Stability Analysis of Periodic Linear Systems
Summary
Periodic linear systems are dynamical models in which the governing matrices vary according to a fixed temporal pattern. Such systems arise in diverse applications, from electrical machines and power converters to synchronised biological rhythms and vibration suppression in mechanical structures. The core challenge is to determine conditions under which solutions remain bounded or converge to an equilibrium despite the inherent time-dependence. Classical Floquet theory converts a periodic problem into an equivalent time-invariant representation via a monodromy matrix, yielding necessary and sufficient tests for exponential stability. Complementary Lyapunov methods construct time-varying or piecewise Lyapunov functions to derive tractable linear matrix inequality conditions. More recent advances exploit matrix polynomial techniques and dwell-time arguments to handle switching among multiple periodic modes, to ensure performance in the presence of delays, uncertainties and external disturbances. Together, these approaches form a cohesive toolkit for analysing robustness and designing controllers that guarantee stability under realistic operating scenarios.
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Stability Analysis of Periodic Linear Systems publication trend
The graph below shows the total number of articles in stability analysis of periodic linear systems across all publications each year (not limited to Nature Index journals).
Technical terms
Periodic linear system: A dynamical system whose coefficient matrices are periodic functions of time, repeating with a fixed period.
Floquet theory: A method that transforms a linear system with periodic coefficients into an equivalent time-invariant form via the monodromy matrix, enabling stability tests.
Lyapunov–Krasovskii functional: An extension of Lyapunov functions that incorporates integral terms to address time delays and piecewise time-varying behaviour.
Dwell time: The minimum time interval that a switched system remains in one mode before transitioning, used to derive stability conditions under switching.
References
- Finite-Time Non-Fragile Extended Dissipative Control of Periodic Piecewise Time-Varying Systems. IEEE Access (2020).
- New Conditions of Analysis and Synthesis for Periodic Piecewise Linear Systems With Matrix Polynomial Approach. IEEE Access (2020).
- Stability analysis of periodically switched linear systems using Floquet theory. Mathematical Problems in Engineering (2004).
- Disturbance Observer-Based Robust Control Design for Uncertain Periodic Piecewise Time-Varying Systems With Disturbances. IEEE Access (2022).
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