Adaptive backstepping control of a class of uncertain nonlinear systems with unknown backlash-like hysteresis

In this note, we consider the same class of systems as in a previous paper, i.e., a class of uncertain dynamic nonlinear systems preceded by unknown backlash-like hysteresis nonlinearities, where the hysteresis is modeled by a differential equation, in the presence of bounded external disturbances. By using backstepping technique, robust adaptive backstepping control algorithms are developed. Unlike some existing control schemes for systems with hysteresis, the developed backstepping controllers do not require the uncertain parameters within known intervals. Also, no knowledge is assumed on the bound of the "disturbance-like" term, a combination of the external disturbances and a term separated from the hysteresis model. It is shown that the proposed controllers not only can guarantee global stability, but also transient performance.


I. INTRODUCTION
Hysteresis exists in a wide range of physical systems and devices, such as biology optics, electromagnetism, mechanical actuators, electronic relay circuits and other areas. Control of such systems is typically challenging. For backlash hysteresis, several adaptive control schemes have recently been proposed; see, for example, [1] and [2]. In [3]- [5], an inverse hysteresis nonlinearity was constructed. An adaptive hysteresis inverse cascaded with the plant was employed to cancel the effects of hysteresis. In [1], a dynamic hysteresis model is defined to pattern a backlash-like hysteresis rather than constructing an inverse model to mitigate the effects of the hysteresis. However, in [1], the term multiplying the control and the uncertain parameters of the system must be within known intervals and the "disturbance-like" term must be bounded with known bound. Projection was used to handle the "disturbance-like" term and unknown parameters. System stability was established and the tracking error was shown to converge to a residual.
In this note, we develop two simple backstepping adaptive control schemes for the same class of nonlinear systems as in [1], with bounded external disturbances included in our case. Besides showing global stability of the system, the transient performance in terms of L2 norm of the tracking error is derived to be an explicit function of design parameters and thus our scheme allows designers to obtain the closed loop behavior by tuning design parameters in an explicit way. In the first scheme, a sign function is involved and this can ensure perfect tracking. To avoid possible chattering caused by the sign function, we propose an alternative smooth control law and the tracking error is still ensured to approach a prescribed bound in this case. In our design, the term multiplying the control and the system parameters are not assumed to be within known intervals. The bound of the "disturbance-like" term is not required. To handle such a term, an estimator is used to estimate its bound. Manuscript  This note is organized as follows. Section II states the problem of this note and assumptions on the nonlinear systems. Sections III presents the adaptive control design based on the backstepping technique and analyzes the stability and performance. Simulation results are presented in Section IV. Finally, Section V concludes this note.

II. PROBLEM STATEMENT
We consider the same class of systems as in [1]. For completeness, the system model is given as follows: where Y i are known continuous linear or nonlinear functions, d(t) denotes bounded external disturbances, parameters ai are unknown constants and control gain b is unknown bounded constant, v is the control input, !(v) denotes hysteresis type of nonlinearity described by d! dt = dv dt (cv 0 !) + B 1 dv dt (2) where ; c, and B 1 are constants, c > 0 is the slope of the lines satisfying c > B1. Based on the analysis in [1], this equation can be solved explicitly The solution indicates that dynamic (2) can be used to model a class of backlash-like hysteresis as shown in Fig. 1, where the parameters = 1; c = 3:1635, and B1 = 0:345, the input signal v(t) = 6:5 sin(2:3t) and the initial condition !(0) = 0. For d 1 (v), it is bounded as shown in [1]. From the solution structure (3) of model (2), (1) becomes where = bc and d(t) = bd1(v(t))+ d(t). The effect of d(t) is due to both external disturbances and bd 1 (v(t)). We call d(t) a "disturbancelike" term for simplicity of presentation and use D to denote its bound. Now, (5) is rewritten in the following form: For the development of control laws, the following assumptions are made.
Assumption 1: The uncertain parameters b and c are such that > 0.

Assumption 2:
The desired trajectory y r (t) and its (n 0 1)th-order derivatives are known and bounded.
The control objectives are to design backstepping adaptive control laws such that • the closed loop is globally stable in sense that all the signals in the loop are uniformly ultimately bounded; • the tracking error x(t) 0 y r (t) is adjustable during the transient period by an explicit choice of design parameters and lim t!1 x(t) 0 y r (t) = 0 or lim t!1 jx(t) 0 y r (t)j 1 for an arbitrary specified bound 1 . Remark 1: Compared with [1], the uncertain parameters and ai are not assumed inside known intervals. The bound D for d(t) is not assumed to be known and it will be estimated by our adaptive controllers. Also the control objectives are not only to ensure global stability, but also transient performance.

III. DESIGN OF ADAPTIVE CONTROLLERS
Before presenting the adaptive control design using the backstepping technique in [6] and [7] to achieve the desired control objectives, the following change of coordinates is made: zi = xi 0 y (i01) r 0 i01; i= 2; 3; . . . ; n where i01 is the virtual control at the ith step and will be determined in later discussion. In the following, two control schemes are proposed.

A. Control Scheme I
To illustrate the backstepping procedures, only the last step of the design, i.e., step n, is elaborated in details.
• Step 1: We start with the equation for the tracking error z 1 obtained from (6) to (8) _ z1 = z2 + 1: We design the virtual control law 1 as where c 1 is a positive design parameter. From (9) and (10) • Step n: From (6) and (8), we obtain _ z n = v(t) + a T Y + d(t) 0 y (n) r 0 _ n01 : Then, the adaptive control law is designed as follows: Then, the derivative of V along with (6) where we have used (11), (13), (21), and the fact that z n d(t) jz n jD to obtain (52).
We then have the following stability and performance results based on this scheme.
• The resulting closed-loop system is globally stable.
• The asymptotic tracking is achieved, i.e., • The transient tracking error performance is given by Proof: From (24), we established that V is non increasing.

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Remark 2: From Theorem 1, the following conclusions can be obtained.
• The transient performance depends on the initial estimate errors e(0);ã(0);D(0), and the explicit design parameters. The closer the initial estimatesê(0);â(0), andD(0) to the true values e; a, and D, the better the transient performance.
• The bound for k x(t) 0 yr(t) k2 is an explicit function of design parameters and thus computable. We can decrease the effects of the initial error estimates on the transient performance by increasing the adaptation gains ; , and 0.
• To improve the tracking error performance we can also increase the gain c 1 . However, increasing c 1 will influence other performance such as k _ x 0 _ yr k2 as shown later.
Since _ V 0, immediately from (22) we know k _ x 0 _ yr k2 =k z2 0 c1z1 k2 k z 2 k 2 +c 1 k z 1 k 2 : Similar to the proof of (29), we can get k z 2 k 2 V (0)= p c 2 and, thus k _ x 0 _ yr k2 1 p c2 + p c1 V (0) From (33), we can see that increasing c 1 also increase the error k _ x 0 _ yr k2. This suggests fixing the gain c1 to some acceptable value and adjust the other gains such as ; , and 0.

B. Control Scheme II
In the previous scheme, a discontinuous function sgn(z n ) is involved in the control and this may cause chattering. To avoid this, we now propose an alternative smooth control scheme.
• The tracking error approaches 1 asymptotically, i.e., lim t!1 jx(t) 0 yr(t)j = 1: Proof: Based on (53), the results can be shown by following similar steps to that of Theorem 1.

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Note that similar remarks made in Remark 2 are also applicable here.
The objective is to control the system state x to follow a desired trajectory yr(t) = 12:5 sin(2:3t) as in [1].
In the simulation of Scheme I, the robust adaptive control law

V. CONCLUSION
This note presents two backstepping adaptive controller design schemes for a class of uncertain nonlinear single-input-single-output system preceded by unknown backlash-like hysteresis nonlinearities, where the hysteresis is modeled by a differential equation, in the presence of bounded external disturbances. In the first scheme, a sign function is involved and this can ensure perfect tracking. To avoid possible chattering caused by the sign function, we propose an alternative smooth control law and the tracking error is still ensured to approach a prescribed bound in this case. Unlike some existing control schemes, the developed backstepping controls do not require the model parameters within known intervals and the knowledge on the bound of "disturbance-like" term is not required. Besides showing global stability, we also give an explicit bound on the L 2 performance of the tracking error in terms of design parameters. Simulation results illustrates the effectiveness of our schemes. To further improve system performance such as the tracking error, especially in the case without using sign functions, it is worthy to take the system hysteresis into account in the controller design, instead of only considering its effect like bounded disturbances. The first step of achieving this is perhaps to obtain an efficient adaptive hysteresis inverse, which is still unclear and currently under investigation.