本文關(guān)鍵詞：不同空間域下擴(kuò)散反應(yīng)方程的邊界控制　出處：《東華大學(xué)》2017年碩士論文　論文類型：學(xué)位論文

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【摘要】：在物理學(xué)、化學(xué)、生物學(xué)等各種工程領(lǐng)域中存在著大量的擴(kuò)散反應(yīng)現(xiàn)象,擴(kuò)散反應(yīng)方程(又稱熱方程)是描述這些工程領(lǐng)域中各種擴(kuò)散反應(yīng)現(xiàn)象的數(shù)學(xué)模型。擴(kuò)散反應(yīng)方程的邊界控制問題在物理學(xué)、生物學(xué)和工程實(shí)踐中有著廣泛的應(yīng)用前景,比如溫度控制、濕紡碳纖維成形過程和化學(xué)反應(yīng)過程等,因此備受廣大研究人員的關(guān)注。擴(kuò)散反應(yīng)方程的邊界控制的主要目標(biāo)是通過賦予邊界控制律,使得不同空間域下的系統(tǒng)能夠在控制律的作用下最終穩(wěn)定至平衡態(tài)。本文主要系統(tǒng)研究了三種空間域下擴(kuò)散反應(yīng)方程的邊界控制問題,針對(duì)具體的擴(kuò)散反應(yīng)系統(tǒng),建立相應(yīng)的PDE模型。在Backstepping控制算法的基礎(chǔ)上,引入Volterra積分映射進(jìn)行可逆變換,建立原系統(tǒng)和穩(wěn)定的目標(biāo)系統(tǒng)的等價(jià)關(guān)系,將原系統(tǒng)轉(zhuǎn)化為指數(shù)穩(wěn)定的目標(biāo)系統(tǒng),利用其等價(jià)性,獲得核函數(shù)方程,通過邊界控制器和核函數(shù)的關(guān)系,得到邊界控制器的精確解,最后基于Lyapunov穩(wěn)定性理論,證明了系統(tǒng)的穩(wěn)定性。首先,研究了n維球?qū)ΨQ空間域下擴(kuò)散反應(yīng)方程的邊界控制問題,因?yàn)樵S多實(shí)際系統(tǒng)可用n維超球坐標(biāo)系來描述,且系統(tǒng)具有球?qū)ΨQ的性質(zhì),所以可通過研究半徑方向的狀態(tài)變化,得到系統(tǒng)的全局動(dòng)態(tài)過程,通過將高維的對(duì)稱系統(tǒng)轉(zhuǎn)化為等價(jià)的徑向一維方程,運(yùn)用Backstepping方法設(shè)計(jì)邊界控制器,利用容易測(cè)量的邊界狀態(tài)值,設(shè)計(jì)了狀態(tài)觀測(cè)器來估計(jì)系統(tǒng)在空間域的所有狀態(tài),從而實(shí)現(xiàn)輸出反饋控制。同時(shí),利用Lyapunov方法,證明了施加反饋控制的閉環(huán)系統(tǒng)具有H1范數(shù)下的指數(shù)穩(wěn)定性。其次,本文還研究了圓柱面空間域下二維擴(kuò)散反應(yīng)方程的邊界控制問題,圓柱的底面是一個(gè)圓形區(qū)域,而圓柱面的邊界就是其表面區(qū)域。狀態(tài)變量關(guān)于圓心旋轉(zhuǎn)對(duì)稱,采用傅里葉級(jí)數(shù)展開法,二維系統(tǒng)可轉(zhuǎn)化為等價(jià)的一維拋物方程,實(shí)現(xiàn)了系統(tǒng)的降維。設(shè)計(jì)穩(wěn)定的目標(biāo)系統(tǒng),引入Volterra可逆積分映射,建立原系統(tǒng)到目標(biāo)系統(tǒng)的等價(jià)關(guān)系,利用原系統(tǒng)與目標(biāo)系統(tǒng)的等價(jià)性,獲得核函數(shù)方程,最后利用邊界控制器和核函數(shù)的關(guān)系,得到邊界控制器的精確解。本文還設(shè)計(jì)了邊界觀測(cè)器,通過邊界微分的測(cè)量值,去估計(jì)整個(gè)系統(tǒng)在圓柱面空間域下的所有狀態(tài)值,實(shí)現(xiàn)全狀態(tài)反饋。最后,本文研究了輸入時(shí)滯系統(tǒng)擴(kuò)散反應(yīng)方程的邊界控制問題,特別對(duì)反應(yīng)系數(shù)依賴于空間變量的時(shí)滯系統(tǒng),通過引入運(yùn)輸方程,和擴(kuò)散反應(yīng)方程系統(tǒng)構(gòu)成級(jí)聯(lián)系統(tǒng),把原系統(tǒng)中的時(shí)滯項(xiàng)轉(zhuǎn)移到運(yùn)輸方程系統(tǒng)中,利用Volterra可逆積分映射,建立了級(jí)聯(lián)系統(tǒng)和穩(wěn)定的目標(biāo)系統(tǒng)的等價(jià)關(guān)系。利用等價(jià)關(guān)系獲得三個(gè)核函數(shù)方程,運(yùn)用分離變量法分別求解出各個(gè)核函數(shù)的解,根據(jù)邊界控制器和核函數(shù)的關(guān)系,最終得到邊界控制器的解。本文運(yùn)用有限差分法進(jìn)行數(shù)值仿真,驗(yàn)證了邊界控制器能使開環(huán)不穩(wěn)定的系統(tǒng)迅速收斂到穩(wěn)態(tài)值。
[Abstract]:There are a lot of diffusion reactions in physics, chemistry, biology and other engineering fields. Diffusion reaction equation (also called thermal equation) is a mathematical model to describe various diffusion reaction phenomena in these engineering fields. The boundary control problem of diffusion reaction equation is in physics. Biological and engineering practices have a wide range of applications, such as temperature control, wet spinning carbon fiber forming process and chemical reaction process. The main goal of the boundary control of diffusion reaction equation is to assign the boundary control law. The system in different space domain can be stabilized to equilibrium state under the action of control law. In this paper, the boundary control problem of diffusion reaction equation in three kinds of space domain is studied systematically. For the specific diffusion reaction system, the corresponding PDE model is established. Based on the Backstepping control algorithm, the Volterra integral mapping is introduced for reversible transformation. The equivalent relation between the original system and the stable target system is established, and the original system is transformed into the exponential stable target system. By using its equivalence, the kernel function equation is obtained, and the relation between the boundary controller and the kernel function is obtained. The exact solution of the boundary controller is obtained, and the stability of the system is proved based on Lyapunov stability theory. Firstly, the boundary control problem of diffusion reaction equation in n-dimensional spherical symmetric space is studied. Because many practical systems can be described in n-dimensional hypersphere coordinate system, and the system has the property of spherical symmetry, the global dynamic process of the system can be obtained by studying the state changes in the radius direction. By transforming the high-dimensional symmetric system into an equivalent radial one-dimensional equation, a boundary controller is designed by using the Backstepping method, and the boundary state value is easily measured. A state observer is designed to estimate all the states of the system in spatial domain, and the output feedback control is realized. At the same time, the Lyapunov method is used. It is proved that the closed-loop system with feedback control has exponential stability under H _ 1-norm. Secondly, the boundary control problem of two-dimensional diffusion reaction equations in cylindrical space is also studied. The bottom surface of a cylinder is a circular region and the boundary of a cylinder is its surface. The state variable is symmetric about the center of the circle and the Fourier series expansion method is used to transform the two-dimensional system into an equivalent one-dimensional parabolic equation. The dimensionality reduction of the system, the design of a stable target system, the introduction of Volterra reversible integral mapping, the establishment of the equivalent relationship between the original system and the target system, and the use of the equivalence between the original system and the target system. The kernel function equation is obtained and the exact solution of the boundary controller is obtained by using the relation between the boundary controller and the kernel function. The boundary observer is also designed and measured by the boundary differential. To estimate all the state values of the whole system in the cylindrical space domain, and realize the full state feedback. Finally, the boundary control problem of the diffusion reaction equation of the input time-delay system is studied in this paper. Especially for time-delay systems whose response coefficients are dependent on spatial variables, transport equations and diffusion reaction equations are introduced to form cascaded systems, and the time-delay terms in the original system are transferred to the transport equation system. The equivalence relation between cascade system and stable target system is established by using the Volterra reversible integral mapping, and three kernel function equations are obtained by using the equivalence relation. According to the relationship between the boundary controller and the kernel function, the solution of the boundary controller is obtained. The finite difference method is used to carry out the numerical simulation in this paper. It is proved that the boundary controller can make the open loop unstable system converge rapidly to the steady state.
【學(xué)位授予單位】：東華大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O231

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