Transformation of Nonlinear Systems into State Affine Control Systems and Observer Synthesis

Copyright <0 IFAC System Structure and Control, Nantes, France, 1998

TRANSFORMATION OF NONLINEAR SYSTEMS INTO STATE AFFINE CONTROL SYSTEMS AND OBSERVER SYNTHESIS. V. Lopez M .• ,1 J. de Leon Morales·· A. Glumineau.·

• Institut de Recherche en Cybernetique de Nantes IRCyN, UMR 6597, BP 92101, 44321 Nantes CEDEX 3, France, Ph. (+33) 2.40.37. 16.36. Fax (+33) 2.40.37.25.22, e-mail: {lopez}[glumineau}@lan .ec-nantes.Jr •• Universidad Autonoma de Nuevo Leon, Facultad de Ingenieria Electrica , Sn . Nicolas de los Garza , Nuevo Lean A .P . 148-F, Mexico . Ph. (+52) 83.76.45.14, e-mail: jleon @ccr.dsi.uanl.mx

Abstract: This paper addresses the synthesis of observers for nonlinear systems and their equivalence to state affine systems (i.e. time-varying linear systems) . Necessary and sufficient conditions are given and then , whether a solution exists, one gets the observer synthesis. These constructive conditions are based on the analysis of the system input/output (I/O) differential equation . The transformed system is again transformed to solved an algebraic Ricatti 's equation instead of a time varying one. Copyright @1998 IFAC Resume: Ce papier porte sur la synthese d 'observateurs pour les systemes non lineaires et leur equivalence en des systemes affines en l'etat (i.e. systemes lineaires variants) . Des Conditions Necessaires et Suffisantes sont donnees et si elles sont satisfaites, la construction d 'un observateur est possible. Ces conditions constructives sont basees sur l'analyse de l'equation differentielle entree-sortie du systeme. Le systeme obtenu est ensuite transforme; pour ne necessiter que la solution d 'une equation de Ricatti algebrique . Keywords: Input-Output injection, Time-varying linearization, Nonlinear Systems, Nonlinear Observers.

1. INTRODUCTION

al., 96) one finds a straightforward approach to verify and compute the linearization condition. This method is fully constructive and in the same aim we propose a solution for a more general class of systems.

For linear time invariant systems, the problem of observer synthesis is completely solved, whereas in the nonlinear case, there is no general method . Nevertheless, some results for almost-linear dynamics are available (Bornard , et al. , 88 ; Viel , 94) . This is why the characterization of these systems has been deeply investigated. Several authors (Krener and Isidori, 83 ; Krener and Respondek, 85; Xia and Gao, 89) investigated the case when a system can be transformed into a linear system up to I/O injection . In (Glumineau ,et 1

The high-gain observers are useful for state affine systems as is shown in (G . Besan<;on and G. Bornard, 96 ; Bornard, et al., 88 ; Hammouri and Gauthier, 88) and the references therein . These observers are based on optimal Kalman's observer and used in physical process, as distilling columns (Viel, 94) .

Supported by CONACyT, Mexico

739

This paper addresses the transformation of a nonlinear system into a state affine one, first proposed and solved using some differential functions in (Hammouri and Gauthier, 92) and more recently in (Hammouri and Kinnaert, 96) . In opposition to the existing method , our approach is based on the analysis of the system I/O differential equation. It allows to give wheter a solution exists, a straightforward Necessary and Sufficient Condition (NSC) . One of the contributions of this paper is the definition of a first algorithm to compute the transformed system functions, from the I/O differential equation.

3.1 The I nput- Output Differential Equation for State Affine Systems

La'

The I/O differential equation for

Pa := yen) = Fn(Al , . . " An-d+ +Fn-dAl "" , An- l ,
2. DEFINITIONS AND NOTATION

~

n1q1

u 1(k,)" '

.•• U

(k m )' m m

,

+ ... + npqp + klll + .. . + kml m = n -

j.

Fn_j is a function involving all monomials of "degree" (n - j) . For instance, Fn verifies

(1)

with x E M where M is an open and dense subset of JRn , U E JRm and y E JR. The entries of f( ·,·) and h( ·) are meromorphic functions of their arguments.

Fn :=y

il1(A1, ··· ,An-d fJi) (A1 ' .. . ,An-2) + 02d ·) f~~) (A 1, . . . , A n - 3) + 03d·)

(5)

f«~=;?l (A 1 , A 2 ) + o(n-2}l ( .) ~ogA1l(n-1 ) + 0 (n-l)10

Let us define the state affine system, considered here

L { ! ~ ~~y(t), u(t))z +
•

such that

Consider the nonlinear system:

"'" {xy == h(x) f(x , u)

La verifies

(2)

where Y := [y(n-1) . .. itl and OfJ 1(-) (2 :5 (3 :5 n1) , involves all the functions not depending on I/O time derivatives of degree (3. Whereas Fn-j verifies

a

where z(t) E JRn, y(t) E JR, u(t) E JR m . When one measures y(t), one can define {) .(y, u) as a new input and as recalled in (Hammouri and de Leon M., 91) if it is regularly persistent (Bornard , et al., 88), thus the system

'" . ._ rn-] .-
j -1) (n-j-2)


. . .
i1j(A j , .. . , A n- 1 ) fJ~) (Aj , ... , A n-2) + 02j(-) f~~)(Aj , .. ',A n-3) +03j(')

(3)

(6)

~ogAjl(n-j) + O(n - m( ' )

is a Kalman-like observer for L:a' Where z(t) E lR" , S(t) E JR+ is a symmetric positive definite matrix and 0 > O. The norm of the estimation error converges locally exponentially to the origin. From now on, L: is supposed to be generically observable (Glumineau, et al., 96) and will be called observable.

for 1 :5 j :5 n - 1 and Fa :=
Let the 3-D nonlinear system %1 = A1 (y, U)Z2 %2 = A2(Y, U)Z3 %3 =
3. PROBLEM STATEMENT. The goal is to find a state coordinates transformation z = ~(x), such that system L: (1) is locally equivalent to system L:a (2, in order to design the observer L:o (3). The approach consists in checking that the I/O differential equation associated to the system L: has the same form than the L:a one. The unicity of this equation for an observable system is shown in (Van der Schaft , 89) .

La'

+
(7)

Its I/O differential equation (4) verifies

Pa

y(3):= F3(Al,A 2) +F2(A 1 ,A2,
:=

where, as defined in (5) and (6), 740

F3:= Y . i(A) ; F2 := cp~2 ) - [r,bl r,ol] ' i(A) i(A)

:=

d Y+ ~ ar,or d 'Pr A r-l [acpr ~- Uj - - ' ay . aUj Ar

[log(Ai A :?) , log'A l -log'A l . IOgA l A2 (

Fl

:= c{;2 - r,0210g

Fa

:= t.p3

A2

OAr ( T dy

(9)

y

i=;Ar)] Ou ' dUj =Wr

(15)

+~ 3=1

3

for 1 ~ r ~ n - 1, Al . A2 ... A n - l . r,on := Pn for r = n .

where Y := [y(2) !i] .

• Theorem 3. : The nonlinear system 2: is locally equivalent to the system 2:a by state coordinates transformation z = ct>(x) if and only if :

3.2 State Affine Transformation Algorithm

This algorithm proceeds in two steps. First, one derives all the Ai functions from Fn in (4) . Then, one finds n - 1 first order partial differential equations and gives a NSC for the existence of a solution. Secondly, substituting them in (4) , one solves the following equation

y(n) _ Fn = Fn-dAl, ... , An-l,r,od +Al Fn-2(A2,"', An- l , r,02) + .. · +Al ··· An-2Fl(An-l,r,on-l) +Al . . . An-lFO(r,on) .

(16)

=0

(17)

dWr

where WIe (1 ~ k ~ n -1), and wr (1 ~ r defined by (ll) and (14) respectively.

(10)

~

2:(1)

• If dWIe /\ dy :I 0 or dWIe /\ du :I 0 then the problem has no solution. • If dWIe /\ dy = 0 and dWIe /\ du = 0

(18)

then let the Ai functions be any solution of

where le-I

Pie = r,ole

IT Ai +

le-I

PIc-l

i=l where P,,(Al,"' , An-l,r,ol, " 'r,on) is the formal I/O equation (4) computed for the system 2:".

for 1 ~ k

Step 2. Substitute all the Ai functions in (4) . For r 1 to n let

=

and let define Kr := Al . A2 ... Ar with An := 1, and the differential form wr as

~ ou;n-r) oP d f;r r

Uj

]

.

(19)

i=l

n , An := 0 and PI := r,ol.

(5),

(13)

Pr := Pr- l - Fn-r+l

~

+ ZIc IT Ai

Proof of Theorem 3 . Necessity: Suppose that Z = ct>(x) transforming system 2: into 2:a exists. Thus P := y(n) can be written P = Pa as in (4) where Fn verifies from

•

1 [ oPr Wr := Kr oy(n-r) dy +

n) are

Thus, if 2: is locally equivalent to 2:a , then the state coordinates transformation z = ct>( x) is computed from 2:,,:

in a similar way as in (Glumineau, et al. , 96) . S.A.T. Algorithm. Step 1. From I/O differential equation of set Po := y(n ). For k = 1 to n - 1 let :

dWIe /\ dy = 0, dWIe /\ du = 0

I;' rn --

~

Oill . .] y (n-l) [Oill - - y. + ~--u3 oy j=l OUj

+

(20)

~

. [OIOgAl (n-l) ologAl (n-l)] A 0 y +~ Ou. U j +~n-l y j=l J

(14)

y

where ~n-l involves all monomials which do not depend on y(n-l) or u(n-l). Then P" verifies

• If dWr ::j; 0 then the problem has no solution. • If dWr = 0, then r,or is a solution of 741

(n-l) [0/11 . + ~ 0111 .. ] + Pa ..- Y 0 Y L., OU' UJ

Y

. [OIOg A l (n-l) y 0 y

Y

j=1

partial equations, one derives all the Ai functions from (12) at the end of the Step 1. Then at the Step 2, one derives 'Pi from (15) since it depends only on the current 'Pi and on Ai previously computed. Then, from (18) one gets ZI = Y and for k = 2 to n,

(21)

J

~ologAl (n-l)] ~ OU ' Uj +

+ L., j=1

J

where ~ involves the monomials of ~n-l plus all monomials of degree ~ n - 1. Apply then the Step 1 of the S.A.T. Algorithm . For k = I, Po = P a and

~

2

0 Po d oiloy(n-l) y O/l1 d

0

2

Po

+ ;=1 L., oil. 'oy(n-l) ;

~ O/lld.

Y

y

which gives the n - 1 first dynamics of I:a (2). To compute the last dynamic in, note that for any k

d

k

Uj y(k)

+ L.,

Zk+l IT Ai

Y

where k-l

k-l

(27)

Pk := 'Pk IT Ai + Pk-l + Zk IT Ai

i=l

i=1

and An := 0, PI := 'PI . Thus, in follows: Zn

in (5). From (6),

Fn

+ Pk·

i=l

Thus, dWl 1\ dy = 0, dWl 1\ du = 0 are verified. Proof for 2 ~ k ~ n - 1 follows the same lines than for k = 1. Substitute the Ai functions in note that Fn-j verifies

=

010gA 1d (22)

OU' U; + 0 j=l; y ologAdy , u)d dl11 (y, U ) + oy Y o

(26)

Zn

=

i n - 1 - 'Pn-1

An (y(n) _

y(n-1) - P n - 1

IT:';ll Ai 1 Fn - 1) IT7:1 Ai 1

-

~-l A-]2 [IT.=1 • n 1 A. [ y (n-1) _ p.n-1 ]ITi-- I '

(28)

---'-:--

D

rn-j

O'Pj (n-;·) +

= -;;-y uy

Y

II

uy

(23)

>:-1 A·]2 [IT.=1 •

i=I'

OIOgAj (n-j) [

~ O'Pj (n-j) L., ou . ui - 'Pi"

~ ologAj

+ L.,

ou'

(n-j )]

ui

i=I'

e .

+

Substituting Pn -

(29)

where e n - j involves monomials not depending on the I/O derivatives of degree (n - j). Apply then the Step 2 of the algorithm r=1 PI := y(n) - Fn ._ O'Pl

.- - - y oy

oy

Thus via (18) one computes the state variables of the system (2) . System (1) is then locally equivalent to system (2) by a state coordinates transformation (18).

(24)

(n-l) + ~ O'Pl (n-l) L., - U · i=l ~Ui •

OlogAl (n-l) y [

from (27), one gets

1

n- ;

-

4. SYNTHESIS OBSERVER FOR STATE AFFINE SYSTEMS

'PI·

The observer solving

~ ologA l (n-l)] e + L., OU . U i + n-l i=l'

I:o

S = -OS -

of

I:a'

can be the realized

AT (t1)S - SA( t1)

+ CT C

(30)

and Kl := Al. Thus, Wl follows

~

l -Wl := K 1 [ 0 oP (n-l) dy + L., Y

=

t

~

oPl

(n-l) dUj

]

The time varying solution is a definite positive matrice, and 0 E !R+. In order to avoid the time varying computation, (Busawon, et al., 97) propose a transformation, ( = nz where

(25)

j=lOUj

n-1

[O'Pl dy + O'Pl dUi _ 'PI . Al oy i=l OU; Al

n:= diag(IT Ar), with Ao:= 1.

(31)

;=0

t

Applying it to system

OAl dy+ OAl dUi)] = d ['PI] ( oy i=l OUi Al

I:a' one gets

( = A( + r( t1, ()

Then ciW l = 0 is verified. The proof for steps 2 ~ r ~ n follows the same lines than for r = 1.

y

= C( = (1

where r(t1,() := n
Sufficiency: If the conditions of Theorem 3 are fulfilled and solving the set of n - 1 first order

742

(32)

A:= nA(t1)n- 1

car . Then. the system I/O differential equation is given by

010 ", 00) 001 ·· ·00

A=

: ". : ;n
nn-

1

n-l

y(:!)

IT Ar
= -a cos y (I m sin y . ii + u _

y)

E. sin

a cosy

(39)

. . F2 := y log AI; FI :=
(41)

Im(l- a cos:? y)

r=l

= Diag (Jog

and for

IT Ar-d , 1 :$ i :$ n .

La' one gets

r=l

where Ao := 1.

where

Assume that 1Ir(19, ()1I :$

KII(lIs

with

1\(lIs .-

(T 5(. Thus it is Lipchitzian with respect to ( ,

uniformly w.r .t . 19 with Lipschitz constant Then consider the system

(= A( + r(19 ,() - S-;'}eT (C( - y)

f{ .

Applying the S.A.T Algorithm. In the Step 1 we have Po := y(2 ) and k=1

(33)

where 5 00 is the algebraic stationary solution of

a2Po 02 Po du aiP dy+ aua ·

a sin y cos y { -1 + a cos 2 y

for 8 large enough . Then , for inputs {} uniformly bounded by some 19 2: 0 : (33) is an observer for (32) , i.e., for 8 large enough

~ dy + {O }du

J

=

The conditions dWI /\ dy 0, dWI verified, so Al is a solution of

a2 Pa

"((t) - ((t)" :$ K(8) exp(-8tj3) 11(0 - (011 (35)

a2 Pa

aiP dy + auay du alogA ld ay y

For the proof and the details cf. (Gauthier,et al., 92). Thus from (31) this observer in the z coordinates have the following form

=

y

FI '. - In rl

c~ X 1 (-al sin X 1 x22 + g cos:>:, sin :>:) - J!.u) m (38)

= Xl,

1(1-

U -

gm sin -;-

y)

(44)

cosy

Then d W1 = 0 is trivially verified and a solution is
X2

acos 2

y (

-a cos Im(l- acos 2 y)

I I ] WI := K1 [aP ay dy + oP ou du = {O}dy + {O}du

The SISO system is a inverse pendulum, as considered in (Besan~on and Bomard, 96). The dynamics of the system are

=

I ; (l-acos2 y)

= Al and wIfollows via (14)

Thus, K r=1

4.1 Physical Example

.

(43)

1 - a cos 2 y

F ._ Al . _ -asiny ,cosY ' 2 2 ·- A Y (1-acos 2 y) y , 1 PI := y(2) - F2

n-1S~lCT(C,i - y) (37)

The transformation n and its inverse is always well defined since the observability assumption.

X2

= 0 are

Step 2: Substituting Al in the function F2 from (41), we obtain PI

substituting (33) in (36) we finally have

=

du

= _ asmycosy dy

A solution is given by: Al (y) =

Xl

1\

= Wl

(36)

i = A(19),i +
(42)

r

-

b. '" -- 0. A, . r1

= 2 : one gets P2 := PI - FI = PI. And W2

xd -

oP ay

aP ou

2d y+2d u= W2 := -

where Xl denotes the angular displacement of the pendulum shaft from vertical position, I is its length, m the mass of the plumb bob, 9 the force of gravity and a the ratio m~M with M the mass of the bearing trolley, u is the force applied to the

d [aucos y - gm sin Im(1 - a cos 2 y)

y]

Thus, d W2 = 0 is verified, then AI

743

(45)

one avoids the time varying computation of the observer gain matrice an then a Kalman-like observer 1:0 (3) for the transformed system 1:" (2) can be directly applied.

The system (38) is then locally equivalent to: i Y

= (~

Y Alci ) ) z

= Cz = Zl

+

(92(~' U))

(46)

and from (18) the state coordinates transformation : Zl = h(x) = Xl ; Z2 = x2(1- acos 2 Xl)!'

6. REFERENCES

£ = A(y)z +
-

n-ls;;;;,lcT(Zl - Zl)

Besan~on,

G . Bornard, State equivalence based observer synthesis for nonlinear control systems. Proc. IFAC 13th Triennial World Congress, (San Francisco, USA 1996) Vol. E, 287-292. G. Bornard, N. Couenne, F. Celle, Regularly persistent observers for bilinear systems Proc. in 29 I. C.N.S., New trends in nonlinear system theory , (Vo1122 Springer Verlag) jurie 1988. K. Busawon, M. Farza H. Hammouri , A simple observer for a class of nonlinear systems, Personal Communication. J .P. Gauthier, H. Hammouri, S. Othman , A simple observer for nonlinear systems applications to bioreactors. IEEE TAC, 37(6,1992) , 885-880 . A. Glumineau, C.H. Moog, F . Plestan, New algebro-geometric conditions for the linearization by input-output injection, IEEE TAC 41(1996),598-603. H. Hammouri, K. Busawon, A global stabilization of a class of nonlinear systems by means of an observer, Appl. Math . Lett. (6 ; 1),(1993) 31-34. H. Hammouri, J .P. Gauthier, Bilinearization up to output injection, Syst. Contr. Lett., (11; 1988),139-149. H. Hammouri, J .P. Gauthier , Global time varying linearization up to output injection, SIAM J. Control Optim., (30; 1992), 1295-1310. H. Hammouri, M. Kinnaert , A new procedure for time-varying linearization up to output injection, Syst. Contr. Lett., (28; 1996) , 151157. H. Hammouri, J . de Leon M., On systems equivalence and observer synthesis. New Trends in Systems Theory (1991), 340-347. A.J . Krener, A. Isidori, Linearization by output injection and nonlinear observers, Syst. Contr. Lett., 3 (1983),47-52. A.J. Krener, W . Respondek, Nonlinear observers with linearizable error dynamics, SIAM J. Contr. Optim. 23 (1985), 197-216. A.J. Van der Schaft, Representing a nonlinear state space system as a set of higher-order differential equations in the inputs and outputs, Syst. Contr. Lett. , 12 (1989) 151-160. F . Viel, Stabilite des systemes controles par retour d'etat estime. Application aux reacteurs de polymerisation et aux colonnes a distiller, These de Doctorat, Universite Claude Bernard-Lyon 1, Lyon, 1994. X.H. Xia, W.B. Gao, Nonlinear observer design by observer error linearization, SIAM J. Contr. Optim. 1 (1989) 199-216.

G.

Thus the observer (37) has the following form (47)

where Soo is the algebraic stationary solution of (33) . Solving for Soo one gets Sl1 = 1/8, S2l = -1/8 2 , S22 = 2/83 . Then S;;;;,lCT = (28 82 )T and n-ls;;;;,lcT = (28 , 82 /Al(Y) )T. 4.1.1. Simulation Results We present the simulation results for a trajectory tracking Yr := Cl sin(c2t) where Cl = .3, C2 = 1, and l = 0.36m , M = 2.4kg" m = 0.23kg , 8 = 50 and zdO) = Z2(0) := 0, idOl = 0.2 , i2(0) = 3.

I

t ,

Figure 1 : Real and estimated state.

Figure 2 : Real and estimated state

In figure 1, we apply a tracking control with the real state and compare them with the estimated one. This error is practically neglectible. In figure 2, we measure only the output and apply the tracking control computed with the estimated states. The stability of the closed loop system using the estimated states, can be easily verified (Hammouri and Busawon, 93) . 5. CONCLUSIONS A NSC for the transformation to the state affine systems of MISO nonlinear systems has been obtained. The main result is stated in terms of the I/O differential equation and some one-forms obtained in a straightforward way and easy to check. With the help of a regular transformation, 744