Problem $3(10$ pts $)$:

The following EOMs describe the behavior of a dynamic system $($a pharmaceutical drug absorption model$).$

$a^{}=-K_{1}a+d_{in}$

$b^{}=K_{1}a-K_{2}b$

$K_{2}b-K_{3}c=c^{}$

Assume $K_1,K_2,K_3$ are constant, and $d_{in}$ is the input.

A$.$ How many states does this system of equations have, and why? $(3$ pts$)$

B$.$ Write the state space form of the system as $x^{}=Ax+Bu.$ You will have to define your state vector and input vector. Clearly identify the matrices. $(4$ pts $)$

C$.$ Assume the desired output of the system is the vector $y$ that consists of a and $K_{2}b$ as individual elements. Write the output equation in the form of $y=Cx+Du.$ Clearly identify the matrices. $(3$ pts $)$