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Finding an equivalent model for the real large-scale systems that is computationally easy to simulate and yields responses similar to the original system is known as model order reduction (MOR). MOR is essential in designing and analyzing large-scale com- plex systems such as robotic systems, hydraulic systems, nuclear reactors, filters, sensor networks, etc. Additionally, the utility of MOR was aided by the growing complexity of mathematical models used to predict real-world systems such as climate systems and the human cardiovascular system, etc. This suggests that the complex models were replaced with a much easier-to-understand and much simpler one. MOR aims to offer an easy replacement for the large-scale stable model that displays the same responses. The goal of reduced-order models (ROMs) is to preserve key characteris- tics of the original large-scale system, such as stability, passivity, and input-output response, etc. MOR has significantly advanced control system design by enabling effi- cient modeling of complex systems. Research has been done over the past few decades on the MOR of large-scale systems, leading to the development of various MOR tech- niques. However, the loss of crucial characteristics of the original system in ROMs, such as stability, significant approximation error, and the absence of a priori error bond limitations, places restrictions on these methods. This thesis aims to explore effective MOR techniques for weighted and limited Gramian-based systems while en- suring the stability of ROMs. Also, this research is extended to nonlinear systems using cross-Gramians. For linear continuous- and discrete-time systems (which incorporate weighted and limited-interval scenarios), the MOR problem is first formulated. Based on weighted and limited intervals Gramians, new methods for linear continuous and discrete-time systems are developed. For specific double-sided weights and intervals of interest, the proposed techniques guarantee the stability of reduced-order models. Furthermore, for the weighted and limited intervals cases, the suggested algorithms provide a priori error-bound formulations. Comparing the proposed strategies to the most advanced model reduction methods already in use, they largely produce low ap- proximation error.