In a recent work, a frequency method based on linear programming was proposed to design fixed-order linearly parameterized controllers for stable linear multi-model SISO systems. The method is based on the shaping of the open-loop transfer functions in the Nyquist diagram under a set of linear constraints guaranteeing a lower bound on the crossover frequency and a linear stability margin. In this paper, this method is extended to guarantee quadratic stability. For this purpose, new linear constraints based on the phase difference of the characteristic polynomials of the closed-loop systems are added in the Nyquist diagram. A simulation example illustrates the effectiveness of the proposed approach.