Blood glucose concentration is regulated primarily by the controlled release of insulin in the pancreas. However, patients with Type I diabetes mellitus are incapable of producing insulin and therefore require shots of insulin to be administered several times a day to regulate the blood glucose concentration.

Assuming that we are on a project to develop a miniaturized implantable insulin pump that is automatic. We will do some preliminary design work to determine the performance characteristics of such a device. As we have discussed in the class, we will need one mathematical model to describe the effect of insulin on glucose and a second mathematical model to describe the effect of a meal on the glucose concentration. We will use Bergman's minimal model (Bergman et al., 1981), described the three differential equations:

where G and I represent the deviation in blood glucose and insulin concentrations, respectively, X is proportional to the insulin concentration in a "remote" compartment. The inputs are Gmeal, (input of glucose due to a meal - considered a disturbance here), and U, the manipulated insulin infusion rate. The blood parameters include p₁, p₂, p₃, n and V₁ (which represents the blood volume). G_b and I_b are the "basal" (baseline) values of blood glucose and insulin concentration.

For the purpose of control system design, a linearized state-space representation with the state variables x₁ = G, x₂ = X, and x₃ = I, input variables u₁ = U - U_b and u₂ = G_meal (glucose disturbance due to meal input) and the output variable y=G can be derived out of the above non-linear model.

Some typical values for the above constants are as follows:

G_b = 4.5 mmol/liter

I_b = 4.5 mU/liter

V₁ =12 liters

p₁ = 0 /min

p₂ = 0.025/min

p₃ = 0.0000013 mU/liter

n = 5/54 /min

The corresponding transfer functions will be (you will have two transfer functions corresponding to the two inputs): and

The transfer function g_p(s) relates the output G to the input I; the transfer function g_d(s) relates the output G to the input G_meal.

The Problem:

Using the Bergman's glucose model:

a. Describe how you would set up the blood glucose control problem as a linear quadratic regulator problem. Determine the appropriate Q and R matrices that you would use and write down an appropriate cost function J that can be optimized.

b. Design a linear quadratic regulator to regulate the level of blood glucose concentration at 70 mg/dL.

Obtain the positive definite matrix P of the Riccati equation and optimal feedback gain matrix K. Assume both blood glucose concentration and insulin concentration are measurable and available for feedback.