This work addresses the problem of adaptive observer design for nonlinear systems satisfying incremental quadratic constraints. The output of the system includes nonlinear terms, which puts an additional strain on the design and feasibility of the observer, which is guaranteed under the satisfaction of an LMI, and a set of algebraic constraints. A particular case where the output nonlinearity matches the unknown parameter coefficient is also discussed. The result is illustrated through a numerical example for the chaos synchronization of the Rössler system.

Induction motors (IMs) experience power losses when a portion of the input power is converted to heat instead of driving the load. The combined effect of copper losses, core losses, and mechanical losses results in IM power losses. Unfortunately, the core losses in the motor, which have a considerable impact on its energy efficiency, are not taken into account by the generally employed dynamic model in the majority of the studies. Due to this, the motor rating often corresponds to the worst-case load in applications, but the motor frequently operates below rated conditions. A hybridized model reference adaptive system (MRAS) with sliding mode control (SMC) is used in this study for sensorless speed control of an induction motor with core loss, allowing the motor to operate under a variety of load conditions. As a result, the machine can run at maximum efficiency while carrying its rated load. By adjusting the ��-axis current in the �� - �� reference frame in vector-controlled drives, the system’s performance is enhanced by running the motor at its optimum flux. Regarding the torque and speed of both induction motors with and without core loss, the Adaptive Observer Sliding Mode Control (AOSMC) has been constructed and simulated in this case. The AOSMC with core loss produced good performance when the proposed controller was tested.