Adaptive Observers for a Class of Nonlinear Systems with Application to Induction Motor

Nonlinear Luenberger‐like observers for nonlinear MIMO systems

“…Proof. Since the system is uniformly observable, we can transfer the system (1), (2) and the observer (18) in the z coordinates by using the transformation z =¯ (x, U ). In z coordinates, the system and the observer are represented bẏ…”

“…To overcome this difficulty,x 1 ,x 2 are initialized to nonzero values at the startup. Simple analysis (for details, please refer [18]) shows that the term M(z, U ) is not globally Lipschitz continuous. Hence the observer proposed in the previous section cannot be a global state estimator for the induction motor system.…”

“…To overcome this difficulty, \documentclass{article}\usepackage{amssymb}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$\hat{x}_1$\end{document}, \documentclass{article}\usepackage{amssymb}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$\hat{x}_2$\end{document} are initialized to nonzero values at the startup. Simple analysis (for details, please refer 18) shows that the term M ( z, U ) is not globally Lipschitz continuous. Hence the observer proposed in the previous section cannot be a global state estimator for the induction motor system.…”

“…To obtain the gain matrix K , consider the observer error dynamics (36) derived in 18. By using the theorem 3.1, the stability condition for error dynamics (36) can be written as where \documentclass{article}\usepackage{amssymb}\usepackage[mathscr]{euscript}\footskip=0pc\pagestyle{empty}\begin{document}$A_T\,{=}\,T(A-KC)T^T$\end{document} and T is a similarity transformation matrix introduced for scaling.…”

“…Notice the state dependency of the nonlinear gain Q −1 (x, U )K of the observer(18), as opposed to the constant gain of the linear case. If the proposed observer is used for the linear system, the observability matrix term Q(x, U ) is independent of states making the nonlinear gain constant.…”