Output-feedback controllers with guaranteed ℒ₂-gain for continuous-time Lur'e systems using noncausal Zames–Falb multipliers

Abstract

This paper focuses on the problem of dynamic output feedback control for continuous-time Lur'e systems with slope-bounded nonlinearities, particularly addressing the closed-loop ${\mathcal{L}}_2$ gain. The synthesis framework is derived using integral quadratic constraints. The following contributions are highlighted: (i) assessment of stability and ${\mathcal{L}}_2$ gain through causal, anticausal, and noncausal Zames–Falb multipliers; (ii) technical enhancements to improve efficiency and to reduce the conservativeness of two different classes of multipliers regarding the ${\mathcal{L}}_1$ norm evaluation; (iii) the synthesis conditions are presented in terms of an iterative procedure based on linear matrix inequalities, allowing arbitrary order for both the controller and the multipliers. The proposed achievements are validated through numerical examples, demonstrating the efficacy and flexibility of the approach compared to existing methods in the literature.

Keywords:

• Continuous-time Lur'e systems
• absolute stability
• Zames–Falb multipliers
• linear matrix inequalities
• ${\mathcal{L}}_2$ gain bounds
• dynamic output-feedback

Acknowledgments

We would like to thank the reviewers for their suggestions, which have significantly contributed to the improvement of the present work.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 When the positivity of the ZF multipliers is assessed by constraints on the matrices of the state space realisation, non-odd nonlinearities can also be considered.

Additional information

Funding

This study was partially supported by Brazilian agencies Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) - Finance Code 001; by Grant 2019/10947-1, São Paulo Research Foundation (FAPESP); by National Council for Scientific and Technological Development (CNPq); by Project Stic-Amsud/CAPES NetConHybSDP, code 22-STIC-09; by the ANR via grant HANDY, No. ANR-18-CE40-0010; and by Science Foundation Ireland under grant number 21/FFP-P/10065.