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Observer Design for Uncertain Nonlinear Systems
5084
robust asymptotic stability with considerably reduced chattering (see Sira-Ramirez et al. 1995; Rios-Bolivar et al. 1995a).
III previous work considering dynCltllical adaptivp backstepping control, full-state measurement has been assumed (Rios-Bolivar et al. I D!Vjb). Here only the output is measured and an observer is designed to generate the unmeasured states. Then, dynamical adaptive backstepping is used to drive the output to t.rack a desired hounded reference signal.
2. PROBLEM STATEMENT The proposed algorithm in Rios- Rolivar et al. (HJ95b) involves nonlirlPa.r transformations of the controlled plant int.o tracking error coordinate::; depending on the control input and its derivatives. It is applicable to the class of observable minimurn-pha..<;e nonlinear systems, dynamically input-output linearizablc arId with constant but unknown parameters, which can be represented by
atiort, when systcllI (1) is transformable into the outp~tt feedback form.. Marino and Tomei (1991) have developed state observers and output-feedback control for systems transformable into the adaptive observer form. The problem or designing ohservers for nonlinear systerns, involving the control input and its derivatives, has been st.udied by Keller and Fritz (1986), and a solution has been proposed, provided that the system can be placed into the GOCF. However, the results were oht.ained for mml i/I par systerns wit.h no uncertainties. In the next two subsections the method proposeu by Keller and Fritz is presented and, it is shown t.hat under certain conditions an adaptive observer can be synthesized from the (~OCF for uncertain systems. The method considered here ilvoids t,he use of filters.
3.1 Observer design for nonlinear systems with derivatives of the input Consider t.he non linear systprTl with no uncertainties
;r = fo(x) + go(x)u
p
x = fO(3:) + go(x)u + ~ (Biri(X) + Bi'lf;;(x)u)
(1)
lJ=h(x)
i=1
.1/
= hex)
'rhe "ame problem has been solv"d by Krstic et al. (1995), using static controllers when system (1) is placed into either pure or st.rict feedback forms. Here the practical restriction of full-state measllrement is relaxed. by assuming that only t.he out.put is lllcasured, while the class or ulIcertain nonlincar systertls corresponds to systems with only out.put-dependent nonlinearities multiplying t.he uncertain parameters. Firstly, an adaptive observer is designed for t.he syst.em placed int.o the Generalized Observer Canonical Form (GOCF) (Keller and Fritz 1!)8li), and then it modified version of the algorithm in Rios-Bolivar rI Ill. (1995b), incorporating llonlinear damping terms, is used to design dynamical adaptive output. tracking control.
where x E ~n is tll" state, u E l}i t.he cont.rol input., lJ E ~ t.he output. It is assumed that (2) is a minimum-phase system with a well-defined relative degree p, i.e. 1 ::; p ::; n. We also assllme that. (2) sat.isfies the observability condition ran k
[. u" Y,y,···,
Zn-l
An appealing problem in control theory is the design of state observers for nonlinear syst.ems (see, for example, Krener and Respondek 1985). This problem become>; more difficult in the presence of uncertain paramet.ers. Various contributions have been reported for uncertain systems. For instance Kanellakopoulos et at. ( 1991 c) and Marino and Tomei (1991) have proposed t.he IIse of stable filters, or a comhinat.ion of stable filt.ers and an observer derived from Cl filtered transform-
5085
=n
= Zn
Y=
OBSERVER DESIGi'\
ch;
Y(n-l)]
(3)
Then (2) can be transformed into the Generalized Observability Canonical Form (GOBCF)
in = :~.
(2)
f(z, U,ll, ....
(4) u(n- p
»)
2'1
This GOBCF was init.ially obtained by Zeitz (1984) for time-variant non linear systems, and then by Fliess (1990) from a differential algebraic viewpoint. A sol ution to the problem of designing state observers for systems transformable into GOBCF, when the full stat.e is not measured, has been given by Keller and Fritz (198li) using the following Generalized Observer Canonical Form. (GOCF)
5086
Note f(z, u, -it) sat.isfies the st.ructural condition (10) cllld the functions ko, kl and k2 are identified as
p
~
= A~ + O'(Y, u, ... , u(n- p ) + L: Bi1/;i(Y)
(26)
i=1
.11= cT~
(20) k o( Zl,
'11,
il) = (1 + Zl)'/1 +
with A, c and 0' defined as in (13) and 1/;i : ~ --+ ~n smooth functions for i = 1, ... , p. Therefore, the following system
zr + it
Then, solving t he partial differential equatiolls (11) (21 ) 0'1(-)
= 111.31
'll
11
-
aoO = -- -
-
ZI
=1
.)
u - .:cl
€= A~ + a(y, U, ___ ,t/n- r ) + K(~n y = cTi,
€n)
+
lJ!(y)T
e
(27)
with
Therefore, the coordinate transformation (28)
(22) places the systCtll (17) into the GOCF
is an observer for (26) and the error system yields -
.
11
~I=-+Y
y
2
+u
-
'/1
t,=tl-lny+~ Y Y€
(23)
=6
and, finally, the observer is
~1
.
T-
F=Aot+W(Y) 8
-e
=
where Ao A - {\- (.T, e = ~ and if = B-() _This error system possesses a strict passivity property from the input iJ to t.he output W(y)e, i.e. t.he nonlinear operator if --+ W(y)e is strictly passive (see Krstic et al. 1995 for details). Therefore, the following parameter estimate
= .::y +1/ 2 + 11 + kl (6 - ~J
.
Il
(,=tl- 1n ~
'
y+-+k 2(6-6) y
(30) (24)
:w =~~ This method is proper when the system order is low. However, with increasing order the structural conditions become stronger (Keller and Fritz 1986).
3_~{
Observer design for a class of uncertain nonlinear systems
Consider now the case with parametric uncertainties. under the condition that the nonlinearities Illultiplying t.he uncertain parameter depend only on the output variable p
:i;=fo(x)+go(x)u+ L:Bi/i(Y)
(29)
(25)
i=1
y=h(x)
will be used as t.he first tuning function.
4. DYNAMICAL BACKSTEPPING CONTR.OL DESIGN In this section a modified version of the algorithm proposed by R.ios-Bolivar et al_ (1995b) is used to design a dynamical adaptive tracking contro\. The differences are due to the incorporation of an adaptive observer to estimate the IInmeasured states, and damping nonlillear terms to compensate the destabilizing effects of the observation errors (similarly to those used by Krstic el al. 1995). Here, for reasons of limited space, we describe the method only for a second order system with relative degree one and in the GOeF
~1
= -0'0(6, tt, ill +
~2
= ~1
-
0'1 (6,
'u)
+
(31)
y€=6
If the output dependence of these nonlinearities is invariant to the nonlinear transformation ~ = ~(x), placing (25) into the GOeF, the system (25) is transformable into the system
5087
By using the observer (27) and considering the parameter estimate
a:-; 0111' first. tlllling functioll, the deriv;divC' of the tracking error Zl = 6 - er is given by
Finally, t.he actueti update law for the unknown parameters ~'ields
Note that. ~,. is t J\(' transformation of th" boullded de(·llordinetlcs. Using sired I'pfpl'pllce signal ;I}, into the the Lya.pullov fUIlct.ion
and the dynamical adaptive control is
e
I., :2
I·
V1=-Z~I+-er
-d 2w"-z:! - 1:2 Z2 = -nu +'Pl0+W(Z2 (Jet 1
-1'
Z1 -
e
2
r = r T > 0, If]
the t i lIle derivettive
= ZI [~I
chi
(-0+
To
+ L"1\c2)
011 r
(44 )
5. EXAMPLE
is obt.aim·d. 'faking the s('('otld tUIling fUlIction as follows
and dell n i np;
+ d 1 le,.
wherp -d'lw 2 Z2 is the second non linear damping term to COrtlpPlIsate th,:, destabilizing effect of the observation ':'rror r: 1.
- nd6 11) + <'-"'2(6)0 - (,. 1
+ZI"I + er- I
+ dd z l +(r)
..
-,-li·+P2T2-e,·
-(Cl
with
(Cl
The following example illustrates the use of this algorithm. Consider
= -,Tl + :r:~ + + = + + Ox~ I.J =
.Tl
second errol' cool'd i neil (' as
11.
X1X2
Bxi (15 )
11
J;')
with
1'1
and d 1 positive design parattletc']'s, VI becomes
')'he limp dprivative of
::'1
Note thClt the nominal system (0 = 0) coincides with the system (17). 'fherefore, applying the transformation (22) on (4 r)) yields
(1 =::+y2+1I+B(y+y4)
is
if .
=
(2 . .(1 - In y
!I~ +wel
U
+ -!I + By
=6
or, fully transformed into the ~ coordinates
with
(1 =11 exp( -6) + exp(261 + u + O(exp(6) + exp(46)) (40)
(2=el - (2+llexp( - 6)+flexp(6) y~
Augmenting t.he Lyapullov function . I')-
. \'1
= +
= e2
( 41)
,11.)
=
- 11
exp( - 6) - exp(261 -
()I(e2,11)=6 -uexp(-6)
'T
=-
')
1'1;;;1-
' ) .
-'
dlz l + 21f1 + er
+Z2 [1'1 -
nn
(-e
+
T1
+ rZ2'P2W)
+ p: g + W(::2 - (1'1 + d l )ZI + ~,.)
[)Ol ---(I
Du
-1
(47)
which is obviously in the GOCF (31) with
1.) -;;;.~ :2 -
the time dpt'ivative is \12
( 46)
+ P2e - e,·' -.(Cl + dder. 1 (42)
5088
\Cl (e2)
= exp(6) + exp(46)
P2(e2)
= exp(6)
11
(48)
Computer simulatiolls were obtained to illustrate the tracking performance of the designed dynamical adaptive backstepping control/observer. Figure 1 shows the tracking performance for a smooth transition of X2 between two equilibrium points.
5089
robust asymptotic stability with considerably reduced chattering (see Sira-Ramirez et al. 1995; Rios-Bolivar et al. 1995a).
III previous work considering dynCltllical adaptivp backstepping control, full-state measurement has been assumed (Rios-Bolivar et al. I D!Vjb). Here only the output is measured and an observer is designed to generate the unmeasured states. Then, dynamical adaptive backstepping is used to drive the output to t.rack a desired hounded reference signal.
2. PROBLEM STATEMENT The proposed algorithm in Rios- Rolivar et al. (HJ95b) involves nonlirlPa.r transformations of the controlled plant int.o tracking error coordinate::; depending on the control input and its derivatives. It is applicable to the class of observable minimurn-pha..<;e nonlinear systems, dynamically input-output linearizablc arId with constant but unknown parameters, which can be represented by
atiort, when systcllI (1) is transformable into the outp~tt feedback form.. Marino and Tomei (1991) have developed state observers and output-feedback control for systems transformable into the adaptive observer form. The problem or designing ohservers for nonlinear systerns, involving the control input and its derivatives, has been st.udied by Keller and Fritz (1986), and a solution has been proposed, provided that the system can be placed into the GOCF. However, the results were oht.ained for mml i/I par systerns wit.h no uncertainties. In the next two subsections the method proposeu by Keller and Fritz is presented and, it is shown t.hat under certain conditions an adaptive observer can be synthesized from the (~OCF for uncertain systems. The method considered here ilvoids t,he use of filters.
3.1 Observer design for nonlinear systems with derivatives of the input Consider t.he non linear systprTl with no uncertainties
;r = fo(x) + go(x)u
p
x = fO(3:) + go(x)u + ~ (Biri(X) + Bi'lf;;(x)u)
(1)
lJ=h(x)
i=1
.1/
= hex)
'rhe "ame problem has been solv"d by Krstic et al. (1995), using static controllers when system (1) is placed into either pure or st.rict feedback forms. Here the practical restriction of full-state measllrement is relaxed. by assuming that only t.he out.put is lllcasured, while the class or ulIcertain nonlincar systertls corresponds to systems with only out.put-dependent nonlinearities multiplying t.he uncertain parameters. Firstly, an adaptive observer is designed for t.he syst.em placed int.o the Generalized Observer Canonical Form (GOCF) (Keller and Fritz 1!)8li), and then it modified version of the algorithm in Rios-Bolivar rI Ill. (1995b), incorporating llonlinear damping terms, is used to design dynamical adaptive output. tracking control.
where x E ~n is tll" state, u E l}i t.he cont.rol input., lJ E ~ t.he output. It is assumed that (2) is a minimum-phase system with a well-defined relative degree p, i.e. 1 ::; p ::; n. We also assllme that. (2) sat.isfies the observability condition ran k
[. u" Y,y,···,
Zn-l
An appealing problem in control theory is the design of state observers for nonlinear syst.ems (see, for example, Krener and Respondek 1985). This problem become>; more difficult in the presence of uncertain paramet.ers. Various contributions have been reported for uncertain systems. For instance Kanellakopoulos et at. ( 1991 c) and Marino and Tomei (1991) have proposed t.he IIse of stable filters, or a comhinat.ion of stable filt.ers and an observer derived from Cl filtered transform-
5085
=n
= Zn
Y=
OBSERVER DESIGi'\
ch;
Y(n-l)]
(3)
Then (2) can be transformed into the Generalized Observability Canonical Form (GOBCF)
in = :~.
(2)
f(z, U,ll, ....
(4) u(n- p
»)
2'1
This GOBCF was init.ially obtained by Zeitz (1984) for time-variant non linear systems, and then by Fliess (1990) from a differential algebraic viewpoint. A sol ution to the problem of designing state observers for systems transformable into GOBCF, when the full stat.e is not measured, has been given by Keller and Fritz (198li) using the following Generalized Observer Canonical Form. (GOCF)
5086
Note f(z, u, -it) sat.isfies the st.ructural condition (10) cllld the functions ko, kl and k2 are identified as
p
~
= A~ + O'(Y, u, ... , u(n- p ) + L: Bi1/;i(Y)
(26)
i=1
.11= cT~
(20) k o( Zl,
'11,
il) = (1 + Zl)'/1 +
with A, c and 0' defined as in (13) and 1/;i : ~ --+ ~n smooth functions for i = 1, ... , p. Therefore, the following system
zr + it
Then, solving t he partial differential equatiolls (11) (21 ) 0'1(-)
= 111.31
'll
11
-
aoO = -- -
-
ZI
=1
.)
u - .:cl
€= A~ + a(y, U, ___ ,t/n- r ) + K(~n y = cTi,
€n)
+
lJ!(y)T
e
(27)
with
Therefore, the coordinate transformation (28)
(22) places the systCtll (17) into the GOCF
is an observer for (26) and the error system yields -
.
11
~I=-+Y
y
2
+u
-
'/1
t,=tl-lny+~ Y Y€
(23)
=6
and, finally, the observer is
~1
.
T-
F=Aot+W(Y) 8
-e
=
where Ao A - {\- (.T, e = ~ and if = B-() _This error system possesses a strict passivity property from the input iJ to t.he output W(y)e, i.e. t.he nonlinear operator if --+ W(y)e is strictly passive (see Krstic et al. 1995 for details). Therefore, the following parameter estimate
= .::y +1/ 2 + 11 + kl (6 - ~J
.
Il
(,=tl- 1n ~
'
y+-+k 2(6-6) y
(30) (24)
:w =~~ This method is proper when the system order is low. However, with increasing order the structural conditions become stronger (Keller and Fritz 1986).
3_~{
Observer design for a class of uncertain nonlinear systems
Consider now the case with parametric uncertainties. under the condition that the nonlinearities Illultiplying t.he uncertain parameter depend only on the output variable p
:i;=fo(x)+go(x)u+ L:Bi/i(Y)
(29)
(25)
i=1
y=h(x)
will be used as t.he first tuning function.
4. DYNAMICAL BACKSTEPPING CONTR.OL DESIGN In this section a modified version of the algorithm proposed by R.ios-Bolivar et al_ (1995b) is used to design a dynamical adaptive tracking contro\. The differences are due to the incorporation of an adaptive observer to estimate the IInmeasured states, and damping nonlillear terms to compensate the destabilizing effects of the observation errors (similarly to those used by Krstic el al. 1995). Here, for reasons of limited space, we describe the method only for a second order system with relative degree one and in the GOeF
~1
= -0'0(6, tt, ill +
~2
= ~1
-
0'1 (6,
'u)
+
(31)
y€=6
If the output dependence of these nonlinearities is invariant to the nonlinear transformation ~ = ~(x), placing (25) into the GOeF, the system (25) is transformable into the system
5087
By using the observer (27) and considering the parameter estimate
a:-; 0111' first. tlllling functioll, the deriv;divC' of the tracking error Zl = 6 - er is given by
Finally, t.he actueti update law for the unknown parameters ~'ields
Note that. ~,. is t J\(' transformation of th" boullded de(·llordinetlcs. Using sired I'pfpl'pllce signal ;I}, into the the Lya.pullov fUIlct.ion
and the dynamical adaptive control is
e
I., :2
I·
V1=-Z~I+-er
-d 2w"-z:! - 1:2 Z2 = -nu +'Pl0+W(Z2 (Jet 1
-1'
Z1 -
e
2
r = r T > 0, If]
the t i lIle derivettive
= ZI [~I
chi
(-0+
To
+ L"1\c2)
011 r
(44 )
5. EXAMPLE
is obt.aim·d. 'faking the s('('otld tUIling fUlIction as follows
and dell n i np;
+ d 1 le,.
wherp -d'lw 2 Z2 is the second non linear damping term to COrtlpPlIsate th,:, destabilizing effect of the observation ':'rror r: 1.
- nd6 11) + <'-"'2(6)0 - (,. 1
+ZI"I + er- I
+ dd z l +(r)
..
-,-li·+P2T2-e,·
-(Cl
with
(Cl
The following example illustrates the use of this algorithm. Consider
= -,Tl + :r:~ + + = + + Ox~ I.J =
.Tl
second errol' cool'd i neil (' as
11.
X1X2
Bxi (15 )
11
J;')
with
1'1
and d 1 positive design parattletc']'s, VI becomes
')'he limp dprivative of
::'1
Note thClt the nominal system (0 = 0) coincides with the system (17). 'fherefore, applying the transformation (22) on (4 r)) yields
(1 =::+y2+1I+B(y+y4)
is
if .
=
(2 . .(1 - In y
!I~ +wel
U
+ -!I + By
=6
or, fully transformed into the ~ coordinates
with
(1 =11 exp( -6) + exp(261 + u + O(exp(6) + exp(46)) (40)
(2=el - (2+llexp( - 6)+flexp(6) y~
Augmenting t.he Lyapullov function . I')-
. \'1
= +
= e2
( 41)
,11.)
=
- 11
exp( - 6) - exp(261 -
()I(e2,11)=6 -uexp(-6)
'T
=-
')
1'1;;;1-
' ) .
-'
dlz l + 21f1 + er
+Z2 [1'1 -
nn
(-e
+
T1
+ rZ2'P2W)
+ p: g + W(::2 - (1'1 + d l )ZI + ~,.)
[)Ol ---(I
Du
-1
(47)
which is obviously in the GOCF (31) with
1.) -;;;.~ :2 -
the time dpt'ivative is \12
( 46)
+ P2e - e,·' -.(Cl + dder. 1 (42)
5088
\Cl (e2)
= exp(6) + exp(46)
P2(e2)
= exp(6)
11
(48)
Computer simulatiolls were obtained to illustrate the tracking performance of the designed dynamical adaptive backstepping control/observer. Figure 1 shows the tracking performance for a smooth transition of X2 between two equilibrium points.
5089