Stability Analysis of Piecewise Affine Systems

Piecewise affine systems are hybrid dynamical models in which the state–space is partitioned into regions, each governed by a distinct affine law. These formulations capture nonlinear behaviour through a mosaic of linear regimes, offering analytical tractability and interpretability. Stability analysis of such systems seeks to establish conditions under which all trajectories converge to an equilibrium or remain bounded, despite mode transitions. Central approaches include construction of common or piecewise Lyapunov functions, solution of linear matrix inequalities, and the use of polyhedral or quadratic functionals to certify robust performance in the presence of uncertainty. Advances in optimisation and algorithmic verification have extended these criteria to high-dimensional settings, enabling applications in robotics, power electronics and biological networks. By combining convex programming with geometric insights, modern methods achieve less conservative bounds on convergence rates and region invariance, supporting deployment in safety-critical contexts.

Research from Nature Portfolio

Recent studies have introduced a data-driven framework for synthesising Lyapunov functions via deep learning, significantly reducing conservatism in high-dimensional piecewise affine models. The resulting neural certificates adapt to regional dynamics, enabling automated stability verification with provable guarantees. Another contribution has advanced parameter-dependent polyhedral Lyapunov functions for uncertain systems, leveraging semi-definite programming to handle polytopic uncertainty without excessive computational burden. This work demonstrates improved scalability over classical techniques, particularly for networks of interacting subsystems. A further development proposes a real-time algorithm for embedded stability checks in power-electronic converters, combining incremental region enumeration with low-complexity updates. This approach ensures rapid detection of instability under switching events, paving the way for more resilient control architectures in variable-frequency drives.

Stability Analysis of Piecewise Affine Systems publication trend

The graph below shows the total number of articles in stability analysis of piecewise affine systems across all publications each year (not limited to Nature Index journals).

Technical terms

Piecewise affine system: A hybrid model partitioning the state–space into regions, each governed by an affine (linear-plus-constant) rule.

Lyapunov function: A scalar function that decreases along system trajectories, used to certify stability or convergence.

Polyhedral Lyapunov function: A piecewise‐linear Lyapunov candidate defined over polyhedral regions, enabling region‐specific stability tests.

Polytopic uncertainty: A structured representation of parameter variations confined within a convex polytope, often used in robust control analysis.

Dwell time: The minimum duration for which a system must remain in one mode before transitioning, employed to guarantee stability under switching.

References

1. On convergence for hybrid models of gene regulatory networks under polytopic uncertainties: a Lyapunov approach. Journal of Mathematical Biology (2021).