In this paper, we propose an autonomous stair-driving system for the stable traversal of stairs by a tracked mobile robot operating in indoor disaster environments. Before developing the system, we conduct dynamic simulations to analyze the requirements for the robot to climb stairs. Simulations are performed under various initial conditions, and based on a detailed analysis of the results, we derive the necessary conditions for the robot's ascent. Using these requirements, we design the autonomous stair-driving system, which includes three main components: stair approach, stair alignment, and stair traversal. First, during the approach stage, we present a strategy for recognizing stairs using an object detection algorithm and generating control inputs for the stair approach motion. Next, in the alignment process, we outline an image processing sequence that extracts the edge contour of the stairs and a method for generating control inputs from the combined contour. Finally, in the traversal sequence, we describe the strategy for driving up the stairs. Additionally, we introduce an integrated ROS system to ensure the sequential execution of each strategy. We also verify the effectiveness of the individual strategies and demonstrate the capability of the proposed system through experiments using mock-up stairs and tracked robots.