Adaptive optimal predictive power control of cellular CDMA systems

Abstract

For direct sequence code division multiple access (DS-CDMA) cellular radio systems, an autoregressive moving-average model with auxiliary input (ARMAX) process is given to model the effects of round-trip delay, channel fading, and interferences (in-cell multiple access interference and out-of-cell interference) upon the power control system. According to the predictive model corresponding to the ARMAX model, a predicitive adaptive power control scheme is proposed to compensate time delay and reduce the affects of channel fading and interference, thus achieving optimal tracking, in the sense of MMSE control error, of the desired signal-to-interference-plus-noise ratio (SINR). Due to the optimal adaptive prediction scheme, the proposed adaptive power control method does not require the statistics of fading and interferences. Several simulation results are given to confirm the performance of the proposed method.

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on Declrlon snd Control
Athens, Greece December 1986
ProccHdlnga of 25th Conference
WAI - 1 :OO
Robustness Optimization
Under Nonlinear Time-Varying Unmodeled Dynamics
In Adaptive Control Systems
Bor-Sen Chen Chiang-Cheng Chiang
Department of Electrical Engineering, Tatung Institute of Technology
40 Chungshan North Road, Sec. 3 , Taipei, Taiwan, Republic of China
Abstract
A new robust stability criterion will be derived
for analyzing the robustness of adaptive control sys-
tems under nonlinear time-varying model
on the concept of the excess robustness of adaptive
systems and the theory of minimum Ha norm.
errors, based
I. Introduction
Adaptive control provides a way of handeling plant
uncertainty by adjusting the controller parameters on-
line to optimize system performance.
have some important papers[1,2,4,8]
robustness properties of adaptive controllers to LTI
unrnodeled error. The robust adaptive control to nonli-
near time-varying model errors is still
this paper, the concept of the excess robustness, the
minimum HOD norm and the conditions of the internal sta-
bility of the closed-loop feedback systems are employed
to treat the problem of the robustness optimization in
adaptive control systems with nonlinear time-varying
model errors.
L * (z) : znU(z-l) where U(z) denotes any polynomial with
-
degree n.
64L): : h e largest singular value of L.
A (z)[A ( z ) ] : the part of the polynomial A(z) with
zeros in I z / < 1 [in / z / : : 11.
Recently, there
addressing the
lacking. In
/ iu(z) 1 1 -
:
sup /u(~)I= sup
izl=l
1u(ejW)/.
W€ [ O 32x1
11. Problem Formulation
In the actual adaptive control system as shown in
Fig. 1, because of nonlinear time-varying unmodeled
dynamics, the actual plant P cannot be perfectly mo-
deled by the LTI parametric model Pg(z) which can be
described as
P , ( z ) =
Be(zj/Ae(z)
where
_i
m
(1)
Ae(z)= 1+ C a i z -: B ( z ) =
i=l
y(k)+aly(k-l)+. . +a y(k-n)= bou(k)iblu(k-l)
biz
-i
(2)
i=O
i.e.,
+ . . . + b,u(k-m)
est,+mates
(3)
Introduce a vector of parameter
e = [al . . . a
and a vector of regressoh
c$(k)= [-y(k-1) . . . -y(k-n) u(k)
While the recursive least-squares estimate
employed to identify the plant by
model, then as k+-, Q(k) approaches a constant 8, and
the actual plant P can be defined by
P= P*(l+AN*(.))
bo ... bml
(4)
. . . u(k-m)]
(5)
scheme is
a LTI parametric
where P,=P&(z)
varying unmodeled dynamics.
Remark: Most of the previous work on adaptive control
systems deals with the case where the nonlinear time-
varying unmodeled dynamics can be neglected and the
order of the plant is exactly known such that 0,
be the true parameter of the plant
ver, it is obvious that the circumstance where these
assumptions are satisfied never actually occurs in
practice.
Let the plant to be controlled at the kth adaptive
step be described by
and A N * ( . ) is the nonlinear time-
will
Howe- i.e., P=P,.
( 6 )
P= P (l+ANk(.))
k
(z) andhNk(.) denotes the nonlinear
time-varylng u modeled dynamics at the kth adaptive
step as shown in Fig. 2. Further, suppose the nonlinear
unmodeled dynamics ANk(.) is bounded and inside the
sector r-6 , S,],
i.e.,
ihk(u(k)) I 5 Jk'u(k) i
k
111. Robust Stability Conditions
Definition 1: The adaptive parametric system as shown
in Fig. 2 is called BIB0 (bounded-input bounded-output)
stable if there exist some finite positive constants
dl, d2, A,, 6 , such that
t 7 )
where P =P
where 6 is a finite posltive constant.
k. 8 (kh
(8)
(i) lu(6)I <dllr(k)l
(ii) /y(k)l <d2ir(k)if& for
Definition 2: The nominal sensitivity function S ( z )
of the adaptive parametric system with nonlinear time-
varying unmodeled dynamics at the kth adaptive step as
shown in Fig. 2, is defined by-l
s (z)= (1+P ( Z ) C ( 2 ) )
where P (z$=Bk(z)/%fz), kand %(z)
the poyynomials of Ae(z) and B,(z) at the kth adaptive
step , respectively.
Lemma 1[61: The norninalgensitivity function S ( z ) of
the adaptive parametric system at the kth adaptive step
(as shown in Fig. 2) is internally stable (or realizab-
le) if and only if
(i) S ( z ) is analytic in I Z I
(ii) kevery zero of the polynomial \ ( z
1 is a zero of S ( z ) of at least
multiplicity
(iii) every zero of the polynomial B ( z )
1 is a zero of I-s (z) of at feas
multiplicity.
Theorem 1: In the adaptive parametric sy
nonlinear time-varvine unmodeled dynamics as
~-
Fig. 2 ,
suppose Sk(z) is internally stable (i.e., the
closed loop of the nominal parametric system as shown
in Fig. 3 is asymptotically stable) and if
for ir;k)I
< =
(9)
(10)
Ir(k)I < 00
(11)
denote and Bk(z)
k
k
1
k
k
t
S
in j z l 2
the same
in l z i
the same
tern with
shown in
(i4 /u(k)I < a3ir(k) 1 fbr ir(k)! < w
(ii) /y(k)j < d 'r(k)l for Ir(k)I < rn
where dg and d4 are finite positive constants.
Remark: In this paper, our design objective is to
synthesize an adaptive controller Ck(z) to satisfy the
robust stability inequality such that as k-. w ,
S,(z)
<
1 / 6 * ; then the adaptive controller can
tolerate nonlinear time-varying unmodeled dynamics
AN,(u(k)) satisfying the property lAN,(u(k))
~ 11-
1 im
5 8,'u(k),
-
IV. Robustness Optimization in Adaptive Control
Let us define the robustness Qk of the adaptive
parametric system to nonlinear time-varying unmodeled
dynamics AN^ as follows:
1
Qk=
( 1 3 )
It is seen that if bk > 5
ric system will be robusty; stabilized under the nonli-
near unmodeled dynamics ANk.
or not Q , > 6 , at every adaptive step, it would be a
then the adaptive paramet-
But if we check whether
I 11-s ( 2 ) ; I@
37
CH2344-0/86/0000-0037 $1.00 @ 1986 IEEE

Page 2

complicated computative work. Fortunately, there exists
a closed-form solution to the minimum Hanorm problem
which will be employed to treat our robustness optimi-
zation problem in adaptive parametric
unmodeled dynamics. Let us define the robustness opti-
mization as
systems with
Let us define
matrix.
has been resolved by several researchers[7,3].
Lemma 2[5 1 :
mizes IIH,(z)( I_, is of an all-,pass form, i.e.,
H (z)= 1-Sk(z) and L(H ) is the Hankel
Fortunarely, the minimum 1 Ikk(z)I lm
problem
(i) The optimal Tk(z),
which mini-
K' '
where m (z) is a strict Hurwitz, monic polynomial with
all zeros in I z / < 1
(ii) min
C , ( Z )
I IHk(z)I I*=
I /Hk(z)
I I@= lpkl
(17)
where / / ? I =
Let e
(2)' B (z)m * ( z ) in / z I 2 1. Then in order to Aatis-
-
H ( z ) must be the following form:
k
N
cy
fL(Hk)).
(2) denote the part of B ( z ) = Bk+(2)Bk-
:
fy t h ! cond!fhion (iii) of the realizability of Sk(z),
. .
k
rJ
k
mk$2 ( 2 ) mkl ( z )
*
Hk(z)= l-Sk(z)= p --
where /3 and m
tion (Yi) of
$ (z)Ai ( 2 ) and
% - ( z ) and let
mk2(z)= 1 +hklz -1 +hk2z-2+. . .+hk lGlz
(18)
-JK+c+l
(19)
where!
Then From the condition (ii) of Lemma 1, it is well
seen
denotes the number of poles of Pk(z) in I z /
1
. .
m l(zi)m (z )-pkm," (zi)mk2
Noklce tkit 4 and # I
uniquely spectfied &'solving
tions. Afterpk and m
robustness optimfzation adaptive control law as
Fk(z)B (z)u(k)=
f (z.)=o;i=1,2,.
i= 1 , '2, . . . , lk-l, can be
pk simultaneous equa-
( z ) are determined, we get the
k2
. , ,&.(22)
To sum up ! h e
for the adaptive controller with robustness optimiza-
tion to nonlinear time-varying unmodeled dynamics is
outlined as follows:
Step 1.
From the recursive least-squares sfheme,- we
evaluate P ( z ) and then factorize Ak(z)=Ak (21%
and B (z)=Bf (z)BI; (z), respectively.
Step 5 ,
equations to 1: and tke coefficients
(22). Then we evaluate Fk(z) by comparin;kg
cients between both sides of (21).
Step 3. Substitute the results obtained in Step 1 and
Step 2 into (23) to get the adaptive control law with
robustness optimization to the nonlinear time-varying
unmodeled dynamics.
above discussion, a design algorithm
( 2 )
Let mf (z)=B- (z) and solve 1
simultaneous
(z) from
coeffi-
OF
mkfi (z)A,+(z)e(k)
(23)
V. Example and Simulation
Example : Now we consider the actual adaptive control
system illustrated in Fig. 1 with the plant
Z + l . 1
P ( z ) =
~ ( u ) ;
(2-0.3)(2-1)
where N(u,t)= O.llsin(t) 1 y(u)+0.5u.
if 1 1 1 1 i 1
(0.3u
10:2u-0.1 if u < -1
+(u)= o 2u+0.1 if u > 1
Notice that after the parameter estimates, the
adaptive parametric model is
-0.25812+0.5636
P*(z)=
Z-1.0058
These simulation results of Fig. 4 show that our
robustness optimization adaptive control law can easily
handel the problem of an adaptive control system
face of nonlinear unmodeled dynamics.
in the
VI. Summary and Conclusion
The main contribution of this paper is the idea of
introducing that the Ha optimization technique can be
applied to the adaptive control problem to
optimal robustnes. Moreover, a simple and direct con-
troller design algorithm has been introduced to systhe-
size robust adaptive controllers which ensure that
practical adaptive control systems will always remain
stable in the face of nonlinear unmodeled dynamics.
achieve
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