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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

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1754 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 11, NOVEMBER 2006

of control values

valid multiaffine feedback control law

the allowed control set

checking the nonemptiness of

duces to checking the feasibility of a set of linear inequalities,

for which there exist several computationally powerful algo-

rithms and software packages (see, e.g., [9] and [17]).

Remark 4 (Constant Feedback Control): An interesting spe-

cial case of Theorem 2 is when

in these sets will lead to a perfectly

by (11). If

, then is a polyhedral subset of

and re-

Anelement

(independent of the current state) control that solves Problem 2.

Note that this is consistent with (11). Indeed, if

for all , then

may be extremely useful in practical situations, where the state

is not available for feedback.

intheaforementionedsetcanbeusedasaconstant

due to (13). This case

VI. EXAMPLE: ANGULAR VELOCITY CONTROL

In this section, we first make the important observation that

the class of systems studied in this paper includes attitude and

angular velocity control systems for aircraft and underwater ve-

hicles. We then show a numerical example for angular velocity

control of an aircraft with gas-jet actuators.

A. Aircraft and Underwater Vehicles

Consider an arbitrarily shaped aircraft with a body fixed

frame

in motion with respect to a world frame

be the inertia matrix of the aircraft with respect to its body

frame and

its mass. Let

which the corresponding control torques

by means of opposing pairs of gas jets. Let

velocity in the body frame,

the translational velocity of the

origin of the body in body coordinates, and

applied to the body at the center of mass expressed in the body

frame. Then, the kinematic equations of the aircraft can be

written as

. Let

be the axes about

are applied

denote the angular

the total force

(18)

(19)

Similarly, for an underwater vehicle modeled as a neutrally

buoyant rigid body submerged in an ideal fluid, if the center

of gravity of the vehicle coincides with the center of buoyancy,

then the equations of motion can be written as [21]

(20)

(21)

where

of the body and the mass of the fluid replaced by the body [21]

andalltheremainingvariableshavethesamemeaningasbefore.

The position and orientation in the world frame

systems described previously are identified with

is an added mass matrix which incorporates the mass

of both

, the Lie

group of rigid body displacements in

(22)

where denotesthedisplacementoftheoriginofthebodyframe

in and its rotation

(23)

The equations relating their positions and velocities are

(24)

(25)

where

If quaternions

sphere in

be written as

is the skew symmetric operator.

( denotes the unit

, (24) can ) are chosen to parameterize

(26)

where

.

There are situations, especially in space missions, in which

one is not interested in controlling the pose (displacement and

rotation) of a spacecraft or underwater vehicle in a reference

frame, but rather in regulating the body velocities of translation

and rotation. In this case, (18) and (19), respectively (20) and

(21), can be seen as control systems with states

controls

. However, there are several situa-

tions in which one is interested in controlling only the attitude

of a vehicle in a given world frame, and then (19) and (26) can

be seen as a control system with state

variables

. The main observation in this sec-

tion is that all control systems mentioned before are affine con-

trol systems with multi-affine drift and constant control distri-

bution as described in (7). The set

trol bounds. Using the results of this paper, we can approach

the rigid body control problem from a totally different perspec-

tive. Our approach is somewhere in between stabilization to a

point and interpolation between two end positions in the con-

figuration space. We propose a feedback control law, that may

containsomediscontinuities,whichallowsfora“maneuvering”

procedure (consisting of continuous trajectories), i.e., driving a

rigid body attitude or angular velocity control system between

arbitrary initial and final regions of the state–space, while satis-

fyingboundsoninputsandstate.Anillustrativetaskthatwecan

solve with this procedure is the following. Given an aircraft or

underwater vehicle with gas jet actuators and physical bounds

on the control torques, which is initially rotating at a certain

angular velocity (not necessarily precisely known), we want to

driveittowardsafinal,desiredangularvelocity.Wealsorequire

are the components of the angular velocity

and

and control

captures the physical con-

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BELTA AND HABETS: CONTROLLING A CLASS OF NONLINEAR SYSTEMSON RECTANGLES1755

that a priorigivenbounds on thevelocityare satisfied during the

transition. After the desired region of the state space is reached,

one can use a locally stabilizing control law [6], [24], if conver-

gence to a specific state is required. Of course we need to make

sure that the local region of attraction includes the target region

of our algorithm. Note that globally stabilizing controllers exist

as well, but using those there is no way one can guarantee that

the trajectories converging to a desired equilibrium satisfy the

required bounds on inputs and state. Especially the possibility

to guarantee that certain bounds on inputs and velocities are re-

spected by the feedback controller, makes the design method

proposed in this paper attractive in a large area of applications.

B. Maneuvering in the Angular Velocity Space

Consider a parallelepiped aircraft with gas-jet actuators. As-

sume that the frame

is fixed at the center of the aircraft and

aligned with its principal axis, so that

Assume that

trollable. Without loss of generality, we will take the control

directions as being the Euclidean basis vectors

and the control will be reparameterized by

tions. Then, the angular control system (19) takes the form of

the known controlled Euler’s equations

.

, i.e., the system is con-

,

along these direc-

(27)

Assuming that the aircraft spans between

direction

(

and along the

) of the body frame , we have

(28)

Finally,thecontrols

trol system (27) is obviously of the form (7) with

multiaffine drift

arelimitedtotakevaluesin.Con-

, the

control directions

.

Consider the following control scenario. Assume that the air-

craftisinitiallyrotatingaroundthe -axisofitsbodyframe

at speed. The goal is to control the aircraft so that it eventu-

ally rotates around its -axis at the same speed and remains in

thisstateforalltimes.Moreover,whiletransitingfromtheinitial

to the final state, the aircraft is forbidden to develop rotational

speed

around its -axis.

To capture the uncertainty on knowledge of the state as well

assensornoise,weallowfordeviationsofamplitude

directions. Under this assumption, the initial state of rotation is

assumedtobethecollectionofallstatesinasmallcubecentered

at

and with side . The amount of allowed speed

, and set of admissible controls

inall

of rotation around the -axis is assumed to be

to drive and keep the system in a small cube centered at

, and with side

Using the results of this paper, we can provide a solution to this

problem in terms of a feedback control law by defining a set of

rectangles in the velocity space and solving control problems of

the type Problems 1 and 2.

Explicitly,accordingtothespecificationsofthetask,consider

a set of four pairwise adjacent rectangles as shown in Fig. 1(a).

The task is accomplished if the following controllers are de-

signed.

• Controller 1: “Drive” the system down along the

while keeping the absolute values of

ThesolutiontothisproblemisfoundbyapplyingTheorem

2 to Rectangle 1 defined by

with exit facet[see Fig. 1(a)].

• Controller 2: “Take the turn” around origin. This control

law can be derived by applying Theorem 2 to Rectangle 2

defined by

[see Fig. 1(a)].

• Controller 3: Drive the system along the

keeping the absolute values of

solution is found by applying Theorem 2 to Rectangle 3

defined by

[see Fig. 1(a)].

• Controller 4: Keep the system in a cubic box centered at

and side. The controller is designed by ap-

plying Theorem 1 to Rectangle 4 defined by

[see Fig. 1(a)].

We used the following numerical data:

and the goal is

, where is a small number.

-axis

and less than .

with exit facet

-axis while

less than . The and

with exit facet

A possible choice of Controllers 1–4 is given later.

represent the controls at the vertices of Rectangle

where Controller is defined, obtained as a solution of the set

of linear inequalities (17) for

,is the feedback control valid everywhere in

the corresponding rectangle, uniquely determined by its values

at the vertices.

1) Controller 1 (Defined in Rectangle 1):

,

and (15) for.

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1756 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 11, NOVEMBER 2006

Fig. 1. (a) Region in the angular velocity space ?? ?? ?? ? corresponding to the maneuvering task. The small rectangle on the ? – axis in the upper part

represents the initial state of rotation about the body ? – axis. The small rectangle on the ? – axis represents the final state of rotation about the body ? – axis.

The thick line represents a closed-loop trajectory starting at ?????? ?. (b) Controls corresponding to the trajectory shown in (a).

2) Controller 2 (Defined in Rectangle 2):

3) Controller 3 (Defined in Rectangle 3):

4) Controller 4 (Defined in Rectangle 4):

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BELTA AND HABETS: CONTROLLING A CLASS OF NONLINEAR SYSTEMSON RECTANGLES1757

Fig. 2. Vector field of the closed-loop system is continuous everywhere, except on the boundary between Rectangles 1 and 2. In Rectangles 2 and 3, the feedback

law is the same. On the common facet of Rectangles 3 and 4 again a switch to another feedback law takes place, but the vector field of the closed-loop system is

continuous here because both feedback laws coincide on this common facet.

It is easily verified that on the common facet of Rectangles 2

and 3, and also on the common facet of Rectangles 3 and 4, the

vector field of the closed-loop system is continuous. In Rect-

angles 2 and 3, the feedback laws are even the same, and no

switch between different feedbacks is required, when the state

trajectory crosses the common facet

gles. On the common facet of Rectangles 3 and 4, i.e., the facet

, the situation is slightly different. Here a switch

from feedback law

to feedback law

feedback laws coincide on the common facet, this does not lead

to a discontinuity in the vector field of the closed-loop system.

Note that a switch from feedback law

required, in order to guarantee that after entering Rectangle 4,

the state trajectory will never leave this rectangle anymore.

On the common facet of Rectangles 1 and 2, i.e., the facet

, thefeedback laws and

to a discontinuity in the vector field of the closed-loop system.

So, in order to avoid ambiguity of the definition of the feed-

back law on this common facet, one has to specify it explicitly.

We choose the feedback law on this common facet to be equal

to

. In this way, it is guaranteed that the constructed feed-

back law solves the given reachability problem. Indeed, feed-

back

on Rectangle 1 guarantees that every trajectory starting

in Rectangle 1 reaches facet

leaving through other facets first. On the common facet

one switches (discontinuously) to feedback law

component of the closed-loop vector field in the direction of

remains negative, the trajectory will cross the common facet

, and feedback

willcrossthecommonfacetofRectangle2andRectangle3,and

reaches Rectangle 4 in finite time. After a (continuous) switch

to feedback law

, the state trajectory will remain in Rectangle

4 forever.

of these two rectan-

occurs, but since both

to feedback lawis

do not coincide. Thisleads

in finite time, without

,

. Since the

guarantees that the trajectory

Note that the feedback law is constructed in such a way

that any state trajectory of the closed-loop system will only

cross the common facet of two rectangles once, because on

both sides of the common facet, the closed-loop vector field

is pointing in the same direction w.r.t. the normal vector of

this common facet.

A trajectory of the closed-loop system in the angular velocity

space

startingfrom

in Fig. 1(a). It can be seen that all specifications are satisfied,

i.e., the trajectory travels through Rectangles 1–3 and stabilizes

in Rectangle 4. The controls

trajectory, which are plotted in Fig. 1(b), are bounded in

as required. It is also interesting to note that the inputs

andare continuous everywhere. This follows from the fact

that on common facets the definition of the feedback laws

for inputs

and coincide. The only discontinuous input

is

; as soon as at Rectangle 2 is reached, it switches

from 0 to 0.5. The (dis)continuity of the closed-loop vector

field and the continuity of the trajectory are also illustrated

in Fig. 2, where the regions around the small Rectangles 2

and 4 are zoomed in.

Remark 5: Note that the overall controller constructed in this

example is a piecewise affine controller. This is a coincidence,

causedbytheparticularchoiceoftheinputvaluesatthevertices.

A different choice of these input values leads to a different con-

trol law, that, in general, will be piecewise multiaffine instead of

piecewise affine.

isshownforillustration

, , andproducing this

VII. CONCLUDING REMARKS

In this paper, we start from the important observation that

a multi-affine function is uniquely determined by its values at

the vertices of a full dimensional rectangle and the restriction

of the function to the rectangle is a convex combination of

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1758 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 11, NOVEMBER 2006

these values. Using these properties, we derive necessary and

sufficient conditions for the existence of a multiaffine feedback

lawkeepingthestateofanaffinecontrolsystemwithmulti-affine

drift and constant control distribution in a rectangle. We also

derive sufficient conditions for driving all state trajectories

of such a system through a desired facet of a rectangle in

finite time. If the control constraints are polyhedral, we show

that all these conditions translate to solving sets of linear

inequalities.

In the future, we will use these results to develop a frame-

work for computationally efficient construction of discrete ab-

stractions for continuous and hybrid systems with multiaffine

dynamics.Specifically,usingiterativerectangularpartitionsand

the results presented in this paper, we want to construct dis-

crete quotients that are either equivalent with continuous or hy-

brid systems with respect to reachability properties, or over-ap-

proximate their reachable sets. Even though the class of sys-

tems that we consider in this paper is rather large, including

Euler–Volterra, and Lotka–Volterra equations, attitude and ve-

locity control systems for aircraft and underwater vehicles, as

well as models of biomolecular networks, in the future we will

try to extend these results to more complicated dynamics, such

as polynomial dynamics.

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EnglewoodCliffs,NJ:

Berlin,

Berlin, Germany:

Calin Belta (M’02) received the B.S. and M.Sc.

degrees in control and computer science from

the Technical University of Iasi, Iasi, Romania,

the M.Sc. degree in electrical engineering from

Louisiana State University, Baton Rouge, and the

M.Sc. and Ph.D. degrees in mechanical sngineering

from the University of Pennsylvania, Philadelphia,

in 1995, 1996, 1999, 2001, and 2003, respectively.

He is currently an Assistant Professor in the

Departments of Manufacturing Engineering and

Aerospace and Mechanical Engineering at Boston

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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.