Some new interpretations of controllability and their practical implications

Annual Reviews in Control PERGAMON

Annual Reviews in Control 23 (1999) 197-206

Some new interpretations of controllability and their practical implications* Michel Fliess Laboratoire des Signaux et Syst~mes C.N.R.S. - Supalec - Universit~ Paris-Sud P l a t e a u de Moulon, 91192 Gif-sur-Yvette, France fliess@iss, supelec, fr

Abstract R.E. Kalman introduced the concept of controllability for finite-dimensional linear systems at the beginning of the sixties. It has played since then a key role. A vast literature has been devoted to various extensions to nonlinear and infinite-dimensional systems. We first interpret Kalman's controllability in a new module-theoretic language, which is independent of any distinction between the system variables. Its nonlinear extension leads to the notion of differentially flat systems, which were introduced in 1992 by J. L~vine, P. Martin, P. Rouchon, and the author. They permit a straightforward t r a j e c t o r y tracking, which has been illustrated by numerous applications. In 1995 H. Mounier worked out the case of linear delay systems and discovered a new type of controllability which is verified by most realistic case-studies. We will conclude with some examples of systems governed by partial differential equations, which were recently studied by H. Mounier, P. Rouchon, J. Rudolph, and the author.

R~sum~.

R.E. K a l m a n a, dans les ann6es soixante, introduit la commandabilit~ des syst~mes lin~aires de dimension finie, concept au r61e cir. Une abondante litt~rature a 6t~ consacr~e b~ diverses g~n~ralisations au non-lin6aire et b. la dimension infinie. Nous interpr~tons d ' a b o r e d la commandabilit~ de K a l m a n dans le langage des modules, ind6pendant de toute distinction entre les variables du syst~me. Le passage au non-lin~aire conduit aux syst~mes diff~rentiellement plats, dfis h J. L6vine, P. Martin, P. Rouchon et l'auteur. Ils permettent un suivi de trajectoires ais~ aux nombreuses applications. En 1995, H. Mounier a d~couvert un nouveau type de commandabilit~ pour les syst~mes lin~aires h retards, v~rifi~e presque toujours en pratique. Nous concluerons par quelques exemples d'~quations aux d~riv~es *Work partially supported by the European Commission's Training and Mobility of Researchers (TMR) Contract ERBFMRXT-CT970137, and by the P.R.C.-G.D.R. A UTOMATIQUE.

partielles, r6cemment ~tudi~es par H. Mounier, P. Rouchon, J. Rudolph et l'auteur.

Key Words Finite-dimensional linear systems, finite-dimensional nonlinear systems, linear delay systems, wave equation, heat equation, controllability, flatness. Introduction A r o u n d 1960, R.E. K a l m a n (see [25, 26, 27]) int r o d u c e d t h e c o n c e p t of controllability for finited i m e n s i o n a l c o n s t a n t linear d y n a m i c s of t h e form

d dt

.

Xn

=A

"

Xn

+B

"

(I)

Um

W e all know t h a t c o n t r o l l a b i l i t y , w h i c h is c h a r a c t e r ized b y r k ( B , A B . . . . , A * ~ - I B ) = n, m e a n s a reachab i l i t y p r o p e r t y , i.e., it is e q u i v a l e n t to t h e p o s s i b i l i t y of j o i n i n g two p o i n t s of t h e s t a t e s p a c e v i a a suita b l e control. I t s key role for u n d e r s t a n d i n g s t r u c t u r a l p r o p e r t i e s a n d c o n t r o l s y n t h e s i s is well d o c u m e n t e d (see, e.g., [28]) a n d h a d an e n o r m o u s influence in t h e d e v e l o p m e n t of f i n i t e - d i m e n s i o n a l n o n l i n e a r c o n t r o l (see, e.g., [24, 43]) a n d in infinite d i m e n s i o n a l linear s y s t e m s (see, e.g., [6, 33, 34]). T h o s e e x t e n s i o n s are all b a s e d on a s t a t e s p a c e d e s c r i p t i o n a n d on a reacha b i l i t y i n t e r p r e t a t i o n of c o n t r o l l a b i l i t y . W e r e p o r t here on a new a p p r o a c h t o c o n t r o l l a bility, w h i c h is b a s e d on t h e following i n t e r p r e t a t i o n of f i n i t e - d i m e n s i o n a l l i n e a r c o n t r o l l a b i l i t y [7]. T h e r e exists a finite set { b l , . . . , bin} such t h a t • a n y s y s t e m v a r i a b l e is a l i n e a r c o m b i n a t i o n of t h e bi's a n d t h e i r d e r i v a t i v e s u p t o s o m e finite order; • a n y bi is a linear c o m b i n a t i o n of t h e s y s t e m variables a n d t h e i r d e r i v a t i v e s u p t o s o m e finite order; • t h e bi's are n o t r e l a t e d b y a n y l i n e a r differential equation.

1367-5788/99/$20 © 1999 Published by Elsevier Science Ltd on behalf of the International Federation of Automatic Control. All rights reserved. PII: S 1367-5788(99)00022-X

198

M. Fliess / Annual Reviews in Control 23 (1999) 197-206

The obvious extension to nonlinear systems is obtained by dropping the above assumption of linearity and leads to (differential) flatness [11, 14]. When coming to linear delay systems, one should add to the time derivations delays and advances. This leads to the notion of 7r-controllability [39, 16], which is verified by most practical examples. The boundary control of the wave equation may be reduced to this case [39, 42]. This approach may be generalized to the heat equations and other partial differential equations [17, 1811. Our paper is organized as follows. We start with the module-theoretic description of finite-dimensional linear systems. Differential flatness of nonlinear systems is then defined via differential and via differential geometry of jets and prolongations of infinite order. Delay systems are then briefly reviewed. The analysis of the boundary control of some simple partial differential equations is finally given. It employs Mikusifiski's version of operational calculus (see [37, 38] and [53]) and Gevrey functions (see [21] and [30, 47]).

1

Finite-dimensional

1.1

• any element of M depends k[d]-linearly on b, • the components of b are k[d]-linearly independent. The rank of this free module is m. Here are some standard properties of finitely generated modules over principal ideal rings (see, e.g., [31]). 1. A finitely generated k[d]-module M may be written M ~- t M @ .~ (4) where t M is the torsion submodule and .T" _~ M / t M is a free module. The rank of M is, by definition, the rank of 9~. 2. For a finitely generated k[d]-module M, the following two properties are equivalent:

linear systems

P r e f a t o r y remarks

• M is torsion,

Consider a controllable and observable SISO system, with input u and o u t p u t y, given by the transfer funcP(s) tion Q---~, where P(s), Q(s) E R[s] are coprime. By B@zout's theorem there exists A, B E R[s] such that A P + B Q = 1. Introduce a new system variable z by

z = A ( ~ t ) Y + B(

)u

(2)

The transfer function of the system with input u and 1 The quantities u, y and z satisfy o u t p u t z is Q--~. u

=

=

Q~z ~dtJ P(

)z

Modules

Let k be a field and k[ d ] be the commutative principal ideal ring of polynomials of the form ~ / i n ~ t e a ~ - d~~, as e k. Let M be a k[d]-module. An element m E M is said to be torsion if, and only if, there exists a polynomial 7c E k[d], 7r ~ 0, such t h a t 7rm -- 0. T h e set t M of all torsion elements of M is a submodule of M; it is said to be trivial if, and only if, t M = {0}. A k[d]-module is said to be 1See, also, [35].

• the dimension dimkM of M as a k-vector space is finite. 3. Any submodule of a finitely generated (free) k[d]-module is a finitely generated (free) d module. Any quotient module of a finitely generated k[d]-module is a finitely generated d module. 4. For a finitely generated k[d]-module M, the following two properties are equivalent:

• M is torsion-free,

(3)

Equations (3) tell us that u and y are expressed as linear combinations of z and its derivatives up to some finite order. According to (2), z is, conversely, expressed as a linear combination of u and y and their derivatives up to some finite order. Those properties are reminiscent to flatness: z is called a flat output. 1.2

torsion if, and only if, all its elements are torsion; it is said to be torsion-free if, and only if, t M is trivial. A finitely generated k[d]-module M is said to be free if, and only if, there exists a basis, i.e., a finite set b = ( b l , . . . , b m ) such that

• M is free.

Remark. All modules considered in the sequel will be finitely generated k[d]-modules. Notation. Write IS] the submodule spanned by a subset S of M. 1.3

Systems

A k-linear system A is a module [7, 9]. A k-linear dynamics is a k-linear system A with an input, i.e., with a finite subset u = ( u l , . . . , u r n ) such that the quotient module A/[u] is torsion. The input u is said to be independent, if, and only if, the submodule [u] is free of rank m. Then, the rank of A is equal to m. A k-linear input-output system is a k-linear dynamics A with an output, i.e., with a finite subset Y = (Yl . . . . , yp) o f A. There exists a short exact sequence 0 - ~ H ~ ~--~ A - ~ 0

M. Fliess / Annual Reviews in Control 23 (1999) 197-206

T h e module ~- is free. The free module J~, which is called sometimes the module of relations, should be viewed as a s y s t e m of equations defining A. Example. Consider the system of equations /z =0

where a ~ c k[d], e = 1 , . . . , v . The unknowns are wl,...,w,. Let ~" be the free module spanned by f l , . . . , f , . Let Af C_ $- be the submodule spanned tt a by }-~=1 ~ f ~ ' The module corresponding to (5) is

7/N.

Controllability

A k-linear system A is called controllable [7, 9] if, and only if, the module A is free. Any basis of A, which may be viewed as a fictitious output, is called a flat, or basic, output. Examples. 1) Consider the classic state-variable representation (1). T h e control variables u = ( u l , . . . , u m ) are assumed to be independent. It follows from [7] that (1) is controllable, i.e., that rk( B , A B , . . . , A n - I B ) = n, if, and only if, the corresponding module A is free. As a m a t t e r of fact, the torsion submodule tA in the decomposition (4) corresponds to the Kalman uncontrollable subspace. 2) Assume that (1) is controllable. There exists a static state feedback which transforms it into the famous Brunovsk~) canonical f o r m (see, e.g., [28]) which reads z ( ' ) = vi, i = 1 , . . . ,m, where the vi's are the new control variables and the ui's the controllability, or K r o n e c k e r , indices. The next property is clear. PROPOSITION. The set z = ( z l , . . . , Zm) is a fiat output. 3) Consider the input-output system

.4

•

Yp

= B

Willems' controllability

T h e set of C°%functions (tl,t2) ~ R, -c<) < tl < t2 _< -I-00, is written C ° ° ( t l , t 2 ) . It is clear that C ° ° ( t l , t 2 ) is a R[d]-module. Note that this module is not finitely generated.

(5)

t~=l

1.4

1.5

199

(ul) •

(6)

~tm

where .4 E k i d ] pxp, d a t a ¢ 0, B E "b[--dlpxm tdtl " It is known that (6) is controllable if, and only if, A and B are left coprime (see, e.g., [23] and [3, 10]). Denote by A its corresponding module. PROPOSITION. The output y = ( Y l , . . . , Yp) is flat if, and only if, the following two conditions are satisfied: 1) the m a t r i c e s A and 13 are left coprime; 2) the s y s t e m is square, i.e., m = p, and the mat~'ix 13 is unimodular. PROOF. The first condition ensures the controllability of (6) and is equivalent to the freeness of A. The second condition is equivalent to ui E [y], i=l,...,m.

Take a R-system A, i.e., a finitely generated R [ ~ ] module A. A (smooth) trajectory of A on the time interval (tl, t2) is a R [ d ] - m o d u l e morphism ~ : A --* C~(tl,t2). The set of trajectories of A on the interval (tl, t2) is, of course, the set Hom(A, C ° ° ( t t , t2) of morphisms A--* C ° ~ ( t l , t ~ ) . Remark. Other spaces of (generalized) functions, such as Sehwartz's distributions, could have been singled out, as long as they may be endowed with a structure of R [ d ] - m o d u l e . For our purpose however the space of C a - f u n c t i o n s seem to be the most appropriate. Set - 0 0 _< tl _< t2 < t3 < t4 < +c~. A trajectory r(t2,ta ) : A --* C ~ ( t 2 , t 3 ) is said to be a restriction of a trajectory "r(tl,t4 ) : A --* C ~ ( t t , t 4 ) if, and only if, for any A E A, the functions T(t2,t3 ) (~) and T(t2,t3 ) (~) coincide on (t2, t3). Set --ec _< t~ < t~ < t~ < t~ < +00. Two trajectories r(tl,t,2) : A ~ C~(t'~,t'2) and T(t,a,t,) : A --* C °° ( t'3, t~) are said to be compatible if, and only if, there exists a trajectory r(tl,t,4) : A --* C (tl, t4) such t h a t ~-(tl,t,2) and T(t,.a,t,4) are restrictions of ~-(ti,t;). ~Ve say that ~-(q,t;) is a past trajectory and ~-(t;,t;,) a f u t u r e trajectory. System A is said to be controllable d la W i l l e m s (compare with [54]) if, and only if, any future traject o r y is compatible with any past trajectory. The next theorem is borrowed from [9]. THEOREM. A R - s y s t e m is controllable d la W i l l e m s if, and only if, it is controllable. Proof. Assume that the R-system A is controllable. Choose a basis b l , . . . , b , ~ of the free R[d~]-module A. Any trajectory T(tl,t2 ) : A ~ C ~ ( t l , t 2 ) is completely determined

b y t h e T(tl,t2)(bi)'s , i = 1 , . . . ,

m.

Well known properties of C~-functions show that the functions T ( t ,1 ,t ,2) (bi) E C oo ( t l1, t 2/) , T(t'~,t,~)(bi) E C~(ff3,ff4), i = 1 , . . . ,m, are restrictions of a function "r(tl ,t, ) ( b~) E C ~ ( t'~ , t'4 ) . Assume now that A is not controllable. Take E t A , ~ ¢ 0. Any trajectory ~(t) of ~ is solution of a linear homogeneous differential equation with constant real coefficients, i.e., P ( ~ ( t ) ) = 0, P E R i d ] , I ! P ¢ 0. Choose a past trajectory on (tl,t2) and a rt'3, t 4J ~ which correspond to diffuture trajectory on ~ ferent Cauchy conditions for some t~, t~ < t~ < t~. Those two trajectories are incompatible. Controllability h la Willems is not satisfied.

M. Fliess / Annual Reviews in Control 23 (1999) 197-206

200 2

Finite-dimensional tems

2.1

nonlinear

sys-

Prefatory remarks

Let us start with a concrete case-study, namely the crane, which is a familiar object of study in robotics and control laboratories (see [11] and the references therein).

system variables may be calculated without integrating any differential equation. This property, which was discovered in [11]. and which may be traced back to Hilbert [22] and E. Cartan [4], has been called (differential) flatness. Many control systems encountered in practice are flat and their control becomes then much simpler. The order of derivation needed for checking flatness is not known a priori. This is why a precise definition of flatness ought to be given in the language of differential algebra or in the differential geometric setting of prolongations and jets of infinite order.

v

The formalism of algebra

2.2

2.2.1

D i f f e r e n t i a l fields

All fields have characteristic zero. An (ordinary) differential field K (see, e.g. [29])2 is a field which is

Z

d equipped with a single derivation d-'~

Va, b E K ,

Figure 1: The two dimensional crane. A dynamic model for the load may be obtained by writing down the equations stemming from the Newton law and the geometric constraints: { m~ m// x z

= = = =

-Tsin0 -Tcos0+rag Rsin0+D R cos 0

(7)

where • (x,z) (the coordinates of the load m), T (the tension of the rope) and 0 (the angle between the rope and the vertical axis OZ) are the unknown variables; * D (the trolley position) and R (the rope length) are the input variables. (7), which combines differential equations and algebraic ones, i.e., equations without derivatives, is an implicit system of equations. It can be shown [15] that the above crane does not possess a classic state variable representation ~ = F(x, u). It is clear that sin 0, T, D and R are functions of (x,z) and their derivatives,i.e., sin0 - x -R- D ' T =

mR(g-5) Z

(5-g)(x-D)

= ~:z, ( x - D ) 2 + z 2 = R 2.

It yields

D

=

R2 =

"'"

such that

Va C K, -da~ = d E K ~d( a + b) = a + b, ~(ab) = ab + a~

A differential field extension L / K is given by two differential fields K, L such that K c_ L. An element in L is differentially algebraic over K if, and only if, it satisfies an algebraic differential equation with coefficients in K. An element in L is differentially transcendental over K if, and only if, it is not differentially algebraic over K. The extension L / K is differentially algebraic if, and only if, any element of L is differentially algebraic over K; L / K is differentially transcendental if, and only if, it is not differentially algebraic. A set ~ -- {~ I i c I} of elements in L is said to be differentially algebraically (in)dependent over K if, and only if, the set of derivatives of arbitrary orders { ~ ) I i E I, vi = 0,1,2 . . . . } is algebraically (in)dependent over K. An independent set which is maximal with respect to inclusion is called a differential transcendence basis of L / K . Two such bases have the same cardinality which is the differential transcendence degree of L / K ; it is written diff tr d o L / K . Notation. The differential subfield of L generated by K and the set ~ is denoted K(~). A differentially transcendental field extension L / K is said to be pure if, and only if, there exists a differential transcendence basis ~ such that L = K(~). Remark. From now on all differential field extensions will be finitely generated.

~z

2.2.2

xz ~ 2 \~ -g/

Let k be a given differential ground field. A k-system is a differential field extension K / k [8, 11]. A kdynamics is a k-system K / k equipped with an input, i.e., with a finite set u = ( u l , . . . , urn) such that

x-2~g z2+(

=

Once the time functions x(t) and z(t) are known as well as their first and second derivatives all the other

Systems

2See [8, 11] for a d e t a i l e d r e v i e w .

M. Fliess / Annual Reviews in Control 23 (1999) 197-206

K/k(u) is differentially algebraic. An output is a finite set y = ( Y l , . . - , Yp) of elements in K/k. 2.3

Equivalence, endogenous ferential flatness

feedback,

dif-

T w o s y s t e m s K J k , ~ = 1, 2, are k-equivalent [11] if, and only if, t h e r e exist differential fields M~ such t h a t M~/K~ is algebraic and t h e two differential extensions M1/k and M2/k are differentially k-isomorphic. In o t h e r words, any variable of one of t h e s y s t e m s m a y be expressed as an algebraic function of t h e variables of t h e o t h e r one and of their derivatives up to some finite order. T h o s e expressions define an endogenous feedback b e t w e e n t h e two s y s t e m s [11]. A s y s t e m K / k is (differentially) fiat [11] if, a n d only if, K / k is k-equivalent to a pure differentially t r a n s c e n d e n t a l extension of k. A differential transcendence basis z = ( z l , . . . , zm) of g / k such t h a t g/k(z) is (non-differentially) algebraic, which m a y be viewed as a fictitious o u t p u t , is called a fiat, or linearizing, output. Recall t h a t a flat s y s t e m is equivalent to a controllable linear system. 2.4

The formalism

2.4.1

of differential geometry

Diflieties

Let I be a c o u n t a b l e set of cardinality ~, which m a y be finite or not. D e n o t e b y R l the set of m a p p i n g s I --* R , where R is t h e real line; R ~ is e q u i p p e d with the p r o d u c t topology, which is Fr6chet. For any open subset V C R l, let C~(~7) be the set of functions V --* R , which only d e p e n d on a finite n u m b e r of variables and are C ~ . A C ~ R C m a n i f o l d (see, e.g., [55]) m a y be defined like in finite dimension via R e_ valued charts. T h e notions of functions, vector fields, differential forms of class C ~ on an open subset are clear. If {x~ [ i E I } are s o m e local coordinates, let us notice t h a t a vector field m a y be given by an infinite expression ~-~.iEl ~ g ~0, , w h e r e a s a differential form E finite wi~...~dxi~ A ... A dx~ is always finite (the ~i's and wh...~ 's are C ~ functions). T h e notion of a (local) C ~ m o r p h i s m b e t w e e n two C ~ R ~ and R e'manifolds, w h e r e g a n d ~' are not necessarily equal, is obvious as well as t h e notion of (local) isomorphism. O n the contrary, t h e non-validity of the implicit function t h e o r e m in these infinite-dimensional Fr~chet spaces is forbidding the usual equivalence between various c h a r a c t e r i z a t i o n s of (local) submersions and i m m e r s i o n s b e t w e e n two finite-dimensional m a n ifolds. We choose t h e following definition: A (local) C ~ submersion (resp. immersion) is a C °~ m o r p h i s m such t h a t t h e r e exist local coordinates where it is a projection (resp. injection). A di~ety 3 J~ is a C ~ R C m a n i f o l d which is e q u i p p e d with a Cartan distribution CTJV[, i.e., a finite-dimensional and involutive distribution. T h e aThis terminology is due to Vinogradov [52].

201

dimension n of CT:~ is t h e Cartan dimension of :M. A n y (local) section of CT~V[ is a (local) Caftan field of 3~. T h e diffiety is called ordinary (resp. partial) if, and only if, n = 1 (resp. n > 1). A differential equation is a diffiety. A C ~ (local) m o r p h i s m between diffieties is said to be Lie-B?icklund if, and only if, it is c o m p a t i b l e with t h e C a r t a n distributions. T h e notions of (local) Lie-B/icklund submersions and immersions are clear. F r o m now on, we will restrict ourselves to ordinary diffieties, i.e., to o r d i n a r y differential equations. E x a m p l e . A f u n d a m e n t a l role is played by the diffiety with global coordinates r1t , Y(~d i ] i = 1 . . . . , m; ~ > 0} and C a f t a n field

i~-i ui>O

oYi

It is w r i t t e n R x R ~ and called a trivial diffiety (see, e.g. [55]) since it c o r r e s p o n d s to t h e trivial equation 0=0. A diffiety :M is said to be (locally) of finite type4 if, and only if, there exists a (local) Lie-B/icklund s u b m e r s i o n :M --* R x R m~ such t h a t the fibers are finite-dimensional; m is called t h e (local) differential dimension of JV[. E x a m p l e . Take the nonlinear d y n a m i c s

= F(x, u)

(8)

where the s t a t e x = ( x l , . . . , x n ) and the control u = ( U l , . . . , U m ) belong to o p e n subsets of R n and R m ; F = ( F 1 , . . . , Fn) is a m - t u p l e of C °c functions of their arguments. Associate to (8) the infinitedimensional manifold © given by the local coordinates {t, xl . . . . ,x,~,ul ~') I i = 1 , . . . , n ; ~ > 0}. T h e C a f t a n distribution is s p a n n e d by the C a r t a n field

d--t = 0-t +

k=l

Fk

+

i=1 ui>0

ui

. (,,)

0hi

A (local) Lie-Biicklund fiber bundle [55] is a triple a = (X,~B, Tr), where 7r : % --~ ~B is a (local) LieB~icklund s u b m e r s i o n b e t w e e n two diffieties. For any b E ~B, 7r-l(b) is a fiber. T a k e a n o t h e r Lie-. Bgcklund a' = (2:',~,~r') w i t h t h e s a m e base ~B. A LieB~icklund m o r p h i s m c~ : a --+ a ' is a Lie-B~icklund m o r p h i s m c~ : ~ --+ ~ ' such t h a t rr -- 7dc~. T h e notion of Lie-B/icklund i s o m o r p h i s m is clear. 2.4.2

Systems

A system [13] is a (local) Lie-B~cklund fiber bundle a - (S, R , T), where • S is a diffiety of finite t y p e where a given C a r t a n field 0s has been chosen once for all 4See [13] for an intrinsic definition.

202

M. Fliess / A n n u a l R e v i e w s in Control 23 (1999) 1 9 7 - 2 0 6

• R is endowed with a canonical structure o f a diffiety, with global coordinate t, and Cartan field __0. Ot '

3 Linear delay systems 3.1 Prefatory remarks

• the Cartan fields 0s and o are v-related.

Consider the elementary linear delay system

A (local) Lie-Biicklund morphism (resp. immersion, submersion, isomorphism) ~ : ( S , R, v) --* (S', R, V') between two systems is a Lie-B~icklund morphism (resp. immersion, submersion, isomorphism) between S and S' such that

Jz(t)=ax(t)+bu(t-h)

a,b, h E R ,

b ~ O , h >O (10) which is related to the classic Smith predictor (see, e.g., [45]). Notice that x plays the role of a flat output if we admit 'an advance, i.e., u(t) = x(t + h) - ax(t + h)

b

• 0s and 0s, are ~-related.

A dynamics is a (local) Lie-B~cklund submersion 5 : (S, R, v) --~ (U, R, it) between two systems, such that the Caftan fields Os and 0g are 5-related. In general, U will be an open subset of a trivial diffiety R × Rm: it plays the role of input and m is the number of independent input channels. Replace, with a slight abuse of notations, 0s and Ov, which play the role of total derivation with respect to t, by ~d . Example. Take a time-dependent dynamics J: = G(t,x, u, it . . . . ,u (~))

(9)

which may contain moreover derivatives of the control variables u = ( u l , . . . , urn) and an analogous infinitedimensional manifold ~ given by the local coordinates {t, x l , . . - , x~, ul "') } where

d-~ = -'~ +

Gk k=l

+

% i = l vi>O

~ (,,) (YUi

The above submersion 7 is given by the projection { t , , u ~ ) } . Notice that the { t , X l , . . . , X n ~ u i ° ('~)~ ] state-space is nothing else that the corresponding fiber.

(11)

This type of parametrization is verified by most concrete examples (see [39, 41]). It permits a straightforward motion planning. 3.2

Mathematical background

Let k[~t, 5] be a ring of polynomials over a field k, in r + 1 indetrminates, where 5 = (51,..., 5r) should be viewed as a set of r delay operators 6. A (linear) delay system [39, 16] is a finitely generated k[~t, 5]-module A. The notion of input, output and dynamics may be defined in the same manner as for finite-dimensional linear systems r. For a k[ d , 6J-module, torsion freeness and freeness are no more equivalent. Here freeness implies torsion freeness. A delay system A is said to be torsion free controllable (resp. free controllable) if, and only if, A is a torsion free (resp. free) module. The next result follows from [51]. THEOREM AND D E F I N I T I O N . Let A be a torsion free controllable delay system. Then there exists ~r E k[5], 7c ~ O, such that the localized k[~,5,~r-1]-module k[ d , 5, 7r-1] ®k[~,61A is free. Then system A is said to be 7r-free controllable. Example. Write again (10) as

= ax + bhu

2.4.3

Equivalence and flatness

Two systems (S, R, v) and (S ~, R, v ~) are said to be (locally) differentially equivalent [13, 14] if, and only if, they are (locally) Lie-B~cklund isomorphic. They are said to be (locally) orbitally equivalent if, and only if, S and S ~ are (locally) Lie-B~icklund isomorphic [13, 14]. The first definition preserves time, whereas the second one does not: it introduces a time-scaling. The triple (R × R ~m, R , pr), where R × R~o = ~, y~ ~ is a trivial diffiety and pr denotes the projection {t,-y~('~) ~ ~-~ t, is called a trivial system. The system (S, R, v) is said to be (locally) differentially flat [13, 14] 5 if, and only if, it is (locally) differentially equivalent to a trivial system; it is said to be (locally) orbitally flat [13, 14] if, and only if, it is (locally) orbitally equivalent to a trivial diffiety. The set Y = (Yl, • - •, Yra) is called a fiat, or linearizing, output. SSee, also, [46, 44].

It is not free controllable, but torsion free controllable. Formula (11) shows that it is 5-free controllable. Remark. See [39, 40] and [16] for more details and related references. Concrete examples may be found in [39, 4!].

4 Partial differential equations 4.1 Prefatory remarks Consider the wave equation

02wl(x,t) Ox 2

02wl(x,t) =

Ot 2

0
t>0

(12)

6 A s s u m e t h a t k is t h e field R or C of reM or c o m p l e x numbers. For any function f : R ~ k, 5~f(t) = f ( t - h~), where h~ > 0, ~ ---- 1 , . . . , r . T h e vector space over the field Q of rational n u m b e r s s p a n n e d by t h e h~ 's is r-dimensional. 7A torsion element of a k [ ~ , 5J-module may be defined as for k[~t]-modules. T h e notion of torsion k [ ~ , 5 ] - m o d u l e is clear.

M. Fliess / Annual Reviews in Control 23 (1999) 197-206

and the heat equation

02w2(z,t) Ox 2

-

Ot

0
T h e initial conditions are

t>0

x, 0) =

(13)

(.,0) = 0

and °w2 (x,0) = 0. The b o u n d a r y conditions are w~(O,t) = 0 and w~(1,t) = u(t), ~ = 1,2, where u(t) designates the control variable. Classic operational calculus replaces (12) and (13) with the ordinary differential equation s) -

s) = o

(14)

where a = s, i f t = 1, and a = v/s, if~ = 2. T h e b o u n d a r y conditions are ~ . ( 0 , s) = 0, ~ ( 1 , s) = fi(s). For the m o m e n t and like usual ~ ( x , s ) and ~(s) are the Laplace transforms of w ~ and u with respect to time, i.e., ~ ( x , s) = f o e-*tw~(x,t) dt and ¢~(s) = f o e-*tu(t) dt" T h e solution of (14) reads

- cosh ax cosha

~(s)

(15)

N o w introduce ¢(s) by

cosh(crx) =

cosh( )

4.2.1

Mathematical

background

T h e set of continuous functions [0, +cx~) --+ R is a c o m m u t a t i v e ring C with respect to the pointwise addition + and the convolution product f , g = g * f = f : o o f ( r ) g ( t - - r ) d r = f+~o g ( ' r ) f ( t - r ) d r . According to a famous theorem due to T i t c h m a r s h (see [37, 38] and [53]), C does not possess zero divisors, i.e.,

¢~

3) The field J~4 contains the subring S of piecewise continuous functions R --+ R with left bounded supports, i.e., for any f E S, there exists a constant E R , such that, for t < /3, f(t) = 0. The translation operator e -hs, h ~ R acts oil f E S by e-hS{f(t)} = { f ( t -- h)}. T h e inverse of e --h~ is e ''~, Notice t h a t e sv/-~ is not an operator, i.e.. does not belong to ~4. A sequence an, n > 0, of operators is said to be operationally convergent [37, 38] if, and only if, there exists an o p e r a t o r p such t h a t the anp's belong to C and converge almost uniformly, i.e., uniformly on any finite interval, to a function in C. A series of operators ~ > 0 b~ is said to be operationally converyent [37, 38] if, and only if, the sequence ~'~=0 b, is operationally convergent. Example. T h e operator e ~'F, ~\ c C, In~w be defined by its Taylor expansion, i.e.,

(is) n>0

(17)

Mikusifiski's operational calculus

f *g = 0

], then { s f } = { / } + {f(0)}. The meaning of operators in the subfield C ( s ) of rational functions in the variable s with complex coefficients is clear. Note also t h a t the fractional derivation ~ appears as the inverse of ~ = { 1

(16)

We see t h a t ~ might play the role of a flat, or basic, output. T h e question then arises how to exploit (16)(17). 4.2

203

f = 0 or g = 0

T h e quotient field A/[ of C is called the Mikusirlski field (see [37] and [53]). Any element of Ad is called an operator. Notation. A function f(t) in C is sometimes written { f ( t ) } when viewed as an operator in A/[. Examples. 1) T h e neutral element 1 for the (convolution) product is the Dirac operator. It is the analogue of the Dirac distribution in Schwartz's distributions theory. T h e Dirac operator 1 should not be confused with the Heaviside function {1}. 2) T h e inverse in ~4 of the Heaviside function {1} is the differentiM operator s. It obeys to the classic rules, i.e., if f c C is C 1, i.e., possesses a derivative

This p r o p e r t y does not hold for the translation operator e -hs. An operational function [37, 38] is a mapping R --~ Ad. One can define the continuity, derivability and integrability of operational functions. Example. To the wave and the heat equations (12)(13) corresponds the operational differential equation (14), where ~b~ and ~ do not indicate any more a Laplace t r a n s f o r m t)ut respectively an operational function and an o p e r a t o r The solution (15) is an operational function. 4.3

Gevrey functions

A complex-valued C ~ - f u n c t i o n f of the real variable t is said to be Gevrey of class or, cr E R, on an interval I (see [21] and [30, 47]) if, and only if, for any integer n>0 lf(n)(t)l <_ Cn+l(n!) a t ~ I or, equivalently,

If(n)(t)l <

""

t

z

where C , C ' > 0 are real constants depending on f and I. Gevrey functions of" class c, are, of course, Gevrey functions of class a', where a ' > a. A function is analytic if, and only if, it is Gevrey of class 1. A Gevrey function of class 1 are of utmost importance.

M. Fliess / Annual Reviews in Control 23 (1999) 197-206

204 Example. The function

Conclusion ift<_O e-~ ift>O 0

¢(t)=

where d > 0, is flats at t = 0, i.e., all its derivatives vanish at t = 0. It is not analytic but Gevrey of class l+d d (see, e.g., [47]). The function 0

ift
fo/Texp(--I/(T(l -- ~-))d)dT ~Td(t) =

aof-1exp(--1/(T(1 -- T))d)dT 1

if t E [0, T] if t > T,

(19) is also Gevrey of class l~-d d " 4.4 A p p l i c a t i o n s 4.4.1

References

The wave equation

Assume that the fiat output ~ is a function R --* R belonging to the Mikusirlski field A4. Then (16)-(17) yield

wl(t,x)

=

u(t)

=

~(t + x) + ¢(t - x) 2 ¢(t+l)+¢(t-1) 2

(20)

The heat equation

In order to justify (20-21), assume that the function ~(t) is flat at t = 0 and Gevrey of class o. THEOREM.

The

series ~-~m>ox2n

~f

and

~-]n>_o ~ are absolutely convergent if, and only if, < 2. Then, if ~ is flat at t = O, their sums are respectivly equal to cosh(xx/~) {~} and cosh(x/~) {~}. Proof. The absolute convergences of the series follow at once from the classic Stirling formula n! e-nnn-½ V ~ , n --* +co. The second part follows from (18). It yields, if a < 2,

r,>_o and

u(t) =

n>_0

[1] Y. Aoustin, M. Fliess, H. Mounier, P. Rouchon, J. Rudolph, Theory and practice in the motion planning and control of a flexible robot arm using Mikusifiski operators, Proc. IFAC-SYROCO Conf., Nantes, 1997. [2] L. Bitauld, M. Fliess and J. L@vine, A flatness based control synthesis of linear systems with an application to windshield wipers, Proc. 4 th Europ. Control Conf., Brussels, 1997.

(21)

For a given x = x0, 0 < x0 _< 1, (20)-(21) determine a linear delay system, which is r-controllable, where r = cosh s. Remark. More concrete examples of the boundary control of the wave equation may be found in [39, 42].

4.4.2

The finite-dimensional flatness, which is easy to teach to engineers (see, e.g., [32, 48, 50]), has already proved to be immensely useful for dealing with concrete topics, concerning the motion planning and the stabilization of nonholonomic mechanical systems [11, 12], magnetic bearings [321, chemical reactors [48, 49], electric motors [5, 36], windshield wipers [2], tracking observers [19], non-minimum phase systems [20], etc. The perspectives with its infinite dimensional analogue, i.e., with this new understanding of controllability also seems very promising; see the control of a vibrating string [42], of a flexible Euler-Bernoulli beam [1, 17] and of a heat reactor [18].

(2n)!

(n)(t) (2n)!

8The word fiat has many different meanings in the mathematical literature.

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[18] M. Fliess, H. Mounier, P. Rouchon and J. Rudolph, Controlling the transient of a chemical reactor: a distributed parameter approach, Proc. CESA '98, Hammamet, 1998.

[35] P. Martin, R.M. Murray and P. Rouchon, Flat systems, Plenary Lectures and Mini-Courses ECC 97, G. Bastin and M. Gevers Eds, Brussels, 1997, pp. 211-264.

[19] M. Fliess and J. Rudolph, Corps de Hardy et observateurs asymptotiques locaux pour syst~mes diff~rentiellement plats, C.R. Acad. Sci. Paris, I I 324, 1997, 513-519.

[36] P. Martin and P. Rouchon, Flatness and sampling control of induction motors, Proc IFAC World Cong., San Francisco, 1996.

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[42] H. Mounier, J. Rudolph, M. Fliess and P. Rouchon, Tracking control of a vibrating string with an interior mass viewed as a delay system, COCV ESAIM, to appear.

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