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Gang Tao. Adaptive Control Design and Analysis (Wiley,2003)(ISBN 0471274526)(T)(637s)_E_.djvu |
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It follows from G.24) and G.26) that
Ф) = ЩШ*)> W = вр(*) ~ %¦ G-27)
Then, we choose the gradient adaptive law for 6p(t):
where
лу*{л\ /i i A31 (-t-\A\(j-\ ('У 0Q^
However, this adaptive law does not ensure that the estimate |6p| > b0 for
some constant bo > 0, which is needed to ensure that k\(t) and &2(t), solved
from the design equation G.23), are bounded...
\&>Vf + hufW + oo(*)- G-48)
From G.29), G.30), and 9p(t) = [hp(t),bp(t)]T, we have
\Ф)\ = ^Mt) < ^A + |У/(*I + M*)l) G-49)
G.50)
for some constant a3 > 0, Then, from G.48)-G.50), we obtain
\yf(t)\ <ai +
+ a4 /*е-«(*-^)Ш Ге-а^-%(а)\с1ас1т G.51)
Jo m(r) Jo
for some constants 04 > 0 and 05 > 0...
Based on the parameter vector 9p(t) obtained from G.70), we identify
[kp~Z0{t), Vl(*)> •¦¦> kPZm-l(t), ^р(*), ~Po(*), -Pl(*), • • • , -Pn-2{t), -pn-i{t)}
= 0E(t) G Rlx(»+m+1) G.77)
306
Chapter 7 Indirect Adaptive Control
and define the parameter estimates of Zi as
Zi(t) = > , i = 0,1,..., m - 1...
Proof: In view of G.82), we define
№#*s(t)F e R2n, G.91)
(t)]r G R2n, G.92)
= [ef,e?,e*0,e*3}TeR2n, G.93)
= »(*) - в\ e(t) = y(t) - ym(t), G.94)
308 Chapter 7 Indirect Adaptive Control
where 6%, i = 1,2,3, and #20 are the nominal plant-model matching parameters
satisfying E.11) with n = n...
Using G.95) and G.110), we obtain
V(t) - vm(t) + p{t)$(t) = Щ + e»(*) + ФУ G.111)
Prom G.62), G.68), and G.78)-G.80), we have
e{t) = kp(t)Z(s,z(t)) [д^уМ] (*) - Н*Ж*)) [л^Iу]] {t)' GЛ12)
where ^ e L2 П L°°, for m(t) in G.71), and k(t) E L2 П L°°, ^(t) E L2 П L°°,
that is, pj(i) E L2 П L°°, i — 0,1,..., m...
This con-
condition is necessary, because a model reference controller is to make the plant
output track the output of a reference system so that the plant zeros are to be
cancelled by a feedback control law, and such a cancellation must be stable...
?
Without the need that the plant zeros are stable as in the model reference
adaptive control case (see Chapters 5 and 6), the following basic assumptions
are needed for solving the pole placement control problem:
(A7.2C): Qm(s)P(s) and Z{s) are coprime...
The singularity of the indirect adaptive pole
placement control law occurs when det[S(p(t), z(t))] = 0, and it is near singu-
singularity when det[S(p(t), z(t))} is close to 0, and either case should be avoided
for implementation of adaptive control...
This modified adaptive law ensures that 9pi(t) G [0%, 9^], Vt > 0, i = 1,..., 2n,
in addition to the properties of the standard gradient algorithm G.164)...
Chapter 7 Indirect Adaptive Control
Using G.185) and G.188)-G.191) with x(t) = u(t) or y(t), we have
1 r_ ;t
(s
u(t)
T
(s + ao)Ac(sr^'-L ^^1[(s + a0)Ac(S)
Introducing uin(t) = ——-\фти>Ш), we have
Letting a2(s) = [1, s, s2,..., s™-1]T, from G.160), we have
Denoting AQ(s) = sn« + X°n _1s™"-1 4 1- Ajs + Ag, we have
A0(s)a2(s)
ai(s)
BQ
sn+nq-l
(s + ao)Ac(s) (s + ao)Ac(s) (s + aQ)Ac(s)'
where ai(s) = [1, s,..., sne+"-2]T as before,
Ag A? ¦¦• A^_x 0 0
n \° \° .....
Then,
for the positive definite function V(z, t) = zTP(t)z, we have
V = -zTz + zTP(t)z < - (a - P\\P(t)\\2) V, G.205)
from which we obtain
V(z(t),t) < V(z@),0)e-at+f}ti^T)hdr...
Model Reference Designs
Design Example
Consider a first-order discrete-time time-invariant plant
y(t + 1) - apy(t) + bpu(t) G.224)
with unknown parameters ap and bp...
G.242)
For m(t) in G.235), we have
rn(t + 1) < 1
332 Chapter 7 Indirect Adaptive Control
and from G.226), it follows that
|Aa(t)r(t)|...
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