2002 Conference on Information Sciences and Systems, Princeton University, March 20–22, 2002
Fliess Operators in Cascade and Feedback Systems W. Steven Gray
Yaqin Li
Department of Electrical and Computer Engineering Old Dominion University Norfolk, Virginia 23529-0246 e-mail:
[email protected]
Department of Electrical and Computer Engineering Old Dominion University Norfolk, Virginia 23529-0246 e-mail:
[email protected]
Abstract — Given two nonlinear input-output systems represented as Fliess operators, this paper considers properties of the cascade and feedback interconnected systems. The cascade system is characterized via the composition product for formal power series. In the multivariable setting this product is shown to be generally well defined and continuous in its first argument. Then a condition is introduced under which the composition product preserves rationality and local convergence. In preparation for the feedback analysis, it is also shown that the composition product produces a contractive mapping on the set of all formal power series using the familiar ultrametric. Next, the feedback connection is considered in the special case where inputs are supplied from an exosystem which is itself a Fliess operator. In particular, a sufficient condition is given under which a unique solution to the feedback equation is known to exist. Then the closed-loop system is characterized as the output of new Fliess operator when a certain series factorization property is available. This leads to an implicit characterization of the feedback product for formal power series.
I. Introduction Let U ' tfc c c 6 denote an alphabet and U W the set of all words over U . A formal power series is any mapping of the form U W p< b , and the set of all such mappings will be denoted by b J U q. For each S M b J U q, one can formally associate a corresponding 6-input, -output operator 8S in the following manner. Let R D and @ K be given. For a measurable function G d@c Ko < b6 , define 88R ' 4@ t8 8R G $ $ 6, where 8 8R is the usual uR -norm for a measurable real-valued function, , defined on d@c Ko. Let u6 R d@c Ko denote the set of all measurable functions defined on d@c Ko having a finite 8 u 8R norm. With |f c A M b fixed and A : f, define inductively for each # M U W the mapping .# G u6 d|f c |f n A o < d|f c |f n A o by . ' , and
.n n4 uuu4 doE| '
|
|3
# MU
u
Fd
y
Fc
Fig. 1: Two Fliess operators connected in cascade.
In many applications input-output systems are interconnected in a variety of ways. Given two Fliess operators, 8S and 8_ with Sc _ M b6 J U q, Figure 1 shows a cascade connection where + ' 8S d8_ doo. It was shown in [3] for the SISO case (i.e., ' 6 ' ) that there always exists a series S _ such that + ' 8S _ do, but a general analysis of this composition product is apparently not available in the literature. So in Section II the composition product is first defined in the multivariable setting and various fundamental properties are presented. Then a condition is introduced under which the composition product preserves rationality and local convergence. In Section III, in preparation for the feedback analysis, it is next shown that the composition product produces a contractive mapping on the set of all formal power series using the familiar ultrametric.
u
n E .n4 uuu4 doE c |f _ c
where f E| . The input-output operator corresponding to S is then
8S doE| '
where 5 ' 4@ t5 c 52 c c 5 when 5 M b , and # denotes the number of symbols in # . These growth conditions on the coefficients of S insure that there exist positive real numbers - and Af such that for all piecewise continuous with 88" $ - and A $ Af , the series defining 8S converges uniformly and absolutely on d|f c |f n A o. Under such conditions the power series S is said to be locally convergent. More recently, it was shown in [7] that the growth condition (1) also implies that 8S constitutes a well defined operator from R6 E- d|f c |f n A o into ^ E7 d|f c |f n A o for sufficiently small -c 7c A : f, where the numbers Rc ^ M dc "o are conjugate exponents, i.e., *R n *^ ' with Ec " being a conjugate pair by convention.
+
Fc
y
Fd
ESc # .# doE|c |f c
which is referred to as a Fliess operator. In the classical literature where these operators first appeared [4, 5, 6, 9, 10, 11], it is normally assumed that there exists real numbers g : f and D such that
ESc # $ g # # -c
# M UWc
(1)
173
Fig. 2: Two Fliess operators connected in a feedback configuration.
Figure 2 shows a feedback connection with the corresponding feedback equation (2) + ' 8S d n 8_ d+ oo
Such a feedback system is said to be well-posed if the support of S and _ each contain at least one word having a nonzero symbol. If, for example, 8S is a linear operator then formally the solution to equation (2) is
+ ' 8S do n 8S 8_ 8S do n u u u
(3)
But it is not immediately clear that this series converges in any manner, and in particular, converges to another Fliess operator, say 8S9_ for some S9_ M b6 J U q. When 8S is nonlinear, the problem is further complicated by the fact that a simple series representation is not apparent. In Section IV the feedback connection is considered in the special case where the inputs are supplied from an exosystem which is itself a Fliess operator as shown in Figure 3. In particular, a sufficient condition is given under which a unique solution + of the feedback equation (2) is known to exist if ' 8K d o. Then + is characterized as the output of a new Fliess operator when a certain series factorization property is available. This leads to an implicit characterization of the feedback product S9_. The section is concluded with various examples.
where ' f for ' c c &, it follows that ?
The following theorem guarantees that the composition product is well defined by insuring that the series (5) is summable. Theorem II.1 Given a fixed _ M b6 J f q, the family of series t# _ G # M f W is locally finite, and therefore summable. Proof: Given an arbitrary # M f W expressed in the form (6), it follows directly that
Fb
u
+
Fc
' # n
#3# {
k
3
'
Lh_E_m c
(7)
where the order of S is defined by
?ut# G # M tTTES G S ' f " G S ' fc
U_ E1 G' t# M f W G E# _c 1 ' f a t# M f W G Lh_E# _ $ 1 t# M f W G # n
# 3#{
Fig. 3: A feedback configuration with a Fliess operator exosystem providing
'
3
Lh_E_m $ 1
Clearly this latter set is finite, and thus U_ E1 is finite for all 1 M f W . This fact implies summability [1].
the inputs.
II. The Composition Product The composition product of two series over an alphabet f ' t%f c % is defined recursively in terms of the shuffle product [2, 3]. For any # M f W and _ M b J f q, let
k
# G # %4 ' f %&f n d_ ww E# _ o G # ' %&f % # c & D fc
(4)
where # %4 denotes the number of symbols in # equivalent to % . For Sc _ M b J f q the definition is extended to
S _'
# Mf
ESc # # _
(5)
For SISO systems it is easily shown that 8S 8_ ' 8S _ . For the multivariable case it is necessary to consider power series of the form _ G f W p< b6 , where f ' t%f c % c c %6 is an arbitrary alphabet with 6 n letters. In which case, the defining equation (4), which preserves the key property 8S 8_ ' 8S _ (see [8]), becomes
# _ '
? n Lh_E_m
'
k
&
and tTTES G' t# M f W G ESc # ' f denotes the support of S. Hence, for any 1 M f W
y
Fd
# _ G'
'
Lh_E# _ ' ?f n & n
Lh_ES ' v
n
# _ ' %?f n n d_n ww %f n4 d_n4 ww u u u %?f 4 n d_4 ww %?f 3 o u u uoo
# G # %l ' fc ' f %&f n d_ ww E# _ o G # ' %&f % # c & D fc ' fc
where _ G 1 p< EE_c 1 , the -th component of E_c 1 . Observe that in general for
# ' %?f n %n %f 4 %n4 u u u %?f 4 %4 %?f 3 c ?n
(6)
174
From the definition, it is easily verified that for any series Sc _c e M
b6 J f q,
ES n _ e ' S e n _ ec
but in general S E_ n e ' S _ n S e. A special exception are linear series. A series S M b J f q is called linear if
tTTES \ t# ' %?f 4 % %?f 3 c M tc 2c c 6c ? c ?f D f Since the shuffle product distributes over addition, given any # '
%?f 4 % %?f 3 :
# E_ n e ' %?f 4 n dE_ n e ww %?f 3 o ' %?f 4 n E_ ww %?f 3 n %?f 4 n Ee ww %?f 3 ' # _ n # e
Therefore,
S E_ n e '
ESc # # E_ n e
# MU
ESc # # _ n ESc # # e ' S _ n S e
'
# MU
A linear series should not be confused with a rational series. The series
S'
"
&
'f
&- %&f %
M b6 J q are locally convergent Theorem II.3 Suppose series with growth constants S S and _ _ , respectively. If is input-limited then is locally convergent with
Sc _
is linear but not rational, while the bilinear series
S' "
"
g c S _ ES _c D gS g_f ES ES n D D -c D M f c where ' 4@ ES c _ . S
E n u u u 4 5f %n u u u %4
&'f 4 ccn 'f
with M b?f? and A f M b?f is rational but not linear. The set b J q forms a metric space under the ultrametric
f
_t|
j
: :
c5
bJf q f bJf q ESc _ p< jLh_ES3_ ,
c
p
