In
control theory
Control theory is a field of mathematics that deals with the control of dynamical systems in engineered processes and machines. The objective is to develop a model or algorithm governing the application of system inputs to drive the system to a ...
,
dynamical systems are in strict-feedback form when they can be expressed as
:
where
*
with
,
*
are
scalars,
*
is a
scalar input to the system,
*
vanish at the
origin
Origin(s) or The Origin may refer to:
Arts, entertainment, and media
Comics and manga
* ''Origin'' (comics), a Wolverine comic book mini-series published by Marvel Comics in 2002
* ''The Origin'' (Buffy comic), a 1999 ''Buffy the Vampire Sl ...
(i.e.,
),
*
are nonzero over the domain of interest (i.e.,
for
).
Here, ''strict feedback'' refers to the fact that the
nonlinear
In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many othe ...
functions
and
in the
equation only depend on states
that are ''fed back'' to that subsystem.
That is, the system has a kind of
lower triangular
In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...
form.
Stabilization
:
Systems in strict-feedback form can be
stabilized by recursive application of
backstepping
In control theory, backstepping is a technique developed circa 1990 by Petar V. Kokotovic and others for designing stabilizing controls for a special class of nonlinear dynamical systems. These systems are built from subsystems that radiate out ...
.
That is,
# It is given that the system
#::
#:is already stabilized to the origin by some control
where
. That is, choice of
to stabilize this system must occur using some other method. It is also assumed that a
Lyapunov function
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s s ...
for this stable subsystem is known.
# A control
is designed so that the system
#::
#:is stabilized so that
follows the desired
control. The control design is based on the augmented Lyapunov function candidate
#::
#:The control
can be picked to bound
away from zero.
# A control
is designed so that the system
#::
#:is stabilized so that
follows the desired
control. The control design is based on the augmented Lyapunov function candidate
#::
#:The control
can be picked to bound
away from zero.
# This process continues until the actual
is known, and
#* The ''real'' control
stabilizes
to ''fictitious'' control
.
#* The ''fictitious'' control
stabilizes
to ''fictitious'' control
.
#* The ''fictitious'' control
stabilizes
to ''fictitious'' control
.
#* ...
#* The ''fictitious'' control
stabilizes
to ''fictitious'' control
.
#* The ''fictitious'' control
stabilizes
to ''fictitious'' control
.
#* The ''fictitious'' control
stabilizes
to the origin.
This process is known as backstepping because it starts with the requirements on some internal subsystem for stability and progressively ''steps back'' out of the system, maintaining stability at each step. Because
*
vanish at the origin for
,
*
are nonzero for
,
* the given control
has
,
then the resulting system has an equilibrium at the origin (i.e., where
,
,
, ... ,
, and
) that is
globally asymptotically stable.
See also
*
Nonlinear control
Nonlinear control theory is the area of control theory which deals with systems that are nonlinear, time-variant, or both. Control theory is an interdisciplinary branch of engineering and mathematics that is concerned with the behavior of dyn ...
*
Backstepping
In control theory, backstepping is a technique developed circa 1990 by Petar V. Kokotovic and others for designing stabilizing controls for a special class of nonlinear dynamical systems. These systems are built from subsystems that radiate out ...
References
{{reflist
Nonlinear control