Differential Algebra and Partial Differential Control Theory

Copyright © IFAC i\'onlinear COlllrol Design. Capri. Ital\" 1989

S~' stems

DIFFERENTIAL ALGEBRA AND PARTIAL DIFFERENTIAL CONTROL THEORY J.

F. Pommaret

Departmellt oJ Mathelllatics , Ecole Xatiollale des Pants et Challssees, EXPC/CERMA, La Cuw·tille, 9316i Xois),-Ie-Gralld Cedex. Fmnce

Abstract. The purpose of this paper is to generalize certain formal problems of classical control theory (controllabilitY,ohservability.invertibility,feedback,equivalence,invariance, linearization,decoupling, . • . ) within the framework of the formal theory of systems of partial differential equations,Lie pseudogroups and differential algebra.The basic idea is to link these problems in control theory to formal problems arising in the study of differential fields,in order to have & clearer picture of the corresponding concepts that can only be expressed with difficulty in the dual language of differential geometry. The main r~sults allow to study control problems involving any number of inputs and outputs, related by non-linear systems of partial differential equations of any order with respect to any number of independent variables )with no reference to any state representation. Keywords. Control theory,partial differential equations,differential algebra , algebraic geometry,Galois theory .

INTRODUCTION In (Pommaret , 1986) we have shown how to apply the methods of differential algebraic geometry to partial differential control theory,namely the study of any input/output relation defined by a linear or non linear system of partial differential equations (PDE) of any order,with any number of independent variables, inputs and outputs . ln particular,we have s hown how to avoid the internal description (state) by means of jet theory . Since the publication of this Note , a few authors have shown how to define and sometimes solve certain formal problems of classical control theory by means of differential algebra (Fliess,1986a,1986b) .

Classical control theory studies the func tional relations existing between input and output signals in complex systems (inertial guidance, • .• ) .This external des cription can be made more precise by means of an internal description exhibiting the state of the system.Two definitions exist: l)A system is said to be controllable if there is an input allowing the system to pass from any given initial state to any given final state in a finite time. 2)A system is said to be observable if one can determine the state at a given moment from the knowledge of the input and output during a finite time .

The purpose of this paper is to extend these results to partial differential equations and to introduce, within this new framework, the Galois theory for systems of POE,also called differential Galois theory (Pommaret,1983) . In particular,we should like to an nounce and illustrate the four followin~ striking consequences that can be obtained by means of finite constructible algorithms: l)Whenever a certain number of variables are related by a system of POE , it is possible to separate them into input and output along a conjecture of Willems (1986). 2)Any classical algebraic control system can be observed by means of a single output. 3)The definition of controllability and observability can be given and tested without any reference to any state or dynamical system. 4)The study of static equivalence , invariance or linearization can be done for any partial differenti~ control system.

However , a careful study shows that the cri teria for testing these two definitions are or intend to be purely formal,that is to say only depend on the equations of the con trol system,without any need to integrate them. These criteria dealing with ranks of matrices are well known for linear (time - independent) control systems and a few people have tried recently to extend them to specific non linear controlsy stems , by using methods from differential geometry (Isidori,198S).

The many examples provided will prove that these re sults cannot be obtained without the new methods quo ted,the main difficulty being to adapt oneself to a new mathematical language.quite far from what is known up to know . For this reason , we shall rather insist on the concepts as the details will be published in a forthcoming book (Pommaret.1988) .

29

In a completely independent waY , after 1960, the formal theory of non - linear systems of POE have succeeded,by means of jet theory , in studying the solution space of systems of POE without integrating them (Spencer, 1972;Pommaret,1978).Thc formal theory of Lie pseudogroups (groups of transformations solutions of systems of POE) allows then t o study the formal integration of a system of POE by means of a cascade decomposition into "simpler" systems related to subpseudogroups of a bigger one called Galois pseudogroup , in a manner similar to that of the classical Galois theory (Pommaret , 1983) . Unhappily , in this formal framework , the two ereceding definitions have no meaning at all and one may therefore question about them.In particular , these two definitions disappear at once in the partial differen tial situation where dynamical systems do

J.

30

F. Pommarel

no t exist.We shall prove that general definitions can be given. showing that the classical ones are only workin g by pure chance .. . though nobody can doubt about their usefulness . In order to escape from the above dilemna,we first need to fix a few definitions and notations for the

new language that must be used. (Pommaret.1988) . A) DIFFERENTIAL GEOMETRY : Let X be a manifold of dimension nand

t

be a fibered

manifold ove r X,of dimension n+m . For simplic ity , in

the sequel,we shall suppose that C =XxY with projec ti on onto the first factor and a sectio n of C will be the g raph of a map from X to Y.We shall denote by Jg(C) the q- jet bundle of E. with local coor dinates (xI.yk) or simply (x.y ) with i=l • .. . • n;k=l •• .. • m and ~

multi-index ~= ( tll " k

k

q

" '~ ) with o,.;I ~I=~i+ ·· · +tI";q • n

n

Yo=y • ~+ li=( l'l" " ' ~i- l. t\+ l, t' i+l' · · · ' ~n ) . Fo r people not familiar with jet theory.it is sufficient to say that (x.y ) transform like the derivatives a fk(x) ~

q

under any transf ormati on of (x.y) whenever y=f(x).We shall denote by llq (X.X)CJ q(X.X) th e groupoid of q-jets of in ver tible m~ps from X t o X defined by the local condition det(y.)iO.Finally.if t and ~ are two fibered manifolds over X.we shall denote by t Xx j their fibered product over X. In the sequel.we shall have 1 =Xx Z and t.. Xx 1-' =XxYxZ . Definition. A non -linear system of PDE of order q on G is a subfibered manifold 11. c J (t ).We shall supq

q

=J ( ~ )()J (t ) r q q+r are fibered ma nif olds 'v'r~ O.The system :R J'q+r

r

q

( t.. »

51

The inductive use of a criterion of formal inte g rabi~

(Pomma ret .1978) may provide.in gen eral.a formally If> (s) q+r +s If> . inte g rabl e system ~ q +r= 'q +r ( ~ q+r+s) WIth the same ~q .by

means of a finite algorithm that

can be used on symbolic computers (MACSYMA) .

2

Example; n=3.m=1.q=2 ~ 2{ Y33 - x Yll=O • Y22= 0 1r-~ $. 2 is surjective but 1\.4~ ~ 3 is no t and it

should not be possible to use a formal compu ter in order to know the formal power series solutions by means of successive differentiations . It is not eas y at all to check that the solution space (state!) of this linear system is a 12 -d imensional vector space .

,

com?letely determined by

ate52)

Jl

.

B) DIFFERENTIAL ALGEBRA : Let K::>Gbe a dif ferential field with derivations A"\

1

n

al • . . . • a .for example "" (x • .•. • x ) .We denote by n K[y] the ring of differential polynomials

K{y}=lim

q - '"

Example. n=2 . m=1.q=2. K=~ 9t 2 ( P l ;; Y22-~(Yll) =0 • P 2 ;; Y12 - Yll=0 The differential ideal generated in K(y} by P P

and l d2P2-dlPl+dlP2=

is not prime as it contains

2 (Yll-l)Ylll

but it contains (Ylll)3 and therefore

its root is prime .

Example . Simila rly. (y

_6y2)4 belongs to the difxx ferential ideal generated by the only differential

polynomial p'(y ) 2 _4y 3 though d P:2y (y x

x

x

xx

_6y2) .

We shall now prove that the main definitio n s of control theory can be understood by means of a new language mixing together A and B.

q

q

is said to be formally integr able if t he morphisms .q+r+l. ~ induced by the canonical q+r . q +r+l ~ q+r projections Iq r+l:J ( E. ) ~ J q+r ( t ) are epimo r q+r q+r+ phisms Vr ;.O

solutions as

it is rather surprising that neither D. C. Spencer for A.nor E. R. Kolchin for B.did find a single link between their respective works during the period 1960 - 197S . Such a link can only be found in our book (Pommaret.1983) and all ows for the first time t o have a precise explicit criterion for knowing when .p. is prime indeed . an absolutely delicate problem in actual practice . even when n=l (Pommaret.1983.p 246) . It is rather striking t o see that such a criterion is just dual to the criterion of formal integrability already quoted .

Definition . A partial differential control system is a mixed system of PDE 5' c. J (t x :1 ) that can be

pose t hat the r-prolongations ell J (J

The analogy between A and B is evident through the notati o ns . In fact.to the projective limit J",( t ) ~ • .. ~ J ( E. )...,. •• • ~ X does correspond by duality q the inductive limit KC ... C K(yq)C __ .CK . However.

q

in the indeterminates y with derivations d

such that i d. IK=a . • d.y~=yk 1 . By quo tient .we may define the difI I I . ~+ i ferential field K=Q(K{y}) of differential rati onal functions in y . If 'p' c K{y} is a prime differential ideal (ab£'p' :::} a or be'p'.'p' is stable by the di).we denote by n= ( n l • . . . • nn) its ge neric zero.image of y under the evaluation homomorphism K{y} ~ K(y}/ .p. .One may then introduce the differential extension L/K with L=Q(K(y}/'p')=K and we shall denote by ( diff ) trd L/K its (differential) trans cendence degree.

q

x

supposed to be formally integrable. In gene r al.t he input y and the output z must be solutions of the respective resolvent systems c J ( E ) and q+r q+r cJ ('j~ ) for a certain r~ . q+r q+r

d6

i

Remark . In practice .one has to eliminate either the y-;;-;-the z by differentiating "conveniently" the finitely many given PDE ~(x.yq.Zq)=O . The study of such a differential correspondence or direc t Backlund problem can be done by means of a finite constructi b le alg o rithm (Pommaret.1983.chap IV B) which uses very sophisticated arguments from the formal theory of systems of PDE that cannot be avoided.even in the simplest situations (ANNOUNCEMENT 1) . We provide five among the best examples we know with slightly different nota ti ons . l 2 3 Example. (x .x .x ).input (u.v).output y 1'2 Y33 - X2Y ll - u =0 Y22 - v =0 There is no res olv ent system for the ou tput (means that no PDE for the output may exist independently of the input ) and a resolvent system of order 6 for the input (independently of the output).In this li near situati o n, one can push u and v into the right members

and the search for comp atibility conditions cannot be done without the full Spencer machinery for cons tr ucti ng differential sequences (the reader can try t o find more than one !)(Pommaret.1978.1983.1988). l 2 3 Example . (x .x .x ) . input y .o utput z 2 2 2 g>2 (2y+zl+z =0 • Yll+2yz +4 y - 2zYl - ~z2=0 ~3 {Ylll+Y2+ 12YY 1=0 ~ 3 { zlll+z2- 6z2z 1=0 Both input and ou tput may be solitons (solitary waves) . Example. (constrained system) The differential con diti on o n the input (p.q.r) in order to have three linearly independent solutions o f the constrained control system (k =1 ,2.3):

~ 3( yk is

_p(x)yk +q (x)yk _r (x)yk= O with xxx xx x 3 9Pxx - 18ppx - 27qx+4(p) - 18pq+S4r=0

31

Differential Algebra and Partial Differential Control Theory Example. (stationary Benard cells) Gravity &=( O,O, - g) 2 3 position (xl ,x ,x ),temperature a ,p ressure ",speed ~ !tl f'"V.v=O ~ ~"':Or ... -+ 7 2 1 , ~v-9g - v.r.=0 , ~6 - v.g=0 (Boussinesq)

Definition. The differential algebraic correspondence is said to be algebraically controllable if K'=K. Otherwise,one will say that K',L',M',N is the minimum realization of K,L,M,N. -------

If one chooses 6 as input,it must satisfy

One has the following key diagram

~~~6 - g

2

---

(a +a )6=0 but it is also possible t o 11 22 3 choose v as another possible input.

/N~ L'

x , input u,output (y1,y2)

L~

u=~

1=0

Though we may feel to be quite far from classical control theorY , we shall now dualize the preceding definition and discover that certain concepts quite difficult to explain by means of A become evident by means of B and conversely; Dual definition . Let L=Q(K{y} /E) and M=Q(K{z} /~) be two differential extension of K.The search for an abstract composite differential field N containing both L and M amounts to find a prime differential ideal r c K{y,z} such that r n K{y}=E ' r n k{z}=~ and to set N=Q(K{y,z}/E.) . Dual remark . A unique field where to put all generic zeros,like the field OC of complex numbers for algebraic equations,does not exist here.Also )when one knows about the resolvent systems , the search for an admissible differential correspondence or inverse Backlund problem may be extremely difficult or even impossible (Pommaret , 1983,1988) . Aplication. In the~urely algebraic case,one knows that G ( 'l2 , V3)= ~ ( I2+V3 ) c. ~ .By analogy,a primitive element also exists for any differential extension N/L such that diff trd N/L=O , the simplest situation with n=l being exactly that of classical algebraiC control theory (Kolchin,1973 , p 103;Pommaret,1983, p 231) (ANNOUNCEMENT 2). Example . independent variable

t

J 1 z=y

1

1 y 2 - u=O , Yx+ 2 y 1 -u=O Yx+

~

Example.

~

1

1

2

y =z , y =u - z

with

z

xx

-z +u -u =0 x

Let now

~

be a family of elements in N-L.

Definition . The differential correspondence is said to be observable by means of ~ if L < ~ > =N.Otherwise, L <~>/ L is the minimum realization of L and ~ in N.

Application . Of course,intuitively,8 control system cannot be controllable if one can find any autonomous output.However,the importance of our definitions lies in the fact that criteria for testing them on linear classical control systems are the same as the well known ones,though striking it may look like,because their pro ofs are quite different (Pommaret,1988) . Also,the concept of autonomous outpu t can be extended to the analytic situation . For eample,the control system uYx-ux=O admits the autonomous output z=y-logu

satisfying

zx=O . More generally,it is

easy to prove ,as an exercise on brackets of vector fields , that an affine classical control system of the form yx=a(y)+b(y)u is contr ollable in ou r sense, according to the Frobenius theorem,if and only if the smallest involutive distribution of vector fields containing b , [a,b),[a,[a,b)) , .. . has maximum rank. We have shown in (Pommaret,1988) that the distribution obtained by certain authors (Isidori,1985) by adding "a" to the latter,does not fit with the linear case and is,in fact, only related to the pseudogroup of output invariance (Pommaret,1988).Similarly,for observability,the misleading property has been that , whenever an input/output system is given,any corresponding internal descripti on by means of a primitive element or a state is automatically observa ble b y means of the output, though this is not held in general by any canonical form (ANNOUNCEMENT 3) .

K= G (x)

t y~_y1+u=0

z=y1+xy2

x

OUTPUT

T/M

K

impossible

always possible

If'>

M'

i~K'/T

INPUT

constraint

of inclusions:

INPUT/OUTPUT

~

y~=O

(x _ 1 )y 1=xz - z+xu x

We shall now discover that many properties of algebraic control systems can be seen from the respective ££!itions of the differential fields K,L , M, N.Such a study should be quite difficult from the differential geometric or functional analysis point of view .In particular , it is as useful to consider the inclusions KC L C N as the inclusions KC MC N though only the first ones are explicitely studied in classical control theory while the second ones are implicit in the Singh and Nijmeijer algorithms. Indeed,it is known (Kolchin,1973,p 102;Pommaret , 1983, p 230) that the set of elements of N that are diffe rentially algebraic (verify at least one algebraiC PDE) over K is a differential intermediate field K' . The composite differential extensions L'=(L , K') and M'=(M,K') of K' are differentially transcendental because,if an element of L' - K' were satisfying an algebraiC PDE over K' , then it should held this property over K too and should be already in K' .In actual practice, looking for K' amounts to look for all single outputs verifying separately at least one algebraiC PDE independent of the inputs, and therefore called autonomous outputs .

We have verifies

z -z=O x

L < ~l > =N

but

z +z-2u= 0

L< ~l +~2 > C

N

because z=yl +y2

.H owever, z=y2_ y l

x

verifies

and is theref ore an autonomous element.It

follows that

y~+/tz-u=O "=cst ,the n

K'=K< ~2 _ ~1 >

and

.finally,in case K'=K

#

N/L' P ;; 2

is defined by

y~+"/ -u

with

"rl

The next definition is the only possible one agreeing with an intuitive use of ope rators in the formal theory of systems of PDE,as a generalization of matrices in linear algebra.It is also the only possible o ne agreeing with the symmetrical use of input and output in the differential algebraic framework. Definition. A differential algebraic correspondence is said to be left (right) invertible if LcM (MCL) . Remark . Our definition of left invertibility agrees with the standard intuiti ve o ne existing in the litterature.It is however more restricti ve than just to set diff trd N/M=O because,whenever L C M then one has automatically O~dif trd N/~diff trd N/L=O in

J.

32

F. Pommaret

classical control theory . (Fliess,1986a).Unlikely,the definition of right invertibility adopted in the lit t er a t ure (Fliess,1986a,1986b),namely that diff trd M/K = nb of outputs must be revisited within our i n trinsic framework as it depends on the generators ( t hough,of course,it may be useful as describing a fo r mal property of control systems) and the words l e ft/ r ight must be used in a coherent way.

Finally,we shall show that the so - called control distribution for affine control systems (lsidori, 1985) comes indeed from the study of output invariance and has only just by chance something to do with controllability as we saw. Let such a system be of the a form: yk=ak(y)+bk(y)u r x

r

but the same technique could be applied to polyno mials in u (Pommaret,1983) . If we look for invariance under the t ransformations Y-?

zl=al(z)+bl(z)u x

r

r

Z

,we must have also

.As before, identifying the jets,we

This system of PDE is not formally integrable . To see such a fact,let us introduce the infinitesimal trans -

Fin a lly,any diffeomorphism of Y,Z or YxZ can be exten de d to Jq( Cxi') by prolongation. It follows that any equivalence problem between mixed systems (such as i n variance,linearization, ... ) leads,in general,to a

new system of PDE for the desired family of diffeomor ph isms,wh ic h ma not be formall inte rable (thus n ot necessarily compatible ! ) by itself Claude,1986; Van d er Schaft,1987) . Such a study is completely solved i n (Pomma r et,1978,1983,1988) by means of finite cons t ructi b le algorithms using projective limits of grou Jq(XxY)Xynq(Y,y) ~ Jq(XxY) and poid ac t ions th e ir g r aphs.The misleading property is tha t ,even if n=l ,one obt a i n s systems of PDE that canno t ~ied wi thout t he formal theory of PDE which is not known by th e mathema t ical community! (ANNOUNCEMENT 4). . 1 2 1 2 Ex amp le . n=l ,l.nput u,outputs (y ,y ) or (z ,z ) The search for a transformation (y1

,y2) ~ (zl,z2)

a llowi ng t o pa ss from the n on - linear PDE y2y1 _u=0 x

t o t he linear PDE zl _u=O leads to the system of PDE 1 -1- 21x 2 oz /oy =y oz /oy =0 which is not formally in t e gr ab le and not even compatible . Indeed,differen t ia t ing the first PDE with respect to y2 ,differen tiating t he s e cond PDE with respect to y1 and sub st r acti ng,on e obtains 0=1 . This is a problem of e quivale nce and a similar situation,but in a problem

of realization,can be found in (Van der Schaft,1987). In or de r to show the power of the formal theory of PDE, we shall consider again the same non - linear given con trol system and ask for its linearization ( which cannot be of the form proposed above as we saw~ For t his,we shall exhibit an arbitrary linear form and shall try to transform it back to the original system by an invertible change of outputs.The same te chnique can be adopted in general but one cannot study t he system of PDE obtained,without the help of t he formal theory.

formations z=y+tn(y)+ . .. for a small parameter t. Linearizing,we obtain: [n,a]=O [n,b]=O The reason for introducing the brackets [a,b] , ... is nothing else than the way to obtain a formally integrable system by "saturating" with the brackets, as we must have indeed the first order PDE coming from the Jacobi identity,namely [n,[a,b]] , .. • . Hence,we are led to introduce the centralizer of the smallest Lie algebra containing a,b,[a,b], . . . . Of cQurse,it is clear that,for linear systems,one has

a(y)=Ay , b(y)=B=cst and there is no reason at all to set y=O in order to make a disa earin ,thou h ~s commonly done in the litterature. In par t i cular, Yx=l with no input should be controllable !) Hence,it is only by chance that invariance is related

to the distribution a,b,[a,b], ••• while controllability is related to the distribution b,[a,b], •••• CONCLUSION We have thus provided our four announcements and hope t hat t h ese new methods,though difficult indeed,will open a new approach to control theory by allowing to deal not only with ordinary differential equations but also with partial differential equations . The situation is even richer because all the algorithms needed are just adapted to symbolic computers for explicit calculations . REFERENCES Claude,D. (1986). Everything you always wanted to know about linearization,in Algebraic and geometriC

methods in non - linear control theory,Reidel Fliess,M . (1986a). Some remarks on non-linear inver tibility and dynamic state feedback,MTNS-85,C. Byrnes,A.Lindquist eds,Elsevier,Amsterdam . Fliess,M . (1986b) . Non - linear control theory and dif ferential algebra,Proc . IIASA Conf . Modelling Adaptative Control,Soprum,Hungary,july.to appear Isidori,A. (1985). Non - linear contr~ systems,an intro duction,Lecture Notes in control and information

science72,Springer Verlag,297 p . Kolchin,E . R. (1973) . Differential algebra and algebraic ~,Academic Press,New York,450 p. Kumpera,A . ;Spencer,D.C. (1972) . Lie equations,Annals of mathematics studies 73,Princeton University Press The gene r al linear form can be written as follows by Pommaret,J.F. (1978) . Systems of partial differential t aking into account the possible exchange of the z: equations and Lie pseudogroups,Gordon and Breach zl+ A(u)z2+ B(u)zl+ C(u)z2+ D(u)=0 Pommaret,J.F. (1983). Differential Galois theory , x x Gordon and Breach,New York,760 p. Aft e r identification of the various jets,one gets: Pommaret,J . F . (1988) . Lie pseudogroups and mechanics, 2 (o z l/oy1+Aoz2/oy1)+y2(Bz1+Cz +D)/u=0,ozl/oy2+Aoz2/oy2=0 Gordon and Breach,New York,600 p. Pommaret,J . F. (1986) . Geometrie differentielle alge De r i ving with respect to u leads to singular transforbrique et theorie du controle,C . R. Acad.Sc . Paris, ma tions,unless A(u)=a=cst and we may choose a=O.In a 302 , I , 15 , p 547 - 550 similar way,one must have B(u)=bu,C(u)=cu,D(u)=du and Van~r Schaft,A . J . (1987) . On realization of non we get: ozl/ oy 1+(bz 1 +cz 2 +d)y2=0 , ozl/ oy 2=0 linear systems •. ,Math . Systems Theory,19,p239-275 Willems,J . C. (1986) . From time series to linear sysSimilarly as ahove,using crossed derivatives,we have: tems,Automatica,22,5,p 561 - 580 2 ( boz 1 /oy2 +coz 2 /oy2)y2+bz 1 +cz +d=0 Such a system is involutive according to the criterion (unless b=c =O,d*O) and we may choose b=O,c= - l,d= - l in or de r t o select the solution :

zl=y1

wh ich is easily seen to be convenient.

z2=(1/y2)_1