Some Implications of Positivity with Interior Equilibria via Asymmetrically Scaled Sectorial Supply Rates

This paper presents some results to develop dissipative systems theory in terms of asymmetrically scaled sectorial supply rates. The supply rates, which describe strict passivity on asymmetric spaces, were proposed recently by the author to simultaneously verify stability and positivity of dynamical systems whose equilibria are interior points of positive state spaces. This paper demonstrates that systems defined by an asymmetrically scaled sectorial supply rate have finite asymmetric gain if their positivity is guaranteed by the supply rate. For cyclic networks whose component systems are defined by asymmetrically scaled sectorial supply rates, this paper shows that the networks are globally asymptotically stable on the positive space whenever positivity is guaranteed by the individual supply rates. The developments deal with general nonlinear systems. It is illustrated that positive systems design benefits a lot from asymmetrically scaled sectorial supply rates and their implications even for affine compartmental systems for which symmetric methods are ineffective.