International Journal of Advanced Robotic Systems Robust Adaptive Control for a Class of Nonlinear Systems Using the Backstepping Method Regular Paper

This paper develops a robust adaptive control for a class of nonlinear systems using the backstepping method. The proposed robust adaptive control is a recursive method based on the Lyapunov synthesis approach. It ensures that, for any initial conditions, all the signals of the closed‐loop system are regularly bounded and the tracking errors converge to zero. The results are illustrated with simulation examples.


Introduction
In recent decades, a large number of papers have studied the problem of robust adaptive control of nonlinear systems (see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and references therein). In [1], a new adaptive law based on an optimal control formulation for the minimization of the 2 L norm of the tracking bounded error is considered. A method for designing a global adaptive neural network controller for a class of uncertain non-linear systems is proposed in [2]. In [3][4][5][6][7][8][9], adaptive control of uncertain nonlinear systems using backstepping is developed. In these papers, the backstepping method guarantees global stabilities, tracking and transient performance for a broad class of strict-feedback system. New adaptive feedforward cancellations (AFC) control providing periodic tracking and/or periodic disturbance rejection is proposed in [10]. In [11], a robust control system combining backstepping and sliding mode control techniques is used to realize the synchronization of two gap junction coupled chaotic FitzHugh-Nagumo (FHN) neurons in the external electrical stimulation. The paper [12] introduces an optimal H  adaptive PID (OHAPID) control scheme for a class of nonlinear chaotic system with uncertainties and external disturbances. In [13], an adaptive backstepping control scheme is proposed for task-space trajectory tracking of robot manipulators. In [14], an adaptive integral backstepping algorithm is proposed as a means to effectuate the attitude control of a 3-DOF helicopter. In [15], a backstepping approach is used for the design of a discontinuous state feedback controller for the controller.
The main contribution of this paper is the design of an adaptive backstepping control method for a class of uncertain single input single output nonlinear systems which can be transformed into a triangular form. Many systems, such as AC motors, spacecraft, magnetic suspension and robot manipulators, possess this structure [14]. These systems can be built from subsystems that can be stabilized [14,15]. In this paper, we assume that the bounds of the uncertainty system parameters are available. The adaptive calculating procedure of the control is a recursive procedure based on the Lyapunov approach. It is composed of several steps. It can start at the known-stable system and back out new controllers that gradually stabilize each outer subsystem. The procedure stops when the final external control is reached. Compared with the adaptive control scheme, the proposed control approach has the advantages of adaptive technique and robust control, which makes this approach attractive for a wide class of nonlinear systems with both uncertain nonlinearities and disturbances. This paper is organized as follows: the formulation of the problem is introduced in Section 2; the controller design and stability analysis are presented in Section 3; the results of the simulation, illustrating the efficiency of the proposed controller, are presented in Section 4.

Problem statement and preliminaries
Let us consider the following nonlinear system where: The parameter 1 c   is selected such that 1 c 0  .

For 1
  , a smooth approximation of the sgn function is The relationship (9) is usually used to reduce the chattering which is caused by the sgn function [16].
Therefore, the equation (7) can be written in the following form Consider the following Lyapunov function candidate By using (1), (5), (6), (9) and (10), the time derivative of The Lyapunov function candidate is defined as: Its derivative is: When i n 1   , (17) can be rewritten as:   n 1 2 n 1 n 1 i i n 1 n 1 n j 1 Step n : In this step the practical control input u will be constructed to make     n 1 n 1 n n n 1 1 n 1 d x x ,c , ,c y as small as possible.
The time derivative of n  is given by: x ,c , ,c y We choose n c   such that We consider the Lyapunov function candidate: At the end of the backstepping procedure, we take: Then, the time-derivative of n V satisfies the following condition: Let us consider: Therefore, we obtain: Theorem 1. Suppose that the proposed control method in this section is applied to the system (1). Then, for any initial conditions, the closed-loop system is globally stable for t 0,       . Moreover, the tracking error converges to zero, i.e.,   e t 0  as t   .
Proof. By means of the Barbalat lemma [17,18], we have This implies that: This completes the proof.

Simulation examples and discussion
In this section, the feasibility of the proposed method and the control performances are illustrated with three examples. The simulation results are carried out using the software MATLAB.

Example 1
For the simulation example 1, we consider the following nonlinear system where: u and y are the control signal and measured output, respectively; The aim of the control is to force the output 1 y x  to asymptotically track a reference signal d y . Using the controller design procedure described in section 3, we can write: Then, the control input becomes: x 0 20,x 0 25    are shown in figures 1-2. It can be seen that the output of the closed-loop system tracks the reference signals. (c) control input u ; (d) tracking error d y y 

Example 2
For the simulation example 2, we consider the nonlinear system, which is described as where: u and y are the control signal and measured output, respectively;   x ,c 1 u 1 6x 5x x 1 x 3x x ,c 10 tanh x For the initial states       1 2 x 0 50,x 0 50    , simulation results are presented in figures 3-6. From the results, we find that the output of the closed-loop system tracks the reference signals. It can be seen that the chattering amplitudes have been effectively reduced where 100  For simulation example 3, we consider the following nonlinear system, which is defined as  x ,x 30   .
The objective of the control is to force the output 1 y x  to asymptotically track a reference signal d y . According to the described controller design procedure in section 3, we have:   x ; (c) control input u ; (d) tracking error d y y 

Conclusion
In this paper, we propose a method of designing an adaptive controller for a class of nonlinear systems using the backstepping technique. The on-line calculation of the control input is obtained using the Lyapunov synthesis approach. The proposed approach guarantees that all the signals of the resulting closed-loop systems are regularly bounded and the tracking error converges to zero. It has advantages, such as a simple structure, easy realization, a good control effect and strong robustness. The efficiency of the proposed control has been demonstrated by simulation studies. Future works could expand the method to be used for a more general class of uncertain nonlinear systems.