W transform and its application in fractional linear systems with rational powers

Abstract

Abstract Fractional linear systems have attracted widespread attention from scholars and researchers due to their excellent performance and potential application prospects. In the analysis and design of fractional linear systems, the solution of fractional linear systems is an important part. So far, the powers of s in the complex‐frequency‐domain equations obtained by the existing fractional Laplace transform are fractions, which makes it difficult to solve algebraic equations formed by multiple fractional powers. To solve this problem, based on the traditional Laplace transform, a new fractional Laplace transform—W transform is proposed, which can make the power of the equation an integer in the W‐domain. The main properties of this transformation are given, and an inversion theorem for W transform is obtained. When this transformation is applied to fractional linear systems with rational powers, the expansion formula will have a large number of terms if the traditional decomposition method is used, which makes the form of time‐domain solutions more complex. Therefore, a partial fraction expansion method in the W‐domain to simplify the form of time‐domain solutions is proposed. On this basis, the general steps of circuit analysis in the W‐domain are given. Finally, examples are used to verify the correctness and feasibility of the application.

Keywords

• frequency‐domain analysis
• Laplace transforms
• linear systems