<i>H</i> _∞ prescribed tracking performance-based fractional-order sliding mode control for disturbed discrete-time systems: an LMI approach

This paper presents a novel approach to the design of a discrete-time Fractional-Order Sliding Mode Control (FSMC) system for Single Input Single Output (SISO) applications, utilising the Autoregressive with exogenous input (ARX) method for system modelling. The proposed methodology emphasises the Prescribed Performance Control (PPC) to ensure minimal tracking errors through a transformation of tracking errors based on a specified performance function. A Fractional-Order Quasi-Sliding Mode Control (FQSMC) is subsequently developed to maintain the tracking error within predetermined bounds, effectively addressing the chattering phenomenon commonly associated with the traditional sliding mode control. The integration of fractional calculus enhances the smoothness of the output while ensuring accurate tracking, making it ideal for real-world applications. Additionally, a novel Linear Matrix Inequality (LMI) observer is introduced to bolster the system's resilience against uncertainties and disturbances without complicating the computational process. The effectiveness of the proposed Prescribed Performance Fractional-Order Quasi-Sliding Mode Control (PP-FQSMC) is validated through extensive MATLAB simulations and experimental studies conducted on a Tank system. Results demonstrate that the PP-FQSMC significantly outperforms traditional methods, with maximum tracking errors of 1.6 × 10−4 compared to 0.2 for the PP-QSMC and a Root Mean Square Tracking Error (RMSTE) of 1.3 × 10−3 compared to 1.2 × 10−4 for the PP-QSMC, ensuring strong tracking performance and stability. The findings indicate that the proposed controller effectively mitigates chattering in control signals while maintaining a smooth output. It showcases its potential for practical applications in environments characterised by model inaccuracies, time delays, and external disturbances.