Problem 1
This is a continuation of Subsection 2.4.2. Consider a satellite of unit mass in earth orbit specified by its position and velocity in polar coordinates \(r, \dot{r}, \theta, \dot{\theta} .\) The input functions are a radial thrust \(u_{1}(t)\) and a tangential thrust of \(u_{2}(t) .\) Newton's laws yield $$ \vec{r}=r \dot{\theta}^{2}-\frac{g}{r^{2}}+u_{1} ; \quad \ddot{\theta}=-\frac{2 \dot{\theta} \dot{r}}{r}+\frac{1}{r} u_{2} . $$ (Compare (2.6) and take \(m_{\mathrm{s}}=1\) and rewrite \(G m_{\mathrm{e}}\) as \(g .\) ) Show that, if \(u_{1}(t)=\) \(u_{2}(t)=0, r(t)=\sigma\) (constant), \(\theta(t)=\omega t\) ( \(\omega\) is constant) with \(\sigma^{3} \omega^{2}=g\) is a solution and that linearization around this solution leads to (with \(x_{1}=r(t)-\) \(\left.\sigma ; x_{2}=\dot{r} ; x_{3}=\sigma(\theta-\omega t) ; x_{4}=\sigma(\dot{\theta}-\omega)\right)\) $$ \frac{d x}{d t}=\left(\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 3 \omega^{2} & 0 & 0 & 2 \omega \\ 0 & 0 & 0 & 1 \\ 0 & -2 \omega & 0 & 0 \end{array}\right) x+\left(\begin{array}{ll} 0 & 0 \\ 1 & 0 \\ 0 & 0 \\ 0 & 1 \end{array}\right) u $$
Problem 2
Given the differential equations $$ \begin{aligned} &\dot{x}_{1}(t)=x_{2}(t) \\ &\dot{x}_{2}(t)=-x_{1}(t)-x_{2}^{2}(t)+u(t) \end{aligned} $$ and the output function \(y(t)=x_{1}(t)\). Show that for \(u(t)=\cos ^{2}(t)\) a solution of the differential equations is \(x_{1}=\sin t, x_{2}=\cos t\). Linearize the state equations and the output function around this solution and write the result in matrix form. Is the linearized system time-invariant?
Problem 5
If \(A_{1}\) and \(A_{2}\) commute (i.e. \(\left.A_{1} A_{2}=A_{2} A_{1}\right)\), then \(e^{\left(A_{1}+A_{2}\right) t}=\) \(e^{A_{1} t} \cdot e^{A_{2} t} .\) Prove this. Give a counterexample to this equality if \(A_{1}\) and \(A_{2}\) do not commute.
Problem 8
We are given the \(n\)-th order system \(\dot{x}=A x\) with $$ A=\left(\begin{array}{ccccc} 0 & 1 & 0 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & \ddots & 0 \\ 0 & \cdots & \cdots & 0 & 1 \\ -a_{0} & -a_{1} & \cdots & -a_{n-2} & -a_{n-1} \end{array}\right) $$ Show that the chanacteristic polynomial of \(A\) is $$ \lambda^{n}+a_{n-1} \lambda^{n-1}+\ldots+a_{1} \lambda+a_{0} $$ If \(\lambda\) is an eigenvalue of \(A\), then prove that the corresponding eigenvector is $$ \left(1, \lambda, \lambda^{2}, \ldots, \lambda^{n-1}\right)^{T} $$
