Robust Positivizing and Stabilizing Controllers for Prescribed Positive Equilibria

Parameters and set pints of external variables in differential equations influence not only equilibria and their stability, but also system positivity. It sharply contrasts with the artificial but standard control formulation in which zero of the state is always an equilibrium, and we concentrate on the stability of the zero equilibrium. This paper deals with non-zero equilibria of nonlinear dynamics variables whose realizable domains are positive, such as mass, concentration, population, and logistic quantities. Once a desired positive point is picked in the state space, we first need to determine set points of input variables to render the point an equilibrium, Then, we design a feedback control to ensure the state variables their positivity and convergence to the equilibrium. Moreover, we want to have the control mechanism equipped with robustness for uncertainties and disturbances. This paper proposes a novel approach to this problem by using positivizing Lyapunov functions formulated with asymmetric supply rates.