Swing-up control for an inverted pendulum with restricted cart rail length

In this paper, we propose a new swing-up strategy for cart inverted pendulums with restricted rail length. The proposed swing-up strategy is derived from a new Lyapunov function. The Lyapunov function is defined as the sum of the square of the pendulum energy and the weighted square of the cart’s velocity. The resulting swing-up strategy is represented in a compact form and has two design parameters. By adjusting these design parameters, we can affect the swing-up strategy such that the restriction on the rail length is satisfied. We also provide a state-dependent transformation to obtain voltage input to a DC motor required to generate the cart’s acceleration obtained from the proposed swing-up strategy. Finally, we illustrate the performance of the proposed swing-up law through simulation and experiments. It is shown that there is quite good correspondence between theory and experiments. Recommended by Editorial Board member Duk-Sun Shim under the direction of Editor Jae Weon Choi. This work was supported by an Inha Research Grant. Ji-Hyuk Yang received the M.Sc. degree in Electrical Engineering from Inha University, Inchon, Korea, in 2008. He is currently pursuing a Ph.D. degree in Electrical Engineering at Inha University, Inchon, Korea. His primary research interest lies in the development of rapid control prototyping environment. Su-Yong Shim received the B.Sc. degree in Electrical Engineering from Inha University, Inchon, Korea, in 2008. He is currently pursuing his M.Sc. degree in Electrical Engineering at Inha University, Inchon, Korea. His research interests are mechatronics and embedded systems. Jung-Hun Seo received the B.Sc. degree in Electrical Engineering from Inha University, Inchon, Korea, in 2008. He is currently pursuing his M.Sc. degree in Electrical Engineering at Inha University, Inchon, Korea. His research interests are mechatronics, embedded systems, and control applications. Young Sam Lee received the B.S. and M.S. degrees in Electrical Engineering from Inha University, Inchon, Korea in 1997 and 1999, respectively. He received the Ph.D. at the School of Electrical Engineering and Computer Science from Seoul National University, Seoul, Korea, in 2003. His research interests include time delay systems, receding horizon control, signal processing, and embedded systems. He is currently with the School of Electrical Engineering, Inha University, Incheon, Korea.     

基于混合遗传算法的力矩受限圆轨二级倒立摆摆起控制

为了实现力矩受限时圆轨二级倒立摆非线性系统的摆起控制,提出了一种新的开环优化控制策略.该策略通过在规定的时间内向系统施加一前馈控制序列,使上、下摆杆从自然悬垂位置摆到倒立点位置并使速度为零.为了获得最优摆起控制序列,采用混合遗传算法进行优化计算,通过编码操作解决控制力矩受限问题.仿真实验证明,该策略是有效和可行的,并为其他非线性多变量系统的控制提供了有效方法.     

Swing-up of the double pendulum on a cart by feedforward and feedback control with experimental validation

The swing-up maneuver of the double pendulum on a cart serves to demonstrate a new approach of inversion-based feedforward control design introduced recently. The concept treats the transition task as a nonlinear two-point boundary value problem of the internal dynamics by providing free parameters in the desired output trajectory for the cart position. A feedback control is designed with linear methods to stabilize the swing-up maneuver. The emphasis of the paper is on the experimental realization of the double pendulum swing-up, which reveals the accuracy of the feedforward/feedback control scheme.     

Stabilisation designs for the inertia wheel pendulum using saturation techniques

In this article, three new global stabilisation designs are presented for the inertia wheel pendulum (IWP) using saturation techniques. The first controller is obtained using a twice differentiable saturation function and the backstepping technique. Without using the backstepping technique, the other two controllers are achieved in such a way that the IWP is equivalently transformed into a feedforward system with higher order terms, and then corresponding saturated controllers are put forward. The central idea of the latter two designs is to attenuate the higher order terms by selecting small saturation levels. For the three designs, global asymptotical stability is proven in a bottom-up manner, by showing that the closed-loop system has no finite escape time and reduces to an asymptotically stable dynamics. Simulation results show that the proposed control laws are effective.     

环形一级倒立摆摆起及稳定控制研究

本文通过拉格朗日方程推导出环形一级倒立摆系统的数学模型,用能量方法对该模型进行摆起控制;在倒立摆成功摆起后,在倒立平衡点对系统模型进行线性化,并用线性二次型最优控制(LQR)算法对倒立摆线性化模型进行稳定控制。仿真结果表明本文给出的控制方法是有效的。     

惯性轮倒立摆的鲁棒镇定

用一个时滞状态反馈控制律镇定惯性轮倒立摆,不仅保证闭环系统全局渐近稳定,还允许闭环系统承受一定的时滞.将惯性轮倒立摆转化为存在高阶非线性的4维积分器链,然后设计一个明确规定了饱和度和时滞参数的饱和控制律.用简单方式证明了闭环的全局渐近稳定性.仿真表明设计是有效的.     

基于RBF的倒立摆摆起角度控制研究

针对小车一级倒立摆的起摆控制,利用径向基函数神经网络的自适应能力,微调系统的控制参数,构造一个具有自调整能力的控制器来增大倒立摆摆起角度范围。结果表明,基于RBF网络的PID控制器较常规PID控制器具有更强的自适应能力、更高控制精度和更好的鲁棒性,能满足控制系统的实时性要求。     

Reaction wheel pendulum control using fourth‐order discontinuous integral algorithm

The fourth‐order model of the reaction wheel pendulum is considered and a fourth‐order discontinuous integral algorithm is used for stabilization and tracking of the system, using a continuous control signal. The states reach the origin or a reference signal in finite‐time, even in presence of uncertain control coefficient and a kind of matched and unmatched uncertainties/disturbances. A homogeneous Lyapunov function is designed to ensure local finite‐time stability of the system, which can be used for designing the controller gains. Simulations and experimental results illustrate the performance and advantages of the presented algorithm.     

DRNN在倒立摆摆起控制中的研究

针对小车一级倒立摆的起摆控制,以DRNN神经网络作为辨识器,在线自适应调整PD控制器的两项参数。在起摆范围相同的情况下,DRNN神经网络控制的倒立摆系统其模型参数变化范围为-50%～30%,传统PD控制倒立摆系统其参数变化范围为-40%～20%。结果表明,基于DRNN神经网络的PD控制器比传统的PD控制器具有较强的抗干扰能力和自适应能力,系统鲁棒性增强,效果明显优于传统的PD控制器。     

Shaping stable periodic motions of inertia wheel pendulum: theory and experiment

We consider an underactuated two‐link robot called the inertia wheel pendulum. The system consists of a free planar rotational pendulum and a symmetric disk attached to its end, which is directly controlled by a DC‐motor. The goal is to create stable oscillations of the pendulum, which is not directly actuated. We exploit a recently proposed feedback‐control design strategy based on motion planning via virtual holonomic constraints. This strategy is shown to be useful for design of regulators for achieving orbitally exponentially stable oscillatory motions. The main contribution is a step‐by‐step procedure on how to achieve oscillations with pre‐specified amplitude from a given range and an arbitrary independently chosen period. The theoretical results are verified via experiments with a real hardware setup. Copyright © 2009 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Modeling and vibration control for a flexible pendulum inverted system based on a PDE observer

This study addresses the problem of trajectory control of a flexible pendulum inverted system on the basis of the partial differential equation (PDE) and ordinary differential equation (ODE) dynamic model. One of the key contributions of this study is that a new model is proposed to simplify the complex system. In addition, this study proposed a nonlinear PDE observer to estimate distributed positions and velocities along flexible pendulum. Singular perturbation method is proposed to solve the coupling system of nonlinear PDE observer. The nonlinear PDE observer is divided into a fast subsystem and a slow subsystem by the use of the singular perturbation method. To stabilise this fast subsystem, a boundary controller is proposed at the free end of the beam. The sliding-mode control method is proposed to design controller for slow subsystems. The asymptotic stability of both the proposed nonlinear PDE observer and controller is validated by theoretical analysis. The results are illustrated by simulation.     

倒立摆的可变增益迭代学习控制算法

对倒立摆系统的平衡控制问题进行研究。在建立系统数学模型的基础上,提出指数变增益迭代学习控制律,并设计了控制器。通过系统仿真实验,结果表明：与常规迭代学习控制律相比较,本文采用的方法收敛速度大大加快,系统动态性能得到很大改善。     

倒立摆的迭代学习控制方法研究

针对倒立摆系统典型的非线性、多变量和自然不稳定的特点,在建立系统数学模型基础上,利用迭代学习控制方法,根据动态指标的要求设计了控制器,并完成了系统仿真,结果表明:与传统反馈控制相比较,所设计的迭代学习控制器使系统的动态性能得到了很大的改善。     

Minimum-time control for an inverted pendulum under force constraints

In order to numerically solve the minimum-time control problem of a linear system, the system is usually discretized with a fixed sampling period. Then the minimum count of control steps is searched to meet the constraints of the final state and the input variables. Since the count is a variable, there is no direct way for handling such problems except by exhaustive iteration. In contrast to the traditional methods, a new numerical technique was developed recently to avoid the exhaustive iteration. In this method, the control step is fixed and the sampling period is treated as a variable. Since this method requires only two iterations, it will reduce the computation time significantly. This paper applies this new numerical technique to generate the minimum-time trajectory between two end-points for an inverted pendulum under force constraints. Two main issues are addressed. The first one is the problem formulation in discrete-time domain and the second one is the generation of feasible solutions for the global search. Simulation examples are included for illustration.     

倒立摆系统的最优控制应用研究

针对倒立摆系统的不稳定性,对最优控制理论在倒立摆控制系统中的应用进行了分析,设计LQR控制器,并在倒立摆实验装置上进行了实验。实验结果表明设计的控制器是有效的,对倒立摆系统的平衡稳定控制效果好,提高了系统的抗干扰能力。     

非线性倒立摆系统平衡点反馈控制参数的优化计算

本文基于圆轨倒立摆非线性系统模型,建立了平衡点状态反馈控制参数的高精度优化算法,编制计算程序确定出反馈控制参数,并对圆轨倒立摆平衡点的实例控制.与将系统线性化后的反馈参数的计算分析相比,本文研究方法具有较大平衡范围.仿真与实例平衡控制都表明了本文研究方法的可行性.     

Rate limited time optimal control of a planar pendulum

Time optimal control problems often admit as solution bang-bang controls but they present several drawbacks in real applications where additional degrees of regularity are desirable. In this paper, the rate limited time optimal swing up of an inverted pendulum is considered and necessary conditions that have to be satisfied by optimal trajectories in the presence of rate constraints are derived. A comparison between this approach and the results from the standard time optimal problem is reported.     

Dynamic modeling and step-climbing analysis of a two-wheeled stair-climbing inverted pendulum robot

ABSTRACT

In this study, the control of a two-wheeled stair-climbing inverted pendulum robot and its climbing motion are analyzed and discussed. The robot adopts a state-feedback controller with a feed-forward constant to stabilize the body and achieve step-climbing motion. The control parameter is considered based on the dynamic model motion on a flat surface and the static model of motion on the step. For climbing stairs with a narrow step tread, a constant torque is applied to reduce the space required for recovering the body stability after climbing. The stability of the robot is numerically analyzed by analyzing the orbital stability of its limit cycle. The stability analysis shows that the control method can achieve a stable stair-climbing motion. The effectiveness of the control method is demonstrated through an experiment. The result indicates that the robot can climb the stairs, and the required time for climbing a single step is approximately 1.8?s.     

The direct Lyapunov method for the stabilisation of the Furuta pendulum

A nonlinear controller for the stabilisation of the Furuta pendulum is presented. The control strategy is based on a partial feedback linearisation. In a first stage only the actuated coordinate of the Furuta pendulum is linearised. Then, the stabilising feedback controller is obtained by applying the Lyapunov direct method. That is, using this method we prove local asymptotic stability and demonstrate that the closed-loop system has a large region of attraction. The stability analysis is carried out by means of LaSalle's invariance principle. To assess the controller effectiveness, the results of the corresponding numerical simulations are presented.     

互联双倒立摆自适应容错模糊控制

针对互联双倒立摆系统,提出了一种基于时滞代换的自适应容错控制方案。该方案用有界的参考信号代换模糊逼近器输入中的未知时滞信号,这使得控制器的设计不再依赖于时滞假设条件。容错反推控制技术和自适应技术相结合来处理代换误差和逼近误差。所提出的方案能有效补偿所有四种类型的执行器故障,同时还可保证闭环系统的全局稳定性。仿真结果进一步验证了该方法的有效性。