Based on an analysis of the continuity of discrete transition functions, this paper presents a new approach for characterizing piecewise affine (PWA) systems’ regions of attraction (RoAs). First, the RoA boundary is proven to consist of the points where the discrete transition function is discontinuous and of the specific parts of switching surfaces. Then, the formulas and numerical algorithm for computing states with discontinuous discrete transition functions are developed. The proposed approach enables the characterization of the entire RoA for both equilibrium points and limit cycles. Finally, the proposed method is used to analyze multiple examples. For clarity, the results of a third-order inductor-capacitor-inductor resonant inverter are provided in a video. Although we focus on PWA systems, the key concept behind continuity and stability also applies to other hybrid dynamical systems, enabling the broader application of the proposed RoA computing method.
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This work was supported in part by the National Natural Science Foundation of China under Grant 61573074, 51277192, 51477020 and by the National High-Tech R & D Program of China under Grant 2015AA010402.
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Chen, Y., Sun, Y., Tang, C. et al. Characterizing regions of attraction for piecewise affine systems by continuity of discrete transition functions. Nonlinear Dyn 90, 2093–2110 (2017). https://doi.org/10.1007/s11071-017-3786-5
- Piecewise affine systems
- Hybrid dynamical systems
- Switched systems
- Region of attraction