Parameterized Synthesis of Discrete-Time Positive Linear Systems: A Geometric Programming Perspective

This letter focuses on optimization problems of discrete-time positive linear systems. To this end, the synthesis problem is presented by introducing parameterized system coefficient matrices and optimizing system parameters directly. Based on results concerning positive linear systems and nonnegative matrix theory, we demonstrate that optimization problems of minimizing the parameter tuning cost while satisfying the
$H_{2}$
norm,
$H_{\infty }$
norm, and
$\ell _{1}/\ell _{\infty }$
Hankel norm constraints can be reduced to corresponding geometric programming problems. In turn, by imposing reasonable assumptions on system matrices, these geometric programming problems can be further transformed into convex optimization problems owing to the convexity of the logarithm transformation on posynomials. Finally, simulation experiments on a numerical example and epidemic spreading process example are used to show the validity of the main results.