In many engineering systems, it is not enough to merge the system paths to zero at infinite time, but the speed of moving these paths to zero is very important. Estimating this speed can be done using exponential functions. This concept is used in exponential stability definition. The purpose of this paper is to design a controller for problem inputs and implement a system of a car with N to a trailer connected to it. This approach is based on the analysis of the Lyapunov stability method. In the given problem, the purpose of conducting and converging the system considering the slip phenomenon as a primitive uncertainty in the system is toward the desired point. Since the trailer tractor system has limitation constraints in the modeling structure, it is difficult to guarantee the stability of a non-holonomic system. Because no controller designed by the control feedback method can continuously and stable ensure the convergence of the system. If this possibility almost dynamic errors, even adaptive controls do not versatile with the operation of the Lyapunov function, especially in the presence of uncertainties, which is a very important factor in system instability, which requires the development of controllers designed to deal with these disturbances. In the simulated results, this paper not only examines the convergence properties, but also shows the ability to control the system by designing a controller in the presence of a slip phenomenon to strengthen the system in the stability debate.

The wheeled mobile robot with differential thrust consists of two independent active wheels and a passive spherical wheel. Assuming its net rolling and non-uncertainty, this robot is a nonlinear system bound to non-holonomic constraints. This system also falls into the category of systems with a lack of operators. Tracing time travel paths is one of the most difficult issues in the field of wheeled robots that we will address in this article. In this regard, first the kinematic model of the system with the presence of uncertainty on the control inputs is expressed in which the linear velocity and angular velocity of the robot are considered system inputs. After determining the desired reference paths, using the linearization of the designed feedback controller ensures the stability of all system state variables globally. The controller is then designed with adaptive rules to solve the problem of tracking time paths based on input-output control in the presence of uncertainties. The stability of this controller is also proven globally. Finally, the performance of the designed controllers to compensate for the uncertainties will be compared by comparing the results.