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Adaptive nonlinear servocompensation of unknown external disturbances
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EX...
14th World Congress of IFAC
1-3b-04-1
Copyright © 1999 IFAC ... "' . 14th Triennial ,"VarId Congress, BelJlng~ P.R. Chlna
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EXTERNAL DISTURBANCES V. o. N ikiforov Department of Automatic and Remote Control Institute of Fine Mechanics a'1ld Optics, SahJinskaya Saint-Petersburg, 197101, RUSSIA [email protected]
14
Abstract: The problem of adaptive compensation of unknown external disturbances is solved for a class of nonlinear systenls which have the globally defined normal form. When a priori unknown external disturbauf;,e can be accurately modelled as the output of a linear cxosystem of the knoV\l'n order but with unknu"vn parameters, the proposed adaptive controller provides global asymptotic tracking. If the disturbance contains unmodeled components l the proposed controller provides boundcdness of all the closed-loop signals and arbitrary slllall tracking error. Copyright @1999 IFAC Keywords: servers
Nonlinear control systems, servomechanism, nonlinear ob-
1. INTRODUCTION The problem of cornplete rejection of external inaccessible disturbances plays an important role in the 1110dern control theory. The known approaches to the problem can be classified depending on a priori required inforluation about the disturbance. In the simplest particular case of deterministic disturbances with a priori known waveform, one can use the farr~ous Internal l'vfodel Princl:ple (J ohnson, 1971; FraIlcis and Wonham 1 1975; Davison 1 1976). In accordance \vith the Principle, the deterIninistic external disturbances are treated as the outputs of linear dynamic systenlS (so-called e3:osystenls).Then the disturbance influence on the plant. response is completely eliminated if the exosystem model is suitably reduplicated in the feedback path of the closed-loop system. In this case it is said about servocornpensation of external disturbances. 'The Internal Model Principle was initially developed for the linear systems, but recent.ly its applic.ations have been extended to the several nonlinear control problems (Desoer and Lin, 1985; Di Benedetto, 1987; Isidori and Byrnes, 1990~ l{halil~ 1994)~
HOVilever, it is more realistically to assume that we known a priori only a class of possible disturbances but not their exact waveform. In this case we are not able to specify beforehand accurate nlodels of the exosystems, and the unknown deterministic disturbances can be treated as the outputs of the linear eXQsystems of the known order but with unknown parameters (Elliot and Good\vin: 1984). Then the problem of counteracting the unknown disturbances can be reformulated as the problem of adaptive servocornpensation. A few ada.ptive controllers of both the discret.e and continuous-time type were designed (EJliot and Goodwin, 1984; Gang Feg and Palanis\vami, 1991; Bonilla et. ai, 1994; Nikiforov, 1995). Recently this approach has been extended to a class of nOlllinear systems (Nikiforov, 1998). However the results obtained in the Inentioned papers are not applicable in the case when unkno,vn disturbance contains unmodeled irregular component. At the sarne time, in most of practical cases only the dOluinant part of the disturbance (but not the disturbance at whole) can be treated as a regular function of tilTIe generated by a linear exosysl
4506
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EX...
tern. Therefore robust Illoditications of adaptive servo1l1echanism controllers are greatly needed.
In the present paper the problen1. of robust adaptive seruoconlpensation of the unknown external disturbances is solved for a class of nonlinear
14th World Congress of IFAC
where d(t) is an unknown bounded function~ while the component t9(t) can be modelled as the output of a linear system (so-called exosystem); this exosystem has t.he form
single-input, single-output systems which have the
gJobally defined normal form~ In contrast to the previous papers, the unkno~"n external distur-
bance is allowed to contain an unmodeled component. In this case 1 the proposed adaptive controller ensures boundedness of all the closed-loop signals and arbitrary disturbance attenuation. If t.he external disturbance generated by an a priori uncertain exosysteol does not contain an unmodeled component, the proposed adaptive controller provides aSylllptotic tracking of a prescribed reference signal.
'l""1he paper is organized as follows: the control problem is posed in Section 11; a ne\:v nonlinear adaptiv~ observer is designed in Section III and in Section 1\1 the proposed adaptive regulator is prf'sp:nted~
2. PROBLE!\-I srrATEMENT
In this paper we consider the
single~input1 single-
output nonlinear systerrl
I(x)
+ g(x)[u + SO]
hex)
y
(1) (2)
Rn input. u, output y and external disturbance 'P. The following assumptions are supposed to be satisfied for the system (1)-(2). \\rith the state x E
l
Plant rn-odel assu1"nptions: i) the vector fields f(x) and g(x) and the function hex) are smooth; ii) for all x E Rn and some integer 1 ~ p ~ n the folloVi.·ing holds
LgLfhCc) == ... -== L.gL~j-2h
Lyh(;l')
D
and
iii) the vector fields
are complete,
== adj-lg,
1 :::; k
(4)
1)
(5)
v,there X E Rq is the state vector of the exosystem , the constant q x q matrix r has all its eigenvalues on the imaginary a..xis and c is a constant vector (without loss of generality the pair (eT, r) is assumed to be observable); v) the dimension q of the exosystem is known, but paraITleters of the matrix r and vector c are unknown; vi) neither disturbance
We consider the problem of designing a statefeedback control to track a reference signal Yr (t) satisfying the following assumption. R,eference signal assumption: vii) the reference signal Yr and its first p derivatives are known and bounded, and y~p) is piecewise continuous.
The considered control problem will be solved by means of adaptive control. When the external disturbance
i-+OG
At the same tin'le, when the disturbance r.p contains an unmodeled component ~r the proposed controller guarantees global boundedness of all the closed-loop signals and arbitrary disturbanc.e attenuation.
Rem,ark 2.1. The plant model assulnptions (i)(iii) mean (see Byrnes and Isidori~ 1991) that the s~ystem (1)-(2) has a uniform relative degree equal
LgLj-l hex) f:. 0;
..\ -k
x
to p and is globaIJy diffeomorphic to the system
::s p
\~lhere
fo(z,e)
~
E'; + dLB( z ~) 1
where y =
ivY the disturbance 9(t) is bounded and can be presented in the fOfln
~l~
= <)(x)
(7)
is the new state
Rn;-p
l
(3)
(Z1~)
+ a (z ~ ~) u + a ( z , ~)
and ~ E RP), terms P(z,£) and u(z,~) are exactly the functions Ljh and LgLj-1h expressed in the new coordinates (i.e., j3(Z7~) == Lj ho~-l (z J;) and a( z, t) LgLj-l ho({>-l (z, ~)) and the p x p matrix E and the constant vector d
(z E
Disturbance lnodel aSS1lmptions:
(6)
z
=
4507
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EX...
with the initial condition 6,,(0) = M X(O) - «0), and q x q matrix ]\;/ is a solution to the matrix equation lwr - G M ~ beT.
are given by 0
E
1 0
0
0 1
0 0
0
1 0
0
0 d
0
0
1
In this case a global coordinate transformation is defined by the following mapping ~(x)
== COl(1Jl(X), ... ,«Pn-p, h(x), L J h(x)1 . .. , LJ-1h(;c))
=
(8)
where the functions
The lemma reduces the uncertainty of the signal iJ to the parametric uncertainty of a constant vec.tor (J associated with the knoVw"n "regressor" C. However, since the signal it is not accessj ble to measurements, the state variable filter (9) is not realiza.ble. We overcome this problem with the use of a new Ilonlinear observer for (. Let us introduce an estimate (" of the state ( in the following form
=
ReUHrrk 2.2. Even if the disturbance 'P does not contain unlllodeled coniponents (i.e., if ~(t) == 0), the assumption (v) means that we know a priori only a class of possible external disturbances, but not their exact ~l'aveform. The problem of complete compensation of such disturbances is not trivial and has theoretical as well as practical significance. In the paper l this problem is solved with the use of a nev,," adaptive nonlinear servocompensation controller. Moreover 1 it will be shown that the proposed c.ontro] systen1 tolerates the presence of an unmodeled component ~(t) and r in this case~ demonstrates an acceptable quality. 0
The control pl"oblem posed will be solved in two steps. First a new nonlinear observer for unkno\vn external dist.urbances will be proposed. Second~ a new stat.e-feedback adaptive regulator \vin be designed and analyzed. 1
(11 ) where the auxiliary vector 17 E Rq is generated by the filter
and the vector-function 1./J(z,~) E Rq satisfies the equation in partial derivatives
o1/J(z, ~) b_1____ a~ d == a(z,~) "
(13)
Then the properties of the nonlinear observer (11)(13) are established by the following theorem.
THEOREM 1. If there exists a vector-full.ction l/J(z,~) satisfying (13)~ then the inaccessible disturbance
3. NONLINEAR OBSERVER
where
The follo''\o~il1g auxiliary lemma reformulates in a 1110re suitable form SOlne results known from the theory of linea.r observers.
LEMMA (NikifofOV, 1998). Along with (4)-(5) consider the dynamic system.
( ==
GC + bt?,
(9)
tvhere ( E Rg is the .state.. and the pair (G , b) is controllable. Then for any q x q Hurwitz matrix G tit ere- exists a ltn'ique constant vector 8 E R~ suclt that the signal {) can be presented in the fornl
(10) lDhere the exponentially decaying 'vector b" obeys the f'q1lalion
<:
is generated by the nonlinear obser'ver
(11)-(13), () is a constant unknown vector and 8 obeys the equation
8
== G6-b~
(15)
with the initial state 6(0) = tt,( z(O)~ ~(O»).
.l\t1X(O) - 1](0) -
Proof. Differentiating Dl.p :::::; ( - TJ - W \vith respect to tinle in vieVv- of (6),(7),(9) and (12)-(13) we obtain
GC: 81j,J
+ brfJ -
Gr] - G1/J
f3
+
81j; E(
8e
+
alP
-/0 + bu + b- - - E { 8z a 8{ 8~
-d[fi + au + o
G((-17-1/J)-b~
a~ -10
az
=
G04'-b~
4508
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EX...
=
with 8'P(O) ((0) - 1}(O) - ,b(z(O), ~(O)). Then substituting ( = 1] + 1P + 6:p = (' + 6:p into (10) Vt,~e obtain T {J =:: e ( + eT(o{J + olfJ)' Introducing the vector D ==: 6f)
+
14th World Congress of IFAC
the observer (11)-(13), and j..t > O. To adjust the vector Bin the presence of unmodeled disturbance ~, we use the following algorithm of adaptation (IoanIlou, 1986)
81,(/, we readily
obtain (14)-(15).
(20)
0
181 <
Corollary. If the disturbance r.p does not contain an unmodeled component (i~ e., ~(t) == 0), then it can be presented in the form. (16) where the exponentially decay'ing vector 6(t) obeys [he eq1Jation iJ === G8, where 8(0) == M X(O) 1] ( 0) -lb ( ~ ( 0) 0) ) . f
e(
R.fluark .1.1. Equation (13) is essentially simplified if the vector b is chosen in the forIll b == col(O, 0, ... ,1) (with matrix G being, for example, in the phase-variables canonical forIll). III this case the vector...function ti'(z, €) can be chosen as -r;:~(.;) = col(O, O~ .. ~ , 1P q (Z, e)) where the function l{)g (z, ~) is a solution to the first order partial differential equation
1
0:( Z, ~) .
(17)
()*
Let us int.roduce the vector of tracking errors ) with coordinates
()*
is any positive constant and T
=
£j
==
~j
_
y~j -
1] ,
THEOREM 2.
Consider the closed-loop system consisting of the plant (1)-(2) subjected to the unknown disturbance (3)-(5), nonlinear observer (11)-(13) and adaptive controller (19)-(21). Let equation (6) be input-ta-state stable when ~ is seen as an input Then under assumptions (i)-(vti): (aj all the closed-loop signals are globally bounded,~ (b) the residual trackin.g error can be made arbitrary small by increasing the feedback gain j-lJ and for all J1. > /la > 0 and any /Jo > 0
= Ec: + d[ - ~PJ + )3(z,e) + cr(z,e)[u
,vhere
El
=Y -
+ (JTe + (JT (j + ~]J,
(18)
Yr.
The proposed adaptive regulator is given by
(Jolt<
> \~ the
lim [y(t) - Yr(t)]
t-oo
(22)
==
aSyTuptotic
O.
Proof. First, we establish boundedness of all the closed-loop signals. Substituting (19) into (18) and differentiating (j == e - (j in vie"" of (20), Vole obtain the following error model
i
e
+ o:d(iI' ( + eT 0 + ~ -jQ;(dT Pc - -P'j(f + J(7(}.
Ae
C~onsider
/-lET
Pda) (23) (24)
the Lyapunov function ;r.
\"..here (j E Rq is the vector of adjustable paranletel's! the vector of feedback gains k -=: coI(k 1 ,k 2 ' . " l k p ) is chosen so that the Inatrix ~4 ~ E - dk T is Hur\vitz! the positive-defined lllatrix P is a solution to the matrix equation .A. T p + P A == -1, the estimate" is produced by
1
..fii"')
where th.e value of K, depends on a preselected con-
It can be sho\vn that the error model, in view of equations (7) and (14)~ takes the form E:
> O.
The stability properties of the closed-loop system are established by the following theorem.
stant ).la; (e) when ~(t) == 0 and tracking is achieved, i. e.
j == 1, 2 ~ ... , p.
(21)
Remark 4.1. In the adaptive controllaVw" (19)-(21) the functions a and {3 can be expressed in any coordinates (x or (z, ~)). Therefore, the argurnent of these functions is omitted. 0
limt_:>:> (c:(t)t :::; E
s 181 ::; 20*
la, > 28*, ""here
4. AD.A.PTI'lE REGULAT'OR C.Ol(El' £2~ . . . ,E p
0*
-
\I e (c,8)
==
1 T 1 nr2 c PC++ 2-yB e.
Its tirne derivative along the solutions of (23)-(24) satisfies:
4509
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EX...
or a(d
-1-l (eT Pda)2 -
T
0"1812 + a(JI'e
P€ -
< _~ _ 0"(8),Op _ (~IET Pdal _ ~6 2
2.Jii
2
-(7(B)(-1_ I01
- vI2 + O"~) 181 2 ::;
)2
14th World Congress of IFAC
Consider the derivative of solutions of (23)
2
V2
+2Je'T PdQ!a~ - 2JllcT PdaJ2. (26)
4tJ
_~\eI2 _ lT~) 101 2+ A,
where the positive constants inequalities T
~o
Taking the boundedness of (" and (j into account)
and Ll obey the
2
1
-
~t5~le o(t)+~(t)[,
eT Pc along the
-lc:l 2 + 21c·T PdallfTBI
<
__1_ tol ) + ~~~
lel;' ==
we can write
1. [2
~~4p.Ac+21e.
\-vhere
J1(JICXJ does not depend on 11. (see equat.ion
(14)) . It. is easy t.o see that V < 0 if JcJ2 > 2~ or l{l~2 2: max{ (28* + 181)2, 2~} (inequality lei 2 ~ (20* + le\)2 guarantees that 101 > 20* and, as a consequence! 0"(0) == 1). Then after simple calculations it can be sho\vn that
(25)
=
where Co = min{l(P), ~}, R 2 max{(20'" + lel)2. 2~}, ~(P) is the minimum eigenvalue of the Inat.rix P. \'ia. condition (25) V·le conclude that £(t) and O(t) are bounded. Since equation (6) has the ISS property, boundedness of means boundedness of ;; and, as a consequence, boundedness of x. rrom ~rheoren1 1 ,ve have boundedness of (, and, therefore, from equation (19) we straightfor\~lard derive boundedness of u, and from equation (12) \Ne obtain boundedness of 'l}~ Thus the boundcdness of all the closed-loop signals is proved.
Denoting
(27) we rewrite (26) as
~ lel1> ::; -lef + 2k T PdalLi" = _).::12 _ 2
e
Noyv \ve prove part (b). By virtue of condition (25) we have that for all t > 0 l-~~(t)
S V ==
max{~~(O),
VD}
~
(v'iJ1cT Pdal _
2 -6 I 1
2,uleT Pdaj2
At ) 2Vii
2
+
D,; 21-"
1 1 12 ~; +d; - ::; --£p+-, 2p Cl 2p
where Cl = l(p) is the maximum eigenvalue of the matrix P. Therefore
!e(t)l:::;
(C1e-2~ltlC:(O)I+J
V~
Cl
2C2P
AE,
=
where C2 J..{P) is the nlinimum eigenvalue of the matrix P. The latter inequality means the validity of (22), where
and, hence,
1 v~ (t) = '2c:(t)T Pc:(t) ]
-
+
1 -... 2.y"'(t)T 8(t)
Taking into account the fa.ct that constants .68. 11(1100, 1':1 and K'2 do not depend on p" we conclud~ that the residual tracking error can be made arbitrary small by increasing J.1. The proof of part (b) . is completed.
-..
21' 8(t)T e(t). Denoting
0'2 ==
2'·/'/, we can write
=
The upper bound 7f might depend on Vu. In turn VD depends on JA. Taking expressions for Vo, Rand ~ into account, we conclude t.hat the following inequality is valid -2
e
<
~1
where
8
1\.2 + -,
=::
G6.
(29)
Consider the Lyapunov function
J-L
where the values of the positive constants fC'] do not depend on the feedback gain p.
Now Vv~e prove part (c). \Vhen ~(t) 0, the dynamic lllodel of the closed-loop system in coordinates € t.akes the fornl
K1
and
-V(£, 6, 0)
1
== 2"c
T
Pe
+ kJo
T
1 --T-''-' P6 D + 2", B {)1
4510
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EX...
"\vhere the positive defined matrix p~ satisfies equation GT Pf; + P 6 G == - I and the positive gain kfJ is equal to (°1 2 / p,. The time derivative of this function along solutions of (24),(28) and (29) has the form
_~lcI2 + eT PdCl'.(I' 0 + eT PdCl'.OT6 -j1(€ T
Pda)~ K)
101 2
0)
-J6J'" p.
_1
.--om"""
()J.
T
u(d PE;
+cr(O)OT (j
_~lcI2_ ~k8'8122
4
(v'liIc
T
PdCl'.l-
1 2 -2!c l -
2~'6If + uco)8Te
3 2 4k616(
--/iT+ 0-(0)0 O.
{}* > IBI (in accordance with the accepted additional assumption), we have tT(e)'OTo ~ 0 for any 6. 'fherefore
Since
14th World Congress ofIFAC
(1994) On the use of adaptive precompensators for rejecting output disturbances in non minimum phase 8150 systems. Prepr. 33rd IEEE Cont. Dec. and Control, Florida, va!' 2, pp.1183-1184. Byrnes, C. 1. and A. Isidori (1991) Asymptotic stabilization of minimum phase nonlinear systems. IEEE Trans. Autom. Contr.~ 36(10), pp.11221136. Davison, E. J. (1976) The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans. A utom. Contr.~ 21(1)~ pp.25-34. Desoer, C. A. and C.-A. Lin (1985) Tracking and disturbance rejection of MIMO nonlinear systems with PI controller. IEEE Trans. Autom. Conir., 30(9), pp.861-867. Elliot, H. and G. C. Good\vin (1984) Adaptive implementation of the internal model principle. Proc. 23rd IEEE CDC, pp.1292-1297. Francis, D. A. and W. 1\1. Wonham (1975) The internal model principle for linear rnultivariable regulators4 Appl. A-fath. Opt., 2: pp.170-191. Gang Feg and M. Palaniswami (1991) Unified treatment of internal model principle based adaptive control algorithms. Int. J. Contr., 54(4) pp.883-901. Ioannou P. (1986) Robust adaptive controller with zero residual tracking error. IEEE Trans. on A utorn. Control, 31(8), pp.773-776. Isidori, A. and C. I~ Byrnes (1990) Output regulation of nonlinear systems. IEEE Trans. A utom# Contr'1 35(2)1 pp.131-140. Johnson) C. D. (1971) Accommodation of external disturbances in linear regulator and servomechanism problems. IEEE Trans. Altto'm. Contr., 16(6), pp.635-644. Khalil, H. K. (1994) Robust servomechanism output feedback controller for feedback linearizable systems. A utQmatica, 30( 10), pp.1587-1599. !(rstic, 1\1., I. Kanellakopoulos and p~ Kokotovic (1995) Nonlinear and Adaptil'e Control Design, New York: John VlilJey and Sons. Nikiforov, O. (1996) Adaptive servocompensation of input disturbances, Prepr. 13th IF:4C World Congress, San-Francisco, USA, vol.K, pp.I75-I80. Nikiforov , V. O. (1998) Adaptive non-linear tracking with complete compensation of unknown disturbances. European Journal of Control, 4(2), pp.132-139. Wonham W. M. (1979) Linear Mu ltiv ariable Control: A Geometric Approach, New York: Springer- Verlag. I
Since x, (, EJ are bounded and a is smooth, we conclude that the right part of (28) is locally Lip-
schitz in
E
uniformly in t. Then, from the LaSalle-
Y"oshiza\i\.'a theol'eln (see Krstic et al., 1995) we cOIlc]ude~ that c:(t) --+ 0 as t --'I- 00. In part.icular it. Illeans that y(t) ~ Yr(t) ~ 0 as t ---;. ()(). 0
5. CONCLUSION The problem of robust adaptjve servocompensation of unkno\'1n bounded external disturbances is solved for a class of nonlinear systems being globa.lly diffeomorphic to the normal form. The main contributions of the paper are the following: i) nonlinear observer for a class of unknown disturbances (Theorem l)j ii) nonlinear regulator ~rith robust adaptive servocompensation of unknO\~ln disturbances CTheorem 2). ,"'hen the a priori unknown disturbance does not contain unmodeled conlponents~ the proposed adaptive regulator provides global asyrnptotic tracking. If the disturbance contains unIllodelcd cOlnponents~ the proposed controller provides boundedness of all the c.losed-loop signals and arbitrary small tracking er~ ror.
'T.
1
REFERENCES Di Benedetto, 1\'1. D~ (1987) Synthesis of an internal model for nonlinear output regulation. Int. J. Contr., 45, pp.l023-1034. BonilIa, M. E., J~ A. Cbeang and R. Lozano
4511
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress of IFAC
1-3b-04-1
Copyright © 1999 IFAC ... "' . 14th Triennial ,"VarId Congress, BelJlng~ P.R. Chlna
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EXTERNAL DISTURBANCES V. o. N ikiforov Department of Automatic and Remote Control Institute of Fine Mechanics a'1ld Optics, SahJinskaya Saint-Petersburg, 197101, RUSSIA [email protected]
14
Abstract: The problem of adaptive compensation of unknown external disturbances is solved for a class of nonlinear systenls which have the globally defined normal form. When a priori unknown external disturbauf;,e can be accurately modelled as the output of a linear cxosystem of the knoV\l'n order but with unknu"vn parameters, the proposed adaptive controller provides global asymptotic tracking. If the disturbance contains unmodeled components l the proposed controller provides boundcdness of all the closed-loop signals and arbitrary slllall tracking error. Copyright @1999 IFAC Keywords: servers
Nonlinear control systems, servomechanism, nonlinear ob-
1. INTRODUCTION The problem of cornplete rejection of external inaccessible disturbances plays an important role in the 1110dern control theory. The known approaches to the problem can be classified depending on a priori required inforluation about the disturbance. In the simplest particular case of deterministic disturbances with a priori known waveform, one can use the farr~ous Internal l'vfodel Princl:ple (J ohnson, 1971; FraIlcis and Wonham 1 1975; Davison 1 1976). In accordance \vith the Principle, the deterIninistic external disturbances are treated as the outputs of linear dynamic systenlS (so-called e3:osystenls).Then the disturbance influence on the plant. response is completely eliminated if the exosystem model is suitably reduplicated in the feedback path of the closed-loop system. In this case it is said about servocornpensation of external disturbances. 'The Internal Model Principle was initially developed for the linear systems, but recent.ly its applic.ations have been extended to the several nonlinear control problems (Desoer and Lin, 1985; Di Benedetto, 1987; Isidori and Byrnes, 1990~ l{halil~ 1994)~
HOVilever, it is more realistically to assume that we known a priori only a class of possible disturbances but not their exact waveform. In this case we are not able to specify beforehand accurate nlodels of the exosystems, and the unknown deterministic disturbances can be treated as the outputs of the linear eXQsystems of the known order but with unknown parameters (Elliot and Good\vin: 1984). Then the problem of counteracting the unknown disturbances can be reformulated as the problem of adaptive servocornpensation. A few ada.ptive controllers of both the discret.e and continuous-time type were designed (EJliot and Goodwin, 1984; Gang Feg and Palanis\vami, 1991; Bonilla et. ai, 1994; Nikiforov, 1995). Recently this approach has been extended to a class of nOlllinear systems (Nikiforov, 1998). However the results obtained in the Inentioned papers are not applicable in the case when unkno,vn disturbance contains unmodeled irregular component. At the sarne time, in most of practical cases only the dOluinant part of the disturbance (but not the disturbance at whole) can be treated as a regular function of tilTIe generated by a linear exosysl
4506
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EX...
tern. Therefore robust Illoditications of adaptive servo1l1echanism controllers are greatly needed.
In the present paper the problen1. of robust adaptive seruoconlpensation of the unknown external disturbances is solved for a class of nonlinear
14th World Congress of IFAC
where d(t) is an unknown bounded function~ while the component t9(t) can be modelled as the output of a linear system (so-called exosystem); this exosystem has t.he form
single-input, single-output systems which have the
gJobally defined normal form~ In contrast to the previous papers, the unkno~"n external distur-
bance is allowed to contain an unmodeled component. In this case 1 the proposed adaptive controller ensures boundedness of all the closed-loop signals and arbitrary disturbance attenuation. If t.he external disturbance generated by an a priori uncertain exosysteol does not contain an unmodeled component, the proposed adaptive controller provides aSylllptotic tracking of a prescribed reference signal.
'l""1he paper is organized as follows: the control problem is posed in Section 11; a ne\:v nonlinear adaptiv~ observer is designed in Section III and in Section 1\1 the proposed adaptive regulator is prf'sp:nted~
2. PROBLE!\-I srrATEMENT
In this paper we consider the
single~input1 single-
output nonlinear systerrl
I(x)
+ g(x)[u + SO]
hex)
y
(1) (2)
Rn input. u, output y and external disturbance 'P. The following assumptions are supposed to be satisfied for the system (1)-(2). \\rith the state x E
l
Plant rn-odel assu1"nptions: i) the vector fields f(x) and g(x) and the function hex) are smooth; ii) for all x E Rn and some integer 1 ~ p ~ n the folloVi.·ing holds
LgLfhCc) == ... -== L.gL~j-2h
Lyh(;l')
D
and
iii) the vector fields
are complete,
== adj-lg,
1 :::; k
(4)
1)
(5)
v,there X E Rq is the state vector of the exosystem , the constant q x q matrix r has all its eigenvalues on the imaginary a..xis and c is a constant vector (without loss of generality the pair (eT, r) is assumed to be observable); v) the dimension q of the exosystem is known, but paraITleters of the matrix r and vector c are unknown; vi) neither disturbance
We consider the problem of designing a statefeedback control to track a reference signal Yr (t) satisfying the following assumption. R,eference signal assumption: vii) the reference signal Yr and its first p derivatives are known and bounded, and y~p) is piecewise continuous.
The considered control problem will be solved by means of adaptive control. When the external disturbance
i-+OG
At the same tin'le, when the disturbance r.p contains an unmodeled component ~r the proposed controller guarantees global boundedness of all the closed-loop signals and arbitrary disturbanc.e attenuation.
Rem,ark 2.1. The plant model assulnptions (i)(iii) mean (see Byrnes and Isidori~ 1991) that the s~ystem (1)-(2) has a uniform relative degree equal
LgLj-l hex) f:. 0;
..\ -k
x
to p and is globaIJy diffeomorphic to the system
::s p
\~lhere
fo(z,e)
~
E'; + dLB( z ~) 1
where y =
ivY the disturbance 9(t) is bounded and can be presented in the fOfln
~l~
= <)(x)
(7)
is the new state
Rn;-p
l
(3)
(Z1~)
+ a (z ~ ~) u + a ( z , ~)
and ~ E RP), terms P(z,£) and u(z,~) are exactly the functions Ljh and LgLj-1h expressed in the new coordinates (i.e., j3(Z7~) == Lj ho~-l (z J;) and a( z, t) LgLj-l ho({>-l (z, ~)) and the p x p matrix E and the constant vector d
(z E
Disturbance lnodel aSS1lmptions:
(6)
z
=
4507
Copyright 1999 IFAC
ISBN: 0 08 043248 4
14th World Congress ofIFAC
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EX...
with the initial condition 6,,(0) = M X(O) - «0), and q x q matrix ]\;/ is a solution to the matrix equation lwr - G M ~ beT.
are given by 0
E
1 0
0
0 1
0 0
0
1 0
0
0 d
0
0
1
In this case a global coordinate transformation is defined by the following mapping ~(x)
== COl(1Jl(X), ... ,«Pn-p, h(x), L J h(x)1 . .. , LJ-1h(;c))
=
(8)
where the functions
The lemma reduces the uncertainty of the signal iJ to the parametric uncertainty of a constant vec.tor (J associated with the knoVw"n "regressor" C. However, since the signal it is not accessj ble to measurements, the state variable filter (9) is not realiza.ble. We overcome this problem with the use of a new Ilonlinear observer for (. Let us introduce an estimate (" of the state ( in the following form
=
ReUHrrk 2.2. Even if the disturbance 'P does not contain unlllodeled coniponents (i.e., if ~(t) == 0), the assumption (v) means that we know a priori only a class of possible external disturbances, but not their exact ~l'aveform. The problem of complete compensation of such disturbances is not trivial and has theoretical as well as practical significance. In the paper l this problem is solved with the use of a nev,," adaptive nonlinear servocompensation controller. Moreover 1 it will be shown that the proposed c.ontro] systen1 tolerates the presence of an unmodeled component ~(t) and r in this case~ demonstrates an acceptable quality. 0
The control pl"oblem posed will be solved in two steps. First a new nonlinear observer for unkno\vn external dist.urbances will be proposed. Second~ a new stat.e-feedback adaptive regulator \vin be designed and analyzed. 1
(11 ) where the auxiliary vector 17 E Rq is generated by the filter
and the vector-function 1./J(z,~) E Rq satisfies the equation in partial derivatives
o1/J(z, ~) b_1____ a~ d == a(z,~) "
(13)
Then the properties of the nonlinear observer (11)(13) are established by the following theorem.
THEOREM 1. If there exists a vector-full.ction l/J(z,~) satisfying (13)~ then the inaccessible disturbance
3. NONLINEAR OBSERVER
where
The follo''\o~il1g auxiliary lemma reformulates in a 1110re suitable form SOlne results known from the theory of linea.r observers.
LEMMA (NikifofOV, 1998). Along with (4)-(5) consider the dynamic system.
( ==
GC + bt?,
(9)
tvhere ( E Rg is the .state.. and the pair (G , b) is controllable. Then for any q x q Hurwitz matrix G tit ere- exists a ltn'ique constant vector 8 E R~ suclt that the signal {) can be presented in the fornl
(10) lDhere the exponentially decaying 'vector b" obeys the f'q1lalion
<:
is generated by the nonlinear obser'ver
(11)-(13), () is a constant unknown vector and 8 obeys the equation
8
== G6-b~
(15)
with the initial state 6(0) = tt,( z(O)~ ~(O»).
.l\t1X(O) - 1](0) -
Proof. Differentiating Dl.p :::::; ( - TJ - W \vith respect to tinle in vieVv- of (6),(7),(9) and (12)-(13) we obtain
GC: 81j,J
+ brfJ -
Gr] - G1/J
f3
+
81j; E(
8e
+
alP
-/0 + bu + b- - - E { 8z a 8{ 8~
-d[fi + au + o
G((-17-1/J)-b~
a~ -10
az
=
G04'-b~
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Copyright 1999 IFAC
ISBN: 0 08 043248 4
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EX...
=
with 8'P(O) ((0) - 1}(O) - ,b(z(O), ~(O)). Then substituting ( = 1] + 1P + 6:p = (' + 6:p into (10) Vt,~e obtain T {J =:: e ( + eT(o{J + olfJ)' Introducing the vector D ==: 6f)
+
14th World Congress of IFAC
the observer (11)-(13), and j..t > O. To adjust the vector Bin the presence of unmodeled disturbance ~, we use the following algorithm of adaptation (IoanIlou, 1986)
81,(/, we readily
obtain (14)-(15).
(20)
0
181 <
Corollary. If the disturbance r.p does not contain an unmodeled component (i~ e., ~(t) == 0), then it can be presented in the form. (16) where the exponentially decay'ing vector 6(t) obeys [he eq1Jation iJ === G8, where 8(0) == M X(O) 1] ( 0) -lb ( ~ ( 0) 0) ) . f
e(
R.fluark .1.1. Equation (13) is essentially simplified if the vector b is chosen in the forIll b == col(O, 0, ... ,1) (with matrix G being, for example, in the phase-variables canonical forIll). III this case the vector...function ti'(z, €) can be chosen as -r;:~(.;) = col(O, O~ .. ~ , 1P q (Z, e)) where the function l{)g (z, ~) is a solution to the first order partial differential equation
1
0:( Z, ~) .
(17)
()*
Let us int.roduce the vector of tracking errors ) with coordinates
()*
is any positive constant and T
=
£j
==
~j
_
y~j -
1] ,
THEOREM 2.
Consider the closed-loop system consisting of the plant (1)-(2) subjected to the unknown disturbance (3)-(5), nonlinear observer (11)-(13) and adaptive controller (19)-(21). Let equation (6) be input-ta-state stable when ~ is seen as an input Then under assumptions (i)-(vti): (aj all the closed-loop signals are globally bounded,~ (b) the residual trackin.g error can be made arbitrary small by increasing the feedback gain j-lJ and for all J1. > /la > 0 and any /Jo > 0
= Ec: + d[ - ~PJ + )3(z,e) + cr(z,e)[u
,vhere
El
=Y -
+ (JTe + (JT (j + ~]J,
(18)
Yr.
The proposed adaptive regulator is given by
(Jolt<
> \~ the
lim [y(t) - Yr(t)]
t-oo
(22)
==
aSyTuptotic
O.
Proof. First, we establish boundedness of all the closed-loop signals. Substituting (19) into (18) and differentiating (j == e - (j in vie"" of (20), Vole obtain the following error model
i
e
+ o:d(iI' ( + eT 0 + ~ -jQ;(dT Pc - -P'j(f + J(7(}.
Ae
C~onsider
/-lET
Pda) (23) (24)
the Lyapunov function ;r.
\"..here (j E Rq is the vector of adjustable paranletel's! the vector of feedback gains k -=: coI(k 1 ,k 2 ' . " l k p ) is chosen so that the Inatrix ~4 ~ E - dk T is Hur\vitz! the positive-defined lllatrix P is a solution to the matrix equation .A. T p + P A == -1, the estimate" is produced by
1
..fii"')
where th.e value of K, depends on a preselected con-
It can be sho\vn that the error model, in view of equations (7) and (14)~ takes the form E:
> O.
The stability properties of the closed-loop system are established by the following theorem.
stant ).la; (e) when ~(t) == 0 and tracking is achieved, i. e.
j == 1, 2 ~ ... , p.
(21)
Remark 4.1. In the adaptive controllaVw" (19)-(21) the functions a and {3 can be expressed in any coordinates (x or (z, ~)). Therefore, the argurnent of these functions is omitted. 0
limt_:>:> (c:(t)t :::; E
s 181 ::; 20*
la, > 28*, ""here
4. AD.A.PTI'lE REGULAT'OR C.Ol(El' £2~ . . . ,E p
0*
-
\I e (c,8)
==
1 T 1 nr2 c PC++ 2-yB e.
Its tirne derivative along the solutions of (23)-(24) satisfies:
4509
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EX...
or a(d
-1-l (eT Pda)2 -
T
0"1812 + a(JI'e
P€ -
< _~ _ 0"(8),Op _ (~IET Pdal _ ~6 2
2.Jii
2
-(7(B)(-1_ I01
- vI2 + O"~) 181 2 ::;
)2
14th World Congress of IFAC
Consider the derivative of solutions of (23)
2
V2
+2Je'T PdQ!a~ - 2JllcT PdaJ2. (26)
4tJ
_~\eI2 _ lT~) 101 2+ A,
where the positive constants inequalities T
~o
Taking the boundedness of (" and (j into account)
and Ll obey the
2
1
-
~t5~le o(t)+~(t)[,
eT Pc along the
-lc:l 2 + 21c·T PdallfTBI
<
__1_ tol ) + ~~~
lel;' ==
we can write
1. [2
~~4p.Ac+21e.
\-vhere
J1(JICXJ does not depend on 11. (see equat.ion
(14)) . It. is easy t.o see that V < 0 if JcJ2 > 2~ or l{l~2 2: max{ (28* + 181)2, 2~} (inequality lei 2 ~ (20* + le\)2 guarantees that 101 > 20* and, as a consequence! 0"(0) == 1). Then after simple calculations it can be sho\vn that
(25)
=
where Co = min{l(P), ~}, R 2 max{(20'" + lel)2. 2~}, ~(P) is the minimum eigenvalue of the Inat.rix P. \'ia. condition (25) V·le conclude that £(t) and O(t) are bounded. Since equation (6) has the ISS property, boundedness of means boundedness of ;; and, as a consequence, boundedness of x. rrom ~rheoren1 1 ,ve have boundedness of (, and, therefore, from equation (19) we straightfor\~lard derive boundedness of u, and from equation (12) \Ne obtain boundedness of 'l}~ Thus the boundcdness of all the closed-loop signals is proved.
Denoting
(27) we rewrite (26) as
~ lel1> ::; -lef + 2k T PdalLi" = _).::12 _ 2
e
Noyv \ve prove part (b). By virtue of condition (25) we have that for all t > 0 l-~~(t)
S V ==
max{~~(O),
VD}
~
(v'iJ1cT Pdal _
2 -6 I 1
2,uleT Pdaj2
At ) 2Vii
2
+
D,; 21-"
1 1 12 ~; +d; - ::; --£p+-, 2p Cl 2p
where Cl = l(p) is the maximum eigenvalue of the matrix P. Therefore
!e(t)l:::;
(C1e-2~ltlC:(O)I+J
V~
Cl
2C2P
AE,
=
where C2 J..{P) is the nlinimum eigenvalue of the matrix P. The latter inequality means the validity of (22), where
and, hence,
1 v~ (t) = '2c:(t)T Pc:(t) ]
-
+
1 -... 2.y"'(t)T 8(t)
Taking into account the fa.ct that constants .68. 11(1100, 1':1 and K'2 do not depend on p" we conclud~ that the residual tracking error can be made arbitrary small by increasing J.1. The proof of part (b) . is completed.
-..
21' 8(t)T e(t). Denoting
0'2 ==
2'·/'/, we can write
=
The upper bound 7f might depend on Vu. In turn VD depends on JA. Taking expressions for Vo, Rand ~ into account, we conclude t.hat the following inequality is valid -2
e
<
~1
where
8
1\.2 + -,
=::
G6.
(29)
Consider the Lyapunov function
J-L
where the values of the positive constants fC'] do not depend on the feedback gain p.
Now Vv~e prove part (c). \Vhen ~(t) 0, the dynamic lllodel of the closed-loop system in coordinates € t.akes the fornl
K1
and
-V(£, 6, 0)
1
== 2"c
T
Pe
+ kJo
T
1 --T-''-' P6 D + 2", B {)1
4510
Copyright 1999 IFAC
ISBN: 0 08 043248 4
ADAPTIVE NONLINEAR SERVOCOMPENSATION OF UNKNOWN EX...
"\vhere the positive defined matrix p~ satisfies equation GT Pf; + P 6 G == - I and the positive gain kfJ is equal to (°1 2 / p,. The time derivative of this function along solutions of (24),(28) and (29) has the form
_~lcI2 + eT PdCl'.(I' 0 + eT PdCl'.OT6 -j1(€ T
Pda)~ K)
101 2
0)
-J6J'" p.
_1
.--om"""
()J.
T
u(d PE;
+cr(O)OT (j
_~lcI2_ ~k8'8122
4
(v'liIc
T
PdCl'.l-
1 2 -2!c l -
2~'6If + uco)8Te
3 2 4k616(
--/iT+ 0-(0)0 O.
{}* > IBI (in accordance with the accepted additional assumption), we have tT(e)'OTo ~ 0 for any 6. 'fherefore
Since
14th World Congress ofIFAC
(1994) On the use of adaptive precompensators for rejecting output disturbances in non minimum phase 8150 systems. Prepr. 33rd IEEE Cont. Dec. and Control, Florida, va!' 2, pp.1183-1184. Byrnes, C. 1. and A. Isidori (1991) Asymptotic stabilization of minimum phase nonlinear systems. IEEE Trans. Autom. Contr.~ 36(10), pp.11221136. Davison, E. J. (1976) The robust control of a servomechanism problem for linear time-invariant multivariable systems. IEEE Trans. A utom. Contr.~ 21(1)~ pp.25-34. Desoer, C. A. and C.-A. Lin (1985) Tracking and disturbance rejection of MIMO nonlinear systems with PI controller. IEEE Trans. Autom. Conir., 30(9), pp.861-867. Elliot, H. and G. C. Good\vin (1984) Adaptive implementation of the internal model principle. Proc. 23rd IEEE CDC, pp.1292-1297. Francis, D. A. and W. 1\1. Wonham (1975) The internal model principle for linear rnultivariable regulators4 Appl. A-fath. Opt., 2: pp.170-191. Gang Feg and M. Palaniswami (1991) Unified treatment of internal model principle based adaptive control algorithms. Int. J. Contr., 54(4) pp.883-901. Ioannou P. (1986) Robust adaptive controller with zero residual tracking error. IEEE Trans. on A utorn. Control, 31(8), pp.773-776. Isidori, A. and C. I~ Byrnes (1990) Output regulation of nonlinear systems. IEEE Trans. A utom# Contr'1 35(2)1 pp.131-140. Johnson) C. D. (1971) Accommodation of external disturbances in linear regulator and servomechanism problems. IEEE Trans. Altto'm. Contr., 16(6), pp.635-644. Khalil, H. K. (1994) Robust servomechanism output feedback controller for feedback linearizable systems. A utQmatica, 30( 10), pp.1587-1599. !(rstic, 1\1., I. Kanellakopoulos and p~ Kokotovic (1995) Nonlinear and Adaptil'e Control Design, New York: John VlilJey and Sons. Nikiforov, O. (1996) Adaptive servocompensation of input disturbances, Prepr. 13th IF:4C World Congress, San-Francisco, USA, vol.K, pp.I75-I80. Nikiforov , V. O. (1998) Adaptive non-linear tracking with complete compensation of unknown disturbances. European Journal of Control, 4(2), pp.132-139. Wonham W. M. (1979) Linear Mu ltiv ariable Control: A Geometric Approach, New York: Springer- Verlag. I
Since x, (, EJ are bounded and a is smooth, we conclude that the right part of (28) is locally Lip-
schitz in
E
uniformly in t. Then, from the LaSalle-
Y"oshiza\i\.'a theol'eln (see Krstic et al., 1995) we cOIlc]ude~ that c:(t) --+ 0 as t --'I- 00. In part.icular it. Illeans that y(t) ~ Yr(t) ~ 0 as t ---;. ()(). 0
5. CONCLUSION The problem of robust adaptjve servocompensation of unkno\'1n bounded external disturbances is solved for a class of nonlinear systems being globa.lly diffeomorphic to the normal form. The main contributions of the paper are the following: i) nonlinear observer for a class of unknown disturbances (Theorem l)j ii) nonlinear regulator ~rith robust adaptive servocompensation of unknO\~ln disturbances CTheorem 2). ,"'hen the a priori unknown disturbance does not contain unmodeled conlponents~ the proposed adaptive regulator provides global asyrnptotic tracking. If the disturbance contains unIllodelcd cOlnponents~ the proposed controller provides boundedness of all the c.losed-loop signals and arbitrary small tracking er~ ror.
'T.
1
REFERENCES Di Benedetto, 1\'1. D~ (1987) Synthesis of an internal model for nonlinear output regulation. Int. J. Contr., 45, pp.l023-1034. BonilIa, M. E., J~ A. Cbeang and R. Lozano
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Copyright 1999 IFAC
ISBN: 0 08 043248 4