Graph

theoretic

methods

for

the

computation

of

disturbance

decoupling

feedback

matrices

for

structured

systems

Report

J.W.

van

der

Woude

Technische

Universiteit

Delft

Delft

University

of

Technology

Faculteit

der

Technische

Wiskunde

en

Informatica

Faculty

of

Technical

Mathematics

and

Informatics

ISSN

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Graph

theoretic

methods

for

the

computation

of

disturbance

decoupling

feedback

matrices

for

structured

systems

J.W.

van

der

Woude

Abstract

Structured

systems

are

considered

for

which

the

disturbance

decoupling

problem

is

known

to

be

generically

solvable

It

is

recalled

how

this

generic

solvability

is

in

one-one

correspondence

with

certain

properties

of

the

graph

representing

the

structure

of

the

system

Further

a

graph

oriented

method

is

derived

for

the

efficient

computation

of

a

feedback

that

actually

solves

the

disturbance

decoupling

problem

It

is

believed

that

the

method

is

especially

useful

in

case

of

sparse

system

matrices

The

method

may

also

be

advantageous

in

case

some

of

the

entries

in

the

system

matrices

have

an

unknown

value

for

the

influence

of

the

various

entries

is

easy

to

establish

Introduction

In

this

paper

we

study

linear

systems

for

which

each

entry

in

the

system

matrices

either

is

fixed

to

zero

or

is

a

free

parameter

We

assume

that

the

free

parameters

in

a

system

are

pairwise

independent

Systems

of

the

above

type

are

often

called

structured

systems

and

can

be

represented

by

means

of

directed

graphs

In

-LSB-

-RSB-

we

developed

a

graph

theoretic

characterization

for

the

generic

solvability

of

the

disturbance

decoupling

problem

formulated

for

structured

systems

Here

generic

solvability

is

to

be

understood

as

solvability

for

almost

all

possible

values

for

the

free

parameters

The

conditions

in

-LSB-

-RSB-

only

state

something

on

the

generic

solvability

and

nothing

is

said

on

how

the

associated

feedback

looks

like

In

this

paper

we

take

the

graph

theoretic

solvability

conditions

of

-LSB-

-RSB-

as

a

starting

point

and

we

derive

an

algorithm

for

the

computation

of

a

feedback

solving

the

disturbance

decoupling

problem

In

the

algorithm

we

explicitly

incorporate

the

graph

that

represents

the

zero-nonzero

structure

within

the

system

matrices

The

underlying

idea

is

that

the

zero-nonzero

strutcure

of

a

system

should

be

used

as

much

as

possible

The

actual

values

of

the

entries

in

the

system

matrices

should

be

used

only

in

some

final

stage

Also

it

may

turn

out

that

not

all

entries

need

to

be

known

Because

of

this

our

algorithm

can

be

used

in

combination

with

a

formula

manipulation

package

in

case

the

values

of

the

free

parameters

are

unknown

or

in

case

these

parameters

are

seen

as

indeterminates

Also

our

algorithm

can

be

advantageous

in

case

the

system

matrices

are

sparse

The

algorithm

developed

here

is

in

principle

a

nonrecursive

version

of

the

well-known

structure

algorithm

-LRB-

cf.

-LSB-

-RSB-

-RRB-

However

the

combination

of

the

algorithm

with

graph

theory

is

believed

to

be

new

The

algorithm

is

a

generalization

of

the

methods

presented

in

-LSB-

-RSB-

and

is

developed

using

ideas

from

-LSB-

-RSB-

The

outline

of

the

paper

is

as

follows

In

section

we

review

some

facts

on

the

disturbance

decoupling

problem

and

we

illustate

the

main

ideas

behind

the

present

paper

In

section

we

present

a

version

of

the

structure

algorithm

together

with

some

modifications

We

continue

modifying

the

algorithm

in

section

Therefore

we

introduce

the

notion

of

essential

rows

We

conclude

section

by

stating

a

version

of

the

algorithm

that

best

suits

our

purposes

and

by

relating

it

to

the

well-known

input-output

decoupling

problem

-LRB-

cf.

-LSB-

-RSB-

-RRB-

In

section

we

describe

how

the

zero-nonzero

structure

in

the

matrices

of

a

structured

system

can

be

represented

by

means

of

a

directed

graph

Further

we

recall

some

important

graph

theoretic

characterizations

of

for

instance

the

generic

rank

of

a

transfer

matrix

and

the

generic

solvability

of

the

disturbance

decoupling

problem

Finally

we

describe

our

main

result

an

algorithm

for

the

efficient

computation

of

a

feedback

solving

the

disturbance

decoupling

problem

explicitely

using

the

zero-nonzero

structure

in

the

system

matrices

We

illustrate

the

algorithm

in

section

by

means

of

two

examples

In

section

we

give

some

proofs

and

in

section

we

conclude

the

paper

with

some

remarks

Disturbance

Decoupling

In

this

paper

we

consider

the

linear

system

-LRB-

x

-LRB-

t

-RRB-

Ax

-LRB-

t

-RRB-

Bu

-LRB-

t

-RRB-

Qd

-LRB-

t

-RRB-

z

-LRB-

t

-RRB-

Hx

-LRB-

t

-RRB-

-LRB-

-RRB-

with

state

x

-LRB-

t

-RRB-

Rn

control

u

-LRB-

t

-RRB-

Rm

disturbance

d

-LRB-

t

-RRB-

Rq

and

output

z

-LRB-

t

-RRB-

Rr

A

B

Q

and

H

are

real

matrices

of

suitable

dimensions

We

recall

that

the

disturbance

decoupling

problem

for

system

-LRB-

-RRB-

amounts

to

finding

a

feedback

u

-LRB-

t

-RRB-

Fx

-LRB-

t

-RRB-

such

that

the

transfer

matrix

of

the

resulting

closed

loop

system

is

zero

i.e.

such

that

H

-LRB-

sI

-LRB-

A+BF

-RRB-

-RRB-

for

all

s.

For

a

fundamental

treatment

of

the

disturbance

decoupling

problem

we

refer

to

Wonham

-LSB-

-RSB-

There

it

is

also

shown

that

a

solution

to

the

problem

exists

if

and

only

if

ImQ

V

where

V

denotes

the

largest

-LRB-

A

B

-RRB-

invariant

subspace

in

KerH

-LRB-

cf.

-LSB-

-RSB-

-RRB-

We

stress

the

fact

that

the

inclusion

ImQ

V

only

is

a

condition

on

the

existence

of

a

feedback

u

-LRB-

t

-RRB-

Fx

-LRB-

t

-RRB-

solving

the

disturbance

decoupling

problem

The

matrix

F

itself

does

not

appear

in

the

condition

In

-LSB-

-RSB-

it

is

shown

that

any

F

such

that

-LRB-

A

BF

-RRB-

V

V

will

solve

the

problem

There

it

is

also

explained

how

first

V

can

be

computed

followed

by

the

computation

of

F

such

that

-LRB-

A+BF

-RRB-

V

V

The

computations

are

based

on

geometric

techniques

requiring

full

knowledge

of

A

B

and

H

In

many

cases

many

of

the

entries

in

the

matrices

are

fixed

to

zero

Also

often

some

of

the

entries

in

A

B

and

H

have

an

unknown

value

The

purpose

of

this

paper

is

to

deal

with

these

cases

in

an

efficient

way

Therefore

we

derive

an

algorithm

for

the

computation

of

a

matrix

F

such

that

-LRB-

A

BF

-RRB-

V

V

in

which

the

zero-nonzero

structure

in

the

matrices

A

B

and

H

is

used

and

in

which

in

first

instance

the

value

of

the

free

parameters

-LRB-

nonzeros

-RRB-

is

not

relevant

In

general

in

the

algorithm

only

a

limited

number

of

the

free

parameters

will

be

actually

needed

The

value

of

the

remaining

free

parameters

is

not

important

in

the

context

of

this

paper

The

algorithm

uses

products

of

the

form

HAiB

and

HAi

The

idea

is

that

these

products

can

be

computed

easily

given

the

zero-nonzero

structure

of

A

B

and

H

Also

in

computing

these

products

the

value

of

only

a

limited

number

of

the

free

parameters

may

required

to

be

known

We

conclude

this

section

by

illustrating

the

ideas

on

which

the

paper

is

based

Therefore

we

consider

a

system

-LRB-

-RRB-

with

a

very

simple

form

A

B

Q

H

h

i

It

is

easy

to

verify

that

for

all

i

i

with

the

disturbance

decoupling

problem

is

solvable

Further

a

feedback

matrix

achieving

decoupling

is

given

by

F

h

i

Clearly

F

is

structured

in

the

sense

that

it

contains

fixed

zero

entries

Moreover

F

only

need

to

depend

on

two

free

parameters

Hence

the

disturbance

decoupling

problem

is

generically

solvable

a

feedback

matrix

achieving

decoupling

may

be

structured

and

only

need

to

depend

on

a

limited

number

of

free

parameters

Structure

Algorithm

In

this

section

we

present

a

nonrecursive

version

of

the

structure

algorithm

-LRB-

cf.

-LSB-

-RSB-

-RRB-

Throughout

this

paper

we

assume

that

rank

H

-LRB-

sI

A

-RRB-

p

i.e.

H

-LRB-

sI

A

-RRB-

has

rank

p

for

almost

all

s

In

section

we

shall

argue

that

in

the

context

of

the

present

paper

this

rank

assumption

can

be

made

without

loss

of

generality

We

now

denote

HB

HAB

HB

HAn

HAn

HB

Rpn?mn

HA

HAn

Rpn?n

The

zeros

in

the

Toeplitz

matrix

denote

zero

matrices

in

Rp?m

A

matrix

F

such

that

-LRB-

A

BF

-RRB-

V

V

can

now

be

computed

by

means

of

the

next

algorithm

Algorithm

a

-RRB-

Compute

a

matrix

K

Rp?pn

such

that

K

-LSB-

L

-RSB-

Rp?mn

with

L

Rp?m

a

full

row

rank

matrix

and

n

zero

matrices

in

Rp?m

b

-RRB-

Solve

F

Rm?n

from

M

LF

with

M

K

Rp?n

Algorithm

will

be

proved

in

section

The

reason

that

we

want

to

use

algorithm

is

that

it

can

be

refined

by

incorporating

additional

information

which

can

be

easily

obtained

from

the

graph

representing

the

zero-nonzero

structure

in

the

system

matrices

To

introduce

these

refinements

we

first

somewhat

rearrange

the

computations

in

algorithm

For

that

purpose

we

denote

col

-LRB-

Tr

-RRB-

for

the

matrix

obtained

by

stacking

the

matrices

Tr

on

top

of

each

other

Hence

col

-LRB-

Tr

-RRB-

Tr

where

the

matrices

Tr

all

have

the

same

number

of

columns

For

instance

we

write

H

col

-LRB-

hp

-RRB-

with

rows

hp

Furthermore

we

denote

for

i

p

i

hiB

hiAB

hiB

hiAn

hiAn

hiB

Rn?mn

i

hiA

hiAn

Rn?n

The

matrix

i

is

a

Toeplitz

matrix

in

which

the

zeros

denote

zero

matrices

in

We

note

that

the

matrices

and

col

-LRB-

p

-RRB-

can

be

obtained

from

each

other

by

elementary

row

permutations

i.e.

there

exists

a

permutation

matrix

P

such

that

P

col

-LRB-

p

-RRB-

The

permutation

matrix

P

is

also

such

that

P

col

-LRB-

p

-RRB-

It

is

now

immediate

that

algorithm

can

be

modified

as

follows

Algorithm

a

-RRB-

Compute

matrices

Kp

Rp?n

such

that

-LSB-

Kp

-RSB-

col

-LRB-

p

-RRB-

Kp?p

-LSB-

-RSB-

Rp?mn

with

Rp?m

a

full

row

rank

matrix

and

n

zero

matrices

in

Rp?m

b

-RRB-

Solve

F

Rm?n

from

with

-LSB-

Kp

-RSB-

col

-LRB-

p

-RRB-

Kp?p

Below

we

shall

use

that

step

a

-RRB-

of

the

above

algorithm

can

be

written

in

more

detail

as

follows

a

-RRB-

Compute

matrices

Kp

Rp?n

such

that

-LSB-

Kp

-RSB-

col

-LRB-

p

-RRB-

Kp

p

Rp?m

-LRB-

n

-RRB-

and

-LSB-

Kp

-RSB-

col

-LRB-

p

-RRB-

Kp

p

Rp?m

with

a

full

row

rank

matrix

where

we

have

denoted

i

h

i

i

i

for

i

p

with

i

hiB

hiAB

hiAn

Rn?m

i

hiB

hiAB

hiB

hiAn

hiAn

hiB

Rn

-LRB-

n

-RRB-

m

Essential

Rows

As

indicated

before

our

goal

is

to

refine

algorithm

by

incorporating

information

about

the

system

that

can

be

-LRB-

easily

-RRB-

obtained

from

its

graph

For

that

purpose

we

need

the

notion

of

essential

row

-LRB-

cf.

-LSB-

-RSB-

-RRB-

First

we

consider

a

general

matrix

M

with

rows

mk

i.e.

M

col

-LRB-

mk

-RRB-

We

call

row

mi

an

essential

row

in

M

if

the

row

mi

can

not

be

written

as

a

linear

combination

of

the

other

rows

mi

mi

mk

in

M

A

row

of

M

that

is

not

an

essential

row

we

call

a

nonessential

row

For

instance

any

zero

row

is

nonessential

Obviously

row

mi

is

an

essential

row

in

M

if

and

only

if

deleting

mi

from

M

would

give

a

drop

in

rank

Let

X

Rt?k

be

a

solution

of

XM

and

denote

X

-LSB-

xk

-RSB-

with

columns

xk

Rt

Then

if

mi

is

an

essential

row

in

M

it

is

necessary

that

xi

Rt

Next

we

consider

in

some

detail

the

structure

of

the

Toeplitz

matrices

i

and

i.

a

-RRB-

First

we

note

that

the

jth

row

of

i

can

be

obtained

from

the

jth

row

of

i

by

simply

deleting

the

leading

-LRB-

row

-RRB-

matrix

hiAj

Now

consider

the

jth

row

of

i

as

a

row

of

col

-LRB-

p

-RRB-

It

is

clear

that

if

the

jth

row

of

i

is

a

nonessential

row

in

col

-LRB-

p

-RRB-

then

the

jth

row

of

i

is

a

nonessential

row

in

col

-LRB-

p

-RRB-

Conversely

if

the

jth

row

of

i

is

an

essential

row

of

col

-LRB-

p

-RRB-

so

is

the

jth

row

of

i

in

col

-LRB-

p

-RRB-

b

-RRB-

Next

we

note

that

the

jth

row

of

i

can

be

obtained

from

the

j

row

of

i

by

simply

adding

p

zeros

to

its

end

This

means

that

if

the

j

row

of

i

is

a

nonessential

row

in

col

-LRB-

p

-RRB-

then

the

jth

row

of

i

is

a

nonessential

row

in

col

-LRB-

p

-RRB-

Conversely

if

the

jth

row

of

i

is

an

essential

row

of

col

-LRB-

p

-RRB-

so

is

the

j

row

of

i

in

col

-LRB-

p

-RRB-

With

the

above

observations

we

obtain

the

following

Proposition

If

the

jth

row

of

i

is

a

nonessential

row

in

col

-LRB-

p

-RRB-

then

the

rows

upto

j

of

i

and

i

are

nonessential

rows

in

col

-LRB-

p

-RRB-

and

col

-LRB-

p

-RRB-

respectively

for

i

j

i

p

and

j

n.

Conversely

if

the

jth

row

of

i

is

a

essential

row

in

col

-LRB-

p

-RRB-

then

the

rows

j

upto

n

of

i

and

the

rows

j

upto

n

of

i

are

nonessential

rows

in

col

-LRB-

p

-RRB-

and

col

-LRB-

p

-RRB-

respectively

for

i

j

i

p

and

j

n.

Recall

that

throughout

this

paper

we

assume

that

H

-LRB-

sI

A

-RRB-

has

full

row

rank

In

-LSB-

-RSB-

it

has

been

shown

that

under

this

rank

assumption

the

nth

row

of

i

is

an

essential

row

in

col

-LRB-

p

-RRB-

for

all

i

p.

Therefore

it

is

possible

to

introduce

the

so-called

essential

orders

defined

as

-LRB-

cf.

-LSB-

-RSB-

-RRB-

i

min

fjj

row

j

of

i

is

essential

in

col

-LRB-

p

-RRB-

g

for

i

p.

By

the

above

proposition

the

rows

i

upto

n

of

i

and

the

rows

i

upto

n

of

i

are

essential

rows

in

col

-LRB-

p

-RRB-

and

col

-LRB-

p

-RRB-

respectively

Further

the

rows

upto

i

of

both

i

and

i

are

nonessential

rows

in

col

-LRB-

p

-RRB-

and

col

-LRB-

p

-RRB-

respectively

Clearly

any

essential

row

must

be

nonzero

To

identify

the

nonzero

rows

in

i

and

i

we

introduce

the

so-called

decoupling

indices

-LRB-

cf.

-LSB-

-RSB-

-RRB-

i

min

fjjhjAj

g

for

i

p.

By

this

definition

the

rows

upto

i

in

i

and

the

rows

upto

i

in

i

are

zero

rows

Also

it

is

clear

that

i

i

n

for

i

p

Now

we

return

to

algorithm

The

product

Ki?i

in

step

a

-RRB-

of

the

algorithm

can

be

computed

by

multiplying

the

jth

column

of

Ki

from

the

right

with

the

jth

row

of

i

and

by

adding

the

obtained

products

In

the

above

we

have

seen

that

the

rows

upto

i

in

i

are

zero

rows

Hence

for

the

product

Ki?i

the

columns

upto

i

of

Ki

are

not

relevant

and

can

taken

to

be

zero

columns

We

claim

that

for

the

product

Ki?i

the

columns

i

upto

n

of

Ki

also

are

not

relevant

since

they

are

zero

cloumns

To

see

this

we

recall

that

the

rows

i

upto

n

of

i

are

essential

rows

in

col

-LRB-

p

-RRB-

Because

the

matrices

Kp

are

to

be

such

that

-LSB-

Kp

-RSB-

col

-LRB-

p

-RRB-

it

necessarily

follows

that

the

columns

i

upto

n

of

Ki

are

zero

columns

With

this

our

claim

follows

The

above

observations

combined

with

step

b

-RRB-

of

algorithm

makes

clear

that

without

changing

the

outcome

of

the

algorithm

we

can

delete

the

rows

upto

i

and

i

upto

n

of

i

and

i

for

i

p

In

the

matrix

Ki

we

then

have

to

delete

the

columns

upto

i

and

i

upto

n

for

i

p.

From

this

it

follows

that

algorithm

can

be

refined

as

follows

Algorithm

a

-RRB-

Preliminary

computation

For

i

p

compute

the

indices

i

i

and

define

i

hiA

i

hiA

i

B

hiA

i

hiA

i

hiA

i

hiA

i

R

-LRB-

i

i

-RRB-

m

and

i

hiA

i

hiA

i

hiA

i

R

-LRB-

i

i

-RRB-

n

where

max

f

-LRB-

i

i

-RRB-

i

pg

b

-RRB-

Compute

Kp

Rp

-LRB-

i

i

-RRB-

such

that

h

Kp

i

col

-LRB-

p

-RRB-

Kp

p

h

i

Rp?mfl

with

Rp?m

a

full

row

rank

matrix

and

zero

matrices

in

Rp?m

c

-RRB-

Solve

F

Rm?n

from

with

h

Kp

i

col

-LRB-

p

-RRB-

Kp

p

The

index

is

introduced

to

avoid

unnecessary

zero

columns

in

the

matrix

col

-LRB-

p

-RRB-

The

reason

for

presenting

algorithm

as

done

above

is

the

fact

that

for

almost

all

cases

the

values

of

the

indices

i

and

i

can

be

extracted

from

the

graph

representing

the

zerononzero

structure

of

the

system

This

graph

together

with

the

combinatorial

aspects

of

how

to

compute

these

indices

will

be

presented

in

the

next

section

After

the

indices

i

and

i

are

determined

the

matrices

i

and

i

have

to

be

computed

The

entries

in

these

matrices

are

of

the

form

hiAkbj

where

B

-LSB-

bm

-RSB-

In

case

the

matrices

A

B

and

H

are

sparse

the

graph

representing

the

zero-nonzero

structure

of

the

system

can

be

helpful

in

efficiently

computing

the

above

products

This

will

be

discussed

in

some

more

detail

in

the

next

section

Special

case

We

conclude

this

section

by

considering

the

case

that

i

i

for

i

p

This

means

that

every

nonzero

row

of

i

is

an

essential

row

in

col

-LRB-

p

-RRB-

Step

a

-RRB-

of

algorithm

yields

that

i

hiA

i

i

hiA

i

for

i

p.

Step

b

-RRB-

then

states

that

there

are

columns

Kp

Rp

such

that

h

Kp

i

col

-LRB-

p

-RRB-

Rp?m

with

a

full

row

rank

matrix

But

this

means

that

the

matrix

col

-LRB-

p

-RRB-

p

has

full

row

rank

and

that

h

Kp

i

is

an

invertible

matrix

From

-LSB-

-RSB-

it

follows

that

the

system

is

so-called

input-output

decoupable

This

means

that

there

exist

matrices

F

Rm?n

and

R

Rm?p

such

that

H

-LRB-

sI

-LRB-

A

BF

-RRB-

-RRB-

is

a

nonsingular

diagonal

transfer

matrix

It

is

easy

to

see

from

step

c

-RRB-

in

algorithm

that

like

in

-LSB-

-RSB-

the

matrix

F

that

suites

our

purposes

can

be

computed

from

the

equation

p

F

p

Graphs

We

recall

that

we

here

consider

linear

systems

of

type

-LRB-

-RRB-

in

which

each

of

the

entries

in

the

system

matrices

either

is

fixed

to

zero

or

is

an

independent

free

parameter

As

mentioned

before

such

systems

can

be

represented

by

means

of

directed

graphs

The

directed

graph

which

represents

a

structured

system

of

type

-LRB-

-RRB-

consists

of

a

set

of

vertices

V

and

a

set

of

directed

edges

E

and

will

be

denoted

by

G

-LRB-

V

E

-RRB-

The

set

V

equals

X

-LSB-

U

-LSB-

D

-LSB-

Z

with

X

xng

the

set

of

state

vertices

U

umg

the

set

of

input

vertices

D

dqg

the

set

of

disturbance

vertices

and

Z

zpg

the

set

of

output

vertices

Clearly

V

consists

of

n

m

q

p

vertices

Denoting

-LRB-

v

-RRB-

for

a

directed

edge

from

the

vertex

v

V

to

the

vertex

V

the

set

of

edges

E

is

described

by

EA

-LSB-

EB

-LSB-

EQ

-LSB-

EH

where

EA

f

-LRB-

xj

xi

-RRB-

jai

j

g

EB

f

-LRB-

uj

xi

-RRB-

jbi

j

g

EQ

f

-LRB-

dj

xi

-RRB-

jqi

j

g

EH

f

-LRB-

xj

zi

-RRB-

jhi

j

g

In

the

latter

for

instance

ai

j

means

that

the

-LRB-

i

j

-RRB-

th

entry

of

A

is

a

free

parameter

For

examples

we

refer

to

section

Now

consider

the

graph

G

-LRB-

V

E

-RRB-

and

let

W

W

be

two

disjoint

nonempty

subsets

in

V

We

say

that

there

exists

a

path

from

W

to

W

if

there

is

an

integer

t

and

vertices

V

such

that

W

wt

W

and

-LRB-

wi

wi

-RRB-

E

for

all

i

t

The

path

is

then

said

to

have

length

t

and

to

consist

of

the

vertices

wt

Occasionally

we

denote

the

path

by

the

sequence

of

directed

edges

it

consists

of

i.e.

by

-LRB-

-RRB-

-LRB-

-RRB-

-LRB-

wt

wt

-RRB-

Two

paths

from

W

to

W

are

called

disjoint

if

the

two

paths

consist

of

disjoint

sets

of

vertices

We

say

that

l

paths

from

W

to

W

are

pairwise

disjoint

if

any

two

of

them

are

disjoint

We

say

that

the

largest

number

of

disjoint

paths

from

W

to

W

is

t

if

there

exists

t

pairwise

disjoint

paths

from

W

to

W

and

any

set

of

-LRB-

t

-RRB-

paths

from

W

to

W

contains

at

least

two

paths

that

are

not

disjoint

Next

let

W

be

an

additional

subset

in

V

and

let

a

set

of

l

pairwise

disjoint

paths

from

W

to

W

be

given

If

the

intersection

of

the

set

W

and

the

set

of

vertices

that

the

l

paths

consist

of

has

cardinality

then

we

say

that

the

total

number

of

vertices

in

W

on

the

l

paths

is

For

a

structured

system

of

type

-LRB-

-RRB-

and

the

associated

graph

G

-LRB-

V

E

-RRB-

we

have

the

following

results

-LRB-

cf.

-LSB-

-RSB-

-LSB-

-RSB-

-LSB-

-RSB-

-RRB-

Theorem

The

generic

rank

of

H

-LRB-

sI

A

-RRB-

is

equal

to

the

largest

number

of

pairwise

disjoint

paths

from

U

to

Z.

For

a

structured

system

the

generic

values

of

the

indices

i

and

i

can

be

characterized

as

follows

-LRB-

cf.

-LSB-

-RSB-

-LSB-

-RSB-

-RRB-

Here

we

denote

Z

n

zi

fz

Zj

z

zig

Theorem

a

-RRB-

The

smallest

total

number

of

vertices

contained

in

X

on

a

path

from

U

to

fzig

is

generically

equal

to

i

for

all

i

p.

b

-RRB-

Let

H

-LRB-

sI

A

-RRB-

generically

have

full

row

rank

equal

to

p

The

smallest

total

number

of

vertices

contained

in

X

on

p

pairwise

disjoint

paths

from

U

to

Z

minus

the

smallest

total

number

of

vertices

contained

in

X

on

-LRB-

p

-RRB-

pairwise

disjoint

paths

from

U

to

Z

n

zi

is

generically

equal

to

i

for

all

i

p

The

generic

solvability

of

the

disturbance

decoupling

problem

for

a

structured

system

of

type

-LRB-

-RRB-

can

be

characterized

as

follows

-LRB-

cf.

-LSB-

-RSB-

-LSB-

-RSB-

-RRB-

Theorem

The

disturbance

decoupling

problem

for

a

structured

system

of

type

-LRB-

-RRB-

is

generically

solvable

if

and

only

if

a

-RRB-

the

largest

number

of

pairwise

disjoint

paths

from

U

to

Z

equals

the

largest

number

of

pairwise

disjoint

paths

from

U

-LSB-

D

to

Z

say

t

and

b

-RRB-

the

smallest

total

number

of

vertices

contained

in

X

-LSB-

U

on

t

pairwise

disjoint

paths

from

U

to

Z

equals

the

smallest

total

number

of

vertices

contained

in

X

-LSB-

U

on

t

pairwise

disjoint

paths

from

U

-LSB-

D

to

Z

Since

we

focus

here

only

on

systems

for

which

H

-LRB-

sI

A

-RRB-

generically

has

full

row

rank

the

next

corollary

easily

follows

Corollary

The

disturbance

decoupling

problem

for

a

structured

system

of

type

-LRB-

-RRB-

is

generically

solvable

if

and

only

if

the

smallest

total

number

of

vertices

contained

in

X

-LSB-

U

on

p

pairwise

disjoint

paths

from

U

to

Z

equals

the

smallest

total

number

of

vertices

contained

in

X

-LSB-

U

on

p

pairwise

disjoint

paths

from

U

-LSB-

D

to

Z.

By

the

above

results

we

can

establish

the

generic

solvability

of

the

disturbance

decoupling

problem

for

a

structured

system

of

type

-LRB-

-RRB-

by

performing

some

computations

in

it

's

graph

We

can

do

this

in

an

efficient

way

by

using

algorithms

from

combinatorics

based

on

the

max

flow

min

cut

theorem

and

on

the

theory

of

minimal

cost

flows

in

transportation

networks

-LRB-

cf.

-LSB-

-RSB-

chapt

-LSB-

-RSB-

chapt

-RRB-

Next

if

required

we

can

determine

from

the

system

's

graph

the

generic

values

of

the

indices

i

and

i.

Also

here

the

above

mentioned

combinatoric

algorithms

can

be

used

Following

algorithm

the

next

step

is

to

compute

the

matrices

i

and

i

containing

entries

of

the

form

hiAkbj

and

hiAk

In

order

to

efficiently

compute

the

products

hiAkbj

-LRB-

similar

remarks

hold

for

the

computation

of

hiAk

-RRB-

we

denote

Pi

j

-LRB-

k

-RRB-

for

the

set

of

paths

from

uj

to

zi

that

have

length

k

From

the

graph

G

-LRB-

V

E

-RRB-

it

is

immediately

clear

that

any

path

in

Pi

j

-LRB-

k

-RRB-

consists

of

the

vertices

uj

zi

and

at

least

one

and

at

most

k

vertices

in

X

Now

consider

a

path

P

in

Pi

j

-LRB-

k

-RRB-

and

let

it

consist

of

the

edges

-LRB-

uj

-RRB-

-LRB-

-RRB-

-LRB-

xok

xok

-RRB-

-LRB-

xok

zi

-RRB-

where

the

indices

ok

are

not

necessarily

distinct

Define

f

-LRB-

P

-RRB-

j

Qkt

aot

ot

hi

ok

In

-LSB-

-RSB-

it

has

been

shown

that

hiAkbj

P

j

-LRB-

k

-RRB-

f

-LRB-

P

-RRB-

Hence

also

for

the

computation

of

the

product

hiAkbj

the

graph

may

be

used

Combining

all

the

above

observations

we

come

to

the

next

algorithm

Let

a

structured

system

of

type

-LRB-

-RRB-

be

given

and

let

its

graph

be

denoted

by

G

-LRB-

V

E

-RRB-

Algorithm

a

-RRB-

Apply

theorem

to

determine

whether

or

not

the

disturbance

decoupling

problem

for

the

system

is

generically

solvable

If

not

then

we

can

stop

else

we

can

proceed

as

described

next

Assume

that

the

largest

number

of

pairwise

disjoint

paths

from

U

to

Z

is

Select

vertices

from

Z

forming

such

that

the

largest

number

of

pairwise

disjoint

paths

from

U

to

is

Renumber

the

vertices

in

Z

in

such

a

way

that

Delete

the

vertices

zp

and

the

edges

incident

with

these

vertices

For

ease

of

notation

drop

the

prime

b

-RRB-

Using

the

-LRB-

reduced

-RRB-

graph

G

now

compute

the

generic

values

of

the

indices

i

and

i

for

all

i

p

and

compute

as

indicated

in

section

the

matrices

i

and

i

for

all

i

p.

For

the

efficient

computation

of

these

matrices

the

graph

G

may

be

used

c

-RRB-

Next

we

perform

the

steps

b

-RRB-

and

c

-RRB-

of

algorithm

to

obtain

a

feedback

matrix

F

that

is

a

solution

to

the

disturbance

decoupling

problem

Note

that

only

in

the

steps

b

-RRB-

and

c

-RRB-

of

algorithm

the

value

of

-LRB-

some

of

-RRB-

the

parameters

is

required

to

be

known

The

graph

obtained

after

step

a

-RRB-

corresponds

to

a

reduced

structured

system

of

the

form

x

-LRB-

t

-RRB-

Ax

-LRB-

t

-RRB-

Bu

-LRB-

t

-RRB-

Qd

-LRB-

t

-RRB-

-LRB-

t

-RRB-

H

x

-LRB-

t

-RRB-

with

H

-LRB-

sI

A

-RRB-

generically

of

full

row

rank

From

section

it

follows

that

the

disturbance

decoupling

problem

for

the

original

system

is

generically

solvable

if

and

only

if

it

is

generically

solvable

for

the

reduced

system

Furthermore

it

is

indicated

there

that

for

any

F

such

that

H

-LRB-

sI

-LRB-

A

BF

-RRB-

-RRB-

also

H

-LRB-

sI

-LRB-

A

BF

-RRB-

-RRB-

and

conversely

Therefore

we

may

concentrate

on

the

reduced

system

and

for

ease

of

notation

we

drop

the

prime

Hence

we

continue

in

step

b

-RRB-

with

a

structured

system

of

type

-LRB-

-RRB-

for

which

the

disturbance

decoupling

problem

is

generically

solvable

and

with

H

-LRB-

sI

A

-RRB-

generically

of

full

rank

p.

Worked

Examples

In

this

section

we

illustrate

algorithm

in

some

detail

We

start

by

reconsidering

the

example

in

section

Hence

n

m

q

p

The

graph

associated

with

the

system

can

be

depicted

as

in

figure

Figure

s

d

s

s

s

su

s

z

Note

that

X

U

D

and

Z

Using

the

graph

it

follows

from

theorem

that

the

disturbance

decoupling

problem

is

generically

solvable

Indeed

there

is

only

one

path

from

U

to

Z

and

the

two

paths

from

U

-LSB-

D

to

Z

are

not

disjoint

Further

the

only

path

from

U

to

Z

contains

vertices

of

X

-LSB-

U

while

any

of

two

the

paths

from

U

-LSB-

D

to

Z

contain

at

least

vertices

of

X

-LSB-

U

By

theorem

it

follows

easily

that

Like

in

the

special

case

discussed

at

the

end

of

section

the

feedback

matrix

F

solving

the

disturbance

decoupling

problem

can

be

computed

by

solving

the

equation

where

H

Hence

F

satisfies

-LSB-

-RSB-

So

F

h

i

as

we

also

have

seen

in

section

Note

that

the

above

answer

is

obtained

by

means

of

formula

manipulation

only

and

that

no

actual

parameter

values

are

required

Next

we

consider

a

more

complicated

system

of

type

-LRB-

-RRB-

with

system

matrices

A

B

Q

H

So

n

m

q

p

The

parameters

are

collected

in

a

vector

The

structure

of

the

system

is

given

by

the

graph

depicted

in

figure

HHHHHHj

aeaeaeaeaeae

ZZZZZZ

aeaeaeaeaeae

ZZZZZZ

Figure

s

s

s

s

s

s

s

s

su

su

s

s

s

In

the

graph

the

largest

number

of

pairwise

disjoint

paths

from

U

to

Z

and

also

from

U

-LSB-

D

to

Z

is

equal

to

This

implies

that

for

almost

all

the

transfer

matrix

H

-LRB-

sI

A

-RRB-

has

rank

Hence

H

-LRB-

sI

A

-RRB-

generically

has

full

rank

Also

it

is

easy

to

see

that

any

pair

of

pairwise

disjoint

paths

from

U

to

Z

and

from

U

-LSB-

D

to

Z

contains

or

more

vertices

in

X

-LSB-

U

From

theorem

it

therefore

follows

that

the

disturbance

decoupling

problem

is

solvable

for

almost

all

Next

we

have

to

compute

the

generic

values

of

the

decoupling

indices

and

the

essential

orders

Note

that

the

shortest

path

in

the

graph

from

U

to

zi

has

one

vertex

in

X

both

for

i

and

i

Furthermore

note

that

the

smallest

number

of

vertices

in

X

in

any

pair

of

disjoint

paths

from

U

to

Z

is

From

theorem

it

now

follows

that

generically

and

Next

the

matrices

and

have

to

be

computed

Note

that

The

entries

in

the

matrices

easily

can

be

determined

from

the

graph

Following

algorithm

step

c

-RRB-

there

remain

steps

b

-RRB-

and

c

-RRB-

of

algorithm

b

-RRB-

Determine

and

such

that

h

i

with

full

row

rank

c

-RRB-

Solve

F

from

where

Here

we

can

for

instance

take

Then

F

such

that

-LRB-

A

BF

-RRB-

V

V

and

consequently

solving

the

disturbance

decoupling

problem

can

be

solved

from

F

Proofs

In

this

section

we

give

mathematical

justification

of

some

earlier

statements

Rank

assumption

We

denote

T

-LRB-

s

-RRB-

H

-LRB-

sI

A

-RRB-

and

S

-LRB-

s

-RRB-

H

-LRB-

sI

A

-RRB-

We

recall

that

throughout

this

paper

we

assume

that

the

disturbance

decoupling

problem

for

system

-LRB-

-RRB-

is

solvable

This

is

equivalent

to

the

existence

of

a

strictly

proper

rational

matrix

X

-LRB-

s

-RRB-

such

that

T

-LRB-

s

-RRB-

X

-LRB-

s

-RRB-

S

-LRB-

s

-RRB-

-LRB-

cf.

-LSB-

-RSB-

-RRB-

From

this

it

follows

that

rank

T

-LRB-

s

-RRB-

rank

-LSB-

T

-LRB-

s

-RRB-

S

-LRB-

s

-RRB-

-RSB-

We

claim

that

within

in

the

context

of

this

paper

the

assumption

that

T

-LRB-

s

-RRB-

has

full

row

rank

can

be

made

without

loss

of

generality

Indeed

suppose

that

T

-LRB-

s

-RRB-

does

not

have

full

row

rank

i.e.

rank

T

-LRB-

s

-RRB-

p

Then

it

is

possible

to

find

a

permutation

matrix

P

such

that

PT

-LRB-

s

-RRB-

T

-LRB-

s

-RRB-

T

-LRB-

s

-RRB-

and

PS

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

with

T

-LRB-

s

-RRB-

having

full

row

rank

The

matrix

P

can

be

simply

determined

by

picking

out

rows

of

T

-LRB-

s

-RRB-

that

are

linearly

independent

The

latter

can

also

be

done

by

collecting

outputs

zi

into

a

vector

such

that

the

transfer

matrix

from

u

to

has

full

row

rank

This

transfer

matrix

can

be

written

as

H

-LRB-

sI

A

-RRB-

with

H

a

n

matrix

composed

of

suitably

choosen

rows

from

H

Clearly

we

then

may

take

T

-LRB-

s

-RRB-

H

-LRB-

sI

A

-RRB-

Obviously

since

T

-LRB-

s

-RRB-

has

full

row

rank

it

follows

that

rank

T

-LRB-

s

-RRB-

rank

-LSB-

T

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-RSB-

Further

since

rank

-LSB-

T

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-RSB-

rank

T

-LRB-

s

-RRB-

rank

-LSB-

T

-LRB-

s

-RRB-

S

-LRB-

s

-RRB-

-RSB-

rank

P

-LSB-

T

-LRB-

s

-RRB-

S

-LRB-

s

-RRB-

-RSB-

the

rows

in

-LSB-

T

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-RSB-

are

linearly

depending

on

the

rows

in

-LSB-

T

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-RSB-

This

means

that

there

is

a

rational

matrix

Y

-LRB-

s

-RRB-

such

that

Y

-LRB-

s

-RRB-

T

-LRB-

s

-RRB-

T

-LRB-

s

-RRB-

and

Y

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-LRB-

s

-RRB-

We

therefore

obtain

that

any

solution

X

-LRB-

s

-RRB-

to

the

equation

T

-LRB-

s

-RRB-

X

-LRB-

s

-RRB-

S

-LRB-

s

-RRB-

is

also

a

solution

to

the

equation

T

-LRB-

s

-RRB-

X

-LRB-

s

-RRB-

-LRB-

s

-RRB-

and

conversely

It

follows

now

from

-LSB-

-RSB-

that

the

disturbance

decoupling

problem

is

solvable

for

x

-LRB-

t

-RRB-

Ax

-LRB-

t

-RRB-

Bu

-LRB-

t

-RRB-

Qd

-LRB-

t

-RRB-

z

-LRB-

t

-RRB-

Hx

-LRB-

t

-RRB-

if

and

only

if

it

is

solvable

for

x

-LRB-

t

-RRB-

Ax

-LRB-

t

-RRB-

Bu

-LRB-

t

-RRB-

Qd

-LRB-

t

-RRB-

-LRB-

t

-RRB-

H

x

-LRB-

t

-RRB-

Using

the

identity

H

-LRB-

sI

-LRB-

A

BF

-RRB-

-RRB-

T

-LRB-

s

-RRB-

-LRB-

I

F

-LRB-

sI

A

-RRB-

-RRB-

F

-LRB-

sI

A

-RRB-

S

-LRB-

s

-RRB-

and

a

similar

identity

involving

H

and

T

-LRB-

s

-RRB-

-LRB-

s

-RRB-

it

easily

follows

from

the

previous

that

if

F

is

such

that

H

-LRB-

sI

-LRB-

A

BF

-RRB-

-RRB-

then

H

-LRB-

sI

-LRB-

A

BF

-RRB-

-RRB-

and

conversely

Hence

for

the

computation

of

a

feedback

matrix

F

solving

the

disturbance

decoupling

problem

we

may

restrict

ourselves

to

the

part

for

which

our

rank

assumption

holds

Factorization

Under

our

rank

assumption

the

matrix

T

-LRB-

s

-RRB-

can

be

factorized

as

T

-LRB-

s

-RRB-

U

-LRB-

s

-RRB-

-LRB-

s

-RRB-

L

-LRB-

s

-RRB-

-LRB-

cf.

-LSB-

-RSB-

-RRB-

In

this

factorization

U

-LRB-

s

-RRB-

is

a

p

p

proper

rational

matrix

that

has

a

proper

rational

inverse

The

matrix

L

-LRB-

s

-RRB-

is

a

p

m

proper

rational

matrix

that

has

a

proper

rational

right

inverse

and

the

matrix

-LRB-

s

-RRB-

diag

-LRB-

s?np

-RRB-

with

np

The

integers

np

are

unique

and

are

called

the

infinite

zero

orders

of

T

-LRB-

s

-RRB-

From

the

behavior

at

infinity

it

easily

follows

that

Ppi

ni

n.

Proof

of

theorem

Next

we

give

a

proof

of

algorithm

starting

from

the

above

factorization

From

the

factorization

it

follows

that

K

-LRB-

s

-RRB-

-LRB-

U

-LRB-

s

-RRB-

-LRB-

s

-RRB-

-RRB-

exists

and

has

a

Laurent

series

expansion

of

the

form

K

-LRB-

s

-RRB-

K?nsn

K

-LRB-

n

-RRB-

sn

K

Let

the

Laurent

series

expansions

of

L

-LRB-

s

-RRB-

and

T

-LRB-

s

-RRB-

be

given

by

L

-LRB-

s

-RRB-

and

T

-LRB-

s

-RRB-

HBs

HABs

respectively

Then

L

-LRB-

-RRB-

has

full

row

rank

equal

to

p

Now

consider

the

identity

L

-LRB-

s

-RRB-

K

-LRB-

s

-RRB-

T

-LRB-

s

-RRB-

for

the

individual

nonnegative

powers

of

s

This

yields

n

equations

which

can

be

written

as

follows

-LSB-

K

K

K?n

-RSB-

HB

HAB

HB

HAn

HAn

HB

-LSB-

-RSB-

-LRB-

-RRB-

Next

define

M

-LRB-

s

-RRB-

K

-LRB-

s

-RRB-

H

-LRB-

sI

A

-RRB-

The

Laurent

series

expansion

of

H

-LRB-

sI

A

-RRB-

is

given

by

H

-LRB-

sI

A

-RRB-

Hs

HAs

Together

with

the

previous

expansion

of

K

-LRB-

s

-RRB-

it

follows

that

the

Laurent

series

expansion

of

M

-LRB-

s

-RRB-

is

of

the

form

M

-LRB-

s

-RRB-

M

-LRB-

n

-RRB-

sn

M

-LRB-

n

-RRB-

sn

M

with

for

instance

M

-LRB-

n

-RRB-

K?nH

M

-LRB-

n

-RRB-

K

-LRB-

n

-RRB-

H

K?nHA

and

K?nHAn

K

-LRB-

n

-RRB-

HAn

K

K

Note

that

M

-LRB-

s

-RRB-

B

K

-LRB-

s

-RRB-

T

-LRB-

s

-RRB-

L

-LRB-

s

-RRB-

is

proper

Hence

M

-LRB-

n

-RRB-

B

M

-LRB-

n

-RRB-

B

M

and

Furthermore

since

H

-LRB-

sI

A

-RRB-

Hs

H

-LRB-

sI

A

-RRB-

it

follows

that

M

-LRB-

s

-RRB-

x

M

-LRB-

s

-RRB-

s

x

for

any

x

KerH

The

above

expansion

of

M

-LRB-

s

-RRB-

therefore

implies

that

M?i?x

M

-LRB-

i

-RRB-

A?x

for

all

x

KerH

and

i

n

Now

recall

that

V

equals

the

set

of

vectors

x

KerH

for

which

there

exists

a

strictly

proper

rational

vector

-LRB-

s

-RRB-

such

that

H

-LRB-

sI

A

-RRB-

x

H

-LRB-

sI

A

-RRB-

-LRB-

s

-RRB-

-LRB-

cf.

-LSB-

-RSB-

-RRB-

Obviously

it

follows

that

V

equals

the

set

of

x

KerH

for

which

there

exists

a

strictly

proper

rational

vector

-LRB-

s

-RRB-

such

that

M

-LRB-

s

-RRB-

x

L

-LRB-

s

-RRB-

-LRB-

s

-RRB-

We

claim

that

V

Tn

i

KerM?i

To

prove

this

claim

let

x

V

and

let

-LRB-

s

-RRB-

be

a

strictly

proper

rational

vector

such

that

M

-LRB-

s

-RRB-

x

L

-LRB-

s

-RRB-

-LRB-

s

-RRB-

Since

L

-LRB-

s

-RRB-

-LRB-

s

-RRB-

is

strictly

proper

it

follows

necessarily

that

M?i?x

for

i

n

Hence

x

Tn

i

KerM?i

Conversely

let

x

Tn

i

KerM?i

Because

has

full

row

rank

the

strictly

proper

rational

vector

-LRB-

s

-RRB-

P

i

is?i

such

that

M

-LRB-

s

-RRB-

x

L

-LRB-

s

-RRB-

-LRB-

s

-RRB-

can

be

defined

recursively

by

Mi?x

-LRB-

Li

-RRB-

This

implies

that

x

V

and

our

claim

is

proved

Recall

that

V

is

an

-LRB-

A

B

-RRB-

invariant

subspace

Hence

for

any

x

V

there

are

y

V

and

u

Rm

such

that

A?x

y

B?u

Given

such

x

y

and

u

it

follows

from

the

previous

that

M?iA?x

M?i?y

M?iB?u

for

i

n

and

Our

goal

is

to

compute

a

matrix

F

such

that

-LRB-

A

BF

-RRB-

V

V

Since

V

Tn

i

KerM?i

such

a

matrix

F

has

to

be

such

that

M?i

-LRB-

A+BF

-RRB-

x

for

all

x

V

and

i

n

Now

observe

that

for

any

x

V

KerH

there

should

hold

M

-LRB-

n

-RRB-

M

-LRB-

A

BF

-RRB-

x

M

-LRB-

n

-RRB-

A

M

x

M

-LRB-

n

-RRB-

B

M

F

x

Since

M?iB

and

M?iA?x

for

all

x

V

for

all

i

n

the

matrix

F

has

to

be

such

that

-LRB-

-RRB-

x

for

all

x

V

Because

has

full

row

rank

such

F

can

taken

to

be

the

solution

of

-LRB-

-RRB-

Recall

that

-LSB-

K

K

K?n

-RSB-

col

-LRB-

HA

HAn

HAn

-RRB-

where

K

K

K?n

satisfy

-LRB-

-RRB-

with

full

row

rank

This

completes

the

proof

of

algorithm

Conclusion

In

this

paper

we

have

studied

the

disturbance

decoupling

problem

We

have

developed

an

algorithm

for

computing

a

feedback

matrix

that

solves

the

problem

in

which

we

explicitly

make

use

of

the

zero-nonzero

structure

present

in

the

system

matrices

The

algorithm

is

based

on

a

representation

of

the

zero-nonzero

structure

by

means

of

directed

graph

and

the

algorithm

developed

in

the

paper

makes

use

of

well-known

and

efficient

algorithms

from

combinatorics

Clearly

the

algorithm

may

be

useful

in

case

the

entries

in

the

system

matrices

are

not

all

exactly

known

References

-LSB-

-RSB-

C.

Commault

J.

Descusse

J.M.

Dion

J.F.

Lafay

M.

Malabre

New

decoupling

invariants

the

essential

orders

Int

J.

Control

Vol

pp.

-LSB-

-RSB-

C.

Commault

J.M.

Dion

A.

Perez

Disturbance

rejection

for

structured

systems

IEEE

Trans

Aut

Control

Vol

pp

-LSB-

-RSB-

M.

Cremer

A

precompensator

of

a

minimal

order

for

decoupling

a

linear

multivariable

system

Int

J.

Control

Vol

pp

-LSB-

-RSB-

J.

Descusse

J.M.

Dion

On

the

structure

at

infinity

of

linear

square

decoupled

systems

IEEE

Trans

Aut

Control

Vol

pp.

-LSB-

-RSB-

P.L.

Falb

W.A.

Wolovich

Decoupling

in

the

design

and

synthesis

of

multivariable

control

systems

IEEE

Trans

Aut

Control

Vol

pp.

-LSB-

-RSB-

M.Gondran

M.

Minoux

Graphs

and

Algorithms

Wiley

Chichester

-LSB-

-RSB-

M.L.J.

Hautus

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disturbance decoupling
feedback matrix
graph theory
structured system