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標題: 不穩定拋物線型偏微分方程不確定系統之邊界控制設計
Boundary Control Design of Unstable Parabolic Partial Differential Equation Systems With Uncertainties
作者: 鄭孟筆
Cheng, Meng-Bi
關鍵字: Distributed-parameter systems;分佈參數系統;Partial differential systems;boundary control;sliding mode control;偏微分方程式;邊界控制;滑動模式控制
出版社: 電機工程學系所
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本論文旨在探討分佈參數系統(distributed-parameter systems)之強健邊界控制問題,其系統模式表示成二階線性拋物線型偏微分方程式(partial differential equations)。此類分佈參數系統又被視為無限維系統(infinite-dimensional systems), 與常見之集中參數系統(lumped-parameter systems)或者有限維系統(finite-dimensional systems) 不同,因其動態模式僅描述成常微分方程式(ordinal differential equations)。明顯地,偏微分方程系統之研究困難度與深度,遠高於常微分方程系統。因此,本論文嘗試採用一般研究常微分方程系統常用觀點如狀態空間(state-space)、取樣資料系統表示(sampled-data system representation)、奇異擾動法(singularly perturbation method)、李阿普諾夫函數(Lyapunov functions)、滑動模式控制(sliding mode control)等,處理該類偏微分方程系統遭受內部模式簡化、參數變動和外部干擾等強健鎮定問題。
在拋物線型偏微分方程系統之中,熱傳導過程(heat conduction process)是最為人所周知,因其系統所有特徵值(eigenvalues)均可在實數軸上從有限實數到負無窮大一系列規則排列,同時其穩定特徵值更是具有快速下降(dropoff)特性,造成該系統動態模式可由有限的慢速模態與無窮個快速模態所組成。藉由直接截斷所有穩定、高頻模式,很自然地可以得到該系統之降階表示式。此種方式數學上等義於奇異擾動系統的慢速近似。針對離散時間控制設計,我們藉由提昇(lifting)技術將該系統重新表示成降階取樣資料系統,該技術可將連續系統切成無窮多個等時間間距的連續點,而所有高頻殘留模式則被集結成為一個直接傳遞項(direct transmission term),包含在該系統之降階有限維模式中。如此一來,一般簡化模式常見的洩漏(spillover)問題可以成功被抑制。同時,藉由非共點(noncollocated)邊界觀測,證明所提出之輸出迴授控制法則對觀測誤差具有強健性。最後,數值模擬驗證出二階取樣資料系統表示法對此無限維系統的可行性。
針對輸出迴授控制問題,我們探討熱傳導過程之最小相位特性(minimum-phase property),引用無限乘積展開(infinite production expansion)與奈奎斯圖形(Nquist plot),指出不同系統輸出情況下最小相位條件,其結果與有限維系統不同。若該系統滿足此一條件,則簡單的比例積分(proportional-integration)邊界控制法則可達成閉迴路系統之穩定性。
此外,利用滑動模式(sliding mode)控制策略來處理拋物線型偏微分方程系統遭受外部干擾以及系統參數不確定所產生的強健鎮定問題。利用沃里泰拉積分轉換(Volterra integral transformation)將原本不穩定系統映射成穩定系統,而所有不穩定因素則集結在新座標系統的邊界上。引用李阿普諾夫方法(Lapunov''s method)建構出一個無限維滑動曲面(sliding manifold),當系統軌跡進入該曲面時,系統能確保其指數穩定性。特別的是,此平面與控制輸入之間的相對階數(relative degree)為零,故可藉由連續邊界控制法則達成控制目的。
本論文所提出方法可拓展至其他更複雜和廣義化的不確定拋物線型系統,其系統參數可隨空間、時間改變。結合切換增益(switching gain)自適應法則,可提出自適應滑動模式邊界控制法則。以上所提強健邊界控制策略可藉由數值模擬方式驗證其可行性,並與其他文獻中成功指標性(benchmark)控制器進行性能比較。

This dissertation addresses the problem of robust boundary control for distributed parameter systems (DPSs) governed by a second-order linear partial differential equation (PDE) of parabolic type. The DPSs are also referred to as infinite-dimensional systems, as opposed to the lumped parameter systems or finite-dimensional systems, which are governed by ordinal differential equations (ODEs). It is clear that the complexity and profoundity of the PDE systems are much higher than those of the ODEs. This dissertation seeks to adapt some of the popular viewpoints from the ODE systems such as the state-space and the sampled-data system formulations, singularly perturbation theory, Lyapunov functions, sliding mode control, …etc., to deal with the boundary control problems of the PDE systems subject to interior model reduction, parametric variations, and exogenous disturbances.
A heat conduction process is one most well-known parabolic PDE system. All its eigenvalues can be arranged orderly from a finite real number through the negative infinity of the real line, in which the stable eigenvalues exhibit a fast dropoff characteristics. Thus, the system dynamics consists of a finite number of slow modes and an infinite number of fast modes. Naturally a reduced-ordered system representation can be immediately obtained by a direct truncation of all the stable and higher modes. Such a truncation is mathematically equivalent to the slow-time approximation of a singularly perturbed system. For discrete-time control design, we propose to reformulate this parabolic PDE system into a truncated, reduced-order sampled-data representation by employing the `lifting'' technique, which chops the continuous-time system response into an infinite number of consecutive equally-spaced time intervals (i.e. the sampling periods). The truncated modes, including all the high-frequency residue dynamics, are lumped into a direct transmission term of this reduced-order finite dimensional model, such that the spillover problem, commonly occurs in the truncated models, can be effectively suppressed. Meanwhile, with the noncollocated boundary observation, the proposed output-feedback control is robust to the observation errors. Simulation results validate the feasibility of second-order sampled-data formulation in this infinite-dimensional plant.
For an output feedback control design, we investigate the minimum-phase property of the heat conduction process. By employing the infinite production expansion and Nyquist plot, the conditions of minimum-phase of the system with different outputs have been verified, which are differ to the results of finite-dimensional systems. Once the system satisfies with this condition, a simple proportional-integration boundary control scheme can achieve closed-loop stability.
The robust stabilization problem of the parabolic PDE system subject to exogenous disturbances and system coefficient uncertainties is addressed by employing the sliding mode control method. A Volterra integral transformation is employed to map the original unstable system into a stable one, with all the unstable factors concentrated in the boundary conditions of the new coordinates. We use the Lypunov''s method to construct an infinite-dimensional sliding manifold. Such a sliding manifold is proven to ensure exponential stability once the PDE system trajectory travels on it. The relative degree of the selected sliding variable with respect to the control input is zero, such that the sliding mode can be achieved with a continuous boundary control law.
The proposed methods can be extended to other more complicated and generalized parabolic systems, in which uncertainties with both spacial and temperal dependencies are present. An adaptive sliding-mode boundary control strategy is developed to incorporate a switching gain adaptation law. The effectiveness and feasibility of the presented robust boundary control strategies are confirmed in numerical simulations and also compared with other well-established benchmark controllers in the literature.
其他識別: U0005-2601201011250400
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