ThMO9
Proceedings of the 38” Conference on Decision & Control Phoenix, Arizona USA December 1999
15:20
l
Adaptive
Robust
Control
without Knowing Variations
Bounds
of Parameter
Bin Yao+
J. Q. Gong
School of Mechanical Engineering Purdue University, West Lafayette, IN 47907, USA + Tel: (765)494-7746 Fax:(765)494-0539 Email:
[email protected]
Abstract A discontinuous projection based adaptive robust control (ARC) design is constructed for nonlinear systems transformable to the semi-strict feedback form without knowing the bounds of parameter variations and uncertain nonlinearities. Departing from the existing robust adaptive control schemes in which the unknown bounds are estimated on-line, the proposed ARC design only adapts actual physical parameters and uses a fixed design bound plus certain robust feedback to account for the possible destabilizing effect of on-line parameter adaptation. The resulting ARC controller achieves a prescribed transient performance and final tracking accuracy in general; the exponential convergence rate of the transient tracking error and the bound of the final tracking error can be adjusted via certain controller parameters in a known form. In addition, in the presence of parametric uncertainties only, if the true parameters fall within the design bound, asymptotic output tracking is achieved. Furthermore, by choosing the design bound appropriately, the controller also has a well-designed built-in anti-integration windup mechanism to alleviate the effect of control saturation. Extensive comparative simulation studies are carried out to illustrate the proposed ARC control strategy.
Keywords: Systems,
Adaptive Uncertainties.
Control,
Robust
Control,
Nonlinear
1 Introduction A high performance robust controller should explicitly take into account the effect of both parametric uncertainties and uncertain nonlinearities. Two conventional approaches exist in dealing with these uncertainties: adaptive control [l, 21 and deterministic robust control [3]. In order to further improve the performance, by effectively combining these two approaches, the adaptive robust control (ARC) has recently been proposed in [4]. The existing ARC controllers [4] assume that the bounds of parameter variations are known. For some applications, this assumption may not be satisfied although actual parameters are usually bounded. Thus, it is of practical interests to relax this assumption and to look for suitable modifications to the above ARC designs to accommodate the lack of this prior information, which is the focus of this paper. The problem of designing robust controllers for systems without knowing the bound of parameter variation is not new and there have been some excellent results published in the literature [5, 6, 7. 81. In [5, 6, 71, in addition to estimating physical parameters, certain u-modification
0-7803-5250-5/99/$10.00
0 1999 IEEE
3334
type of parameter adaptation laws is also used to obtain the estimate of the unknown bounds of the parameters. Although theoretically these schemes guarantee the global stability and the asymptotic convergence of the tracking error to a small value, however, as pointed out and experimentally verified in [9, lo], these gain (or bound)-based robust adaptive control schemes may suffer from the following practical implementation problems. Firstly, if the on-line gain or bound estimate is small due to the use of a small initial estimate or a strong a-modification, then, the resulting actual feedback gain might not be large enough during the transient period and a large transient tracking error may exhibit. On the other hand, if the on-line gain or bound estimate becomes too large due to the presence of measurement noise or other neglected factors, then the resulting actual feedback gain might be so large that the close-loop system becomes unstable due to the unavoidable neglected dynamics in implementation. In viewing these practical problems associated with gain-based robust adaptive control schemes, in this paper, a different design philosophy will be adopted: parameter adaptation will only be used for estimating the actual physical parameters to achieve a better model compensation for an improved performance. The effect of unknown parameter bound shall be handled effectively via certain robust feedback. Under this design philosophy, by introducing a pre-set design bound representing the level of parametric uncertainties that one would like to compensate for dynamically, the discontinuous projection based ARC design [4] will be generalized to solve the problem of unknown parameter bound. The resulting ARC controller achieves a prescribed transient performance and final tracking accuracy in general. In addition, in the presence of parametric uncertainties only, if the true parameter falls within the design bound used for the parameter projection, asymptotic tracking is also achieved-a performance that the existing robust or robust adaptive control schemes cannot have. Furthermore, by choosing the design bound of the parameter projection mapping appropriately, the controller also has a well-designed built-in anti-integration windup mechanism to alleviate the effect of control saturation. Comparative simulation results will be shown to illustrate the proposed ARC strategy.
2 Problem As a starting point, let first-order SISO system
d = @(z,t)B
Formulation us consider
+ A(z,t)
where (a(~,t)=[4,(~:,t), of known basis functions,
the
following
+ U, y = 2
‘.., q&.(~,t)]~ 2 is the state
is the variable,
simple
(1) vector 0 =
paunknown system [ 01, .. ‘, 4 IT .IS the constant rameter vector, A(x,t) represents the lumped uncertain disturnonlinearities due to approximation error, external bances, etc., 11 is the input to the system, and y is the assumptions output of the system. The following practical are made:
law for the estimate of the bound is turned on only when the estimate of the parameter “tries to” overwhelm the estimate of the corresponding bound. In this sense, s is called a switching function matrix. It can also be easily seen that inequality ]0i] < ,?i always holds and &‘s are nondecreasing functions of time.
Assumptions
Theorem law
Al
System bounds
parameters are bounded may not be known, i.e.,
where A2
IQiI < Pi,
i=
pi are unknown
positive
Uncertain nonlinearity known function with
1,2;..,r
Proof.
(2)
By
choosing
law (5) with the adaptation are bounded and asymptotic
a Lyapunov
guments,
by an un-
Theorem
t)l I
8(x,
0
derivation,
control law (5) may suffer from possible instability due to the unboundedness of the adaptation law. It is seen from (6) that b keeps increasing whenever the corresponding adaptation law is turned on and the tracking error is non-zero. In practice, the tracking error will always be non-zero due to other neglected factors. It is thus likely that @ will become larger and larger in implementation, and an unstable closed-loop system may result.
G is used to represent
the esti-
p = [ ~1, . ., ,o,. IT. The control objective is to make the output y to track a desired trajectory yd(t) as quickly and accurately as possible. For system
of *, and
(l),
let xd = yd and the tracking
3 Robust
Adaptive
error
6). Transient
be e = x--;~d.
the following
switching
=
id - ke - a’si
4(x,
where u,l and u,z following conditions
and the parameter
-
adaptation
b = rls@e, I’1 = diag 721, .‘.,
where design
I
(DRC)
(5)
j = l?z(I - s)j@llel
{ 711, ~2,. }
controller
e[A(x, t) + ~s~l 5 d2(t)c2
(6) e2(t)
where diag{
robust
Control
+ U,l] 5 1 + KIIPII- Il~(O)ll)l 2E1 (8) Ili(O)ll
e[-QTs(0)
Let the switching , s, }, Consider
t)d(t)sign(e)
Robust
is unknown.
u = id - ke - apT(x, t)e(O) + u,l + u,z
function
V(e, 6, ,6) = $(e2 + gTI’;‘e + p”rr;‘p), it can be shown that G < -ke’. Using the standard adaptive control ar-
constants.
A(x,~) is bounded known shape, i.e.,
all signals is achieved.
is used,
output tracking
their
although
3.1 If the control
(6)
. 71,. } > 0 and 0, are the adaptation
I?2 = rate where
matrices, I@] = [ 1411, ..‘, I#+. IT, and I is the identity matrix of dimension T. It is noted that d(t) is assumed to be known temporally and this assumption will be removed in the later discussion.
exp(-2kt)e2(0)
norm of d(t).
Proof.
Chose
viewing
+ t
l+J(IIPII-IIP(o)ll)l 2~~+lldll~~~ IIP(O)lI
the L,
In
It is seen from (6) that adaptation for the parameter estimate is turned on only when the estimate of the parameter is within the estimate of the bound, and the adaptation
c =
to), and if IBil 5 ii(O), Vi, then, in addition to the results in A, asymptotic output tracking is achieved. 0
details
5.1
adaptation
dynamics
t) + u,l + 21,~
t)e” + A(x,
5.1 Let r = cPe. With
e2(t)
Robust
error
signals are bounded. Furthermore, tracking error exponentially converges value and is bounded above by
ill].
5 Adaptive
the resulting
- @(x,
and parameter hold:
hl L IlW~,t)ll (ll~(o)ll+ Ilb(O)ll) andh2 2 J(x,t), respectively
law,
and
and 21,~ = -&hie functions
With
u,l
satisfies
the following
I(IIPII- Iliv)ll)l 2tl Ilb(O)II
~$1 and u,2 are required conditions, eu,; < 0, i
to satisfy
= 1,2.
I(IIPII- Il~(O)ll)l 2E1= E1 Ilb(O)ll
Remark 6 According to Theorem 5.1, theoretically, the fixed design bound p(O) should be chosen as large as possible. However, in practice, this should not be overdone, especially in the presence of control saturation. Due to the use of the project mapping in the parameter adaptation law (13)‘ the proposed ARC law has a built-in anti-integration windup mechanism [4], Furthermore, by choosing the preset bound p(O) properly based on the control saturation level, the resulting ARC law should be able to handle the control saturation well. One practical way to choose b(O) is as follows. It is reasonable to expect that for all 6 satisfying (14), the nominal value of the model compensation part Us is within Ihe control saturation limit. In other words, 1;i.d -
(16)
are
which is the same as that in [4], This indicates that all the results in [4] are recovered when the bounds are known.
law + 11,~
variations becomes
aT(xd,
t)eI
5 %nax
- 6,
(19)
where IL,,, is the control saturation level (i.e., u(t) 5 umaz), and t, < u,,, is the margin left for robust control term. Assuming that lid(t)I < id,,,,, , and ll@(xd, t)II < @mar I Vt, from (14). we have Iid
(17) Thus,
- @-(xd,t)el
5 ,dm,,
+ %naz~~~(O)~~
(20)
we may choose
the trivial (21)
6 ARC of SISO Semi-Strict
Nonlinear Feedback
Systems Forms
in
the following the previous
functions steps
for
step
j
(j
> 2) from
those
in
In this section, the ARC scheme proposed in Section 5 will be extended to SISO nonlinear systems transformable to the following semi-strict feedback form [14]:
di=~i+~ +~‘T(21,...,zi,t)B+A,(~,t), &=a(x)u where
+ gy(Xl, 2
i I: n - 1
. ,xn,t)Q+&(x:,t),!J=
21
T
=
[
21,
...>
2,
]
are
state
(22)
7-j(Zj, e,q= Tj-1+ WjZjQj
vector,
Xi,b) E R’, i=
l,..., n are the known shape are assumed to be sufficiently smooth. 0 Assumption Al and Ai(x are assumed to be by
@T(x1;-, functions, satisfies bounded
which
jAi(x,t)l
5 &(zi,t)di(t),
i = l;..,n
where UJU~> 0 is a weighting is bounded by &(x,t)l
pi = [ ~1,
..., are known, xi IT, &(c,t) but bounded disturbances.
di(t) are unknown
and
Jj(Zj,t)
The controller design follows the discontinuous projection based backstepping ARC procedure in [4] but with the modification presented in Section 5 for the relaxed assumptions on the bounds of parameter variations.
Let first
A,(x,t)
function
il = xz + !P:(xl,t)B+
= cPl(zl,t). as
A,(x,t)
maxEz({dl(t)}.
=
the following
virtual i=i-1
=
-F*j-l
-
k,jZj
+
>
kj
+
law
aai-l x
x6+1
JTXPj+ ‘Yjl + ‘yjz
+ at hi
C
control
I=1
&j-l --
The
(24)
satisfying
‘=“-‘{~s,(2,,t)}}),
3
= A,(z,t), and \Irt(sl,t) of (22) can be rewritten
&i(~,t)
(32)
smooth
{ dj(zj,t),maxl,,
1
equation
(23),
< sj(zj,t)&(t) any
At step j 5 i, choose
GJj
Step
is
mu2
he( and ij(t)
6.1
From
(23) where
where
factor.
(31)
(33)
Q;II
(34)
where kj, cgi, and cliij are positive design parameters, and ajz are robust control functions satisfying
Let d(z:,,t) = 6( xl,t). Then, lAl(x,t)l 5 6 xl,t dl(t). Thus, in principle, the ARC scheme develope 6 in 3 ection 5 can be applied to (24) to synthesize a virtual control law LYE for x2 so that ~1 can track its desired trajectory old with the desired properties stated in Section 5. Considering the particular difficulties and addressing strategies for the backstepping ARC design with discontinuous projection [4], the following ARC control function is suggested
zplp+
Zj[&
“jl]
