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):

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

x₀ = corticotropin-releasing hormone

x₁ = corticotropin

x₂ = free cortisol

x₃ = albumin-bound cortisol

x₄ = corticosteroid-binding globulin

d₀ = an external generating factor

Express the system in the form ẋ = Ax + Bu.

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dxo = a00xo + a02x2 + do dt dx = a10xo + a11x1 + a12x2 dt dx2 = a20xo + a21X1 + a22x2 + a23X3 + a24X4 dt dx3 = a32X2 + a33.X3 dt dx4 = a42X2 + a44.X4 dt