| 摘要: | 篇論文主題有三： (一).現有時延系統之穩定度分析方法之探討。且針對特定時間延遲情況，提出更寬鬆的穩定度條件，及更新的解決方法。 (二).針對以 T-S 模糊模型表示之時延非線性系統，分別設計出狀態迴授控制器、靜態輸出迴授控制器及濾波器，並使之達成 H2 或 H∞ 性能要求。 (三).在設計靜態輸出回授控制器時，解決雙線性矩陣不等式 (BMI) 之求解的問題。 T-S 模糊模型的優點在於其可完全代表一個非線性系統，因此針對一個非線性時間延遲系統，且透過 Lyapunov 定理，T-S 模糊模型提供一套系統化的方法來研究該非線性時間延遲系統之穩定度問題與性能要求。在使用這套方法時，為避免模糊規則數太多所造成控制器或濾波器求解不易的問題，非線性時延系統與 T-S 模糊模型之間會存有誤差，為使此誤差不至於影響所求得控制器或濾波器對原有非線性時延系統之控制或估測效益，本論文引用強健控制概念將誤差以參數不確定項 (norm bounded uncertainty) 來表示，因此可確保原系統與模型之間的誤差為零，即非線性時延系統可精確的表達成具有不確定項的 T-S 時延模糊模型。 現有的時延系統的分析方法主要分成兩大部分： 1. Lyapunov-Razumikhin 法。 2. Lyapunov-Krasovskii 法。 Lyapunov-Razumikhin 法是最早用於時延系統的分析方法，然由於其本身條件限制，已漸漸不受青睞。 Lyapunov-Krasovskii 法是目前最常見的分析方法，本論文也將採用此方法用於穩定度分析。除此之外，本論文也將提出一套方法來修正採用 Lyapunov-Krasovskii 法時所造成之延遲時間之變動率(rate of varying on delay-times)受限的問題。另外，本論文也將提出一套有別於前述兩種方法之新的穩定度分析技巧。再者，為了便於完整表達時延系統之分析技巧，延遲時間可分成以下四種類別： 1. 延遲時間固定且已知 (fixed and known delay time)。 2. 延遲時間固定且未知 (fixed but unknown delay time)。 3. 延遲時間變動且變動率小於一 (time-varying delay with rate of varying less than 1)。 4. 延遲時間變動且變動率不受限 (time-varying delay with rate of varying being unlimited)。 在連續系統中，由於固定的延遲時間之已知或未知並不影響分析方法，故一併討論。且在現有的文獻中，延遲時間皆受限於變動率小於一之限制，也就是這些文獻只能處理微小變動的延遲時間問題。因此，本論文提出一套新的方法移除此項限制，也就是說大變動的延遲時間是允許的。 至於在離散系統中，由於不涉及微分動作，故類別 2. 3. 4. 之解法相同，因此僅討論延遲時間固定且已知，及延遲時間未知兩種情況。在延遲時間固定且為已知的情況下，本論文亦提出一個新的方法可使穩定條件比現有方法之條件更加寬鬆。 針對上述具有不確定項的 T-S 時延模糊模型，本論文根據平行分散式補償器 (Parallel Distributed Compensation, PDC) 的概念，分別設計狀態回授控制器、靜態輸出回授控制器及 Luenberger 濾波器，並探討其性能控制或性能估測問題。 狀態回授控制問題中，本篇採用一般常見之 PDC 形式的狀態回授控制器，根據 Lyapunov 定理及重新定義變數後，推導出可使系統穩定並滿足性能要求之線性矩陣不等式形式 (LMI) 的充分條件，利用電腦求出所需之控制器增益。控制系統中，當系統狀態無法完全獲知時，則必須採用估測器獲得所需的資訊或直接以輸出回授做控制，因此本論文另一個課題即是探討靜態輸出回授控制問題。在靜態輸出回授控制中，最大的問題在於所推導出的穩定或性能條件是以雙線性矩陣不等式 (BMI) 的形式呈現，而 BMI 無法如同 LMI 般可輕易地由現有工具程式求解。因此，此部份的重點放在如何求解 BMI 的問題上，本論文利用以下兩種方法來解決其 BMI 求解問題： 1. V-K 疊代線性矩陣不等式 (V-K iterative LMI)。 2. 系統相似型轉換 (Similarity Transformation)。 透過這些方法，可將輸出回授性能控制問題中的 BMI 條件轉換為 LMI 形式來求解，或者經由假設特定變數並反覆求解某些 LMI 來達到求解 BMI 的目的。另外，在文末亦說明了動態輸出回授控制與靜態輸出回授控制間的關連性，所以上述方法亦可應用於求解動態輸出回授的控制器增益。 在濾波器的設計部分，採用的是 Luenberger 形式，同樣地透過 Lyapunov 定理及上述時延系統的分析方法，同樣地各提出一組以線性矩陣不等式 (LMI) 形式表示的充分條件，可同時使誤差系統穩定，並達成所要之性能需求。 上述之穩定度與性能問題乃是針對具有參數不確定項之 T-S 時延模糊模型所做的探討，無論連續或離散時間系統皆考慮其中。由於具有不確定項的 T-S 時延模糊模型可精確的表達非線性時間延遲系統，因此，所設計的控制器或濾波器將可直接用於穩定或估測原有非線性時延系統，並達到所要求的性能限制。 In this dissertation, several methods for a class of uncertain Takagi-Sugeno fuzzy models with time delays are proposed. Via Lyapunov theory, the asymptotical stability of delayed fuzzy systems is assured. In addition to stability issue, this dissertation also investigates the system performances including H∞ , H2 and mixed-norm considerations. Applying Lyapunov-Krasovskii approach rather than Lyapunov-Razumikhin to analyzedelayed systems is the most popular method nowadays due to reason that the performance considerations can be incorporated. Moreover, the characteristics/classifications of delay time is one of the most important and interesting subjects in delayed schemes. For the sake of completeness, delay time can be classified into four categories such as (a). fixed and known delay time, (b). fixed but unknown delay time, (c). time-varying delay with rate of varying less than 1, (d). time-varying delay with rate of varying unlimited. For discrete-time systems, Lyapunov-Krasovskii approach generates almost the same derivational techniques for case (b), (c) and (d). Similarly, for continuous-time systems, the techniques for case (a), (b) and (c) are the same. Therefore, the discussions of delay systems are organized in four parts such as 1. discrete-time systems with fixed and known delay time, 2. discrete-time systems with varying or unknown delay time, 3. continuous-time systems with fixed or slow-varying delay time, 4. continuous-time systems with fast-varying delay time. In the first case, a new method is proposed in this dissertation. Via a theoretical proof, the relaxation of our approach in comparison with existing ones is guaranteed. As for the third case investigating the fixed or slow-varying delay time for continuous-time systems, the existing literature have a limitation on derivative of delay time being less than one. This is why slow-varying case is named after. However, this limitation is not always satisfied, or the information of it is not available at all time, especially in filtering frameworks. Therefore two approaches are offered to remove this restriction. In other words, analyzing a nonlinear system subject to fast-varying delay time is allowed. We will refer to this case as the fourth case. Moreover, the stabilization and estimation are also included in this dissertation.Although the state feedback controller is a typical design approach. the system states are not always completely available, thus an idea of stabilizing systems via output signals arises and is known as static output feedback regulators. It is well known that the output feedback control is a bilinear matrix inequality (BMI) problem and that the feasible solutions can not be obtained via existing convex algorithms. A reasonable approach known as iterative LMI (ILMI) is utilized to solve this BMI problem but the solvability may depend on the initial guess. Therefore the second method, similarity transformation, is proposed to convert the BMI problems into an LMI. Since a system with or without similarity transformation applied does not change its stabilizability. Therefore regulators determined from transformed systems can stabilize the original systems successfully. Since dynamic output feedback controllers can be converted into static output feedback regulators via augmenting some state vectors, the proposed methods for static output feedback controls can be directly implied to dynamic controllers. Although there are two basic types of filters, the Luenberger H∞ filters are more robust than the Kalman-type since the advantage of using a Luenberger filter in comparison with a Kalman filter is that the former needs no statistical assumption on the exogenous signals. Therefore a Luenberger filter is employed in this dissertation to achieve our objectives in the filtering problems. Since nonlinear systems can be exactly represented by a class of Takagi-Sugeno fuzzy models with norm-bounded parametric uncertainties, the proposed approaches can directly be extended to stabilize/estimate such retarded nonlinear systems. |
