Multilayer Decomposition Techniques for the Identification of High-Order Nonlinear Systems

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Project title

Code/計畫編號

MOST103-2221-E019-022

Translated Name/計畫中文名

高階非線性系統鑑定之多層解構技術

Project Coordinator/計畫主持人

Ching-Hsiang Tseng

Funding Organization/主管機關

National Science and Technology Council

Department/Unit

Department of Electrical Engineering

Website

https://www.grb.gov.tw/search/planDetail?id=8351587

Year

2014

Start date/計畫起

01-08-2014

Expected Completion/計畫迄

31-07-2015

Bugetid/研究經費

600千元

ResearchField/研究領域

電信工程

本計畫的目標是要開發使用伏特拉級數(Volterra series)來鑑定高階非線性系統的實用方法。伏特拉級數已經被廣為使用來成功地解決科學及工程上遭遇到的許多非線性問題。以伏特拉級數將非線性系統模式化時牽涉到系統之時域或頻域的伏特拉核心(Volterra kernel)鑑定。即使是對一個相對低階的伏特拉模型而言，所需要的核心係數數目亦經常很大，這使得伏特拉核心的鑑定成為一件極具挑戰性的任務，在非線性階數高時則更是如此。在文獻中我們可以找到許多鑑定非線性系統伏特拉核心的方法，多數這些方法僅能用在相對低階之非線性系統，而無法直接推廣適用於高階非線性系統。雖然有些方法的確理論上可應用於高階非線性系統，但它們在處理高階非線性系統時所需之超高複雜度卻使這些方法在實際上不可行。因此開發適用於高階非線性系統之實用伏特拉核心鑑定技術能顯著提昇伏特拉模型用於解決科學及工程上高階非線性問題時之便利性。有鑑於此，我們在本計畫中提出高階非線性系統之多層解構(multilayer decomposition)技術，這個技術的構想是基於我們近期所開發之頻譜凹口(spectral notch)法的推廣。此一多層解構的效果是透過設計各式具頻譜凹口的輸入訊號去激發非線性系統來達成。藉由使用這些輸入訊號，我們證明了一個高階非線性系統鑑定任務可以被系統化地解構成多個低階非線性系統鑑定子任務。此一解構有將伏特拉核心鑑定任務之複雜度降為原有複雜度之1/O(M2)的效果。此外，這個程序可以持續進行以達成將高階非線性系統多層次解構的目的。由於每一層解構都可以將複雜度降為上一層之1/O(M2)，我們可以依需要增加解構層數，直到所得之子任務的複雜度降至可接受的範圍為止。在本計畫中我們於高階非線性系統之實數及複數伏特拉模型皆有探討。由於實數及複數之伏特拉模型有不同之架構，因此也會有不同之多層解構結果。我們推導出多層解構程序之細節，並以電腦模擬驗證其正確性。此外，我們亦將開發之多層解構技術應用於鑑定正交分頻多工系統之非線性通道。我們發現，由於正交分頻多工系統之數位本質及某些特性，我們所提出的多層解構技術必須要再客製化才能適用於正交分頻多工系統。我們也以電腦模擬驗證了所得客製化多層解構技術的正確性及效能。本計畫之成果不僅使基於伏特拉級數之非線性系統鑑定理論與實務更加鞏固，亦促使以往無法達成之高階非線性系統鑑定任務成為易事。The goal of this project is to develop a practical method for identifying high-order nonlinear systems using the Volterra series. The Volterra series has been widely used to successfully solve many nonlinear problems encountered in science and engineering. Modeling a nonlinear system with a Volterra series involves identifying the time-domain or the frequency-domain Volterra kernels of the system. The number of the required kernel coefficients is often very large even for a relatively low-order Volterra model. This makes the identification of the Volterra kernels a very challenging task, especially when the order of the nonlinearities is large. Various methods for identifying the Volterra kernels of nonlinear systems can be found in the literature. Most of these methods can only be used for systems with relatively low order nonlinearities, they can not be easily extended to suit high-order nonlinear systems. Although some of these methods are indeed applicable to high-order nonlinear systems in theory, their prohibitively high complexity in dealing with high-order nonlinear systems makes them infeasible in practice. Therefore, the development of a practical Volterra kernel identification method for high-order nonlinear systems can greatly facilitate the usage of the Volterra model to solve scientific and engineering problems involving high-order nonlinearities. Bearing this goal in mind, we develop a multilayer decomposition technique for identifying high-order nonlinear systems in this project. The idea is based on a generalization of the spectral notch method we have recently developed. The way to accomplish the multilayer decomposition is by designing various input signals with spectral notches to excite the nonlinear system. By using these input signals, we show that a high-order nonlinear system identification task can be systematically decomposed into low-order nonlinear system identification subtasks. Such a decomposition has an effect of lowering the complexity of the Volterra kernel identification task by a rate of 1/O(M2). Furthermore, this process can be successively conducted so that a multilayer decomposition of the high-order nonlinear system identification task can be achieved. Since each layer of decomposition reduces the complexity by a rate of 1/O(M2) with respect to its preceding layer, one can increase the layers in the decomposition as desired until the complexity of the resulting subtasks becomes acceptable.

Keyword(s)

非線性系統
伏特拉模型
系統鑑定
多層解構
高階頻譜
頻譜凹口
頻域分析
正交分頻多工
通道估測
nonlinear system
Volterra kernel
system identification
multilayer decomposition
higher-order spectrum
spectral notch
frequency-domain analysis
OFDM
channel estimation

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Multilayer Decomposition Techniques for the Identification of High-Order Nonlinear Systems

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