Abstract. In this paper, the discontinuous projection based adap- tive robust control (ARC) scheme is generalized to a class of nonlinear systems in an extended ...
Proceedings of the 38” Conference on Decision & Control Phoenix, Arizona USA December 1999
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Observer Based Adaptive Robust Control of a Class of Nonlinear Systems with Dynamic Uncertainties Li Xu
Bin Yao
School of Mechanical Engineering Purdue University, West Lafayette, IN 47907, USA
Abstract
unknown nonlinear functions such as disturbances and modeling errors. The following practical assumptions are made:
In this paper, the discontinuous projection based adaptive robust control (ARC) scheme is generalized to a class of nonlinear systems in an extended semi-strict feedback form by incorporating a nonlinear observer and a dynamic normalization signal. The unmeasured states associated with the dynamic uncertainties are assumed to enter the system equations in an afine fashion. A novel nonlinear observer is first constructed to estimate the unmeasured states for a less conservative design. Estimation errors of dynamic uncertainties, as well as other model uncertainties, are dealt with effectively via certain robust feedback control terms for a guaranteed robust performance. In contrast with existing conservative robust adaptive control schemes, the proposed ARC method makes full use of the available structural information on the unmeasured state dynamics and the prior knowledge on the bounds of parameter variations for high performance. The resulting ARC controller achieves a prescribed transient performance and final tracking accuracy in the sense that the upper bound on the absolute value of the tracking error over entire time-history is given and related to certain controller design parameters in a known form. Furthermore, in the absence of uncertain nonlinearities, asymptotic tracking is also achieved.
1
Problem
The following
Assumption E
R(? i! {e:
E
flA e {ii:
Ai
E
Ra k {Ai em,,,
emin 16(2,q,u,t)j :
lAi(x,q,
(2)
5 @fi,t)} u,t)l
< %(~i,t)}
b(?i, t) and & (zc~,t) are known.
0
0
stable.
The objective is to design a control input u such that the output y tracks a desired trajectory Ed as closely as possible in spite of various model uncertainties.
2
State
Estimation
Motivated by [l], we first introduce a transformation of coordinate. Define a vector < = q--w(P~), where W(ZL)is a vector of design functions yet to be determined. For state estimation we employ the following filters
where A(%l) = G,(z~) - &~p$, [, E R”, F,j represents the jth column of F,,, and vei,j the jth element of the vector 90;. The state estimate can thus be represented by i = co + 4-e (4) where < = [cl,. . ,&,I E Rmxp. Then, the state estimation error E = i - [ would be governed by the following dynamic system
system is considered
0 1999 IEEE
6~and uncer-
Assumption 2 The q-subsystem, with 11 as the state and %:l(t) as the input, is bounded-input-bounded-state
where %l = [z~,...,zI]~ E R’, Z, = [~l,...,xi]r E R’, and z = [xi,...,~~]r E R”. y E R and u E R are the control input and the output, respectively. 7 E ]R” represents the unmeasured states, and 0 E RP is a vector of unknown constant parameters. F7) E lRmxp, G, E IR mXm, q~i E ]Rp and ‘pVZE R” are matrices or vectors of known smooth functions, which are used to describe the nominal model of the system. d and A, represent the 0-7803~5250-5/99/$10.00
e A
where &in,
Statement
nonlinear
1 Parametric uncertainties & and Ai satisfy
tain nonlinearities
i = A(z~)E + A,,
A,=-B+CzAi
I i=l
aw z
(5)
Assumption 3 The observer error dynamics (5) is exponentially input-to-state stable, i.e., there exists a Lyapunov function V, such that
72
where -yl, 72 and ~c are class Kw functions, cc 20 and The unperturbed system of d. 2 0 are two constants. o (5) is assumed to be exponentially stable. Lemma 1 1 If (6) holds, then, for any constants z c (O,cc), any initial condition &o = c(O) and ra >0, for any function ~ such that ~(il) z ~=(litl), there ezists a finite 2’0 = T“(C, .O, co) ~ O, a nonnegative function D(t) defined for all t ~ O and a dynamic signal described by i = —Er+ ~(fil) +de,
r(o) = To
for all t ~ O where the solutions
D(t)
(8)
are defined.
3
~ts
=
‘kis
Zi,
Cei12 + lc4irOi12 +cel@i12
ki$ ~gi+l~
where gi and ce are positive constants, Cei and C+a are positive definite constant diagonal mat rices, ~(1–l)c
t–l = Xj=l
acY/_~ V(%+l
‘(i-l)c
=
~~=~
*{xj+l
+@+
LJ(zi))}+
+ ~T!f%j) + @Tw.i *i+
&;_~
(9)
i. ii.
The control law and parameter update law are summarized in the following. The detailed design procedure and the proof of theorem can be obtained from the authors.
A(Q’i.92
– @’@t
+
Ela,_~ ~
+ WTj(C’
(15)
*LO i>l
control
term satisfying
– @~C + Ai)
< Ci(l
+ ~2)
Z%atsz ~ o
where Ci is a positive
constant
(16)
and p(t)~ ~; 1(2D(t)).
Theorem 1 Let the parameter estimates be updated by the adaptation law
step 1< i
