We propose a novel approach to construction of randomized algorithms algorithms. Random sample points are generated not in the coefficient space, but in so-called Fam-Medich parameter space (see e.g. [1], Lemma 3.3). This algorithm generates stable discrete-time polynomials, and for each of them we find the nearest polynomial in the subspace of characteristic polynomials corresponding to low-order controllers. If this polynomial is stable, the stabilizing controller is found. Otherwise we proceed to generate stable polynomials. If we achieved no success, several “most promising” candidates are found and we try to improve them locally by algorithms proposed in [2]. Numerical simulation demonstrated high efficiency of the approach.