【正文】 基于 LQR 的二級(jí)倒立擺控制系統(tǒng) 學(xué) 院 專 業(yè) 班 級(jí) 學(xué) 號(hào) 姓 名 指導(dǎo)教師 負(fù)責(zé)教師 沈陽(yáng)航空 航天大學(xué) 2021 年 6 月沈陽(yáng)航空航天大學(xué)畢業(yè)設(shè)計(jì) {論文 } I 摘 要 倒立擺系統(tǒng)是一個(gè)典型的多變量、非線性、強(qiáng)耦合和快速運(yùn)動(dòng)的高階不穩(wěn)定系統(tǒng)，它是檢驗(yàn)各種新的控制理論和方法有效性的典型理想模型。在其控制過(guò)程中，能有效地反映諸如鎮(zhèn)定性、魯棒性、隨動(dòng)性以及跟蹤等許多關(guān)鍵問(wèn)題。 本文主要研究二級(jí)倒立擺 LQR 控制方法。首先建立了二級(jí)倒立擺的數(shù)學(xué)模型，然后對(duì)二級(jí)倒立擺的數(shù) 學(xué)模型進(jìn)行控制設(shè)計(jì)，應(yīng)用 遺傳算法 確定系統(tǒng)性能指標(biāo)函數(shù)中的加權(quán)陣 Q， R 得到系統(tǒng)狀態(tài)反饋控制矩陣 。用 MATLAB 進(jìn)行了系統(tǒng)仿真。在幾次湊試 Q 矩陣值后系統(tǒng)的響應(yīng)結(jié)果都不盡如人意，于是采用遺傳算法對(duì) Q矩陣優(yōu)化。仿真結(jié)果證明：經(jīng)過(guò)遺傳算法優(yōu)化后的系統(tǒng)響應(yīng)能更加滿足設(shè)計(jì)要求。最后在實(shí)驗(yàn)臺(tái)上進(jìn)行了試驗(yàn)，驗(yàn)證了 matlab中仿真的結(jié)果。 關(guān)鍵詞 ： 二級(jí) 倒立擺 ； LQR 控制 ； 遺傳算法 基于 LQR的二級(jí)倒立擺控制系統(tǒng) II Research on Double Inverted Pendulum Control System based on LQR Abstract The inverted pendulum is a typical high order system, with multi variable, nonlinear, strongcoupling, fleet and absolutely instable. It is representative as an ideal model to prove new control theory and techniques. During the control process, pendulum can effectively reflect many key problems such as equanimity, robust, followup and track, Therefore， this paper studies a control method of double inverted pendulum LQR. First of all, the mathematical model of the double inverted pendulum is established, then make a control design to double inverted pendulum on the mathematical model, and determine the system performance index weight matrix Q, R by using geic algorithm in order to attain the system state feedback control matrix. Finally, the simulation of the system is made by MATLAB. After several test matrix Q value the results are not satisfactory response, then we optimize Q matrix by using Geic Algorithm. Simulation results show: that the system response can meet the design requirements effectively after Geic Algorithm optimization. Key words： Double inverted pendulum。 LQR control。 Geic Algorithm. 沈陽(yáng)航空航天大學(xué)畢業(yè)設(shè)計(jì) {論文 } III 目 錄 摘 要 ...................................................................................................................................... I 1 緒論 ................................................................................................................................... 1 引言 ............................................................................................................................. 1 倒立擺設(shè)備簡(jiǎn)介 ......................................................................................................... 1 國(guó)內(nèi)外研究情況 .......................................................................................................... 5 論文 的主要內(nèi)容 .......................................................................................................... 8 2 倒立擺數(shù)學(xué)模型的建立與分析 ........................................................................................ 9 數(shù)學(xué)建模的方法 ..................................................................................................... 9 二級(jí)倒立擺的結(jié)構(gòu)和工作原理 .................................................................................. 9 拉格朗日運(yùn)動(dòng)方程 .................................................................................................... 10 推導(dǎo)建立數(shù)學(xué)模型 .................................................................................................... 11 二級(jí)倒立擺系統(tǒng)性能分析 ....................................................................................... 18 能控性分析 ........................................................................................................ 18 能觀性分析 ........................................................................................................ 18 3 線性二次最優(yōu)控制算法簡(jiǎn)介 ......................................................................................... 20 線性二次型最優(yōu)調(diào)節(jié)器原理 .................................................................................... 20 加權(quán)陣 Q、 R 的選擇 ................................................................................................. 22 4 遺傳算法 ......................................................................................................................... 23 遺傳算法的基本理論 ............................................................................................... 23 基本遺傳算法 ........................................................................................................... 24 基本遺傳算法的局限性 ............................................................................................ 24 遺傳算法 的求解步驟 ................................................................................................ 25 遺傳算法的特點(diǎn) ........................................................................................................ 26 遺傳算法優(yōu)化加權(quán)陣 ................................................................................................ 28 5 二級(jí)倒立擺 LQR控制設(shè)計(jì)及仿真 ................................................................................. 29 二級(jí)倒立擺控制系統(tǒng)框圖 ....................................................................................... 29 任選加權(quán)陣的 LQR 最優(yōu)控制仿真 ......................................................................... 29 用遺傳算法優(yōu)化 Q 陣的仿真結(jié)果 ............................................................................ 33 實(shí)物演示 ................................................................................................................... 35 基于 LQR的二級(jí)倒立擺控制系統(tǒng) IV 結(jié)論 ..................................................................................................................................... 37 致 謝 ................................................................................................................................... 38 參考文獻(xiàn) ............................................................................................................................. 39 附錄（程序清單） ............................................................................................................. 41