Digital Pendulum Control Experiments


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1 EE341L CONTROL SYSTEMS LAB 2013 Digital Pendulum Control Experiments Ahmed Zia Sheikh M. Salman Khalid Suleman Belal Kazi
2 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 EXERCISE 7 PID CONTROL OF CART MODEL POSITION (SIMULATED) EXERCISE 8 PID CONTROL OF CART MODEL POSITION (REALTIME) EXERCISE 9 MODEL SWING UP CONTROL (SIMULATION): EXERCISE 10 SWING UP CONTROL (REALTIME): EXERCISE 11 PENDULUM STABILIZATION USING THE NONLINEAR MODEL (SIMULATION) EXERCISE 12 PENDULUM REAL TIME STABILIZATION EXERCISE 13 CRANE CONTROL EXERCISE 14 SWING UP AND HOLD CONTROL EXERCISE 15 UP AND DOWN
3 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.. 2
4 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 (NonLinear) 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) 3
5 Figure 2 Initial Angle of 0.01 (Very slight disturbance from vertical) 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 3 Pendulum Simulation Parameters 4
6 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 nonlinear equations for both theta=0 and theta=pi i.e. Inverted and hanging respectively. Results for both are shown below: Figure 4 Linear Model for initial angle =π (Crane Mode) 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 5
7 Figure 5 Linear Model for inverted Pendulum theta=0 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 6 Static Friction Offsets 6
8 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: 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 7
9 After using the realtime model to get the required data, Matlab identification interface was used to get the transfer function of the cart. Figure 8 Figure 9 8
10 Figure 10 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. 9
11 Figure 11 Crane Identification experiment The transfer function derived in Matlab for the crane mode: Figure 12 Measured Vs. Estimated Response 10
12 Figure 13 Peak Shows Natural Frequency Figure 14 Pole Zero Plot: Poles slightly to right of jw axis show error 11
13 Exercise 6 Inverted pendulum linear model identification Figure 15 Response of inverted Pendulum 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 ed 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 prespecified 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. 12
14 Figure 16 Unmodified Root Locus Figure 17 Root Locus Of the Cart System after adjustment 13
15 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 14
16 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 15
17 Exercise 9 Model swing up control (SIMULATION): Figure 20 Simulation of Model Swing up control with default parameters Exercise 10 Swing up control (REALTIME): Figure 21 Real Time Swing up Control 16
18 Exercise 11 Pendulum stabilization using the nonlinear model (SIMULATION) Figure 22 Pendulum Stabilization (Simulation) Assuming pendulum is at 0 degrees whilst starting Exercise 12 Pendulum real time stabilization Figure 23 Real Time Pendulum Stabilization 17
19 Exercise 13 Crane control SIMULATION: Figure 24 Desired Angle is pi (perfectly still pendulum) 18
20 Figure 25 Simulation for another control Signal REALTIME: Figure 26 Default Settings: Sine wave 0.1 Hz, 0.2v Amplitude 19
21 Figure 27 Sawtooth 0.2Hz 0.4v Amplitude Exercise 14 Swing up and hold control Figure 28 Square 0.1Hz, 0.2 Amplitude 20
22 Figure 29 Simulation Results Figure 30 Real Time Trial 1 (FAILURE) Cart hits end of track 21
23 Figure 32 First Success Swing up and Hold in Real Time 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 31 Swing up and hold with decreased initial voltage, extremely stable 22
24 Exercise 15 Up and down Figure 33 Graph of Final Experiment 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. 23
Contents. PART I METHODS AND CONCEPTS 2. Transfer Function Approach Frequency Domain Representations... 42
Contents Preface.............................................. xiii 1. Introduction......................................... 1 1.1 Continuous and Discrete Control Systems................. 4 1.2 OpenLoop
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