Engineering domain: Control Design

Magnetic levitation systems suspend a ferromagnetic ball beneath an electromagnet by balancing gravitational and magnetic forces.  The fundamental nonlinear dynamics are given by
\[
m\ddot y=mg-\frac{kI^2}{y^2},
\]
where \(y(t)\) is the ball’s vertical displacement and \(I(t)\) the coil current.  Around an operating point \((y_0,I_0)\), a first‑order Taylor expansion and appropriate nondimensionalization yield a normalized second‑order linear model suitable for controller design.  In this task, you are required to analyze that linearized model and then validate the controller on the original nonlinear dynamics.

To simplify the problem, consider the following equation:
\[
\ddot y=1-\frac{u^2}{y^2}.
\]

\begin{enumerate}
  \item Linearize the nonlinear equation around the operating point $y = 1, u=1$. You need to give $a_{11},a_{12},a_{21},a_{22},b_{11},b_{21}$, where
  \[\dot{x}=\begin{bmatrix}
      a_{11}&a_{12}\\
      a_{21}&a_{22}
  \end{bmatrix}x+\begin{bmatrix}
      b_{11}\\
      b_{21}
  \end{bmatrix}u.\]
  \item Determine a feedback gain $K=(k1\ k2)$ that places the closed‑loop poles at \(-1\pm j1\).
  \item The ball position $x_1$ can be measured using a photocell, but the velocity $x_2$ is more difficult to obtain. Therefore, suppose the output is $y=x_1$. Design a full-order observer $L = (l1\ l2)'$ having eigenvalues at $-5, -6$, and use the observer feedback to produce closed-loop eigenvalues at $-1\pm j1, -5, -6$. 
  \item Simulate your controller for the nonlinear model
  \[
    \ddot{y} = 1 -\frac{u^2}{y^2},
  \]
  with nominal equilibrium \(y_0=u_0=1\). With the initial condition $(x(0),\dot{x}(0))$ set as $(1,0)$ for the observer, you need to identify whether the following initial conditions $(x(0),\dot{x}(0))$ for the real system are within the region of asymptotic stability -- the set of initial states from which the closed‑loop system converges to the equilibrium. You need to use $1$ as yes and $0$ as no.
  \begin{enumerate}
      \item $(2, 2)$
      \item $(2.5, 0)$
      \item $(0.8, 0)$
      \item $(0.9, -1)$
  \end{enumerate}
\end{enumerate}