Interconnected Systems

\underline{Problem 1}: Consider the following interconnected system:

Suppose System 1 is modeled as:

\begin{aligned}\frac{d}{dt} [ \begin{matrix}X_{1} \\ X_{2} \end{matrix} ] & = & A_{1} [ \begin{matrix}X_{1} \\ X_{2} \end{matrix} ] + B_{1}e \\ U & = & C_{1} [ \begin{matrix}X_{1} \\ X_{2} \end{matrix} ] + D_{1}e \end{aligned}  (1)

and System 2 is modeled as:

\begin{aligned}\frac{d}{dt} [ \begin{matrix}X_{3} \\ X_{4} \end{matrix} ] & = & A_{2} [ \begin{matrix}X_{3} \\ X_{4} \end{matrix} ] + B_{2}U \\ Y & = & C_{2} [ \begin{matrix}X_{3} \\ X_{4} \end{matrix} ] + D_{2}U \end{aligned}  (2)

Compute the overall system with input Y_{sp} and output Y for the following values of the matrices:

\begin{aligned}A_{1} & = & [ \begin{matrix}1 & 2 \\ 3 & 8 \end{matrix} ] \\ B_{1} & = & [ \begin{matrix}1 \\ 2 \end{matrix} ] \\ C_{1} & = & [ 1 1 ] \\ D_{1} & = & [1] \\ A_{2} & = & [ \begin{matrix}−1 & 2 \\ 4 & −5 \end{matrix} ] \\ B_{2} & = & [ \begin{matrix}1 \\ 0 \end{matrix} ] \\ C_{2} & = & [ −1 2 ] \\ D_{2} & = & [0] \end{aligned}  (3)

\underline{Problem 2}: Consider the following interconnected system: Suppose System 1 is modeled as:

\begin{aligned}\frac{d}{dt} [ \begin{matrix}X_{1} \\ X_{2} \end{matrix} ] & = & A_{1} [ \begin{matrix}X_{1} \\ X_{2} \end{matrix} ] + B_{1}E \\ U & = & C_{1} [ \begin{matrix}X_{1} \\ X_{2} \end{matrix} ] + D_{1}E \end{aligned}  (4)

System 2 is modeled as:

\begin{aligned}\frac{d}{dt} [ \begin{matrix}X_{3} \\ X_{4} \\ X_{5} \end{matrix} ] & = & A_{2} [ \begin{matrix}X_{3} \\ X_{4} \\ X_{5} \end{matrix} ] + B_{2}U \\ Y & = & C_{2} [ \begin{matrix}X_{3} \\ X_{4} \\ X_{5} \end{matrix} ] + D_{2}U \end{aligned}  (5)

System 3 is modeled as:

\begin{aligned}\frac{d}{dt} [ \begin{matrix}X_{6} \\ X_{7} \end{matrix} ] & = & A_{3} [ \begin{matrix}X_{6} \\ X_{7} \end{matrix} ] + B_{3}Y \\ Q & = & C_{3} [ \begin{matrix}X_{6} \\ X_{7} \end{matrix} ] + D_{3}Y \end{aligned}  (6)

System 4 is modeled as:

\begin{aligned}\frac{d}{dt} [ X_{8} ] & = & A_{4} [ X_{8} ] + B_{4}Y_{sp} \\ P & = & C_{4} [ X_{8} ] + D_{4}Y_{sp} \end{aligned}  (7)

Compute the overall system with input Y_{sp} and output Y for the following values of the matrices:

\begin{aligned}A_{1} & = & [ \begin{matrix}1 & 2 \\ 3 & −4 \end{matrix} ] \\ B_{1} & = & [ \begin{matrix}1 \\ 0 \end{matrix} ] \\ C_{1} & = & [ 1 0 ] \\ D_{1} & = & [1] \\ A_{2} & = & [ \begin{matrix}−1 & 2 & 0 \\ 4 & −5 & 1 \\ 0 & 3 & 8 \end{matrix} ] \\ B_{2} & = & [ \begin{matrix}1 \\ 1 \\ 0 \end{matrix} ] \\ C_{2} & = & [ 1 1 0 ] \\ D_{2} & = & [0] \\ A_{3} & = & [ \begin{matrix}−1 & 0 \\ 3 & −2 \end{matrix} ] \\ B_{3} & = & [ \begin{matrix}1 \\ 1 \end{matrix} ] \\ C_{3} & = & [ 1 0 ] \\ D_{3} & = & [0] \\ A_{4} & = & [5] \\ B_{4} & = & [1] \\ C_{4} & = & [3] \\ D_{4} & = & [0] \end{aligned}  (8)

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