Stabilization of SISO plants by use of low-order controllers is one of fundamental problems in linear control theory still remaining open. Its importance is based on the well-known fact that the majority of practically exploited controllers are PID-regulators. However the standard tools to design stabilizing PID-controllers (or even to check their existence) are lacking. The attempts to construct randomized algorithms for this purpose were unsuccessful, because the stability domain in the space of controller coefficients is usually small enough.We propose a novel approach to construction of such 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. Several extensions of the algorithms (for continuous-time systems, for robust design) are also considered.