PID controller design for output PDFs of stochastic systems using linear matrix inequalities

Area of Science:

• Control Systems Engineering
• Stochastic Systems Analysis
• Probability Theory

Background:

• Stochastic systems often exhibit non-Gaussian behavior, posing challenges for traditional control methods.
• Controlling the probability density functions (PDFs) of system outputs is crucial for advanced applications.
• Existing methods for PDF control have limitations in handling complex stochastic systems.

Purpose of the Study:

• To develop a novel pseudo proportional-integral-derivative (PID) tracking control strategy for general non-Gaussian stochastic systems.
• To enable precise control of conditional output PDFs to follow a specified target function.
• To improve robustness and tracking convergence in stochastic system control.

Main Methods:

• A linear B-spline model is employed to approximate output probability density functions (PDFs).
• The control structure, a pseudo PID controller, is designed prior to the PDF controller.
• The problem is reformulated as a weight-tracking problem based on the B-spline approximation.
• Controller solvability is analyzed using matrix inequalities, accommodating model uncertainties.

Main Results:

• The proposed strategy successfully controls conditional PDFs of system outputs to match target functions.
• A convex optimization-based design procedure enhances controller robustness and guarantees tracking convergence.
• Simulations demonstrate the effectiveness and efficiency of the developed control approach.

Conclusions:

• The pseudo PID tracking control strategy offers an effective solution for controlling non-Gaussian stochastic systems.
• The B-spline PDF modeling and convex optimization approach provide a robust and convergent control design.
• The findings present a significant advancement in the field of stochastic system control and PDF tracking.