Homework 4

Due at March 10 at 11:59pm. Submit to https://canvas.uw.edu/courses/1862386/assignments/11194461.

Problem 1

Consider the block diagram for the following system:

a) Compute the transfer function from the input \(r\) to the output \(y\).

b) Show the following state space system has the same transfer function, with the appropriate choice of parameters:

\[\begin{align} \frac{d}{dt} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} &= \begin{bmatrix} 0 & 1 \\ -a_2 & -a_1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} r \\ y & = \begin{bmatrix} b_2 & b_1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}. \end{align}\]

Find the values of \(a_1,a_2,b_1,b_2\) that correspond to the transfer function you computed in (a).

c) Computer the transfer function between \(r\) and \(z\). Hint: it’s not \(1\).

Problem 2

A commonly seen transfer function is the second order function:

\[P(s) = \frac{K}{s^2+b_1 s+ b_2}\]

for some positive \(K, b_1, b_2\).

a) Suppose the input to the system is a complex sinusoid, \(e^{j\omega t}\). Find the value of \(\omega\) that leads to largest ouput.

b) Now we interpret the transfer function as coming from the swing equation. The input is still \(e^{j\omega t}\). Find \(\omega\) that leads to the largest frequency excursion at the output. Hint: think about that the output of \(P(s)\) is, and how to get frequency from that.

Problem 3

The PLL transfer function we derived in class has the following form:

\(G(s)= \frac{V(k_p s + k_i)}{s^2 + V k_p\,s + V k_i}\) It is more common to write it into this form \(G(s) = \frac{2\zeta\omega_n s + \omega_n^2} {s^2 + 2\zeta\omega_n s + \omega_n^2}\)

a) Find the relationships between \(k_p\), \(k_i\), \(\zeta\) and \(\omega_n\).

b) Plot what happens when the input is a unit step. That is, \(U(s)=\frac{1}{s}\). You can use a computer to find the time domain response.

c) Normally, we want \(k_p\) and \(k_i\) to take some intermediate values. What can go wrong if they are very small (small control effort)? Or very large (large control effort)? Hint: look at the step response.

Problem 4

a) The \(\mathcal{H}_2\) norm of a system is defined as

\(\Vert G \Vert_2=\left\{\frac{1}{2\pi} \int_{-\infty}^{\infty} G(j\omega)^* G(j\omega) d\omega\right\}^{1/2}\).

This represents how much energy a input would “gain” after it passes through the system. Find \(\Vert G \Vert^2\) for \(G(s)=\frac{1}{s+1}\).

b) We can define another norm, called \(\Vert G \Vert_\infty= \max \vert G(j\omega) \vert\). Give an interpreation of this norm.

Problem 5

Consider a system described by

\[\frac{dy}{dt}-y=\frac{du}{dt} - u.\]

If we try to find the transfer function, we get

\[\frac{Y(s)}{U(s)}=\frac{s-1}{s-1}=1.\]

Is this correct? Namely, does it mean \(y(t)=u(t)\)? Explain what is going on here.