Proceedings of the American Control Conference
Philadelphia, Pennsylvania * June 1998
Nonlinear Invariance: Cross-Positive Vector Fields
DAVIDG. MEYER* AND TERIL. PIATT
Nonlinear and Real- Time Control Laboratory
Electrical & Computer Engineering Department
University of Colorado, Boulder, CO 80309-0425
HAYDNN. G. WADLEY AND RAVIVANCHEESWARAN
Intelligent Processing of Materials Laboratory
Materials Science and Engineering Department
University of Virginia
Charlottesville, VA 22901
ROBERTKOSUT
SC Solutions, Inc.
3211 Scott Blvd
Santa Clara, CA 95054
Abstract
on the solution curve can be inferred from properties of the vector field?
This paper introduces the concept of crosspositivity for vector fields and explores ramifications of cross-positivity relative to solutions of ordinary differential equations. The motivation is to
understand the dynamic behavior of systems where
the natural physics place constraints on the vector
field. Our results extend to nonlinear systems several well-known results about properties of x = An:
when A satisfies special conditions.
1
Introduction
In many systems of interest the natural physics constrains the vector field to satisfy certain properties.
For example, if a certain physical quantity s is nondecreasing in a process, then the constraint
0
applies. A question then of interest both in pathplanning and regulation is exactly what constraints
%>
>
We have found in our work [MVW92, MW93,
VMW971 on the control of consolidation of
*Researchsupported in part by National Institute of
Standards under contract 50SBNBC8517, by the Defense Advanced Projects Research Agency under contracts MDA972-93-H-0005 and MDA972-95-1-0016, by
the National Science Foundation under grant DMS9615854, as part of the Virtual Integrated Prototyping
Initiative, and award ECS-9455705, and by Hughes Research Labs.
0-7803-4530-4198
$10.000 1998 AACC
In the realm of linear systems it has long been
understood that certain structure in the matrix A
results in strong constraints on eAt, for example,
[Ost37, Yor67, Rob66, BP791. The most famous
of these is the result that non-negative initial conditions yield non-negative solutions if Ai,
0 for
i # j. Several researchers have suggested exploiting these types of results for the purposes of control of linear systems by designing feedbacks to give
the closed-loop A matrix a specific structure. Results and work along these lines include those in
[TB93, VHB88, BH931 among others. The results
in this paper begin to directly extend these well
appreciated linear system ideas into the realm of
nonlinear systems.
titanium-alloy, ceramic fiber composite materials,
that exploiting the physical constraints on the vector field is crucial to good control; however, the
exploitation was on an ad-hoc basis. The work presented here is the beginning for a systematic framework for mathematically understanding our success
with metal matrix composites and migrating those
techniques to new and different problems.
308
1.1
Elementary Background on Cones
It is a simple matter to verify that C+ is a cone;
moreover C+ is always a closed subset of R”. See
[Lue69] for more information. The positive conjugate of a polyhedral cone is again polyhedral, and
on polyhedral cones, positive conjugation is idempotent, i.e. (C+)+== C. The next lemma, whose
proof is left to the reader, shows this explicitly for
polyhedral cones.
Definitions and results in this paper can, with just
a little care, be extended to more general settings,
e.g. to vector fields on a smooth manifold, but
here we present results only for R”,the usual real
n-dimensional vector space. We will use (.IT to
denote the transpose of a vector or matrix.
Let C be a nonempty subset of R”. If aC+pC C
C for all a > 0 , p 2 0, then C is called a cone. A
cone C is pointed if C n -C = (0) and is reproducing if C - C = Rn. Though the results in this
paper extend to general cones, we shall, for clarity
of exposition and conservation of space, restrict our
attention to ‘%polyhedral”cones as defined next.
Lemma 1.3 Let C = Cone(P). Then Cs =
Cone(P-T).
Notice that for polyhedral cones C , yTx = 0 , x E
C , y E C+ happens if and only x E dC and y E
dCs. This is a well-known result in duality theory
and is true for more general cones as well (see, e.g.
[Fen53]).
Generalizing the notion a cross-positive matrix
introduced by Schneider and Vidysagar in [SV70],
we make the following definition.
D e f i n i t i o n 1.1 Let P be a non-singular n x n matrix. Then the polyhedral cone defined b y P is the
set
Cone(P)A x E R” I P x 2 0 }
{
Polyhedral cones are closed, pointed, reproducing
cones and they are exactly the convex hull of their
”edges” as the next lemma, whose proof is very
straightforward, shows.
Lemma 1.2 Let (41,q 2 , . . . , qn} be a set of n linearly independent vectors in R”, and let Q =
[ q1 42
qn ] be the non-singular matrix
formed using the qi as columns. Consider the set of
“non-negative rays” emanating from the qi defined
as
emanating along the qi. Then C o n e ( Q - l ) =
C o ( R ) where “CO”denotes convex hull.
The polyhedral cone defined by independent vectors {qi}y==l is the same as the “non-negative” span
of the qi found by using only non-negative coefficients while taking linear combinations of the qi.
In other words, we might say that { q i } is a set of
generators for Cone(&-’)
D e f i n i t i o n 1.4 Let C be a cone. A continuous
vector field, f ( x ) : R” + R” is said to be:
0
0
0
c
cross-positive on
if yTf(x) 2
C , y E Cs such that yTx = 0.
o for all x E
strongly cross-positive on C z$ f (x) is crosspositive on C and for each x E dC there is a
y E C+ such that yTx = 0 and yTf (x)> 0.
strictly cross-positive on C i # y T f ( x )
all x E C , y E C+ such that yTx = 0.
> o for
If C = Cone(P) is a polyhedral cone and x E dC,
then the number of components where Px is identically zero is exactly the number of independent vectors in dC which annihilate x. These independent
vectors can be found1 as yj = P T e j and, in fact,
following Lemma 1.2 a vector y E x’nC+, the
orthogonal complement to x intersect the positive
conjugate, can be written as y = &cujyj cuj 2 0.
These facts imply the existence of separating hyperplanes as given in the next lemma which is stated
here for completeness. The proof is very straightforward and left to the reader.
Lemma 1.5 Let C =: Cone(P) C R” be a polyhe1.2
dral cone and f (x) be vector j e l d which is strongly
cross-positive on
Let x b E
and 1 5 k 5 n be
Cross-Positivity of Vector Fields
The positive conjugate, C+, of a cone C is defined
as
C+ = y E Rn I yTx 2 OVx E C }
{
c.
ac
4
an index so that e r P z b = 0. If x, C i s such that
so that yTXb = 0,
e T P x , < 0 then there is y E
yTx, < 0, and yTf (21,)> 0.
309
An invariance result arrived at in [SV70] is that for
a square matrix A and a cone C, eAtxO E C for all
- 2 0 E C if and only if A is a cross-positive matrix
on C. For C = Cone(1) this is a classic result
from the theory of Ad-matrices ([Rob66, RP791) and
invariance results for x = Ax of this type have a
long history [Ost37, Yor671.
One aim is to generalize results of this type to
nonlinear autonomous systems = f(x). The result of Schneider and Vidyasagar can be stated, using the definitions of this paper, that the cone C
is forward invariant for 2 = Ax if and only if the
vector field f ( x ) = Ax is cross-positive on C. A
simple nonlinear extension of this type of result is
the lemma below.
x
Lemma 1.6 Let C be a cone. Iff(.) : Rn + Rn
as strongly cross-positive on C , then C as a forward
invariant set for x = f (x).
4
+
+
Y T ( E ) Q Z 0 ( t * ) = 0; YT(€)QZ0(t*
As in the first case, integrating
inner product with Y(E) gives
*
E)
<0
and
yT(e)f ( ~ , , ( t * )>) o for
PROOF:
Let Qa,,(t) denote any solution to IE. =
f ( x ) , z ( O )= 2 0 . It needs to be shown that xo E
C
aZ,(t)E C V t 2 0. It is most convenient and
illustrative to break the proof into two cases, each
of which proceeds by contradiction.
For the first case, assume zo E int (C), the strict
interior of C, and suppose there is some time, tl
for which QZo(tl)
4 C . Let
T=
the left side of (1) is strictly negative. This is a
contradiction, so we conclude 7 = 0 as wanted.
For the second case, assume xo E BC and let 3 be
the (nonempty) set of indices for the components
of Pxo which are zero. If aZo(t)
E int (C) for any
time t > 0, then the first case of the proof shows
that Q Z o ( tnevers
)
leaves cone C thereafter. So, it
can be assumed that Qa,,(t)E dC until the first
exit time, t*.
Since QZ,(t)is C1 and t* is the first exit time, there
is an €1 so that E < €1 implies azo
(t*+ E )
C. This
implies, again by continuity, that there is an €2 5 €1
and an index IC E J so that ezPQ,,(t*
E ) < 0 for
E < €2. Applying Lemma 1.5 yields a Y ( E ) E dC+
satisfying
As before, for
E
< €2
$ and taking the
smaller than €2 and small enough,
the right hand side of (2) is positive by continuity. However, the left hand side is negative. This
gives the desired contradiction and the proof is
complete.
{ tl@Z,(t) 6 c }
and t* = inf 7 . As ‘7- # 8, t* is well-defined and
is the “first exit time” so that (a,,(t) E int (C) for
t < t*. Moreover, by continuity, Q,,(t*) E dC, and
as f is strongly cross-positive on C, there is a y E
C+ so that yTQ,,,(t*) = 0 and y*f (@,,(t*))> 0.
Now then, for all 0 < E < t*
E
The cone C = Cone(1) and the vector field
f ( x ) : R2 -+ R2 given by
shows that the hypothesis of Lemma 1.6 cannot be
relaxed to cross-positivity without additional assumptions on f. Any cone C and the vector field
f ( x ) = 0 shows that the converse of Lemma 1.6 is
not true.
If we relax our expected conclusion to weakly invariant (meaning, roughly, that some solution of
i = f(z) stays in C) then we can relax to crosspositivity. Alternatively, if we assume the vector
field is such that j: = f(x) has unique solutions on
an open set containing the cone, we can relax to
cross-positivity. Either way, we get necessary and
and so
e
where g(7)
yTf
is a continuous realvalued function satisfying g ( t * ) > 0. It follows that
for E > 0 small enough, the integral on the right
side of (1) is positive. However, the quantity on
310
>0
sufficient conditions for forward invariance. This
theorem is given next. Its proof is delayed until
the next section.
So, assume some ~ i say
,
€1, is positive. Define E
by
Theorem 1.7 Let C C Rn be a cone. If f(x) is
cross-positive on C , then C is weakly forward invariant.
That this E is the one sought can be easily verified.
First, as qi was a set of generators, any y E C+ can
be written Caiqi, ai 2 0. So,
2
Relations to Subtangentiality
Recall (see, e.g. EFen53, Yor671) the following definit ion.
Definition 2.1 Let S C Rn and x E S . Then a
vector v E Rn is subtangential to S at a point x if
and only if
lim inf
d(S,x
t
t-iO+
+tu) = o
and clearly for t < E , Ciaioi(t)2 0 and this implies xo t f (xo)E C for t < E as desired.
+
i
The main theorem of this section is the following
Theorem 2.3 Let C be a cone and f (x)a continuous vector field. Than f (x)is subtangential to C
for all x E C i f and only if f (x)is cross-positive on
C.
(3)
where d(S, z ) denotes the (Euclidean) distance from
S to the point z Note that (3) is trivially satisfied
if x E i n t ( S ) .
PROOF:
Clearly we need only consider points on the
boundary of C;hence let xo be an arbitrary point
on aC. Using duality we have
The following lemma is needed for the main theorem
Lemma 2.2 Let C be a cone, let xo E aC and let
f ( x ) be a vector field cross-positive on C . Then
there exists E > 0 so that for all t < E
d ( C , z ) = max -y T z
VEC+
(4)
llYllI1
and so
PROOF:
Let (41, q 2 , . . . ,qn} be a set of generators for C+.
Consider the function
(5)
Let yo be any unit nolrm vector in dC+ satisfying
yrxo = 0 (there is at ].east one) and notice
Define
to conclude the “only if” direction.
The “if” direction follows immediately from
Lemma 2.2. This completes the proof.
I
Because zo E C , qTx0 is positive or zero. If qTx0
is zero, then qTJP(z0) 2 0 because f ( x ) is crosspositive on C. We conclude cri(0) 0, ~i is welldefined, and either ~i > 0 or $ZO = q F f ( z g ) = 0.
If ei = 0 for each i then qTzo = q T f ( x 0 ) = 0 for
every generator qi in C+ which implies z o and f (zo)
are zero, and so 20 t f ( z 0 ) 0 E C V t and any
positive number works for E .
The proof of Theorem 1.7 follows from Theorem 2.3 and Theorem 3.3 in [Yor67].
>
+
Remark 2.4 One at first might be surprised that
cross-positivity is “onr!y” a subtangentiality condition, and hence that a large portion of results in the
control theory literature on invariance conditions
=
31 1
for linear systems is just a corollary of classical
differential equations results ([Ost5’7, Nag42,’) from
the 1940’s and earlier. W e don’t thank this is a good
way to look at things. The point is that subtangentiality is not easy to directly compute whereas the
condition of cross-positivity, which involves minimization over very spec@ convex sets, should be
quite amenable to computation. Thus, the authors
hope that the theory of cross-positivity leads to practical control theoretic application of powerful classic
results.
3
Conclusions
We have introduced the notion of a cross-positive
vector field and related this condition to (forward) invariance of the associated differential equation. Cross-positivity turns out t o be equivalent
to subtangentiality for cones and this can mean
more practical (computational) application of certain classical results in the theory of differential
equations.
4
Acknowledgements
The authors gratefully acknowledge several very
helpful comments from the anonymous reviewers.
[Rob661
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A. Berman and R. J. Plemmons, Nonnegative matrices an the mathematical sciences,
Academic Press, New York, New York,
1979.
[Nag421
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312