A linear, time-invariant model of the

Chapter 3, Problem 23

(choose chapter or problem)

A linear, time-invariant model of the hypothalamic pituitary-adrenal axis of the endocrine system with five state variables has been proposed as follows (Kyrylov, 2005):

\(\begin{aligned} & \frac{d x_0}{d t}=a_{00} x_0+a_{02} x_2+d_0 \\ & \frac{d x_1}{d t}=a_{10} x_0+a_{11} x_1+a_{12} x_2 \\ & \frac{d x_2}{d t}=a_{20} x_0+a_{21} x_1+a_{22} x_2+a_{23} x_3+a_{24} x_4 \\ & \frac{d x_3}{d t}=a_{32} x_2+a_{33} x_3 \\ & \frac{d x_4}{d t}=a_{42} x_2+a_{44} x_4 \end{aligned}\)

where each of the state variables represents circulatory concentrations as follows:

\(x_0\)= corticotropin-releasing hormone

\(x_1\)= corticotropin

\(x_2\)= free cortisol

\(x_3\)= albumin-bound cortisol

\(x_4\)= corticosteroid-binding globulin

\(d_0\)= an external generating factor

Express the system in the form \(\dot{\mathbf{x}}=\mathbf{A x}+\mathbf{B u}\).