平面倒立摆建模与最优控制算法的研究

倒立摆系统是一个多变量、非线性、高阶次、强耦合、欠驱动的自然不稳定系统,是自动控制理论教学和研究中典型的物理模型。本文以平面倒立摆系统为研究对象,使用拉格朗日方程建立了多级平面倒立摆的数学模型。采用线性二次最优控制算法,分别设计了LQR及LQR-模糊控制器,实现平面多级倒立摆的平衡控制,结论证明了本文设计的LQR控制器有很好的稳定性、鲁棒性和适应性。     

一级倒立摆状态反馈控制系统设计

倒立摆控制系统是一个复杂的、不稳定的、非线性系统,对倒立摆系统的研究能有效的反映控制中的许多典型问题。对一级直线型倒立摆,首先运用牛顿运动定律建立倒立摆系统的运动方程,进而求出系统的状态空间表达式,建立数学模型。其次运用状态反馈极点配置法,以小车的位移、速度,摆杆与竖直向上的偏角、摆角变化速度作为四个状态变量,由给定的控制要求求出状态反馈增益矩阵,将极点配置在控制要求的位置。另外考虑到系统的某些状态如小车速度和摆杆角速度不容易直接测量等,本文分别基于小车和摆杆子系统设计了两个全维观测器,分别对状态量进行了重构并给出了仿真结果分析。     

非线性倒立摆起摆和稳定控制研究

该文深入研究了直线倒立摆的摆起和稳定控制。通过两种方法对倒立摆非线性模型进行线性化得到线性化模型。第一种方法是利用微分几何李导数理论，对坐标和输入输出进行变换，然后在局部进行线性化得到一个线性化模型；另一种方法直接在平衡点附近进行线性化得到线性化模型。最后用MATLAB仿真和实物控制分别证明了能量反馈理论和稳定控制理论具有良好的控制效果，且通过Lie理论对系统进行坐标变换得到线性化模型比略去高阶项得到的线性化模型具有更好的鲁棒性和控制效果。     

三级倒立摆的LQR方法优化参数控制

倒立摆是智能控制的理想对象。使用拉格朗日方程建立三级倒立摆系统的非线性数学模型,在平衡点处对其线性化,利用LQR（Linear Quadratic Regulator）最优控制理论,导出控制规律。通过对三级倒立摆一系列稳定摆动和加扰实验仿真曲线的分析,明确了加权矩阵Q中各权系数对系统稳定性控制的重要性,由此来优化权系数的选择。实验表明,系统显示出较好的鲁棒性和动态性能。     

Optimal PID Control of Spatial Inverted Pendulum With Big Bang – Big Crunch Optimization

As the extension of the linear inverted pendulum (LIP) and planar inverted pendulum (PIP), this paper proposes a novel spatial inverted pendulum (SIP). The SIP is the most general inverted pendulum (IP) than any existing IP. The model of the SIP is presented for the first time. The SIP inherits all the characteristics of the LIP and the PIP, which is a nonlinear, unstable and underactuated system. The SIP has five degrees of motion freedom and three control forces. Thus, it is a multiple-input and multiple-output (MIMO) system with nonlinear dynamics. To realize the spatial trajectory tracking of the SIP, the control structure with five PID controllers will be designed. The parameter tuning of the multiple PIDs is a challenging work for the proposed SIP model. To alleviate the difficulties of the parameter tuning for the multiple PID controllers, optimal PIDs can be achieved with the help of Big Bang-Big Crunch (BBBC) optimization. The BBBC algorithm can successfully optimize the parameters of the multiple PID controllers with high convergence speed. The optimization performance index of the BBBC algorithm is compared with that of the particle swarm optimization (PSO). Simulation results certify the rightness and effectiveness of the proposed control and optimization methods.      

倒立摆的自适应积分反步控制策略

针对直线单级倒立摆在模型参数不确定和外部扰动情况下的稳定控制问题,提出一种自适应积分反步控制策略。采用拉格朗日方程建立倒立摆系统的运动学模型,为减少稳态误差,将误差积分项引入反步法,设计了倒立摆的控制器;对含有未知参数的系统非线性状态微分方程,设计适当的Lyapunov函数推导出系统未知参数的自适应更新律,削弱了参数不确定性的影响。将自适应积分反步控制与一般的反步法控制、模糊控制及神经网络控制的仿真结果进行了对比,并在LabVIEW开发环境下进行了实物实验。结果表明,自适应积分反步法可以较为迅速且精确地完成稳定控制,较好地克服系统参数不确定及外部扰动的影响,具有较强的鲁棒性。     

二级倒立摆状态反馈控制器设计优化方法

设计了一种二级倒立摆的状态反馈控制器。研究了一种基于单纯形算法的参数优化方法。首先给出了二级倒立摆系统模型并分析了系统特性,设计了状态反馈控制器,提出基于ITAE(Integrated Time&Absolute Error)性能指标应用单纯形优化方法对状态反馈控制参数进行优化设计,以快速准确地得到状态反馈阵,并利用固高GLIP2002型直线倒立摆系统进行仿真分析和二级倒立摆系统控制实验,实现了二级倒立摆优化控制。提出的一种如何寻找使系统稳定且具有较好性能的状态反馈控制参数这一控制领域重要问题的解决方法,避免了极点配置法手动试凑方法调试参数困难的问题,对非线性、强耦合运动体的控制具有理论意义和应用价值。     

A state space embedding approach to approximate feedback linearization of single input nonlinear control systems

This contribution presents a numerical approach to approximate feedback linearization which transforms the Taylor expansion of a single input nonlinear system into an approximately linear system by considering the terms of the Taylor expansion step by step. In the linearization procedure, higher degree terms are taken into account by using a state space embedding such that the corresponding system representation has not to be computed in every linearization step. Linear matrix equations are explicitly derived for determining the nonlinear change of coordinates and the nonlinear feedback that approximately linearize the nonlinear system. If these linear matrix equations are not solvable, a least square solution by applying the Moore–Penrose inverse is proposed. The results of the paper are illustrated by the approximate feedback linearization of an inverted pendulum on a cart. Copyright © 2006 John Wiley & Sons, Ltd.     

一种基于非线性滤波的倒立摆控制方法研究

对于倒立摆这样的强非线性系统,采用传统的BP算法存在着收敛速度慢、易陷入局部极小值的缺陷,而采用卡尔曼滤波方法则会带来很大的模型误差。为了解决上述问题,提出了基于粒子滤波优化神经网络的方法。首先建立了倒立摆神经网络控制器的物理模型并将模型粒子化,而后用粒子滤波算法对粒子进行优化估计,将估计结果作为网络的权值应用到倒立摆控制中,采用离线训练方式,仿真比较了卡尔曼滤波和粒子滤波两种方法控制效果,结果表明,新算法较卡尔曼滤波方法在控制性能上有明显提高。     

不确定平面二级倒立摆的鲁棒自适应控制

为提高倒立摆控制系统的抗扰动能力,降低其对未建模动态等的敏感度,研究了不确定平面二级倒立摆的鲁棒自适应控制器的设计方法。把倒立摆动力学模型分解为确定和不确定两部分,用一个非线性参数化模糊逻辑系统逼近平面二级倒立摆的不确定动态,采用李雅普诺夫稳定性理论推导出使平面二级倒立摆的状态误差渐近收敛的鲁棒控制器及自适应律。理论分析和仿真结果表明所提出的控制算法是有效的。     

Optimal Control of Nonlinear Inverted Pendulum System Using PID Controller and LQR: Performance Analysis Without and With Disturbance Input

Linear quadratic regulator(LQR) and proportional-integral-derivative(PID) control methods, which are generally used for control of linear dynamical systems, are used in this paper to control the nonlinear dynamical system. LQR is one of the optimal control techniques, which takes into account the states of the dynamical system and control input to make the optimal control decisions.The nonlinear system states are fed to LQR which is designed using a linear state-space model. This is simple as well as robust. The inverted pendulum, a highly nonlinear unstable system, is used as a benchmark for implementing the control methods. Here the control objective is to control the system such that the cart reaches a desired position and the inverted pendulum stabilizes in the upright position. In this paper, the modeling and simulation for optimal control design of nonlinear inverted pendulum-cart dynamic system using PID controller and LQR have been presented for both cases of without and with disturbance input. The Matlab-Simulink models have been developed for simulation and performance analysis of the control schemes. The simulation results justify the comparative advantage of LQR control method.     

单级倒立摆稳定控制及其联合仿真

针对倒立摆系统的稳定控制,提出了一种基于广义扩展线性化的非线性控制方法,设计了非线性状态反馈控制器.同时利用ADAMS软件建立单级倒立摆虚拟样机模型,通过其输入输出接口实现对Matlab的通信,并进行了ADAMS/Controls控制模块与Matlab联合仿真分析.仿真结果验证了控制方法的有效性,且更接近物理样机控制效...     

Dynamic analysis of a multibody open-chain system

The free plane motion of a compound pendulum is studied using the Lagrange equations of motion. The specific compound pendulum consists of a main plane disk from which are suspended two chains, each of which is composed of m and n rigid plane disks, respectively. A nonlinear system of m + n + 1 ordinary differential equations (ODEs) with respect to the properly selected generalized coordinates, is obtained. For small-amplitude swing the above system becomes a linear differential system. The eigenvalues and eigenvectors constitute the eigenfrequencies and the modeshapes, respectively, of the free swing of the considered compound pendulum. Also, relations are extracted which ensure a unique swing with no relative rotation between any two disks. Finally, several numerical results are given.     

DETERMINATION OF FEEDBACK FOR HUMAN POSTURE CONTROL WITHOUT PHYSICAL INTERVENTION

Abstract

This paper is a study of a method to determine from external observations only, potential sets of feedback gains that can be used by humans to control their postural stance.

These gains amplify sensor signals in order to produce torques at the joints. The method consists of modeling the human biped in two ways-as a one degree-of-freedom inverted pendulum with a vestibular estimator and as a two degree-of-freedom compound inverted pendulum for which the vestibular system is approximated. The feedback gains in these models are systematically changed until the computer simulated motion of the nonlinear model is close enough to observed human motion. Comparisons are made between these results and those of other experimental investigations.

An analysis of minimal sets of gains and sensitivity of the gains to measurement errors in weights, lengths, and moment of inertia are discussed.     

Design of optimized cascade fuzzy controller based on differential evolution: Simulation studies and practical insights

In this study, we discuss a design of an optimized cascade fuzzy controller for the rotary inverted pendulum system and ball & beam system by using an optimization vehicle of differential evolution (DE). The structure of the differential evolution optimization environment is simple and a convergence to optimal values realized here is very good in comparison to the convergence reported for other optimization algorithms. DE is easy to use given its mathematical operators. It also requires a limited computing overhead. The rotary inverted pendulum system and ball & beam system are nonlinear systems, which exhibit unstable motion. The performance of the proposed fuzzy controller is evaluated from the viewpoint of several performance criteria such as overshoot, steady-state error, and settling time. Their values are obtained through simulation studies and practical, real-world experiments. We evaluate and analyze the performance of the proposed optimal fuzzy controller optimized by Genetic Algorithm (GA), and DE. In this setting, we show the superiority of DE versus other methods being used here as well as highlight the characteristics of this optimization tool.     

平面二级倒立摆的圆周行走与镇定控制

采用拉格朗日方程建立了平面二级倒立摆的非线性动力学模型, 将其在平衡位置线性化, 得到系统在两个正交控制方向解耦的近似模型. 针对每一个方向上由互相耦合的基座小车定位子系统和摆杆镇定子系统所组成的六阶欠驱动系统, 设计了自适应滑模模糊控制器, 实现了基座小车沿圆周行走条件下摆杆的镇定控制, 验证了该控制算法对欠驱动、不稳定、多变量耦合系统的有效性.     

On the generation and solution of the symbolic,open-loop characteristic equation for a double inverted pendulum

Symbolic algebra manipulation is applied computationally to the mechanical system described by the inverted pendulum of two links, to yield analytic roots of the open-loop characteristic equation obtained from the linearized, symbolic dynamic equations of motion of the system. Detailed analysis of the characteristic polynomial form is presented, together with conditions relating polynomial root properties to physical parameter inequalities. Finally, features of the theoretical form of the characteristic polynomial for the general n-link system are proposed, by simple explication of results provided by this low order system, and also those existing for the corresponding single-link case which may be obtained and checked directly.     

RBF-ARX模型在倒立摆系统建模中的应用

针对直线一级倒立摆控制系统的非线性特性,采用RBF-ARX模型对倒立摆系统的全局非线性动态特性进行建模.讨论了RBF-ARX模型结构的选取,模型参数辨识,RBF参数优化等问题.并且分别比较了该倒立摆系统的RBF-ARX模型与全局线性ARX模型,以及将RBF-ARX在某一工作点局部线性化后的模型与局部线性ARX模型的预测输出和模型误差,验证了RBF-ARX模型在倒立摆系统建模和辨识中的有效性.     

三级倒立摆的自动摆起与稳定控制

采用非线性逆系统轨迹控制实现三级倒立摆的自动摆起,并设计了变增益LQR控制器将其稳定在竖直倒立位置.首先,三级倒立摆从静止下垂状态摆起到竖直倒立位置的过程,从数学角度看是一个两点边值问题,通过求解该两点边值问题获得摆起的标称轨迹,利用逆系统方法设计前馈控制,同时结合增益调度反馈控制使摆起过程稳定;其次,在稳定控制阶段,...     

A neural network-based approximation method for discrete-timenonlinear servomechanism problem

A feedback controller that solves the discrete-time nonlinear servomechanism problem relies on the solution of a set of nonlinear functional equations known as the discrete regulator equations. The exact solution of the discrete regulator equations is usually unavailable due to the nonlinearity of the system. The paper proposes to approximately solve the discrete regulator equations using a feedforward neural network. This approach leads to an effective way to practically solve the discrete nonlinear servomechanism problem. The approach has been illustrated using the well-known inverted pendulum on a cart system. The simulation shows that the control law designed by the proposed approach performs much better than the conventional linear control law.