Controlling a Class of Nonlinear Systems on Rectangles

Abstract

In this paper, we focus on a particular class of nonlinear affine control systems of the form xdot=f(x)+Bu, where the drift f is a multi-affine vector field (i.e., affine in each state component), the control distribution B is constant, and the control u is constrained to a convex set. For such a system, we first derive necessary and sufficient conditions for the existence of a multiaffine feedback control law keeping the system in a rectangular invariant. We then derive sufficient conditions for driving all initial states in a rectangle through a desired facet in finite time. If the control constraints are polyhedral, we show that all these conditions translate to checking the feasibility of systems of linear inequalities to be satisfied by the control at the vertices of the state rectangle. This work is motivated by the need to construct discrete abstractions for continuous 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

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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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1750IEEE 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 the are
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
andfor, 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
ofto

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BELTA AND HABETS: CONTROLLING A CLASS OF NONLINEAR SYSTEMSON RECTANGLES 1751
III. PROBLEM FORMULATION
With the notation and definitions introduced in the previous
section,wearenowreadytoformulatetheproblemswestudyin
this paper. As already outlined in the Introduction, we consider
control systems of the form:
(7)
where the state
as defined in (1) and the input
. The vector field
defined in (6) and
rections.Notethatthesystemsweconsiderareaparticularclass
of nonlinear affine control systems [16], which have the general
form
, where
is a matrix spanning the control distribution. Therefore, in
this paper, we consider a particular class of drift, and constant
control distributions.
We first consider the problem of designing boundedfeedback
control laws that keep the state trajectories of the closed-loop
system in the rectangle
.
1) Problem1(RectangularInvariant): Determineafeedback
control law
for system (7), such that the corre-
sponding closed-loop system is positively invariant on the rec-
tangle
.
Thepositiveinvarianceconditionintheaboveproblemmeans
that, if a state trajectory
of the closed-loop system satisfies
, thenfor all
We then consider the problem of controlling system (7) so
that in finite time the state is driven to a desired facet of
without leaving
before this desired facet is reached.
2) Problem 2 (Control to a Facet): Determine a feedback
control law
for system (7) such that, indepen-
dent of the initial state, all state trajectories of the closed-loop
system leave
through a desired facet in finite time, mean-
while guaranteeing that a trajectory does not leave the rectangle
through any of the the remaining facets.
To solve Problems 1 and 2, we restrict our attention to mul-
tiaffine feedback controllers
law is automatically continuous and bounded on
closed-loop system
is restricted to a rectangular region
is constrained in a convex set
is assumed to be multi-affine as
is a constant matrix of control di-
of
is a “drift” vector field and
.
,
. In this case, the feedback
, and the
is multiaffine.
IV. MULTIAFFINE FUNCTIONS ON RECTANGLES
In this section, we state and prove an interesting property of
multiaffine functions on rectangles: a multiaffine function (6)
defined on an
-dimensional rectangle (1) is uniquely deter-
mined by its values at the vertices. Moreover, inside the rec-
tangle, the function is a convex combination of its values at the
vertices.Theseresultsconstitutethebasisforthemaintheorems
statedandprovedinSectionV,whichprovidesolutionstoProb-
lems 1 and 2.
Lemma 1: Let
be an-dimensional rectangle with
as vertex set. Let
assume that
be a multiaffine function, and
(8)
Then
.
Proof: (Byinduction).
is affine, then
Induction
:Ifand
and.
Step:
Thereexist multiaffine
such that
functions
and
Then, for all vertices
of the-dimensional rectangle
we have
(9)
Subtractionofbothequations
, and since
for all
yields
, we obtain
.
This implies that also
for all
. By the induction hypothesis
.
be an-dimensional rectangle in
be a map, relating every vertex
. Then there exists a unique multiaffine
such that
and, hence
Proposition 1: Let
, and let
to a vector in
function
of
(10)
Moreover, if for every
underis denoted by
the image of
and,
, is given by (3), then the multi-affine map
realizing (10) is given by (11), as shown at the
bottom of the page.
(11)

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1752IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 11, NOVEMBER 2006
Proof: It follows from (3) that, for every
, the product
contains either a factoror a factor
; if, then the first factor is present,
, then the second factor is present. This proves
defined in (11) is multiaffine. Furthermore, for every
and if
that
fixed
if
if
So, indeed
all
If
then
for
.
is a multiaffine function satisfying (10),
is multiaffine, and
. By Lemma 1,
for all
, hence,defined in
(11) is unique.
Proposition 2: In every point
multi-affine function
of the values of
Proof: According to Proposition 1 we have (12), as shown
at the bottom of the page, and by applying the same proposition
to the identity function
, which is of course a multiaffine
function from
to
, the value
is a convex combination
.
of a
at the vertices of
(13)
Since
for all
, the product
, it follows that for
(14)
Hence, (12) and (13) show that
bination of the values of
Corollary 1: Let
on the
-dimensional rectangle
and let
be the face of
is an element. Then,
combination of the values of
is a convex com-
. at the vertices of
be a multiaffine function
. Let
of lowest dimension of which
,
is a convex
at the vertices of.
Lemma 2: Let
everywhere in
and. Then,
if and only if
.stands for any of
, for all
.,,,,
Proof: The necessity follows immediately from the fact
that the vertices
belong to
also immediate from the fact that
function and, therefore, its restriction to the rectangle
convex combination of its values
tices
.
It is easy to see that Lemma 2 remains valid if
to a facet
of.
. The sufficiency is
is a scalar multiaffine
is a
at the ver-
is restricted
V. CONTROL OF MULTIAFFINE SYSTEMS ON RECTANGLES
The following theorem gives a complete description of the
solution to Problem 1 under the assumption that the feedback
controllers are restricted to multiaffine functions of the state.
It basically states that there exists a multiaffine feedback con-
troller
solving Problem 1 if and only if
such that, at each vertex
so that the velocity of the closed-loop system
at the vertex has negative projec-
tionsalongtheouternormalsofallfacetscontainingthatvertex.
Formally, we have the following.
Theorem 1 (Equivalent Condition for Problem 1): There ex-
ists a multiaffine feedback control law
system (7) such that all state trajectories of the corresponding
closed-loop system that start in the rectangle
rectangle
for all times if and only if the following sets are
nonempty:
,, andare
, we can choose a control
for
, remain in the
(15)
for all.
Proof: For sufficiency, if all the sets
nonempty, than we can choose arbitrary
and let
on
taking the values
function can be constructed using (11). By Proposition 2,
is a convex combination of
since
andis convex,itfollows that
.
The vector field
is a multiaffine function on
tices
side (15) state that, for an arbitrary vertex
are
be the unique multiaffine function
at the vertices. Such a
everywhere in, and
,
of the closed-loop system
with values at the ver-
. The inequalities in-
,
has a negative projection along
the outer normals of all facets containing the vertex. This is
(12)

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BELTA AND HABETS: CONTROLLING A CLASS OF NONLINEAR SYSTEMSON RECTANGLES 1753
equivalent to saying that, for an arbitrary facet, the vector field
of the closed-loop system is oriented inside the facet at the
vertices. Formally, for any facet
, we have
for all
conclude that
with. From Lemma 2, we
(16)
for all
nuity of the velocity vector field
guarantees that the state of the closed-loop system cannot leave
the rectangle through any of the facets (see, e.g., [13, App. A]
forasimilarproofincaseofsystemswithaffinedynamics).This
proves the first part of the equivalence.
For necessity, assume there exists a multiaffine control law
solving Problem 1. Then, we take
and we will prove that
Of course,
. We only need to show that
. In combination with the Lipschitz conti-
, condition (16)
.
for all
by contradiction that there exists a vertex
direction
(i.e., satisfied with “ ”), then by continuity this implies that
there exists a whole neighborhood of
which
has a strictly positive projection along
.Then,therewillexisttrajectoriesofthesystemleaving
the rectangle through facet
and the theorem is proved.
Next, we give sufficient conditions for the existence of
a solution to Problem 2: If
at each vertex
, we can choose a control
so that the velocity of the closed-loop
system
positive projection along the outer normal of the exit facet
and a negative projection along the outer normals of all facets
containing that vertex different from the exit facet, then we can
construct a solution
of Problem 2. Formally, we have the
following.
Theorem 2 (Sufficient Conditions for Problem 2): There ex-
istsamultiaffinefeedbackcontrollaw
(7) such that all state trajectories of the corresponding closed-
loop system that start in the rectangle
arbitrary facetin finite time, without crossing other
facets first, if the following sets are nonempty:
and. If we assume
and a
so that the previous inequality is false
inin
. This gives a contradiction
,, andare such that,
at the vertex has a strictly
forsystem
are driven through an
and
for all(17)
for all vertices.
Proof: Choose arbitrary
betheuniquemultiaffinefunctionon
at the vertices as shown in (11). By Proposition 2,
is a convex combination of
and since
, it follows that
First, using arguments similar to those in the proof of The-
orem 1, we note that the state of the closed-loop system cannot
leave the rectangle through any of the facets different from
. Indeed, from the second line of (17), we have
and let
takingthevalues
everywhere in,
,.
for all
field corresponding to the closed-loop system has negative pro-
jection along the outer normals of all
from the exit facet
and the one opposite to it in the
th direction. Using the convexity property of multiaffine func-
tions in the form of Lemma 2, and the fact that the vector field
isLipschitzcontinuous,weconcludethatthestate
oftheclosed-loopsystemcannotleavetherectanglethroughany
ofthesefacets.Forthefacetoppositeto
normalis
,theinequalityisstrictaccordingtothefirst
line of (17). Therefore, the state trajectory of the closed-loop
system can only leave through
Since
,, which means that the vector
facets different
,sinceitsouter
.
for all
exists an
where in
system havea strictly positivespeed in thedirection of
and the Theorem is proved.
Remark 1: Under the conditions of Theorem 2, the state of
the closed-loop system leaves the rectangle the very first time
it hits the exit facet. On the exit facet, trajectories cannot turn
back into the rectangle
.
Remark2(NecessaryConditionsforControltoaFacet): The
sufficient conditions in Theorem 2 are somewhat stronger than
necessary ones. For example, if one additionally requires that
the property described in Remark 1 has to be satisfied, one can
easily prove along the same lines that the sufficient conditions
becomenecessaryifwerelaxtherequirementthatatthevertices
opposedtotheexitfacettheprojectionoftheclosed-loop vector
field along the outer normal of the exit facet is only positive as
opposedtostrictlypositive.Ontheotherhand,itisalsopossible
to relax the property described in Remark 1 that all trajectories
leave
immediately upon reaching the exit facet. Instead,
one may allow that some trajectories turn back into
theyleavethe rectangle through therequired exit facet on a later
occasion. In this case, it is not necessary that in all vertices of
the exit facet the vector field of the closed-loop system has a
positive component in the direction of
Remark 3 (Computational Issues): The sets
Theorem 1 andin Theorem 2 represent allowed sets
forcontrolsatthevertices.Ifthesesetsarenonempty,anychoice
, by Lemma 2, we conclude that there
such that
. Therefore,thestatetrajectories oftheclosed-loop
every-
before
.
in