AME 302 Homework 6

University of Notre Dame
Aerospace and Mechanical Engineering

AME 302: Differential Equations, Vibrations and Control
Homework 6

B. Goodwine
Spring, 2004

Issued: March 30, 2004
Due: April 2, 2004

It is now 2008, and you work for the Q-Branch of the British Secret Service. In your fight against the Special Executive for Counterintelligence, Terrorism, Revenge and Extortion (SPECTRE) your first assignment is to work on the control system for the Illudium PU-36 Explosive Space Modulator. In particular, Q has asked you to determine and plot how the poles of the transfer function $\frac{Y(s)}{R(s)}$ for the system illustrated in the following figure change as as varies from to $+\infty$ . (10 points)

Q is happy to allow you to check your answers using Matlab, but ever since SPECTRE's infiltration of Mathworks (the maker of Matlab) last year, he insists that you you also verify each aspect of the plot (asymptote angles, asymptote intersection point, break in and break out points, arrival and departure angles of complex conjugate poles and zeros, etc.) by hand.
Recalling that the the angle between the imaginary axis and a complex conjugate pole is related to the damping ratio of a system, Q wants you to determine the value of that corresponds to the case when $\zeta = 0.5$ .
A helpful hint may be to observe that

$\begin{displaymath} 1 + K G(s) = 0 \qquad \Longleftrightarrow \qquad G(s) = -\fr... ...ft\vert G(s) \right\vert = \left\vert \frac{1}{K} \right\vert. \end{displaymath}$

If we write

$\begin{displaymath} G(s) = \frac{(s-z_1)(s-z_2)\cdots(s-z_m)}{(s-p_1)(s-p_2)\cdots(s-p_n)} \end{displaymath}$

then

$\begin{displaymath} \left\vert G(s) \right\vert = \frac{\vert s-z_1\vert\vert s-... ...vert}{\vert s-p_1\vert\vert s-p_2\vert\cdots\vert s-p_n\vert}, \end{displaymath}$

i.e., the magnitude of is the product of the distances of from all the zeros to divided by the product of distances from all the poles to . (20 points)
A block diagram of the internal control mechanism for the latest version of the wristwatch produced by Q-Branch that contains a miniature howitzer is illustrated in the following Figure. Plot the root locus for this system. (10 points)
The main control loop for the ejection seat in the Aston Martin DB5 customized by Q-Branch is illustrated in the following figure.
- Plot the root locus for this system. (10 points)
- By considering only the two complex conjugate poles closest to the imaginary axis, determine a value that will result in a rise time of approximately less than seconds. (5 points)
- Using the Matlab step() command, verify that the step response of the system has the desired rise time. Note that the step() command need the whole transferfunction, Y/R, not just the part we commonly call G(s). (5 points)
- Recalling that the formulas for the time domain specifications were derives for second order systems only, explain why this approach worked for this system which is not second order. (10 points)
Q-Branch is designing a briefcase that will explode if not opened in the vertical position. A critical component of the locking/trigger mechanism for the briefcase in its horizontal position is illustrated in the following figure. Plot the root locus for this system. (10 points)
The block diagram describing the guidance system for the missile launcher for the Aston Martin DB5 is illustrated in the following figure. Proportional plus derivative (PD) control is utilized in the first block where and . Plot the root locus for this system. (10 points)
Instead of using simple PD control for the missile launcher, Q suggests using lead compensator of the form

$\begin{displaymath} C(s) = \frac{s+z}{s+p} \qquad p > z. \end{displaymath}$

Plot the root locus for the system using simple proportional control as illustrated in the following figure. (10 points)

Now plot the root locus for the system using a lead compensator as illustrated in the following figure and explain why in this example adding this form of compensation increased damping for the system. (20 points)