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An extended direct scheme for robust adaptive nonlinear control
0005-1098/91 $3.00 + 0.00 Pergamon Press plc © 1991 International Federation of Automatic Control
Automat/ca, Vol. 27, No. 2, pp. 247-255, 1991 Prin~d in Great Britain.
An Extended Direct Scheme for Robust Adaptive Nonlinear Control* I. K A N E L L A K O P O U L O S , t §
P. V. K O K O T O V I C t
and R. MARINO~t
A new direct adaptive control scheme is developed for nonlinear systems satisfying an extended matching condition, and shown to be robust with respect to unmodeled dynamics. Key Words--Adaptive control; nonlinear systems; extended matching; robustness; unmodeled dynamics.
growth" condition. To stress the practical importance of nonlinearities which are not globally Lipschitz, let us remind the reader that they are c o m m o n in mechanical systems with centrifugal forces, in electrical systems with flux-current or flux-speed products, in chemical kinetics, etc. The most recent works of P o m e t and Praly (1989b), Bastin and C a m p i o n (1989) and Campion and Bastin (1990) have m a d e significant progress toward removing the global Lipschitz conditions for systems without unmodeled dynamics. Unfortunately, the applicability of the simple direct scheme, unlimited by the type of nonlinearity, is limited in its dependence on the unknown constant parameters. While, as in most other schemes, this dependence is assumed to be linear, a further restriction is that the unknown parameters appear only in system equations with control variables. Only a narrow class of nonlinear systems satisfies this strict matching condition exactly. To broaden this class, Taylor et al. (1989) use matched reduced-order models and treat the unmatched terms as u n m o d e l e d dynamics. For example, a matched reducedorder model of an electric m o t o r with uncertain load is only its mechanical equation, while all the electrical p h e n o m e n a are to be treated as unmodeled dynamics. This p a p e r introduces an extended matching condition which further broadens the applicability of the simple direct scheme without sacrificing any one of its advantages. T h e unknown constant p a r a m e t e r s are now allowed to appear also in equations separated from the control variables by one integration. As an illustration, an electrical equation can now be added to the above-mentioned model of an electric motor. This m a k e s the effects of unmodeled dynamics less significant.
Abstract--The proposed adaptive scheme achieves regulation for a class of nonlinear systems with unknown constant parameters and unmodeled dynamics. The scheme does not employ overparametrization and does not restrict the class of nonlinearities by any growth conditions. Instead, the dependence on the unknown parameters is restricted by an extended matching condition, which, however, is satisfied in many systems of practical importance, such as most types of electric motors. 1. INTRODUCTION THE DIRECt adaptive regulation scheme of Taylor et al. (1989) is, whenever applicable, a simpler alternative to m o r e elaborate schemes ( N a m and Araposthathis, 1988; P o m e t and Praly, 1989a; Sastry and Isidori, 1989). In addition to its simplicity, the direct scheme has also a robustness property with respect to u n m o d e l e d dynamics which are present in most engineering applications. An overview by Sastry and Kokotovic (1988) shows that similar robustness properties are yet to be established for other nonlinear adaptive schemes. A n o t h e r m a j o r advantage of the simple direct scheme is its applicability to systems with nonlinearities which are not globally Lipschitz, like x 2 or XlX2. In this regard, other schemes are m o r e restrictive because they assume that the nonlinearities are globally Lipschitz or satisfy some "linear * Received 30 March 1989; revised 2 February 1990; received in final form 15 June 1990. The original version of this paper was presented at the IFAC Symposium on Nonlinear Control System Design which was held in Capri, Italy during June, 1989. The published proceedings of this IFAC Meeting may be ordered from: Pergamon Press plc, Headington Hill Hall, Oxford OX3 0BW, U.K. This paper was recommended for publication in revised form by Associate Editor C. C. Hang, under the direction of Editor P. C. Parks. ? Coordinated Science Laboratory, University of Illinois, 1101 W. Springfield Ave., Urbana, IL 61801, USA. Dipartimento di Ingegneria Elettronica, Seconda Universita di Roma "Tor Vergata", Rome 00173, Italy. §Author to whom all correspondence should be addressed. 247
I. KANELLAKOPOULOS et al.
248
The extended direct scheme presented in this paper is a nonlinear version of a direct scheme proposed for linear systems of relative degree two; see, e.g. the recent book by Narendra and Annaswamy (1989). However, there is a major difference between the linear and nonlinear versions. In the linear case, the relative-degreetwo scheme is a step toward a more general scheme. In the nonlinear case no general scheme is yet available which can handle as broad a class of nonlinearities as the extended direct scheme. The organization and style of this paper stress simplicity and applicability, rather than theoretical novelty, of the proposed extended direct scheme. Although the same scheme can be developed for multi-input systems (Kanellakopoulos et al., 1989, 1990), the presentation is focused on the single-input case. As a further simplification, the scheme is developed for systems in a special form in which the meaning of the extended matching condition is obvious. A more general class of nonlinear systems, transformable into this special form, is defined in the Appendix. The stability proofs in this paper make use of Lyapunov functions which, since an early paper by Parks (1966), have been a standard adaptive control tool. The tutorial character of the paper is enhanced by a position control example which illustrates analytical derivations and gives a hint of potential applications. 2. THE EXTENDED DIRECT SCHEME The nonlinear plant with an unknown constant parameter vector a = [aa . . . . . ap] r is assumed to be in the form
P =f0(z) + ~ aif~(z) +go(Z)U, i=1
(2.3)
Before we employ (2.3), we need to know that the mapping x , ~ $ , is one-to-one, onto and continuous. Hence, we assume that there exist Ba c ~P and a constant 6 > 0 such that
1 + ~ a l aWli(X) >-6, V x ~ B x , V a ~ B a . i=1 aXn Then, the last two equations of (2.2) rewritten as
fCn-1 = ~n "31-(a -- ~)rwa(x)
(2.4) are (2.5)
Xn = [V + arw2(X)](1 + ~IT aWl(X)~ Ox,, I (n~l aWl(X) aWl(X) ) + ~r - xj+l + arwl(x) j=l axj axn-~ + ~w,(x) = [v + arw2(x)]t~(x, ~) + + gtrw4(x)a + ~rw~(x).
tVw3(x) (2.6)
Note that fl(x, 4) and the elements of w3(x), w4(x) defined by (2.6) are smooth functions on
l<-i<--n-2
P .iOn-1 =Xn + ~ aiwu(x) =Xn +arwl(x) i=1 P .iOn= v + Z aiwzi(x) = v + arw2(x), i=1
P .rn =Xn + ~'~ ~iWli(X) =Xn +arWl(X). i=1
(2.1)
where f0, fl . . . . . fp, go are smooth vector fields on B, and go(z) is bounded away from zero on B~, a subset of R". Proposition 1 in the Appendix gives necessary and sufficient conditions for the existence of a change of coordinates x = $ ( z ) and a state feedback control u = o¢(z) + fl(z)v, where v is a new control variable, which transform (2.1) into the special form ~=x,-+~,
Note that w~(x) and w2(x ) are smooth vector fields on Bx = #p(Bz). The meaning of the extended matching condition (A.3), formulated in the Appendix, is clearly displayed in (2.2). The original system (2.1) has been transformed into a chain of n - 2 integrators and two nonlinear equations with unknown parameters. The control variable v enters only the last equation and, hence, the strict matching condition of Taylor et al. (1989) is not satisfied unless the vector field wl(x) is identically zero. In the case of extended matching w~(x) is not required to vanish and, hence, unknown parameters are allowed to appear in the last two equations of (2.2). Note also that none of the functions appearing in (2.1) and (2.2) are required to be globally Lipschitz or to satisfy some other linear growth or sector conditions. Instead of the true parameter values a = [al . . . . . ap] r, which are unknown, a controller will be designed using parameter estimates = [ ~ . . . . . ~p]r. The first step in this direction is to introduce a new state in, instead of Xn, using the expression
(2.2)
where the expressions for wx(x) and wz(x) are given by (A.12) and (A.13) in the Appendix.
Bx X Ba. We now proceed to find a control which renders equations (2.5)-(2.6) linear in the parameter error a - t~ and makes them otherwise independent of the unknown parameter vector a. On closer inspection of (2.6), we see that these tasks will be accomplished by a control of the
Robust adaptive nonlinear control form
249
parameter update law 1
v = -a~w2(x)
- [to,x, + . . . #(x, 4)
+ k._,x._,
+ k . ( x . + a~w,(x)) + d~w,(x)l,
"~ a T w 3 ( x ) "~" a T w 4 ( x ) a
(2.7)
which, in addition to nonlinearity cancellation terms, contains a linear state feedback part with constant gains kl . . . . . k,. These gains are chosen to place at some desired stable locations the roots of the characteristic polynomial
s" + k~s "-l +. • • + k2s + kl = 0.
~, = [fl(x, a )WT(X) + flrw4(x)](a -- 4) - - klX 1 . . . . . k,:2,. introduce
= ~?(x, 4) =
the
following
(2.9) compact
x._, x~ + arw,(x) l
rw~(x, a)P~p(x, a),
P A + Arp
= -L
(2.14)
To confirm that the update law (2.13) leads to a differentiator-free implementation of the feedback control (2.7), let us rewrite (2.7) as an explicit function of the available signals x and t~:
o
=
-a~w2(x)
1
l~(x, 4) × [i,x, + . . . + k.(x, + a~w,(x)) + a~w3(x) + a~w,(x)a
+ w~(x)rw~(x, a ) e v ( x ,
4)1. (2.15)
Our next task is to prove stability and a convergence property of the adaptive scheme consisting of the state equation (2.12) and the parameter update law (2.13).
Lemma 1. The equilibrium :2 = 0, a = a, of the scheme (2.12)-(2.13) is stable for every a e Bo. Moreover, there exists a set k~ c R "+p such that from all (:2(0), 4(0)) e ~ the state :2(0 converges to zero: t~--oo
,
ix._,/ L :2. _l
(2.13)
where F is a positive definite matrix ("adaptation gain") and P > 0 is chosen to satisfy the Lyapunov equation
lim :2(0 = O.
[ Xl 1F I =
ti)P:2 =
(2.8)
To verify that the control law (2.7) is implementable on Bx x B~, first observe that I/~(x, 4)1-> a by assumption (2.4). Second, and this is a characteristic of the extended direct scheme, note the presence of the time derivative ,~ of the parameter estimate 4. It may appear that the implementation of (2.7) would require differentiators. Fortunately, this is not so. As we shall see, the parameter update law for t~ will furnish ~ as a known explicit function of available signals. To design this update law, we substitute the control (2.7) into (2.6) and obtain
Then we notation:
= r::(:2,
(2.16)
Proof. Differentiating the Lyapunov function V(:2, a) =:2rp:2 + (a - a ) r F - l ( a - 4) (2.17) along the solutions of (2.12)-(2.13), we obtain
(: =:2r(ArP + PA):2 + 22rPI¢~:(:2, fi)(a - 4) A=[!
I],
(2.10)
- 2t~rF-l(a - 4) -- -11:2112+ 2[:2rP14:(:2, d ) r - ~ r l r - ~ ( a - 4)
--
1
....
kn
-- -11:2112-<0.
g,(:2, 4) = ~ ( ~,(x, 4), 4) = w(x, 4)
(2.18)
In this notation, the system with feedback control (2.7) is rewritten as
This proves the stability of the equilibrium :2 = 0, a = a. The convergence result (2.16) now follows from LaSalle's invariance theorem, by which (:2(0, a(t))--->M as t---),o, where M is the largest invariant set of (2.12)-(2.13) contained in the set {(:2, 4) ::2 = 0} where 12 = 0. Finally, in view of (2.18), the function V(:2(t), a(t)) is nonincreasing and, hence, a subset of t) is the set
:~ = A:2 + W ( i , a)(a - 4).
~ v = {(:2, 4) : V(:2, 4) - c},
I
=
0 0
... .-.
0 0
]
w~(x) /~(x, a ) . , f ( x )
+
. (2.11)
a~w.(x)l
(2.12)
This "error form" shows that asymptotic stability is achieved when the estimate t~ is correct, a - t~ =- 0, because (2.8) is the characteristic polynomial of A. We now prove that, even when ~a, stability will be achieved with the AUTO
27:2-C
(2.19)
where c is the largest constant such that
f~v = {(x, a): v(~e(x, a), 4) <_ c} c Bx x Bo. (2.20) []
I. KANELLAKOPOULOSet al.
250
Our final task is to prove the same stability and convergence properties for the actual adaptive system
Yci=xi+l,
lira xn-l(t) = xn_l(0) + lira
(2.26) By Barbalat's lemma [see, e.g. Sastry and Bodson (1989, p. 19)], the uniform continuity and integrability of 2,-1(0 imply
1
2. = (a -
8) + . . • + k . x , + k,8rw
(x)
(2.21)
lim ~ _ l ( t ) = lim (x~(t) + arw~(x( t) ) ) = O. t---~
+ 8 w (x) + 8 +w
2~_1(~) d r = 0.
l<-i<-n-2
YC,_l = x, + arwx(x)
x
have
(x)rw
(x,
(2.27) 8)]
= V W r ( x , 8)PW(x, 8). This system differs from the scheme (2.12)(2.13) only in the last state, which in the scheme is $,, while here it is x,. To determine the equilibrium x = x ~, 8 = a of the system (2.21) which corresponds to the equilibrium ~ = 0 , 8 = a of the scheme (2.12)-(2.13), we note that at x~ = x2 . . . . . x,_~ = 0 the equation
.fc,_~ =x,, + a r w ~ ( x ) = O
(2.22)
has, because of (2.4), a unique solution x~, for each a • B~. A direct substitution proves that X =X e
t---~ oo
w,(x)8
~ r, = [0 • • " 0 x~]
8 =a
(2.23)
is the equilibrium of (2.21) corresponding to :f=0, 8=a.
Theorem 1. The equilibrium x = x ~, 8 = a of the adaptive system (2.21) is stable for every a e B, and the set ~2v defined in (2.20) is a subset of its region of attraction. Moreover, for all (x(0), 8(0)) e Qv, the state x(t) converges to its equilibrium value x e, that is, lim x(t) = x e.
(2.24)
Proof. Lemma 1 proves the stability of $ = 0, 8 = a . On the other hand, the mapping (x, 8 ) ~ ( $ , 8 ) is one-to-one, onto and continuous and it maps the point (x e, a) to the point (0, a). This proves the stability of x = x e, 8 = a, and, furthermore, implies that the solution (x(t), 8(0 ) of (2.21) for any ( x ( 0 ) , 8 ( 0 ) ) • g 2 v is uniformly bounded and remains in Bx x Ba for all t ->0. Differentiation of kn-i = x , + arwl(x) in (2.2) and the use of (2.6) gives xn-1 = [v + arw2(x)]fl(x, a) + arw3(x) + arwa(x)a.
(2.25)
Recall that all the functions appearing in (2.25) are smooth on B~ × Ba, including, because of (2.4), the control v as expressed by (2.15). Therefore, Y,_~(t) it also uniformly bounded and ~,-l(t) is uniformly continuous. From (2.16) we
Combining (2.16), (2.22) and (2.27) with the fact that Wl(X) is a smooth vector field, we conclude that lim x, (t) -- x,.e (2.28) [] 3. ROBUSTNESS TO U N M O D E L E D DYNAMICS
The stability results in the preceding section assume that, up to a set of unknown constant parameters, an exact model of the plant is available. A crucial question to be addressed in this section is whether, and to what extent, these stability properties are preserved in the presence of unmodeled dynamics. As in some earlier works on adaptive control, such as Ioannou and Kokotovic (1983) and Taylor et al. (1989), the unmodeled dynamics are assumed to be fast and are treated as singular perturbations, that is, the change of model order is parameterized by a scalar #. When # > 0, the unmodeled dynamics are included and the model order is higher than assumed in the adaptive scheme design. When = 0, the unmodeled dynamics vanish and the model reduces to the assumed design model. In applications, the parameter # is used to represent small inertias, inductances, time constants etc., as illustrated by many physical examples in Kokotovic et al. (1986). Starting with the design model in the special form (2.2), we perturb it in the following way: 3Ci---~Xi+l,
l <_i<_n-2
X n - 1 = Xn -[- a T W l ( X )
Yc, = arw2(x) + r r ( x ) ~ + g l ( x ) u
(3.1)
~ = Q ( x ) ~ + gz(x)u, where ~ • R v is the state of the unmodeled dynamics, and the entries of r(x), Q ( x ) , gl(x) and gz(x) are smooth functions on Bx. Within the framework of singular perturbations, the simple form of unmodeled dynamics included in (3.1) is sufficiently general, because, as shown by Kanellakopoulos (1989), other perturbed versions of the general model (2.1) can be transformed into (3.1). The relationship of (3.1)
Robust adaptive nonlinear control with the unperturbed design model (2.2) will become clear after the change of variables
= h(x, u) + rl,
(3.2)
which exhibits the function
h(x, u ) = - a - l ( x ) g 2 ( x ) u
Re~{a(x)}<--al,
VxeBx.
(3.4)
This, in turn, assures the existence of Q - l ( x ) in (3.3). Along with the change of state variables (3.2), we introduce the following change of the control variable v = [gl(x) - rr(x)a-l(x)g2(x)]u = g(x)u
(3.5)
and assume that there exists a2 > 0 such that Ig(x)l - 32
Yx • Bx.
These changes of variables perturbed model (3.1) into
transform
ffi=Xi+l, l<--i<--n - 2 2,-1 = x, + arwl(x) 2, = v + arw2(x) + rr(x)r/
(3.6) the
and see that/~ = 0 at } = 0, d = 0. On the other hand, for t / = 0 the first two equations of (3.9) represent the unperturbed scheme (2.12)-(2.13), which has an equilibrium at 2 = 0, a = a. For simplicity, we now make the additional assumption that wl(0)= w2(0)= 0. Under this assumption 2 = 0 implies x = 0, that is, the equilibria of the adaptive scheme (3.9) and of the adaptive system are the same, and, furthermore, W(O, fi) = 0 for all a • B,. As in Taylor et al. (1989), the stability of the equilibrium 2 = 0, ~ = a, r/= 0 of (3.9) will be investigated using the composite Lyapunov function
+ c2r/r/3r (2, t~)t/,
(3.8)
= A2 + 1?¢(2, a)(a - ~) + R(2, d)~l
a)P2
/u//= 0 ( 2 , a)r/-/u/~(2, a, 77, a),
(3.11)
where c~ and c 2 are positive constants and ~ ( 2 , ~) is the positive definite solution of
is expressed as a function of x, a, r/ and a. The expression for 2 as a function of x, a, t / a n d v is given by (3.7), while the expressions for v and 1) as functions of x, t~, r/ and a can be obtained from (2.15) and (2.13). It is now obvious that for r/= 0, the perturbed model (3.7) reduces to the assumed design model (2.2). However, when /z > 0 , we cannot expect that r/(t)-~ 0 even if t/(0)= 0, because, in general, h(x, a, r/, a) is not zero. As we shall see, the stability properties established by Theorem 1 will be preserved if the term /~/~(x, d, t/, a) is sufficiently small. The analysis leading to such a result retraces the derivations of Section 2 for the perturbed model (3.7). The adaptive scheme for the perturbed model (3.7) employs the same control (2.15), the same update law (2.13) and the same notation (2.10), (2.11) as in Section 2. The resulting adaptive scheme with unmodeled dynamics is
=
i(2, ~, rl, a) = f~(2, a)} + ~a(2, a)d
~ ( 2 , a)0.(2, d) + QT(2, ~)~(2, ~) = - I . hu i)
(3.10)
(3.7)
where the time derivative
hx - h, --~ gx J¢ +
(v)
/~(2, a) = h X,g--~
Vc(£, a, tl) = cl[£rP£ + (a - fi)rF-l(a - ,~)1
~il = a(x)rl - Mi(x, a, ri, a),
t;t =
where 0 ( 2 , d) = Q(x), /~(~, a) = [ 0 . . . 0 r(x)] r and /~(2, a, r/, a) = h(x, a, r/, a). To verify that 2 = 0, a = a, r/= 0 is an equilibrium of (3.9), we return to (3.3) and (3.8). Then, with the help of (2.15), we define
(3.3)
as the quasi-steady-state of ~, and r/ as its fast transient. The unmodeled dynamics are assumed to be asymptotically stable for all fixed x • Bx, that is, there exists ol > 0 such that
251
(3.9)
(3.12)
The time derivative of Vc along the solutions of (3.9) is
(Z = c , [ - 2 r 2 + 22rp#(2, a)r/] + c= - -~ ,7",7 + ,7 PAx, ~, ,l, a),7 m
2r/rPi(2, a)/~(2, ~, ~/, a)].
(3.13)
The function /~, as defined in (3.10), satisfies /~(0, a, 0, a) = 0 for all a, a • B,. Hence, it is bounded by
11i(2, a, n, a)ll--~P111211 +02 Ilnll (3.14) where the constants Pl, P2 are such that for all x • Bx, ~ • B., a • B. the following inequalities hold:
IIh~(~, a)~(~, a)(a -~)11 ~ po 11211 (3.15) IIh~(~, ~)A +/~a(2, ~)Fff'r($, ~)ell + p o ~ p l (3.16)
IIh~(.e, a)#(.,L a)ll-
(3.17)
Furthermore, we choose constants Cl, c2 and c3
252
I. KANELLAKOPOULOSet al.
which satisfy 2
IIPA~, a)11 pl ~
Cl
(3.18)
2 Ileg(~, 8)11-< c2 2 IIP:(~, a)ll 02+ I1~(~, 8, rl, a)[[-c3
(3.19) (3.20)
for all x • B~, 8 • Ba, rI • Bn, a e B~. A stronger effect of the fast variable r/ manifests itself as an increase in the values of c~, c2 and c3. This is particularly clear for c2, because c2 bounds the matrix/~ through which 7/ enters into the adaptive scheme (3.9). Constants c~ and ca show the effects of r/ through the properties of the Lyapunov matrix ~. Using the bounds (3.14)-(3.20), we obtain from (3.13):
V'c------[llxll[l"[I][
c1
--CLC2 1
L ClC c2( [ II~ll].
(3.21)
x LIInllJ
This bound leads to the following robustness result: Theorem 2. The equilibrium x = 0, 8 = a, r! = 0 of the perturbed adaptive system (3.7) with the feedback (2.15) and the update law (2.13) is stable for every a e Ba and for every/~ satisfying
1 0 < #
ClC2 -I-C3
(3.22)
with Cl, c2, c3 as defined by (3.18)-(3.20). A subset of its region of attraction is the set ~ = {(x, 8, 7/): Vc0p(x, 8), 8, r/) <--c}, where
(3.23)
c is the
largest constant such that Moreover, for all (x(0), r~(O), r/(O)) • f2~, and /~ • (0,/~*), the state (x(t), rl(t)) converges to its equilibrium value, that is ~ c c Bx x Ba x B~.
limx(t) = 0 ,
lim r/(t) =0.
(3.24)
Proof. If (3.22) is satisfied, the matrix in (3.21)
is positive definite and thus l?c is negative semidefinite. This proves the stability of the equilibrium £ = 0, 8 = a, r/= 0 of the perturbed adaptive scheme (3.9). Hence, the equilibrium x = 0, 8 = a, T/= 0 of the perturbed adaptive system is stable and an estimate of its region of attraction is the set f2c in (3.23). Furthermore, the largest invariant set of (3.9), contained in the set where l?c = 0, is the set {($, 8, T/) : $ = 0, ~/= 0}. The convergence result (3.24) then follows from LaSalle's invariance theorem and the fact that lim $ ( t ) = 0 implies lim x ( t ) = O. t----~ t----~ []
It may appear that Theorem 2 requires a detailed knowledge of the unmodeled dynamics. In fact, even when nothing else is known about the unmodeled dynamics other than that they are fast and stable in the sense of (3.4), Theorem 2 still provides conceptual robustness bounds. Whatever the unmodeled dynamics, some strictly positive constants Cl, c2, c3 and, hence, #* exist so that the stability of the closed-loop system and the regulation property (3.24) are preserved for all # - #*. In applications, a more detailed description of the unmodeled dynamics may be available. For example, the dynamics of some known parts of the system (e.g. actuators and sensors) may have to be neglected in order to make the system appear in the special form (2.2). (In the next section, a motor position control system is brought to the special form (2.2) by neglecting the power amplifier dynamics.) For a specific application, the time constant is known and the bounds (3.14)-(3.22) can be used to first determine a region Bx × Ba x B~, and then evaluate the set 9c in which the proposed adaptive design is applicable. On the other hand, if g2c is a design specification, then the above bounds can be used to determine the required time constant /~. It is pointed out, however, that these bounds are conservative. Following the outlined procedure, tighter bounds can be obtained by taking into account details of the problem at hand. 4. DISCUSSION AND EXAMPLES
In this section we stress the practicality of the results in the preceding two sections by applying them to a common control problem. We then discuss a further extension of these results. E x a m p l e 1. Position control o f a D C - m o t o r with uncertain load torque and u n m o d e l e d dynamics.
In normalized units, the position control system is described by
-dO -~(3) dt dto
P
- - = i + Y~ ajwl/(O, to) dt j=~
di
T,,,
(4.1)
~=-~-e (--to--i + ~)
dg .-dr= - ~ + gu, where 0, to, i and ~ are the motor position, speed, and armature current and voltage, respectively, while the control variable is the input u into the amplifier with constant gain g.
Robust adaptive nonlinear control The uncertain load torque is represented by a weighted sum of some known nonlinear functions wls(O, to) with unknown constant weights as, ] = 1 , . . . , p. For illustrative purposes we take p = 1 and w11(0, t o ) = to2. The ratio TflTm of the motor electrical and mechanical time constants was neglected in Taylor et al. (1989) in order to satisfy the strict matching condition. Here this quantity is retained in the model, and the singular perturbation parameter is the amplifier time constant/z. The control objective is to regulate the motor position and speed to the desired constant values 0 = 0 d ~ and t o = 0 . Denoting X l = 0 - - 0 d ~ , X2 = tO, X3 = i, we rewrite (4.1) in the form of the perturbed model (3.1):
253
The perturbed adaptive scheme (3.9) is 21 = X 2
x2 = 23 + (ax - t~i)x22 "~3 = - k l X l
- k 2 x 2 - k 3 x 3 + ( a l - 41)
Tm X 2~1x 3 + ~ - r/
(4.7)
al = F l ~ r ( 2, al)P2 u O = - ,1 - k a .
The largest invariant set of the scheme (4.7), contained in the set where l?c = 0 (see (3.21)), is the set {(2, 41, r / ) : 2 = O , T/=O}. Thus, our scheme achieves regulation of the state (2(0, 7/(t)). 1. The adaptive scheme presented in this paper can be further extended to include the case where unknown parameters enter the control vector field. (In the above example, this would occur if the ratio T~/Tm = a 3 were also unknown.) A detailed analysis of this was presented in Kanellakopoulos et al. (1989). The main difference from the case treated in this paper is that the update law (2.13) depends on the control variable v. Therefore, v and ~ are defined implicitly, and can be expressed as explicit functions of 2 and ~ only in a region S around the equilibrium 2 = 0 , d=a. The boundaries of this solvability region are the manifolds on which the Jacobian of the implicit function defining v is singular. As an illustration, consider the following example. Remark
3C1 = X 2 fC2 : X 3 +
alx 2
(4.2)
Tm
= T , (-x2 - x3 + # ~ = - ~ + gu.
The change of variables (3.2) is now ~ = gu + rl, so that h = gu, kt = gfi, and the perturbed model (4.2), rewritten in the form (3.7), becomes .~1 = X2 fC2 = X 3 +
alx 2
Tm
(4.3)
Tm
3=T, ( - x 2 - x 3 +gu + /M/= - r / - / ~ g t i .
E x a m p l e 2. The system
As our design model (2.2) we use the first three equations with T/-= 0. The meaning of this design model is that the amplifier is replaced by its constant gain g and the voltage gu is applied to the motor armature. The design now proceeds as in Section 2. Using
(4.4)
3~3 = X 3 + a l x 2
and noting that assumption (2.4) is trivially satisfied, we design the control (2.7):
3C1 = ( 1 0 "~ a ) x 2
is controllable for a ~ e - 1 0 . Following development of Section 2, we use 22 = (10 + ~)x2 v
~t
2
k2x2 + k323 + 2alx223 + alx2].
(4.5) This results in the update law (2.13) with
ai)= W(x, 40=
[°1
the (4.9)
1 (10 + 4) 2 [-Xl - (10 + a)x2 - tlXz + (10 + t~)x~]
(4.10)
and reduce (4.8) to the error form (2.12) .~1 = x2 + x2(a - 4)
v = -[kzXl +
(4.8)
x2 = -x32 + (10 + a ) v
x2 = - x l - 22 + (10 + a ) v ( a - ~), which is exponentially stable for a = a . Lyapunov function
(4.11) The
V(2, t~) = 1.5x21 + XlX 2 "1- 2 2 + (a -- 4) 2 (4.12) results in the update law
.
L2alXLl
(4.6)
~t = 1.5x~x2 + 0.5(10 + a)x 2 + (10 + 4)
x [0.5Xl + (10 + a)x2]v.
(4.13)
254
I. KANELLAKOPOULOSet al.
Equations (4.10) and (4.13) implictly define v and ~ as functions of x and ~. Eliminating t~, we get (10 + ~)[(10 + ~) + xz(0.5Xl + (10 + d)x2)]v = - x l - (10 + a)x2 - 1.5XlX~ + 0 . 5 ( 1 0 + d ) x 3.
(4.14)
To obtain v explicitly, the term multiplying it must be nonzero, which means that the adaptive scheme is defined only in a solvability region S c Bx x Ba. By inspection of (4.14), a subset of S is, for example
{(xl, x2, ~) : Ix~l < 2, Ix21 < 2, 15 + ~1 < 3}.
(4.15) []
R e m a r k 1 (cont'd). In general, the solvability
condition for systems mentioned in R e m a r k 1 requires that in the formulation of all the results of this paper, starting with the region of attraction estimate ~ v in (2.20), the set Bx x B, be replaced by the solvability region S c Bx x B,. An approach which avoids the implicit definition of v and ~ was proposed by P o m e t and Praly (1989b). For (4.8) this approach consists in treating the p a r a m e t e r a appearing in the term (10 + a ) v as a second unknown p a r a m e t e r b to be estimated separately as/~. As a result of this overparametrization, the update law for ~ no longer depends on v, which now appears in the definition of ~ only. R e m a r k 2. For simplicity, all the results in this
paper are given for the regulation problem. However, with standard technical modifications, the extended direct scheme can be used for the tracking of a given reference Ydes(/) by the output y = x l of the system (2.2). Moreover, zero dynamics can be added to (2.2) under the same assumptions as in Sastry and Isidori (1989). Of course, the invariance theorem of LaSalle does not apply to this time-varying problem, and the proofs of the tracking analogs of L e m m a 1 and T h e o r e m 1 employ the standard uniform continuity argument and Barbalat's lemma. Details and further extensions can be found in Kanellakopoulos (1989). To achieve robustness of the tracking scheme to unmodeled dynamics and bounded disturbances, one of the update law modifications proposed in adaptive linear control, such as the a-modification of Ioannou and Kokotovic (1983), can be used. If this simple a-modification scheme is used, robustness properties analogous to those established for the linear case in T h e o r e m s 5.2.1 and 6.5.1 of Ioannou and Kokotovic (1983) can be shown to hold in the nonlinear case.
5. CONCLUSIONS The goals of the adaptive scheme proposed in this paper--simplicity, applicability to a wider class of nonlinearities and robustness to unmodeled d y n a m i c s - - h a v e been achieved under the extended matching condition. Although this condition restricts the dependence on the unknown parameters, it is satisfied by m a n y systems appearing in engineering applications. In a manner similar to that illustrated by E x a m p l e 1, the extended adaptive scheme is applicable to switched reluctance (Taylor, 1988) and permanent magnet stepper (Bodson and Chiasson, 1989) motors, and to other electromechanical or electrohydraulic systems with load and power input uncertainties. It would seem, therefore, that when the extended direct scheme is applicable, it should be given preference over more elaborate schemes. For problems to which this scheme is not applicable, the choice, or the development, of an appropriate scheme is a topic of ongoing research (Sastry and Isidori, 1989; Pomet and Praly, 1989a, b; Bastin and Campion, 1989; Campion and Bastin, 1990). Acknowledgements--This work was supported in part by the National Science Foundation under Grant ECS 88-18166 and in part by the Air Force Office of Scientific Research under Grant AFOSR 90-0011. REFERENCES Bodson, M. and J. Chiasson (1989). Application of nonlinear control methods to the positioning of a permanent magnet stepper motor. Proc. 28th CDC, Tampa, FL, 531-532. Bastin, G. and G. Campion (1989). Indirect adaptive control of linearly parametrized nonlinear systems. Preprints of the 3rd IFAC Symposium on Adaptive Systems in Control and Signal Processing. Glasgow, UK.
Campion, G. and G. Bastin (1990). Indirect adaptive state feedback control of linearly parametrized nonlinear systems. Int. J. Adaptive Control Signal Proc., 4, 345-358. Ioannou, P. A. and P. V. Kokotovic (1983). Adaptive Systems with Reduced Models. Springer, New York. Kanellakopoulos, I. (1989). Adaptive feedback linearization: stability and robustness. M.S. Thesis, University of Illinois, Urbana. Kanellakopoulos, I., P. V. Kokotovic and R. Marino (1989). Robustness of adaptive nonlinear control under an extended matching condition. Preprints IFAC Syrup. on Nonlinear Control System Design, Capri, Italy, 192-197. Kanellakopoulos, I., P. V. Kokotovic and R. Marino (1990). Robust adaptive nonlinear control under extended matching conditions. Technical Report UILU-ENG-902202 (DC-115), Coordinated Science Laboratory, University of Illinois, Urbana. Kokotovic, P. V., H. K. Khalil and J. O'Reilly (1986). Singular Perturbation Methods in Control: Analysis and Design. Academic Press, New York.
Nam, K. and A. Arapostathis (1988). A model reference adaptive control scheme for pure-feedback nonlinear systems. 1EEE Trans. Aut. Control, AC-33, 803-811. Narendra, K. S. and A. M. Annaswamy (1989). Stable Adaptive Systems. Prentice-Hall, Englewood Cliffs, NJ. Parks, P. C. (1966). Lyapunov redesign of model reference adaptive control systems. IEEE Trans. Aat. Control, AC-11, 362-367. Pomet, J. B. and L. Praly (1989a). Adaptive nonlinear control: an estimation-based algorithm. In Descusse, J.,
Robust adaptive nonlinear control M. Fliess, A. Isidori and D. Leborgne (Eds), New Trends in Nonlinear Control Theory. Springer, Berlin. Pomet, J. B. and L. Praly (1989b). Adaptive nonlinear regulation: equation error from the Lyapunov equation. Proc. 28th CDC, Tampa, FL, 1008-1013. Sastry, S. S. and M. Bodson (1989). Adaptive Control: Stability, Convergence and Robustness. Prentice-Hall, Englewood Cliffs, NJ. Sastry, S. S. and A. Isidori (1989). Adaptive control of linearizable systems. I E E E Trans. Aut. Control, AC-34, 1123-1131. Sastry, S. S. and P. V. Kokotovic (1988). Feedback linearization in the presence of uncertainties. Int. J. Adaptive Control Signal Proc., 2, 327-346. Su, R. (1982). On the linear equivalents of nonlinear systems. Syst. Control Lett., 2, 48-52. Taylor, D. G. (1988). Feedback control of uncertain nonlinear systems with applications to electric machinery and robotic manipulators. Ph.D. Thesis, University of Illinois, Urbana. Taylor, D. G., P. V. Kokotovic, R. Marino and I. Kanellakopoulos (1989). Adaptive regulation of nonlinear systems with unmodeled dynamics. IEEE Trans. Aut. Control, 34, 405-412.
255
if and only if the feedback linearization (A.2) and the extended matching (A.3) conditions are satisfied.
Proof Sufficiency. It is proven in Su (1982) that condition (A.2) is sufficient for the existence of a function Ol(z): U--* • with the properties (1) (ddp1, ad~-Igo) ¢:0 in U, (d~b~, X ) = O V X e ~.-z. (2) (q~l, Lf0q~,. . . . . L~-lq~,) r = (qh . . . . . q~.)r = q~r(z ) is a change of coordinates in U, such that the system 2 =fo(Z) + u(t)go(Z )
:q=xi+ p l < i < - n - 1 ~. = L~odpl(z) + LeoL~_ldpl(z)u(t).
1 u = LsoL~o_,dpa(z ) ( - L ~ I ( Z )
z ¢ a"
% = sp
(A.1)
f/(x)=[0...0
O<-i<-n - 1 (A.2)
Extended matching condition. l<-i<-p.
(A.3)
Proposition 1. There exists a state feedback control u = oc(z) + fl(z)v, fl(z) ~ 0 in U1, and a state space change of coordinates x = O(z) in U = U1, with ~(0)= 0, such that the system P i=I
becomes in the x coordinates ici = xi+l, l <-i <-n - 2 (m.5)
((i)(i)1 0
,
(A.10)
w,i(x )
w2i(x)] r,
l<-i<--p, (A.11)
Wu(X) = Lf~L~o-Eqal((a-l(x))
(A.12)
w~(x) = Lf~L~o-lq~l(~ - l(x)).
(A. 13)
In conclusion, the state feedback (A.8), applied to the system (A.1), results in
L~ ¢p1(z) n tz~ + 1 2 =fo(z) - LgoL~o_ldpx(z ) ~o~ p LsoL~o-ldpl(z) go(z) v p
+ ~ aifi(z ).
=f0(z) + go(z)ot(z) + go(z)fl(z)v + ~ alfii(z ) (A.4)
p f(n-I = Xn 4- Z a i w l i ( x ) /ffil p Yen : V 4- Z aiw2i(x) i=1
(A.9)
where
are involutive and of constant rank i + 1 in U1.
f~e~,
l<-i<--n-1
and condition (A.3) implies that
Feedback linearization condition. The distributions ad~go},
(A.8)
In the new coordinates x = q~(z), the distribution ~ is
with f0(0) = 0, g(0) :~ 0, fo, fl . . . . . fp, go smooth vector fields in /]1, a neighborhood of the origin.
~ = s p { g o , adfogo . . . . .
4- 11) = ~ ( z ) 4- f l ( z ) v ,
the system (A.6) becomes
p
2 =f0(z) + ~ aifi(z ) + u(t)go(Z),
(A.7)
By using the state feedback (recall that L~oL~o-ld~l(z)~ 0 in U)
:q=xi+x, YCn=V. APPENDIX: THE EXTENDED MATCHING CONDITION Consider the system
(A.6)
becomes in the new coordinates
(A.14)
i=1
When expressed in the new coordinates x = q~(z), the system (A.14) becomes (A.5). Necessity. If there exist a state feedback and a change of coordinates transforming the system (A.1) into the system (A.5), one can directly verify that the conditions (A.2) and (A.3) are satisfied for the system (A.5). Since the conditions (A.2) and (A.3) are invariant under state feedback transformations and changes of coordinates, they are satisfied for the system (A.1). to
Automat/ca, Vol. 27, No. 2, pp. 247-255, 1991 Prin~d in Great Britain.
An Extended Direct Scheme for Robust Adaptive Nonlinear Control* I. K A N E L L A K O P O U L O S , t §
P. V. K O K O T O V I C t
and R. MARINO~t
A new direct adaptive control scheme is developed for nonlinear systems satisfying an extended matching condition, and shown to be robust with respect to unmodeled dynamics. Key Words--Adaptive control; nonlinear systems; extended matching; robustness; unmodeled dynamics.
growth" condition. To stress the practical importance of nonlinearities which are not globally Lipschitz, let us remind the reader that they are c o m m o n in mechanical systems with centrifugal forces, in electrical systems with flux-current or flux-speed products, in chemical kinetics, etc. The most recent works of P o m e t and Praly (1989b), Bastin and C a m p i o n (1989) and Campion and Bastin (1990) have m a d e significant progress toward removing the global Lipschitz conditions for systems without unmodeled dynamics. Unfortunately, the applicability of the simple direct scheme, unlimited by the type of nonlinearity, is limited in its dependence on the unknown constant parameters. While, as in most other schemes, this dependence is assumed to be linear, a further restriction is that the unknown parameters appear only in system equations with control variables. Only a narrow class of nonlinear systems satisfies this strict matching condition exactly. To broaden this class, Taylor et al. (1989) use matched reduced-order models and treat the unmatched terms as u n m o d e l e d dynamics. For example, a matched reducedorder model of an electric m o t o r with uncertain load is only its mechanical equation, while all the electrical p h e n o m e n a are to be treated as unmodeled dynamics. This p a p e r introduces an extended matching condition which further broadens the applicability of the simple direct scheme without sacrificing any one of its advantages. T h e unknown constant p a r a m e t e r s are now allowed to appear also in equations separated from the control variables by one integration. As an illustration, an electrical equation can now be added to the above-mentioned model of an electric motor. This m a k e s the effects of unmodeled dynamics less significant.
Abstract--The proposed adaptive scheme achieves regulation for a class of nonlinear systems with unknown constant parameters and unmodeled dynamics. The scheme does not employ overparametrization and does not restrict the class of nonlinearities by any growth conditions. Instead, the dependence on the unknown parameters is restricted by an extended matching condition, which, however, is satisfied in many systems of practical importance, such as most types of electric motors. 1. INTRODUCTION THE DIRECt adaptive regulation scheme of Taylor et al. (1989) is, whenever applicable, a simpler alternative to m o r e elaborate schemes ( N a m and Araposthathis, 1988; P o m e t and Praly, 1989a; Sastry and Isidori, 1989). In addition to its simplicity, the direct scheme has also a robustness property with respect to u n m o d e l e d dynamics which are present in most engineering applications. An overview by Sastry and Kokotovic (1988) shows that similar robustness properties are yet to be established for other nonlinear adaptive schemes. A n o t h e r m a j o r advantage of the simple direct scheme is its applicability to systems with nonlinearities which are not globally Lipschitz, like x 2 or XlX2. In this regard, other schemes are m o r e restrictive because they assume that the nonlinearities are globally Lipschitz or satisfy some "linear * Received 30 March 1989; revised 2 February 1990; received in final form 15 June 1990. The original version of this paper was presented at the IFAC Symposium on Nonlinear Control System Design which was held in Capri, Italy during June, 1989. The published proceedings of this IFAC Meeting may be ordered from: Pergamon Press plc, Headington Hill Hall, Oxford OX3 0BW, U.K. This paper was recommended for publication in revised form by Associate Editor C. C. Hang, under the direction of Editor P. C. Parks. ? Coordinated Science Laboratory, University of Illinois, 1101 W. Springfield Ave., Urbana, IL 61801, USA. Dipartimento di Ingegneria Elettronica, Seconda Universita di Roma "Tor Vergata", Rome 00173, Italy. §Author to whom all correspondence should be addressed. 247
I. KANELLAKOPOULOS et al.
248
The extended direct scheme presented in this paper is a nonlinear version of a direct scheme proposed for linear systems of relative degree two; see, e.g. the recent book by Narendra and Annaswamy (1989). However, there is a major difference between the linear and nonlinear versions. In the linear case, the relative-degreetwo scheme is a step toward a more general scheme. In the nonlinear case no general scheme is yet available which can handle as broad a class of nonlinearities as the extended direct scheme. The organization and style of this paper stress simplicity and applicability, rather than theoretical novelty, of the proposed extended direct scheme. Although the same scheme can be developed for multi-input systems (Kanellakopoulos et al., 1989, 1990), the presentation is focused on the single-input case. As a further simplification, the scheme is developed for systems in a special form in which the meaning of the extended matching condition is obvious. A more general class of nonlinear systems, transformable into this special form, is defined in the Appendix. The stability proofs in this paper make use of Lyapunov functions which, since an early paper by Parks (1966), have been a standard adaptive control tool. The tutorial character of the paper is enhanced by a position control example which illustrates analytical derivations and gives a hint of potential applications. 2. THE EXTENDED DIRECT SCHEME The nonlinear plant with an unknown constant parameter vector a = [aa . . . . . ap] r is assumed to be in the form
P =f0(z) + ~ aif~(z) +go(Z)U, i=1
(2.3)
Before we employ (2.3), we need to know that the mapping x , ~ $ , is one-to-one, onto and continuous. Hence, we assume that there exist Ba c ~P and a constant 6 > 0 such that
1 + ~ a l aWli(X) >-6, V x ~ B x , V a ~ B a . i=1 aXn Then, the last two equations of (2.2) rewritten as
fCn-1 = ~n "31-(a -- ~)rwa(x)
(2.4) are (2.5)
Xn = [V + arw2(X)](1 + ~IT aWl(X)~ Ox,, I (n~l aWl(X) aWl(X) ) + ~r - xj+l + arwl(x) j=l axj axn-~ + ~w,(x) = [v + arw2(x)]t~(x, ~) + + gtrw4(x)a + ~rw~(x).
tVw3(x) (2.6)
Note that fl(x, 4) and the elements of w3(x), w4(x) defined by (2.6) are smooth functions on
l<-i<--n-2
P .iOn-1 =Xn + ~ aiwu(x) =Xn +arwl(x) i=1 P .iOn= v + Z aiwzi(x) = v + arw2(x), i=1
P .rn =Xn + ~'~ ~iWli(X) =Xn +arWl(X). i=1
(2.1)
where f0, fl . . . . . fp, go are smooth vector fields on B, and go(z) is bounded away from zero on B~, a subset of R". Proposition 1 in the Appendix gives necessary and sufficient conditions for the existence of a change of coordinates x = $ ( z ) and a state feedback control u = o¢(z) + fl(z)v, where v is a new control variable, which transform (2.1) into the special form ~=x,-+~,
Note that w~(x) and w2(x ) are smooth vector fields on Bx = #p(Bz). The meaning of the extended matching condition (A.3), formulated in the Appendix, is clearly displayed in (2.2). The original system (2.1) has been transformed into a chain of n - 2 integrators and two nonlinear equations with unknown parameters. The control variable v enters only the last equation and, hence, the strict matching condition of Taylor et al. (1989) is not satisfied unless the vector field wl(x) is identically zero. In the case of extended matching w~(x) is not required to vanish and, hence, unknown parameters are allowed to appear in the last two equations of (2.2). Note also that none of the functions appearing in (2.1) and (2.2) are required to be globally Lipschitz or to satisfy some other linear growth or sector conditions. Instead of the true parameter values a = [al . . . . . ap] r, which are unknown, a controller will be designed using parameter estimates = [ ~ . . . . . ~p]r. The first step in this direction is to introduce a new state in, instead of Xn, using the expression
(2.2)
where the expressions for wx(x) and wz(x) are given by (A.12) and (A.13) in the Appendix.
Bx X Ba. We now proceed to find a control which renders equations (2.5)-(2.6) linear in the parameter error a - t~ and makes them otherwise independent of the unknown parameter vector a. On closer inspection of (2.6), we see that these tasks will be accomplished by a control of the
Robust adaptive nonlinear control form
249
parameter update law 1
v = -a~w2(x)
- [to,x, + . . . #(x, 4)
+ k._,x._,
+ k . ( x . + a~w,(x)) + d~w,(x)l,
"~ a T w 3 ( x ) "~" a T w 4 ( x ) a
(2.7)
which, in addition to nonlinearity cancellation terms, contains a linear state feedback part with constant gains kl . . . . . k,. These gains are chosen to place at some desired stable locations the roots of the characteristic polynomial
s" + k~s "-l +. • • + k2s + kl = 0.
~, = [fl(x, a )WT(X) + flrw4(x)](a -- 4) - - klX 1 . . . . . k,:2,. introduce
= ~?(x, 4) =
the
following
(2.9) compact
x._, x~ + arw,(x) l
rw~(x, a)P~p(x, a),
P A + Arp
= -L
(2.14)
To confirm that the update law (2.13) leads to a differentiator-free implementation of the feedback control (2.7), let us rewrite (2.7) as an explicit function of the available signals x and t~:
o
=
-a~w2(x)
1
l~(x, 4) × [i,x, + . . . + k.(x, + a~w,(x)) + a~w3(x) + a~w,(x)a
+ w~(x)rw~(x, a ) e v ( x ,
4)1. (2.15)
Our next task is to prove stability and a convergence property of the adaptive scheme consisting of the state equation (2.12) and the parameter update law (2.13).
Lemma 1. The equilibrium :2 = 0, a = a, of the scheme (2.12)-(2.13) is stable for every a e Bo. Moreover, there exists a set k~ c R "+p such that from all (:2(0), 4(0)) e ~ the state :2(0 converges to zero: t~--oo
,
ix._,/ L :2. _l
(2.13)
where F is a positive definite matrix ("adaptation gain") and P > 0 is chosen to satisfy the Lyapunov equation
lim :2(0 = O.
[ Xl 1F I =
ti)P:2 =
(2.8)
To verify that the control law (2.7) is implementable on Bx x B~, first observe that I/~(x, 4)1-> a by assumption (2.4). Second, and this is a characteristic of the extended direct scheme, note the presence of the time derivative ,~ of the parameter estimate 4. It may appear that the implementation of (2.7) would require differentiators. Fortunately, this is not so. As we shall see, the parameter update law for t~ will furnish ~ as a known explicit function of available signals. To design this update law, we substitute the control (2.7) into (2.6) and obtain
Then we notation:
= r::(:2,
(2.16)
Proof. Differentiating the Lyapunov function V(:2, a) =:2rp:2 + (a - a ) r F - l ( a - 4) (2.17) along the solutions of (2.12)-(2.13), we obtain
(: =:2r(ArP + PA):2 + 22rPI¢~:(:2, fi)(a - 4) A=[!
I],
(2.10)
- 2t~rF-l(a - 4) -- -11:2112+ 2[:2rP14:(:2, d ) r - ~ r l r - ~ ( a - 4)
--
1
....
kn
-- -11:2112-<0.
g,(:2, 4) = ~ ( ~,(x, 4), 4) = w(x, 4)
(2.18)
In this notation, the system with feedback control (2.7) is rewritten as
This proves the stability of the equilibrium :2 = 0, a = a. The convergence result (2.16) now follows from LaSalle's invariance theorem, by which (:2(0, a(t))--->M as t---),o, where M is the largest invariant set of (2.12)-(2.13) contained in the set {(:2, 4) ::2 = 0} where 12 = 0. Finally, in view of (2.18), the function V(:2(t), a(t)) is nonincreasing and, hence, a subset of t) is the set
:~ = A:2 + W ( i , a)(a - 4).
~ v = {(:2, 4) : V(:2, 4) - c},
I
=
0 0
... .-.
0 0
]
w~(x) /~(x, a ) . , f ( x )
+
. (2.11)
a~w.(x)l
(2.12)
This "error form" shows that asymptotic stability is achieved when the estimate t~ is correct, a - t~ =- 0, because (2.8) is the characteristic polynomial of A. We now prove that, even when ~a, stability will be achieved with the AUTO
27:2-C
(2.19)
where c is the largest constant such that
f~v = {(x, a): v(~e(x, a), 4) <_ c} c Bx x Bo. (2.20) []
I. KANELLAKOPOULOSet al.
250
Our final task is to prove the same stability and convergence properties for the actual adaptive system
Yci=xi+l,
lira xn-l(t) = xn_l(0) + lira
(2.26) By Barbalat's lemma [see, e.g. Sastry and Bodson (1989, p. 19)], the uniform continuity and integrability of 2,-1(0 imply
1
2. = (a -
8) + . . • + k . x , + k,8rw
(x)
(2.21)
lim ~ _ l ( t ) = lim (x~(t) + arw~(x( t) ) ) = O. t---~
+ 8 w (x) + 8 +w
2~_1(~) d r = 0.
l<-i<-n-2
YC,_l = x, + arwx(x)
x
have
(x)rw
(x,
(2.27) 8)]
= V W r ( x , 8)PW(x, 8). This system differs from the scheme (2.12)(2.13) only in the last state, which in the scheme is $,, while here it is x,. To determine the equilibrium x = x ~, 8 = a of the system (2.21) which corresponds to the equilibrium ~ = 0 , 8 = a of the scheme (2.12)-(2.13), we note that at x~ = x2 . . . . . x,_~ = 0 the equation
.fc,_~ =x,, + a r w ~ ( x ) = O
(2.22)
has, because of (2.4), a unique solution x~, for each a • B~. A direct substitution proves that X =X e
t---~ oo
w,(x)8
~ r, = [0 • • " 0 x~]
8 =a
(2.23)
is the equilibrium of (2.21) corresponding to :f=0, 8=a.
Theorem 1. The equilibrium x = x ~, 8 = a of the adaptive system (2.21) is stable for every a e B, and the set ~2v defined in (2.20) is a subset of its region of attraction. Moreover, for all (x(0), 8(0)) e Qv, the state x(t) converges to its equilibrium value x e, that is, lim x(t) = x e.
(2.24)
Proof. Lemma 1 proves the stability of $ = 0, 8 = a . On the other hand, the mapping (x, 8 ) ~ ( $ , 8 ) is one-to-one, onto and continuous and it maps the point (x e, a) to the point (0, a). This proves the stability of x = x e, 8 = a, and, furthermore, implies that the solution (x(t), 8(0 ) of (2.21) for any ( x ( 0 ) , 8 ( 0 ) ) • g 2 v is uniformly bounded and remains in Bx x Ba for all t ->0. Differentiation of kn-i = x , + arwl(x) in (2.2) and the use of (2.6) gives xn-1 = [v + arw2(x)]fl(x, a) + arw3(x) + arwa(x)a.
(2.25)
Recall that all the functions appearing in (2.25) are smooth on B~ × Ba, including, because of (2.4), the control v as expressed by (2.15). Therefore, Y,_~(t) it also uniformly bounded and ~,-l(t) is uniformly continuous. From (2.16) we
Combining (2.16), (2.22) and (2.27) with the fact that Wl(X) is a smooth vector field, we conclude that lim x, (t) -- x,.e (2.28) [] 3. ROBUSTNESS TO U N M O D E L E D DYNAMICS
The stability results in the preceding section assume that, up to a set of unknown constant parameters, an exact model of the plant is available. A crucial question to be addressed in this section is whether, and to what extent, these stability properties are preserved in the presence of unmodeled dynamics. As in some earlier works on adaptive control, such as Ioannou and Kokotovic (1983) and Taylor et al. (1989), the unmodeled dynamics are assumed to be fast and are treated as singular perturbations, that is, the change of model order is parameterized by a scalar #. When # > 0, the unmodeled dynamics are included and the model order is higher than assumed in the adaptive scheme design. When = 0, the unmodeled dynamics vanish and the model reduces to the assumed design model. In applications, the parameter # is used to represent small inertias, inductances, time constants etc., as illustrated by many physical examples in Kokotovic et al. (1986). Starting with the design model in the special form (2.2), we perturb it in the following way: 3Ci---~Xi+l,
l <_i<_n-2
X n - 1 = Xn -[- a T W l ( X )
Yc, = arw2(x) + r r ( x ) ~ + g l ( x ) u
(3.1)
~ = Q ( x ) ~ + gz(x)u, where ~ • R v is the state of the unmodeled dynamics, and the entries of r(x), Q ( x ) , gl(x) and gz(x) are smooth functions on Bx. Within the framework of singular perturbations, the simple form of unmodeled dynamics included in (3.1) is sufficiently general, because, as shown by Kanellakopoulos (1989), other perturbed versions of the general model (2.1) can be transformed into (3.1). The relationship of (3.1)
Robust adaptive nonlinear control with the unperturbed design model (2.2) will become clear after the change of variables
= h(x, u) + rl,
(3.2)
which exhibits the function
h(x, u ) = - a - l ( x ) g 2 ( x ) u
Re~{a(x)}<--al,
VxeBx.
(3.4)
This, in turn, assures the existence of Q - l ( x ) in (3.3). Along with the change of state variables (3.2), we introduce the following change of the control variable v = [gl(x) - rr(x)a-l(x)g2(x)]u = g(x)u
(3.5)
and assume that there exists a2 > 0 such that Ig(x)l - 32
Yx • Bx.
These changes of variables perturbed model (3.1) into
transform
ffi=Xi+l, l<--i<--n - 2 2,-1 = x, + arwl(x) 2, = v + arw2(x) + rr(x)r/
(3.6) the
and see that/~ = 0 at } = 0, d = 0. On the other hand, for t / = 0 the first two equations of (3.9) represent the unperturbed scheme (2.12)-(2.13), which has an equilibrium at 2 = 0, a = a. For simplicity, we now make the additional assumption that wl(0)= w2(0)= 0. Under this assumption 2 = 0 implies x = 0, that is, the equilibria of the adaptive scheme (3.9) and of the adaptive system are the same, and, furthermore, W(O, fi) = 0 for all a • B,. As in Taylor et al. (1989), the stability of the equilibrium 2 = 0, ~ = a, r/= 0 of (3.9) will be investigated using the composite Lyapunov function
+ c2r/r/3r (2, t~)t/,
(3.8)
= A2 + 1?¢(2, a)(a - ~) + R(2, d)~l
a)P2
/u//= 0 ( 2 , a)r/-/u/~(2, a, 77, a),
(3.11)
where c~ and c 2 are positive constants and ~ ( 2 , ~) is the positive definite solution of
is expressed as a function of x, a, r/ and a. The expression for 2 as a function of x, a, t / a n d v is given by (3.7), while the expressions for v and 1) as functions of x, t~, r/ and a can be obtained from (2.15) and (2.13). It is now obvious that for r/= 0, the perturbed model (3.7) reduces to the assumed design model (2.2). However, when /z > 0 , we cannot expect that r/(t)-~ 0 even if t/(0)= 0, because, in general, h(x, a, r/, a) is not zero. As we shall see, the stability properties established by Theorem 1 will be preserved if the term /~/~(x, d, t/, a) is sufficiently small. The analysis leading to such a result retraces the derivations of Section 2 for the perturbed model (3.7). The adaptive scheme for the perturbed model (3.7) employs the same control (2.15), the same update law (2.13) and the same notation (2.10), (2.11) as in Section 2. The resulting adaptive scheme with unmodeled dynamics is
=
i(2, ~, rl, a) = f~(2, a)} + ~a(2, a)d
~ ( 2 , a)0.(2, d) + QT(2, ~)~(2, ~) = - I . hu i)
(3.10)
(3.7)
where the time derivative
hx - h, --~ gx J¢ +
(v)
/~(2, a) = h X,g--~
Vc(£, a, tl) = cl[£rP£ + (a - fi)rF-l(a - ,~)1
~il = a(x)rl - Mi(x, a, ri, a),
t;t =
where 0 ( 2 , d) = Q(x), /~(~, a) = [ 0 . . . 0 r(x)] r and /~(2, a, r/, a) = h(x, a, r/, a). To verify that 2 = 0, a = a, r/= 0 is an equilibrium of (3.9), we return to (3.3) and (3.8). Then, with the help of (2.15), we define
(3.3)
as the quasi-steady-state of ~, and r/ as its fast transient. The unmodeled dynamics are assumed to be asymptotically stable for all fixed x • Bx, that is, there exists ol > 0 such that
251
(3.9)
(3.12)
The time derivative of Vc along the solutions of (3.9) is
(Z = c , [ - 2 r 2 + 22rp#(2, a)r/] + c= - -~ ,7",7 + ,7 PAx, ~, ,l, a),7 m
2r/rPi(2, a)/~(2, ~, ~/, a)].
(3.13)
The function /~, as defined in (3.10), satisfies /~(0, a, 0, a) = 0 for all a, a • B,. Hence, it is bounded by
11i(2, a, n, a)ll--~P111211 +02 Ilnll (3.14) where the constants Pl, P2 are such that for all x • Bx, ~ • B., a • B. the following inequalities hold:
IIh~(~, a)~(~, a)(a -~)11 ~ po 11211 (3.15) IIh~(~, ~)A +/~a(2, ~)Fff'r($, ~)ell + p o ~ p l (3.16)
IIh~(.e, a)#(.,L a)ll-
(3.17)
Furthermore, we choose constants Cl, c2 and c3
252
I. KANELLAKOPOULOSet al.
which satisfy 2
IIPA~, a)11 pl ~
Cl
(3.18)
2 Ileg(~, 8)11-< c2 2 IIP:(~, a)ll 02+ I1~(~, 8, rl, a)[[-c3
(3.19) (3.20)
for all x • B~, 8 • Ba, rI • Bn, a e B~. A stronger effect of the fast variable r/ manifests itself as an increase in the values of c~, c2 and c3. This is particularly clear for c2, because c2 bounds the matrix/~ through which 7/ enters into the adaptive scheme (3.9). Constants c~ and ca show the effects of r/ through the properties of the Lyapunov matrix ~. Using the bounds (3.14)-(3.20), we obtain from (3.13):
V'c------[llxll[l"[I][
c1
--CLC2 1
L ClC c2( [ II~ll].
(3.21)
x LIInllJ
This bound leads to the following robustness result: Theorem 2. The equilibrium x = 0, 8 = a, r! = 0 of the perturbed adaptive system (3.7) with the feedback (2.15) and the update law (2.13) is stable for every a e Ba and for every/~ satisfying
1 0 < #
ClC2 -I-C3
(3.22)
with Cl, c2, c3 as defined by (3.18)-(3.20). A subset of its region of attraction is the set ~ = {(x, 8, 7/): Vc0p(x, 8), 8, r/) <--c}, where
(3.23)
c is the
largest constant such that Moreover, for all (x(0), r~(O), r/(O)) • f2~, and /~ • (0,/~*), the state (x(t), rl(t)) converges to its equilibrium value, that is ~ c c Bx x Ba x B~.
limx(t) = 0 ,
lim r/(t) =0.
(3.24)
Proof. If (3.22) is satisfied, the matrix in (3.21)
is positive definite and thus l?c is negative semidefinite. This proves the stability of the equilibrium £ = 0, 8 = a, r/= 0 of the perturbed adaptive scheme (3.9). Hence, the equilibrium x = 0, 8 = a, T/= 0 of the perturbed adaptive system is stable and an estimate of its region of attraction is the set f2c in (3.23). Furthermore, the largest invariant set of (3.9), contained in the set where l?c = 0, is the set {($, 8, T/) : $ = 0, ~/= 0}. The convergence result (3.24) then follows from LaSalle's invariance theorem and the fact that lim $ ( t ) = 0 implies lim x ( t ) = O. t----~ t----~ []
It may appear that Theorem 2 requires a detailed knowledge of the unmodeled dynamics. In fact, even when nothing else is known about the unmodeled dynamics other than that they are fast and stable in the sense of (3.4), Theorem 2 still provides conceptual robustness bounds. Whatever the unmodeled dynamics, some strictly positive constants Cl, c2, c3 and, hence, #* exist so that the stability of the closed-loop system and the regulation property (3.24) are preserved for all # - #*. In applications, a more detailed description of the unmodeled dynamics may be available. For example, the dynamics of some known parts of the system (e.g. actuators and sensors) may have to be neglected in order to make the system appear in the special form (2.2). (In the next section, a motor position control system is brought to the special form (2.2) by neglecting the power amplifier dynamics.) For a specific application, the time constant is known and the bounds (3.14)-(3.22) can be used to first determine a region Bx × Ba x B~, and then evaluate the set 9c in which the proposed adaptive design is applicable. On the other hand, if g2c is a design specification, then the above bounds can be used to determine the required time constant /~. It is pointed out, however, that these bounds are conservative. Following the outlined procedure, tighter bounds can be obtained by taking into account details of the problem at hand. 4. DISCUSSION AND EXAMPLES
In this section we stress the practicality of the results in the preceding two sections by applying them to a common control problem. We then discuss a further extension of these results. E x a m p l e 1. Position control o f a D C - m o t o r with uncertain load torque and u n m o d e l e d dynamics.
In normalized units, the position control system is described by
-dO -~(3) dt dto
P
- - = i + Y~ ajwl/(O, to) dt j=~
di
T,,,
(4.1)
~=-~-e (--to--i + ~)
dg .-dr= - ~ + gu, where 0, to, i and ~ are the motor position, speed, and armature current and voltage, respectively, while the control variable is the input u into the amplifier with constant gain g.
Robust adaptive nonlinear control The uncertain load torque is represented by a weighted sum of some known nonlinear functions wls(O, to) with unknown constant weights as, ] = 1 , . . . , p. For illustrative purposes we take p = 1 and w11(0, t o ) = to2. The ratio TflTm of the motor electrical and mechanical time constants was neglected in Taylor et al. (1989) in order to satisfy the strict matching condition. Here this quantity is retained in the model, and the singular perturbation parameter is the amplifier time constant/z. The control objective is to regulate the motor position and speed to the desired constant values 0 = 0 d ~ and t o = 0 . Denoting X l = 0 - - 0 d ~ , X2 = tO, X3 = i, we rewrite (4.1) in the form of the perturbed model (3.1):
253
The perturbed adaptive scheme (3.9) is 21 = X 2
x2 = 23 + (ax - t~i)x22 "~3 = - k l X l
- k 2 x 2 - k 3 x 3 + ( a l - 41)
Tm X 2~1x 3 + ~ - r/
(4.7)
al = F l ~ r ( 2, al)P2 u O = - ,1 - k a .
The largest invariant set of the scheme (4.7), contained in the set where l?c = 0 (see (3.21)), is the set {(2, 41, r / ) : 2 = O , T/=O}. Thus, our scheme achieves regulation of the state (2(0, 7/(t)). 1. The adaptive scheme presented in this paper can be further extended to include the case where unknown parameters enter the control vector field. (In the above example, this would occur if the ratio T~/Tm = a 3 were also unknown.) A detailed analysis of this was presented in Kanellakopoulos et al. (1989). The main difference from the case treated in this paper is that the update law (2.13) depends on the control variable v. Therefore, v and ~ are defined implicitly, and can be expressed as explicit functions of 2 and ~ only in a region S around the equilibrium 2 = 0 , d=a. The boundaries of this solvability region are the manifolds on which the Jacobian of the implicit function defining v is singular. As an illustration, consider the following example. Remark
3C1 = X 2 fC2 : X 3 +
alx 2
(4.2)
Tm
= T , (-x2 - x3 + # ~ = - ~ + gu.
The change of variables (3.2) is now ~ = gu + rl, so that h = gu, kt = gfi, and the perturbed model (4.2), rewritten in the form (3.7), becomes .~1 = X2 fC2 = X 3 +
alx 2
Tm
(4.3)
Tm
3=T, ( - x 2 - x 3 +gu + /M/= - r / - / ~ g t i .
E x a m p l e 2. The system
As our design model (2.2) we use the first three equations with T/-= 0. The meaning of this design model is that the amplifier is replaced by its constant gain g and the voltage gu is applied to the motor armature. The design now proceeds as in Section 2. Using
(4.4)
3~3 = X 3 + a l x 2
and noting that assumption (2.4) is trivially satisfied, we design the control (2.7):
3C1 = ( 1 0 "~ a ) x 2
is controllable for a ~ e - 1 0 . Following development of Section 2, we use 22 = (10 + ~)x2 v
~t
2
k2x2 + k323 + 2alx223 + alx2].
(4.5) This results in the update law (2.13) with
ai)= W(x, 40=
[°1
the (4.9)
1 (10 + 4) 2 [-Xl - (10 + a)x2 - tlXz + (10 + t~)x~]
(4.10)
and reduce (4.8) to the error form (2.12) .~1 = x2 + x2(a - 4)
v = -[kzXl +
(4.8)
x2 = -x32 + (10 + a ) v
x2 = - x l - 22 + (10 + a ) v ( a - ~), which is exponentially stable for a = a . Lyapunov function
(4.11) The
V(2, t~) = 1.5x21 + XlX 2 "1- 2 2 + (a -- 4) 2 (4.12) results in the update law
.
L2alXLl
(4.6)
~t = 1.5x~x2 + 0.5(10 + a)x 2 + (10 + 4)
x [0.5Xl + (10 + a)x2]v.
(4.13)
254
I. KANELLAKOPOULOSet al.
Equations (4.10) and (4.13) implictly define v and ~ as functions of x and ~. Eliminating t~, we get (10 + ~)[(10 + ~) + xz(0.5Xl + (10 + d)x2)]v = - x l - (10 + a)x2 - 1.5XlX~ + 0 . 5 ( 1 0 + d ) x 3.
(4.14)
To obtain v explicitly, the term multiplying it must be nonzero, which means that the adaptive scheme is defined only in a solvability region S c Bx x Ba. By inspection of (4.14), a subset of S is, for example
{(xl, x2, ~) : Ix~l < 2, Ix21 < 2, 15 + ~1 < 3}.
(4.15) []
R e m a r k 1 (cont'd). In general, the solvability
condition for systems mentioned in R e m a r k 1 requires that in the formulation of all the results of this paper, starting with the region of attraction estimate ~ v in (2.20), the set Bx x B, be replaced by the solvability region S c Bx x B,. An approach which avoids the implicit definition of v and ~ was proposed by P o m e t and Praly (1989b). For (4.8) this approach consists in treating the p a r a m e t e r a appearing in the term (10 + a ) v as a second unknown p a r a m e t e r b to be estimated separately as/~. As a result of this overparametrization, the update law for ~ no longer depends on v, which now appears in the definition of ~ only. R e m a r k 2. For simplicity, all the results in this
paper are given for the regulation problem. However, with standard technical modifications, the extended direct scheme can be used for the tracking of a given reference Ydes(/) by the output y = x l of the system (2.2). Moreover, zero dynamics can be added to (2.2) under the same assumptions as in Sastry and Isidori (1989). Of course, the invariance theorem of LaSalle does not apply to this time-varying problem, and the proofs of the tracking analogs of L e m m a 1 and T h e o r e m 1 employ the standard uniform continuity argument and Barbalat's lemma. Details and further extensions can be found in Kanellakopoulos (1989). To achieve robustness of the tracking scheme to unmodeled dynamics and bounded disturbances, one of the update law modifications proposed in adaptive linear control, such as the a-modification of Ioannou and Kokotovic (1983), can be used. If this simple a-modification scheme is used, robustness properties analogous to those established for the linear case in T h e o r e m s 5.2.1 and 6.5.1 of Ioannou and Kokotovic (1983) can be shown to hold in the nonlinear case.
5. CONCLUSIONS The goals of the adaptive scheme proposed in this paper--simplicity, applicability to a wider class of nonlinearities and robustness to unmodeled d y n a m i c s - - h a v e been achieved under the extended matching condition. Although this condition restricts the dependence on the unknown parameters, it is satisfied by m a n y systems appearing in engineering applications. In a manner similar to that illustrated by E x a m p l e 1, the extended adaptive scheme is applicable to switched reluctance (Taylor, 1988) and permanent magnet stepper (Bodson and Chiasson, 1989) motors, and to other electromechanical or electrohydraulic systems with load and power input uncertainties. It would seem, therefore, that when the extended direct scheme is applicable, it should be given preference over more elaborate schemes. For problems to which this scheme is not applicable, the choice, or the development, of an appropriate scheme is a topic of ongoing research (Sastry and Isidori, 1989; Pomet and Praly, 1989a, b; Bastin and Campion, 1989; Campion and Bastin, 1990). Acknowledgements--This work was supported in part by the National Science Foundation under Grant ECS 88-18166 and in part by the Air Force Office of Scientific Research under Grant AFOSR 90-0011. REFERENCES Bodson, M. and J. Chiasson (1989). Application of nonlinear control methods to the positioning of a permanent magnet stepper motor. Proc. 28th CDC, Tampa, FL, 531-532. Bastin, G. and G. Campion (1989). Indirect adaptive control of linearly parametrized nonlinear systems. Preprints of the 3rd IFAC Symposium on Adaptive Systems in Control and Signal Processing. Glasgow, UK.
Campion, G. and G. Bastin (1990). Indirect adaptive state feedback control of linearly parametrized nonlinear systems. Int. J. Adaptive Control Signal Proc., 4, 345-358. Ioannou, P. A. and P. V. Kokotovic (1983). Adaptive Systems with Reduced Models. Springer, New York. Kanellakopoulos, I. (1989). Adaptive feedback linearization: stability and robustness. M.S. Thesis, University of Illinois, Urbana. Kanellakopoulos, I., P. V. Kokotovic and R. Marino (1989). Robustness of adaptive nonlinear control under an extended matching condition. Preprints IFAC Syrup. on Nonlinear Control System Design, Capri, Italy, 192-197. Kanellakopoulos, I., P. V. Kokotovic and R. Marino (1990). Robust adaptive nonlinear control under extended matching conditions. Technical Report UILU-ENG-902202 (DC-115), Coordinated Science Laboratory, University of Illinois, Urbana. Kokotovic, P. V., H. K. Khalil and J. O'Reilly (1986). Singular Perturbation Methods in Control: Analysis and Design. Academic Press, New York.
Nam, K. and A. Arapostathis (1988). A model reference adaptive control scheme for pure-feedback nonlinear systems. 1EEE Trans. Aut. Control, AC-33, 803-811. Narendra, K. S. and A. M. Annaswamy (1989). Stable Adaptive Systems. Prentice-Hall, Englewood Cliffs, NJ. Parks, P. C. (1966). Lyapunov redesign of model reference adaptive control systems. IEEE Trans. Aat. Control, AC-11, 362-367. Pomet, J. B. and L. Praly (1989a). Adaptive nonlinear control: an estimation-based algorithm. In Descusse, J.,
Robust adaptive nonlinear control M. Fliess, A. Isidori and D. Leborgne (Eds), New Trends in Nonlinear Control Theory. Springer, Berlin. Pomet, J. B. and L. Praly (1989b). Adaptive nonlinear regulation: equation error from the Lyapunov equation. Proc. 28th CDC, Tampa, FL, 1008-1013. Sastry, S. S. and M. Bodson (1989). Adaptive Control: Stability, Convergence and Robustness. Prentice-Hall, Englewood Cliffs, NJ. Sastry, S. S. and A. Isidori (1989). Adaptive control of linearizable systems. I E E E Trans. Aut. Control, AC-34, 1123-1131. Sastry, S. S. and P. V. Kokotovic (1988). Feedback linearization in the presence of uncertainties. Int. J. Adaptive Control Signal Proc., 2, 327-346. Su, R. (1982). On the linear equivalents of nonlinear systems. Syst. Control Lett., 2, 48-52. Taylor, D. G. (1988). Feedback control of uncertain nonlinear systems with applications to electric machinery and robotic manipulators. Ph.D. Thesis, University of Illinois, Urbana. Taylor, D. G., P. V. Kokotovic, R. Marino and I. Kanellakopoulos (1989). Adaptive regulation of nonlinear systems with unmodeled dynamics. IEEE Trans. Aut. Control, 34, 405-412.
255
if and only if the feedback linearization (A.2) and the extended matching (A.3) conditions are satisfied.
Proof Sufficiency. It is proven in Su (1982) that condition (A.2) is sufficient for the existence of a function Ol(z): U--* • with the properties (1) (ddp1, ad~-Igo) ¢:0 in U, (d~b~, X ) = O V X e ~.-z. (2) (q~l, Lf0q~,. . . . . L~-lq~,) r = (qh . . . . . q~.)r = q~r(z ) is a change of coordinates in U, such that the system 2 =fo(Z) + u(t)go(Z )
:q=xi+ p l < i < - n - 1 ~. = L~odpl(z) + LeoL~_ldpl(z)u(t).
1 u = LsoL~o_,dpa(z ) ( - L ~ I ( Z )
z ¢ a"
% = sp
(A.1)
f/(x)=[0...0
O<-i<-n - 1 (A.2)
Extended matching condition. l<-i<-p.
(A.3)
Proposition 1. There exists a state feedback control u = oc(z) + fl(z)v, fl(z) ~ 0 in U1, and a state space change of coordinates x = O(z) in U = U1, with ~(0)= 0, such that the system P i=I
becomes in the x coordinates ici = xi+l, l <-i <-n - 2 (m.5)
((i)(i)1 0
,
(A.10)
w,i(x )
w2i(x)] r,
l<-i<--p, (A.11)
Wu(X) = Lf~L~o-Eqal((a-l(x))
(A.12)
w~(x) = Lf~L~o-lq~l(~ - l(x)).
(A. 13)
In conclusion, the state feedback (A.8), applied to the system (A.1), results in
L~ ¢p1(z) n tz~ + 1 2 =fo(z) - LgoL~o_ldpx(z ) ~o~ p LsoL~o-ldpl(z) go(z) v p
+ ~ aifi(z ).
=f0(z) + go(z)ot(z) + go(z)fl(z)v + ~ alfii(z ) (A.4)
p f(n-I = Xn 4- Z a i w l i ( x ) /ffil p Yen : V 4- Z aiw2i(x) i=1
(A.9)
where
are involutive and of constant rank i + 1 in U1.
f~e~,
l<-i<--n-1
and condition (A.3) implies that
Feedback linearization condition. The distributions ad~go},
(A.8)
In the new coordinates x = q~(z), the distribution ~ is
with f0(0) = 0, g(0) :~ 0, fo, fl . . . . . fp, go smooth vector fields in /]1, a neighborhood of the origin.
~ = s p { g o , adfogo . . . . .
4- 11) = ~ ( z ) 4- f l ( z ) v ,
the system (A.6) becomes
p
2 =f0(z) + ~ aifi(z ) + u(t)go(Z),
(A.7)
By using the state feedback (recall that L~oL~o-ld~l(z)~ 0 in U)
:q=xi+x, YCn=V. APPENDIX: THE EXTENDED MATCHING CONDITION Consider the system
(A.6)
becomes in the new coordinates
(A.14)
i=1
When expressed in the new coordinates x = q~(z), the system (A.14) becomes (A.5). Necessity. If there exist a state feedback and a change of coordinates transforming the system (A.1) into the system (A.5), one can directly verify that the conditions (A.2) and (A.3) are satisfied for the system (A.5). Since the conditions (A.2) and (A.3) are invariant under state feedback transformations and changes of coordinates, they are satisfied for the system (A.1). to