Positivity-Preserving <i>H</i>_∞ Model Reduction for Discrete-Time Positive Systems via a Successive Convex Optimization Algorithm

Abstract

This paper considers the positivity-preserving model reduction for discrete-time positive systems. Given a stable high-order positive system, we aim to find a reduced-order model such that the approximation error is minimized within a prescribed H∞ performance and positivity is preserved. Regarding the bounded real lemma, the sufficient and necessary condition for the existence of a reduced-order model is established in terms of bilinear matrix inequality and convex semi-definite constraint, which ensures that the reduced-order system is positive and the resulted error system is stable and has an H∞ performance level. Based on the inner-approximation strategy, we approximate the bilinear constraints with convex ones, under which an iterative procedure is provided to calculate the desired reduced-order model. Finally, an example is provided to demonstrate the effectiveness and potential benefits of the presented results.

Keywords

• positive systems
• <i>H</i>_∞ model reduction
• bilinear matrix inequalities
• successive convex optimization algorithm