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This paper presents Gradient-based Iterative (GI) and Two-Stage Gradient-based Iterative (2S-GI) identification algorithms for the Controlled Auto-Regressive Moving Average (CARMA) form of a multivariable tumor model. The mathematical proof of the 2S-GI algorithm for multivariable CARMA systems is provided, demonstrating its effectiveness in parameter estimation. The step-by-step introduction of the algorithm facilitates further studies and implementation. A comprehensive comparison between the GI and 2S-GI algorithms is conducted, evaluating their performance in terms of convergence rate and estimation accuracy. The introduced multivariable tumor model serves as a testbed for the algorithms’ effectiveness. The results of the comparison, supported by simulated data, demonstrate the superiority of the 2S-GI algorithm in accurately estimating the parameters of the CARMA system. This research provides valuable insights into the application of gradient-based iterative algorithms in controlling multivariable tumor models, paving the way for improved control strategies in cancer treatment. [ABSTRACT FROM AUTHOR]

Since system identification plays a crucial role in controlling systems, it is essential to have access to appropriate identification methods. In this paper, two novel identification methods are proposed for estimating Controlled Auto-Regressive Moving Average (CARMA) systems: the multi-innovation gradient-based iterative algorithm and the two-stage multi-innovation gradientbased iterative algorithm. Our primary objective is to estimate the unknown parameters of a tumor model using these methods. To evaluate the effectiveness of the proposed methods, various factors are considered, such as convergence rate and estimation error. By conducting simulations, the practical applicability and performance of the introduced algorithms are demonstrated. The obtained results are presented through tables and figures, providing a comprehensive analysis of the estimation outcomes. The multi-innovation gradient-based iterative algorithm and the twostage multi-innovation gradient-based iterative algorithm offer valuable contributions to the field of system identification, particularly in the context of CARMA systems. These methods offer an innovative approach to estimate the parameters of complex systems, specifically focusing on tumor models. The convergence rate and estimation error analysis highlight the reliability and accuracy of the proposed methods, indicating their potential for practical implementation. In conclusion, this paper presents novel identification methods for estimating CARMA systems in the context of tumor models. The proposed algorithms demonstrate promising results in terms of convergence rate and estimation accuracy. These findings contribute to the development of effective and reliable identification techniques, offering valuable insights for controlling and understanding complex systems. [ABSTRACT FROM AUTHOR]