| Volume 14, Number 4, 2024, Pages - DOI：10.11948/JAAC-2023-0361 |
| Uniformly exponentially stable approximation for the transmission line with variable coefficients and its application |
| Bingfeng Zhang |
| Keywords:Transmission line, Exponential stability, State reconstruction Semi-discretization, Average central-difference |
| Abstract: |
| We analyze an ideal transmission line, which is defined by the telegraph equation with variable coefficients, from the perspectives of numerical analysis and control theory in this note. Because the spatially semi-discrete scheme of the original system is insufficient for discussing uniform exponential stability, we apply a similar transform to the continuous system and produce an intermediate system that may be easily analyzed. To begin, we discuss uniform exponential stability for the intermediate system using an so called average central-difference semi-discrete scheme and the direct Lyapunov function approach. The proof is the same as in the continuous case. The Trotter-Kato Theorem is used to demonstrate the stability and consistency of numerical approximation scheme. Finally, we propose a semi-discrete strategy for the original system through an inverse transform. All results on intermediate system are then translated into the original system. The numerical state reconstruction problem is addressed as an essential application of the main results. Furthermore, several numerical simulations are used to validate the effectiveness of the numerical approximating algorithms. |
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