2013

Digital Pendulum Control Experiments

Ahmed Zia Sheikh 2010030

M. Salman Khalid 2010235

Suleman Belal Kazi 2010341

EE-341L CONTROL SYSTEMS LAB

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TABLE OF CONTENTS

ABSTRACT ..........................................................................................................................................2

PENDULUM OVERVIEW ......................................................................................................................3

EXERCISE 1 – NONLINEAR MODEL .......................................................................................................3

EXERCISE 2 – LINEAR MODEL ..............................................................................................................5

EXERCISE 3 – STATIC FRICTION COEFFICIENTS......................................................................................6

EXERCISE 4 – CART MODEL IDENTIFICATION .......................................................................................7

EXERCISE 5 – CRANE LINEAR MODEL IDENTIFICATION .........................................................................9

EXERCISE 6 – INVERTED PENDULUM LINEAR MODEL IDENTIFICATION ............................................... 12

EXERCISE 7 – PID CONTROL OF CART MODEL POSITION (SIMULATED) ................................................ 12

EXERCISE 8 – PID CONTROL OF CART MODEL POSITION (REALTIME) .................................................. 15

EXERCISE 9 – MODEL SWING UP CONTROL (SIMULATION): ................................................................ 16

EXERCISE 10 –SWING UP CONTROL (REAL-TIME): .............................................................................. 16

EXERCISE 11 – PENDULUM STABILIZATION USING THE NONLINEAR MODEL (SIMULATION) ................ 17

EXERCISE 12 – PENDULUM REAL TIME STABILIZATION ...................................................................... 17

EXERCISE 13 – CRANE CONTROL ....................................................................................................... 18

EXERCISE 14 – SWING UP AND HOLD CONTROL ................................................................................. 20

EXERCISE 15 – UP AND DOWN .......................................................................................................... 23

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ABSTRACT

The Digital Pendulum is a modern version of a classical control problem; that of erecting and balancing a

free swinging pendulum in its inverted position or moving a hanging pendulum in a controlled manner.

The cart on the track is digitally controlled to swing up and to balance the pendulum into an upright

sustained position or to move the cart with pendulum in an unperturbed down position. The cart track is

of limited length, imposing constraints on the control algorithm.

In pendulum mode the system is used to control the twin arm pendulum from an initial position,

hanging at rest with the cart in the center of its travel along the track, to a final position with the

pendulum upright and the cart restored to its central position. In crane mode the control problem is to

move the position of the cart without undue movement of the pendulum. This problem is typical of that

experienced when controlling a gantry crane.

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PENDULUM OVERVIEW

Exercise 1 – Nonlinear model

In this experiment, the pendulum is modeled in Simulink using it’s equations of motions and then

simulated to see the output.

Equations of the pendulum (Non-Linear) from Instruction manual:

In the simulation no voltage is given to the motor, and only an initial angle is set, below are the results

Figure 1 Results for an initial angle of zero (Inverted Pendulum)

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The pendulum parameters in the simulation were adjustable; all of the graphs were obtained using the

approximate values specified in the datasheet as shown below

Figure 2 Initial Angle of 0.01 (Very slight disturbance from vertical)

Figure 3 Pendulum Simulation Parameters

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Exercise 2 – Linear model

For small values of theta, the equations of motion in Exercise 1 take on the form:

In this experiment, the pendulum is modelled using non-linear equations for both theta=0 and theta=pi

i.e. Inverted and hanging respectively. Results for both are shown below:

As can be seen the linear model applies very well to small oscillations about the theta=pi position.

The next graph shows the linear model for a small disturbance for theta=0

Figure 4 Linear Model for initial angle =π (Crane Mode)

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In this case the linear model fails as the system is unstable and a small disturbance causes a large change

in theta thus making our assumptions void.

Exercise 3 – Static Friction Coefficients

Figure 5 Linear Model for inverted Pendulum theta=0

Figure 6 Static Friction Offsets

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Because of the static friction forces (also called stiction); with very small values of control signal the cart

will not move. Furthermore the static friction force may not be symmetrical. We can compensate for

that simply by adding an additional voltage offset value to the cart control signal. Measurement of the

value of the offset is the purpose of this exercise.

RESULTS: Static friction in: Positive Direction: 0.2 Negative direction: -0.4757 PROBLEMS: Several problems were faced in this experiment, according to the manual the values of the friction in both directions should have been almost the same however even after numerous attempts and readjustments of the pendulum our values differed by almost 100%, moreover upon further experimentation it was found that the pendulum was behaving erratically using these values, so the values used in all subsequent experiments were: Positive Direction: 0.1 Negative directions: 0.4

Exercise 4 – Cart model identification

In this experiment we identify the linear model between the control signal u and the cart position x

Figure 7 Experiment To Identify Cart Model

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After using the real-time model to get the required data, Matlab identification interface was used to

get the transfer function of the cart.

Figure 8

Figure 9

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The pendulum had to be immobilized or else the cart would veer off to one direction due to excess

swing.

Finally, the transfer function for the cart, generated using Matlab was:

Exercise 5 – Crane linear model identification

“Crane” refers to the pendulum in the stable position (theta=pi), in this experiment a transfer function

for the crane mode is found.

The excitation signal is applied for the first 10 seconds then the pendulum is allowed to swing freely for

80 seconds to identify the damping effect.

Figure 10

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The transfer function derived in Matlab for the crane mode:

Figure 11 Crane Identification experiment

Figure 12 Measured Vs. Estimated Response

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Figure 13 Peak Shows Natural Frequency

Figure 14 Pole Zero Plot: Poles slightly to right of jw axis show error

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Exercise 6 – Inverted pendulum linear model identification

In the case of this experiment, a vital file “Mextract.m” which was required for extracting the transfer

function of the inverted pendulum was missing; we tried to use a file of the same name from the Maglev

experiment but failed.

We emailed Feedback systems about this omission, they have not replied till the publishing of this

report.

Since we could not develop a model for the inverted pendulum, due to which some of the further

experiments of the inverted pendulum control were performed using the pre-specified model.

Exercise 7 – PID control of cart model position (SIMULATED)

Using the cart model identified in previous experiments, a PID controller was designed for controlling

the cart position. Various values of PID were tested for different experiments.

Figure 15 Response of inverted Pendulum

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Figure 17 Root Locus Of the Cart System after adjustment

Figure 16 Unmodified Root Locus

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Figure 16 Simulated Step Response of the adjusted system

P=27.84 I=36 D= 4.5

Figure 17 Response of the PID Compensated system

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Exercise 8 – PID control of cart model position (REALTIME)

Figure 18 Real Time Results with P=10 I=25 D=2

Figure 19 Real Time Results with P=27.84 I=36 D=4.5

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Exercise 9 – Model swing up control (SIMULATION):

Exercise 10 –Swing up control (REAL-TIME):

Figure 20 Simulation of Model Swing up control with default parameters

Figure 21 Real Time Swing up Control

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Exercise 11 – Pendulum stabilization using the nonlinear model (SIMULATION)

Exercise 12 – Pendulum real time stabilization

Figure 22 Pendulum Stabilization (Simulation) Assuming pendulum is at 0 degrees whilst starting

Figure 23 Real Time Pendulum Stabilization

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Exercise 13 – Crane control

SIMULATION:

Figure 24 Desired Angle is pi (perfectly still pendulum)

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Figure 25 Simulation for another control Signal

REAL-TIME:

Figure 26 Default Settings: Sine wave 0.1 Hz, 0.2v Amplitude

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Exercise 14 – Swing up and hold control

Figure 27 Sawtooth 0.2Hz 0.4v Amplitude

Figure 28 Square 0.1Hz, 0.2 Amplitude

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Figure 29 Simulation Results

Figure 30 Real Time Trial 1 (FAILURE) Cart hits end of track

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In the real time trials it was observed that as the initial voltage of the square wave which slowly

increases the pendulum’s oscillations is increased two things occur:

1. The pendulum reaches the top point more quickly.

2. The pendulum becomes less stable at the top inverted position, and in extreme cases fails to

balance

Figure 32 First Success Swing up and Hold in Real Time

Figure 31 Swing up and hold with decreased initial voltage, extremely stable

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Exercise 15 – Up and down

This is the final experiment in the study of the inverted pendulum and involves a combination or

culmination of all previous experiments into one. First the pendulum is started from a rest position and

made to balance in the inverted position after which it is allowed to fall down into the crane mode and

settled as quickly as possible.

Figure 33 Graph of Final Experiment