2 Basic Interconnections

2

Basic Interconnections

2.1

INTRODUCTION

We suggested in the introduction to Chapter 1 that the idea of a system as “an interconnection of subsystems” is very fundamental. Going along with this, there is perhaps no problem in systems theory more important than the investigation of interconnections and decompositions of systems. I t is of great importance to understand how the properties of an interconnection of systems are related to the properties of the systems interconnected. Also, it is fundamental that systems theory explain how a given system may be decomposed into subsystems, i.e., how it may be realized as an interconnection of other systems. As a background for such considerations here, we need to develop the concept of interconnections within our formalism. I n Chapter 1 , we introduced the concept of a T-processor as a special case of the T-process and suggested that the former serve as our main concept of a “general system.” Consistent with this point of view, it follows that “interconnections” in our theory will be a set of operations which are defined on and yield T-processors. We actually consider five operations on T-processors for this purpose. Two of these are binary operations(one T-processor is produced from two given ones) and the remaining three are unary (one T-processor is produced by modifying another). 41

2.

42

HASIC INTERCONNECTIONS

.\lthoiigh, of course, the concept of interconnections is very ell knoM-n throughout systems theory, there has been relatively little \vork done on formalizing and studying them in such a general case as \ve no\v have the capability for. We shall draw upon the results ohtained in this chapter in our subsequent considerations extensively. 2.2

PROCESSORS IN SERIES

In this section and the next, \ r e introduce two of the very basic interconnections of T-processors. I n keeping with many precedents? in systems theory, we call these particular interconnections, ‘‘ber 1es’ ’ and ‘‘parallcl .’ ’

2.2.1

Definition. If P and Q are T-processors, then the series

inttvcorrnertion of P and Q is the set

I’LQ

I’

‘~ (1

:{uz

(3y):uytP&yzEQ;

is said to be properf if? P” C Q1.

I W . ~ . I I ~ K .Clearly,

P

0

0

is generally a T-process. Moreover,

it is a T-processor since, as is obvious, ( P (2) C PlQz. 0

2.2.2 Lemma. I f P and

\\

here the synibol I>IH)OF.

0

011

0 are

T-processors, then

the right is composition.

\Ye have

-I_ .llthoiigh i i l i n o s t evei-1 t)riinch of systems theory discusses “series” a n d “parallel” ititri-coiiiir(.tioiis, there is some :itnhiguity in what is generally meant by these terms. \\‘hat is nieant b y “parallel” is different in automata theory than what is nieiiiit i n linear system theory and there is more t h a n one notion of a “series” ~ i i t e t . ~ ~ i i ~ i iii e c both t i ~ ~ areas. i 1 ’l‘liis requireiiicmt is h i g h l y intuitive. I t says that every o u t p u t of P must b e :I11 IIl~”1t of 0 .

2.2.

43

PROCESSORS IN SERIES

REMARK. Lemma 2.2.2 makes clear the fact that the use of the symbol 0 for both composition of relations and for series interconnections of T-processors is justified, i.e., the dual of series interconnection is composition applied to the relations P , and Q.+ .

T h e following conditions are immediate by Corollary 1.6.12:

2.2.3 Corollary. If P, Q, and R are T-processors, then Po(Q0R) = (P0Q)oR

If P and Q are functional [bifunctional], then P [bifunctional]. Also ( p o Q)-1 = Q-1 o P-1

0

0

is functional

2.2.4 Lemma. If P and Q are T-processors, then ( P 0)' C Pl and (P C Q2. If P 0 Q is proper, then ( P 0)' = Pl. 0

0

T h e first two conditions are obvious. If P 2 C Q', then

PROOF.

u E Pl

0

3 3

(3y): uy E P 3 (3y): uy E P & y t P2 (3y):uy E P & y E Q l (3y)(3z):uy E P & y z E Q 3 (32): uz E P o 0 3 u E ( P o Q)l

that is, P1C ( P 0 Q)I. Thus, ( P o Q)'

=

Pl.

I

2.2.5 Corollary. Let P and Q be T-processors. If P o Q is proper, then ( P Q)l is an uncoupled T-processor iff Pl is. Also, P Q is free iff P is free. 0

0

2.2.6 Lemma. Let P, Q, R, and S be T-processors. If P C R and Q C S , then ( P o 0) C (R S ) . 0

PROOF.

uz E

We have

(P 9)

(3y):uy E P & y z E Q

0

2

3

(3y):uy E R & y z

E

S

I

uzE(R0 S)

2.2.7 Lemma. If P is a T-processor, then both I ( P ' ) and I ( P 2 ) are T-processors and I(P') P 0

~

I-'

=

P I(P) 0

where both interconnections are proper.

44

2. Obvious.

PROOF.

2.2.8

BASIC INTERCONNECTIONS

If P and

Lemma.

IP

0 are

c I[)

T-processes, then

cQ

I’

-?-

Obvious.

PIWOF.

L e m m a . Let P be a 7’-processor and let Q be a T-process. P C P and

2.2.9

T h e n IQ

L

(Kl .

1~~001.

PI*

-

PXiQ

Recall P*lQ = { ( u , y ) ,u P * y & u € Q ]

JYe h a l e U ( K 1 P)* J

I’ 3 (3z):u 3 t r,Q & 2-y F P z & zy t I’ @+uu E 1Q & uy E P \> ( 3 z ) :uu’t 1,Q & 2L u?, t I Q

>

u

-+

F

y

0& uP,y

Q)y

> u(P,

that is, (10 P)k P , Son, ( P * 1.6.1 1, \\e see ’ P C P. I

ro

0.

~

0)C P ,

. ‘l’hus, by Lemma

K r v w k . ‘Thus, the series interconnection c P serves to denote the “restriction” of the T-processor P on the input set \\here 0 is ‘1 qiveii 7’-process and 0 C Pl.

0,

2.2.10

Lemma.

PK~OE.

1’2

f=

If P

l(P) Q &

0

0

Let P m d be 7’-processors. If P is functional.

functional, then Z(P‘)

0

C Q is

0 is functional, we hax.t

>‘ZL

i

[(P) 0

y y E I ( P ) & 3’2c Q & y“L E Q Y i PZ & y 3 r C,, & J ’ W E ,Q ( 3 )UJ :

i I’&y,-ty&4’ZL i Q

( 3 u ) : u z i I ’ ~ ~ & u z l !9 ~P ->z

T h i s proves I ( P ) [I

is

functional.

I

w

2.3. 2.3

PROCESSORS IN PARALLEL

45

PROCESSORS IN PARALLEL

V’c continue with a similar elementary investigation of the “parallel” interconnection of T-processors.

2.3.1 Definition. If P and Q are T-processors, then the parallel interconnection of P and Q is the set P//Q

-=

{(uu)(Yz)1

UY E

P & uz E 0 )

REMARK. We see P / / Q C (P1Q1)(P2Q2), and it follows that P / i Q is a T-processor. Moreover, (P//Q)‘ C PlQl and ( P / / Q ) 2C P2Q2; hence, both ( P , / Q ) land ( P / i Q ) 2are T-processors. Thus, P / / Q is always multivariable.

2.3.2 Lemma. If P and Q are T-processors, then (P,’/Q)l= P’Q’ and (P/lQ)2= P2Q2. PROOF.

We see

(I’i/Q)l = { u u , ( 3 y ) : u y € P & ( 3 z ) : U Z € Q } { u v ~ u r P l & v r Q 1 } = ply‘ 1

T h e condition ( P / l Q ) 2= p2Q2 is similar.

[

By Theorem 1.7.2, we have:

2.3.3 Corollary. If P and 0 are T-processors, then (P //Q)l and (P//Q12 are 7‘-processors. Moreover, ( P / / Q ) l and (Pl’’Q)2are uncoupled. 2.3.4 Lemma. If P and Q are T-processors, then ( P / i Q )is free iff both P and Q are free. PROOF. If u and v are T-time functions, then uu is constant iff both u and v are. In fact,

(a)@)

=

(a, b)

Now, since (P/’Q)l PIQ1, it follows that (P,’!Q)l contains precisely one constant T-time function iff both P’ and Q1 do. [ ~

46

2. IlASIC Lemma. If I’ and

2.3.5

INTERCONNECTIONS

0 are

(P/‘Q)-’ iwooi~.

2.3.6

=

T-processors, then P-yIg-1

For the reader.

Lemma. If P and 0 are nonempty T-processors, then is functional [bifunctional] iff both P and Q are functional c t i o nal] .

IW)OF. I,et P and Q be functional. Choose ( u v ) ( y z )E ( P / / Q ) and ( u ‘ v ’ ) ( y ’ z E’ )(1’ 0). Clearly, uy, u‘y’ E P and DZ, v’z’E Q. I~-sing1,emma I .5.15 - fL’V’

1(7)

-. u

-z u‘

&v

=

@’

3

y

.-:

y‘ & z

=

z‘

=> y z = y’z’

that is, ( P i j Q ) is functional. Conversely, let ( P / / Q )be functional. F P. Sincc 0 is nonempty, there exists some vz E Q. 0)is functional, u = u‘

-:. uv

=

yz

u‘t1 ~2

~

y‘z

3

y

= y‘

that is, P is functional. Similarly, Q is functional. T h c condition for the bifunctional case follows from Lemmas 1.7.7 and 2.3.5.

2.3.7 Lemma. Let P, 0,K, and S be T-processors. If P C R and Q C S, then ( P i ‘ Q )C ( R S). PIU)OI;.

Obvious.

1 , e r n n ~2.2.7 inakcs clear the fact that any T-processor admits (tri\,ial) series ctccompositions which are proper. ’l’here is an issile i n the par;dlel case which we can easily settle:

2.3.8 Theorem. If P is a noncmpty T-processor, then the follov in? statements are equivalent: (1)

(11)

’l’hcre exist 7’-processors R and Q such that P Q//Ii. 1’1 m d P’ are 7’-processors and I’ 1 1 I/, where ~

L

I

{ u y , (32)(32): ~

{vz

,

( u v ) ( v z )t C’]

, ( 3 u ) ( 3 y ) : (.v)(yz)

E PI

2.3.

47

PROCESSORS IN PARALLEL

(iii) P’ and P2 are T-processors, and (uv’)(yz’)E P & (u’.)(y’x)

P * (u.)(yx)

E

t

P

(ii) 3 (i) is trivial. (i) 3 (ii). If P = Q / / R , then obviously both P’ and P2 arc T-processors. Now, since P is nonempty, both Q and R are V. nonempty. I n this case, clearly, Q = U and R (ii) 3 (iii). If P = U / / V ,then PROOF.

2

(uv’)(yz’)E P & (u’.)(y’.)

E

P

3

uy E

u& 7Jx E v

-. (u.)(y.)

(iii)

3

E

P

-

( u v ) ( y x )E ( U / / V )

Given (iii), U / / V C P. I n fact,

(ii).

( u v ) ( y z )E (L’//L’) * uy E U & U Z E v

3

(3v’)(32’): (uv’)(yz’)E P

& (3uf)(3y’):(u’v)(y’z)E P

2

( u v ) ( y z )E 1’

But, also, P C ( U / / V ) .T h a t is, (u.)(yz)

Hence, P

2.3.9

=

E

P

* (uy) E U & (u.)

U//V.

E

I; * (u.)(yz)

E

(U//l.)

I

Theorem, If P, Q, R , and S are T-processors, then ( P / / Q ) (RIIS) = ( P R ) / / ( Q S) O

O

We conclude this section by proving:

2.3.10 Theorem. If P and Q are nonempty T-processors, then both P and Q are images of P / / Q .

2.

48

Consider the function

IW)OF.

Clc,irly, t i 7’,

BASIC INTERCONNECTIOSS

/I:

/I(P

t ( ( U Z ’ ) ( yz)

0)

+

I/)

(t((IW)(JJ2)))h ~

that

/I IS, (U~)(J
Hence I’

I5

~

/ [ I J . Yo\+, for all ( u z ~ ) ( ytz )P / Q and all

( ( l u , f7’),

(l(UV),

( t y , tz))h

t(yz))A

(tu, t y )

t(uy)

uy. ‘l’hus, since Q is iionenipty,

an image of I’

0.Similarly, 0 is an image of P / , Q . I

PROCESSOR PROJECTIONS

2.4

I -processes, though highly generalized and abstract, are still concrete models for real dynamical proccsscs. It is, of course, in the realin of real dynamical processes that one must look to decide bvhich operations on 7-processors correctly interpret to be interconnections and which do not. If it is obvious that the above series and parallel interconnections of T-processors are correct formalizations o f corresponding notions of interconnections in real physical processes, it is equally u n c l t m what other elementary operations must be added i n order to have a “complete” set of interconnection operations for 7‘-processors, i.e., in the sense that any “real” interconnection can be represented algebraically. It is our guess at this point in time that we get approximately “ a complete’’ set when we add to the above series and parallel opcrxtions (\vhich are of course binary operations) three unury operations. One of thesc unary operations which we call the “closed loop” operation (and which is introduced in the next section) is employed to account for “feedback.” Feedback which is definitely distinct from series and parallel interconnection is one of the most important concepts in systems theory froin the point of view of practical engineering. Fccdback accomplishes many very ,

I

2.4.

PROCESSOR PROJECTIONS

49

important tasks; most especially, it permits some quite complex systems to be constructed from simple subsystems. I n other words, one suspects at the outset that the “closed loop” operation applied to a processor may yield a processor with radically different properties than the processor to which the operation is app1ied.t T h e other two unary operations we believe are fundamental here are quite unsophisticated projection operations:

2.4.1 Definition. If P is a T-processor, then the sets P:

=

{u(uy)j uy E P }

: p = {@Y)Y I UY

E

P>

are the feedforward and the feedback of P, respectively. REMARK. P : C PIP, so P : is a T-processor. Moreover, ( P : ) 2C P, so P : is multivariable. :PC PP2,which implies :P is a T-processor. Also, (:P)’ C P so :P is likewise multivariable. We note

u(.y)

E

P:

0

uy E P & u

(UY)ZE : P - uy E P & y

=

=

x

z

2.4.2 Lemma, If P is a T-processor, then (P:)l= Pl, (P:)2= P, (:P)I = P, and ( : P ) 2-=P2. PROOF.

Obvious.

2.4.3 Lemma. If P and Q are 7’-processors, then ( P : ) C ( Q : ) o P C Q e( : P ) C ( : Q ) PROOF. Clearly, if P C Q , then ( P : )C ( Q : ) and ( : P ) C ( : Q ) . Conversely, if ( : P )C (:Q), we have

u y ~ P a ( u y ) y ~ : P( u* y ) y ~ : Q - u y ~ Q

and if ( P : )C (Q:), uy E P

u(uy) E P:

that is, in either case, P C Q. -I. Unfortunately,

* u(uy) €9: * uy E Q

I

by the same token, we should not expect to prove a great

deal about the closed loop operation in general; i.e., what we are mostly doing here is showing preservation of various properties under operations.

50

2.

BASIC INTERCONNECTIONS

2.4.4 Corollary. If P and Q are T-processors, then ( P : ) = ( 0 : )e P

=

Q 0 ( : P )= (:Q)

2.4.5 Theorem. If P is a T-processor, then the following statements are equivalent: (i) There exists a T-processor Q such that P (ii) P2 [P'] is a T-processor, and u =z

u(zy)E P

[(uy). E P

(iii) P 2 [PI]is a T-processor and P

=

3

y

= Q: [ P =

:Q];

= z]

( P z ) :[P = :(P1)].

PROOF. Wc shall treat the latter case and leave the former as an exercise.

(i) 3 (ii) is obvious. (ii) 3 (iii). If (ii) holds, then uy E Pl

Thus, P :(I"). (iii) 3 (i) is trivial.

0

(32): (uy).

E P e (uy)y E

P

~~

I

2.4.6 Lemma. If P is a T-processor, then P : is free iff P is free. :P is functional. PROOF.

2.4.7

Obvious.

Lemma. If P is a T-processor, then P

FKOOF.

=

P: :P 0

LVc have :P 0 ( 3 4 3 2 ) : u(s2) E P: & u = x & (.z)y E :P & z = y -a u(uy) E P: & ( u y ) yE :P cr> uy E P & uy E P 0 uy E P

uy E P: 0 : P c- (3x)(3z): u(xz) E P: & ( x z ) y E

that is, P

-

P: o :P.

I

2.4.

51

PROCESSOR PROJECTIONS

Lemma 2.4.7 shows in the given weak sense, for any REMARK. T-processor P, P : and :P are “inverses” of each other. This interesting property leads to some other useful identities. 2.4.8

Lemma. If P, Q, R, and S are T-processors, then (POQ) = ( P o Q : ) :Q 0

( P o 8)o ( R o S ) = ( P PROOF.

0

(8

0

R):)0 (:(Q R) S ) 0

0

Application of Lemma 2.4.7 gives (PoQ)=Po(Q:o:Q)

=(PoQ:)o:Q

( P o Q ) o ( R o S ) = P o ( Q 0 ( R 0 S ) ) = P O ( @0 R ) 0 S )

(((Q R ) : :(Q R ) ) 0 S ) = P ((Q R): (:(Q R ) S ) ) = ( P (Q R ) : ) (:(Q R ) S ) =P

0

0

0

0

0

0

0

0

0

0

0

0

0

0

where we have repeatedly used the associative law established in Corollary 2.2.3. I 2.4.9 Theorem. If P is a T-processor, then P , P : , and :P are pairwise isomorphic. PROOF.

We note

Therefore, consider the relation h

Clearly, h: GlP (uy)y. Hence,

--f

4, ( ( a , b), 6)) I a ( @ W }

= {((a,

Gl(:P) (1 : 1 onto) and for all uy

E P,

uy 0 h

=

52

2.

BASIC INTERCONNECTIONS

and h is an isomorphism from P to :P. Similarly, the relation R

~

{ ( ( a ,( a , b ) ) , (66)) I a ( W b 1

is an isomorphism from P : to P. Finally, g 0 h is an isomorphism from P: to :P. I 2.5

CLOSED LOOP PROCESSORS

Algain,“feedback” is one of the most intriguing concepts in the entire systems area. In this section, we introduce the notion of the “closed loop” of a T-processor. T his can be regarded as a direct attempt to formalize the concept of feedback in as general a \lay as possible still retaining the flavor of the concept as it is employed in engineering. T h e closed loop of a T-processor is u hat that T-processor “looks like” with feedback introduced in the form of a direct connection from output to input. As we shall show, the closed loop operation and the feedback operation introduced in Section 2.4 arc intimately related. 2.5.1 Definition. I,et P be a 7’-processor. If P’ is a T-processor, then the closed loop of P is the set #P

=

{UY,(U,V)Y EP:

KFVAHK.In general, # P C PI. Thus, # P is itself a T-processor. C (P’)’ and (#P)2 C n P2.Also, # P C Pl o I(P2),

\Ye see (#P)’ 1.e.,

u?/r#P~~(u3’)yEP~uy€P’&y€P2 f’

u-y E

P‘ & y y

E

I ( P ) 2 uy € P’ Z(p2) 0

2.5.2 Lemma. Let P and Q be T-processors with P1 and Q1 IikeLvise T-processors. Then, PCQ

=“

#PC#Q

PROOF. Obvious. REMARK. ‘l’hc converse of Lemma 2.5.2 fails and this is important.

2.5.

53

CLOSED LOOP PROCESSORS

Next, we see how the closed loop and feedback operations are related: 2.5.3 Theorem. If Q is a T-process, then #(:Q) = Q. If P is a T-processor with Pl a T-processor, then :(#P) C P. PROOF.

We note (:Q)l is a T-processor, so #( is: well (I defined. )

We have

#(:~)={~Y/(~~)YE:~}={~YI~YE~}=Q Next, we see (.Y)Y

that is, :(#P) C P.

E

:(#P)

UY E

#P * (.Y>Y

Ep

I

REMARK. Theorem 2.5.3 interprets to say :(#P) is the least T-processor whose closed loop is the same as that of P, i.e., #(:(#P)) = #P. Thus, for a T-processor R, :R is the least “open loop” of R. :(#P) also interprets to be that part of P that can be “observed” or “measured” under closed loop conditions.

An interesting special case, namely where :(#P) characterized as follows:

=

P can be

2.5.4 Theorem. Let P be a T-processor. If Pl is a T-processor, then the following statements are equivalent: (i) :(#P) = P. (ii) P = :(I“). (iii) U P = 21ZP1. (iv) (uy)z E P * y = z . (v) P = : ( P l ) & # P = P’.

-

(ii) is by Theorem 2.4.5. (i) (iii). I n the proof of Theorem 2.4.9, wc showed 2aQ for any T-processor Q. Thus, here PROOF.

(ii)

3

OlP

(iii)

3

(iv).

(uy) z E P

=

a(:@=

@(:(Pl))= 2aPl

Given (iii),

* (Vt): t((uy)z)E OlP * (Vt): t((uy)z)E 2 a r ’ 3

(Vt): ((tu, ty), t z ) E 2@P1

(Vt): ty

=

tz

2

y

=

z

54

2.

BASIC INTERCONNECTIONS

(iv) 3 (v). Given (iv), P :(P1)by Theorem 2.4.5. But then by Theorem 2.5.3, #P = #(:(PI)) = Pl 1

so (v) holds.

(v)

3

(i). We have :(#P)

=

:(PI)= P

I

Another interesting special case is to characterize when # P For this, we have:

=

Pl.

2.5.5 Theorem. Let P be a 7'-processor. If Pl is a T-processor, then the following statements are equivalent: (i) (ii) (iii) (iv) (v)

# P = Pl. uy E Pl u (uy)y uy E P 1 3 (uy)y :(PI)C P. P' c #P.

IWWF.

(i)

3

(ii). uy

E E

P. P.

Given (i), E

Pl

0

uy E # P u (uy)y € P

(ii) * (iii) is trivial. (iii) 3 (iv): ( u y ) z t :(PI) 3 u y E P ' & y

=

z

3

( u y ) y t P & y = z =. ( U Y ) X € P

(iv) (v). P' = #(:(P1))by Theorem 2.5.3. Using Lemma 2.5.2 then, if :(PI)C P, we see P1 C #P. (v) 3 (i). I n general, # P C PI. I Ifre have a number of identities involving the closed loop operation:

2.5.6

Lemma. If P, Q, and R are 7'-processors and R1 is

T-processor, then

#((PiiQ) R ) = P #((W2)//Q) R) O

O

O

a

55

EXERCISES PROOF.

We see

uy E #((P//Q)0 R ) 0 (uyly E (PllQ) R 0

P / / Q & (xz) y E R P & y z E Q & (xz) y E R

u (3x)(3z): (uy)(xz)E

o (3x)(3z): ux E

o (3x)(3z): ux E P & x E P2 & y z E Q & (xz) y

ER P & xx € I ( P 2 )& y z E Q & ( x z ) y E R o (3x)(3z): ux E P & (xy)(xz) E Z(P2)//Q& (xz)y E R

u (3x)(3z): ux E

u (3x): ux E

-

P & (xy) y E (Z(P2)//Q) o R

o (3x): ux E P & xy E #((Z(P')//Q) 0 R )

uy E p

O

#((V2)//Q) R) O

2.5.7 Lemma. If P and Q are T-processors with Q1 a T-processor, then #(PllQ) = Po (#Q): PROOF.

4 ~ E4#(PI@)

0

( u ( y z ) ) ( y z E) P / / Q

-

UY E

P&(

~ 4 EO

#Q e uy E P & y ( y z ) E (#Q): e (3x): ux E P & x ( y z ) E (#Q): 0 u(y.) E P (#Q):

o uy E P & y z E

0

2.5.8 Lemma. If P and Q are T-processors with Q' a T-processor, then P o #Q = # ( p / / Q ) :(#el 0

PROOF. Combining Lemma 2.5.7 and the first condition of Lemma 2.4.8,

(P

O

#Q)

=

(P

O

(#!a:):(##!a = # ( P / / Q ) :(#!a I O

O

Exercises 2-1. If P is a T-processor and Q is a T-process, prove ( P O Q)* = (V*)-*/Q)Y

2-2. Prove for any T-processors P and Q that P / / Q and PQ are isomorphic.

56

2.

DASIC INTERCONNECTIONS

2 - 3 . P r m c or give a counterexample: For any T-processors P and 0,P Q is uncoupled iff both P and Q are uncoupled.

2-4.

ProLC for any T-processors P and (2, P, iL,

2-5.

r-

(P//Z(Q')) ( I ( P 2 ) / / Q )

Prove for any T-processor P, #((P:)-1

2-6.

0

P)

=

P

Let P be a T-processor with both P' and P2 T-processors, and let lT = {uy , (3v)(31): (uv)(y.) € P )

I-

= (u2

(%)(3y): (uv)(yz)E P }

Prove that if C' is functional and V is uncoupled, then Pp is u n co u p 1e d .

2-7. Prove for any 7'-processors P and Q,

P.0: = { u ( y z ) ~ 2 I ~ y E P & y S E Q ) 2-8.

Give necessary and sufficient conditions on a T-processor P that there exist T-processors 0 and R such that P = Q R : , where Q 9 R: is proper. 0

2-9.

Let I' be a T-processor with P' a T-processor. Prove

(P'): P 0

=

{uy ,

(32):( u 2 ) y

E P)

2-10. Let P be a T-processor xvith Pl a 7'-processor. Prove if ( P I ) : P is functional that # P is functional.

-

2- 1 1. Prove if P is a T-processor with P' a T-processor that

#r = ( ( P I ) :

0

P : ) :(:(PI)) 0

and hence that # P may always be represented as a (often improper) series interconnection.

EXERCISES

57

2-12. (Feedback compensation problem.) If P and Q are T-processors, prove that the following statements are equivalent: (i) There exists a T-processor R with R‘ a T-processor such that P = #(R 0 Q), where R2 C Q1 and R1= P. (ii) P 2 C Q2. (iii) P = # ( ( : P 0 Q-l) 0 Q). 2-13. Develop and prove a theorem analogous to that of Exercise 2-12 for series compensation of T-processors, i.e., for the case of P = R Q. 0

2-14. In automata theory, the following operation on T-processors is defined and called “parallel interconnection”: P$Q ={u(~z)Iu~EP&uzEQ}

Has this operation been subsumed in our theory of interconnections ? 2-15. Work out an elementary theory of the interconnection given in Exercise 2-14.