This article presents a unified understanding and judgement of the stability and convergence of a general self-tuning control (STC) system, which consists of arbitrary control strategy, arbitrary parameter estimation algorithm and a deterministic/stochastic linear time-invariant (LTI) plant. The necessary conditions required for the global stability and convergence of a general STC system are relaxed, i.e. the convergence of parameter estimates is removed for both deterministic and stochastic STC schemes. To reach this goal, 'virtual equivalent system (VES)' concept and methodology is adopted. With the help of VES, the original nonlinear dominant ( nonlinear in structure) problem is converted to a linear dominant ( linear in structure) problem. The results developed in this article show that STC systems are stable and convergent for the abundance of control strategies and parameter estimation algorithms, which will provide great flexibility in the applications of STC.