IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 10, OCTOBER 2000
1893
Dynamic Surface Control for a Class of Nonlinear Systems
D. Swaroop, J. K. Hedrick, P. P. Yip, and J. C. Gerdes
Abstract—
A new method is proposed for designing controllers with ar-
bitrarily small tracking error for uncertain, mismatched nonlinear systems
in the strict feedback form. This method is another “synthetic input tech-
nique,” similar to backstepping and multiple surface control methods, but
with an important addition,
I
low pass filters are included in the design
where
is the relative degree of the output to be controlled. It is shown that
these low pass filters allow a design where the model is not differentiated,
thus ending the complexity arising due to the “explosion of terms” that
has made other methods difficult to implement in practice. The backstep-
ping approach, while suffering from the problem of “explosion of terms”
guarantees boundedness of tracking errors globally; however, the proposed
approach, while being simpler to implement, can only guarantee bounded-
ness of tracking error semiglobally, when the nonlinearities in the system
are non-Lipschitz.
Index Terms—
Integrator backstepping, nonlinear control system design,
semiglobal tracking, sliding mode control, strict feedback form.
I. INTRODUCTION
Tremendous strides have been made in the past 25 years in the area
of controller design for nonlinear systems. Variable structure control or
sliding mode control [5], [25], uses a discontinuous control structure to
guarantee perfect tracking for a class of systems satisfying “matching”
conditions. Retaining the concept of an “attractive” surface but elimi-
nating the control discontinuities, the method of sliding control [21] is
currently being applied in many different applications.
The works of Brockett [2], Hunt
et al.
[9], Isidori [10], Jakubczyk
and Respondek [12] initiated a surge of interest in feedback lineariza-
tion and more generally in the application of differential geometry to
nonlinear control [10], [18], [20].
Recently, the area of robust nonlinear control has received a great
deal of attention in the literature. Many methods employ a synthesis
approach where the controlled variable is chosen to make the time
derivative of a Lyapunov function candidate negative definite. Corless
and Leitmann [3] have applied this approach to open-loop stable mis-
matched nonlinear systems.
A design methodology that has received a great deal of interest re-
cently is “integrator backstepping.” The recent book by Krstic
et al.
[16] develops the backstepping approach to the point of a step by step
design procedure. The integrator backstepping (IB) technique suffers
from the problem of “explosion of terms.” The following example il-
lustrates the backstepping approach as well as the difficulty that this
paper seeks to solve:
_
x
I
=
f
I
(
x
I
)+
x
P
+
f
I
(
x
I
)
_
x
P
=
u
(1)
Manuscript received March 20, 1998; revised August 15, 1999. Recom-
mended by Associate Editor, M. Krstic.