A formula for iterated derivatives along trajectories of nonlinear systems

Systems & Control Letters 11 (1988) 1-7 North-Holland

1

A formula for iterated derivatives along trajectories of nonlinear systems F. L A M N A B H I - L A G A R R I G U E Laboratoire des Signaux ei Systemes, E.S.E., Plateau du Moulon, 91190 Gif-Sur-Yvette, France

P.E. C R O U C H Department of Electrical and Computer Engineerin$ Arizona State University, Tempe, AZ 85287, U.S.A.

Received 20 July 1987 Abstract: The paper presents a formula for the n-th order total derivative of the output function of a nonlinear system, in terms of

derivatives of the inputs. The formula is given in terms of noncommuting variables and the shuffle product and has potential applications to input-output decoupling and inversesof nonlinear systems. Keywords: Nonlinear system,Iterated derivatives,Combinatorics, Identities.

1. Introduction

In this paper we derive a formula for the n-th order total time derivative of the output function of a nonlinear system, affine in the controls, in terms of derivatives of the inputs. The formula should be of interest in various nonlinear control problems, but especially in the theory of inverse systems and input-output decoupling. For example, Nijmeyer and Respondek [9], have recently proposed a constructive algorithm for solving the input-output decoupling problem using dynamic c~mpensation which requires the computation of the ranks of matrices whose components are obtained by repeated differentiation of certain outputs of the system. An earlier, less computationally attractive algorithm, was given in Descusse and Moog [3]. See also Fliess [5]. Although the computations in the above papers may be carried out in si,.,aple examples, there is no way of doing this for general classes of systems because of the complexity of the algebra involved. The result presented in this paper may be viewed as a first step toward reducing this complexity, and thus enabling general computations as above to be contemplated. However, direct implementation of the results certainly requires much further research. The formula we present is of independent interest, in that it involves a mix of commutative and noncommutative algebra and in particular the shuffle product. The shuffle producL first introduced to s3,stems theory by Fliess [4] is usually found in the theory of abstract combinatorics. However, in the paper by the authors [2], further relations between combinatorial identities, primarily displayed in Ree [10], and problems in systems theory are given. Other examples may be found in the paper by Leroux and Viennot [7], which is in turn partially motivated by a paper [6], concerned with explicit computation of input-output maps of nonlinear systems expressed in terms of series of iterated integrals. In contrast this paper deals with properties of iterated derivatives. Finally, we remark that although no direct use is made of the constructs appearing in Cartier and Foata [1], there are undoubted similarities which may be useful in further analysis of nonlinear systems using abstract combinatorics. 0167-6911/88/$3.50 © 1988 ElsevierScience Publishers B.V. (North-Holland)

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F. Lamnabhi.Lagarrigue, P.E. Crouch / Iterated derivatives

2. The main formulas

In this section we introduce the main results and give the proofs in subsequent sections. The first result concerns a nonlinear control system of the form m

~m:

X=

E

gi(x)lli(t)

'

y=h(x),

i=1

where x belongs to a differentiable manifold M (for instance R Jr), gi, i = 1 , . . . , m, are vector fields on M, and h is a real valued function. All data are assumed to be real analytic. We let ( x l , . . . , x jr) denote local coordinates on M, about x. We write Ai for gi viewed as a differential operator; in local coordinates

A, = E o g (x)(a/axk). We denote by ([~ ""{~) two sequences of positive integers ordered so that the top satisfies jl ~J2 ~ "'"

<~jq. If then

(+,,

....

-.. ~,,--

,

~

,

Pl

~

P2

Ps

where a~ < ~2 < ' ' ' < O/s are distinct integers appearing with multiplicities p~, P2,..., Ps respectively, the bottom sequence is ordered so that

i1<~

...

<~ipl ,

ipl+l~

.'.

~ipE,...,ips_l+l~

...

<~im-i o.

To each set of integers ({~... "'"~, J, ) ordered as above we may write for r = 1 .... , s,

(B,,..., where fl~ < fl: < . . . < fl~ are distinct integers appearing with multiplicities #1,..-, # r Clearly t, fli and #i are functions of r. For simplicity we denote by #(r)! the product # l ( r ) ! . . . i~tto(r)! while we denote by #! the product/~(1)! . . . / ~ ( s ) ! Our main theorem may now be stated as follows: Theorem 1. Let yt")(t) be the n-th order total time derivative of the output function y(t) of "~m" Then n

YO')(t) = E

q=l

E

la,.,!¢h

1 ~]

[i]j.,...,jqh)(x(t))u(j,)(t)... iq ~,

~'

q ....

q

/"

"

u(.jq)(t )' Zq "~

~lI "" lq I q .

~.~=lJ,=n-q il ..... iq~{1 ..... m}

where E J~,..'"J~,, is the differential operator obtained by substituting A~ for Z~, i - 1, . .., m, and the identity for X o in the expression

Here Z~, i = 1,..., m, and X o are noncommuting indeterminates and W denotes the shuffle product (see Section 3). Let Zo,,,, denote the following system obtained from ~',,,~-t by setting u 1 ~

0,

U i -~--Ui_l,

2 ~< i ~< m + 1:

n!

y(t)=h(x). i=1

A o will now denote go viewed as a differential operator. F r o m Theorem 1 we easily derive the following important special case.

F. Lamnabhi-Lagarrigue, P.E. Crouch / Iterated derivatives

3

Corollary 1. Let y(n)(t) be the n-th order total time derivative oJ lhe output function y(t) of Zo, m. Then n

n--q

~

~

q=l

k=0

y(")(t)=A~h(x(t))+

1

k

--~.( F/q.~:'/qqh)(x(t))u}~)(t)...

~

u(.Jq)ft),

¢~l...Jq~ ~,ll ... I q

iq

•

i i ..... iqE {1 . . . . . m }

where k I"J~it."'..i~A is the differential operator obtained by substituting A i for Zi, i = 0,.. ., m, and the identity for X o in the expression Zko 12 --I z h1 """ ""Yq lq *

3. Algebraic preliminaries Consider an alphabet P - { Xo, Z I , . . . , Z,,, } in m + 1 noncommuting indeterminates, and P* the free monoid generated by P. The neutral element of P * is the empty word denoted by 1; and the product is just concatenation. Let U be the algebra over R generated by P * , and introduce the shuffle product on U (see Lothaire [8]) defined on basis elements by

1WZ-ZWI=Z;

1121-1,

VZ~P, vz,

z'

Lemma 1. Let j l , . . . , j, be q integers such that q <~n and 5".~=]j i = n - 1, and let i i ..... , i q ~ {1, .... m }.Then ...

-

X

,,,

[n-l)(n-2-J°tq))'"(n-q+l-Eq=3j°u)

o~S(q) ~ Jo(q)

Jo(q-q)

J'o(2,

(1) Zi°°''"

Here S(q) is the permutation group on q letters { 1 , . . . , q } and (7,)= n ! / k ! ( n -

Zi"(q'"

k)!.

Proof. The elements in the shuffle product

(XdlZil) ]1] ( Xd2Zi2) 12"'" ~1 ( XdqZiq)

(2)

fall into q[ distinct groups, each group defined by a distinct permutation o ~ S(q), and each element in the group has the form

(3)

. . . xbz,.,,,

with 11 + 12 + "'" + iq - J l +J2 + "'" +Jq = n - 1. Identity (1) will be proven if we demonstrate that the number of elements in each group (3) is

n- 1

n -- 2 --Jo(q)

Jo(q)

J o ( q - 1)

.

.

.

n - q + 1 - Xq__3jou) Jo(2)

"

However this is a simple combinatorial result obtained by noting that the shuffle product (Z1 Z2 "'" " 1) words in P* of which ("-1 s ) words appear with Z. as the Z._s_ 1)12 (Z.-~ • "" Z,,_ 1 Z,,) is a sum of (~+ last letter. The following result may be obtained in a routine manner.

Lemma 2. m

E

it . . . i q = l f(iq . . . . . i q ) = 0

m

z,z,...z,=

E

!

E

iq''.iq=l PI! "'" ps! oEStq) ii <~i2<~... <~iq f(iq . . . . . i q ) = 0

4

F. Lamnabhi-Lagarrigue, P.E. Crouch / Iterated derivatives

where the summations are constrained by a S(q)-invariant function f; f(iq,..., iq)-- f(io(l),..., io(q) ) for every o ~ S(q) and for a given set of indices il,..., iq satisfying il <~i2 <~ "'" <~iq we write (il"'"iq)=(~

l'''''al'_ . a 2 ' ' ' " a 2 " ' " - . a s ' ' ' " a s ) Pl

P2

Ps

where a I < az < "'" < as are distinct integers appearing with multiplicities Pl, P2,..., Ps respectively. 4. Formula for derivatives In order to get the main formula we first prove an intermediate result. Consider the single input-tingle output nonlinear system

y(t)-h(x). We let A~j) denote the differential operator uCJ)(t)gl, J = O, i , . . . , with A~°) = u(t)A 1. Proposition 1. For system ,S 1,

Y(n)(t) = q--lkq=! E E kq_~ffiO E "'"

k =0

kq

kq-1

j

k2

"( Ai'-q-r'q":k')A~ k2, " " " A~kOh)(x(t)).

(4)

Proof. By induction on n. For n = 1 we have y°)(t)ffi u(t)(glh)(x(t))ffi(A(°)h)(x(t)). result is true for n and calculate

d ' ( d y (t)) d" y("+l)(t ) = ~ "~ = "~(u(t)(Alh)(x(t))

We assume the

).

Using the Leibnitz formula we get

y("+')(t) = A l h ( x ( t ) ) u ( n ) ( t ) +

Y'

kq+, (A,h(x(t))(n-kq")u(kq÷')(t) •

kq+l--0

We now use the induction hypothesis to derive a formula for (Alh(x(t))(n-kq ÷i), kq+ 1 • 0 , . . . , n - 1 , treating (Alh) as an output for ,YI:

Y(n+l)(t)-Alh(x(t))u(n)(t)

+

E

kq+ 1

kq+l--O

•

ffi

E kqffiO

E kq_ I •0

"'"

E k2ffi0

kq

E Lamnabhi-Lagarrigue, P.E. Crouch / Itera:ed derivatives

5

n-I Using the identity Ek~÷,ffiOEq=~÷' ffi Eqffi,ET,~-qffio we get n

n-q

n-q-kq+l

Y(n+')( t) ffi A(qn) + E E q=l kq+t •O

E kqffiO

n - 1 - kq+ 1 •

i o e

kq

n-q-V~+~k~

I!

E kaffiO

kq+1

"'"

)

n - q + 1 - r"-'i=3 q+lk i k2

s(n-q-rJ21k,,~tk2, . . . ..~q .~tl [al A(kq) ~1 d(k¢+l) /(h)(x(t)). I

On setting q equal to q - 1 in the above expression we obtain the expression (4) with n replaced by (n + 1). We now generalize this result for system ,~, and let A~j) denote the differential operator u~J)g~, j - 0, 1,..., with --~At°)-- u~(t)A~, i -- 1, ... , m. Proposition 2. For system "~m, Yt")(t) ffi E

E

"'"

q=l i l . . - i , = l j, ..... j¢ffiO

.(a(JOA(J2) ._,

•

Jq

Jq-1

J2

.. A(J¢)h ~f ,, ,, x (,t)).

(5)

Proof. Simply apply Proposition I and the fact that the sets {(Jl, J2,---,Jq)"1 ~kq>~O, p-kq>~kq_1 >~0,...,p-~,q=3ki>~k2>~O, kl =p-~q=2ki} concide. Proof of Theorem I. From Proposition 2 and L e m m a 2, the n-th order totaltime derivativeof y(t) may be written as n m n-q 1 Y ( n ) ( t ) ffi E E E ! qffiffil iq"'iqffil Jl ..... jqffiffiO Pl! " " Ps. ~q.lj~-n-q

E

• eS(q)

Jo(q)

" " "

Jo(q)

Jo(2)

~,

io(I)

io(2)

"

to(q)

where (Jl, J2,..., Jq) = ( a l , . . . , al, a 2 , . . . , or2,..., a s , . . . , as) (multiplicities P l , . . . , P, respectively). Using the definition of A}~' and the commutivity of the functions u~J)(t) we may rewrite the second line of (6) as

E

a~S(q)

(nl)(n2 q) Jo(q)

n

q-1

• i,

Jo(2)

Jo(q)

• u fi !J O ( t ) u ~ 2 ) ( t ) ' ' " u(.Jq)(t)(Ai.o,Aiot2,'" Aio(¢,h)(x(t)), Iq ~,

which using Lemma 1 and the notation introduced in Section 2 may be expressed as E J t...'/q~(h "')(x(t))u~:O(t)q

. . . u(.jq)(t).,q

(7)

6

F. L a m n a b h i ~ L a g a r r i g u e , P.E. C r o u c h / I t e r a t e d d e r i v a t i v e s

Thus equation (6) may be rewritten as m

n

~q '/l""/qth) u~{')( • u(Yq)(t). , , F6...iq " . ( x ( t ) ) t).. iq , Pl. " "" Ps. 6 " " ' =1

n-q

Y(')(t)= E q=l

1

E

.il..... A = 0

(8)

zq.lj,=n-q

A ~J2~ "'" ~Jq Now m

zJ'"'Jqtl(J')rt) i "" i d il x

E

"'"

u(Jq)(t) iq

i 1 • • • iq = 1 m

=

E

1

E o

=a

~m

Ill ip~+l

1

""-"ip~=l

Z{:~;:SJo'PI'UJ'°"'(t) "'" uJ°'"'(t)

s(p,)

~.~

Zyo,,,+t,...,,p,,uJo,p,+,,(t~

/~(2)! a ~s( Pa)

" uJo,p,,(t)

'°'"*"""

il,~+ l <~ " " " <~ipa m

• ""'

1

E

it,, i+1 . . . i p = l

E

)!

~t(S

ZJ.'°'Ps-,+""'J.'otps,uJ.°'Ps-'+"(t) I o ( P s _ l + l ) "'"

Io(ps )

'''

IO(Ps-l+l)

U.Y°"- ( t ) ,

(9)

I ° ( p s)

o~S(ps)

ipl+l <~ . . . <~ips

where we have used Lemma 2 and the notation of Section 2. Using the cormnutativity of the shuffle product, expression (9) becomes m

i,...io= , E

Pl! ~(1)!

p V .....

l~(s)! " _ _ z. J ' "•'"J qt.q u .( IJI '. ) (.t ) .

ld('Jq)(t)" iq

i1<~i2~< . . . <~ip~ .,,

iPs-I + ! <~ "'" <~ips

Combining this result with the expression (8) and the fact that p ! - - # ( 1 ) ! . . . #(s)! gives the result of Theorem 1.

5. Examples The main use of Theorem I and Corollary 1 will be the calculation of the coefficients of terms such as

it,~tJ~)(t) ... utqJO(t), Eqffilj, ffi n - q in the expression for a derivative yt')(t). We illustrate this with two examples. Example 1. Find the coefficient of u2u2ut21) in y(5)(t), where y(t) is the output of ,S 2. Clearly q ffi 4,

i1

i2

i3

and hence s = 2 ,

i4

1

1

2

2 '

P l = 3, P2 = 1, and #! =2!. Thus the coefficient of

UlU2U2" 2 . . t~)

is obtained from

zOO01 1122 = Z1 w Z 1 w Z 2 w XoZ 2 by setting Z1 = A1, Z 2 = A 2, X0 - id. The resulting coefficient is easily calculated

to be

l(F ,12h)(x(t)) 0001

= {7A2A2 +6AIA2A1A2 + 5A1A~A1 + 5A2A21A2 + 4 A 2 A I A 2 A 1 + 3A~A~ } ( h ) ( x ( t ) ) .

F. Lamnabh~.-Lagarrigue, P.E. Crouch / Iterated derivatives

Example 2. Find the coefficient of •~ 22, Clear!y q ~ 1 (1) in ytS)(t) where y ( t ) is the output of -Y02, n - q - k = 1, so k - 1 and

iI

i2

i3

and hence s = 2, P2

2

2

7

3,

--

1 '

7 0 0221 1 2, p l = 1 and/z! = 2l. Thus the coefficient of u22u~1) is obtained from Zo W L, Z o ll.i Z 2 ll.I Z 2 U.!XoZ 1 by setting Z o = Ao, Z 1 = A1, Z2 = A2 and X o = id. The resulting coefficient is easily calculated to be =

-

1

2"~.(1F°°~h )l' x ( t ) )

=

-

{ A1A,~A o + 2A2A1A2A o + 3A2A1Ao + 4A2AoA1 + A1A2AoA 2 + 2A2AIAoA 2 + 3A2AoA1A 2 + 4A2AoA2AI + AaAo A2 + 2AoA1A 2 + 3A,~A2A1A 2 + 4AoA2AI } ( h ) ( x ( t ) ) .

References [1] P. C~xtier and D. Foata, Problems Combinatoires de Communication et Rearrangements, Lecture Notes in Mathematics No. 85 (Spria~;er, Berlin-New York :'~9). [2] P.E. Crouch and F. Latm=d0~,i-Lagarrigue, Algebraic and multiple integral identities, Preprint (1987). [3] J. Descusse and C. Moog, L.xzoupling with dynamic compensation for strong invertible affine nonlinear systems, Internat. J. Control 42 (1985) 1387-1398. [4] M. Ffiess, Fonctionelles causales non lineares et indeterminees non commutatives, Bull. Soc. Math. France 109 (1981) 340. [5] M. Fliess, A new approach to the structure at infinity of nonlinear systems, Systems Control Left. 7 (1986) 419-421. [6] M. Fliess and F. Lanmabhi-Lagarrigue, Application of a new functional expansion to the cubic anharmonic oscillator, J. Math. Phys. 23 (1982) 495-502. [7] P. Leroux and G. Viennot, Combinatorial resolution of systems of differential equations, I. Ordinary differential equations, in: G. Labelle and P. Leroue, Eds., Combinatoire Enumerative, Lecture Notes in Mathematics No. 1234 (Springer, Berlin-New York, 1986). [8] M. Lothaire, Combinatorics on words, in: G.C. Rota, Ed., Encyclopedia of Mathematics and its Applications, Vol. 2 (AddisonWesley, Reading, MA, 1983). [9] H. Nijmeijer and W. Respondek, D~namic input-output decoupling of nonlinear control systems, Preprint (1987). [10] R. Ree, Lie elements and an algebra associated with shuffles, Ann. Math. 68 (1958) 211-220.