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Sub-optimal control of fuzzy linear dynamical systems under granular differentiability concept.

Mehran Mazandarani¹, Naser Pariz²

¹Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam; Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam.

ISA Transactions

|March 20, 2018

Summary

This study presents a sub-optimal control method for fuzzy linear dynamical systems with uncertain parameters. The approach ensures system stability and optimal performance, outperforming traditional fuzzy standard interval arithmetic methods.

Keywords:

Fuzzy arithmetic Fuzzy differential equations Fuzzy optimal control Granular differentiability Horizontal membership functions Multidimensional RDM fuzzy arithmetic RDM arithmetic UBM phenomenon

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Area of Science:

Control Theory
Fuzzy Systems
Dynamical Systems

Background:

Fuzzy linear dynamical systems present challenges due to inherent uncertainties in coefficients and initial conditions.
Traditional control methods struggle with the granular nature of fuzzy derivatives and interval arithmetic.

Purpose of the Study:

To develop a sub-optimal control strategy for fuzzy linear dynamical systems that ensures state variables remain close to zero.
To address limitations of existing fuzzy interval arithmetic approaches in accurately determining system eigenvalues and control laws.

Main Methods:

Utilizes relative-distance-measure (RDM) fuzzy interval arithmetic and calculus of variations to derive an optimal control law.
Defuzzification of fuzzy feedback gains to obtain a practical sub-optimal control law.
Introduces granular eigenvalues, granular controllability, and granular stabilizability concepts for fuzzy systems.

Main Results:

The proposed RDM-based approach accurately determines system eigenvalues, unlike fuzzy standard interval arithmetic (FSIA) methods.
Demonstrates effective sub-optimal control for a Boeing 747 longitudinal model and an uncertain bus suspension system.
Validates the robustness of the control strategy under uncertain initial conditions and parameters.

Conclusions:

The RDM fuzzy interval arithmetic offers a more accurate and reliable method for controlling uncertain fuzzy linear dynamical systems.
The developed sub-optimal control law is effective in practical applications, including aerospace and automotive systems.
The study advances the understanding of granular properties like controllability and stabilizability in fuzzy dynamical systems.