Types of Pendulum Research Paper Example

1. Introduction

There are two types of pendulum. A simple pendulum is an ideal body consisting of a particle suspended by an inextensible wire and negligible mass. When removed from its position of equilibrium and loose, the pendulum will oscillate in a vertical plane under the action of gravity; the movement is periodic and oscillatory.

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The other type of pendulum is called an inverted pendulum and for understanding an analogy can be made with a plate or rope tightrope, where each one seeks to control the position of the center of gravity by keeping it on top of the rope or not dropping the dishes. For the study of control algorithms there are some plant options for implementing the inverted pendulum and in general the most used is the one that has as actuator element a car that moves in the X axis and has fixed a shaft with free axis. As can be observed the inverted pendulum is a naturally unstable system whose control can only be exercised in a small region, it is only possible to establish the control of the pendulum if the variation of its vertical position is very small, this means that if there is a would not be possible to keep it upright. Mathematically the goal is to keep the rod angle close to zero, this control is done through the car movements that seek to balance the rod.

The inverted pendulum is a classic control problem, usually covered in introductory classes of controls and dynamics, and is well known for its excellent analogy for the design of a vibration controller on platforms for the launch of a rocket, as well as for the stabilization of cranes, buildings, robotics and especially for didactic applications, for being an excellent means of checking and evaluating the different control methodologies. Many modern inverted pendulums use gyroscopic sensors, optical encoders with microprocessors or complete computers to implement their control algorithms. While the sophistication of these sensors and the power to estimate these devices have their advantages, they generate a control problem that is not accessible to the student without knowledge of advanced control techniques.

The problem of the inverted pendulum, although it treats a very simple mechanical system, represents several practical situations that can be analyzed from the concepts involved in its study. For example, biomechanical models of the walking mode of human beings, allowing applications in areas such as prostheses and robotic arms, since the standing and stable position of a person when walking is very close to the inverted pendulum (Ribeiro, 2007). In particular, the concept of a mobile inverted pendulum has been used in several applications, such as a human transport vehicle, which in the future may present as an alternative to urban transport (Tirmant et al., 2002), including its use by disabled people. In addition, studies of autonomous robots based on this concept have been performed by NASA, presenting significant results, such as those found in (Ambrose et al., 2004). According to Miller III et al. (1995), the inverted pendulum system has inherently unstable dynamical characteristics, which makes it possible to study different types of controllers, from classical to hybrid controllers and / or based on Artificial Intelligence (AI) thus serving as the target system for such techniques to be put to the test.

Due to their relative ease of implementation, Proportional-Integral-Derivative (PID) compensators are still widely used in industry in SISO (Single Input Single Output) systems. However, these controllers are based on linearized models of the plant to be controlled, which, according to Drummond et alii (1999), may represent loss of important information for systems with high levels of requirements. Normally, such a controller is designed to work at a certain point of operation (set point) by adjusting its parameters (proportional, integral and derivative gains). Therefore, if changes occur that modify the operating point of the system, such as changing the initial condition, variations of plant physical parameters and external disturbances, the PID controller may no longer perform satisfactorily.

To remedy such deficiencies and improve the performance of the control system, hybrid schemes can be used that add an AI-based Intelligent Control System (SCI), such as neural networks, to the PID compensator. Some works in this line have presented satisfactory results, such as those developed by Jung and Kim (2008), Cho and Jung (2003) and Miyagawa and Ishida (1995). According to Yoneyama and Nascimento Jr. (2000), the use of neural networks is very promising and attractive in cases that present non-linearity, uncertainties, parameter variations or even when the dynamics of the plant are unknown.

The Inverted Pendulum is a system formed by a pendulum mounted on a car driven by an actuator. The objective to achieve is to keep the pendulum in vertical position as much as possible and have control over the position of the car. The system can be modeled as a linear system considering that the angular deviations of the pendulum are very small. The goodness of the design is given by the movements that the car needs to do to position itself without the pendulum deviating from its vertical position more than a certain angle, which must occur in a reasonable time. The control system will have two outputs: pendulum inclination and carriage position. Said outputs will be processed by the control system to be designed which will give as response the force and the sense to apply to the movement of the car, therefore the application of a model in the space of states will be appropriate. However, for didactic purposes, emphasis has also been placed on conventional control.

For the present work, the objective is the Comparison of control methods for an inverted pendulum System. For this, in spite of the limiting characteristics that involve the use of this technique, it uses classic linear controllers PID, H-infinity controller and H2 controller. In this way, the objective is to evaluate the system through some performance criteria, such as peak and as-set times, as well as robustness and sensitivity. In addition, it is intended to raise possible positive and negative points of such technique when applied to this problem with unstable characteristics.

The inverted pendulum is known to be one of the most important and classic problems of control theory. It is an unstable and non-linear control. It is often used as an academic example, mainly for being a more accessible control system, and on the other hand, it allows to show the main differences of control in open loop and its stabilization in closed loop. The inverted pendulum is a servo mechanism that consists of a system in which a pendulum that can rotate freely is mounted. As the purpose of this work is to give the possibility of executing the control algorithm in a real system, it implies that a system can move without any limitation, that is, if it were mounted on a rail, it will have no obstacles.

If the pendulum is considered separated from the system, it has two equilibrium points: a stable one, below; and another unstable, above. The objective of the control is to change the dynamics of the system so that in the vertical position, above, there is a stable equilibrium point. In other words, the idea is to find the force to be applied to the system so that the pendulum does not fall, even if it is disturbed by a ladder or impulse type push. Fuzzy control incorporates expert knowledge within its structure. What allows to have...