Chapter X Interconnections of Subsystems, Decompositi0N, and Decoupling

Chapter X

INTERCONNECTIONS OF SUBSYSTEMS, DECOMPOSIT10 N, AND DECOUP11NG

An important application of general systems theory is in the large-scale systems area. Essential in this application is the ability to deal with a system as a family of explicitly recognized and interconnecting subsystems. Actually, this is how the so-called “systems approach” is defined in many instances in practice, and the entire systems theory concerned with a system as an indivisible entity is viewed as nothing but a necessary prerequisite for the consideration of the large-scale and complex problems of real importance. In this chapter, we shall be concerned with various questions related to interactions between subsystems that are interconnected and constitute a given system ;the objective is to provide a foundation for new developments in systems theory aimed at large-scale systems applications. To demonstrate the breadth of possible applications, specific problems in two different areas are considered. Conditions for decoupling of a multivariable system by means of a feedback are given in terms of the functional controllability of the system; both general and linear systems cases are considered. Conditions for decomposition of a finite discrete time system into a special arrangement of simpler subsystems are given.

1. CONNECTING OPERATORS

Interconnection of two or more systems is an extremely simple operation. All one needs to do is to connect an output ofa system with the input ofanother system or to apply the same input to two systems; in practice, e.g., that might I70

1 . Connecting Operators

171

simply mean “connecting the wires” as indicated. Unfortunately, formalization of that simple notion is rather cumbersome primarily because of the “bookkeeping” difficulties,i.e., a need to express precisely what is connectable with what and indeed what is actually being connected. To avoid being bogged down by unduly pedantic definitions, we shall introduce first a class of connectable systems and then define various connecting operations in that class. x we shall denote by E the family of x For any object = component sets of F, E = { Y,, . . . , Let Si c Xi x be a general system with the objects

v vl

vn, vn}.

xi =X(Xij:jEZx,},

v

yi = x { q j : j € Z , , }

In general, some, but not all, of the component sets of Xi are available for connection. Let Z,, denote the Cartesian product of such component sets, i.e., XijE Z,, means that Xij is a component set of X i and is available for connection. Denote by Xi* the family of all component sets not included in Z,,, Xi* = {Xij:Xij$Z,,}, and by Xi* the Cartesian product of the members from Xi*, Xi* = x { X i j : X i j ~ X i *The } . input object can then be represented as the product of two composite components Xi = Xi* x Z,,. Analogously, let Z,, be the Cartesian product of the output components available for connection. Given a system Si c X ix Yi, there can be defined, in general, many “different” connectable systems Si, c (Xi* x Z,,) x x Z,,,), depending upon the selection of Z,, and Z,, . The relationship and Z,, is between Si as defined on Xi, and Si, as defined on Xi*, Z,, , obvious. In both cases, we have essentially the same system except for an explicit recognition of the availability of some components for connection. We can now define a class of connectable systems

(x*

x*,

s, = { S i z : S i zc (x: x Z,,) x (YT x Z,,)} and the connection operations within that class.

0

is termed the cascade (connecting) operation.

Chapter X

172

Definition 1.2. Let

s 3

c

+ : 3,

(X1* x

x

S,

+

x,* x

Subsystems, Decomposition, Decoupling

S , be such that S ,

Z) x

(Yl

z,

x Y,),

and ((X,,XZ,Z),(Yl,Y,))ES3 -((x19Z),Yl)Esl

+ S2 = S3,where =

z,, = z

&((X29Z),Y,)ES2

+ is termed the parallel (connecting)operation. Definition 1.3. Let 9 be the mapping 9: 3, + 3, such that 9 ( S , ) = S2, where S, c (X* x Z,) x ( Y * x Z,), S 2 c X* x Y*, Z , = Z, = Z, and (x, Y ) E s2 * (3z)(((x,4, (Y, 4) E Sl)

9is termed the feedback (connecting)operation. Examples of the application of the connection operators are given in Fig. 1.1. It should be noticed that the connection operators could be also defined in alternative ways. For example, rather than defining a feedback using a single system and connecting an output with an input as shown in Fig. 1.1, another subsystem could have been assumed in the feedback path as shown in Fig. 1.2. However, the three basic operations as given in Definitions 1.1 to 1.3 cover in combination most of the cases of interest, and in this sense they can be considered as the primary connecting operators. For example, the connection for the case in Fig. 1.2 is given by 9 ( S 1 S2), as shown in Fig. 1.3. Notice that the operations 0 , +, and 9 are defined as partial functions. Although it is possible to make these functions total, we shall not do that for the sake of simplicity. When (S, S,) S3 is defined, we have ( S , S,) S3 = S , ( S , o S3). As a matter of fact, if S1, S2, and S , are defined as 0

0

then

0

0

0

0

173

1. Connecting Operators

+

Parallel connection

Feedback operation

FIG.1.1

FIG. 1.2

Chapter X

174

Subsystems, Decomposition, Decoupling

FIG.1.3

Similarly, we have (S,

+ S,) + s3 = s, + ( S , + S 3 )

if both sides are defined. The operation 0 does not have identity element. However, in both the input and the output objects, one can define identities I, t X x X , I , = { ( x , x ) : x ~ X }and , I , c Y x Y, I , = { ( y , y ) : yY~} . For a given system S c X x Y, it is possible then to define the left inverse -IS c Y x X and the right inverse S-' c Y x X such that - ' S o S = I,,

and

SoS-'

=

I,

A function f defined on a family of subsets X i in X such that f ( X i )E X i is called a choice function. We have then the following immediate propositions. Proposition 1.1. A system S c X x Y has a right inverse S-' c Y x X if and only if there exists a choice functionf: { S ( x ):x E 9(S)} 3 W ( S )such that { f ( S ( x ) ) )n S(x') = 0 for any x' # x, where S(x) = {y : ( x ,y) E S } . Proposition 1.2. Suppose Y = W(S). Then a system S c X x Y has a left inverse - ' S c Y x X if and only if there exists a choice function f : {(y)S:y E W ( S ) }-+ B(S) such that { f ( ( y ) S ) )n (y')S = 125 for any y' # y where (y)S = { x : ( x , y S) ~} .

For the operation + ,the empty system 0is the identity. The three operations are interrelated ; for instance, 9 ( S 1 s,) = 9 ( S , 0 s,) holds if both sides are defined. 0

I . Connecting Operators

175

We shall now consider how some properties of the subsystems are affected by interconnections.First, we shall look into the question of nonanticipation. In this respect, we have the following propositions.

Proposition 1.3. If S, c X, x (Y, x Z) and S, anticipatory, so is S3 = S, 0 S,.

c

(X,x Z) x Y, are non-

PROOF. Let ( Y , 4 = (PlY(C13 X i ) , P l Z ( C 1 9 X i ) ) and Y , = P Z ( C 2 , (x27 4) be nonanticipatory global state representations. Suppose (x, ,x,) 1 T' = (x,', x,') I T'. Then y , I T' = y,' I T' and z I T' = z' I T', where (y,', z') = (ply(cl,xi'), plz(c,, x,')), and consequently, y, I T' = y,' 1 T', where Q.E.D. y,' = p,(c,, (xz', z')). Hence S, S, is nonanticipatory. 9

0

Proposition 1.4. If S, c (X,x 2)x Y and S2 c (X,x Z) x Y are nonanticipatory systems, so is S3 = S1 S , .

+

Proposition 1.5. Suppose S c (X x Z)x ( Y x Z) is nonanticipatory such that (y, z') = (JI (c, x, z), p,(c, x, z)) is a nonanticipatory systems-response function. If z 1 $' = p,(c, x, z) 1 T' has a unique solution z I T' for each (c, x) and c, t4en d ( S ) is also nonanticipatory. PROOF. It follows from the definition of %(S) that

(x, Y ) E P(S)* ( W ( ( X , z), (Y, 4)E S)

* (34(3c)(y= py(c,x, 4 8z z

= PAC, x, 4)

Since z = p,(c, x, z) has a unique solution for each (c, x), let z = +(c, x) t+ z = p,(c, x, z). Then (x, Y ) E

*( W Y

= py(c,x,

&, x)))

Let p(c, x) = py(c,x, +(c, x)). If p(c, x) is nonanticipatory, the desired result follows immediately. Let R I T' = R' I T', 2 = pz(c,R,2), and 2' = &, 2',2'). Since p, is nonanticipatory, p,(c, 2 , 2 ) 1 T' = p,(c, A, 2) 1 T'. Hence, &,2')I T' = +(c, 2) I T'. Since py is nonanticipatory, we have p(c, 3) I T' = p(c, 2)I T'. Q.E.D. As an example of the application of Proposition 1.5, consider the system given in Fig. 1.4 and defined by ((x, z), (y, 2')) E s * ( ~ Y ( O ) )f i t ) = ~ ' ( = t ) fio)

+

176

Chapter X

Subsystems, Decomposition, Decoupling

FIG.1.4

The global-state response function for z is then given by pZ(c,X, Z) = Z'

C)

z'(t) = c

+

[

(X

+ Z) dt

that is, z I T' = p&, x, z) I T' has a unique solution z I T' for each (c, x) and t. Therefore, by Proposition 1.5 the system is nonanticipatory even after the feedback is closed. This can easily be verified by observing that the system after the feedback is closed is defined by

y(t) = z(t) = y ( o ) -dy/dt

=x

+

+y

The linearity of a system is preserved by the three interconnecting operations. Recall that a linear system is defined on a linear space. Proposition 1.6. Suppose S, and Sz are linear systems. Then S, Sz, S, Sz, and 9 ( S , ) are linear when S, Sz, S , + S 2 , and 9 ( S , ) are defined.

+

0

0

177

1. Connecting Operators

PROOF.We shall consider only the case of feedback. Let S , c ( X x Z ) x ( Y x Z).If (x,y ) E 9( S s , )and (x’,y’) E 9 ( S , ) , then there exist z E 2 and z‘E Z such that (x,z,y, z)t E S1 and (x’,z’,y’, z’)E S , . Since S , is linear, (x x‘, z z’, y y’, z z‘) E S , . Hence, (x x’, y y’) E 9 ( S , ) . Similarly, Q.E.D. (ax,a y ) E F(S,),where a is a scalar multiplier.

+

+

+

+

+

+

In subsequent sections we shall consider functional systems. Recall that :

(i) a system S c X x Y is functional if and only if

(x,y) E S & (x,y‘) E S

-+

y = y’

(ii) a system S c X x Y is one-to-one functional if and only if S is functional and

(x,y) E s & (x‘,y) E s -+ x

= x’

Proposition 1.7. If S , and S2 are functional, then S , S2 and S , + S2 are functional if they are defined. Furthermore, the one-to-one functionality is also preserved by the cascade and parallel operations. 0

In general, the functionality is not preserved by the feedback operation. Proposition 1.8. Suppose S c ( X x 2) x ( Y x 2) is functional. Let

S(X) = {z :(3Y)((X, z,y, z ) E S>>

S(x, y)

=

{ z ; (3z’)((x, z , y , z’)E S)}

Then 9 ( S ) is functional if and only if for each x E X (C1)

(3Y)(S(X)

= S(x, Y ) )

holds. In particular, if S satisfies the relation

(x,z,y, z)E S & (x,z’,y’, z’) E s -+ z = z’ F(S)is also functional. PROOF. Suppose condition (Cl) holds. Suppose (x,y) E 9 ( S ) and (x,y ’ ) ~ 9 ( S ) .Then there exist z and z’such that (x,z,y, z)E Sand (x,z’,y’, z’) E S hold. Therefore, z E S(x) and z’ E S(x). It follows from condition (Cl) that there exists j such that z E S(x,9) and z’ E S(x, 9) hold ; that is, (x,z, j , 2)E S and (x,z’,j , Y ) E S hold for some 2 and 9’. Since S is functional from assumption, (x,z,y, z ) E S and (x,z, 9 , 2 ) E S imply that y = 9. Similarly, we have y‘ = 9. Consequently, y = y’.

t For notational

convenience, we shall write (x, z, y, z) instead of ((x, z), (y, z)).

118

Chapter X

Subsystems, Decomposition, Decoupling

Conversely, suppose F(S)is functional. Let 2 E X be arbitrary. If there is no such j as (2,9) E F(S),then S(2) = 0. Then S(2) c S(2, y) trivially holds. Suppose (2, j)E P(S).Let z E S(2) be arbitrary. Then there exists y such that (2, z, y, z) E S . Since F(S)is functional, y = 9 holds. Consequently, z E S(2,j). Q.E.D. 2. SUBSYSTEMS, COMPONENTS, AND DECOMPOSITION

There are many ways in which the subunits of a system can be defined. We shall introduce only some of the notions that seem to be among the most interesting in application. Definition 2.1. Let S c X x Y be a general system. A subsystem of S is a subset S’ c S , S‘ c X x Y. A component of S is a system S* such that S can be obtained from S* (possibly .in conjunction with some other systems) by the application of the basic connecting operators. It should be noticed that in application, the term “subsystem” is used for the whole range of different concepts, including the notion of a component as given in Definition 2.1. This should be borne in mind in the interpretation of the statements from this chapter. We shall fdst consider the component-system relationship. Given two systems S , c X , x Y, and S , c X , x Y,, let n, and n, be the projection operators, x (Yl x

n,:(X,

x

n,:w,

x X,) x

x2)

Y2) - + W l

x Y,)

and (Yl

x

Y2) -+

(X,x

Y2)

such that n,(x,,x,,y,,y,)

=

(xl,yl),

and

n2(xl,x2,yl,~2) =(

~ 2 ~ ~ 2 )

Let S c (XI x X , ) x (Yl x Y,). Definition 2.2. Two systems S1 = n,(S) and S , noninteracting (relative to S ) if and only if S = S , is called a noninteractive decomposition of S .

=

n,(S) are considered

+ S , , and (n,(S),n,(S))

Definition 2.3. A noninteractive decomposition ( S , , . . . , S,), where S , + = S is called a maximal noninteractive decomposition if and only if no Si has a (nontrivial) noninteractive decomposition.

..’ + S ,

2 . Subsystems, Components, and Decomposition

179

It is an interesting question whether or not a system has a maximal noninteractive decomposition. It is obvious that if a system has finitely many components, i.e., X = X , x ... x X , and Y = Y, x ... x Y, for some n and rn, it has a maximal noninteractive decomposition. This problem will be considered in relation to the feedback in Section 4. Let us consider basic decompositions by components. Proposition 2.1. Every system S c ( X , x X,) x (Y, x Y,) can be decomposed as S = %(S, S,), that is, in cascade and feedback components, where S , c ( X , x Z , ) x (Y, x Z , ) and S2 c ( X , x Z,) x (Y, x Zl), and Z1and 2, are auxiliary sets. (See Fig. 2.1.) 0

FIG.2.1

Proposition 2.2. Every system S c ( X , x X,) x (Y, x Yz) can be decomposed into cascade components as shown in Fig. 2.2.

180

Chapter X

Subsystems, Decomposition, Decoupling

FIG.2.2

Proposition 2.3. Let S c (X, x X , ) x (Y, x Y2) and S(x) = {y : (x, y) E S}, where X = X , x X, and Y = Yl x Y,. Let n l ( y l ,y2),= y1 and n2(y1,y2) = y,, Then, if and only if S(x) = n,(S(x)) x n,(S(x)) for every x E 9(S), S can be decomposed by some S, and S , as shown in Fig. 2.3.

FIG.2.3 PROOF. Suppose S(x) = nI(S(x)) x n2(S(x))for every x ~ 9 ( S )Let . S, c X x Yl be such that

(X,Yl)ESl ~ ( ~ Y , ) ( ( X , Y l ~ Y , ) ~ S )

181

2. Subsystems, Components, and Decomposition and S, c X x Y, be such that ( x ,Y,)

E Sz c-) V Y , ) ( ( XY ,I

Y Z )E S )

Apparently (X,Y,,YZ)ES

+

(x,Y,)ES,&(x,Y,)ES*

Conversely, suppose ( x ,y , ) E S , & ( x ,y,) E S , . Then y , E n , S ( x ) and y 2 E n,S(x); that is, ( y , , y , ) ~ S ( x because ) S(x) = ll,(S(x))x ll,(S(x)). Consequently, ( x ,Y , ,Y z ) E s. Suppose there exist S , c X x Y, and S , c X x Y, such that the above decomposition is realized. Apparently, S(x) c ll,(S(x))x H,(S(x)). Let y , E II,(S(x))and y , E lT,(S(x))be arbitrary. Then ( x ,y,) E S , and ( x ,y,) E S2 Q.E.D. should hold. Consequently, ( x ,y , ,y,) E S. Proposition 2.4. Every system S c X x Y can be decomposed by some S‘ as shown in Fig. 2.4, where

--

( x ,x’)E Ex

S ( X ) = S(X’)

(Y, Y‘) E E ,

(Y)S = ( Y ’ P

-

S(x) and (y)S are defined in Propositions 1.1 and 1.2, and q x : X X I E X , q, : Y -+ Y / E , are the natural mappings ; that is, ( [ y ] ,y ) E q i l b]= qy(y). -+

Suppose 2 E [ x ] and 9 E [ y ] such that (a,9)E S. Such elements exist for ( [ x ][, y ] )E S’ by definition. Since x’ E [XI, we have S(2) = S(x‘). Furthermore, 9 E S(2) implies (x’,9) E S. Consequently, x‘ E (9)s.On the other hand, y’ E b] and 9 E [ y ] imply (y‘)S = (9)s.Hence, we have x‘ E (y‘)S,that is, (x’,y’) E S. Q.E.D. In conclusion, we shall briefly consider the subsystem-system relationship. Definition 2.4. Let Si (i E I ) be a functional subsystem of S c X x Y; i.e., Si c S and Si:9(S) -+ Y. If Si = S, then {Si : i E I } is called a functional

system decomposition.

uieI

182

Chapter X

Subsystems, Decomposition, Decoupling

A functional system decomposition is apparently equivalent to a globalstate representation. Such a decomposition is, therefore, always possible and is not unique. Definition 2.5. Let S, = { Sai: i E I,} and S, =_ { Sp,; i E I,} be two functional system decompositions of S c X x Y. Let S, I S , if and only if S, c S,. Then if a decomposition Sosatisfies the relation that for any functional system decomposition S,

s, I So + So = S, So is called a minimal functional system decomposition of S . It can be shown quite readily that every finite system has a minimal functional system decomposition.

3. FEEDBACK CONNECTION OF COMPONENTS

In this section, we shall consider the feedback system as given in Fig. 3.1. The overall system is defined as

s c(X

x Z,) x ( Y x Z,)

and the feedback component itself is

Denote by S , the set of all input-output pairs of the system S in X x Y, i.e.,

and by S , the class of all possible feedback components

S, = { S , : S , c z,x Z,} Let

be the map e : S , + S such that %(Sf) = F ( S S,) 0

183

3. Feedback Connection of Components X

*Y

zY

FIG.3.1

For any given feedback component, Fsgives the resulting overall system. % therefore indicates the consequences of applying a given feedback. Some conceptually important properties of % are given by the following propositions.

Q.E.D.

Proposition 3.2. Let 4 :lI(S,)

+

sfbe such that

(z’,z) E %(Sf) * (3(x, Y ) E S’)(((X,

4,( Y , z‘))

E S)

Then S’ c E(gs(S’)) holds. If S satisfies the relation

(C1) then

s

(x, z, y, z’) E & (a,z,

*((Y(S’))

=

9,z’) E s + (x, y) = (a,9) S’ * s’E a(*)

184

Chapter X

PROOF.

Subsystems, Decomposition, Decoupling

In general, since S’ c S,, ( x ,Y )E S’

+

+

(3(z,z’))((x,z, Y , z? E s & (z’, z ) E %(V) ( x ,Y ) E %(%S’))

Suppose condition (Cl) holds. Obviously, Fs(%(S)) = S’ + S’ E a(%). Conversely, we have that (x, Y ) E

Y(S”

W’,z))((z’,z ) E %(Sf) & ( x ,z, y , z’) E S )

+

(3(z’,z))(3(%9))((%9) E S’ & (?, z , 9, z‘)

+

E

s

& ( x ,z, Y , z’) E S ) +

(x, Y ) = (299)E S’

(because of the condition stated in the proposition). Consequently, since S’ c %(%(s’)) generally holds, we have S’ = %(%(Sf)). Q.E.D. Let

sf, = ((z’,z) :(3(x,y))((x,z, y , z’) E S ) } c Z, Sf* = (sf* : sf*c Sf,}

x 2,. Let

Proposition 3.3. Suppose S satisfies the relation ( x ,z, y, z’)

E

s & (x, 2, y, 1’)E s + (z, z’) = (2, 1’)

The restriction of % to Sf*, S’ = %(Sf*),then Sf* = %‘(S‘). PROOF.

--

Sf* is then a one-to-one mapping, and if

Notice that if S’ = %(Sf*)holds, we have

(z’,4 E SS(S’)

(3(x,Y ) ) ( ( X , Y ) E S‘ & ( x ,z, y , z’) E S )

( Y x ,Y ) ) W ’ , 2))@’, 2) E Sf*& (x, 2, Y , 1’)E s 8t ( x ,z, y, z’) E S )

(z’, z ) = (2’, 2) E s,* Q.E.D.

Consider now conceptual interpretation of Propositions 3.1, 3.2, and 3.3. Important properties of a feedback and the consequences of applying feedback to a given system (e.g., for the purpose of decoupling) can be studied in reference to the properties of Fs. Proposition 3.1 indicates that the application of any feedback results in a restriction in S B .To illustrate the importance of Proposition 3.2, consider a feedback connection defined on a linear space as given in Fig. 3.2, i.e., S c (X x Z,) x ( Y x Z,), Z , = X , Z , = Y, and (x, 2, y , z’) E

s

-

z’ = y & ( x

+ z, y ) E s,

4 . Decoupling and Functional Controllability

185

FIG.3.2

Since X is linear, the addition operation is defined in X . Assume that S is functional (e.g., an initial state is fixed) and the input space is reduced such that S1 is a one-to-one mapping. The conditions from Propositions 3.2 and 3.3 are then satisfied. Suppose S’ is an arbitrary subsystem of S,, S’ c S,. Proposition 3.2 gives, then, the conditions when S’ can be synthesized by a feedback. S‘ can be obtained from S by applying feedback if and only if %(%@’)) = S’ ; furthermore, if S’ can be synthesized, the required feedback is given by $(S’). This will be studied further in Section 4. are given in the following propositions. Additional properties of

Proposition 3.4. FSis an order homomorphism, i.e., Sf c Sf’

-+

%(Sf)c

%(SO. Proposition 3.5. 4 is an order homomorphism, i.e., S’ c $I -+ 4(S)c %($I). Proposition 3.6. Suppose S satisfies the condition of Proposition 3.3. Then the class of the systems that can be synthesized by feedbacks forms a closure system on S,. % o 4 is the closure operator for that closure system.

% :n(S,)--+ II(S,). It follows from Propositions 3.4 PROOF. Let J = Proposition $). 3.2 shows that and 3.5 that if S’ c 9’ c S,, then J(S’) e .I( S’ c J ( S ) for any S’ c S,. Furthermore, since %(%(S’)) E a(%) for any S’, it follows from Proposition 3.2 also that JJ(S’) = J ( S ) for any S’ c S B . Since J(S’) = S’ C I S’ E a(%), a(%)forms a closure system on S,. Q.E.D. 0

4. DECOUPLING AND FUNCTIONAL CONTROLLABILITY

The objective of this section is to give necessary and sufficient conditions for the decoupling of a system via feedback. It represents an illustration of

186

Chapter X

Subsystems, Decomposition, Decoupling

the use of the general systems framework developed in this chapter. Many other systems problems that are essentially structural or algebraic in nature can be treated in a similar way. (a) Decoupling of Functional Systems

We shall consider a system S c ( X x Z,) x ( Y x Z,) with a feedback component Sf c Z , x 2, connected as shown in Fig. 4.1. Furthermore, it will always be assumed that the following condition is satisfied :

(PI)

(x, zx, Y, zy) E

s

+

Y = zy

n

FIG.4.1

i.e., 2, = Y. We shall consider, therefore, the feedback system as defined on ( X x 2,) x Y rather than on ( X x Z,) x ( Y x Z,). The notation F(S Sf), where S, c Y x Z , , is then not quite consistent with the previous definitions. However, this inconsistency should be allowed here for simplification of the notation. Before considering the question of decoupling explicitly, some additional properties of the feedback connection need to be established. 0

Proposition 4.1. Consider a feedback system S c ( X x Z,) x Y whose feedback component is Sf c Y x Z , . Suppose

(i) Sf and *(Sf) are functional, i.e., Sf : ( Y )+ Z,, R(Sf):(X)+ Y (ii) S satisfies the condition ( x , z, y ) E s & (x’, z , y ) E s

(P2) Then * ( & ) : ( X )

+

Y is a one-to-one functional.

x = x’

4 . Decoupling and Functional Controllability PROOF.

187

Suppose %(S,)(x) = y and %(S,)(x’) = y . Then (W((X,

z, Y ) E

s

=

S,(Y))

and (3z’)((x‘,z’, y ) E S & z’ = S,(y))

Consequently, we have z = z’. Hence, (P2) implies x

=

x’

Q.E.D.

Using Proposition 4.1 we shall show a basic result.

Proposition 4.2. Suppose (i) S is functional, S :B(S) + Y; (ii) S satisfies (P2); (iii) S satisfies the condition (x, z, y) E S & (x, z‘, y ) E

(P3)

s

+

z = z’

Let S, c I l ( Y x Z,) be the set of all functional feedback components defined in Y x Z,, i.e., Sf:9(Sf)4 Z , , such that B(S.S,) is functional. Let be an arbitrary system 3 c X x Y, and K ( X , Y )is a family of subsystems of 3 such that K ( X , Y) = {S’ :S’ c & S’ is one-to-one functional} Then

%(sf) = K(X, Y )

if and only if

3 = SB

where SB =

{(& Y ) :(3z)((x,z , Y ) E S ) }

PROOF. We consider the only i f part first. Suppose %(Sf)= K ( X , Y) holds. Let (x, y ) E 3 be arbitrary. Since %(Sf) = K ( X , Y) and since U K ( X , Y) = 3, there exists S, E Sf such that (x,y ) %(Sf); ~ that is, (3z)((y,z ) E S, &(x, z, y) E S). Hence, (x, y ) E S,. Let (a,9)E S , be arbitrary such that (%)((a,&j ) E S). Let S, = {(9,2)}. We shall show that S, E Sf. Notice that if (x, y ) E B(S .S,) and (x, y’) E B(S S,), then y = 9 = y’ holds; that is, B(S S,) is trivially functional. Furthermore, S , is functional. Therefore, S , E S , . Naturally, (a,9)E %(Sf).Since %(Sf)= K ( X , Y), (a,?)E 3. Next, we consider the if part. Suppose = S , holds. Let S’ E %(Sf)be arbitrary. Then there exists S, E S, such that S = %(Sf).S‘ is functional due to the assumption of and satisfies the relation S’ c SB = 3 because S‘ = B ( S S,). Since S‘ is one to one from Proposition 4.1, S’ is an element of K ( X , Y).

-

s,

Chapter X

188

Subsystems, Decomposition, Decoupling

Let S’ E K ( X , Y ) be arbitrary. Since

s = S,,

we have S’ c S,. Let Sf =

gS(S‘); that is,

S,

=

( ( Y ,z ): W

( ( X , Y ) E S’

& ( x , z, Y ) E 3

1

Suppose ( y , z ) E Sf and ( y , z’)E S,. Then (3x)(3x’)((x,y ) E S‘ & (x’,y ) E S’ & (x, z, y ) E s & (x’, z’, y ) E S )

Since S‘ is one to one, we have x = x’. Consequently, (P3) implies that z = z’. Therefore, S , is functional. We can show 9 ( S . S,) = S’ because

F(S * S,) 3 ( x , Y ) * ( 3 z ) ( ( x z, , Y ) E s & (Y, 4 E S,)

* ( 3 z ) ( y = S(x, z ) %6 ( 3 2 ) ( y = S(2)& y

s(a)& y

4-+

( 3 z ) ( 3 2 ) ( y= S(x, z ) & y

4-+

( 3 z ) ( y = S(x, z ) & y = S’(x))

=

= =

S(2,z))) S(2, z ) )

because (P2) implies x = 2 - ( x , y ) ~ S ‘ , and because S’ c S,. Therefore, B(S . S,) = S‘, where S’ is functional by definition. Consequently, S , E S , , Q.E.D. which implies S‘ E %(Sf). We shall apply Proposition 4.2 to decoupling of multivariable systems. We shall first introduce the formal definitions of decoupling by a feedback and functional controllability. Definition 4.1. A general system S c X x Y is functionally controllable if and only if W Y E Y)(3X E X)((X,Y ) E S )

Definition 4.2. AmultivariablegeneralsystemS c ( X I x . .. x X , ) x Z, x (Y, x . . x Y,) is called decoupled by feedback if and only if there exists a feedback system S , c
B(S S,) = S , 0

where Si c X i x

+ ... + S ,

( i = 1,. . . ,n) is functionally controllable.

Decoupling as defined by Definition 4.2 means that after the appropriate feedback Sf is applied, each of the output components, for example, y i , can be changed by changing solely the corresponding input x i , while no other output is affected by such changes of x i . The functional controllability, on the other hand, means that any output ( y l , . . . ,y,) E Y can be reproduced by an appropriate selection of the input x . Definitions 4.1 and 4.2 are given for general systems. If a general system is functional as considered in this section, the necessary modifications are

4. Decoupling and Functional Controllability

189

obvious. In particular, if S is a multivariable functional system, i.e., S:(X1 x ... x X,) x Z X + ( Y 1x . . . x

r,)

a feedback system Sf associated with S is always assumed functional; i.e., S , : Yl x . . . x Y, -+ 2,. Relations between the two concepts are given by the following propositions. Proposition 4.3. Let S be a multivariable functional system S :(X x 2,) -+ Y, whereX = X, x ... x X, , Z , = Z,, x ... x ZXn,andY = Yl x ... x r,. If S can be decoupled by a feedback S , , then S is functionally controllable, i.e., W Y E Y)(%

4Ex

x ZXNY = S(X, 4)

PROOF. Let S , : Y -+ 2, be the feedback that yields the decoupling. Let j = (j,,. . . ,9,)E Y be arbitrary. Let 2iE Xi be such that j i = Si(ai).This is possible because Si is functionally controllable. Then F ( S S,)(2) = j,where x = ( x l , .. . ,2,). This is also true because F ( S S,) = S , + . . . + S,. Consequently, it follows from the definitions of the feedback that there exists 2 E 2, such that j = S(2, 2) and 2 = S,@) hold. Q.E.D. 0

*

A

0

In application, the problem of decoupling is of interest in a more specialized setting. Namely, there is given a system So :X , ---* Y, and the question is asked whether a feedback input can be combined, in a given sense, with the external input so that the resulting feedback system is decoupled. To provide for the possibility to combine the external inputs and with feedback inputs, an additional component H is given, resulting in the overall system as shown in Fig. 4.2. H will be referred to as the input port, H : X x Z , -+ X,. A S = H o So

- - - - - - - - -- - - - ------ - - -- I

I

FIG.4.2

problem of decoupling is then the following. Given a system So, an input port of a given type H , and a class of feedbacks Sf,what conditions So should be satisfied so that there exists a feedback S , E S, such that %(Sf)is decoupled, where S = H So? The answer to this question is given by the following definition. 0

Chapter X

190

Subsystems, Decomposition, Decoupling

Definition 4.3. An input port H : X x Z , + X , is termed complete if and only if it satisfies the following condition : ( V x E X ) ( V X , € XO)(3Z€ Z,)(X,

= H(x,z))

Proposition4.4. Let X = X I x . . . x X , , Y = Yl x ... x Y,, and Z , = z, x . ’ . x Z,,. Suppose (i) S , : X , + Y is one to one, i.e., reduced with respect to the input; (ii) H is complete and furthermore satisfies the conditions H ( x , z ) = H(x’, z ) + x = x’,

H ( x , z ) = H ( x , z’)

--f

z

= z’

(iii) S, c I l ( Y x Z,) is the set of all functional systems Sf such that F ( ( H 0 So) S,) is functional ; (iv) the cardinality of yi is equal to that of X i for each i. 0

Then So is functionally controllable if and only if S decoupled by a feedback from S,.

=

H 0 So can be

PROOF. We consider the if part first. Suppose H So can be decoupled by a feedback from S,. Then Proposition 4.3 implies that H 0 So is functionally controllable, i.e., 0

(W(Wm 9 = ( H

O

So)(%

2))

Let 9, = H ( 2 , 2 ) . Then the above condition implies (V9)(390)(9 = So(2,))

that is, So is functionally controllable. Next, we consider the only i f part. Suppose So is functionally controllable. Let 2 E X and 9 E Y be arbitrary. Since So is functionally controllable, there exists 2, E X such that 9 = So(?,). Furthermore, since H is complete, there exists I E Z , such that 2, = H(2,I). Therefore, we have Let

(V(x9 Y ) ) ( W ( X ,z, Y ) E H

s=

SB

= { ( x ,y ) : ( 3 z ) ( ( x ,z, y ) E H

0

So)

So)} =

xx

Y

Proposition 4.2 implies, then, that %(Sf)= K ( X , Y ) . Suppose Si:Xi--* yi is one-to-one functional and functionally controllable for each i I n. Since the cardinality of yi is equal to that of Xi, such Si exists. Then S’ = S1 + . . . + S , c 3 because of the structure of 3. Furthermore, S’ is one-to-one functional. Consequently, %(Sf)= S1 . . + S , for some feedback S, E S,. Q.E.D.

+

We shall consider now linear systems. The counterpart of Proposition 4.2 for linear systems is given by the following proposition.

4 . Decoupling and Functional Controllability

191

Proposition 4.5. Suppose S satisfies conditions (i), (ii),and (iii) from Proposition 4.2 and furthermore

(iv) S is a linear system. Let SF be a subset of Sf defined in Proposition 4.2, S: c S, such that S , E SfLif and only if S , is linear. Let 9 c X x Y be an arbitrary linear system and L ( X , Y ) a subset of the family of subsystems defined in Proposition 4.2, L(X, Y) c K ( X , Y ) such that S' E L ( X , Y ) if and only if S' is linear. Thene(S$) = L ( X , Y ) if 9 = S B where SB

= (3z)((x,z, y ) E

s))

The proof of Proposition 4.5 is almost the same as that of Proposition 4.2 except that when we define a feedback component S , in the proof, we have to show that it is linear. The proof is left as an exercise. For the linear systems, the following mapping H : X x Z , -+ X , is conventionally used for an input port : H ( x , z ) = x + z. It is easy to see that H is complete and that condition (ii) of Proposition 4.4 is apparently satisfied. Then the counterpart of Proposition 4.4 for linear systems is given by the following proposition. Proposition4.6. Let X = X 1 x . . . x X,, Y Z,, x ... x Z,,. Suppose

=

Yl x ... x

x , and

Z,

=

(i) So:X , -+ Y is linear and reduced with respect to the input ; (ii) S: c I I ( Y x Z,) is the set of all linear functional systems S , such that 9 ( ( H So) 0 S,) is functional ; (iii) for each pair ( X i , there exists a linear one-to-one correspondence si:xi-+ y i . 0

x),

Then So is functionally controllable if and only if S = H So can be decoupled by a feedback from S:. 0

We shall now consider time systems. Our first concern is with nonanticipation. Recall that a functional time system S , : X , -+ Y is nonanticipatory if and only if (Vt)(Vx)(VA)(x 1

T'

=

2 I T'

-+

S,(X)

I T'

=

&(A) I T')

Consequently, a functional time system So is nonanticipatory if and only if So I T' is functional for each t. We have then the following proposition. Proposition 4.7. Let S c ( X x Z,) x Y. Let S , c Y x Z , and F ( S . S,) = %(Sf) be functional and nonanticipatory. If S satisfies the condition that :

192

Chapter X

Subsystems, Decomposition, Decoupling

for each t E T (x, z, y) E s & (2, I , 9)E s & (z, y) 1

T'

= (2,j?) 1

T' -+

x1

T'

=2I

T'

then %(Sf)1 T' is one-to-one functional. PROOF. Since %(Sf) is functional and nonanticipatory, %(Sf)[T' is functional. Suppose y = %(S,)(x) and 9 = %(Sf)(2).Suppose y ) T' = 91 T'. We shall show that x I T' = 2 I T'. It follows from the definition of $(Sf) that

(3z)(32)((x, z, y) E s & (a,239) E s & z = S,(y) & 2 = S,(j))

Since yl T' = j1 T' and since S , is nonanticipatory, we have z I T' = I I T'. Then, since (z, y) I T' = ( 2 , j )1 T', we have x 1 T' = 2 I T'. Hence, %(Sf) I T' is one-to-one functional. Q.E.D. The following is the counterpart of Proposition 4.2 for nonanticipatory linear time system. Proposition 4.8. Suppose

(i) S c ( X x Z,) x Y is linear, nonanticipatory, functional, and furthermore satisfies the conditions that for each t E T

T'

=

S(jZ,2) 1 T' & z 1 T'

S(x, z) I

T'

=

S(2,2)I T' & x I T' = 2 I T'

=

21

T' -+ x I T'

S(x, z ) 1

=

21

=

2 I T'

T'

and +z

I T'

(ii) S, c n(Y x Z,) is the family of linear nonanticipatory functional systems such that if S , E S,, then 9 ( S S,) is functional and nonanticipatory. 0

Let $ = S , = {(x, y) :(3z)((x, z, y) E S)} and let L(X, Y ) be the family of all linear subsystems S' of 3 such that S' I T' is one-to-one functional for each t. Then %(Sf)= t ( X , Y ) . PROOF. Let S' E %(Sf)be arbitrary. Then there exists S , E S, such that S' = %(Sf) = B(S S,). Since S and S , are linear, S' is linear and functional due to condition (ii). Furthermore, since S' = F ( S Ss), we have S' c SB = 3. Since S' 1 T' is one-to-one functional due to Proposition 4.7, S' is an element of L ( X , Y). Let S'EL(X, Y ) be arbitrary. Since 3 = S,, we have S' c S,. Let 0

0

Sf = {(Y,

2) : (

3 X m Y ) E S'

lk (x, z, Y) E S}

=

%(S')

S , is apparently linear. Suppose (y, z) E S , and ( j , 2 )E S,. Then (3x)(32)(y = S ' ( X ) & j = S ' ( i ) & y = S(X,Z)&j = S ( 2 , i ) )

4. Decoupling and Functional Controllability

193

Suppose y I T' = j I T'. Since S' 1 T' is one-to-one functional, x I T' = 2 I T' holds. Therefore, z I T' = 2 I T'. Hence, S , is linear, nonanticipatory, and functional, Since F ( S 0 S,) = S' holds due to Proposition 4.2, 4 ( S S,) is functional and nonanticipatory. Consequently, S, E S,. Q.E.D. 0

The relation between the functional controllability and the decoupling for linear nonanticipatory time systems is given in the following, which is the counterpart of Proposition 4.4. Let H : X x Z , + X , be the same as used for Proposition 4.6. Then we have the following proposition. Proposition 4.9. Let X = X , x ... x X,, Y Z,, x . . . x Z,.. Suppose

(i) S , : X , condition :

-+

=

Y, x ... x Y,, and Z ,

=

Y is linear, nonanticipatory, and satisfies the following (Vt)(Vx)(S,(x)lT'= 0

-+

xI

T'

= 0)

(ii) S, c n(Y x Z,) is the family of linear nonanticipatory functional systems such that if S, E S,, then F ( ( H So) S,) is functional and nonanticipatory ; (iii) for each pair ( X i , there exists a linear mapping S i : X i -+ such that Si I T' is a one-to-one correspondence for each t E T. 0

0

x),

x

Then So is functionally controllable if and only if S = H 0 So can be decoupled by a feedback from S,. If we relax the conditions of Definition 4.1 (for instance, condition (iii) is omitted or the domain of Si is allowed to be a proper subset of X i ) ,condition (iv) of Proposition 4.4, condition (iii) of Proposition 4.6, and condition (iii) of Proposition 4.9 can be relaxed. Propositions 4.8 and 4.9 are presented for linear nonanticipatory time systems. However, it is apparent that they also hold for general nonanticipatory time systems with slight modifications. (b) Decoupling of General Systems Following the reasoning in subsection (a),we shall investigate in the sequel decoupling of general systems. Figure 4.3 is a schematic diagram of systems considered in this section, where S c X , x Y is a general system, H:X x Z X , is an input part, and S , : Y Z is a feedback functional system. -+

-+

Chapter X

194 X

-

Subsystems, Decomposition, Decoupling

H

=

-

s -

-Y

.-

FIG.4.3

The following two propositions are of conceptual importance.

+

Proposition 4.10. Let S, c X , x Y , , S , c X , x Y,, and S = S1 S , c ( X , x X , ) x (Y, x Y2). Then S is functionally controllable if and only if both S , and S , are functionally controllable. Proposition 4.11. Let S c X , x Y and 3 c X x Y be systems and let H :X x Z + X , be an input port. Suppose there exists S,: Y -+ Z such that s^ = F ( H 0 S 0 S,). Then S is functionally controllable if 5 is so.

Q.E.D.

PROOF.The assertion follows from W ( s ) c 9 ( S ) .

Proposition 4.12. Let S c X , x Y be a functionally controllable system and let H :X x Z -+ X , be an input port. Then, given a functionally controllable system c X x Y, there exists a feedback component S , : Y -+ Z such that 9 ( H S 0 S,) = 9 if and only if for any y E Y there exists z E Z such that 0

(i) f N Y ) 3 , Z ) = ( y ) S ; (ii) ~ ( xz ,) E ( y ) + ~ x c ( y ) s;

where (y)S = { x :(x, y) E S } and ( y ) s = { x :( x ,y ) E 3) are defined in Section 1. PROOF.

We shall first prove the ifpart. Let V ( y )be defined by z

E

VY)

++

(H((Y)S,4 c (Y)S

and

H ( x , z ) E (Y)S

+

-

x E (Y)Q

By the premise V(y) # 6.Using the correspondence y V ( y )and the axiom of choice, we can construct a function S , : Y + Z . We claim 3 = F ( H S 0 S,). c F ( H S 0 S,) is clear. To prove 3 3 F ( H S S,), let (x, y ) E F ( H 0 S 0 Sf). Then there exist z E Z , x, E X , such that H ( x , z ) = x,, (x,, y ) E S , and S,(y) = z . Hence, H ( x , z ) E (y)S, which implies x E ( y ) s . Thus (x, y ) E 3. 0

0

0

0

195

4. Decoupling and Functional Controllability To prove the only i f part, suppose

3=9(H

0

S 0 S,). Then (y)g =

{x : H(x, S,(Y)) E (Y)S). Therefore, H((Y)S, S,(Y)) c ( Y E Let H ( x , S,(Y)) E ( Y V . Then (x, y ) E 9 ( H 0 S 0 S,) = 3. Thus, x E (y)$. Q.E.D.

Proposition 4.13. Let S c X , x Y be a system and let H : X x Z X , be an input port. Suppose that S , c X , x Y, and S , c X , x Y2 are controllable systems such that there exists a system c X x Y which satisfies the following conditions : -+

s

(i) S = S , + s,; (ii) for any y E 9 ( S ) , there exists z E Z such that ( a ) H((y)$,z ) c (y)S,and (/$ H ( x , 4 E ( Y P -+ x E (Y)S. Then S can be decoupled into S , and S , by a functional feedback if and only if S is functionally controllable. PROOF.The only if part follows from Proposition 4.11 since S , and S , are functionally controllable (hence so is S , + S , by Proposition 4.10). The if Q.E.D. part follows from Proposition 4.12. Propositions 4.12 and 4.13 are also true for linear systems as follows. Proposition 4.14. Let S c X , x Y be a functionally controllable linear system and let H :X x Z -+ X , be a linear input port. Then, given a functionally controllable linear system 3 c X x Y, there exists a linear functional feedback component S, : Y -+ Z such that 9 ( H 0 S 0 S,) = 3 if and only if for any y E Y, there exists z E Z such that H ( ( Y ) ~ , zc) (YP;and (ii) ~ ( xz ), E ( y ) -+ ~ x E (y)S.

6)

PROOF.The only if part can be proved by the same reasoning as in the proof of Proposition 4.12. To prove the if part, it is enough to show that we can construct a linear S, : Y -+ Z . Let V ( y )be as in the proof of Proposition 4.12. We claim z E V(y), and 2 E V ( j ) az fl2 E V ( a y p j ) -+

+

+

for scalars a and p. To see this, assume that H(x, az

+

ZE

V(y),2~ V ( j ) .First, we shall prove

E (ay

+ Bj)S

-+

x E(ay

+ Pj)S

Let x, E X be chosen so that H ( x , , z ) E (y)S. Let x, = (I/p)(x - ax,), where we assume that p # 0,because if fl = 0,then the statement is clear. Then

Chapter X

196

x

=

Subsystems, Decomposition, Decoupling

axl + fix,. Hence, H ( x , az

+ p2) = a H ( x , , 2 ) + PH(x,,2)

By the assumption, ( a w l z) Y

+ BWx,

9

21, cry

+ Pj) E s

( a W ,z), cry) E S Hence, ( p H ( x 2 ,2), p j ) E S. Therefore, H ( x , , 2) E @)S. By the assumption, we have x1 E ( y ) 3 and x2 E (j)$.Thus, x E (cry + 89)s. To prove that “Y

+ B9)S

+

c

fX.2

(cry + P9)S

let x E (cry + pj)s^. Since 3 is functionally controllable, there exists 2 E X such that ( 2 , j )E 3. Hence, (x - pa, cry) E 3. Thus, H ( x , az

+ pi) = aH((l/a)(x - pa),z ) + PH(2, 2) E (cry + p j ) S

proving that N a y

+ 89K

f

82) c b Y + B9)S

where we assume a # 0,because if cr = 0,the assertion is clear. Let 3, c Y x Z be defined by (Y, z ) E 3,

-

zE UY)

Then 3, is a linear system. Therefore, by using Zorn’s lemma we can show that there exists a linear map S , : Y -+ 2 such that { ( y ,S,(y)) :y E Y } c 3,. Q.E.D. S , is a desired linear map. Proposition 4.15. Let S c Xi x Y be a linear system and let H :X x Z -+ X , be a linear input port. Suppose that S1 c X , x Y, and S, c X, x Y, are functionally controllable linear systems such that there exists a system 3 c X x Y which satisfies the following conditions: (i) 3 = s1+ s,; (ii) for any y E W(S),there exists z E 2 such that (a) H((y)$, z ) c (y)S, and (8)H(x, 4 E ( Y P x E (Y$. +

Then S can be decoupled into S1 and S , by linear functional feedback if and only if S is functionally controllable. PROOF. The only ifpart follows from Proposition 4.11 since S, and S, are functionally controllable (hence so is S, + S , by Proposition 4.10). The if part follows from Proposition 4.14. Q.E.D.

197

4 . Decoupling and Functional Controllability As an example, let us consider a system S c X , x Y defined by A y Bxo, where

+

X , = L,(o, m),

p=

B : n x n nonsingular matrix

A : n x n matrix,

Then S is functionally controllable. Let us examine the feedback system in Fig. 4.4, where X = L,(o, m). Let Si c X i x (i = 1,. . . , n) be defined by pi = aiyi p i x i , where x i and y i are scalar functions oft, and a i , pi are con-

+

I

x

xo -- * s

; =

--

-Y

I

I

I I I I

I

I

I

I

I I L

-

-

I

-I

Sf

is functionally controllable. Suppose that S is decoupled into S , ,. . . ,S,. Then for any y there exists z such that H((y)s,z) c (y)S.Therefore, if xiand y i satisfy pi = aiyi p i x i , then

+

-0

-0

Yi

+ B(K

0

XI 0

+ zi)

-0

Hence 0

B-'(aiz - A ) Yi .o

'0'

+ (PiB-'

-

K)

Xi

.o.

-

zi

Chapter X

198

Subsystems, Decomposition, Decoupling

zi must be independent of xi. Therefore,

for each i. Thus

l o

Hence we have K

=

B-

'8, S ,

=

B- ' ( a - A), where

In fact, Scan be decoupled into S , , . . . ,S , if K = B - ' f i and S , = B - ' ( a - A). Furthermore, if rank B = n (where B is an n x m matrix), then the analogous argument can be applied. Proposition 4.12 is applicable, with additional conditions, to nonanticipatory systems as follows. Proposition 4.16. Let S c X , x Y be a functionally controllable system with a nonanticipatory systems-response function p and let H : X x Z + X , be an input port. Let c X x Y be a given functionally controllable nonanticipatory system with a nonanticipatory systems-response function p. Assume that p , p , and H satisfy the following conditions, respectively :

s

(la) ( 1b) (2a) (2b) (3)

p(c, x) 1 T' = p(c, 2) 1 T' + x 1 T' = 2 1 T' (VC)(VY E W)) ( 3x1(P(C9 x) = Y ) p(c, x) I T' = p(c, a) I T' -, x I T' = i I T' WCWY E W(S)I(~X)(~(C, X) = Y ) H ( x , z ) I T' = H ( 2 , 2 ) 1 T' & x 1 T' = i I T' -, z I T'

=

2 I T'

5 . Abstract Pole Assignability

199

Then there exists a nonanticipatory feedback component S,: Y + Z such that B ( H 0 S 0 S,) = if and only if for any y E Y, there exists z E Z such that

s

6) H((Y13, z ) = (Y)S ~,x E ( y ) S (ii) ~ ( xz ,) E ( y ) PROOF. The only if part is clear. We know that if (i) and (ii) hold, there exists a feedback component S f :Y -+ Z such that 9 ( H 0 S 0 S,) = Therefore, to prove the ifpart, it is enough to show that if F ( H 0 S S,) = and if the given conditions hold, then S , is nonanticipatory. Suppose y, 9 E W($.Then there exist c, d, x, and 2 such that

s.

0

Y

= b(d, x) = P ( C ,

Suppose yl T' and

=

H ( x , Sf(y))),

91 T'. Then

and

=

E = D V , 2) = P ( C , H(% Sf(9)N

by the assumptions, we have x I T'

H ( x , Sf(Y))I T' Hence, S,(y) I T'

s

=

=

21 T'

H ( 2 , Sf(9))I T'

S , ( j ) I T'. Therefore, S, is nonanticipatory.

Q.E.D.

5. ABSTRACT POLE ASSIGNABILITY

The problem of pole assignability has been considered for the class of linear time-invariant ordinary differential equation systems, and the equivalence between pole assignability and state controllability is a well-known result for that class of systems. This is at least conceptually interesting and worthy of being studied in our abstract framework. However, the result is deeply rooted into specific structures of differential equations ;in particular, the matrix description of a system is directly related to the definition of pole assignability. For abstract developments, therefore, we shall first modify the concept of pole assignability in a way suitable for our abstract framework. The type of feedback system considered in this section is shown in Fig. 5.1, where (i) S is a strongly nonanticipatory time-invariant linear dynamical system (P, (ii) The state space is of finite dimension and is taken for the output of S, i.e., B = C and Y c C T . (iii) The input alphabet A is equal to the state space C, i.e., A = C. (iv) The feedback Sfis a linear static functional system, i.e., Sf :C + C( = A) and SXy)(t) = S,(y(t)). The class of S , in consideration in this section is denoted by S,. (v) The input port H is defined by H ( x , z ) = x + z.

a.

200

Chapter X

Subsystems, Decomposition, Decoupling

z

FIG.5.1

(vi) A system S in consideration is assumed to satisfy the following conditions ; for any S, E S f , c E C, x E X , and t E T (a) po(c,x

(8)

P,(C,

x

+ S,(y)) I T, = y I T, has a unique solution y ;

+ S,(y)) = y & p,(c, 2 + S , ( j ) ) = 9 & x' = 2 + j'

Due to assumption (vi) (tl), a function follows : $,t(C,

x')

=

4ot(C,

x'

$ot

:C x X'

+

-

= 3'.

C can be defined as

+ (Sf(Y))I T')

where y is the solution of po(c,x' . x, + S,(y)) = y, and x, is arbitrary. : t E T } satisfies the comFurthermore, it is not difficult to show that position property, i.e.,

$of(c,x' . x,,,) = $or($ot(C, xf),F-Y&,)) where z = t' - t. This can be proven by the time invariance of $ and assumptions (iv)and (vi). [Notice that S, is commutative with F', i.e., F f ( S f ( y ) )= S,(F'(y)).] will be referred to as the state-transition function of the feedback system. As usual, we shall write $,r(C,

x') =

$lO,(C)

+ $2ot(xf)

Under the above assumptions, the pole assignability is defined in the abstract framework as follows. Definition 5.1. Let S be a linear dynamical system specified as above. Let f ( w ) = w" + a,-lw"-l + . . . + a, be an arbitrary monic polynomial with the indeterminate w, where the coefficients are elements of the field d (over which S is defined), a, is not zero, and n is equal to the dimension of C.The system S is then said to be pole assignable if and only if for any t > 0 there exists a feedback S, E S, such that

+ a n - i ( $ i o r Y 1 + . . . + a , ~= o

f($iot) = ( $ i o t ~

6. Simplification Through Discrete- Time Dynamical Systems

20 1

holds, where pointwise addition, pointwise scalar multiplication, and multiplication by composition are assumed for and I : C + C is the identity mapping. The relationship between pole assignability and state controllability is given by the following proposition. Proposition 5.1. Let S be a linear dynamical system specified as above whose state space C is of dimension n. Assume that the field d has more than (n + 1) elements and that the state response 4 1 0 t : C+ C of S is an isomorphism. Then, if S is pole assignable, the system S is controllable from zero.

Let E E C be arbitrary. Since is an isomorphism, there exists # o such that ( I $ ~ ~ , ) ” c1,I = L, is an isomorphism. Choose Sf such that satisfies ($lo,)” c1,I = 0. Then we have ($1,$“ - (410r)” = -L,.

PROOF. c1,

+

+

Since L, is an isomorphism, there exists c E C such that - ($lot)”)(c)= 2 ; that is, $loi(c) - 410i(c)= i? (due to the state composition property), where 1 = nt. Furthermore, we have from the definition of that for any c E C and Sf E Sf, 4loi)(c) = 4oi(o, SAY)’) holds, where y satisfies the relation po(c, Sf(y)) = y. Consequently, ($102

-

Q.E.D. In the above proposition, 410ris assumed to be an isomorphism. As Proposition 4.3, Chapter VII, shows, this holds under rather mild conditions ; in particular, this is true for a linear time-invariant differential equation system. 6. SIMPLIFICATION THROUGH DECOMPOSITION OF DISCRETE-TIME DYNAMICAL SYSTEMS?

In the preceding sections, we have considered the decomposition of a system, primarily into two subsystems of a rather general kind. In the present section, we shall consider decomposition of a system into a family of subsystems of a prescribed, much simpler type. We shall consider time-invariant discrete-time dynamical systems defined by the maps p,:C x XI 3

t See Hartmanis and Stearns [I21

r,,

& ‘ : C x X t I ,+

c

Chapter X

202

Subsystems, Decomposition, Decoupling

where T = (0, 1,2,. . .}. To simplify the developments, the following will be assumed : (i) If the system is (strongly)nonanticipatory, which is the most interesting case in practice, the dynamic behavior of the system is completely represented by 4 = { q5ttp:t, t' E T ) . In this section, we shall consider, therefore, only $, refer to it as a state-transition system, and be concerned with the decomposition of 6. (ii) Since the system is time invariant and discrete time, $is characterized by a single mapping, namely, c$ol :C x A + C ;+,,, or in general 4tr,,can be determined uniquely from 4ol by using the composition property of the state-transition function. We shall denote 4ol by 4 and refer to 4 itself as a state-transition system. X'. If a composition operation is (iii) Let X = AT and X = Ute, defined on X as :X x + X such that x' P'= x' F'(P'),where F' is the shift operator, X is essentially a free monoid on A ; the null element of that monoid will be denoted by xo. In considering the structure of X later on in this section, we shall treat X as a monoid in the above sense. 0

0

0

For the sake of notational convenience and in order to follow the conventional literature, we shall define a special form of the cascade connection for the state-transition systems. Definition 6.1. Let r$l :Cl x A + C1 and 4 2 : C 2 x ( C , x A) + C2 be two state-transition systems, where A is the input alphabet for t$l, while C1 x A is the input alphabet for 4,. The cascade connection r$ of 41 and 42 is then

$:(C1 x C,) x A + C1 x C2 such that &(i.19

4, 4 = (41(c19 4 , 4 2 ( c 2 ,(c19 4))

Definition 6.2. A state-transition system 4 :C x A -,C is decomposed into a cascade connection of two state-transition systems 41:C1 x A -,C1 and b2:C, x (C, x A ) + C 2 if and only if there exists a mapping R :C, x C2 -, C such that R is onto and the following diagram is commutative:

(C, x C , ) x A R

I

/ I

c

'C1 x IR

c2

6. Simplification Through Discrete-Time Dynamical Systems

203

Intuitively, 4 is decomposed into the cascade connection of 4, and 42 ifand only if 4 = 4, . 4 2. R = .R holds. (Strictly speaking, some notational modifications of 4, and 42 are necessary for 4, . b 2 .R to be compatible with the cascade operation introduced in Section 1.) The following is a basic method to decompose a state-transition system.

4

Proposition 6.1. Let 4 : C x A --+ C be a state-transition system. Given a class of subsets { C , :a E C , } of C such that

(i)

(4

c = U a e C 1 c,

(VCa)(Va)(3CB)(4(Ca, a) = C,)

Then there exists a set C2 and two state-transition systems 4, :C, x A -, C , and 42:C2 x (C, x A ) -, C2 such that 4 is decomposed into the cascade connection of 4, and 42. c C, x C be a general system such that (a, c) E R H c E C,. PROOF. Let Let a set C2 be a global state of 8 ;that is, there exists a mapping R : C , k C2 --+ C such that c E C , H (a, c ) E ff ++ (3c2)(c= R(a,c2)). Let 4, :C1 x A C1and 42:C, x (C, x A ) -, Czbesuch that cjl(a, a) = B--+ &C,, a) c C, and

R(+,(a, a), ~ 2 ’ ) There may be more than one fi that satisfies the relation 4 ( C a ,a) c C,. In that case, any p satisfying the condition can be used to define 4, , As for 42, we shall show that there exists at least one cz’ that satisfies the relation $(R(a,c2),a) = R($,(a, a), c2‘).The definition of R implies R(a,c 2 )E C,. Let $,(a, a) = fi. The definition 4, then implies +(C,, a) c C,. Consequently, $(R(a,c2),a) E C,. Therefore, there exists c2‘E C2 such that $(R(a,c2),a) = R(B, cz’). If there are more than one cz’ satisfying the condition, any one of them can be used for 42. Let 4 2 ( ~ 72 (a, a)) = ~ 2 + ‘

&,3

CZ), a) =

c2), a) = (41(c1, a),42(c2,

(c13

4))

We shall show that R[&(c,, c,),a)] = &R(c,, c2),a) holds. Let 42(c2,(c,, a)) = c2‘.The definition of 42implies then that +(R(c,,c2),a) = R(4,(c,,a), c2’). Therefore, R[&(c,, c2),a)] = &R(c,, cz), a) holds. Furthermore, since C, = C , R is onto. Q.E.D.

uxsC,

Let us apply Proposition 6.1 and its proof to the case when the state space C of a state-transition system is finite. Let C, = C - { a } , where a E C. Then it is easy to show that { C , : a E C} satisfies the conditions of Proposition 6.1. In other words, a finite state-transition system always has a cascade decomposition. In order to discuss stronger results on decomposition of the finite case, we shall first introduce the following definitions.

204

Chapter X

Subsystems, Decomposition, Decoupling

Definition 6.3. Let 4 : C x A -, C be a state-transition system. For each U E A , let 4 a : C + C be such that 4,(c) = +(c, a). If 4, is onto, the input a is called a permutation input. If 4ais a constant function, the input a is called a reset input. If is the identity function, the input a is called an identity input. Definition 6.4. Let 4 :C x A -, C be a state-transition system. If all the inputs of 4 are permutation, 4 will be referred to as a P system. If all the inputs of 4 are a reset or an identity, 4 will be referred to as an R system. If all the inputs of 4 are either permutation or reset, will be referred to as a P-R system. Let us return to the finite case where C, = C - { a } . If the input a is a permutation for 4, it is also a permutation for the component subsystems as given in Proposition 6.1. (Refer to the proof of Proposition 6.1.) If the input a is not a permutation, there exists C , E { C, : a E C ) such that d(C*,a) c C , for all a E C , = C . Consequently, the input a is a reset for Therefore, 4l is a P-R system. Notice that the cardinality of C , can be less than that of C , or C . If we apply repeatedly the procedure of the cascade decomposition given in Proposition 6.1 to a state-transition system, the system is decomposed finally into the cascade connection of P-R systems, because a state-transition system whose state space has two elements has to be a P-R system. Consequently, we have the following proposition. Proposition 6.2. A finite state-transition system can be decomposed into a cascade connection of P-R systems. In the obvious manner, we can extend the domain of a state-transition system 4 :C x A -+ C to C x X,where X is the free monoid of A ; that is, $(c, x‘) = q50t(~,x‘). Let 4x:C + C be such that 4x(c) = &c, x). It is easy to show that 4x,. 4x = c # ~ ~ ,. where ~, 4x,. 4x is the usual composition of the two functions 4xand . Consequently, we can consider x E X itself as a mapping, i.e., x :C -, C such that x(c) = 4x(c) and A(c) = c, where A = xois the identity element of the monoid X. From now on X will be considered a monoid of functions in the above sense. Let a relation E c X x X be defined by c#J~,

(x,x’) E E 4+ (Vc)(x(c)

=

x’(c))

E is apparently an equivalence relation. Moreover, E is a congruence relation with respect to the monoid operation of Let X / E = {[XI : x E X}.A composition operation on X / E , therefore, can be defined as

x.

[XI *

[x’]= [x.x’]

X / E is also a monoid with respect t o the new operation.

6 . SimpliJication Through Discrete-Time Dynamical Systems

205

Proposition 6.3. Let 4 : C x A -, C be a state-transition system. Let C1 be a group in X I E , where the group operation is the monoid operation of X / E . The state-transition system 4 then can be decomposed into the cascade connection oftwo state-transition systems 4 , :C , x A -+ C , and 42:C x (C, x A ) -, C given by

where c; PROOF.

is the inverse of c1 in the group C , . Let R :C1 x C

-+

C be defined by R k , , c)

=

Then,

4)) = 41(C1¶4(42(C9(c19 4))

R($((c, c ) , d = R(4,(c,3 a),42tc, (c1 3

I

= (c1. [ a l k ) = 4(R(c,3

Furthermore, if c 1 = [A] E C, ,we have cl(c)

=

c. Hence, R is onto.

Q.E.D.

Let us consider a finite state-transition system again. Proposition 6.2 shows that every finite state-transition system can be decomposed into a cascade connection of P-R systems. Let us apply Proposition 6.3 to a P-R system. Let P be the set of all permutation inputs. Let P* be the free monoid on P . Then it is easy to show that P*/E c X / E is a finite group. Furthermore, since every element of P* is an onto mapping, if an input a is a reset, we have [a]$ P*/E. Consequently, if a is a reset, 42(c, (c, ,a)) of Proposition 6.3 does , is the identity mapping, not depend on c, and if a is a permutation, 4 2 ( ~ ( c 1a)) where C1 = P*/E. Therefore, 42 is an R system. Proposition 6.4. Every finite state-transition system can be decomposed into a cascade connection of P systems and R systems.

We shall consider 4 , of Proposition 6.3. Let us recall the following definitions. Definition 6.5. A subgroup H of a group G is called normal subgroup if and only if aH = H a for every a E G, where aH = { x : ( 3 y ) ( yE H & x = a . y ) } .

206

Chapter X

Subsystems, Decomposition, Decoupling

Definition 6.6. A group G is a simple group if and only if it has no nontrivial (i.e., different from G itself or the identity) normal subgroup. Definition 6.7. Let 4 :C x A + C be a state-transition system, where C is a group. If C is simple, the system is called simple.

Let 4 : C x A 3 C be a state-transition system whose state space C is a group. Let G be a group and let $ :C + G be a group homomorphism. Let E , c C x C be such that (c, c') E E ,

* $(c)

=

W')

Then E , is apparently a congruence relation. Let H , = {c : $(c) = e } ,where e is the identity element of G. Proposition 6.5. Let $ : C condition I&)

-,G be a

= $(c') + $(&, a)) =

group homomorphism such that the

$(4(c', a))

for every a E A

holds. Then 4 :C x A 4 C can be decomposed into the cascade connection of 41: ( C / E , ) x A + (C/E,) and 4 2 : H , x ((CIE,) x A ) 4 H,.

uaEC

PROOF. Let C, = [a] E C/E,, where a E C. Then C , = C . Furthermore, we can show that for any C , and a E A , # C a , a) c C,, where /?= &a, a). Let c E C , be arbitrary ;that is, $(c) = $(a). Then the property of $ assumed in Proposition 6.5 implies that $(4(c, a)) = $(&a, a ) ) ; i.e., 4(c, a ) E C,, where p = &a, a). Let h : C/E, + C be a choice function; i.e., h([a])E C,, where a E C. Let R :(C/E,) x H , + C be defined by R(C,, y) = h(C,) y , where y E H,. Then

c E c,

Jl(4 = $(a)

c-)

++

(3Y)(Y E H , Lk c

= EY)

- ( 3 y ) ( 3 j ) ( y E H ~ & 9 E H , & c = a9. j - ' y )

where h([a])= a j

(~Y)(E Y H , & C = R(Ca, Y ) ) The final result can be proven in the same way as Proposition 6.1. Let us return to 4 , :C 1 x A -+ C 1 of Proposition 6.3. c-)

Q.E.D.

Proposition 6.6. Let (6, :C1x A + Cl be the cascade component of Proposition 6.3. Let $ :C , -+ G be an arbitrary group homomorphism. Then $(cl) = $(c,') implies that $(&(c, a)) = $(41(cl', a)) for every a E A . Consequently, if C1 is not a simple group, the state-transition system 41can be decomposed into the cascade connection of : ( C , / H ) x A + ( C l / H )and 412:H x ( ( C J H ) x A ) + H where H c C 1 is a normal subgroup.

207

6 . Simplijkation Through Discrete-Time Dynamical Systems PROOF. Suppose $(cl) = $(cl‘). Suppose [a] E C1.The definition of implies that

4) = $(c1 . [ a ] ) = $(Cl).$([aI) If [a] $ C1,then $(41(C1?

$(41(c13

4) = $(Cl)

=

= $(Cl’)

41

Wl‘).$ ( [ a ] ) = $(41(Cl’? a)) = $(41@1’,4)

SupposeH is a normal subgroup of C,. Then there exists another group G and a group homomorphism $ :C, -+ G such that H = H , and aH = [a], where a E C1. The final result follows by Proposition 6.5. Q.E.D. As we mentioned in Proposition 6.6, the state-transition system 4 , has a peculiar property ; i.e., for every group homomorphism $,

4) = $ ( 4 1 ( C l ’ ? 4) for every a E A. This property is also preserved by 4, and 4, when 41,and 412are defined in the same way as 4 , and dZin Proposition 6.1. Notice that if 4 :C x A -+ C is a finite state-transition system, C1of c$l of Proposition 6.3 $(Cl)

= $(c,’)

+

$(4l(Cl?

is finite. Consequently, we have the following proposition.

Proposition 6.7. If 4 :C x A C is a finite state-transition system, the system can be decomposed into a cascade connection of state-transition systems which are either simple or reset.

A reset system has a parallel decomposition. We shall define a parallel decomposition of a state-transition system in the following way. Definition 6.8. Let 4 :C x A -+ C be a state-transition system. Let { 4i:Cix A + Ci:i= 1,. . . ,n) be a family of state-transition systems. If there exists a mapping R :C1 x . . . x Cn+ C such that R is onto and the following diagram is commutative : (C, x

I

... x C,)

x A

I

m

bC1

x

.”

I

x

c,

208

Chapter X

Subsystems, Decomposition, Decoupling

Proposition 6.8. Let 4 :C x A C be a reset system. Given a family of sets {Ci: i = 1,. . . ,n ) and an onto mapping R :C1 x . . . x C, + C. Then there exists a family of state-transition systems { 4 i : C ix A -+ Ci : i = 1,. . . , n } such that 6 = { 4 i :i = 1,. . . ,n } is a parallel decomposition of the reset system 4. -+

PROOF.

Let a state-transition system +i :Ci x A

4i(Cit

a) =

ci, c,i

'

-+

Ci be given by

if a is an identity input, i.e., 4, is the identity if a is a reset input

where

. . , can)E c, x . . . x c, c, = such that R(E,) = d(c, a ) for every c E C. Since R is onto, when a is a reset input, it is clear that there exists E , E C, x . . . x C, such that R(E,) = #(c, a) for every c. If there are more than two ?, that satisfy the above condition, any one of them can be used to define 4i.We shall show that R satisfies the commutative diagram of Definition 6.8. Suppose a is an identity input. Then R(W(c,,.. ., c,),

4) = R(c1, . . ., c,) = 4(W, . . ,c,), 4 7 .

Suppose a is a reset input. Then R(Nc12 . . . 9 c,),a)) = R(?,) = $(C' =

4

4(R(cl,. . . ,C,), a )

Q.E.D.

Suppose the state space C of a reset system is finite. There exist then a positive integer n and a mapping R :(0, l}" -+ C such that R is onto. Consequently, it follows from Proposition 6.8 that every finite reset system 4: C x A -+ C has a parallel decomposition { f j i : { o 11 , x A

-+

{0,1}: i = 1,. . . , n }

Definition 6.9. If the state space C of a state-transition system 4 has two elements, I) is called a two-state transition system. Combining Definition 6.9 with Proposition 6.7 by using Proposition 6.8, we have the following proposition. Proposition 6.9 (Krohn-Rhodes [ 171)lf 4 :C x A -+ C is a finite state-transition system, the system can be decomposed into a cascade and parallel connection of the state-transition systems which are either simple or two states.

6: Simplification Through Discrete-Time Dynamical Systems

209

The results of the principal decomposition theorems in this section are shown diagrammatically in Fig. 6.1. Finite state discrete-time dynamical systeiii (FSDTD)

I

Proposition 6.2

0

...

- +.+.+.+*

++ ...*

U U

Proposition 6.4

Proposition 6.1

* *

0

Proposition 6.9

...* P-R: P-R system P: Psystem R : R system

S: Simple system 2: Two-state system

FIG.6.1

0