TMlO 2 1 0
Proceedings of the 36th
Conference on Decision & Control
San Diego, California USA December 1997
Common Factors and Controllability of Nonlinear Systems
Yufan Zheng
Institute of Systems Science, East China Normal University
Shanghai 200062, CHIK-4
email: yfzhengQeuler.mat h .ecnu .edu.cn
J.C. Willems
Institute of Mathematics and Computing Science, University of Groningen
9700 AV Groningen, The Netherlands
email: J.C.Willems@math.rug.nl
Cishen Zhang
Department of Electrical and Electronic Engineering, The University of Melbourne
Parkville, VIC. 3052, AUSTRALIA
Fax: 61-3-9344 6678, email: c.zhangQee.mu.oz.au
Abstract: This paper presents a new necessary and
sufficient condition for controllability of nonlinear systems. Over the differential field of the nonlinear system, a polynomial equation is derived to present the
system behavior. The condition for controllability is
then presented in terms of the common factor of the
polynomial equation. This condition extends the well
known result for linear system controllability to nonlinear systems, and provides an effective procedure for
examining controllability of nonlinear systems.
effective procedure for examining controllability of nonlinear systems with the aid of the Euclidean algorithm
The rest of the paper is organized as follows. Section
2 extends the behavioral approach to linear systems
to present a definition for controllability of nonlinear
systems. Section 3 presents properties of the differential field. Section 4 uses the properties of Section 3 to
present the nonlinear system behavior and the condition for controllability of nonlinear systems. Section 5
further provides a procedure for examining controllability of nonlinear systems and examples of the procedure.
1 Introduction
Controllability is a fundamental property of systems
and has been well studied for linear systems. For controllability of nonlinear systems, there have been a
number of results, e.g. [9, 4, 5, 7, 11, but the subject
still requires further study. In this paper, we extend
the behavioral approach to linear system controllability [lo, 11, 121 to present a new necessary and sufficient
condition for controllability of nonlinear systems.
We show that the nonlinear system behavior can be
presented by a polynomial equation over the differential field of the nonlinear system [3, 2, 131. The system controllability condition is then provided in terms
of the common factors of the two polynomials of the
equation. Specialized to linear systems, this condition
is consistent with the resulk on linear system controllability.
The necessary and sufficient condition can provide an
'Author for correspondence
0-7803-3970-8197
$10.00D 1997 IEEE
2 Problem statement
In this paper, d denotes the differential operator that
satisfy the Leibniz rule d(a1az) = a l d a z (da1)az for
two functions al and a 2 . s = -$ denotes the derivative
operator with respect to time t such that sa = a = a(')
and s i , = a(;) for a function of time a ( t ) and i 2
0.
denotes the partial derivative operation with
respect to a variable a. deg p ( s ) denotes the degree of
a polynomial p ( s ) in s.
+
&
We consider the following scalar nonlinear system:
where f : Pf'
x E'' -+ LY is a meromorphic function, and U and y are the system input and output,
respectively. Let
w
=
( ) be the trajectory of the
system. This equation defines the dynamical system
C = ( E ,E 2 ,a,) with behavior
2584
Bj
use this property to define the controllability for nonlinear systems as follows.
= { W E C"(R,IR2) I (1) holds}
Definition 1: The nonlinear system (1) is controllable
if there exists no autonomous function v such that that
system behavior is expressed as
In order to motivate the definition of controllability
that will be used, we firstly consider that (1) is a linear
system. In such a case, we can write (1) as
P(S)Y
+d S ) U = 0
21
(2)
w(t)=
2.
U('))
=0
(7)
&
there exists a w E B(p,q)
Then the problem of nonlinear system controllability is
to examine whether (7) is satisfied for some functions
v and h.
for t < 0
for t 2 r
{ ti!:)-
3 Differential field and the Ore ring
In this section, we present some fundamentals of differential field [2, 31 and the Ore ring [8, 61 that we will
use to present the behavior and controllability of the
nonlinear system (1).
In [ll], condition 1 is referred to as controllability.
There is however a third equivalent property that we
will use as the concept for controllability in this paper.
Let j j ( s ) , QI(s) E R[s],and define the mapping L ( P _ , :~ )
-+ C " ( R , R)by
L?(p,q)
L(P_i)(Y,
U ) = $(S)Y
+ QI(s)u
(3)
Now consider the behavior L? = L ( P _ , ~ ) L ? (It~ ,fol~).
lows from 1113 that fi is itself a differential system
such that either L? = C"(R, R)or L? is autonomous
(i.e. W I , w:! E
and w l ( t ) = wz(t) for t < 0 imply
11.1 = wz). The later will be the case if and only if there
esists a polynomial p E R[s] such that
P(+(P_,c)4P>9)
=0
(4)
If either B(s) or g(s) # 0 and if deg p ( s ) > 0 , (4) must
imply that p ( s ) is acommonfactor of { p ( s ) , q ( s ) } such
that
ds) = p ( s ) @ ( s )
= P(S)?qS),
e ,
where v : R'+l x R
' -+ R,h : 83"' -+ R,are meromorphic maps with
# 0 and 1 > 0.
p and q have no common factors.
P(S)
y ( ' ) , . . . , y('), U ,U P ) , . . . , &-I))
h ( v ,v('),
where p , q E R[s]and R[s]denotes the polynomial ring
over the real field. Let L?(p,q) be the behavior of (2). It
is shown in [ll] that the following two conditions are
equivalent:
1.
For all wl,wz E L?(,,,),
and a T 2 0 such that
=
(5)
A differential field K is a field together with a derivative
operation ( ) : K -+ K that satisfies the following
rules.
(K1
+
and
(KIKZ)
for
Kl,K2
Kz)
= kl
+ kz
= klK2 + KliE:!
E IC.
A uector space A over differential field 5 as a vector
space together with a derivative operation ( ) : A -+ A
that satisfies the following rules.
(a
+ b) = u + b
(.a)
= i u + nu
and
Hence conditions 1 and 2 are equivalent to:
for a, b E A and
There exists no F ( s ) , @ ( sE) R [ s ]such that (5)
holds for some p ( s ) E R [ s ]with deg p ( s ) > 0.
Suppose that the nonlinear system (1)can be explicitly
written in terms of the highest derivative y(") as
3.
Accordingly, the system controllability means that
there are no 23, g, p E R[s] with deg p ( s ) > 0 such that
the behavior of (2) can be expressed as
,("I
K
E IC.
+ d(y, y ( l ) ,. . .,y("-'),u,
&),
. . . ,J " - - l ) ) = 0
(6)
where : la" x R" + R is a meromorphic function.
Let K be the field of meromorphic functions of y(') for
0 5 i 5 n - 1 and u ( i ) ,for j 2 0.
\Ve note that (6) can be interpreted as that v is an autonomous variable and cannot be controlled. We will
To verify that the K is a differential field, let function
$ ( y , y ( l ) ,. ., y("-'), U , U ( ' ) , . . . , ~ ( ~ be
1 )an element of
h', we have
v = p(s)y
+
@(.)U
p(s)v = 0
-
2585
c o m m o n left f a c t o r of p1 (s) and p2 ( s ) if cleg pc (s) is the
greatest of all the common left factors of ( p l ( s ) - p z (s)).
The well known Euclzdean Algorithm is applicable for
finding the greatest common left factor of two polynomials. Specifically, for deg p l ( s ) = d l and deg p 2 ( s ) =
dz with d l > d2, two polynomials p 3 ( s ) and 71( s ) ,with
deg p 3 ( s ) 5 d2 - 1 and deg n ( s ) = d l - d2, can be
computed such that
+
We can furtlier have ($1
$2) = $1+&
and ( $ l * $ ~=
)
&$z
$ l d z . Thus IC is a differential field defined by
the nonlinear system (1).
+
Over the differential field IC, we define a vector space
The Euclidean Algorithm is to continuous the computation to obtain
PZ(s)
Pk-Z(s)
Then it can be easily verified that V* is a differential
vector space over IC defined by the nonlinear system
(1) ~ 3 141.
,
The differential field X: and the operator s induce a
polynomial ring, denoted by X:[s].A polynomial p ( s ) E
K[s]is written as
P(S)
= y m s m +ym-lSm-'
Pk-l(S)
=
p3(S)YZ(s)
+ P4(s)
= Pk-l(S)Yk-Z(S)
= Pk(S)Yk-l(S)
+Pk(s)
(11)
The algorithm terminates in a finite number of steps
and, as a result, the greatest common left factor of
p l ( s ) and p 2 ( s ) is p k ( s ) . p k ( s ) and two polynomials
p l ( s ) and Fz(s) are obtained such that
P 1 ( s ) = Pk ( S ) i j l (SI ,
Pz (SI = Pk (s)Pz(SI
+ - . . + c ~ I s + c ~ o(8)
The degree of the polynomial in (8) is m if y m # 0,
and the polynomial p ( s ) is called monic if y m = 1.
While all the other algebraic operations in the ring satisfy the operations in the function field ic, the multiplication between the derivative operator s and an element p E X: obeys the following rule.
If deg p k ( s ) = 0 , then the polynomials { p l ( s ) , p z ( s ) }
have no common left factor and are called coprime.
4 A condition for controllability
This section presents a condition for controllability of
the nonlinear system (1) in terms of the polynomials
in the Ore ring X:[s].For the nonlinear system ( I ) , we
can apply the derivative operation to obtain
The ring K [ s ]thus defined is called the Ore ring and it
is a non-commutative ring [8, 61.
In the differential field IC, there is no non-zero zero elem e n t in the sense that if ~ 1K Z, E IC with ~1 # 0, I E #
~ 0
then K l Q # 0. It follows that for three polynomials
~ ( s )PI(.),
,
p 2 ( s ) E K [ s ] ,with deg p l ( s ) = d l > 0 and
deg P Z ( S ) = d2 > 0 , such that p ( s ) = p 1 ( s ) p z ( s ) , the
degree of p(s) satisfies
deg
P(S)
= deg
PI(S)
+ deg PZ(S) = d l + dz
Let
(10)
where p1 ( s ) is called the left divzsor and pz(s) is called
the rzght dzvisor of p ( s ) , and p ( s ) is called left divisible by p l ( s ) and right divisible by p z ( s ) . If for
p l ( s ) , pz(s) E K[s] such that p c ( s ) is a left divisor
of ( p l ( s ) - p z ( s ) ) , then p c ( s ) is called the c o m m o n left
f a c t o r of p l ( s ) and p z ( s ) . p,(s) is called the greatest
p ( s ) , q ( s )E IC[s].We now use d y ( ' ) = s i d y and d u ( j ) =
s j d u in the differential vector space to write (12) as
2586
p(s)dy
Let dw = d
( )
Let w = fi(s)dy - c(s)du. We obtain
+ q(s)du = 0
(15)
p(s)w = 0
be the system trajectory in the dif-
We can further apply the result of [l]on nonlinear system controllability to (17) and show that there exist
function U , h E K for the nonlinear system ( l ) ,where
U is an autonomous function and U and h can be written in the form (7)[14, 151. Hence the system is not
controllable by Definition 1. 0
ferential form. Then (15) presents the nonlinear system
behavior in terms of the polynomials { p ( s ) ,q ( s ) } over
the differential vector field K .
For the nonlinear system (1) with the behavior (15),
we present the controllability condition of the system
in the following theorem.
Specialized to linear systems, Theorem 1 clearly yields
the result on coprimeness of { p ( s ) ,q ( s ) } as elements in
nZ[s],that is a well known condition for linear system
controllability.
Theorem 1: The nonlinear system (1) is controllable
in the sense of Definition 1 if and only if the polynomials { p ( s ) ,q ( s ) } have no common left factors.
In the result of Theorem 1, we have limited our attention to the case of scalar input/output systems. This,
however, is a restriction that is not particularly important. A generalization that suggests itself is to systems
in which, in the spirit of behavioral approach, no distinction is made between input and output.
Proof:
Necessity: Suppose that the nonlinear system (1) is
not controllable. According to Definition 1, there exists
functions w , h E X: such that (7) is satisfied.
In the differential vector space, we can apply the the
differential operation to the functions v and h in (7)
and use dy(i) = sidy, du(j) = sjdu and dh(") = smdh
t o obtain
5 Computation procedure and examples
Theorem 1 provides an effective test for the controllability of nonlinear systems. This can be carried out
by examining whether the two polynomials in K[s] of
the nonlinear system have a common left factor by the
Euclidean Algorithm (11). We provide a pseudo-code
for examination of the nonlinear system controllability, that works very effectively when f is rational or
polynomial function.
dv
= p(s)(@(s)dy- i(s)du) = 0
where
&,
6E K , @(s), i ( s ) , p(.)
Input the function f ;
(i)
E K[s].
(ii)
0sj
Since {dy('), du(j); 0 <
- i 5 n - 1,O 5 j 5 n - 1) is a set
of independent vectors in the differential vector space
[13, 14, 151, we can match the equations (16) and (15)
to obtain
Compute
5 72- 1.
pi and
It further follows from (10) that
+ deg F,
deg q = deg p
& and
for 0
5 i 5
n and
(iii) Obtain the polynomials { p ( s ) q, ( s ) } in the form
(13) and (14);
(iv)
deg p = deg p
(17)
Let p1 = p and p 2 = q , then compute recursively
yi-2 for i 2 3 of the following equation
+ deg i
I n (7) the 1 > 0, then deg p > 0. Hence the polynomials { p ( s ) , q ( s ) }has a common left factor p ( s ) , i.e.
{ p ( s ) ,q ( s ) } are not coprime.
Stificiency: Suppose that the polynomials { p ( s ) ,q ( s ) } ,
with deg p ( s ) = n and deg q ( s ) 5 n - 1, has a comnon left factor p ( s ) , with deg p ( s ) > 0, such that the
equation (15) can be written as
P(S)dY - q(s)du = p(s)(@(s)dy- G(s)du) = 0
(v)
Terminate the algorithm when pi = 0. If
deg pi-1 = 0, then { p ( s ) , q ( s ) }have no common left
factors. Otherwise, p i - 1 is the greatest common left
factor of { p ( s ) ,q ( s ) } .
We now demonstrate the procedure with two examples
as follows.
Example 5.1:
2587
Consider the following nonlinear system.
Therefore, this system (19) is uncontrollable in the
sense of Definition 1.
We can follow (13) and (14) to obtain the polynomials
{ p ( s ) , q ( s ) }of the system as
p ( s ) = s2 -
I
2jr+y-us
y 2 - yu
Y
Y2
Y+Y
=s- Y
Directly applying the Euclidean algorithm yields the
greatest common left factor of { p ( s ) ,q ( s ) } that is
q(s)
4s)
= Y(S
1
+
+
We now derive the polynomials { p ( s ) ,q ( s ) } of the system and show that { p ( s ) q, ( s ) } have a common factor.
We use (21) to obtain
- 1)
Using the derivative operation rule: s'p = ps
easy to have
4s) =SY -Y -Y
and then we have
P(S)i
Let W ( t ) = ( Y l ( t ) , U l ( t ) ) T and m2(t) = (yz(t),uz(t))'
be two solutions of (19). It can be shown that if
b(jrl(0))
C ( Y l ( O ) , ~ l ( O ) ) # b(Yz(0))
C(YZ(O)I~2(0)),
then the two solutions cannot be concatenated. Thus,
under the uniqueness condition for the solution of differential equations, the behavior of the system is also
uncontrollable in the sense of [ll].
a4-- -ac
+ - ab
+ + it is
ay av
ay
a4
aZc
a2c
- = -y+
ay
-U + - -
ayau
ay2
ab ac
away
= ds)
Hence, the nonlinear system (18) is not controllable.
Furthermore, by the result of [14,151, we can find an
one-form w satisfying (17) that
p(s)w = (s
Applying the differential operation to (19) yields
- 1)d-Y - L ,
$-
Y
Then the system has an autonomous function
v=-
a4
-dy
a4 + -du
a4
ay + -sdu
ai^
au
=0
This provides the polynomials { p ( s ) ,q ( s ) } of the system in the following form
jl-U
Y
and a function h such that
h(w, 21) = 21 - v = 0
q(s)
a4
= -s
a;
ad
+au
(27)
Example 5.2:
Substituting the equations (22)-(25) into { p ( s ) ,q ( s ) } in
(26) and (27) and using the derivative operation rule
scp = 'ps +, we have
Consider the nonlinear system
y(2)
+ 4(& y, iL,
+
U)
=0
(19)
ac = --sac
.L\ssume that the system can be alternatively written
as
4
i
l
Y, U ) = Y
+ C(Y,
aZc.
aZc
+
( a y + -4)
ay ay ay ayau
sU)
h(w, G) = G + b ( v ) = 0
(20)
and
s-
It can be verified that (20) holds if and only if
ac . ac .
+ -Uau + b ( y + c(yl U ) )
d(Y9 Y, G I U ) = -Yay
(21)
dc
au
dc
d2C .
02c
-A)
au + (-y+
ayau
au2
= -s
Thus, it is obtained that
2588
[5] A., Isidori, Nonlinear control systems: an introduction, 2nd Edition, Springer-Verlag, Berlin, 1989.
(s+
2) ( s +
e)
[GI C. Kaith, Algebra I: Ring, Modules and Categories, Springer-Verlag, 1980.
[7] H. Nijmeijer and A.J. van der Schaft, Nonlznear
Dynamzcal Control Systems, Springer-Verlag, 1990.
[8] 0. Ore, Theory of non-commutative polynomials,
Ann. of Mathematics, Vol. 34, pp.480-508, 1933.
-
[9] H. Sussmann and V. Jurdjevic Controllability of
nonlinear systems, J. Differential Equatzons, Vol. 12,
pp. 95-116, 1972.
(s+ab)ac
av
Bu
This result shows that the polynomials { p ( s ) ,q ( s ) } has
a common left factor (s
[lo] J.C. Willems, Models for dynamics, Dynamzcs
Reported, Vol. 2, pp. 171-269, 1989.
6 Conclusion
[ll] J.C. Willems, Paradigms and puzzles in the theory of dynamical systems, IEEE Trans on Automatic
Control, Vol. 36, pp. 259-294, 1991.
+ E).
This paper presents a new condition for controllability
of nonlinear systems. The condition is in terms of the
common factor of the system polynomials defined over
the differential field that provides an effective procedure for examining the system controllability. Given
that the nonlinear system controllability is more complicated than that of linear systems, the condition, that
is consistent with the result for linear systems, and the
approach used in this paper can provide new insights
into nonlinear system properties and make impact on
further study of nonlinear systems.
Acknowledgment
[12] J.C. Willems, On interconnections, control, and
feedback, IEEE Transactions on Automatzc Control,
Vol. 42, pp. 326-339, 1997.
[13] Y.F. Zheng, On transfer functions of nonlinear
systems, J. East China Normal University (English
version), V01.2, No.2, 1995.
[14] Y.F. Zheng and C. Zhang, Invariant structure
and integral chains of nonlinear multivariable control
systems, to appear 36th IEEE CDC, 1997.
[15] Y.F. Zheng, P. Liu, A. Zinober, C. Moog, What
is the dimension of the minimal realization of a nonlinear system, Proc. the 34th IEEE CDC, pp. 4239-4244,
1995.
This work was completed when the first two co-authors
visited the Department of Electrical and Electronic Engineering at The University of Melbourne, and was s u p
ported by National Science Foundation of China, Australian Research Council and CSSIP at the University
of Melbourne.
References
[l] E. Aranda-Briaire, C.H. Moog, and J.B. Pomet,
A linear Algebraic Framework for Dynamic Feedback
Linearization, IEEE Trans. Automatic Control, Vol.
40, pp.127-132, 1995.
[2] M.D. Di Benedetto, J.W. Grizzle and C.H. Moog,
Rank invariants of nonlinear systems, SIAM. J. Control
and Optimization, Vol. 27, pp. 658-672, 1989.
[3]
M. Fliess, Generalized controller canonical forms
for linear and nonlinear dynamics, IEEE Trans. Auto-
matic Control, Vol. 35, pp. 994-1001, 1990.
[4] R. Hermann and A.J. Krener, Nonlinear controllability and observability, IEEE Trans. Automatic Control, Vol. 22, pp. 728-740, 1977.
2589