Piecewise-Constant Stabilization.
With the help of topological necessary conditions for continuous stabilization it is shown that, in general, in order to stabilize continuous- and discrete-time systems one has to use time-dependent or discontinuous feedback controls. On the other hand, the criterion of stabilization in the class of piecewise-constant feedbacks is established. In the context of this paper a piecewise-constant feedback is associated with a piecewise-constant function of the form The piecewise-constant feedback synthesis outlined here has several attractive features. First, it can be effectively applied to design feedback stabilizers subjected to control constraints. Second, the designed feedback laws do not cause sliding mode or chattering behavior in the closed loop system; i.e., on a finite interval of time the control in the closed loop system may have only a finite number of jump discontinuities.
Introduction
. Stabilization of dynamical systems is one of the basic problems
in systems theory. In [9], [10] and [11] it is shown that many nonlinear systems
are not stabilizable by any continuous feedback of the form For the purpose
of illustration we give a geometrical interpretation of the results presented in [9], [10]
and [11]. In particular, one can see from these geometrical illustrations that on a
compact simply-connected manifold a nonlinear system can not be globally stabilized
at any of its equilibria by a continuous feedback of the form We use this
fact as a motivation for our work over the main contribution of this paper, criterion of
stabilization by means of piecewise-constant feedbacks that do not cause sliding mode
and/or chattering behavior. In other words, on a finite interval of time the control in
the closed loop system may have only a finite number of jump discontinuities.
This paper is dealing with a dynamical nonlinear system having either the form
(1)
or
(2)
(R n denotes n-dimensional Euclidean space) and u 2 U ae R
Let pr x denote the projection of R n
x \Theta R n
It will be
shown that (1) (or (2)) is stabilizable at an equilibrium
or
over a compact set K ae R n if, and only if, one can steer the system from any initial
point x 2 K into x   with the help of piecewise-constant control inputs and system
This paper was written May 16, 1997.
y Department of Mathematics, Arizona State University, Tempe, AZ 85287-1804.
nikitin@asu.edu and http://lagrange.la.asu.edu
(1)(or (2)) satisfies the following condition (of some sort of stability) at
neighborhood W of x   (open connected set containing x   ) there
exists another neighborhood V of x   ; such that one can move the system (1)(or (2))
from any point in V into x   without leaving the neighborhood W: Although, we prove
all our results for systems defined on R n ; their reformulation for systems on a smooth
manifold is straightforward and does not seem to bring anything new except some
changes in phraseology.
The use of continuous stationary feedbacks of the form does not allow
to solve the problem of stabilization for many nonlinear systems. This fact was
understood by many system researchers (see, e.g. [1], [2], [7]). Thus, in order to
stabilize a nonlinear system in many practical and theoretical situations one needs to
design time-dependent or discontinuous feedbacks instead of continuous feedbacks of
the form u(x). The use of time-dependent continuous feedback laws was consid-
ered, for instance, in [4], [12] [13], [14] and discontinuous feedback controls have been
discussed in, e.g., [9], [10].
The idea of using discontinuous stabilizers instead of continuous is not new and
was broadly discussed in the framework of variable structure systems (see, e.g., [5]).
The mathematical foundation of variable structure systems was developed in [15].
Necessary conditions of smooth stabilization underline the fact that the use of
non-stationary and discontinuous feedback stabilizers is unavoidable in many applications
of control theory. The commonly accepted starting point for the discussion
on necessary conditions of smooth stabilization is the classical result of [2].
Theorem 1. ( R.W.Brockett [2]) If the system -
continuously
stabilizable at
(bi) all the modes of its linearization -
positive real parts are
(bii) there exists some neighborhood Q of x   2 R such that for each y 2 Q one can
find a control
which steers the system from y at to x   at
(biii) the mapping f(x; u) : R n
x \Theta R m
every neighborhood of (x   ; u   )
onto a neighborhood of zero.
Theorem 1 was one of the first necessary conditions for smooth stabilizability of
nonlinear systems. It played an outstanding role in the development of nonlinear control
theory. For example, Theorem 1 was successfully used to establish that many of
drift-free nonholonomic nonlinear control systems can not be continuously stabilized.
Nevertheless, necessary condition (biii) suffers deficiency of being generic, i.e., (biii) is
almost always satisfied. The result obtained in [3] strengthens (biii), but nevertheless
it is still generic, and therefore, fulfilled for almost every smooth nonlinear system
(1).
In [10], [11] the first non-generic topological necessary conditions have been ob-
tained. Here we give geometrical illustration of these conditions and generalize them
to the class of discrete-time systems having form (2). These necessary conditions
imply that on a smooth compact manifold neither any continuous nor discrete-time
system is globally stabilized by a continuous feedback of the form
prove the criterion of stabilization in the class of piecewise-constant feedbacks and
discuss synthesis procedures for this class of feedback control laws. The motivation
for studying piecewise-constant feedbacks is threefold. First, a piecewise-constant
feedback is meant to be implemented on digital processors and does not require quantization
of input-output signals. Second, the control theory ideology and, in particular,
piecewise-constant feedback stabilization may serve as a framework for software development
projects where a software package is treated as a feedback control. Third,
as you can see from the results of this paper, any system, which is stabilized by a
continuous feedback can be stabilized by a piecewise-constant feedback.
We hope that we have convinced our reader that, in some situations where stabilization
of a nonlinear system is concerned, a piecewise-constant feedback could be
preferable to continuous one.
The structure of the paper is as follows. The first section is introductory. The
second section presents geometrical interpretation of the necessary conditions obtained
in [11] and generalizes them to the class of discrete-time systems of the form (2). The
third section contains the criterion of stabilization in the class of piecewise-constant
feedbacks. The fourth section is reserved for conclusion.
2. Topological necessary conditions of continuous stabilization. This
section presents topological necessary conditions for continuous stabilization. It also
contains geometrical interpretations of the results obtained in [11], as well as, new
formulation and proof of these results for discrete-time systems.
2.1. Continuous-time systems. Consider the system:
is a complete C 1 vector field on R n for every
The set
is called the equilibrium set of the control system.
A system is said to be continuously stabilizable at
K ae R n if there exists a continuous function u(x), such that u(x
is an asymptotically stable equilibrium of the closed loop system
and the domain of x   -attraction contains K; i.e.,
lim
e tf
where e tf is the flow generated by the vector field f(x; u(x)). We say that x   is
stable if for any neighborhood W of x   (open connected set containing x   ) there is a
neighborhood V of x   ; such that
A system which is continuously stabilizable at
to be continuously stabilizable in the large.
2.1.1. Basics of function degree. For the sake of completeness and reader
convenience, we briefly recall some facts about the degree of continuous functions.
Additional details can be found in [6], [8]. We use the following notation.
(i) M is a compact, n - dimensional, oriented C 1 manifold. Its interior is denoted
by Int(M ), its boundary by @M:
x is also used for local coordinates on M with a fixed orientation. If local
coordinates are given, then @
@x f(x) is the Jacobian matrix and det( @
is the Jacobian determinant of f at x:
has continuous first order partial derivatives
in M: f 2 C(M) if f is a continuous function from M into R
(iv) Given a real positive number r and y 2 R (y) is the closed ball center
y, radius r :
(v) An immersion
x \Theta R m
homeomorphically into its
image i(M) ae R n
x \Theta R m
u with topology induced by R n
x \Theta R m
u is called a natural
embedding.
are used for
respectively, where P are the projections: P x (x;
Definition 1. Suppose OE 2 C 1 (M); p 62 OE(@M) and p is not a critical value
of OE on M;i.e.,
det( @
@x
Define the degree of OE at p relative to M to be d(OE; M; p); where
@x
OE(x))]:
If OE 2 C(M); then the degree of OE can be defined as the degree of a sufficiently
good C 1 approximation of OE (for details, see [6], [8]).
Definition 2. Suppose that OE 2 C(M) and p 62 OE(@D): Define d(OE; M; p) to be
d(/; M; p); where / is any function in C 1 (M) satisfying
where ae(x; is not a critical value of / on M:
Recall that if X and Y are topological spaces, two continuous functions f and g
are said to be homotopic (f - g) if there is a continuous function (homotopy)
such that
We will need the following properties of degree.
Theorem 2.
(1) If H(t; x) j h t (x) is a homotopy and p 62 h t (@M) for
is independent of t 2 [0; 1]:
(2) If a continuous feedback stabilizes the system -
(3) Suppose OE 2 C(M): If d(OE; M; p) is defined and non-zero, then there is
such that
(4) If a continuous feedback stabilizes the system x at
where Idn is the identity n \Theta n-matrix.
For a proof, see [6], [8].
Lemma 1. Let f : R n
x \Theta R m
be a smooth function, and let ! be a bounded
connected component of f \Gamma1 (0) such that
@
@x
@
If continuous function such that
x \Theta R m
x \Theta R m
then
are the projections: P x (x;
This lemma is proved in [10], [11].
2.1.2. Geometrical illustrations of necessary conditions. We start with
the following necessary condition of continuous stabilization.
Theorem 3. If f : R n
x \Theta R m
is a smooth function and f \Gamma1 (0) is bounded,
then the system -
u) is not continuously stabilizable in the large at any point
For a proof of this theorem see [10], [11].
For geometrical illustrations consider a nonlinear single-input system of the form
x
Fig. 1. If f \Gamma1 (0) is a closed curve, then the graph of a continuous feedback
where
x \Theta R u
Theorem 3 asserts that the system can not be continuously
stabilizable. The reason is that the graph of any continuous feedback
stabilizing the system has at least two points of intersection with the equilibrium set
depicts a geometrical illustration of this fact.
Example 1.
Consider the system
The equilibrium set for this system is defined by
The system is locally continuously stabilizable at any point of its equilibrium set
with x 2 6= 0 (see [10] for further details). But it is not continuously stabilizable at
any compact set containing the entire
circle
It happens because on a compact set containing the circle
the graph of any continuous feedback locally stabilizing the system at (x
2 ) has at
least two points of intersection with this circle (Fig.2).
As illustrated by Example 1, system (1) is not continuously stabilizable at any
point of a connected bounded component of its equilibrium set. This leads us to the
following theorem.
Theorem 4. Let f : R n
x \Theta R m
be a smooth function, and let ! be a
connected component of f \Gamma1 (0) such that
@x
Fig. 2. Over a compact set containing the circle (x
a graph of a continuous
feedback stabilizing the system at has at least two points of intersection with the circle.
If the system -
smoothly stabilizable in the large at a point
then ! is unbounded.
The proof of this theorem is given in [10] and [11].
Theorems 3 and 4 give us the topological necessary conditions that are stable
under perturbations which are small in fine Whitney topology. Moreover, these conditions
remain valid for a system on a compact smooth simply connected manifold,
and since on the compact manifold the equilibrium set f \Gamma1 (0) is evidently bounded,
we conclude that the system can not be continuously stabilizable over this manifold.
Thus a smooth generic system is never globally continuously stabilizable on a compact
simply connected manifold.
Theorems 3 and 4 may give an impression that the topological barrier for
continuous stabilization is provided only by the fact that either the equilibrium set
or one of its connected components is bounded. However, stabilization also depends
upon the way in which one connected component of the equilibrium set loops around
another. Let us consider a single-input nonlinear system. We introduce the normal
parameterization of f \Gamma1 (0) as follows.
Definition 3. Let i ae R n
x \Theta R u be a segment of a connected component of
@
@x
@
Then a parameterization
x \Theta R u
of the segment will be called normal if
det
d-
d-
We can formulate the following necessary condition of smooth stabilization in the
large.
Theorem 5. Let -
be a smooth single-input system such that
@
@x
@
segment with the normal parameterization
x \Theta R u
Then the system is not continuously stabilizable in the large at a point
whenever there is a connected component ~
such that one can find points
! such that
and
The proof of this theorem is given in [10], [11].
If the system -
continuously stabilizable at (x   ; u   ); then without
loss of generality [8], we can assume that f(x; u(x)) is differentiable at x   and
@
where
be a segment with the normal parameterization
x \Theta R u
and -   2 R; x i (-
Then it follows from the definition of natural parameterization that there exists
a positive real number ff such that
d
d-
d-
@x
dx
d-
@
@x f(x; u) @
z
Fig. 3. If f \Gamma1 (0) has two connected components knotted as shown, then the graph of a continuous
feedback show the normal parameterization of
i:
@
@x (f(x; u(x))) @
@x
since
@x
Thus
and
Taking these inequalities into account and using the fact that ~
! is a connected component
of the equilibrium set we conclude that under the conditions of Theorem 5
the graph of the feedback u(x) has at least two points of intersection with the
One of these points of intersection is (x   ; u   ) and another one
is on the connected component ~
defined in Theorem 5. A geometrical illustration of
this fact is Fig.3.
Example 2. Consider the system
Fig. 4. Due to the topological structure of the equilibrium set the continuous stabilization at the
origin over a set containing the unit disk x 2
impossible.
The equilibrium set of this system is defined by
It is easy to show that the system is locally stabilizable at the origin by a linear feed-back
law. On the other hand, our topological considerations, in particular Theorem
5, show that this system is not continuously stabilizable over any set containing the
unit disk centered at the origin (Fig.4).
The discussion presented in this subsection shows that many systems are not
continuously stabilizable. Hence, it is reasonable and even sometimes necessary to
design piecewise-constant stabilizers.
2.2. Discrete-time systems. This subsection shows that many discrete-time
systems are not continuously stabilizable. Consider a discrete-time system of the form
where u
Let pr x denote the projection of R n
x \Theta R m
denotes the equilibrium set of system (4),
x \Theta R m
System (4) is said to be continuously stabilizable at
over a set K ae R n if there exists a continuous function u(x), such that u(x
and x   is an asymptotically stable singular point of the closed loop system
and the domain of x   -attraction contains K; i.e.,
lim
x is the solution of the closed loop system (5) with the initial condition
We say that x   is stable if for any neighborhood W of x   (open connected
set containing x   ) there is a neighborhood V of x   ; such that
A system, which is continuously stabilizable at
is said to be continuously stabilizable in the large.
The analogue of Theorem 3 for discrete-time systems is as follows.
Theorem 6.
x \Theta R m
is a smooth function and (pr x
bounded, then the system x u) is not continuously stabilizable in the large
at any point (x   ; u
Proof. If (pr x there is a positive real number
that
Hence
and properties (1), (3) (Theorem 2 ) imply
is a continuous function, while
whenever is a continuous feedback stabilizing the system in the large. Thus
the system can not be continuously stabilized in the large at any point (x   ; u
hence the theorem is proved.
The analogue of Theorem 4 for discrete-time systems is as follows.
Theorem 7. Let f : R n
x \Theta R m
be a smooth function, and let ! be a
connected component of (pr x
@
@x
@
where Idn is the identity matrix with n columns and n rows.
If the system x continuously stabilizable in the large at
Proof. If is a continuous feedback stabilizing in the large the system
u) at a point is bounded, then there is B r (0) such
that
d
Note that
x \Theta R m
x \Theta R m
and
Thus making use of Lemma 1, we obtain the contradiction which proves the theorem.
Theorems 6 and 7 have the same geometrical sense as Theorems 3 and 4 (Fig.1).
In order to formulate the analog of Theorem 5 for discrete-time systems we need
to change the definition of natural parameterization.
Definition 4. Let i be a segment of a connected component of (pr x
and let
@
@x
@
Then a parameterization
x \Theta R u
of the segment will be called normal if
det
d-
d-
The analogue of Theorem 5 for discrete-time systems is as follows.
Theorem 8. Let f(x; u) be such that
@x
segment with a normal parameterization
x \Theta R u
Then the system x u) is not smoothly stabilizable in the large at a point
there is a connected component ~
(0)ni such that
one can find points (~x 2 ; ~
! such that
and
The proof and geometrical illustration for this theorem coincide with that of
Theorem 5 (Fig.3).
After a few technical modifications all topological necessary conditions of continuous
stabilization also remain valid for discrete-time systems. Thus, many discrete-time
systems are not continuously stabilizable, and therefore, we need to consider either
piecewise-continuous or non-stationary stabilizing feedbacks. This fact serves as one
of the main motivations for establishing the result presented in the next section. More-
over, the approach outlined in the next section can be effectively applied to design
feedbacks subjected to state and control constraints.
3. Piecewise constant stabilization criterion. The goal of this section is to
give the criterion of piecewise-constant stabilization. The proof of this criterion is
constructive and can be used for feedback synthesis. Moreover, it is based on the
general topology, and therefore, valid both for discrete- and continuous-time systems.
For the sake of brevity, only the criterion for continuous-time systems is presented.
Consider the system
is a subset in R m : f(x; u) is a complete C 1
vector field on R n for every u 2 U ae R m fixed.
PC(U) is the set of all piecewise-constant mappings form R n into U ae R m . A
function called piecewise-constant on some set Q if there exists a covering
the closure Int V i of the interior Int V i coincides with
the restriction uj V i
of u to V i is a constant from U:
Let u(x) 2 PC(U): Then we define the solution for the initial value problem
as follows.
Definition 5. A function of time x u (t; x 0 is called a solution
for the initial value problem (8), if the following conditions hold:
x u (t; x 0 ) is a continuous function of time;
the derivative d
dt x u (t; x 0 ) exists and
d
dt
x
either for all t 2 [0; T ] or for all t 2 [0; T ] excluding a finite number of points.
Having fixed the feedback u(x) such that u(x) 2 PC(U) and the solution for
exists for all x 0 2 R n we obtain the flow e tf generated by the closed loop system
evolving over R n . e tf x 0 denotes the point into which the flow e tf steers x 0 and
g. V is called an invariant set of the system, iff e tf
Definition 6. A system -
is said to be piecewise-constantly stabilizable
at a domain K ' R n , if there exists a piecewise-constant
feedback from PC(U); such that u(x   is an asymptotically stable
equilibrium of (8) and K ' D(x   ); where D(x   ) is the domain of x   -attraction, i.e.,
for every x 0 2 D(x   ) the solution e tf x 0 of the closed loop system exists for all t - 0
and lim t!1 e tf x
If
called piecewise-constantly stabilizable
at or in the large). If there exists a neighborhood
O(x   ) and -
piecewise-constantly stabilizable in (x   ; u
O(x   ), then -
is said to be locally piecewise-constantly stabilizable at
A control said to be piecewise constant, if there exist time
points
Definition 7. We say that a point p can be piecewise-constantly steered into a
point q, if there exist
such that the solution x u (t; p) of the initial value problem
exists on the time interval [0; T ], is unique and x u q. If for every point
such that p is piecewise-constantly steered into
q; then the set V is called piecewise-constantly accessible from the set K:
The piecewise-constant accessibility has a very important property formulated in
the following lemma.
Lemma 2. Let K be a compact subset in R be an open subset.
Then if V is piecewise constantly accessible from K; then there exist a natural number
real number T ? 0 and
such that for each point p 2 K there exist a point
T , such that
denotes the vector field f(x; u
Proof. The set
is open for all . The piecewise constant accessibility
of V from K implies
Thus it follows from the compactness of K that there exists a natural number - such
that
where Therefore we can take
and
To formulate the main result of this paper we need the following analogue of the
Liapunov asymptotic stability.
Definition 8. An equilibrium
is said to be
asymptotically stable if there exists \Delta ? 0 such that for any 0 - \Delta
one can find ffi ? 0 such that x   is piecewise-constantly accessible from B ffi (x   ) without
leaving can be 1) and a
piecewise-constant control such that the solution x u (t; p) of the initial
value problem
exists on [0; T
lim
x
x*
x*
e
e
Fig. 5. The closed loop system is supposed to move with respect to the sets K;
n=0 in
accordance with the arrows.
Theorem 9. A system -
piecewise-constantly stabilizable at its
a compact set K if, and only if,
asymptotically stable and x   is piecewise-constantly accessible from K:
Proof. Necessity. If u(x) 2 PC(U) stabilizes -
at
over K, then (x   ; u   ) is evidently Liapunov asymptotically stable and x   is piecewise
constantly accessible from K:
Sufficiency. Since asymptotically stable, then there
exists a sequence of positive real numbers f" n g 1
n=0 such that
lim
and for any is accessible from leaving
Roughly speaking, the main idea of the proof is to design piecewise constant
feedbacks which steer the systems in accordance with the
arrows marked in Fig. 5.
The point x   is piecewise-constantly accessible from K; and therefore, for any
constantly accessible from K: Lemma 2 with V
being the interior
of the ball
us the existence of a natural
number N; real number T ? 0 and
such that for each point p 2 K there exist a point
such that
denotes the vector field f(x; u
Consider the family of open sets generated by
Due to Lemma 2
But K is a compact set, and hence, one can choose a finite number of N-tuples
j=1 such that
The feedback u 0 (x) 2 PC(U) which steers any point of K
into the interior
of the set
can be designed in the following way
f
and so on until
Thus u 0 (x) is defined on the set
\Theta (
If
then the construction of u 0 (x) is completed. Otherwise, there
exists
We put
Thus we have defined on the set
\Theta (
If
constructed. Otherwise, there exists
j such that j 6=
. We can define the feedback
on the set
\Theta (
in the same way as it has been done on the set \Xi 2 .
The existence of a natural number -, such that K
Thus the construction of u 0 (x) is completed after a finite number of steps. Let   0
be the domain where u 0 (x) is defined. By construction, if a trajectory of the system
closed by u 0 (x) starts at a point of   0 ; then it will reach the set
in a finite time.
The set therefore, we can employ the
method used to construct u 0 (x) for designing which steers any point
We denote by   1 the domain
where asymptotically stable one can
construct
a trajectory of the
system closed by u 1 (x) starts at a point of   1 ; then it will reach the set
in finite time.
We proceed in this way and obtain for each the feedback un (x) 2
PC(U) defined on
un (x) steers the system from any state in   n into
Hence, the function u(x) 2 PC(U) which stabilizes -
at x   over the
compact set K can be defined as
lim
wn
where wn (x) is given by
wn un (x) for x 2   n
wn
wn
wn
The proof is completed.
Definitions 6, 7, 8 and Lemma 2 admit natural formulations for discrete-time
systems. Moreover, the main ideas in the proof of Theorem 9 are of topological
nature and, after minor technical adjustments, they lead us to the following analogue
of Theorem 9 for discrete-time systems.
Theorem 10. A system x piecewise-constantly stabilizable at
its equilibrium a compact set K if, and only if,
is Liapunov asymptotically stable and x   is piecewise-constantly accessible from K:
The proof of Theorem 9 remains valid for much more general objects than discrete-time
systems. The method of the proof outlines a general framework which leads to
the synthesis of control systems defined over a finite algebra, in particular Boolean
algebra. Such systems are very often called algorithms and arise in numerous software
engineering problems. In order to demonstrate the use of this framework in software
engineering the author developed a simple game "Corners" (Java version of "Corners"
is posted at "http://lagrange.la.asu.edu/docs/Corners"). The analysis of algorithms
from the point of view of a control theorist is beyond the scope of this paper. Here
we only present some examples of the synthesis of piecewise constant stabilizers.
Although, in the next example the construction of the piecewise-constant feedback
seems completely ad hoc, it contains an effective recipe which, if combined with
topological methods developed in [10], leads us to an effective stabilizer synthesis
procedure for two-dimensional affine nonlinear systems.
Example 3. Consider the linear system
where System (10) is continuously stabilizable at the origin
by a linear feedback. Hence, by Theorem 9 it is stabilizable by a piecewise constant
feedback as well. We look for a piecewise constant feedback stabilizer having the
following form.
and
where '(-) is the function defined as
If the real numbers the inequality
then the feedback (11) globally stabilizes the system (10) at the origin. Indeed, one
can show that the function
is monotonically decreasing along the trajectories of the system (10) closed by the
feedback (11).
Example 3 combined with the topological methods from [10] allows us to construct
a piecewise-constant stabilizer for any controllable affine nonlinear system
satisfying the following conditions.
are continuously differentiable functions

where
and
In this example the system
is globally stabilized at the origin by means of a piecewise-constant feedback. We
assume that for all x 2 R 2 the functions a(x) and b(x) satisfy the following inequalities
It is known [10], that system (13) can be stabilized by a time-dependent continuous
feedback. Hence, by Theorem 9 there exists a piecewise-constant stabilizer for (13).
Choose positive real numbers ff; fi so that
and
Then the piecewise-constant feedback
with '(-) being defined in (12) and
figlobally stabilizes the system at the origin.
Indeed, the function
is monotonically decreasing along the trajectories of the closed loop system, and
therefore, the origin is globally asymptotically stable.
Using the topological methods based on, so called, stable covering (see [10]) one
can construct piecewise constant stabilizers for a generic nonlinear affine system with
an equilibrium set having several connected components.
The next example illustrates the use of Theorem 9 in the stabilizability analysis
of a nonholonomic system.
Example 5. Consider the system
It is stated in [2], that system (14) is not continuously stabilizable at the origin. Our
goal is to analyze its piecewise-constant stabilizability.
Let us show that the origin is Liapunov asymptotically stable. Indeed, for any
point 3 the piecewise-constant control
r
r
r
r
steers system (14) from the point P into the origin. First, this control moves the
system from P into (0; 0; x 3 ) along the straight line. Then traces the
boundary of the square with the side of length
2 and moves system (14) from
(0; 0; x 3 ) into the origin.
Thus, the origin is Liapunov asymptotically stable for system (14) and all conditions
of Theorem 9 are satisfied. Hence, system (14) is piecewise-constantly stabilizable
at the origin over any compact set from R 3
The next example illustrates the use of Theorem 10.
Example 6.
Consider a discrete time system of the form
Let us analyze stabilizability of system (15) at the origin. By Theorem 8 system (15)
is not continuously stabilizable at the origin. On the other hand, for any x 1 one can
find -
u such that sin(x 2
on the next step
(15) to the origin. Thus, all conditions of Theorem 10 are satisfied, and therefore,
system (15) is piecewise-constantly stabilizable at the origin over any compact set
from
4. Conclusion. This paper presents geometrical illustrations of the topological
necessary conditions of continuous stabilization obtained in [10], [11]. It follows from
these conditions that many systems are not stabilizable by continuous stationary
feedbacks. Since in the majority of control applications one needs to implement a
control system with the help of digital processors, it is natural to design piecewise-constant
stabilizing feedbacks. We have proved the criterion of stabilization in the
class of piecewise-constant feedbacks and discussed their synthesis for affine nonlinear
two-dimensional systems.
It follows from Theorem 9 that any system which is stabilizable by a continuous
feedback can be also stabilized by a piecewise-constant feedback of the
Moreover, the proof of Theorem 9 leads to the synthesis procedure for
stabilizing feedbacks subjected to state and control constraints.
The approach proposed in this paper can be applied to deal with a wide variety of
control objects that are much more general than those described by differential and/or
discrete-time equations. For example, one can use this approach for software development
projects. In fact, we tested already the methodology provided by the proof of
Theorem 9 and developed a simple computer game (see "http://lagrange.la.asu.edu/docs/
Corners"). In order to use this methodology for computer algorithm synthesis one
needs to choose some natural topology and then follow the proof of Theorem 9. In
a computer game and/or a program where one needs to minimize some cost function
this topology is naturally introduced by this cost function. In general the choice of an
appropriate topology raises a number of difficult questions which can serve as topics
for further research.
5.

Acknowledgement

. The author is grateful to Anton Nikitin for his editorial
remarks and reading the early version of this manuscript. The author is much obliged
to anonymous referees for valuable suggestions. Some ideas presented in this paper
have been generated during (or short after) the author's visit to the University of
Gent in June, 1996. The author would like to thank Professor D. Aeyels for his kind
support of this research and his hospitality.



--R

Stabilization of a class of nonlinear systems by a smooth feedback control
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Necessary Condition for
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"World Scientific Publishing Co Pte Ltd"
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Control of chained systems application to path following and time-varying point- stabilization of mobile robots
Differential equations with discontinuous right-hand sides
--TR
discrete-time system;degree of functions;nonlinear system;feedback stabilization