Question 2 (a) The oral absorption of a particular drug can be modelled using the following second order differential eq

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Question 2 A The Oral Absorption Of A Particular Drug Can Be Modelled Using The Following Second Order Differential Eq 1 (81.89 KiB) Viewed 16 times

Question 2 (a) The oral absorption of a particular drug can be modelled using the following second order differential equation: d² 5 y(t) + (k₂ + k) d dt y(1) + (k₂ k.) y(1) = */ /u(1) (1) dr² where ka and ke are rate constants, V is an apparent volume, y(t) denotes the plasma concentration (in mg/L) of the drug and u(t) denotes the input (a bolus injection of D mg of drug). Assuming that the plasma concentration of the drug is measured rewrite the system Equation (1) in state-variable form: x(t) = Ax(1) + Bu(t), x(0) = x₁₂ (2) y(t) = Cx(t) for appropriate matrices A, B and C. [7 marks] (b) For the particular case where ka = 3 s¹, ke = 2 s¹ and V = 2.5 L, determine the eigenvalues of the system matrix A and hence determine the behaviour of the free response (i.e., when u(t) = 0) as t tends to infinity. [6 marks] (c) The system matrix A found in Part (a) is in companion form. By applying a suitable transformation x = V z, that diagonalises the system defined by Equation (2), obtain the modes of the system. [10 marks] (d) Using your answer to Part (c), or otherwise, determine whether the system is controllable, and whether it is observable. [2 marks]