IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 51, NO. 11, NOVEMBER 20061749
Controlling a Class of Nonlinear Systems
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 ? ?
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
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
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.
? ? ?
is constant, and the controlis
Index Terms—Aircraft control, convex analysis, hybrid systems,
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: firstname.lastname@example.org).
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: email@example.com).
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
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
control systems in , and for nonlinear systems in an abstract
categorical context in . However, in , 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 , multi-rate automata , ,
and rectangular automata , , or by restricting the dis-
crete behavior, as in order-minimal hybrid systems , .
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 , , or using polynomial functions as in
 and . However, to derive the transitions of the discrete
procedures such as quantifier elimination for real closed fields
and theorem proving , which severely limit the dimensions
of the problems that can be approached.
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 
and Lotka-Volterra  equations, attitude and velocity control
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) ,
. 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
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
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 –. 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.,  and ), rectangular grids are easier
to work with, certainly in problems of higher dimension.
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.
. A full dimensional polytope
hull of at least
is the intersection of
hyperplanes. More generally, a face of
with several of its supporting hyperplanes. If the dimension
of the intersection is
-face. In particular, all facets of
-dimensional rectangle in
with the property that
and consider the -dimensional Euclidean space
is defined as the convex
affinely independent points in
with one of its supporting
is the intersection of
) the face is called a
-faces, and theare
is characterized by two
The set of vertices of
is denoted by, and may be char-
Letwith. Then every
, characterized by
isomorphic with an
particularly interested in facets. For
and for, is
-dimensional rectangle. We are
denote the indicator function
hasfacets described by
,. The outer normal of facet
is given by
denote the Euclidean basis of
We end the discussion on rectangles by noting that an ar-
. Moreover, for an arbitrary vertex
facets containing it are given by
Definition 1 (Multiaffine Function): A multiaffine function
, with the property that the degree of
, in any of the indeterminates
than or equal to 1. Stated differently,
, , where,
) is a function in which each
is a polynomial in the indeter-
has the form
convention that if
For example, for
Finally, note that if
for alland using the
and arbitrary , all multiaffine func-
is a multiaffine function on an
-face of , then the
BELTA AND HABETS: CONTROLLING A CLASS OF NONLINEAR SYSTEMSON RECTANGLES1751
III. PROBLEM FORMULATION
With the notation and definitions introduced in the previous
this paper. As already outlined in the Introduction, we consider
control systems of the form:
where the state
as defined in (1) and the input
. The vector field
defined in (6) and
of nonlinear affine control systems , which have the general
is a matrix spanning the control distribution. Therefore, in
this paper, we consider a particular class of drift, and constant
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
for system (7), such that the corre-
sponding closed-loop system is positively invariant on the rec-
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
before this desired facet is reached.
2) Problem 2 (Control to a Facet): Determine a feedback
for system (7) such that, indepen-
dent of the initial state, all state trajectories of the closed-loop
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
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-
is a “drift” vector field and
. In this case, the feedback
, and the
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
lems 1 and 2.
Lemma 1: Let
be an -dimensional rectangle with
as vertex set. Let
be a multiaffine function, and
is affine, then
Then, for all vertices
of the -dimensional rectangle
Subtraction of bothequations
, and since
, we obtain
This implies that also
. By the induction hypothesis
be an -dimensional rectangle in
be a map, relating every vertex
. Then there exists a unique multiaffine
and , hence
Proposition 1: Let
, and let
to a vector in
Moreover, if for every
under is denoted by
the image of
, is given by (3), then the multi-affine map
realizing (10) is given by (11), as shown at the
bottom of the page.
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
is a multiaffine function satisfying (10),
is multiaffine, and
. By Lemma 1,
, hence,defined in
(11) is unique.
Proposition 2: In every point
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
, the value
is a convex combination
at the vertices of
, the product
, it follows that for
Hence, (12) and (13) show that
bination of the values of
Corollary 1: 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
of lowest dimension of which
is a convex
at the vertices of.
Lemma 2: Let
if and only if
. stands for any of
, for all
Proof: The necessity follows immediately from the fact
that the vertices
also immediate from the fact that
function and, therefore, its restriction to the rectangle
convex combination of its values
It is easy to see that Lemma 2 remains valid if
to a facet
. The sufficiency is
is a scalar multiaffine
at the ver-
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-
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-
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
for all times if and only if the following sets are
, , andare
, we can choose a control
, remain in the
Proof: For sufficiency, if all the sets
nonempty, than we can choose arbitrary
taking the values
function can be constructed using (11). By Proposition 2,
is a convex combination of
andis convex,itfollows that
The vector field
is a multiaffine function on
side (15) state that, for an arbitrary vertex
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
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
with. From Lemma 2, we
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]
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
. We only need to show that
. In combination with the Lipschitz conti-
, condition (16)
by contradiction that there exists a vertex
(i.e., satisfied with “ ”), then by continuity this implies that
there exists a whole neighborhood of
has a strictly positive projection along
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
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
Theorem 2 (Sufficient Conditions for Problem 2): There ex-
(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
so that the previous inequality is false
. This gives a contradiction
, , and are such that,
at the vertex has a strictly
are driven through an
for all (17)
for all vertices.
Proof: Choose arbitrary
at the vertices as shown in (11). By Proposition 2,
is a convex combination of
, 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
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
line of (17). Therefore, the state trajectory of the closed-loop
system can only leave through
,, which means that the vector
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
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
field along the outer normal of the exit facet is only positive as
to relax the property described in Remark 1 that all trajectories
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
, by Lemma 2, we conclude that there
. Therefore,thestatetrajectories oftheclosed-loop
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.,  and ).
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
(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.
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
in motion with respect to a world frame
be the inertia matrix of the aircraft with respect to its body
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
be the axes about
denote the angular
the total force
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 
of the body and the mass of the fluid replaced by the body 
The position and orientation in the world frame
systems described previously are identified with
is an added mass matrix which incorporates the mass
, the Lie
group of rigid body displacements in
in and its rotation
The equations relating their positions and velocities are
be written as
is the skew symmetric operator.
( denotes the unit
, (24) can ) are chosen to parameterize
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
. 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
. 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
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-
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
are the components of the angular velocity
captures the physical con-
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 , , 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
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-
Assuming that the aircraft spans between
and along the
) of the body frame , we have
trol system (27) is obviously of the form (7) with
Consider the following control scenario. Assume that the air-
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
to the final state, the aircraft is forbidden to develop rotational
around its -axis.
To capture the uncertainty on knowledge of the state as well
directions. Under this assumption, the initial state of rotation is
and with side . The amount of allowed speed
, and set of admissible controls
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.
a set of four pairwise adjacent rectangles as shown in Fig. 1(a).
The task is accomplished if the following controllers are de-
• Controller 1: “Drive” the system down along the
while keeping the absolute values of
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
[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
[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.
and less than .
with exit facet
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.
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):
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
. In this way, it is guaranteed that the constructed feed-
back law solves the given reachability problem. Indeed, feed-
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
reaches Rectangle 4 in finite time. After a (continuous) switch
to feedback law
, the state trajectory will remain in Rectangle
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
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
and coincide. The only discontinuous input
; 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,
A different choice of these input values leads to a different con-
trol law, that, in general, will be piecewise multiaffine instead of
, , 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
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
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
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
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
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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
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
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,
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.