4.1.5. Stability Analysis. The fixed-base pendulum linear dynamics may be written in state space form as...

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4.1.5. Stability Analysis. The fixed-base pendulum linear dynamics may be written in state space form as follows: where x = A Ap = x Y = = 0 1 [;)], B Bp 0 Apx + Bpu Cpx + Dpu = [ 0 C = [10], D=0 (8) 1) Obtain the initial condition response when the initial angle (0) is 0.0349 rad (2) using MATLAB for the cases where the length of the pendulum is l = 10 m, 5 m, 3 m, 2 m, and 1 m. Include the figure in your report. Hint: Use the provided m-file template pendulum_ol_cl_analysis.m. 2) Is the system stable, unstable, or marginally stable? 3) What is the time to double (from 2 (0.0349 rad) to 4 (0.0698 rad)) corresponding to each of the pendulum lengths 1 = 10 m, 5 m, 3 m, 2 m, and 1 m? Hint: Use the figure you obtained in MATLAB. Left clicking on a curve brings up a data cursor you can drag to get plot values. 4) What are the plant poles corresponding to each pendulum lengths 1 = 10 m, 5 m, 3 m, 2 m, and 1 m for the chosen parameters m = 0.5331 kg, g = 9.8 m/s? Hint: Plug into (6). 5) With decreasing pendulum length (10 m to 1 m), does the plant become more unstable, less unstable, or does the stability remain same? Does this agree with the initial condition response results (i.e., did time-to-double become worse (decrease) with worsening stability (more unstable))? Remark: Physically, the unstable pole is due to the fact that the inverted pendulum falls when perturbed from the vertical. Balancing a broomstick is not too difficult; balancing a pencil is very difficult. 4.1.5. Stability Analysis. The fixed-base pendulum linear dynamics may be written in state space form as follows: where x = A Ap = x Y = = 0 1 [;)], B Bp 0 Apx + Bpu Cpx + Dpu = [ 0 C = [10], D=0 (8) 1) Obtain the initial condition response when the initial angle (0) is 0.0349 rad (2) using MATLAB for the cases where the length of the pendulum is l = 10 m, 5 m, 3 m, 2 m, and 1 m. Include the figure in your report. Hint: Use the provided m-file template pendulum_ol_cl_analysis.m. 2) Is the system stable, unstable, or marginally stable? 3) What is the time to double (from 2 (0.0349 rad) to 4 (0.0698 rad)) corresponding to each of the pendulum lengths 1 = 10 m, 5 m, 3 m, 2 m, and 1 m? Hint: Use the figure you obtained in MATLAB. Left clicking on a curve brings up a data cursor you can drag to get plot values. 4) What are the plant poles corresponding to each pendulum lengths 1 = 10 m, 5 m, 3 m, 2 m, and 1 m for the chosen parameters m = 0.5331 kg, g = 9.8 m/s? Hint: Plug into (6). 5) With decreasing pendulum length (10 m to 1 m), does the plant become more unstable, less unstable, or does the stability remain same? Does this agree with the initial condition response results (i.e., did time-to-double become worse (decrease) with worsening stability (more unstable))? Remark: Physically, the unstable pole is due to the fact that the inverted pendulum falls when perturbed from the vertical. Balancing a broomstick is not too difficult; balancing a pencil is very difficult.