Black Box Levant

1812

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 11, NOVEMBER 2005

Quasi-Continuous High-Order Sliding-Mode Controllers
Arie Levant
Abstract—A universal finite-time-convergent controller is developed
capable to control the output of any uncertain single-input-single-output
system with a known permanent relative degree . The tracking error is
_ ...
steered to zero by means of acontrol dependent only on
and continuous everywhere except the set = _ =
=
= 0.
A robust output-feedback controller version provides for the tracking
accuracy proportional to the sampling noise magnitude.
Index Terms—Finite-time stability, high-order sliding mode, output feedback control, robustness.

I. INTRODUCTION
Sliding mode control remains one of the most robust and effectivetools to cope with heavy uncertainty conditions [22]. The main drawback of the standard sliding modes is mostly related to the so-called
chattering effect caused by the high-frequency control switching
[7], [8].
Let s be the output variable of an uncertain single-input–singleoutput (SISO) dynamic system and w(t) be an unknown-in-advance
smooth input, both available in real time. The task is toestablish
and keep  = s 0 w(t) = 0. The standard sliding-mode control
u = 0k sign is applicable if the relative degree is 1, i.e., if 
_
0
explicitly depends on the control u, and u > 0. Higher-order sliding
_
mode [13], [16] is applicable for controlling SISO uncertain systems
of arbitrary relative degrees [3], [6], [11], [14]–[16], [20]. The corresponding finite-time-convergentcontrollers (r -sliding controllers)
[13], [16] require actually only the knowledge of the system relative
degree r . The produced control is a discontinuous function of the
tracking error  and of its real-time-calculated successive derivatives
_
; ; ; . . . ; (r01) . The accuracy is improved in the presence of
switching delays, and the chattering effect is successfully treated,
provided thecontrol derivative is used as a new control input [3], [13].
The discontinuity set of controllers [15], [16] is a stratified union of
manifolds with codimension varying in the range from 1 to r , which
causes certain transient chattering. To avoid the chattering one needs
to increase artificially the relative degree r , inevitably complicating
the controller implementation [15], [16]. Thefinite-time-stable exact
tracking is lost with alternative controllers developed in [2] and [21]
for r = 3 and r = 2, respectively.
A sliding-mode controller of a new type is proposed in this note,
being a feedback function of ; ;  ; . . . ;  (r01) , continuous every_
where except the manifold defined by the equations

 =  =  = 1 1 1 = (r01) = 0
_


(1)

of the r -sliding mode. Themode   0 is established after a finite-time
transient. In the presence of errors in evaluation of the output  and its
derivatives, a motion in some vicinity of (1) takes place. Therefore,
control is practically a continuous function of time, for the trajectory
never hits the manifold (1) with r > 1. The controller design is based

on the homogeneity reasoning [18], [19], and can beconsidered as a
demonstration of the principles [18].
Combining with the recently proposed robust exact finite-time-convergent differentiator [16] an output-feedback controller is obtained
providing for exact tracking   0 if the measurements of the tracking
error  are exact, and for  proportional to the maximal measurement
error otherwise. Its transient features are much better than those of theknown r -sliding controllers [15], [16] (Section VI). Simulation demonstrates the practical applicability of the new controller.
II. PRELIMINARIES AND THE PROBLEM STATEMENT
Consider a smooth dynamic system with a smooth output function  ,
and let the system be closed by some possibly-dynamical discontinuous
feedback and be understood in the Filippov sense [5]. Then, provided
that...