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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 11, NOVEMBER 20061749
Controlling a Class of Nonlinear Systems
on Rectangles
Calin Belta, Member, IEEE, and Luc C. G. J. M. Habets
Abstract—In this paper, we focus on a particular class of non-
linear affine control systems of the form ? ?
thedrift
isamulti-affinevectorfield(i.e.,affineineachstatecom-
ponent), the control distribution
constrained to a convex set. For such a system, we first derive nec-
essary and sufficient conditions for the existence of a multiaffine
feedbackcontrollawkeepingthesysteminarectangularinvariant.
We then derive sufficient conditions for driving all initial states in
a rectangle through a desired facet in finite time. If thecontrol con-
straints are polyhedral, we show that all these conditions translate
tocheckingthefeasibilityofsystemsoflinearinequalitiestobesat-
isfied by the control at the vertices of the state rectangle. This work
is motivated by the need to construct discrete abstractions for con-
tinuous and hybrid systems, in which analysis and control tasks
specified in terms of reachability of sets of states can be reduced to
searches on finite graphs. We show the application of our results to
the problem of controlling the angular velocity of an aircraft with
gas jet actuators.
? ? ?
, where
is constant, and the controlis
Index Terms—Aircraft control, convex analysis, hybrid systems,
nonlinear systems.
I. INTRODUCTION
T
reachability analysis is to construct the set of states reached
by trajectories of the system originating in a given (possibly
uncountable) initial set. Safety verification is the problem of
proving that a system does not have any trajectory from a given
initial set to a given final (unsafe) set. For discrete systems with
a finite number of states, these problems are decidable, i.e., can
be solved by a computer in a finite number of steps. For contin-
uous and hybrid (i.e., described by both continuous and discrete
dynamics) systems, these problems are very difficult (in general
undecidable) because of the uncountability of the state space.
One way to solve formal analysis problems for continuous
and hybrid systems is to construct the set of states reached by
the system, or an over-approximation of this set, by working
directly in the continuous state space. Such methods are called
HE central problems in formal analysis of systems are
reachability analysis and safety verification. The goal of
Manuscript received December 8, 2004; revised November 2, 2005. This
work was supported in part by the National Science Foundation CAREER
Award 0447721 and the National Science Foundation under Grant 0410514 at
Boston University, Boston, MA.
C. Belta is with the Center for Information and Systems Engineering, the
Departments of Manufacturing and Aerospace and Mechanical Engineering,
Boston University, Brookline, MA 02446 USA (e-mail: cbelta@bu.edu).
L. C. G. J. M. Habets is with the Department of Mathematics and Computer
Science, Technische Universiteit Eindhoven, NL-5600 MB Eindhoven, The
Netherlands, and also affiliated with the Center for Mathematics and Computer
Science (CWI), Amsterdam, TheNetherlands (e-mail: l.c.g.j.m.habets@tue.nl).
ColorversionsofFigs.1and2areavailableonlineathttp://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAC.2006.884957
direct and are not the subject of this paper. Our work is related
to the group of indirect methods, where the main idea is to map
the continuous or hybrid system to a discrete transition system
through an iterative partitioning procedure producing finer and
finer quotients, until the initial system and the discrete quotients
become equivalent with respect to reachability properties. This
procedureiscalledabstractionandthecorrespondingalgorithm
is called the bi-simulation algorithm. If such an iterative refine-
ment procedure terminates, then the initial continuous or hybrid
systems and their discrete quotient are called bi-similar and the
reachabilityproblemiscalleddecidable.Thebi-simulationrela-
tionwasfirstintroducedin[28],[23],formallydefinedforlinear
control systems in [27], and for nonlinear systems in an abstract
categorical context in [14]. However, in [15], it has been shown
that reachability is undecidable for a very simple class of hybrid
systems. Several decidable classes have been identified though
by restricting the continuous behavior of the hybrid system, as
in the case of timed automata [3], multi-rate automata [1], [25],
and rectangular automata [15], [29], or by restricting the dis-
crete behavior, as in order-minimal hybrid systems [18], [19].
All thesedecidable classes are too weak torepresent continuous
and hybrid system models that arise in practice. Then one might
be satisfied with sufficient abstractions, i.e., with a discrete quo-
tient that can be used to over-approximate the reachable set of
the initial system. But even finding the discrete quotient is not
at all trivial. Related work focuses on partitioning using linear
functions of the continuous variables, as in the method of pred-
icate abstractions [2], [30], or using polynomial functions as in
[30] and [10]. However, to derive the transitions of the discrete
quotient,onehastobeabletoeitherintegratethevectorfieldsof
theinitialsystem[2],orusecomputationallyexpensivedecision
procedures such as quantifier elimination for real closed fields
and theorem proving [30], which severely limit the dimensions
of the problems that can be approached.
Inthispaper,wefocusonaparticularclassofnonlinearaffine
control systems of the form
is a multi-affine vector field ( i.e., affine in each state compo-
nent), the control distribution
constrained to a convex set. This class of continuous dynamics
is rather large, and includes the celebrated Euler–Volterra [31]
and Lotka-Volterra [22] equations, attitude and velocity control
systemsforaircraft[26]andunderwatervehicles[4](inthiscase
the control directions capture the axes about which the control
torques are applied), and models of genetic regulatory networks
(where product type nonlinearities model mass action kinetics
and the elements of
capture permeability of membrane) [7],
[5]. For such systems, we define rectangular partitions of the
state space and use the relationship between the structure of the
, where the drift
is constant, and the controlis
0018-9286/$20.00 © 2006 IEEE
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1750 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 11, NOVEMBER 2006
vector fields and the shape of the regions to solve two prob-
lems: Problem 1: Keep the system in a rectangle for all times
and Problem 2: drive the system through an exit facet in finite
time. In this paper, we show that if the control constraint set
is polyhedral, then the solutions to the above problems can be
parameterizedbypolyhedralsets.Themainideainconstructing
solutionstoProblems1and2isusingaveryinterestingproperty
of multiaffine functions on rectangles: a multiaffine function is
uniquely determined by its values at the vertices of a rectangle
and its restriction to the rectangle is a convex combination of
these values. The solutions to Problems 1 and 2 enable one to
construct computationally efficient characterizations of decid-
ability of such systems. Indeed, a partitioned continuous system
is bisimilar with the discrete quotient produced by the partition
if and only if all initial states in a region either stay in the region
forever or transit in finite time to just one neighbor.
This work draws inspiration from [11]–[13]. In these works,
the authors study affine continuous dynamics on simplices. The
starting point for their results is an observation similar to the
one we use in this paper: an affine function is uniquely deter-
mined by its values at the vertices of a simplex and its restric-
tion to the simplex is a convex combination of these values. In
this paper, we extend these results to a larger class of contin-
uous dynamics, i.e., we allow for product type nonlinearities.
Moreover, we focus on a different partition geometry, which is
more attractive for large dimensional problems. Although trian-
gulations may be carried out in Euclidean spaces of any finite
dimension (see e.g., [20] and [8]), rectangular grids are easier
to work with, certainly in problems of higher dimension.
Therestofthepaperisstructuredasfollows.InSectionII,we
introduce the notation and give some basic definitions, before
we formally state the problems in Section III. The interesting
properties of multi-affine functions on rectangles enabling the
framework of this paper are presented in Section IV. Based on
this, in Section V, we present the main theorems providing so-
lutions to the problems stated in Section III. Our approach is
illustrated in Section VI by an application to the control of an
aircraft with gas jet actuators. We conclude in Section VII with
final remarks and directions for future work.
II. PRELIMINARIES
Let
. A full dimensional polytope
hull of at least
facet of
is the intersection of
hyperplanes. More generally, a face of
with several of its supporting hyperplanes. If the dimension
of the intersection is
(with
-face. In particular, all facets of
vertices of
are 0-faces.
An
-dimensional rectangle in
vectors
with the property that
for all
and consider the -dimensional Euclidean space
is defined as the convex
affinely independent points in
with one of its supporting
is the intersection of
. A
) the face is called a
-faces, and theare
is characterized by two
and,
:
(1)
The set of vertices of
acterized as
is denoted by, and may be char-
(2)
Letwith. Then every
, characterized by
-face
equations of
of the
-dimensional rectangle
the form
or
... ...
or
where
isomorphic with an
particularly interested in facets. For
and for, is
-dimensional rectangle. We are
, let
denote the indicator function
(3)
Then,
hasfacets described by
(4)
for all
,. The outer normal of facet
is given by
(5)
for all
denote the Euclidean basis of
We end the discussion on rectangles by noting that an ar-
bitrary facet
has
. Moreover, for an arbitrary vertex
facets containing it are given by
Definition 1 (Multiaffine Function): A multiaffine function
(with
of the
components
minates
, with the property that the degree of
, in any of the indeterminates
than or equal to 1. Stated differently,
, , where,
.
vertices, with
, the
.,
) is a function in which each
is a polynomial in the indeter-
,
is less
has the form
(6)
with
convention that if
For example, for
tionshavetheform
where
Finally, note that if
restriction
-dimensional rectangle.
for alland using the
, then
and arbitrary , all multiaffine func-
.
,
,.
is an
is a multiaffine function on an
-face of , then the
of to
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BELTA AND HABETS: CONTROLLING A CLASS OF NONLINEAR SYSTEMSON RECTANGLES1759
University, Boston, MA. His research interests include verification and con-
trol of hybrid systems, robot planning and control, and gene and metabolic
networks.
Dr. Belta received an NSF CAREER award in 2005, a Fulbright study award
in 1997, and was the Valedictorian of his class in 1995. He received the Best
Poster Award at the International Conference on Systems Biology in 2004 and
was a finalist for the ASME Design Engineering Technical Conference Best
Paper Award in 2002 and for the Anton Philips Best Student Paper Award at the
IEEE International Conference on Robotics and Automation in 2001.
Luc C.G.J.M. Habets received the M.Sc. degree
degree in applied mathematics and the Ph.D. degree,
both from Eindhoven University of Technology,
Eindhoven, The Netherands, in 1989 and 1994,
respectively.
He spent three years at the Institute for Dynam-
ical Systems at Bremen University, Germany, and re-
turned to Eindhoven in 1997 to become a Lecturer at
the Department of Mathematics and Computer Sci-
ence. Since 2000, he has also been affiliated as a Re-
searcher with the Center for Mathematics and Com-
puter Science (CWI), Amsterdam, The Netherlands. His main research interests
include hybrid systems, time-delay systems, behavioral theory, and algebraic
and computational aspects in systems and control.