Based on the superiority of adaptive filtering algorithms designed with hyperbolic function-like objective functions, this paper proposes generalized spline adaptive filtering (SAF) algorithms designed with hyperbolic function-like objective functions. Specifically, a series of generalized new SAF algorithms are proposed by introducing the q-deformed hyperbolic function as the cost function, named SAF-qDHSI, SAF-qDHCO, SAF-qDHTA & SAF-qDHSE algorithms, respectively. Then, the proposed algorithm is theoretically demonstrated with detailed mean convergence and computational complexity analysis; secondly, the effect of different q values on the performance of the new algorithm is verified through data simulation; the new algorithm still has better performance under the interference of Gaussian noise and non-Gaussian noise even when facing the system mutation; finally, the new algorithm is verified through the measured engineering data, and the results show that the new algorithm has better convergence and robustness compared with the existing algorithm. In conclusion, the generalized algorithm based on the new cost function proposed in this paper is more effective in nonlinear system identification.