Relative Flatness and Flatness of Implicit Systems

Copyright <:> IFAC Nonlinear Control Systems Design, Enschede, The Netherlands, 1998

RELATIVE FLATNESS AND FLATNESS OF IMPLICIT SYSTEMS Paulo Sergio Pereira da Silva . ,1 Carlos Correa Filho'

• Escola PoliUcnica da USP - PEE Cep 05508-900 - Siio Pav.lo - SP - BRAZIL Email: pau1o~lac.usp.br

Abstract: In this work we will define the concept of relative flatness of a system with respect to a subsystem. A sufficient condition of relative flatness based on a relative derived flag will be presented. We will show that the notion of relative flatness is useful for studying the structure of implicit systems. In fact, a necessary condition for the flatness of a class of nonlinear implicit systems will be easily obtained as a consequence of our first result. Copyright © 1998IFAC Keywords: Nonlinear systems, implicit systems, flatness, relative flatness, feedback linearization.

1. INTRODUCTION AND MOTIVATION

and Sluis, 1993), (Aranda-Bricaire et aL , 1995) , (van Nieuwstadt et al., 1994» .

The problem of feedback linearization is an important structural problem in control systems theory. This problem was completely solved by staticstate feedback (Jakubczyk and Respondek, 1980; Hunt et aL, 1983) but the necessary and sufficient conditions for the solvability of this problem by dynamic state feedback is yet an open problem (see for instance (Charlet et al., 1989; Shadwick, 1990; Charlet et al., 1991; Sluis, 1993) for some results).

Linear singular (or implicit) systems is an important class of control systems and many papers and books on this subject are found in the literature (Campbell, 1982; Dai, 1989; Campbell, 1990; Lewis, 1986). It is interesting to point out that the rrwdule theoretic approach of (Fliess, 1990) is also valid for implicit systems. The solvability of nonlinear implicit differential equations is considered in (Brenan et al., 1995; Campbel and Griepentrog, 1995; Rheinboldt, 1991; Liu, 1995a; Liu, 1995b). Other problems like controllability (Lin and Ahmed, 1991), disturbance decoupling (Liu, 1996) stabilization (McClarnroch, 1990; Chen and Shayman, 1992; Kumar and Daoutidis, 1994), canonical forms (Fliess and Levine, 1995) have been already considered.

The notion of differential flatness, introduced by Fliess et al (Fliess et al., 1992; Fliess et al., 1993; Fliess et al., 1995a; Fliess et aL, 1995b) , which is also applicable to implicit systems, is strongly related to the problem of feedback linearization. It is important to point out the usefulness of the techniques of exterior calculus (Briant et al., 1991) for the problem of exact linearization (see (Shadwick, 1990), (Gardner and Shadwick, 1992), (Sluis, 1992), (Martin and Rouchon, 1993), (Murray, 1993), (Tilbury et al., 1995), (Shadwick

The aim of this paper is to present the notion of relative flatness with respect to a subsystem. We will show that this concept may be useful for control systems theory, in particular for studying the structure of nonlinear implicit systems.

1 This research was partially supported by CNPq under the grant 300492/95-2.

495

et al. , 1993; Fliess et al., 1995b; Fliess et al., 1997; Pomet, 1995). We have chosen to present a simplified exposition. For a more complete and intrinsic presentation the reader may refer to the cited literature.

Our approach will follows the infinite dimensional geometric setting recently introduced in control theory (Fliess et al., 1993; Fliess et al., 1995b; Fliess et al., 1997) in combination with the ideas presented in (Shadwick, 1990; Sluis, 1992; Shadwick and Sluis, 1993; Tilbury et al., 1995). Our sufficient conditions for the flatness of implicit systems may be regarded as a generalization of the conditions obtained in (Sluis, 1992; Shadwick and Sluis, 1993) for explicit systems.

Diffieties. A diffiety M is a JRA manifold equipped with a distribution of ~ of finite dimension r, called Cartan distribution. A section of the Cartan distribution is called a Cartan field. An ordinary diffiety is a diffiety for which dim ~ = 1 and a Cartan field OM is distinguished and called the Cartan field. In this paper we will only consider ordinary Diffieties that will be called simply by Di.ffieties.

The problem of feedback linearization of implicit systems has been considered for instance by (Liu, 1993; Kawaji and Taha, 1994). The existing literature considers the problem of finding state transformations and state feedbacks such that the closed loop system is a linear singular system. In this work we will tackle the problem of finding sufficient conditions for the flatness of a class of implicit systems of the form 2

x=

I(x) + g(x)u

y= hex) = 0

A Lie-Backlund mapping 4> : M t--> N between Diffieties is a smootl! mapping that is compatible with the Cartan fields'; i.e. ,4>.OM = ON 04>. A LieBiicklund immersion (respectively, submersion) is a Lie-Backlund mapping that is an immersion (resp., submersion). A Lie-B8.cklund isomorphism between two diffieties is a diffeomorphism that is a Lie-Backlund mapping.

(1.1a)

(Ub)

Context permitting, we will denote the Cartan field of an ordinary diffiety M simply by Given a smooth object 4> defined on M (a smooth function, field or form) , then L.!L4> will be denoted

Our point of view will be different, since we do not talk about feedback linearization, but only the flatness of the implicit system. Anyway, it seems that the concept of state-feedback for implicit systems of the form (l.la-l.lb) must be defined with care. In fact, there may exist nontrivial differential relations linking the components of the input u and the (pseudo )-state x (see (Fliess, 1989), (Fliess and Glad, 1993), (Glad, 1988), (Willems, 1992) for notions of input and state).

1t.

•

by

dt

4> and L nd 4> = 4>(n) , n

E IN.

Tt

Systems. The set of real numbers R have a trivial structure of diffiety with the Cartan field given by the operation of derivation of smooth functions. A system is a triple (S, lR, 7) where S is a diffiety equipped with Cartan field Os and 7 : S t--> JR is a Lie-Backlund submersion. The global coordinate function t of JR represents time, that is chosen for once and for all.

fr.

Most of the techniques that are necessary for the proof of our sufficient condition of relative flatness are very similar to the techniques of the proof of the main result of (Pereira da Silva, 1996a; da Silva, 1997). An application of a simplified version of the results presented here to robotics can be found in (Pereira da Silva, 1996b).

A Lie-Bii.cklund mapping between two systems (S, JR, 7) and (S', JR, 7') is a time-respecting LieBiicklund mapping 4> : S t--> S', i.e. , 7' = 704>. Context permitting, the system (S, lR, 7) is denoted simply by S.

The field of real numbers will be denoted by lR. The set of real matrices of n rows and m columns is denoted by F x rn. The matrix MT stands for the transpose of M. The set of natural numbers {I, . . . , k} will be denoted by lkl-

State Space Representation and Outputs. A local state representation of a system (S, JR, 7) is a local coordinate system, 'IjJ = {t, x, U} where x = {Xi, i E lnl} U = {ujk)l,j E lml, k E lV} where 70'IjJ-1(t, x, U) = t. The set of functions x = (Xl, . .. , Xn) is called state and u = (Ub ... , Urn) is called input. In these coordinates the Cartan field is locally written by

We will use the standard notations of differential geometry and exterior algebra in the finite and infinite dimensional case (Warner, 1971; Briant et al., 1991; Zharinov, 1992).

2. DIFFIETIES AND SYSTEMS

d _ 0

dt -

~ i=l

In this section we recall the main concepts of the infinite dimensional geometric setting of (Fliess

0

at + L..,; li Ox . + 1

"'""

L..,;

kEN,

(k+l)

uj

0 . ---w(2.2. OUj

jELrnl

Note that li may depend on t, X and a finite number of elements of U. In this sense, the state representation defined here is said to be general-

This class is more general than the one considered by (Liu, 1993; Kawaji and Taha, 1994).

2

496

Relative static-state feedbacks. We will consider now a special case of Lie-Biicklund Transformation that will be called by Relative StaticState Feedback. We stress that this transformation is a particular case of quasi-static state feedback (Delaleau and Fliess, 1993; Rudolph, 1995; Delaleau and Pereira da Silva, 1997) (see also (Perdon et aL, 1990; Perdon et al., 1993) for related concepts).

ized, since one accepts that fi may depend on the derivatives of the input. H the fi depend only on (x, u) for i E Ln1, then the state representation is said to be classical A state representation of a system S is completely determined by the choice of the state x and the input U and will be denoted by S: (x, u) or simply by (x, u). An output y of a System S is a set of functions defined on S. System associated to differential equations. Now assume that a control system is given by a set of equations

i = 1 Xi = fi(t,x,u, ... ,u(Cti ), i E Yj

=

'7j(x, u, ... , u(o;»), j E

Lnl lPl

Definition 1. Let S be a system and Si be a suI>system of S as described by the equations (3.4a)(3.4b). Let:E = span {dt,dx a, (dua(j) :j EN)}. Assume that dimx a = na, dimxb = nb, dimu a = ma, dim Ub = mb. Let Zb and Vb be sets respectively of of nb and mb smooth functions. Then the states (xa, Xb) and (X a,4) are linked by a relative static-state transformation with respect to the subsystem Si if span { dx a } + :E = span {dz b} +:E. Furthermore, «Xa,Xb),(Ua,Ub)) and «Xa,Zb),(Ua,Vb)) are linked by a relative static-state feedback with respect to the subsystem Si if the states (xa, Xb) and (xa, Zb) are linked by a relative static-state transformation and span {dx b, dUb} + :E = span {dzb, dVb} + :E

(2.3)

One can always associate to these equations a diffiety S of global coordinates 1/1 = {t, x, U} and Cartan field given by (2.2). Flatness. We present now a simple definition of flatness in terms of coordinates 3. A system S equipped with Cartan field and time function t = T is locally flat around ~ E S if there exists a set of smooth functions Y = (Yb"" Ym), called flat output, such that the set {t, y~) li ELm1, j E 1N} is a (local) coordinate system of S around

1t

~

E S, where y;j)

=

1t Yi. j

Remark. The definition above implies that, for Za = Xa and Va = Ua then {t ,zG,zb ,(vij) ,v~;) :jEN)} is a local coordinate system for which

Note that the Cartan

field is locally given by :

!!: _~ "" dt - at + L.J L.J

(J+l)~ (j)

Yi

jElViELml

Za

Oyi

=

Xa

Va = U a ( ) Zb = Zb(t, Xb, Xa, Ua, . .. , Ua 0 ) Vb = Vb(t, Xb, Ub, X a, Ua, ... , Ua(P»))

(3.5)

3. SUBSYSTEMS and represents an inversible (time-varying) feedback because

Subsystems. Given a system S, a subsystem Si of a system S is a system Si such that there exists a surjective 4 Lie-Biicklund submersion 7r : S ..--.

Xa Ua Ua Ub

Si . State equations adapted to subsystems. Assume that there exists a local classical state representation (x, u) of a system S ofthe form

Xa = fa(xa, u a) Xb = fb(x a, Xb, u a, Ub) .

=

Za

= Va

= x 2 (t, Zb, Za, Va, . . . , (v a)( 'Y) )

(3.6)

= u 2(t, Zb, Vb, Za, Va, . .. , (Va)(c5»)

where a, {3, ,,(,6 are convenient integers. Output Subsystem. Given a system S with output Y, an output subsystem is a subsystem 7r : S ..--. Y such that 7r"T"Y = span {dt, dy(k) : k E 1N}. It is not difficult to show that output subsystems are unique up to Lie-B8cklund isomorphisms.

(3.4a) (3.4b)

where x = (Xa, Xb) and u = (u a, Ub). In this case (3.4a) is the state equations of a subsystem Si and 7r : S ..--. Si is such that 7r(t, x, U) = (t,xa,Ua), where U denotes the set (u(j)lj E 1N) and Ua denotes the set (u a (j) Ij E IV). A state representation of S the form (3.4a)-(3.4b) is said to be adapted to the subsystem S.

State equations adapted to output su~ systems. The next theorem shows that we can generically construct the output subsystem and it admits adapted state equations.

Theorem 1. Given a system S with output y one can (generically) define a local state representation of the form

For more intrinsic definitions and some variations, see (Fliess et al., 1993; Fliess et al., 1994; Fliess et al., 1997). 4 Since submersions are open maps, one can always consider that SI = 1I"(S). 3

(3.7a) 497

~=h(x,y,y,u)

(3.Th)

where the state is z = (x,11), the input is e and span {dy( i ) liElV}= span { dY,dYb)liElV}.

Assume that the codistributions span {[(k), dt}

=

are involutive and that dimi(k)(q)/J(q) is (locally) constant for kEN, and i(N) = J for N big enough. Then the system S is relatively flat w.r.t SI .

(y, u)

We state now the result that assure that a subsystem is generically represented by state equations of the form (3.4a)-(3.4b).

5. IMPLICIT SYSTEMS REGARDED AS LlFrBACKLUND IMMERSIONS

Proposition 1. Assume that SI is a subsystem of S and there exist local state representations for SI and S around every point of SI and S. Then, generically, there exists a state representation of S of the form (3.4a)-(3.4b) in a way that (3.4a) is a state representation of SI'

In this section we will consider implicit systems of the form (1.1a)-(1.1b). We will show that, under some regularity assumptions, the system (1.1a)(1.1 b) may be regarded as a system r that is immersed in S. 'h\ other words, let S be the nonconstrained syst~ defined by (1.1a). Using theorem 1 we will construct a system r and a a Lie-Biicklund immersion L : r t-+ S such that every solution a(t) of S respecting the constraints y(t) == 0 is of the form a(t) = LO,(t) for a suitable solution ')'(t) of r.

4. RELATIVE FLATNESS We will state now the concept of relative flatness. Definition 2. Let S be a system and SI and S2 11"1 and 11"2 be the corresponding Lie-Backlund submersions respectively of S onto S1 and S2 . The system S is said to be decomposed by SI and S2 if around every point ~ of S there exists local coordinates (t , Xl) for SI , (t , x2) for S2 and (t , Xl , x2) for 5 S such that 1I"i (t , Xl, x2) = (t, Xi ), i = 1, 2. A subsystem S is said to be relatively flat with respect to a subsystem S1 if there exists a flat subsystem S2 such that S is decomposed by SI and S2.

be two subsystems of SI. Let

Consider the explicit (nonconstrained) system S defined by (l.la) with output y = h(x) with global coordinates

Proof. Take as a flat output of S the union of flat outputs of S1 and S2. The details are left to the reader. 0

AI. Regularity Assumption. Let r = {~ E Sly(k)(~) = 0 for all k E lV}. Around all ~ E r then theorem 1 holds.

A sufficient condition for relative flatness.

A2. Existence and Uniqueness Assumptions. Let { E r and let 11" : V C S t-+ Y be the correspond output subsystem constructed in a neighborhood V of {. Assume that for all 1I E V there exist a unique (t, ia, Oa) such that y(k)(t,ia,Oa) = 0 for all admissible t (In other words the subsystem Y admits a unique solution (t, ia, Oa) obeying the restriction y(k) = 0, kEN.

Theorem 2. Consider the system S of equations (3.4a)-(3.4b). Let X = (xa, Xb). Define

= =

span {dx - xdt}

+J

span {w(P)lw E

j(k-l)

t ==

and

dw(P) mod /(k- 1

Under the assumptions AI-A2, the set r n V may be endowed with the structure of a Frechet manifold by defining the local chart cfJ : r n V t-+ lR Ab where cfJ(t, ia , Xb, Oa , Ub) = (t, Xb, Ub). We can define a Cartan field on r by the equation

0, } k E IN

and J = span {(dx a - xadt), (du a (j) - U a (i+l)dt)lj

5

We denote

zi 01ri

simply by

zi

IN) } and Car-

Consider now the following assumptions :

Consider a system S and a subsystem SI of S given by the equations (3.4a)-(3.4b) where (3.4a) represents the subsystem S1. Let dimx a = na, dimxb = nb, dim U a = ma , dim Ub = mb.

/(0)

E

tan field (2.2). By theorem 1, we can construct around a (generical) point { E S, the output subsystem Y and a time respecting LieB8cklund projection 11" : S t-+ Y such that 11". (T·Y) = span {dt, dy(k) : k E 1Nl Furthermore, from theorem 1, this system admits a state space representation of the form (3.4a)-(3.4b) such that span {dxa , dUa} = span {dy(k) : k E IN} and span {dxa , dxb,dua , dub} = span { dx , du, dy, .. . , dy(r') }. In these coordinates, 1I"(t, Xa , Xb , Ua , Ub) = (t, Xa , Ua ).

Proposition 2. Let SI be a flat subsystem of a system S. Assume that S is relatively flat with respect to SI. Then S is flat.

/(k)

{x, (uP) :i ELm1;j

EN} .

(abU8e of notation) .

498

Proposition 9. Let r be the system constructed above. Then the insertion map L : r n V t-+ S is a Lie-Biicklund immersion. Furthermore, all the solutions ~(t) of (1.1a) obeying the restriction (Llb) and inside V are of the form ~(t) = LOll(t) where lI(t) is a solution of r n V.

Proof. Straightforward consequence of theorem 1 (see also equations (3.7a)-(3.7b» and corollary 2. o Theorem 4. Let S be the system defined by (1.1a). According proposition 3, the equations (1.1a)-(1.1b) define a system r that is immersed in S. Suppose that assumptions A1-A2 of the last section holds for the system (1.1a) with the constraints Llb. Assume that the conditions of theorem 3 are satisfied around ~ E S. Then the implicit system r is locally flat around ~.

6. FLATNESS OF IMPLICIT SYSTEMS

In this section we will derive a sufficient condition of flatness of implicit systems. Let us begin with an auxiliary result :

Proof. Straightforward consequence of proposition 3, theorem 3 and proposition 4. 0

Proposition 4. Let r, Sand Y be systems, where r an immersed system in Sand Y a subsystem of S and let L : r t-+ Sand 7r : S t-+ Y be respectively the corresponding Lie-Biicklund immersion and submersion. Assume that there exists local coordinates (t, 1') of r, (t, 1', y) of S and (t, y) of Y such that L(t,1') = (t, 1', 0) and 6 7r(t, 1', y) = (t , y). Assume that S is relatively flat with respect to Y. Then r is flat.

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Corollary 1. Assume that S is flat with flat output Y = (Yl,"" Ym). Consider the constrained system obtained by adding to S the constraints Yl = ... = Yr = O. Let ~ be a point such that y;j) = 0 for j E Nand i E Lr1. Then the implicit system r is locally flat around 7 ~ with flat output Yr+l,· ··Ym· Now let r be an implicit system (1.1a)-(1.1b) as constructed in the preceding section. We have the following sufficient condition for relative flatness with respect to the output subsystem Y.

Theorem 9. Suppose that assumptions Al holds for the system (1.1a) with the constraints (Llb). Let j(O) = span {dx - fdt} + J, where J = span {dy(k-l) - y(k)dt: k E 1N}. Consider the relative derived flag

j
j
dw mod (j
I

=O}

(6.9)

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