A combination of two drugs (hydrochlorothiazide and oxybutynin) is commonly used to regulate blood pressure in elderly patients. These two drugs mainly affect two physiological variables of the patient (blood pressure and urine production rate). Since the goal is to regulate both variables with these two drugs, interaction analysis has to be performed to design two SISO control loops. For the following model, there are two inputs and two outputs (Ogunnaike and Ray, Process Dynamics, Modeling, and Control, Oxford University Press, 1994, p. 771.):
\left[\begin{array}{l}y_{1} \\y_{2}\end{array}\right]=\left[\begin{array}{cc}\frac{-0.04 e^{-0.1 s}}{0.11 s+1} & \frac{0.0005 e^{-0.15 s}}{0.21 s+1} \\\frac{0.22}{0.12 s+1} & \frac{-0.02}{0.21 s+1}\end{array}\right]\left[\begin{array}{l}u_{1} \\u_{2}\end{array}\right]
where
\begin{aligned}&y_{1}=\text { normalized (dimensionless) blood pressure } \\&y_{2}=\text { normalized urine production rate }\\&u_{1}=\text { rate of hydrochlorothiazide ingestion } \\&u_{2}=\text { rate of oxybutynin ingestion }\end{aligned}(a) Calculate the relative gain array.
(b) What loop pairing would you suggest?
