Homework 3

Create a main document that contains all your answers and shows your work. Use either a program on your tablet like Notability, etc., or a Word document on your PC. In addition to your main document that contains all the answers, upload all Matlab and Simulink files that you used to Blackboard.

Problem 1

For the transfer function shown below plot the time response of a unit step input using (a) commands in MATLAB and (b) Simulink. Include screenshots of the Matlab code and Simulink code and the plot for (a) and (b) in your main document.

$$T(s) = \frac{5}{(s + 3)(s + 6)}$$

Problem 2

Find the time response of a unit step input to the SSR shown below using (a) commands in MATLAB and (b) Simulink. Include screenshots of the Matlab code and Simulink code and the plot for (a) and (b) in your main document.

$$\dot{\mathbf{x}} = \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -3 & -2 & -5 \end{bmatrix} \mathbf{x} + \begin{bmatrix} 0 \\ 0 \\ 10 \end{bmatrix} r$$

$$y = \begin{bmatrix} 1 & 0 & 0 \end{bmatrix} \mathbf{x}$$

Problem 3

Use Simulink (integrator block method) to simulate a non-linear model of the hydraulic tank system shown below. Assume the equation is

$$C\dot{P} = Q_{in} - K_{T}\sqrt{P - P_{atm}}$$

The input is $Q_{in}$ and the output is $P$, tank base pressure.

Assume the following:

• $C = 0.0002$ m$^3$/Pa
• $K_T = 4 \times 10^{-4}$ m$^{3.5}$/kg$^{0.5}$
• $Q_{in} = 0.052$ m$^3$/s
• $P_{atm} = 1.0133 \times 10^{5}$ N/m$^2$
• $P_0 = 1.15 \times 10^{5}$ N/m$^2$

Note the $P_0$, which is the initial pressure in the tank.

Run the simulation to $t = 1400$ sec. Plot the output, $P$, as a function of time. Include the plot and a screenshot of the Simulink drawing in your main document.