本论文主要内容是：建立了n 级平面倒立摆模型，利用模糊推理建模法和边缘线性化方法对其进行线性化处理，应用变论域自适应模糊控制理论和线性二次最优法对其进行控制器设计, 完成了三级平面倒立摆仿真和硬件系统。主要内容如下：1. 建立了n 级平面倒立摆的物理模型。首先对n 级平面倒立摆进行了结构描述，设定了记号标记，然后应用多刚体力学的基础知识，分析了万向节和摆杆的运动特性，给出了n 级平面倒立摆的总动能和总势能的表达式，并由此推导出该系统的Lagrange 函数和Lagrange 方程，得到了n 级平面倒立摆的物理模型的方程表达式。2. 将模糊推理建模法应用到n 级平面倒立摆建模当中，并具体给出了一级平面倒立摆模糊推理模型与其物理模型的比较；由于模糊推理模型是变系数非线性模型，应用模糊推理建模的边缘线性化方法，消去变系数非线性模型中的变量交叉项，得到边缘线性化模型，并给出了一级平面倒立摆边缘线性化模型与其物理模型的比较；而当平面倒立摆摆杆级数增加时，其变量数大量增加，这时在对其模型进行边缘线性化操作时，其各个变量的论域划分所得到的片数急剧增加，达到了上百万、上千万片，出现了“维数灾”。针对该问题，本文用局部边缘线性化方法，对三级平面倒立摆进行局部边缘线性化处理，既减少了“维数灾”，又能最大程度的保持对原物理模型的最大逼近，最后给出了三级平面倒立摆局部边缘线性化与其物理模型的比较图。3. 基于n 级平面倒立摆物理模型，给出了n 级平面倒立摆在平衡点x0 = 0 附近的整体线性化模型，进而求得其线性状态方程式。证明了n 级平面倒立摆在平衡点x0 = 0 附近是能观的。对于三级平面倒立摆的边缘线性化模型非线性特性，即状态方程x = Ax+b+Bu 中含有非线性项b，要分析其系统性能很困难，对其近似逼近的线性系统进行分析，得到其近似线性系统的可控可观性。4. 采用变论域自适应模糊控制理论和线性最小二乘法设计了三级平面倒立摆控制器。首先，根据大量实验数据，给出三级平面倒立摆状态变量θ11 、θ12 、dθ11、dθ12在可控范围情况下的估测论域，然后对论域进行分割TΘ11、TΘ12、TdΘ11、TdΘ12。根据θ11 、θ12 、dθ11、dθ12活动范围将三级平面倒立摆进行整体线性化和局部边缘线性化；然后，应用变论域自适应模糊控制理论和线性最小二乘法，分别对三级平面倒立摆的整体线性化和局部边缘线性化部分进行控制器设计。5. 采用变论域自适应模糊控制理论和线性最小二乘法实现了三级平面倒立摆控制系统仿真。该倒立摆系统不仅有良好的稳定性和鲁棒性，还可以使倒立摆走到指定位置。然后，介绍了三级平面倒立摆硬件控制系统的结构，分析了三级平面倒立摆硬件系统面临的困难，给出了三级平面倒立摆主要性能指标，并实现了对三级平面倒立摆硬件系统的控制。

This paper mainly involves the contents as followings: the modeling of n-order spherical inverted pendulum（NSIP）, the linearization of NSIP by using modeling method based on fuzzy inference(MMFI)，the designing of the controller of triple spherical inverted pendulum by adaptive fuzzy controllers based on variable universe, and implementation of the simulation and hardware experiment of the triple spherical inverted pendulum.1. The detailed process of inferring the physical model of NSIP is proposed. The expression of total kinetic energy and potential energy of NSIP are offered. Based on the energy expressions, the Lagrange function and the Lagrange equations of NSIP are presented. The differential equations describing NSIP are given.2. First, The modeling method of fuzzy inference is applied to NSIP, and a nonlinear model with variable coefficients is proposed. Then Marginal linearization method (MLM) is given to turn the nonlinear model with variable coefficients into a linear model with variable coefficients. Because the state variable of spherical inverted pendulum is more, MLM causes to the explosion of numbers of fuzzy inference rules. A Sub-marginal linearization is applied and numbers of fuzzy inference is decreased. The sub-marginal linearization model approximates well the physical model of triple spherical inverted pendulum.3. Based on the physical equations of NSIP, the whole linearization model around the equilibrium point x0 = 0 of NSIP is proposed and the observability is proved. The marginal linearization model of triple inverted pendulum is given as x = Ax + b + Bu with the nonlinear part of b . Through dealing with x = Ax + b + Bu by linear Least Squares method, linearization model is propose and is proved with the controllability and observability.4.The controller of triple spherical inverted pendulum is given by using variable universe adaptive fuzzy control theory and linear Least Squares. According to data by experimenting, the range of state variables θ11 、θ12 、dθ11、dθ12 and the universes of θ11 、θ12 、dθ11、dθ12 are separated respectively. In the different range of state variables triple spherical inverted pendulum is linearised respectively by whole linearization and sub-marginal linearization method. 5. Simulation experiment of triple spherical inverted pendulum is very well implemented by using variable universe adaptive fuzzy control theory and linear Least Squares. The result shows that it not only has quite good stability and robustness, but also is able to make the cart of the pendulum moving to the appointed place. Then the model and the usage of the main components of the triple spherical inverted pendulum hardware system are briefly introduced. The difficulties of hardware system of triple spherical inverted pendulum are given. The parametric value of the hardware system are also given and the controlling of the hardware system of triple spherical inverted pendulum is implemented successfully.