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The stability and convergence analysis of a multivariable stochastic self-tuning system (STC) is very difficult because of its highly nonlinear structure. In this paper, based on the virtual equivalent system method, the structural nonlinear or nonlinear dominated multivariable self-tuning system is transformed into a structural linear or linear dominated system, thus simplifying the stability and convergence analysis of multivariable STC systems. For the control process of a multivariable stochastic STC system, parameter estimation is required, and there may be three cases of parameter estimation convergence, convergence to the actual value and divergence. For these three cases, this paper provides four theorems and two corollaries. Given the theorems and corollaries, it can be directly concluded that the convergence of parameter estimation is a sufficient condition for the stability and convergence of stochastic STC systems but not a necessary condition, and the four theorems and two corollaries proposed in this paper are independent of specific controller design strategies and specific parameter estimation algorithms. The virtual equivalent system theory proposed in this paper does not need specific control strategies, parameters and estimation algorithms but only needs the nature of the system itself, which can judge the stability and convergence of the self-tuning system and relax the dependence of the system stability convergence criterion on the system structure information. The virtual equivalent system method proposed in this paper is proved to be effective when the parameter estimation may have convergence, convergence to the actual value and divergence.

From Nonlinear Dominant System to Linear Dominant System: Virtual Equivalent System Approach for Multiple Variable Self-Tuning Control System Analysis