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# EE 350 Control Systems Assignments

## Assignment 4: State space equation and it's solution

4.10 Determine the rank of the following matrix a)$\begin{pmatrix}1 \\ 3j \\ -1\end{pmatrix}$; b)$\begin{pmatrix}1 & 4 & -5 \\ 7 & 0 & 2 \end{pmatrix}$

4.11 For a system $\dot{x}=Ax;A=\begin{pmatrix}1/2 & -1 & 0 \\ 0 & -1 & 0 \\ 0 & 0 & -2\end{pmatrix}$ Solve the equation for initial conditions: $a) x(0)=\begin{pmatrix}1 & 1 & 1\end{pmatrix}^T$; b) $x(0)=\begin{pmatrix}2/3 & 1 & 0\end{pmatrix}^T$. Plot the states in figure for both the initial condition.

4.12 For $A=\begin{pmatrix}a & b \\ -b & a \end{pmatrix}$, Prove that $\phi=e^At=e^{at} \begin{pmatrix} cosbt & sinbt \\ -sinbt & cosbt\end{pmatrix}$.

4.13 4.13 For a system $A=\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$; $B=\begin{bmatrix} 1 \\ 1 \end{bmatrix}$; and $x(0)=\begin{bmatrix}1 & 0\end{bmatrix}^T$; \begin{align}\label{eq:N_q_n} \begin{split} u(t)&=1 \quad \quad \quad \quad \text{for}\quad t\geq0 \\ &=0 \quad \quad \quad \quad \text{elsewhere} . \end{split}\\ \end{align} Solve the state space equation. Plot the states in a figure.

4.14 For a system $\dot{x}=Ax+Bu, y=Cx,$ Where $A=\begin{bmatrix} 0 & 1 & 0 & 0 \\ 3 & 0 & 0 & 2 \\ 0 & 0 & 0 & 1 \\ 0 & -2 & 0 & 0 \end{bmatrix}$; $B=\begin{bmatrix} 0 & 0 \\ 1 & 0 \\ 0 & 0 \\ 0 & 1 \end{bmatrix}$; $C=\begin{bmatrix}1 & 0 & 1 & 0\end{bmatrix}$. Determine the TF of the system.

4.15 For a system $\dot{x}=Ax+Bu, y=Cx,$ Where $A=\begin{bmatrix} 0 & 0 & 1 & 0 \\ 3 & 0 & -3 & 1 \\ -1 & 1 & 4 & -1 \\ 1 & 0 & -1 & 0 \end{bmatrix}$; $B=\begin{bmatrix} 0 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{bmatrix}$; $C=\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$.
a) Determine the TF of the system.
b) For $x(0)=\begin{bmatrix} 1 & 1 & 1 & 1 \end{bmatrix}^T$ and $u(t)=\begin{bmatrix}1 & 1\end{bmatrix}^T$, $t\geq 0$. Determine the solution of the system. Plot the states using computer also plot the measurement.