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The proposed nonlinear backstepping controller is not only to stabilize the robot system, but also to drive the trajectory tracking errors to converge to zero exponentially.
That is to say, the proposed backstepping controller is not only to stabilize the flexible robotic manipulator, but also to drive the trajectory tracking error and tip-deflection to converge to zero asymptotically.
According to the Lyapunov stability criterion, the wheeled mobile manipulator is divided into several stabilizing subsystems, and an adaptive law is designed to estimate the general nondeterminacy, which make the controller be capable to drive the trajectory tracking error of the mobile manipulator to converge to zero even in the presence of perturbations and mathematical model errors.
However, in model C, we had two different performance errors that drive the two networks: difference between performance and desired trajectory to drive the trajectory controller, and the difference between performance and the expected output to drive the plant estimator.
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The designed tracking controller can not only suppress the mismatched perturbations when the controlled dynamics (master slave) are in the sliding mode, but also drive the trajectories of synchronization errors into a small bounded region whose size can be adjusted through the designed parameters.
In this model, the stochastic time trajectories of CI switch between two stable equilibria if the noise amplitude is sufficient to drive the trajectories occasionally out of the basin of attraction of one equilibrium into the basin of attraction of the other equilibrium (see Figure 4(a)).
The proposed AFBC offers a design approach to drive the chaotic trajectory to track a desired trajectory, and it is comprised of a fuzzy backstepping controller and a robust controller.
This RISMC system offers a design approach to drive the state trajectory to track a desired trajectory, and it is comprised of an adaptive RCMAC and a robust controller.
The corresponding reaching law is designed, which can drive the state trajectory of the closed system onto the sliding surface within limited time.
One of the most important optimal control problems is how to drive the corresponding trajectory of the equation from an initial state to a given target set in the shortest time, through applying constrained controllers.
Thirdly, a SMC law is synthesized to drive the system trajectories onto the predefined switching surface in a finite time.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com