Homework 6, due March 29, 2017

Due at the beginning of class on Wednesday, March 29, 2017.
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goodwine
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Homework 6, due March 29, 2017

Post by goodwine »

Reading: Chapter 8 except section 8.4 and Chapter 9 through section 9.4.

Most of this problems this week will cover the concepts from the lectures for the last week or so, specifically directed toward the inverted pendulum problem.
  1. Consider m x''(t) + b x'(t) + k x(t) = f(t).
    1. If k < 0, then it is like the inverted pendulum with damping. Explain why this is the case.
    2. Assign m, b and k values from your NDID number in the following manner. Let -k be the last digit, 10 b the second to last and m the third to last. If any of them are zero, then use to the next non-zero digit instead. For example, if you NDID is 90000330405, then k = -5, b = 4/10 and m = 3, and if it is 900009283756, then k = -6, b = 1/2 and m = 7.
    3. Compute by hand the solution to your pendulum with f(t) = 0 and x(0) = -1 and x'(0) = 1. This is not a transfer function problem because f(t) is zero and also there are nonzero initial condition. Is the solution what you expect for this system? Explain your reasoning.
    4. Determine the transfer function X(s)/F(s). If F(s) is a unit step input, determine X(s) and then use partial fractions to determine x(t). Is the solution what you expect? Explain your reasoning. Determine the location of the poles for the transfer function. Explain how the solution you obtained could be predicted by the pole values.
    5. If f(t) = cos(w t) where w is the fourth non-zero digit in your NDID from the left, change k to be positive and determine X(s) and use partial fractions to compute x(t). Only use positive k for this part of the problem.
    6. Build a simulink model of the system and verify each of the parts above for which you computed a solution.
    Hints for Part 1:
    1. It is ok to use decimals. You don't have to carry square roots of thirty five trillion through the problem.
    2. The simulink in this part will not have a feedback loop: only the transfer function with an input and output.
    3. For part f, you don't have to validate part c in simulink. Putting non-zero initial conditions in simulink is a hassle. It's designed for transfer functions and zero initial conditions.
    4. Whenever you have a quadratic in the denominator, you want to factor it (or use the quadratic formula) before you do partial fractions because that will tell you whether you have two real poles or complex ones. That difference will matter in the partial fraction expansion.
  2. Initial control attempts for the pendulum.
    1. Represent the inverted pendulum with proportional feedback control for the torque with a block diagram.
    2. Use simulink to represent the system. Use small and large gains (k-values) to attempt to stabilize the system with a unit step input. Hint: very small gains should not work. Very large ones should work. Just by experimenting with simulink, try to find the gain value that is the boundary between stable and unstable behavior.
    3. By hand compute the transfer function from the reference input, R(s), to the output, X(s). For small gain values, where are the poles? For large gain values, where are the poles? Do these locations correspond to the response behavior you saw in the previous part?
    4. Use the Routh criterion to determine the range of k values for which the system with proportional control will be stable.
    5. For 3 different values of k for which the proportional controlled system has complex poles, compare the rise times, percentage overshoot and settling times and relate them to the corresponding pole locations. You may either use the step() command on Matlab or a simulink model to compute the responses.
    Hints for part 2:
    1. Use the same numbers as for part 1b.
    2. For part 2b, use a step input and change the defaults to do the following: keep the step time at 1, but change the step from starting at zero and going to one to be starting at one and going to zero. What will happen is the input of one between zero and one second will push the pendulum off the inverted zero position, but then starting at time one, the reference will switch back to zero, and if the controller can stabilize it, it will move back to the zero position.
    3. For any simulink stuff, print the model and also the plots that verify your answers.
Bill Goodwine, 376 Fitzpatrick
smayers

Re: Homework 6, due March 29, 2017

Post by smayers »

Dr. Goodwine,

For 1D. are we assuming that our equation is no longer homogeneous? i.e. (mx" + bx' + kx = f(t))?
goodwine
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Re: Homework 6, due March 29, 2017

Post by goodwine »

smayers wrote: ↑Wed Mar 22, 2017 10:29 pm For 1D. are we assuming that our equation is no longer homogeneous? i.e. (mx" + bx' + kx = f(t))?
Yes, f(t) not zero (but zero initial conditions now).
Bill Goodwine, 376 Fitzpatrick
akiess1

Re: Homework 6, due March 29, 2017

Post by akiess1 »

Dr. Goodwine,

For part 2e, does a system with just proportionality control have rise times/percent overshoot/settling times, or do we need to add a damping term to make it PD control in order for it to settle to a constant value?
goodwine
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Re: Homework 6, due March 29, 2017

Post by goodwine »

akiess1 wrote: ↑Mon Mar 27, 2017 10:52 pm Dr. Goodwine,

For part 2e, does a system with just proportionality control have rise times/percent overshoot/settling times, or do we need to add a damping term to make it PD control in order for it to settle to a constant value?
There should be some damping in the system due to the b term from the very beginning, so you don't have to add a derivative control term. If you did add one, that would make it better, but it is not part of this problem.
Bill Goodwine, 376 Fitzpatrick
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