An Algebraic Fuzzy Pole Placement Approach to Stabilize Nonlinear Mechanical Systems

Based on the general structure of mechanical systems described by their state-space representation, the Takagi–Sugeno fuzzy modeling, and the controllability property of fuzzy systems, an algebraic and practical approach to computing the fuzzy gain capable of ensuring the stability property of the Takagi–Sugeno fuzzy model is proposed in this article. The main idea consists of finding a continuous fuzzy gain such that any linear behavior, defined by the adequate selection of eigenvalues, is induced in the closed-loop fuzzy system. Therefore, by continuity, if the fuzzy model is an approximation sufficiently close to the mechanical system, then such a nonlinear system is also stabilized by the fuzzy controller. A notable advantage of the proposed method, when compared with similar approaches, is the simplicity of the resulting gain. The validity of the approach is illustrated through the numerical simulation of a sufficiently complex nonlinear system.