以延遲相關方式探討二維狀態延遲系統之強健濾波與控制問題

Abstract

本論文主要以延遲相關方式探討二維狀態延遲系統的強健濾波和控制問題。一個系統的動態行為若受到兩個獨立整數變數i 和j 影響，則稱此系統為二維系統。二維信號和系統經常被廣泛應用在影像處理、數位信號處理和程序控制方面，因此二維系統的研究近年來逐漸受到重視。二維系統中的特例，二維狀態延遲系統，常見於許多實際應用中，譬如物質輾壓過程、偏差分方程式、圖像資料處理和傳輸等方面。因此，二維狀態延遲系統的分析和設計問題成為值得研究的課題。本研究的分析和設計是採用線性矩陣不等式的架構。 首先，本論文探討二維狀態延遲系統的漸進穩定性，利用線性矩陣不等式的推導技術，得出一個充分穩定性條件，此條件與水平和垂直方向延遲大小有關。其次，提出具延遲相關的H∞ 和H2 性能準則。然後利用該準則發展出一套解決強健H∞，H2 與混合H∞/ H2 濾波器的設計方法。有別於傳統的相關研究，本論文針對整個參數不確定性空間多面體的不同頂點，採用不同的李雅普諾夫矩陣設計強健濾波器，此種設計方法可得到較不保守的設計結果。 最後，本論文針對二維狀態延遲系統，探討狀態回授控制器的設計問題。為了使整個設計過程建立在線性矩陣不等式的架構上，另外推導出延遲相關之強健穩定性條件，然後利用此條件發展出強健穩定化方法，找出一個狀態回授控制器使得整個閉迴路系統不只強健穩定，且不受系統參數不確定性影響。

This dissertation studies the robust filtering and control problems for two-dimensional (2-D) state-delayed systems in the Fornasini-Marchesini second model by using a delay-dependent approach. A 2-D system is one that has dynamics depending on two independent integer variables i and j. 2-D signals and systems have become more and more important in the fields like image processing, digital signal processing, and process control. The study of 2-D systems has attracted increasing attentions in recent years. A particular case of 2-D systems, 2-D state-delayed systems, can be found in many practical applications such as the material rolling process, partial difference equation modeling, and image data processing/transmission. Thus the analysis and synthesis of 2-D state-delayed systems are worthwhile investigation issues. The main focus of this research is the use of linear matrix inequality (LMI) techniques for both analysis and synthesis problems. Firstly, a computationally tractable sufficient condition for the asymptotic stability of 2-D state-delayed systems, which depend on the size of delays in both horizontal and vertical directions, are derived in terms of LMIs. Then, delay-dependent H-infinity performance and H-2 performance criteria are proposed. Based on the results, efficient methods to solve the robust H-infinity filtering, H-2 filtering, and mixed H-2/H-infinity filtering problems are developed. Differently from the quadratic stability framework, the filter design methods in this dissertation adopt the parameter-dependent Lyapunov function approach, which utilizes different Lyapunov matrices in the entire polytope domain and produces less conservative design results. Finally, the state feedback controller synthesis problem for the system is also considered. A new delay-dependent robust stability condition is derived, and used to develop a robust stabilization method. The goal is to find a state feedback controller such that the closed-loop system is robustly stable for all admissible uncertainties.