Modern Problems of Control Theory

Practice and Homework 1.

1. Simple Pendulum Control

2. Pendulum on the Cart Control

3. Invert Pendulum on the Cart Control

4. Car Automatic Cruise Control

5. DC Motor Speed Control

6. DC Motor Position Control

7. Bus Suspension Control

8. Aircraft Pitch Control

9. Ball & Beam Control

10. Train System Control

See http://ctms.engin.umich.edu/CTMS/index.php

A Simple Pendulum

Key Topics: Modeling Rotational Mechanical Systems, Nonlinear Systems, Underdamped Second-Order Systems, System Identification

Contents

• Equipment needed

• Purpose

• Modeling from first principles

• Linearization

• System identification experiment

• Model validation

• Extensions

Equipment needed

• Arduino board (e.g. Uno, Mega 2560, etc.)

• simple pendulum (slender metal bar with end weight) with clamp or stand

• rotary potentiometer (e.g. 10K-Ohm linear taper potentiometer)

The orientation of the simple pendulum will be measured employing a rotary potentiometer. The Arduino board is simply employed for data acquisition (and to supply excitation for the potentiometer). Specifically, an Analog Input on the Arduino board is employed to read the potentiometer output which is then fed to Simulink for visualization and for comparison to our resulting simulation model output.

Purpose

The purpose of this activity with the simple pendulum system is to demonstrate how to model a rotational mechanical system. Specifically, the theory of modeling is discussed with an emphasis on which simplifying assumptions are appropriate in this case. The associated experiment is employed to demonstrate how to identify different aspects of a physical system, as well as to demonstrate the accuracy of the resulting model.

Modeling from first principles

First we will employ our understanding of the underlying physics of the simple pendulum system to derive the structure of the system model. We will term this process, "modeling from first principles." In this example we employ the following variables.

• (m) mass of the pendulum bar

• (M) mass of the pendulum end weight

• (l) length to end weight center of mass

• (theta) pendulum angle from vertical (down)

To begin, we first draw the free-body diagram where the forces acting on the pendulum are its weight and the reaction at the rotational joint. We also include a moment due to the friction in the joint (and the rotary potentiometer). The simplest approach to modeling assumes the mass of the bar is negligible and that the entire mass of the pendulum is concentrated at the center of the end weight.

The equation of motion of the pendulum can then be derived by summing the moments. We will choose to sum the moments about the attachment point since that point is the point being rotated about and since the reaction force does not impart a moment about that point.

(1)

Assuming that the mass of the pendulum is concentrated at its end mass, the mass moment of inertia is . A more accurate approach would be to consider the rod and end mass explicitly. In that case, the weight of the system could be considered to be located at the system's mass center . In that case, the mass moment of inertia is . Depending on the parameters of your particular pendulum, you can assess if this added fidelity is necessary.

For the experiment we will perform shortly, the simple pendulum employed consists of a rod of length and mass with an end mass of . Therefore, the difference between and is significant enough to include. The difference between and is also significant enough to include.

We will also initially assume a viscous model of friction, that is, where is a constant. Such a model is nice because it is linear. We will assess the appropriateness of this model later. Sometimes the frictional moment is not linearly proportional to the angular velocity. Sometimes, the stiction in the joint is significant enough that it must be modeled too.

Taking into account the above assumptions, our equation of motion becomes the following.

(2)

Linearization

Some parameters of the given pendulum system are relatively easy to measure, for example, things such as the pendulum length or the pendulum mass. Other parameters, such as the viscous coefficient of friction , are not as easy to measure directly. Therefore, we will perform a simple experiment to help identify . This experiment will also help to validate some of the simplifying assumptions we made in the process of generating our model.

The fact that the system model we generated is nonlinear makes the parameter identification process a little more challenging. However, we can use a linearized version of the model to help us in the identification process. Presuming that for our experiment the pendulum swings through small angles (about ), we can use the approximation that . Therefore, our linearized model becomes the following.

(3)

Examining the above, the linearized model has the form of a standard, unforced, second-order differential equation. Matching this equation to the canonical form, , we can see how the various system parameters influence the free response of the pendulum system. More specifically:

(4)

(5)

If we had simplified the pendulum as having its mass concentrated at its end, then .