VTT

*2111

~NN

T21

~N

2121

*2111

*2111

*21

)()~

(

~~

RHV

SVU

RHT

SNN

VUNNNT

TU

11


Now V1 is a function of W ,

where is outer, is inner ( ),

and

where is the proper stable part of .Then

Norm Minimization (Norm Minimization (22))

222

11*12

11*121

*12

)()~

(

~~

RHV

SWUU

RHT

SNN

WUUNNVN

K

K

)(~

])[(

12

12

11

12121 WU

UN

N

N

SUYNXDV

K

U 12~N INN

1212~~

S KNN 11*12

~

.222

21 VTV

12


Optimal ControllerOptimal Controller

To summarize,

where only the final term

depends on W.

Thus the optimal controller C0 corresponds to

hence

and.2

22

12 TTT

222

21

2 VTTT

,0V

SWUUV

)( 11S0

USUCC

13


State Space SolutionState Space Solution

The strategy to minimize 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 as a function of W has no linear term.

Solution of two algebraic Riccati equations

is used in the process.

T

2T

14


Operations with SystemsOperations with Systems

JHGF

JGFsIHS :)( 1

21121

21121

22

21

0

JJHHJJGFHG

GFSS

2121

22

11

21 00

JJHHGFGF

SS

Denote a system transfer function

Then

TT

TTT )(:

JG

HFsSS

11

111

JHJ

GJHGJFS

15


Construction of right coprime, proper stable factors

where L is a matrix such that F + G2L is stable.

Plant Matrix Fractions (1)Plant Matrix Fractions (1)

,])([

2222

21

2 wGLGFsILIu

ILGLGF

D

wGLGFsIHy

HGLGF

N

0

21

22

2

2222

)(

w u yxG2

F

L

H2

uPuDNy 221

2222

16


Construction of left coprime, proper stable factors

where K is a matrix such that F + KH2 is stable.

x̂u

y

G2

F

K

H2

–e

Plant Matrix FractionsPlant Matrix Fractions (2) (2)

yKKHFsIHIuGKHFsIHe

IHKKHF

DH

GKHFN

])([)(

2

222

2

2222

122

0

21

22

uPuNDyyDuN 22221

2222220

17


Solution pairs to the Bézout equations

can be explicitly constructed as

Proof by direct verification,

using the rules for operations with systems.

Bézout EquationBézout Equationss

02

22

LKKHF

Y

ILGKHF

X

IYNXDIYNXD 22222222

02

2

2

LKLGF

Y

IHKLGF

X

18


InnerInner FunctionsFunctions

I

IGHJXGL

HJXGLFHHFXXF

GF

JJGHJ

JHFHH

GF

JH

GF

JG

HFNN

L

LLLLLLLL

L

L

LLLL

L

L

LLL

T21

T12

T2

T1

T12

T2

TT1

T1

T2

12T12

T21

T12

12T1

T1

T1

2

121

2

T12

T2

T1

T

1212

)(

)(

0

0

)(,0 1T12

T21

T1

T HJXGLHHFXXF LLLLLLL

Select L so as to make inner

IX

I

L

0similarity transformation

on putting

12N

LJHH

LGFF

L

L

1211

2

:

:

with the notation

19


Strictly Proper Stable RationalStrictly Proper Stable Rational FunctionsFunctions

2RH

2T2

1T

T2

1T

1

T21

T12

T1

T1

1

1

1

T12

T2

T1

T

1112

RH0

00

0

0

0

0

0

0

G

GXF

G

GXF

GF

GHJ

FHH

GF

H

GF

JG

HFNN

LLLL

L

L

LLL

L

L

LLL

For such a gain L, belongs to

By duality, a gain K can be selected so as

to make inner and belong to .

1112 NN

IX

I

L

0similarity transformation

T21N

2111 NN K2RH

LJHH

LGFF

L

L

1211

2

:

:

with the notation

20


Now

and

yield

so that minimum is achieved for W = 0.

The optimal controller results in the observer-based form

applying the rules for operations with systems.

Optimal ControllerOptimal Controller

2

1

2

211121

RHRH

VNNTN

02211

S LKKHLGF

XYYXC

22

1112112

RHRH

WNNVN K

WNNT K 211

211

2

21


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 and ,

while the state-space approach

shapes these matrices to make them inner by selecting K and L.

This selection makes and belong to ,

hence no need to extract their proper stable parts S and .

ComparisonComparison (1) (1)

2RH

21N

S

12N

2111 NN KNN 1112

22


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.

ComparisonComparison (2) (2)

23


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.

Computational AspectsComputational Aspects

24


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.

Comparison (3)Comparison (3)

25


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.

Comparison (Comparison (44))

26


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.

Comparison (Comparison (55))

27


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.

ReferencesReferences

Documents

The H 2 Control Problem A Detailed Comparison of State-Space and Transfer-Function Solutions Vladimír Kučera Czech Technical University in Prague SWAN