A General and Efficient Low-Order Approximation Method for Fractional-Order Systems With Improved Accuracy–Complexity Tradeoff

Fractional order filters and compensators are natural solutions to modeling complex physical phenomena or to improved design of closed loop performance compared to traditional integer order transfer functions. Despite manifold advantages, a caveat in their breakthrough in the systems and control engineering community is their unlimited memory, which makes them difficult to implement in practice. To preserve the benefits of fractional order filters, the implementation must meet both accuracy and simplicity from user expectations point of view. The traditional implementation is based on integer order approximations in a bounded frequency range. The continuous-time approximations are either simple or accurate, but rarely both simultaneously and focus on simple fractional order differentiator and integrator. In this paper, a weighted least-squares method is proposed based on a simple algorithm, making it an efficient and elegant solution. The method is suitable for all structures of fractional order systems and general non-rational transfer functions, being more versatile compared to existing approaches. Our solution also produces low-order approximations without compromising accuracy. Guidelines for a proper use of the method are provided. Several numerical examples, including fractional order controllers, general fractional order transfer functions and dead-time systems are illustrated. A comparison with the classical Oustaloup Recursive Approximation method demonstrates its added value with respect to accuracy and efficiency.