Describe how you would set up the blood glucose control problem as a linear quadratic regulator problem.

Describe how you would set up the blood glucose control problem as a linear quadratic regulator problem.

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). Gb and Ib 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–Ub and u2=Gmeal (glucose disturbance due to meal input) and the output variable y=G can be derived out of the above non-linear model.

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