System Dynamics
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Problems

Problem 11.1 (SCALLYWAG)

Use a computer to solve this problem. Consider the transfer function \[\begin{aligned} H(s) &= \frac{10 (s + 3)} {(s+2) (s^2 + 8 s + 41)}. \end{aligned}\]

  1. What are poles and zeros of \(H\)?

  2. Comment on the stability of the system described by \(H\) (justify your comment).

  3. Construct a pole-zero plot.

  4. Use a function like the Python control package function step_response() to simulate the unit step response of the system and plot it for \(t\in [0,3]\) seconds.

Problem 11.2 (SWASHBUCKLING)

Consider a system with linear state-space model matrices $$\begin{align} A &= \begin{bmatrix} -1 & 4 \\ 0 & -3 \end{bmatrix} & B &= \begin{bmatrix} 1 \\ -1 \end{bmatrix} \\ C&= \begin{bmatrix} 1 & 0 \end{bmatrix} & D &= \begin{bmatrix} 0 \end{bmatrix}. \end{align}$$

  1. Derive the transfer function \(H(s)\) for the system. Express it as a single ratio in \(s\).

  2. What are the poles and zeros?

  3. Compare the poles to the eigenvalues of \(A\).

  4. Draw or sketch a pole-zero plot.

  5. With reference to the pole-zero plot, comment on the stability and transient free response characteristics of the system.

  6. Use the inverse Laplace transform \(\mathcal{L}^{-1}\) to find the system’s forced response \(y(t)\) to step input \(u(t) = 9\,u_s(t)\).

Problem 11.3 (BORIS)

Consider a mass-spring-damper system with mass \(m\), spring constant \(k\), and damping coefficient \(B\) with the I/O ODE \[\begin{aligned} \ddot{y} + \frac{B} {m} \dot{y} + \frac{k} {m} y &= \frac{1} {m} u \end{aligned}\] for input force \(u(t) = F_S(t)\) and output position \(y(t) = x_m(t)\).

  1. Find the corresponding transfer function \(H(s) = Y(s)/U(s)\).

  2. Find the natural frequency and damping ratio in terms of system parameters \(m\), \(k\), and \(B\).

  3. What are poles and zeros of \(H\) in terms of system parameters \(m\), \(k\), and \(B\)?

  4. For system parameters \(m = 10\) kg, \(k = 1\cdot10^5\) N/m, and \(B = 500\) N\(\cdot\)s/m, construct a pole-zero plot.

  5. Comment on the stability of the system described by \(H\). Are there any values of system parameters \(m\), \(k\), and \(B\) for which the system is marginally stable or unstable?

  6. For system parameters \(m = 10\) kg, \(k = 1\cdot10^5\) N/m, and \(B = 500\) N\(\cdot\)s/m, use a function like the Python control package function step_response to simulate the unit step response of the system and plot it for \(t\in [0,0.3]\) s.

Online Resources for Section 11.6

No online resources.