Fixed-order controller design for LTI-SISO polytopic systems is investigated using rank deficiency constraint on the controller Sylvester resultant matrix. It is shown that the non-convexity of fixed-order controller design problem can be gathered in a rank deficiency constraint on Sylvester resultant matrix of the controller. Then, an improved convex approximation of the rank deficiency constraint is used and hence, the whole design problem is represented via LMIs. The effectiveness of the proposed method is shown by applying it to an experimental system.