External Positivity of Discrete-Time Linear Systems: Transfer Function Conditions and Output Feedback

Abstract

This paper develops an optimization-based synthesis procedure for fixed-order output-feedback controllers, ensuring a stable closed-loop system with a monotonic step response and rejection of persistent disturbances. This result is based on a unified set of external positivity conditions for discrete-time linear systems. The conditions are derived using an alternative technique for the inverse <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\mathcal {Z}}$</tex-math></inline-formula> -transform of rational transfer functions based on Bell polynomials and a convenient recurrence relation for the impulse response. Our approach yields a necessary and sufficient condition expressed as polynomial inequalities in the transfer function coefficients. Under additional assumptions, we derive a sequence of weaker but numerically more tractable conditions, leading to an efficient controller synthesis procedure.