From Nonlinear Dominant System to Linear Dominant System: Virtual Equivalent System Approach for Multiple Variable...

Area of Science:

• Control Engineering
• Systems Theory
• Nonlinear Dynamics

Background:

• Multivariable stochastic self-tuning systems (STC) present significant challenges in stability and convergence analysis due to their inherent nonlinearity.
• Existing methods often struggle with the complexity of these systems, limiting analytical tractability.

Purpose of the Study:

• To develop a simplified approach for analyzing the stability and convergence of multivariable stochastic self-tuning systems.
• To address the difficulties arising from the nonlinear structure of these control systems.

Main Methods:

• Introduction of the virtual equivalent system method to transform nonlinear STC systems into structurally linear or linear-dominated systems.
• Development of four theorems and two corollaries to analyze parameter estimation convergence cases (convergence, convergence to actual value, divergence).

Main Results:

• The virtual equivalent system method effectively simplifies stability and convergence analysis for multivariable STC systems.
• Parameter estimation convergence is shown to be a sufficient, but not necessary, condition for STC system stability and convergence.
• The proposed theorems and corollaries are independent of specific controller designs and parameter estimation algorithms.

Conclusions:

• The virtual equivalent system theory provides a robust framework for judging STC system stability and convergence, relying on system nature rather than specific control strategies.
• This method relaxes the dependence of stability convergence criteria on detailed system structure information.
• The effectiveness of the virtual equivalent system method is demonstrated across various parameter estimation behaviors, including convergence and divergence.