Homework 2 - Due: 02/01

Problem 1

Convert each of the following higher-order differential equations into state-space form:

  • System (a): \[\ddot x = \dot u + 3 u\]
  • System (b): \[\dfrac{d^3 x}{dt^3} - 2 \dfrac{d^2x}{dt^2} + x = u\]
  • System (c): \[\ddot x = \sin \left( \dot x - x \right)\]

Problem 2

Find the transfer function for the following systems:

  • System (I): \[ \begin{bmatrix} \dot x_1 \\ \dot x_2 \end{bmatrix} = \begin{bmatrix} 0 & -a_1 \\ 1 & -a_2 \end{bmatrix} \begin{bmatrix}x_1 \\ x_2\end{bmatrix}+ \begin{bmatrix} b_0 \\ b_1 \end{bmatrix} u; \qquad y = \begin{bmatrix} 0 & 1 \end{bmatrix}\begin{bmatrix}x_1 \\ x_2\end{bmatrix} \]

  • System (II): \[ \begin{bmatrix} \dot x_1 \\ \dot x_2 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ -a_1 & -a_2 \end{bmatrix} \begin{bmatrix}x_1 \\ x_2\end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} u; \qquad y = \begin{bmatrix} b_0 & b_1 \end{bmatrix}\begin{bmatrix}x_1 \\ x_2\end{bmatrix} \]

Problem 3

Let \(A\) be the linear operator in the plane corresponding to the counter-clockwise rotation around the origin by some given angle \(\theta\). See Problem 2 in Homework 1. Compute the matrix of \(A\) relative to the basis:

\[ \left\{ \begin{bmatrix} -1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \end{bmatrix} \right\} \]

Problem 4

Let us revisit Problem 5 from Homework 1. Alice and Bob have reason to believe that Cheng’s performance scores and bonus amount that he gave them were incorrect. They still want to figure out their unknown weights, but now they have to use only their own scores and bonus amounts, without Cheng’s.

  1. Set up this problem as solving a linear equation of the form \(Ax = b\) for the unknown vector \(x\). The new matrix \(A\) will be \(2 \times 3\)

  2. Let \(A^\dagger\) be a right inverse for \(A\), which by definition is a matrix such that \(AA^\dagger=I\). Express all solutions of \(x\) in the equation in (a) above in terms of such a right inverse \(A^\dagger\), the vector \(b\) and the nullspace of \(A\). Carefully state and justify your answer.

  3. Find a particular solution \(x^*\) of the equation in (a). Hint: Use your solution from Homework 1. Verify numerically that \(x^∗\) indeed belongs to the solution space you described in (b) using some CAS.

Problem 5

Let \(A\) be a real valued \(n \times n\) matrix. Suppose that \(\lambda + i \mu\) is a complex eigenvalue of \(A\) and \(x + iy\) is a corresponding complex eigenvector. Here \(\lambda, \mu \in \mathbb{R}\) and \(x, y \in \mathbb{R}^n\).

  1. Show that \(x-iy\) is also an eigenvector with eigenvalue \(\lambda - i \mu\).

  2. Let \(V\) be the 2-dimensional subspace over \(\mathbb{R}\) spanned by \(x\) and \(y\). In other words, \(V\) is the set of linear combinations, with real coefficients, of the real-valued vectors \(x\) and \(y\). Show that \(V\) is an invariant subspace for \(A\), in the sense that for every \(z \in V\) we have \(Az \in V\).

Problem 6

Consider the following matrix, whose exponential we derive in class:

\[ A = \begin{bmatrix} a & b \\ -b & a \end{bmatrix} \qquad a, b \in \mathbb{R} \]

  1. Re-derive an expression for \(e^{At}\) by diagonalizing \(A\) over \(\mathbb{C}\) and using the formula \[e^{a \pm ib}:= e^a \cos b \pm ie^a \sin b\] (treat this as the definition of the complex exponential).
  2. Using the formila \(e^{A(t+\sigma)} = e^{At}e^{A \sigma}\) and the result of (a), for suitably chosen values of \(a\) and \(b\), verify the trigonometric identities: \[ \begin{align} \cos (t+\sigma) &= \cos t \cos \sigma - \sin t \sin \sigma \\ \sin (t + \sigma) &= \sin t \cos \sigma + \cos t \sin \sigma \end{align} \]