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- Title
Stability radius formulation of L<sub>σ</sub> ‐gain in positive stabilisation of regular and time‐delay systems.
- Authors
Shafai, Bahram; Naghnaeian, Mohammad; Chen, Jie
- Abstract
This study initially considers the relationship between stability radius and Lσ ‐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 closed‐loop system with uncertainty structure represented by the unknown static output feedback. This makes it possible to relate the Lσ ‐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 Lσ ‐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 Lσ ‐gains and the unique commonality among the optimal state feedback gain matrices in obtaining Lσ ‐gains of the stabilised system.
- Publication
IET Control Theory & Applications (Wiley-Blackwell), 2019, Vol 13, Issue 15, p2327
- ISSN
1751-8644
- Publication type
Article
- DOI
10.1049/iet-cta.2018.5296