Blood glucose concentration is regulated primarily by the controlled release of insulin in the pancreas. However, patients with Type I...

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 G_meal, (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