摘要线性二次型最优控制器的设计是最优控制理论在实践中的一个重要应用。其中需要控制的对象是一个线性系统，而需要的优化的性能指标是包含系统的状态变量和(或)控制变量的二次型函数的积分。能够优化性能指标的最优解可以通过求解矩阵黎卡提方程获得。在实践中，能够优化性能指标的控制器通常通过设计基于矩阵黎卡提方程的解的线性状态反馈回路来实现。由于这类控制器通常能够兼顾多项性能指标，因此得到了特别的重视。目前已经在工业生产、航空和航天等领域取得了广泛的应用。84426

      单极倒立摆系统是一个典型的不稳定系统，具有非线性、强耦合性等特点，且影响系统的变量较多。由于这些特点，单极倒立摆系统的控制方法在航天、机器人，军工和一般过程控制等领域有广泛的应用。本文针对单极倒立摆系统进行数学建模和能控分析，采用基于线性二次型最优控制的方法为单极倒立摆系统设计最优控制器，并通过MATLAB 对设计出的控制器进行仿真，通过设定不同的参数以验证该控制器对系统平衡控制的有效性，并找到其中的最优控制方法。

毕业论文关键词：数学建模；最优控制；倒立摆系统；系统仿真

Abstract The design of linear quadratic optimal controller is an important application of the optimal control theory in practice。 Given a linear system, the performance index that needs to be optimized is the integral of a quadratic function that depends on the state variable of the system and (or) the control variable。 The optimal solution can be obtained by solving the matrix Riccati equation。 In practice, the controller that can optimize the performance index is realized by designing a linear state feedback loop which is based on the solution of the matrix Riccati equation。 Because this kind of controller is able to take into account a number of performance indexes, researchers have paid special attention to its design and analysis。 At present, it has been widely used in the fields of industrial production, aviation and spaceflight。

The unipolar inverted pendulum system is a typical unstable system, which has the characteristics of being nonlinear, strong coupling and so on。 Because of these characteristics, the control methods of unipolar inverted pendulum system have been extensively used in the fields of aerospace, robotics, military and general process control。 In this thesis, based on mathematical analysis and modeling of a unipolar inverted pendulum system, an optimal controller is designed using the principles of the linear quadratic control theory。 The performance of the controller is simulated and evaluated with MATLAB。 The parameters are varied to evaluate the effectiveness of the controller and find the optimal control method。       

Keywords: Mathematical modeling; Optimal control; Inverted pendulum system; System simulation

目录

第一章 绪论 1

1。1引言 1

1。2国内外研究现状 1

1。3倒立摆的特点和应用价值 4

1。4 倒立摆的控制方法 5

第二章 单级倒立摆模型的建立 7

2。1引言 7

2。2 直线一级倒立摆 7

2。3单级倒立摆系统数学模型的建立 10

2。4本章小结 12

第三章 线性二次型最优控制理论 13