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Stabilization of Nonholonomic Systems


The stabilization problem of nonholonomic systems, for many reasons, has been an active research topic for the last three decades. A key motivation for this research stems from the fact that nonholonomic systems pose considerable challenges to control system designers. Nonholonomic systems are not stabilizable by smooth time-invariant state-feedback control laws, and hence, the use of discontinuous controllers, time-varying controllers, and hybrid controllers is needed. Systems such as wheeled mobile robots, underwater vehicles, and underactuated satellites are common real-world applications of nonholonomic systems, and their stabilization is of significant interest from a control point of view. Nonholonomic systems are, therefore, a principal motivation to develop methodologies that allow the construction of feedback control laws for the stabilization of such systems.

In this dissertation, the stabilization of nonholonomic systems is addressed using three different methods. The first part of this thesis deals with the stabilization of nonholonomic systems with drift and the proposed algorithm is applied to a rigid body and an extended nonholonomic double integrator system. In this technique, an adaptive backstepping based control algorithm is proposed for stabilization. This is achieved by transforming the original system into a new system which can be asymptotically stabilized. Once the new system is stabilized, the stability of the original system is established. Lyapunov theory is used to establish the stability of the closed-loop system. The effectiveness of the proposed control algorithm is tested, and the results are compared to existing methods.

The second part of this dissertation proposes control algorithm for the stabilization of drift-free nonholonomic systems. First, the system is transformed, by using input transformation, into a particular structure containing a nominal part and some unknown terms that are computed adaptively. The transformed system is then stabilized using adaptive integral sliding mode control. The stabilizing controller for the transformed system is constructed that consists of the nominal control plus a compensator control. The Lyapunov stability theory is used to derive the compensator control and the adaptive laws. The proposed control algorithm is applied to three different nonholonomic drift-free systems: the unicycle model, the front-wheel car model, and the mobile robot with trailer model. Numerical results show the effectiveness of the proposed control algorithm.

In the last part of this dissertation, a new solution to stabilization problem of nonholonomic systems that are transformable into chained form is investigated. The smooth super twisting sliding mode control technique is used to stabilize nonholonomic systems. Firstly, the nonholonomic system is transformed into a chained form system that is further decomposed into two subsystems. Secondly, the second subsystem is stabilized to the origin using the smooth super twisting sliding mode control. Finally, the first subsystem is steered to zero using the signum function. The proposed method is applied to three nonholonomic systems, which are transformable into chained form; the two wheel car model, the model of front-wheel car, and the firetruck model. Numerical computer simulations show the effectiveness of the proposed method when applied to chained form nonholonomic systems. This research work is mainly focused on the design of feedback control laws for the stabilization of nonholonomic systems with different structures. For this purpose, the methodologies adopted are based upon adaptive backstepping, adaptive integral sliding mode control, and smooth super twisting sling mode control technique. The control laws are formulated using Lyapunov stability analysis. In all cases, the control laws design for the transformed models is derived first, which is then used to achieve the overall control design of the kinematic model of particular nonholonomic systems. Numerical simulation results confirm the effectiveness of these approaches.

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