Open AccessStability radius formulation of L σ ‐gain in positive stabilisation of regular and time‐delay systems
Author(s) -
Shafai Bahram,
Naghnaeian Mohammad,
Chen Jie
Publication year - 2019
Publication title -
iet control theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.059
H-Index - 108
eISSN - 1751-8652
pISSN - 1751-8644
DOI - 10.1049/iet-cta.2018.5296
Subject(s) - mathematics , control theory (sociology) , linear matrix inequality , positive systems , regular polygon , perturbation (astronomy) , linear system , stability (learning theory) , convex optimization , computation , stability theory , linear programming , convex combination , mathematical optimization , nonlinear system , computer science , control (management) , mathematical analysis , algorithm , physics , artificial intelligence , machine learning , geometry , quantum mechanics
This study initially considers the relationship between stability radius and L σ‐gain of linear time‐invariant positive systems. The L 1‐, L 2 ‐, 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 closed‐loop 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 L 2 ‐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.