Adaptive Weighted Control for A Class of Nonlinear Systems

Optimized indirect adaptive robust control for a class of uncertain nonlinear systems with time‐varying parameters

“…Therefore, (1) covers any function that may be encountered in practice. It must be stated that if the functions of class c are absent, the nonlinear robust feedback terms adopted in the work of References 1, 9–11, 17, 18 is not necessary. The instances will be shown in the Section 4.…”

“…In the same way, Assumption 3 is also reasonable. It is noted that the derivatives of unknown parameters are assumed bounded in the work of References 1 and 18. It means that these parameters must not only be bounded but also be smooth, and the mutation is not considered.…”

“…For the vector to be estimated $$$ \bullet ={\left[{\bullet}_1\kern0.5em \dots \kern0.5em {\bullet}_s\right]}^T $$$, the smooth projection can be defined as the work of References 18, 19, 33, and: $$$$ \boldsymbol{\pi} \left(\boldsymbol{\tau} \right)={\left[{\pi}_1\left({\tau}_{\bullet_1}\right)\kern0.5em \dots \kern0.5em {\pi}_s\left({\tau}_{\bullet_s}\right)\right]}^T $$$$ where $$$ \boldsymbol{\tau} ={\left[{\tau}_{\bullet_1}\kern0.5em \dots \kern0.5em {\tau}_{\bullet_s}\right]}^T $$$ is the tuning function. Let $$$ {\boldsymbol{\varepsilon}}_{\bullet }={\left[{\varepsilon}_{\bullet_1}\kern0.5em \dots \kern0.5em {\varepsilon}_{\bullet_s}\right]}^T $$$ be a vector consist of arbitrary small positive real numbers.…”

“…In order to overcome this defect, researchers mainly use two methods to eliminate the effect of unstructured uncertainties. Nonlinear robust controller can suppress the error caused by modeling uncertainties, such as adaptive robust control (ARC) adopted in the work of References 1, 9–11, 17, 18, and the robust integral of the sign of the error (RISE) control adopted in the work of References 5 and 35. Disturbance observer can learn the main uncertainties and use them for compensation, such as extend state observer (ESO) adopted in the work of References 6, 7, 19 and neural network adopted in the work of References 9, 10, 14, 15.…”

“…Structured uncertainties consist of known functions and parameter uncertainties, as well as unstructured uncertainties are studied together. Among them, the parameters in matrix form are researched, and the unsmooth and discontinuous parameters assumed to be excluded in the work of References 1 and 18 are also taken into account. A modular approach based on the Z ‐swapping technique is exploited so that the parameters multiplied before the derivative of the state variables can also be estimated.…”