On the Robustness to Unmatched Disturbances of Finite-Time Dynamic Surface Control (FT-DSC): A Comparison with Second-Order TSM in a Third-Order System

Controlling high-order nonlinear systems in the presence of disturbances—especially unmatched disturbances (those acting at channels different from the control input)—is a challenging problem. For third-order systems, the system structure itself creates additional difficulties in designing controllers that guarantee both performance and stability. Dynamic Surface Control (DSC) is a recursive backstepping-based technique that circumvents the derivative-explosion problem of traditional backstepping by introducing first-order filters. When nonlinear (exponential) terms are used inside these filters, they form finite-time systems without incurring the computational burden of high-order derivatives. Such finite-time DSC (FT-DSC) not only preserves the advantages of standard DSC but also enhances its rejection of mismatched disturbances compared with conventional finite-time sliding controllers.