## 6 Linear and Non-Linear Insulin Infusion Control Schemes
### Adaptive Control  
This control method is used by the controllers which must adapt to a controlled system with varying parameters which are initially uncertain. For adaptive modeling, the \"_Minimal Model of Bergman\"_[135] is commonly used due to its conceptual simplicity. Most of the existing T1 DM models are designed via the Bergman model. The T1 DM model can be extended for the T2 DM with ease. The model has two inputs: (i) glucose rate \(p(t)\), and (ii) the subcutaneously injected insulin flow \(u(t)\); at the same time this input is the control input as well. The output of the model is the plasma glucose level \(G(t)\). The model has three state variables that are connected to the blood plasma which are: (i) the blood glucose concentration \(G(t)\), (ii) insulin-excitable tissue glucose uptake activity \(X(t)\), and (iii) the blood insulin concentration \(I(t)\). Figure 11 shows the detailed model of an adaptive control using the T1 DM model.  
\[\dot{G}(t)=-(p1+X(t))\ G(t)+p1G_{\mathrm{B}}\ +p(t) \tag{9}\]  
\[\dot{X(t)}=-p2X(t)+p3[I(t)-I_{\mathrm{B}}] \tag{10}\]  
\[I(t)=-\mathrm{n}[I(t)-I_{\mathrm{B}}]+u(t) \tag{11}\]  
where \(G_{\mathrm{B}}\) and \(I_{\mathrm{B}}\) are the basal pre injection level of glucose and insulin in blood, p1: insulin-independent rate constant of glucose uptake in muscles and liver in (1/min), p2: rate for decrease in tissue glucose up take ability (in 1/min), p3: insulin-dependent increase in glucose uptake ability in tissue per unit of insulin concentration above the basal level in ((\(\mathrm{(}\mathrm{(}\mathrm{uU/mL)^{-1}\ min^{-1}}\)).  
Figure 11: The schematic structure of the T1 DM model used [135].  
Figure 10: Self-tuning controller with the help of estimates and adjustments [134].  
The controller reacts promptly to large and rapid variations in the insulin action [108]. The absorption model is approximated as the exponential equation given below.  
\[D(t)=\frac{D_{g}(t)Ag\ t\ e^{\frac{-t}{tmax,G}}}{t^{2}max,G} \tag{12}\]  
where \(D_{g}(t)\) represents the time function of the external glucose input.  
Then, a robust fixed point transformation (RFPT) method is applied to overcome problems like reliability of the model parameters and environmental disturbances [136]. The RFPT method is an alternative for the model reduction techniques. Only the response of the system to the control signal is observed. The deformed input is used to calculate this signal to approximate the model for already defined \"desired system response\" and \"purely kinematic terms\" are used to determine the desired response without using any information on the system's dynamics. Fixed point theorem is used to map the control signal and the system's response generated in the control cycle and actual desired response for single- input/single-output (SISO) is as follows:  
\[\mathrm{r_{n+1}=G(r_{n}r^{Des})=(r_{n}+K_{c})^{*})\times\{1+B_{c}[tanh(A_{c}(f( r_{n})-r^{Des}))]-K_{c} \tag{13}\]  
where K\({}_{c}\), A\({}_{c}\), B\({}_{c}\) = \(\pm\) 1 are the adaptive control parameters. For the adaption of the RFPT method a route for control signal is elaborated which determines control actions and parameters. Bergman equations and RFPT equations are solved to obtain the desired values.  
\[\mathrm{u^{Desired}=\ -\frac{G\widetilde{(t)}}{p3G_{b}}+\ \mathrm{Additive\ term}} \tag{14}\]  
The control parameters can be set without any optimization. Controller is efficient to control BG levels that are very close to basal value for a patient.