Reduced Order Positive Filter Design for Positive Uncertain Discrete-Time Linear Systems

The problems of reduced order \pmb
H
₂
and \pmb
H
_∞
positive filter design for positive uncertain discrete-time linear systems are investigated in this letter. Due to the positivity constraint on the matrices of the filter, optimal \pmb
H
₂
and \pmb
H
_∞
filters cannot be obtained through standard linear matrix inequality (LMI) methods, even in the context of full order filtering for precisely known systems. Therefore, new sufficient LMI conditions are proposed for \pmb
H
₂
and \pmb
H
_∞
positive filter design for positive discrete-time linear systems, having as main advantage the fact that the filter matrices are variables of the problem. In this case, no structural constraints on the optimization variables (source of conservativeness) are needed to ensure positivity. Thanks to a relaxation in the stability of the filter, an iterative algorithm with a feasible initial condition is proposed, allowing the search for positive filters that assure an \pmb
H
₂
or \pmb
H
_∞
guaranteed attenuation level for the filtered system. The conditions can deal with full or reduced order filtering, polytopic type uncertainty and structural constraints. Examples inspired on models borrowed from the literature illustrate the results.