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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.

Tractor-trailer wheeled robots (TTWRs) are highly nonlinear and underactuated dynamical systems. It is necessary to use nonlinear control methods, for the control of wheeled robots. Back-stepping method is a Lyapunov-based systematic technique for designing nonlinear control algorithms. In this paper, an adaptive back-stepping controller is proposed for the TTWRs. The proposed algorithm uses an adaptive layout for the compensation of the system wheel slips, which updates controller parameters based on a combination of error signals and estimated uncertainties. This paper is one of the firsts to propose a control algorithm for off-axle TTWRs in the presence of wheel slips. The control algorithm is designed to track the reference trajectories and make the robot asymptotically stable around the reference trajectories. The stability of the method is proved using Lyapunov theory. In order to compensate the sliding of wheels as the system uncertainties, appropriate adaptive rules have been investigated. Obtained results demonstrate the efficiency of the proposed method. The results for the tracking control of the TTWR in the presence of wheel slips show that the slip effects are effectively compensated using the proposed adaptive back-stepping control algorithm.

SUMMARY The purpose of this paper is to design a stabilizing controller for a car with n connected trailers. The proposed control algorithm is constructed on the Lyapunov theory. In this paper, the purpose of navigating the system toward the desired point considering the slip phenomenon as a main source of uncertainty is analyzed. First mathematical models are presented. Then, a stabilizing control approach based on the Lyapunov theory is presented. Subsequently, an uncertainty estimator is taken into account to overcome the wheel slip effects. Obtained results show the convergence properties of the proposed control algorithm against the slip phenomenon.

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