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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 RECTANGLES1751

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

, then for 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:

Thereexistmultiaffine

such that

functions

and

Then, for all vertices

of the -dimensional rectangle

we have

(9)

Subtraction of bothequations

, 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

under is 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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1752 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 11, NOVEMBER 2006

Proof: It follows from (3) that, for every

, the product

contains either a factor or 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 RECTANGLES1753

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 facet in 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

, , and are 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 and in Theorem 2 represent allowed sets

forcontrolsatthevertices.Ifthesesetsarenonempty,anychoice

, by Lemma 2, we conclude that there

such that

. Therefore,thestatetrajectories oftheclosed-loop

every-

before

.

in