Versatile video coding (VVC) aims to achieve high compression but also issues like varying content/network conditions. Existing rate control (RC) methods struggle to achieve optimal quality under these complex scenarios. This paper proposes a novel RC scheme for VVC based on a linear model. The Lagrange minimization multiplier is introduced under bit budget constraints, allowing optimized bit allocation. RC optimization is formulated as a convex solution, and is derived into the optimal quantization parameter (QP) for RC. Experimental analysis demonstrates the proposed linear model-based RC algorithm performances are better compared to other state-of-the-art methods due to their use of a linear model and optimal QP determination.