An Improvement in a Local Observer Design for Optimal State Feedback Control: The Case Study of HIV/AIDS Diffusion

The paper addresses the problem of an observer design for a nonlinear system for which a preliminary linear state feedback is designed but the full state is not measurable. Since a linear control assures the fulfilment of local approximated conditions, usually a linear observer is designed in these cases to estimate the state with estimation error locally convergent to zero. The case in which the control contains an external reference, like in regulations problems, is studied, showing that the solution obtained working with the linear approximation to get local solutions produces non consistent results in terms of local regions of convergence for the system and for the observer. A solution to this problem is provided, proposing a different choice for the observer design which allows to obtain all conditions locally satisfied on the same local region in the neighbourhood of a new equilibrium point. The case study of an epidemic spread control is used to show the effectiveness of the procedure. The linear control with regulation term is present in this case because the problem is reconducted to a Linear Quadratic Regulation problem. Simulation results show the differences between the two approaches and the effectiveness of the proposed one