Finite Input/Output Representative of a Class of ... - Semantic Scholar
Auromatica.
Pergamon
Printed
Pn: SOOOS-1098(96)00178-l
Vol. 33, No. 2. PP. 257-262, 1997 @ 1997 Elsevier Science Ltd m Great Britain. All rights reserved CQJS-109w97 f17.00+0.00
Brief Paper
Finite Input/Output
Representative of a Class of Volterra Polynomial Systems S. KOTSIOS *
Key Word.. Algebraic Volterra series.
approaches;
factorization
methods;
Abstract-In this paper the problem of finite input/output representation of a special class of nonlinear Volterra polynomial systems is studied via the notion of linear factorization of b-series. This is an algebraic method based mainly on the star-product operation and on a related Euclidean-type algorithm. 01997 Elsevier Science Ltd. All rights reserved.
in order y(t)
+
Ul,‘(t
transformations;
an input/output
discrete
representation
of the form
- 1) + ’ ’ . + UkJ’(t -k) c
biU(t - il)U(t
~ i2)
U(t - in), (2)
n=Oi=(i~.i*.....l.)EI.
where I, is tinite, for each n = 0,. , m. The advantages of the expression (2) are obvious. Firstly, stability can be easily checked, due to the existence of a certain number of useful theorems. Secondly, this finite expression can be easily transformed to a linear regression form y(t) = BT+(t), e = [-al,-a2...., -ak. bi,. , bijn.. ,biliz...in17 4 = [~‘(l l),y(t-2) ,.... y(t-k).u(t-I) ,__., Ia-IM-j) ,..., u(ti1)u(t - i2) ’ . . ~(t - in)], suitable for identification, which can be carried out by solving a linear system of equations, via one of the popular algorithms such as the RLS, LMS (Kalouptsidis and Theodoridhs, 1993). The identification of these systems has been studied earlier in Netravali and de Figueiredo (1971). A wide variety of tools have been used for the description of discrete systems of the form (I) z-transforms (Alper, 1965; Rugh, 1981), difference polynomials (Cohn, 1965), formal power series (Fliess, 1987a), to mention but a few. Each of these alternative representations has its advantages, depending on the problem to be solved. In order to describe and solve our problem we use the notion of a &operator. The set A of &operators, as well as their star-product, are algebraic tools introduced in Kotsios and Kalouptsidis (1993a) to cope with the model matching problem for a class of nonlinear discrete systems. Initially, the &operators had a computational orientation but gradually it became clear that they also had some particular properties. The distributive property, for instance, does not apply in our case and therefore the set (A, *. +) is not a ring, in contradiction to the rings of difference or differential polynomials (Cohn, 1965; Ritt, 1950) and the rings of formal power series (Isidori, 1995). The distributive property is a strong linear property and consequently all the methodologies which make use of it may face nonlinear problems which have ‘slight linear behaviour’ within their structure. The h-operator and the star-product seem to be capable of describing ‘deeper’ nonlinearities. The focus of our investigation here is to explore how this is possible and under what conditions. Some efforts have been made in this direction and despite the complexity of the matter some preliminary results already exist. Details can be found in Kotsios and Kalouptsidis (1993a), Kotsios and Kalouptsidis (1993b), Kotsios and Kalouptsidis (1997) Kotsios (1997a) and Kotsios and Lappas (1997). In this paper we wish to find &polynomials A(b). B(b); A(6) linear, so that the system A(d)y(t) = E(d)u(t) will yield the same output with (1) under identical inputs. Our main objective is to devise a factorization of the original system C, of the form C = L * B, where L is a linear series and B is a finite nonlinear polynomial. Having established a theorem which guarantees that the inverse of L is a finite linear polynomial, then obviously we can transform the infinite system
Generally, the modelling problem is one of the most important in the study of systems theory. Models can be classified as either implicit or explicit. Implicit models are those in which the system response is expressed as an implicit operation on the system input. Explicit models are those in which the response is expressed as an explicit operation on the system output. The question of which model is best depends on the particular questions we wish to ask and how well we understand the systems operation. VolterraAViener representations of nonlinear systems are explicit models. They are based on the Volterra series functional representation which derives from mathematics. Quite often, especially for computational reasons, we study discrete Volterra systems. Most of the results of continuous-time systems can also be developed for discrete-time systems. They have been described in Rugh (1981) and Schetzen (1980). A serious problem arises when we want to transform a Volterra system, described through infinite Volterra series, to a finite input/output representation using finite operators. A part of this problem has been studied in Fliess (1987a,b, 1989), Frazho (1982, 19X0), Rugh (1981), Pearlman (1980) Schetzen (1980) and Sontag (1970, 1977), in parallel with the realization problem, where bilinear input/output systems had the same dynamic behaviour as those with infinite Volterra systems. In the present paper, we deal with discrete infinite Voltetra systems of the form m
theory;
to establish
=f
1. Introduction
m
y(t) = 1 1 . . . ,~oQ,i,...inU(t - il) ’ ’ ’ U(t - in), n=Oi, =o t =o. I,“...
realization
(1)
Received 31 January 1995; revised 18 December 1995; revised 24 June 1996; received in final form 9 September 1996. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor Tamer Basar. Corresponding author Dr Stelios Kotsios. Tel. +30 1 766 5803; Fax +30 1 722 8981; E-mail [email protected]. + Department of Electronic and Electrical Engineering, University College Dublin, Belfield, Dublin 4, Ireland. Current address: Department of Informatics, Division of Communications and Signal Processing, Faculty of Science, Panepistimiopolis, TYPA Buildings, 157 71 Athens, Greece. 251
258
Brief Papers
y(r) = Cu(t) to the finite L-‘y(r) = Bu(r) input/output form. At this point we wish to illustrate that the inverse transform from finite to infinite representation can be easily performed (Kotsios and Kalouptsidis, 1993a). In order for our factorization to succeed we adopt the LFalgorithm of &series. This is an extension of the corresponding algorithm used for the factorization of &polynomials (Kotsios and Kalouptsidis, 1993a), and is based on an Euclidean-type division by means of the star-product. Let us recall that the set (A, *. +) is not a ring and thus we must approach our problems by using new notions. In this paper we study &polynomials which can be factorized in the product L * B, where L is a linear polynomial. The existence of the linear factor L plays a crucial role in the develompent of our theory but finally restricts the set of the systems that can be faced by our method. For more general systems we require a new factorization, capable of including and handling more complicated products.
(,it,.._,, il:fz ,...., i2;..., f,,,,...., iY)andwesetjei,=j+iL.The kl -tiIIIeS
k&neS
k,,,-times
oroof of this formula can be found in Kotsios and Kalountsidis il993a) and arises from a simple substitution of B into-d. Since star-product represents composition of maps, the following properties hold: (i) [A + B] * C = A * C + B * C; (ii) A * (B * C) = (A * B) * C, (iii) A * B # B * A, unless A and B are linear; and (iv) d(A * B) = d(A) + d(B). Remark 1. We can easily check that the distributive property A * [B + C] = A * B + A * C is not valid, except in the case when A is linear. This particular feature endows the set of 6-
operators with some special properties. The method outlined here, therefore, differs from the classical approaches of Ritt (Cohn, 1965), z-transform (Rugh, 1981) and formal power series (Isidori, 1995). A more detailed discussion can be found in Kotsios (1997a).
Let A
2. The Soperaror
Example 2. and B = 63
Let us recall the notions of &series and B-polynomials, which first appeared in Kotsios and Kalouptsidis (1993a). These algebraic tools are proper for the description of polynomial discrete systems.They are just another language closely related to the notions of z-transform (Rugh, 1981; Schetzen, 1980) and formal power series (Sontag, 1970, 1977; Fliess, 1987a,b, 1989) but suitable for computational applications. Let y(r) be a real sequence defined over the set of integers 2 and let F be the set of causal sequences. Thus, if y(r) E F, then y(r) = 0 for r < 0 Let i be an integer. We define the 6i operator as the i-shift 6; : F * F, ai{y(r)} = {y(t - i)}. A multindex im of order m is an element (il. iz, , i,,,) of the set Zm. The operator 6r,,, - : F +. F is defined as 6i,{y(r)} = 6i, bi*6ij.. 6j, {y(r)] = y(r-il)y(r--iz). y(r-i,).
of second order, then d(A)
* B+*..
+63
=
1
2
+ * * *
= 1, d(B) = 2 and
+6162*B+6263*B+...
+(65+66+67+‘.‘+6465+b566+“.) +(&
+ 67 + 68 + ’ ”
+ 6566
+ 6667
+ ”
+(a,
•‘- 65 + 66 + ’ ’ ’ + 6364
+ 6465
+ * * *)
x(65
+ 66 + 67 + ”
’ + 6465
+ 6566
+ ”
‘)
+(65
+ 66 + 61 + . ’ ’ + 6465
+ 6566
+ ‘.
‘1
‘) + ’ ”
x(66+67+68+“‘+6566+d6b7i”‘)+“’ = 64 + 265 +6364
+ 366
+ 36465
+a46465
+ 367
•‘- 6466
+ 646566
+656&i +
S
6263
two &series
+ - - * be
=(64+65+66+“‘+b364+6465+“‘)
the multindex i= = ( 1, I, 2,2,2) we have = .56:6iy(r) = 530 - l)y3(r - 2) and for the (I, 1.2,2,3. 3,3) we have 6(1,1,z2,3,3,3)y(r) = - l)y2(r - 2)y3(r - 3).
Given hi,,, and 6,s,,we define their sum as (bi,+6,c,) {y(r)} 6i,,, An expression of the form - (y(l) ] +a, {y(r)}. m
+ 6364
A*B=61*B+&*B
Note that if iI = iz = i3 = . . . = im = i we have hi,,, = 67. By convention we set &Iy(t)} = { 1) for each r E Z. Example I. For 561b162626zy(r) multindex im = 6:&5&W = J(r
= 61 + 62 + 63 + * * . + 6162 +
+ 64 + 65 + ’ * * + 6263
” ’
+ . ’ ’ + 36566
+ ’ ’ .
+ 656465 f
63646465
+ 63648566
+ ’ ’ ‘.
=
aidi.
We say that the &series A is right-invertible if we can find a &series B such that A * B = 60, A is left-invertible if we can find a series B such that B * A = 60. The right and left inverses do not coincide (Kotsios and Kalouptsidis, 1993a).
n=O iE1”
where I, = Z”, Is = {e} is called a b-series. These series are along the traditional path of Volterra series and formal power series (Isidori, 1995). The quantity ,‘&r, ai6i is called the linear part of the series, Obviously for I, and n finite we get a bpolynomial. For each &series A = C ai& f 0 we define d(A) as follows: d(A) = min{min(it, iz,, _. , i,,), i = (il, iz,. , i,) E Z” such that ai # 0, for n = I, 2,. , k}. Let A and B be g-series. Their star-product is the composition operator A * B = A 0 B, where 0 is the usual compositibn map among operators. Since 6, * d,{y(r)} = 6, * {I},= 11) = b,{y(r)} we conclude that b, * 6, = 6,. Let A = 6;;, B = 6;; be two operators. Then, their star-product is given by the formula 6:; * 6{: = 6{:$,. In the more general case where A = ~~=o &K, arc&, B = E.,“=o Eiern bt& are two b-series their star-product is given by the formula
Theorem 1. (Kotsios and Kalouptsidis, 1993a) If A is a series with nonzero linear part, and with constant term zero, then A is right(left)-invertible.
The following theorem plays an important role in the construction of the finite input-output representation. Theorem 2. Let A = I:, ri6i be a linear b-series with rg # 0. The inverse of this series is a linear polynomial if and only if there is a constant number m such that
5 m=l
1
1
a&j,
b,
=
0,
Vk>m,
where
'
A*B=a&,+
-‘e
zkc
=
ro rl
0 ro
::
Li
/ rm
rm-I
00.. 00..
ro
0
rl
r0
. . . bj,,,‘Jjek,
kEKmjE(UJnP
where a is the constant term of the series A, j = (jl, j2, , j,) E (U,I,Y = U,l,XUnInX~ . . X U,I, m-times, and the pointwise sum j 8 ira is defined as follows: let j = (jt, j2,. , j,) E Ik, x Ikl x x Ik,,, c z k, +kz+...k, , we extend im to the vector i; =
e=[l,O,O,O
,... O]f,,,
rl Zk = Irk.
rk-
ro I# I,
, rk-m
Prooj: The inverse of A is a linear series X = xEoxiSi that A * X = 60 or equivalently
I.
such
Brief Papers ro 0 rl ro r2 r3
rl r2
0 0
0 0
r0 rl
0
259
6)
(3)
m
ro
A = ~aoid&i
/
+
i=O
Since we want X to be a linear polynomial, there must be a number m such that Xj = 0 for j > m. Defining Y = [x0, XI. x2,. , x,,,lT, (3) becomes c r,+l
rm r,+l
pm+2
rl r2
P 0 0
0 0 ro
ro rl
[.
:II
UOij6Ob;dj
+ ’ ’ .
+ (ii.iz.....
2 i,)=(O.O.....O)
aOi,i*...in606i,
6i*
∈
(ii)
1 0 0 0
=
f, (i.j)=(O.O)
(iii)
[1
or GE = e,
[ rm+i rm
where some of 1, are finite.
rm rl [If0 [ rm+l r,-j
rl
ro]
r2
rm+2
[
g
I
ro]
= 0,
=0
0 = 0 for each k > m and finally
and thus CZ = e , Zk% ZkC-‘e=Ofork>m. Conversely, (3) may be rewritten as follows: Cf = e,
zk?
+ rk-m-IX,,,+]
f
rk-m-ZXm+2
f
. . ’ + rOXk
= 0,
k=m+l,m+2,...
or Zkc-‘e + rk-m-tX,,,+t f rk-,-2X,,,+2i . . . + rOXk = 0, k = m + 1, m + 2. Since zkC-‘e = 0, we easily conclude that .vk = 0, k = m + 1, m + 2, ... and thus X is a polynomial. n
Example 3. Let L = (0.5)60 + 6263 then the left inverse of L is easily shown to be L’ = 260 - 86263 + 128&&5s + . . . while the right inverse of L is L” = 260 - 46263 + 8646:6,5 + . . . ForthelinearseriesL=60+261+362+..~+(i+l)&+... the inverse is the linear polynomial L-’ = 60 - 261 + 62.
Proof We prove the result by contradiction. Suppose that there is a linear series L = ET=, bi.di, , bij f 0 , 6i, = 60 , such that A = L * D for some non 1mear polynomial D. Then A = biodio * D + bi,6i,
In this section we present and study the notion of the linearly factorizable (LF) series. This notation is essential in the solution of our problem because it permits the analysis of certain series to simpler factors through a proper factorization. Definition 1. Let A be a &polynomial, with d(A) z 0 We say A is linearly factorizable (LF) if there is a nontrivial linear b-polynomial L, L # cbo and a b-polynomial D such that
n
= L * D = L’ * D’ such that D and = CL, D’ = C-ID for some constant
number c. There exist a linear series M such that L = M * L’, hencewehaveM*L’*D= L’*D’ = L’*(M*D-D’) = 0 and thus M * D - D’ = 0. If M is an infinite linear series then D’ must be infinite - a contradiction. If M is a linear polynomial D’ must be LF - a contradiction too. So the only remaining case is M = C&J, for some constant number c, from which we take the required relations. Proof
A=L*D.
n
Dejnifion 2. Let A be a b-series, which does not contain linear terms with d(A) 2 0. We say A is linearly factorizable (LF) if there is a nontrivial linear &series L and a b-polynomial D which is not LF such that A = L * D. E.wmple 4. The &polynomial A = 6 162 + 6263 is LF since it can be written A = (61 + 62) * 6061. The &polynomial A = 606i62 +dad: is not LF. The &seriesA = badi +6182+6263+ is LF, since A = (60 + 61 + 6 + . .) * 6061, but the A = ~06]+6062+‘~‘+~Obi+“‘+606]+6~62+”’+6~bj+’~’ 3
is not.
Next we develop an algorithm structurally similar to the Euclidean division algorithm, first introduced in Kotsios and Kalouptsidis (1993a), which will be needed later. in the construction of linear factorizations. Let us recall some standard definitions. 3. Let S be an operator of the form S = oiii,,,,i, 6j, 6i, ’ ’ * 6i,. The number n is called the degree of S and is denoted by deg(S) = n. Definition
Consider two ordered operators S, S’ of the form # 0, i] 5 i2 2 i3 5 5 in and S' ii 5 ii 5 5 i; such that deg(S) = deg(S’) = n and ii ‘1 d(S) < d(S’) = ii. We say that S’ is reducible with respect to S if there is an integer h >_ 0 suchthat (il+h,iZ+h ,..., i,+h)=(i;,i; ,.._, i;), Definition
The following proposition B-series not LF: Proposirion
* D + .*.
(i) Since A consists of terms containing zero delays we conclude that D contains such ‘zero-delay’ terms as well. Therefore, the operation bi,6i, * D will produce, for instance, terms of the form bi,aoidi,6i+i, , ii # 0 which are not present in A, hence bi, = 0 or aor = 0 which is a contradiction. (ii) From the form of A we conclude that it contains an infinite number of terms in an increasing order, i.e. quadratic, cubic and so on. Since D is a polynomial it contains only a finite number of terms. Let diiir,,,ik6ii . . . 6c be the maximum degree term of D, then all the terms of A with order higher than k cannot be produced from the product L * D, which is a contradiction. (iii) Suppose that I,,, is finite, for some specific m, and i,,,) E I,. Then, D ai,i*...i,~il . * . 6r,,, E A, i = (ii. iz,.. must contain a term of order m, and the product L *D will create an infinite number of these terms, which leads to a contradiction. Proposition 2. Suppose A D’ are not LF. Then, L’
3. The LF-series
* D + . ’ . + bi,,di,,
provides us with some classes of
I. The following series are not LF:
4.
S = al,Iz. ,.
6i, bi, ' ' 6jn, Clj,j*,,,jn = a.7 .I .I 6! 6' . . . 6': , !,!>..h I, 12
Brief Papers If S’ is reducible with respect to S, then we can write a,, I = z*, ai1i2.J~
s’
* s,
Definition 5. S’ is irreducible with respect to S if it is not reducible. 6. The maximum term of a nonlinear polynomial D is the maximum degree term of D. If more than one term with the same maximum degree exists, we select that term which contains the maximum number of maximum delays. If more than one such term exists we choose the term which contains the maximum number of the maximum of the remaining delays and so forth. Defiition
For instance, if D = 606s + 6e606i + 6&&&, then b&6e6e is the maximum term. If D = 26060 + 3606061 + 26t616z + 56&6t, then the maximum term is 626261. If D = 6060 + 6e6c6i + 6&& + 615766 then the maximum term is d&69. Finally the maximum term of D = 6060 +
638362
+ 636361
is 636362
Definition 7. A series R is called irreducible with respect to the polynomial D iff either R contains no terms of degree equal to the maximum degree of D, or each term of R with degree equal to the degree of the maximum term S of D is irreducible with respect to S.
Theorem 3. Let A be a series and D be a nonlinear polynomial containing only nonlinear terms with D # 0. There is a linear series L and a nonlinear series R, where either R = 0 or R is irreducible with respect to D such that A=L*D+R
(4)
Moreover, series L and R are uniquely determined. Let S = diiir..,i,dii 6i, 6i, be the maximum term of (i) Suppose that A does not contain terms of degree equal to the degree of S. Then, A = (060) * D + A. Thus, L = 060, R = A are solutions of the problem. (ii) Suppose next that each term of A with degree equal to deg(S) is irreducible to S. Then, L = 06,, R = A are again solutions. The only remaining case is A contain at least one term S’ = a~I,rpl” d J 5~I, 6 ‘2 J . . . 6g which is reducible with respect to S; then Proof:
D.
-aA, ”
=
* s
dili2...i,
for some integer ht L 0. We define the series
If RI is irreducible with respect to D the procedure terminates. Otherwise the above procedure is repeated by setting b,,
0.
dili2...i.
I
b,, + SdA2
I
3.1, The LF-algorithm.
Step I: Factorize A = ~6, $ d , a = d(A) If d contains only zero terms then stop else go to step 2. Step 2: Let DO be the polynomial consisting of all zero terms of d. Applying the procedure developed in Theorem (3) we find a linear series LJJand a nonlinear series RI such that d = b * DO + RI obviously d(4) = 0. Step 3: Go to step I, setting RI instead of A. 2. Working in a similar way to that of Kotsios and Kalouptsidis (1993a) we can easily derive that O=d(De)
Pergamon
Printed
Pn: SOOOS-1098(96)00178-l
Vol. 33, No. 2. PP. 257-262, 1997 @ 1997 Elsevier Science Ltd m Great Britain. All rights reserved CQJS-109w97 f17.00+0.00
Brief Paper
Finite Input/Output
Representative of a Class of Volterra Polynomial Systems S. KOTSIOS *
Key Word.. Algebraic Volterra series.
approaches;
factorization
methods;
Abstract-In this paper the problem of finite input/output representation of a special class of nonlinear Volterra polynomial systems is studied via the notion of linear factorization of b-series. This is an algebraic method based mainly on the star-product operation and on a related Euclidean-type algorithm. 01997 Elsevier Science Ltd. All rights reserved.
in order y(t)
+
Ul,‘(t
transformations;
an input/output
discrete
representation
of the form
- 1) + ’ ’ . + UkJ’(t -k) c
biU(t - il)U(t
~ i2)
U(t - in), (2)
n=Oi=(i~.i*.....l.)EI.
where I, is tinite, for each n = 0,. , m. The advantages of the expression (2) are obvious. Firstly, stability can be easily checked, due to the existence of a certain number of useful theorems. Secondly, this finite expression can be easily transformed to a linear regression form y(t) = BT+(t), e = [-al,-a2...., -ak. bi,. , bijn.. ,biliz...in17 4 = [~‘(l l),y(t-2) ,.... y(t-k).u(t-I) ,__., Ia-IM-j) ,..., u(ti1)u(t - i2) ’ . . ~(t - in)], suitable for identification, which can be carried out by solving a linear system of equations, via one of the popular algorithms such as the RLS, LMS (Kalouptsidis and Theodoridhs, 1993). The identification of these systems has been studied earlier in Netravali and de Figueiredo (1971). A wide variety of tools have been used for the description of discrete systems of the form (I) z-transforms (Alper, 1965; Rugh, 1981), difference polynomials (Cohn, 1965), formal power series (Fliess, 1987a), to mention but a few. Each of these alternative representations has its advantages, depending on the problem to be solved. In order to describe and solve our problem we use the notion of a &operator. The set A of &operators, as well as their star-product, are algebraic tools introduced in Kotsios and Kalouptsidis (1993a) to cope with the model matching problem for a class of nonlinear discrete systems. Initially, the &operators had a computational orientation but gradually it became clear that they also had some particular properties. The distributive property, for instance, does not apply in our case and therefore the set (A, *. +) is not a ring, in contradiction to the rings of difference or differential polynomials (Cohn, 1965; Ritt, 1950) and the rings of formal power series (Isidori, 1995). The distributive property is a strong linear property and consequently all the methodologies which make use of it may face nonlinear problems which have ‘slight linear behaviour’ within their structure. The h-operator and the star-product seem to be capable of describing ‘deeper’ nonlinearities. The focus of our investigation here is to explore how this is possible and under what conditions. Some efforts have been made in this direction and despite the complexity of the matter some preliminary results already exist. Details can be found in Kotsios and Kalouptsidis (1993a), Kotsios and Kalouptsidis (1993b), Kotsios and Kalouptsidis (1997) Kotsios (1997a) and Kotsios and Lappas (1997). In this paper we wish to find &polynomials A(b). B(b); A(6) linear, so that the system A(d)y(t) = E(d)u(t) will yield the same output with (1) under identical inputs. Our main objective is to devise a factorization of the original system C, of the form C = L * B, where L is a linear series and B is a finite nonlinear polynomial. Having established a theorem which guarantees that the inverse of L is a finite linear polynomial, then obviously we can transform the infinite system
Generally, the modelling problem is one of the most important in the study of systems theory. Models can be classified as either implicit or explicit. Implicit models are those in which the system response is expressed as an implicit operation on the system input. Explicit models are those in which the response is expressed as an explicit operation on the system output. The question of which model is best depends on the particular questions we wish to ask and how well we understand the systems operation. VolterraAViener representations of nonlinear systems are explicit models. They are based on the Volterra series functional representation which derives from mathematics. Quite often, especially for computational reasons, we study discrete Volterra systems. Most of the results of continuous-time systems can also be developed for discrete-time systems. They have been described in Rugh (1981) and Schetzen (1980). A serious problem arises when we want to transform a Volterra system, described through infinite Volterra series, to a finite input/output representation using finite operators. A part of this problem has been studied in Fliess (1987a,b, 1989), Frazho (1982, 19X0), Rugh (1981), Pearlman (1980) Schetzen (1980) and Sontag (1970, 1977), in parallel with the realization problem, where bilinear input/output systems had the same dynamic behaviour as those with infinite Volterra systems. In the present paper, we deal with discrete infinite Voltetra systems of the form m
theory;
to establish
=f
1. Introduction
m
y(t) = 1 1 . . . ,~oQ,i,...inU(t - il) ’ ’ ’ U(t - in), n=Oi, =o t =o. I,“...
realization
(1)
Received 31 January 1995; revised 18 December 1995; revised 24 June 1996; received in final form 9 September 1996. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Henk Nijmeijer under the direction of Editor Tamer Basar. Corresponding author Dr Stelios Kotsios. Tel. +30 1 766 5803; Fax +30 1 722 8981; E-mail [email protected]. + Department of Electronic and Electrical Engineering, University College Dublin, Belfield, Dublin 4, Ireland. Current address: Department of Informatics, Division of Communications and Signal Processing, Faculty of Science, Panepistimiopolis, TYPA Buildings, 157 71 Athens, Greece. 251
258
Brief Papers
y(r) = Cu(t) to the finite L-‘y(r) = Bu(r) input/output form. At this point we wish to illustrate that the inverse transform from finite to infinite representation can be easily performed (Kotsios and Kalouptsidis, 1993a). In order for our factorization to succeed we adopt the LFalgorithm of &series. This is an extension of the corresponding algorithm used for the factorization of &polynomials (Kotsios and Kalouptsidis, 1993a), and is based on an Euclidean-type division by means of the star-product. Let us recall that the set (A, *. +) is not a ring and thus we must approach our problems by using new notions. In this paper we study &polynomials which can be factorized in the product L * B, where L is a linear polynomial. The existence of the linear factor L plays a crucial role in the develompent of our theory but finally restricts the set of the systems that can be faced by our method. For more general systems we require a new factorization, capable of including and handling more complicated products.
(,it,.._,, il:fz ,...., i2;..., f,,,,...., iY)andwesetjei,=j+iL.The kl -tiIIIeS
k&neS
k,,,-times
oroof of this formula can be found in Kotsios and Kalountsidis il993a) and arises from a simple substitution of B into-d. Since star-product represents composition of maps, the following properties hold: (i) [A + B] * C = A * C + B * C; (ii) A * (B * C) = (A * B) * C, (iii) A * B # B * A, unless A and B are linear; and (iv) d(A * B) = d(A) + d(B). Remark 1. We can easily check that the distributive property A * [B + C] = A * B + A * C is not valid, except in the case when A is linear. This particular feature endows the set of 6-
operators with some special properties. The method outlined here, therefore, differs from the classical approaches of Ritt (Cohn, 1965), z-transform (Rugh, 1981) and formal power series (Isidori, 1995). A more detailed discussion can be found in Kotsios (1997a).
Let A
2. The Soperaror
Example 2. and B = 63
Let us recall the notions of &series and B-polynomials, which first appeared in Kotsios and Kalouptsidis (1993a). These algebraic tools are proper for the description of polynomial discrete systems.They are just another language closely related to the notions of z-transform (Rugh, 1981; Schetzen, 1980) and formal power series (Sontag, 1970, 1977; Fliess, 1987a,b, 1989) but suitable for computational applications. Let y(r) be a real sequence defined over the set of integers 2 and let F be the set of causal sequences. Thus, if y(r) E F, then y(r) = 0 for r < 0 Let i be an integer. We define the 6i operator as the i-shift 6; : F * F, ai{y(r)} = {y(t - i)}. A multindex im of order m is an element (il. iz, , i,,,) of the set Zm. The operator 6r,,, - : F +. F is defined as 6i,{y(r)} = 6i, bi*6ij.. 6j, {y(r)] = y(r-il)y(r--iz). y(r-i,).
of second order, then d(A)
* B+*..
+63
=
1
2
+ * * *
= 1, d(B) = 2 and
+6162*B+6263*B+...
+(65+66+67+‘.‘+6465+b566+“.) +(&
+ 67 + 68 + ’ ”
+ 6566
+ 6667
+ ”
+(a,
•‘- 65 + 66 + ’ ’ ’ + 6364
+ 6465
+ * * *)
x(65
+ 66 + 67 + ”
’ + 6465
+ 6566
+ ”
‘)
+(65
+ 66 + 61 + . ’ ’ + 6465
+ 6566
+ ‘.
‘1
‘) + ’ ”
x(66+67+68+“‘+6566+d6b7i”‘)+“’ = 64 + 265 +6364
+ 366
+ 36465
+a46465
+ 367
•‘- 6466
+ 646566
+656&i +
S
6263
two &series
+ - - * be
=(64+65+66+“‘+b364+6465+“‘)
the multindex i= = ( 1, I, 2,2,2) we have = .56:6iy(r) = 530 - l)y3(r - 2) and for the (I, 1.2,2,3. 3,3) we have 6(1,1,z2,3,3,3)y(r) = - l)y2(r - 2)y3(r - 3).
Given hi,,, and 6,s,,we define their sum as (bi,+6,c,) {y(r)} 6i,,, An expression of the form - (y(l) ] +a, {y(r)}. m
+ 6364
A*B=61*B+&*B
Note that if iI = iz = i3 = . . . = im = i we have hi,,, = 67. By convention we set &Iy(t)} = { 1) for each r E Z. Example I. For 561b162626zy(r) multindex im = 6:&5&W = J(r
= 61 + 62 + 63 + * * . + 6162 +
+ 64 + 65 + ’ * * + 6263
” ’
+ . ’ ’ + 36566
+ ’ ’ .
+ 656465 f
63646465
+ 63648566
+ ’ ’ ‘.
=
aidi.
We say that the &series A is right-invertible if we can find a &series B such that A * B = 60, A is left-invertible if we can find a series B such that B * A = 60. The right and left inverses do not coincide (Kotsios and Kalouptsidis, 1993a).
n=O iE1”
where I, = Z”, Is = {e} is called a b-series. These series are along the traditional path of Volterra series and formal power series (Isidori, 1995). The quantity ,‘&r, ai6i is called the linear part of the series, Obviously for I, and n finite we get a bpolynomial. For each &series A = C ai& f 0 we define d(A) as follows: d(A) = min{min(it, iz,, _. , i,,), i = (il, iz,. , i,) E Z” such that ai # 0, for n = I, 2,. , k}. Let A and B be g-series. Their star-product is the composition operator A * B = A 0 B, where 0 is the usual compositibn map among operators. Since 6, * d,{y(r)} = 6, * {I},= 11) = b,{y(r)} we conclude that b, * 6, = 6,. Let A = 6;;, B = 6;; be two operators. Then, their star-product is given by the formula 6:; * 6{: = 6{:$,. In the more general case where A = ~~=o &K, arc&, B = E.,“=o Eiern bt& are two b-series their star-product is given by the formula
Theorem 1. (Kotsios and Kalouptsidis, 1993a) If A is a series with nonzero linear part, and with constant term zero, then A is right(left)-invertible.
The following theorem plays an important role in the construction of the finite input-output representation. Theorem 2. Let A = I:, ri6i be a linear b-series with rg # 0. The inverse of this series is a linear polynomial if and only if there is a constant number m such that
5 m=l
1
1
a&j,
b,
=
0,
Vk>m,
where
'
A*B=a&,+
-‘e
zkc
=
ro rl
0 ro
::
Li
/ rm
rm-I
00.. 00..
ro
0
rl
r0
. . . bj,,,‘Jjek,
kEKmjE(UJnP
where a is the constant term of the series A, j = (jl, j2, , j,) E (U,I,Y = U,l,XUnInX~ . . X U,I, m-times, and the pointwise sum j 8 ira is defined as follows: let j = (jt, j2,. , j,) E Ik, x Ikl x x Ik,,, c z k, +kz+...k, , we extend im to the vector i; =
e=[l,O,O,O
,... O]f,,,
rl Zk = Irk.
rk-
ro I# I,
, rk-m
Prooj: The inverse of A is a linear series X = xEoxiSi that A * X = 60 or equivalently
I.
such
Brief Papers ro 0 rl ro r2 r3
rl r2
0 0
0 0
r0 rl
0
259
6)
(3)
m
ro
A = ~aoid&i
/
+
i=O
Since we want X to be a linear polynomial, there must be a number m such that Xj = 0 for j > m. Defining Y = [x0, XI. x2,. , x,,,lT, (3) becomes c r,+l
rm r,+l
pm+2
rl r2
P 0 0
0 0 ro
ro rl
[.
:II
UOij6Ob;dj
+ ’ ’ .
+ (ii.iz.....
2 i,)=(O.O.....O)
aOi,i*...in606i,
6i*
∈
(ii)
1 0 0 0
=
f, (i.j)=(O.O)
(iii)
[1
or GE = e,
[ rm+i rm
where some of 1, are finite.
rm rl [If0 [ rm+l r,-j
rl
ro]
r2
rm+2
[
g
I
ro]
= 0,
=0
0 = 0 for each k > m and finally
and thus CZ = e , Zk% ZkC-‘e=Ofork>m. Conversely, (3) may be rewritten as follows: Cf = e,
zk?
+ rk-m-IX,,,+]
f
rk-m-ZXm+2
f
. . ’ + rOXk
= 0,
k=m+l,m+2,...
or Zkc-‘e + rk-m-tX,,,+t f rk-,-2X,,,+2i . . . + rOXk = 0, k = m + 1, m + 2. Since zkC-‘e = 0, we easily conclude that .vk = 0, k = m + 1, m + 2, ... and thus X is a polynomial. n
Example 3. Let L = (0.5)60 + 6263 then the left inverse of L is easily shown to be L’ = 260 - 86263 + 128&&5s + . . . while the right inverse of L is L” = 260 - 46263 + 8646:6,5 + . . . ForthelinearseriesL=60+261+362+..~+(i+l)&+... the inverse is the linear polynomial L-’ = 60 - 261 + 62.
Proof We prove the result by contradiction. Suppose that there is a linear series L = ET=, bi.di, , bij f 0 , 6i, = 60 , such that A = L * D for some non 1mear polynomial D. Then A = biodio * D + bi,6i,
In this section we present and study the notion of the linearly factorizable (LF) series. This notation is essential in the solution of our problem because it permits the analysis of certain series to simpler factors through a proper factorization. Definition 1. Let A be a &polynomial, with d(A) z 0 We say A is linearly factorizable (LF) if there is a nontrivial linear b-polynomial L, L # cbo and a b-polynomial D such that
n
= L * D = L’ * D’ such that D and = CL, D’ = C-ID for some constant
number c. There exist a linear series M such that L = M * L’, hencewehaveM*L’*D= L’*D’ = L’*(M*D-D’) = 0 and thus M * D - D’ = 0. If M is an infinite linear series then D’ must be infinite - a contradiction. If M is a linear polynomial D’ must be LF - a contradiction too. So the only remaining case is M = C&J, for some constant number c, from which we take the required relations. Proof
A=L*D.
n
Dejnifion 2. Let A be a b-series, which does not contain linear terms with d(A) 2 0. We say A is linearly factorizable (LF) if there is a nontrivial linear &series L and a b-polynomial D which is not LF such that A = L * D. E.wmple 4. The &polynomial A = 6 162 + 6263 is LF since it can be written A = (61 + 62) * 6061. The &polynomial A = 606i62 +dad: is not LF. The &seriesA = badi +6182+6263+ is LF, since A = (60 + 61 + 6 + . .) * 6061, but the A = ~06]+6062+‘~‘+~Obi+“‘+606]+6~62+”’+6~bj+’~’ 3
is not.
Next we develop an algorithm structurally similar to the Euclidean division algorithm, first introduced in Kotsios and Kalouptsidis (1993a), which will be needed later. in the construction of linear factorizations. Let us recall some standard definitions. 3. Let S be an operator of the form S = oiii,,,,i, 6j, 6i, ’ ’ * 6i,. The number n is called the degree of S and is denoted by deg(S) = n. Definition
Consider two ordered operators S, S’ of the form # 0, i] 5 i2 2 i3 5 5 in and S' ii 5 ii 5 5 i; such that deg(S) = deg(S’) = n and ii ‘1 d(S) < d(S’) = ii. We say that S’ is reducible with respect to S if there is an integer h >_ 0 suchthat (il+h,iZ+h ,..., i,+h)=(i;,i; ,.._, i;), Definition
The following proposition B-series not LF: Proposirion
* D + .*.
(i) Since A consists of terms containing zero delays we conclude that D contains such ‘zero-delay’ terms as well. Therefore, the operation bi,6i, * D will produce, for instance, terms of the form bi,aoidi,6i+i, , ii # 0 which are not present in A, hence bi, = 0 or aor = 0 which is a contradiction. (ii) From the form of A we conclude that it contains an infinite number of terms in an increasing order, i.e. quadratic, cubic and so on. Since D is a polynomial it contains only a finite number of terms. Let diiir,,,ik6ii . . . 6c be the maximum degree term of D, then all the terms of A with order higher than k cannot be produced from the product L * D, which is a contradiction. (iii) Suppose that I,,, is finite, for some specific m, and i,,,) E I,. Then, D ai,i*...i,~il . * . 6r,,, E A, i = (ii. iz,.. must contain a term of order m, and the product L *D will create an infinite number of these terms, which leads to a contradiction. Proposition 2. Suppose A D’ are not LF. Then, L’
3. The LF-series
* D + . ’ . + bi,,di,,
provides us with some classes of
I. The following series are not LF:
4.
S = al,Iz. ,.
6i, bi, ' ' 6jn, Clj,j*,,,jn = a.7 .I .I 6! 6' . . . 6': , !,!>..h I, 12
Brief Papers If S’ is reducible with respect to S, then we can write a,, I = z*, ai1i2.J~
s’
* s,
Definition 5. S’ is irreducible with respect to S if it is not reducible. 6. The maximum term of a nonlinear polynomial D is the maximum degree term of D. If more than one term with the same maximum degree exists, we select that term which contains the maximum number of maximum delays. If more than one such term exists we choose the term which contains the maximum number of the maximum of the remaining delays and so forth. Defiition
For instance, if D = 606s + 6e606i + 6&&&, then b&6e6e is the maximum term. If D = 26060 + 3606061 + 26t616z + 56&6t, then the maximum term is 626261. If D = 6060 + 6e6c6i + 6&& + 615766 then the maximum term is d&69. Finally the maximum term of D = 6060 +
638362
+ 636361
is 636362
Definition 7. A series R is called irreducible with respect to the polynomial D iff either R contains no terms of degree equal to the maximum degree of D, or each term of R with degree equal to the degree of the maximum term S of D is irreducible with respect to S.
Theorem 3. Let A be a series and D be a nonlinear polynomial containing only nonlinear terms with D # 0. There is a linear series L and a nonlinear series R, where either R = 0 or R is irreducible with respect to D such that A=L*D+R
(4)
Moreover, series L and R are uniquely determined. Let S = diiir..,i,dii 6i, 6i, be the maximum term of (i) Suppose that A does not contain terms of degree equal to the degree of S. Then, A = (060) * D + A. Thus, L = 060, R = A are solutions of the problem. (ii) Suppose next that each term of A with degree equal to deg(S) is irreducible to S. Then, L = 06,, R = A are again solutions. The only remaining case is A contain at least one term S’ = a~I,rpl” d J 5~I, 6 ‘2 J . . . 6g which is reducible with respect to S; then Proof:
D.
-aA, ”
=
* s
dili2...i,
for some integer ht L 0. We define the series
If RI is irreducible with respect to D the procedure terminates. Otherwise the above procedure is repeated by setting b,,
0.
dili2...i.
I
b,, + SdA2
I
3.1, The LF-algorithm.
Step I: Factorize A = ~6, $ d , a = d(A) If d contains only zero terms then stop else go to step 2. Step 2: Let DO be the polynomial consisting of all zero terms of d. Applying the procedure developed in Theorem (3) we find a linear series LJJand a nonlinear series RI such that d = b * DO + RI obviously d(4) = 0. Step 3: Go to step I, setting RI instead of A. 2. Working in a similar way to that of Kotsios and Kalouptsidis (1993a) we can easily derive that O=d(De)
