Direct Transformation of Nonlinear Systems into a State Affine Form

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DIRECT TRANSFORMATION OF NONLINEAR SYSTEMS INTO A STATE AFFINE FORM I. Souleiman ** A. Glumineau ••• G. Schreier .... IRCCyN, UMR C.N.R.S. 6597 Ecole Centrale de Nantes , UniversiU de Nantes , Ecole des Mines 1 rue de la Noe, BP 92101 , 44321 Nantes Cedex 3, France Phone (+33) 2-40-37-69-13. Fax (+33) 2-40-37-69-30 E-mail:

•• [email protected]·fr ••• [email protected] .... [email protected]

Abstract: A transformation of a non linear system into a special state affine system is given in order to design an observer. Necessary and sufficient conditions are directly obtained i.e. without computing the input-output differential equations of the non linear systems by solving the state e ilnination problem. Copyright ~ 2001lFAC Keywords: Observers, nonlinear systems, differential algebra

The paper is organized as follows . In Section 2, the problem under interest is stated. Some preliminaries are given in section 3. In the next section, necessary and sufficient conditions and a constructive algorithm is given. The algorithm is illustrated by an example in section 5. Finally, a conclusion and prospects are drawn .

1. INTRODUCTION

In the linear case, the observation problem has its standard solutions in Kalman's and Luenberger's works. In the nonlinear case, a fruitful observer design technique consists in transforming this nonlinear system into a state affine form . In the last decade, two frameworks have been used to tackle this problem: the geometric approach ((Besan<;on et al., 1996), (Hammouri and Kinnaert , 1996)) and more recently the algebraic one ((L6pez-M. et al., 1998)), where input-output differential equations were used. Thus, as a preliminary, the problem of state elimination must be solved. Transformations into state affine systems are studied in (Besan<;on et al. , 1996) , (Hammouri and Kinnaert, 1996), (L6pez-M. et al., 1998). However, in (L6pez-M. et al., 1998), NSC and an algorithm are given for the transformation into a special observable state affine form using state elimination techniques. This last result is constructive.

2. PROBLEM STATEMENT Consider the nonlinear system

i = y

j(i , u)

= h(i)

(1)

where i E M is the state (M is an open and dense subset of Rn) , U E Rm is the input , y E R is the measured output, j and hare meromorphic functions of their arguments and L is supposed to be generically observable (Conte et al. , 1999) , i.e. , this property is supposed to be satisfied locally around a regular point Xo of M.

The goal of this paper is to give a constructive algorithm which transforms a nonlinear MISO system into the same special state affine systems than in (L6pez-M . et al. , 1998) without solving the state elimination problem. In (De Leon et al. , 2000) , using an extended state affine form of the induction motor, a flux - torque non linear cascade observer is designed.

We make the assumption, that the output and the input are known. First, to simplify the study without loss of generality, we rewrite the nonlinear system (1) using a state coordinate transformation x := X(i) (i.e. rank~~ = n):

507

where xT := [X2 ... xn] . The system equations can be expressed using the new coordinates as

~ :

{.:i: = y =

f(x , y, u) Xl

... , db p } . There exists locally two functions t/J(a , b) and T( a , b) such that T · t/J

(2)

L :: .dai =

(3)

i=l

{

such that ~ is locally equivalent to the MISO state affine system i = A(u , y)z Ya = Cz

0 ... 0 al(u, y) 0 a2(u, y) ...

o .

0

+ rP(u , y)

0 0

.'.:

o 0 o 0 C=[lO ... O] [

0

•

w

if and only if

dw 1\ db l

1\ . . . 1\

(7) db p = O.

Definition 1. For all k E [1, n], the Ek space is formally defined as d y , dU, ... , dY (k-l) , du (k-l)} . E k -- SpanK {J=.
(4)

1

This space is used to identify the elements of the matrix A(u, y) and to compute the state coordinate transformation (3).

, <1>=

.. 0 an-l(u,y)

.. .

. ..

0

Definition 2. For all k E [1, n], the [k space is formally defined as "ck _- SpanK {d y , dU, .. . , dY (k-l) , du (k-l ) } .

Note that z E ~n and that TT = [Tl , TT] . The stated problem is of major importance since an observer for (4) can be designed as follows

i= : R= {

, .. . ,

In section 4, the given algorithm uses the spaces defined below.

where

A=

= O. (6)

)..

T :

: {

dw 1\ w

a differential I-form da)..}. There exists locally a function 1/(a, b) such that

w E SpanK(a,b) {oo l

Zl

~a

if and only if

Lemma 3. Consider

In this paper, the problem under interest is the search for a state coordinate transformation

T :

=w

This space is used to find the different elements of the input-output injection rP( u, y) .

A(19)z+rP(19)-R- l CT (Cz_y) -OR-AT(19)R-RA(19)+CT C(5) with 19 = (u,y) .

4. MAIN RESULT In this paper, the main tool consists in analysing the differential of the nth time derivative of the output y :

where R is a real symmetric, positive definite matrix. 0 is a positive constant and u must be 0 strictly persistent (Bornard et al., 1988) (i.e. 3 0 such that the system is observable). In (L6pez-M. et al., 1998), the model of an inverted pendulum is transformed into a state affine system and an observer is designed.

y=l1(x,y, u)

11 811 . + 811. 1( - . . ) y.. = 88x 1-(-x , y,U ) + &;y 8u u := 2 x , y,y , U,U (8) n-2

y

3. SOME PRELIMINARlES

(n) = 8ln-l 1-(8i

x , y, u

) + " [8 In ~ k=O

From (8) , the differential

With K the field of meromorphic functions of a and b (a E IRA, b E IRP) , consider a differential I-form W E SpanK(a ,bd 001 , . . . , da).. , db l , ... , db p } . The differential I-form w is a closed form if dw = O. A differential I-form w is an exact form if there exists a function t/J(a , b) such that w = dt/J. The exterior product is denoted 1\ .

1

&y( k) Y

dy(n)

(k+l )

+ 8ln-l (k + l )] 8u(k ) u

can be written as

(9) using the following definitions of PlO , Pu and Py

Lemma 1. (Poincare's lemma). A differential I-form w E SpanK(a,b) {oo l , . . . , da).., db l , ... , db p } where a E IR).. and b E IRP is locally exact if and only if dw = O. Lemma 2. (Conte et al. , 1999) Consider a differential I-form w E SpanK(a,b){dal , ... , da).. , db l , 508

Pick functions aj', I such that

Furthermore, we want to decompose P u into P llu and P2u (respectively P y into P lly and P 2y ), where only P Uu and P lly depends explicitly on x. For that reason, Pllu and Plly are defined as follows:

m

1=1

Define the differential one-form Wj as

= L L~du~j)

with

;=1 j=O

n-l P lly = L ptdy(j)

with

m

_.

iYy

=

Wj

J~ oJYy L

ox

k=2

)=0

+ P ll

where P ll = P llu and P2

dXk

the differential of the can be rewritten:

nth

-

aj'duI

-a

Wj

oHj+l = Tj+l(x) (~ L,. (n-j) dUI + 1=1

HI

= PI (x, U, ... , u(n-l), y, ... ,y(n-l))(lO)

=

aj

wj.

Then, determine

+P2(u, ... , u(n-l), y, ... , y(n-l))

WHl = Wj - d [HHITHll

The same analysis is made for the formal equivalent system (4): dy~n)

OHHl) oy(n- Jl dy

vU I

where HHl is defined in appendix A. and TH 1 (x) are solutions of

time output derivative

Wj

dy(n)

+ aJdy.

Check dW j I\Wj 1\ dUI 1\ . .. 1\ dUm = 0 and dW j 1\ Wj 1\ dy = O. Otherwise, stop! wj is computed from wj in the same way. Moreover

k

+ P lly

= Pu + Py

=L 1=1

where integration constants equal to zero. Afterwards, using the definitions PI := PlO

1, .. . ,m and

j aj'dui n - ) - ardy(n- j ) E en-j .

Wj - L

n-l

m

P llu

ar

= P 1(T(x), U, . . . , u(n-l), y, . .. , y(n-l) (11) +P2(u, . .. , u(n-l), y, ... , y(n-l))

2 nd Remark 1. If a solution of the state transformation problem exists, that means:

If the condition Wj+l = 0 for j = n - 1 is not fulfilled, stop the algorithm! Otherwise, elements aj determined in this 1st step can be now substituted in P!f which will be used in the 2 nd step. Step: Computation of
OJ - Lf3j'duin-j) - f3;dy(n- j ) E En-j .

To find transformation (3), we introduce a direct algorithm that checks the existence of functions aj and
1=]

Define a differential one-form

6] =

and with the formal system (4), we derive the same decomposition

= Pl(T(x), U, . .. , u(n-l), Ya, ... , y~n-l))

1=1 vU I

+p2a( u, ... , u (n-l) ,Ya, ... ' Ya(n-l))


n

= L(H;dz, + z;dH;) + dG .

•

i=2

1

st

Lf3j'dul +f3jdy.

Verify conditions d8 j l\dull\ . . . 1\ dUm = 0 and d8 j I\dy = o. If it is not satisfied, stop the algorithm! Using the same technique, oj and 8j can be defined from state affine system ~a, then m -a L oGn - j oGn _ j d 0). = y ,. ( n -) ) dUI + 0 y ( n -) )

= PlO + Pu + P2 = PI (x , U, .. . ,u(n-l), y, . .. ,y(n-l)) (12) +P2(u, .. . , u(n-l), y, ... , y(n-l))

dy~n)

as

1=]

Algorithm 1: Initial step: Compute dy(n)

8j

m

Remark 2. If Wj is equal to zero, we can deduce that aj is a constant, but we will use the following formula to find the transformation term TH 1

Step: Computation of aj and Ti Set Wl = PI and w'1 = Pf. j= 1, ... ,n-1

509

+ [o.la2 + ala2]T3(x)} d {
PI = d {al(u , y)T2(X) P; =

1 st Step j = 1 (search of ai , T 2 ) - - Set Wl = PI. Then , we pick functions 0 1 and such that

where K j and Qj are defined in appendix A. We will choose, in this case, without loss of generality aj = 1 because the special state affine systems considered here are observable when all aj terms are different to zero.

or

or

We obtain 01 = ye Xl and = ue X2 • Let's define a differential one-form Wl as

Denote I\du := dUI 1\ ... 1\ dUm . Theorem 1. The system I: is locally equivalent to I:a if and only if, for k E [1 , n]

Wl = oi'du + ordy

= e X2 (udy

d8 k

1\

du = 0, d8 k

dW k 1\ Wk

1\

1\

dy = 0

du = 0, dWk

1\ Wk 1\

+ ydu).

Conditions dW l 1\ Wl 1\ du = 0 and dW l 1\ wll\dy = 0 are verified . In the same way, using the possible equivalent state affine system, we have

dy = O.

•

Proof. The proof of theorem 1 is developed in appendix B.

W

5. EXAMPLE

1=

pt'

We pick functions 0'lu and o~Y such that

Consider the following academic single input single output system I: exl : Then, we define a differential one-form w'l as

This system is observable. We will research the diffeomorphism such that I:exl is locally equivalent to Using the equality w'l trivially

(14)

= Wl and H2 = ai,

Thus algorithm 1 is applied. Initial step: From I: exl , we have

dy(3) =

Thus

PI =

j = 2 yields T3(X) = In(x3 - X2) , and a2 = -u. - - It is easy to check that W3 = o. So the algorithm can be continued by step 2. nd Step The element ai determined in the 1st 2 Xl d (uuy + u 2y) In(x3 - X2)} step can be now substituted in P!J that will be used in the 2nd step. 2y4 +d {u 2y + (uysin(y) + (uysin(y) + u )} . j = 1 (search of 'PI) - - 81 = P2 E £3 = SpanK {du, dy, du , di; , dii. , dY} . Xl Let's pick functions (3'1 and (3r such that d (uuy + u 2y) In(x3 - X2)}

{uye {uye

P2 = d {u 2y + (uysin(y)

+ (uysin(y) + u 2y4 )} .

From the second output time derivative of we can express

Then, we define a differential one-form

I: ex1a ,

81

+ [o.la2 + al·a2]T3(x)} +d {'PI + 0. 1'P2 + al ;P2 + al a2'P3} .

as

dy(3) = d {al(u , y)T2(x)

This one-form satisfies the following conditions: d8 l l\du = 0 and d8 l l\dy = o. 81 is computed in the same way on dyi2 ) .

From (14) , we deduce Pi and P!J as follows . 510

-a

<>1 =

a'Pl aC Z aC Z ay dy + aii. du = ay dy

From the equality

+

a'Pl au du o

81 = 81 and Cl =
'PI can be derived 'PI = uZy . j = 2 yields 'Pz = sin(y). j = 3 yields 'P3 = _y3.

expressed as follows C = L:~=l C n - k with Cn-k involving all monomials in C depending on (y(r))q and (U)vtl)SI such that rq + L:;:l VlSI = n - k . Remark 3. Cn-k , k > 1 depends on ('Pk. ... , 'Pz , ab aZ , . .. , ak-d and their time derivatives. k = 1 and k = n represent special cases, where C n - l = 'PI(n-l) an d C 0 = K n-l'Pn·

The non linear functions 'Pj and ai of state affine system (14) are determined, due to the conditions of theorem 1 are satisfied. Finally, the equivalent state affine system is

Remark 4. Hi could depend on (al , . .. , ai-d and their time derivatives. For example, Hz can be expressed explicitely as Hz = a~n-l) . However, H i is limited to monomials depending on (y(r))q and (U)vtl)SI such that rq + L:;:I VISI = n - i + 1. n

Thus , we can compute the differential of yi ) as n follows : dyi ) = L:~~(Hidzi + zidHi) + dC . Let 's define the following differential forms :

with the transformation

6. CONCLUSIONS

P1a

( X, - U , Y ,···, tL (n-l)

, Y (n-l))..=

'"'(H ~ i dTi+ T i dHi ) 1=2

New necessary and sufficient conditions for the transformation of a MISO nonlinear system into a class of state affine systems have been obtained in a direct and constructive way without solving the state elimination problem. The given algorithm can be implemented in symbolic computation systems. An observer for the transformed system can be computed without time-varying computation of the observer gain matrix (L6pez-M. et al., 1998). An interesting extension of this work would be to find the transformation for a more general class of MISO state affine systems.

na ( u , y , . .. ,tL (n-l) , y (n-l)) := r2

dG .

These forms are used in the algorithm 1. Let's notice that Pf and P2 are exact differential one-forms. Then, we have: dPf = 0, dP2 = O. Appendix B: Proof of theorem 1 Necessity: Suppose that the transformation (3) which transforms (2) into (4) exists. 1stStep In the first step, the necessity of the conditions dW j A Wj A dUI A ... A dUm = 0, dWj A Wj A dy = 0 will be proved. j1h iteration We pick o:jl (I E [I , m]) and o:J such that

Appendix A : About the state affine differential equations

m

W - '"' o:u1du(n- j ) - o:Ydy(n- j ) E E"-j J L J I J •

Let us analyse the differential of the nth time output derivative for the state affine systems (4) . To compute the nth time output derivative of this class of systems, we will use the following equations given in (L6pez-M. et al., 1998) , for k E [1 , n-l]

+ Qk. Qk = Kk-l'Pk + Qk-l + Kk-lZk

y(k) = Kk zk+l

1=1

The term in Wj depending on du)n-j) and dy(n- j ) comes from Hj+lTj+l(x) , And we have

(16) (17)

where Kk = 07=1 al (with the definition an = 0) and Ql = 'PI ' The nth time output derivative can be derived using these iterative equations. y

(n)

= Qn

= Kn-I'Pn

.

.

+ Qn-I + Kn-lzn

(18)

Afterwards, we will decompose the nth time output derivative as follows: y(n) = L:~z(HiZi ) + C . where H i involves all terms multiplied by Zi and C corresponds to the rest . This rest can be 511

We can easily verify dW j /\Wj /\ dUI /\ .. . /\ dUm = 0 and dW j /\ Wj /\ dy = O. 2 nd Step. This step proves the necessity of the following conditions of theorem 1: d8j /\ dUl/\ . . . /\ dUm = 0 and d8 j /\ dy = O. pt iteration. P2 contains the (n - l)th derivative of 'PI ' Using P2 , 81 selects terms with u(n-l) and y(n-l) . The terms in 01 depending on du}n-l) and dy(n-l) derives from G n - l = 'Pin-l)(u, y). We obtain 81 = d'Pl . Then d8 1 /\ dUI /\ . . . /\ dUm = 0 and d8 1 /\ dy = O. The next iterations follow the same lines. /h iteration. G n _ j is given as a sum of monomials depending on (y(r »)q and (U}Vd) SI such that rq + 2:;:1 1/1 SI = n - j . We obtain

= Wj - d(Tj+lHj+d . Using the results of t he first step, functions al (u, y), ... , aj-l (u, y) can be substituted in P!f . 2nd Step. The conditions d8 j /\ dUI /\ .. . /\ dUm = o and d8j /\ dy = 0 are verified, then 8j = 2:;:1 (u, y)duI + (u , y)dy where are function depending only on U and y. For j = 1 to n /h iteration Note that

Wj+l

.B.t

.BJ

.B't ,.BJ

m

6j

=

L.Bt (u, y)dul +.BJ (u, y)dy 1=1

then 'Pj is solution of

-

OJ =

LT»

8u

1=1

Note We can easily compute d8 j /\ dUI /\ . .. /\ dUm = 0 and d8j /\ dy = O. If the transformation (3) exists, the conditions of theorem 1 must be satisfied. Thence, the necessity of theorem 1 is proved.

8G n _

that

j

8G

-(--.) dUI n-)

_

n j + -(--.) dy 8y n-)

I

the

input-output

injections

('P2, . . . , 'Pj-l) as well as all elements ai of matrix A are known at the jth iteration and that the function G n _ j depends on ('P2 , .. . ,'Pj,al , ... , aj-d

and their time derivatives. So, only the inputouput injection 'Pj is unknown and can be determined. If the conditions of theorem 1 are satisfied, the functions ai , Ti and 'Pi can be well estimated . So, the sufficiency of theorem 1 is then proved.

Sufficiency: Suppose that the conditions of theorem 1 are verified. Applying algorithm 1, we show that the new state affine system and the transformation can be determined: aj(u, y) , 'Pi(U , y) and Ti(X).

1 st Step. The following conditions dW j /\Wj /\dUl/\ ... /\ dUm = 0 and dW j /\ Wj /\ dy = 0 are satisfied, then Wj = Tj+l(X)Oj(u,y) where OJ(u , y) is a differential one-form on U and y . 1st iteration. From (12), we obtain

7. REFERENCES

Besan<;on, G ., G. Bornard and H. Hammouri (1996). Observers synthesis for a class of nonlinear control systems .. European Journal of Control 3(1), 176-193. Bornard, G ., N. Couenne and F . Celle (1988) . Regularly persistent observers for bilinear systems. 29 I .C.N.S., New trends in nonlinear system theory 122, 287-292. Conte, G ., C.H. Moog and A.M. Perdon (1999).

Wl = T 2 (x)Ol(U,y) .

From the formal state affine system, one gets Z2dal(U , y) = T2(x)dal(U,y) . The equality Wl = wl' leads to the expression of al (u , y) and T 2(x) . Then, W2 can be computed as follows: W2 = Wl - d(T2(X)H2 ) , where H2 = a~n-l ) /h iteration. Pick functions and aJ such that

wl' =

a'/

Wj -

L

a;lduln-j) - a~dy(n-j) E

Nonlinear Control Systems - An Algebraic Setting. Springer-Verlag.

En-j.

De Leon, J. , A. Glumineau and I. Souleiman (2000) . Nonlinear observer for induction motors : a benchmark test . Proc. of IEEE CCA pp. 146-15l. Hammouri, H. and M. Kinnaert (1996) . A new procedure for time-varying linearization up to output injection. Syst. Contr. Lett. 28, 151157. Lapez-M. , V., J. de Lean Morales and A. Glumineau (1998). Transformation of nonlinear systems into state affine control systems and observer synthesis. IFAC CSSC, Nantes ,

1= 1

Define a differential one-form Wj as m

Wj =

L a'/duI + aJdy . 1=1

As told before, since dW j /\ Wj /\ dUI /\ .. . /\ dUm = 0 and dW j /\ Wj /\ dy = 0, we have Wj = Tj+l(X)Oj(u , y) from the real system. The same selection applied to the formal state affine system gives

France.

By identifying Wj = wj, we obtain Tj+l(X) and aj(u , y) . Finally, Wj+l can be computed as follows : 512