Pedulum Experiment

Pendulum Controller Experiment

Laboratory Report

Submitted

2007

Pendulum Controller Experiment

Laboratory Report

Summary

This is an investigation into the use of state-space dynamic models using state-feedback and pole-placement techniques. State-space design is a major method of designing feedback control systems. Hence, it is very important to understand the principle behind it and have knowledge of their limitations.

The following experiments look at the control of a crane and an inverted pendulum. Their step responses using various feedback gains of the state variables are analysed and compared to a linear model created in MATLAB.

It was found that the system poles were very sensitive to the feedback gain values. The linear models devised gave good approximations to stable crane systems but could not be used to model systems on the brink of becoming unstable or when large displacements were involved. The non-linear friction acting on the inverted pendulum caused limit cycling.

Table of Contents

Summary 2

Table of Contents 3

1. Introduction 4

2. Apparatus and Method 4

3. Theory 4

From Appendix I: section A.2 Inverted pendulum model 4

From Appendix I: A.5 Closed-loop characteristic equation 5

4. Results, Observations and Calculations 7

4.1 Crane Control: 7

4.1.1 Friction and Stiction 7

4.1.2 Carriage Controller: with only CP and CV feedback 7

4.1.3 Carriage Controller: with CP, CV and PP feedback 8

4.1.4 Pole-placement 8

4.1.5 Variation of p2 9

4.2. Inverted Pendulum: 9

4.2.1 No carriage feedback 9

4.2.2 Pole placement 10

4.2.3 Limit Cycles 10

4.2.4 No pendulum feedback 10

5. Analysis 10

4.1 Crane Control: 11

Figure 1: Crane system step response with p = [0.35 0.14 0.32 0.01] 11

Figure 2: Crane system step response with p = [0.84 0.77 0.29 0.63] 11

Figure 3: Crane system step response with p = [0.84 0.46 0.29 0.63] 11

4.2 Inverted Pendulum: 12

Figure 4: Inverted pendulum system step response with p = [0.25 0.32 0.35 0.28] 12

Figure 5: Inverted pendulum system step response with p = [0.23 0.20 0.63 0.40] 12

Figure 6: Inverted pendulum system step response with p = [0.23 0.78 0.63 0.40] 12

6. Conclusion 13

APPENDICES…………………………………………………………………………………………...…12

Figure 1: Crane system step response with p = [0.35 0.14 0.32 0.01]..............................13

Figure 2: Crane system step response with p = [0.84 0.77 0.29 0.63] .............................14

Figure 3: Crane system step response with p = [0.84 0.46 0.29 0.63] .............................15

Figure 4: Inverted pendulum system step response with p = [0.25 0.32 0.35 0.28].........16

Figure 5: Inverted pendulum system step response with p = [0.23 0.20 0.63 0.40] .........17

Figure 6: Inverted pendulum system step response with p = [0.23 0.78 0.63 0.40] .........18

Appendix I: Laboratory Handout……………………………………………………………………19