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# EE 350 Control Systems Assignments

## Assignment 5: Design using state space

5.5 Consider the state equation which is represented by $\dot{x}=Ax+Bu$, where $A=\begin{pmatrix}-0.5 & 0 \\ 0 & -1 \end{pmatrix}$ and $B^T=\begin{pmatrix}0.5 & 1 \end{pmatrix}$.
a) If $x^T_0=\begin{pmatrix}a & b \end{pmatrix}$, derive an input that will drive the state to zero point in T sec.
b) For $x^T_0=\begin{pmatrix}5 & -5 \end{pmatrix}$, plot $u(t)$, $x_1(t)$, $x_2(t)$ for $T=1,2,5$ second. Comment on the magnitude of the input in your results.

5.6 For a plant $\dot{x}=Ax+Bu$, where $A=\begin{pmatrix}0 & 1 \\ -2 & -3 \end{pmatrix}$, $u=\begin{pmatrix}0 \\ 2 \end{pmatrix}$, design a state feedback control to place the closed loop eigenvalues at $-2 \pm2j$.

5.7 For a plant $\dot{x}=Ax+Bu$, where $A=\begin{pmatrix}0 & 1 & 0\\ 0 & 0& 1\\ -15 & -23 & -9 \end{pmatrix}$, $u=\begin{pmatrix}0 \\ 0 \\ 4 \end{pmatrix}$, design a state feedback control to place the closed loop eigenvalues $-3, -4, -5$.

5.8 A model of aircraft roll control system is shown in figure. Design a state feedback to place the closed loop eigenvalues at $-5, -5, -50$.

Fig 5.1 Model of aircraft roll control

5.9 A multiple input system is modeled as $\dot{x}=\begin{pmatrix}-3 & 2 \\ 4 & -5 \end{pmatrix}x + \begin{pmatrix}1 & 0\\ 0 & 1 \end{pmatrix}u$. Find the feedback gain which places the pole at -4, -9 position.

5.10 Design the observer gain matrix $L$ to estimate the state of the system, $\dot{x}=\begin{pmatrix}-1 & 1 \\ 0 & -4 \end{pmatrix}x + \begin{pmatrix} 0\\ 4 \end{pmatrix}u; y=\begin{pmatrix} 1 & 0 \end{pmatrix}x$. Place the eigenvalues of the observer at $-10\pm10j$. Plot the truth states of the system assuming the initial values to be unity. Also plot the states estimated from the observer. Assume the initial states something different than the truth initial states. Plot the error.
Design the observer by placing the set of poles at three different positions. Is there any correlation between the pole positions of the observer and error plots. You may use MATLAB for plotting purpose.

5.11 Consider a system $\dot{x}=\begin{pmatrix}0 & 1 & 0 \\0 & -5 & 5 \\ 0 & 0 & -20\end{pmatrix}x + \begin{pmatrix} 0 \\ 0 \\ 80 \end{pmatrix}u; y=\begin{pmatrix} 1 & 0 & 0 \end{pmatrix}x$. Our objective is to redesign the system so that the closed loop poles are at -1, and $-2\pm4j$. Now let us assume all the states are not available for measurement and feedback. Only the first state is available for measurement. Find L for an observer with the eigenvalues at -6, and $-6\pm6j$ which could reconstruct all the states .