Nonlinear Smooth Sliding Mode Control Framework for a Tumor-Immune Dynamical System Under Combined Radio-Chemotherapy

Sliding mode control (SMC) is a robust nonlinear control framework that enforces system trajectories onto predefined manifolds, providing strong robustness guarantees against uncertainties. However, SMC inherently introduces unwanted transients or chattering in system state trajectories, which may cause issues especially for sensitive applications such as regulation of drug administration. This paper proposes a multi-input smooth sliding mode control (MISSMC) strategy that combines radiotherapy and chemotherapy for a nonlinear tumor–immune dynamical system described by ordinary differential equations. The closed-loop system is first analyzed to establish key qualitative properties: all state variables remain positive and bounded, the sliding surfaces exhibit asymptotic convergence, and explicit analytical upper bounds on the cumulative therapy doses are derived under clinically motivated constraints. On this basis, a smooth hyperbolic-tangent sliding manifold and associated control law are designed to regulate the radiation and drug infusion rates. While the use of a hyperbolic-tangent smoothing function effectively suppresses chattering, it introduces a small steady-state error due to the presence of a boundary layer. To address this limitation, integral action is incorporated into the sliding surfaces, ensuring asymptotic convergence of tumor state and reducing residual steady-state error, while enhancing robustness against model uncertainties and parameter variations. Numerical simulations, based on a brain-tumor case study, show that the proposed smooth SMC markedly suppresses transient overshoots in both states and control inputs, while preserving effective tumor reduction. Compared with a conventional (non-smooth) SMC scheme, the MISSMC controller reduces baseline radiation and chemotherapy intensities on average by roughly 70%. Similarly, MISSMC lowers the overall cumulative doses on average by about 40%, without degrading the therapeutic outcome. The resulting integral smooth SMC framework therefore offers a rigorous nonlinear-systems approach to designing combined radio-chemotherapy protocols with guaranteed positivity, boundedness, and asymptotic stabilization of the closed-loop system, together with explicit bounds on the control inputs.