SYSC3600-L3

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SYSC3600B - Lab 3 Control of an Inverted Pendulum Nicholas Nemec 101211060 12/1/2023

1: Introduction The purpose of this laboratory is to observe the behavior of an inverted pendulum, an unstable, nonlinear system. We were tasked with simulating the integration of a proportional-plus-derivative(PD) controller design modeled using Simulink and MATLAB to study its ability to stabilize the inverted pendulum system based on various factors. 2: Controller Design Figure 1: Inverted pendulum system[Lab Manual]

Figure 2: Simulink model to simulate the zero-input response of the inverted pendulum and the cart[Lab Manual] Figure 3: Simulink model that implements the PD controller to simulate the non-linear dynamics of the cart/pendulum[Lab Manual]

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Figure 4: Simulink subsystem that implements the non-linear dynamics of the angle of the pendulum Θ o (t) and the position of the cart x(t)[Lab Manual]

3: Pre-Lab Transfer function of the system: 𝐻(?) = Θ ? (?) Θ ? (?) = 𝐵 ? 2 + ? ? ?? ?+[ ? ? −(?+?)? ?? ] General transfer function: 𝐻(?) = ?ω ? ? 2 +2ζω ? 2 +ω ? 2 Using the given values, M=1000 kg, m=200 kg, l=10 m, ω n =0.5 rad/sec, ζ=0.7 and g=9.81 m/s 2 , the values of k p and k d were calculated to be:

4: Lab 4.1: Inverted pendulum demo in Simulink In your report, discuss why an overall underdamped response in balancing the inverted pendulum is preferred over a critically damped or an overdamped response. It was found that an overall underdamped response is more favored over critically/overdamped response because the behavior of an underdamped system contains an oscillating pattern with a small amount of overshoot. It’s this behavior that allows the system to achieve a balanced state whereas in an overdamped or critically-damped system, the cart would never be able to balance the inverted pendulum due to the lack of overshooting. The result is the cart attempting to balance the inverted pendulum forever and the cart will never stop moving. 4.2: Testing the pendulum Figure 5: variation of x (t) when Θ = 20 ॰

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Figure 6: variation of Θ o (t) when Θ = 20 ॰ Describe the zero-input response of the pendulum as well as what happens to the position of the cart. How is this affected by the starting angle of the inverted pendulum? When the system was tested with zero-input and a starting Θ of 20, the observed response was the sinusoid shown in Figure 2. The position of the cart tends to simply move back and forth. When the starting Θ was decreased, the maximum Θ value increased and the cart seems to move slower. However, when the starting Θ increases the maximum Θ value reached is reduced, as well as the carts motion becomes faster.

4.3: Simulation of PD controlled non-linear inverted pendulum 4.3.1: Problem of derivative with the PD controller In your report you should include the transfer function for your lowpass filter with cutoff frequency of ω cf = 100 rads/s, along with the Bode plots for the filter. Make any observations you can about the lowpass filter from the bode plots. Low-pass filter transfer function: , where ω cf = 100 rads/s 𝐻 ?𝑃 (?) = 1 ? ω ?? +1 𝐻 ?𝑃 (?) = 1 0.01?+1 Figure 7: Bode plot and phase diagram of the lowpass filter when ω cf = 100 rads/s As can be observed in Figure 3, the slope of the bode plot is -20 dB/decade. The magnitude of the plot approaches -60 dB with a phase angle of -90 o . The -3 dB cutoff frequency has a phase angle of roughly -45 o .

4.3.3: Simulating the PD controlled inverted pendulum Figure 8: variation of Θ o (t) when Θ = 5 ॰ Figure 9: variation of Θ o (t) when Θ = -30 ॰

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Figure 10: variation of Θ o (t) when Θ = 65 ॰ Figure 11: variation of Θ o (t) when Θ = 75 ॰

In your report you should include plots of the angle Θ o (t) of the inverted pendulum. For each plot, describe what is happening in the physical system to the inverted pendulum. Does the PD controller fail to keep the pendulum inverted in any of the cases? If so, discuss why this might happen. In the cases where the starting angles were 65 ॰ and 75 ॰ the PD controller failed to keep the pendulum inverted and they showed overdamped behavior as can be observed in Figure 10 and Figure 11. The system fails to stabilize, and this is caused by the large starting angle. As for the two cases where the starting angle is 5 ॰ and -30 ॰ , we can observe the underdamped behavior with the small overshooting present in the response curves as observed in Figure 8 and Figure 9. Both of these cases are able to stabilize within 25 seconds. 4.4: Additional questions 1. What if the reference angle Θ r is set to something other than zero? Will the closed-loop system keep the pendulum at that angle? Try it for a few small angles Θ r before coming to a conclusion. The closed-loop system can only keep the pendulum at the reference angle if it's close enough to 0. This is because when the system reaches the reference angle it stops trying to balance the pendulum, since the reference angle is not vertical for non-zero values the mass of the pendulum would then push it down. This process would repeat indefinitely as the system attempts to stabilize the pendulum's position. Thus we can conclude in order to have a balanced system the reference angle would either have to be 0 or very close to 0. 2. How many initial conditions should exist for Eq. 13 and what is the purpose for each initial condition? Are there any initial conditions missing in the realization shown in Fig. 10? Are any of the initial conditions assumed to be zero in Fig. 10? There should be two initial conditions for Eq. 13 in the lab manual. The first is the starting angle of the pendulum, Θ o . The other initial condition which is missing in the realization of Fig. 10 is the initial angular velocity,Θ o ’. In the implementation of Fig. 10 the initial angular velocity is assumed to be 0 as it is not being accounted for.

5: Conclusion In conclusion, we modeled an inverted pendulum system in MATLAB and Simulink. The inverted pendulum was tested before(open-loop) and after(closed-loop) the addition of a lowpass filter to introduce closed-loop control into the system to observe the effect it has on the ability of the system to stabilize based on varying starting angles. As was observed in Figure 10 and Figure 11, if the starting angle was too large the closed-loop system would fail to stabilize.

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