Sliding Mode Control Design Using PIR Sliding Surface for Second Order Systems

This study proposes a new Sliding Mode Control (SMC) approach in which a proportional–integral–retarded (PIR) structure is employed as the sliding surface for second-order dynamical processes. Unlike conventional sliding surfaces that rely on proportional–derivative structures, the proposed PIR sliding surface integrates proportional, integral, and intentional delay terms directly into the surface definition, thereby eliminating explicit derivative action while improving transient shaping capability and noise robustness. A complete analytical formulation is provided, including the derivation of the PIR-based sliding surface, the associated closed-form SMC control law, and a Lyapunov-based stability analysis ensuring finite-time convergence and closed-loop stability. The controller parameters are optimally tuned using the MATLAB fmincon algorithm based on four integral performance criteria, namely ISE, IAE, ITSE, and ITAE. The proposed PIR-SMC scheme is evaluated through comprehensive simulation studies on a second-order electromechanical system. The results demonstrate fast reference tracking, reduced overshoot, and accurate steady-state performance. Robustness is further assessed by applying ±10% variations to the process parameters without retuning, where consistent dynamic behavior is maintained. Additionally, by delivering an external disturbance at various time instants, the suggested controller's disturbance rejection capability is examined. In all cases, the PIR-SMC-controlled system suppresses the disturbance effectively and restores the output to the reference value within 0.5–1 s without steady-state error. Comparative analyses indicate that ISE-based optimization provides superior transient performance, while ITAE-based optimization minimizes long-duration errors. Overall, the proposed PIR sliding surface–based SMC offers a robust and flexible control framework suitable for second-order processes and represents a viable alternative to conventional sliding surface based designs.