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 of-a 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 T-related 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 input-output 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 input-output 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) F-1(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 I-I. 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(i-1) 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 Cl-functions 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~o-vector 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 n-dimensional 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 T-related 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 T-related 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 Co-dlffeomorphism 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 J-feedback related on X to (F, H, X, where JJ^oJ. Thus, the J-feedback 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 J-feedback 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 input-output 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 input-output 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) y-j(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 input-output 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 (n-r) 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,(n-r) 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,(n-r) 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,(n-r) 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 T-1. 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,..., y-r ) 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,(n-r) 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) = f-m+ (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 T-related 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 J-feedback 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 input-output 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! (i-k))) 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 = n-p. 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 for-each 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 (n-d,-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 J-feedback 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(J-I)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+,(j-di-1) 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+p-I; (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 X-X'. 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 (k-1)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 Input-output 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 r-J(. 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,- = (T-1, 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 T-l, 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 T-1 - 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 closed-loop 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 one-to-one 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, ( * xl-x3) 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)' e-iX 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) = e-i 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. e-X 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. e-l cos x2), 2(X3) (3.5.35) p(x) = [ 1( e-l 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 (e-Xlsinx2. 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(j-1)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: lu-u*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 u-4 (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- (t-T)/x IS(r, X)I d' + 12L IA e-f-T 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)1-B,(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, Q-I(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 trade-off 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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