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Observer design for nonlinear systems
Systems & Control Letters 13 (1989) 135-1,12 North-Holland
135
Observer design for nonlinear systems John TSINIAS
Department of Mathematics, National Technical University, Zografou Campus, 157 73 Athens, Greece Received 6 December 1988 Revised 5 February 1989
Abstract: This paper deals with the observer design problem of a wide class of nonlinear systems subjected to bounded nonlinearities. A sufficient Liapunovlike condition is provided and the proposed dynamic observer is a direct extension of the one in linear case.
Keywords: Nonlinear control system; observer design; Liapunov conditions.
1. Introduction The observer design of a general nonlinear system is a difficult p r o b l e m of central importance in control theory and has received m u c h attention by m a n y authors in the last few years (e.g. [2,3,5-12, 14,17,19,21]). F o r instance, in [3,8,10,11,19] the nonlinear plants are transformed into simpler forms where observer design is performed. In [2] a nonlinear observer is constructed such that the eigenvalues of the linearized error equation are placed at specific values that are locally invariant with respect to the operating point of the system. Walcott and Zak [17] present a discontinuous generalized d y n a m i c observer, where the exact knowledge of the system nonlinearities is not required and the estimate error approaches zero as t ~ + o~. T o avoid the undesirable "chattering' p h e n o m e n o n resulting from the discontinuity of the observer, they propose a continuous observer using techniques from the variable structure system theory, where the error converges to some preferably small neighborhood to zero. We also mention the contributions [6,9,12,19,21], where asymptotic observers for a wide class of systems are explored. In this work sufficient conditions and an extremely simple a p p r o a c h for the observer design are provided. The main sufficient condition we
propose (given in Section 2 as (A1)) is of L i a p u n o v type and it turns out to be also necessary in the linear case. It must pointed out that Liapunovlike conditions of the same kind have been also successfully used to determine stabilizing feedback laws for nonlinear systems [15,1]. O u r main result presents some advantages when c o m p a r e d to the previous mentioned contributions, because it is a direct extension of the observer design in the linear case [20], it is m u c h simpler to follow and the observer's dynamics are at least continuous mappings, whereas the error equation is globally asymptotically stable at the origin. Hence the trajectories e(t) of the error equation are defined for all positive t approaching zero as t ~ + o~ for any initial e(0) and the observer performs globally the state determination of the system. O n the other h a n d our a p p r o a c h requires the exact knowledge of the plant's nonlinearities. We shall consider nonlinear systems of the form .¢=f(x,
y = Hx,
u),
x~l
n, u ~ U c x R m,
y ~ R k,
(la) (lb)
where R" and R k are the state and the o u t p u t space of the system, H is a constant real matrix of dimension k × n and the input space ,~r consists of all measurable mappings of time with values on a subset U of R m. We assume that for any input u ~ U and initial state x o ~ R " the c o r r e s p o n d i n g trajectory x(t, Xo, u) of (1) is defined for all positive t. Finally the m a p p i n g R" × R m ~ (x, u) --~f(x, u) is continuously differentiable and we shall denote by D x f ( x , u) its derivative with respect to x. A d y n a m i c observer of (1) is a control system that requires the input u and the o u t p u t y of the original system (1) to p e r f o r m its state determination. We seek an observer in the following form: =f(z,
u)+R(u)y,
z ~ a n,
(2)
and we wish to select continuous m a p p i n g s f ( z , u) and R ( u ) so that the composite control system of
0167-6911/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
J. Tsinias/ Observerdesignfor nonlinearsystems
136
(1) a n d (2) o p e r a t i n g in R " × R" is a s y m p t o t i c a l l y stable u n i f o r m l y in u (in the sense of [16], Def. 8) with respect to the closed region
M={(~, z)~R'×R': x=z}. This r e q u i r e m e n t implies positive invariance of M, f r o m which it follows
f(x, u)=f(x, u)- R(u)Hx,
V(x, u)~R"x U. H e n c e (2) takes the form
= f ( z , u) + R ( u ) ( y -
Hz).
(3)
F u r t h e r m o r e the trajectories
e(t, e0; x(t, Xo, u), u) = x ( t , x o, u ) - - z ( t , Zo; h ( x ( t , x o, u)), u) of the error e q u a t i o n d=(f(x,
u)-f(z,
u))-R(u)ne,
Remark 1. Assumption (A1) implies
e'P{Dxf(O,O)}e< - k 111ell 2,
A s we shall establish in T h e o r e m 3, the a b o v e c o n d i t i o n is equivalent to the fact that the linearized p a i r ( Dxf(O, 0), H) is detectable. R e m a r k 2. A n o t h e r i m p o r t a n t r e m a r k is that conditions (A1) a n d (A2) are invariant u n d e r a n y linear c h a n g e of coordinates. I n d e e d , s u p p o s e that (A1) a n d (A2) are valid in the a b o v e c o o r d i n a t e system of x. U n d e r the linear h o m e o m o r p h i s m R n ~ x - - - , z = T x ~ R n, (1) takes the f o r m ~ = f(z, u) with o u t p u t y=I2Iz, where f ( z , u ) = Tf(T-lz, u) a n d I?I=HT -1. If we define / 3 = T '-1 PT -1, then /3 is positive definite a n d for each d ~ ker H - {0} it follows T - I ~ ker H (0}. H e n c e
w,/3(
e=x-z,
Ve~kerH.
(4)
u) } w
=w'T'
a p p r o a c h i n g zero as t ~ + oo, for every x o ~ R " a n d for each e 0 = x o - z o in a vicinity of 0 ~ R".
l?(Dxf(x , u)}r-lw
_< -k~llwll 2, for any (z, u ) E R ' X with
U a n d w locally a r o u n d $,
2. Main results
k~=llT' lllllT-allk~>0. O u r m a i n p u r p o s e is to show that state det e r m i n a t i o n of (1) is possible if the following two c o n d i t i o n s are fulfilled:
Similarly
,w ( A I ) There exists a positive definite (symmetric) m a t r i x P of d i m e n s i o n n × n a n d a positive constant k~ such that for a n y e~kern-
{0}
t^
^
u)}wl<_
V(x, w, u ) ~ R " x R " x
(u)llwll 2 U,
with
p ( u ) = lIT'-1 II IIT-] lip(u)-
there is a n e i g h b o r h o o d S e of e such that
v'P{ Dxf(x, u)}v < - k I IIvll =, V(x, v, u ) ~ R " x s e x U, where ' stands for t r a n s p o s e a n d II II is the usual E u c l i d e a n norm.
R e m a r k 3. Let us assume that (1) is affine in control:
f ( x , u) = F ( x ) + G ( x ) u , a n d there are positive c o n s t a n t s M 1 a n d M 2 so that
(A2) T h e r e exist a c o n t i n u o u s real function p : R " ---, R a n d a positive c o n s t a n t k z such that p(u) > k2, V u E U, and
II P ( D F ( x ) )
u)).l _
IIP ( D G ( x ) )
Iv'P{Dxf(x,
V(x, v, u ) ~ R " × R " × U.
+ (DF(x))'P
II -< M1,
VX EE R',
(5) + (DG(x))'PII < M2,
Vx ~ R'.
(6)
J. Tsinias / Observer design for nonlinear systems
for every (q, u) ~ R" x U and e ~ M. Let us consider the m a p R ( u ) = c p ( u ) P - 1 H ' and the L i a p u n o v function V( e ) = ~e 1 ,Pe. The resulting error equation is written
Then
!.'P{Dj(x, u)}vl <(3//1+)142)(1+ Ilull) Ilvll % V ( x , v, u ) e R " x R " x
137
U;
e = ( f ( x , u) - f ( x
-- e, u))
- cp(u)p-1H'He
(10)
hence (A2) is fulfilled with
p(u) = ( M 1 +
M2)(1 + Hull).
and the time derivative 1) of V for (10) is
It is interesting to note that we can select p constant:
15"(e) = e ' P ( f ( x ,
u) - f ( x
- e, u))
- c p ( u ) [l He[[ 2 p - - sup (M1 + M 2 ) ( 1 +
Ilull),
u~U
provided that U is bounded. Similarly (A2) is satisfied with p independend on u, provided that (5) holds and G is independent of x. If for instance (1) is linear, f ( x , u ) = A x + Bu, it follows Dxf(X, u) = A and therefore (A2) is valid with no any additional requirement. The next theorem is the main result. T h e o r e m 1. Suppose that (1) satisfies (A1) and (A2), then: (i) For any sufficiently large positive constant c, the error equation (4) with
R(u) =cp(u)P-aH"
e ( t , e0; x ( t , Xo, u), u ) + O
ast+
+o¢
for any initial conditions eo, x o and for any u ~ . Furthermore the convergence will be exponential. (ii) For any pair x~ and x b of indistinguishable states by some input ~ ~ ~ , namely
, u)}e
- c p ( u ) [I He II 2 + kl IIe II 2 is negative Ve ~ R ~ and (q, u) ~ R ~ x U provided that c is a sufficiently large positive constant. Let us first consider e ~ • ~ - {0} such that there exist a positive constant 1 and v belonging to aS A M with e ---/v. Then according to (9) we have
F ( e , q, u ) = 1 2 ( v ' P { D x f ( q ,
u)}v
- c p ( u ) II Hv [I 2 + kl II v II 2) <_12(v'P{Dxf(q, u ) } v + k ~ Ilvll 2) _<0 for every (q, u ) ~ R n × U. The other case is e R " - (0} with e = / v for some positive 1 and v a S - ( a S n M). Then taking into a c c o u n t (A2) it follows
F ( e , q, u ) = 12F(v, q, u) <_: p ( u)( L , - cL2) < 0
Vt>_O, for each (q, u ) ~ R ' × L1 L21, where
it holds x ( t , x~, ~ ) - x ( t ,
F ( e , q, u ) = e ' P { D x f ( q
(7)
where p and P are given in (A1) and (A2) performs the state determination of (1). In particular
y ( t , x~, F~)=y(t, x b, Ft),
We shall establish that there exists a positive constant c such that l ) ( e ) < - k l l l e l l 2 for any e ~ R ~. It suffices to show that the expression
xb, t t ) ~ O
ast--+ +oo.
(8) L~ =
(i) We consider an open sphere S of radius r > 0, centered at zero. Let Proof.
sup
v'P{DJ(q,
provided
u)}v
p(u)
that
c>
klllvll z
+ -
p(u)
(q,u)~R~XU
_< r2(1 + k;
C --- aS (~ ker H,
U
kl) < +
aS being the b o u n d a r y of S. T h e n C is c o m p a c t and by use of (A1) there exists an o p e n set M such that C c aS n M and
and
e'P{ D ~ f ( q , u ) } e < - k I [lell z
We conclude l)'(e)_< - k I 11e 1[2 for any e, x and
(9)
Lz=
min
v~as-(aSNM)
II Hv II z > 0.
138
J. Tsinias / Observer destgn for nonlinear systems
u. Since V is quadratic and positive definite the previous L i a p u n o v inequality implies that (10) is globally asymptotically stable at zero uniformly in u and x (see for instance [16]), whereas
for every Zo~ and Zo2. Setting z o = Zol = 2 0 2 and taking into account (11) and (12) we obtain
IIx(t, xo, ~ ) - x ( t ,
xb, ~)il
_
e ( t ) = e ( t , e0; x ( t , Xo, u), u) is defined for all t > 0. The last assertion follows from the stability of (10) and the continuity of the mapping
x a, u)ll
+llz(t, z0; y, ---~0
)il
as/----~ + ~ ,
and the p r o o f is completed.
[]
clef
E ( e , x, u ) = f ( x ,
u)-f(x-e,
u)
It should be noticed that for linear systems
Yc=Ax+Bu,
-cp(u)P-lH'He with respect to (e, x, u). In particular for any W c R" c o m p a c t and for each x 0 and u E o~,
II g ( e ,
x ( t , Xo, u), u)II < q~(x(t, x o, u), u),
V e ~ W, t > O ,
y = Hx,
(x, u)~R'×R
q~(x, u ) = m a x (
(13b)
the property (ii) of T h e o r e m 1 is equivalent to detectability of the pair (A, H ) [20]. Indeed, consider x a and x b ~ R" that are indistinguishable:
A
Then
l i E ( e , x, u)ll, e ~ W } .
Then q, is continuous with respect to (x, u) ~ R" × R " , and hence ep(x(t, Xo, u), u), x o E R ' , u o~-, is measurable on t. Furthermore,
(e, t)~E(e,
(13a)
y ~ R k,
x . - - X b ~ k e r [ H ' , A H, , . . . . .
where
'~,
x ( t , x o, u), u),
X o E R n, U ~ o ~ ,
x ( t , x a, u ) -
x ( t , x b, u) = H e m ( x a -*0,
xh)
t---~ + ~
for all u, if and only if the pair (A, H ) is detectable. Furthermore we have: Theorem 2. The following statements are equiv-
is a Carath6odory map, and thus (similar to [4,15]) one can show that e ( t ) is defined for all t > 0 approaching zero as t --, + ~ . Moreover a positive constant L can be found with
alent: (i) The pair (A, H ) is detectable. (ii) The system (13) satisfies (A1). In particular there exists a positive definite matrix P and a positive constant k 1 such that
II e ( t ) tl < L 11e0 Ilexp( - k , I[P II it)
e ' P A e < - k l llell2,
Ve~kerH.
(14)
for every e0, x 0, u ~.~- and t > 0. (ii) Consider a pair x~ ~ x b of indistinguishable states by some input ~ ~ ~ . It follows
(iii) There is a positive constant c such that the linear system
y ( t ) d e = f h ( x ( t , X~, ~t))
2 = Az + cP-1n'(y
- H z ) + Bu
(15)
is an observer for (13). = h(x(t,
x b, ?t)),
Vt > O.
(11)
O n the other hand, as t --* + ~ ,
IIz(t, z0 ;
h(x(t,
x~, Ft)), Ft) -x(t,
x~, ?4)I! ~ 0,
(12a)
e ' P ( A - R H ) e < - k 1 II e II 2,
[]z(t, z02; h(x(t, xb, ~)), ~)
-x(t,
Proof. (i) ~ (ii). Since the pair (A, H ) is detectable there exists a real matrix R of dimension n × k, a positive definite matrix P and a positive constant k I so that
0,
(12b)
Ve ~ R"
(see for instance [18] and [20]), which obviously implies (14).
J. Tsinias / Observerdesignfor nonlinearsystems
(ii) =, (iii). The implication is an immediate consequence of T h e o r e m 1 and R e m a r k 3. (iii) = (i). Since (15) is a linear observer for (13) it is well k n o w n from linear control theory that (A, H ) will be detectable. []
139
c we c o m p u t e L 1 and L2 as in the p r o o f of T h e o r e m 1. Let S be the sphere of radius r = 1, centered at zero and M=
(v=(va,
v2)'~R2:
-2VlV2 + 101021 < - k l II v II 2 }. We illustrate the nature of our results by two interesting examples.
Obviously ker H -
{0) c M and
OS-(OSAM) Example 1. We consider the system
= {v~R2:
2+3x+sinx=u,
x~R,
with output y = x control system
2, or equivalently the planar
x2
=
x2)'~R
y=(1
-1) (X xzl ) .
+ u
2,
1 '
u~R,
(16a) (16b) (16c)
Let us set
f ( x , u) = ( x 2, - 3 x 1 - sin x 1 -Jr- U ) ' , H=(1
-1)
and
Ilvll--1, -2v~v2+ Ivlv21 >_ - k ~ ) .
Then
-3x 1-sinxl
x=(x,,
u~U=R,
P=
0
1 "
L2=
min
oEOS--(aSNM)
(vl
- - V 2 ) 2 ~--- 1
Zl=(1+3k;1) and consequently 3k~-a)/(1 - 2ka).
we
can
:~2
=
+U
--sin x~
x=(x,,x2)'~R
2,
cos x~
IlP( D:,.f(x, u)} Jl = IIP( D:,f( x, 0)} II
satisfies (A1) and (A2) with
V(x, u)~R2XR
e'P(D~f(x, u)}ele,=e2
c > (1 +
P=(0
,
u~U=[-1,1],
y = x 1 +X
and for each 0 4= e ~ ker H we have e~ = e 2, so that
choose
Example 2. Similarly it is shown that the p e n d u lum system [4]
We can easily check that conditions (A1) and (A2) are fulfilled. In particular
<3,
- 2kl,
2,
3 0) 1
and p constant; thus we can apply the observer design m e t h o d of T h e o r e m 1.
= - - 2 e 2 - e l 2 cos x 1_< --el2 3. Bilinear systems
=-ka(e2+e2)le,=e2 where 0 < k 1 < ½. Since Dxf(x, u) is independent of u, there is a n e i g h b o r h o o d S e of e such that
In this section we shall specialize T h e o r e m 1 to bilinear control systems. Let us consider the single input system
v'P{ Dxf(X, u)} v < - k , I[ v II 2,
~c=Ax+u(b+Bz),
V ( X , V, U ) ~ a n X S e X ~ .
y = Hx,
According to T h e o r e m 1 the system (3) with
f ( z , u) = ( z 2, - 3 z 1 - sin z~ + u ) ' ,
and assume that there exist a positive definite matrix P and a positive constant k a such that for every e ~ ker H it follows
H = ( 1 , - 1 ) , p = 3 and R given b y (7) is an observer for (16). In order to estimate the constant
e'PAe < - k 1 II e II 2
y ~ R k,
x ~ R n, u ~ U ,
(17a) (17b)
(18)
140
J. Tsinias / Observer design f o r nonlinear systems
and
e'PBe = 0.
(19)
Obviously then (17) satisfies (A1) provided that U is bounded. Indeed, if we set
f(x, u)=Ax +u(b+ Bx) we have
e'P{Dxf(x,
= e'PAe + ½ue'(PB + B ' P ) e =e'PAe<_ - k 111ell 2, V e e K e r H , and using the fact that U is bounded,
<
-
1
~kl
lul Iv'Pnvl
II v II
2
PB i + B ' P = O ,
V i = l . . . . . m,
the system (3) with f and R as before performs the state determination of (20) for every u. An interesting problem is to find sufficient conditions for the observation of bilinear systems (20), where each component u, of the input u is a positive measurable function of time. Bilinear systems with positive controllers are frequently used to describe a lot of chemical, physical and biological mechanisms (see for instance [13,19]). The proof of the next proposition follows similar arguments as in the proof of Proposition 3, and thus it has been omitted.
u)}e
v'P{ D x f ( x , u ) } v < v ' P Z v +
and R = cP i l l ' is an observer, provided that U is a bounded subset of R m. I f in addition
,
f o r any v i n a v i c i n i t y o f e e k e r H - { 0 } . N o t i c e
finally that (5) and (6) of Remark 3 are fulfilled and since U is bounded, (A2) is valid with p independent of u. Hence according to Theorem 1 we can find c > 0 such that (3) with
Proposition 4. Given the system
(20) with
U,o a, and under the assumption
kerH-{0}c{e: e'PAe+ kl llell2_
f ( z , u) = A z + u(b + Bz)
(q(
~)
(e:e'PBie
(21)
and R = c P - 1 H ' is an observer of (17). For U being unbounded the same result holds if instead of (19) we assume PB + B'P = 0. The previous analysis can be trivially extended to the multi-input case. Thus we have:
where P is a positive definite matrix and k I > 0, there exists a positive constant c such that the observer governed by (3) with
Proposition 3. Consider the bilinear system
f(z, u)=Az+
l <_i<<.rn
m
Y'~ ( b i + B i 2 ) u i i=1
=Ax + ~. (hi+ Bix)ui,
(20a)
and
i=1
IlulI)P-'H"
xeR",
u=(u, ..... um)'eU,
(20b)
R(u) =c(1
y = Hx,
y ~ ~*,
(20c)
performs the state determination of (20).
and suppose that there exist a positive definite matrix P and a positive constant k I with kerHc
{e: e ' P A e + k I
¢q(
(']
Ilel12_<0}
ker(PBi+B~P)).
l
Then there is a positive constant c such that the system (3) with
+
Remark 4. Similar to Remark 2 it can be shown that (21) is invariant under any linear change of coordinates. The following result provides necessary and sufficient conditions for observation of planar hilinear systems with positive controls by employing the result of Proposition 4.
m
f(z, u)=Az+
E i=1
(bi+Biz)ui
Proposition 5. Consider a single input, single-output planar bilinear system with positive controls, and
J. Tsinias / Observer design for nonlinear systems
suppose that there exists a change of coordinates such that the system takes the f o r m (17) with
A =
01
H=(0
alE a22 1),
'
B = ~ b2 ~ x=(xl,
bE2
x2)'~B
E andy~a.
Then the system satisfies (21), if and only if one of the following two conditions holds: (i) b H > 0 , b 2 1 > 0 , (ii) b H < 0. P r o o f . Let b l l a n d b21 satisfy o n e of the a b o v e c o n d i t i o n s . Both cases c a n b e simultaneously, h a n d l e d as follows. W e c o n s i d e r r 1 > 0,
r=
( - rtbtx - e)/b21 -1
if b21 4= 0, ifb21=0,
r
is positive definite a n d for a n y 0 ¢ x E k e r H we obtain
x ' P B x = (rib11 + rb21)x ~ f -ex 2 < 0
if b21 4= 0,
[ rlbllx 2 < 0
if b21 = 0,
(22)
and
x ' P A x = rx~ = r II x II 21 x ~ k e r . < 0.
R e m a r k 5. I n case (i) of P r o p o s i t i o n 5 the system is observable (in the sense of [19]) for all u > 0, while in case (ii) it is o b s e r v a b l e for all u > 0 except for the c o n s t a n t i n p u t ~ = - b ~ 1 p r o v i d e d that b21 < 0. F o r u = ~ the c o r r e s p o n d i n g u n o b s e r v a b l e linear system is fibll 0
a12 + ub12 a22 +-UbE2 | x +
Y = x2"
Several o t h e r different a p p r o a c h e s have b e e n d e v e l o p e d d e a l i n g with the o b s e r v e r design p r o b lem of b i l i n e a r systems. F o r instance, a c c o r d i n g to [19], if the p l a n a r b i l i n e a r system of P r o p o s i t i o n 5 is c o m p l e t e l y o b s e r v a b l e o r e q u i v a l e n t l y bE1 = 0 ( L e m m a 1 in [19]) a n o n l i n e a r o b s e r v e r can be constructed. O n the o t h e r h a n d the general a p p r o a c h o f [19] p r e s e n t s a d i s a d v a n t a g e in p r a c t i c a l a p p l i c a t i o n s since the o b s e r v e r p a r a m e t e r s d e p e n d o n the time derivatives of the i n p u t u. F o r p l a n a r systems the o b s e r v e r design requires the exact k n o w l e d g e of (u, fi).
Sufficient L i a p u n o v l i k e c o n d i t i o n s for the o b server design have been p r o p o s e d . T h e a p p r o a c h we d e v e l o p e d is a p p l i c a b l e to a r e a s o n a b l y wide class on n o n l i n e a r systems i n c l u d i n g b i l i n e a r control systems. T h e result of T h e o r e m 1 ( s t a t e m e n t (i)) can be easily e x t e n d e d to n o n l i n e a r systems Yc = f ( x , u) with n o n l i n e a r o u t p u t y = h ( x ) , where h = R " ~ • k is a c o n t i n u o u s l y d i f f e r e n t i a b l e m a p ping. I n p a r t i c u l a r if (A1) a n d (A2) are satisfied with H = Dh(O), the system (3) with R given b y (7) p e r f o r m s the state d e t e r m i n a t i o n for x a n d e 0 b e l o n g i n g to some p r e f e r a b l y small n e i g h b o r h o o d of zero. Hence, in the general case of n o n l i n e a r o u t p u t , the c o r r e s p o n d i n g result will p r e s e n t a local nature.
(23)
F r o m (22), (23) a n d R e m a r k 4 it follows that the s y s t e m satisfies (21). Conversely, if we a s s u m e that (21) is valid, it is n o t difficult to see that rib11 + rb21 < 0 for s o m e real c o n s t a n t s r~ > 0 a n d r < 0; h e n c e either (i) o r (ii) is satisfied. []
f¢ =
Since Ubll < 0 we c a n easily verify that (8) h o l d s for a n y i n t i s t i n g u i s h a b l e states x a a n d x b.
4. Concluding remarks
a n d r 2 > rE//rl. T h e n for each case a n d for bE1 :/: 0 a positive e can b e f o u n d such that r > 0. T h e p r e v i o u s choice o f rl, r E a n d r implies that the matrix
,=(r;
141
References [1] A. Andreini, A Bacciotti and G. Stefani, Global stabilizability of homogeneous vectors fields of odd degree, Systems Control Left. 10 (1988) 251-256. [2] W. Baumann and W. Rugh, Feedback control of nonlinear systems by extended linearization, IEEE Trans. Automat. Control 31 (1986) 40-47. [3] D. Bestle and M. Zeitz, Canonical form observer design for non-linear time variable systems, Internat. J. Control, 38 (1983) 419-431. [4] M.J. Corless and G. Leitmann, Continuous state feedback guaranteering uniform ultimate boundedness for uncertain dynamic systems, IEEE Trans. Automat. Control 26 (1981) 1139-1144. [5] I.A. Derese, Bilinear Observers for bilinear systems, IEEE Trans. Automat. Control 26 (1981) 590-592.
142
J. Tsinias / Observer design for nonlinear systems
[6] J.P. Gauthier and D. Kazakos, Observability and observers for non-linear systems, Proc. 25th Conf. Decision Contr., Athens (1986). [7] O.M. Grassall and A. Isidori, An existence theorem for observers of bilinear systems, I E E E Trans. Automat. Control 26 (1981) 1299-1300. [8] H. Hammouri and J.O. Gauthier, Bilinearization up to output injection, Systems Control Lett. 11 (1988) 139-149. [9] S.R. Kou, D.L. Elliot and T.J. Tam, Exponential observers for nonlinear dynamic systems, Internat. J. Control 29 (1975) 204-216. [10] A.J. Krener and W. Respondek, Nonlinear observers and linearizable error dynamics, S l A M J. Control Optim. 23 (1985) 197-216. [11] J. Levine and R. Marino, Nonlinear system immersion, observers and finite-dimensional filters, Systems Control Lett. 7 (1986) 133-142. [12] S. Nicosia, P. Tomei and A. Tornamb~, Approximate asymptotic observers for a class of nonlinear systems, Proc. of the 26th CDC, Los Angeles (Dec. 1987) 157-162. [13] J.F. Selgrade, Asymptotic behavior of solutions to single
[14] [15] [16]
[17]
[18] [19]
[20] [21]
loop positive feedback systems, J. Differential Equations 38 (1980) 88-103. A.J. van der Schaft, On nonlinear observers, I E E E Trans. Automat. Control 30 (1986) 1254-1256. J. Tsinias, Stabilization of affine in control nonlinear systems, Nonlinear Anal 12 (1988) 1283-1296. J. Tsinias and N. Kalouptsidis, Prolongations and stability analysis via Liapunov functions of dynamical polysystems, Math. Systems Theory 20 (1987) 215-233. B.L. Walcott and S.H. Zak, State observation of nonlinear uncertain dynamical systems, I E E E Trans. Automat. Control 32 (1987) 166-170. J.C. Willems, Stability Theory of Dynamical Systems (Wiley, New York, 1970). D. Williamson, Observation of bilinear systems with application to biological control, Automatica 13 (1977) 243-254. W.M. Wonham, Linear Multivariable Control (Springer, New York, 1979). X. Xia and W. Gao, On exponential observers for nonlinear systems, Systems Control Lett. 11 (1988) 319-325.
135
Observer design for nonlinear systems John TSINIAS
Department of Mathematics, National Technical University, Zografou Campus, 157 73 Athens, Greece Received 6 December 1988 Revised 5 February 1989
Abstract: This paper deals with the observer design problem of a wide class of nonlinear systems subjected to bounded nonlinearities. A sufficient Liapunovlike condition is provided and the proposed dynamic observer is a direct extension of the one in linear case.
Keywords: Nonlinear control system; observer design; Liapunov conditions.
1. Introduction The observer design of a general nonlinear system is a difficult p r o b l e m of central importance in control theory and has received m u c h attention by m a n y authors in the last few years (e.g. [2,3,5-12, 14,17,19,21]). F o r instance, in [3,8,10,11,19] the nonlinear plants are transformed into simpler forms where observer design is performed. In [2] a nonlinear observer is constructed such that the eigenvalues of the linearized error equation are placed at specific values that are locally invariant with respect to the operating point of the system. Walcott and Zak [17] present a discontinuous generalized d y n a m i c observer, where the exact knowledge of the system nonlinearities is not required and the estimate error approaches zero as t ~ + o~. T o avoid the undesirable "chattering' p h e n o m e n o n resulting from the discontinuity of the observer, they propose a continuous observer using techniques from the variable structure system theory, where the error converges to some preferably small neighborhood to zero. We also mention the contributions [6,9,12,19,21], where asymptotic observers for a wide class of systems are explored. In this work sufficient conditions and an extremely simple a p p r o a c h for the observer design are provided. The main sufficient condition we
propose (given in Section 2 as (A1)) is of L i a p u n o v type and it turns out to be also necessary in the linear case. It must pointed out that Liapunovlike conditions of the same kind have been also successfully used to determine stabilizing feedback laws for nonlinear systems [15,1]. O u r main result presents some advantages when c o m p a r e d to the previous mentioned contributions, because it is a direct extension of the observer design in the linear case [20], it is m u c h simpler to follow and the observer's dynamics are at least continuous mappings, whereas the error equation is globally asymptotically stable at the origin. Hence the trajectories e(t) of the error equation are defined for all positive t approaching zero as t ~ + o~ for any initial e(0) and the observer performs globally the state determination of the system. O n the other h a n d our a p p r o a c h requires the exact knowledge of the plant's nonlinearities. We shall consider nonlinear systems of the form .¢=f(x,
y = Hx,
u),
x~l
n, u ~ U c x R m,
y ~ R k,
(la) (lb)
where R" and R k are the state and the o u t p u t space of the system, H is a constant real matrix of dimension k × n and the input space ,~r consists of all measurable mappings of time with values on a subset U of R m. We assume that for any input u ~ U and initial state x o ~ R " the c o r r e s p o n d i n g trajectory x(t, Xo, u) of (1) is defined for all positive t. Finally the m a p p i n g R" × R m ~ (x, u) --~f(x, u) is continuously differentiable and we shall denote by D x f ( x , u) its derivative with respect to x. A d y n a m i c observer of (1) is a control system that requires the input u and the o u t p u t y of the original system (1) to p e r f o r m its state determination. We seek an observer in the following form: =f(z,
u)+R(u)y,
z ~ a n,
(2)
and we wish to select continuous m a p p i n g s f ( z , u) and R ( u ) so that the composite control system of
0167-6911/89/$3.50 © 1989, Elsevier Science Publishers B.V. (North-Holland)
J. Tsinias/ Observerdesignfor nonlinearsystems
136
(1) a n d (2) o p e r a t i n g in R " × R" is a s y m p t o t i c a l l y stable u n i f o r m l y in u (in the sense of [16], Def. 8) with respect to the closed region
M={(~, z)~R'×R': x=z}. This r e q u i r e m e n t implies positive invariance of M, f r o m which it follows
f(x, u)=f(x, u)- R(u)Hx,
V(x, u)~R"x U. H e n c e (2) takes the form
= f ( z , u) + R ( u ) ( y -
Hz).
(3)
F u r t h e r m o r e the trajectories
e(t, e0; x(t, Xo, u), u) = x ( t , x o, u ) - - z ( t , Zo; h ( x ( t , x o, u)), u) of the error e q u a t i o n d=(f(x,
u)-f(z,
u))-R(u)ne,
Remark 1. Assumption (A1) implies
e'P{Dxf(O,O)}e< - k 111ell 2,
A s we shall establish in T h e o r e m 3, the a b o v e c o n d i t i o n is equivalent to the fact that the linearized p a i r ( Dxf(O, 0), H) is detectable. R e m a r k 2. A n o t h e r i m p o r t a n t r e m a r k is that conditions (A1) a n d (A2) are invariant u n d e r a n y linear c h a n g e of coordinates. I n d e e d , s u p p o s e that (A1) a n d (A2) are valid in the a b o v e c o o r d i n a t e system of x. U n d e r the linear h o m e o m o r p h i s m R n ~ x - - - , z = T x ~ R n, (1) takes the f o r m ~ = f(z, u) with o u t p u t y=I2Iz, where f ( z , u ) = Tf(T-lz, u) a n d I?I=HT -1. If we define / 3 = T '-1 PT -1, then /3 is positive definite a n d for each d ~ ker H - {0} it follows T - I ~ ker H (0}. H e n c e
w,/3(
e=x-z,
Ve~kerH.
(4)
u) } w
=w'T'
a p p r o a c h i n g zero as t ~ + oo, for every x o ~ R " a n d for each e 0 = x o - z o in a vicinity of 0 ~ R".
l?(Dxf(x , u)}r-lw
_< -k~llwll 2, for any (z, u ) E R ' X with
U a n d w locally a r o u n d $,
2. Main results
k~=llT' lllllT-allk~>0. O u r m a i n p u r p o s e is to show that state det e r m i n a t i o n of (1) is possible if the following two c o n d i t i o n s are fulfilled:
Similarly
,w ( A I ) There exists a positive definite (symmetric) m a t r i x P of d i m e n s i o n n × n a n d a positive constant k~ such that for a n y e~kern-
{0}
t^
^
u)}wl<_
V(x, w, u ) ~ R " x R " x
(u)llwll 2 U,
with
p ( u ) = lIT'-1 II IIT-] lip(u)-
there is a n e i g h b o r h o o d S e of e such that
v'P{ Dxf(x, u)}v < - k I IIvll =, V(x, v, u ) ~ R " x s e x U, where ' stands for t r a n s p o s e a n d II II is the usual E u c l i d e a n norm.
R e m a r k 3. Let us assume that (1) is affine in control:
f ( x , u) = F ( x ) + G ( x ) u , a n d there are positive c o n s t a n t s M 1 a n d M 2 so that
(A2) T h e r e exist a c o n t i n u o u s real function p : R " ---, R a n d a positive c o n s t a n t k z such that p(u) > k2, V u E U, and
II P ( D F ( x ) )
u)).l _
IIP ( D G ( x ) )
Iv'P{Dxf(x,
V(x, v, u ) ~ R " × R " × U.
+ (DF(x))'P
II -< M1,
VX EE R',
(5) + (DG(x))'PII < M2,
Vx ~ R'.
(6)
J. Tsinias / Observer design for nonlinear systems
for every (q, u) ~ R" x U and e ~ M. Let us consider the m a p R ( u ) = c p ( u ) P - 1 H ' and the L i a p u n o v function V( e ) = ~e 1 ,Pe. The resulting error equation is written
Then
!.'P{Dj(x, u)}vl <(3//1+)142)(1+ Ilull) Ilvll % V ( x , v, u ) e R " x R " x
137
U;
e = ( f ( x , u) - f ( x
-- e, u))
- cp(u)p-1H'He
(10)
hence (A2) is fulfilled with
p(u) = ( M 1 +
M2)(1 + Hull).
and the time derivative 1) of V for (10) is
It is interesting to note that we can select p constant:
15"(e) = e ' P ( f ( x ,
u) - f ( x
- e, u))
- c p ( u ) [l He[[ 2 p - - sup (M1 + M 2 ) ( 1 +
Ilull),
u~U
provided that U is bounded. Similarly (A2) is satisfied with p independend on u, provided that (5) holds and G is independent of x. If for instance (1) is linear, f ( x , u ) = A x + Bu, it follows Dxf(X, u) = A and therefore (A2) is valid with no any additional requirement. The next theorem is the main result. T h e o r e m 1. Suppose that (1) satisfies (A1) and (A2), then: (i) For any sufficiently large positive constant c, the error equation (4) with
R(u) =cp(u)P-aH"
e ( t , e0; x ( t , Xo, u), u ) + O
ast+
+o¢
for any initial conditions eo, x o and for any u ~ . Furthermore the convergence will be exponential. (ii) For any pair x~ and x b of indistinguishable states by some input ~ ~ ~ , namely
, u)}e
- c p ( u ) [I He II 2 + kl IIe II 2 is negative Ve ~ R ~ and (q, u) ~ R ~ x U provided that c is a sufficiently large positive constant. Let us first consider e ~ • ~ - {0} such that there exist a positive constant 1 and v belonging to aS A M with e ---/v. Then according to (9) we have
F ( e , q, u ) = 1 2 ( v ' P { D x f ( q ,
u)}v
- c p ( u ) II Hv [I 2 + kl II v II 2) <_12(v'P{Dxf(q, u ) } v + k ~ Ilvll 2) _<0 for every (q, u ) ~ R n × U. The other case is e R " - (0} with e = / v for some positive 1 and v a S - ( a S n M). Then taking into a c c o u n t (A2) it follows
F ( e , q, u ) = 12F(v, q, u) <_: p ( u)( L , - cL2) < 0
Vt>_O, for each (q, u ) ~ R ' × L1 L21, where
it holds x ( t , x~, ~ ) - x ( t ,
F ( e , q, u ) = e ' P { D x f ( q
(7)
where p and P are given in (A1) and (A2) performs the state determination of (1). In particular
y ( t , x~, F~)=y(t, x b, Ft),
We shall establish that there exists a positive constant c such that l ) ( e ) < - k l l l e l l 2 for any e ~ R ~. It suffices to show that the expression
xb, t t ) ~ O
ast--+ +oo.
(8) L~ =
(i) We consider an open sphere S of radius r > 0, centered at zero. Let Proof.
sup
v'P{DJ(q,
provided
u)}v
p(u)
that
c>
klllvll z
+ -
p(u)
(q,u)~R~XU
_< r2(1 + k;
C --- aS (~ ker H,
U
kl) < +
aS being the b o u n d a r y of S. T h e n C is c o m p a c t and by use of (A1) there exists an o p e n set M such that C c aS n M and
and
e'P{ D ~ f ( q , u ) } e < - k I [lell z
We conclude l)'(e)_< - k I 11e 1[2 for any e, x and
(9)
Lz=
min
v~as-(aSNM)
II Hv II z > 0.
138
J. Tsinias / Observer destgn for nonlinear systems
u. Since V is quadratic and positive definite the previous L i a p u n o v inequality implies that (10) is globally asymptotically stable at zero uniformly in u and x (see for instance [16]), whereas
for every Zo~ and Zo2. Setting z o = Zol = 2 0 2 and taking into account (11) and (12) we obtain
IIx(t, xo, ~ ) - x ( t ,
xb, ~)il
_
e ( t ) = e ( t , e0; x ( t , Xo, u), u) is defined for all t > 0. The last assertion follows from the stability of (10) and the continuity of the mapping
x a, u)ll
+llz(t, z0; y, ---~0
)il
as/----~ + ~ ,
and the p r o o f is completed.
[]
clef
E ( e , x, u ) = f ( x ,
u)-f(x-e,
u)
It should be noticed that for linear systems
Yc=Ax+Bu,
-cp(u)P-lH'He with respect to (e, x, u). In particular for any W c R" c o m p a c t and for each x 0 and u E o~,
II g ( e ,
x ( t , Xo, u), u)II < q~(x(t, x o, u), u),
V e ~ W, t > O ,
y = Hx,
(x, u)~R'×R
q~(x, u ) = m a x (
(13b)
the property (ii) of T h e o r e m 1 is equivalent to detectability of the pair (A, H ) [20]. Indeed, consider x a and x b ~ R" that are indistinguishable:
A
Then
l i E ( e , x, u)ll, e ~ W } .
Then q, is continuous with respect to (x, u) ~ R" × R " , and hence ep(x(t, Xo, u), u), x o E R ' , u o~-, is measurable on t. Furthermore,
(e, t)~E(e,
(13a)
y ~ R k,
x . - - X b ~ k e r [ H ' , A H, , . . . . .
where
'~,
x ( t , x o, u), u),
X o E R n, U ~ o ~ ,
x ( t , x a, u ) -
x ( t , x b, u) = H e m ( x a -*0,
xh)
t---~ + ~
for all u, if and only if the pair (A, H ) is detectable. Furthermore we have: Theorem 2. The following statements are equiv-
is a Carath6odory map, and thus (similar to [4,15]) one can show that e ( t ) is defined for all t > 0 approaching zero as t --, + ~ . Moreover a positive constant L can be found with
alent: (i) The pair (A, H ) is detectable. (ii) The system (13) satisfies (A1). In particular there exists a positive definite matrix P and a positive constant k 1 such that
II e ( t ) tl < L 11e0 Ilexp( - k , I[P II it)
e ' P A e < - k l llell2,
Ve~kerH.
(14)
for every e0, x 0, u ~.~- and t > 0. (ii) Consider a pair x~ ~ x b of indistinguishable states by some input ~ ~ ~ . It follows
(iii) There is a positive constant c such that the linear system
y ( t ) d e = f h ( x ( t , X~, ~t))
2 = Az + cP-1n'(y
- H z ) + Bu
(15)
is an observer for (13). = h(x(t,
x b, ?t)),
Vt > O.
(11)
O n the other hand, as t --* + ~ ,
IIz(t, z0 ;
h(x(t,
x~, Ft)), Ft) -x(t,
x~, ?4)I! ~ 0,
(12a)
e ' P ( A - R H ) e < - k 1 II e II 2,
[]z(t, z02; h(x(t, xb, ~)), ~)
-x(t,
Proof. (i) ~ (ii). Since the pair (A, H ) is detectable there exists a real matrix R of dimension n × k, a positive definite matrix P and a positive constant k I so that
0,
(12b)
Ve ~ R"
(see for instance [18] and [20]), which obviously implies (14).
J. Tsinias / Observerdesignfor nonlinearsystems
(ii) =, (iii). The implication is an immediate consequence of T h e o r e m 1 and R e m a r k 3. (iii) = (i). Since (15) is a linear observer for (13) it is well k n o w n from linear control theory that (A, H ) will be detectable. []
139
c we c o m p u t e L 1 and L2 as in the p r o o f of T h e o r e m 1. Let S be the sphere of radius r = 1, centered at zero and M=
(v=(va,
v2)'~R2:
-2VlV2 + 101021 < - k l II v II 2 }. We illustrate the nature of our results by two interesting examples.
Obviously ker H -
{0) c M and
OS-(OSAM) Example 1. We consider the system
= {v~R2:
2+3x+sinx=u,
x~R,
with output y = x control system
2, or equivalently the planar
x2
=
x2)'~R
y=(1
-1) (X xzl ) .
+ u
2,
1 '
u~R,
(16a) (16b) (16c)
Let us set
f ( x , u) = ( x 2, - 3 x 1 - sin x 1 -Jr- U ) ' , H=(1
-1)
and
Ilvll--1, -2v~v2+ Ivlv21 >_ - k ~ ) .
Then
-3x 1-sinxl
x=(x,,
u~U=R,
P=
0
1 "
L2=
min
oEOS--(aSNM)
(vl
- - V 2 ) 2 ~--- 1
Zl=(1+3k;1) and consequently 3k~-a)/(1 - 2ka).
we
can
:~2
=
+U
--sin x~
x=(x,,x2)'~R
2,
cos x~
IlP( D:,.f(x, u)} Jl = IIP( D:,f( x, 0)} II
satisfies (A1) and (A2) with
V(x, u)~R2XR
e'P(D~f(x, u)}ele,=e2
c > (1 +
P=(0
,
u~U=[-1,1],
y = x 1 +X
and for each 0 4= e ~ ker H we have e~ = e 2, so that
choose
Example 2. Similarly it is shown that the p e n d u lum system [4]
We can easily check that conditions (A1) and (A2) are fulfilled. In particular
<3,
- 2kl,
2,
3 0) 1
and p constant; thus we can apply the observer design m e t h o d of T h e o r e m 1.
= - - 2 e 2 - e l 2 cos x 1_< --el2 3. Bilinear systems
=-ka(e2+e2)le,=e2 where 0 < k 1 < ½. Since Dxf(x, u) is independent of u, there is a n e i g h b o r h o o d S e of e such that
In this section we shall specialize T h e o r e m 1 to bilinear control systems. Let us consider the single input system
v'P{ Dxf(X, u)} v < - k , I[ v II 2,
~c=Ax+u(b+Bz),
V ( X , V, U ) ~ a n X S e X ~ .
y = Hx,
According to T h e o r e m 1 the system (3) with
f ( z , u) = ( z 2, - 3 z 1 - sin z~ + u ) ' ,
and assume that there exist a positive definite matrix P and a positive constant k a such that for every e ~ ker H it follows
H = ( 1 , - 1 ) , p = 3 and R given b y (7) is an observer for (16). In order to estimate the constant
e'PAe < - k 1 II e II 2
y ~ R k,
x ~ R n, u ~ U ,
(17a) (17b)
(18)
140
J. Tsinias / Observer design f o r nonlinear systems
and
e'PBe = 0.
(19)
Obviously then (17) satisfies (A1) provided that U is bounded. Indeed, if we set
f(x, u)=Ax +u(b+ Bx) we have
e'P{Dxf(x,
= e'PAe + ½ue'(PB + B ' P ) e =e'PAe<_ - k 111ell 2, V e e K e r H , and using the fact that U is bounded,
<
-
1
~kl
lul Iv'Pnvl
II v II
2
PB i + B ' P = O ,
V i = l . . . . . m,
the system (3) with f and R as before performs the state determination of (20) for every u. An interesting problem is to find sufficient conditions for the observation of bilinear systems (20), where each component u, of the input u is a positive measurable function of time. Bilinear systems with positive controllers are frequently used to describe a lot of chemical, physical and biological mechanisms (see for instance [13,19]). The proof of the next proposition follows similar arguments as in the proof of Proposition 3, and thus it has been omitted.
u)}e
v'P{ D x f ( x , u ) } v < v ' P Z v +
and R = cP i l l ' is an observer, provided that U is a bounded subset of R m. I f in addition
,
f o r any v i n a v i c i n i t y o f e e k e r H - { 0 } . N o t i c e
finally that (5) and (6) of Remark 3 are fulfilled and since U is bounded, (A2) is valid with p independent of u. Hence according to Theorem 1 we can find c > 0 such that (3) with
Proposition 4. Given the system
(20) with
U,o a, and under the assumption
kerH-{0}c{e: e'PAe+ kl llell2_
f ( z , u) = A z + u(b + Bz)
(q(
~)
(e:e'PBie
(21)
and R = c P - 1 H ' is an observer of (17). For U being unbounded the same result holds if instead of (19) we assume PB + B'P = 0. The previous analysis can be trivially extended to the multi-input case. Thus we have:
where P is a positive definite matrix and k I > 0, there exists a positive constant c such that the observer governed by (3) with
Proposition 3. Consider the bilinear system
f(z, u)=Az+
l <_i<<.rn
m
Y'~ ( b i + B i 2 ) u i i=1
=Ax + ~. (hi+ Bix)ui,
(20a)
and
i=1
IlulI)P-'H"
xeR",
u=(u, ..... um)'eU,
(20b)
R(u) =c(1
y = Hx,
y ~ ~*,
(20c)
performs the state determination of (20).
and suppose that there exist a positive definite matrix P and a positive constant k I with kerHc
{e: e ' P A e + k I
¢q(
(']
Ilel12_<0}
ker(PBi+B~P)).
l
Then there is a positive constant c such that the system (3) with
+
Remark 4. Similar to Remark 2 it can be shown that (21) is invariant under any linear change of coordinates. The following result provides necessary and sufficient conditions for observation of planar hilinear systems with positive controls by employing the result of Proposition 4.
m
f(z, u)=Az+
E i=1
(bi+Biz)ui
Proposition 5. Consider a single input, single-output planar bilinear system with positive controls, and
J. Tsinias / Observer design for nonlinear systems
suppose that there exists a change of coordinates such that the system takes the f o r m (17) with
A =
01
H=(0
alE a22 1),
'
B = ~ b2 ~ x=(xl,
bE2
x2)'~B
E andy~a.
Then the system satisfies (21), if and only if one of the following two conditions holds: (i) b H > 0 , b 2 1 > 0 , (ii) b H < 0. P r o o f . Let b l l a n d b21 satisfy o n e of the a b o v e c o n d i t i o n s . Both cases c a n b e simultaneously, h a n d l e d as follows. W e c o n s i d e r r 1 > 0,
r=
( - rtbtx - e)/b21 -1
if b21 4= 0, ifb21=0,
r
is positive definite a n d for a n y 0 ¢ x E k e r H we obtain
x ' P B x = (rib11 + rb21)x ~ f -ex 2 < 0
if b21 4= 0,
[ rlbllx 2 < 0
if b21 = 0,
(22)
and
x ' P A x = rx~ = r II x II 21 x ~ k e r . < 0.
R e m a r k 5. I n case (i) of P r o p o s i t i o n 5 the system is observable (in the sense of [19]) for all u > 0, while in case (ii) it is o b s e r v a b l e for all u > 0 except for the c o n s t a n t i n p u t ~ = - b ~ 1 p r o v i d e d that b21 < 0. F o r u = ~ the c o r r e s p o n d i n g u n o b s e r v a b l e linear system is fibll 0
a12 + ub12 a22 +-UbE2 | x +
Y = x2"
Several o t h e r different a p p r o a c h e s have b e e n d e v e l o p e d d e a l i n g with the o b s e r v e r design p r o b lem of b i l i n e a r systems. F o r instance, a c c o r d i n g to [19], if the p l a n a r b i l i n e a r system of P r o p o s i t i o n 5 is c o m p l e t e l y o b s e r v a b l e o r e q u i v a l e n t l y bE1 = 0 ( L e m m a 1 in [19]) a n o n l i n e a r o b s e r v e r can be constructed. O n the o t h e r h a n d the general a p p r o a c h o f [19] p r e s e n t s a d i s a d v a n t a g e in p r a c t i c a l a p p l i c a t i o n s since the o b s e r v e r p a r a m e t e r s d e p e n d o n the time derivatives of the i n p u t u. F o r p l a n a r systems the o b s e r v e r design requires the exact k n o w l e d g e of (u, fi).
Sufficient L i a p u n o v l i k e c o n d i t i o n s for the o b server design have been p r o p o s e d . T h e a p p r o a c h we d e v e l o p e d is a p p l i c a b l e to a r e a s o n a b l y wide class on n o n l i n e a r systems i n c l u d i n g b i l i n e a r control systems. T h e result of T h e o r e m 1 ( s t a t e m e n t (i)) can be easily e x t e n d e d to n o n l i n e a r systems Yc = f ( x , u) with n o n l i n e a r o u t p u t y = h ( x ) , where h = R " ~ • k is a c o n t i n u o u s l y d i f f e r e n t i a b l e m a p ping. I n p a r t i c u l a r if (A1) a n d (A2) are satisfied with H = Dh(O), the system (3) with R given b y (7) p e r f o r m s the state d e t e r m i n a t i o n for x a n d e 0 b e l o n g i n g to some p r e f e r a b l y small n e i g h b o r h o o d of zero. Hence, in the general case of n o n l i n e a r o u t p u t , the c o r r e s p o n d i n g result will p r e s e n t a local nature.
(23)
F r o m (22), (23) a n d R e m a r k 4 it follows that the s y s t e m satisfies (21). Conversely, if we a s s u m e that (21) is valid, it is n o t difficult to see that rib11 + rb21 < 0 for s o m e real c o n s t a n t s r~ > 0 a n d r < 0; h e n c e either (i) o r (ii) is satisfied. []
f¢ =
Since Ubll < 0 we c a n easily verify that (8) h o l d s for a n y i n t i s t i n g u i s h a b l e states x a a n d x b.
4. Concluding remarks
a n d r 2 > rE//rl. T h e n for each case a n d for bE1 :/: 0 a positive e can b e f o u n d such that r > 0. T h e p r e v i o u s choice o f rl, r E a n d r implies that the matrix
,=(r;
141
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