NONLINEAR DECOUPLING THEORY
WITH APPLICATIONS TO ROBOTICS
by
In Joong Ha
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Computer, Information; and Control Engineering)
in The University of Michigan
1985
Doctoral Committee:
Professor Elmer G. Gilbert, Chairman
Professor N. Harris McClamroch
Professor Semyon M. Meerkov
Professor William L. Root
Associate Professor Carl P. Simon
ABSTRACT
NONLINEAR DECOUPLING THEORY WITH APPLICATIONS TO ROBOTICS
by
In Joong Ha
Chairman: Elmer G. Gilbert
Some theoretical results on nonlinear decoupling theory are
presented and their applications to robotic manipulator control are
discussed.
First, refinements and extensions of some known results on
feedback decoupling of nonlinear systems are given. Precise
definitions of decoupling and decomposition are stated. Some
conditions under which the two definitions are equivalent for
nonlinear systems are found. A previously known condition is
shown to be necessary as well as sufficient for a system to be
decouplable or locally decomposable.
Second, we obtain new results which characterize the whole
class of nonlinear feedback control laws which decouple or
decompose. These results are important from both mathematical
and engineering viewpoints. For instance, there exist systems
where our results allow the stable decoupling ofa decouplable
system, while former results do not. The class of decoupling
control laws is characterized by solutions of certain first order
partial differential equations. The class of decomposing control
laws is characterized by simple feedback laws applied to a
standard decomposed system (SDS). The SDS is similar to the
decomposed system of Isidori, Krener, Gori  Giorgi, and Monaco but
has finer structure. These new results are provided by a
generalization of ideas used by Gilbert for linear systems.
Third, we discuss a form of approximate decoupling. We
neglect fast dynamics of a system to obtain a computationally
simple control law. It is shown that when the neglected dynamics
are sufficiently fast, the simplified law decouples the actual
system "approximately" in a certain sense.
Finally, these results are applied to decoupled control of
robotic manipulators. Two cases are considered. In the first
case, actuator dynamics are completely neglected. In the second
case, the dynamics of a significant class of actuators are taken
into account. Our formulas for the complete class of decoupling
control laws unify and generalize previous results on the decoupled
control of robotic manipulators. For example, it is possible to
achieve decoupled control of the end  effector.
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ACKNOWLEDGMENTS
First of all, I would like to thank my advisor, Professor
Elmer G. Gilbert. During my doctoral study, he has devoted a
great amount of precious time to my academic disciplines and
helped me in many ways. He has contributed much to my
academic achievement by his sincere and excellent guide.
I am indebted to my wife, In Hye who has made untold
sacrifices so that I could concentrate only on study. I am also
grateful to my and her parents for their unfailing encouragement.
I would like to thank the faculty members of the Computer,
Information, and Control Engineering Program for offering
excellent courses and giving helpful advice. Special thanks go to
Professsors W. L. Root, N.H. McClamroch, F. J. Beutler, and R. A
Volz. Mr. D. W. Johnson has been a supportive friend.
I wish to thank the members of my dissertation committee
for their Interest in my work and their helpful comments.
Finally, the financial support of the Air Force office of
Scientific Research, Air Force Systems Command, USAF, Grant No.
F 49620  82  C  0089 is appreciated.
iii
TABLE OF CONTENTS
DEDICATION...i.. 9..... 11....I
ACKNOWLEDGMENTS.....I
LIST OF FIGURES......... vi
CHAPTER
1. INTRODUCTION...1...... 1
2. MATHEMATICAL BACKGROUND 9
2. 1. General Notation and Definitions
2. 2. Basic Concepts of Differential Geometry
2. 3. Some Fundamental Results
3. NONLINEAR DECOUPLING THEORY. 30
3. 1. Definitions
3. 2. Decoupling and Decomposition
3. 3. Decouplabillty and Decomposabllity
3. 4. The Whole Class of Decoupling and
Decomposing Control Laws
3. 5. Examples
3. 6. Conclusion
4. APPROXIMATE DECOUPLING.... 127
4. 1. Notation and Assumptions
4. 2. Result for Approximate Decoupling
5. APPLICATIONS TO ROBOTICS.......... 140
iv
5. 1. Decoupled Control of Robotic Manipulators
5. 2. Decoupled Control of Robotic Manipulators
with Significant Actuator Dynamics
6. CONCLUSION.......... 158
REFERENCES * *.............. 160
V
LIST OF FIGURES
Figure
3. 1. 1. (,i,H ) Is Trelated on X to (F,H,KX).... 34
3. 1.2. (F, H, ) Is J feedback related on X
to (F,H,X).*....... *......36
3. 2. 1. Summary of main results In Section 3. 2 showing
assumptions required for each Implication.... 66
3. 3. 1. A standard decomposed system (F H, T()) is
J  feedback related on ~ to the system (F, H, ~) I. 82
3. 3. 2. Summary of main results in Section 3. 3 showing
assumptions required for each Implication....... 90
3. 4. 1. Relationships between F, H, X ), (F, H, X),
and (F*, H*,X)................. 98
3. 4. 2. Relationships between ( F, H, K), (F, F*, H* ),
(F,H, H)t, and (F,H,T T)) i....... 102
3. 4. 3. A schematic description of Theorem 3. 4. 5, where
u = <(x) + p(x) u, u = <() + p(B) u are control laws In
8"((F,H,) ), 8S ((F,H,T(K)), respectively. 109
3. 4. 4. Summary of main results In Section 3. 4 showing
assumptions required for each implication....... 112
vi
CHAPTER I
INTRODUCTION
In this dissertation, we present some theoretical results on
nonlinear decoupling theory and discuss their applications to robotics.
Let us begin the introduction with a simple discussion of the main
ideas.
Suppose we have a nonlinear system:
(1.1) x(t)  f(x(t), u(t)), y(t)  h(x(t)),
where x(t) e R", u(t) e Rm, y(t) e Rm. A new, closed  loop, system
is obtained by using a nonlinear feedback control law u = K(x, u)
(1.2) x(t) = f(x(t), K(x(t), (t))), y(t) = h(x(t)),
where K: R" x Rnm 4 Rm, u(t) e Rm. This system, with input u,
has different dynamics and inputoutput characteristics than (1.1).
Roughly speaking, the system, (1.1) is decoupled if for i = 1,..., m, the ith component ui of u effects only the ith component
yi of y. If the system (1.2) is decoupled by a control law K, we
say (1.1) is decouplable. In particular, if the input output map
of the system is described by yi = i( u1,..., um ), i = 1,..., m,
1
2
decoupling requires *( u1,..., m )  (u1),  1,..., m. The
control law K which decouples (1.1) is called a decoupling control
law. The concept of decoupling can be easily generalized for the
case y(t) e R1, where I > m and y is partitioned into m
subvectors. But, in this dissertation, we consider only the most
common case, I = m.
Some applications of decoupling theory are found in robotics
([Fre.2, Fre.3, Nij.5, Sin.4, Tar.l, Yua. 1). A simple illustrative
example is as follows. The rigid body equations of motion for a
mechanical manipulator with D.C. motor drives can be described by
(1.3) M(q)q * F(, q) = u, y q,
where q, u e Rm and M(q) is an (m x m) nonsingular matrix and we
have simplified the notation by not showing the explicit dependence
on t. Then, applying the nonlinear feedback control law u = M(q)u
+ F(q, q) leads to a simple decoupled linear system
(1.4) y
(1.4) q = u, y = q.
The system, (1.4) may be decoupled in a stable way by a linear
control law u = T ql + 2 q + r3 u, where u(t) e Rm is the new
closed loop input and',, Y2, 3 are appropriate (mx m) diagonal
constant matrices. If there are additional dynamics representing
actuators or structural flexibility, a solution of the decoupling
problem may not be so straightforward. This motivates a more
3
general and deeper investigation into nonlinear decoupling theory.
With respect to (1.2), there are four questions of obvious
importance:
(a) Under what condition, is decoupling possible?
(b) What is the class of control laws which decouple?
(c) What is the class of decoupled closed  loop systems?
(d) What is the correspondence between elements of the
classes mentioned in (b) and (c)?
If a given system can be decoupled but the decoupled system is
not internally stable, decoupling does not make sense.
Furthermore, the decoupled system may need to have desirable
input output characteristics. These problems can be fully
investigated only by characterizing the whole class of decoupling
control laws. Thus, question (b) is important in decoupling theory.
The questions (c), (d) are related to the structural aspects of
decoupled systems.
Decoupling theory was first developed for linear systems of
the form:
(1.5) f(x,u)' Ax + Bu, h(x) ^ Cx
(1.6) K(x, u)' Fx + Gu,
where A, B, C, F, and G are constant matrices with appropriate
dimensions. For question (a), Morgan [Mor.l] first presented a
concrete definition of decoupling with a sufficient condition for
decoupling. Then, a complete answer to question (a) was
established by Falb and Wolovich ( [Fab.l] ). The remaining questions
(b), (c), and (d) were answered first by Gilbert([Gil.l, Gil.2]).
4
Wonham & Morse considered the general case, I * m, using a novel
geometric approach ( [Mos.l, Mos.2, Won. I, Won.21).
The literature on nonlinear decoupling Is more recent. The
case which has been considered extensively is:
(1.7) f(x, u),(x) + I, f(x),,
(1.8) K(x,u) = <(x) + (x)u,
where f,: Rn  R", i  O,..., m, c: Rn  R", and y: R" R> I ".
For this nonlinear system, the theory is still incomplete compared
with linear decoupling theory. All of our following discussion
applies to systems and feedback control laws of the forms (1.7),
(1.8). Clearly, the system (1.3) can be written in the form (1.7)
if M(q) is nonsingular, q e Rm.
For question (a), the earliest works are [Naz. 1, Maj.l, Por.l,
51n 1] with [Cla.1, Fre. 1, Sh. 1] appearing later. These papers
present nonlinear versions of Falb and Wolovich's necessary and
sufficient condition for linear decoupling, where the definition of
decouping is based on the input  output behaviour of systems.
Later, authors consider decomposition of the above class of
nonlinear systems ([Isi. 1, Nlj.2, Res. 1]). Decomposition concerns
dynamic structure of the systems in state space. The system,
(1.1) Is decomposed If In an appropriate system of coordinates,
(1.1) appears as a system having m independent subsystems such
that for the ith subsystem, the input and the output are the ith
components of u and y, respectively. In other words, the system
5
(1.1) Is decomposed If there exists a mapping T: R" 4 R" such
that through the state transformation  T(x), (1.1) Is expressed as
(1.9) (t) ((t)(t) (t)),  1,..., m,
xm,(t) fl(x(t), t)),
where x = (x,.., m, xm ). If the system (1.2) is
decomposed by a control law K, we say (1.1) is decomposable.
The control law K is called a decompoing control law. In [Isi.l],
decomposition is called onintercting control Decomposition is a
strong definition for nonlinear decoupling. It is clear that
conditions for decomposition are also sufficient conditions for
nonlinear decoupling. In most of the papers on decomposition, the
philosophic approach is to generalize to nonlinear systems the line
of attack introduced by Wonham and Morse.
For question (b), some partial results are found in [Cla.l,
Fre.l, Maj.l, Sin.1, Sin.2, Sin.3, Sih 1]. The class of decoupling
control laws in these papers is given by linear or nonlinear
functions of outputs and their time derivatives. As will be
shown later, this is not the most general form of decoupling
control law. For question (c), (d), no results have been presented.
In this dissertation, we give precise definitions of decoupling
and decomposition. Then, we present various detailed results
concerning questions (a), (b), (c), and (d). This is our main
contribution. Next, we consider "approximate decoupling" for
systems with "fast" dynamics. Finally, we apply these results to
decoupled control of robotic manipulators.
6
Now, we describe in greater detail the major contributions
and organization of the dissertation. Chapter 2 contains the
general mathematical background on which the development in later
ctapters is based. In particular, some elements of differential
geometry are reviewed. For example, Lie algebraic tools are
Introduced because they play the same role in the treatment of
nonlinear systems that linear algebra plays in the treatment of
linear systems.
Chapter 3 is the main part of this dissertation. We begin
by defining decoupling ( Definition 3. 1.3 ) and decomposition
(Definition 3. 1. 5 ). Our definition of decoupling is based on the
concept of inputoutput map. It is an extension of Hirshorn's
definition of disturbance decoupling ( [Hir.2 ). An earlier origin
may be found in the work of Silverman and Payne([Sil.l]).
We give algebraic conditions for decoupling ( Theorem 3. 2. 1 )
and decomposition ( Theorem 3. 2. 2 ). Theorem 3. 2. 1 is a minor
extension of the results on disturbance decoupling in [Hir.2, Isi.l].
But, we believe ourproof is clearer and simpler. With additional
steps, Theorem 3. 2. 2 is implied by arguments contained in [Isi. I.
The conditions for decomposition are more complex than those for
decoupling. But, in Thorem 3. 2. 3, we present some conditions
under which two concepts are equivalent.
For question (a), we prove rigorously that a nonlinear version
of Falb & Wolovich's condition is both necessary and sufficient for
a system to be decouplable ( Theorem 3. 4. 1 ). This has been
considered in [Maj.l, Sin.l] but with unclear proofs. The nonlinear
version of Falb & Wolovich's condition is also a necessary and
7
sufficient condition for a system to be'locally decomposable'
(Theorem 3. 4 2 ). The proof is a refinement of an argument
contained in [Isi.ll An important implication of Theorem 3. 3. 1
and Theorem 3. 3. 2 is that decouplability and decomposability are,
under the hypotheses which they share, equivalent.
For question (b), we characterize the whole class of
decoupling control laws ( Theorem 3. 4 1 ) and decomposing control
laws (Theorem 3. 42 ). In the case of linear systems, (1.5), (1.6),
the characterizations reduce to a single result contained in [Gil.1].
Through an example( Example 3. 5. I ), we illustrate that while
previous work on the class of decoupling control laws may not
allow a system to be decoupled in a stable way, our characterization
of the whole class of decoupling control laws may. We show
that the class found by previous authors can be the whole class
of decoupling control laws only under very restrictive assumptions
(Remark 3. 4 7 ).
For questions (c), (d), first, we introduce a standard form of
decoupled systems ( Definition 3. 3. 1 ). It is a nonlinear version of
the form proposed by Gilbert([Gil.l]) in the case of linear
systems. Then, we show that a class of decoupled systems has
the standard form in an appropriate state representation (Theorem
3. 3. 3 and Theorem 3. 3. 4). For this class of systems, we obtain
answers to questions, (c), (d)( Theorem 3. 4 5). The underlying
idea is to characterize the whole class of the control laws which
decouple or decompose the standard form.
Chapter 4 concerns approximate decoupling. We neglect fast
dynamics of a system to obtain computationally simple control laws.
8
They decouple the simplified model but do not decouple the actual
system. It is shown that when the neglected dynamics are
sufficiently fast, the simplified law decouples the actual system
approximately" in a certain sense ( Theorem 4 2. 1 ).
In Chapter5, the results of earlier chapters are applied to
decoupled control of robotic manipulators. Two cases are
considered. In Section 5. 1, actuator dynamics are completely
neglected. In Section 5. 2, the dynamics of a significant class of
actuators are taken into account. Our general formulas give a
unified and generalized framework for previous results on the
decoupled control of robotic manipulators.
Finally, Chapter 6 contains a brief summary of the results
presented in the previous chapters and discuss some of their
possible extensions.
CHAPTER 2
MATHEMATI CAL BACKGROUND
In this chapter, we present the general mathematical
background on which our development in later chapters is based.
In Section 2. 1, general notation and definitions of differential
calculus are introduced. In Section 2. 2, some basic concepts of
differential geometry are introduced. Section 2. 3 contains some
theorems from differential geometry. Readers who are familiar
with differential geometry can use this chapter as a reference
for notation and proceed directly to the following chapters. For
full details, see [Boo.l, Die.l, Mun.l, Wag.l, War.l].
2. 1. General Notation and Definitions
Let Nl [0Q, 1,2,' ). Let p, q e N. Then, npp denotes
the set [j e N: p ~ j I q). For i e l,, M denotes the set
(j: j el but j ~ i }. The real line, its upper half line [0, 0)
are denoted by R, R+, respectively. The (p x p) identity matrix
is denoted by Ip. The transpose of a (p x q) matrix Q is QT.
In this paragraph, X, i, i are topological spaces. Let F
be a mapping from X into X ( if %  R, F is a function ). The
9
10
image of a subset U of X by F, denoted by F(U), is defined by
(2.1.1) F(U) [x e x = F(x), x e U).
The inverse image of a subset V of X by F, denoted by F'(V),
is defined by
(2.1.2) F1(V)  (x e X: x = F(x), 2 e V).
Let H be a mapping from P into x The composition of H and
F, denoted by HoF, Is defined by HoF(x) = H(F(x)), x e X A
A
mapping F: X  is invertible if there exists a mapping G: X
4 X such that F o G and Go F are the identity mappings on the
sets X, X, respectively. Since 6 is unique, it is called the
inverse mapping of F and denoted by F'. Let U be an open
subset of X If for each open subset V of X, the Intersection
un F'(V) is open in U, F is continuous on U. If F is continuous
on X and has a continuous inverse mapping F1, F is a
homeomorphism on X
A topological space X is Hausdorff if for each pair x, z of
distinct points of X, there exist open neighborhoods U, V of x,
z, respectively, that are disjoint. A topological space X is
connected if the only subsets of X that are open and closed in
X are the empty set and X itself.
The vector space of n tuples of real numbers with
componentwise addition and multiplication is denoted by R". The
11
element x s R" Is written as a column vector. The transpose of
x, denoted by xT, stands for Its expression as a row vector. Note
that R" with a norm I I Is a Banach space.
Finally, we Introduce some basic definitions of differential
calculus on Banach spaces, which are found In [Dle.l, Wag.11 From
now on until the end of this section, X,, are Banach spaces.
The set of all continuous linear mappings from X Into ~ is
A A
denoted by ( X; ). Then, It can be shown that b(X;X)
with Its Induced norm, III II sup ItxI; I xl I I Is a Banach
space. For simplity, In the rest of this section, the norms of
Banach spaces are Identically denoted by II. Let F be a
A
continuous mapping from an open subset S of X Into %. Let x,
s ScX. If there exists a v s S(X; X) such that
(2.1.3) llm IF(x)  F(xO)  v(x x) /I x  x = 0,
X4
x e S  (xJO
then, F Is differentiable at xO. The mapping v Is usually denoted
by DF(xo) and Is called the first derivative of F at xo since It Is
actually unique. When X RI" and X ^ R", DF(xo) Is a (q x n)
matrix and Is called the Jacobian matrix of F at xO. Then, the
rank of F at xO e X Is the rank of its Jacobian matrix at xo. If
F is differentiable at each xo s S, F is differentiable on S.
Then, DF is a mapping from S Into S(X; i,). If It is
continuous on 5, F Is continuously differentiable on S.
12
Suppose that F is continuously differentiable on S and DF
is differentiable at x0. Then, F is twice differentiable at xO.
The derivative of DF at xo is called the second derivative of f
at x0 and denoted by D2F(xo). If F is twice differentiable at
each xo e S, it is twice differentiable on S. Then, D2F is a
mapping from S into I(X; X;X)). If it is a continuous
mapping from S Into i(X;(X; X;i), F is twice continuously
differentiable on S. Inductively, we can define higher order
derivatives. The details are omitted. If F is p times
continuously differentiable on S, we write F 6 CP on S.
Particularly, when F is continuous (infinitely continuously
differentiable or smooth), we write F e C~( C' ) on S. If F e
Coo on S and at each xo e S, there exists a neighborhood U of xo
such that U is open in S and F can be expanded on U as an
infinite Taylor series, F is real analytic(CW) on S.
Now, suppose that X is the product space of two Banach
spaces X,, X2; X X x. For each xo ( a ) e X,
we can consider the partial mappings x, I  F(x,, a2) and x2 1
A
F(a, x2) of open subsets of X, and X2, respectively, into X. If
the partial mapping x, I  F(x,, a2) ( x2 I  F(a1, x2)) is
differentiable at al (a2 ), F is differentiable with respect to the
first( second ) argument at xo. The derivative of that mapping,
which is an element of I(X,; X) (B(X2; X)) is called the
13
first partial derivative of F at x0 with respect to the first( second)
argument and written as D1F(a,, a2) or (aF/X1)x=xo( D2F(a1,a2) or
(aF/x2)x=xo ). Inductively, we can define the second and higher
order partial derivatives. Details are omitted. Note that
DDjF(x, x2) e S(X; (; X) ), i, j e 12. Note that any bilinear
mapping in S(X; V(Xj;)) can be identified with a bilinear
A
mapping in B(X, xXj; X), i, j e t2. Therefore, DDjF(x,, x2) is
written (v, w) I  D,DF(x,, x2) [v] [w]. In particular when n,, n2 e
N, X,1 Rnl, X2 A Rn2, and; 4 R, we have for 1, J e,2,
(2.1.4) DDjF(x, x2) [v] [w] = w D(DjF(x, x2))T v,
where Di(DjF(x,, x2))T is an (nj x ni) matrix.
Now, let [0, L) be an interval of the real line R. Suppose
that there is a partition of [0, L) such that 0 = to < tI < < t
= L. Let F be a mapping from [O, L) into X. Let i 6 il.q
The mapping F has an extension ( F, U1 ) on the interval [ tol), t )
If there exists a mapping F1 from an open interval Ui into X such
that [t( i_), ti) c Ui and F,(t) = F(t), t e [t(i1) ti). The mapping
F is piecewise C~ on [ O, L ) if on each interval [ t) t( i ), ) i e Hl.
it has an extension ( F, Ui ) such that F. is bounded and C~ on U.
More generally, when on each interval [ t(j_,), tj), i se 1,a, F has
14
an extension (F., Ui) such that F, is bounded and CP (C~) [ CI]
instead of CO on Ui, F is piecewise CP (C" )[ CW] on [O, L). Note
that piecewise C' and C" mappings can actually be discontinuous
at the points tj. Thus, derivatives of F in usual sense are not
defined at the points ti. However, we will find it convenient to
define the kth derivative of F, DkF in the following way: for t e
[ t(i,_l t), DtF(t) = DF(t).
2. 2. Basic Concepts of Differential Geometry
A manifold X of dimension n, or n  dimensional manifold is
a topological space with the following properties:
(1) X is Hausdorff,
(2) At each p e X, there is a pair (U, S) such that U is
an open neighborhood of p and * is a homeomorphism from
U onto an open subset of R",
(3) X has a countable basis of open sets.
The pair (U, 4) is called a coordinate neighborhood or chart. Charts
(U, ), (V, p) are C~compatible if UF V nonempty implies that
composite functions ol, I o ~ are CO  diffeomorphisms of the
open subsets *(Un V)and YI(U V) of Rn. A smooth structure or C'atlas on a manifold X is a family A = ( U, 4 i of charts such that
(1)' The U. cover X,
15
(2)' For any c, p, the charts (U,, *) and (Vi, ^ ) are
C~ compatible,
(3)' The collection A is maximal: any chart (V,,) Co 
compatible with every (U, ) e A is itself in A
A smooth manifold is a manifold with a CO atlas.
If in the previous paragraph we replace "Cw and smooth" by
CW and real analytic", we obtain the definitions of C@ 
compatibility, C  atlas, and real analytic manifold instead of Co
compatibility, Cw atlas, and smooth manifold. Clearly, any
open subset of R" is a real analytic manifold.
Let X, X be smooth manifolds of dimension n, m,
respectively. A mapping T from X into X is Co if for each p
e X, there exist charts (U, )) of p and ( V, A) of T(p) with
T(U) c V such that the mapping =T  poTo  1 from +(U) into'V) is Co in the sense defined in Section 2. 1. The rank of T
at p is the rank of T at +(p) (see Section 2. 1 ). Note that
A
the rank is independent of the choice of charts. If T: X  X
is a homeomorphism and T'1:X X is Co, it is a COdiffeomorphism on X Suppose that n i m. If T: X  X has a
rank n at all points of X, it is an immersion of X in X If T:
X > is an immersion and one to  one on X, it is an one 
to  one immersion.
Let S be a set. An n  ary operation on S is a mapping
from S" into S. A system consisting of a set and one or more
16
n  ary operations on the set is an algebraic system or simply
algebra. An algebraic system Is usually denoted by <(X, f,., fk >
where S Is a nonempty set and ft,', fk are operations on S.
Given any point p E X, we define < C((p), +,  > as the
algebra of Co0 functions whose domain of definition includes
some open neighborhood of p. Here, the binary operations +,
are the usual addition, multiplication of two functions,
respectively. Any two functions are considered equal if they
agree on any open neighborhood of p.
We define the tangent space T(X) to X at p e X to be
the set of all mappings Yp:CO(p)  R satisfying, for any, lP 6
C(p), the following three conditions:
(i) Yp( +I) =Yp + YP.
(ii) Yp = 0 if * is a constant mapping,
(iii)Yp(+ I) = (Yp ) iIp) + (Yp) +(p),
with the vector space operations in T(X) over R defined
by (Yp + Zp) Yp + Zp, (aYp ) a(Yp ) for Yp,
Z e Tp(X) and for a e R.
A tangent vector to X at p e X is any Yp e T(X).
A cotangent space T *(X) to X at p e X is the dual space
to 7T(X) at p e X, defined by the set of all linear mappings dp
from Tp(X) into R with the vector space operations in 7T*(X)
17
such that for Yp e Tp(X), a e R, and d, p e 7'*(),
(i)' (d' +* p)Y, d Yp + pYp,
(ii)' (aofp )Yp = a pYp.
A cotangent vector to X at p e X is any p e Tp*(X).
A vector field Y on X is a mapping assigning to each point
p e X a tangent vector Ype p(X). A covector field or one
form d on X is a function assigning to each point p e X a
cotangent vector dp e T*(X).
Any function * from X into R defines a covector field,
denoted by d, on X by the formula:
(2.2.1) d Yp = Yp, p e X for any vector field Y on X
This covector field d_ is called the differential of * and d p, its
value at p, the differential of * at p. We may often write as
Y(p) ( o(p)) the tangent vector Y ( the cotangent vector d )
assigned to a point p e X by a vector field Y (a covector field
d ). Similarly, we often write d (p), Y4(p), dY(p) instead of
d, Yp, da V
The vector fields Yi, i e tn, on X are linearly
independent on an open subset W of X if at each p e W, the
18
tangent vectors (Y)p, I el, are linearly Independent. The covector
fields fi, I e 6t,l on X are linearly independent on an open
subset W of X if at each p e W, the cotangent vectors (o'j), i e
1,K are linearly independent. Note that if the vector fields are
not linearly independent at a point p eX, they are not linearly
independent on any open neighborhood of p. Let 1, 1e t1,b be
Clfunctions from X into R. Let *4( 1..., ) The functions
i, i eln, are functionally independent ((Gou. 1, Hil.1]) on an open
set W of X if there does not exist any C'  function': *(W)  R
such that'Po (x)= O, x e W but P is not identically zero on +(W).
It is easy to show that if d *, i e tlk, are linearly independent
on W, then,,, 1 e tt, are functionally independent on W. But,
the converse statement is not necessarily true. A simple example to
show this is l(xP,x2) = x1 sin x2, *2(x,,x2) = x1 cos x2, (x,, x2) e R2
Let T be a Co mapping from an n  dimensional smooth
manifold X into an m dimensional smooth manifold ~. For each
psX, it induces a linear mapping:Tp (X) 4 TT(p)(X), defined by
(2.2.2) T=p(Y) = Yp( oT) for C C(T(p)), Yp 6 sp(X).
The mapping T% is often called the differential of T at p and
denoted by dTD.
19
The dimensions of the tangent space Tp(X) and the
cotangent space,tX() to X at p e X are the same as that of
X Therefore, at each p e X, there is a set of n linearly
independent tangent vectors which span Tp(X). It is called a
basis of T(X). Since'(X) is the dual space to Tp(X), the
basis of (X) is uniquely determined by a basis of Tp(X). So,
it is called the dual basis of (X). Suppose that X is an
open subset of R". Let (x,, *, x ) be the coordinate vector.
Then, the n vector fields a/ax,, 1 e n,, are linearly independent
on X and at each point p e X, ((a/ax,), i e 1, ) is a basis of
T (X). We call this basis a canonical basis. The canonical
dual basis dxi i e,, ) is determined by
(2.2.3) (dx)p (a/axj)p ^ (axi/aXj)p = 6j, p e X, i, j e ^
where 6..i 1 if i = j, S. * 0, otherwise.
Now, consider an n dimensional manifold X which is not
necessarily an open subset of R". Let (, U) be a chart at p e
X Then, by the definition of smooth manifold and the
observations in the previous paragraph, it follows that there
exist vector fields E,, i e,l on U such that
(a) E,, i e H,, are linearly independent on U,
20
(b) At each q e U, ((E,)q, 1 e 1,,n) is a basis of Tq(X).
One possible choice is
(2.2.4) (Ej)q ^ f*q)((a/ax)q)), i e,, q e U.
Correspondingly, there exist n covector fields W, i e t,,,on U
such that
(a)' (i)q (E)q = j, q e U, i,j es t,l
(b)' At each q C U, (i,)q, i 6e 1, ) is a basis of t'(X).
Using the above notations, any vector field Y and covector
field a' on X can be locally represented, respectively, by
In
(2.2.5) Yq a I a,(q)(E,)q, q e U,
(2.2.6) o ^ b (q) (i)q, q s U,
q (
where a,, b, are functions from U into R. When X is an open
subset of R", any vector field Y and covector field d on X are
globally identified by (2.2.5) and (2.2.6) with E, = a/axi, Jw = dx,,
i e tl.^ When the a,, b1 are C0 on U, Y and f' are,
respectively, a C~ovector field on U and a C~~ covector field on
U. If at each p e X, there exists a chart (U, ) such that Y,
rf are Co on U, they are respectively a C0 vector field on X
and a C~o covector field on X5
If in the previous paragraph, Co~ is replaced by Ca and X
21
is a real analytic manifold, then Y, of are a C  vector field, a
C  covector field, respectively.
A Lie algebra is a vector space L over R which, in addition
to its vector space structure, possesses a product [, ] satisfying
the following properties:
(1) L is closed under the product: [Y,Z] e L If Y,Z e L,
(2) The product is bilinear over R: for a, b e R and for X, Y,
Z e L,
(2.2.7) [aX + bY, Z] = a[X, Z + b[Y,Z],
(2.2.8) [X, aY + bZ] = a[X,Y] + b[X,Z],
(3) The product is skew commutative:
(2.2.9) [Z,Y] = [Y,Z] for Y,Z e L,
(4) The product satisfies the Jacobi identity:
(2.2.10) [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0.
Let C~(X) (Cw(X)) be the set of all Co~ (C ) functions
from X into R. Let V"(X) ( VW(X)) be the set of all Co (C")
vector fields on X Then, V~(X) is a vector space over R
and a module over C~(X). We define a product [Y, Z ] of Y, Z e
V~(X) by
(2.2.11) [Y, Z ] = Y, (Z4)  Z, (YV), * E C (p), p E X
22
The product defined by (2.2.11) is called the Lie bracket of Y, Z.
Clearly, it satisfies the properties (1)  (4) in the previous
paragraph and so V*(X) with the Lie bracket is a Lie algebra.
Let (U, ) be a chart at p e X Then, for Y  2 ai()E,, Z
X bi( )E,, [Y,Z] can be expressed locally on U as
i=i
n 1. n
(2.2.12) [Y, Z q  (2 aj(q) (Ej)q b  2 b(q) (E)q ai (E,)q, q e U.
i "=I,i=j, j=1 jq
Of course, when X is an open subset of Rn, this expression
holds globally on X with E= a/ax, i e,tl.
If a subset E of V'(X) is closed under the Lie bracket, it
is involutive. A subalgebra of V((X) is an involutive linear
subspace of V((X) over R. A distribution A on X is a mapping
which assigns to each p e X, a subspace Ap of T(X). If Y is a
vector field such that Yp eAp, p e X, we write Y e A on X A
distribution A on X is involutive if for all vector fields Y, Z
such that Y, Z e A on X, [Y, Z] e A on X If dimension of Ap
is k, p e X, A has a dimension k on X A distribution A on
X is CO~ (C ) if at each p e X, there exist an open neighborhood
U and k(p) linearly independent C~ (C )vector fields Y,, i e
Htt, on U such that at each point q e U, {(Yj)q, i e ttk spans
A q Note that a subspace of V~~(X) ( e.g. a subalgebra of V~~(X))
generates a C~  distribution on X. Thus, the subspace may
23
have a dimension on X Note this dimension is a pointwise
concept, not a function space concept. Note that in general, the
dimension of a C0 distribution A may not be defined on X (the
number of basis vectors for Aq may depend on q ). Let V*(X)
be the set of all Co covector fields on X The C~codistribution ALof a Ca distribution A on X is defined by
(2.2.13) A [(o' e (X): dYp = 0 Yp e Ap ), p e
Let Z be a vector field on X A distribution A on X is
Z invariant if [Z, Y] e A whenever Y e A. The codistribution
A of a distribution A on X is Z invariant if for any e
C((X), d0 e A always implies that dZ~ 6 A'. Simple
calculations show that a distribution A on X is Z invariant if
and only if the codistribution A of A is Z  invariant.
Let T be a C.0 mapping from an n  dimensional smooth
manifold X into an m  dimensional smooth manifold X. A
vector field Y V~~(X) is T related on X to a vector field Y s
V"(X) if
(2.2.14) YT( = Yp(~oT), 6 CI(T(p)), p e X.
A function * e IC(X) is T related on X to a function 4 e
C~(X) if
(2.2.15) (p)= oT(p), p
(2.2.15) Y(p) = Y 0oT(p), p c %
24
Other definitions such as integral curve and Lie derivative
will be Introduced in the next section.
2. 3. Some Fundamental Results
We state some well known results without proofs. As
in Section 2. 2, X, ~ are smooth manifolds of dimensions n, m,
respectively. Using the definitions in Section 2. 2, the following
facts may be easily verified.
Fact 2.3.1 ([Boo.1],p. 155). For, l e C~(X) and Y, Z e
V'(X), the following equality holds for all p e X
(2.3.1) [ YY, Z lp = (P) ()[ Y Z + ) (,  Ip)(Z) Yp. 0
A A
Fact 2.3.2([Boo.1], p. 154). If Y,Z e V(0() are Trelated on X to Y, Z e V~(X), respectively, then [Y, Z] is Trelated on X to [ Y, Z ]. 0
Fact 2. 3. 3. If e C~(X), Y e V*~(X) are T related on
X to E C~~(X), Y e Vo(X), respectively, then Y is T related
on X to Y4. D
Let X 6 V~(X). If a C~o mapping F from an open
25
interval J of R into X satisfies
(2.3.2) Ft((a/3r)) = XF(o, t e J,
the mapping F is an integral curve of X. Customarily, we write
F(t) instead of F*t((a/ar)t). The following theorem is concerned
about the existence of integral curves for a given vector field X.
It is essentially a restatement of the existence theorem for
ordinary differential equations.
Theorem 2. 3. 1 ([Boo. 1], p. 132). Let X e VO(X). Then, for
each p eX, there exist an open neighborhood U of p, a real number
6(p) > O, and a Co~ mapping X: ( , 6) x U  X satisfying
(2.3.3) eX(t,q) = Xx(tq)j, X(o, q) = q, q e U. 0
When we emphasize that BX(t, q) is a function of q for a
fixed time t e R, we may write &\q).
Theorem 2. 3. 2 ([Boo. I], p. 133). Let X e VO(X). Then,
for each p e X, there is a maximal open interval I P {a(p)< t <
b(p) ) containing t = 0, on which the integral curve BX(, p) of X
passing through p at t = 0 is defined. Moreover, the integral
curve 8x(, p) of X passing through p at t = 0 is unique on ID. ~
26
By Theorem 2. 3. 2, for each t e R, we can define a subset D:<
of X by
(2.3.4) DI4 { Lp e X:t e Ip).
A vector field X on X is complete if l = X for all t e R.
Theorem 2.3.3 ([War. 1], p. 37 ). Let X e V~'(X). Then, the
following properties hold.
(i) f is open for each t e R,
(ii) U Dt = X,
t>o
(iii) For each t e R, B is a C~ diffeomorphism from Li
onto OX with inverse 8X
t 4'
(iv) On the domain of 8Xo 8,
5 1'
(2.3.5) 8o O8 = 8X 0
S I t+S
Vector fields can be differentiated with respect to a vector
field. The vector field LyZ, called the Lie derivative of Z with
respect to Y at p e X is defined by
(2.3.6) (LyZ) = lim ((8tY)*((ZY)  Zp i / t.
The following result connects the Lie
The following result connects the Lie bracket with the Lie
27
derivative we defined just above.
Theorem 2.3.4 ([Boo. I], p. 153 ). Let Y, Z e V'(X). Then,
(2.3.7) (LyZ)p = [Y, Z ], p e X
By Theorem 2. 3. 4, we shall confuse LyZ with [Y, Z] and
define the successive Lie brackets of Y, Z e V**(X) by
(2.3.8) Ly Z [Y,L(k)z], k e,
where
(2.3.9) LyZ  Z.
Next, consider
Theorem 2. 3. 5 ( Cambell  Baker  Hausdorff Formula ). Let Y,
Z be C" vector fields on X Then, at each p e X, there exists a
real number 6> 0 such that
(2.3.10) (8) pZp = I (( t)/ k (YZ)) t e ( 6 ).
Although this results appears in many places, it is remarkable that
no proof is given in the standard references. The proof follows
from Theorem 2. 3. 4 and, while not exactly obvious, is not too
28
difficult. Note that if Y. Z are C", this formula does not
necessarily hold. Most of all the results which we derive in
future, where real analyticity is required, come from Theorem
2.3.5.
Now, we state two Inverse Function Theorems, the Constant
Mapping Theorem, and the Frobenius Theorem.
Theorem 2.3.6 (Local Inverse Function Theorem, [War. 1],
p. 30 ). Let T be Co' ( C ) mapping from X Into C Suppose
that at a point p e X, T% is an isomorphism from T?(,) onto
T,)(X). Then, there is an open neighborhood U of p such that T is a
CO ( CI)  diffeomorphism from U onto the open subset T(U) of X.
Theorme 2.3.7 ( Gui. 1, p. 18 ). Let T be a C (C) 
mapping from an open subset W c X into X Then, T is a Co
( C)  diffeomorphism from W onto T(W) C X if and only if
(1) At each point p e W, T% is an isomorphism from Tp(X)
onto Tp(X),
(2) T Is one  to  one on W. 0
Theorem 2. 3. 8 (The Constant Mapping Theorem, [War. 11, p. 18 ).
Let T be a Co mapping from X into KX Suppose that X is
connected and T=.  0, p e X. Then, there exists a constant c e
29
R such that
(2.3.11) T(x) = c, x e X. 0
Before we state the Frobenius Theorem, we define an
(immersed) submanifold and integral manifold. Let W be a
subset of X W is an (immersed) submanifold of X if there
exist an r  dimensional smooth manifold N and an one  to  one
immersion T: N  X such that r ~ n and W = T(N). An
integral manifold of a Co0 distribution A is a connected
submanifold E of X with the property that Ap = Tp(E), p e E
For a more general definition of integral manifold, see [Boo.l].
A Co distribution A on X of dimension k is completely
integrable on X if each point p e X has a chart (U, ) such
that the k vector fields E,  d '(a/ax,), i e l,k are a local
basis on U for A, where x,,., xn are the local coordinates.
In this case, an integral manifold E of A through q e U is
(2.3.12) ~E '({ x e +(U): xk.1 = ak'.', Xn = a ),
where ( a1,, a,) ^ (q).
Theorem 2. 3.9 ( Local Frobenius Theorem, [Boo. I, p. 159 ). Let
A be a C  distribution on X with dimension k. Then, A is
involutive on X if and only if it is completely integrable on X. D
CHAPTER 3
NONLINEAR DECOUPLING THEORY
This chapter contains results on decoupling and decomposition.
In Section 3. 1, further notation and definitions on systems are introduced
on the basis of the general notation and definitions in Chapter 2.
In particular, the precise definitions of decoupling (Definition 3. 1. 3)
and decomposition (Definition 3. 1. 5) are proposed. In Section 3. 2, we
present the results on decoupling ( Theorem 3. 2. 1 ) and decomposition
(Theorem 3. 2. 2 and Theorem 3. 2. 3 ). In Section 3. 3, the results on
decomposability ( Theorem 3. 3. 1 ), decouplability ( Theorem 3. 3. 2 ), and
the standard decomposed system ( Theorem 3. 3. 3 and Theorem 3. 3. 4)
are presented. In Section 3. 4, we characterize the whole class of
control laws which decouple or decompose nonlinear systems ( Theorem
3. 4 1  Theorem 3. 4 4 ). Then, for a class of nonlinear systems,
we discuss the class of closed  loop decoupled systems generated
by the whole class of decoupling control laws ( Theorem 3. 4. 5 ).
In Section 3. 5, three examples are considered which illustrate the
significance of the results developed in the previous sections.
Section 3. 6 makes comments on the results discussed in this chapter.
3. 1. Definitions
Recall the system (1.1), (1.7) of Chapter 1. We now give
30
31
It an alternative abstract formulation. For each I e n,, we
may view f1 as the coordinate representation of a vector field X,
on R" in the canonical basis ( a/xj, j e In,, such that
(3.1.1) X ~ Z f..() a/axj,
where fij is the jth component of fi, j e l,,. Then, we can
write (1.1), (1.7) as the vector field representation:
(3.1.2) x = F( x, u) ^ Xo(x) + 2 X,(x) u, y  H(x).
Here, x(t) e R", x is interpreted as x(t) = x*/'r)t), u,(t) e R is
the ith component of u, and H = h. Conversely, suppose that in
(3.1.2), Xi, i E o, are vector fields defined, more generally, on
an n  dimensional manifold X Then, at each p e X, there exists
a chart (U, ) such that in the coordinates *, the system (3.1.2) has
the form of (1.1), (1.7). We shall denote by F,H, X) the system
(3.1.2) defined on an ndimensional manifold K Its local representation
(1.1) defined on U is denoted by (f,h, U. Note that if X is an
open subset of R", then h = H and U = X Through the vector
field representation, we can tackle abstract systems defined on
manifolds which are not necessarily an open subsets of RI. Moreover,
as will be seen later, the vector field representation of the system
gives an efficient notation for handling the complex differentiations
involved in our developments. Also, it is easier to compare our
32
results with results In the prior literature. We denote by Yi, hi, Hi
the ith scalar components of y, h, H, respectively.
Let Us be the set of all piecewise Co  mappings from R+
into Rm. We say the system (F, H, X is smooth if
(I) Xs e Vi(X), 1e to,
(I) u e Us,
(iii)H: X  Rm is Co0,
(iv) X is a n dimensional smooth manifold.
To simplify our definitions, all systems considered in this section
are assumed to be smooth. At the end of the section, we will
indicate the appropriate extensions to real analytic systems.
Consider the local representation (1.1), (1.7) of (3.1.2). For
u eUB0 and t eR+, f(, u(t)) is C. On the other hand, for x 6X,
f(x, u()) is piecewise C'. These observations with well known
results ( [Hal.l, War.1, Var.l]) on the existence of solutions of
differential equations imply the following. For each x(O)  xo e
X and each u e U", there exists a maximal interval (0, L) e R+,
L = L (x0, u), such that (3.1.2) has a unique solution x [ O, L)  X
which is continuous but piecewise Co. Both x, y are not
differentiable in the usual sense but they are differentiable in the
peculiar sense discussed in Section 2. 1. As will be seen later,
some proofs of our results utilize piecewise constant inputs.
This is the main reason for the introduction of piecewise
differentiability in Section 2. 1 and the set Us in this section.
Define the set If by
33
(3.1.3) I"4 [(y,L): L > 0 and y: [O,L) > Rm Is continuous
and piecewise C ]).
Then, we can view the input  output behavior of smooth system
(F, H, X) as a mapping O from Ut x X into AI'. Similar
concepts are found in [Gil.3, Sus.21. Explicitly, we write (y, L)
 ( (u, xO), L (u, x) ) for an input u and an initial state x(O) ^ xO.
Let xO e X and u,'u U~0. Then, since L(xo, u), L(xO,'u) are
not necessarily equal, the comparisons of the outputs y =4 (u, xo),
y = ( u, x ) are restricted to their common interval [ 0 L ), where
L ^ min. L(xo, ). L(xo, u)). For Instance, we write *( u, x )
(u, x ) if they are equal on [ O, L). Similarly, we write u =
if they are same on [0, L).
The following definition concerns state transformations
between systems.
Definition 3. 1. 1. Suppose for two systems (F, H, X ),
(F, H, X, there exists a C" diffeomorphism T:X 4 X such that
A
(i) Xj is T  related on X to Xi, i e no,
(ii) Hi is Trelated on X to Hi, i e n,,.
Then, (F, H, X) is T  related on X to (F, H, X. O
The intuitive idea of this definition is that we obtain
F(x, u), H(x) from F(x, u), H(x) by the'substitution' of variables x
34
= T(x), u * u. See Fig. 3. 1. 1 for a schematic representation.
The definition yields the following obvious conclusion. if
([, H, } Is T related on X to (F,H, X, then for any input u e
Ut and any Initial state x(O)  xo e X,
(3.1.4) ( u, xo )  ( u, T(x) ).
A definition similar to Definition 3. 1. 1 is found in [Sus.2].
Next, we introduce a general relation between systems, which
takes into account both state and input  feedback transformations.
Let T, <, p be mappings from X into C, Rm, and Rmx m
respectively, such that P(x) is nonsingular, x s X Define a
mapping J: X x Rm > x Rm by
(3.1.5) J(x, u)  T(x) j, (x, u) e X x Rm.
[8(x)]l(x) + [p(x)]l u
x = T(x)
r —~~....... —— ~ —~i r'..... —— I
X X
u — I F, u)  T  Tl t — H y
x= F(j, U) y y= ()
_i _ _ __ __ _ J,  I
Figure 3.1.1. {F, H, X} is Trelated on X to {F,H,X}
35
We often write J  I(, p, T). We denote by (F, H,X)4I the
feedback system of (F, H,X) corresponding to a control law u(x) + p(x) u In other words, (F, H, YX) stands for the system
x F(x, u) i F(x, (x) + (x) u), y  x) H(x). Or, ( F, H, X )]1
= (F,, ). A control law u = <(x) + (x) 0 is smooth if <: X 
Rm and p: RX" " are Ca. All control laws considered in
this section are assumed to be smooth.
Definition 3. 1. 2. Suppose there exists a Codlffeomorphism
J: X x Rm 4 x Rm defined by (3.1.5) such that (F, H,) is Trelated on X to the system (F, H, X 1. Then, (F, H, X is Jfeedback related on % to { F, H, X). 0
The intuitive idea of this definition is that we obtain
F(x, u), H(x) from F(x, u), fkx) by the'substitution" of variables x
= T(x), u = [(x)]1( u  (x)). See Fig. 3. 1.2 for a schematic
representation. The definition is a nonlinear version of the
Control law equivalence used for linear systems in [Gil.1].
Similar definitions are found in [Bro.1, Hun.1, Hun.2, Hun.3, Isi.2,
Mey.1, Jak.1, Sur. 1, where the systems do not have outputs, i.e,
they are a pair (F, X).
The J  feedback relation Is actually an equivalence relation
on the set of all smooth systems defined on n  dimensional
36
smooth manifolds. Consider three systems (F,H,X), (,fi, Xi,
( F, H, X. Suppose (F, H, X) is J  feedback related on X to
[F,H,X) and (F,H,X is 3 feedback related on X to (F, H,.
Then, it is easy to see that (F, H, X is Jfeedback related on
X to (F, H, X, where JJ^oJ. Thus, the Jfeedback relation is
transitive. It is obvious that the J  feedback relation is
symmetric and reflexive. Two systems belonging to the same
equivalence class are the same with respect to what can be
accomplished by feedback. This fact motivates much of our
later work.
In order to make precise definitions of decoupling and
decomposition, the following technical details are needed. Let
xo X and i e Sl. Let 0. be the ith component of ~.
x = T(x)
— ____ — _
1. — I — I — 1 I —— 3
U  x x_
U = F(x, u) T 1  H Y
^=F( ) J= T,I,)
Figure 3.. 2. F, H, X is J  eedback reat
Figure 3. 1.2. (F, H, X] is Jfeedback related on X to ( I H X
37
Suppose ( U, xo ) * *( i, xo ) for all inputs u, T e U' such that
u Zi. Then, y, Is decoupled at xO. If yi is decoupled at
every xO E X, yj is decoupled on X. A similar definition for
disturbance decoupling is found in [Hir.2]. The intuitive idea of
decoupling for y, is that y, is not'connected' to uj, j e ft. If
y, is decoupled for i e,,, the system is decoupled. The
following definition makes this notion precise.
Definition 3. 1.3.
(1) (F, H,X) is decoupled at x eX if yi is decoupled at x0, i e
t'~,l,. If (F,H, X is decoupled at each x ea X, (F, H, X is
decoupled on X
(2) (F, H, X) is decouplable at x0 e' if there exists a control
law u = ((x) + P(x) iu such that (F, H, X, )} is decoupled at xo.
( F, H, X) is decouplable on' if there exists a control law u =
<(x) + (x)3( such that (F, H,X Y is decoupled on X 0
For some applications, we may need a stronger definition
of decoupling. Let i e ttl, and xo e X Let i.(, Xo).i(^, x0) for all inputs i, tu s It such that ui u but u. = u, j
e f,. Then, y, is connected at x0 to u,.
38
Definition 3. 1.4.
(1) If F,H,X) is decoupled at xo e X and y, is connected at xO
to u, i 6 Blin, (F,H,X is inputoutput decoupled at xO. If
(F, H, X) is input  output decoupled at each xO e X, it is inputoutput decoupled on X
(2) (F,H, X) is inputoutput decouplable at xO e X if there
exists a control law u  c(x) + P(x) 0 such that (F, H, X )}' is
input  output decoupled at xO. (F, H, X] is input  output
decouplable on X if there exists a control law u = <(x) + p(x)
such that F, H, X ])b is input  output decoupled on X O
These definitions of decoupling and decouplability are based
entirely on the input  output maps for the systems. There is a
different concept of decoupling, which is based on the structural
forms of state equations. For this idea of decoupling, we use
the term decomposition
Definition 3. 1.5.
(1) (F,H, X is decomposed at xO e X if there exist: (a) an
open neighborhood ~ of xo; (b) an open subset X of R"; (c) a
Cr diffeomorphism T: > X, (d) Integers s, 2 1, i e.,,
7an +1
ands..~ 2 0 satisfying n = E sj; and (e) a system (F,H,X]
39
which is T related on E to (F, H, ) such that Its coordinate
representation (,,X) has the form
(3.1.6) f() + gj(x1)u, y hf(x), i ean,
xM+ fm+() +
where xi(t) e R'A, i e H,,,, and x' (x,, *, X1 x+ ). If E
 X In the above statement, [F, H, X) Is decomposed on X
(2) F,H, X) is decomposable at xO e X if there exists a control
law u cc(x) + (x)i such that (F, H, X )1 is decomposed at x.
(F, H, X is decomposable on X if there exists a control law
u = c(x) + p(x) such that (F, H, X ]1 is decomposed on X. 0
Note from these definitions that if (F,H, X is decomposed
at xO e X, then there exists an open neighborhood E of xo such
that [F, H, E) is decoupled on E The converse statement is not
necessarily true. It is obvious from [Gil.l] that a linear
system is decoupled on R" if and only if it is decomposed on R".
In Section 3. 2, we show that if (F, H, X) is real analytic
and is decomposed at each xO e X, it is decoupled on X
Unfortunately, it is not clear that the same result holds for
smooth systems.
Definitions similar to Definition 3. 1. 5 are found in many
papers including [lsi.1, Nij.2, Res.l]. Some of these papers use
the terminology noninteracting control for decomposed control.
40
Thus, papers which concern noninteracting control apply to
decomposition (not to decoupling).
Next, we define some distributions based on the smooth
system (F,H,XI. Let L~((F,HX)) LX,, ks ni,., ltI,, l,
where xX, X. Define L( F,H,X)) as the smallest subalgebra
which contains ~( ( F, H, X ) ). We say ( F, H, X satisfies the
controlability rank condition If
(3.1.7) dimension of b( (F, H, X)) = n, p s X
For each i e n,,,, let
(3.1.8) A( (FH,X)) t ( LXILXi LX.X: r (r o, 1),
r e l,, k a noa, and j e J ),
where LXLx Li Xj ^ X; if k=O. Define Ai({F,H,X)) as the
smallest subalgebra containing A( (F, H, X)).
Some insight about these definitions may be gained by
considering the linear system (f, h, R"n in (1.5). Let [/ax,..
a/ax,] be the n row vector of the vector fields a/ax,, 1 e tL,.
Recall that for w e R", W ^ [a/ax a/ax,] w = w, a/ax, s a
vector field In V~(R"), where wi is the ith component of w.
Let BI be the ith column of B. Then, the distributions L, A,
take on familiar forms:
41
(3.1.9) ( (F, H, X)}) * span ([t /x, * * a/ax,] AkB,,
1 e,,,.m k e t,o L,
(3.1.10) A( { F, H, X )  span [d/ax, * * /axn] A^j,
j e Mf, k e t0, ).
We conclude this section with definitions of invertibility,
reachability, and the precise concept of real analytic systems.
Let x eK ({F,H,X) is invertible at xo if *( u, xO) ( u, x )
for all distinct inputs u, Zu e . If (F,H, X) is invertible at
each xo e X, it is invertible on K Similar definitions appear
in the literature on invertibility of systems (see for instance,
[Sil.l, Hir.t] ).
We say([Her., Sus.l]) that x, e X is reachable from xO e
X at time t, > 0 if there exists u e U1 such that the solution
of (3.1.2), x(t) e X, t e [0, t, ], x(O)  xO, satisfies x(t,) x,.
We denote by.L( xo, t, ) the set of all points in X which are
reachable from xO e X at t t,.
If Co in the definitions of the sets Ut, ~ is replaced by
CW, we obtain the definitions for the sets UW, te. Similarly,
if V"(X), UI, C','smooth manifold" in the definitions of smooth
system and smooth control law are replaced, respectively, by
VW(X), tUt, CW, "real analytic manifold", we obtain the definitions
of real analytic system and real analytic control law.
42
3. 2. Decoupling and Decomposition
By the definitions of decoupling and decomposition, it is
clear that a decomposed system Is always decoupled. In this
section, we show that the conditions for decomposition are more
complex than those for decoupling. But, we present some conditions
under which the two concepts are at least locally equivalent.
To state our results a variety of assumptions are needed.
To simplify the presentation we list them together here.
(A. 1) The system F,H, X) is smooth,
(A. 1)' The system F, H, X) Is real analytic,
(A 2) For each constant input u(t) 6 RP", the vector field F(, u)
is complete,
(A. 3) The system F, H,X) satisfies the controllability
rank condition on X,
(A 4) The codistribution t( F, H,] ) has constant
dimension p, 2 1 on X, i e ttl,.
We begin by giving a necessary and sufficient condition for
decoupling.
Theorem 3. 2. 1. Suppose that (Al)', (A2) are satisfied.
Then, (F, H, X is decoupled on X If and only if
(3.2.1) dH, e ( UF,H,X)) on X, i e,m. D
43
Theorem 3. 2. 1 is important because it gives an algebraic
condition for nonlinear decoupling. Before presenting its proof,
some discussion of the Theorem may be useful. The condition in
(3.2.1) simply requires that YH, = 0 on X for all Y e A,(( F, H, X ).
As will be seen in the examples of Section 3. 5, the distribution
A, is, in most cases, spanned by a finite number of vector fields.
Thus, (3.2.1) does not necessarily require an infinite number of
calculations. Suppose that (F,H, X) is a linear system, (1.5)
such that X = Rn. Then, by (3.1.10), the condition in (3.2.1)
becomes
(3.2.2) CABj = 0, k e n. 1, e ns l, If j,
where C, is the ith row of C. It is not difficult to show, by
using the well known expansion for (sIn A )' ([Gan.l] ), that
(3.2.2) is a necessary and sufficient condition for the Laplacetransform transfer function matrix of the linear system to be
diagonal. This is the definition of linear decoupling given in
[Gil.l] and it is equivalent to saying that the (linear) inputoutput map is diagonal. For other, essentially equivalent,
definitions of linear decoupling, see [Fal.1, Sil.l, Won.2].
We begin our proof of Theorem 3. 2. 1 with the following
two lemmas. The first lemma is a well known result on the
reachable set, R( x0, t ).
44
Lemma 3. 2. 1. (Theorem 4. 5 in [Sus. 1]) Suppose that (A. 1)',
(A.2) are satisfied. Let x e X and t> 0. Let I(L [( F, H, X ), x,)
be the largest integral manifold of L((F, H,X)) passing through a
point x, e R( x0, t ). Then, R( x,, t) c I( L (F, H, X)), x1 ).
Moreover, the interior of t( xO, t) relative to I( U( F, H, X ]), x )
is dense in R.( x, t) and not empty. O
To state the second lemma, we need the following notation.
Let i e,m,. Define a multiindex I, by any finite sequence of
integers taken from on, such that at least one of its elements
must belong to the set,1. For such a multiindex Ii i ( i 2,
* i, i), let XIj be defined by
(3.2.3) Xji X X X.. X
Then, for each 1 e e.m, define
(3.2.4) D (i a (: aY ae R and. is any finite collection
TEI
of multiindices Ii}
Using these notation, we can state the following result.
Lemma 3. 2. 2.
(i) Let 4 e C~(X). Then,
45
(3.2.5) d e 4i ( F, H, X) on X
if and only If
(3.2.6) 0o = 0 on X, e D,( (F, H, X)).
(ii) ( F, H, X ) is Xo  invariant and X;  invariant on X
Proof. Consider (i). It is clear that (3.2.6) implies (3.2.5).
We prove (3.2.5) implies (3.2.6) by induction. By (3.2.5), we have
(3.2.7) Xj = XjXo  XjXi  0 on X if j e;,.
Suppose that for k = 0, 1,',L,
(3.2.8) XjXi X.. Xi = 0 on X, j e,, ire o, i).
Then, by (3.2.5) and (3.2.8),
(3.2.9) XjXi X Xjl Xij+I, I (f1)(LX LXj. LX, Xj) 0,
if j e, and r 6 (o, ).
Thus, we have for any k e nos,
(3.2.10) jXixi: Xi = 0 if J e n and ir e (o,i).
From this, (3.2.6) follows immediately. Part (ii) is an easy
46
consequence of (i). 0
Note that A, " Dj because, is not a set of vector fields.
Now, using these Lemmas, we prove Theorem 3. 2. 1.
Proof of Theorem 3.2. 1. First, assume (3.2.1) holds. Fix i
8 ilj. Define a multiindex J, by any finite sequence of integers
taken from ( o, i. For such a multiindex Ji (i, i, 2, ik, define
(3.2.11) X1 ) Xi Xi Xj Xi,
(3.2.12) yJ_ = Xj; Hi(x),
where yj, = Hi(x) if Ji = 0. Let tY be the set of all Yjh
defined by (3.2.12). Differentiating yj,(t) with respect to t in
the sense described at the end of Section 2. 1, we obtain, by
(3.2.1) and Lemma 3. 2. 2,
(3.2.13) yji(t) = Xj1 Hi(x(t)),
ji(t) = Xo Xj Hi(x(t)) + 2 uj(t) X Xj Hi(x(t))
 XO Xj, Hi(x(t)) + ui(t) Xi Xji Hi(x(t))
yJO(t) = X Xj Hi(x(t)) + ui(t)Xi X, XXj Hi(x(t)) +
u,(l1(t) X1 Xj, Hi(x(t)) + ui(t) X, Xi Xo Xji Hi(x(t)) +
(ui(t))2 X XXj Hi(x(t)),
47
Note that
(3.2.14) yj(t) is a finite linear combination of some y e6 J
such that each coefficient is 1 or some monomial in u,
and the time derivatives of u1 at t.
Let u, U be any two inputs belonging to U" such that
(3.2.15) Ui  u.
Let x(O) ^ xo e X. We shall denote by x, x, the solutions of
(F, H, X corresponding to u, u, respectively. Similarly, yj in
(3.2.12) corresponding to R, x are denoted by y., yj,, respectively.
Let L min (L( x, u), L( xo, ) ). Since u,'U are piecewise real
analytic, there exists a partition of [0, L) such that 0 = to < t,
< t2 <... < tr = L and on each interval [tj t(j+l)), j e o(),
both u and Z' are real analytic. Then, by (A1)',
(3.2.16) yj, y~j are continuous and piecewise C" on [O, L) such
that on each interval [tj, t(j+), j eo,(r ),they are C".
Now, using the above facts, we prove by induction on the
intervals [0, tj), j e l1r that
(3.2.17) y~j(t) = y.k(t), t e [O, L) for all yj, e hj.
48
This with Jj In (3.2.12) Implies
(3.2.18),( u, x,) = i( x, x) on [0,L),
which is what we need to complete the first part of our proof.
First, consider the time interval [0, t). By (3.2.14) and
(3.2.15),
(3.2.19) yj()(0) = ) (k)(0), k e o._
Thus, by (3.2.16) and analytic continuation [Die.1], we obtain
(3.2.20) yj(t) = yj.(t), t e [0, t ).
Next, suppose that for 1 < j < r,
(3.2.21) yj(t) = yj(t), t e [O, tj).
Then, by (3.2.16), (3.2.21) holds on 0, tj]. By (3.2.14) and
(3.2.15), this implies
(3.2.22) ij(k)(tj) ='(k)(t), k e no,.
Then, by (3.2.16) and analytic continuation, we obtain
(3.2.23) yj,(t) = y7j(t). t e [tj,, tjl ).
49
(3.2.24) yj(t)  yj(t), t e [ 0, tj+ ).
Thus, we have shown (3.2.17).
Now, assume that (F, H,X) Is decoupled on. We denote
by x( t, u, xo ) the state response of (F, H, X) to an input u and
an initial state x(O) = xO. By (A.2), given x e K, q e i1,, ck 6
R, k e. q, we can choose real numbers k > 0, k e o0, and xo e
X such that x(t, u*, x) e X for all t e [ tq] and x(tq, u, Xo) = x,
Fr'
where tr  T., r et1,q u* A 0,O j e6 fi, and uj* is given by
(3.2.25) u*(t) ci, t,1 i t e tr, r = 1,,q.
Let (F,H, X}i be the system obtained from (F,H, X by
letting u, = O. Clearly, for any given xO e X, there exist to > O,
o E X, u~ e Ut with u,~ = 0 such that xo = x( to, u~. ). Then,
by Lemma 3. 2. 1, R(xo, to) c I( f( F, H, X i), x). Let (xo, to) be
the interior of t(xo, to) relative to I( L( F, H, X)i), xo). Then, there
are two cases: (i) xo e S.(o, to) and (ii) xo e R(Xo, to)  &(xO, to )
First, we consider the case (i). Since (i) holds and Xj e
U[F, H, X]i ), j e fL,, we can choose > 0 such that
(3.2.26) 8 )X(xo) e t(X ), 6 e ( ,' ).
50
Then, for any 6 e ( Y, ) and j e n1,, there exists an input u'
with uiS * 0 such that
(3.2.27) x( to, u, ) 8XJ(xo).
Now, construct two piecewise real analytic inputs u, u as follows:
(3.2.28) tu(t) = uj(t) O^, 0 t to
ul*(t  t o), to to + t,,
(3.2.29) ij(t) ^! uj~(t). 0 t t s to, j e n.0, 0 0 t t to + tq,
(3.2.30) uj(t) ujS(t), 0 K t K to, j G M
0, to, t i to + tq.
Let X Xk + ck X, k e n1,. Consider the response of the
original system IF, H, X}. By the above construction of u, x =
x( to + tq u, x ). Since (F, H,X) is decoupled on X, we have
(3.2.31) Hi(x) = Hi( x( to tq, u, x ) )
Hj( oXq o...o0 1 j e X o..o.8Xq(x)).
i Tq IT  Tq
Note that (3.2.31) holds for all 6 e (,Y). Therefore,
differentiating (3.2.31) with respect to 6 and letting = 0 yields
(3.2.32) 0 = dH,(x)(BX).. (891)*X(B X o...o.X)).
51
Applying the Cambell  Baker  Hausdorff formula ( Theorem 2. 3. 5 )
to (3.2.32) successively q times leads to the following fact:
there exists t1 > 0 such that
1,OaqZG.1.. dH()L... L; Xj(x),
(3.2.33) 0=2 ***, dH (o)...!X.()
for all'k < 1, k e,iq.
Note that q1 depends only on x, Xj, and Xk e f,1q. Thus, when
we construct u* in (3.2.25), r, k e 1l,q can be assumed to be
chosen smaller than S. Small variations of rk, k e,q in
(3.2.33) yields
(3.2.34) dHi(x) L^q... LXJXj(x) = 0, q e o., k e Hlq, j e M,.
Now, we go back to the case (ii). Note that (ii) does
not necessarily imply (3.2.26). To show (3.2.34) is still true
for the case (ii), we need a slight modification of the above
arguments. By Theorem 2. 3. 3, there exist open neighborhoods
U, V of xo, x, respectively such that x( tq, u*, ) is a C 
diffeomorphism from U onto V. Let U be the intersection of U
and.( Xo, to). Then, since xo is in the closure of U, there
exists a sequence { x(p) ) converging to xo such that
(3.2.35) x,(p) e U c f.(x, t ), p e,.
52
Let x(p) x( tq, u*, xo(p)), p e l,.. Fix p e,1... By
(3.2.35), all arguments and equations following (3.2.26) do not
change if xo, x are replaced by xo(p), x(p), respectively. In
particular, we have
(3.2.36) dHi(x(p)) LX'. LX Xj(x(p)) = 0, for all e o,. and j e M,.
But since dHi, Lq" LXXj are continuous on X and x(p) > as p
> 00 (3.2.36) implies (3.2.34). Thus, we have shown that
(3.2.34) holds for both of the cases (i), (ii). Finally, since q, x,
and ck, k e n,, are chosen arbitrarily, (3.2.34) implies
(3.2.37) Hi = 0 on X, o e D( F, H, X)).
Then, Lemma 3. 2. 2 completes the proof. D
Remark 3. 2. 1. Theorem 3. 2. 1 is a minor generalization of
results for disturbance decoupling of real analytic nonlinear
systems, which are stated in [Hlr.2, Isi.1]. The first (sufficiency)
part of our proof is entirely different from those in [Hlr.2, Isi.l].
Our second ( necessity) part adopts its main idea from [Hir.2].
We feel that some of the arguments in the proofs by these
authors are incomplete. For instance, the details for the case
when input is piecewise real analytic are not given ( [Hir.2, isi.1])
and the fact that countably infinite intersections of open sets
53
are not necessarily open is not taken into account( [Hr.2] ). We
believe our proof is simpler and clearer. Finally, it is
interesting to note that while in [Hir.2], X is required to be
connected, it need not to be connected in [Isi.1] and here. 0
Theorem 3. 2. 1 concerns decoupling but not input  output
decoupling. We can easily show under the hypotheses of Theorem
3. 2. 1 that (F, H, X) is input  output decoupled on X if and only
if (3.2.1) and the following condition are satisfied:
(C) For each i e slm, the single input  single output system
(F, H,, X }, obtained from the original system F, Hi, X) by
setting uj = 0, e M, is invertible on X
Algebraic conditions which are either necessary or sufficient for
(C) have been obtained. But, those which are both necessary
and sufficient have not yet been presented in the literature. A
special case of invertibility of nonlinear systems is considered in
[Hir.1, Nij.1].
Some results for inputoutput decoupling of smooth nonlinear
systems are stated without proof in [Nij.3, Nij.4]. Their
validity is in doubt for the following reasons. The first of
two necessary and sufficient conditions for input  output
decoupling is similar to (3.2.1), although the assumption made is
(A.l). But recall that (A. 1)' is crucial in tne derivation of
(3.2 1). The second condition is for (C). Bu;, it is not clear
it is both necessary and sufficient for (C).
54
Another condition corresponding to (3.2.1) appears in [Tar.l]
and is used there as a definition of decoupling for smooth systems
(assumption (A l)). The connection between it and our definition
of decoupling would require (A.1)' instead of (Al).
Note that (A2) is used only in the necessity part of the
proof, where it is required to apply Lemma 3. 2. 1. But, (A2) can
be greatly relaxed. For instance, in Lemma 3. 2. 1 and, hence
Theorem 3. 2. 1, (A.2) can be replaced by
(A2)' There exists a locally path  connected subset 0 of Rm such
that for each constant input u(t) e 0, the vector field
F(, u ) is complete ( Sus. 1]).
Next, we consider necessary and sufficient conditions for
local decomposition.
Theorem 3. 2. 2. Suppose that (A. 1) is satisfied. Then,
(F,H,X} is decomposed at xO e X if and only if there exist an
open neighborhood E of xo and m involutive distributions A* on E
which have dimension ri < n such that on E,
(3.2.38) (i) dHi e (A,) c i e,',
(ii) (Ai*Yis Xo, Xj  invariant, i 6 11.m
(iii) (A*), i e im are mutually disjoint at each x eE.
Note that although Theorem 3. 2. 2 requires only the assumption
55
of smoothness, the conditions for decomposition are more complex
than those for decoupling. In Section 3. 1, we pointed out that if
a system (F,H,X} is decomposed at x0, then there exits an open
neighborhood E of xO such that (F, H, E is decoupled on f A
comparison of the conditions in Theorem 3. 2. 1 and Theorem
3. 2. 2 suggests that the converse is not necessarily true.
Theorem 3. 2. 2 is implied by Theorem 5. 1 in [Isi. 1], where
conditions similar to those in (3.2.38) are stated as being necessary
and sufficient for (F, H, X) to be decomposable on X. However,
the conditions in [Isi.l] do not necessarily imply the existence of T
which is a Co diffeomorphism on X Thus, they are necessary
and sufficient conditions for (F,H, X to be decomposable at each
xo e X rather than on X. In this sense, Theorem 3. 2. 2 may be
viewed as a corrected version of the result in [Isi.l].
We omit the proof of Theorem 3. 2. 2 since it can be obtained
from [Isi.1] and utilizes some ideas contained in the proof of Theorem
3. 2. 3. Note that it is not easy to check for the existence of
Ai*, i e?tM satisfying conditions specified in Theorem 3. 2. 2.
This is in contrast with the ease of applying the decoupling conditions
in (3.2.1). In this respect, the following Corollary is valuable.
Corollary 3.2. 1. If (F,H,X) satisfies (A.1)' and is
decomposed at each xO e X, it is decoupled on X.
Proof. The given hypotheses imply (3.2.38) (i) and thus
(3.2.1) holds. Since the sufficiency part of the proof for
56
Theorem 3. 2. 1 does not require (A.2), (A. )' and (3.2.1) imply that
(F, H, X) is decoupled on X D
It is uncertain that this Corollary is true for smooth
systems. This motivates the following Theorem.
Theorem 3. 2.3. Suppose that (A.1), (A.3), (A4) are
satisfied. Then, F, H, X is decomposed at each xO 6 X if and
only if (3.2.1) holds. 0
Apart from giving an easily verified condition for
decomposition, this result has other important implications. It
shows that under assumptions (A1), (A3), (A4), the condition for
decomposition of smooth systems is reduced to that of decoupling
of real analytic systems. Consequently, we see from Theorem
3. 2. 1 and Theorem 3. 2. 3 that under assumptions (A 1)', (A2), (A3),
(A4), the concepts of decomposition and decoupling are equivalent.
There are several circumstances where the assumptions of
Theorem 3. 2. 3. hold. The most obvious is the case of controllable
linear systems. For real analytic systems that satisfy (A.3),
Theorem 3. 2. 3 holds on a submanifold. In particular, when X is
connected, we can show by analytic continuation that there exists
a submanifold XO of X such that Xo is open and dense in X and
the assumption (A.4) is satisfied on X.
We believe Theorem 3. 2. 3 and the equivalence of decoupling
and decomposition is new. The only similar result we know of
57
is in [Nij.4], where the structure of a decoupled system with m 
2 was investigated. Although assumptions similar to ours were
made, the structure in (3.1.4) was not obtained.
In order to prove Theorem 3. 2. 3, we need the following three
Lemmas. Particularly, Lemma 3. 2. 5 is the key to Theorem 3. 2. 3.
Lemma 3.2. 3. Let A be an involutive C~o distribution on X
with dimension r < n. Then, at each point p eX, there exist an
open neighborhood U of p and (nr) C~ functions 8j, j e n,~),
from U into R such that dB., j e 1 (n ), are linearly independent
on U and at each q s U, ( dS.(q), j 6 l(nr) is a basis for A.
Proof. Since A is involutive and has dimension r on X,
Theorem 2. 3. 9 ( Frobenius Theorem ) applies and at each p 6 U,
there exists a chart ( U, )) such that if x1,., xn denote the
local coordinates, then ((E,) A *q)((a/axi)(q)) ) 6 M l.r ) is a
local basis of Aq, q e U. Let 8j  J(r+j)' j 6 l,(nr) where k
is the kth component of *. Then, since
(3.2.39) E, j ='(a/axi) (r+j) = (/axi) (r+j) )
= a(rj)/axi = 0 on U i e tr, jr e ^l,(nr)
the desired result foflows immediately. O
58
Lemma 3. 2. 4. Let A be an involutive CO  distribution on X
with dimension r < n. Let j, j 6 tl,(nr) be any C0 functions
from an open subset U of X into R such that ( d.(q), j 6sl(n )
is a basis of Aq, q s U. These functions exist by Lemma 3. 2. 3.
Then, for any CO function't from U into R satisfying
(3.2.40) dT e ~ on U,
there exist an open subset E of U and a CO function g, defined
on an open connected subset of R(n), such that
(3.2.41) I(x) = g( 8,(x),, (n)(x) ), x e E.
Conversely, if (3.2.41) holds on U, then (3.2.40) holds.
Proof. Fix p e U. Since ( dj(q), j (nr) is a basis
of Aq, q e U, we can choose r C~ functions 8 U  R, j e
lr 5so that
(3.2.42) rank of T = ( 8,, n ) at p = n
By Theorem 2. 3. 6, there exist an open neighborhood U c U of p
such that
(3.2.43) T is a C" diffeomorphisrr on U.
59
Choose an open neighborhood E c U of p so that
(3.2.44) T(~) is open and connected.
Define g T(E) 4 R by
(3.2.45) g o T1.
Then, er = goT. Since (d8S(q), j e ll,, ) Is a basis of aq, q
e U, it follows that
(3.2.46) Djg( 9(q), 8 (q) ) = 0, q e ~ if j > n  r.
By Theorem 2. 3. 8 and (3.2.44), this implies g( yl,..., Y ) =
c( Y1,..., yr ) on T(~) and (3.2.41) follows. Next, suppose that
(3.2.41) holds on U. Then, di!(x) is a linear combination of d9j(x), j
e 1,(nr) at each point x e U. Thus, (3.2.40) holds. 0
Lemma 3.2.5. Suppose that (A.1), (A.3), (A4) are satisfied.
1n
Then, at each point xO s X, there exist ( I pi ) C~ functions ti.,
j=1
j e l pi, i e Ht, from an open neighborhood V of xo into R such that
(I) dt,, j e j lpif, e i,m are linearly independent on V,
(11)d 1, e t((F,H,X}) on V, j e 1lpi, 1 e I,.
60
Proof. Fix xO e X By Lemma 3.2.3, (Al) and (A4) imply
that for each 1 e Im, there exists an open neighborhood 1E of xo
and C*functlons tJ, J eSl pi from E; into R such that
(3.2.47) dtj, J e l, pi are linearly independent on ~,
(3.2.48) dtij e ( [ F, H,X)) on ~, j e Ptlp.
Let V  E ~1( " ~.I. Then, (3.2.48) implies (ii). Now, we
show that (A3) implies (i). Suppose that at a point xo e V,
there exist constants.j(o)., j e lptp,, 1 e Sm such that
Let 11  2 2 J (^) t on V. Fix (,e M,.. Let 1l=i=t 3=I Jcl 0'1J.Z _(xo) 0 on V. Define a multiindex k by any finite sequence
of integers taken lo, such that at least one of its elements is
t. For such a multiindex I  (i1,I, ik), define a vector field
YI  LXi... LXi,, Xi Let k be the set of all such vector fields
YI. Then, from (3.2.49), (ii), and Lemma 3. 2. 2, it follows that
(3.2.50) YII(,) = YITI(Ro) = 0, Y1i e '
On the other hand, by (ii) and the definition of tl,
61
(3.2.51) ZZ(o) = 0. Z 6 ^((FH,X,).
By the definition of L~( F,H, X)), (3.2.50) and (3.2.51) imply
(3.2.52) Y fl(xo) = O, Y e L~( (F,H, X )
By the definition of L( F, H, X.), (3.2.52) holds for all Y e
(F, H, X)). By (A.3), this implies
(3.2.53) 0 = d(l(O) = X 00) d 0(0),
and from (3.2.47),
(3.2.54).j(NO) = 0, j e t,
Since 1 was chosen arbitrarily, we conclude that
(3.2.55) ij(to) = 0, j e lp i e
and our proof of (i) is complete. O
It is interesting to note that although the distributions Ai,
i e l,m do not satisfy the conditions of Respondek, Tarn and
others ( Res. l, Tar. ] ), this lemma shows that these distributions
are still simultaneously integrable in their terminology. Now,
we present the proof of Theorem 3. 2. 3.
62
Proof of Theorem 3.2.3. Assume (F, H, X) is decomposed
at each xo X. Fix xO e X. Then, by Definition 3. 1. 5, there
exist: (a) an open neighborhood E of xO; (b) an open subset X of
Rn; (c) a C  dlffeomorphism T: E ; (d) integers s, 2 1,
m+l
i e,I, and sn 2 0 satisfying n = Z si; and (e) a system
1=1
F, HX) which is T related on E to F, H, E) such that its
coordinate representation (f,h, X has the form (3.1.6). Let T
= ( TI,Tm, Tm, ) and Ti (T,, Tl si )' i 6 1.
Let Xi, i a e om be the vector fields corresponding to (F,H, ).
Fix i e fl. Let i be any CO function of xi only. Then,
by the special structure of (f,h,X), (3.1.1), and Definition 3. 1. 1,
we obtain
(3.2.56) (L Li,.. Lx Xj )i = 0 on T(E),
ij e [, i, j e H,, k e no..Since Xi is T  related on E to Xi, i e no,) Fact 2. 3. 2 and
(3.2.56) imply
(3.2.57) (LXi.LXij X )( oTT) = 0 on E
if i e (, i), j e M,, k e no..
On the other hand,
63
(3.2.58) Hi(x)  Hi(Ti(x)), x e ~, i e.tl
By the definition of A( [F,H, X), (3.2.57) and (3.2.58) imply
(3.2.59) YH1 = 0 on E, Y e A1( F,H,X)), i e tn.
Since xO can be arbitrarily chosen, (3.2.59) implies (3.2.1).
Now assume (3.2.1) holds. By (A.1), (A3), (A4), Lemma
3.2.5 may be applied. Fix xO e X. We use the same notation
in Lemma 3. 2. 5 except that tij is denoted by Ti j, j e 11 p,,
i s lt,. Let T ^ ( Tj, i..., T ) i i n. Let Pm+l i
n  2 pi. If pm+i > 0, it is possible to choose other C*functions Tm,. j e nl, Pn+ from V into RPm+' so that T 
( T1' T m+l ) has rank n at x where Ti = ( Tm+ 1,'
T.l, p, ). Then, by Theorem 2. 3. 6, there exists an open
neighborhood W c V of xO such that T is a C  diffeomorphism
from W into R". It then follows that there exist C~ functions
fm+j' bij i nl J e s 1 pm such that
(3.2.60) XO T+l j(x) = fm+ (T(x)), XiTm+ j(x) = bi.(T(x)), x e W.
On the other hand, for each i e tI,, Lemma 3. 2. 4 holds
with r n  p and 8 T jei 1, Thus, by (3.2.1) and Lemma
j iaj',i''
64
3. 2. 4, there exist an open neighborhood U c W of xo and Cfunctions hi, defined on an appropriate subset of RPi such that
(3.2.61) Hi(x) = hi(Ti(x)), x e U, ie,n.
Next, by Lemma 3. 2. 2  (ii) and Lemma 3. 2. 5  (ii),
(3.2.62) dXoTi,j, dXiTij e a(( F, H, X)) on U, j e lp, i e n 1.
Consequently, applying Lemma 3. 2. 4, it follows that there exist
an open neighborhood E c U of xO and C functions f.j, ij., j 6,I I,, I e l, such that
(3.2.63) X Tj(x) = fj(T,(x)), Xi Tij(x) = gij(T(x)), x e E
Let X  T(E). Let x (xl,, xm )  ( T(x),,
T~,(x)). Let f; (fij,. fi,pi ), g= ( g9i, 1 *, gip), i e..n
Letfm+ ( f, 1 fm+l,pm+ 1 ), bj (b i,1' bi,pm+l ) i e,
Let H 4 h = (6h,,hm). Define vector fields Xi, i e lo by?, P; P??,
(3.2.64) X,(x)   Z f (x.) (/aj) fm+j(x) (a/aX+),
P. m= j j=+M
Pi
(3.2.65) X(x) gij(xi)(a/ax.) + bi.j(R)(a/axm+ ), i e i.
j J j=1
Note from (3.2.60), (3.2.62), (3.2.63), that Xi, i seom, and Hi, i e
t,, are Trelated on ~ to X1, i e 1~,' and H1, i 6e 1,
65
respectively. Let ( FH,X) be the system constructed as above.
Then, it is an easy consequence that the above, ( F,H,) with si
~ Pl, 1 e 11sX+) meet the requirements of Definition 3. 1. 5. D
In this section, we have elaborated on the difference
between decoupling and decomposition and have presented algebraic
conditions related to them. Examples of systems which are
decoupled and decomposed are easy to construct See for example,
the standard decomposed system of Definition 3. 1. 5. It is not
easy to give an example of a system which is decoupled but not
decomposed. The various results are connected to one another In
the way described in the Figure 3. 2. 1.
66
F, H, X ] is decoupled on X
Th. 3.2.1 (A1)', (A.2)
(A2)1 dHi 6 (F,H, X)) Cor. 3. 2.1 (A1)'
Th. 3.2. (AI), (A3), (A4)
iF, H, X is decomposed at each x e X
Th. 3.2.2 (A 1)
3 m involutive distributions A,* satisfying
(i) dimension A,* = ri < n on X,
(ii) dHi e (A*^ c 4 on X,
(iii)(Ai* is Xo invariant and X  invariant on X,
(iv) (A,*t i eflH, are mutually disjoint at each x eX.
Figure 3. 2. 1. Summary of main results in Section 3. 2 showing
assumptions required for each implication.
67
Section 3. 3. Decouplability and Decomposability
We now turn to the question of when a system is decouplable
by a control law. Results concerning local decomposability ( when
a system is decomposable at a point xo e X) are found In [Isl.l,
Nij.2], where Wonham and Morse's geometric approach ([Mos.l, Mos.2,
Won. 1, Won.2]) is generalized to nonlinear systems. They are
sufficient conditions for local decouplability since a locally
decomposable system is always locally decouplable. Little has been
done on the question of global decomposability( when a system is
decomposable on X). Results concerning global decouplability (the
system is decouplable on X) are found in [Cla.l, Fre.1, Maj.l,
Por.l, Sin.l, Sih.1]. These results extend Falb and Wolovich's
result on linear decoupling to nonlinear systems. The earliest of
all the above papers is by Singh and Rugh([Sin. 1). Subsequent
papers add relatively little to their results.
In this section, we discuss global decouplablity based on our
precise definition of decoupling (Definition 3. 1.3) and investigate
the connections between decouplability and decomposability. We
add the following assumptions to the list we made in Section 3. 2.
(A. 5) The control law u =(x)+ 3(x)i is smooth,
(A. 5)' The control law u = ((x) + p(x) u is real analytic,
(A. 6) P(x) is nonsingular on X,
(A. 7) There exist di e N, i e PM^tm such that the following mrow vector conditions are satisfied
68
(3.3.1) [ XXok Hi(x). XmXok Hi(x) O, x e X, k e no.(dil),
(3.3.2) Di*(x) [ XXodi Hi(x). XmXodiHi(x)] 0", x e X
We begin this section by giving a necessary and sufficient
condition for global decouplability. When (A7) is satisfied, let
D*(x) and A*(x) denote, respectively the (m x m) and (m x 1)
matrices of functions defined by
(3.3.3) D*(x) 4 R*(x), A*(x) Xo(d1+ Hi(x)
Q:(x) j Xo(dm+ 1) Hm(X)
Theorem 3.3. 1. Suppose [ F, H, X) satisfies (A. 1), (A.7)
and the class of control laws satisfies (A5), (A6). Then,
IF, H, X is decouplable on X if and only if
(3.3.4) D*(x) is nonsingular at each x e X
Furthermore, u = [D*(x)]( G A(x)) decouples F, H, X) on X
That is, for ((x) ^ [D*(x)]^A*(x) and p(x) e [D*(x)]', the system
{ F, H, X is decoupled on X 0
This is actually a nonlinear version of the well known result
by Falb and Wolovich on linear decoupling ([Fal.1]). The sufficiency
part of Theorem 3. 3. 1 is proved in [Cla.1, Fre.l, Maj.l, Por.l, Sin.l,
Sih. 1]. Its necessity part is stated in [Maj.1, Sin. ], but the
arguments are not entirely clear. We shall prove the necessity
part rigorously, based on our precise definition of decoupling.
69
We need the following lemma for the proof of Theorem 3. 3. 1.
Lemma 3. 3. 1. Suppose that (F, H,X) satisfies (A1) and
(A7). Consider control laws u = <(x) + p(x) which satisfy (A.5)
and (A.6). Let (F,H,X) be J feedback related on X to (F,H,X)
by J = (T,<,B). Then:
A.(i) (A.7) is satisfied on T(X) with d, = di, i e M,m
(ii) D*(T(x)) = D*(x) B(x), *(T(x)) A*(x) + D*(x) cx(x), x e X,
(iii) Xok (T(x)) = Xk H(x), x e X, k e fo.di
Proof. Let Io be the identity mapping from X onto X
We show that for two special cases, J = (I o, c, } and J = ( T, 0,
I ), (i) and (ii) hold. Then, by the transitivity of J  feedback
relations, (i) and (ii) hold for the general case, J=(T, x, ].
First, we consider the case of J = ( Io,, i }. Then,
(F,H,X) = (F,H, X )1^ with X = X. Let Xi, i 6e oom be the
vector fields corresponding to (F,H,X). Then, for all x e X,
(3.3.5) Xo = Xo + i.i( )Xi,
A'A
(3.3.6) Xj = j( )Xi, Hi = H, j e t,
where a. is the ith component of:, Bij is the (i, j)th component
of p. From this and the definition of (d,, i eM:m), 1: follows
70
that for all 1 e?t,, and x e X,
(3.3.7) X'k = Xk Hi, k e noi,
(3.3.8) Xj k  Xj Xo Hi O, j e l,,. k e o,(di_).13.) 0 1H JO 61
XjXodiHi = Pkj( )XkXodiHip, j E 6l.
By (3.3.8),
(3.3.9) [i 1 diAi(x)... mXdi Hi(x)] = D*(x) p(x) on X.
By (A.6) and (3.3.2), this implies
(3.3.10) [IXOdHij(x)... XmXodHi(x)] 0, x e X.
The definition of (d,, i e,tm for (F, H,., (3.3.7), (3.3.8), and
(3.3.10) imply (i), (ii), and (iii).
Next, we consider the case of J = T, 0, I ) Then, F, H, X is
T  related on X to (F, H, X). Therefore, by Definition 3. 1. 1 and
Fact 2. 3. 3, we have
(3.3.11) Xok Hi(T(x)) = Xk Hi(x), x e X, k e Ko., ie mI,
(3.3.12) XjXokHi(T(x)) = XjXokHi(x), XeX, kett i, i,j elm.
This implies (i), (ii), and (iii). 0
Lemma 3. 3. 1 is a nonlinear version of the invariant property
71
of the Integers, d(, 1 e tm, on linear systems, which is shown in
[Gil. 1]. The case of J  (I, l,Bp) and X 4 R" Is proved In [Por. ].
Now, we prove Theorem 3. 3. 1.
Proof of Theorem 3. 3. 1 First, assume (3.3.4) holds. Let
(t, A, x ^ ( F, H, X )1P with q(x)  [D*(x)]IA*(x) and p( [D(x) [Dx)]Then, (F, H, X) is Jfeedback related on X to I F, H, X by J =
Io, A*, D*). The vector fields associated with (F, H, ) are given
by (3.3.5) and (3.3.6). By these observations and Lemma 3. 3. 1,
direct computation shows that
(3.3.13) Xok H,(x) = Xk H(x), k e,
x t(di+ jx) = 0,
(3.3.14) XjXkHi(x) = 1 if j = i and k = di,
1 0 otherwise.
Let yj be the ith output of IF, H,X. Then, by (3.3.13) and (3.3.14),
differentiating y9 (di+l)times with respect to t leads to
(3.3.15) i(t) = A,(x(t)),
i(1)(t) = ifRi(x(t)) + 2 uj(t)ifi,(x(t))
jft J
=i u(t), d. = 0,
0 1(x(t)),, 0,
72
(3.3.16) 9(di+t)(t) = (t),
where initial conditions are given by
(3.3.17) (k)(o) = i (x(O)), k 6 o.di
By Definition 3. 1. 3, this implies F, H, X) is decoupled on X
Next, assume that there exists a control law u = (x) + (x)
such that (F,H,X)1P is decoupled on X Let F, HA,) =
(F, H, X)a'. Let Qj, 1 e n, be the vector fields corresponding
to (F,H, X). Let y9 be the ith output of (F,H,X). Since
F, H, X) is J  feedback related on X to (F, H, %) by J =
(10., p) i, we have by Lemma 3. 3. 1,
(3.3.18) k(t) y = kH(x(t)), k e o,
(di+)(t) = X(di+)(x(t)) + u(t) X Xdi H(x(t)).
J ^ J
But, by Definition 3. 1.3, it follows that for any initial state
x(O) e6 x and for any two Inputs u, u e U' with u, = u,,
(3.3.19) Ay, 4 (u,( x,)  j(ui x) = 0.
This implies
(3.3.20) ay4k)(O) = 0, k e ~_,.
73
Then, by this and (3.3.18), we must have
(3.3.21) Ai(di+)(O) = uij(0)  Xu(xo) X= O
J Al
Since we can choose uj(O), uJ(O), j e, and xo arbitrarily,
(3.3.21) implies
(3.3.22) Xj oiH = 0 on X, j e.
By Lemma 3. 3. 1  (1), and the definitlon of di, 1 e n, },
(3.3.22) implies
(3.3.23) X (x) 4 XodiH(x) W 0, x e X, i e 1,.
On the other hand, by Lemma 3. 3. 1  (ii), (3.3.22), and (3.3.23),
(3.3.24) D*(x) (x) = diag X(x), x e X.
Then, (3.3.4) is a direct consequence of (A6), (3.3.23), and (3.3.24).
Because of its importance in our subsequent developments,
we henceforth reserve the notation (F*,H*,X) for the system
{ F, H, X, where <(x)  [D*(x)]lA*(x), p(x) ^ [D*(x)]1.
Remark 3. 3. 1: The input  output map for ( F*, H*, X) is
determined by equations of the form (3.3.16). By Definition 3. 1.4,
74
this implies that F*, H*, ) is also inputoutput decoupled onX.
Thus, under the assumptions (A.), (A5), (A6), and (A7), (33.34) is
a necessary and sufficient condition for both decouplablity and
input  output decouplability. Our result on input  output
decouplability of smooth systems is stronger than the one which
appears in [Nij.3, Nij.4]. The result there is local. Moreover, it
is derived on the basis of algebraic conditions for decoupling
whose validity is not clear, as was discussed in Section 3. 2. O
Next, we consider decomposability.
Theorem 3. 3. 2. Suppose that the hypotheses in Theorem
3. 3. 1 are satisfied. Then, F, H, %) is decomposable at each xo
e X if and only if (3.3.4) holds. O
Theorem 3. 3. 1 and Theorem 3. 3. 2 have the important
implication that under the assumptions (Al), (A5), (A.6), and (A7),
decouplability and decomposability are equivalent. In [Isi.I], the
sufficiency of Theorem 3. 3. 2 follows under additional assumptions,
which are basically equivalent to assuming that dX0kHi(x), k e
ttli, i e ll are linearly independent on X But, as will be
shown in Lemma 3. 3. 3, this is automatically implied by (A7) and
(3.3.4). We believe that the necessity of Theorem 3. 3. 2 is new.
The following Lemmas are needed for the proof of Theorem 3. 3. 2.
75
Lemma 3.3.2. Let te N, * e eC(X). Let Y, Z e V"(X). Ir
(3.3.25) dZY O on X, k e no1,
then,
(3.3.26) (LiyZ) y(i) = (l)iZyj on X, j e o,(Sl' i e n,.
Proof. From [Var.l], we have
(3.3.27) LiyZ  (1)i (1 i!/(k! (ik))) e no.
K=0O
Postmultiplying Y(Ji)+ on the both sides of (3.3.27) and using (3.3.25)
yield (3.3.26). 0
Lemma 3. 3. 3. Suppose that a system [ F, H, X satisfies
(A7) and (3.3.4). Then, dXokH,, k E ^,di i e 1, are linearly
independent on X
Proof. Let (F,,X)  [F*,H*, X). Then, (3.3.14) holds.
By Lemma 3. 3. 2, this implies
(3.3.28) (LJoXj)Aori  (l)k if j  i, k  di r, r e'odi
0 otherw ise.
76
By using (3.3.28), we now show that dX H,(x), k e n,,o, 1 e n,
are linearly Independent at each point x e X
Suppose that at a point xO e X, there exist constants r i(xo), j
e ntodi i 6 ft1, such that
(3.3.29) 2 X J (xo) dXo (xo) O.
Define a CO function q1 from X into R by
AmJi
(3.3.30) t = 22 (xo) oji.
I=Ij=o iJ
Then, by (3.3.29),
(3.3.31) (Loj) I(Xo) = 0, k e oh,, 6 Pl,,.
Applying (3.3.28) to (3.3.31) and choosing k, j appropriately lead to
(3.3.32) rij(xo) =0, j nodi, i e A,.
This with (3.3.13) completes the proof. 0
Before presenting the proof of Theorem 3. 3. 1, we give some
comments on Lemma 3.3.3. In [Sin.1, Fre.1], it is shown that under
the same assumptions, XokHi, k e,odi, i e,m are functionally
independent on X But as mentioned in Section 2. 2, this does not
necessarily imply that dXdkH,, k e n.,i, i E l,, are linearly
77
independent on x The converse is, however, always true. In
[Isi.l], linear independence of dXokHi, k e tod, i e lM is assumed
in addition to (A.7) and (3.3.4). Now, we prove Theorem 3. 3. 2.
Proof of Theorem 3. 3. 2 Suppose there exists c, P such that
I F, H, %)' is decomposed at each xo e X Let {F, F,'X)
{ F, H, X ]. Then, by Theorem 3. 2: 2,
(3.3.33) dFi e 4 ( F. HX)) on X, ie tt.
By Lemma 3. 3. 1 (i), d = di. This with (3.3.33) and Lemma 3. 2. 2
shows that (3.3.22) holds for each i e?l m. The remaining
arguments are exactly the same as those following (3.3.22).
Next, assume (3.3.4). Let ( Fi,,X) I ( F*, H*, X I. Then,
(3.3.13) and (3.3.14) hold. Let Tij ^ o(j') R j e l(d+l) i e, lm
Then, by (3.3.13) and (3.3.14), the following equations hold on X:
(3.3.34) XoT T(k+), k E ldi, i e,
0, k = d+ l, i e Bt',
(3.3.35) jT, = 1, j = i, k = d + 1, i e ttl m
0, j e tt, k e l, (,+l) i e tl,m.
(3.3.36) Hi = Tj., ie t,.
On the other hand, by Lemma 3. 3. 3 and (3.3.13), dikHi, k e
VII LII VCIILI IIY IIU IUJ LLIIIIIU J.J. ~ UIU Y0 1'
78
od, 1 e m, are linearly independent on X Let T, T (T1,,
Tl(dM+) ) i 1 mn'. Let p (di+l) and Pm+l = np. Fix xo E X
Because dXoi,, k e o0dm, 1 e nlm are linearly independent on X, it
is possible to choose a C~ mapping Tm+i: X  RPm+1 such that T 4
( T1, T., T, m ) has rank n at x. Then, by Theorem 2. 3. 6, there
exists an open neighborhood E of xo such that T is a C~diffeomorphism from ~ into Rn. Consequently, there exist C0functions fm+,j, bij, i e l,,n, j e Nlpm+ defined on appropriate
subsets of R" such that
(3.3.37) XoTm+j(x) = fm+ (T(x)), XiTm+ (x) = b, (T(x)), E
Now, Let x 4 (,, x', xm m )  ( ), T(x),, T +x) T (x)).
Let X % T(~). Define vector fields Xj, i e Nom and functions H, i
e Ntll by
() All 7J 4x
(3.3.38) X(x) = 2.2 xj.+l(a/ax.j) + I f+ () / (m+),
0= I j=1
(3.3.39) X(x) a/ax i(di+) i e l,m'
(3.3.40) Hi(x)  x,, i 8 t,m
where, xj is the jth component of x. Let F, H, X be the system
constructed as above. Then, the above ~, X, and IF,H,X) with s.
dj+l, ie mL, s,,m = Pm+i meet the requirement of Definition 3. 1.5.
79
In particular, its coordinate representation (f,h, ) has the form
(3.1.6) such that foreach 1 6 ni,,
(3.3.41) (x) () x Bi, hi(xi) = Cixi,
where A, o Id 1, B = o,a C [ 1 o.. o 1
Since [F*,H, X) is decomposed at each x e X, F, H, X) is
decomposable at each xo e X by the control law u = [D*(x)l( u 
A^(x) ).
Remark 3. 3. 2. We applied Theorem 3. 2. 2 to prove the
necessity part of Theorem 3. 3. 2. Theorem 3. 2. 2 also yields an
alternative proof of the sufficiency part, which is basically the one
given in [Isi.l]. The argument goes as follows. Under the
hypotheses of Theorem 3. 2. 2, it is not difficult to find the
distributions A*, i e n,, required in Theorem 3. 2. 2. For
(F*,H*, }, choose A*, i e n,, by
(3.3.42) A,* = (Y e V~(X): dXokHIY = 0 on X, k e no).
Then, by (3.3.13), (3.3.14), and Lemma 3. 3. 3, has a constant
dimension (nd,l) on X and (Ai,*) i e tL, are mutually
disjoint at each x e X Moreover, A** is involutive on X By
80
(3.3.13), (3.3.14), and Lemma 3. 2. 2, (3.2.1) holds for F*, H*, X).
Thus, (3.2.38)(1) is implied. Since by (3.3.13) and (3.3.14), A* Is
A
XO  invariant and X  invariant on X, It automatically follows that
A A 
(Ai* is X, invariant and Xi invariant on X Thus, the distributions
a,*, i e T1 meet all requirements for Theorem 3. 2. 2 to hold.
Consequently, (F*, H*, X} is decomposed at each x0 e X 0
By adding further assumptions to those in Theorem 3. 3. 2, we
can obtain a more detailed structure for (f,h, ) than the one in
(3.1.6) and (3.3.41). First, we define a standard decomposed system.
Definition 3. 3. 1. Let X be an open connected subset of R".
A system (F,H, } is a standard decomposed system if its
coordinate representation [f,h, X) has the following properties
(1) There exist nonegative integers d,, i e n, m and pD, i e,i.ml, satisfying n = X p, and pi d + 1, i e L,.m so that
(f, h, ) has a form
(3.3.43) xi = fi(xi, Ui) = A x + B, yi h) = Cj, i e m
(3.3.44) m+l = m + ()
where:xj(t) s RP', e lm+l X ( X,' Xml) E R"; A B i Ci
are respectively (d+ 1) x, (dj+l) x 1, 1 xpj matrices such that
81' o Id,: B]'C o 1 o * o ]
(2) Let 5 = ({x: x i ( X,..., x,+) 6e ). Each subsystem
(fi, hj, X, i e tlm, in (3.3.43) satisfies the controllability
rank condition on XV,
(3) dim. A( F, FH, ) = pi on R, i e nIm.
Remark 3. 3. 3. The standard decomposed system In Definition
3. 3. 1 is a nonlinear version of the system obtained by Gilbert ( Gil. 1).
It is worth noting that properties (2), (3) together imply the standard
decomposed system F, H, X) satisfies (A3). When F, H, X) is a
linear system, it can be shown that property (3) Is equivalent to
condition (iv) in Definition 6 of [Gil.1]. O
Now, we are ready to state the following result.
Theorem 3. 3. 3. Suppose that the hypotheses in Theorem 3. 3. 1,
(3.3.4), and (A3) are satisfied. Further, assume that (F*, H*, X)
satisfies (k4). Then, at each xO e X, there exist: (a) an open
neighborhood E of xo; (b) an open connected subset R of R"; (c) a
C"  diffeomorphism T:  X; and (d) the system ( F, R,X), which
is T related on E to (F*, H*, ~E, is a standard decomposed system
"nq
with di = di, Pi = pi, i s t1m, and Pl = P:Pm+^n pi, where the
pj and d1 appear in (A.4) and (A.7). C
82
See Figure 3. 3. 1 for a schematic description of the result of
Theorem 3. 3. 3. Since ( F, H, E) and F, H, T(E) ) are J  feedback
related, they are equivalent with respect to what can be accomplished
by feedback ( recall Section 3. 1 ). Thus, the value of Theorem 3. 3. 3
lies In that the class of decoupling control laws can be characterized
by looking at the standard decomposed system instead of the general
system. This motivates some results in Section 3. 4. For the proof
of Theorem 3. 3. 3, we need the following Lemma.
Lemma 3.3.4. Suppose that (F, H,X) satisfies (Al) and
(P, H,* ) is Jfeedback related on X to (F,H,X) by J = [T,, ),
where J satisfies (A5) and (A6). Then, If (F,H,X)} satisfies (A3)
on X, (F, H, X) satisfies (A3) on T(X).
Proof. First, we consider the case of J = Io,0,B ) Then, we
have
J =, (D*)'A*, (D*)'
(F, H, E) F, H, T(E))
standard decomposed system
J*4 I, (D*)'A*, (D*)1) A ( T, 0, Im)
I F*, H*, E)
Figure 3. 3. 1. A standard decomposed system I F, H, T(~) ) is J 
feedback related on ~ to the system { F, H, E).
83
(3.3.45) X0 = 0 + I,( *)i
(3.3.46) X ij(), j e m
where xi is the ith component of x and jij is the (i,j)th component
of p. Then, by Fact 2. 3. 1, these imply
(3.3.47) Lp((F,H,X)) c Lp((F,A,) ), p sX
By (A3), this implies that dim.L( F, H, X)) = n, p e X
Next, we consider the case of J = {T, 0, 1Im } Then, by
Definition 3. 1. 1, i, Is T  related on X to Xi, i e om. By Fact
* AAA A
2. 3.2, each vector field Y e L F, HX} ) is T  related on X to
a vector field Y e U ( F, H, X) ). Since T is a Co diffeomorphism
on X, this implies that at each p e X, LT(p)( ( F, H, X) is
isomorphic to Lp( F, H, X) ). Thus, dim.Lq( F,H, X ) = n, q e
T(X). Our assertion follows easily from the two cases of J and
the transitivity of J feedback relations. O
Now, we prove Theorem 3. 3. 3.
Proof of Theorem 3.3.3. Let ( F, H, X) = ( F*, H*, X. By
given hypotheses and Lemma 3. 3. 4, (F, H, X satisfies (A.3). Fix
xo e X Then, by Lemma 3. 2. 5, there exist an open neighborhood
84
V of xo and ( pi ) C functions *.i: V  R, j 6e lp, i 6
int, such that on V,
(3.3.48) di.j, j 6 nli, i 6 Ml, are linearly independent,
(3.3.49) dij s ( ( F,,X ),  j 6 l.pi, i 6 1.'
As was shown in the proof of Theorem 3. 3. 2, 1, f, X} is
decomposed at xO. Therefore( see the proof of Theorem 3. 2. 3
and (3.2.59)), there exists an open neighborhood V c V of xO
such that
(3.3.50) dFi e 6( t, FR,X)) on, i e 1m.
This and Lemma 3. 2. 2  (ii) implies
(3.3.51) dk A ~ ^ A A ^ o A
(3.3.51) dXkH1 e Ad( ( F H, X ) on V, k e no, i e M.
This, (3.3.12), and Lemma 3. 3. 3 shows
(3.3.52) pi 2 di + I, i e nt1m.
Next, we show that there exists an open neighborhood W c
V of xo and a basis of ( (F, H,X)) on W which contains dXokHi,
k e Mo.di By (3.3.48), (3.3.49), (3.3.51), and Lemma 3. 2. 4, for
A
each i e Mt.r, there exist an open neighborhood V, c V of x
85
and C — functions pij from an appropriate subset of RPI into R, j
ePl1(d+l) such that
(3.3.53) Tij(x) X(JI)Hi(x) = ( ij,(x), i, (x) )
x e VI) j 8e,(dM+)1
Then by Lemma 3. 3. 3, (3.3.12), and (3.3.48), D'Pj( 1 il(xo),,
ii(xo) ), j e iMldi+l) are linearly independent (1 x pi) row vectors.
Now, for each i 6 1,t let ri A p  di 1 and choose ri (1 xPi)
row vectors i.j such that
(3.3.54) ^ Q DJi( (),... (x.))
i(di+ 1)(, 1 (Xo)','*. (Xo) )
Ti.1
tli,ri
is a nonsingular (p, x pi) matrix. Let
(3.3.55) Ti ( i,l'.' ipi )' Ti,(di+l+j) =l T, e el,ri, i e LI,m.
Then, by the construction of T.j, j e l pi, i e tm,
(3.3.56) dTi j(xo), j e lD' 1ii e l m are linearly independent,
86
(3.3.57) dTj e Zt (f,,X}) on V, J e l,,,, 1 e n,.
Let V' V,n.' Vm and Pm+l I n  Z Pi. If p,+1 1, choose
a Cm mapping Tm+i from V into RPm+l such that T has rank n at
xo, where
(3.3.58) T ^= ( T,, i Tm, T ), Ti (T,, T,, i).
Then, by Theorem 2. 3. 6, there exists an open neighborhood W c
V of xo such that
(3.3.59) T is a C0 diffeomorphism on W,
(3.3.60) (dT j(p), j eM,, ) is a basis of ()p( ( F H, X) ), p W.
Now, using (3.3.59) and (3.3.60), we show property (1) of
Definition 3. 3. 1. By Lemma 3. 2. 2  (ii), (3.3.60) implies
(3.3.61) dXoT j, dRiTij e [( F, H,X ) on W, j e li, i e lP.
Then, by (3.3.60), (3.3.61), and Lemma 3. 2. 4, there exist an open
neighborhood E c W of xO and C  functions 6.J, T from
appropriate open connected subsets of RP into R, j e t1.p,, i e
nm such that
(3.3.62) XT' ++(x) = 8 (T (x)), X T (d,+i+)(x) = (T,(x)), x e ~.
T~d~lj I
87
On the other hand, by (3.3.59), there exist CO  functions fm+l.
bij i e l,1, j e pm+l defined on appropriate subsets of R"
such that
(3.3.63) i Tm+l,(x) = f+(T(x)) XiTm+l,j() bj(T(x)), xe
Let X T(E). Let x  (x,,(T(x)xm T M+(x),
Let ei(,,' " ), ri (I, "'i I,), i e H,.m Let fT
n(f. Define
(fm+1.l,l' m+lpm+l ) bi = ( bl'',, bPm+l), e 1. Define
vector fields Xi, i e ttom by
(3.3.64) X(x)  X (X x,(j+) a/_a 8.i () /a ) i
I j= j ji+,(jdi1)
J= m+ Ij m+(j)a/aM'
(3.3.65) X(x) a/axi.( ) + I () a/ax. ) +
J,(di~l i i j
2 bi.j(X) a/am+j i e
(3.3.66) Hi(x) = xj, ie l'g
where, xij is the jth component of xi. Let { F, H, X be the
system constructed as above. Then, the coordinate representation
(f,h, X} of (F,H,X} has the form indicated in (1), where di =
d,, i e,l m and Pi = pi, i e Lm+l.
Let Yi be a C~ vector field in Ai( (F, H, X ). Then, using
7n+pI;
(3.3.64) and (3.3.65), we can show that if Y. = ].( ) a/xk
~.~i jk jk
88
Is a local representation of Ye on X,
(3.3.67)',(x) = 0, X 6 X, k e8 n,.
By Lemma 3. 3. 4, (F, H, X) must satisfy (A.3). Thus, (3.3.67) implies
property (2) of Definition 3. 3. 1. Property (3) follows from the
fact that by (3.3.59), (A,)p( (F, f1, ~ ) and (A,)Tp)( (F, H, X) are
isomorphic at each p e E. 0
Remark 3. 3. 4. The system (F, H, X is locally J  feedback
related to a standard decomposed system, where J ~ 
(T', A*o T1, D* oT1. As is shown in the proof of Theorem
3. 3. 3, the choice of T is not unique. Thus, there are infinitely
many standard decoupled systems which can be J  feedback related.
to (F,H,X). 0
Finally, we state a converse result of Theorem 3. 3. 3.
Theorem 3. 3. 4. Suppose that ( F, H, X satisfies (A. 1) and
the class of control laws satisfies (A.5), (A.6). Suppose further
that at each xo e X, there exist: (a) an open neighborhood ~ of xO;
(b) an open connected subset X of R"; (c)mappings T: 4E X, 2:
~  Rm, B: ~  Rmm; and (d)the system F,H, X, which is Jfeedback related on ~ to (F, H, ~E) by J (lT,c, B ), is a standard
decomposed system. Then, the following properties hold:
89
(1) (F,H,X) satisfies (3.3.4), (A3), and (A.7) with d,~d, 1 8e,
(II) (F*, H* X) satisfies (A4) with p, P, 1 e tl,
(iii) ((x) = [D*(x)lA*(x) and p(x) [D*(x)]1.
Proof. By Remark 3. 3. 3, F, H, X) satisfies (A3). By
Lemma 3. 3. 4, this implies that F,H,X) satisfies (A3). Direct
computation shows that (F, H,X) satisfies (A.7) with D*(x) = Im
and A*(R) = O. By this, Lemma 3. 3. 1, and (A5), we see that
(F, H, X) satisfies (3.3.4), (A7) with d, = di, i e?11,, and,
furthermore (iii). Since ( F, H, X) is J  feedback related on E to
(F,H, E by J ( T, (, ), (iii) implies that (F,H,X) is T  related
on E to F*, H*, X. Consequently, (A,)q( F*,, H*, X) and
(A)T(q)( I F, H, X)) are isomorphic at each q E This implies (ii). O
In this section, we have shown that (3.3.4) is a necessary and
sufficient condition for both decouplability and decomposability.
We have also specified a class of nonlinear systems which are
J  feedback related to standard decomposed systems. See Figure
3. 3. 1 for a schematic description of the results obtained in this
section.
Finally, we remark that (A7) can be weakened by
(A7)' There exist d1 e N, i e t,, satisfying (3.3.1) and
(3.3.2)' There is at least a point x0 e X such that
90
[ Xi1Xo H(Xo).o XmXod Hi(X) ] 0.
If (F,H, X) is smooth, (A7)' implies that there exists an open
neighborhood X' cX of xO such that (A7) is satisfied on X' instead
of K Thus, when (A7) is replaced by (A7)', all results in this
section hold with XX'. In other words, they are locally valid.
( F, H X ) is decouplable on X (F, H, X is decomposable at each xO e X
Th. 3. 3. 1 (A ), (A5), (A6), (A7) Th. 3. 3.2
D*x) is nonsingular at each x0 e X
Th. 3. 3. 31 (A. I),(A3), (A4),
(A5), (A.6), (A7)
At each x0 e X, 3 an open neighborhood E of x0 and a C"diffeomorphism T on ~ such that F,H,T(E) I, which is J 
feedback related on E to f F, H, E 1 by J=1 T,  (D*)'A*, (D*)1)
is a standard decomposed system.
Figure 3. 3. 2. Summary of main results in Section 3. 3 showing
assumptions required for each implication.
91
Section 3. 4. The Whole Class of Decoupling and
Decomposing Control Laws
In this section, we consider the class of control laws which
decouple or decompose a nonlinear system and thus obtain some
answers to questions (b), (c), (d) in Chapter 1. We believe our
results are new and are important contributions. We begin by
discussing at some length the significance of characterizing the
whole class of decoupling control laws.
Let (F, H, X be a system which satisfies the hypotheses in
A A
Theorem 3. 3. 1 and (3.3.4). Let F, H, X) ( F*, H*, X}. Then,
the input output map for (F, H,X Is determined by (3.3.16) and
(3.3.17). Now, suppose we choose the following control law for
A
(F,H,X)
(3.4.1)  = Fi 9(k) +
= Fjk X (k1)H(x) + Cj U i l,m6
where the Fk are constants, the ci are nonzero constants, and the
last equality comes from (3.3.13) and (3.3.18). Note that this
procedure corresponds to choosing for the original system
(F, H, X) a control law of the form u = ((x) + p(x) u where
(,+l
(3.42) >(x) ^= [D*(x)]1' F,, XO(kL)H,(x)'  A*(x) l,
92
By (3.3.16) and (3.4.1), the input output map for (F, H, X)} is
given by (3.3.17) and
Ji+l
(3.4.4) y(di+l)(t) 2 Fi y ) + ci U i e o n.
T i
Therefore, (F, H, X)},1 is decoupled on X and the control law
(3.4.2), (3.4.3) is a decoupling control law for (F,H,X).
Moreover, appropriate selection of the constants Fik and c, gives
good input  output dynamic characteristics. The class of
decoupling control laws (3.42), (3.4.3) was considered in [Fre.l,
Sin.l, Sih. 1 ].
A nonlinear control law more general than the one in
(3.4.1) is:
(3.4.1) 0, =?Y)'I' ( )+ 9 (d,..., y) ui,
where *i,'P are arbitrary C"  functions of their arguments
The corresponding control law u = ~(x) + (x) for (F, H, X} is
(3.4.5) >(x) = [D*(x)rl { lIl(X)  A*(x)},
(3.4.6) p(x) = [D*(x)]  diag X(x),
where
93
(3.4.7) i(x) i i( Hl(x), XoH (x),..., Xodi H(x) ),
(3.4.8) X(x)''l( Hi(x), XoHi (x),..., Xoi Hi(x)).
Now, by (3.3.16) and (3.41)', the Inputoutput map for (F, H, X)
is given by (3.3.17) and
(3.4.2)''(dj+1)(t) * *( 9(di) 9A )'+ l( Y(di) Yi ) u, i e
We see that the new feedback system F, H, XY) is still
decoupled on X. Thus, the control law (3.4.5) (3.4.8) is a
decoupling control law and is more general than the control law
(3.4.2), (3.4.3). The class of decoupling control laws (3.4.5)(3.4.8) was suggested in some examples which appear in [Cla.1,
Sin.2].
Can we find a still more general class of decoupling control
laws?. Knowledge of a more general class of decoupling control
laws allows more flexibility in choosing a decoupling control law.
For an instance, as will be shown later by an example (Example
3. 5. 1 in Section 3. 5 ), a decoupling control law (3.4.2), (3.4.3)
may not decouple a system in a "stable" way but it may be
possible by finding a more general decoupling control law. Thus,
characterizing the whole class of decoupling control laws is a
significant question both from engineering and mathematical
viewpoints. For future purposes, we define several classes of
control laws.
94
Definition 3. 4. 1. L ( [ F, H. X.) )( ( F, H, X ) ) Is the class
of control laws u <((x) + p(x) satisfying (A5) ((A5) ), (A6), and
(3.45)  (3.4.8).
Definition 3. 4. 2.'( ( F, H, X ) ( [ F, H, X ) ) is the class
of control laws u = <(x) + p(x) u satisfying (A.5) ((A5)' ), (A6),
(3.45), (3.4.6), and
(3.49) d, dX s ( ( F*, H*, X)) on X. O
Remark 3.4. 1. By (3.3.13) and (3.3.51), the smooth functions
i, X in (3.47), (3.4.8) satisfy (3.49). Thus,
(3.4.10) Sr c Sa and e c S".
In general, SEt (S) is a very limited subset of' (S ). When n
= Z (di +l), it is usually true that S'( F, H,X})= S( (F, H, X).
But, as will be seen in Example 3. 5. 2 of Section 3. 5, this is not
always so. Further discusssion will be given in Remark 3. 4 7. 0
The following theorem shows that S" is actually the whole
class of real analytic decoupling control laws for a real analytic
nonlinear system.
Theorem 3.4. 1. Suppose that (F, H,X) satisfies (Al)', (A.7).
95
Suppose that class of control laws satisfies the following
assumptions: (A5)', (A.6), and for u = c(x) +(x) u in the class,
(F, H, X)4 satisfies (A2). Then, the control law u = <(x)+ (x)u
decouples (F, H, X) on X if and only if it belongs to gS( 1 F. H. X] )
and (3.3.4) holds. 0
Remark 3. 4. 2. The condition (3.4.9) is equivalent to
(3.411) t = YA, = 0 on X for all Y e AlF*,H*,X).
Thus, Theorem 3. 4. 1 reduces the problem of characterizing the
whole class of decoupling control laws to that of finding all
solutions of the set of the first order linear partial differential
equations specified by (3.411). When (F,H,X) is a linear system
and the class of control laws is restricted to be linear (e.g., (1.5)
and (1.6)), (3.4.11) is reduced to a set of linear algebraic
equations. Moreover, Theorem 3. 4.1 is reduced to a result
contained in [Gil.l]. 0
We need the following Lemma for the proof of Theorem 3. 4. 1.
Lemma 3. 4. 1. Let i, lJ, ij, j M,,, i e l tti be any
Cm functions from X into R such that
(3.4.12) dt,, dcp, d~,, e &( {F,H,X] ) on X, i e m,.
96
Define C'vector fields X, 1 ie?t, by
(3.413) X' Xo 0+ Iji
j=I J'
(3.4.14) I (')Xj i e 1.
Let i e t.m. Let k be any finite nonnegative integer. Then,
if ij 6 (oi), j 6,
(3.4.15) dX1l2 X e lk A (F,H, X}) on.
Proof. By Lemma 3. 2. 2  (ii),
(3.4.16) dXoL,, dXitj e i( {F,H,X ) on X.
Recall that Xj e Ai, j e M if j i. This with (3.4.12) (3.416)
implies
(3.4.17) dXot, dXi t s ( { F,H,X)) on X.
Successive application of this result yields (3.4.15), immediately. C
Now, we prove Theorem 3. 4 1.
Proof of Theorem 3. 4. 1. Let u = c(x)+ B(x)u be a control
law which satisfies (34.5), (3.46), (34.9), and (3.410). Let X., 1
97
e no, be vector fields corresponding to [F, H, X) ). Let X. 1 e
noHm be vector fields corresponding to (F*, HX). Then, we
have
(3.4.18) X0 = + (),
j=1
(3.4.19). = Xj(*)j. j en
Let i e 1,. Let k be any finite nonnegative integer. Then, by
(3.3.33), (3.4.9), (3.4.18), (3.4.19), and Lemma 3. 4. 1, we see that if
iq eo,i), q e,1,
(3.4.20) dXilXi2' XikHi e 4( F* H*,X)) on X.
But note that j e A( (F*,H*,X)) on X, j eni. Hence, (3.419)
and (3.420) imply that if iq e o, i 1, q e %,; and j e i,,
(3.4.21) Xj Xi X i2''XikHi = 0 on X.
This with Lemma 3. 2. 2  (i) and Theorem 3. 2. 1 implies that
( F, H, X }) is decoupled on X
Next, suppose that u = ((x) + P(x) u decouples ( F, H, X} on X
Let (F,H,X)  (F,H,X)}. Let X, i e Kom be vector fields
corresponding to I F, H, X. Then, by Theorem 3. 2. 1 and Lemma
3. 2. 2 (i),
98
(3.4.22) Xj Xl, X2 XikHi  0 on X,
iqe (o,i), q e ti,3 k e kO,, j e tl, i e li,.
On the other hand, by Theorem 3. 3. 1, (3.3.4) holds. Let Xi, i e
fol be vector fields corresponding to (F*, H*, X. Define C0mappings: X R Rm, r: X  Rmxm by
(3.4.23) t(x) D*(x)~(x) + A*(x), r(x) D*(x)(x)
Then, we see from Fig 3. 4. 1 that
(3.4.24) XO = Q + ai( )Xj,
0 j=f j
(3.4.25) Xj rJ(. i j e 1
where rj is the (i, j)th component of r. On the other hand, by
Lemma 3. 3. 1  (i), direct computation with (3.4.24) shows that
J^(Io)
( F, H,X ) (> (F, H, X)
J*OIlo, A*, D*)\ / JoOJ*Ii,,r}
( F*, H*, X)
Fig 3. 4. 1. Relationships between ( F, H, X ), t F, H, X ), and ( F*, H*, X )
99
(3.4.26) ~ =k k
(3.4.26) X0 Hi, k e,odl,, m1 eM,.
This with (3.3.13), (3.3.14), (3.4.24), and (3.4.25) yields
(3.427) Xo(di+l)H =, XiXodiH = ri on X, ie,.
This with (3.4.22) and Lemma 3. 2. 2  (i) shows that
(3.4.28) dtl, dFjr e (F,H,X.) on X, i e 61m.
Note that by (3.3.23), (3.3.24), and (3.4.23), we must have
(3.4.29) r. (x) = 0, x e X, i * j,
(3.430) X(x) r..(x) * O, x e X, i e lm.
Consequently, we can write (3.4.24), (3.4.25) as
(3.4.31) X =  2 (
0'
(3.4.32) Xj = ( /.( ) Xj j e,.
Since dX e U( IF,H,X)) implies d(A)' e A (F, H,X) ), these
equations with (3.4.22), (3.4.28), and Lemma 3. 4. 1 lead to
A /A ~ A A'(3.4.33) dXjXi...Xi jk. dXjX12  XikX e {F,FHX) on X,
iqe{O,i}, q El,k' k 8 o. j e,, i E, Xl.
Recall that Xj e A,( (F, H, X) on X. j e t,. Therefore, (3.432)
J~~~~~~~ 1L.Teeor,(..2
100
and (3.433) Imply
(3.434) XJXH) AX.. iTh  Xjii i2XV. =X 0 on X.
XjXil Xi2. Xilj i' X. Xi2".Xi k  0 on xi
iq e(o,i], q e t,1k, k e6 tL0, j e 1, i e n^.
This and Lemma 3. 2. 2  (i) complete the proof. 0
Remark 3. 4. 3. A result on the characterization of
decoupling control laws is found in [Sin.l], where it is shown that
if a smooth control law u = <(x) + p(x) u decouples (F, H, X) on
X, then
(3.435) Xjok Hi = on X, k e o. if i j,
where Xj. j e lo are vector fields corresponding to (F, H,) Y
and Hi ^ Hi, i e l,. A more complete result is that under
hypotheses of Theorem 3. 2. 1, a control law u  c(x) + P(x)
decouples (F,H, X) on X if and only if (3.4.22) holds. But,
(3.4.22) is an implicit and complex condition for u = c(x)+ P(x) u to
be a decoupling control law. It results in high order partial
differential equations. On the other hand, the condition given by
Theorem 3. 4. 1 is explicit and involves only the first order partial
differential equations. Thus, (3.4.35) is not so useful for
characterizing the class of decoupling control laws as our
condition. O
101
Unfortunately, we are not able to prove that Theorem 3. 4. 1
Is valid for smooth systems and smooth control laws. But, we
can show that S"( (F, H, X)) is the whole class of smooth
decomposing control laws for a special class of smooth systems.
Theorem 3. 4. 2. Suppose the hypotheses for Theorem 3. 3. 3
are satisfied. Then, a control law u = ( (x)+ (x) u decomposes
(F, H,X) at each x e X if and only if it belongs to (( F, H, ) X
and (3.3.4) holds.
Proof. Suppose a control law u = ((x) + P(x) u decomposes
(F,H,X) at each xoeX. Let (F,H,X)(F,HH,%X). Let X*, i e
nom be vector fields corresponding to (F,H,X). Then, by (3.2.38)
 (1) and Lemma 3. 2.2  (1), (3.4.22) holds. Then, the remaining
arguments are exactly the same as those following (3.4.22) except
that (3.3.4) holds by Theorem 3. 2. 2 Instead of Theorem 3. 3. 1.
Next, suppose that a control law u = (x)+ p(x)u belongs to
S'( (F, H, % ). Fix xo e X Then, by Theorem 3. 3.3, there exist
an open neighborhood E of xO and a mapping T: E  R" such that
F, H, X) which is J  feedback related on E to ( F, H, E by J 
[T, (D*)'A*, (D*)' is a standard decomposed system with X 
T(E), di= di, 1 e n,n. and pi = p,, 1 e 1n,. The mapping T
constructed by (3.3.53), (3.3.55), and (3.3.58) satisfies (3.3.59) and
(3.3.60) on ~. Then, by (3.4.9) and Lemma 3. 2. 4, there exist an
102
open neighborhood K c E of xO and C'functlons T, A, defined on
appropriate subsets of RI, i e l, such that
(3.4.36) t,()  j(T,(x)), X(x)  (T,(x)), x e K
Let tf 4 (,' i"',m) and Fr diag X. Let J, (T, OIIm and
J2 (I[, il, diag X). Then, as can be seen from Figure 3. 4 2,
(F, H, KH ) is J3  feedback related to F, H, T(K)) on T(O) by
J = J2 J,1. Direct computation shows that J2~J, = (T1, ir 1.
The form of the standard decomposed system, the form of tl, r,
and Definition 3. 1. 5 imply (F, H, X EI is decomposed at xO. 0
( F, H, )
Jo _^ [ Ioj (D.)'A. (D (D*)I
(F*, H*, K
J1 =IT,O, 0 I / 2 [ It, diag X)
F, H, T(HK) > F, H, H )1
J3 = J2 J1 = J Tl, r
Fig. 3. 4. 2. Relationships between F, H, K }, ( F*, H*, K F, H, HK )},
and (F, H,T(H)).
103
This result has other implications. Recall that If (F,H,,X)
is decomposed at xO e X, then, there exists an open neighborhood
E of xo such that (F, H, E) is decoupled on E Therefore, Theorem
3.4.2 shows that under its hypotheses, r"((F,H,X) is a class of
smooth control laws which decouple (F,H,X) at least locally
around each point xO e X
If (F, H, ) is a standard decomposed system, we might
expect intuitively from its special structure that its decoupling
control laws are of the form u = (x) + X(xj)ui, i e fl t. In
the next Theorem, we show that this is really the case. Before
doing so, we formalize the class of control laws.
Definition 3. 4.3. Let (F,H, X } be a standard decomposed
system. 2"( {F, H, X i ) ( (F, R, ) ) is the class of control
laws =(x) + Bp(Rx) satisfying (A5) ((A.5) ), (A6), and
(3.437);[(x)  fii(xr), p(x) = diag X(xR),
where., X are functions from XX into R, i e1t,. 0
Theorem 3. 4. 3. Let ( F, H, X) be the standard decomposed
system in Definition 3. 3. 1. Suppose that (F, H, X satisfies (A. 1)'.
Suppose that class of control laws satisfies the following
104
assumptions: (A5)', (A6), and for u  (x) + (jx) u in the class,
F, H, ) satisfies (A.2). Then, the control law u = (C) + (R) u
decouples (F, H, on X if and only if it belongs to'( ( F, H, ) ).
Proof. Suppose that a control law I = (Rx() i+ p belongs
to SV( (F, H, X ). Then, since (F, H, fX ) is decomposed on X, it
is decoupled on x
Next, suppose u = <(x)+ p(x) u decouples (F, H, X) on X.
Direct computation shows
(3.4.38) D*(x) = Im, A*(x) = 0 on X
By Theorem 3. 4. 1, this implies that i, p must have the
following properties:
(3.4.39) t(X) = l(xl) p(x) = diag X(x), on,
(3.4.40) di, dX s6 ( (FH,.X ) on X, i s tIl,
(3.4.41) X(x) 0, xi 5s, i e 1m.
Direct computation using the property (1) of IF,H,X) in Definition
3. 3. 1 shows that if Y X ~,() a/a3x belongs to
j=( K=l Jk j
aj( { F, H,,), (3.3.67) must hold. This and the property (3) of
F, H,X) imply that if a covector field fi  f t j, (') dx.
~~~~~~~~~~~~'; =. (<=1 J.k d Jk
105
belongs to (F, H, R } ) on X,
(3.442) 6J x) = 0. x e X, k e t11pj if j ~ i.
Since X is connected, this and Theorem 2. 3. 8 imply that any C 
function IP from X into R satisfying (3.4.40) must be the function
of x only. D
Note that the property (3) of the standard decomposed
system is essential in obtaining Theorem 3. 4. 2. Just as with
Theorem 3. 4. 1, we are not able to prove that Theorem 3. 4. 3
extends to smooth systems and smooth control laws. But, we can
show that S~( (F, H, X)) is the whole class of decomposing
control laws for the smooth standard decomposed system F, H, X).
Theorem 3. 4. 4. Let IF, H, X) be a standard decomposed
system in Definition 3. 4. 1. Suppose that ( F, H, X) satisfies (A.1)
and class of control laws satisfies (A.5) and (A.6). Then, the
control law u = &(() + p(x) decomposes ( F, H, X) on X if and
only if it belongs to S( F, H,X)).
Proof. Let u = >(x) + (x)' be a smooth control law in
S( ( F H,, ) ). Obviously, [ F, H,'X), is decomposed on X
Next, suppose that a control law u = >(x) + ~(x) i' decomposes
106
(F, HX on X. This Implies that (F, H,,X) is decomposed at
each xo e. Let (F, HX) F, H, ). Let i, 1 6 o, be
vector fields corresponding to (F,, X). Then, by Theorem 3. 2. 2
and Lemma 3.2.2  (1),
(3.4.43) Xj Xi X12 XiH1 0 on X,
iq e (o, i], q s tlA, k e., j e p, i e lrn,
Using this and (3.4.38), we can show that (3.4.39)(3.4.41) hold.
The arguments are very similar to those following (3.4.22) except
for minor differences in notation. Once (3.4.39)(3.4.41) hold, the
remaining arguments are exactly the same as those following
(3.4.41). 0
The control laws In the sets sI ( F, H, X)), "( F, H, X )
are closely related to those in the sets W({ F, H, X)), ({ F, H, X) ).
We show that for a class of nonlinear systems, there is a one to
 one correspondence between them.
Theorem 3. 4. 5. Suppose that the hypotheses of Theorem
3. 3. 3 are satisfied. Let xo 6 X. Let E, T, (F, H,IX be the
open neighborhood of x, the mapping, and standard decomposed
system given by Theorem 3. 3. 3. Then, there exist an open
neighborhood K c ~ of xo such that
(i) For every u =>(x)+ j(x)u in S'( { F, H, ) ), there exists
107
a unique control law UT <(x) + px) In ( F, H, T(LK) )
such that ( F, H, T(K) J] is T  related on K to ( F, H, IH 1".
Conversely, for every u a(x) + P(x) u in s "(F, H, T(K) )),
there exists a unique control law u = <(x)+ B(x) In
S((F, H, K ) such that (F, H, K ) is T1  related on T(K)
to (F, RH, T(K) ],A
(ii) Let u  cc(x) + p(x), = c(x) + p(x) i be control laws in
( ( F, H, K ) ), S'( (F. H, T(K) ) ), respectively. Suppose they
are in the one  to  one correspondence described In (1).
Then,
(3.4.44) <(x) = [D*(x)]'((T(x))  A*(x)), p(x) = [D*(x)]p(T(x)).
(iii) In particular, when T is a C'  diffeomorphism on X
and X is connected, the above (1), (11) hold with K = X
Proof. First consider (i). Suppose u = <(x) + (x) u belongs
to S( (F, H,K) ). Then, following the second part of the proof
for Theorem 3. 4. 3 leads to the fact that there exist an open
neighborhood K c ~ of xo and C functions Ij, X, defined on
appropriate subsets of RPI, i e6 fl such that (3.4.36) holds.
Note that given T and K, the i and X are unique. Define <
(',,...,, ) and p  diag X.. Then, u = (x) + p(x) u belongs to
108
r ( F, H, T(K ) ). Furthermore, (F, H, T(K) J]E Is T  related on K
to (F, H, K )E. Next, consider the converse statement. Suppose
u = (x) + p(x) U belongs to S( [ F, H, T(K) ). Define <, p by
(3.4.44). Then, by (3.3.60), it follows that u <(x)+ p(x)u belongs
to'( (F, H, T()) ). Clearly, (F, H, K ])e Is T  related on K
to (F, H, T())J.
Part (ii) has been shown implicitly above. Part (iii) follows
from that given hypotheses imply that (3.3.59), (3.3.60) hold on X
and T(X) is connected. By the arguments similar to those
following (3.2.44), (3.4.36) holds globally on X. 0
Remark 3. 4. 4. See Figure 3. 4. 3 for a schematic description
of Theorem 3. 4. 5 Systems (F, H, T(K) )}e described in (i) of
Theorem 3. 4. 5 have the forms:
(3.445), =A(x,.,),,' +(x,),,),
(.4)g (x)2 ) XjX,) j(X)t1
(3.4.46) X=f = )+ib1i,511ux) + ";l"' bxu.
Thus, part(i) characterizes the class of closedloop locally decomposed
or decoupled systems. Part (ii) shows connection between a given
closed  loop system and a feedback control law. 0
109
Remark 3. 4. 5. Since (F, H, T(K ) })1 Is T  related on IL to
( F, H, It ), the solutions of the differential equations for the two
syatems are related by T ( i. e., (t) = T(x(t))). Also the two
systems have the same input output maps. When K  X, these
results are valid globally on X O
Remark 3.4. 6. For a standard decomposed system (F, H, Xi
in Definition 3. 3. 1, let
(3.4.47). *  { (Xi....,xi ) e +1:
x A (,,. —, X m, Xl ) e. i e t,.
Define Si( {F,H,X )(S (( F,H,X )) by a set of all control laws
J (Io, pI}
J= Io), p)
(F, H, T(it) >; ( F, H, T(H) Fi
standard decomposed system
Fig. 3. 4. 3. A schematic description of Theorem 3. 4. 5, where u =
<(x) + p(x) O, I = (x) + (x) u are control laws in
S( ( F, H, K ) ), S;( [ F, H, T(t) ) ), respectively.
110
u = a() + p(x) satisfying (A5) ( (A5)' ), (A6), and
(3.4.48) 4( 7 ),(XI....X1+ ),
1m( YXm 1 * Xm n+l )
(3.449) p(x) diag X( x..,. x ^,+l )
where A, X are arbitrary functions from X* into R. Clearly,
r(tF, H, ) is a subset of "(f F, H, ). Let ( F, H, X be a system
which satisfies the hypotheses of Theorem 3. 4 5. All statements
in Theorem 3.4.5 still hold with V((F, H, tK ), ~( ( F, H, T(K)})
replaced by S' ( F, H, I )), S (( F, H, T(K)} ), respectively. 0
Remark 3. 4. 7. Suppose the hypotheses of Theorem 3. 3. 1,
(3.3.4), and n = 2(dd+l ) are satisfied. Then, the hypotheses of
7=i
Theorem 3. 4 5 are satisfied trivially. In particular, p, = d + 1, i e
ln,, and T is given by T ^ (Ti,...,Tm), where Ti E (Ti,..'
T. d+ ), and Ti. A Xo(1) H.. Then, (ii) of Theorem 3. 4. 5 shows
that at least locally, S( F, H, X ) = ( F, H, X ). When T is
a C diffeomorphism on X and X is connected, (iii) confirms that
S( {F, H, %) ) = ( ( F, H,.%} ). Note that for this case, we do
not need to solve the partial differential equations (3.4.11) to
characterize S"( t F, H, X ). But, if T is not a C'diffeomorphism,
S~((F, H, X] ) = S'( (F, H, X ) is not necessarily true. This will
be shown through Example 3. 5. 2 in Section 3. 5. 0
1ll
In this section, we have presented results concerning questions
(b), (c), (d) In Chapter 1. They are described in Figure 3. 4. 4 n a
schematic way. The simplicity of the results for standard
decomposed systems, together with Remark 3. 4. 4 and 3. 4. 5,
suggests that in system design it may be easier to deal with the
standard decomposed system than with the original system. But,
it should be noted that in order to transform the original system
into the standard decomposed system, we have to compute a
mapping T ( see Theorem 3. 3. 3 ). Computing the mapping T is
usually a difficult job since it is basically equivalent to solving a
set of the first order linear partial differential equations.
3. 5. Examples
In this section, we present three examples which illustrate
the significance of the results developed in the previous sections.
Example 3. 5. 1 is a real analytic system (F, H, R3 ) which is
decouplable and decomposable on R3. For this example, S?( {F, H, X } )
is a proper subset of Sw( {F, H, X ). While there is no control
law in S8( ( F, H, R3 I) which decouples ( F, H, R3 ) on R3 with Bounded
Input  Bounded State ( BIBS ) stability, there are many control laws in
Sw( [ F, H, R3 ) which decouples ( F, H, R3 } on R3 with BIBS stability.
Example 3. 5. 2 shows that n = E ( di+1 ) does not necessarily imply
S( { F, H, X ) = 8( { F, H, X) ). For this example, T defined in Remark
3.4.7 is a C"diffeomorphisrn locally at each point of R3 but not globally
112
u a <(x) + p(x) u decouples u a ((x) + p(x) i decomposes
(F, H,X) on I(F,H, X) at each x eX
Th. 3. 4. (A. 1 )',(A2),(5)', Th. 3. 4 (A. 1 ),(A.3),(A4),
(A6),(A7) (A5),(A6),(A7)
<(x) = [D*(x)]1 ( I(X)  Ax) ), u = (x) + p(x) 0
P(x) = D*(x)] 1 diag X(x),  belongs to
where dl., dX e a( F*, H*, X)). Sor r
Th. 3. 4. 5 onetoone correspondence (A 1 ) (A.3),(A4)
(A5),(A6),(A.7)
between (,BP and.,
(x) = ( 1(X),...,(Xm) ) u = (x) + P(x) u
(x) = diag X(xj). belongs to SW or r
Th. 3. 4. 3 (A 1 )',(A2), Th. 3. 4 (A 1 ),(A5),
(A.5)',(A6) (A.6),
U = ) + p(x) u'K' decouples u = (x) + P(M) I decomposes
a standard decoupled system a standard decoupled system
(F, H,X) on. (F,,X) on X
Figure 3. 4. 4. Summary of main results in Section 3. 4 showing
assumptions required for each implication.
113
on R3. Thus, this example shows that If T is not a C" diffeomorphism,
S( ( F, H, R3 ) ) = S F [ F, H, R3 ) is not necessarily true.
Example 3. 5. 3 was considered in [Sin. 1] We show that for
this example, T defined in Remark 3. 4. 7 is a C  diffeomorphism
on R3 and hence SI(( F, H, R3 ) =?(( F, H, R3 }). In [Sin. 1], a necessary
condition for a control law to decouple ( F, H, R3) is given in a form
of partial differential equations and a class of decoupling control laws
is specified. We give a more complete solution for this example.
Example 3. 5. 1. Let us consider a real analytic system
(F,H, R3 with m  2 and
(3.5.1) Xo(x)  ( 2 + x 3) /x2,
(3.5.2) X1(x) ^ a/ax1 + (1 + x  x3) ad/x2  a/ax3,
(3.5.3) X2(x) = a/x, + (1 x3) a/ax2,
(3.5.4) H1(x)  x1, H2(x) I x + x3.
Direct computation shows that all hypotheses in Theorem
3. 3. 1 and (3.3.4) are satisfied with
(3.5.5) d1 = = 0, D*(x) = 1 1, A*(x) = 0.
L 0 1
Thus, by Theorem 3. 3. 1 and Theorem 3. 3. 2, (F, H, R3 is
decouplable on R3 and decomposable at each xo e R3.
To characterize the whole class of real analytic decoupling
114
control laws, we have to compute A( 1 F", H*, R3)), 1 e.,2.
Let j, i e to.2 be the vector fields corresponding to the
decoupled system { F", H*, R3). Then, by (3.5.5), we have
(3.5.6) o(x) (x2 + x, x3) a/ax2,
(3.5.7),(x) = a/ax, ( * xlx3) a/ax2 a/a3,
(3.5.8) X2(x) =  x a/ax2 + a/ax3.
From these, we can compute
(3.5.9) L%(x) = L^ (x) = L'I(x) =0 L0'L^X1(X) = 0,
(3.5.10) L'sX1(x) =  a/ax2, L^Xl(x) = a/ax2.
From these, it is easy to see that on R3,
(3.5.11) A( F*, H*, R3 ) span X2 ),
(3.5.12) A2( I F*, H*, R3) = span {X1, L^1 ).
These with (3.5.5) determine Sw' ( F, H, R3 ) ).
Note (3.5.7) (3.5.10) imply (A.3). On the other hand, by
(3.5. 11), (3.5.12),
(3.5.13) dim. A([ F*, H*, R3) = 2, dim.( ( F*,H*,R3}) =.
Thus, (A.4) is satisfied by p, = 2, P2 = 1. Consequently, all
hypotheses of Theorem 3 4 5 are satisfied. Define Cu functions
115
TJ, J e n,,, I e n, by
(3.5.14) T1,l(x) H,(x), T,l(x) i x2 + x3, T2.1(x) H2(x).
Let T A (T.T1, T, ). Then, we can easily show that T is a
CW diffeomorphism from R3 onto R3 and (dTij(q), j B li, ) is a
basis of ()( ( F*, H*, R3), q e R3, i n, Let [F, H. R3
be a standard decoupled system whose coordinate representation is
(3.5.15) [X0,1 [ 0 + ul Y = X
X2 U2, Y2 X2
Then, we can check that the above T and {F, HR3 ) with ~ = R3
are those described in Theorem 3. 3. 3 and Theorem 3. 4. 5.
By (3.444) and (3.5.5), Se( F, H, R3 ) is given by
(3.5.16) c(x) = (I(x,, x2 + xx x3) 2(x, + x3)
2(1 + x3)
(3.5.17) P(x) = (' (xl, x2+ x x3): 2(X + Xg)
O: 2(x1+ x3)
where *i,'i, i elt.2 are arbitrary C"functions of their arguments
such that,(z,, z2) o0, (z,, z2) e R2 and T,(z,) O, z, e R. On the
116
other hand, by Definition 3. 4. 1 and (3.5.5), S(( F, H, R3 ) is
given by
0O
(3.5.18) &(x) =' t(x) 2x + x3)
(3.5.19) p(x)= l(x,): 2(X1x3)',
0 2(x1 + X3)
0 3
where i.,'P, i e t12 are arbitrary CW functions of their
arguments such that rp(z) O, z e R, i e L1,2. From (3.5.16)
 (3.5.19), we see that Sw( [ F, H, R3 ) ) = S( ( F H, R3 ) ) but
So( (F, H, R3) ) c S( F, H, R3 ).
By Theorem 3. 4. 3, S( (F, H, R3 ) is given by
(3.5.16)' C(x) = I,(x1., x1),
( J2(X2)
(3.5.17)' (xR) = C1(x1.1 x12) 0
0 2(X2)
where i,'i, i etl,2 are arbitrary C functions of their arguments
such that 1(x,, x2) 0, (pX, x2) e R2 and I2(x3) 0, x3 e R. Note
that as is indicated by Theorem 3. 4. 5, there is one  to  one
117
correspondence between the control laws of Sw ( F. H, R3)) In
(3.5.16), (3.5.17) and those of'( (F,H, R3 ) in (3.5.16)', (3.5.17)'.
Using the standard decomposed system, it is easy to see how
to choose control laws which decouple { F, H, R3) in a stable way.
Suppose we want to decouple F, H, R3) on R3 with BIBS stability.
First, consider = (x) + p(x) G where a, p satisfy (3.5.16)', (3.5.17)'.
Let ( f,, R3) be the coordinate representation of ( F, H, R3 ].
Then, fi, R3 is described by
(3.5.20) x1. (= X xx) x 1u Y, X,
X2 l1,2 + l(X.l' X1.2 I (x 1U X 1.2
2 *2('X2) + ('2) Y2 X2
Note from property (2) of Definition 3. 3. 1 and (3.1.9) that
i1 hl, R2), f2, h2 R ) in (3.5.15) are controllable linear systems.
Therefore, there are many choice of i, +2 so that IF.R3)' is
decoupled on R3 with BIBS stability. For such a control law u =
>(x) + p(x) u, choose a control laws u = >(x) + p(x) u by (3.443).
Then, { F, H, R3 ]4 is T  related on R3 to ( F, H, R3 ) h. Recall
that T is a C@ diffeomorphism on R3. Furthermore, by a special
form of T in (3.5.14), it follows that for any constant b,
(x e R3: IT(x)l b b ) is bounded. These observations imply that
(F, H, R3)3 is decoupled on R3 with BIBS stability. Thus, we
have shown that there are many control law u =:(x) + p(x) u in
118
S( [F, H, X])) which decouple (F, H, R3) on R3 In a stable way.
Next, consider u = () + p(x) i in i( ( F, H, R3 )). Then,,, p
have the forms
2(2) j ~ (2)
where *, i'P, 1i e H are arbitrary C4 functions of their
arguments such that I(z), 0, z e R, i e?1 2. Now, let u  (x)
+ p(x) u be a control law in S?( (F, H, R3 ). By Theorem 3. 4. 5 and
Remark 3. 4. 1, we know that for each u = (x) + p(x)u in
S( { F, H, R3), there is a unique control law u = (x) + (x) tu in
S( ( F, H, R3 ) )such that (F, H, R3 ), is T related on R3 to
(F, H, R3 )1. Let ( f, h, R3) be the coordinate representation of
F, H, R3,p. Then, (,, R3 is described by
(3.5.20)';, = Y,' =11,
L 12 1.1 1 i' L'
X1,2 X1.2 + ( ) (,l )
0
X2 352(2) w+ 2( t2 Y2n
From (3.5.20)', we see that there is no * and' such that for
every bounded u, xX,2 is bounded. By the special structure of T
in (3.5.14), this implies that there is no u = &:(x)+ p(x)ui in
119
S( ( F, H, R3 ) which decouples ( F, H, R3 ) on R3 in a stable way. O
Example 3. 5. 2. Let us consider a real analytic system
(F,H,R3) with m = 2 and
(3.5.22) X0(x)  a/ax2,
(3.5.23) X1(x)' cos x2 a/ax, + sin x2 a/ax2, X2(x) = a/ax3,
(3.5.24) H,(x)' eiX sin x2 H2(x) = x.
Direct computation shows that all hypotheses of Theorem 3. 3. 1 and
(3.3.4).are satisfied and
(3.5.25) d1 = 1, d2 = 0, D*(x) = ei 0 A*(x) = ei sin x2,
0 1. 0
Let Xi, i e o.2 be the vector fields corresponding to the
decoupled system ( F*, H*, R3). Then, by (3.5.25), we have
(3.5.26) XO(x) = (.5 sin 2x2) a/ax, + (cos x2)2 a/ax2,
(3.5.27) i(x) = (eXi cos x2) a/ax, + ( eXi sin x2) a/ax2,
(3.5.28) X2(x) = a/ax3
Note that since 3 = 2 di + 1, (A.4) is satisfied by p, = 2, 2 = 1.
Define functions Ti: R3  R, j e6 NM(di+,' i E6 M12 by
120
(3.5.29) T, (x) = x,, Tl2(x) ^ x2, T2.(x) ^ x3.
We can check easily that at each q e R3, (dTij(q), j e t1,l ) is a
basis of (A)( ( F*, H*, R3 ), i e t1.2 and T ( T11,, Tl.2. T2 ) is a
Cu diffeomorphism from R3 onto R3. Thus, for each i e 1,,
Lemma 3. 2. 4 holds with r = n  p, A = A,, Sj. Tij, X = U = E =
R3. From these observations, Definition 3. 4. 2, and (3.5.25),
"( ( F, H, R3 ) is given by
(3.5.30) ((x)= (ex Il(x. 2) sin X, p(x) = eX1 X(x,. x) o
(l2(X3) Xo L (X3)
where ti. X., i M1.2 are arbitrary C functions of their arguments
such that Xl(x1, x2) * 0, (xI. x2) e R2 and ((x3) ( 0, x3 e R.
But, (3.5.30) can be more simply described by
(3.5.31) <(x)  l(xl'x2), p(x)'0,(Xi.x2) o
2(x3) o 2(X3)
where 6i. Ti, i e,t.2 are arbitrary CO functions of their
arguments such that Al(x,. x2) * 0, (x,, x2) e R2 and T2(x3) 0, X3
e R. But, by Definition 3. 4. 1 and (3.5.25), S( ( F, H, R3 ) ) is
given by
121
(3.5.32) &:(x) eX IT( e il sin x2. eX cos x2)  sin x2},
2(x3)
(3.5.33) p(x) = eXi X( ex sin x. el cos x ) o
o 4(X3)
where.. X, i et1t are arbitrary C"functions of their arguments
such that XL(x,.X2) = 0, (x,.x2) e R2 and A(x) 0, x3 e R.
But, (3.5.32), (3.5.33) can be more simply described by
(3.5.34) >(x) = I( exl sin x2. el cos x2),
2(X3)
(3.5.35) p(x) = [ 1( el sin x, e'l cos x2):
0: (X3)
where $. p, i e tL, are arbitrary C  functions of their
arguments such that P(xl. x2) * 0, (xX. x2) e R2 and 2(x3) O, X3
e R. Note that F(x,, x2) a (eXlsinx2. el icosX2) is a C@diffeomorphism locally at each point (xl, x2) e R2 but not a C@diffeomorphism globally on R2. Therefore, there does not exist a
C@ function Ah such that +)( e'l sin x2. e'l cos x2) = x2', x2
R. Thus, although the class of control laws given by (3.5.31) is
122
locally equivalent to the one given by (3.5.34), (3.5.35), they are
not globally equivalent. Thus, we have shown that although, (dl+l) * 3, S?((F,H,R3)) is only a proper subset of
t(IF H, R3)). 0
xample 3. 5. 3. Consider a real analytic system F, H, X)
with m  2, X xA (x(xl, x2, x3) e R3: 2>0), and
(3.5.36) X(x)' x1x2 a/x3,
(3.5.37) X,(x) a/ax,, Xz(x) a/ax2,
(3.5.38) H,(x) x2, H2(x) x3.
Then, we have
(3.5.39) d  O, d2 1, D*(x)  1, A*(x) O.
x2 xi
Note that X is connected and I (d + 1)  3.
Define functions Tj: R3  R, j 6 1da+1). i 8 112 by
(3.5.40) T1.(x) ^ x2, T2,(x) = x3, T22(x)' xx2.
Note that Tj = Xo(j1)H, j 6e np, i e,. Clearly, T,(TT,,
T21, T22) ls a C'  dlffeomorphlsm from X onto X  1( e R3:
123
x, >). By Remark 3.4.7 and (3.5.39), P(IF,H,X)) ?([F, H,X)) and is given by
(3.5.41) <(x) a' t2( xX2, x3 )  x, 1(x2) / x2',
(3.5.42) j(x)  x xl(2)/xX X 3)/x
x,(x2) 0
where i,. i,, 1 11.2 are arbitrary C@ functions of their
arguments such that XA(x,) O, xe R2 and A2(x2. X) O, (x2 x3) e R.
To compare our solution with the one given by Singh and
Rugh([Sin.1]), we consider the partial differential equations given
by (3.435). It should be noted that as is pointed out in Remark
3. 4 3, (3.435) is not a sufficient but a necessary condition for a
control law to decouple a system. Through some calculation, we
can obtain that c, p solve (3.435) if and only if they satisfy
(3.5.41), (A5), and
(3.5.43) p(x)  x xx, x2, x3 )/x2 2( xl, x2, X3 )/X2 1;k1( xp, x2, x3 ) 0
where X, i e 1n are arbitrary C" functions of their arguments
such that (x1, x x2,)O, ( x1, xi, x3) e R3. A control law u =
124
(x) + (x)t satisfying (3.5.41), (3.5.43) Is not necessarily a
decoupling control law. This can be verified by comparing (3.442)
and (3.443). In [Sin. 1, the following class of control laws Is
proposed as a class of decoupling control laws:
(3.5.41)' <(x)' {2(X1X2)  xI it(x2)) / x2'
~L hq(X2)
(3.5.42)' p(x)  x1/X2 l/x2,
L I 0
where I. I e n,2 are arbitrary C" functions of their arguments.
Clearly, this class of decoupling control laws is a proper subset
of S( { F, H, X)) In (3.5.41), (3.5.42). This example shows that
the condition on < which (3.435) yields is the same as Theorem
3. 4. 1 does. But, this may not be generally true. 0
3. 6 Conclusion
In this chapter, we have presented various results on
decoupling and decomposition of nonlinear systems. Some of them
are refinements or elaborations of previously known results. They
are: (a) the definitions of decoupling( Definition 3. 1.3 ) and
decomposition ( Definition 3. 1. 5 ); (b) a necessary and sufficient
condition for decoupling ( Theorem 3. 2. 1 ); (c) a necessary and
125
sufficient condition for decomposition ( Theorem 3. 2. 2, Theorem
3. 2. 3); (d) a necessary and sufficient condition for decouplability (
Theorem 3. 3.1 ); and (e) a necessary and sufficient condition for
decomposabllty ( Theorem 3. 3. 2). We have clarified and / or
simplified these known results. This includes the elimination of
redundant conditions and proofs for the necessity parts of some of
the theorems.
Completely new results are: (1) the characterization of a
class of nonlinear systems which are J  related to the standard
decomposed systems ( Theorem 3. 3. 3 and Theorem 3. 3. 4); (2) the
characterization of the whole class of decoupling control laws
(Theorem 3. 4. 1 ) and decomposing control laws ( Theorem 3. 4. 2);
(3) the characterization of the class of decoupled closed  loop
systems ( Theorem 3. 4. 3  3. 4 5). We have distinguished them in
the summary Figures 3. 2. 1, 3. 3. 2, and 3. 4 4 with an asterisk.
The new results contribute to the questions (b), (c), (d) in Chapter 1.
They provide a deeper and clearer understanding of nonlinear
decoupling theory. They provide information about the flexibility
we can have in the design of decoupled control systems.
A difficulty exists. In most of these results, it is generally
required to solve a set of the first order linear partial differential
equations. It is not always possible to find the closed forms of
solutions of these partial differential equations. This difficulty is
shared with all other literature on the differential geometric
approaches.
Finally, we would like to emphasize again the practical
126
importance of a standard decomposed system. Suppose we have a
system and there exists a control law such that through an
appropriate Input and state transformation ( a J  feedback relation,
Definition 3. 1.2 ), the system with the control law can be described
as a standard decomposed system. In this case, the design of
decoupled control systems becomes much easier since we can deal
with the standard decomposed system instead of the original system.
This advantage comes from the simplicity of the results for
standard decomposed systems, as Is pointed out in the last
paragraph of Section 3. 4. Specifically, the class of decoupling
control laws for the standard decomposed system is given by
(3.437) ( see Theorem 3. 4. 3) and for each decoupling control law in
this class the decoupling control law for the original system can be
obtained through the J  feedback relation ( see Theorem 3. 4 5 ).
In general, the J  feedback relation which transforms the original
system into the standard decomposed system requires the solutions
of a set of first order partial differential equations. However, in
some applications the J  feedback relation may be found by
inspection or rather simple manipulation of the dynamic equations
for the original system. This is the case for the robotic
manipulators in Chapter 5.
CHAPTER 4
APPROXIMATE DECOUPLING
In practice, some degree of modelling error is unavoidable.
Therefore, it may be impossible to achieve'exact" decoupling in
the sense of Section 3. 1. Even when the exact model is
available and decouplable, it may require decoupling control laws
which are computationally complex. Thus, it may be more
practical to have control laws which require less computation but
decouple the system "approximately" in some sense. In this
chapter, we neglect fast dynamics of a system to obtain simpler
decoupling control laws and investigate the effect of the neglected
fast dynamics on the decoupling of the actual system. Section
4. 1 contains notation and assumptions, under which we state a
result on approximate decoupling in Section 4. 2.
4. 1. Notation and Assumptions
In the previous chapters, we have considered systems defined
on manifolds, which are not necessarily open subsets of Rn. To
simplify developments in this chapter, we consider only the class
of nonlinear systems defined on open subsets of R". Consider the
following system, denoted by ZI,
127
128
(41.1) x  bo(x) +2 gj(x) zj + bj(x) U, y h(x),
(41.2) X  A() z + Bo(x,X) + I Bj(x, X) j,
where: X is an open subset of R" containing the origin; X is a
0
positive constant scalar and X e [0, X]; gj: X 4 RI, j e ttl
bj: X R", j e,,; Bj: X x [0,X] > Rr, j e tom; A:
[O, ] 4 R"!; h: X 4 Rm. We assume:
(B.1) A(O) is a stable matrix.
The degenerate system of 2, denoted by 20, is, rr i
(41.3) x = bo(x) + I gJ(x) z + b(x)uj, y = h(x),
 j= 1( J j= i J
(41.4) 0 = A(O)z + BO(x,O) + Bj(x, )uj.
By (B.1), A(O) Is nonsingular. Consequently, 10 can be written as
3n
(4.1.5) = f(x) + i f(x)u, y = h(x),
where
(4.1.6) f,(x) b,(x)  [ g(x)  gq(x) ][A(O)]'B(x, 0), i e no,.
Note that even when Zx is not decouplable on X x pR, Zo may be
decouplable on XW Suppose that the degenerate system 10 of of
129
Is decouplable on X Let u = ((x)+ p(x) u be a control law which
decouples 2I on X Let 25 be the feedback system of 2,,
corresponding to the control law u  c(x)+ p(x) u. Then, we can
describe lt by
(4.1.7) x = gA(x)+ 4 i (x)Z 2 fy h(x),
(41.8) XA = A(X)z + Be(x, ) + Bj(x, X)uj,
where
(41.9) g,(x) = bo(x) + [ b,(x)' bM(x) ]I(x),
(41.10) tj(x) [b(x) bi(x)(x) b (X j(x), j e,
(4.1.11) BO( (x, ) [ BB(x, ) + [B(x,) Bm X)]((X),
(4. 1.12) B,(xX,[
(4.112) Bj(x, ) [B,(x,X) B m(x,X)]j(x), j e In,
(4.1.13) h(x)  h(x), j(x) gj(x), j e l,
and pj is the jth column of B. Clearly, the degenerate system
of 2j, denoted by 29. Is decoupled on X but 2J may not be
decoupled on XxRr. Let L be a positive constant. Let F e Rr.
Define norms I I1, I ILI III by
(4.1.14) Ix(t) l^( Ix(t)12 )/2, Ix max. [Ix(t)l: t e [O,L]),
II F II = max. (IFzl: z e Rr, Izl = 1 }.
130
Let x*: [0, L]X be a nominal solution of IZ for a nominal
initial state x*(O) = x* e X and a nominal input u* e Ut. For
positive real numbers X and'j, i e t 4, define sets Uu., Ri,
zot, and Rt, by
(41.15) Uu* 4 [u e U: luu*lL. 21, lulL } 22,
(41.1.6),' (X e X: Ixx* I 3 ),
(4.1.17) LZ0 (ze Rr: Izl i i4), R. X (X eR: 0 X i) }.
We further assume
(B.2) There exists 05 > 0 such that for all x(O)e lx, and u e su,,
X2' has a solution x: [0, L] 4 X satisfying I x IL 5
(B.3) (1)bJ BJ, j es ot are C,
(2)gj, j e l,. are C',
(3) A, h are C.
Note from [Gil.3] that (B.3) implies that (B.2) is true for sufficiently
small 2V, 3. We denote by *( 0, x, zo, X) the ith output, jy(t) of
Co for an input 0 e Uu., initial conditions x(0) ^ xo e.,, z(0)
 z e zR., and X e R.' Now, we are ready to state a result
for approximate decoupling.
131
4 2. Result for Approximate Decoupling
Theorem 4. 2. 1. Suppose that (B.I), (B.2), and (B.3) are
satisfied. Suppose that a control law u = ((x) + p(x) satisfies
(A5) and decouples 20. Then, there exist positive real numbers a,
X, such that for every X e RX,, every input u e U,,, and any
initial conditions x(O) 4o xs., z(O) A z e I,z
25 has the following properties:
(i) 25 has a solution (x, z): [0, L]  X x R,
(ii) For each i e l, and for any two inputs u, u' U, such
that Uj = Ui,
(4.2.1) IIU Xo Zo )  A ( U x Z,,, X)1 < X. o
This Theorem shows that if X is sufficiently small, the
control law which decouples the degenerate system still decouples
the original system in an approximate way described in (ii). Now,
we give the proof of Theorem 4. 2. 1.
Proof of Theorem 4. 2. 1. Let x, z be the solutions of the
degenerate system 2^ in (4.1.7), (4.1.8) for x(O) 4 xo e RA and
O = u e u4
(4.2.2) x = g(X) + (X) + f(x) u, x(O) = x
j, I j J
132
(4.2.3) z =  [A(0)]1( Boi, 0) +. 8j(x, 0) u].
Here, and often in the future, the explicit dependence on t is not
shown. From (4.2.3),
(42.4) X z = A(X) + B0(x, X) + I Bj(x, X) u + X Ko(x(t), u(t), u(t), X),
where
A A
(42.5) K,(x(t), u(t), u(t), X) ^  [A(O)]1 { DB(x,0) +J DB(x,O) u)
1(X) + ^ gjf(x) z + () u )  Z [A(O)tl.x, 0) u +
[ A(0) A(X) I)) + X( Bo(, 0)  o(x, X)l +
X"1; (8j(x, 0)  Bj(xX))Uj.
For simplicity of notation, we henceforth write Ko(t,X) instead of
Ko( (t), u(t), u(t), X). This kind of notational abuse will often
appear in what follows.
Let tI be the solution of the following differential equation
(4.2.6) 1 = XA' A(O) T, 1(0) 4 z(0) z(0).
The solution is
(4.2.7) I(t) = eA(O) VX(z(0)  2(0).
But by (B. 1), there exist od,, > 0 such that
133
(42.8) II e^A(~) II s 6e
Let x, z be the solutions of 2 in (4.1.7), (4.1.8) for x(0) 
x0, z(O) z0, and u = u. Let V  x  x, S 4 z  z  1. It
should be clear that the variables V, S are the functions of time t
depending implicitly on X, xo, zO, and u. From (4.2.4), (4.2.7),
(428), we obtain the following differential equations.
(42.9) V=W( t, V, 5, ) + K,(t,X), V(0)=0,
(42.10) S =' A(O) 5 + A1 W2(t, V, 5, X) + K2( t, X), S(0)= 0,
where
(4.2.11) W,(t,V,S, A)(A(x+V)^(X)] + gj(V)()
j= 1 s J J'
(Zj+ lj) +~ A(+ V) Sj + [ j(X + V) fj(R) 1Uj,
(4.2.12) KI( t,, ) = Z 9j(X) tlj,
(42.12) K+t.X^S ^(x)^.
(4.2. 13) W2(t, V, S, X) ^ { A(X)  A(O) S + [ Bo(x + V, )  Bj(x, X) +
j= (Bj(x+, X)  Bj(, )) uj,
(4.2.14) K2( t, X) = X1 A(X)  A(0)  Ko( t, X).
Choose 7 > 0 so that
(4.2.15) X(t)* V(t) e X, t e [0, L] if IVI,  ~, and IXI, ~ 5.
134
Define the sets Av, 1t by
(4.2.16) Vt ^ (it R": Itl < 7}, = (t 6 Rr: It1 < l 7).
We show that the solutions V,S of (4.2.9), (4.2.10) can be kept
within YV, ss, respectively. Then, as long as the trajectories of
V, S stay in the regions Rv, Is, respectively, by (B.2), (B.3),
(42.3), (42.7), (4.2.9), (42.11), and (4.2.12), there exists 28 > 0
uniformly with respect to 9l,. 9.,, U,., tv, and R9 such that
(4.2.17) IV IL 2Define V, S by
(4.2.18) V(tX) t JK K1(r,X) di,
(42.19) S(t, X) A to e(O) (  K)/ 2(, X) d'.
Then, by (B.2), (B.3), (42.5), (4.2.7), (42.8), (4.2.12), and (4.2.14),
there exist 2g, lo > 0 uniformly with respect to Rax, RZo, UUM.,
RO,'v, and.ts such that
(4.2.20) I VIL X9, 151 L A 2,.
135
On the other hand, by (B.2), (B.3), (4.2.11), (4.2.13), there exist
I11' 212 > 0 uniformly with respect to;:, O,Z Uu1, Rt, RV,
and Is such that
(4.2.21) IWl(t.V,SA)1I i?1(IV(t,X)I + IS(tA)I).
(42.22) IW2(t, V, S X)I 1 2(I V(t, X) I + IS(t, ) I).
Finally, we will need some constants related to those we
have introduced so far. Take 2 such that
(42.23) 0 < < min.(a, d/12.
Define df, i e 1,6 by
(4.2.24) o 2 2, 2f  2,, ( 1 + 2 / d ),
(4.2.25) O'3 A212 8 / Ofd + L + 12 12 28/ d2 + )
(4.2.26) o'4,,f2 L + 9' o5 4 eU2L,
(4.2.27) 0 16  A211212 o / o' + 3.
Then, choose A1 so that
(4.2.28) 0 < < min. {, 7/'5t, 7/'6
First, consider part (i). We show that the following
136
statement is true:
(S) If X e X,, for any Input u s Uu. and any initial states x(O)
1 X 6 e.ll, z(O) 4 zo e 0I, the solutions V, S of (4.2.9),
(4.2.10) exist on [0, L] and stay in Rt, Rs, respectively.
Then, this will imply part (). We prove (S) by contradiction.
Suppose the contrary of (S):
(S)' There exist X e tL, x 6 ULU', xz e zo F, and to e
(0, L) such that both V, S stay in Ra, Rs, respectively, only
during the time interval [0, to).
By (5)', (4.2.9), (4.2.10), (4.2.18), and (4.2.19), the following
Volterra Integrals must hold for all t s [0, t):
(4.2.29) V(t, X) = W1(', V(r, X), S(r, X), X) dT + V(t, X),
(4.2.30) S(t, X)  X'1 eA(O) (t  T/ W2(r, V(', X), S(', X), X ) d + S(t, X).
Then, by (4.2.8) and (4.2.20)  (4.2.22), the following inequalities
hold for all t e [, to):
(4.2.31) I V(t, ) I, IV(t, X)l dr + I1,, IS(, X)l dr + X g.
(4.2.32) I S(t, X) I i 12 J e (tT)/x IS(r, X)I d' +
12L IA efT IV(, )I di +X + 10.
137
By (4.2.17), (4.2.32) implies
(4.2.33) IS(t, X)I l, 2g e(t'Tx IS(', X)ldr +
( 21 / or)lV(t, X)l + ( 212g28 / d2+ 10), t 6 [0o t).
Multiplying both sides of (4.2.33) by et/x, applying Gronwall's
Lemma([Die.l]) and then dividing the result by eat/,
(4.2.34) 1 S(t, X) I i 2121 J e(~x 2Xt T)x T ( 212 / o') I V(T, X) I +
A( 122 / ~2 + 1)1] d, t e [O, to).
By (4.2.17), (4.2.23), and (4.2.28), (4.2.34) implies
(4.2.35) I S(t, X) I i ( I11212 / o' ~f) I V(t, X) I + X, t e [, to).
Substituting (4.2.35) into (4.2.31) and applying Gronwalls Lemma
leads to
(4.2.36) IV(t,X)I X'4( I + 2jte 2(t d  r ) X d5, t [0, to).
By (4.2.27) and (4.2.36), (4.2.35) implies
(4.2.37) S(t, X) I A X 6 te[O, to).
Thus, (4.2.28), (4.2.36), and (4.2.37) show that
(4.2.38) I V(t,X)I, I S(t, ) <,, t e[0, t).
138
This with (42.5), (42.7), (42.10), (42.13), and (42.14) shows that
there exists 13(X) > 0 such that
(42.39) IS(t,X) < 13' t 6 [0 to).
This and (4.2.17) imply that the sequences (S(tr, X)], {V(t, X))
are convergent sequences in RI, R", respectively, if lim tr = to and
0 < tr i to, r e nl.. Let (t,,X) = lim S(tr,X) and V(to,) =
lim V(t,, X). Then by (4.2.38),
(42.40) I V(to, X) I, I S(to., ) I <?7.
This implies that the solutions V, S will continue beyond to.
This violates the assumption (S)'. Thus, we have shown that X X,
guarantees the existence of solutions V: [0, L] > 2v, S: [0, L] > As.
Next, we prove part (ii). Note that (4.2.36), (4.2.37) hold on
[0, L] uniformly with respect to I,, Rzo Uu*, R,.l This fact
with (B.3)  (3) shows that there exists }14 > 0 such that
(4.2.41) I (x)  h(x) I i }14X
holds uniformly with respect to R., Rzo, RAu, and HX,.
Let ~~(u, x,) denote the ith output of,PD for an input u e U, and
139
an initial state x(O)' xo e 1%.. Let u, u e Uu, be two distinct
Inputs with U, = u. Then, since 28 is a decoupled system on X,
by (4.2.41), the following inequality holds:
(42.42) I I(U. x0. Zo A)  X,(u, x0. z0. ) I ((u. )  (u. x0) L +
I j(U, xo z, X x z x) I+ I (', xo) ,(u xo, zo, X)
2 14X,
for all x0 R.,, zo, RZo, and X e CR,.
Remark 4. 2. 1. The proof is a straighforward extension of
well  known singular perturbation techniques for systems without
inputs and outputs ([Hop. 1, Hop.2, Kok.l, Lev.l, Sab.l, Tih.1, Vas.l]).
Our proof follows closely the one given in [Lev.l]. But in [Lev.l],
part (i) of Theorem 4. 2. 1 was implicitly assumed rather than
proven. 0
Remark 4. 2.2. A concept similar to our approximate
decoupling appears in [Wil.l, You.l], where asymptotic(which
corresponds to "approximate", here) disturbance decoupling of
linear systems was considered. D
CHAPTER 5
APPLICATIONS TO ROBOTICS
In this chapter, the results developed in the previous
chapters are applied to decoupled control of robotic manipulators.
In Section 5. 1, actuator dynamics are completely neglected but in
Section 5. 2, the significant part of actuator dynamics are taken
into account.
5. 1. Decoupled Control of Robotic Manipulators
Consider the following system:
(5.1.1) M(q) q + N(q, q) = L(, q) u, y = C(q),
where: q e Rm, E is an open connected subset of R2m; Q 
(q e Rm: (q, q) 6 E); M: Q > R"; N: E Rm; C: Q 
Rm; L: E > Rmxm. The rigid body dynamics of a robotic
manipulator can be described by the above second order differential
equation when actuator dynamics are neglected.
We assume
(C.I) M, N, L, C are C",
(C.2) M(q), DC(q) are nonsingular, q 6 Q,
140
141
(C.3) L(CI, q) is nonsingular, (q, q) e E
We may need the following stronger assumptions
(C.1)'M, N, L, C are C@,
(C.4) C is one to  one on Q.
Let x,4 q, x2^ q, and x (x1, x2). By (C.2), we can write the
system (5.1.1) into the following form:
(5.1.2) O = f(x) + X fi(x), y = h(x),
where
(5.1.3) f(x)' [M(x2)'N(x,, X2) h(x) C(x2),
(5.1.4) fi(x) [M(x2)rlL(xl, x2), i e li.M
0
Here, L, is the ith column of L. We denote the system (5.1.2) by
(f, h,~]o. In the following theorem, we consider the decoupling
of [ f, h, E),.
Theorem 5. 1. 1. Suppose that for each of the following
parts, (C.1)  (C.3) are satisfied. (i) The system rf, h, E} is
decomposable at each x, c E and decouplable on E with d1 = 1, i e
142
ttlj,. Moreover, the control law u = ((, q) + p(, q) u decouples
(f, h,o E on E If, p have the following forms on E:
(5.1.5) C((q, q) = [L(q, q)]( M(q)[DC(q)]( n(, q)  Qo(q, q)q ) + N(q,q),
(5.1.6) (q, q) = [L(q, q)]M(q) [DC(q)]' r(q, q),
where
(5.1.7) I(q, q)^ 1( C,((q), DC1(q) ), Qo(q, q) I ITD(DC(q))T,
m( Cm(q). DCm(q) ) J lTD(DCm(q))T
(5.1.8) r(q, q) 4 diag?( Ci(q), DCi(q) q ),
and *i, li are arbitrary C  functions of their arguments such that
r(q, q) is nonsingular, (q, q) e E
(ii) Suppose that (C.4) is satisfied and the class of control laws
satisfies (A5) and (A.6) of Section 3. 3. Then, f, h,o E is
decomposable on.  The class given by (5.1.5)(5.1.8) is the
whole class of smooth decomposing control laws.
(iii) Suppose that (C.1)' is satisfied. Suppose that class of control
laws is real analytic and for every control law in the class,
{f, h, E)}'P satisfies (A2) of Section 3. 2. Then, the class given
by (5.1.5)(5.1.8) is the whole class of real analytic decoupling
control laws.
Proof. First consider part (i). Let Xi, i e f1m be the
143
vector fields corresponding to (f,h,E)o. Fix 1 e ni,.
Straightforward computation shows that
(5.1.9) Xo H(x) = DC(x2) x1,
(5.1.10) Xo2 H(x)  x D(DCi(x2))T x  DCt(x2)[M(x2)]lN(x, x2),
(5.1.11) XjHi(x) = 0, j s8,,,
for all x e E On the other hand, by (C.2) and (C.3),
(5.1.12) Xj X H,(x) = DCi(x2) [M(x2)]Lj(x, x2) " 0,
for all x e E and i,j 6 H11. Thus, d = 1, i e 1,M and
(5.1.13) D*(x)  DC(x2) [M(x2)L(x, x2),
(5.1.14) A*(x) = 0(x, x2)  DC(x2) [M(x2)rlL(xl, x2).
By (C.2), (C.3), (5.1.12), Theorem 3. 3. 1, and Theorem 3. 3. 2, (f, h, ~ )0
is decouplable on E and decomposable at each xO E
S"( f, h, E~,) is given by (5.1.5) and (5.1.6).
Now, consider part (li). Note that 2m = (d. + 1). Define a
C mapping T from E into R2m by
(5.1.15) T ^ ( T1,..., ), T. (T,,T, Ti. C1x,,2
T^2 n Dq (x,)x,. i e,^
144
By (C.2), DT(x) Is nonsingular, x e E By Theorem 2. 3. 7, this
with (C.4) implies that T is a C —diffeomorphism on I From
this, Theorem 3. 4 4, and Remark 3. 4 7, part (i) follows easily.
Part (ill) follows from Theorem 3. 4. 1, Remark 3. 4. 7, and the fact
that that T is a C" diffeomorphism on E O
Before making remarks on Theorem 5. 1. 1, we consider the
following system, denoted by 2:
(5.1.16) M(q)q + F(, q) = T, y  C(q),
(5.1.17) v =A(X) v + B,(q, q, X) + 5 B( q, X) u, T = G(q, q)v,
where: q, M, E Q are defined as in (5.1.1); F: E  R"; X is a
positive constant scalar and X e [0,X]; A: [0,]  R'G; G:
Rmr; Bj: x [0,Xo]  R', j e r. The dynamics of a
robotic manipulator with D. C. drives ( [Asa. 1, Erl. 1, Daz.1]) or
electro  hydraulic actuators ([Mcc. 1, Mer.l] ) can be described by
the above equations. Then, (5.1.16) represents the dynamics of a
robotic manipulator, where q is the vector of generalized joint
coordinates; M is a generalized inertia matrix; F is the vector
equivalent forces due to Coriolis and centrifugal effects, friction
forces, and gravitation; and y is the output to be controlled (e. g.,
the position and orientation of the end  effector ). The system
(5.1.17) represents additional actuator dynamics, where u is the
electrical control input to actuators and T is the output torque ( or
145
force) generated by the actuators.
In the modelling process, when X Is very small (which means
that the additional actuator dynamics (5.1.17) are very fast,
relatively to the mechanical dynamics (5.1.16)), the additional
actuator dynamics are usually neglected. In other words, for
simplicity it is assumed that X = 0. We denote this system by
2:. If (B.1) in Section 4 1 is assumed, we can write the
degenerate system 2, of 2X as (5.1.1), where
(5.1.18) N(q, q) a F(q, q) + G(q, q) [A(O)1B,(q, q, 0),
(5.1.19) L(q, q) 4  G(I, q) [A(0)]1[ B1(q, q, 0) B, q 0)].
Thus, we have shown that when actuator dynamics are neglected,
the dynamics of a robotic manipulator can be described by (5.11).
Remark 5. 1. 1. Theorem 5. 1. 1  (i) includes previous results
([Bej., Fre.2, Fre.3, Hew.l, Mar., Pau.1, Rai., Sin.4, Tar. ]) as
special cases. For instance, in [Bej.1, Mar.l, Pau.1, Rai.1],
(5.1.20) m=6, C(q) ^ q, L(qq) I, E = R12
In [Fre.2, Fre.3],
(5.1.21) m 3, L(q, q)I, C(q) (q1cos, q1sin q2q3), ~R3xQ, QI(q1,q2,q3)eR3:0<q1 <(, o<q<21f, q3eR 1.
146
It can be shown that these problems satisfy the assumptions
required for Theorem 5. 1. 1. The case of (5.1.20) is called joint
coordinate control. The case of (5.1.21) is called hand coordinate
control The hand coordinate system is the Cartesian coordinate
system fixed on the gripper or the end  effector. A more
general form of the hand coordinate control can be described by
(5.1.22) m ^ 6, C(q) a p(q)
*(q)
8(q), q)
where p(q) e R3 is the position of the origin of the hand coordinate
system from the inertial reference coordinate system; 8, 8, are
Euler angles of of the hand coordinate system with respect to the
inertial reference coordinate system. For the case of (5.1.22), the
hypotheses of Theorem 5. 1. 1 hold with ~ = R6 x Q, where Q is an
open subset of R6. The details are omitted. 0
Remark 5. 1. 2. We believe that Theorem 5. 1. 1  (ii), (iii)
are new. The class of decoupling control laws the above authors
consider is, in (5.1.7), (5.1.8),
(5.1.23) i( Ci(q), DC,(q) q ) = ~,1 Ci(q) + i.2 DCi(q) q,
(5.1.24) p,( Ci(q), DCi(q)q ) = 6, Ci(q) + 6,2 DCi(q) q
where,, 2, 6t 6 2 are real constants. It is obvious that
1(i'1 I ^' I 1' * *4
147
ours is a more general class of control laws which decouple. It
is not so obvious that the class is the most general class. 0
Remark 5. 1. 3. In the conventional approaches to control of
robotic manipulators ([Luh.2, Luh.3, Mar., Pau.2]), the case of
(5.1.20) is extensively studied and the design is based on singleinput, single  output models for each joint coordinate, treating
coupling effects between joint coordinates as disturbance inputs.
Though corrections for varying inertias and gravitational loads are
sometimes introduced in these approaches, precise and high speed
control is difficult to achieve. In the decoupled control
investigated in [Fre.2, Fre.3, Hew.l, Rei.l, Pau.1] and here, it is
possible. The disadvantage of decoupled control is that it
requires a,large amount of computation. But methods for reducing
the computational complexity and the use of special processors
have been investigated by some authors ( [Hol. 1, Luh. 1, Wal. 1, Tur. 1]).
Although these computational methods are proposed originally for
the case of (5.1.20), they are also applicable for the general
problem considered here. O
Remark 5. 1. 4. An alternative and perhaps more straightforward
derivation of Theorem 5. 1. 1 (i) is as follows. Differentiating y
in (5.1.1) twice with respect to t and, in the resulting equation,
replacing q by the expression obtained from (5.1.1), we can obtain
(5. 1.25) y = C(q)[M(q)]l L(q, q) u  N(c, q) ) + Q(, q) q
148
By (5.1.25), the control law u = (r(, q) + p(q q) 0 satisfying (5.1.5)(5.1.8) with n=0 and r = I leads to
(5.1.26) y = U.
Thus, (f,h, E) is decouplable on E This alternative approach is
implied in [Gil.41. It does not require knowledge of vector fields and is
based on the special structure of (5.1.1). The characterization of the
entire class of decoupling control laws follows from Remark 3. 4. 4
or Remark 3. 4. 7 ( see also the last paragraphs of Section 3. 4, 3. 5 ). 0
Next, let us consider the effect of the neglected fast
dynamics (5.1.17) on decoupling of the original system 2x. Let u
= ((q, q) +(q, q) be a control law satisfying (5.1.5)(5.1.8).
We denote by 4a, 2V,, respectively, the feedback systems of 27,
2I corresponding to the control law u = (q, q) + p(q, q). For
the following result, we need
(C.5) F, A, G and Bj, j e to, are C.
Theorem. 1.2. Suppose that (C.1)  (C.3) and (C.5) are
satisfied. Suppose that 2^I satisfies (B. ) and(B.2) of Section 4. 1.
Then, 21P has the properties (i), (ii) in Theorem 4. 2. 1 with X =.
The theorem shows that although a control law which
149
decouples X0 on ~ may not decouples 2, on ~xRr, it does
approximately. Theorem 5. 1. 2 is a direct consequence of Theorem
4. 2. 1 and Theorem 5. 1. 1.
5. 2. Decoupled Control of Robotic Manipulators with
Significant Actuator Dynamics
Consider the following system:
(5.2.1) M(q) q + N(q, q) = g,( v, i, q ), y = C(q),
(5.2.2) v = a(v, q, q) +. a( v, q, q ) uj
where: X is an open connected subset of R3; q, v, e Rm; E
{(q,q) (v,q, q) e X); Q q: (q(, q) e); M: Q  Rmxm; N:
 Rm; C; gC:: Rm; a: X  Rm, i e o.
The dynamics of a robotic manipulator can be described as above
when significant actuator dynamics are taken into account. Except
for the increased complexity, development in this section is quite
similar to that in Section 5.1. In addition to (C. 1) (C.3) and
(C.4), we assume
(D.1) M, N, C, go, a,, i e Hom are C'.
(D.2) Q,(v, q, q)  Dlgo(v,, q) [ a1(v, E,q) " am(v, C, q)] is
nonsingular, (v, q, q) e X
150
Let V ^ (v e Rm: (V,,q) e X}. We may need the following
stronger assumptions:
(D.1)' M, N, C, g, a,, i e fCo are Cw.
(D.3) g(', q, q) is one to  one on V for each (q, q) e E
Let x v, x q, x3 ^ q, and x = (x, x2 x3). By (C.2),
we can write the system (5.2.1), (5.2.2) as
(5.2.3) = fo(x) + 5 fi(x)ui, y = h(x),
where
(5.2.4) f,(x) = aO(x,, x2 x3)
[M(x3)]'t go(X1, X2, 3)  N(x2, x3)
X2
(5.2.5) f,(x)^ a,(xx2,X3) i e X 1.
0
0
Let ( f, h, X j denote the system (5.2.3). Under the above assumptions,
we consider decoupling of I f, h, X lo. We use the following notation
( see Section 2. 1 for the definition of the third order derivative )
(5.2.6) Qo(l(q, q) ( [Dr'(q) qT Q 11(qq qT) _ rTD(DC(q))T,
[DMm(q) q]T i T D(DCm(q))T
151
(5.2.7) Qg(q, q)  DC(q)[M(q)]' N(q, q)  QH(q, q),
(5.2.8) Q8(q, q) = DC(q) [M(q)]1 D1N(q, q)  2 Q11(, q),
(5.2.9) Q7(q, q) (Q(, q)  DC(q)[M(q)]' Ql(, q))[M(q)]'
(5.2.10) Q6(q, q) ^ DC(q) [M(q)]1 D2N(, q) q + Q7(, q) N(q, q) 
D3C(q) [q] [q] [ql],
(5.211) Q5(v, q, q) [M(q)]' go(v, q, q)  N(, q),
(5.2.12) Q4(v, q, q)' D1 g,(v, q, q) aO(v, q, q),
(5.2.13) Q3(v, q, q) Q4(v, q, q) + D2 go(v, q, q) Q5(v, i q) +
D3 g(v, q q) q,
(5.2.14) Q2(v,., q) ^ Q(, q) Q5(v, q, q) + Q6(q q) 
Q7(v,, q) go(v, q, q),
(5.2.15) Ql(v, q q) 3 Q l(q, q) s((v, q, q) + D3C(q) [q] [q] [q],
(5.2.16) i(v, q, q) q D2N(q, q) q  Q4(v, q, q)  D3 g(v, q, q) +
{ D1N(q, q) + Q1o(, q)  D2 go(v, q, q)) Q5(v, q),
(5.2.17) Vi(v, q,q) = DCi(q) Q(v, q q) + T D[DCi(q)]Tq, i e l,,
where MiT(q) is the transpose of the ith row of M(q) and C, is
the ith component of C. Note that
(5.2.18) Q8(, q) = DQg(q, q), Q6(q, q) = D2Qg(, q)q.
Theorem 5. 2. 1. Suppose that for each of the following
parts, If, h, X, satisfies (C.2), (D. 1), and (D.2). (i) Then, { f, h, X })
152
is decomposable at each xO e X and decouplable on X with d = 2,
1 e t,,. Moreover, a control law u =:(v, q, q) + p(v, q, q) u
decouples (f, h,] X on X if (, p have the forms on X:
(5.2.19) ((v, q, q)' [Q1(v, q, q)]'lf^(v, q, q) +
M(q) [DC(q)]l' r(v, q, q) ,1(v, q, q) ],
(5.2.20) p(v, q, q) A [Ql(v, q, q)]r'M(q) [DC(q)]l r(v, q, q),
where
(5.2.21) n(v,, q) ({1(C,(q), DC,(q) q, V,(v, q, q) )
Im( Cm(q), DCm(q) q, Vm(v, q, q))
(5.2.22) r(v,, q) = diag P( C,(q), DCj(q) q, V,(v, q, q)),
and i,'P are arbitrary C — functions of their arguments such that
r(v, q, q) is nonsingular, (v, q, q) e X
(ii) Suppose that (C.4), (D.3) are satisfied and the class of control
laws satisfies (A5), (A6) of Section 3. 3. Then, I f, h, X is
decomposable on X The class given by (5.2.19)  (5.2.22) is the
whole class of smooth decomposing control laws.
(iii) Suppose that (D.1)' is satisfied. Suppose that class of control
laws is real analytic and for every control law in the class, f, h, X 1)R
satisfies (A.2) of Section 3. 2. Then, the class given by (5.2.19)(5.2.22) is the whole class of real analytic decoupling control laws. 0
153
Proof. First consider part (i). Let
(5.2.23) W,(x3) DC(x) [M(x3)], I e,j.
Then, we can derive
(5.2.24) [ D(Wi(x3))Tx2 ]T = (Q7)i(X2 X3), i e,
where (Q7)i is the ith component of Q7. Let Xj, j e nm be the
vector fields corresponding to (f, h, X ). Straightforward
computation with (5.2.23) and (5.2.24) shows that
(5.2.25) X H,(x) = DCi(x3) x2,
(5.2.26) Xo2 Hi(x) = Vi(X1, X2 X3) = W1(X3) go(xl, x2, X3)  (Qg)(, x 3),
(5.2.27) Xo3 Hi(x) = Wi(x3) Q4(X 2, 3) + ( Wi(x3) D2 go(xl, x2, X3) 
D,()i,(x2, x3) )] Q(x, X2, 3)  D2(9Q)i(x2, X3) X2 +
(Q7)i(x2, x3) go(x1, x2, x3) + Wi(x3) D3 go(x1, x, x3) x2'
where (Qg), is the ith component of Qg. Note that
(5.2.28) XjHi(x) = XXoHi(x) = 0, x e X, i, j e l,'
But (D.2) implies
(5.2.29) D,*(x) = W(x3) Q,(xl, x2, X3) O, x e X, i 6e l,If.
Thus d1 = 2, i e t1,,. By (C.2), (D.2), Theorem 3. 3. 1, and
154
Theorem 3. 3. 2, (f, h, X )o Is decouplable on X and decomposable at
each x, e XK ((f, h,X o ) is given by p(v,, q) in (5.2.20) and
(5.2.30) c(v,q,q) = [Ql(v,q,q)r'(l(q) [DC(q)r1(Q2(vq,q) +
n(v,q) )  3(v,,q)).
But, since
(5.2.31) M(q) [DC(q)]lQ2(v, q, q) = M(q) [DC(q)r' Q8(I, q) Q5(v, q, q) 
07(q, q) M(q) Q5(v, q, q)  D3C(q) [lq] + D2N(q q) q
=  1(q) [DC(q)r l(v, q, q) + DN(q, q) Q5(v, q, q) +
D2N(q, q) q + Qo(q, q) Q5(v, q, q),
(5.2.30) can be reduced to (5.2.19).
Consider part(ii). Note that 3m = (di+1 ). Define a
"=t
mapping T from X into R3m by
(5.2.32) T (T1,..., Tm), Ti (Ti1, T.2, Ti.3), Ti.1(Xl X2, x3) C1(x3)'
Ti.2(xx2,x3) DCi(x3)x2, Ti.3(X,2,X3) Vi(X1,X2X3), ie l.
By (C.2) and (D.2), DT(x) is nonsingular, x s X. By Theorem 2. 3. 7,
this with (C.4), (D.3) implies that T is a C"  diffeomorphism on X
From this, Theorem 3. 4. 4, and Remark 3. 4. 7, part (ii) follows easily.
Part (iii) follows from Theorem 3. 4. 1, Remark 3. 4. 7, and the
fact that T is a C  diffeomorphism on X El
155
Remark 5. 2. 1. Using the special structure of (f, h, X Xo, there
is an alternative and perhaps more straightforward way to show
that tf, h, X ) Is decouplable on X and that the control law u =
<(v, q, q) + p(v,, q) satisfying (5.2.19), (5.2.20) decouples
(f, h, X), on X Differentiate both sides of the first equation of
(5.2. 1) with respect to t. Then, in the resulting equation, replace q, v
by expressions obtained from (5.2.1), (5.2.2). Then, we can obtain
(5.2.33) M(q) q = Q1(v, q, q) u  Q2(v, q, q).
On the other hand, differentiating the second equation of (5.2.1)
three times with respect to t leads to
(5.2.34) y = a,(v, q, q) + DC(q)q.
From (5.2.33) and (5.2.34),
(5.2.35) y' = DC(q) [M(q)]' ( Q(v, q, q) u  a2(v, q, q)) + Ql(v, q, q).
From (5.2.35), it is clear that the control law u = c(v, q, q) +
p(v, q, q) t satisfying (5.2.19), (5.2.20) with 1= 0 and r = Im gives
(5.2.36) y = u.
Thus, [ f, h, X) is decouplable. The characterization of the entire
class of decoupling control laws follows from Remark 3. 4. 4 or
Remark 3. 4. 7 ( see also the last paragraphs of Sections 3. 4, 3. 5). D
156
Remark 5. 2. 2. Nijmeijer ([Nij.4]) considered decoupling of
the system In (5.2.1), (5.2.2) with m = 2, go(v, q, q) = v, and C(q) =
q. In [Yua.1], the dynamics of a robotic manipulator with D.C.
drives were linearly perturbed around an equilibrium point. Then,
the decoupled control of the linearly perturbed system was considered.
Thus, the nonlinearity of the system was not fully taken into account. 0
Consider the following system, denoted by 2:
(5.2.37) M(q)q + N(l, q) = T, y = C(q),
(5.2.38) v = bo(v, q q) + 2 g( v, q q )z, + 2 bj( v, q, q )uj,
= g( v,, q ),
(5.2.39) Xz = A(X) z + B( v, q, q, X) + B( v, q, q, X)u,
~~~0 ~ j=I J
where: X, E, Q, q,v, M, N are defined as in (5.2.1), (5.2.2); gj:
X' Rm, j e o j:f X  Rm, j 6e om X; is a positive
constant scalar and A e [O, A]; A: [O. Ao]  R(rX; Bj X x
[0,A ]  R', j e nom. AS in Section 5. 1, the system (5.2.37)
represents the dynamics of a robotic manipulator. Here, the
additional actuator dynamics are grouped into two subsystems
(5.2.38), (5.2.39). The system (5.2.38) ((5.2.39)) represents the
slow (fast) part of the additional actuator dynamics. Suppose
that we neglect the fast dynamics by letting X = 0. Then, the
resulting system is the degenerate system 2, of 1 and consists
157
of the systems (5.2.37), (5.2.38), and
(5.2.40) 0 = A(0)z + B( v,, q, 0) + B( v,, q, 0) uj.
Jj
If we assume (B.1), Zo can be written as (5.2.1), (5.2.2) with
(5.2.41) a,(v, q, q) 4 b,(v,, q)
[g,(v,, q) * * gq(v,, q)] [A(0)1 B.( v, q, q,, i e Ho,.
Thus, we have shown that the dynamics of a robotic manipulator
with the actuator dynamics can be described as (5.2.1), (5.2.2)
when the slow part of the actuator dynamics are taken into account.
Finally, we consider the effect of neglected fast part of
the actuator dynamics on decoupling of the original system 7.
Let u = x(v, q, q) + p(v,, q) u be a control law satisfying (5.2.19)
(5.2.22). We denote by x9,.AP, respectively, the systems 2,
Y, with the control law u = ~(v, q, q) + P(v, q, q). To apply
the theory of Chapter 4 we need:
(D.4) A, Bi, b,, g,, i e tLn are C'.
Theorem 5.2.2. Suppose that (C.2), (D.1), (D.2), and (D.4)
are satisfied. Suppose that A.P satisfies (B.1) and (B.2). Then,
properties (i), (ii) in Theorem 4. 2. 1 hold. O
CHAPTER 6
CONCLUSION
In the previous chapters, we have addressed various
theoretical issues of decoupling and decomposition and their
applications to robotics.
In Chapter 3, the major portion of well known results on
linear decoupling have been extended to nonlinear systems. Since
in Section 3. 6, our main contributions have been summarized and
some concluding remarks on them have been given, we shall not
repeat the same discussion here. Those results contribute to a
deeper and clearer understanding of nonlinear decoupling theory.
They supply full information about the flexibilities we can have
in the design of decoupled systems.
In Chapter 4, a tradeoff between the exact decoupling of
systems and the computational complexity of decoupling control
laws has been considered. We have shown that neglecting the
fast dynamics of the systems leads to control laws which require
less computation but decouple the systems in an approximate way.
In Chapter5, these results have been applied to the
decoupled control of robotic manipulators. Two cases have been
considered. In the first case, actuator dynamics are completely
neglected. In the second case, the dynamics of a significant
158
159
class of actuators are taken Into account. We have shown that
our formulas for the complete class of decoupling control laws
unify and generalize previous results on the decoupled control of
robotic manipulators ( see comments in Remark 5. 1. 1, 5. 1. 2, and
5. 2. 2 ). For example, it is possible to achieve decoupled control
of the end  effector.
Some of our results may be extended with increased
complexity to the general case where the numbers of inputs and
outputs are not necessarily equal or the systems do not have the
form in (1.7). All our results can be easily extended to time
varying nonlinear systems since they can be changed into time
invariant nonlinear systems by assigning a new state x,, to the
time variable t.
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