<p>This paper presents an optimal proportional-integral (PI) disturbance observer design for relative degree-one systems, formulated as a quadratically constrained convex optimization problem (QCQP), and demonstrates its application to a surface-mounted permanent magnet synchronous motor (SPMSM) back-electromotive force (EMF) estimation system. By deriving the transfer function between actual and estimated back-EMF, explicit pole-zero relationships can be expressed in terms of observer gains. Performance requirements, such as bandwidth and attenuation levels of the observer system, are translated into convex equality and inequality constraints. A phase-delay-minimizing cost function is introduced to select the gain that minimizes phase delay within the feasible solution set. Both simulations and experimental validations on an SPMSM confirm the effectiveness of the proposed gain tuning method.</p>