Stability radius formulation of L σ ‐gain in positive stabilisation of regular and time‐delay systems

Abstract

This study initially considers the relationship between stability radius and -gain of linear time-invariant positive systems. The -, -, and -gains of an asymptotically stable positive system are characterised in terms of stability radii and useful bounds are derived. The authors show that the structured perturbation of a stable matrix can be regarded as a closed-loop system with uncertainty structure represented by the unknown static output feedback. This makes it possible to relate the -gains in terms of closed-form expression available for stability radii of Metzler matrices. The authors generalise the above connection for positive-delay systems as well. Performance characterisation and computation of -gains are also given based on linear programming for and linear matrix inequality (LMI) for . The importance of this characterisation becomes evident when state feedback controllers are designed for regular and time-delay systems with positivity constraints. In particular, they show that positive stabilisation with maximum stability radius for the case of can be considered as an -gain minimisation, which can be solved by LMI. This inherently achieves the performance criterion and establishes a link to the reported iterative convex optimisation approaches that have been developed for the cases of and . A significant result of this study is the derivation of bounds for -gains and the unique commonality among the optimal state feedback gain matrices in obtaining -gains of the stabilised system.