Pedulum Experiment
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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.

Summary
1. Introduction
2. Apparatus and Method
3. Theory
From Appendix I: section A.2 Inverted pendulum model
From Appendix I: A.5 Closed-loop characteristic equation
4. Results, Observations and Calculations
4.1 Crane Control:
4.1.1 Friction and Stiction
4.1.2 Carriage Controller: with only CP and CV feedback
4.1.3 Carriage Controller: with CP, CV and PP feedback
4.1.4 Pole-placement
4.1.5 Variation of p2
4.2. Inverted Pendulum:
4.2.1 No carriage feedback
4.2.2 Pole placement
4.2.3 Limit Cycles
4.2.4 No pendulum feedback
5. Analysis
4.1 Crane Control:
Figure 1: Crane system step response with p = [0.35 0.14 0.32 0.01]
Figure 2: Crane system step response with p = [0.84 0.77 0.29 0.63]
Figure 3: Crane system step response with p = [0.84 0.46 0.29 0.63]
4.2 Inverted Pendulum:
Figure 4: Inverted pendulum system step response with p = [0.25 0.32 0.35 0.28]
Figure 5: Inverted pendulum system step response with p = [0.23 0.20 0.63 0.40]
Figure 6: Inverted pendulum system step response with p = [0.23 0.78 0.63 0.40]
6. Conclusion
Introduction
State-space design is a major method of designing feedback control systems. It organizes the differential equations describing a dynamic system as a set of first-order vector differential equations and the solution can then be visualised as a trajectory of the state vector in space. Dynamic models are used in many areas of engineering and it is therefore important to understand the principles of analysing and ultimately, how to designing them.

The following experiments will aim to use the control of a crane/inverted pendulum to illustrate:
State-space dynamic models
State-feedback and pole-placement
Limit cycles and describing functions
Apparatus and Method
See laboratory handout (Appendix I).
Theory
(See Appendix I for detailed theory)
From Appendix I: section A.2 Inverted pendulum model
If we assume equations (14) and (15) in Appendix I are correct, they will linearize to equations (16) and (17). These are two simultaneous 2nd order o.d.e.s equivalent to a 4th order o.d.e. in either x or, but they can also be written as a first order o.d.e. as follows.

Solving for and, we get
(15)
(17)
(14)
(16)