Analysis of Positive Systems With Input Saturation: Invariant Hyperpyramids and Hyperrectangles

This article addresses the problem of estimating the domain of attraction of positive systems under a saturated linear feedback. Compared with the analysis of general saturated linear systems, one remarkable difference is that only the domain of attraction inside the first orthant is of interest. Hence, we tackle this problem by exploiting different shapes of invariant sets confined in the first orthant, including hyperpyramids and hyperrectangles, which are, respectively, the level sets of sum-separable and max-separable Lyapunov functions. For an open-loop positive system, a sufficient condition is given to ensure that a pyramid (or a hyperrectangle) is contractively invariant for the closed loop. This condition is nonconservative for the single-input case. Moreover, for a system without open-loop positivity, we establish conditions such that the closed loop has a hyperrectangular invariant set. Examples show the superiority of the results, in comparison with those existing ones using invariant ellipsoids.