University of Tripoli Faculty of Engineering Department of Electrical and Electronic Engineering Inverted pendulum on the cart عبدالرحيم محمد ابشير عمورة 22170969 محمد الدردار.د Abstract : This work describes the use of observer-based and full state feedback H2 controllers for the stability control of an inverted pendulum on a cart with disturbance forces. The system equation of motions has been modeled using the Lagrangian equation, which linearized the system to the unstable upward position. By comparing the suggested controllers using Matlab/Scripts, the system stability has been simulated, and a .promising outcome has been successfully examined Introduction : The inverted pendulum on a cart system is a classic example in control theory and dynamics, widely studied in engineering, physics, and robotics. It serves as a fundamental model for understanding the principles of stabilization, control, and stability analysis in dynamic systems. This system consists of a pendulum mounted on a cart, where the cart is free to move along a horizontal track, while the pendulum is free to rotate in the vertical plane. A pendulum system that has its center of mass above its pivot point is known as an inverted pendulum on a cart. In the absence of an extra feedback control system, it is unstable and nonlinear. The inverted pendulum stability, which is employed as a controller testing technique, is a significant issue in dynamics and control strategy. By positioning the pole along a fixed axis of rotation, the majority of applications in this system provide one degree of freedom. A feedback control theory-based controller, which adjusts the pendulum's angle and moves the mass of the pendulum as the system begins to fall over, can be used to improve the stability of an inverted pendulum in its inverted state Description of the System: 1. Cart: The cart is a mobile platform that can move horizontally along a track. It serves as the base for the entire system and provides the means for controlling the position of the inverted pendulum. 2. Inverted Pendulum: The inverted pendulum is a rigid rod attached to the cart at one end, with a point mass (or a concentrated mass) at the other end. The pendulum is free to rotate about the cart's axis of rotation. 3. Actuators and Sensors: Actuators, such as motors or electromagnetic devices, are used to apply forces to the cart, allowing control over its position. Sensors, such as encoders or accelerometers, provide feedback on the system's state, including the position and velocity of the cart and the angle of the pendulum. Dynamics of the System: The dynamics of the inverted pendulum on a cart system are governed by the principles of classical mechanics and control theory. The system's motion is influenced by several factors, including: Gravity: The gravitational force acts on the pendulum, causing it to tend towards the vertical downward position. Inertia: The mass of the pendulum and the cart contribute to the system's inertia, affecting its response to external forces. Friction: Friction between the cart and the track, as well as air resistance, can dampen the system's motion and affect its stability. Control Forces: Actuators apply control forces to the cart based on feedback from sensors, aiming to stabilize the pendulum in the upright position or achieve desired trajectories. Challenges and Control Strategies: The inverted pendulum on a cart system presents several challenges in terms of control and stability: Balancing: The primary objective is to balance the pendulum in the upright position while keeping the cart within a specified range of motion. Nonlinear Dynamics: The system's nonlinear dynamics pose challenges for control design, requiring sophisticated control strategies to achieve stable performance. Sensitivity to Disturbances: The system is sensitive to external disturbances, such as uneven surfaces or sudden pushes, requiring robust control algorithms to maintain stability. Applications : The inverted pendulum on a cart system has several practical applications across various fields, leveraging its dynamics and control principles to address specific challenges. Some notable applications include: Robotic Manipulation and Balancing: The principles learned from the inverted pendulum on a cart system are directly applicable to the design and control of humanoid robots and robotic manipulators. Humanoid robots often have segmented limbs resembling inverted pendulums and require sophisticated control algorithms to maintain balance while performing tasks such as walking, running, or manipulating objects. Segway and Personal Mobility Devices: The Segway Personal Transporter and similar self-balancing scooters utilize concepts similar to the inverted pendulum on a cart system. By dynamically adjusting the position of the cart (in this case, the platform on which the rider stands), these devices maintain balance and stability, enabling intuitive and efficient personal transportation. Inverted Pendulum Cart Systems in Education: Inverted pendulum on a cart systems are widely used as educational tools in engineering and robotics curricula. Students can learn about control theory, feedback systems, and stability analysis by designing and implementing control algorithms for these systems in laboratory settings. Transportation and Autonomous Vehicles: The control strategies developed for the inverted pendulum on a cart system find applications in autonomous vehicle technologies. By stabilizing the vehicle's orientation and trajectory, these control algorithms contribute to the safe and efficient operation of autonomous cars, drones, and other unmanned vehicles. Industrial Automation and Material Handling: Inverted pendulum on a cart systems can be adapted for material handling and automation tasks in industrial settings. By equipping carts with inverted pendulum mechanisms, manufacturers can develop agile and precise material handling systems that optimize production efficiency and flexibility. Aerospace and Satellite Attitude Control: The dynamics of the inverted pendulum on a cart system are relevant to the attitude control of satellites and spacecraft. Control algorithms inspired by the principles of stabilizing the inverted pendulum can be employed to adjust the orientation and stabilize the attitude of satellites for optimal performance in space missions. Medical Robotics and Prosthetics: The principles of balance and stability derived from the inverted pendulum on a cart system are applicable to the design of medical robots and prosthetic devices. Prosthetic limbs and robotic exoskeletons can benefit from advanced control algorithms inspired by the dynamics of the inverted pendulum to provide users with improved stability and mobility. Objectives : Stabilization: The primary objective of a pendulum on a cart system is often to stabilize the inverted pendulum in an upright position. This involves maintaining the pendulum at a desired angle while preventing it from falling over. Balancing: Another key objective is to balance the cart and pendulum system to keep the center of mass within a stable region. This ensures that the system remains in equilibrium and does not exhibit erratic or unstable behavior. Controlled Motion: Depending on the application, the system may be required to move the cart in a controlled manner while maintaining stability. This could involve following a predefined trajectory or responding to external disturbances while keeping the pendulum upright. Energy Efficiency: In some cases, the objective may be to minimize energy consumption while achieving stabilization and controlled motion. This involves optimizing the control strategy to achieve the desired performance with minimal energy input. Robustness: The system may need to be robust against uncertainties such as changes in mass distribution, friction, or external disturbances. Robust control strategies aim to ensure that the system remains stable and performs well under varying conditions. Performance Metrics: Depending on the application, specific performance metrics may be defined to evaluate the system's effectiveness. These could include measures such as settling time, overshoot, tracking accuracy, or energy consumption. Safety: Ensuring the safety of the system and any surrounding environment is often a critical objective. This involves designing control algorithms and mechanical components to prevent hazardous situations, such as the pendulum swinging uncontrollably or the cart moving unexpectedly. Adaptability: In some applications, the system may need to adapt to changing conditions or requirements. This could involve adjusting control parameters, switching between different control modes, or accommodating variations in the environment. Nonlinear system and stability The inverted pendulum on a cart system is a classic example of a nonlinear dynamic system, exhibiting complex behavior due to the nonlinear coupling between the motion of the cart and the rotation of the pendulum. In this chapter, we explore the nonlinear dynamics and stability analysis of this system, which are essential for understanding its behavior and designing effective control strategies Nonlinear Dynamics of the System: A system of coupled nonlinear ordinary differential equations (ODEs) controls the dynamics of an inverted pendulum on a cart system. These equations account for variables like gravity, friction, and control inputs to describe the motion of the cart and the pendulum. Mathematical terms in the equations which represent the sine and cosine of the pendulum angle are the source of the nonlinearities. Equations of Motion: The equations of motion for the inverted pendulum on a cart system can be derived using principles of Newtonian mechanics and Lagrangian dynamics. These equations typically involve terms representing the position and velocity of the cart, as well as the angle and angular velocity of the pendulum. Nonlinear terms arise due to the trigonometric functions involved in describing the pendulum's motion .
