The paper presents a systematic method of stability analysis for high-dimensional dynamic systems involving a time delay and some unknown parameters. Here, the term "unknown" means that the parameters are constants but yet to be determined. The analysis focuses on the stability switches of those systems with increase of the time delay from zero to infinity. On the basis of the generalized Sturm criterion, the parameter space of concern is divided into several regions determined by a discrimination sequence and the Routh-Hurwitz conditions. It is found, as the time delay increases, that the system may undergo no stability switch, exactly one stability switch, or more than one stability switches when the parameters are chosen from different regions. To demonstrate the approach, a detailed analysis of the stability switches is made in the paper for an active vehicle suspension equipped with a delayed "sky-hook" damper and a four-wheel steering vehicle with time delay in driver's response, respectively.