2.8 points: Sharks and Sardines: Given the system
$$
\dot{x}_1(t) = ax_1(t) - cx_1(t)x_2(t) - u(t)
$$
$$
\dot{x}_2(t) = -bx_2(t) + fx_1(t)x_2(t)
$$
$$
y(t) = x_1(t)
$$
where
* $x_1(t)$ and $x_2(t)$ are the populations of sardines and sharks respectively
* $u(t)$ is the fishing activity on sardines
* $y(t)$ is the output representing the population of sardines.
Find a linear state space representation for this sytem:
1. Find a nominal solution $\dot{x}(t) = \begin{bmatrix} \dot{x}_1(t) \\ \dot{x}_2(t) \end{bmatrix}$ when $\dot{u}(t) = 0$ and $\dot{x}(t) = 0$.
2. Compute the Jacobians $\frac{\partial f}{\partial x}$, $\frac{\partial f}{\partial u}$, $\frac{\partial g}{\partial x}$, $\frac{\partial g}{\partial u}$
3. Construct the linear state space representation in terms of matrices A, B, C and D
4. Use MATLAB to compute the impulse and step response of the linear system you derived for
a = 1, b = 2, c = 3, d = 4 and f = 4.
