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.
Some typical values for the above constants are as follows:
p1 = 0 /min
p2 = 0.025/min
p3 = 0.0000013 mU/liter
The corresponding transfer functions will be (you will have two transfer functions corresponding to the two inputs):
The transfer function gp(s) relates the output G to the input I; the transfer function gd(s) relates the output G to the input Gmeal.
Using the Bergman’s glucose model
- 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.
- 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