Summary: This paper introduces the design of the second-order GCF (Generalized Comb Filter) compensator based on $L_1$ optimization. The problem is represented by a set of nonlinear equations solved by the Newton-Raphson algorithm. The proof of the uniqueness of the solution is also provided. One can reduce the complexity of the design approximating the original set of nonlinear equations by a set of linear equations which depends only on the passband edge frequency $ω_p$. A design example demonstrates the effectiveness of the proposed design.