On the Radius of Convergence of Interconnected Analytic Nonlinear Systems ∗ Makhin Thitsa Department of Electrical and Computer Engineering Old Dominion University Norfolk, Virginia USA RPCCT 2011 San Diego, California ∗ This research is funded in part by the NSF grant DMS 0960589
RPCCT 2011 Workshop Overview ∗ 1. Introduction 2. Mathematical Preliminaries 3. Radius of Convergence 3.1 The Cascade Connection 3.2 The Self-excited Feedback Connection 3.3 The Unity Feedback Connection 4. Conclusions and Future Research ∗ See www.ece.odu.edu/ ∼ sgray/RPCCT2011/thitsaslides.pdf 2
RPCCT 2011 Workshop 1. Introduction • For each c ∈ R ℓ �� X �� , one can associate an m -input, ℓ -output operator F c in the following manner: ◮ With t 0 , T ∈ R fixed and T > 0, define recursively for each η ∈ X ∗ the mapping E η : L m 1 [ t 0 , t 0 + T ] → C [ t 0 , t 0 + T ] by � t E x i ¯ η [ u ]( t, t 0 ) = u i ( τ ) E ¯ η [ u ]( τ, t 0 ) dτ, t 0 η ∈ X ∗ and u 0 ( t ) ≡ 1. where E ∅ = 1, x i ∈ X , ¯ ◮ The input-output operator corresponding to c is the Fliess operator � y = F c [ u ]( t ) = ( c, η ) E η [ u ]( t, t 0 ) . η ∈ X ∗ 3
RPCCT 2011 Workshop • If there exist real numbers K c , M c > 0 such that | ( c, η ) | ≤ K c M | η | c | η | ! , ∀ η ∈ X ∗ , (1) where | η | denotes the number of symbols in η , then c is said to be locally convergent. The set of all such series is denoted by R ℓ LC �� X �� . • If c ∈ R ℓ LC �� X �� then F c : B m p ( R )[ t 0 , t 0 + T ] → B ℓ q ( S )[ t 0 , t 0 + T ] for sufficiently small R, T > 0, where the numbers p , q ∈ [1 , ∞ ] are conjugate exponents, i.e., 1 / p + 1 / q = 1 (Gray and Wang, 2002). • In particular, when p = 1, the series defining y = F c [ u ] converges provided 1 max { R, T } < M c ( m + 1) LC �� X �� → R + take c to the smallest possible geometric • Let π : R ℓ growth constant M c satisfying (1). 4
RPCCT 2011 Workshop • In this case, R ℓ LC �� X �� can be partitioned into equivalence classes, and the number 1 /M c ( m + 1) will be referred to as the radius of convergence for the class π − 1 ( M c ). η ∈ X ∗ K c M | η | n ≥ 0 K c M n c n ! x n • For example, c = � c = � 1 , ¯ c | η | ! η are in the same equivalence class. • This definition is in contrast to the usual situation where a radius of convergence is assigned to individual series. • In practice, it is not difficult to estimate the minimal M c for many series, in which case, the radius of convergence for π − 1 ( M c ) can be easily computed. • If there exist real numbers K c , M c > 0 such that | ( c, η ) | ≤ K c M | η | c , ∀ η ∈ X ∗ , then c is said to be globally convergent. The set of all such series is denoted by R ℓ GC �� X �� . 5
RPCCT 2011 Workshop v u F d F c y (a) cascade connection u F c y + F d (b) feedback connection Fig. 1 The cascade and feedback interconnections 6
RPCCT 2011 Workshop • It is known that the cascade connection of two locally convergent Fliess operators always yields another locally convergent Fliess operator (Gray and Li, 2005). • Every self-excited feedback interconnection ( u = 0) of two locally convergent Fliess operators has a locally convergent Fliess operator representation (Gray and Li, 2005). • Lower bounds on the radius of convergence were given by Gray and Li (2005) for the cascade and self-excited feedback connections. 7
RPCCT 2011 Workshop Problem Statement Compute the radius of convergence of the • cascade • self-excited feedback • and unity feedback interconnection of two input-output systems represented as locally convergent Fliess operators. Remarks: • The Lambert W-function plays the key role throughout the computations. • The unity feedback system has the same generating series as the Faa di Bruno compositional inverse, i.e., c @ δ = ( − c ) − 1 . 8
RPCCT 2011 Workshop 2. Mathematical Preliminaries Definition 1: (Fliess, 1981) A series c ∈ R ℓ �� X �� is said to be exchangeable if for arbitrary η, ξ ∈ X ∗ | η | x i = | ξ | x i , i = 0 , 1 , . . . , m ⇒ ( c, η ) = ( c, ξ ) . Theorem 1: If c ∈ R ℓ �� X �� is an exchangeable series and d ∈ R m �� X �� is arbitrary then the composition product can be written in the form ∞ � � ( c, x r 0 0 · · · x r m m ) D r 0 ⊔ D r m c ◦ d = x 0 (1) ⊔ ⊔ · · · ⊔ x m (1) . k =0 r 0 ,...,rm ≥ 0 r 0+ ··· + rm = k 9
RPCCT 2011 Workshop c ∈ R ℓ Definition 2: A series ¯ LC �� X �� is said to be a locally maximal series with growth constants K c , M c > 0 if each component of (¯ c, η ) is K c M | η | c | η | !, η ∈ X ∗ . An analougus definition holds when ¯ c ∈ R ℓ GC �� X �� . n ≥ 0 a n /n ! z n be analytic in some Theorem 2: (Wilf, 1994) Let f ( z ) = � neighborhood of the origin in the complex plane. Suppose a singularity of f ( z ) of smallest modulus be at a point z 0 � = 0, and let ǫ > 0 be given. Then there exists N such that for all n > N , | a n | < (1 / | z 0 | + ǫ ) n n ! . Furthermore, for infinitely many n , | a n | > (1 / | z 0 | − ǫ ) n n ! . 10
RPCCT 2011 Workshop 3. Radius of Convergence 3.1 The Cascade Connection Theorem 3: Let X = { x 0 , x 1 , . . . , x m } . Let c ∈ R ℓ LC �� X �� and d ∈ R m LC �� X �� with growth constants K c , M c > 0 and K d , M d > 0, respectively. If b = c ◦ d then | ( b, ν ) | ≤ K b M | ν | b | ν | ! , ν ∈ X ∗ (2) for some K b > 0, where M d M b = �� , � � M c − M d 1 1 − mK d W mK d exp mM c K d where 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 (2). 11
RPCCT 2011 Workshop Two lemmas are needed for the proof of Theorem 3. The following lemma can be proved inductively. Lemma 1: Let X = { x 0 , x 1 , . . . , x m } and c, d ∈ R ℓ �� X �� such that η ∈ X ∗ | ( c, η ) | η . Then for any fixed ξ ∈ X ∗ it | c | ≤ d , where | c | := � follows that | ξ ◦ c | ≤ ξ ◦ d . c and ¯ Remark: If ¯ d are maximal series with growth constants K c , M c and K d , M d , respectively, it can be shown through the left linearity of c ◦ ¯ the composition product and Lemma 1 that | c ◦ d | ≤ ¯ d . 12
RPCCT 2011 Workshop c ∈ R ℓ Lemma 2: Let X = { x 0 , x 1 , . . . , x m } . Let ¯ LC �� X �� and ¯ d ∈ R m LC �� X �� be locally maximal series with growth constants K c , M c > 0 and K d , M d > 0, respectively. If ¯ c ◦ ¯ b = ¯ d , then the sequence (¯ b i , x k 0 ), k ≥ 0 has the exponential generating function K c f ( x 0 ) = 1 − M c x 0 + ( mM c K d /M d ) ln(1 − M d x 0 ) for any i = 1 , 2 , . . . , ℓ . Moreover, the smallest possible geometric growth constant for ¯ b is M d M b = �� . � � M c − M d 1 1 − mK d W mK d exp mM c K d 13
RPCCT 2011 Workshop Proof of Lemma 2 (outline): There is no loss of generality in assuming ℓ = 1. First observe that ¯ c is exchangeable, and thus, from Theorem 1 it follows that ∞ ⊔ ( x m ◦ ¯ ⊔ r 0 ⊔ r m ⊔ d ) ⊔ k ! x ¯ � � K c M k 0 b = ⊔ . . . ⊔ ⊔ c r 0 ! r m ! k =0 r 0 ,...,rm ≥ 0 r 0+ ··· + rm = k ∞ ⊔ k , � ⊔ � M c ( x 0 + mx 0 ¯ � = K c d 1 ) k =0 from which the following shuffle equation is obtained ¯ b = K c + M c [¯ ⊔ ( x 0 + mx 0 ¯ b ⊔ d 1 )] . (3) 14
RPCCT 2011 Workshop Let b n := max { (¯ b, ν ) : ν ∈ X n } . Then it can be shown using (3) that b n satisfies the following recursive formula n − 2 � � n b i mK d M ( n − i − 1) � b n = M c ( n − i − 1)! + b n − 1 M c (1 + mK d ) n, (4) d i i =0 n ≥ 2, where b 0 = K c and b 1 = K c M c (1 + mK d ). Remark: When all the growth constants and m are unity, b n , n ≥ 0 is the integer sequence shown in Table 1. Table 1: Sequence satisfying (4) with all constants set to unity sequence OEIS number n = 0 , 1 , 2 , . . . b n A052820 1 , 2 , 9 , 62 , 572 , 6604 , 91526 , . . . 15
RPCCT 2011 Workshop It is easily verified that the sequence b n , n ≥ 0 has the exponential generating function K c f ( x 0 ) = 1 − M c x 0 + ( mM c K d /M d ) ln(1 − M d x 0 ) . Since f is analytic at z 0 = 0, by Theorem 2 the smallest geometric growth constant is M b = 1 / | x ′ 0 | , where x ′ 0 is the singularity nearest to the origin � � � M c − M d ��� 1 1 x ′ 0 = 1 − mK d W mK d exp . M d mM c K d Thus, the lemma is proved. Remark: The proof of Theorem 3 follows directly from Lemmas 1 and 2. 16
RPCCT 2011 Workshop Theorem 4: Let X = { x 0 , x 1 , . . . , x m } . Let c ∈ R ℓ GC �� X �� and d ∈ R m GC �� X �� with growth constants K c , M c > 0 and K d , M d > 0, c and ¯ respectively. Assume ¯ d are globally maximal series with growth constants K c , M c > 0 and K d , M d > 0, respectively . If b = c ◦ d and ¯ c ◦ ¯ b = ¯ d then | ( b, ν ) | ≤ (¯ b i , x | ν | 0 ) , ν ∈ X ∗ , i = 1 , 2 , . . . , ℓ, where the sequence (¯ b i , x k 0 ), k ≥ 0 has the exponential generating function � mK d exp( M d x 0 ) + M d x 0 − mK d � f ( x 0 ) = K c exp . M d /M c Therefore, the radius of convergence is infinity. Remark: Consistent with the known fact that global convergence is not preserved under the cascade connection. 17
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