ABSTRACT: Glucose tolerance test is advised to detection of pre-diabetics and mild diabetics in the medical practice. Several mathematical models such as glucose model, were proposed for mimicking the blood glucose-insulin interaction. To predict accurate insulin requirement for each glucose concentration, we need to solve optimal control problems. In this model, constraints are linear and nonlinear forms of cost function. Although ordinary methods can be used in glucose model, but non-negative natures of medical variables made them difficult to use. To finding a new approximate solution of glucose model, Parameterization method with non-negative coefficients in polynomial was used. In this state Parameterization method, we use polynomials that ensure that the control variable is non-negative in this model. And increases the time for the model solution to be non-negative compared to conventional methods. The simplicity, lower requirement for mathematical calculations and more compatibility with variables are positive aspects of our Parameterization method.

In bang-bang optimal control problems, the control function is inherently piecewise constant. This feature creates substantial difficulties for the standard Legendre-Gauss-Radau pseudospectral method, which relies on polynomial approximation for the control function. This study introduces a simplified approach that seamlessly integrates sigmoid-based control Parameterization with the traditional Legendre-Gauss-Radau pseudospectral method. This integration enables precise approximation of discontinuous control profiles while maintaining the polynomial approximation for state variables. The proposed method significantly minimizes the number of decision variables in the optimization problem while precisely determining both the number and locations of switching points. This leads to notable enhancements in computational efficiency and solution accuracy. Numerical experiments conducted on two benchmark problems, a bridge crane system and a robotic arm control problem, demonstrate the exceptional precision and efficiency of the proposed method. Despite its simplicity, the method delivers results that are on par with those produced by more advanced and intricate techniques.

To develop a system for specific purpose, it needs to estimate its parameters (Parameterization). It can be used in different fields like engineering, industry etc. In this work, authors used adaptive algorithm to model a system that is applicable in industry for control. This adaptive model is non-linear where its estimation is based on kernel based Least-mean square (LMS) algorithm. The kernel used as Polynomial and Gaussian. As the system is nonlinear polynomial kernel-based algorithm fails to prove its efficacy, though it is of low complexity approach. Gaussian kernel-based application for nonlinear system control performance better as compared to polynomial kernel. Further its complexity is reduced and used for faster performance. The result shows its performance in form of MSE, MAE, RMSE for identification and control that is very useful in industrial application.