An introduction to bounded rate systems - Springer Link

AN I N T R O D U C T I O N BOUNDED

TO

RATE SYSTEMS

C.Bruni Istituto

- G.Koch di A u t o m a t i c a

Universit~ Via Eudossiana,

di Roma

18 - 00184

Roma

A. Germani CSSCCA - CNR

SUMMARY

In this work which

a new class

the d e n o m i n a t i o n

appears physical

to be quite phenomena

of n o n l i n e a r

"bounded

relevant

systems

rate systems"

for its c a p a b i l i t y

in d i f f e r e n t

is introduced,

is proposed. of m o d e l i n g

field such as biology,

This

for

class

important

ecology,engine-

ering. Bounded trol systems, vestigated

rate systems

situate

so that a bounded

and d e v e l o p e d

between

rate

exploiting

bilinear

and l i n e a r - i n - c o n

system theory may be u s e f u l l y already

available

results.

in-

472

I~ A NEW CLASS

OF N O N L I N E A R

SYSTEMS

The need of i n t r o d u c i n g

nonlinear

cal w o r l d

is p r e s e n t l y

an u n q u e s t i o n e d

important

nontechnical

fields

systems

in m o d e l i n g

fact e s p e c i a l l y

like biology,

ecology,

It is equally well know that the m a i n o b s t a c l e is the d i f f i c u l t i e s linear

classes

situation

classes

by the a n a l y t i c a l

study of a general

no~

motivates

the actual

to spot out specific

which possibly

analytical

study

couple

to the a b i l i t y

step in this d i r e c t i o n

of b i l i n e a r

applicative number

systems

trend,

the advantages of m o d e l i n g

of

relevant

of phenomena.

A noteworth

systems,

Another

theoretical

significant

class

relevant

and optimal

This paper

results

is intended

the i n t r o d u c t i o n quite valid

class

of

the

from an

to achieve

a

[1,2].

of this trend was that

the study of a more

is the systems

are a v a i l a b l e

linear

expecially

in co~

on t h e c o n t r o ~

[1,3,4,5,6]. to give a futher

contribution

a new class of n o n l i n e a r

by x the n - d i m e n s i o n this

results

systems,

control

me line by i n t r o d u c i n g

put vector,

was

in fact appears

example

of n o n l i n e a r

for w h i c h

lability

which

point of v i e w and in the m e a n time a l l o w e d

of i m p o r t a n t

general

usual

socio-economics. in this d i r e c t i o n

presented

of n o n l i n e a r

not too d i f f i c u l t

trol

of

system.

This

class

the physi-

in a number

systems.

along

the sa

Denoting

as

state v e c t o r (t:) and by u the p - d i m e n s i o n

is d e f i n e d

i~

as follows:

o

x(t)

where

the operator

= {(x)

+ Nxu

~:~ ÷ R n, ~ open

cally L i p s c h i t z i a n (2) w i t h at m o s t

11+(x) i!
0

which

is in c o n t r a s t By a s i m i l a r

changes

its

sign

with

proof,

(2.18). we can also

in ~ t h r o u g h

b)

x2 ({) = 0

Recalling

(2.13)

reject

a horizontal

,

the p o s s i b i l i t y tangent

flex

that

x2 ([) < 0

(2.22)

this means:

xl (t) >_ul ([) = s O

This

implies

the e x i s t e n c e

(2.13)

we then

(2.23)

of a 0 < t'
0 , e n t h a l p y v a r i a t i o n p,Cp

, density

and s p e c i f i c

We n o w d e f i n e

in the r e a c t i o n , h e a t of the input

assumed

exothermic

flow.

the c o n s t a n t s :

F = U__U__ (-AH_____~); K4 = ~E Ks = ~ ; Ks V p C p ; K3 = P Cp

(2.48)

and the s t a t e a n d input v a r i a b l e s ;

(2.49)

Ul u =

l

u2

u3

T h e n eq.

(2.47)

=

Ic°

(2.50)

TO

Tk _

take the form:

x = F ( x ) x + Bu

(2.51)

where: ~(KI + K o e - K---i-4 x2 ) F (x) =

0

_ K4 K3 K o e

x~

)I (2.52)

- (Ks + K2

484

-KI B

0

(2.53)

=

,_0

The

KI

function

in the o p e n bounded

0

set

rate

K2 i

F is l o c a l l y

~ = {x@R2

system,

we n o w

S = {x~R2;

for any

choice

itself.

first

equation

in

" It

xl (t o ) e

=

)to(KI+Ko

t

-~ ~t

+I

and

therefore As

> t

-

the

Je

2.6.

e

0}

(2.56)

(which

solution

are of

the o n l y

(2.51)

physi-

stays

(2.51) m a y be i n t e g r a t e d K4 x2 (

imply

xl (t) ~ 0 .

if we a s s u m e

that

there

exists

a time

that:

Other ded

a

o

K1u2 (~) + K 2 U 3

which

bounded range

is a c t u a l l y

K4

(KI + K

xt (t o ) ~ 0

x2 ({) from

(2.54)

(2.55)

e

far as x2

such

o

t

uniformly

that

0 }

any p o s s i b l e

_

xl (t)

with

defined:

in S and of u in U

choices), the

that,

> 0, u2 (t)>0,

of X(to)

Indeed,

show

xl >_ 0, x2>

U = {u : ul (t)

cally meaningful

Lipschitzian

: x2 > 0} . To p r o v e

in

we get:

(2.59)

(t) < 0

with

the

Kinetics

important

examples

are

(2.58)

(2.51)

reactor

systems

< 0

assumed

positiveness

of q u a d r a t i c

the u s u a l l y

adopted

systems

models

of u2 (t) , u3 (t) .

which

are

also

for the n u c l e a r

boun

reactor

kinetics. As

is e x t e n s i v e l y

of a p o i n t region

with

reactor Newton

with

reported one

cooling

[16,17]

group is:

a model

of d e l a y e d

for the

neutrons

free

and one

response feedback

485

(T-T o) + = c =

9 - Ic

= K(v

where

~ is

average Vo'

Co -

the

neutron

reactor I-X

= temperature

density,

To"

= reactor

I/7

= mean

of

fraction

decay

constant

time

c is

with

neutron

heat

- TO )

the

precursor

equilibrium

density,

values

T

is

an

respectively

Furthermore:

generation

I/K

- Vo ) - ? ( T

coefficient

I = precursor 1 = neutron

(2.60)

temperature,

Vo'

8 = delayed

~, + Ic

1

reactivity

time

capacity

for

heat

transfer

to

the

coolant

Defining:

X

=

[xl II X2

=

x3

equations

(2.60)

x

=

(2.61)

CO

_

take

To

the

form:

F(x) x

(2.62)

where: ~B

-

F(x)

The to p r o v e that

=

system is

purpose

the let

S =

~

y

-~

0

K

0

-y

(2.62) uniform us

C~

y

is

y(Vo+Xl)

clearly

the

xl > -

set

Vo,

(2.63)

a quadratic

boundedness

define

{xER3:

-

of

its

one

possible

so

that

all

we

trajectories.

have For

S:

x2 > - C o }

(2.64)

486

and the scalar function V on S:

I

V(X)

= vO

X[

[ F
t . --

0

PROOF.

First

and uniformly

of all,

bounded

(to,Xo) E RIxS~

there

note

that,

being

i n ~ D S, a n d b e i n g exists

a RO =

F,

f locally

u continuous

Lipschitzian

for any

{ ( t , x ) : I t - t o l £a,l[x-XoH ! ~}

such

that:

a) b)

R

C o the R H S

of

t for e a c h Therefore

(1.6)

Furthermore (1.2)

(1.6) ~ in Ro, fixed

is u n i f o r m l y

x and uniformly

admits

t h e R H S of

a local (1.6)

solution

continuous in x.

through

is c o n t i n u o u s

in

(to,Xo).

and recalling

is s u c h that:

l[F(x(t))x(t) + f(x(t)) + N x ( t ) u ( t )

+

{INiJ flu(t)[I

KI (t) ~ K2 (t) a r e

+ Bu(t) [[ t --

RE~£ARK I. A l l systems que

satisfy

solution

REMARK proved of Thm. ses

the p r e v i o u s l y

the a s s u m p t i o n s

3.1.

for

all

solution

of

.

examples

3.1.

and

of b o u n d e d

therefore

admit

rate a uni

in large.

2. A s i m i l a r

in [21 ] . The

in w h i c h

local

o

mentioned

of Thm.

z(t o) E R and any

and both

theorem,

for

set of a s s u m p t i o n s

somehow

more

systems seems

restrictive

are a p p l i c a b l e .

evolving

on group,

to be s i m i l a r

than

the

latter,

is

to that in t h o s e

one ca

490

4. C O N C L U S I O N S

In the previous of b o u n d e d

rate

section we e n l i g h t e n e d

systems.

We showed

processes

may be included

in that class.

From

the t h e o r e t i c a l

point

ded rate

systems

the theory our m i n d

was given

this

should be an important

In p a r t i c u l a r troi would

results

be useful.

b u t i o n was given

in [9]

where

system as a bilinear This

bounded

rate

for b i l i n e a r

systems

are already

[3,4,5,6,21,24]

available

which

a controllability

rive

is concerned,

hopefully

theory

controllability

properties

of the state equation. systems, available

Under

for which [I].

will

for b o u n d e d

some papers

already

exploiting

contr~

for the immune

structure

in a u n i f o r m l y

for the class

We also m e n t i o n

bilinear

stability

the general

a first

bounded

feedback

theoretical already

res-

of a b o u n d e d loop.

results

available

for

results

systems.

As far as c o n t r o l l a b i l i t y sults

and o p t i m a l c o n

is concerned,

of how to achieve

by suitably

and in

task to be pursued.

asymptotic

system

is a good example

result for h o u ~ point out that

on stability, c o n t r o l l a b i l i t y

exploiting

physical

is still to be d e v e l o p e d

As far as s t a b i l i t y

ponse model was p r o v e d rate

3. We must however

of systems

of the class

of important

of v i e w a first general

in Sect.

for such a class

the r e l e v a n c e

how a number

a number

of linear

be a useful rate

[25,26]

starting

re-

systems

point to build

systems. in w h i c h

by looking

it is shown how r o d e

only at the u - d e p e n d e n t p a r t

this aspect b o u n d e d some results

of important

in c o n t r o l

rate

systems

on c o n t r o l l a b i l i t y

behave as

theory

are

491

APPENDIX

In o r d e r

to p r o v e

THEOREM.

Given an open subset

condition

eq.

(I .3)

we m a y

state

following

~ of R n a necessary

~ : ~ + R n to be

for a function

the

and sufficient

locally Lipschitzian

with the

growth property:

II~ ( x ) II ~ C ,

IlxlI + C 2

where CI ,C2 are nonnegative

= F(x)x

Sufficiency.

PROOF.

~

(A.I)

is that there exist

locally Lipschitzian

two functions

and with uniformly

such that:

(x)

bounded

¥x~

constants,

f : ~ + R n which are

F : ~ ÷ R n×n,

bounded range

,

subset

MC~

we

+ f(x)

If

,

(A.2)

VxC

(A.2)

holds,

then

f o r xl , x2

in a n y

closed

have:

11% (xl) - ~ (x2) II= IIF(xJxl

+ f(xl ) - F ( x 2 ) x 2

< liE(x1 )[I Ilxl-x2 tI+IIF(x:)-F(x2)U < maxIIF(xl)II x~ ~ M

- f(x2) II

tlx2 II+[tf(xl )-f(x2)I1

IIx~-x2 II+ maxllx2 II L F x2 E M

II x l - x 2 If+ Lf~xl-x211