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Fuzzy observer design using linear matrix inequalities for fuzzy closed-loop control systems. (English) Zbl 1079.93030
The study is focused on the design of fuzzy observers for fuzzy Takagi-Sugeno models and fuzzy controllers. The fuzzy observer under discussion is governed by a collection of “if-then” ruies assuming the following form $\text{if }y_1(t)\text{ is }A_{i1}\text{ and }y_p(t)\text{ is }A_{ip}\text{ then }d\widehat{\mathbf x}/dt = A_i,\widehat{\mathbf x} + B_i{\mathbf u}+ L_i ({\mathbf y} - \widehat{\mathbf y})$ where “$$p$$” denotes the number of measured outputs, $${\mathbf y} =C_i{\mathbf x}$$ is the output of the $$i$$-th rule of the system, $$\widehat{\mathbf y}$$ denotes a global output estimate while $$L_i$$ is the $$i$$-th local observer gain matrix. $$A_{i1}, A_{i2},\dots,A_{ip}$$ are the corresponding fuzzy sets defined in the output space.
The design of the fuzzy observer exploits the linear matrix inequalities (LMI) formulation of the stability conditions for closed-loop fuzzy models. A separation property for the observer and controller is formulated. Both continuous-time and discrete-time fuzzy observers are introduced and discussed. A numerical example presented in the study is concerned with a model of an inverted pendulum.
##### MSC:
 93C42 Fuzzy control/observation systems
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