On the realization problem for stationary, homogeneous, discrete-time systems

Autoraatica, Vol. 14, pp. 357 366

PergamonPressLtd. 1978.Printedin Great Britain © InternationalFederationof AutomaticControl

0005-1098/78/07014)357502.00/0

On the Realization Problem for Stationary, Homogeneous, Discrete-Time Systems*t STEVEN J. CLANCY:~ and WILSON J. RUGH~

By introducing the regular kernel and regular transfer function representations for degree 2 homogeneous systems, minimal realizability of state-affine difference equations is simply characterized along with a convenient method for constructing such realizations. Key Word lndex--Bilinear control; difference equations; discrete time systems; minimal realization; nonlinear systems; system theory; z-transforms; Volterra series.

Summary--For degree 2, stationary, homogeneous systems, we consider realizations in the form of state-affine difference equations, and in the form of feedback-free interconnections of stationary linear systems, adders, and multipliers. Realizability conditions are presented in terms of the symmetric transfer function (or kernel), and in terms of a new representation called the regular transfer function (or kernel). A procedure is given for constructing a minimal dimension state-affine realization, and the corresponding interconnection.

For simplicity we will consider only degree 2 homogeneous systems. Our results generalize to higher degree cases, although the development becomes much more complicated. The standard time-domain representation for a stationary, degree 2, homogeneous system takes the form

I. INTRODUCTION

y(k) = ~ ~ h ( k - i, k-j)u(i)u(j),

k

k

i=Oj=0

VOLTERRA series representation provides a potentially powerful tool for the study of nonlinear, discrete-time, stationary systems. Early studies in this area focused on the multivariable z-trarrsform and its properties with regard to system representation[i,2]. More recently, attention has shifted to consideration of bilinear discrete-time state equations where progress has been made in answering several questions of system theoretic importance[3-5]. Also the more general case of polynomial discrete-time state equations has been treated[6-8-]. All of this later work is concerned with the realization problem and related concepts. We will discuss the realization problem for the special case of stationary, homogeneous, discretetime systems. One general reason for interest in this case arises in the context of system identification. When a system is described by a finite Volterra series, the different amplitude dependencies of the homogeneous subsystems can be used to separate the response contributions of each subsystem. Thus the identification and realization properties of homogeneous systems are central to constructing a model of the system. THE

k=O, 1,...

(1)

where u(k) and y(k) are the input and output signals (real sequences), and the symmetric kernel h(i,j)(=h(j,i)) is real and defined for integers i,j >0. In the frequency domain, the symmetric transfer function H(zl, z2) given by the 2-variable z-transform of h(i,j) is often used. As defined in [1] and [2], H(zl,z2) is the power series in complex variables Zl and z 2 given by

H(z,,z2)=~

~, h(i,j)z-(iz2 j

(2)

i=0 j=o

within the region of convergence. The symmetric transfer function representation is quite useful in describing the response to sinusoidal or Gaussian input signals. However, to compute the response to a general input, the required 'association of variables' technique is rather complicated. In addition to the symmetric kernel and transfer function, we will use a new type of representation for the degree 2 system. Aside from providing the foundation for our realization resuits, this representation is of interest because it exhibits in an explicit fashion the input/output behavior of the system, and avoids association of variables. We begin by examining the response of the system to an input of finite length L + 1. From (1) the z-transform of the response can be written in

*Received April 15, 1977; revised October 17, 1977. The original version of this paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by associate editor J. Ackermann. tReseareh sponsored by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant No. AFOSR-76-3001. 1:Electrical Engineering Department, The Johns Hopkins University, Baltimore, MD 21218, U.S.A.

357

358

STriVEN J. CLANCY

the form

Y(z)= ~

and

WILSON

J. RUGH

inputs. Let do(k) be the unit pulse, ao{0)= 1 and 6 o ( k ) = 0 for k4:0. Then the input u(k)=doIk gives

y(k)z k

k=O

•

Y (z)= 2fo(Z)

rain [k, L] min [k, L]

=E k=O

E

E

i=O

and the input u(k)=do(k)+iJo(k-i ) gives

j=O

× h ( k - i, k-j)u(i)u(j)z k

(3)

Separating out those terms for which i=j, and rearranging the remaining terms using the symmetry of h(i,j) permits (3) to be written in the form L

h(],j)z -j ~ u2(n)z -''

Y(z)= j=O

n-O L

+ Z

~.

Z 2h(i+j,J) z-~

i=1 j=O L

x ~ u(n)u(n-i)z-"

(4)

n=i

Now define the regular kernelf(i,j) as follows in terms of the symmetric kernel:

f(i,j)=h(i+j,j)6

,(i), i,j>O

(5)

where 6_ ~(i)is the special unit step function ,

i<0

1/2,

i=0

,

i>0

0

6_,(i)= LI

Y (z) = 2fo (z) + 2z-% (z) + 2z-!ritz), i = 1 , 2 ....

F(zl, z2) = ~ f.(z2 )z, '

(7)

i=0

where .z

f~(z2) = ~ f(i,j)zzJ, i=O,l ....

(8)

j=O

I1. T H E Z - T R A N S F O R M

For the purposes of this section, suppose that f(i,j) is an arbitrary, real, doubly indexed sequence defined for i,j>O. We view the basic definition of the z-transform off(i,j),

i=0

"

(12)

as a formal nonpositive power series in the indeterminates z 1 and z2, rather than a convergent power series in complex variables. We say that F(z,, z2) is a proper rational z-transform if there exist real-coefficient polynomials P(zl, z 2) and non-zero Q(z~,z2) such that P(Zl, 22)= Q(zI,

z2)F(Zl,

z2)

where the degree of P(z,,zz) in z, or z2 is (necessarily) no greater than the corresponding degree in z~ or z2 of Q(z1,22). With some abuse of notation we write, for a proper rational ztransform,

F(zl,z2)Y ( z ) = 2 ~ f i ( z ) ~ u(n)u(n-i)z

~.f(i,j)z,-'z~ j

i=0 j=O

For then (4) becomes L

(11

Thus each f/(z) in (7), (8) can be found from the response to unit pulse inputs. Our approach to the realization problem for degree 2 homogeneous systems is based upon algebraic properties of the 2-variable z-transform, particularly in regard to the regular transfer function. Thus we will review the basic development of the z-transform from an algebraic viewpoint. The style of the development follows that in [9], where the single-variable case is discussed in elementary terms.

F(z,,z2)= ~ (6)

Furthermore, define the regular transfer function F(zl, z2) as the z-transform of f(i,j) similar to (2). However it is convenient to write F(zl,z2) in the form

L

(10

P(zl, z2 ) Q(zl, zz)

(13)

(9)

n=i

The input/output representation (9) permits the easy calculation of the response to any given input of length L + 1 from F(z,,zz). Moreover it indicates how the regular transfer function can be identified from the responses to a simple set of

Furthermore, a z-transform will be called recognizable if it is proper rational and the denominator polynomial can be written as a product of single-variable real-coefficient polynomials, Q(z,, z z ) = Q, (z,)Qz(z2). It will be of interest in the sequel to have conditions for recognizability of a z-transform.

On the realization problem for stationary, homogeneous, discrete-time systems To this end we write (12) in the form

F(z,,zz)= ~ fj(zl)z; j j=O

(14)

Equating coefficients of like powers of of z2 on both sides of (21) shows that each fj(zl) is a proper rational function and that m-1

fm+k+l(Zl) = E --q2,.ifk+J+l(Zl)' j=O

where o0

f j ( z l ) = ~ f(i;j)z;i,j=O, 1,...

i=0

~-f(1,j)

f(2,j)

f(3,j)

"'l

I f(2,j)

f(3,j) f(4,j) f(4,j), f(5,j).

"t

k=0,1,...

(15)

For a single-variable formal series such as fj(zl), a well-known condition equivalent to proper rationality is that the Hankel matrix

Hf~=[f(3,j).

359

(22)

Thus all the rank conditions are satisfied. Now assume that the rank conditions are satisfied. Since each H:j has finite rank, each fi(zl) is proper rational. Since ~:(21) has finite rank, say m, there exist real numbers ao ..... a,,_ 1 such that

(16)

m-1 fm+k+l(Zl) = ~ - - a j f k + j + x ( z l ) , k = O , 1....

j=O

(23) have finite rank. Assuming that each fj(zl) in (15) is proper rational, we define a Hankel matrix of rational functions: f,(zl)

r

f~(zl)

f~(zl)

...

, |f2(z1) f3(z1) f4(z1) • °'~f(ZlJ:~f3(z1). f4(z1), fs(z1)

t

(17)

Let

bo(z 1) = aofo(zl ) +... + a,,_ lf,,-1 (zl) + f,,(zl ) bl (z1)=alfo(zl )+... +am- lfm-2(Z1 ) +tim- 1(Z1) (24) b,,,_ 1(zl ) = a,,,_ lfo(z 1)+fl (z1)

b~(zl)=fo(zl) Theorem 1. The z-transform F(za, z2) in (12) is recognizable iff the associated Hankel matrices H:: j>O, and ~¢g:(Zl) all have finite rank over the real field. Proof Suppose F(zl, z2) is recognizable. Then there exist polynomials n-1 Ql(Zl)--z]q- E ql.i zil'Q2(z2) i=0

F(Zl,Z2) =

P(zl, Z2) O.1(zl)9.~(z~)

(18)

P(Zl, z2) = B(zl ) ~ bj(zl )z½ j=O

Ql(Zl)Q2(z2)=B(Zl) zT+ E ajz j=o

and

P(Zl, Z2) = ~ Pj(Zl)ZJ2 j=o

(19)

pj(zl)= ~ pi.jzil, j=O, 1,...,m

(20)

where

i=0

so that

P(Zl, 22) = Q1 (zl)Q2(z2)F(zl, z2)

= Q, (z~)Q2(z~) Y', fj(z~)z~J j=O

(25)

where

ra--1

= z"i + E q2.:{ j=0

and let B(z~) be the least common multiple of their denominators. Then

(21)

(26)

•

If the condition on J~f:(Zl) in Theorem l is weakened to rank finiteness over the field of rational functions, then a necessary and sufficient condition for proper rationality of F(zl,z2) is obtained. This is analogous to the continuoustime case.[10]. W e note that an alternative necessary and sufficient condition for recognizability of F(zl, z2) can be developed by writing the formal series as in (7) and (8). Define a Hankel matrix H f, for each fi(zz) in (8) and, assuming each fi(z2) is proper rational, define a Hankel matrix of rational functions ~:(z2) similar to (17). Then F (Zl , z2) is recognizable iff H y ,, i > O, and ,,ug:( z 2 )

360

S T E V E N J. C L A N C Y a n d

all have finite rank over the real field. The proof is similar to that for Theorem 1. It will be useful in the sequel to have conditions under which a doubly indexed real sequence f (i,j) takes the exponential Jbrm

f(i,j) =k

I r

lq=l

v=l

~,~,q ak,

WILSON

J. R U G H

state-affine realization of a given degree 2 homogeneous system. Then there is another Ndimensional state-affine realization of the system in the block form

i

i--r+l

x ~"- v + l J J~i-r+ , k1~--v+

(27)

for all i,j>O. The coefficients in (27) may be complex, but they must satisfy well-known conjugacy conditions since f(i,j) is real. If, say, "~k =0, we assume implicitly that ti ( - ri+ 1 P'~k~']i-r+1AI~= v when r > i + l . Theorem2. A doubly indexed real sequence f(i,j) has the exponential form (27) for all i,j>O iff the corresponding z-transform is recognizable. Proof Suppose f(i,j) has the form (27) for all i,j>O. Then the z-transform of the general term in (27) is r,v~ ak,qZ, l Z2 (zl - 2k)r(z2 - ~q)~

(28)

Using linearity of the z-transform and placing all terms over a common denominator gives that F(z~, z2) is recognizable. Now suppose F(zl, zz) is recognizable. Then F(zl, zz) is a sum of terms of the form

pijz'l z~

ct2]z(k) +[c21

y(k)=[O

O]z(k)u(k)+du2(k), (31)

z(0)-0

where All is nxn, n = r a n k [btAbl...AN-lbl], A22 is ( N - n ) x (N-n), and the remaining coefficient matrices are partitioned accordingly. Proof We first decompose the N-dimensional state space into the direct sum RN=LI@L2, where L t = s p a n {bl, Abl,..., AN-lbl}. Now let {0~1 . . . . , ~ n ] be a basis for L 1 and {0~n+ 1. . . . . ~N] be a basis for L z. Since L~ is an A-invariant subspace which contains bt, changing variables to z(k)=[~laz...eN] x(k) yields the block form state-affine realization

(29)

Q~(z~)(2~(z2) so that applying the single-variable inverse ztransform to each factor gives the result. III. S T A T E - A F F I N E A N D I N T E R C O N N E C T I O N REALIZATIONS

We will be interested principally in so-called state-affine difference equation realizations of the form

x(k + 1) = Ax(k ) + Dx(k )u(k ) + b~u(k ) +b2u2(k),k=O, 1.... y(k)

=clx(k)+c2x(k)u(k )

(30)

+du2(k),x(O)=O

y(k)=[Cll

Cz23z(k)u(k)

+ du 2(k), z(0) = 0

(32)

with the dimensions as claimed. Now computing y(0), y(1),.., and imposing the requirement that only degree 2 terms can be present gives k

i-1

~ c12AJz-ziD21A~-~J-tbxxu(k-i)u(k-j)

y(k)= ~ where the (finite) dimension of the state vector x will be called the dimension of the realization. Of course not every difference equation of the form (30) corresponds to a degree 2 homogeneous system. Our first step is to choose a convenient structural form for those that do. Lemmal. Suppose (30) is an N-dimensional

cl2]z(k)+[c21

i=2 j=l k

+ ~ c21Aij-llbllu(k-i)u(k) i=l k + E c12AJ21b22u2(k-j)+du2(k) j=l

(33)

On the realization problem for stationary, homogeneous, discrete-time systems Thus we can take the submatrices D~, D12, D22, b21 , Cll , and c22 to be zero, yielding the block form (31). • The condition for state-affine realizability in terms of the regular transfer function representation is quite simple• Theorem 3. The regular transfer function F(zl, z2) is state-affine realizable iff it is recognizable. Proof If F(zl,zz) is state-affine realizable, then by Lemma 1 we can assume a realization in the form (31) with a corresponding input/output relation in the form (33). Computing the response y(k) to u(k)=6o(k ) via (33) and comparing with (10) shows that

~

Now using the identities

n-1

~, ql,iz~

i=O

z~

----1

Ql(zl)

Ql(Zl)

m--I Z~' Qz(z2)

Z qz,JzJ2

1

i:o Q~(z~)

(38)

we can write (37) in the form

1 n-

ra-1

~,__Z° Y ~,,/,z~ F(zl,z2)-

j=o Ql(Zl)Qz(Zz)

ln-1 ,j=O

+

(34)

f(0'J)='~ 1 . ~ ~c12AS221bz2,j>O

361

|m-1

2i~=0 'iZi

Q~(z,)

4

2j~=O ~jzj2

Q~(z~)

+~-

(39)

where Similarly, letting u(k)=bo(k)+bo(k-i) and using (11) gives

in (33)

ffi,j = Pi,j -- Pn,jql, i -- Pi,mq2,j + P.,.,ql, iq2,j fii= Pi, m -- Pn,mql,i

f(i,j) = ½[y(i +j) - 2f (0,j) - 2f(0, i+j)] =f½cz1Aixll b11, i > 0 , j = 0 [~c12A22 D21All b11,t,J>0 1

j - I

i-1

-

•

Pj=Pn,j--P.,mq2,j (35)

Then a realization of the form (31) for (37) is specified by

Thus the regular transfer function is given by

--o

F(z,,z2)= ~ ~f(i,j)z'[iz2 j

All =

i=O j=0

d 1 1 =~+~C12[Z21-- A22]- b22

"'"

0

--q2,0 -~

Ira- 1 (36)

i i

--q2,m-l J

FPo, 0

Pl

L /~O,m- 1

/~1,,~-1

90

ii:

D2 ~ = [/~i.s] v ---

hll ~

I

-qx,.- 1

A22 = I 0

½P(zl,z2) F(z.zz)= Q,(zl)Q2(z2)

q~,~zi~ "

--q1,1

..•

which Clearly is recognizable when all terms are placed over a common denominator• If F(zl, z2) is a recognizable regular transfer function, then it can be written in the form

i=0

1.-1

i. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

-All]-lbll

z"~+

i 1 l E

--ql,o

+lc 12[Z2I-- A22]- 1D21[z lI

m

i l )

0

q'-lC21[Zll -- A11] - lbll

2 i=o s=o

(40)

1

b22

qz,sz

(41)

07)

c12=[0...0 1], c21=[0o

...

Pn-1],

d=p,,,,

STEVEN J. CLANCY and WILSON J. RUGH

362

as is readily verified by substitution of (41) into (36). • The question of state-affine realizability in terms of the symmetric transfer function representation is more difficult. Our approach is to first establish the relationships between the symmetric and regular transfer functions, based upon the relationship between the symmetric and regular kernels in (5). Perhaps it should be emphasized here that both the symmetric and regular transfer functions correspond to nonpositive power series. There are, for example, symmetric, proper rational functions which do not correspond to nonpositive power series of the form (12), and thus are not symmetric transfer functions. Of course this situation is detected readily by division. Lemma2. The symmetric transfer function H(zl,z2) corresponding to the regular transfer function F(z 1, z2) is given by

H(Zl, Z2)=F(zl,zlz2)+F(zz, zlz2)

(42)

Proof By the definition of the z-transform we can write x,

H(z~,z2)= ~, ~ h(i,j)z[~z)-s i=0 j=o ~tj

=~

/_

~, [h(i,j)6_l (i-j)

i=o j=o

+h(/,i)6_l(]-i)]z['z2 s

(43)

where 6_ l(k) is defined in (6). Changing variables gives

H(z,,z2)= ~ k=0

F(Zl, 2"2) corresponding to the symmetric transfer

function H(z~, z 2 ) is given by

F(zI,z2J=[H(zI,z2/zl)] t--'

(45)

Proof Changing variables in (44) shows that H(zl, z2/zl)= ~

~ h(k+j,j)6 ,(k)z;kz2 .i

k=0 i=0

+

~ h(m+i, i)6_l(m)z"~z2 u+m) m=0

i=0

(46) Clearly the first series in (46) is F(zl, 2"2) while the second series involves only positive powers of z~, plus 1/2 of each z ° term. Deleting this second series gives (45). • In the case where the symmetric transfer function H(Zl, Z2) is given in proper rational form, we proceed as follows. The proper rational function H(zl, z2/zl) can be viewed as a rational function in z2 with coefficients which are polynomials in z~. Division with respect to zz yields a nonpositive power series in z 2 with coefficients which are (not necessarily proper) rational functions in Zl. Now write each of these coefficients as a strictly proper rational function in zi plus a polynomial in z~. In each polynomial delete all the terms with positive powers of z~, and divide the z ° term by 2. Then from Lemma 3 it is clear that the result is the regular transfer function in the form

~ h(k+j,j)6 l(k)2"1k(Z1z2) -j j=O

f ( z l , z2)-= 2 J)(Zl)zzJ=[H(z1,2.2/zl)]l-) j=O

(47)

+ ~, ~ h(m+i, i)6_,(m)z2"(z,z2) -i m=O

i=0

(44) Using (5) it is clear that this can be rewritten as (42). • Now suppose H(z~,z2) is a symmetric transfer function as defined in (2). A simple variable transformation yields H(Zl,Z2/Zx). Although H(zl,zz) corresponds to a nonpositive power series in zl and z 2, H(zl,z2/zl) corresponds to a series which contains positive powers of z 1. By deleting all terms in the series with positive powers of zl, and multiplying each term containing z ° by 1/2, we form a nonpositive power series which we denote by [H(Zl, z2/zl )]~-!. Lemma 3. The regular transfer function

where each f~(zl) is a proper rational function. Theorem4. The symmetric transfer function H(zl, z2) is state-affine realizable iff it is proper rational and the Hankel matrix :h~:(zl) corresponding to (47) has finite rank over the real field. Proof If H(Zl, Z2) is state-affine realizable, it follows from Theorem3 and Lemma3 that the regular transfer function given by (47) is recognizable. Lemma 2 then shows that H(zl, z2) is proper rational, and Theorem1 shows that )f:(zl) has finite rank over the real field. On the other hand, if H(zl,z2) is a proper rational transfer function, then it follows from (47) that each f~(zl) is a proper rational function. Thus the corresponding Hankel matrices H:: j >0, all have finite rank. If in addition the

On the realization problem for stationary, homogeneous, discrete-time systems Hankel matrix ~ : ( Z l ) corresponding to (47) has finite rank over the real field, then Theorem 1 gives that F(Zl, Z2) is recognizable. State affine realizability then follows from Theorem 3. • Examples. The symmetric proper rational function 1/(Zx+Z2) does not correspond to a nonpositive power series. Thus it is not a symmetric transfer function. The symmetric proper rational function

The question of minimality of dimension for state affine realizations of degree 2 homogeneous systems is answered very simply in terms of the regular transfer function. We say a regular transfer function is reduced if there is no (nonconstant) factor common to both the numerator and denominator polynomials. Theorem 5. Suppose

F(zl, z2)=

2z z -z z2-z,z H(zl, z2) = 2 1aZ a2 - - z2z2-zlz2+i~ 1

(48)

does correspond to a nonpositive power series and thus is a symmetric transfer function of a degree 2 system. A short computation gives ! 0k/(Z1, Z 2 / 2 1 ) ] ( - ) =

1

1 "nt- ~-~- Z 2 1 nt-_~- Z 1 2 Z1 Z1

1

Z;3 +...

(49)

from which it follows that rank H : ( z , ) = oc. Thus the system is not state-affine realizable. (The corresponding regular transfer function is

F(z,, z2) = z l z2/(zl z2 - 1 ), which is not recognizable.) Finally,

H(zl, z2)-

zlz2

(50)

Z1Z 2 -- 1

corresponds to a nonpositive power series. In this case

+½Z;3+...

P(zl,z2) Ql(zl)Q~(z2)

(52)

is a reduced regular transfer function with n =degree Ql(zl) and re=degree Qz(z2). Then a minimal state-affine realization of F(Zl, Z2) has dimension n + m. Proof Suppose the regular transfer function (52) satisfies the hypotheses of the Theorem but that there exists a state-affine realization of, dimension n + m - q, q > 0. By Lemma 1 we can assume this realization is in the block form (31). But then a calculation of the corresponding regular transfer function via (36) shows that the sum of the degrees of the denominator polynomials is n + m - q , which contradicts the hypothesis that F(zl, z2) is reduced. • Given a reduced regular transfer function, the proof of Theorem3 shows how to construct a minimal state-affine realization. We note also that there is a realization of the same dimension in the form Of a feedback-free interconnection of adders, multipliers, and linear subsystems. To show this, first write (39) in the form n-1

1 r

~

i=0

F(z"z2)=S,~-

[H(zD Z2/Z1 )](-) = 2 - -±1 2 Z 2 _1 - I +12.22

363

i l)i, kZ1

Q,(z,)

m-1

(51)

E wk.:

j=o

which gives immediately that rank H:(z 1)= 1 and we have state-affine realizability. (The corresponding regular transfer function is

Q2(z2)

1n--1

Z_°

k- t

Q~(Zl)

1 m--1

F(zl, z2)= 1/2 z2/(z2 - 1).) State-affine realizability conditions in terms of the regular and symmetric kernel representations for degree 2 systems follow easily from our development. By Theorems 2 and 3, a regular kernel f(i,j) is state-affine realizable if it has the exponential form (27) for all i,j>O. Using Theorem 1, this can be phrased in terms of rank conditions. Conditions for state-affine realizability of a symmetric kernel h(i,j) follow from the fact that h(i,j) is of exponential form for all i>j>O iff f(i,j) is of exponential form for all i,j>O.

+

j=o

Qz(zz)

~_Pn,r,

2

(53)

by performing the full-rank factorization P = V W, where P = [/~i.j], V = [vi. j], W = [wi. j] and rank P = r a n k V = r a n k W = r . Now Fig. 1 shows an interconnection realization for F(z1,z2) which corresponds to the coefficient matrices in (41). Example. Suppose we are given the symmetric transfer function H(z 1, z2)= N(z I , z2)/D(zx, z2)

364

STEVEN J. CLANCY and WILSON J. RUGH

$ 7 8 - 0 0 4 3 -VA-I

FIG.1. Interconnectionrealizationcorrespondingto (53). where

and division yields

N ( z 1 , 7 ` 2 ) = z a4.z 22 + z ~2z 24. +

3 3 - 4 z a z 32 2ZlZ2

2 - 4 z a z 22 3

H(zl,zzlzl)=

+ 3z~z~ + z~z~ + ~, z~ - =, ~

O(zi, z2) =

z1z24. 3 -t- z 12z3 4_

+ 6 z ~ z 2 + 6z~z32 + z'~z z + z 1z'~ - 3 z ~ z 2 - z3z2 - z, z 3 - 4z~z2 - 4ZlZ 2 - z 3 - z 3

3zl

-

37,2 +

(Zl--1)2 z°

+ /F z, + 1+

2z~z22_2z2z~_ 5z~z~

+ 7ZlZ2 + 3z 2 + 3z 2 -

[Zl ]

2 z l - 1 ] -1

G--i)Jz~

[

3zl - 2 -]z_ 2

+Lazl+2z, + 2 + ( T C i p j

2

1 (54)

Then

+~3z 3+3z 2+3z~ + 3 4Z 1 --

3] _

(56)

4 ( < ~ i 7 2 ] Z 2 3-{-'''

N(Zl,Z2/Zl)=Zlz4

+ (2Zl3 --4Zi2 )Z23

where the pattern is clear from these first four terms. Now applying Lemma 3 to (56) we find that

+ (7`,~ -4zI + 3? + z~)7`~ +

(zl -

z 3)z~

D ( z l , z z l z l ) = ( z l - 1 )2z~ +

F(z',z2)=[H(z,,zz/z,)]t-)=~[-(z_ezl I

]z o

[(zl L 1 )2 1F(z~-l)+2z,l

x (z 2 - 3 z , -

)z

,

i<--/7 jz2

1)]z 3 + [ ( z 1 - 1) 2

1 F2(z2-1)+2z,1

-2

× (-2z 3+2zi+3)]z22+[(z1-1) 2

+2L ~ll:i) g

x (z4 + z 3 - 3 z 2 ) ] z 2 - z 3 ( z , - 1) 3

+ 1F3(z~- 1 ) + 2 z , ]

(55)

IJ z2 3

2L (Z--l~ -Jz2 +...

(57)

On tlae realization problem for stationary, laomogeneous, chscrete-t~me systems and forming the Hankel matrix ougs(zl) for (57) shows that it has rank 2. Specifically, fk+3(Zl)= --fk+l(Zl)+2fk+2(Zl), k=O, 1....

(58) Performing the calculations outlined in the proof of Theorem 1 shows that bo(zl)=0 1 z 2 - 2zl - 1

b' (Zl)=2

(59)

i~ll ~ 1)T

1 2z 1 bz(Zl)=2 (z I - 1 ) 2

and thus

F(zl,z2)-

1 2zlz 2 + z2z2 -2ZlZ 2 - 2

(Z1-- 1)2(Z2-- 1) 2

Z 2

(60)

which is clearly reduced. Writing (60) in the form (39) gives

F(zl, z2) =

1 4ZlZ2-2z

1 --2Z 2

2 (z 1 - 1)2(z2 - 1) 2 1

2zl

-F2(z _1)~

1 z2 ~ 2 ( z 2 _ 1)2

(61)

from which we see that a minimal state-affine realization of the form (31) is given by

Ale=I?

1 123, A22=[01

o [o 71 2

bEE

'bll=

21],

'

365

in conjunction with the realization problem for homogeneous systems. Perhaps more importantly, they yield a simple formula for computing input/output behavior. The notions of recognizability and rationality we have used for (commutative) formal power series agree with those used in the more general (noncommutative) setting in [[-1]. However the conditions we have derived are much different. This largely accounts for the fact that our realization theory differs sharply from those obtained by specializing [3], [4], or [6] to the case of homogeneous systems. Of course it would be of interest to generalize our approach at least to the case of finite Volterra series, where a fairer comparison with other approaches could be made. Although from a practical viewpoint it seems to be most feasible to work on a homogeneous term-by-homogeneous term basis when modeling or analyzing a nonlinear system. We should emphasize that a minimal stateaffine realization of a degree 2 system may not be a minimal realization in the larger sense. In 1-12] it is shown that the dimension of minimal realizations of a degree 2 homogeneous system is given by the degree (in either variable) of the denominator polynomial of the reduced symmetric transfer function. Using Lemma 2 it can be shown that this degree can be less than the sum of the denominator degrees in a reduced, recognizable, regular transfer function. Finally we note that a more detailed analysis of state-affine realizations using concepts of span reachability and observability shows that any two minimal state-affine realizations of a degree 2 system are related by a linear change of variables. This analysis is omitted since it is similar to that for internally bilinear systems in [3].

Lid

REFERENCES

A corresponding interconnection realization in the form of Fig. 1 follows easily by taking the full-rank factorization D21=D21 I. We should point out that if the symmetric transfer function corresponding to (62) is computed using (60) and (42), it will become clear that the given H(zl,z2) contained an 'unrealizable' factor (z 1 + z 2 - 1 ) in both the numerator and denominator. An advantage of the formal series approach is that common factors are removed in the division process.

[-1] P. ALPER: A consideration of the discrete Volterra Series. IEEE Trans. Aut. Control AC-10, 322-327 (1965). [2] A' M" Busm S°me techniques f°r the synthesis °fn°nlinear systems. MIT-RLE Report No. 441 (1966). [3] A. ISIDORI: Direct construction of minimal bilinear realizations from nonlinear input-output maps. IEEE Trans. Aut. Control AC-lg, 626-631 (1973). [4] M. FLIESS: On the realization of bilinear dynamic systems. C. R. Acad. Sci. Paris, Ser. A, 277, 923-926 (1973). [5] T. J. TARN and S. NONOYAMA: Realization of discretetime internally bilinear systems. Proc. 1976 Conf. on Decision and Control. Clearwater, Florida, pp. 126-133

(1976). IV. CONCLUSIONS

Our results indicate that the regular kernel and regular transfer function representations should prove to be useful tools in nonlinear system theory. They clearly are natural representations

[6] E. O. SONTAG: O n the internal realization of polynomial

response maps. Ph.D. Dissertation, University of Florida, Gainesville, Florida (1976). [-7] T. J. TARN, S. NONOYAMA and L. S. OEI: Algebraic structure of discrete-time polynomial systems. Proc. 1976 Conf. on Decision and Control. Clearwater, Florida, pp. 826-834 (1976).

366

STEVEN J. CLANCY a n d WILSON J. R U G H

[8] E. G. GILBERT: Bilinear and 2-power input/output maps: finite dimensional realizations and the role of functional series. Proc. 1977 Joint Automatic Control Conj. San Francisco, Calif., pp. 917 924 (1977). [9] W. J. RUGH: Mathematical Description O] Linear Systems. Marcel Dekker, New York (1975). [10] G. E. MITZEL and W. J. RUGH: On a multidimensional s-transform and the realization problem for homogeneous nonlinear systems. IEEE Trans. Aut. Control

AC-22, (5), (1977). [11] M. FLIESS: Matrices de Hankel. J. Math. Pures AppL 53, 197 224, 1974. [12] E. G. GILBERT: Minimal realizations for nonlinear I/O maps: the continuous-time, 2-power case. Proc. 1977 Co~f on lnJormation Sciences and Systems. The Johns Hopkins University, Baltimore, MD, pp. 308 319 (1977).