Designing Robust Software Sensors for Nonlinear Systems via Neural Networks and Adaptive Sliding Mode Control

The paper’s proof of Theorem 1 hinges on an incorrect inequality for the tanh nonlinearity (claiming s^T tanh(s) ≥ σ||s||^2 with a uniform σ>0), and imposes gain conditions that depend on the unknown estimation error ||e_k||, making the conditions non-operational. These issues occur in Step 8 and the subsequent conditions (30)–(33), and they undermine the claimed Lyapunov decrease and the stated exponential convergence result . By contrast, the model’s solution uses standard LTV-Lyapunov arguments, bounds tanh conservatively via ||tanh(s)|| ≤ ||s||, and obtains practical exponential stability to a bounded region; under decaying NN error it correctly tightens to exponential convergence to 0, provided a standard small-gain margin is ensured. The model’s argument is consistent with the paper’s own Assumption 3 (uniform observability implying the existence of a stabilizing observer gain) and avoids the paper’s flawed tanh step.