A Study on Asymptotic Stability of TSK Fuzzy PID and TSK Fuzzy State Feedback Control Systems
- A Study on Asymptotic Stability of TSK Fuzzy PID and TSK Fuzzy State Feedback Control Systems
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- In this thesis, we analyze the asymptotic stability of nonlinear systems controlled with Takagi-Sugeno-Kang (TSK) fuzzy PID controller and state feedback controller using circle criterion. We first study a linear system controlled with TSK fuzzy PID controller. We divide the TSK fuzzy PID controller into linear dynamic parts and a memoryless nonlinear function, and merge the linear dynamic parts with the linear plant under control and form a new interconnected form of the augmented linear plant and the memoryless nonlinear function. We then derive the transfer function matrix of the augmented linear plant and the sector condition of the memoryless nonlinear function. Based on the derived transfer function matrix and the sector condition, we determine sufficient conditions that guarantee asymptotic stability of the control system by using circle criterion.
The derived result can be adopted only for the linear system. However, TSK fuzzy PID controller is a nonlinear controller which is more appropriate for nonlinear system control. Therefore, we extend the analysis to the case that the plant under control is a second order nonlinear motion system. We use loop transformation to transform the nonlinear system to a linear system.
Next, we study asymptotic stability of a class of nonlinear system controlled with TSK fuzzy state feedback controller on a given domain. State feedback controller is as well-known as PID controller and hence, we consider TSK fuzzy state feedback controller. This result shows that our analysis can be applied to TSK fuzzy controller that has any number of premise variables.
Finally, we study asymptotic stability of a class of nonlinear system controlled with a conventional state feedback controller on a given domain. We commonly analyze the stability of this control system using linear stability analysis techniques by linearizing the nonlinear system at the operating point or using Lyapunov stability analysis technique. However, the linear stability analysis using the linearized system is valid only for a small region around the operating point and the Lyapunov stability analysis is sometimes difficult because of choosing Lyapunov function candidate. We present a new approach that guarantee asymptotic stability of the nonlinear system controlled with the conventional state feedback controller on a given domain by using loop transformation and circle criterion.
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