A fast graph theoretic algorithm for the feedback decoupling problem ...

A FAST GR,~,~,HT ~ O R E T I C ALGORITHM FOR THE FEEDBACK D ECOUPLING PROBLEM OF + NONLINEAR SYSTEMS A. KASINSKI

and

J. LEVINE

ABSTRACT : The feedback decoupling problem of nonlinear systems is actually well understood in a theoretic point of view. However, to compute the decoupling feedbacks, the only method known by the authors, consists in using a formal derivation program as check if differential expressions are null [3]. We give a generic interpretation of these expressions in terms of the graph of the system in the sense of [6], and deduce a faster algorithm using the minimal length of the paths joining one of the inputs to the i th output.

(*)

Institut Automatyki Politec~mika Poznanska 6 0 % 5 POZNAN, ul. Piotrowo, 3A, POLAND.

(**-) Centre d'Automatique et Informatique Ecole Nationale Sup6rieure des Mines de Paris 35, rue St Honor6 77305 FONTAIN~BLK~.U - FRANCE. (÷)

The first author was supported by a scholarship of the Minist~re des Relations Ext4rieures de la R~publique Fran~aise.

I - The fee dbag,k deceupling problem We consider a linear-analytic i = fo(X)+

(D

system, given in local coordinates, by

N

M

S uif.(x)+ i=I l

Z wigi(x) i=I

Yi = h (x) ' i = I .....

:

p

z i = ki(x) , i = I, ..., q where

x

belongs to a connected

n-dimensional

are the input functions,

h I, ..., hp

analytic on

:

X, and where

n f.j Fi(x) = r. (x) j=1 i n

~x.

and

analytic manifold

k I, ..., kq

, i = 0 ....

X, u = (uI ,...~U~ w

are the output functions,

N

(~)

~x , i = I, ..., M J~ J are analytic vector fields on X. The feedback decouolin~ problem i--J ,...,N, j=1, o..,N,

consists in finding analytic functions

eventually defined on an open subset

(} of

X

i j (~ ,~i)~

such that

551

the feedback control : N

ui(~)= i(~) ÷~ makes the

p

outputs

We shall denote

~ij(~j

YI' "''' Yp

(2)

, i=I, °..,N

locally independent of

i w , i = I, ..., M.

Fi" i = O, ..., N, the vector fields obtained by the feedback (2) : N

n

~o(x) =fo (~) + zi:~i(~)fi(~) ' ~°(~): 7j:~9~(~)o~Xi N

(3)

n

The problem is actually well understood and the differential geometric methods

[4]

of the

together with the algebraic ones

[I ] draw an almost complete picture

thboretic solution. In the geometric approach of the "structural" decoupling,

we imtroduce the maximal involutive distribution ~ of constant rank, which is (Fo,FI ,-.. ,FN)-invariant. Isidori, Krener, Gori-Giorgi snd Monaco [4] have proved the follow~ng : Theorem I : The structural decouDling problem has a local solution if and only if : span { GI, .... GMI c Furthermore,

~

=

~

ker dh..

(4)

~ can be obtained by the following induction (see [5]) :

J)O = span {dhI ,...,dhp]

(5)

N

~k = i~ ~i ~-I +~-I where

2.

(6)

is the Lie derivative with respect to the vector field

Fi, and

1

~- (kUo~) ~ . ,

(T)

The algebraic methods, using Fliess' input-output map reprensentation, give a "functional" point of view : in place of a distribution, one looks for a module of vector fields, playing the same role as the distribution ~ with a non constant rank

(see [I ]).

Theorem 2 : The outputs Yi' "''' Y~

but eventually

Claude [I ] has proved the following : are decoupled with respect to

M w , ..., w ,

if and only if there exists an analytic module ~ which is also a Lie subalgebra of vector fields on Vi:[Fi, ~ with

F. I

c ~W, and defined

Furthermore,

~

X

such that

span{G1, .... GMI

by and

: c

~c

P ~ ker dh. i=i l

(3)-m ~

can be computed in a purely algebraic way (that is to

say without solving differential or partial differential equations) by the procedure described hereafter.

(8)

552

For this purpose, we need the • Definition I : The characteristic number

Pi

of order

i

is the unique integer

satisfying : ~j E {I

.,N} : F FPih. ~ 0 , ~**

.j 0

and :

Vj £ {I, .... NI,Vm £{O,...,pi-1 } If

F Fmh joi

=- 0

order of

Vj, Vm, we set

F2h i = F o ( F o l h i

Remark that m, and

that

(9)

i

)

F.Fmh. ---O.

(10)

~ o I

Pi = + o=

and i£

is a polynomial

~

: Fjh i ~ O, Pi = O,

of differentials of

FOhi ---hi" --Pi can be interpre ted

up to the

integrations one of ............ the u ..] . . . .such . . tha% . . . Yiw . . is . . affected . . . . by .

To compute

~

and

#, we introduce the following quantities

A~(X) : Fj(x)F~ i (x)hi(x)

,

:

i = I .... , p , j = I ..... N

P ~i(x) = ~i(hi(x)' Fo(X)hi(x)' .... Poi(x)hi(x))-

(11)

FOpi+1(x),i=1 '°'" , p

,~(~) = ~J(h ( x )•, F ~ o(~)h'(x)'~ ..., Fo~i (~)hi(x)) , i = I , ...,~ j=1, with

hi

as the minimal number

~i

and

(13)

...,N

~Jl arbitrary analytic functions.

Let us call : A element is

(12)

Aj

the ,

p × N

matrix-valued analytic function

~ = ( ~ ,... ,~ )T , and

~

the

p × N

whose

(i,j) th

matrix-valued analytic

i

function whose

(i,j) th element ~s

@J.

Theorem ~ : if

GkFoh i ---0 Vk £{I ..... M1 • Vm ~ Pi ' a necessary and sufficient

l

condition for that

(~,~)

to realize the local functional decoupling of

(~), is

(~,#) locally solve the system : Aa = ~

(~4)

a# = $ In this case, the change of variables X i = h i , ..., X i = F~ i h i o Pi

,

: i -- I, ..o, p ,

(15)

puts the system (~) locally into the form : "i i = X~ O

~i

=xi

Pi -I

(16)

Pi

~i i xi = ~i(Xo .... , % ) Pi Yi = Xi o

i = I, ...,p. |

N

+ S j=1

~(~,

..

x i )v. °'

Pi

S

553

Clearly,

this procedure involves a huge amount of formal calculus,

determine the characteristic numbers times the expressions and then check if

Pi ' i = I, ..., p : one must differentiate

Pi

hi, Fohi~ etc..., whose complexity is growing very fast,

F.Fmh.

Claude and Dufresne

especially to

is null or not=

A program has been d e v e l o p p e d b y

[3], using the language MACSYMA,

to compute these formal

expressions. The aim of this paper is to introduce a faster method to compute the minimal number of formal differentiations that the numbers

Pi

Pi

with

: for this purpose, we shall prove

can generieally be very easily obtained on the system's graph•

We shall also give a lower bound

vi

for

Pi

in the non-generic

obtained from the graph, and prove that either

v. < n-1

or

case, still

vi = Pi = + ~

These results are finally synthesized in an algorithm to compute

(a,8).

II-The system's graph As in

[6], we introduce the following system's graph :

Definition 2 : We call

F

the systemts graph of

with given local coordinates,

fallows

in a given open subset

the oriented graph whose input-nodes

(uI ,...,uN, w I ,...,wM), whose intermediate-nodes and whose output-nodes are

~

(Yl ' " " Y p ) "

of

are

are the state variables

The oriented arcs of

~

(x I ,. ..,Xn),

F are obtained as

:

o There exists an oriented arc joigning i = I .... ,N, k=1,...,n,

and joigning

ui

to

xk

iff

fk(x) ~ 0 in O

wi

to

xk

iff

g~i(x) ~ 0

in C)

i = I, ..., M, k = I, ..., n. There exists an oriented arc joining

x k to x

•

iff 8fDo(x) ~ 0 3

inO,

j, k = I ,

..., n.

. There exists an oriented are joining inO,

i = I, ..., p, k = I, ••., n. •

Definition 3 : We call an oriented path joining An introductory example

xk

to

Yi

iff

~h. i (x) ~ 0 6xk

d(u!,Yi) the minimal number of oriented arcs of u3

to

Yi' and

d. = Min d(uJ,Yi) l 1 ~ d I

implies

G1h m 0

and

back to (9) and (~0), and compute p.. We first bh I h x (G.h)(x) = g.(x)'~" (X_) -- O. Then :(F )( ) = I I ax~ z o o

3

o

I

and thus p = I ; Finally, we also have

P1=dl - 2 = I, and that

GiFoh -- O. To check our assertion,

2

check that (F_h)(x)--T~ (x) ~ (x^) toO, I x Oh x +f~ x ~h sx? z f ( ) ~ (^) ( .,x~ (x.) o ox I ~ o ] ~ u~-2 L

1 O

OX I

I

bf2 8h f2 GIFoh = gl (~--~3 8x2 + o

Thus, almost without computations,

PI

let us go

2

b2h 6x36x2, ~ 0 ,

as claimed above.

and the relations

G I h =- 0

and

GIFoh-= O, can be deduced from the system's graph (we only need to compute

FIFoh !).

Clearly, the system's graph synthesizesothe structure o f the interactions o f the input and output variables versus integration Qf the state variables.

Thus, it

is not surprising thatx in genera lt but generically 9nly, the minimal length represents the minimum number of integrations namely

f9 r the inputs t9 affec~

d.

l

Yi'

Pi' up to a constant equal to 2 since the first and last arcs do not

represent Remark

integrations.

: in

depend on

F, we do not take into account the fact that x I , ..., x n

or not.

f1' "''' fN' g1' "''' gM

For our purpose these interactions do not play

any role in generic situations and, if they play a role in non-generic cases, the profit of the graph's

method vanishes, as will be seen after.

III-The characteristic numbers

Pi ' their lower bounds

Besides the characteristic numbers defined as follows Definition ~

k z

bf ° ~-I

ke

....

k

6f o o

£ {I, ...,nl bh. ~

vi

~0

such that

in

fJ

8 fkr- I

bf k o bh.

~

~',

°''r

(18)

o

6- vi' that there exists a (non minimal) oriented path from one of the

uJ

to

Yi

of length

Unfortunately, this is not true, as the next example shows

:

Pi+2.

559

i xl = x4 + uzl x2 x2w - x2u ~

-x3w

x4 = x 5

~5 = u y = XlX2X 5

We have

v = 0, whereas

P = I :

~ (~x2~)-x 2 ~~

Fh=x

I ~

Fo h = x2x3x4' FIFoh =

-r-

(x~x~x3)~o, %h = ~2 ~a

(~x2~3)-x3~-~- (~x2x3)~°'

-x 2 ~ x 4 ~ O .

But it can be seen that, in

F, there is no oriented path joining

length equal to

The only path of length larger than 2 is

p + 2 = 3.

(u, x5, x 4, x I ,y)

of

length 4. Thus, if

give anymore information on

Pi"

r + 2

to

y

with

Pi > vi' we see that the graph does not

~owever, to compute

and if there is no path of length

u

in

F3.Frh. o i

with

r > vi'

F, it is no need to compute the

first term of (26) (with r in place of v. ) since if there w e r e a non z e r o expres1 sion in this term, there should exist a path of length r + 2,which contradicts our assumption,

m

Remark ~ : the two preceding examples give a good illustration of non-generic systems

: in both there were orthogonality relations between

the expressions non generic, for we obtain

~(k O)

are

~ O, but their sum is

0.

: F1h = (1+E)XlX2-XlX 2 = ~xlx 2 ~ O .

and

h, so that

Of course, this is

if we change, for example in Remark I,

Remark 4 : It is worth noting that if

FI

ux I

in

(1+E)ux I ,

•

r < vi, we necessarily have

F.Frh - O Yj = 1, o.., N. In the same way, going back to the system (17) of 3 o 1 the introductory example~ we have d(w,y I ) = 4, and thus GiFrh -= O Vr < 4-2 = 2. Also, this remark is useful to avoid computing a number of formal expressions

:

Jl

d., is obtained only for paths joining u , ...,u Jr l v to Yi" one can be sure that F~F i h. ---0 %~ ~ J1' "''' Jr' and one needs to E O l v@he~k only those expressions F. FVih., ..., F. F Zh. for minimal paths. • $I o l ~r o z

if

vi

or,

more precisely

iV - Description of the .algorithm. All the following computations

must be done formally,

for example with the

languages MACSYMA or REDUCE. I. The graph

F

To avoid a complete construction of

F

with a number of useless

n~des and arc~s~

560 one can determine

d. = v. + 2, and i

U. the subset of the

i

to the minimal paths, directly from the data of method

[u~ .... u')corresponding

i

~, and by a dynamic programming

:

6h.

° Starting from

Yi(i = I, ..., p), we build every incident arc with:

Then, for every

~x v

such that

(x~ ,yi ) £ F,

0

(uJ,x k )

in

r

~

,i ,~x-

is an arc

o

0

by

~jo ~ O.

If

(uJ,x k ) E F, then

0

di=2, v i = 0

and u j £ U i"

0

• If (uJ,Xko) ~ F xk

we test if there

0

by

Vj, we change

~fk°o

Yi

into

~]Q , and build every incident arc to

~ 0 ; then again, for ever~

Xkl

such that

(Xkl ,xk ) E F, O

we test if there I. is an arc (u~,x k ) is no arc from

u S,

Vj,

to

in

r

by

fj

everylpath of length

~0,

~n-1,

and so on. I

The same procedure can be done in parallel to determine and in

~i

t~e subset of the

(w I , ..., w M)

If there

v. = + ° %

then

.

Min d(wD,Yi)-- = ~ , I ~j~M

corresponding to the

(~i+2)

length

r.

2. CQmputation Qf

Pl

and the matrix

A.

• We first compute

F.F Vih. Vj such that u J" E ~.. i S o I Two cases can happen : v. • either FjFolhi ~ 0 for at least one j with u j g U i. v. Pi = vi' A lj = PjPol h i

Then

= 0

Vj

such that

uJ £ ~i

Vj

such that

u j ~U.. l

If

V~ = + ~, then

p4 = + ~

and the

i th line of

A

can be deleted.

vi . or

F.F h. -= 0 S o •

Then

p~ > v~

gj

such that

u" E ~. l

and one must compute

i

F.~rh. .lol

Vr > v.

VJ = I , ..., N, until the moment when one of these expressions becomes non 0 (Pi

is then equal to the corresponding

expression is null If

p~

is finite, the

every expression If

i t llin

(11)

for

3. The comparison between

Pi

and

of the matrix

i th line of

Pi < ~i' we have

• If

Pi > ~i' we have to look further if

G .pmh. ---0 jo l

GjFo~ih i

G ~mh.

- 0

= 0

G~Fnmh i_

Two cases can happen 0

r = n-1

if every

m 0

A is obtained by computing

A.

~i"

o If

if

or until

j = I, ..o, N.

Pi = + ~' one can delete the

and after

r)

(then ~. = 2 ~)o

Vj

Vm ~ Pi'

such that

gj.

w j £ ~i'

Vm = ~i_ + I, ..., Pi_ '

Vj.

:

Vm ~ p~, Vj, 1

then the decoupling problem has a local

561

solution iff the system o ~jo~{1,

(14)

.... M1 , Zm o < 0 i

has a local solution m

(a,~).

such that G4Fo ° hi ¢o, t~en the

decoupling problem has no solution, and the system is finitely decoupied up to the order

m

4. Inversion of the,,,s,ystem ( 1 4 ) Remark 5 : If evaluation of

vi < Pi' and if FjFomhi with

m + 2 joining

u j to

necessarily null.

o

,

V(~,~).

Same as in [ 3 ] . ~

(see [2]). '

has a large dimension, it can be useful, in the

m > v i, to remark that if there is no path of length Yi

in F, every expression

Thus, we eliminate this way

n

~m3 ~i(ko' ...,k m)

is

formal differentiations in

F .Fmh o • SOl

Remark 6 : It is clear that this method is more efficient for larger

v's and l 1/. does n o t c o n t a i n t o o many e l e m e n t s , J_ we need a very low number of formal derivations and the efficiency of thismethod is

l a r g e r n , N, M, p. the h i g h e s t .

If

'~i = P i

and i f

On t h e o t h e r h a n d , i f

vi < Pi'

since a minimal length in

F is

computed much faster than a formal derivation, the economy of time grows with V-

v.. ! •

C0nclusion. We have pried

that the feedback decoupling method of Claude and Dufresne

can be significantly simplified by the introduction of the system's graph. graph has the property that the minimal length

di

Pi

This

between the i th output

and the inputs (uI , ..., aN), is generically equal to the number

[3]

plus 2, and in general smaller or equal to

i th characteristic

Pi + 2.

This property

can be used to avoid a number of formal computations and is all the more efficient as

d. i

is large.

Acknowledgement.

The authors are indebted to P. Willis and F. Oeromel, of Eeole Polytechnique,

that have successfully realized

the programming

work.

REFERENCES

[I] Do CLAUDE. Decoupling of nonlinear systems. Syst. and Con~.

Letters. Volo I,

nO4 (1982), 242-248. [2] D. CLAUDE. Decouplage des syst&mes

" du lin4aire au nonlin4air% in :

Develcppement et utilisation d'outils et modules math4matiques en automatique, analyse des syst~mes et traitement du signal. Colloque National CNRS, Sept. 82, Belle-lie, France.

[3] ~)o CLA~E, P. DUFRESNE. An application of Macsyma to nonlinear systems decoupling. Ehropean Conference on Computer Algebra, Ap ril 82, Marseille, France .

562

[4] A. ISIDORI, A. KRE}~R, C. GORI-GIORGI, S. MONACO. Nonlinear decoupling via feedback. IEEE Trans. AC. Vol. AC26, n°2 (1981), 331-345. [5] A. ISIDORI. The geometric approach to nonlinear feedback control : a survey. Analysis and Optimization of Systems. Lecture Notes in Control and information sciences n°44, Springer, 1 9~2. [6] D. SILJAK° On teachability of dynamic systems. Int. J. Syst. So. Vol. 8, n°3, (1977), 321-358.