Non-fragile guaranteed cost fuzzy control for nonlinear first-order hyperbolic partial differential equation systems

Abstract

This paper concerns the non-fragile guaranteed cost control for nonlinear first-order hyperbolic partial differential equations (PDEs), and the case of hyperbolic PDE systems with parameter uncertainties is also addressed. A Takagi–Sugeno (T–S) fuzzy hyperbolic PDE model is presented to exactly represent the nonlinear hyperbolic PDE system. Then, the state-feedback non-fragile controller distributed in space is designed by the parallel distributed compensation (PDC) method, and some sufficient conditions are derived in terms of spatial differential linear matrix inequalities (SDLMIs) such that the T–S fuzzy hyperbolic PDE system is asymptotically stable and the cost function keeps an upper bound. Moreover, for the nonlinear hyperbolic PDE system with parameter uncertainties, using the above-design approach, the robust non-fragile guaranteed cost control scheme is obtained. Furthermore, the finite-difference method is employed to solve the SDLMIs. Finally, a nonlinear hyperbolic PDE system is presented to illustrate the effectiveness and advantage of the developed design methodology.

Keywords:

• T–S fuzzy hyperbolic PDE system
• non-fragile guaranteed cost control
• parameter uncertainties
• asymptotically stable
• linear matrix inequalities

2010 AMS Subject Classifications::

• 37B25
• 93C20
• 93C42
• 93D20

Acknowledgement

This work is supported by the National Natural Science Foundation of China (No. 60974139) and the Special research projects in Shanxi Province Department of Education (No. 2013JK0578).