A key observation in this analysis was that certain combinatorial integer sequences cataloged in the Online Encyclopedia of. Integer Sequences (OEIS) naturally ...
45th Southeastern Symposium on System Theory Baylor University, Waco, TX, USA, March 11, 2013
System Interconnections and Combinatorial Integer Sequences W. Steven Gray
Makhin Thitsa |η|, is the number of letters in η, while |η|xi is the number of times the letter xi appears in η. The set of all words with length k is denoted by X k . The set of all words including the empty word, ∅, is designated by X ∗ . It forms a monoid under catenation. Any mapping c : X ∗ → Rℓ is called a formal power series. The value of c at η ∈ X ∗ is written as (c, η) and called the coefficient of P η in c. Typically, c is represented as the formal sum c = η∈X ∗ (c, η)η. Given ∗ a subset L ⊆ X P , the characteristic series of L is defined by char(L) = η∈L η. The collection of all formal power series over X is denoted by Rℓ hhXii. It forms an associative R-algebra under the catenation product and a commutative and associative R-algebra under the shuffle product, that is, the R-bilinear mapping Rℓ hhXii × Rℓ hhXii → Rℓ hhXii uniquely specified by the shuffle product of two words
Abstract— Given two nonlinear input-output systems which are analytic in the sense that they have convergent Fliess operator representations, it is known that their interconnection in almost any fashion will produce another system in this same class. Recent work has focused on characterizing the radius of convergence for a variety of such interconnections. A key observation in this analysis was that certain combinatorial integer sequences naturally appear. The first goal of this paper is to gather from the literature all the known relationships between system interconnections and such sequences and organize them in a coherent manner. In the process it becomes clear that Stirling numbers play a central role in the most nontrivial types of system interconnections, namely, cascade and feedback connections. The second goal is to describe these relationships.
I. I NTRODUCTION Given two nonlinear input-output systems which are analytic in the sense that they have convergent Fliess operator representations, it is known that their interconnection in almost any fashion will produce another system in this same class [10], [11], [16], [19]. Recent work has focused on understanding the precise nature of the convergence of such interconnected systems using a notion of radius of convergence [12], [13], [19], [21], [22], [24]. A key observation in this analysis was that certain combinatorial integer sequences cataloged in the Online Encyclopedia of Integer Sequences (OEIS) naturally appear [18]. This paper has two general goals. The first is to gather from the literature all the known relationships between system interconnections and such sequences and organize them in a coherent manner. The primary focus will be on the cascade and feedback connections since parallel connections are trivial by comparison. The supposition is that this will eventually lead to a better understanding of the underlying combinatorial structures that are inherent in system interconnection problems. The second goal is to take a first step in this direction and show that Stirling numbers play a central role in the characterization of system interconnections. Since Stirling numbers of the first and second kind are used to enumerate different types of set partitions, it appears that the combinatorics of set partitions and interconnections are closely linked. The paper is organized as follows. In the next section a brief overview of Fliess operators and their interconnections is given. In addition, the relevant facts about Stirling numbers are summarized. In Section III the cascade connection and its associated integer sequences are presented. The feedback connection is addressed in Section IV. The paper’s conclusions are summarized in the final section.
(xi η) ⊔⊔ (xj ξ) = xi (η ⊔⊔ (xj ξ)) + xj ((xi η) ⊔⊔ ξ) and η ⊔⊔ ∅ = η for all η, ξ ∈ X ∗ [7]. A. Fliess Operators and Their Convergence One can formally associate with any series c ∈ Rℓ hhXii a causal m-input, ℓ-output operator, Fc , in the following manner. Let p ≥ 1 and t0 < t1 be given. For a Lebesgue measurable function u : [t0 , t1 ] → Rm , define kukp = max{kui kp : 1 ≤ i ≤ m}, where kui kp is the usual Lp -norm for a measurable real-valued function, ui , defined on [t0 , t1 ]. Let Lm p [t0 , t1 ] denote the set of all measurable functions defined on [t0 , t1 ] having a finite k · kp norm and Bpm (R)[t0 , t1 ] := {u ∈ Lm p [t0 , t1 ] : kukp ≤ R}. Assume C[t0 , t1 ] is the subset of continuous functions in ∗ the map Lm 1 [t0 , t1 ]. Define iteratively for each η ∈ X m Eη : L1 [t0 , t1 ] → C[t0 , t1 ] by setting E∅ [u] = 1 and letting Z t ui (τ )Eη¯[u](τ, t0 ) dτ, Exi η¯[u](t, t0 ) = t0
∗
where xi ∈ X, η¯ ∈ X , and u0 = 1. The input-output operator corresponding to c is the Fliess operator X Fc [u](t) = (c, η) Eη [u](t, t0 ) (1) η∈X ∗
[7], [8]. If there exists real numbers Kc , Mc > 0 such that |(c, η)| ≤ Kc Mc|η| |η|!, η ∈ X ∗ ,
then Fc constitutes a well defined mapping from Bpm (R)[t0 , t0 + T ] into Bqℓ (S)[t0 , t0 + T ] for sufficiently small R, T > 0, where the numbers p, q ∈ [1, ∞] are conjugate exponents, i.e., 1/p + 1/q = 1 [15]. (Here, |z| := maxi |zi | when z ∈ Rℓ .) The set of all such locally convergent series is denoted by RℓLC hhXii. In particular, when p = 1, the series (1) converges if
II. P RELIMINARIES A finite nonempty set of noncommuting symbols X = {x0 , x1 , . . . , xm } is called an alphabet. Each element of X is called a letter, and any finite sequence of letters from X, η = xi1 · · · xik , is called a word over X. The length of η, The authors are affiliated with the Department of Electrical and Computer Engineering, Old Dominion University, Norfolk, Virginia 23529-0246, {sgray,MThitsa}@odu.edu.
978-1-4799-0038-1/$31.00 ©2013 IEEE
(2)
max{R, T } < 135
1 Mc (m + 1)
TABLE I M AXIMAL SERIES WITH UNITY GROWTH CONSTANTS alphabet X = {x1 } X = {x0 , x1 }
locally convergent P cLC = k≥0 k! xk1 P dLC = η∈X ∗ |η|! η
Observe dLC =
globally convergent P cGC = k≥0 xk1 P dGC = η∈X ∗ η
∞ X
k
k! char(X ) =
k=0
∞ X
k=0
and thus, using the fact that Fc Fd = Fc ⊔⊔ d [7], ∞ ∞ X X k Fchar(X) ⊔⊔ k = Fchar(X) FdLC = k=0
k=0
[3], [4]. It is important in applications to identify the smallest possible geometric growth constant, Mc , in order to avoid over restricting the domain of Fc . So let π : RℓLC hhXii → R+ ∪ {0} take each series c to the infimum of all Mc satisfying (2). Then RℓLC hhXii can be partitioned into equivalence classes, and the number 1/Mc (m + 1) will be referred to as the radius of convergence for the class π −1 (Mc ). This is in contrast to the usual situation where a radius of convergence is assigned to individual series. When c satisfies the more stringent growth condition
=
1
0 0 1
0
1
0
1 1
0 1
1 3
7 15
1 6
25
1 10
1
and read row-by-row, the sequence is given the name A048993 in the OEIS as shown in Table IV. This sequence is known to satisfy the identity ! n n o ∞ X X n k xn t(ex −1) e = t . (5) k n! n=0 k=0 Pn Setting t = 1, it follows that k=0 nk = Bn , where Bn denotes the Bell numbers, namely sequence A000110 in the OEIS. On the other hand, a cyclic partition of a set is a partition in which the elements of each partition element are arranged into cycles. Two cyclic partitions having the same set of subsets are distinct if any of their cycles are distinct. The Stirling cycle number, nk , is the number of ways to partition a set of n elements into k nonempty cycles. For example, the set A = {a, b, c} has a single partition with one nonempty subset, namely, the set A itself, so 31 = 1. But A has two distinct cyclic partitions of cardinality 3, specifically, {a → b → c} and {a → c → b}. Thus, 31 = 2. This integer sequence corresponds to A132393 in the OEIS when written in triangular form
B. State Space Realizations of a Fliess Operator A Fliess operator Fc defined on Bpm (R)[t0 , t0 + T ] is said to be realized by a state space realization when there exists a system of n differential equations and ℓ output functions (3a)
i=1
(3b)
where each gi is an analytic vector field expressed in local coordinates on some neighborhood W of z0 , h is an analytic function on W such that (3a) has a well defined solution z(t), t ∈ [t0 , t0 +T ] in W for any given input u ∈ Bpm (R)[t0 , t0 + T ], and
1 0 0
Fc [u](t) = h(z(t)), t ∈ [t0 , t0 + T ]
0
[7], [8], [15]. PExample 1: Consider the locally maximal series dLC = η∈X ∗ |η|! η. It is not difficult to produce a one dimensional state space realization of FdLC : u 7→ y using the identity char(X k ) =
(4)
C. Stirling Numbers The number of ways to partition a set containing n elements into k nonempty subsets is a Stirling number of the second kind [17], [26]. This integer sequence is frequently denoted by both S(n, k) and nk . When these numbers are arranged in triangular form
and u[t0 ,t1 ] denotes the restriction of u to [t0 , t1 ] [15]. The set of all such globally convergent series is designated by RℓGC hhXii. The following definitions will be important. Definition 1: For a given alphabet X, a series c¯ ∈ RℓLC hhXii is said to be locally maximal with growth con|η| stants Kc , Mc > 0 if each component of (¯ c, η) is Kc Mc |η|!, ∗ ℓ η ∈ X . A series c¯ ∈ RGC hhXii is said to be globally maximal with growth constants Kc , Mc > 0 if each component |η| of (¯ c, η) is Kc Mc , η ∈ X ∗ . Specific examples of maximal series are given in Table I for the case where all the growth constants are unity.
y = h(z),
.
Realizations for the maximal series in Table I are given in Table II.
m m Lm p,e (t0 ) := {u : [t0 , ∞) → R : u[t0 ,t1 ] ∈ Lp [t0 , t1 ], ∀t1 ∈ (t0 , ∞)},
gi (z) ui , z(t0 ) = z0
1 − Fchar(X)
z˙ = z 2 + z 2 u, z(0) = 1, y = z.
the series (1) defines an operator from the extended space Lm p,e (t0 ) into C[t0 , ∞), where
z˙ = g0 (z) +
1
It follows directly that FdLC is realized by
|(c, η)| ≤ Kc Mc|η| , η ∈ X ∗ ,
m X
char(X) ⊔⊔ k ,
0 0
2 6
24
1 1
1 3
11 50
(6)
1 6
35
1 10
1
and read row-by-row. It satisfies the identity ! ∞ n h i X X 1 1 n k xn t(ln( 1−x )) = . = t e (1 − x)t n! k n=0
char(X) ⊔⊔ k , k ≥ 0. k!
k=0
136
(7)
TABLE II S TATE SPACE REALIZATIONS OF MAXIMAL SERIES IN TABLE I alphabet X = {x1 } X = {x0 , x1 }
locally convergent
globally convergent
z˙ = u, z(0) = 0, y = (1 − z)−1 z˙ = z 2 u, z(0) = 1, y = z z˙ = 1 + u, z(0) = 0, y = (1 − z)−1 z˙ = z 2 + z 2 u, z(0) = 1, y = z
z˙ = u, z(0) = 0, y = ez z˙ = zu, z(0) = 1, y = z z˙ = 1 + u, z(0) = 0, y = ez z˙ = z + zu, z(0) = 1, y = z
TABLE III S TIRLING TRANSFORMS OEIS sequence
an , n ≥ 0
OEIS sequence
bn = S(a), n ≥ 0
A000012 A000027 A000142 A000142 A133942 A007840
1 n+1 n! a0 = 1, (n − 1)!, n ≥ 1 (−1)n n!, n ≥ 0 1,-1,3,-14,88,-694,6578
A000110 A000110 A000670 A000629 A033999 A133942
1,1,2,5,15,52,203,877,. . . 1,2,5,15,52,203,877,4140,. . . 1,1,3,13,75,541,4683,. . . 1,1,2,6,26,150,1082,9366,. . . (−1)n , n ≥ 0 (−1)n n!, n ≥ 0
Pn It is easy to verify by setting t = 1 that k=0 nk = n!, which is sequence A000142 in the OEIS. The Stirling num bers of the first kind are defined as s(n, k) = (−1)n−k nk . Given an integer sequence an , n ≥ 0, its Stirling transformation, b = S(a), is the integer sequence bn =
n X
u
Fd
Fig. 1.
S(n, k) ak , n ≥ 0.
Cascade connection of Fliess operators
Dxi : RhhXii → RhhXii : e 7→ x0 (di ⊔⊔ e), s(n, k) bk , n ≥ 0.
where i = 0, 1, . . . , m and d0 := 1. Assume D∅ is the identity map on RhhXii. Such maps can be composed in the obvious way so that Dxi xj := Dxi Dxj provides an R-algebra which is isomorphic to the usual R-algebra on RhhXii under the catenation product. The composition product of a word η ∈ X ∗ and a series d ∈ Rm hhXii is defined as
k=0
A table of Stirling transforms is given in Table III. The exponential generating function of a is by definition fa (x) =
∞ X
an
n=0
xn . n!
(xik xik−1 · · · xi1 ) ◦ d = Dxik Dxik−1 · · · Dxi1 (1) = Dη (1). | {z }
In which case,
η
x
fb (x) = fS(a) (x) = fa (e − 1).
For any c ∈ Rℓ hhXii the definition is extended linearly as X c◦d= (c, η) η ◦ d.
Observe that (5) corresponds to setting fa (x) = etx . One ˜ as can also define a signed Stirling transformation, S, ˜bn =
y
The cascade connection of two Fliess operators as shown in Fig. 1 was found by Ferfera in [5], [6] to always yield an input-output system having a Fliess operator representation. To describe its generating series explicitly, let d ∈ Rm hhXii and define the family of mappings
According to [2], if an is the number of elements in some set with distinct labels 1, 2, . . . , n, then bn is the number of elements labeled 1, 2, . . . , n with repetitions allowed. It can be directly verified that the inverse Stirling transform, a = s(b), is an =
Fc
III. C ASCADE C ONNECTIONS
k=0
n X
v
n X
η∈X ∗
(−1)n−k S(n, k) ak , n ≥ 0.
k=0
It is immediate that
fa (x) = fS˜−1 (˜b) (x) = f˜b ln
1 1−x
.
(8)
Therefore, (7) corresponds to setting f˜b (x) = etx . 137
In which case, for any c ∈ Rℓ hhXii and d ∈ Rm hhXii, the identity Fc ◦ Fd = Fc◦d is satisfied. It is known in general that the composition product is associative, distributive to the left over the shuffle product, and for any c ∈ Rm hhXii, the mapping d 7→ c ◦ d is a contraction on Rm hhXii in the ultrametric sense [11]. The following example illustrates how the Stirling numbers of the first kind appear naturally in such cascade structures. k PExample 2: Let X = {x0 , x1 }, c = x1 and dLC = |η|! η. It is not difficult to show that Fc is realized η∈X ∗
TABLE IV I NTEGER SEQUENCES FROM THE OEIS GENERATED BY CASCADE INTERCONNECTIONS OEIS sequence A132393 A048993 A007840 A002866 A003320 A052820
description n (by row) k n (by row) k S −1 (A133942) an = 2n−1 n!, n ≥ 1 maxk=0,...,n kn−k S˜−1 (A072597)
A000110
S(A000012)
an , n = 0, 1, 2, . . .
interconnection
1,0,1,0,1,1,0,2,3,1,0,6,11,6,1,. . . 1,0,1,0,1,1,0,1,3,1,0,1,7,6,1,. . . 1,-1,3,-14,88,-694,6578,-72792,. . . 1, 1, 4, 24, 192, 1920, 23040, . . . 1, 1, 1, 2, 4, 9, 27, 81, 256, 1024, . . . 1, −2, 9, −62, 572, −6604, 91526, . . .
xk1 ◦ dLC xk1 ◦ dGC xk1 k! ◦ dLC fcLC ◦ FcLC [14], [23] FcLC ◦ FcLC [20] FdLC ◦ FdLC [19], [21], [22], [24] FdGC ◦ FdGC [19], [21], [22], [24] fdGC ◦ FdGC [14], [23]
1, 1, 2, 5, 15, 52, 203, 877, 4140, . . .
and W denotes the Lambert W -function, namely, the inverse of the function g(W ) = W exp(W ). Furthermore, no smaller geometric growth constant can satisfy (9), and thus, the radius of convergence is 1 Mc − Md 1 1 − mKd W exp . Md (m + 1) mKd mKd Mc
by
1 k z . k! In which case, using (4) and (7), the zero-input response of the composition of Fc and FdLC is k X ∞ h i n n t 1 1 ln = . Fc◦dLC [0](t) = k n! k! 1−t n=0 z˙ = u, z(0) = 0, y =
When m and all the growth constants are unity, the theorem is proved by showing that the Taylor series for the output function of this interconnection with the fastest growing coefficients is the zero-input response ∞ X 1 xn . (dLC ◦ dLC , xn0 ) 0 = f (x0 ) := n! 1 − x + ln(1 − x0 ) 0 n=0
Thus, Fc◦dLC [0] is the exponential generating function of the Stirling cycle numbers for fixed k. The global counterpart of this example produces the exponential generating function for the Stirling numbers of the second kind. Specifically, using the first state space realization for FdGC in Table II and (5), observe ∞ n o n k X n t 1 t e −1 = . Fc◦dGC [0](t) = k n! k! n=0
The integer sequence (dLC ◦ dLC , xn0 ), n ≥ 0 is a simple grammar known as A052820 in the OEIS (see Table IV). It is related to another combinatorial sequence as described next. Theorem 2: The exponential generating function f (x) = 1/(1 − x + ln(1 − x)) is the inverse signed Stirling transformation of n X 1 (n − k + 1)k , n ≥ 0, an = n! (10) k!
In [19], [21], [22], [24] it was shown, utilizing a theorem about the growth rate of coefficients of analytic functions [26, Theorem 2.4.3], that the radius of convergence for the cascade connection of two locally convergent systems or two globally convergent systems is determined exactly by the zero-input response of an interconnection of systems having maximal generating series. The fact that the zeroinput responses in the previous example generate both sets of Stirling numbers strongly suggests that these integers play a direct role in determining the radius of convergence for cascade connections. The specific hypothesis for the local case is that the inverse Stirling transform of some other fundamental combinatorial integer sequence will appear in the analysis since this transform is in terms of Stirling numbers of the first kind. The analogous claim for the global case is that the Stirling transform of some other significant sequence should be present. This is in fact the case as described next. The first two theorems address the locally convergent case. Theorem 1: [19], [21], [22], [24] Suppose X = {x0 , x1 , . . . , xm }. Let c ∈ RℓLC hhXii and d ∈ Rm LC hhXii with growth constants Kc , Mc > 0 and Kd , Md > 0, respectively. If b = c ◦ d then |ν|
|(b, ν)| ≤ Kb Mb |ν|!, ν ∈ X ∗
k=0
which is sequence A072597 in the OEIS. Proof: The claim follows from (8) since it is known that the exponential generating function of the sequence (10) is fa (x) = 1/(e−x − x) [18].
The singularity of fa (x) = 1/(e−x − x) nearest the origin is x = W (1), the so called omega constant or golden ratio for exponentials. This implies the minimum geometric growth constant for A052820 is 1/(1−W (1)). In the case of globally convergent series, an analogous situation exists. The known result is presented first. Theorem 3: [19], [21], [22], [24] Suppose X = {x0 , x1 , . . . , xm }. Let c ∈ RℓGC hhXii and d ∈ Rm GC hhXii with growth constants Kc , Mc > 0 and Kd , Md > 0, respectively. Assume c¯ and d¯ are globally maximal series with the same growth constants as c and d, respectively. If b = c ◦ d and ¯b = c¯ ◦ d¯ then |ν| |(b, ν)| ≤ (¯bi , x ), ν ∈ X ∗ , i = 1, 2, . . . , ℓ, 0
(9)
where the sequence (¯bi , xk0 ), k ≥ 0 has the exponential generating function mKd exp(Md x) + Md x − mKd . f (x) = Kc exp Md /Mc
for some Kb > 0, where Mb =
1 − mKd W
Md 1 mKd
exp
Mc −Md mKd Mc
,
138
Therefore, e is the solution of the equation e = (c ◦ d) ◦ e. In the case of unity feedback systems (or equivalently, if c ◦ d is redefined as c), this equation reduces to e = c ◦ e. The exponential generating functions underlying the radius of convergence for unity feedback systems are considered next. The known result for locally convergent systems is presented first. Theorem 4: [12], [13], [19], [24] Suppose X = {x0 , x1 , . . . , xm }. Let c ∈ Rm LC hhXii with growth constants Kc , Mc > 0. If e = c@δ then
Since f is entire, the radius of convergence of b is infinity. In the case where m = 1 and all the growth constants are unity, the central generating series for this interconnection reduces to ∞ X xn (dGC ◦ dGC , xn0 ) 0 = exp(exp(x0 ) + x0 − 1). f (x0 ) := n! n=0 This is the exponential generating function for the Bell numbers, left shifted by one. As shown in Table III, this is an elementary Stirling transform pair, namely the image of an = n + 1, n ≥ 0 under S. The section is concluded by noting that a number of other known combinatorial sequences can be generated by cascade structures. The situation is summarized in Table IV along with the appropriate citations.
|(e, η)| ≤ Ke (α(Kc )Mc )|η| |η|!, η ∈ X ∗ , for some Ke > 0, where α(Kc ) =
IV. F EEDBACK C ONNECTIONS
u
Fc
+
1 . 1 − mKc ln (1 + 1/mKc )
Furthermore, no geometric growth constant smaller than α(Kc )Mc is possible, and thus, the radius of convergence is 1/α(Kc )Mc (m + 1). In the case where m = Kc = Mc = 1, the theorem is proved by showing that the Taylor series for the output function of this interconnection with the fastest growing coefficients is the zero-input response
y
Fd
f (x0 ) :=
∞ X
(dLC @δ, xn0 )
n=0
Fig. 2.
Feedback connection of Fliess operators
The integer sequence (dLC @δ, xn0 ), n ≥ 0 is A112487. The first few entries of this sequence are given in Table V. This sequence can be derived from the Stirling cycle numbers by first defining for any k ≥ 0 the exponential generating function for the k-th diagonal (left to right) of the triangle in (6), namely, n ∞ X x n+1 fk (x) := . n + 1 − k n! n=0
Consider two Fliess operators interconnected to form a feedback system as shown in Fig. 2. The output y must satisfy the feedback equation y = Fc [u + Fd [y]] for every admissible input u. It was shown in [11], [16] that there always exist a unique generating series e so that y = Fe [u]. In which case, the feedback equation becomes equivalent to
It is then known that each of these functions can be written in the form fk (x) = ex gk (x), where
Fe [u] = Fc [u + Fd◦e [u]].
g0 (x) = 1
The feedback product of c and d is thus defined as c@d = e. Specifically, e is the unique fixed point of the contractive iterated map
g1 (x) = x +
x2 2!
x2 x3 x4 +5 +3 2! 3! 4! x3 x4 x5 x6 g3 (x) = 6 + 26 + 35 + 15 3! 4! 5! 6! x5 x6 x7 x8 x4 g4 (x) = 24 + 154 + 340 + 315 + 105 4! 5! 6! 7! 8! .. . g2 (x) = 2
S˜ : ei 7→ ei+1 = c˜◦(d ◦ ei ), where ˜◦ denotes the modified composition product. That is, the product X (c, η) η˜◦d, c˜◦d = η∈X ∗
˜ where η˜◦d = D(1) with ˜ x : RhhXii → RhhXii : e 7→ xi e + x0 (di ⊔⊔ e) D i
−1 xn0 = . n! 1 + W (−2 exp(x0 − 2))
[18]. If the coefficients of these polynomials are arranged in the triangular form:
and d0 := 0 [11]. Therefore, e = c@d satisfies the fixed point equation e = c˜◦(d ◦ e). In the case of a unity feedback system, denoted by c@δ, this equation reduces to e = c˜◦e. The output of a self-excited feedback loop is described by the fixed point e ∈ Rm [[X0 ]], X0 := {x0 }, of the contractive iterated map
1 1 2 6 24
1 5
26 154
3 35
340
(11) 15
315
105
the integer sequence A112487 corresponds to the row sums of this triangle. Some intriguing connections between (11)
S : ei 7→ ei+1 = (c ◦ d) ◦ ei . 139
TABLE V I NTEGER SEQUENCES FROM THE OEIS GENERATED BY FEEDBACK INTERCONNECTIONS an , n = 0, 1, 2, . . .
OEIS sequence
description
A000110
S(A000012)
1, 1, 2, 5, 15, 52, 203, 877, 4140, . . .
F(1+x1 x∗ )@δ [9]
A000111
Euler numbers
1, 1, 2, 5, 16, 61, 272, 1385, 7936, . . .
F(1+x1 +x2 )@δ [9]
A001147 A112487 A000142 A000629
(2n − 1)!!, n ≥ 0 Derived from unsign Stirling numbers 1st kind ˜ S(A007840) S(A000142(n − 1))
1, 1, 3, 15, 105, 945, 10395, 135135, . . . 1, 2, 10, 82, 938, 13778, 247210, . . . 1, 1, 2, 6, 24, 120, 720, 5040, 40320, . . . 1, 2, 6, 26, 150, 1082, 9366, 94586, . . .
and the twisted Mellon transform are given in [25]. An interpretation of A112487 in terms of combinatorics on finite groups is described in [1]. Next the global case is considered. The following result is known. Theorem 5: [12], [13], [19], [24] Let X = {x0 , x1 , . . . , xm } and c ∈ Rm hhXii with growth GC constants Kc , Mc > 0. If e = c@δ then |(e, η)| ≤ Ke (γ(Kc )Mc )|η| |η|!, η ∈ X ∗ ,
(12)
1 . ln(1 + 1/mKc )
Furthermore, no geometric growth constant smaller than γ(Kc )Mc can satisfy (12), and thus, the radius of convergence is 1/γ(Kc )Mc (m + 1). As in the previous cases, the proof is constructed by showing that the zero-input response defines the radius of convergence for the interconnection, which is generally finite even when both subsystems are globally convergent. When m = Kc = Mc = 1, this response is given by f (x0 ) :=
∞ X
n=0
(dGC @δ, xn0 )
0
1
FcLC @δ [12], [13], [19] FdLC @δ [12], [13], [19], [24] FcGC @δ [19] FdGC @δ [9], [13], [19], [24]
[3] L. A. Duffaut Espinosa, Interconnections of Nonlinear Systems Driven by L2 -Itˆo Stochastic Processes, Doctoral Dissertation, Old Dominion University, 2009. [4] L. A. Duffaut Espinosa, W. S. Gray, and O. R. Gonz´alez, On Fliess operators driven by L2 -Itˆo random processes, Proc. 48th IEEE Conf. on Decision and Control, Shanghai, China, 2009, pp. 7478-7484. [5] A. Ferfera, Combinatoire du Mono¨ıde Libre Appliqu´ee a` la Composition et aux Variations de Certaines Fonctionnelles Issues de la Th´eorie des Syst`emes, Doctoral Dissertation, University of Bordeaux I, 1979. [6] , Combinatoire du mono¨ıde libre et composition de certains syst`emes non lin´eaires, Ast´erisque, 75-76 (1980), 87-93. [7] M. Fliess, Fonctionnelles causales non lin´eaires et ind´etermin´ees non commutatives, Bull. Soc. Math. France, 109 (1981), 3-40. , R´ealisation locale des syst`emes non lin´eaires, alg`ebres de [8] Lie filtr´ees transitives et s´eries g´en´eratrices non commutatives, Invent. Math., 71 (1983), 521-537. [9] W. S. Gray, On the rationality of the feedback connection, Proc. 19th Inter. Symp. Mathematical Theory of Networks and Systems, Budapest, Hungary, 2010, pp. 1035-1040. [10] W. S. Gray and L. A. Duffaut Espinosa, A Fa`a di Bruno Hopf algebra for a group of Fliess operators with applications to feedback, Systems Control Lett., 60 (2011), 441-449. [11] W. S. Gray and Y. Li, Generating series for interconnected analytic nonlinear systems, SIAM J. Control Optim., 44 (2005), 646-672. [12] W. S. Gray and M. Thitsa, On the radius of convergence of self-excited feedback connected analytic nonlinear systems: The local case, Proc. 42nd IEEE Southeastern Symp. on System Theory, Tyler, Texas, 2010, pp. 116-121. , On the radius of convergence of self-excited feedback connected [13] analytic nonlinear systems, Proc. 49th IEEE Conf. on Decision and Control, Atlanta, Georgia, 2010, pp. 7092-7098. , A unified approach to generating series for cascaded nonlinear [14] input-output systems, Inter. J. Control, 85 (2012), 1737-1754. [15] W. S. Gray and Y. Wang, Fliess operators on Lp spaces: convergence and continuity, Systems Control Lett., 46 (2002), 67-74. [16] , Formal Fliess operators with applications to feedback interconnections, Proc. 18th Inter. Symp. Mathematical Theory of Networks and Systems, Blacksburg, Virginia, 2008. [17] K. H. Rosen, ed., Handbook of Discrete and Combinatorial Mathematics, CRC Press, Boca Raton, Florida, 2000. [18] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, available at www.research.att.com/ ∼njas/sequences. [19] M. Thitsa, On the Radius of Convergence of Interconnected Analytic Nonlinear Systems, Doctoral Dissertation, Old Dominion University, 2011. [20] , unpublished, 2011. [21] M. Thitsa and W. S. Gray, On the radius of convergence of cascaded analytic nonlinear systems: The SISO case, Proc. 43rd IEEE Southeastern Symp. on System Theory, Auburn, Alabama, 2011, pp. 32-38. , On the radius of convergence of cascaded analytic nonlinear sys[22] tems, Proc. 50th IEEE Conf. on Decision and Control and European Control Conf., Orlando Florida, 2011, pp. 3830-3835. [23] , On the radius of convergence of mixed cascades of analytic nonlinear input-output systems, Proc. 44rd IEEE Southeastern Symp. on System Theory, Jacksonville, Florida, 2012, pp. 231-236. [24] , On the radius of convergence of interconnected analytic nonlinear input-output systems, SIAM J. Control Optim., 50 (2012), 27862813. [25] Z. Wang, Spectral Properties of K¨ahler Quotients, Doctoral Dissertation, MIT, 2008. [26] H. S. Wilf, Generatingfunctionology, 2nd Ed., Academic Press, San Diego, 1994.
for some Ke > 0, where γ(Kc ) =
interconnection
ex 0 xn0 . = n! 2 − ex 0
The sequence (dGC @δ, xn0 ), n ≥ 0 is equivalent to the OEIS sequence A000629 as shown in Table V. It is easy to verify that this sequence is the Stirling transform of the factorial sequence. Several other known results along these lines for feedback systems are also given in Table V. V. C ONCLUSIONS This paper summarized from the source literature all the known relationships between system interconnections and combinatorial integer sequences and organizes them in a coherent manner. The focus was on the cascade and feedback connections since the parallel connections are trivial. It became apparent in the process that the Stirling numbers of the first kind are closely related to problems involving local convergence, while Stirling numbers of the second kind are central to problems involving global convergence. R EFERENCES [1] R. Bacher, Counting packings of generic subsets in finite groups, http://arxiv.org/abs/1011.0975v2, 2011. [2] M. Bernstein and N. J. A. Sloane, Some canoncial sequences of integers, Linear Algebra Appl., 226-228 (1995), 57-72
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