Stability radius formulation of L_σ-gain in positive stabilisation of regular and time-delay systems

This study initially considers the relationship between stability radius and L_σ-gain of linear time-invariant positive systems. The L₁-, L₂-, and L_∞-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 closedloop system with uncertainty structure represented by the unknown static output feedback. This makes it possible to relate the L_σ-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 L_σ-gains are also given based on linear programming for σ = 1,∞ and linear matrix inequality (LMI) for σ = 2. 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 σ = 2 can be considered as an L₂-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 σ = 1 and σ = ∞. A significant result of this study is the derivation of bounds for L_σ-gains and the unique commonality among the optimal state feedback gain matrices in obtaining L_σ-gains of the stabilised system.