- Email: [email protected]
Stabilization, Regulation, and Optimization of Multirate Sampled-Data Systems
STABILIZATION, REGULATION, AND OPTIMIZATION OF MULTIRATE SAMPLED-DATA SYSTEMS Patrizio Colaneri Riccardo Scattolini Nicola Schiavoni D i p a r t i m e n t o di E l e t t r o n i c a e I n f o r m a z i o n e P o l i t e c n i c o di Milano Milano, I t a l y
I. I N T R O D U C T I O N
In c l a s s i c a l digital control systems, it is usually a s s u m e d t h a t both the p l a n t i n p u t s - u p d a t i n g and t h e p l a n t o u t p u t s - m e a s u r e m e n t a r e p e r f o r m e d at a unique c o n s t a n t r a t e and in a s y n c h r o n o u s fashion. However, t h i s h y p o t e s i s is sometimes not realistic, for economical and/or t e c h n o l o g i c a l r e a s o n s , and, f u r t h e r m o r e , r e l a x i n g it o f t e n allows the designer to obtain improved control p e r f o r m a n c e s . Hence, one is lead to c o n s i d e r the s o - c a l l e d multirate sampled-data control systems, which are c h a r a c t e r i z e d by the f a c t t h a t each input is u p d a t e d a t an i t s own r a t e and each o u t p u t is m e a s u r e d at an its own rate. The a n a l y s i s and the design of such s y s t e m s has recently received a great deal of attention. For an o v e r v i e w of the a r e a see, e.g., [11-[31. T h e r e a r e t w o p r i m a r y r e a s o n s of i n t e r e s t in m u l t i r a t e CONTROL AND DYNAMIC SYSTEMS, VOL. 71 Copyright 9 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.
95
96
PATRIZIO COLANERI ET AL.
digital control. A f i r s t s t r o n g m o t i v a t i o n behind t h e i r use is due to the possible presence of t e c h n o l o g i c a l c o n s t r a i n t s which enforce the use of control schemes where sensor m e a s u r e m e n t s and c o n t r o l c a l c u l a t i o n s have to be p e r f o r m e d at different sampling rates, see, e.g., [4]-['/]. This t y p i c a l l y o c c u r s in one of the f o l l o w i n g cases" (i) Some s e n s o r s r e q u i r e a s i g n i f i c a n t t i m e b e f o r e t h e y s u p p l y the m e a s u r e m e n t s of the p l a n t o u t p u t v a r i a b l e s to the regulator. For example, such a s i t u a t i o n o c c u r s in c o n t r o l l i n g chemical p l a n t s w h e r e e x p e n s i v e c h r o m a t o g r a p h s are used to measure composition products. These m e a s u r e m e n t s a r e t h e n i n f r e q u e n t and d e l a y e d w i t h r e s p e c t to those of other variables measured by s e n s o r s not s u f f e r i n g of such a l i m i t a t i o n . (ii) A small n u m b e r of s e n s o r s is used to m e a s u r e a l a r g e n u m b e r of o u t p u t v a r i a b l e s at d i f f e r e n t times, or t h e s e n s o r s allow one to m e a s u r e all the p l a n t o u t p u t s a t the s a m e r a t e and time, but h a r d w a r e c o n s t r a i n t s p r e v e n t one f r o m t r a n s m i t t i n g d a t a s i m u l t a n e o u s l y f r o m all the s e n s o r s to t h e c o n t r o l p r o c e s s i n g unit. (iii) The p l a n t o u t p u t s a r e all m e a s u r e d a t the s a m e r a t e and time, but t h i s r a t e is less t h a n t h a t of t h e p l a n t i n p u t s u p d a t i n g allowed by the c o n t r o l a p p a r a t u s e s . (iv) Some a c t u a t o r s a r e m a n i p u l a t e d less f r e q u e n t l y t h a n o t h e r s in o r d e r to r e d u c e the e f f o r t of t h e s e a p p a r a t u s e s . As a second r e a s o n , it has been shown t h a t the use of multirate and periodically time varying controllers can significantly improve the closed-loop performance of a s a m p l e d - d a t a s y s t e m in t e r m s of model m a t c h i n g , s e n s i t i v i t y r e d u c t i o n , d i s t u r b a n c e r e j e c t i o n , pole and z e r o a s s i g n m e n t with state feedback, see, e.g., [1], [8], [9]. However, these promising results usually refer to the sampled v e r s i o n of the s y s t e m , while p a r t i c u l a r c a r e has also to be paid to t h e i n t e r s a m p l e b e h a v i o r which can be s i g n i f i c a n t l y d e t e r i o r a t e d by the m u l t i r a t e input u p d a t i n g , see [10]. A deep d i f f e r e n c e e x i s t s in the two f o r m e r c l a s s e s of applications of multirate controlwhen a multirate approach is used to improve c o n t r o l performances, the frequencies and phases of inputs-updating and outputs-measurement are free design parameters to be d e t e r m i n e d by the c o n t r o l s t r a t e g y in o r d e r to o p t i m i z e the
STABILIZATION, REGULATION, AND OPTIMIZATION
97
r e q u i r e d p e r f o r m a n c e s . On the c o n t r a r y , when a m u l t i r a t e solution is enforced by technological c o n s i d e r a t i o n s , the same p a r a m e t e r s are problem d a t a and must be faced by the adopted control s ynt he s i s technique. In t hi s paper, a t t e n t i o n will be focused on this last s i t u a t i o n . Hence the p r o b lem a d d r e s s e d will be to design a m u l t i r a t e d i g i t a l regulator once the i n p u t s - u p d a t i n g and o u t p u t s - s a m p l i n g m e c h a n i s m s ar e fixed. R e s e a r c h in m u l t i r a t e control can be t r a c e d back to the l a t e f i f t i e s [11]-[13]; however, it has received m ore and m o r e a t t e n t i o n only in the pa s t decade. For an overview of the most s i g n i f i c a n t r e s u l t s of the a r e a the r e a d e r is r e f e r r e d to [3], [14]. In [ 1 4 ] a t t e n t i o n is focused on the a n a l y s i s of a control s t r u c t u r e where a d i f f e r e n t sam pl i ng r a t e is a s s o c i a t e d with any pair of i n p u t - o u t p u t v a r i a b l e s , then impulse modulation models are developed and c r i t e r i a to assess closed-loop stability are presented. Several s y n t h e s i s a l g o r i t h m s have r e c e n t l y been proposed in a l i near time-invariant setting. Among them, the p o l e - p l a c e m e n t appr oa c h has been considered in [15]-[21]. The Linear Quadratic Gaussian (LQG) technique has been applied in [4], [22]-[26], while some s y n t h e s i s a l g o r i t h m s based on c o s t - f u n c t i o n minimization have been p r e s e n t e d f o r c o n t r o l l e r s with a p r e s c r i b e d s t r u c t u r e [5], [27]-[29]. Some predictive and self-tuning multirate control a l g o r i t h m s have been proposed in [6], [7], [30], [31]. The o u t p u t r e g u l a t i o n problem, t h a t is the problem of z e r o i n g the s t e a d y - s t a t e e r r o r to the maximum possible e x t e n t in p r e s e n c e of exogenous signals of p r e s c r i b e d dynamics has been t r e a t e d in [32]-[34]. Finally, the case of com pl et el y a s y n c h r o n o u s sampling has been t r e a t e d in [35]. The aim of this paper is to review the main a s p e c t s concerning the application of popular s y n t h e s i s techniques, namely the p o l e - p l a c e m e n t approach and the LQG method, to the m u l t i r a t e control problem. Specifically, the a s s u m p t i o n is made t h a t the plant under control is a d i s c r e t e - t i m e l i near t i m e - i n v a r i a n t system. It is also assumed t h a t each o u t p u t has its own f r e q u e n c y and phase of m e a s u r e m e n t and each input has its own f r equency and phase of updating. The p ap er is organized as follows. In Section 2, the discrete-time linear time i n v a r i a n t model of the p l a n t under control is introduced and the sampling and u p d a t i n g
P A T R I Z I O C O L A N E R I ET AL.
98
m e c h a n i s m s a r e given a p r e c i s e m a t h e m a t i c a l f o r m u l a t i o n . Further, it is a l s o s h o w n how m u l t i r a t e s y s t e m s c a n be c a s t e d into t h e w i d e r c l a s s of p e r i o d i c s y s t e m s . Then, in Section 3 some preliminary results on the structural properties ( st ab i I i zab i I i ty, detectability and zeros ) of the m u l t i r a t e system are given in terms of the o r i g i n a l plant under control. Section 4 deals with the pole-placement and LQG methods when the system state is assumed to be available f o r control, while, since our main goal is to design output feedback controllers, in Section S the problem of state reconstruction is considered. In p a r t i c u l a r , t w o s t a t e o b s e r v e r s a r e p r e s e n t e d : in t h e f i r s t one t h e p o l e - a s s i g n m e n t t e c h n i q u e is a g a i n a p p l i e d , w h i l e t h e s e c o n d is d e r i v e d by r e s o r t i n g to t h e K a l m a n f i l t e r i n g approach. In Section 6 the previous results on state f e e d b a c k and s t a t e observers are joint t o g e t h e r with the a i m of d e r i v i n g s t a b i l i z i n g f e e d b a c k c o n t r o l l a w s . F i n a l l y , t h e c l a s s i c a l o u t p u t r e g u l a t i o n p r o b l e m is f a c e d in S e c t i o n 7 w h e r e , u n d e r s o m e p a r t i c u l a r a s s u m p t i o n s on t h e i n p u t s updating, a suitable regulation structure is presented w h i c h g u a r a n t e e s t h e a s y m p t o t i c t r a c k i n g of given r e f e r e n c e s i g n a l s in s p i t e of t h e p r e s e n c e of p e r s i s t e n t d i s t u r b a n c e s and plant u n c e r t a i n t i e s .
II.
THE PLANT, THE INPUTS-HOLDING AND THE OUTPUTS-SAMPLING MECHANISMS
Let the system under control be described by the following discrete-time linear time-invariant stochastic model
:p.
x(t+l) y(t)
w h e r e A~Rn'
= Ax(t) = Cx(t) n
, B~R n ,
+ Bu(t)
(1.b)
+ M w (t) 2
m
2
, M ~R n ' q , C~ R
p' n
, M ~R p ,
1
w
2
are
uncorrelated
(1.a)
+ M w (t) a 1
zero-mean
identity covariance matrices,
p
and w
2
i.e.,
gaussian
white
and 1
noises
with
,--~
~
{~
~
9
~"
~-
"~i
.
9
~,
"~i
~
~
,--~
.
("+
"~i
I
~-'o
::7
0
~
~ s
..
o cr
~*
. . . .
o~
~,..
A --]~
IA
O
~,,..0
(1)
. . . . +
~
~ ~-~
~
"d
+
~
~
~
o
~. ~
I
.]~
~*
~
-.~
(1)
o-"
9 --.
~"
(1)
~
"'~
O
-
~
~.
-.
"~
~ ~
~
+
-.
~ ;~
~
~'~
~
~-'
~.
~ "-
~
~
o
~ 9
cr
C::
~*
~ -.
=g
"
~ (1)
0~
-.
~ .-.
_ O
~.
~
'E~
~0
el)
~
-~.
0
(Tq
""
~
~
~
~-
~
..
o
m
o
1~
~
~
Ln
F)"
~.
~,
(]:)
9:
~-
~
-
o
<:
~
~
:
~'
~
.
~,
(~
~
0
~
0
I
~
=-
~
~
.
"
~
~
~
.
~.
~
~
~
.
~.
w ~" ~
~
~
g"
~"
(I)
~
..
~
~
.~-
~.
o
~ ~ ~
~
~
o
~-~
~
~ ' ~
__
~*o
g.~
~ O
~ ~
"~
--"
cL <
~.~
-
O
~
o
--~ )
0
>
o
r"
Z
o
r"
~
HI +
Nt ,,..,
,1-]l +
,-,it ,....
",i
,i,-,+
0
i-.
~" ,,
m (~
Z ~.
:~ =.
~*~ -,
Z ._.
~
~.
0
~
,~
~
~-.
==
""
i-.o
,-4"
I~ O
~
C~
13o"
~
:~
~
~
~ . ~~
=~"
~I~
.-
,-It A
T~ d,,~
o
~'%
~ m ' "
o
~
~
o
~
+
~,
~
""
-,
.,
0
""
+
o
~,
~
- ~.
,_.
~
(1)
I~
- -
~
~
=
~
:~ =.
-
~
~
~
~,.~
~+
"li
~
~+
~ ..
-"
~
~"
~
~
u~
0
O
~
"
,,~
i
-~_.
I
~r
-
o ~
m
~i'~
~
o ~= ~
, ~.
('l)
,,--] t
~
~ ~
~ ~+
9
4~
.~
9
.~
"m (I)
"E~
+
X
pl,.
m
9
9
t
r~
0
+
"-~
<:
II
+
'--'
.-4
~
..
~
..
~"
0
r-,"
~
I:::L O'-" ~,.
~
II ~-.
'-b
~ .~
-~
~
~
0
o
::~
<: ~(I)
o"
Z
>
0
O
N
"~ >.
--
-~
z
x
.,.~
i[i~
9
i[-
II~ 1 7 6
~
~
9-
~
~
~
II
d~T
I~) ~
~
5.~
~
.,-., o
,.c::
i
,[.-,
o
o
o"
,,iq
gl
", i oi i l .iw~e o
~,iii.i
~D
II
~
"" 0
~ ,
~
~
+
(._,
~
o
:>.,
"~
~-
+
<:!
~
+
~-
f~
X
II
00
~,"
m
<
v
,
I
l' . .
9.
tJ
I
'~
.
,->
II 9.
o
-
o
.
I
,
.
I
.
"o
~
~
0
~
~ ' ~
~~
~
II
z
~
N
o
C~ "
+~[i~
II..
,
. ~2
"--" Z
~.
~ ..--, ~~
o-~
L,
o~
+.,
.
~
+a
:~
.
.~
e X N ,.~
r A "~
~
.0
o
~1
X
r .,-,
"'-'
~
9 ~
0
..
.
~
"~
(D
~-o
b,~
~
+-~
o~
~
~
~.
..X
~ ~o
~
o
~
"F~ ~
~'~ _
o r.., ..
~,
~
o~
o.~
-o
~
1~ ~)
~
.~
t~ o
r.., Ei
c"l:l
"~ " " ~
N
"0
o
~
~
1~ .i~
. ~ .,-,
= ~'~ I~ ~-~
~
"~--'
+'2~
~-&~ I~ "~
~ ~.~
,i-a
0"~ ~
~
~,
+.,
,,
d
~ ' B ~
El m ~
-~
~.
~
,-:
~
txo
~
~
+
~-~
~
~
II
~
c r ~u'}
~
~
~"
~.
(I)
b0
,_
~i,. ~
9
C~
""
-.
~
u~
1-11
-
~
I~ '-b ~
o
~
0
m
~
~
{A F ,~
~
_.N
~
0
"o
~
~
,.
o-
~
~
B'~+~
~ "
~
C~
'
""
1~,~
{A
~
,.
~"
~"
~
m
C
o
~.T;"
~"
o
~,
~
.~
o
~"
'~
.,--
9
~
DJ
~"
'-"~
~
~
..
+
~
~
+
~
~
I~
.~
~
~
=
I ~ ~_.,o
o
'-"<
~
~
~
o
~
~
9
~ ,,
~..
m
%
~
~
0
CL.
~
I~
<1
~
~
"-~
~@
~
~
~+
~
.~
~
~
~
~
-"
~
,_..
~
m
-
-cr
~
.
.
.
-
~
9
--
-'.
.
o
~
~
I~ ~
~
o
..
~
~
0
0
"
~,
:~
~
o ~
u~ ~...,.
~..
o
~
.
~-~
~
~
u~
~
~
o ~
"
.~
,-,
._.
c
,_,
~
-
~
.~
~
~
m
C)
m
•
~;
9
,_..~-
"0 "
-I,
..
o
..
I
-
"--"
.-.
"o
.
o-
0.
,~
-~
cr
0
o
0
""
o
~D
0
.-0
u
~
II
~
'-~
o
(~
--]n 9
o
CI
~
~
~
~
~
~r
~'~
r~ r~
~~
~~
~"
~
r~
t~
o
--,
-.
0
c~
c ~ ,-..
N
~"
c~
_
~'~
~
~
~~
~
~
-~'~ ~ ~
~
~ ~
T
~'(~
~ ~
~
~ ~>
~-~ . 0 ~,
0 ~ .----'~
~
~ o
,_., ~
9
=.
~
~
~
,.
~
~'~
~=
~ ~
~.
~
~
z
>+
,
~
~
I
~e~
Cr
N
~r" ~,.,.
,
"
~, .-.
-~
~
0
i/l
II
r
--
~
=
Z:r
Z::r
~,.0
>
0 "--b
m
Cr
N
I-+
'--
~
00
,--.
,.
~"
~ 0
~
'-"
~v
,v
~
;--"
--
~.~o
-.
"~
~
(~
~
,~
~
00
O0
.
'-h
1::::
m
~"
,<
0
~r
0
~"
~
;'-'"
~N
~
o -
~"
"~ ~ ~. ' o ~
~.~
,~
~
~"
g
I:~
0
9
0
"~I
0
CO
~
"I./I " i-I. 0
~.q~
-h
'-1
~
0
~"
~o ~
~.~o~. o o ~ o~~-~.
~oo
0
~ ~-.o--= ~.-~
~=
~_.~'~
N
~
~
<
,'-".
~
~
o
--
9
N
0
o
z
~o
0
0
N CD
0
~=.~.
,.-!
~r'
e-+
'-o
0
N (I) ,.-I 0
0
~=..o
m
~,,..
=~ ~ ~
0
"
-.~"
O~~
'-I
~~
~~~
~_
~ ~~;-.~ ~
-~
r~
~ =('Dq -~'~~ . ~ . . o ~ . ~ - ~
('D
..~
~
.,~
~~
r-~
0~
""
0~ " ~
~"
. ~
~
~ ~
~ ~
~.~
~
,..~. 0
~-
~.~
"-~~
~ ~ ~~ =~ ~ ~ ~
~=
~..
~= "~ ~- ~,
'--' " 1 ; " - - '
,_,
.,~
~9 , ~ . ~
0
~'
~
.7,
~; .~ ~ ~
o.~ ~"
0"
~
"-~
0
CD
0
.---...~
'~
~"~ . o~ ~
~o
~.~
~
0~
~.g
="~ ~ "~ ~~'~a~~.. ~
(Y~
0
o~
-....I
~o , -- ' ~ "1
0
@
0
9 ~ ~ ~~
~,=<~=~
?~
nn
N
s 9
N-
~
Z
""
>,_.
-.
~--
~
""
--]1
,I ~'~
~-
~
IV
--
('1
o
cD i./1
.,r
('1)
~...o
~e
-.
(')
~"
~
~. e
,...
~
~.>
~,
~
~
-----~o
~e
.---..
~.
,....
b.,.~ ~
"I
o
~
- . _-:. ~ ~
.
-z~~
~,.~
~-~
9
9
~.~~ ~
o
-~a~T
.
.~.~o~_
~-
[--,
5
N
-
105
STABILIZATION, REGULATION, AND OPTIMIZATION
If lar is a t r a n s m i s s i o n z e r o of ~, w i t h o u t b e i n g r an e i g e n v a l u e of ~P, t h e n t h e r e e x i s t s ~, such t h a t ~'=/a, w h i c h is a t r a n s m i s s i o n z e r o of ~o.m
(ii)
In t h e s p e c i a l c a s e w h e r e S ( . ) - 0 , i.e., t h e i n p u t s a r e u p d a t e d a t any t i m e i n s t a n t , s y s t e m (3) s i m p l y r e d u c e s to u(t)=r(t). This allows one to neglect Eq. (3.a) and c o n s i d e r s y s t e m (12), as f a r as t h e z e r o s a r e c o n c e r n e d , as t h e c a s c a d e c o n n e c t i o n of ~P and N. T h e o r e m 5 [33] If S ( . ) = 0 , t h e n t h e s e t of t r a n s m i s s i o n zeros b e l o n g s to t h e s e t of t h e t r a n s m i s s i o n z e r o s of P ' I I
of
(12)
IV. STATE-FEEDBACK CONTROL LAWS In this section, we extend the main classical stabilization techniques, namely Pole-Placement (PP) and Linear-Quadratic (LQ) c o n t r o l , to t h e c a s e of m u l t i r a t e s y s t e m s . To t h i s end, we c o n s i d e r t h e T - p e r i o d i c s y s t e m (7.a) and a s s u m e that its s t a t e ~(.) is a v a i l a b l e for control. Since our ultimate goal is to design output f e e d b a c k c o n t r o l l e r s , ~ we m a k e r e f e r e n c e in t h e s e q u e l to p e r i o d T, i n s t e a d of T. N
A. POLE-PLACEMENT The TIR of s y s t e m
(7.a) w i t h w =0, 1
is e a s i l y
determined
as
~(k+l)-
~ ~(k) + r"
(13)
r'(k)
where (k) .= ~(kT) , r(k) "= [ r ( k T ) ' a n d $, 1~ a r e r e s u l t holds.
suitable
r(kT+l)'
matrices,
for
... r ( k T + T - 1 ) ' ] ' which
the
following
,--
Ir
0
II
J~
,0
o"
"-'
o
+
,-+
,0,~
~,
~
F
_,
N
~-
~-.~
~ ~
o-
- r+
o ~ ('!)
0
o 0~
o
m
~
m
~
~
0
~
,-0
0
~
('1)
0
~
--m
~-oo
,~
m
o
o
~Z
~,
0
~
II
,~
o
~,
~
~
c~
~
~'
-i
~,
9
o~'
~
9
.+
~ ~ ~
0 m
~
5. ~
~ ~
~ ' ~~.
~"!~ "1
o '-"b
~
~"
+
~
1~ ~
~ "
~
~
o
~
~ ~ ,-.
~ (1)
'-'"
~
=~"
0
"~ ~ "1
,-""
~ @
r~ ""
,. II
0
"
~
~ ~
~-' ~ ~
+
+
9
~
-.
~"
~-'
o ~ ~ ~ 0 ,-...
o ~-'(1) 0
'-'~
~'~
m
X
~
C~ O~ ~ ~ ,-~ "1
~
n)
"1
~O.~o
('1) ~ < m
~,~
~
~ ~"
7'
~.
~,
,...
"o
@
~.. I 0 0
~" ~
1~
~
m cr
~0
~
0
~
-"
-"
~
~
(~
Cr
z o
I
~
'-"
-
T~
-
,-. "~
0
,-. c r
too.
,_..
~'c~
'~"
.-1~
~.~
,e,~4 ~9
,-.
o ~
~'.~ ,-.
~ o .__.
.___,
o"
r~
'-'"
~
-
o
~
0
~"
~--
~,--. ~I~"
o- b
~
m ~.~
m
~
~;
~
=
~
o~
=~
~
~
x~
m
~
r~ ~..
~
~
~
~"
~
o ~"
o
~-
~.
~
,.
o
~--
o
~
~"
0
~ ~
~
.. Z
~
~
0
~-.
"~' ' m
~-o
~-
a..
a=
-H
~," ~" E9 ~
~
~
m
.~"
(~
~ o -~
---~ " ~'~
~
o
o
~r
e-,
0,,.., ~
N
m
cr
0
~
0
~
~,
~
~',H ~ " ;~'~ ~
L
~
i
"~i
,0
'-]
~
,0
~
~
I
m
B
m
i-,~
m
0
I-~
~
0
~
~
~
~
0
o
o
..
-
;~,
'-1>
+
;~,>
{/1
~'0
~
~
"-~
o
o
0
--.
~
0
~e "
0
'-1 0
~> '-'1> ~>
~
'--] ~,
+
~'~
+
"-]
~
~'~
"]
~
,--,.
~
B ~m
~
~
II
II
~ ~
"
.:
"
m
o
m
~
0
"
~
9
0
o
m
"-' a'~ ,.-.
~
~
~
II
~,_.
,~5
;~+m
~]N 0 + ,,--. ~ ,.T3 ~ "
m
~o
9
~ 0
:.
"
'~
~-" ~
~ ~
0
4
..,u
II .,0
~>
"~i
"-"
..
~
m
9
N
0
> z
0 z
0
~q
G~
n
~..
'--'0
~
+
~
~
~
>
"~i
'-"
0
0(1)
II
N 0 ~
~"~
0
~"" ~ e-+
0
~-'" ~'~
r~
~
V
~"~
0
"I ~ 0
o
m
0
0
I
"~
1~
"~
~ ~ ~ ~
CT ~ .
o"*
~)
+
0"
~-+ 0 0
~)
,.
(1)
~ ~.. 0
.~ ~ F"
~Oo
-~
~'~-
C~cr
+
t~
C~
0
o
~
~
I
~
0
~
~--
~'~
~
-.
N
'-'
~
~)
~o
+
II
9.:
~
g
o
U1
0,-.,-,
(/1 0
'-I
0
,'~
0
0
o
~ ~ CL~:J
~-~
00
0
0 0
9
~
9 "
0
0
0 + '-1
i""
i""
+ '-1
7"
o
0 ~
._ ~
0
O"
9
~n
0
0
C~
,.~
~" ~
e-~
,-'. ('1)
0
0.,.~
~00~,
0
~ ~ 0 ~ a..~'-~
~-~"
~
O0 -~.
-]
,--~.
II ~
"~
~
~
0
~
~
~
0" ~
0
0
1~
~
(1)
0
0 ~
,---
0
C7" ~._~
~.~m ~
~,"0
~ ~'~"~
~
~=
~1 _~ (1)
'-'"
_]
9
,-,. ~
,.--
~
[-1"1
~o
>
N
oo~
0--o
~
,-..
,-,.
8
0
0
~
,,.-,,
0
I--,.,' 9
r~
g)
0
b,,-, 9
1: 0
0
~0
I--'.
'--'
0
0
~
~
~
o
I--.'-
0
,~
~
o 0
~_~
o
~
~o
~=
"b
~.
~....
~--...
,,-j
(I)
0
8
c)
<: ['11
zr
"3"
0
I"11
c-+
0
rll
-] >
~
m r-+
o
@ B
cr
"~ o
-] =r
m 9
~-"
~'
~"
9
ID.
~ re 0"
0
~
0
~
~
~ "1
0
-"
~=~
~
~
~D
0
~'
'--'- ~ 9 ~ ~'"
~
,~,~
~~
m ~.
0
o,
~
~
o
~
(1) "'b
(1)
---
~--
0
o
2
I
o ,--- ~
[",3
r*
o
"4,----
~
~
'1~
r-~
cr
~.. N
~-.
~
--.
~~ ' ~ "
~
"~
(1)
o
0 "~
~'~"
~
~
~
0
o
o
II
re
r
~
=~
0
~~-*
'-"
re
"1
(1) 0
~
II
0
o
--.
~
"" ~-,
[%1
+
"0
"1
+
~
D~
'--J
~~
+
+
~
~~
~
"" ~
0 13.
~D
I
0 CD
"1
~
rn
0
9 Z
N
9
Z
9
r"
X~
z
[',4
~o
?'*
o
Zr
--"
=
"-m
0
o
o
.~. ,<
-
o"
c::
--.
_.
o
13
=" (1)
~, o"
=
o
~
,_,.
ro
-o
""
ct) o
-m
~
~
.
o
o
~
~
~
"0
~
""
~"
I:~
~'0
~-c~
F'~"
0
~ m cr --
,0 m ~I ,...
~
9
~,
~
;~
~,r"
.
.
--"
o ~
~
~
~<~
('1)
~
~
o~,_,,.
~
0
~
"~
~ ,-,.
I::> )0r ~
.
. [--,
9 ~ ,
"
~
o
-.
~ +
-. r9 ' ,
, ,.-,
o r',
~
~
,-.
~
• :~
m
=r o
-
3
,-.
~
=
c% ~..
I~
~ I~
r
,-,. ~'m,
~:~
~
o
~
,
~'
~
~ ~-'
i:>,
r',
+
~
+
II
+ . ~
u~ (1)
~
:::Y'
-
-" ~
~
~ 4~
~'-"
~ --
+
'--]
~
o,-,
-
.
+
~
~
-I~
~
-~, ~
~,
o
o ~
~
~:
~
="
~,
:~
~ ~ ('t)
o
~
r~ ~-.
~ -. ~
~ ,-,"
cT
~"
o
III
~
~"
III 0
~
:~
~"~
~ ~ (I)
~
o
Z ~
~
rm'
-~
;::;-
::::r'
g
~
o
'-"
=r
"
cry.
~
~
~~"
,-"
o=~
~-~ =~'
~
"~
o ~ = o
~o
~
< ~ - ~ ~
F'~'~
~
--o
~ 0 ~--, "Z:~ c r 9
_
cD
~
o - ~ ' = # ~
,-" ~- "
u~
~',~
o,9.,, ~: ,-,- ~
@~
o
~+
ct)
~
o
0
o
m -]
7.
0
0-
"~
o
~'~ ~ o
~
,,
l.n
Oh
c~'~H
('I)
r-l.
CZ, "
~ r~
u'l ~ (1)
~.,
--,
i-,o
0., Z:r ~
t~
~n
.
~~,o
0
0
i,--o
I~
~
0
~
~'!~
,-+
0n ~ ' o
"Cl ~ r-t. ,_,. ~).0 ~. ~
O~
c~ r ~
~
~
o "~
o
0..
Zr 0
0 ,-I
~==,o
0
0 --,
~" ('1)
~
~
..o
'-'"
"~ -I~ .
0
0
0
~-'
0
I~'~
'
+ r',
,_,
~
, ........
~
+
+
~
I
"+
+
_.
~" "~
~r
--]
II
w--.o
~
x
--
--
ox
II
~=.o
,_
~
.'~
~.
,,-.o
c~
,_.
~
~
, ~
r~
.-~
,,
+
~e~
~
~
~"
m
ox
r-~
--]
c~
""
0 ~
~
,-~
'-1
~
Im
o
.
0
,,
~-,
I
~.]
""
--,
~
~
~
~
9
c~
"q
~o
U~
"" C)
I
U1
('I)
l,n
0
~ ~..
~ o 0~ .~.~.~
~.
~ Oo
i.,]
~
~
C)
u~
i...~ 9
~
~ o~
0~
~.
=.
~'
~',
o
,
i~"
-~~
CI)
~
'-'"
0n
~
r,
~ ,-..
~
~
0
o
~Z
m
.,-i o,..i
o
..
o
[--
o-~
<
~: ~ ~
""
Z 0
--.~
~
o
.~
ffl
~-, ~-~
o
_~
o
~
~
o
o
9 0
(D
I~
o
~
i
..
"
cr
,_._.,
II ,i.l
~
o
0
+'~
o
t_~
~
,.~
O4
,.--i
0
~ ,,,,-I
~o
~
~ o
"~
0
::~
"-I~
~
""
(p
~
II
' ,..-1
o~
~
0 .,
II
[--
~
i~
0
~
~,
.
~,
o
-~
0
= ~ o
0 0
--'
o
"C
0
..~
~
~
..~
t~
.~
~
:
o
"~ b "S o ~ =
=
-,-,
t~
-~.-
~ t~
I./1
"~'~
"'~
~ o
'+~
o
~
~I . ~
r .~ gl ....~ . ~
:n
o
~
'-"
~"
~,_,
t~
o
"
o
~'~
0
"~
~
~ 0 ..0 ~
~bo~
; ~
~
I~ ..-.
~'~
L~)
bO O0
m
~
e.+
r
I
+
"-"
o,q
,_,
~o
Z
Zr.~
0
0
~~
o
-x
o
~~
~
C~
C~
,-..
~
0
~ :~
~
V
~
o
~
i
II
~
0
~
0
0
~
o~
.~:~
~D
~<
0
m
~'o
~
r~
Cr. (D
~'~'~
~ ~m
e
o
""
~
o~~
--.~
~ m
~"
--.
~"
~.
m
~
o
~
~
o
N
~
o
~
~
~
~.,
Z
~
~
g
~
~-.
~.
,
-
o
~
i
I
0
~
&
&
..-:
:~
0
.
| ~.
o
X
2.
~
~.
~
0
,,~
"
-~
~.
:~
:~
..q
~
~-
~
~,
~
~ ~
Oo C~' "b ~,
~-~
"~
~-~"
ocr ~
~"
m
~
-o
""
~
0
0
oc
--"
cvo
~
o
P
o
,_, v
~
~
~
o
~ ~ 0 Y
o e't" ~ cr'~ "-'s "-s
=
~
:~
~
~ ~ ~
,-, F~" ~ ~
~
_
0
> z
o
~>
t-.n 0 c~
o
(1)
a)
0
(1)
~
CY
~
,,-+ (1)
,X:D
0
0"
~ -0 . ~,-~
r-+ Ul
~" ~ .
~o
{~
CI)
0
'-'"
~
~" ~
(A .-..
o
.~
(I)
~.~~
cz
0
O~
~
,... ~ "
~ "
~c~
~-~
~l~ C1)
,--
c~
~
9
(1)
~ ~ P ,.... u~
:~
"o"
c)
'~- "
--zr
~ o
~.
~o.~ - .o . ~~
o ,--'
r+
C)
~-~ ~
a) O~
< ,_,
,.+ o
'~~
~-
~
~ .---. '-"
"
-.
0 c-F
Cl)
~
c)
'-"
".-" (1)
~,
~
4"
o
0
~"
0
0 "I (I)
~
Cl)
cI)
"-b .. o~.
~ ,..
'-'" CL
ul
0
0
,v~"
I
~a)
~ ,-..
..
,.~--]
~ ~
~
Oc~ '~"
~
'-"~
~
.--.
~
(D" "rl
'-"
e~
~
X
~
I~
o
0
-.
--]
-
Ixc' :::~' ~ "
c~ ,-..
~~
CI) "S
Zr" Cl)
(~"
(-,)
I
~D
~:~
{n
,-4'< ~
C3
~
~
-~
C
"D
9
"1
ca
r-~
o
~v
"~
~~" -
",I
8
,~
o " _ ,I,,'--"
'-'" '--' Cb
,e,
"I
w,..
zr' (I)
r
C:z.
(I)
cr"
CL (I) r (I) o r-+
u'}
t-,.
9
~ ~.~..--~ zr' 9
~- ~.
~'..~
"-"
~o "+ 9
C: ,-..
"J u}
0
o
,-b o
(I)
0
(I)
.-j
cI)
r-F
i,-.-9 U')
,I,,
(I)
:::r"
('I)
~" cD
-]
o
o
0
~
0rr~
II
""
,-+
'=r'J
~,=,,.
C:Z. r-+ '-'" 0
o 0
I~
~ Zr (1)
~
(,J 0
~-+
'---'
I~" "~' r'+
"n
.-,
-I'-
,~,
,.-.-,
t
4"
'0' ~ ~ ~
r-*
(3
0
~.
II
I~
~
,-+ + .,I
"~
0 CL i,.-,9
(I) '-I
I
..]
CI)
0
'-'b (I) "I CI)
cI)
o
~
~ 0 ..--.
ci)
'~
~C)
.-,
'~'I
.~,
I>'
"D
I:>
4-
'---'
~"
D-
"D "11
I
]i
.~.
rD
(I) "J
i,-,~
> (~
m
el)
(-i,.
m
('I)
N m O"
~..I~
t--tin CV
o
-
=r
"1 N ('I)
~..,~
0
('b
i...~ ~
<~
@
Cr' ,-'.
9
"1
~
r-+ ~ ~ 0
,-~. " ~
l-e
-,1
,---~ ~ '-~ ~
0~ ~:: 0~ ~
CI)
'-~
~
o
0
"I I,~
~" 0
o
{11 CI) "0
~'u
~,-.~
~
@
0 "1
=r' 0 ,-.
.
I
'
,-,.
,_.
:
N
,,
,
!
,._. ,...,
N~r~
.
~
I1}
0
~
~1~
,--,
0
(1)
(I)
~.,,~
m
cl)
IXI
~.., ,,
(I)
m
u~
0
o
I~ h-K u~
>q"
+ (1) ~ '-J>O (,~
.=. , e , , ~ =:1
9
~~ ' ~.....-~ o ~'~
O ~
~ 9I d
~'~
x 0 O~
o
N
"~
+
~"
~
+
r'~
I
,-
+
~"
T
§
~
,~1
.-:
"~
~
~-
0
o
""
0
o.
cr
~ .
(1)
(~
~
r-+
t-'+ 0
~
::
~,
~
0
c~
z
~
"~
t-t.]
1~1
~,
~ (I)
9
~
i
r+
m
'-m
m
m
c F.~ o_ -I
=.
0
(1)
8. o
~ '~
'--
_ ~:~~
~
~.
IN
9
z
0 Z
N
(9 m" 0
~-~~
T~
r-+
X
m r-+
GO 4~
CT
GO 4~
I
F"
r-e
r-+
r-+
r-+ +
o
,.
,_..
~-~
~
~~-~
~-,. ~
Cr
0 ~ CT
~=,.,
m'~
~"
..
o
0
~=.~o
0 "
e-+
~
m
~.
-I
"
~-"
o
8
o
0
+
>
+
+
~r~
8
~'~
""
0
X
c)
0 "'1
,"4"
'-'I
,-.-
0
0
~-
";~
0
~
>
,0
>
o~
~
~"
-j
m
m"
Cr
m
~
"/-:~" 9
F"
~
~
'-I
,...
ill
~'~..-:
(~
-~
~-~
~
~ ~
~.
~..
"~
o
~~
0
'~
"~
~o~
~
=~'~"-~
~.. "~
~'~
~-~
.~.~o
2.~
~"
~o ~vO ~ ~
~ o ~'J i.-.o
=r
~ x~
~o ~
~
CT
m
~--
'~
~
~o
-
~. ~
~
~'~
~-
="
~
II
II
~
-
-
~
'-"
"
II
~:~
~
.
~. ~..
~
0
m
0
(~
~
(~
0
o
..
0
0
>
Z
-]..
"~
~.
cr ~" ~
0
~
CD
,-t
~..
~-.
~D
m
(,,,,) r~j
o
0
,-..
m 0
~-
0
i,--.
0
~ E ~'"
~
(-,)
-4
~"
+
'--'
""
o~"
~-~
:_.,
~.-,
0"~
0
I
o
~I
~"
~
~,.~
0 9 CL,.Q
~
'-'1
03 0'~ : : el) Ca :~-'*
('I)
('1)
('1)
~D
~
0
~'~'~
~
o
~"
~
"b
O0 (J1 ~
~
~
~
~
~ ,.+
,
~'~.
:::r (1)
II
,
(I)
....
~:r
~"
~'~
(I)
~'~
"I 't~
(1)
0
('I) ~1
c~
~'o
~..
CD
~ ,-j
0
o
~
1:::
"
"~
X
~+
~
m
0
o.
0
0
9
"~
0
0
0 ('~ 0~
,-'-
"~
J:~
~I~
~
~
0
0
co
'-~"
(~
~-'~
@
,-I
O
M '~
(I)
"r
~PE
0
0"
0
=r ~ r+
Z
(1) 0
~ ~
,-.~
0
~r
c-~ N
~
g~o
D
~-..
I
~
I t--
I
"li
"~
o
I
,
+
0
~a
ca
,
~-~
+ "-J
~.~
-
e~
(1)
03
~"
+
II
+ ~
O~
o~-
EL
.~'~
O"
O
r+
O O
r" O
~--
r+
o
~
'
O
03
03
I
O 2:
N
E
E; O
m
9
"-'"
'-"
CY'
B
O
O
O
,_,. ~:r
5
N
~"'.
~
'Ea
o..
-- e
,.~
~.
o
~'P==
O
"~
O
~
,.~
o~,
t-+
,-.. ~
;:::r
O
0
o
,.-..
9
~/)
i
?+
ao~" 0 ~:~ ~
~
~"
.,,o
~
~
--
~
n
"1
"O
e-+ =r' CD
O"
N
N
O"
O" i.,-,. ,,, i,-,...
i.-,, 9
~'-"' 9
"""
CT
i,--.
~''
I~
~
mF
..
=-
=r
'-'"
L0 00
?-+
~
~
~
,-'] ,--
o
~' ?..+
::::r.
S"
,--.
a
,-,.
"'1
o
~-"
0 ~....
2
c::
v
;:>
0 v
4-
v
v
,~
0
o
88
+
~-~
09
,~
~-~
oo
~
o
~-.
G3
(1)
('1)
~"
~_+" ~
0
_
"
~r~
~'~ ~
o
~
~
~
~-.. 0
0 c-+
~
~
v
'~
~_~
~
o
8
§
8
4,-.
+
r+
[vj
,--,
C9
of ~
4,
~-~ o
0
~
,._.
9
9
119
STABILIZATION. REGULATION, AND OPTIMIZATION
(iii)
Do not A., J
(iv)
exist
two
distinct
eigenvalues
of
A,
A
i
and
1~ I>-1 {~j I->1 such that AT .= AT.; i
Do not
'
exist
'
l
eigenvalues
A of
J
A,
A~l,
I Al=l,
such
that AT=I; (v) The pair (~(.),F(.)) is detectable; Then (a) The solution of the LQG problem (7), (33) e x i s t s and is given by system (34); (b) The closed-loop T-periodic system is a s y m p t o t i c a l l y stable and given by eqs. (35), (36); (c) The optimal p e r f o r m a n c e index is given by (38), w h e r e F(.) is the unique T-periodic positive semidefinite solution of the T-periodic d i f f e r e n c e Lyapunov equation
C37).ii VII. OUTPUT REGULATION The classical robust output regulation problem consists of determining a suitable regulator which guarantees the asymptotic tracking of given r e f e r e n c e signals in spite of the presence of p e r s i s t e n t d i s t u r b a n c e s and plant u n c e r t a i n t i e s . For this kind of problem to have a solution, a well known f a c t is t h a t the control signals must be f r e e to cover the same functional class as t h a t of the r e f e r e n c e and d i s t u r b a n c e signals. It is then a p p a r e n t t h a t problems g e n e r a l l y a r i s e when dealing with n o n s t a n d a r d u p d a t i n g mechanisms, a p a r t from the p a r t i c u l a r case w h e r e the exogenous signals are constant functions. Strictly speaking, the exact solution of the output regulation problem does not exist, and only p a r t i a l solutions can be achieved [33]. Hence, f r o m now on, it will be assumed t h a t the plant input is updated at any time instant. A. STATEMENT The plant described by
P
under
control
is
assumed
square
and
2
~
o
C~ ,,<
~.
0
"1
~
e
0
m t< m e-~
o
c+ ~" (1)
o
m t< m r-+
II
o
,/~
('D
~"
~
-,
~ ('D
0
e.+
0
~"
~"
(1)
D" (1)
0
~
I
"li
~,
,..
=
o '--'
o
m
('D
~
.
m
~ I:~
0
m
('D
I'D
~
~
=
C/~
0
~
~
:~
~
~
(1)
tl
~
"I
,-1 ~
C~.
m
~
o '-~
~-
~
:::r ~ ~
""
CY"
o
6"~
~
i.--i
~ ""
~
~--
'l'l]l 0~
,.... ~
CD
,
""
0
'-'"
0
~+ '-"
~
m
~
0
~
~ ,-'-
~
i--0
"I
~'"
0
~
(I)
.
>'
~
.
9
('~
~
r-~
~
(-1
~
t~
r+
~
II
+ '--
CD
o
m
~
().q
o
0
m
,_~
~
'--b
.~'~:
o~"
~=
~
I::L " ~
.-~
I~1
E
o
~ '-'"
~-.
9
~
"~
~-
"0
"I
~
'I:::I
CD
:~
"
,--..
9
('1 fll
-.
:~
-
9
m
"1 ~
~
I~
+
X
II
+
,~.
+
tm -] > t-"
> Z rrl
0
STABILIZATION, REGULATION. AND OPTIMIZATION
e(t) "= y~
121
- y(t)
the problem considered in the p a p e r is"
Output Robust A s y m p t o t i c Regulation Problem (ORARP) Find a c o n t r o l l e r ~' such that: (i) The closed-loop system (P,N,~') is a s y m p t o t i c a l l y stable; (ii) The output r e g u l a t i o n c o n s t r a i n t lim
t-)oo
e(t)
= 0
holds t r u e f o r any y~ any x (0), in a
and d(t), g e n e r a t e d by g with robust way, i.e., for all
e
perturbations
of
matrices
p r e s e r v e the a s y m p t o t i c s y s t e m (~P,N,E).m The block scheme r e p o r t e d in Fig. 1.
of
A
B, C, D
'
stability
the
of
overall
I
and the
D
2'
which
closed-loop
control
system
is
B. SOLUTION Denote by 2~ the set
of the distinct
eigenvalues
of 8
e
and
by
2~ the
union
(without
repetitions)
of
Je
and
the
e
set of the d i s t i n c t eigenvalues of system ~P. Then, l e t t i n g 0 denote the zero m a t r i x of any size, the following solvability condition f o r ORARP can be s t a t e d :
Theorem 15 [32], [34] Suppose that: (i) The p a i r (A,B) is stabilizable; (ii)
The p a i r (A,C) is detectable; A-)t I
B
e
(iii)
Det (
)~'0, C
0
e
e
P A T R I Z I O C O L A N E R I ET AL.
122
(iv)
There
does
not
exist
a
couple
of
and ~ j' L~i h-~1 I~ j I'-I ' ~uch that ~ -i ~ '
elements
of
~,
A
j
T h e n , ORARP a d m i t s a s o l u t i o n . l l x
I
(0)
~
g
.....
Y
I ~
0
+
N~
)0 e~
-
I
u
v
P
>
g
1
N
Fig. 1" The block s c h e m e of t h e o v e r a l l c o n t r o l s y s t e m .
This theorem supplies a sufficient condition for the e x i s t e n c e of g; in o r d e r to s t a t e a r e s u l t s p e c i f y i n g i t s structure, some definitions are necessary. L e t t h e m i n i m a l p o l y n o m i a l of m a t r i x A be e
V Z
V-1 +
a
z el
V-2 +
a
z
+
...
+
e2
Then, d e f i n e t h e m a t r i c e s
a
z eV-1
+
a eli
1l)
,-..
o
o
~f)
o
o
<
o
~
~
o"
..
(m
'-I
9
r
o
o _
o
-~
~ T
ID.(~
0
~o ~ o~ ~'~ ,_.. ~ o
~
0
""
~
~
~ ~ . ~ ~
~
o
9
,_..
~
~
~'~.
~
0,,-. 9
m
~'
o
-
~.
~....
~-~ o ~
~
o
c~
-
~
~
~
~
~
~
~Zr
' < o ~
00
~@
.
~
~ ~ ~
w
~ ~ ~~- -" ~ ,
~
~ ~
~ ~ ~ ~U ~
=o~
~
o
o
~r~
~
~
~-.. Cr
,~
>
~-,r~
C~
<
+
~x
~
~
<
~--
~r+
>
,
X
"~
~
~--
~
~I
/11
~
~
fll
~
x
13"
I~
. . . . . . 0~ 0~
~
/ll
x
0~
13"
>
!
'
I
i
i ~
,
~
|
~
C"~I
!
'
0
0
0
I~I
i
'
,
I
~
i
i ~
~
,"
0
0
0
-"
''
>I
0
0
0
0
0
9
9
9 Z
N
C
> 7. U
9
m
9
N
124
PATRIZIO C O L A N E R I ET AL.
m u l t i r a t e s y s t e m s can be used to s y n t h e s i z e s y s t e m ~r f o r instance the pole-placement or t h e LQG m e t h o d s . These t e c h n i q u e s should be applied to t h e c a s c a d e c o n n e c t i o n of s y s t e m s ~ , :P and ~V, w i t h input v and o u t p u t e~=-~. Hence, t h e o u t p u t t r a c k i n g c o n s t r a i n t is r o b u s t also w i t h r e s p e c t to v a r i a t i o n s of o r d e r and p a r a m e t e r s of ~r as long as a s y m p t o t i c s t a b i l i t y is p r e s e r v e d . The s a m e p r o p e r t y holds f o r as c o n c e r n s v a r i a t i o n s of the o r d e r n of t h e p l a n t ~P. It is w o r t h n o t i c i n g t h a t , t h o u g h the input of ~' is t h e v a r i a b l e e~, the c o n t r o l l e r is able to a s y m p t o t i c a l l y b r i n g o
to z e r o t h e d i f f e r e n c e e b e t w e e n t h e r e f e r e n c e s i g n a l y and t h e p l a n t o u t p u t y at all t i m e i n s t a n t s , not only a t t i m e s w h e r e the o u t p u t is m e a s u r e d . The block s c h e m e of Fig. 1 shows t h a t the s t r u c t u r e of t h e o v e r a l l c o n t r o l s y s t e m is v e r y s i m i l a r to a p o s s i b l e one f o r monorate systems. However, by c o m p a r i n g the s u f f i c i e n t c o n d i t i o n s of T h e o r e m 1 f o r the s o l v a b i l i t y of ORARP w i t h t h e d i s c r e t e - t i m e v e r s i o n of t h e n e c e s s a r y and sufficient conditions for the solvability of the same p r o b l e m in t h e m o n o r a t e case, it may be o b s e r v e d t h a t c o n d i t i o n (iv) h e r e does not have a c o u n t e r p a r t t h e r e . As a matter of f a c t , this condition guarantees (but is not n e c e s s a r y f o r ) the d e t e c t a b i l i t y f r o m y of t h e s t a t e of t h e s y s t e m (~t,~P,JV), along w i t h the s t a n d a r d c o n d i t i o n s (ii) and (iii). The consequence of t h a t is the impossibility of a s y m p t o t i c a l l y z e r o i n g t h i s way the s y s t e m e r r o r f o r some e x o g e n o u s s i g n a l s f o r whom it can be b r o u g h t to z e r o when the outputs are always measured.
VIII. CONCLUDING REMARKS T h i s p a p e r has r e v i e w e d some r e c e n t results about stabilization and regulation of multirate sampled-data systems. F i r s t , a p r e c i s e m a t h e m a t i c a l f o r m u l a t i o n of t h e input and o u t p u t mechanisms has been given in t e r m s of a discrete-time periodic system. Then, its structural p r o p e r t i e s have been i n v e s t i g a t e d and r e l a t e d to t h o s e of t h e u n d e r l y i n g t i m e - i n v a r i a n t plant. The c l a s s i c a l p o l e - p l a c e m e n t and LQ t e c h n i q u e s have
125
STABILIZATION. REGULATION, AND OPTIMIZATION
then been used for deriving stabilizing state feedback c o n t r o l l a w s and s t a b l e s t a t e o b s e r v e r s . F i n a l l y , it has been shown how to s e l e c t a p r o p e r regulator structure, which, in some significant cases, guarantees zero-error regulation in t h e face of wide c l a s s e s of e x o g e n o u s s i g n a l s , d e s p i t e t h e p o s s i b l e lack of i n f o r m a t i o n due to t h e o u t p u t s - s a m p l i n g .
APPENDIX Let
S be
the
discrete-time
T-periodic
system
described
by
x(t+l) - A(t) x(t) + B(t) u(t) y(t) = C(t) x(t) + D(t) u(t) w h e r e t e Z , A(t)ER n'n B(t)eR n'm C(t)ER p'n D(t)ER p'm Denote by ~ (t,z) t>z, the transition matrix ~
A
i.e.,
~
*
of
A(.)
~
~ (t,z):=A(t-l)A(t-2)...A(z).
Matrix
A
~ (z+T,z)
is
A
the so-called m o n o d r o m y matrix associated with A(.). Its eigenvalues do not depend on z and are called c h a r a c t e r i s t i c m u l t i p l i e r s of ~. S y s t e m 5e is a s y m p t o t i c a l l y stable if and only if all its c h a r a c t e r i s t i c multipliers a r e inside t h e open u n i t disk. Now, d e f i n e t h e l i f t e d input, t h e s a m p l e d s t a t e and t h e l i f t e d o u t p u t as u(k) "= [u(kT+T)'
u(kT+T+I)'
...
u(kT+T+T-I)']'
y(kT+z+l)'
...
y(kT+z+T-l)']'
^
x(k) "= x(kT+z) y(k) "= [y(kT+T)' respectively, (0-
w h e r e 1: is a given i n i t i a l sampling x(T)=x(O). The l i f t e d or t i m e - i n v a r i a n t 5e a s s o c i a t e d w i t h 5~ is e a s i l y o b t a i n e d above as
x(k+l) = A x(k) + 13 u(k)
time (TIR) from
.j .< H, LM Z < 0 L; 9 N
~r
X
II
,U
~
~ ~, ~ ..~
I--,
+ t~ ~9, II .
"~ .
.
l:I:1
.-,
I--
L)
" ,,
o,.~
I--,
..
9
II
,._., .
.
,L)"
9
.
"
.
9
,---,
,~ II
,._., .
f-
II
o,--~
H"
II
+
.-, I
cq
-,--~
.II . . .
e,
+ I~ ~_.
II
U
""' + I~
+
"~
E
.,.._,
+ I~ ~L. 1~
'~ II
o,.~
,I~
o,,
A
i
+
o,.--~
+
o,-.~
II
i + I~
,..-i ~
I + I~
,..-I ~
i
.-~
+ '~' U
II
~176 o'--i o,,
V
C)
o
""
".~ ~
+~ I:I::I
t~ ~
.,.~
N ._-
~~1~ o o
I~
~
o
~L,.
.Eo
I:Z,
I::::
+_,
,,--i
~. #~
+-'
o,--~
-~~'-
~I
o .
"D
~O " ~~ o . ~L~-,. ~ N . ~ ~....N O ~
I.~ ~ ~
O r I.-,
O'~ +'; I-, ~ O 1:~ I/~ ~-,
o~ ~ ~,o~-.a O
L)
I-, o ~) ..~
~
"
1~ o
1~ 6
o
~
o ;:;m .,.~
..-,
"
o
1~
00
o6
k.,
~
0
"T:
.._,
o ~ lD ,_,
~
o,--q
,~~o
,---,
+-'
(1.) ,-~ .D
c/l
.l_l
o
"~
~I
,---.,
STABILIZATION. REGULATION. AND OPTIMIZATION
127
feedback controller based on m u l t i r a t e sampling of the plant output," I E E E T r a n s . A u t o m a t . Contr., AC-33, 812-819, 1988. [3] M. Araki, "Recent development in digital control theory," Proc. 12th IFAC W o r l d Congr., 9, 251-260, 1993. [4] D.P. Glasson, "Research in m u l t i r a t e estimation and control," The A n a l y t i c S c i e n c e Corp., Rep. TR1356-2, 1980. [5] P. Colaneri, R. Scattolini and N. Schiavoni, "A design technique for m u l t i r a t e control with application to a distillation column," Proc. 12th IMACS W o r l d Congr., 589-591, 1988. [6] R. Scattolini, "Self-tuning control of systems with infrequent and delayed output sampling," Proc. I E E P a r t D, 135, 213-221, 1988. [7] J.H. Lee, M.S. Gelormino and M. Morari, "Model predictive control of m u l t i - r a t e s a m p l e d - d a t a systems" a s t a t e - s p a c e approach, 55, 153-191, 1992. [8] P.T. Kabamba, "Control of linear systems using generalized s a m p l e d - d a t a hold functions," I E E E T r a n s . A u t o m a t . Contr., AC-32, 772-783, 1987. [9] T. Mira, Y. Chida, Y Kaku and H. Numasato, "Two-delay robust digital control and its a p p l i c a t i o n s - avoiding the problem of unstable limiting zeros", I E E E t r a n s . A u t o m a t . Contr., AC-35, 962-970, 1990. [10] K.L. Moore, S.P. Bhattacharyya and M. Dahleh, "Capabilities and limitations of multirate control schemes," A u t o m a t t c a , 29, 941-951, 1993. [11] G . M . Kranc, "Input-output analysis of multirate feedbac system," I R E T r a n s . A u t o m a t . Contr., 3, 21-28, 1957. [12] R.E. Kalman and J.E. Bertram, "A unified approach to the theory of sampling systems," J. F r a n k l i n I n s t . , 267, 405-436, 1959. [13] E. I. Jury, "A note on m u l t i r a t e sampled-data systems," I E E E T r a n s . A u t o m a t . Contr., Ac-12, 319-320, 1967. [14] M. Araki and K. Yamamoto, "Multivariable m u l t i r a t e sampled-data systems: state space description, t r a n s f e r c h a r a c t e r i s t i c s , and Nyquist criterion," I E E E T r a n s . A u t o m a t . Contr., AC-31, 145-154, 1986.
128
PATRIZIO COLANERI ET AL.
[15] A.B. Chammas and C.T. Leondes, "On the design of linear time invariant systems by periodic output feedback: P a r t I-II. Discrete-time pole assignment," I n t . J. Contr., 27, 885-903, 1978. [16] A.B. Chammas and C.T. Leondes, "Pole assignment by piecewise constant output feedback," I n t . J. Contr., 29, 31-38, 1979. [17] P.P. Khargonekar, K. Poolla and A. Tannenbaum , "Robust control of linear time invariant plants using periodic compensation," I E E E T r a n s . A u t o m a t . Contr., AC-30, 1088-1096, 1985. [18] M. Araki and T. Hagiwara, "Pole assignment by multirate sampled-data output feedback," Int. J. Contr., 44, 1661-1673, 1986. [19] T. Hagiwara and M. Araki, "Design of a stable feedback controller based on m u l t i r a t e sampling of the plant output," I E E E T r a n s . A u t o m a t . Contr., AC-33, 812-819, 1988. [20] P. Colaneri, R. Scattolini and N. Schiavoni, "Stabilization of multirate sampled-data systems," A u t o m a t i c a , 26, 377-380, 1990. [21] P. Colaneri, R. Scattolini and N. Schiavoni, "Regulation of Multirate Sampled-Data Systems," C-TAT, 7, 429-441, 1991. [22] N. Amit, "Optimal control of m u l t i r a t e digital control systems," Stanford Univ, Rep. N. SUDAAR 523, 1980. [23] T. S/Sderstr~m and B. Lennartson, "On linear optimal control with infrequent output sampling," Proc. 3rd I M A t o n i . on Control T h e o r y , 605-624, 1981. [24] M.C. Berg, N. Amit and J.D. Powell, "Multirate digital control system design," I E E E T r a n s . A u t o m a t . Contr., AC-33, 1139-1150, 1988. [25] H . M . A1-Ramany and G.F. Franklin, "A new optimal multirate control of linear periodic and t i m e - i n v a r i a n t systems," I E E E T r a n s . A u t o m a t . Contr, 35, 406-415, 1990. [26] P. Colaneri, R. Scattolini and N. Schiavoni, "LQG optimal control of m u l t i r a t e s a m p l e d - d a t a systems, I E E E T r a n s . A u t o m a t . Contr., AC-37, 675-682, 1992. [27] J.R. Broussard and N. Halyo, "Optimal m u l t i r a t e output feedback," Proc. o l 23rd C o n l . D e c i s i o n Contr., 1984, 926-929.
STABILIZATION.REGULATION.ANDOPTIMIZATION [28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
129
R. Scattolini and N. Schiavoni, "Design of m u l t i r a t e control systems via p a r a m e t e r optimization," Proc. o f 26th C o n f . D e c i s i o n Contr., 1556-1557, 1987. M. E. Sezer and D.D. Siljak, "Decentralized m u l t i r a t e control, I E E E Trans. Automat. Contr., AC-35, 60-65, 1990. P. Carini, R. Micheli and R. Scattolini, "Multirate s e l f - t u n i n g predictive control with application to a binary distillation column," Int. J. S y s t e m Sc., 21, 51-64, 1990. R. Scattolini, "A m u l t i r a t e s e l f - t u n i n g c o n t r o l l e r for multivariable systems," Int. J. System Sc., 23, 1347-1359, 1992. P. Colaneri, R. Scattolini and N. Schiavoni, "Stabilization and regulation of multirate s a m p l e d - d a t a systems," Proc. Int. Syrup. MTHS-91, 511-516, 1991. P. Colaneri, R. Scattolini and N. Schiavoni, "The output control problem for m u l t i r a t e sampled-data systems", Proc. 31 C o n f . Dec. Contr., 1768-1773, 1992. R. Scattolini and N. Schiavoni, "On the output control of m u l t i r a t e systems subject to a r b i t r a r y exogenous signals," I E E E Trans. Automat. Contr., 643-646, 1993. V.S. Ritchey and G.F. Franklin, "A s t a b i l i t y c r i t e r i o n for asynchronous multirate linear systems," IEEE Trans. Automat. Contr., AC-34, 529-535, 1989. R.A. Meyer and C.S. Burrus, "A unified analysis of multirate and periodically time-varying digital f i l t e r s , " I E E E Trans. C i r c u i t s S y s t . , CAS-22, 162-167, 1975. S. Bittanti, "Deterministic and stochastic linear periodic systems," T i m e S e r i e s and L i n e a r S y s t e m s , Springer Verlag, 141-182, 1986. P. Bolzern, P. Colaneri and R. Scattolini, "Zeros of d i s c r e t e - t i m e linear periodic systems," I E E E T r a n s . Automat. Contr., AC-31, 1057-1058, 1986. O.M. Grasselli and S. Longhi, "Zeros and poles of linear periodic d i s c r e t e - t i m e systems," C i r c u i t s S y s t . S i g n a l P r o c e s s . , 7, 361-382, 1988. O.M. Grasselli and S. Longhi, "The geometric approach for linear periodic discrete-time systems," Linear A l g e b r a Appl., 158, 22-60, 1991.
130
[41]
[42]
PATRIZIO COLANERI ET AL.
S. Bittanti, P. Colaneri and G. De Nicolao, "The difference periodic Riccati equation for the periodic prediction problem," IEEE Trans. Automat. Contr., AC-33, 706-711, 1988. P. Colaneri and G De Nieolao, "Optimal s t o c h a s t i c control of m u l t i r a t e s a m p l e d - d a t a systems," Proe. 1st E u r o p e a n Contr. C o n f . , ?.519-2523, 1991.
ACKNOWLEDGEMENTS This paper has partially been supported by CNR (Centro di Teoria dei Sistemi) and MURST (407~ and 60Z funds).
I. I N T R O D U C T I O N
In c l a s s i c a l digital control systems, it is usually a s s u m e d t h a t both the p l a n t i n p u t s - u p d a t i n g and t h e p l a n t o u t p u t s - m e a s u r e m e n t a r e p e r f o r m e d at a unique c o n s t a n t r a t e and in a s y n c h r o n o u s fashion. However, t h i s h y p o t e s i s is sometimes not realistic, for economical and/or t e c h n o l o g i c a l r e a s o n s , and, f u r t h e r m o r e , r e l a x i n g it o f t e n allows the designer to obtain improved control p e r f o r m a n c e s . Hence, one is lead to c o n s i d e r the s o - c a l l e d multirate sampled-data control systems, which are c h a r a c t e r i z e d by the f a c t t h a t each input is u p d a t e d a t an i t s own r a t e and each o u t p u t is m e a s u r e d at an its own rate. The a n a l y s i s and the design of such s y s t e m s has recently received a great deal of attention. For an o v e r v i e w of the a r e a see, e.g., [11-[31. T h e r e a r e t w o p r i m a r y r e a s o n s of i n t e r e s t in m u l t i r a t e CONTROL AND DYNAMIC SYSTEMS, VOL. 71 Copyright 9 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.
95
96
PATRIZIO COLANERI ET AL.
digital control. A f i r s t s t r o n g m o t i v a t i o n behind t h e i r use is due to the possible presence of t e c h n o l o g i c a l c o n s t r a i n t s which enforce the use of control schemes where sensor m e a s u r e m e n t s and c o n t r o l c a l c u l a t i o n s have to be p e r f o r m e d at different sampling rates, see, e.g., [4]-['/]. This t y p i c a l l y o c c u r s in one of the f o l l o w i n g cases" (i) Some s e n s o r s r e q u i r e a s i g n i f i c a n t t i m e b e f o r e t h e y s u p p l y the m e a s u r e m e n t s of the p l a n t o u t p u t v a r i a b l e s to the regulator. For example, such a s i t u a t i o n o c c u r s in c o n t r o l l i n g chemical p l a n t s w h e r e e x p e n s i v e c h r o m a t o g r a p h s are used to measure composition products. These m e a s u r e m e n t s a r e t h e n i n f r e q u e n t and d e l a y e d w i t h r e s p e c t to those of other variables measured by s e n s o r s not s u f f e r i n g of such a l i m i t a t i o n . (ii) A small n u m b e r of s e n s o r s is used to m e a s u r e a l a r g e n u m b e r of o u t p u t v a r i a b l e s at d i f f e r e n t times, or t h e s e n s o r s allow one to m e a s u r e all the p l a n t o u t p u t s a t the s a m e r a t e and time, but h a r d w a r e c o n s t r a i n t s p r e v e n t one f r o m t r a n s m i t t i n g d a t a s i m u l t a n e o u s l y f r o m all the s e n s o r s to t h e c o n t r o l p r o c e s s i n g unit. (iii) The p l a n t o u t p u t s a r e all m e a s u r e d a t the s a m e r a t e and time, but t h i s r a t e is less t h a n t h a t of t h e p l a n t i n p u t s u p d a t i n g allowed by the c o n t r o l a p p a r a t u s e s . (iv) Some a c t u a t o r s a r e m a n i p u l a t e d less f r e q u e n t l y t h a n o t h e r s in o r d e r to r e d u c e the e f f o r t of t h e s e a p p a r a t u s e s . As a second r e a s o n , it has been shown t h a t the use of multirate and periodically time varying controllers can significantly improve the closed-loop performance of a s a m p l e d - d a t a s y s t e m in t e r m s of model m a t c h i n g , s e n s i t i v i t y r e d u c t i o n , d i s t u r b a n c e r e j e c t i o n , pole and z e r o a s s i g n m e n t with state feedback, see, e.g., [1], [8], [9]. However, these promising results usually refer to the sampled v e r s i o n of the s y s t e m , while p a r t i c u l a r c a r e has also to be paid to t h e i n t e r s a m p l e b e h a v i o r which can be s i g n i f i c a n t l y d e t e r i o r a t e d by the m u l t i r a t e input u p d a t i n g , see [10]. A deep d i f f e r e n c e e x i s t s in the two f o r m e r c l a s s e s of applications of multirate controlwhen a multirate approach is used to improve c o n t r o l performances, the frequencies and phases of inputs-updating and outputs-measurement are free design parameters to be d e t e r m i n e d by the c o n t r o l s t r a t e g y in o r d e r to o p t i m i z e the
STABILIZATION, REGULATION, AND OPTIMIZATION
97
r e q u i r e d p e r f o r m a n c e s . On the c o n t r a r y , when a m u l t i r a t e solution is enforced by technological c o n s i d e r a t i o n s , the same p a r a m e t e r s are problem d a t a and must be faced by the adopted control s ynt he s i s technique. In t hi s paper, a t t e n t i o n will be focused on this last s i t u a t i o n . Hence the p r o b lem a d d r e s s e d will be to design a m u l t i r a t e d i g i t a l regulator once the i n p u t s - u p d a t i n g and o u t p u t s - s a m p l i n g m e c h a n i s m s ar e fixed. R e s e a r c h in m u l t i r a t e control can be t r a c e d back to the l a t e f i f t i e s [11]-[13]; however, it has received m ore and m o r e a t t e n t i o n only in the pa s t decade. For an overview of the most s i g n i f i c a n t r e s u l t s of the a r e a the r e a d e r is r e f e r r e d to [3], [14]. In [ 1 4 ] a t t e n t i o n is focused on the a n a l y s i s of a control s t r u c t u r e where a d i f f e r e n t sam pl i ng r a t e is a s s o c i a t e d with any pair of i n p u t - o u t p u t v a r i a b l e s , then impulse modulation models are developed and c r i t e r i a to assess closed-loop stability are presented. Several s y n t h e s i s a l g o r i t h m s have r e c e n t l y been proposed in a l i near time-invariant setting. Among them, the p o l e - p l a c e m e n t appr oa c h has been considered in [15]-[21]. The Linear Quadratic Gaussian (LQG) technique has been applied in [4], [22]-[26], while some s y n t h e s i s a l g o r i t h m s based on c o s t - f u n c t i o n minimization have been p r e s e n t e d f o r c o n t r o l l e r s with a p r e s c r i b e d s t r u c t u r e [5], [27]-[29]. Some predictive and self-tuning multirate control a l g o r i t h m s have been proposed in [6], [7], [30], [31]. The o u t p u t r e g u l a t i o n problem, t h a t is the problem of z e r o i n g the s t e a d y - s t a t e e r r o r to the maximum possible e x t e n t in p r e s e n c e of exogenous signals of p r e s c r i b e d dynamics has been t r e a t e d in [32]-[34]. Finally, the case of com pl et el y a s y n c h r o n o u s sampling has been t r e a t e d in [35]. The aim of this paper is to review the main a s p e c t s concerning the application of popular s y n t h e s i s techniques, namely the p o l e - p l a c e m e n t approach and the LQG method, to the m u l t i r a t e control problem. Specifically, the a s s u m p t i o n is made t h a t the plant under control is a d i s c r e t e - t i m e l i near t i m e - i n v a r i a n t system. It is also assumed t h a t each o u t p u t has its own f r e q u e n c y and phase of m e a s u r e m e n t and each input has its own f r equency and phase of updating. The p ap er is organized as follows. In Section 2, the discrete-time linear time i n v a r i a n t model of the p l a n t under control is introduced and the sampling and u p d a t i n g
P A T R I Z I O C O L A N E R I ET AL.
98
m e c h a n i s m s a r e given a p r e c i s e m a t h e m a t i c a l f o r m u l a t i o n . Further, it is a l s o s h o w n how m u l t i r a t e s y s t e m s c a n be c a s t e d into t h e w i d e r c l a s s of p e r i o d i c s y s t e m s . Then, in Section 3 some preliminary results on the structural properties ( st ab i I i zab i I i ty, detectability and zeros ) of the m u l t i r a t e system are given in terms of the o r i g i n a l plant under control. Section 4 deals with the pole-placement and LQG methods when the system state is assumed to be available f o r control, while, since our main goal is to design output feedback controllers, in Section S the problem of state reconstruction is considered. In p a r t i c u l a r , t w o s t a t e o b s e r v e r s a r e p r e s e n t e d : in t h e f i r s t one t h e p o l e - a s s i g n m e n t t e c h n i q u e is a g a i n a p p l i e d , w h i l e t h e s e c o n d is d e r i v e d by r e s o r t i n g to t h e K a l m a n f i l t e r i n g approach. In Section 6 the previous results on state f e e d b a c k and s t a t e observers are joint t o g e t h e r with the a i m of d e r i v i n g s t a b i l i z i n g f e e d b a c k c o n t r o l l a w s . F i n a l l y , t h e c l a s s i c a l o u t p u t r e g u l a t i o n p r o b l e m is f a c e d in S e c t i o n 7 w h e r e , u n d e r s o m e p a r t i c u l a r a s s u m p t i o n s on t h e i n p u t s updating, a suitable regulation structure is presented w h i c h g u a r a n t e e s t h e a s y m p t o t i c t r a c k i n g of given r e f e r e n c e s i g n a l s in s p i t e of t h e p r e s e n c e of p e r s i s t e n t d i s t u r b a n c e s and plant u n c e r t a i n t i e s .
II.
THE PLANT, THE INPUTS-HOLDING AND THE OUTPUTS-SAMPLING MECHANISMS
Let the system under control be described by the following discrete-time linear time-invariant stochastic model
:p.
x(t+l) y(t)
w h e r e A~Rn'
= Ax(t) = Cx(t) n
, B~R n ,
+ Bu(t)
(1.b)
+ M w (t) 2
m
2
, M ~R n ' q , C~ R
p' n
, M ~R p ,
1
w
2
are
uncorrelated
(1.a)
+ M w (t) a 1
zero-mean
identity covariance matrices,
p
and w
2
i.e.,
gaussian
white
and 1
noises
with
,--~
~
{~
~
9
~"
~-
"~i
.
9
~,
"~i
~
~
,--~
.
("+
"~i
I
~-'o
::7
0
~
~ s
..
o cr
~*
. . . .
o~
~,..
A --]~
IA
O
~,,..0
(1)
. . . . +
~
~ ~-~
~
"d
+
~
~
~
o
~. ~
I
.]~
~*
~
-.~
(1)
o-"
9 --.
~"
(1)
~
"'~
O
-
~
~.
-.
"~
~ ~
~
+
-.
~ ;~
~
~'~
~
~-'
~.
~ "-
~
~
o
~ 9
cr
C::
~*
~ -.
=g
"
~ (1)
0~
-.
~ .-.
_ O
~.
~
'E~
~0
el)
~
-~.
0
(Tq
""
~
~
~
~-
~
..
o
m
o
1~
~
~
Ln
F)"
~.
~,
(]:)
9:
~-
~
-
o
<:
~
~
:
~'
~
.
~,
(~
~
0
~
0
I
~
=-
~
~
.
"
~
~
~
.
~.
~
~
~
.
~.
w ~" ~
~
~
g"
~"
(I)
~
..
~
~
.~-
~.
o
~ ~ ~
~
~
o
~-~
~
~ ' ~
__
~*o
g.~
~ O
~ ~
"~
--"
cL <
~.~
-
O
~
o
--~ )
0
>
o
r"
Z
o
r"
~
HI +
Nt ,,..,
,1-]l +
,-,it ,....
",i
,i,-,+
0
i-.
~" ,,
m (~
Z ~.
:~ =.
~*~ -,
Z ._.
~
~.
0
~
,~
~
~-.
==
""
i-.o
,-4"
I~ O
~
C~
13o"
~
:~
~
~
~ . ~~
=~"
~I~
.-
,-It A
T~ d,,~
o
~'%
~ m ' "
o
~
~
o
~
+
~,
~
""
-,
.,
0
""
+
o
~,
~
- ~.
,_.
~
(1)
I~
- -
~
~
=
~
:~ =.
-
~
~
~
~,.~
~+
"li
~
~+
~ ..
-"
~
~"
~
~
u~
0
O
~
"
,,~
i
-~_.
I
~r
-
o ~
m
~i'~
~
o ~= ~
, ~.
('l)
,,--] t
~
~ ~
~ ~+
9
4~
.~
9
.~
"m (I)
"E~
+
X
pl,.
m
9
9
t
r~
0
+
"-~
<:
II
+
'--'
.-4
~
..
~
..
~"
0
r-,"
~
I:::L O'-" ~,.
~
II ~-.
'-b
~ .~
-~
~
~
0
o
::~
<: ~(I)
o"
Z
>
0
O
N
"~ >.
--
-~
z
x
.,.~
i[i~
9
i[-
II~ 1 7 6
~
~
9-
~
~
~
II
d~T
I~) ~
~
5.~
~
.,-., o
,.c::
i
,[.-,
o
o
o"
,,iq
gl
", i oi i l .iw~e o
~,iii.i
~D
II
~
"" 0
~ ,
~
~
+
(._,
~
o
:>.,
"~
~-
+
<:!
~
+
~-
f~
X
II
00
~,"
m
<
v
,
I
l' . .
9.
tJ
I
'~
.
,->
II 9.
o
-
o
.
I
,
.
I
.
"o
~
~
0
~
~ ' ~
~~
~
II
z
~
N
o
C~ "
+~[i~
II..
,
. ~2
"--" Z
~.
~ ..--, ~~
o-~
L,
o~
+.,
.
~
+a
:~
.
.~
e X N ,.~
r A "~
~
.0
o
~1
X
r .,-,
"'-'
~
9 ~
0
..
.
~
"~
(D
~-o
b,~
~
+-~
o~
~
~
~.
..X
~ ~o
~
o
~
"F~ ~
~'~ _
o r.., ..
~,
~
o~
o.~
-o
~
1~ ~)
~
.~
t~ o
r.., Ei
c"l:l
"~ " " ~
N
"0
o
~
~
1~ .i~
. ~ .,-,
= ~'~ I~ ~-~
~
"~--'
+'2~
~-&~ I~ "~
~ ~.~
,i-a
0"~ ~
~
~,
+.,
,,
d
~ ' B ~
El m ~
-~
~.
~
,-:
~
txo
~
~
+
~-~
~
~
II
~
c r ~u'}
~
~
~"
~.
(I)
b0
,_
~i,. ~
9
C~
""
-.
~
u~
1-11
-
~
I~ '-b ~
o
~
0
m
~
~
{A F ,~
~
_.N
~
0
"o
~
~
,.
o-
~
~
B'~+~
~ "
~
C~
'
""
1~,~
{A
~
,.
~"
~"
~
m
C
o
~.T;"
~"
o
~,
~
.~
o
~"
'~
.,--
9
~
DJ
~"
'-"~
~
~
..
+
~
~
+
~
~
I~
.~
~
~
=
I ~ ~_.,o
o
'-"<
~
~
~
o
~
~
9
~ ,,
~..
m
%
~
~
0
CL.
~
I~
<1
~
~
"-~
~@
~
~
~+
~
.~
~
~
~
~
-"
~
,_..
~
m
-
-cr
~
.
.
.
-
~
9
--
-'.
.
o
~
~
I~ ~
~
o
..
~
~
0
0
"
~,
:~
~
o ~
u~ ~...,.
~..
o
~
.
~-~
~
~
u~
~
~
o ~
"
.~
,-,
._.
c
,_,
~
-
~
.~
~
~
m
C)
m
•
~;
9
,_..~-
"0 "
-I,
..
o
..
I
-
"--"
.-.
"o
.
o-
0.
,~
-~
cr
0
o
0
""
o
~D
0
.-0
u
~
II
~
'-~
o
(~
--]n 9
o
CI
~
~
~
~
~
~r
~'~
r~ r~
~~
~~
~"
~
r~
t~
o
--,
-.
0
c~
c ~ ,-..
N
~"
c~
_
~'~
~
~
~~
~
~
-~'~ ~ ~
~
~ ~
T
~'(~
~ ~
~
~ ~>
~-~ . 0 ~,
0 ~ .----'~
~
~ o
,_., ~
9
=.
~
~
~
,.
~
~'~
~=
~ ~
~.
~
~
z
>+
,
~
~
I
~e~
Cr
N
~r" ~,.,.
,
"
~, .-.
-~
~
0
i/l
II
r
--
~
=
Z:r
Z::r
~,.0
>
0 "--b
m
Cr
N
I-+
'--
~
00
,--.
,.
~"
~ 0
~
'-"
~v
,v
~
;--"
--
~.~o
-.
"~
~
(~
~
,~
~
00
O0
.
'-h
1::::
m
~"
,<
0
~r
0
~"
~
;'-'"
~N
~
o -
~"
"~ ~ ~. ' o ~
~.~
,~
~
~"
g
I:~
0
9
0
"~I
0
CO
~
"I./I " i-I. 0
~.q~
-h
'-1
~
0
~"
~o ~
~.~o~. o o ~ o~~-~.
~oo
0
~ ~-.o--= ~.-~
~=
~_.~'~
N
~
~
<
,'-".
~
~
o
--
9
N
0
o
z
~o
0
0
N CD
0
~=.~.
,.-!
~r'
e-+
'-o
0
N (I) ,.-I 0
0
~=..o
m
~,,..
=~ ~ ~
0
"
-.~"
O~~
'-I
~~
~~~
~_
~ ~~;-.~ ~
-~
r~
~ =('Dq -~'~~ . ~ . . o ~ . ~ - ~
('D
..~
~
.,~
~~
r-~
0~
""
0~ " ~
~"
. ~
~
~ ~
~ ~
~.~
~
,..~. 0
~-
~.~
"-~~
~ ~ ~~ =~ ~ ~ ~
~=
~..
~= "~ ~- ~,
'--' " 1 ; " - - '
,_,
.,~
~9 , ~ . ~
0
~'
~
.7,
~; .~ ~ ~
o.~ ~"
0"
~
"-~
0
CD
0
.---...~
'~
~"~ . o~ ~
~o
~.~
~
0~
~.g
="~ ~ "~ ~~'~a~~.. ~
(Y~
0
o~
-....I
~o , -- ' ~ "1
0
@
0
9 ~ ~ ~~
~,=<~=~
?~
nn
N
s 9
N-
~
Z
""
>,_.
-.
~--
~
""
--]1
,I ~'~
~-
~
IV
--
('1
o
cD i./1
.,r
('1)
~...o
~e
-.
(')
~"
~
~. e
,...
~
~.>
~,
~
~
-----~o
~e
.---..
~.
,....
b.,.~ ~
"I
o
~
- . _-:. ~ ~
.
-z~~
~,.~
~-~
9
9
~.~~ ~
o
-~a~T
.
.~.~o~_
~-
[--,
5
N
-
105
STABILIZATION, REGULATION, AND OPTIMIZATION
If lar is a t r a n s m i s s i o n z e r o of ~, w i t h o u t b e i n g r an e i g e n v a l u e of ~P, t h e n t h e r e e x i s t s ~, such t h a t ~'=/a, w h i c h is a t r a n s m i s s i o n z e r o of ~o.m
(ii)
In t h e s p e c i a l c a s e w h e r e S ( . ) - 0 , i.e., t h e i n p u t s a r e u p d a t e d a t any t i m e i n s t a n t , s y s t e m (3) s i m p l y r e d u c e s to u(t)=r(t). This allows one to neglect Eq. (3.a) and c o n s i d e r s y s t e m (12), as f a r as t h e z e r o s a r e c o n c e r n e d , as t h e c a s c a d e c o n n e c t i o n of ~P and N. T h e o r e m 5 [33] If S ( . ) = 0 , t h e n t h e s e t of t r a n s m i s s i o n zeros b e l o n g s to t h e s e t of t h e t r a n s m i s s i o n z e r o s of P ' I I
of
(12)
IV. STATE-FEEDBACK CONTROL LAWS In this section, we extend the main classical stabilization techniques, namely Pole-Placement (PP) and Linear-Quadratic (LQ) c o n t r o l , to t h e c a s e of m u l t i r a t e s y s t e m s . To t h i s end, we c o n s i d e r t h e T - p e r i o d i c s y s t e m (7.a) and a s s u m e that its s t a t e ~(.) is a v a i l a b l e for control. Since our ultimate goal is to design output f e e d b a c k c o n t r o l l e r s , ~ we m a k e r e f e r e n c e in t h e s e q u e l to p e r i o d T, i n s t e a d of T. N
A. POLE-PLACEMENT The TIR of s y s t e m
(7.a) w i t h w =0, 1
is e a s i l y
determined
as
~(k+l)-
~ ~(k) + r"
(13)
r'(k)
where (k) .= ~(kT) , r(k) "= [ r ( k T ) ' a n d $, 1~ a r e r e s u l t holds.
suitable
r(kT+l)'
matrices,
for
... r ( k T + T - 1 ) ' ] ' which
the
following
,--
Ir
0
II
J~
,0
o"
"-'
o
+
,-+
,0,~
~,
~
F
_,
N
~-
~-.~
~ ~
o-
- r+
o ~ ('!)
0
o 0~
o
m
~
m
~
~
0
~
,-0
0
~
('1)
0
~
--m
~-oo
,~
m
o
o
~Z
~,
0
~
II
,~
o
~,
~
~
c~
~
~'
-i
~,
9
o~'
~
9
.+
~ ~ ~
0 m
~
5. ~
~ ~
~ ' ~~.
~"!~ "1
o '-"b
~
~"
+
~
1~ ~
~ "
~
~
o
~
~ ~ ,-.
~ (1)
'-'"
~
=~"
0
"~ ~ "1
,-""
~ @
r~ ""
,. II
0
"
~
~ ~
~-' ~ ~
+
+
9
~
-.
~"
~-'
o ~ ~ ~ 0 ,-...
o ~-'(1) 0
'-'~
~'~
m
X
~
C~ O~ ~ ~ ,-~ "1
~
n)
"1
~O.~o
('1) ~ < m
~,~
~
~ ~"
7'
~.
~,
,...
"o
@
~.. I 0 0
~" ~
1~
~
m cr
~0
~
0
~
-"
-"
~
~
(~
Cr
z o
I
~
'-"
-
T~
-
,-. "~
0
,-. c r
too.
,_..
~'c~
'~"
.-1~
~.~
,e,~4 ~9
,-.
o ~
~'.~ ,-.
~ o .__.
.___,
o"
r~
'-'"
~
-
o
~
0
~"
~--
~,--. ~I~"
o- b
~
m ~.~
m
~
~;
~
=
~
o~
=~
~
~
x~
m
~
r~ ~..
~
~
~
~"
~
o ~"
o
~-
~.
~
,.
o
~--
o
~
~"
0
~ ~
~
.. Z
~
~
0
~-.
"~' ' m
~-o
~-
a..
a=
-H
~," ~" E9 ~
~
~
m
.~"
(~
~ o -~
---~ " ~'~
~
o
o
~r
e-,
0,,.., ~
N
m
cr
0
~
0
~
~,
~
~',H ~ " ;~'~ ~
L
~
i
"~i
,0
'-]
~
,0
~
~
I
m
B
m
i-,~
m
0
I-~
~
0
~
~
~
~
0
o
o
..
-
;~,
'-1>
+
;~,>
{/1
~'0
~
~
"-~
o
o
0
--.
~
0
~e "
0
'-1 0
~> '-'1> ~>
~
'--] ~,
+
~'~
+
"-]
~
~'~
"]
~
,--,.
~
B ~m
~
~
II
II
~ ~
"
.:
"
m
o
m
~
0
"
~
9
0
o
m
"-' a'~ ,.-.
~
~
~
II
~,_.
,~5
;~+m
~]N 0 + ,,--. ~ ,.T3 ~ "
m
~o
9
~ 0
:.
"
'~
~-" ~
~ ~
0
4
..,u
II .,0
~>
"~i
"-"
..
~
m
9
N
0
> z
0 z
0
~q
G~
n
~..
'--'0
~
+
~
~
~
>
"~i
'-"
0
0(1)
II
N 0 ~
~"~
0
~"" ~ e-+
0
~-'" ~'~
r~
~
V
~"~
0
"I ~ 0
o
m
0
0
I
"~
1~
"~
~ ~ ~ ~
CT ~ .
o"*
~)
+
0"
~-+ 0 0
~)
,.
(1)
~ ~.. 0
.~ ~ F"
~Oo
-~
~'~-
C~cr
+
t~
C~
0
o
~
~
I
~
0
~
~--
~'~
~
-.
N
'-'
~
~)
~o
+
II
9.:
~
g
o
U1
0,-.,-,
(/1 0
'-I
0
,'~
0
0
o
~ ~ CL~:J
~-~
00
0
0 0
9
~
9 "
0
0
0 + '-1
i""
i""
+ '-1
7"
o
0 ~
._ ~
0
O"
9
~n
0
0
C~
,.~
~" ~
e-~
,-'. ('1)
0
0.,.~
~00~,
0
~ ~ 0 ~ a..~'-~
~-~"
~
O0 -~.
-]
,--~.
II ~
"~
~
~
0
~
~
~
0" ~
0
0
1~
~
(1)
0
0 ~
,---
0
C7" ~._~
~.~m ~
~,"0
~ ~'~"~
~
~=
~1 _~ (1)
'-'"
_]
9
,-,. ~
,.--
~
[-1"1
~o
>
N
oo~
0--o
~
,-..
,-,.
8
0
0
~
,,.-,,
0
I--,.,' 9
r~
g)
0
b,,-, 9
1: 0
0
~0
I--'.
'--'
0
0
~
~
~
o
I--.'-
0
,~
~
o 0
~_~
o
~
~o
~=
"b
~.
~....
~--...
,,-j
(I)
0
8
c)
<: ['11
zr
"3"
0
I"11
c-+
0
rll
-] >
~
m r-+
o
@ B
cr
"~ o
-] =r
m 9
~-"
~'
~"
9
ID.
~ re 0"
0
~
0
~
~
~ "1
0
-"
~=~
~
~
~D
0
~'
'--'- ~ 9 ~ ~'"
~
,~,~
~~
m ~.
0
o,
~
~
o
~
(1) "'b
(1)
---
~--
0
o
2
I
o ,--- ~
[",3
r*
o
"4,----
~
~
'1~
r-~
cr
~.. N
~-.
~
--.
~~ ' ~ "
~
"~
(1)
o
0 "~
~'~"
~
~
~
0
o
o
II
re
r
~
=~
0
~~-*
'-"
re
"1
(1) 0
~
II
0
o
--.
~
"" ~-,
[%1
+
"0
"1
+
~
D~
'--J
~~
+
+
~
~~
~
"" ~
0 13.
~D
I
0 CD
"1
~
rn
0
9 Z
N
9
Z
9
r"
X~
z
[',4
~o
?'*
o
Zr
--"
=
"-m
0
o
o
.~. ,<
-
o"
c::
--.
_.
o
13
=" (1)
~, o"
=
o
~
,_,.
ro
-o
""
ct) o
-m
~
~
.
o
o
~
~
~
"0
~
""
~"
I:~
~'0
~-c~
F'~"
0
~ m cr --
,0 m ~I ,...
~
9
~,
~
;~
~,r"
.
.
--"
o ~
~
~
~<~
('1)
~
~
o~,_,,.
~
0
~
"~
~ ,-,.
I::> )0r ~
.
. [--,
9 ~ ,
"
~
o
-.
~ +
-. r9 ' ,
, ,.-,
o r',
~
~
,-.
~
• :~
m
=r o
-
3
,-.
~
=
c% ~..
I~
~ I~
r
,-,. ~'m,
~:~
~
o
~
,
~'
~
~ ~-'
i:>,
r',
+
~
+
II
+ . ~
u~ (1)
~
:::Y'
-
-" ~
~
~ 4~
~'-"
~ --
+
'--]
~
o,-,
-
.
+
~
~
-I~
~
-~, ~
~,
o
o ~
~
~:
~
="
~,
:~
~ ~ ('t)
o
~
r~ ~-.
~ -. ~
~ ,-,"
cT
~"
o
III
~
~"
III 0
~
:~
~"~
~ ~ (I)
~
o
Z ~
~
rm'
-~
;::;-
::::r'
g
~
o
'-"
=r
"
cry.
~
~
~~"
,-"
o=~
~-~ =~'
~
"~
o ~ = o
~o
~
< ~ - ~ ~
F'~'~
~
--o
~ 0 ~--, "Z:~ c r 9
_
cD
~
o - ~ ' = # ~
,-" ~- "
u~
~',~
o,9.,, ~: ,-,- ~
@~
o
~+
ct)
~
o
0
o
m -]
7.
0
0-
"~
o
~'~ ~ o
~
,,
l.n
Oh
c~'~H
('I)
r-l.
CZ, "
~ r~
u'l ~ (1)
~.,
--,
i-,o
0., Z:r ~
t~
~n
.
~~,o
0
0
i,--o
I~
~
0
~
~'!~
,-+
0n ~ ' o
"Cl ~ r-t. ,_,. ~).0 ~. ~
O~
c~ r ~
~
~
o "~
o
0..
Zr 0
0 ,-I
~==,o
0
0 --,
~" ('1)
~
~
..o
'-'"
"~ -I~ .
0
0
0
~-'
0
I~'~
'
+ r',
,_,
~
, ........
~
+
+
~
I
"+
+
_.
~" "~
~r
--]
II
w--.o
~
x
--
--
ox
II
~=.o
,_
~
.'~
~.
,,-.o
c~
,_.
~
~
, ~
r~
.-~
,,
+
~e~
~
~
~"
m
ox
r-~
--]
c~
""
0 ~
~
,-~
'-1
~
Im
o
.
0
,,
~-,
I
~.]
""
--,
~
~
~
~
9
c~
"q
~o
U~
"" C)
I
U1
('I)
l,n
0
~ ~..
~ o 0~ .~.~.~
~.
~ Oo
i.,]
~
~
C)
u~
i...~ 9
~
~ o~
0~
~.
=.
~'
~',
o
,
i~"
-~~
CI)
~
'-'"
0n
~
r,
~ ,-..
~
~
0
o
~Z
m
.,-i o,..i
o
..
o
[--
o-~
<
~: ~ ~
""
Z 0
--.~
~
o
.~
ffl
~-, ~-~
o
_~
o
~
~
o
o
9 0
(D
I~
o
~
i
..
"
cr
,_._.,
II ,i.l
~
o
0
+'~
o
t_~
~
,.~
O4
,.--i
0
~ ,,,,-I
~o
~
~ o
"~
0
::~
"-I~
~
""
(p
~
II
' ,..-1
o~
~
0 .,
II
[--
~
i~
0
~
~,
.
~,
o
-~
0
= ~ o
0 0
--'
o
"C
0
..~
~
~
..~
t~
.~
~
:
o
"~ b "S o ~ =
=
-,-,
t~
-~.-
~ t~
I./1
"~'~
"'~
~ o
'+~
o
~
~I . ~
r .~ gl ....~ . ~
:n
o
~
'-"
~"
~,_,
t~
o
"
o
~'~
0
"~
~
~ 0 ..0 ~
~bo~
; ~
~
I~ ..-.
~'~
L~)
bO O0
m
~
e.+
r
I
+
"-"
o,q
,_,
~o
Z
Zr.~
0
0
~~
o
-x
o
~~
~
C~
C~
,-..
~
0
~ :~
~
V
~
o
~
i
II
~
0
~
0
0
~
o~
.~:~
~D
~<
0
m
~'o
~
r~
Cr. (D
~'~'~
~ ~m
e
o
""
~
o~~
--.~
~ m
~"
--.
~"
~.
m
~
o
~
~
o
N
~
o
~
~
~
~.,
Z
~
~
g
~
~-.
~.
,
-
o
~
i
I
0
~
&
&
..-:
:~
0
.
| ~.
o
X
2.
~
~.
~
0
,,~
"
-~
~.
:~
:~
..q
~
~-
~
~,
~
~ ~
Oo C~' "b ~,
~-~
"~
~-~"
ocr ~
~"
m
~
-o
""
~
0
0
oc
--"
cvo
~
o
P
o
,_, v
~
~
~
o
~ ~ 0 Y
o e't" ~ cr'~ "-'s "-s
=
~
:~
~
~ ~ ~
,-, F~" ~ ~
~
_
0
> z
o
~>
t-.n 0 c~
o
(1)
a)
0
(1)
~
CY
~
,,-+ (1)
,X:D
0
0"
~ -0 . ~,-~
r-+ Ul
~" ~ .
~o
{~
CI)
0
'-'"
~
~" ~
(A .-..
o
.~
(I)
~.~~
cz
0
O~
~
,... ~ "
~ "
~c~
~-~
~l~ C1)
,--
c~
~
9
(1)
~ ~ P ,.... u~
:~
"o"
c)
'~- "
--zr
~ o
~.
~o.~ - .o . ~~
o ,--'
r+
C)
~-~ ~
a) O~
< ,_,
,.+ o
'~~
~-
~
~ .---. '-"
"
-.
0 c-F
Cl)
~
c)
'-"
".-" (1)
~,
~
4"
o
0
~"
0
0 "I (I)
~
Cl)
cI)
"-b .. o~.
~ ,..
'-'" CL
ul
0
0
,v~"
I
~a)
~ ,-..
..
,.~--]
~ ~
~
Oc~ '~"
~
'-"~
~
.--.
~
(D" "rl
'-"
e~
~
X
~
I~
o
0
-.
--]
-
Ixc' :::~' ~ "
c~ ,-..
~~
CI) "S
Zr" Cl)
(~"
(-,)
I
~D
~:~
{n
,-4'< ~
C3
~
~
-~
C
"D
9
"1
ca
r-~
o
~v
"~
~~" -
",I
8
,~
o " _ ,I,,'--"
'-'" '--' Cb
,e,
"I
w,..
zr' (I)
r
C:z.
(I)
cr"
CL (I) r (I) o r-+
u'}
t-,.
9
~ ~.~..--~ zr' 9
~- ~.
~'..~
"-"
~o "+ 9
C: ,-..
"J u}
0
o
,-b o
(I)
0
(I)
.-j
cI)
r-F
i,-.-9 U')
,I,,
(I)
:::r"
('I)
~" cD
-]
o
o
0
~
0rr~
II
""
,-+
'=r'J
~,=,,.
C:Z. r-+ '-'" 0
o 0
I~
~ Zr (1)
~
(,J 0
~-+
'---'
I~" "~' r'+
"n
.-,
-I'-
,~,
,.-.-,
t
4"
'0' ~ ~ ~
r-*
(3
0
~.
II
I~
~
,-+ + .,I
"~
0 CL i,.-,9
(I) '-I
I
..]
CI)
0
'-'b (I) "I CI)
cI)
o
~
~ 0 ..--.
ci)
'~
~C)
.-,
'~'I
.~,
I>'
"D
I:>
4-
'---'
~"
D-
"D "11
I
]i
.~.
rD
(I) "J
i,-,~
> (~
m
el)
(-i,.
m
('I)
N m O"
~..I~
t--tin CV
o
-
=r
"1 N ('I)
~..,~
0
('b
i...~ ~
<~
@
Cr' ,-'.
9
"1
~
r-+ ~ ~ 0
,-~. " ~
l-e
-,1
,---~ ~ '-~ ~
0~ ~:: 0~ ~
CI)
'-~
~
o
0
"I I,~
~" 0
o
{11 CI) "0
~'u
~,-.~
~
@
0 "1
=r' 0 ,-.
.
I
'
,-,.
,_.
:
N
,,
,
!
,._. ,...,
N~r~
.
~
I1}
0
~
~1~
,--,
0
(1)
(I)
~.,,~
m
cl)
IXI
~.., ,,
(I)
m
u~
0
o
I~ h-K u~
>q"
+ (1) ~ '-J>O (,~
.=. , e , , ~ =:1
9
~~ ' ~.....-~ o ~'~
O ~
~ 9I d
~'~
x 0 O~
o
N
"~
+
~"
~
+
r'~
I
,-
+
~"
T
§
~
,~1
.-:
"~
~
~-
0
o
""
0
o.
cr
~ .
(1)
(~
~
r-+
t-'+ 0
~
::
~,
~
0
c~
z
~
"~
t-t.]
1~1
~,
~ (I)
9
~
i
r+
m
'-m
m
m
c F.~ o_ -I
=.
0
(1)
8. o
~ '~
'--
_ ~:~~
~
~.
IN
9
z
0 Z
N
(9 m" 0
~-~~
T~
r-+
X
m r-+
GO 4~
CT
GO 4~
I
F"
r-e
r-+
r-+
r-+ +
o
,.
,_..
~-~
~
~~-~
~-,. ~
Cr
0 ~ CT
~=,.,
m'~
~"
..
o
0
~=.~o
0 "
e-+
~
m
~.
-I
"
~-"
o
8
o
0
+
>
+
+
~r~
8
~'~
""
0
X
c)
0 "'1
,"4"
'-'I
,-.-
0
0
~-
";~
0
~
>
,0
>
o~
~
~"
-j
m
m"
Cr
m
~
"/-:~" 9
F"
~
~
'-I
,...
ill
~'~..-:
(~
-~
~-~
~
~ ~
~.
~..
"~
o
~~
0
'~
"~
~o~
~
=~'~"-~
~.. "~
~'~
~-~
.~.~o
2.~
~"
~o ~vO ~ ~
~ o ~'J i.-.o
=r
~ x~
~o ~
~
CT
m
~--
'~
~
~o
-
~. ~
~
~'~
~-
="
~
II
II
~
-
-
~
'-"
"
II
~:~
~
.
~. ~..
~
0
m
0
(~
~
(~
0
o
..
0
0
>
Z
-]..
"~
~.
cr ~" ~
0
~
CD
,-t
~..
~-.
~D
m
(,,,,) r~j
o
0
,-..
m 0
~-
0
i,--.
0
~ E ~'"
~
(-,)
-4
~"
+
'--'
""
o~"
~-~
:_.,
~.-,
0"~
0
I
o
~I
~"
~
~,.~
0 9 CL,.Q
~
'-'1
03 0'~ : : el) Ca :~-'*
('I)
('1)
('1)
~D
~
0
~'~'~
~
o
~"
~
"b
O0 (J1 ~
~
~
~
~
~ ,.+
,
~'~.
:::r (1)
II
,
(I)
....
~:r
~"
~'~
(I)
~'~
"I 't~
(1)
0
('I) ~1
c~
~'o
~..
CD
~ ,-j
0
o
~
1:::
"
"~
X
~+
~
m
0
o.
0
0
9
"~
0
0
0 ('~ 0~
,-'-
"~
J:~
~I~
~
~
0
0
co
'-~"
(~
~-'~
@
,-I
O
M '~
(I)
"r
~PE
0
0"
0
=r ~ r+
Z
(1) 0
~ ~
,-.~
0
~r
c-~ N
~
g~o
D
~-..
I
~
I t--
I
"li
"~
o
I
,
+
0
~a
ca
,
~-~
+ "-J
~.~
-
e~
(1)
03
~"
+
II
+ ~
O~
o~-
EL
.~'~
O"
O
r+
O O
r" O
~--
r+
o
~
'
O
03
03
I
O 2:
N
E
E; O
m
9
"-'"
'-"
CY'
B
O
O
O
,_,. ~:r
5
N
~"'.
~
'Ea
o..
-- e
,.~
~.
o
~'P==
O
"~
O
~
,.~
o~,
t-+
,-.. ~
;:::r
O
0
o
,.-..
9
~/)
i
?+
ao~" 0 ~:~ ~
~
~"
.,,o
~
~
--
~
n
"1
"O
e-+ =r' CD
O"
N
N
O"
O" i.,-,. ,,, i,-,...
i.-,, 9
~'-"' 9
"""
CT
i,--.
~''
I~
~
mF
..
=-
=r
'-'"
L0 00
?-+
~
~
~
,-'] ,--
o
~' ?..+
::::r.
S"
,--.
a
,-,.
"'1
o
~-"
0 ~....
2
c::
v
;:>
0 v
4-
v
v
,~
0
o
88
+
~-~
09
,~
~-~
oo
~
o
~-.
G3
(1)
('1)
~"
~_+" ~
0
_
"
~r~
~'~ ~
o
~
~
~
~-.. 0
0 c-+
~
~
v
'~
~_~
~
o
8
§
8
4,-.
+
r+
[vj
,--,
C9
of ~
4,
~-~ o
0
~
,._.
9
9
119
STABILIZATION. REGULATION, AND OPTIMIZATION
(iii)
Do not A., J
(iv)
exist
two
distinct
eigenvalues
of
A,
A
i
and
1~ I>-1 {~j I->1 such that AT .= AT.; i
Do not
'
exist
'
l
eigenvalues
A of
J
A,
A~l,
I Al=l,
such
that AT=I; (v) The pair (~(.),F(.)) is detectable; Then (a) The solution of the LQG problem (7), (33) e x i s t s and is given by system (34); (b) The closed-loop T-periodic system is a s y m p t o t i c a l l y stable and given by eqs. (35), (36); (c) The optimal p e r f o r m a n c e index is given by (38), w h e r e F(.) is the unique T-periodic positive semidefinite solution of the T-periodic d i f f e r e n c e Lyapunov equation
C37).ii VII. OUTPUT REGULATION The classical robust output regulation problem consists of determining a suitable regulator which guarantees the asymptotic tracking of given r e f e r e n c e signals in spite of the presence of p e r s i s t e n t d i s t u r b a n c e s and plant u n c e r t a i n t i e s . For this kind of problem to have a solution, a well known f a c t is t h a t the control signals must be f r e e to cover the same functional class as t h a t of the r e f e r e n c e and d i s t u r b a n c e signals. It is then a p p a r e n t t h a t problems g e n e r a l l y a r i s e when dealing with n o n s t a n d a r d u p d a t i n g mechanisms, a p a r t from the p a r t i c u l a r case w h e r e the exogenous signals are constant functions. Strictly speaking, the exact solution of the output regulation problem does not exist, and only p a r t i a l solutions can be achieved [33]. Hence, f r o m now on, it will be assumed t h a t the plant input is updated at any time instant. A. STATEMENT The plant described by
P
under
control
is
assumed
square
and
2
~
o
C~ ,,<
~.
0
"1
~
e
0
m t< m e-~
o
c+ ~" (1)
o
m t< m r-+
II
o
,/~
('D
~"
~
-,
~ ('D
0
e.+
0
~"
~"
(1)
D" (1)
0
~
I
"li
~,
,..
=
o '--'
o
m
('D
~
.
m
~ I:~
0
m
('D
I'D
~
~
=
C/~
0
~
~
:~
~
~
(1)
tl
~
"I
,-1 ~
C~.
m
~
o '-~
~-
~
:::r ~ ~
""
CY"
o
6"~
~
i.--i
~ ""
~
~--
'l'l]l 0~
,.... ~
CD
,
""
0
'-'"
0
~+ '-"
~
m
~
0
~
~ ,-'-
~
i--0
"I
~'"
0
~
(I)
.
>'
~
.
9
('~
~
r-~
~
(-1
~
t~
r+
~
II
+ '--
CD
o
m
~
().q
o
0
m
,_~
~
'--b
.~'~:
o~"
~=
~
I::L " ~
.-~
I~1
E
o
~ '-'"
~-.
9
~
"~
~-
"0
"I
~
'I:::I
CD
:~
"
,--..
9
('1 fll
-.
:~
-
9
m
"1 ~
~
I~
+
X
II
+
,~.
+
tm -] > t-"
> Z rrl
0
STABILIZATION, REGULATION. AND OPTIMIZATION
e(t) "= y~
121
- y(t)
the problem considered in the p a p e r is"
Output Robust A s y m p t o t i c Regulation Problem (ORARP) Find a c o n t r o l l e r ~' such that: (i) The closed-loop system (P,N,~') is a s y m p t o t i c a l l y stable; (ii) The output r e g u l a t i o n c o n s t r a i n t lim
t-)oo
e(t)
= 0
holds t r u e f o r any y~ any x (0), in a
and d(t), g e n e r a t e d by g with robust way, i.e., for all
e
perturbations
of
matrices
p r e s e r v e the a s y m p t o t i c s y s t e m (~P,N,E).m The block scheme r e p o r t e d in Fig. 1.
of
A
B, C, D
'
stability
the
of
overall
I
and the
D
2'
which
closed-loop
control
system
is
B. SOLUTION Denote by 2~ the set
of the distinct
eigenvalues
of 8
e
and
by
2~ the
union
(without
repetitions)
of
Je
and
the
e
set of the d i s t i n c t eigenvalues of system ~P. Then, l e t t i n g 0 denote the zero m a t r i x of any size, the following solvability condition f o r ORARP can be s t a t e d :
Theorem 15 [32], [34] Suppose that: (i) The p a i r (A,B) is stabilizable; (ii)
The p a i r (A,C) is detectable; A-)t I
B
e
(iii)
Det (
)~'0, C
0
e
e
P A T R I Z I O C O L A N E R I ET AL.
122
(iv)
There
does
not
exist
a
couple
of
and ~ j' L~i h-~1 I~ j I'-I ' ~uch that ~ -i ~ '
elements
of
~,
A
j
T h e n , ORARP a d m i t s a s o l u t i o n . l l x
I
(0)
~
g
.....
Y
I ~
0
+
N~
)0 e~
-
I
u
v
P
>
g
1
N
Fig. 1" The block s c h e m e of t h e o v e r a l l c o n t r o l s y s t e m .
This theorem supplies a sufficient condition for the e x i s t e n c e of g; in o r d e r to s t a t e a r e s u l t s p e c i f y i n g i t s structure, some definitions are necessary. L e t t h e m i n i m a l p o l y n o m i a l of m a t r i x A be e
V Z
V-1 +
a
z el
V-2 +
a
z
+
...
+
e2
Then, d e f i n e t h e m a t r i c e s
a
z eV-1
+
a eli
1l)
,-..
o
o
~f)
o
o
<
o
~
~
o"
..
(m
'-I
9
r
o
o _
o
-~
~ T
ID.(~
0
~o ~ o~ ~'~ ,_.. ~ o
~
0
""
~
~
~ ~ . ~ ~
~
o
9
,_..
~
~
~'~.
~
0,,-. 9
m
~'
o
-
~.
~....
~-~ o ~
~
o
c~
-
~
~
~
~
~
~
~Zr
' < o ~
00
~@
.
~
~ ~ ~
w
~ ~ ~~- -" ~ ,
~
~ ~
~ ~ ~ ~U ~
=o~
~
o
o
~r~
~
~
~-.. Cr
,~
>
~-,r~
C~
<
+
~x
~
~
<
~--
~r+
>
,
X
"~
~
~--
~
~I
/11
~
~
fll
~
x
13"
I~
. . . . . . 0~ 0~
~
/ll
x
0~
13"
>
!
'
I
i
i ~
,
~
|
~
C"~I
!
'
0
0
0
I~I
i
'
,
I
~
i
i ~
~
,"
0
0
0
-"
''
>I
0
0
0
0
0
9
9
9 Z
N
C
> 7. U
9
m
9
N
124
PATRIZIO C O L A N E R I ET AL.
m u l t i r a t e s y s t e m s can be used to s y n t h e s i z e s y s t e m ~r f o r instance the pole-placement or t h e LQG m e t h o d s . These t e c h n i q u e s should be applied to t h e c a s c a d e c o n n e c t i o n of s y s t e m s ~ , :P and ~V, w i t h input v and o u t p u t e~=-~. Hence, t h e o u t p u t t r a c k i n g c o n s t r a i n t is r o b u s t also w i t h r e s p e c t to v a r i a t i o n s of o r d e r and p a r a m e t e r s of ~r as long as a s y m p t o t i c s t a b i l i t y is p r e s e r v e d . The s a m e p r o p e r t y holds f o r as c o n c e r n s v a r i a t i o n s of the o r d e r n of t h e p l a n t ~P. It is w o r t h n o t i c i n g t h a t , t h o u g h the input of ~' is t h e v a r i a b l e e~, the c o n t r o l l e r is able to a s y m p t o t i c a l l y b r i n g o
to z e r o t h e d i f f e r e n c e e b e t w e e n t h e r e f e r e n c e s i g n a l y and t h e p l a n t o u t p u t y at all t i m e i n s t a n t s , not only a t t i m e s w h e r e the o u t p u t is m e a s u r e d . The block s c h e m e of Fig. 1 shows t h a t the s t r u c t u r e of t h e o v e r a l l c o n t r o l s y s t e m is v e r y s i m i l a r to a p o s s i b l e one f o r monorate systems. However, by c o m p a r i n g the s u f f i c i e n t c o n d i t i o n s of T h e o r e m 1 f o r the s o l v a b i l i t y of ORARP w i t h t h e d i s c r e t e - t i m e v e r s i o n of t h e n e c e s s a r y and sufficient conditions for the solvability of the same p r o b l e m in t h e m o n o r a t e case, it may be o b s e r v e d t h a t c o n d i t i o n (iv) h e r e does not have a c o u n t e r p a r t t h e r e . As a matter of f a c t , this condition guarantees (but is not n e c e s s a r y f o r ) the d e t e c t a b i l i t y f r o m y of t h e s t a t e of t h e s y s t e m (~t,~P,JV), along w i t h the s t a n d a r d c o n d i t i o n s (ii) and (iii). The consequence of t h a t is the impossibility of a s y m p t o t i c a l l y z e r o i n g t h i s way the s y s t e m e r r o r f o r some e x o g e n o u s s i g n a l s f o r whom it can be b r o u g h t to z e r o when the outputs are always measured.
VIII. CONCLUDING REMARKS T h i s p a p e r has r e v i e w e d some r e c e n t results about stabilization and regulation of multirate sampled-data systems. F i r s t , a p r e c i s e m a t h e m a t i c a l f o r m u l a t i o n of t h e input and o u t p u t mechanisms has been given in t e r m s of a discrete-time periodic system. Then, its structural p r o p e r t i e s have been i n v e s t i g a t e d and r e l a t e d to t h o s e of t h e u n d e r l y i n g t i m e - i n v a r i a n t plant. The c l a s s i c a l p o l e - p l a c e m e n t and LQ t e c h n i q u e s have
125
STABILIZATION. REGULATION, AND OPTIMIZATION
then been used for deriving stabilizing state feedback c o n t r o l l a w s and s t a b l e s t a t e o b s e r v e r s . F i n a l l y , it has been shown how to s e l e c t a p r o p e r regulator structure, which, in some significant cases, guarantees zero-error regulation in t h e face of wide c l a s s e s of e x o g e n o u s s i g n a l s , d e s p i t e t h e p o s s i b l e lack of i n f o r m a t i o n due to t h e o u t p u t s - s a m p l i n g .
APPENDIX Let
S be
the
discrete-time
T-periodic
system
described
by
x(t+l) - A(t) x(t) + B(t) u(t) y(t) = C(t) x(t) + D(t) u(t) w h e r e t e Z , A(t)ER n'n B(t)eR n'm C(t)ER p'n D(t)ER p'm Denote by ~ (t,z) t>z, the transition matrix ~
A
i.e.,
~
*
of
A(.)
~
~ (t,z):=A(t-l)A(t-2)...A(z).
Matrix
A
~ (z+T,z)
is
A
the so-called m o n o d r o m y matrix associated with A(.). Its eigenvalues do not depend on z and are called c h a r a c t e r i s t i c m u l t i p l i e r s of ~. S y s t e m 5e is a s y m p t o t i c a l l y stable if and only if all its c h a r a c t e r i s t i c multipliers a r e inside t h e open u n i t disk. Now, d e f i n e t h e l i f t e d input, t h e s a m p l e d s t a t e and t h e l i f t e d o u t p u t as u(k) "= [u(kT+T)'
u(kT+T+I)'
...
u(kT+T+T-I)']'
y(kT+z+l)'
...
y(kT+z+T-l)']'
^
x(k) "= x(kT+z) y(k) "= [y(kT+T)' respectively, (0-
w h e r e 1: is a given i n i t i a l sampling x(T)=x(O). The l i f t e d or t i m e - i n v a r i a n t 5e a s s o c i a t e d w i t h 5~ is e a s i l y o b t a i n e d above as
x(k+l) = A x(k) + 13 u(k)
time (TIR) from
.j .< H, LM Z < 0 L; 9 N
~r
X
II
,U
~
~ ~, ~ ..~
I--,
+ t~ ~9, II .
"~ .
.
l:I:1
.-,
I--
L)
" ,,
o,.~
I--,
..
9
II
,._., .
.
,L)"
9
.
"
.
9
,---,
,~ II
,._., .
f-
II
o,--~
H"
II
+
.-, I
cq
-,--~
.II . . .
e,
+ I~ ~_.
II
U
""' + I~
+
"~
E
.,.._,
+ I~ ~L. 1~
'~ II
o,.~
,I~
o,,
A
i
+
o,.--~
+
o,-.~
II
i + I~
,..-i ~
I + I~
,..-I ~
i
.-~
+ '~' U
II
~176 o'--i o,,
V
C)
o
""
".~ ~
+~ I:I::I
t~ ~
.,.~
N ._-
~~1~ o o
I~
~
o
~L,.
.Eo
I:Z,
I::::
+_,
,,--i
~. #~
+-'
o,--~
-~~'-
~I
o .
"D
~O " ~~ o . ~L~-,. ~ N . ~ ~....N O ~
I.~ ~ ~
O r I.-,
O'~ +'; I-, ~ O 1:~ I/~ ~-,
o~ ~ ~,o~-.a O
L)
I-, o ~) ..~
~
"
1~ o
1~ 6
o
~
o ;:;m .,.~
..-,
"
o
1~
00
o6
k.,
~
0
"T:
.._,
o ~ lD ,_,
~
o,--q
,~~o
,---,
+-'
(1.) ,-~ .D
c/l
.l_l
o
"~
~I
,---.,
STABILIZATION. REGULATION. AND OPTIMIZATION
127
feedback controller based on m u l t i r a t e sampling of the plant output," I E E E T r a n s . A u t o m a t . Contr., AC-33, 812-819, 1988. [3] M. Araki, "Recent development in digital control theory," Proc. 12th IFAC W o r l d Congr., 9, 251-260, 1993. [4] D.P. Glasson, "Research in m u l t i r a t e estimation and control," The A n a l y t i c S c i e n c e Corp., Rep. TR1356-2, 1980. [5] P. Colaneri, R. Scattolini and N. Schiavoni, "A design technique for m u l t i r a t e control with application to a distillation column," Proc. 12th IMACS W o r l d Congr., 589-591, 1988. [6] R. Scattolini, "Self-tuning control of systems with infrequent and delayed output sampling," Proc. I E E P a r t D, 135, 213-221, 1988. [7] J.H. Lee, M.S. Gelormino and M. Morari, "Model predictive control of m u l t i - r a t e s a m p l e d - d a t a systems" a s t a t e - s p a c e approach, 55, 153-191, 1992. [8] P.T. Kabamba, "Control of linear systems using generalized s a m p l e d - d a t a hold functions," I E E E T r a n s . A u t o m a t . Contr., AC-32, 772-783, 1987. [9] T. Mira, Y. Chida, Y Kaku and H. Numasato, "Two-delay robust digital control and its a p p l i c a t i o n s - avoiding the problem of unstable limiting zeros", I E E E t r a n s . A u t o m a t . Contr., AC-35, 962-970, 1990. [10] K.L. Moore, S.P. Bhattacharyya and M. Dahleh, "Capabilities and limitations of multirate control schemes," A u t o m a t t c a , 29, 941-951, 1993. [11] G . M . Kranc, "Input-output analysis of multirate feedbac system," I R E T r a n s . A u t o m a t . Contr., 3, 21-28, 1957. [12] R.E. Kalman and J.E. Bertram, "A unified approach to the theory of sampling systems," J. F r a n k l i n I n s t . , 267, 405-436, 1959. [13] E. I. Jury, "A note on m u l t i r a t e sampled-data systems," I E E E T r a n s . A u t o m a t . Contr., Ac-12, 319-320, 1967. [14] M. Araki and K. Yamamoto, "Multivariable m u l t i r a t e sampled-data systems: state space description, t r a n s f e r c h a r a c t e r i s t i c s , and Nyquist criterion," I E E E T r a n s . A u t o m a t . Contr., AC-31, 145-154, 1986.
128
PATRIZIO COLANERI ET AL.
[15] A.B. Chammas and C.T. Leondes, "On the design of linear time invariant systems by periodic output feedback: P a r t I-II. Discrete-time pole assignment," I n t . J. Contr., 27, 885-903, 1978. [16] A.B. Chammas and C.T. Leondes, "Pole assignment by piecewise constant output feedback," I n t . J. Contr., 29, 31-38, 1979. [17] P.P. Khargonekar, K. Poolla and A. Tannenbaum , "Robust control of linear time invariant plants using periodic compensation," I E E E T r a n s . A u t o m a t . Contr., AC-30, 1088-1096, 1985. [18] M. Araki and T. Hagiwara, "Pole assignment by multirate sampled-data output feedback," Int. J. Contr., 44, 1661-1673, 1986. [19] T. Hagiwara and M. Araki, "Design of a stable feedback controller based on m u l t i r a t e sampling of the plant output," I E E E T r a n s . A u t o m a t . Contr., AC-33, 812-819, 1988. [20] P. Colaneri, R. Scattolini and N. Schiavoni, "Stabilization of multirate sampled-data systems," A u t o m a t i c a , 26, 377-380, 1990. [21] P. Colaneri, R. Scattolini and N. Schiavoni, "Regulation of Multirate Sampled-Data Systems," C-TAT, 7, 429-441, 1991. [22] N. Amit, "Optimal control of m u l t i r a t e digital control systems," Stanford Univ, Rep. N. SUDAAR 523, 1980. [23] T. S/Sderstr~m and B. Lennartson, "On linear optimal control with infrequent output sampling," Proc. 3rd I M A t o n i . on Control T h e o r y , 605-624, 1981. [24] M.C. Berg, N. Amit and J.D. Powell, "Multirate digital control system design," I E E E T r a n s . A u t o m a t . Contr., AC-33, 1139-1150, 1988. [25] H . M . A1-Ramany and G.F. Franklin, "A new optimal multirate control of linear periodic and t i m e - i n v a r i a n t systems," I E E E T r a n s . A u t o m a t . Contr, 35, 406-415, 1990. [26] P. Colaneri, R. Scattolini and N. Schiavoni, "LQG optimal control of m u l t i r a t e s a m p l e d - d a t a systems, I E E E T r a n s . A u t o m a t . Contr., AC-37, 675-682, 1992. [27] J.R. Broussard and N. Halyo, "Optimal m u l t i r a t e output feedback," Proc. o l 23rd C o n l . D e c i s i o n Contr., 1984, 926-929.
STABILIZATION.REGULATION.ANDOPTIMIZATION [28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
129
R. Scattolini and N. Schiavoni, "Design of m u l t i r a t e control systems via p a r a m e t e r optimization," Proc. o f 26th C o n f . D e c i s i o n Contr., 1556-1557, 1987. M. E. Sezer and D.D. Siljak, "Decentralized m u l t i r a t e control, I E E E Trans. Automat. Contr., AC-35, 60-65, 1990. P. Carini, R. Micheli and R. Scattolini, "Multirate s e l f - t u n i n g predictive control with application to a binary distillation column," Int. J. S y s t e m Sc., 21, 51-64, 1990. R. Scattolini, "A m u l t i r a t e s e l f - t u n i n g c o n t r o l l e r for multivariable systems," Int. J. System Sc., 23, 1347-1359, 1992. P. Colaneri, R. Scattolini and N. Schiavoni, "Stabilization and regulation of multirate s a m p l e d - d a t a systems," Proc. Int. Syrup. MTHS-91, 511-516, 1991. P. Colaneri, R. Scattolini and N. Schiavoni, "The output control problem for m u l t i r a t e sampled-data systems", Proc. 31 C o n f . Dec. Contr., 1768-1773, 1992. R. Scattolini and N. Schiavoni, "On the output control of m u l t i r a t e systems subject to a r b i t r a r y exogenous signals," I E E E Trans. Automat. Contr., 643-646, 1993. V.S. Ritchey and G.F. Franklin, "A s t a b i l i t y c r i t e r i o n for asynchronous multirate linear systems," IEEE Trans. Automat. Contr., AC-34, 529-535, 1989. R.A. Meyer and C.S. Burrus, "A unified analysis of multirate and periodically time-varying digital f i l t e r s , " I E E E Trans. C i r c u i t s S y s t . , CAS-22, 162-167, 1975. S. Bittanti, "Deterministic and stochastic linear periodic systems," T i m e S e r i e s and L i n e a r S y s t e m s , Springer Verlag, 141-182, 1986. P. Bolzern, P. Colaneri and R. Scattolini, "Zeros of d i s c r e t e - t i m e linear periodic systems," I E E E T r a n s . Automat. Contr., AC-31, 1057-1058, 1986. O.M. Grasselli and S. Longhi, "Zeros and poles of linear periodic d i s c r e t e - t i m e systems," C i r c u i t s S y s t . S i g n a l P r o c e s s . , 7, 361-382, 1988. O.M. Grasselli and S. Longhi, "The geometric approach for linear periodic discrete-time systems," Linear A l g e b r a Appl., 158, 22-60, 1991.
130
[41]
[42]
PATRIZIO COLANERI ET AL.
S. Bittanti, P. Colaneri and G. De Nicolao, "The difference periodic Riccati equation for the periodic prediction problem," IEEE Trans. Automat. Contr., AC-33, 706-711, 1988. P. Colaneri and G De Nieolao, "Optimal s t o c h a s t i c control of m u l t i r a t e s a m p l e d - d a t a systems," Proe. 1st E u r o p e a n Contr. C o n f . , ?.519-2523, 1991.
ACKNOWLEDGEMENTS This paper has partially been supported by CNR (Centro di Teoria dei Sistemi) and MURST (407~ and 60Z funds).