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Wavelet Analysis for System Identification∗ Wavelet analysis for system identification∗ 大阪教育大学・数理科学 芦野 隆一 (Ryuichi Ashino) Mathematical Sciences, Osaka Kyoiku University 大阪電気通信大学・数理科学研究センター 萬代 武史 (Takeshi Mandai) Research Center for Physics and Mathematics, Osaka Electro-Communication University 大阪教育大学・情報科学 守本 晃 (Akira Morimoto) Information Science, Osaka Kyoiku University Abstract By the Schwartz kernel theorem, to every continuous linear sys- tem there corresponds a unique distribution, called kernel distribution. Formulae using wavelet transform to access time-frequency informa- tion of kernel distributions are deduced. A new wavelet based system identification method for health monitoring systems is proposed as an application of a discretized formula. 1 System Identification A system L illustrated in Figure 1 is an object in which variables of different kinds interact and produce observable signals. The observable signals g that are of interest to us are called outputs. The system is also affected by external stimuli. External signals f that can be manipulated by the observer are called inputs. Others are called disturbances and can be divided into those, denoted by w, that are directly measured, and those, denoted by v, that are only observed through their influence on the output. Mathematically, a system can be regarded as a mapping which relates inputs to outputs. Hereafter, we will use the terminology “system” rather ∗This research was partially supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Scientific Research (C), 15540170 (2003– 2004), and by the Japan Society for the Promotion of Science, Japan-U.S. Cooperative Science Program (2003–2004). 1 Unmeasured Measured disturbance v disturbance w System L Output g Input f Figure 1: System L. than “mapping”. A system L is said to be linear if L[α1f1 + α2f2] = α1L[f1] + α2L[f2], for arbitrary constants α1, α2 and arbitrary inputs f1, f2. To choose a model set for linear systems, the most general setting should be given by using the Schwartz kernel theorem [Tr67]. Let us denote by D(Rn), the space of compactly supported C∞ functions with the canonical LF -topology [Tr67]. A distribution T in Rn is a continuous linear form on D(Rn). The set of all the distributions in Rn is denoted by D 0(Rn) and the duality of distributions is denoted by h·, ·i*, that is, the duality between T ∈ D 0 and φ ∈ D is written as * * T (φ) = hT, φi = hT (x), φ(x)ix. The space of C∞ functions in Rn with the canonical Fr´echet-topology is denoted by E (Rn) and its topological dual space is denoted by E 0(Rn), which is the space of compactly supported distributions in Rn. On the other hand, as the inner product of L2 is denoted by h·, ·i, we have hf, gi* = hf, gi, f, g ∈ L2. The space of C∞ functions in Rn rapidly decreasing at infinity is called Schwartz space and denoted by S (Rn). The topological dual space of S (Rn) is denoted by S 0(Rn) and an element of S 0(Rn) is called tempered distribu- tion. Theorem 1 (The Schwartz kernel theorem) Let L : D(Rn) → D 0(Rn) be a continuous linear system. Then, there is a unique distribution k ∈ D 0(R2n) such that * n L[f](x) = hk(x, y), f(y)iy, f ∈ D(R ). The distribution k is called kernel distribution of L. If L : S (Rn) → S 0(Rn), then k ∈ S 0(R2n). 2 Invariance under fundamental operators Let us define three fundamental operators for time-frequency analysis. They are unitary operators when they act on L2(Rn). Define the translation oper- ator Ta by n Taf(x) = f(x − a), a ∈ R , the modulation operator Mξ by ixξ n Mξf(x) = e f(x), ξ ∈ R , and the dilation operator Dρ by −n/2 −1 Dρf(x) = ρ f(ρ x), ρ ∈ R+ := { x ∈ R ; x > 0 }. For simplicity, we consider Dρ only for ρ > 0 although we can consider Dρ for ρ ∈ R\{0}. A continuous linear system L : D(Rn) → D 0(Rn) is said to be translation- n n invariant if TaL[f] = L[Taf] for every f ∈ D(R ) and every a ∈ R .A continuous linear system L : D(Rn) → D 0(Rn) is said to be modulation- n n invariant if MξL[f] = L[Mξf] for every f ∈ D(R ) and every ξ ∈ R .A continuous linear system L : D(Rn) → D 0(Rn) is said to be dilation-invariant n if DρL[f] = L[Dρf] for every f ∈ D(R ) and every ρ ∈ R+. In this section, we will give necessary and sufficient conditions on the kernel distribution k(x, y) corresponding to a continuous linear system L : D(Rn) → D 0(Rn) for invariance under the three fundamental operators. The proofs can be found in [AMM2]. Proposition 1 Let L : D(Rn) → D 0(Rn) be a continuous linear system and k(x, y) be its kernel distribution. The system L is translation-invariant if 0 n and only if there exists a unique h ∈ D (R¡ ) such¢ that k(x, y) = h(x−y), that is, L[f] = h ∗ f. As a result, we have L D(Rn) ⊂ E (Rn). The distribution h is called the impulse response of L. n 0 n 0 n If L is¡ continuous¢ from S (R ) to S (R ), then h ∈ S (R ), and hence n n we have L S (R ) ⊂ OM (R ), where OM is the space of slowly increasing C∞ functions. Proposition 2 Let L : D(Rn) → D 0(Rn) be a continuous linear system and k(x, y) be its kernel distribution. The system L is modulation-invariant if and only if there exists a unique g ∈ D 0(Rn) such that L[f] = gf for every f ∈ D(Rn). Proposition 3 Let L : D(Rn) → D 0(Rn) be a continuous linear system and k(x, y) be its kernel distribution. The system L is dilation-invariant if 1 and only if k(ρx, ρy) = k(x, y) for every ρ ∈ R . ρn + 3 Stability Let L : D(Rn) → D 0(Rn) be a continuous linear system, and k be its kernel distribution. L is said to be Lp-stable (1 ≤ p ≤ ∞) if there exists a constant n C such that kL[f]kLp ≤ CkfkLp for every f ∈ D(R ). If p < ∞ and L is Lp-stable, then L can be extended to a bounded linear operator from Lp(Rn) to Lp(Rn). Here, we concentrate on the case when p = 2. When the kernel distribution k is locally integrable, the following is well- known. (For example, [La02], Theorem 2 and Theorem 3, §16.1.) Proposition 4 (1) If k ∈ L2(R2n), then L is L2-stable, and 2 n kL[f]kL2(Rn) ≤ kkkL2(R2n)kfkL2(Rn) for every f ∈ L (R ). (1.1) (2) Assume that there exist constants M1, M2 such that Z Z n |k(x, y)| dx ≤ M1, |k(x, y)| dy ≤ M2 for a.e. x ∈ R . (1.2) Rn Rn Then, L is L2-stable, and p 2 n kL[f]kL2(Rn) ≤ M1M2kfkL2(Rn) for every f ∈ L (R ). Let L be translation-invariant and h be its impulse response. We have the following ([St70], IV, §3.1; [H¨o60]). Proposition 5 L is L2-stable ⇐⇒ hˆ ∈ L∞(Rn), where hˆ is the Fourier transform of h. Causality Causality is natural for a physical system in which the variable is time. It means that the response at time t depends only on what has happened before and at t. In particular, a system does not respond before there is an input. Thus causality is a necessary condition for a system to be physically realizable. Let L be a continuous linear system D(Rn) → D 0(Rn), and k ∈ D 0(R2n) be its kernel distribution. A continuous linear system L is said to be causal if n supp L[f] ⊂ supp f + R+ n for every f ∈ D(R ). Here, A + B := { a + b ; a ∈ A, b ∈ B } and R+ := [0, ∞). We simply write a + B for {a} + B. A distribution f ∈ D 0(Rn) is n said to be causal if supp f ⊂ R+ . When L is translation-invariant, we have the following lemma. 4 Lemma 1 Let h ∈ D 0(Rn). If L[f] = h ∗ f for f ∈ D(Rn), then the following three conditions are equivalent. (a) L is causal. (b) If f is causal then L[f] is causal. (c) h is causal. 2 Wavelet Analysis of Linear Systems Assume that ψ ∈ L2(Rn) satisfy Z ∞ |ψb(sξ)|2 0 < Cψ := ds < ∞, (2.1) 0 s where Cψ is independent of ξ. Note that this condition is satisfied if ψ(6= Z0) is a radially symmetric continuous function with compact support, and ψ(x) dx = 0. For simplicity we further assume that kψkL2(Rn) = 1. Rn The continuous wavelet transform of f ∈ L2(Rn) with respect to ψ is defined by Z ³ ´ −n/2 x − b Wψf(b, a) : = |a| f(x) ψ dx = hf, TbDaψiL2(Rn), Rn a (2.2) n (b, a) ∈ Hn := R × R+. 2 n+1 It is well-known that Wψf ∈ H1 := L (Hn; dbda/a ) and 2 n 2 n hWψf, WψgiH1 = Cψhf, giL (R ) for every f, g ∈ L (R ). (2.3) (See, for example, [Gr01]. As for a group representation theoretic approach to wavelet transforms, see [Wo02].) Let L be a bounded linear operator from L2(Rn) to L2(Rn). We state theorems concerning the interaction between L and Wψ. The proofs can be found in [AMM2].
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