空间多级倒立摆非线性控制方法研究
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摘要
本文以实现空间四级倒立摆实物系统稳定控制为核心,针对多级倒立摆的控制方法及其硬件结构的优化进行了研究.首先,建立了空间n级倒立摆的数学模型,提出了变增益LQR控制方法,讨论了滤除倒立摆系统中白噪声的控制方法；其次,基于变增益LQR控制方法完成了导轨受限情况下二级倒立摆逐级摆起控制仿真实验及实物实验；第三,对三级倒立摆的自动摆起问题在理论上进行了探讨,完成了三级倒立摆的自动摆起控制仿真实验；最后,对空间四级倒立摆控制系统进行了定性分析,设计了空间四级倒立摆系统的硬件结构并进行了优化,实现了世界上首例空间四级倒摆实物控制.
本文主要内容如下：
1.采用Lagrange方法,分别建立了空间一、二、三级倒立摆的数学模型,并给出了n级空问倒立摆系统的数学模型的统一表达式；通过对空间四级倒立摆系统建模,验证了n级空间倒立摆系统的数学模型统一表达式的正确性.
2.针对一类自治非线性控制系统,提出了变增益LQR控制方法,并成功应用于多级倒立摆的实物控制系统.该方法是一种基于状态的增益调度反馈控制方法,其主要特点是在控制过程中,实时计算非线性控制系统的雅可比矩阵,实时求解代数黎卡提方程,进而得到随状态变化的实时反馈增益.基于变增益LQR控制方法,对LQG控制器进行了改进,仿真和实物控制实验均取得了较好的控制效果.
3.变增益LQR控制器在倒立摆自动摆起控制中的应用.首先,实现了导轨受限情况下二级倒立摆非线性系统的逐级自动摆起控制.其次,改进了逆系统轨迹控制器,实现了基于逆系统轨迹控制的三级倒立摆自动摆起控制的仿真实验.
4.空间四级倒立摆实物控制系统.普通倒立摆的摆杆只能在一个铅垂面内摆动,而空间倒立摆的摆杆可以在三维空间内自由转动,自由度增加一倍.因此,从控制理论和控制工程的意义上讲,空间倒立摆实物系统控制的实现要比普通倒立摆的实现困难得多,不仅因为这样的系统其变量、非线性程度及不稳定性等因素成倍增加,而且有关机械结构的设计和电子器件的选择也遇到瓶颈性的困难.本文对空间四级倒立摆系统进行了定性分析,并设计了与模型匹配的杆系结构和驱动平台.最后,实现了空间四级倒立摆控制的实物实验.实验结果表明：空间四级倒立摆系统不但具有良好的稳定性和鲁棒性,还可使倒立摆小车行走到指定的位置.
In this paper, some control methods for mult-rod inverted pendulum system are studied and the physical structure of inverted pendulum system is optimized, which is aimed to implement the stabilization of the spherical quadruple inverted pendulum sys-tem. Firstly, the mathematical model of spherical n-rod inverted pendulum is derived; the variable gain linear quadratic regulator(VGLQR) control technique is proposed, and control method for filtering the white noise from the inverted pendulum control system is discussed. Secondly, the swing-up control of nonlinear double inverted pendulum un-der restricted rail length is realized based on VGLQR control technique. Thirdly, the swing problem of triple inverted pendulum is theoretically discussed, and the simulation for swing-up of a triple inverted pendulum is achieved. Finally, the qualitative analy-sis of spherical quadruple inverted pendulum is discussed; the structure of the spherical quadruple inverted pendulum system is designed and optimized; the spherical quadruple inverted pendulum is firstly stabilized in the world.
The main works in the thesis are as follows:
1. The detailed process of inferring the mathematical model of spherical single/ double/triple inverted pendulum is proposed, and the model of spherical n-rod inverted pendulum(SNIP) is conclude. In addition, the model of the spherical quadruple inverted pendulum is derived via the Lagrangian method, by which the correctness of the mathe-matical model of SNIP is Verified.
2. For a class of autonomous nonlinear control systems, the VGLQR control tech-nique is proposed, and the controller of VGLQR is successfully applied to stabilizing a multi-rod inverted pendulum. This method is a kind of gain-scheduled feed-back control technique based on states variables. The main feature of VGLQR is to obtain the Jacobi matrix and resolve the algebraic Riccati equations at each sampling time online, and a more precise real-time feedback gain matrix, which is changing with respect to states, is obtained. Based on VGLQR technique, the controller of LQG is improved, and the results of simulation and experiments illustrate the new LQG controller is of significant performance.
3. The VGLQR control technique is applied in control systems for swing-up inverted pendulum. At first, based on VGLQR technique, the swing-up of a double inverted pendulum under restricted rail length is realized. Then, the inversion-based controller is improved, and the simulation experiment for swinging-up of the triple inverted pendulum is achieved.
4. Physical experiment for the spherical quadruple inverted pendulum is imple-mented. Because the rod of ordinary inverted pendulum can only rotate in a plumb plane, and the rod of spherical inverted pendulum can free rotate in three-dimensional space, the degrees of freedom of the spherical inverted pendulum are as twice as that of ordinary inverted pendulum. Therefore, both in theory and practice, it is more difficult to stabilize a spherical inverted pendulum than to stabilize a ordinary inverted pendulum because not only the factors of state variables, nonlinearity and instability fold increase in spherical inverted pendulum systems but also great difficult is encountered for the design of physical construction and the component selection. In this paper, the qualitative anal-ysis of spherical quadruple inverted pendulum is discussed, and the physical construction, which is matched with the mathematical model, is designed. Finally, the physical exper-iment is very well implemented, and the result illustrates that it not only has quite good stability and robustness, but also is able to make the cart of the pendulum moving to the place where it is appointed in advance.
本文主要内容如下：
1.采用Lagrange方法,分别建立了空间一、二、三级倒立摆的数学模型,并给出了n级空问倒立摆系统的数学模型的统一表达式；通过对空间四级倒立摆系统建模,验证了n级空间倒立摆系统的数学模型统一表达式的正确性.
2.针对一类自治非线性控制系统,提出了变增益LQR控制方法,并成功应用于多级倒立摆的实物控制系统.该方法是一种基于状态的增益调度反馈控制方法,其主要特点是在控制过程中,实时计算非线性控制系统的雅可比矩阵,实时求解代数黎卡提方程,进而得到随状态变化的实时反馈增益.基于变增益LQR控制方法,对LQG控制器进行了改进,仿真和实物控制实验均取得了较好的控制效果.
3.变增益LQR控制器在倒立摆自动摆起控制中的应用.首先,实现了导轨受限情况下二级倒立摆非线性系统的逐级自动摆起控制.其次,改进了逆系统轨迹控制器,实现了基于逆系统轨迹控制的三级倒立摆自动摆起控制的仿真实验.
4.空间四级倒立摆实物控制系统.普通倒立摆的摆杆只能在一个铅垂面内摆动,而空间倒立摆的摆杆可以在三维空间内自由转动,自由度增加一倍.因此,从控制理论和控制工程的意义上讲,空间倒立摆实物系统控制的实现要比普通倒立摆的实现困难得多,不仅因为这样的系统其变量、非线性程度及不稳定性等因素成倍增加,而且有关机械结构的设计和电子器件的选择也遇到瓶颈性的困难.本文对空间四级倒立摆系统进行了定性分析,并设计了与模型匹配的杆系结构和驱动平台.最后,实现了空间四级倒立摆控制的实物实验.实验结果表明：空间四级倒立摆系统不但具有良好的稳定性和鲁棒性,还可使倒立摆小车行走到指定的位置.
In this paper, some control methods for mult-rod inverted pendulum system are studied and the physical structure of inverted pendulum system is optimized, which is aimed to implement the stabilization of the spherical quadruple inverted pendulum sys-tem. Firstly, the mathematical model of spherical n-rod inverted pendulum is derived; the variable gain linear quadratic regulator(VGLQR) control technique is proposed, and control method for filtering the white noise from the inverted pendulum control system is discussed. Secondly, the swing-up control of nonlinear double inverted pendulum un-der restricted rail length is realized based on VGLQR control technique. Thirdly, the swing problem of triple inverted pendulum is theoretically discussed, and the simulation for swing-up of a triple inverted pendulum is achieved. Finally, the qualitative analy-sis of spherical quadruple inverted pendulum is discussed; the structure of the spherical quadruple inverted pendulum system is designed and optimized; the spherical quadruple inverted pendulum is firstly stabilized in the world.
The main works in the thesis are as follows:
1. The detailed process of inferring the mathematical model of spherical single/ double/triple inverted pendulum is proposed, and the model of spherical n-rod inverted pendulum(SNIP) is conclude. In addition, the model of the spherical quadruple inverted pendulum is derived via the Lagrangian method, by which the correctness of the mathe-matical model of SNIP is Verified.
2. For a class of autonomous nonlinear control systems, the VGLQR control tech-nique is proposed, and the controller of VGLQR is successfully applied to stabilizing a multi-rod inverted pendulum. This method is a kind of gain-scheduled feed-back control technique based on states variables. The main feature of VGLQR is to obtain the Jacobi matrix and resolve the algebraic Riccati equations at each sampling time online, and a more precise real-time feedback gain matrix, which is changing with respect to states, is obtained. Based on VGLQR technique, the controller of LQG is improved, and the results of simulation and experiments illustrate the new LQG controller is of significant performance.
3. The VGLQR control technique is applied in control systems for swing-up inverted pendulum. At first, based on VGLQR technique, the swing-up of a double inverted pendulum under restricted rail length is realized. Then, the inversion-based controller is improved, and the simulation experiment for swinging-up of the triple inverted pendulum is achieved.
4. Physical experiment for the spherical quadruple inverted pendulum is imple-mented. Because the rod of ordinary inverted pendulum can only rotate in a plumb plane, and the rod of spherical inverted pendulum can free rotate in three-dimensional space, the degrees of freedom of the spherical inverted pendulum are as twice as that of ordinary inverted pendulum. Therefore, both in theory and practice, it is more difficult to stabilize a spherical inverted pendulum than to stabilize a ordinary inverted pendulum because not only the factors of state variables, nonlinearity and instability fold increase in spherical inverted pendulum systems but also great difficult is encountered for the design of physical construction and the component selection. In this paper, the qualitative anal-ysis of spherical quadruple inverted pendulum is discussed, and the physical construction, which is matched with the mathematical model, is designed. Finally, the physical exper-iment is very well implemented, and the result illustrates that it not only has quite good stability and robustness, but also is able to make the cart of the pendulum moving to the place where it is appointed in advance.
引文
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