Feedback control of affine nonlinear singular control systems

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Feedback control of affinenonlinear singular controlsystemsLiu Xiaoping & Sergej CelikovskyPublished online: 08 Nov 2010.

To cite this article: Liu Xiaoping & Sergej Celikovsky (1997) Feedback controlof affine nonlinear singular control systems, International Journal of Control,68:4, 753-774, DOI: 10.1080/002071797223325

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Feedback control of a� ne nonlinear singular control systems

LIU XIAOPING² and SERGEJ CÏ ELIKOVSKYÂ ³

This paper discusses feedback control problems for a� ne nonlinear singularsystems. Namely, the problems of regularization (regular singular systems havefor a particular initial state and input, a unique solution), non-interaction and exactlinearization. First, an algorithm providing necessary and su� cient conditions forthe regularizability of the a� ne nonlinear singular systems is introduced. Thisalgorithm resembles the well-known constrained dynamics algorithm for a� nenonlinear systems. For regularizable a� ne nonlinear singular systems, necessaryand su� cient conditions for the solvability of the non-interacting control problemare derived. They are based on another algorithm producing a sequence of integers,the generalization of the well known notion of relative degree for usual a� nenonlinear systems. Finally, the problem of feedback exact linearization of thedynamic part of singular systems is brie¯ y addressed. An example is provided toillustrate the main results.

1. Introduction

In this paper, the following a� ne nonlinear singular control system

Çx = f1(x) + p1(x)z + g1(x)u

0 = f2(x) + p2(x)z + g2(x)u} (1)

will be considered, where x Î Rn is the vector of di� erential variables, z Î Rs is thevector of algebraic variables, u Î Rm is the vector of inputs, pi(x) and gi(x) , i = 1,2,are matrix-valued smooth functions having appropriate sizes, f1(x) and f2(x) arevector-valued smooth functions with dimensions n and s. Notice, that p2(x) andg2(x) are identically zero in some practical models, such as constrained robotics,constrained mechanical systems, etc.

Systems of the form (1) are also known as so-called di� erential-algebraic systems.Although systems of the form (1) are a special kind (i.e. a� ne in algebraic variables)of singular nonlinear systems, there exist a large number of physical systems that canbe modelled according to (1). Among them, constrained robot systems (McClamrochand Wang 1988), mobile robots (d’Andrea-Novel et al. 1991), constrained mechan-ical systems (Bloch et al. 1992 and You and Chen 1993), chemical processes (Kumarand Daoutidis 1995), etc.

In recent years, there has been substantial progress in dealing with many aspectsof singular systems. Some of these successes have been recently listed in the books by

0020-7179/97$12.00 Ñ 1997 Taylor & Francis Ltd.

INT. J. CONTROL, 1997, VOL. 68, NO. 4, 753± 774

Received 10 January 1996. Received in ® nal revised form 9 April 1997. Communicated byProfessor V. KucÏ era.

² Department of Automatic Control, Northeastern University, Shenyang, 110006, P RChina. e-mail: [email protected].

³ Control Theory Department, Institute of Information Theory and Automation, Acad-emy of Sciences of the Czech Republic 182 08 Prague 8, Czech Republic. e-mail: [email protected].

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Campbell (1982) and Dai (1989), and in survey papers by Campbell (1990) and Lewis(1986). For the case of linear singular systems, many papers and works dealing withthe various control problems, which include solvability, controllability, observ-ability, stability, pole-placement, observer design, decoupling, optimal control, etc,have appeared. For a fairly comprehensive introduction to linear singular systems,see the books by Campbell (1982) and Dai (1989). For nonlinear singular systems,there has been a considerable amount of earlier work on solvability and numericalsolution, see Brenan et al. (1989), Campbell and Griepentrog (1995), Liu (1995 a,1995 b), and Rheinboldt (1991).

However, recently only limited attention has been devoted to studying controlproblems like controllability (Lin and Almed 1991), linearization (Kawaji and Taha1994 and Liu 1993 a), input± output decoupling (Liu 1993 b), feedback stabilization(McClamroch 1990 and Chen and Shayman 1992), output tracking (Krishnan andMcClamroch 1994), Kronecker’s canonical forms (Rouchon et al. 1992), etc.

The main topic of this paper is the non-interacting control problem (or, also, theinput± output decoupling problem) for system (1). The non-interacting controlproblem is one of the major control design problems, since it aims to reduce amulti-input multi-output system to a set of single-input single-output systems, thusgreatly facilitating the control strategy. The problem of non-interacting control hasbeen studied in great detail for non-singular systems, see Wonham (1985), Isidori(1995) or Nijmeijer and Schaft (1990) and references therein for a survey. Withregard to the non-interacting control problem of singular systems, the respectivepublished results are limited. The ® rst results on this subject were presented in themiddle of the 1980s (Christodoulou and Paraskevopoulos 1984) for singular linearsystems. For the case of singular nonlinear systems, the same problem wasinvestigated by Liu (1993 b). The results in Liu (1993 b) have the drawback thatthe conditions under which the non-interacting control problem is solvable dependon an unknown matrix which needs to be chosen ® rst. Related results can be foundin Delaleau and Silva (1995).

The problem is solved in two steps. First, a generalized constrained dynamicalgorithm is used to regularize the system. This means that the system is changed byusing feedback in order to have a unique solution for any continuous input and aconsistent initial state. Regularization is considered as an inevitable part of anyfeedback design for system (1). Secondly, given a regularizable system, anotheralgorithm is proposed, which produces a sequence of integers. This second algorithmprovides a condition for existence as well as a procedure for construction of aninput± output decoupling feedback law. It is important to underline that allalgorithms operate with the data of the original systems only and provide criteriathat may be easily checked for the system in question.

Finally, as an application of the results on the non-interacting control problem,the so-called dynamic equation exact linearization problem is addressed. It consistsof ® nding feedback and coordinate transformations such that the dynamic part ofthe corresponding closed-loop system is linear. Numerous aspects of the exactlinearization problem have been studied, to a large extent, for non-singular systems,see for example monographs by Isidori (1995) or Nijmeijer and Schaft (1990) andsurvey papers by Claude (1986), CÏ elikovskyÂ (1995), Respondek (1985). Nevertheless,substantially less results are known for singular systems, see Kawaji and Taha(1994), Liu (1993 a), and references therein.

The paper is organized as follows. In § 2 we give necessary de® nitions and

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problem formulations. Section 3 addresses the regularization problem. The non-interacting control problem is investigated in § 4 while the problem of dynamicequation exact linearization is discussed in § 5. Section 6 provides an example toillustrate the main results. Some conclusions and outlooks are contained in § 7.

Notation

For vector-valued smooth functions f (x) = [f1(x) , . . . , fn(x)]T and g(x) =[g1(x) , . . . ,gn(x)]T, a smooth function h(x) and a matrix-valued smooth functionp(x) = [p1(x) , . . . ,pr(x)] with pi(x) = [pi

1(x) , . . . ,pin(x)]T, the following notation

will be used in the rest of the paper:

dh(x) = [ ¶ h(x)¶ x1

, . . . , ¶ h(x)¶ xn ]

L 0f h(x) = h(x) , L f h(x) = dh(x) f (x) , L k+ 1

f h(x) = L f ( L kf h(x) )

L gL f h(x) = L g( L f h(x) )

L p L kf h(x) = [L p1 L k

f h(x) , . . . , L pr L kf h(x)], k = 0,1, . . .

In a di� erential-geometric setting, dh is called the di� erential of h, and L f h, the Liederivative of h along f and they may also be given by coordinate-independentde® nitions. See Isidori (1995) and Nijmeijer and Schaft (1990) for other di� erentialgeometric notions as well as for an exposition of the di� erential-geometric theory ofnonlinear control systems.

2. De® nitions and problem formulation

Similar to the case of linear systems, the theory of the singular system (1) dependson what is meant by its solution; throughout this paper we adopt the followingde® nition.

De® nition 1: A pair of functions x( t) and z( t) , de® ned on an interval [0, T ) of time,is a solution of the system (1) with initial condition x0 and continuous input u if

(1) x( t) is di� erentiable and z(t) is continuous;(2) x( t) and z( t) satisfy (1) for all t Î [0, T ) ;(3) x(0) = x0. u

To be more general, one may relax in De® nition 1 the trajectories’ qualities up toabsolutely continuous x( t) , Lebesgue integrable z( t) ,u( t) and ask (1) to be valid foralmost all t Î [0, T ) . Nevertheless, since this paper aims to study and use structuralproperties of (1) to characterize its regularization, non-interacting control andlinearization, the quality of trajectories is not important here.

Even if system (1) has solutions for a given initial condition x0, it solutions neednot be unique. This gives rise to the notion of a regular singular system according tothe following de® nition.

De® nition 2: The system (1) is said to be regular for continuous u at x0 if it has aunique solution for the initial condition x0 and the continuous input u. It is said to beregular for continuous u on a manifold N Ì Rn if it is regular for the continuous u atany x0 Î N. u

Control of a� ne nonlinear singular control systems 755D

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The de® nitions given above imply some regularity properties of the input u andan appropriate relation between the behaviour of u(0) and z(0) and x0. So in order tolift limitations on the input, the following de® nition is necessary.

De® nition 3: The system (1) is said to be strongly regular at x0 if it is regular at x0

for any continuous input u. It is said to be strongly regular on a manifold N Ì Rn ifit is strongly regular at any x0 Î N. u

One may expect that a given, possibly non-regular, system would become regularafter introducing suitable feedback transformation. This leads to the notion of aregularizable singular system.

De® nition 4 Ð Regularization problem: The system (1) is said to be regularizable atx0 if there exists a smooth regular feedback of the form

u = a (x) + b (x)v + g (x)z (2)

de® ned in a neighbourhood U of x0 with b (x) non-singular on U, such that thecorresponding closed loop system

Çx = f1(x) + g1(x) a (x) + [p1(x) + g1(x) g (x)]z + g1(x) b (x)v

0 = f2(x) + g2(x) a (x) + [p2(x) + g2(x) g (x)]z + g2(x) b (x)v} (3)

is strongly regular at x0. The system (1) is said to be regularizable on a manifold N ifit is regularizable at any x0 Î N. u

Once the question of regularity of a particular system is settled, the regularizedsystem may be subjected to further feedback and coordinate transformations toachieve some additional desired properties, e.g. non-interacting property and/orlinearity. These desired properties of the transformed system correspond to variousfeedback control problems. In the following, these control problems will bemathematically de® ned.

De® nition 5 Ð Non-interacting control or input± output decoupling problem: Given ana� ne nonlinear singular system (1) with output y = h(x) Î Rm and an initial pointx0, ® nd a regular feedback of the form (2), de® ned in a neighbourhood U of x0 withb (x) non-singular on U, such that the corresponding transformed system (3) has thefollowing properties:

(a) it is strongly regular at x0

(b) it has non-interacting property at x0. More precisely, for the initial state x0

and for any pair of input functions va( t) ,vb( t) , t ³ 0, such that(va( t) ) i = (vb( t) ) i, " t ³ 0, the corresponding outputs ya( t) and yb( t) satisfy" t ³ 0 (ya( t) ) i = (yb( t) ) i . Moreover, for the initial state x0 and fori = 1, . . . ,m there exist va and vb such that for the corresponding outputs(ya) i /= (yb) i . Here, ( ) i stands for the ith component of ( ) . u

Remark 1: It is appropriate to underline here that the feedback-transformed systemin De® nition 5 is required to have both strong regularity and a non-interactingproperty. For example, the system


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Çx1 = x2

Çx2 = z1

0 = x2 - u1

Çx3 = x4

Çx4 = z2

0 = x4 - u2

y1 = x1

y2 = x3

obviously has the non-interacting property since Çyi = ui for i = 1,2. However, it isnot strongly regular and therefore the non-interacting control problem in the senseof De® nition 5 needs to be investigated further. In fact, it is easily seen that thefeedback

u1 = z1 + x2 - v1

u2 = z2 + x4 - v2

transforms the above system into the following form

Çx1 = x2

Çx2 = v1

0 = z1 - v1

Çx3 = x4

Çx4 = v2

0 = z2 - v2

y1 = x1

y2 = x2

which has a non-interacting property and is strongly regular. u

De® nition 6 Ð Dynamic equation exact linearization problem: Given an a� ne non-linear singular system (1) and an initial point x0, ® nd a regular feedback of the form(2), de® ned in a neighbourhood U of x0 with b (x) non-singular on U, and a smoothcoordinate transformation x = U (x) also de® ned on U, such that the correspondingclosed-loop system (3) in the coordinates x takes the form

Çx = Ax + Bv

0 = f2( x ) + p2( x )z + g2( x )v} (4)

with p2( x ) non-singular on U and (A,B) controllable. u

Remark 2: Note the non-singularity of p2( x ) implies that (4) is index one (Campbell1990). u


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3. Regularization problem

This section is devoted to the regularization problem. The following algorithm issimilar to the constrained dynamics algorithm (Nijmeijer and Shaft 1990) and thezero dynamics algorithm (Isidori 1995).

Algorithm 1:

Step 0.Assume that [p2(x) g2(x)]has constant rank q0 in a neighbourhood U0 of x0 in

Rn. If [p- 02 (x) g- 0

2 (x)]is the submatrix formed from the ® rst q0 independent rows of[p2(x) g2(x)], then there exists an s ´ s non-singular matrix R0(x) such that

R0(x)[p2(x) g2(x)]= [ p- 02 (x) g- 0

2 (x)

0 0 ]where [p- 0

2 (x) g- 02 (x)]has q0 rows. It is convenient to partition R0(x) f2(x) as

R0(x) f2(x) = [ f 02(x)

~f 0

2(x) ]where f 0

2(x) has q0 rows.Premultiplying the algebraic equation of (1) by the matrix R0(x) gives

0 = [ f 02(x)

~f 0

2(x) ] + [ p- 02 (x)

0 ]z + [g- 02 (x)

0 ]u

It is obvious that a necessary condition for the existence of solutions to the system (1)is that the initial condition x0 belongs to the set M0 = {x Î U0 :

~f 0

2(x) = 0}.Further, assume the di� erential d

~f 0

2(x) of the mapping~f 0

2(x) has constant rank t0

in a neighbourhood U0 of x0. Then, locally around x0, M0 is an (n - t0) -dimensionalsubmanifold. Without loss of generality, assume that the ® rst t0 rows of d

~f 0

2(x) areindependent (otherwise, reorder the rows of

~f 0

2(x) ) , and denote

~f 0

2(x) = [ u 0(x)~u 0(x) ]

where u 0(x) has t0 rows. Then M0 = {x Î U0 : u 0(x) = 0}.In other words, the system (1) has been equivalently (regarding the regulariz-

ability property) transformed into the following

Çx = f1(x) + p1(x)z + g1(x)u

0 = f 02(x) + p- 0

2 (x)z + g- 02 (x)u

0 = u 0(x)

üïïýïïþ

(5)

with x0 Î M0 being obviously a neccessary condition for the regularizability.

Step 1.Consider system (5). Di� erentiation of the last equation of (5) with respect to

time and substituting for Çx gives

0 = L f1 u 0(x) + L p1 u 0(x)z + L g1u 0(x)u

Assume that[ p02(x) g0

2(x)

L p1 u 0(x) L g1u 0(x) ] has constant rank q1 in a neighbourhood U1 of


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x0 in the relative topology of M0 induced by Rn (i.e. U1 Ì M0!) . If [p12(x) g1

2(x)]is

the submatrix formed from the ® rst q1 independent rows of [ p02(x) g0

2(x)

L p1 u 0(x) L g1u 0(x) ],

then there exists an s ´ s non-singular matrix R1(x) such that

R1(x)[ p02(x) g0

2(x)

L p1u 0(x) L g1

u 0(x) ] = [ p12(x) g1

2(x)

0 0 ]Denote

R1(x)[ f 02(x)

L f1 u 0(x) ] = [ f 12(x)

~f 1

2(x) ]where f 1

2(x) has q1 rows.Now, it is easily seen that the algebraic equations become as follows

0 = [ f 12(x)

~f 1

2(x) ] + [ p12(x)0 ]z + [g1

2(x)0 ]u

It is obvious now that a necessary condition for the existence of solutions to thesystem (1) is that the initial condition x0 belongs to the set M1 = {x Î U1 :~f 1

2(x) = 0}.Further, assume the di� erential of the mapping col [u 0(x) , ~

f 12(x)]has constant

rank t0 + t1 in a neighbourhood U1 of x0. Then, locally around x0, M1 is an(n - t0 - t1)-dimensional submanifold. Without loss of generality, assume that the® rst t0 + t1 rows of col [du 0(x) ,d~

f 12(x)]are independent (otherwise, reorder the rows

of~f 1

2(x) ), and denote

~f 1

2(x) = [ u 1(x)~u 1(x) ]

where u 1(x) has t1 rows. Then M1 = {x Î U1 : u 1(x) = 0}= {x Î U0 : u 0(x) =u 1(x) = 0}.

It is easily seen that the system (1) is equivalent to the following

Çx = f1(x) + p1(x)z + g1(x)u

0 = f 12(x) + p1

2(x)z + g12(x)u

0 = u 1(x)

üïïýïïþ

(6)

with x0 Î M1.

Step k + 1.Consider the following system

Çx = f1(x) + p1(x)z + g1(x)u

0 = f k2(x) + pk

2 (x)z + gk2(x)u

0 = u k(x)

üïïýïïþ

(7)

which could be de® ned by induction, where x0 Î Mk .


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Di� erentiating the last equation of (7) gives

0 = L f1 u k(x) + L p1 u k (x)z + L g1u k (x)u

Assume that [ pk2(x) gk

2 (x)

L p1 u k(x) L g1u k (x) ] has constant rank qk+ 1 in the neighbourhood

Uk+ 1 of x0 in relative topology of Mk induced by Rn (i.e. Uk+ 1 Ì M0!) . If[pk+ 1

2 (x) gk+ 12 (x)] is the submatrix formed from the ® rst qk+ 1 independent rows of

[ pk2(x) gk

2 (x)

L p1u k(x) L g1

u k (x) ], then there exists an s ´ s non-singular matrix Rk+ 1(x) such

that

Rk+ 1(x)[ pk2 (x) gk

2 (x)

L p1 u k(x) L g1 u k (x) ] = [ pk+ 12 (x) gk+ 1

2 (x)

0 0 ]Denote

Rk+ 1(x)[ f k2 (x)

L f1 u k (x) ] = [ f k+ 12 (x)

~f k+ 1

2 (x) ]where f k+ 1

2 (x) has qk+ 1 rows.Now, it is easily seen that the algebraic equations become as follows:

0 = [ f k+ 12 (x)

~f k+ 1

2 (x) ] + [ pk+ 12 (x)

0 ]z + [ gk+ 12 (x)

0 ]u.

It is obvious that a necessary condition for the existence of solutions to the system (1)is that the initial condition x0 belongs to the set Mk+ 1 = {x Î Uk+ 1 :

~f k+ 1

2 (x) = 0}.Further, assume the di� erential of the mapping col [u 0(x) , u 1(x) , . . . ,

u k (x) , ~f k+ 1

2 (x)] has constant rank ( t0 + t1 + ´´´+ tk + tk+ 1) in a neighbourhoodUk+ 1 of x0. Then Mk+ 1 is an (n - t0 - t1 - ´´´- tk - tk+ 1) -dimensional submanifold.Without loss of generality, assume that the ® rst t0 + t1 + ´´´+ tk + tk+ 1 rows ofcol [du 0(x) ,du 1(x) , . . . ,du k(x) ,d~

f k+ 12 (x)] are independent (otherwise, reorder the

rows of~f k+ 1

2 (x) ) , and denote

~f k+ 1

2 (x) = [ u k+ 1(x)~u k+ 1(x) ]

where u k+ 1(x) has tk+ 1 rows. Then Mk+ 1 = {x Î Uk+ 1 : u k+ 1(x) = 0}={x Î U0 : u 0(x) = u 1(x) = ´´´ = u k (x) = u k+ 1(x) = 0}.

Accordingly, the system (1) has been equivalently changed into the following

Çx = f1(x) + p1(x)z + g1(x)u

0 = f k+ 12 (x) + pk+ 1

2 (x)z + gk+ 12 (x)u

0 = u k+ 1(x)

üïïýïïþ

(8)

with x0 Î Mk+ 1.

Remark 3: Similar algorithms have been used to develop the state-space descriptionfor di� erential-algebraic equation systems (Kumar and Daoutidis 1994 and Chenand Shayman 1992), to solve the disturbance decoupling problem for a� ne non-


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linear singular systems (Liu 1996), and to derive the canonical form of a general classof nonlinear singular systems (Rouchon et al. 1992). u

It is worth noting here that if~f k

2 (x) /= 0 for some k, then there does not exist anysolution x( t) with x(0) = x0 at all. So, the equality

~f k

2 (x0) = 0 at every step ofAlgorithm 1 is necessary for the system (1) to have a solution. If for a given x0 Î Rn

the two constant rank assumptions and the equality~f k

2 (x0) = 0 are satis® ed at everystep of Algorithm 1, the point x0 is called a regular point of Algorithm 1.

Remark 4: Observe that the sequence tk is non-increasing and bounded by 0, i.e.

t1 ³ t2 ³ ´´´ ³ tk+ 1 ³ ´´´ ³ 0

and the sequence qk is non-decreasing and bounded by s, i.e.

q1 £ q2 £ ´´´ £ qk+ 1 £ ´´´ £ s.

Therefore, Algorithm 1 will terminate after a ® nite number of steps, say k* + 1(bounded by n), with tk* > 0 and tk*+ 1 = 0. In addition, by construction, it followsthat

t0 £ s - q0, tk+ 1 £ tk - qk+ 1 + qk, k > 0. (9)

u

The following properties of the algorithm are very useful in discussing theregularization problem.

Lemma 1: The integers qi and ti, i = 0, . . . , and the vectors~f i

2(x) and u i(x) ,i = 0, . . . , are invariant under feedback of type (2).

Proof: The proof is by induction on the steps of the algorithm. For i = 0, the resultis obvious. Assuming that the result holds for i = k, then it is easily seen that thefollowing relation is satis® ed.

[ pk2(x) gk

2 (x)

L p1 u k(x) L g1u k (x) ] ® [ pk

2 (x) + gk2 (x) g (x) gk

2 (x) b (x)

L p1 u k (x) + L g1u k(x) g (x) L g1

u k (x) b (x) ]= [ pk

2 (x) gk2 (x)

L p1 u k(x) L g1u k (x) ][ I 0

g (x) b (x) ]which implies that Rk+ 1(x) ® Rk+ 1(x) and qk+ 1 ® qk+ 1 because of the non-singularity of b (x) where ® denotes the action of feedback. In addition

[ f k+ 12 (x)

~f k+ 1

2 (x) ] = Rk+ 1(x)[ f k2 (x)

L f1 u k (x) ] ® Rk+ 1(x)[ f k2(x)

L f1 u k(x) ] + Rk+ 1(x)[ gk2(x)

L g1u k(x) ] a (x)

= [ f k+ 12 (x) + gk+ 1

2 (x) a (x)~f k+ 1

2 (x) ]which implies

~f k+ 1

2 (x) ®~f k+ 1

2 (x) , thus tk+ 1 ® tk+ 1 and u k+ 1(x) ® u k+ 1(x) . Now theproof is completed. u

Theorem 1: Assume that x0 is a regular point of Algorithm 1. Then, the system (1) isregularizable at x0 if and only if qk*+ 1 = s.

Proof:

If. It follows from qk*+ 1 = s that the matrix [pk*+ 12 (x) gk*+ 1

2 (x)]has full row rank,


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which means that there exists a matrix g (x) such that pk*+ 12 (x) + gk*+ 1

2 (x) g (x) is non-singular. Imposing the feedback u = v + g (x)z on system (1), it follows from Lemma1 that by using Algorithm 1 the corresponding closed-loop system (1) can beequivalently changed into

Çx = f1(x) + [p1(x) + g1(x) g (x)]z + g1(x)v

0 = f k*+ 12 (x) + [pk*+ 1

2 (x) + gk*+ 12 (x) g (x)]z + gk*+ 1

2 (x)v} (10)

with x0 Î Mk*+ 1. It is obvious that system (10) has a unique solution, i.e. it isregularizable at x0.

Only if. Assume that qk*+ 1 < s. Then performing Algorithm 1 produces thefollowing system

Çx = f1(x) + p1(x)z + g1(x)v

0 = f k*+ 12 (x) + pk*+ 1

2 (x)z + gk*+ 12 (x)v

where [pk*+ 12 (x) gk*+ 1

2 (x)] has full row rank qk*+ 1 < s, x Î Mk*+ 1. According toLemma 1, it follows that the rank of pk*+ 1

2 (x) can be increased to at mostqk*+ 1 < s by using the feedback. As a result, the number of constrained equationsis always less than the number of the variable z no matter what feedback is imposed.So, z can never be determined uniquely, i.e. the system is not regularizable at x0.u

Remark 5: It is easily seen from the proof of Theorem 1 that the number qk*+ 1

constructed in Algorithm 1 is the largest number os components of z which can beuniquely determined through the regular feedback. u

It is worth noting that for any regular point x0 of Algorithm 1, a regularizablesystem can be equivalently described as

Çx = f1(x) + p1(x)z + g1(x)v

0 = f k*+ 12 (x) + pk*+ 1

2 (x)z + gk*+ 12 (x)v} (11)

where [pk*+ 12 (x) gk*+ 1

2 (x)] has full row rank which equals the number of thealgebraic variable z, x Î Mk*+ 1. More precisely, for x0 Î Mk*+ 1 the sets of solutionsof (11) and (1) coincide.

4. Non-interacting control problem

This section will be concentrated on the non-interacting control problem forsystems that are supposed to be regularizable. Namely, from the results of previoussection, the systems of the form (11) restricted to the smooth manifold Mk*+ 1 areregularizable. Moreover, due to the local considerations of this paper, without anyloss of generality the following assumption may be imposed during the rest of thepaper.

Assumption 1: The matrix [p2(x) g2(x)]has the full row rank s at x0.

It is not di� cult to see that, if Assumption 1 is valid, then system (3) is stronglyregular at x0 if and only if p2(x) + g2(x) g (x) is non-singular at x0. As a consequence,z can be uniquely determined from the second equation of (3) as

z = - [p2(x) + g2(x) g (x)]- 1[f2(x) + g2(x) a (x) + g2(x) b (x)v].


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Substituting this into the ® rst equation of (3) yields

Çx = f (x) + g(x) a (x) + g(x) b (x)v (12)

withf (x) = f1(x) - [p1(x) + g1(x) g (x)][p2(x) + g2(x) g (x)]- 1f2(x)

g(x) = g1(x) - [p1(x) + g1(x) g (x)][p2(x) + g2(x) g (x)]- 1g2(x) .

It is easily seen that the closed-loop system (3) with y = h(x) has the non-interactingproperty if and only if there exist a (x) , b (x) and g (x) such that p2(x) + g2(x) g (x) isnon-singular at x0 and the system (12) with y = h(x) has the non-interactingproperty, that is, the system

Çx = f (x) + g(x)w

y = h(x) } (13)

can be input± output decoupled by the feedback w = a (x) + b (x)v.From Proposition 5.3.1 of Isidori (1995), the following conclusion can be easily

drawn.

Lemma 2: Assume that Assumption 1 is satis® ed, there exists a g (x) such thatp2(x) + g2(x) g (x) is non-singular at x0, and for such a g (x) , there exist integersq 1, . . . , q m so that for all x in a neighbourhood of x0

L gL kf hi(x) = 0, k = 0, . . . , q i - 2, i = 1, . . . ,m

andL gL q i- 1

f hi(x0) /= 0, i = 1, . . . ,mThen the non-interacting control problem is solvable is and only if the system (13) has avector relative degree ( q 1, . . . , q m ) at x0, i.e. the matrix

A(x) =

L gL q 1- 1f h1(x)

..

.

L gL q m- 1f hm(x)

éêêêë

ùúúúû

is non-singular at x0.

Apparently, the conditions of this lemma are not checkable because they depend onthe unknown matrix g (x) which need not be unique. In order to obtain the easily-checkable conditions for the solvability of the problem, the following algorithm isproposed. Notice that this algorithm operates with the original singular system (1)only and will be later used to get a clear and checkable criterion for the solvability ofthe non-interacting control problem. Moreover, it will be shown at the end of thissection that this algorithm may be even used to compute explicitly the non-interacting control feedback.

Algorithm 2:

Step 1.Set u

0i (x) = hi(x) and calculate L p1 u

0i (x) , L g1

u0i (x) and L f1 u

0i (x) . If the matrix


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[ p2(x) g2(x)

L p1 u0i (x) L g1

u0i (x) ]

has constant rank s in a neighbourhood of x0, then by Assumption 1, there exists aunique vector-valued smooth function E0

i (x) of dimension s such that

[L p1 u0i (x) L g1

u0i (x)]= E0

i (x)[p2(x) g2(x)]Denote u

1i (x) = L f1 u

0i (x) - E0

i (x) f2(x) . Otherwise, set ri = 1 and quit the algorithm.

Step k + 1.Assume that we have de® ned a sequence of u

0i (x) , . . . , u

ki (x) . Now calculate

L p1 uki (x) , L g1

uki (x) and L f1 u

ki (x) . If the matrix

[ p2(x) g2(x)

L p1 uki (x) L g1

uki (x) ]

has constant rank s in a neighbourhood of x0, then there exists a unique vector-valued smooth function Ek

i (x) of dimension s such that

[L p1 uki (x) L g1

uki (x)]= Ek

i (x)[p2(x) g2(x)] (14)

Denote uk+ 1i (x) = L f1 u

ki (x) - Ek

i (x) f2(x) . Otherwise, set ri = k + 1 and quit thealgorithm.

Remark 6: Algorithm 2 can be easily extended to the system with outputsyi = hi(x) + qi(x)z + di(x)u, i = 1, . . . ,m. u

Remark 7: It can easliy be seen that if s = 0, i.e. the system (1) becomes atraditional nonlinear system, then the integer ri is equal to the minimal integer suchthat L g1 L k

f1hi(x) º 0 in a neighbourhood of x0 for all 0 £ k £ ri - 2, i.e. the

characteristic number (Nijmeijer and Shaft 1990). So, we may say that Algorithm2 is the generalization of the de® nition of the characteristic number to the singularnonlinear systems. Note that ri is also a relative degree when L g1 L ri- 1

f1 hi(x0) /= 0(Isidori 1995). u

Performing Algorithm 2 for i = 1, . . . ,m, we can obtain integers r1, . . . , rm . Now,let us introduce the following matrices

D(x) =L f1 u

r1- 11 (x)

´´´L f1 u rm- 1

m (x)

éêë

ùúû

Dz(x) =L p1 u

r1- 11 (x)

´´´L p1 u rm- 1

m (x)

éêë

ùúû

Du(x) =L g1 u r1- 1

1 (x)

´´´L g1 u rm- 1

m (x)

éêë

ùúû

u

Assumption 2: The matrix

[ p2(x) g2(x)

Dz(x) Du(x) ]is non-singular at x0.

Lemma 3: Suppose that Assumption 1 holds and the matrix

[ p2(x) g2(x)

L p1 uri- 1i (x) L g1

uri- 1i (x) ]


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is of full row rank at x0 for all i = 1, . . . ,m. Then for any matrix g (x) which rendersp2(x) + g2(x) g (x) non-singular at x0

L gL kf hi(x) = 0, k = 0, . . . , ri - 2, i = 1, . . . ,m

for all x in a neighbourhood of x0, and

L gL ri- 1f hi(x0) /= 0, i = 1, . . . ,m

Proof: For the proof, see Appendix A. u

Lemma 4: Suppose that Assumption 1 holds. Then:

(1) if Assumption 2 is satis® ed, then for any g (x) which renders p2(x) + g2(x) g (x)invertible at x0, the system (13) has a vector relative degree (r1, . . . , rm ) ;

(2) if there exists a g (x) such that p2(x) + g2(x) g (x) is non-singular at x0 and thesystem (13) has a vector relative degree ( q 1, . . . , q m ) , then Algorithm 2terminates with ri = q i and Assumption 2 is satis® ed.

Proof: For the proof, see Appendix B. u

The main result of this section is the following theorem providing necessary andsu� cient conditions for solvability of the non-interacting control problem.

Theorem 2: Suppose that Assumption 1 holds and the matrix

[ p2(x) g2(x)


uri- 1i (x) ]

is of full row rank at x0 for all i = 1, . . . ,m. Then the non-interacting control problem issolvable if and only if Assumption 2 is satis® ed.

Proof: The proof is an immediate consequence of Lemmas 2, 3 and 4. u

Up to now, Algorithm 2 has been used to obtain the checkable criterion for thesolvability of a non-interacting control problem. Nevertheless, this algorithm alsoenables us to construct the feedback giving a non-interacting property in an explicitway. To show this, the following lemma will be useful.

Lemma 5: Suppose that Assumptions 1 and 2 hold. Then the vectors

du01, . . . ,du

r1- 11 , . . . ,du

0m , . . . ,du

rm- 1m

are linearly independent at the point x0.

Proof: For the proof, see Appendix C. u

Now, using Lemma 5 and Algorithm 2, the feedback solving non-interacting controlproblem may be computed in the following way.

First, by Lemma 5 one can always select (n - r) smooth functions, sayy 1(x) , . . . , y (n- r) (x) , such that the vectors

du01, . . . ,du

r1- 11 , . . . ,du

0m , . . . ,du

rm- 1m ,dy 1, . . . ,dy (n- r)

are linearly independent at x0, that is

col ( u01, . . . , u

r1- 11 , . . . , u

0m , . . . , u

rm- 1m , y 1, . . . , y (n- r) )


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constitutes a local coordinates transformation, where r = r1 + ´´´+ rm . Further,denote new coordinates as

x ki = u

k- 1i (x) , k = 1, . . . , ri, i = 1, . . . ,m

h i = y i(x) i = 1, . . . ,n - r

h = [h 1, . . . , h n- r]T

x = [x 11, . . . , x r1

1 , . . . , x m , . . . , x rmm ]T

then the system (1) takes, in these new coordinates, the following form:

Çh = f1( h , x ) + p1( h , x )z + g1( h , x )u0 = [f2(x) + p2(x)z + g2(x)u]x= u - 1 ( h ,x )

Çx 1i = x 2

i

..

.

Çx ri- 1i = x ri

i

Çx ri = [L f1 uri- 1i (x) + L p1 u

ri- 1i (x)z + L g1

uri- 1i (x)u]x= u - 1( h ,x )

yi = x 1i

i = 1, . . . ,m

üïïïïïïïïïïïïïïïïïïïïïýïïïïïïïïïïïïïïïïïïïïïþ

(15)

Finally, selecting arbitrary g (x) such that p2(x) + g2(x) g (x) is non-singular at x0 andimposing the following feedback law:

u = - {Du(x) - [Dz(x) + Du(x) g (x)][p2(x) + g2(x) g (x)]- 1g2(x)}- 1

´ {D(x) - [Dz(x) + Du(x) g (x)][p2(x) + g2(x) g (x)]- 1f2(x)}+ {Du(x) - [Dz(x) + Du(x) g (x)][p2(x) + g2(x) g (x)]- 1g2(x)}- 1v + g (x)z (16)

one would immediately see that the corresponding system becomes as follows:

Çh = f1( h , x ) + p1( h , x )z + g1( h , x )v

0 = f2( h , x ) + [p2(x) + g2(x) g (x)]x= u - 1( h ,x ) z + g2( h , x )v

Çx 1i = x 2

i

..

.

Çx ri- 1i = x ri

i

Çx ri = vi

yi = x 1i

i = 1, . . . ,m

üïïïïïïïïïïïïïïïïïïïïïïýïïïïïïïïïïïïïïïïïïïïïïþ

(17)

The structure of these equations readily shows that the non-interaction requirements


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have been achieved. As a matter of fact, the input v1 controls only the output y1,through a chain of r1 integrators, the input v2 controls only the output y2, through achain of r2 integrators, etc. In addition, the non-singularity of the matrixp2(x) + g2(x) g (x) guarantees that the system (17) has the property (a) given inDe® nitions 5. Therefore, the feedback (16) solves the non-interacting controlproblem.

5. Dynamic equation exact linearization problem

This section will shortly address the dynamic equation exact linearizationproblem. With the discussion similar to that before Lemma 2, it is easily seen thatthe problem in question is solvable if and only if there exists a matrix g (x) such thatp2(x) + g2(x) g (x) is non-singular at x0 and the dynamic equation of (13) islinearizable at x0. According to Isidori (1995), the following results are easilyobtained.

Lemma 6: Suppose that Assumption 1 holds. Then the dynamic equation exactlinearization problem is solvable if and only if there exists a g (x) such thatp2(x) + g2(x) g (x) is non-singular at x0 and, for such a g (x) , the distributionsD1, . . . ,Dn are all involutive and non-singular at x0 with

Dk (x) = span {ad rf g(x) , r = 0, . . . ,k - 1}, k = 0,1, . . .

where f ,g are as in L emma 2.

The same observation as that regarding Lemma 2 is true here, namely, to checkthat a particular system does not have a linearizable dynamic part, one has toexclude conditions of this lemma for all described g (x) . Since there are in® nitelymany of them, this is not realistic. Nevertheless, results of the previous section on thenon-interacting control problem enable us to overcome this obstacle. The followingtheorem provides su� cient and necessary conditions for the solvability of thedynamic equation exact linearization problem, which may be, contrary to those ofLemma 6, easily checked.

Theorem 3: Suppose that Assumption 1 holds. Then the dynamic equation exactlinearization problem is solvable if and only if for any particular choice of g (x) whichmakes p2(x) + g2(x) g (x) invertible, the distributions D1, . . . ,Dn are all involutive andnon-singular at x0.

Proof: The if part’ can be easily deduced from Lemma 6. Now, let us prove the`only if ’ part. Assume that the dynamic equation exact linearization problem issolvable. Then there exists a matrix g (x) such that p2(x) + g2(x) g (x) is non-singularat x0 and the dynamic equation of (13) is linearizable at x0, which implies that (13)has a vector relative degree ( q 1, . . . , q m) with q 1 + ´´´+ q m = n (Isidori 1995). Itfollows from Lemma 4 that Assumption 2 is true, so (13) has a vector relative degree( q 1, . . . ,pm ) with q 1 + ´´´+ q m = n for any g (x) which makes p2(x) + g2(x) g (x)invertible. Therefore, for any g (x) which makes p2(x) + g2(x) g (x) invertible, thedistributionsD1, . . . ,Dn are all involutive and non-singular at x0. This completes theproof. u

Remark 8: It is easily seen from the proof of Theorem 3 that if the dynamicequation exact linearization problem is solvable, then for any g (x) which makes


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p2(x) + g2(x) g (x) invertible, the distributions D1, . . . ,Dn are all involutive and non-singular at x0. As a consequence, to check the linearizability of the dynamic part ofthe system in question, one has just to select arbitrary g (x) making p2(x) + g2(x) g (x)invertible and then to check regularity and the involutivity requirement for thischoice. u

6. Example

In this section, we consider the following singular nonlinear system

Çy =

y 2 + ( y 3)2 + y 2( y 4) 2e y 3

y 1 y 4 - 2y 2( y 3)2

y 2 y 3

- ( y 4) 3 + y 1[y 2 + ( y 3) 2]- 2y 1 y 2 + y 2( y 4)

2(1 - 2y 1ey 3 ) - 2y 1( y 3)

2 + y 6 + ( y 5)2

y 1 y 3 y 4 - 2y 5{- 2y 1 y 2 + y 2( y 4) 2(1 - 2y 1ey 3 ) - 2y 1( y 3) 2 + y 6 + ( y 5) 2}

éêêêêêêêêêêë

ùúúúúúúúúúúû

+

0 0

1 0

y 2 y 3 1

y 1 y 3

0 0

y 2 y 3 1



z +

0 e y 3

e y 3 - 2y 3 - 2y 3ey 4

1 + y 3ey 3 e y 4 - 2

y 3ey 3 0

0 1 - 2y 1ey 3

y 3ey 3 - 2y 5(1 - 2y 1e

y 3 )



u

0 = [ y 2( y 4)2

( y 1) 2 + y 5 ] + [ 0 0

0 0 ]z + [ 0 10 0 ]u

y1 = y 1

y2 = y 3

Evidently the system does not satisfy Assumption 1. However, by using Algorithm 1,it follows that under the coordinates transformation

x1 = y 1

x2 = y 2

x3 = y 3

x4 = y 4

x5 = ( y 1)2 + y 5

x6 = y 6 + ( y 5)2

the original system becomes


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Çx1

Çx2

Çx3

Çx4

éêêêë

ùúúúû

=

x2+ (x3) 2+ x2(x4) 2ex3

x1x4 - 2x2(x3) 2

x2x3

- (x4) 3+ x1[x2+ (x3)2]

éêêêë

ùúúúû

+

0 01 0

x2x3 1x1 x3

éêêêë

ùúúúû

z+

0 ex3

ex3 - 2x3 - 2x3ex4

1+ x3ex3 ex4 - 2

x3ex3 0

éêêêë

ùúúúû

u

Çx5 = x6

Çx6 = x1x3x4 + x2x3z1 + z2 + x3ex3 u1

0 = [x2(x4) 2

x5 ] + [ 0 00 0]z + [ 0 1

0 0 ]u

y1 = x1

y2 = x3

üïïïïïïïïïïïïïïïïïïïïïïýïïïïïïïïïïïïïïïïïïïïïïþ

(18)

So for any initial state x0 which belongs to the manifold M = {x Î R6: x5 =

x6 = 0}, the system (18) is equivalent to the following:

Çx1

Çx2

Çx3

Çx4

éêêêë

ùúúúû

=

x2+ (x3)2+ x2(x4)

2ex3

x1x4 - 2x2(x3)2

x2x3

- (x4)3+ x1[x2+ (x3) 2]

éêêêë

ùúúúû

+

0 01 0

x2x3 1x1 x3

éêêêë

ùúúúû

z+

0 ex3

ex3- 2x3 - 2x3ex4

1+ x3ex3 ex4- 2

x3ex3 0

éêêêë

ùúúúû

u

0 = [ x2(x4) 2

x1x3x4 ] + [ 0 0x2x3 1 ]z + [ 0 1

x3ex3 0 ]u

y1 = x1

y2 = x3

üïïïïïïïïïïïïïïïýïïïïïïïïïïïïïïïþ

(19)

It is easily seen that this system satis® es Assumption 1. In order to check whether ornot the system can be input± output decoupled, we may perform Algorithm 2 and testAssumption 2. Appropriate calculations show that

u01(x) = x1

E01(x) = [ex3 0]

u11(x) = x2 + (x3) 2

u02(x) = x3

Dx(x) = [ x1x4

x2x3 ]Dz(x) = [ 1 + 2x2(x3) 2 2x3

x2x3 1 ]Du(x) = [ [2(x3) 2 + 1]ex3 - 4x3

1 + x3ex3 ex4 - 2 ]

It is easy to check that Assumption 2 is satis® ed. So the original system can be input±output decoupled. In fact, under the coordinates transformation


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x 1 = x1

x 2 = x2 + (x3) 2

x 3 = x3

h = x4

the original system becomes as follows:

y1 = x 1

Çx 1 = x 2

Çx 2 = x1x4 + [1 + 2x2(x3)2]z1 + 2x3z2 + [2(x3)

2 + 1]ex3u1 - 4x3u2

y2 = x 3

Çx 3 = x2x3 + x2x3z1 + z2 + (1 + x3ex3 )u1 + (ex4 - 2)u2

Çh = - h 3 + x 1 x 2 + x 1z1 + x 3z2 + x 3ex 3u1

0 = [ x2(x4) 2

x1x3x4 ] + [ 0 0x2x3 1 ]z + [ 0 1

x3ex3 0 ]u

In addition, by using the feedback

[ u1

u2 ] = [ x1x3x4 - x2[x3 + (x4) 2(2 - ex4]x1x4[1- 2(x3) 2]- x2(x4) 2(1- 4x3) + {x1x3x4- x2[x3+ (x4) 2(2- ex4 )]}ex3 ]

+ [ 0 01 0 ]z + [ 0 1

- 1 0 ]v

the closed loop system takes the form of

y1 = x 1

Çx 1 = x 2

Çx 2 = v1

y2 = x 3

Çx 3 = v2

Çh = - h 3+ x 1 x 2+ x 3{x 1 x 3 h - [x 2- ( x 3) 2][x 3+ h 2(2- e h )]}e x 3 + x 1z1+ x 3z2+ x 3ex 3v2

0 = [x 1 h [1- 2( x 3)2]+ 4[x 2- ( x 3) 2]x 3 h 2+{x 1 x 3 h - [x 2- ( x 3)

2][x 3+ h 2(2- e h )]}e x 3

x 1 x 3 h + x 3{x 1 x 3 h - [x 2 - ( x 3) 2][x 3 + h 2(2 - e h )]}e x 3 ]+ [ 1 0

[x 2 - ( x 3) 2]x 3 1 ]z + [ - 1 00 x 3e

x 3 ]v

üïïïïïïïïïïïïïïïïïïïïïïïïýïïïïïïïïïïïïïïïïïïïïïïïïþ(20)

which possesses the properties (a) and (b) in De® nition 5.It is worth noting that this system can be input-output decoupled with stability.

In fact, solving z from the algebraic equation of (20) and substituting it into the 6thequation of (20), then letting x 1 = x 2 = x 3 = 0, it follows that the zero-dynamic of the


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system becomes Çh = - h 3, which is asymptotically stable at h = 0. Therefore, thenon-interacting control problem with stability is solvable for the system.

7. Conclusions and outlooks

The regularization problem has been discussed for a� ne nonlinear singularsystems. By using an algorithm similar to the zero-dynamic algorithm and con-strained dynamic algorithm, necessary and su� cient conditions for the solvability ofthe regularization problem has been derived. Based on this result, the non-interacting control problem has been investigated in detail for the same kind ofsystems. A new algorithm has been proposed, which provides a condition forexistence as well as a procedure for explicit construction of an input-outputdecoupling feedback law. By using the results on the non-interacting controlproblem, the condition for the solvability of the dynamic equation exact lineariza-tion problem has been given.

There are still interesting open problems related to the presented results. One ofthem is the already mentioned problem of input-output decoupling with stability fora� ne singular control systems. This and other problems, like robust stabilizationand robust tracking, are matters of ongoing research.

ACKNOWLEDGMENTS

Liu Xiaoping is supported by grants from NSF of China, Fok Ying TungEducation Foundation, NSF of Liaoning, China, Trans-Century Training ProgramFoundation for Talents, State Education Commission of China, Laboratory ofRobotics, and the Chinese Academy of Sciences. Sergej CÏ elikovskyÂ is supported bythe Grant Agency of the Academy of Sciences of the Czech Republic through grantNo. 275702.

Appendix A: Proof of Lemma 3

It follows from (14) that for k = 0, . . . , ri - 2, i = 1, . . . ,m and x Î U, aneighbourhood of x0

[L p1 uki (x) lg1

uki (x)][ I 0

g (x) I ] = Eki (x)[p2(x) g2(x)][ I 0

g (x) I ]which implies

Eki (x) = [L p1 u

ki (x) + L g1

uki (x) g (x)][p2(x) + g2(x) g (x)]- 1 (21)

for any g (x) such that [p2(x) + g2(x) g (x)] is non-singular at x0. By induction, itfollows from (21) that for i = 1, . . . ,m and x Î U

L kf hi(x) = u

ki (x) , k = 0, . . . , ri - 1 (22)

L gL kf hi(x) = 0, k = 0, . . . , ri - 2 (23)

L gL ri- 1f hi(x) = d i(x) (24)

withd i(x) = L g1 u

ri- 1i (x) - [L p1 u

ri- 1i (x) + L g1 u

ri- 1i (x) g (x)][p2(x) + g2(x) g (x)]- 1g2(x)

In addition


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rank [p2(x) g2(x)

L p1u

ri- 1i (x) L g1

uri- 1i (x) ] = rank {[ p2(x) g2(x)


uri- 1i (x) ][ I 0

g (x) I ]´ [ I - [p2(x) + g2(x) g (x)]- 1g2(x)

0 I ]}= rank [ p2(x) + g2(x) g (x) 0

L p1 uri- 1i (x) + L g1

uri- 1i (x) g (x) d i(x) ]

= s + 1

which implies that L gL ri- 1f hi(x) /= 0. u

Appendix B: Proof of Lemma 4

B.1. Proof for Part 1

It follows from (24) that

A(x) =

L g1 ur1- 11 (x) - [L p1 u

r1- 11 (x) + L g1

ur1- 11 (x) g (x)][p2(x) + g2(x) g (x)]- 1g2(x)

..

.

L g1u

rm- 1m (x) - [L p1 u

rm- 1m (x) + L g1

urm- 1m (x) g (x)][p2(x) + g2(x) g (x)]- 1g2(x)

éêêë

ùúúû

= Du(x) - [Dz(x) + Du(x) g (x)][p2(x) + g2(x) g (x)]- 1g2(x) (25)

Since the relation

rank [ p2(x) g2(x)

Dz(x) Du(x) ] = rank {[ p2(x) g2(x)

Dz(x) Du(x) ][ I 0g (x) I ]

´ [ I - [p2(x) + g2(x) g (x)]- 1g2(x)

0 I ]}= rank [ p2(x) + g2(x) g (x) 0

[Dz(x) + Du(x) g (x)] A(x) ]holds for any g (x) which renders p2(x) + g2(x) g (x) invertible, it follows fromAssumption 2 that A(x) is non-singular, that is, (13) has a vector relative degree( r1, . . . , rm ) for any g (x) which renders p2(x) + g2(x) g (x) invertible.

B.2. Proof for Part 2

Assume that there exists a g (x) such that p2(x) + g2(x) g (x) is non-singular at x0

and (13) has a vector relative degree ( q 1, . . . , q m) . Then, according to the de® nition ofthe vector relative degree and the relation


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rank [ p2(x) g2(x)

L p1 L kf hi(x) L g1 L k

f hi(x) ] = rank {[ p2(x) g2(x)

L p1 L kf hi(x) L g1 L k

f hi(x) ][ I 0g (x) I ]

´ [ I - [p2(x) + g2(x) g (x)]- 1g2(x)

0 I ]}= rank [ p2(x) + g2(x) g (x) 0

L p1 L kf hi(x) + L g1 L k

f hi(x) g (x) L gL kf hi(x) ]

it is not hard to verify the following relations

Eki (x) = [L p1 L k

f hi(x) + L g1 L kf hi(x) g (x)][p2(x) + g2(x) g (x)]- 1

uk+ 1i (x) = L f1 L k

f hi(x) - Eki (x) f2(x) = L k+ 1

f hi(x)

üýþ

(27)

that isEk

i (x) = [L p1 uki (x) + L g1 u

ki (x) g (x)][p2(x) + g2(x) g (x)]- 1

uk+ 1i (x) = L f1 u

ki (x) - Ek

i (x) f2(x) } (28)

with u0i (x) = hi(x) for k = 0, . . . , q i - 2, i = 1, . . . ,m.

From the non-singularity of A(x) and the relation (26), it is easily seen thatAlgorithm 2 ends with ri = q i and Assumption 2 is satis® ed. u

Appendix C: Proof of Lemma 5

It follows from Lemma 4 that the system (13) has a vector relative degree(r1, . . . , rm ) . So it follows from Lemma 1.2 of Isidori (1995) that the vectors

dh1(x) ,dL f h1(x) , . . . ,dL r1- 1f h1(x) , . . . ,dhm(x) ,dL f hm (x) , . . . ,dL rm- 1

f hm(x)

are linearly independent at x0. The conclusion can be drawn from (22). u

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774 Control of a� ne nonlinear singular control systemsD

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Feedback control of affine nonlinear singular control systems