Control Theory for Switched Linear Systems
Summary
Switched linear systems comprise a finite collection of linear subsystems together with a rule that governs transitions between them. Such systems capture a wide range of practical processes, from power converters and networked control platforms to autonomous vehicles and robotic manipulators. The central challenge lies in ensuring stability and performance despite abrupt mode changes. Stability analysis typically relies on multiple Lyapunov functions or a common Lyapunov function, coupled with constraints on the frequency or pattern of switching—most notably dwell-time and average dwell-time concepts. Controller synthesis often exploits linear matrix inequality (LMI) techniques to derive conditions for robust performance under parameter uncertainty, external disturbances and constrained actuators. Advanced designs incorporate H∞ and H2 optimisation to attenuate worst-case disturbances, while observer-based schemes provide state estimation in the face of incomplete measurements and noise. Recent advances have extended classical results to singular and neutral switched systems, embracing time-varying delays and stochastic perturbations. Practical implementations demonstrate improved resilience in automotive path-tracking, power electronics regulation and fault-tolerant control, emphasising the global significance of this theory for complex, safety-critical applications.
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Recent studies have reinforced robust stability criteria for discrete-time switched systems under uncertainty by introducing novel matrix inequalities that systematically handle state parameter variations. A new switching-law design based on these inequalities yields sufficient conditions for both stability and stabilisation, verified through constructive examples. Concurrent work on switched singular systems has demonstrated that proportional-derivative (P-D) state feedback, in conjunction with multiple Lyapunov functions and free-weighting matrices, can eliminate impulsive behaviour at switching instants and guarantee normal and stable closed-loop responses. This synchronous P-D design improves dynamic performance compared with stepwise methods, as confirmed by numerical simulations. Furthermore, observer-based robust control for switched neutral systems with interval time-varying delays has been advanced by combining the average dwell-time approach with the Yakubovich lemma. This framework provides LMI-based criteria for simultaneous observer and controller design, ensuring exponential stability despite coupled uncertainties and delay effects. Collectively, these contributions broaden the applicability of switched linear theory to more complex system classes and practical scenarios.
Control Theory for Switched Linear Systems publication trend
The graph below shows the total number of articles in control theory for switched linear systems across all publications each year (not limited to Nature Index journals).
Technical terms
Switched linear system: A dynamic system composed of several linear subsystems and a rule determining switching between them.
Lyapunov function: A scalar function used to assess and guarantee the stability of an equilibrium point in dynamical systems.
Dwell time: The minimum time interval required between consecutive switches to ensure system stability.
Linear matrix inequality (LMI): A convex constraint expressed in matrix form, widely used for control synthesis and stability analysis.
P-D state feedback: A control law combining proportional and derivative actions on the state vector to improve transient performance.
References
- Robust $H_\infty$ Fault-Tolerant Observer-Based PID Path Tracking Control of Autonomous Ground Vehicle With Control Saturation. IEEE Open Journal of Vehicular Technology (2024).
- Robust stability and stabilization of uncertain switched discrete-time systems. Advances in Continuous and Discrete Models (2012).
- On Stabilization of Linear Switched Singular Systems via P-D State Feedback. IEEE Access (2020).
- Observer-Based Robust Control Method for Switched Neutral Systems in the Presence of Interval Time-Varying Delays. Mathematics (2021).
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