This study presents an analysis and controller design for dual-output single inductor DC-DC buck converter. We derive a linear model of the system and subsequently analyzing its properties using multivariable control techniques such as Relative Gain Array, Participation Matrix, and Singular Value Decomposition. Interactions among different system components are identified, and the most influential inputs are highlighted. Based on these findings, a Linear Quadratic Integral controller is proposed and compared with a decentralized Proportional Integral controller present in the literature. Simulations results demonstrate a clear improvement in the performance using the proposed controller, indicating its effectiveness in addressing the cross-regulation problem in the converter. This study provides a significant contribution to the design and control of dual-output DC-DC converters with a single inductor.