Adaptive Nonlinear Control
Adaptive Nonlinear Control
Adaptive Nonlinear Control
Abstract
In this chapter, passivation and small gain techniques are used as two
fundamental tools to systematically design stabilizing adaptive controllers for
new classes of nonlinear systems. We first show that, for a class of linearly
parametrized nonlinear systems with only unknown parameters, the concept of
adaptive passivation can be used to unify and extend most of the known
adaptive nonlinear control algorithms based on Lyapunov methods. Then, we
consider the global robust adaptive control problem for a broader class of
nonlinear systems with time-varying and dynamic uncertainties in addition to
parametric uncertainties. Small gain arguments are used to provide a robus-
tification methodology for prior backstepping-based adaptive controllers.
6.1 Introduction
The area of adaptive nonlinear control has moved on quickly since the early
1990s- see the survey paper [37] and two more recent textbooks [26, 32]. We
observe that the development of most adaptive nonlinear controller designs
was based on Lyapunov methods. In this chapter, we shall approach this field
from the somewhat different input/output viewpoint using the concepts of
passivation and small gain. Our contributions are twofold: (1) we exploit recent
advances on the feedback stabilization of nonlinear systems via passive systems
theory and apply some of these useful results to formulate a passivation
framework for adaptive nonlinear stabilization; (2) we employ small gain
techniques as a means to study robustness issues in adaptive systems with
unmodelled dynamics. The latter topic has received less attention in the
120 Adaptive nonlinear control: passivation and small gain techniques
literature and the results presented here are a substantial development beyond
earlier results.
In the first part, after reviewing some needed definitions and properties of
passivity and passive systems, we briefly state the breakthrough made by
Byrnes, Isidori and Willems [2] on the feedback equivalence of nonlinear
systems to passive systems (or simply, passivation). Roughly speaking, a
nonlinear control system can be transformed into a passive system via a
change of feedback if and only if it is minimum phase and of relative degree
one. As shown in [2], this theorem together with other passivity tools allow
unification of early global stabilization results. Using this as a starting point,
we first establish an adaptive version of a basic result in [2], that is, a linearly
parametrized nonlinear system is feedback equivalent to a passive system if and
only if it is adaptively stabilizable. Then, we show this property can be
propagated each time we add a feedback passive system with linearly appearing
unknown parameters. This recursive passivation design procedure is different
to the currently popular adaptive backstepping with tuningfunctions introduced
in [26] in several respects. We were motivated by a passivity-aimed adaptive
design strategy. Some elementary examples illustrate this point in the case of
output feedback passivation [15]. Our passivation-based adaptive controller
design reduces to the above-mentioned Lyapunov-type adaptive scheme in the
case of strict-feedback structures, but the construction turns out to be simpler
and easier to understand by means of feedback passivation. To be more
precise, we first augment the system (with unknown parameters) under
consideration by adding a new parameter update system with adaptive law
as input. Then, we only need to render the augmented system passive by a
change of feedback at the levels of both control input and adaptive law. The
immediate benefit is that we can do it recursively and shed light on how
overparametrization is avoided.
The second part of the chapter collects a series of nonlinear small gain
techniques recently presented in our papers [14, 16, 20, 17, 21, 36] on the basis
of Sontag's input-to-state stability (ISS) concept [42, 44]. As demonstrated in
these papers and other references therein, nonlinear small gain theorems have
proved to be powerful design tools for interconnected nonlinear systems with
complex structure. For a class of feedback linearizable systems with various
disturbances including parametric uncertainty and unbounded nonlinear
unmeasured dynamics, we show that these techniques are of paramount
importance in designing a robust adaptive controller in the presence of
unbounded unmodelled dynamics. Adaptive feedback designs without and
with dynamic normalization will be proposed and compared in two elementary
examples including a physical example of a simple pendulum.
The rest of the chapter is organized as follows: Section 6.2 states needed
definitions and some known basic results. Section 6.3 is devoted to the adaptive
passivation development for interconnected nonlinear systems. Section 6.4
Adaptive Control Systems 121
presents a novel small-gain based adaptive scheme for nonlinear systems with
unbounded unmodeled dynamics. Section 6.5 offers some brief concluding
remarks.
6.2.2.2 Feedbackpassivation
As seen in the preceding subsection and demonstrated by numerous research-
ers, passive systems enjoy many desirable properties which turn out to be very
useful for practical control systems design. Naturally, this motivated people to
address the feedback passivation issue. That is, when can a nonlinear control
system in the form (6.1)--(6.2) be rendered passive via a state feedback
transformation? This question had remained open until [2] where Byrnes,
Isidori and Willems nicely provided a rather complete answer by means of
differential geometric systems theory.
Definition 2.4 (Passivation) A system (6.1)-(6.2) is said to be feedback
(strictly) C-passive if there exists a change of feedback law
u = #l (x) + #2(x)v (6.8)
such that the system (6.1)-(6.2)-(6.8) with new input v is (strictly) C-passive.
Theorem 2.2 121 Consider a nonlinear system (6.1)-(6.2) having a global
normal form
~, = q(z, y)
(6.9)
j, = b(z, y) + a(z, y)u
where a(-) is globally invertible. Then system (6.1)-(6.2), or (6.9) is globally
feedback equivalent to a (resp. strictly) C2-passive system with a positive
definite storage function, if and only if it is globally weakly minimum phase
(resp. globally minimum phase).
As an application of this important result to the global stabilization of
cascaded nonlinear systems, it was shown in [2] that several previous stabil-
ization schemes via different approaches can be unified. In particular, combin-
ing Theorems 2.1 and 2.2, the following general result can be established.
Theorem 2.3 [2, 341 Consider a cascaded nonlinear system of the form
+A (r x)y
= f ( x ) + a(x)u (6.1 O)
y =
124 Adaptive nonlinear control: passivation and small gain techniques
~c = f ( t , x , u ) (6.11)
y=h(t,x,u)
where x E R n is the state, u E Rm~ is the control input and y E Rm2 is the system
output. Notice that many dynamical systems subject to exogenous disturbances
can be described by a differential equation of the form (6.11).
Roughly speaking, the property of input-to-state stability (ISS) says that the
ultimate bound of the system trajectories depends only on the magnitude of the
control input u and that the zero-input ISS system is globally uniformly
asymptotically stable (GUAS) at the origin. More precisely,
Definition 2.5 A system of the form (6.11) is said to be input-to-state stable
(ISS) if for any initial conditions to > 0 and X(to) and for any measurable
essentially bounded control function u, the corresponding solution x(t) exists
for each t > to and satisfies
Ix(t)l ___#(Ix(t0)l, t - to)+ 7(llutto,,]ll) (6.12)
where 13 is a class KL-function and 7 is a class K-function.
Adaptive Control Systems 125
An output version of the ISS property was given in [21, 16] and is recalled as
follows.
Definition 2.6 A system of the form (6.11) is said to be input-to-output
practically stable (lOpS) if for any initial conditions to > 0 and x(t0) and for
any measurable essentially bounded control function u, the corresponding
solution x(t) exists for each t > to and satisfies
ly(t)l <_ ~(Ix(t0)l, t - to) + 7(llut,0,,lll) + d (6.13)
where/3 is a class KL-function, 7 is a class K-function and d is a non-negative
constant.
If d = 0 in (6.13), the IOpS property becomes 10S (input-to-output stabil-
ity). Moreover, the system (6.11) is input-to-state practically stable (ISpS) if
(6.13) holds with y = x.
An obvious property of an IOpS system of form (6.11) is that the system is
bounded-input bounded-output (BIBO) stable. In addition, for any lOS system
(6.11), a converging input yields a converging output.
Among various characterizations of the ISS property is the notion of an ISS-
Lyapunov function which was introduced by means of some differential
dissipation inequality [46], [28]. In Section 6.4, we consider a class of uncertain
systems with persistently exciting disturbances. As a consequence, an extension
of the ISS-Lyapunov function notion turns out to be necessary.
Definition 2.7 A smooth function V is said to be an ISpS-Lyapunov function
for system (6.11) if
9 V is proper and positive definite, that is, there exist functions ~bl, ~2 of class
Koo such that
~ (Ixl) _< V(x) <_~2(Ixl), Vx E a n (6.14)
9 there exist a positive-definite function c~, a class K-function X and a non-
negative constant c such that the following implication holds:
OV
{Ixl ___x(lul)+ c} ~ .-~x(X)f(t,x,u) < -~(Ixl) (6.15)
When (6.15) holds with c = 0, V is called an ISS-Lyapunov function for system
(6.11) as in [28].
With the help of arguments in [46, 47, 28], it is not hard to prove the
following fact.
Fact: If a system of form (6.11) has an ISpS- (resp. ISS-) Lyapunov function
V, then the system is ISpS (resp. ISS).
(finite-) gain theorem [5] which were recently proposed in the work [21, 20, 16].
They will be used to construct robust adaptive controllers and develop the
stability analysis in Section 6.4. The interested reader is referred to [4, 6, 20, 21,
49] and references therein for applications of the nonlinear small gain theorems
to several feedback control problems.
Consider the general interconnected time-varying nonlinear system
Yr = fl(t, Xl,Y2, Ul), Yl = hl(t, xl,Y2, Ul) (6.16)
Yc2 = f2(t, x2,Yl,U2), Y2 = h2(t, x2,Yl,U2) (6.17)
where, for i = 1,2, xl E ~ ' , u~ E Rm~' and yi E Rm2~,andS, hi are C l in all their
arguments. It is assumed that there is a unique C l solution to the algebraic
loop introduced by the interconnection functions hi, h2.
Theorem 2.4 121, 16] Assume that the subsystems (6.16) and (6.17) are IOpS
in the sense that, for all 0 <_ to <_ t,
[yl(t)[ _< fll ([Xl (to)[, t - to) + 3"~l([[Y2[to,tll[) + ~i'(llull,o,,lll) § d~
(6.18)
]y2(t)l _</~2(IX2(t0)), t -- to) -I- ~2 (l[Yl[to,t]lJ) -}- "y~(llu2/,0,,]ll) + d2
Assume further that the subsystems (6.16) and (6.17) are also ISpS (resp. ISS).
If there exist two class Koo-functions p~ and p2 and a nonnegative real number
st satisfying:
(I + p2)o ~2 o (I + pl)o ~ ( s ) < ~ ]~
- Vs > se (6.19)
(I + p~) o ~ o (I + p2) o ~2 (~) -< 9 f -
then the interconnected system (6.16)-(6.17) with (yl, y2) as output is ISpS
(resp. ISS) and IOpS (resp. IOS when ae - di = 0 for i - 1,2).
Motivated by the above Fact, we consider properties of the feedback system
derived from existence of ISpS-Lyapunov functions on each subsystem.
Following step by step the proof of Theorem 3.1 in [20], we can prove the
following result.
Theorem 2.5 Consider the interconnected system (6.16)-(6.17) with yl = xl
and Y2 = x2. Assume that, for i = 1,2, the xrsubsystem has an ISpS-Lyapunov
function V~ satisfying the properties
(1) there exist class Koo-functions Oil and Oa such that
0.(Ixtl) < Vt(x~) < ~,2(Ix~l), V x~ ~ a ~' (6.20)
(2) there exist class Koo-functions ai, class K-functions Xi, O't and some
constant cs _> 0 so that Vl (xl) _> max {xI(V2(x2)),'Yl (lul [) + cl } implies
V(x, O) > O, 0 and ~-, with V(0, 0 ) = 0 V0, and a CO function h such that the
resulting system with adaptive feedback
then system (6.34) is strongly adaptively feedback passive with a proper storage
function V.
Proof By a Converse Lyapunov Theorem [27], there exists a C ~ Lyapunov
function U which is positive definite and proper satisfying
OU
'~;.. ( r < -r/CO (6.36)
t OU OW
_< -n(r + ~ ~ (r162 ~)yO, + - - ~ (~)f(~)
1--1
OW
Notice that ~ (x)f(x) is negative definite in x.
For each ((,x), denote the m x I matrix c((,x) as
Since ~ is proper in its argument (r x, 0), it follows from (6.43) that all the
solutions (r of the closed loop system (6.34), (6.41) and (6.42) are
well-defined and uniformly bounded on [0, oo).
Furthermore, a direct application of LaSalle's invarianee principle ensures
that all the trajectories (r ~(t)) converge to the largest invariant set E
contained in the manifold {((,x,8)I(ff, x ) = (0,0)}. Therefore, x(t) tends to
zero as t goes to oo. In other words, the cascade system (6.34) is globally
adaptively stabilized by (6.41) and (6.42). Finally, Proposition 3.1 ends the
proof of Theorem 3.1
Remark 3.2 It is of interest to note that, if rank {g(0)(gl(0) + Ag(0))} = 1 =
dim 0, then
lim ]O(t) - 01 = 0
t-*+oo
Indeed, on the set E, we have a(o)(a, (o) + Aa(o))(o - O(t)) - 0 which, in turn,
implies that ~(t) = 0. So, E = { (0, 0, 0) }.
The following corollary is an immediate consequence of Theorem 3.1 where
the (-system in (6.34) is void.
Corolla~ 3.1 Consider a linearly parametrized nonlinear system in the form
.~ = f (x) + A f (x)O + g(x)(u + Ag(x)O)
y = h(~) (6.44)
which implies hi1 (~, 0, 0) is affine in 0. Then, there exist smooth functions/~ll
and Ah~ such that
hll (~, 0, 0) =/~ll (t~,/~) + Ahll (~, 0)(0 -/~) (6.49)
Adaptive ControlSystems 133
Consider the nonnegative functions
v2 = v~ (+, +) + 89ly - o~ (+, O)l2 (6.50)
"~2 <( --~1 (~, O) "[- h~l (~, O, O)y -~- (OVl
k,"~ (~' ~) "~"(~ - O)r r _ l ) "rl
001
(7"1 + 'TI) Oq~91(A0(~) +j~ (~')0 + (GIo(~) + AGI(~)O)Yj+I)] (6.52)
o~ --~-
We wish to find changes of feedback laws which are independent of 0
u = O2(x, ~) + v
(6.53)
r0 - rl + ~1 = r2 + "~2
such that V2 satisfies a differential dissipation inequality like (6.47).
To this end, set
An = r ( ~ + Tr)y (6.54)
"~ = At2 + ~2 (6.55)
where f12, T2 and ~2 ~ R m• are defined by
~I/2 = (00,
"-~ (6, [~)AGI! (~)y,..., 001 (~',0 ) A G , , ( ~ ) y ) ( 6 . 5 6 )
00t
ft2 =J~(x) + Ahll(~,/~) - - ~ f l ( ~ ) - ~2(x,/~) (6.57)
+ (~ - o) ~(a~ + T~)y
+ \ - ~ ((, ~) + (~ - O) )
l e2 (6.59)
134 Adaptive nonlinear control." passivation and small gain techniques
and that
OV2 OVI ~T O~I
0/~ (x, 0) = - - ~ (~, 0) - --~ (~,/~), (6.60)
~ (7"1+.AT2) "4-G2(x,O)v.+.G2(x,0)~92
q. ~T []~11(~, 0) +f20(X) +f2(x)0 -" "001
:= -n~(~,o) + (ov~
\'-~- (x, ~) + (~- o) ~r- ' ) ~ + yrO2(x, o)v
( orv' )
+yT tg2(x,O)+G2(x,O)'O2(x,O)+Y~r 0~ (~,0) (6.61)
Observing that T2 depends on 02, the following variable is introduced to split
this dependence
p=rOTVl,, (~,1~) E R I
O0
With (6.58)
Or Vl
G2(x, 0)02(x, 0) + T2F 0 0 (~j'/~) = G~(x,0 + p)O2(x,O) (6.62)
we obtain
P~ <_-,nCx, #) + h~Cx,#, o) (v) (6.65)
with ~ = 711(~, 0) "~-jy[2. The first statement of Proposition 3.2 was proved.
The second part of Proposition 3.2 follows readily from our construction
and the main result of Sontag [43].
On the basis of Corollary 3.1 and Proposition 3.1, a repeated application of
Proposition 3.2 yields the following result on adaptive backstepping
stabilization.
Adaptive Control Systems 135
= + y, - - > 0
9 o
It was shown in [40, p. 67] that any linear output feedback u = - k y + o, with
k > 0, cannot render the system (6.72) passive. In fact, Byrnes and Isidori [1]
proved that the system (6.72) is not stabilizable under any C l output feedback
law. As a consequence, this system is not feedback passive via any C l output
feedback law though it is feedback passive via a C ~ state feedback law.
However, system (6.72) can be made passive via the C O output-feedback
given by
l 3
u = -ky~ + o, k > 24/3 (6.73)
_< -(1 _
(3)
k_24/3 /3 (6.79)
138 Adaptive nonlinear control: passivation and small gain techniques
Letting ,.q.,,
~ _< - ( l - ~)z 4 -
(3)
k - 24/3e~/3 r + y~ + (~ _ o)rr-~e (6.81)
In other words, the system (6.76) is made passive via adaptive output-feedback
law (6.'/3)-(6.80). In particular, the zero-input closed-loop system (i.e.
(~, e) ~- (0, 0)) is globally stable at (z, ~, O) = (0, 0, O) and, furthermore, the
trajectories (z(t),((t)) go to zero as t goes to ~ .
y=Xl
where u in R is the control input, y in R is the output, x = (xl,... ,xn) is the
measured portion of the state while z in Rn~ is the unmeasured portion of the
state. 0 in R I is a vector of unknown constant parameters. It is assumed that the
Al's and q are unknown Lipschitz continuous functions but the ~o~'sare known
smooth functions which are zero at zero.
Adaptive Control Systems 139
The following assumptions are made about the class of systems (6.82).
(A1) For each 1 _< i _< n, there exist an unknown positive constant p* and two
known nonnegative smooth functions ~n, ~,2 such that, for all (z, x, u, t)
IA~(x,z,u,t)l <_p;'~P,(l(x~,... ,x~)l) (6.83)
Without loss of generality, assume that ~,2(0) = 0.
(A2) The z-system with input xl has an ISpS-Lyapunov function V0, that is,
there exists a smooth positive definite and proper function Vo(z) such
that
OVo
Oz (z)q(t,Z, Xl) <_ -o~0(Izl) -F ,),0(lxl l) -t- do V (Z, Xl) (6.84)
where F > 0, A > 0 are two adaptation gains, r/is a smooth class-Koo function
to be chosen later, p __
> max J',,*
w+ ,yt,,*2 I1 -< i _< n} is an unknown constant and the
140 Adaptive nonlinear control: passivation and small gain techniques
where r/(x~) is the value of the derivative of r/at x 2. In the sequel, r/is chosen
such that r/' is nonzero over R+.
Since q~ll is a smooth function and @ll([Xl[) = !hi (0) + ]xll f0l ~l(slxl[) ds,
given any el > O, there exists a smooth nonnegative function ~l such that
p;,t(~)lxtlr < p,f(x2)x2~,,(x,)+ e,r 2, Vxt E n (6.88)
By completing the squares, (6.87) and (6.88) yield
1)'1 ___./(X~I)XI (X2 "~-oT~I(XI)+pxl~l(x,)+p~x,r/'(x2))+ (0--O)rr-l~
1
+~ - p ) ~ + qh2([z[)2 + CiVil (0) 2 (6.89)
Define
Fn'(x2)xl~pl (xl)
rl = -Ftro0 + (6.90)
w, = -Aapp+ Ax~l((bl(xl ) + ]ff(x2))ff(x 2) (6.91)
Ol = - x l g , ( ~ ) -Or~o,(x=) -p(x,~,(x,) + ~x,r/'(x~)) (6.92)
w2 = x2 - 01 (xl, 0,/3) (6.93)
where go, av > 0 are design parameters, Ul is a smooth and nondecreasing
function satisfying that =/1(0) > 0.
Consequently, it follows from (6.89) that
~}'1~ --'/']'X~IVl (X~l) "t- T~XIW2 -- 0"0(0 -- o)TO -- O'p([7 --p)p -[- (0 -" o)TF-'('O -- 3"1)
1
+~ - p ) ( ~ - tvl) + qh2(lz[) 2 + elr (0) 2 (6.94)
It is shown in the next subsection that a similar inequality to (6.94) holds for
each (xl,..., xi)-subsystem of (6.82), with i = 2,..., n.
1 < i _< k, there exists a proper function Vi whose time derivative along the
solutions of (6.82) satisfies
i
f/i <<_-~f x~l (v, (x~l ) - i + 1) - ~-~(cy- i+j)w~ + wiwi+, -ao(O-O) r O-ap(~ - p)~
j=2
i i
+ ~-~j~(,_j+,p(Izl) 2 + ~-~j~i_j+l~b(i_j+l)l(O) 2 (6.95)
j=l j=l
In (6.95), 6j > 0 (1 < j < i) are arbitrary, cj > n - j (2 < j < i) are design
parameters, Oj (1 < j < i), ri and wi are smooth functions and the variables
w/'s are defined by
It is further assumed that 0/(0,..., 0,0,/~)= 0 for each pair of (0,/~) and all
l<i<k.
The above property was established in the preceding subsection with k = 1.
In the sequel, we prove that (6.95) holds for i = k + 1.
Consider the Lyapunov function candidate
In view of (6.95), differentiating Vk+l along the solutions of system (6.82) gives
k
f/k+l <_~--~'X~l (Vl ( ~ ) - k + 1 ) - ~ ' ~ ( c j - k + j)wf + WkWk+, --ao(O--O) r O--ap(~--p)~
/=2
+ - ~ wj o0 +
j=2 op /
Ix k 0Ok r
(x.. j+ Jl- o O_-ff
k k
+ ~-~flP(k-j+l)2(]z[) z + ~--~jek-y+l~P(k-y+l)l(0) 2 (6.98)
/=1 j=l
Recalling that p > max .,fn* n . 2 ]1 < i < n}, by virtue of assumption (A1), we
tri ,ri
_ _
142 Adaptive nonlinear control: passivation and small gain techniques
have
+ lwk+,I(P2+l'/'(k+,),
(l(x,,...,
xk+,)I)
k ,Io~kl,~, ) k+, )2
+ 'Jl l' <l<x,,...,x,)l) +
(6.99)
From (6.93) and (6.96), it is seen that (wl,..., Wk, Wk+l) = (0,..., 0, 0) if and
only if (Xl,...,Xk, Xk+l)= (0,...,0,0). Recall that ff(x~)~0 by selection.
With this observation in hand, given any ek+l > 0, lengthy but simple
calculations imply the existence of a smooth nonnegative function ~k+l such
that
k
Iw~+,l ~+,~(k+,),(l(x,,...,xk+,)l) + ~P]
j=l
k k+l
< pW2k+I+k+l (WI, W2, 99 9 Wk+l,O,#) -~"X~lflI (~1)-I-E W]j"1-E 6j~jl (0) 2 (6.100)
j:2 j=l
Combining (6.98), (6.99) and (6.100), we obtain
k
j=2
+ O-o)~r-'-~]~joO:
j=2 j=2
x k OOk
"I"W k-l-I k-l-2"I-W k "I-OTqOk.l-I"- E ~]Xj(Xj'I'I"l-OT~oj)
j=l
(1
-t-~Wk+l "~-I-'~~_l,-~j, '
1 ~--,(OOk~2
-i"+k+l
) -- GQ--
OOk~ OOk/~]
~ --~ j
k+l k+l
9
+ ~JV)(k-j+2)2(Izl) 2 + Ejek_j+2V)(k_j+2),(O)2 (6.101)
j---I j=l
Inspired by the tuning functions method proposed in [26, Chapter 4] for
Adaptive Control Systems 143
~+~ = rk + r
( ~'k+~ ~k" O
~ =l ~ k )
-- . _ ~ o x j qOj wk+l
(6.102)
j=l
k ig~k
~k+l = --Ck+l ~k+l -- Wk + ~-ff~Xy Xj+l
- o - wj oO r
j=2 j=l
0Ok 0Ok
+ 0---ffrk+l +--~-Wk+lWk+2 (6.104)
~'n _< --~(X~l)X~l(Pl (X~l)--n -[- l)--~--~(c,- n + i)w~i -- era(O-- o)To -- O't,(~- p)~
i=2
H n
(6.106) yields:
~Fn<( -cVn + ~l (Izl:) + ~l (6.114)
where
tr0 ) , trpA; 2 <
c = min {2cl, 2(el - n + i), Am~x(F_l _ i<
_ n} (6.115)
n
~_2 (6.116)
81 -- T 1012 "~"T p -I" E ign-i+lr (0) 2
i=1
Let _a~ and ~v be two class-Koo functions such that
~(Izl) _< Vo(z) <_ ~o(Izl) (6.117)
Given any 0 < e2 < c, (6.114) ensures that
~'. < - 6 ~ v . (6.118)
Adaptive Control Systems 145
whenever
OVo
cgz (z)q(z, xl) <_ -e3~0(Izl) + 7(7 -I o e370(lxll)) + e3do (6.120)
Vo < - e 3 e 4 ~ o ( l z l ) (6.121)
as long as
- >_ max
V0 e3~v o OtO10 1 -2 ~4.,.),(7_10')'o(IXll)), E3~v 00tO 1 1 - Ea
(6.122)
To check the small gain condition (6.23) in Theorem 2.5, we select any class-
Koo function 7 such that
7(s)<
I--E4
2 a0o~-loa_v
( ~ ~ i- l ( c)- E)2
4 s , Vs>0 (6.123)
Finally, to invoke the Small Gain Theorem 2.5, it is sufficient to choose the
function r/appropriately so that
7 -l o 70(Ix~l) <_ 89 + ~5 (6.124)
where e5 > 0 is arbitrary. In other words,
7 -~ o 7o(Ix~l) _< V.(x~,x2,...,x,,O,p)+s5 (6.125)
Clearly, such a choice of the smooth function r/ is always possible.
Consequently,
Vo _<-e364c~o(lzl) (6.126)
as long as
- >_ m a x
V0 e3fly 0 ~olO 1--~4
2 ,),(2Vn)' 6"3flyo o~01o 1 2 e4 -l 2~4
(6.127)
Under the above choice of the design functions ~ and Vl, the stability properties
of the closed loop system (6.82), (6.106) and (6.107) will be analysed in the next
subsection.
146 Adaptive non/inear control." passivation and small gain techniques
Similarly, with the help of (6.126) and (6.127), a gain for the ISpS z-subsystem
with input Vn and output f'o is given by
2
X 2 ( S ) ~'~ r o OtO 1 o i g47(2s ) (6.129)
As it can be directly checked, with the choice of 7 as in (6.123), the small gain
condition (6.23) as stated in Theorem 2.5 is satisfied between Xl and X2. Hence,
a direct application of Theorem 2.5 concludes that the solutions of the
interconnected system are uniformly bounded. The second statement of
Theorem 4.1 can be proved by noticing that the drift constants in (6.119)
and (6.127) can be made arbitrarily small.
Remark 4.1 It is of interest to note that the adaptive regulation method
presented in this section can be easily extended to the tracking case. Roughly
speaking, given a desired reference signal y,(t) whose derivatives y!/)(t) of order
up to n are bounded, we can design an adaptive state feedback controller so
that the system output y(t) remains near the reference trajectory y,(t) after a
considerable period of time.
Remark 4.2 Our control design procedure can be applied mutatis mutandis
to a broader class of block-strict-feedback systems [26] with nonlinear
Adaptive Control Systems 147
unmodelled dynamics
= q ( t , Z , Xl)
~l --" m 1 2 ( 6 - 60)
~2 = m12 6 + m
( k (~- ~~ (6.132)
to transform the target point (6, 6) = (6o, O) into the origin (~1,~2) = (0, 0).
It is easy to check that the pendulum model (6.131) is written in ~-co-
ordinates as
k
~l =~jz - m ~ l (6.133)
Since the parameters k, l and m are unknown and the angular velocity 6 is
unmeasured, the state ~ = (~1,~2) of the transformed system (6.133)-(6.134) is
therefore not available for controller design. We try to overcome this burden
with the help of the 'Separation Principle' for output-feedback nonlinear
systems used in recent work (see, e.g., [23, 26, 32, 36]).
Here, an observer-like dynamic system is introduced as follows
[_,, 1] ,, ~
__,, #2~,_ 0 J r - -~ ~l - mgl sin
( 6o + ~-~1
1 ) --I-IAo(t)
A
(6.137)
For the purpose of control law design, let us choose a pair of design parameters
#l and t'2 so that A is an asymptotically stable matrix.
Letting xl = 6 - 6o, x2 = ~2 and z = e/a with
a = max{Jr 2 - r - kl2[, Jr 2- r lao } (6.138)
Adaptive ControlSystems 149
x2 = u + r - ml2)xl + r
y=xl
1
Since the unknown coefficient ~-~, referred to as a 'virtual control coefficient'
[26], occurs before x2, this system (6.139) is not really in the form (6.82).
Nonetheless, we show in the sequel that our control design procedure in
subsection 6.4.3 can be easily adapted to this situation.
Let P > 0 be the solution of the Lyapunov equation
PA + A r P = - 2 I (6.140)
Then, it is directly checked that along the solutions of the z-system in (6.139)
the time derivation of Vo = z rPz satisfies
1 1
VI = 2m/2X~l +~-~(s 2 (6.142)
where
p > m a , x{a2 1 1 k2 02 }
(6.143)
_ , m l 2 , m 2 l 4 , m 2 , m 214
It is important to note that we have not introduced the update parameter 01 for
the unknown but negative parameter - k / m because the term - ( k / m ) x l is
stabilizing in the xl-subsystem of (6.139).
The time derivative of Vl along the solutions of (6.139) yields:
1
~/'1 <-~ --1)1 X2 + X I W 2 -- O'p (.P -- P ).P "l" -~ (,P -- p ) (i3 -- '{X71)"~'89 2 (6.144)
150 Adaptive nonlinear control: passivation and small gain techniques
w2 = x2 - Ol (xl,/~) (6.147)
(u (v +~) ( 1 k 1 2)x,)
+W2 +g2(l--ml2)Xl+g2azl+ 1 -~X2----XIm +'m-Faz +~--f
(6.149)
From the definition of x2 in (6.147) and p as in (6.143), we ensure that
(
W2 VI "~"
(I ())
~ X 2 <_ p VI "~" "~" VI "~"~~ 4 w~ + - ~ (6.150)
W2
2
In other words
cr~ _2
f," <_ -cV +-~t, (6.156)
with
c min {(vl -- 1)ml2,v2,0.StrpA, O.SAm=(V)-l }
Finally, from (6.156), it is seen that all the solutions of the closed loop system
are bounded. In particular, the angle 6 eventually stays arbitrarily close to the
given angle 6o if an a priori bound on the system parameters m, I are known
and the design parameters el, v~, trp and A are chosen appropriately.
1 (~ -
7x2~l) (6. 62)
7
where r/stands for the derivative of r/.
By choosing the adaptive law and adaptive controller
= -a070 + 7x2r/(x 2) (6.163)
u = - x v ( x 2) - Ox - 8 8 2) (6.164)
where o'o > 0 and v(.) > 0 is a smooth nondecreasing function, it holds:
I~ < -x2~7~v(x2) - 89ao(O - O) 2 + z 4 + 89aoO 2 (6.165)
Select v so that
F~'(F)v(F) > ,I(F) (6.166)
Then (6.165) gives:
< -~W+z 4 +-~
ao 02 (6.167)
u = - x - ~ x - ~ 4 x3 (6.172)
In view of (6.159) and (6.167), a direct application of the Small Gain Theorem
2.5 concludes that all the solutions (x(t),z(t),O(t)) are bounded over [0, oo).
In the sequel, we concentrate on the Method II: robust adaptive control
approach with dynamic normalization as advocated in [17,18].
To derive an adaptive regulator on the basis of the adaptive control
algorithm in [17], [18], we notice that, thanks to (6.159), a dynamic signal
r(t) is given by:
= -0.8r + x 2, r(O) > 0 (6.173)
The role of this signal r is to dominate Vo(z) - the output of the unmodelled
effects- in finite time. More precisely, there exist a finite T ~ 0 and a
nonnegative time function D(t) such that D(t) = 0 for all t >_ T ~ and
where Ao > 0. A direct application of the adaptive scheme in [18] yields the
I poq;akll o; sau!l paqsep eq; al!qM II poq;ayV
o; sajas sau/I p/Ios aq; :(~)a ; n o q # ~ I poq;akll snssaA (l)a q#M II poq;Qkl/ Z'9 e . m w
9 g t, ~ I. 0
...... | |
I
I
o~
I
I oc
I
I
t lye
i
t
og
11
i!d
1 i | i i
09
s~ee
9 t ~ 9 t ~, , O_u
Oo o ~ - -
,. ... ... J
r
o
C
6
00L-
09"
J
. . . . . "" 0 g
9 1,
. . . . . . . . . . . . "
0
0~"
t, Oo
i z o
IP
g
I=
r-
"*
....9
u = -
5 1) x
-~ + - ~
A0
- Ox - -l-~ x r (6.177)
6.5 Conclusions
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