ESC794
Mathematics for Control Theory
Paper Analysis Project 2. Preparatory Reading
Instructions: You earn points incrementally with equal weight in all asignments and
tests. You are not required to turn in all items.
This assignment is based on the paper by Khammash, M. and Pearson, J.B, “Robust
Disturbance Rejection in l1 -Optimal Control”, Systems and Control Letters, 14 (1990), pp.
93-101. The paper is available through the Michael Schwartz Library. As supplementary
reference materials, sections of Dahleh, M. and Pearson, J.B. Jr.,“Optimal Rejection of
Persistent Disturbances, Robust Stability, and Mixed Sensitivity Minimization”, IEEE
Transactions on Automatic Control, v. 33, n. 8, 1988, pp. 722-731 will be used, along with
sections of the book “Multivariable Feedback Control: Analysis and Design”, by Skogestad
and Postlethwaite, Wiley 1997. Additional information and guidelines will be given in class.
Specific questions and/or numerical calculations will be assigned on Oct. 15, and the
assignment will be due on Oct. 29.
1. Read sections 2.2 to 2.6.1, 2.6.4 and 2.7 in the book by Skogestad and Postlethwaite,
focusing on the mixed sensitivity approach.
2. Read the paper by Khammash and Pearson up to and including Section 4. Material
from Section 5 until the end of the paper is not part of the assignment, but the proofs
of Corollary 4.2 and Theorem 4.3 are.
3. Read sections VIII A and B up to, but not including, Theorem 8 in the paper by
Dahleh and Pearson, focusing on what Theorem 7 says and the mixed sensitivity
approach described in VIII B.
Systems & Control Letters 14 (1990) 93-101
North-Holland
93
Robust disturbance rejection in ~1 optimal
control systems*
Mustafa KHAMMASH
a n d J.B. P E A R S O N
Department of Electrtcal and Computer Engineering, Rtce Umverstty, P 0 Box 1892, Houston, TX 77251-1892, U S A
Received 28 August 1989
Revised 4 November 1989
Abstract Given a class of plants formed by perturbing a normnal discrete-time hnear sinft-lnvarlant plant with norm bounded
unstructured perturbation, the problem of finding a single compensator that wdl stablhze all plants m thas class and at the same time
rmmmtze the worst case norm of a certain performance measure is addressed Assurmng the performance measure of interest is the
sensitivity function, first an expressmn for the supremum, over all plants in tins class, of such a measure ~s derived for a system with a
robustly stabllizang compensator Tins Is acbaeved by first showing that the proposed expression is an upper bound for the worst case
sensitivity norm, and then by constructing a plant in the allowable class that wdl acineve this upper bound_ Combined with already
known necessary and sufficient conditions for robust stabdlty, the derived expression provides an effective way of comb~mng both
robust stability and performance in one, easy to compute, measure_ Secondly, a scheme for finding a controller that mtntmt,zes the
worst case norm of the sensitivity function whale achieving robust stability, is provided Tbas makes posstble the Incorporation of
stability and performance robustness requirements m one destgn procedure
Keywords Stabdity robustness, performance robustness, optimal disturbance rejection, sensitivity immm~atlon, gal control systems
1. Introduction
W h e n m o d e h n g physical systems as h n e a r plants for the p u r p o s e o f designing f e e d b a c k controllers that
m a k e the closed l o o p system achieve c e r t a m specifications, one c a n n o t escape the m o d e l i n g uncertmnties
that are inherent in such a process E v e n if the u n d e r l y i n g physical system co u l d be m o d e l e d exactly at one
time, p a r a m e t e r variations that c o u ld a p p e a r for any o n e of m a n y reasons e v e n t u a l l y take their toll on the
system and render the m o d e l inaccurate. F o r this reason, a c o n t r o l l e r that achieves g o o d p e r f o r m a n c e
w h e n controlling the model, m i g h t n o t p e r f o r m so well when used to c o n t r o l the actual plant and could
even m a k e the system unstable Therefore, robustness of the c o n t r o l system to variations m the p l an t are of
great practical i m p o r t a n c e . Stability robustness can be achieved if the c o n t r o l l e r can be m a d e to stabilize a
whole family of plants. P e r f o r m a n c e robustness, on the other hand, can be achieved if in a d d m o n the
controller can be chosen so as to give ' g o o d ' p e r f o r m a n c e for each one of the m e m b e r s of the p l an t class.
Stability robustness is therefore required for p e r f o r m a n c e robustness In this respect, recent work by
D a h l e h and O h t a [1] provides necessary and sufficient c o n d i n o n s for B I B O stablhty robustness Th e p l an t
p e r t u r b a t i o n s considered m [1] take the f o r m of m u l t i p h c a t i v e or additive p e r t u r b a t i o n s with a b o u n d e d
n o rm . In addition, the p e r t u r b a t i o n s are allowed to be t i m e - v a r y i n g or nonlinear.
This p a p e r considers p e r f o r m a n c e robustness w h e n the p e r f o r m a n c e criterion is f ~ d i st u r b an ce
r e j e c u o n G o o d p e r f o r m a n c e , in this case, translates i n t o small n o r m s for certain l o o p functions, e.g. the
sensitivity function. Accordingly, in the case of sensitivity, robust p e r f o r m a n c e can be achieved if the n o r m
of the sensitivity function can be m a d e small for all p e r t u r b e d plants, an o b j e c t i v e that can be achieved by
rmnimazlng, with the p r o p e r choice of a robustly stabihzing controller, the w o r s t case n o r m of this
function.
* This research was supported by the N S.F under grant ECS-8806977
0167-6911/90/$3 50 © 1990, Elsevier Science Publishers B V (North-Holland)
94
M Khammash, J B Pearson / Robust d~sturbance rejectton
2. Notation
R q The space of q-tuples of real numbers. If x = ( x 1, . , Xq) E ~ q , then I x l ~ = m a x , Ix, [.
g~_ Space of all b o u n d e d sequences of real n u m b e r s , Le x = ( x ( k ) } ~ = o ~ l
~ if and o n l y if
sup~ I x ( k ) I < ~z. If x ~ [ ~ then 11x II ~ = sup~ I x ( k ) [
{q~ Space of q-tuples of elements of [ ~:. If x = ( x 1. . . . . Xq) ~ g ~ then II x II ~ = m a x , I1x, 11~g ~- Space of absolutely s u m m a b l e sequences If x ~ [~ then l[ x II1 = E ~ 0 I x ( k ) IE~×q Space of p × q matrices w~th entries in {~
P N The t r u n c a t i o n o p e r a t o r on sequences H e n c e ff x = ( ( k ) } k = 0 is any sequence, then PNX =
{x(O), x(1), . , x ( N ) , O ,
)
P~:([~) T h e set of all x ~ E q ~ such that x , ( k ) = O V k > N
and 1 < t < q
~op×q
r v × q then
TV T h e space of all b o u n d e d linear causal o p e r a t o r s m a p p i n g tq~ to fp:~ If R ~ -cap
[I R II ' : sup_~. 0 II R x I[ ~ / I I x II ~ which is the i n d u c e d o p e r a t o r n o r m Each R m (PP'<q
~TV
can be c o m p l e t e l y
charactemzed b y its block l o w e r - t r i a n g u l a r pulse response matrix
R,j R E(PP×q~Tv can be naturally divided up into its p - q c o m p o n e n t s R,j where R , j - g~: ~ g~: d e n o t e s
the c o m p o n e n t of R m a p p i n g the j - t h i n p u t to the t-th o u t p u t
R, WhenR~aP×q~'rv, R , . C q - - * g ~ denotes the , th c o m p o n e n t of R, l e i f R ( x ~ ,
.,Xq)=(y~,.
,yp),
then R , ( x 1. . . . xq) = y , .
£~(~×q. S u b s p a c e of ~ o ~ q consisting of time m v a r m n t o p e r a t o r s F o r each R ~ £ , a ( ~ q c o r r e s p o n d s a
u m q u e r in ~a;×q where r,s IS the impulse response of R,s.
I[ R II ~ If R ~ £,a(~×q, then II R 11~, is ~ts r e d u c e d o p e r a t o r n o r m which also equals max~ _<,_<p [I R, II =
max~ <_t<p E J=l
q I[ r,s [I a"
[ x / y ] If x ~
p and 0 ~ y = ( y ~ ,
.,yq)~q
a n d ff j ~s the smallest indexing integer such that
I_v/] >-- I Y, ] for all l = 1 . . . . q then [x/y] is defined to be the real p × q m a t r i x f o r m e d b y setting its j - t h
c o l u m n to be ( 1 / ) ) ) x a n d all of the other columns to zero A consequence of ttus d e f m m o n is that
[x/yly= x
3. Problem statement
Let P0 be a given n o m i n a l discrete-time plant. P0 ~s assumed to be hnear, shlft-mvariant, a n d strictly
causal with q inputs a n d p outputs.
D e n o t e b y S(Po) the set of all h n e a r s h l f t - l n v a n a n t d i s c r e t e - t i m e controllers with the a p p r o p r i a t e
d i m e n s i o n that stablhze P0 W e now define a family of p l a n t s f o r m e d b y a d d i n g weighted m u l t l p h c a t w e
p e r t u r b a t m n s to the n o m i n a l plant. Let
rt-= (P_ P = & ( I +
W A)}
where W~ ~A°vq( q and A [ f ---, Eq~ is causal with IJ AII -= s u p ~ . o II Ax I[ ~/11 x II ~ < 1 So A 1s allowed
to be txme-varylng or n o n h n e a r W e also define
q" = { C ~ S( Po) C s t a b f l i z e s a l l P ~ H }
W h e n p e r f o r m a n c e is m e a s u r e d b y the n o r m of the weighted sensitivity functmn, the p r o b l e m of
achieving robust p e r f o r m a n c e and stability can now b e stated as follows.
C~'/"
(z+ cP)
w ll = voo,
where W2 E ~ q l y q.
It is therefore desired to c o m p u t e ]'opt and to find a controller C ~ '/' that will m a k e the q u a n t i t y
supp ~ ~i [I( I + C P ) - 1W2 [I arbitrarily close to -/op~
95
M Khammash, J B Pearson / Robust dtsturbance rejectton
4. Problem solution
Theorem 4.1, to be presented next, is essentially the key to solving the problem posed earlier. Together
with Corollary 4.2 and Theorem 4.3, it forms the main result in this paper. The proof of Theorem 4 1
requires a few prehrmnary propositmns and will be postponed until after it is demonstrated how these
results can be utilized to solve the stated problem
Theorem 4.1. Let T and S both be m ~q~q with Tsattsfymg [IT l[ ~e< 1, Then
sup I1(I+ TA)-ISII = max
IIS, II~,
Acausal
l<,--<q 1 -- liT, ll~e
0All_<1
Corollary 4.2. Let C ~ S( Po) such that t l ( I + CPo)-ICPoW1 II~< 1. Then
sup 11(1 + cp)-lw=ll- - max
l<-,~ql-
P~II
((i+Ceo)-
CeoWl) i .
Proof. Expand (I + CP)-IW2 as follows
(I + C P ) - ' W 2 --- ( I + CPo + CPoW1A )-Iw2
= ((I + CPo)(I + ( I + CPo)-ICPoW1 k ) ) - ' W 2
= ( I + (Z+ CPo)-1CPoW1A)-I(I+ C P o ) - ' W :.
Now apply Theorem 4.1 with T=(I+CPo)-aCPoW1 and S = ( I + C P o ) - I W 2 .
In the next theorem we will denote by [(I + CPo)-aW2
[]
~,(I + CPo)-ICPoW1] the matrix formed from
( I + CPo)-1W2 and 7 ( 1 + CPo)-ICPoW1
Theorem 4.3. Let C ~ S( Po), and let "y > O. Then
C stabthzes every P ~ I-[ and
sup ]l(I + CP)-IW2II < Y
II
P~H
If and only If
Ill(/+ CPo)- w2
7(1+ CPo)-ICPoW1] IIj< Y
Proof. From [1] we have that
c~a/
** i i ( i + C p o ) - l c p o w 1 L < l
(1)
Using Corollary 4.2, together with (1) and the fact that the ~ - n o r m of a transfer funcuon m a m x is
equal to the maximum ag-row-norm, we can write
C ~ •
and P~r/Isup
[ ( I + C P ) - 1W:[I < 7
¢=, ]I(I+CPo)-1CPoWIL<I
((I+CP°)-IW2)'
and max
'
1-
~
((I+CPo)-'CPoW1),
<3'
M Kharnrnash, J B Pearson / Robust disturbance rejecnon
96
**
CPo) -1 CPoW1) ' < 1
((I+
and
((I+CP°)-iW2)'
1-
[(I+ Ceo)
' W2
<7,
t=l,
. ,q
((I + CPo)-ICPoW1),
+ v ((1+ C e o ) - a C e o < ) ,
Ceo) -a
~
7(I+CPo)-~CPoW~]
<7
<v,
,=1,
. ,q
[]
Theorem 4.3 suggests a way to mlninuze the quantity supe ~ 17 II ( I + CP)- IW2 I[ subject to robust
stablhty, by which 3`op~ can be approached arbitrarily closely and a controller that achieves this can be
found Prowded robust stability can be achieved, the following simple iteratlve scheme accomphshes the
desired rmnimlzatmn'
Step 1 Solve using the methods of [3,7,8] the following.
nun
I(I + CPo)-ICPoW l[Le= a.
C ~ S ( Po)
If a ¢ 1 stop_ robust stability cannot be achieved and it is meaningless to try to achieve robust
performance.
Step 2. If a < 1, using the controller found in Step 1 set "/upper equal to
((1+ CPo)-aW2), J
max
l<_,<_q 1 - ((1+ Cpo)- ceow ),
Clearly 3`upper lS an upper b o u n d for 7opt- Since zero is a lower b o u n d to Yopt set 3'lowe~equal to zero.
Step 3. Set 333equal
man
I
to 2(3`upper + 3331.... )_ N o w solve, again using the methods m [3,7,8], the following
[(I + CPo) ~Wz 7(I + CPo)-'CPoW1]
= YLmp-
C~S(Po)
Step 4. If 7trap < 3`, set "[upper equal to 7; otherwise set ~%wer equal to ~ G o to Step 3.
It is clear that this iteration converges to 3`opt- Furthermore, when a 7 close enough to 3'opt has been
reached, a controller that achieves this 3' can be computed. The optlmazauon problems which must be
solved m steps 1 and 3 have been studied in [3,7,8] and numerical algonthms for thmr solution are
avadable The algorithms involve only linear p r o g r a m m i n g The solution of the resulting linear p r o g r a m
gives the value of the m i n i m u m n o r m and is easily u u h z e d to c o m p u t e the optimal controller
5. Underlying theory
This section will present a few lemmas and p r o p o s i u o n s that will be required in the proof of Theorem
4.1. We start with the following lemmas which will be used to prove Proposition 5.4.
Lemma 5.1. Let (sl,.
, S q ) ~ q such that max,s,>O
sequence ( )% } ~=l of real numbers defined as
k, = m a x ( s , ) ,
Xk=max{t,X~_l+s,},
k>_2,
t
converges. Furthermore, lira k ~ ~?t~ = m a x , s , / ( 1 - t, )
Also let (t I . . . . t q ) ~ R q with O<_t,<l. The
M Khammash, J B Pearson / Robust dtsturbance rejectton
97
(pl×q, X " " "YGolXqn
5 = ( z I. . . . . z , ) ~
L e m m a 5.2. Let R = ( R 1. . . . R , ) belong to *"-'TI
'~TI
" Also suppose
P N ( t q ~ ) × - - - XPN([q~ ). Then gwen e > 0 and f l = ( f l l . . . . . fl~) such that fl,>__ [[z,l[~ for t = l . . . . n,
there extst an mteger N > N and (zl . . . . Z,) ~ P~( gq~) × . . . ×P~({q~) such that
PN~., = Z , ,
1 = 1 .....
El,
t=l,....n,
II 5,[I ~ = fl,,
IIP~(R~e~ + - . . + R , ~ , ) I [ ~ >- II R~ II ~,/3~ + - . . + II R , II ~'/L - e
Proof. Without loss of generahty, we m a y assume that each of the R , ' s belongs to L P ~ ~ since otherwise
we can divtde an R, ~ £ , o ~ q , with q, > 1 into its q, c o m p o n e n t s each of wtuch belonging to .LP~-~~. Of
course we should also divide the corresponding z, ~ PN(Eq~,) into its q, c o m p o n e n t s in PN(d°°), as well as
duplicate the corresponding fl, to m a t c h the expanded c o m p o n e n t s of z, T h e fact that the n o r m of an
R, ~ . ~ - ( q ' ~s equal to the sum of the n o r m s of its individual c o m p o n e n t s in .5¢~-~~ makes this expansion
yield the desired results, as m a y be easily checked.
So, let r, = {r,(k))]°=0 ~ yl be the pulse response of R, for t = 1 . . . n T h e n for each such l, there
exists a positive integer, N, such that
l
I r ' ( k ) l<- 2 n ( N +el ) f l ,
V k >-N , ,
and
~
k=0
,E
Ir,(k) l _> IIR,
IP~¢- 2nil----7, W > N , .
N o w 57 and the Y,'s can be defined as follows:
57=N+
max Nj+I,
1~j<n
(z,(k)
~,(k) =/fl'
if0<k<N,
if N < k < N
and r , ( N -
k ) > 0,
-- fll
if N < k < N and r , ( . N - k )
0,
otherwise.
< O,
It follows from these deflnltmns that
PNZI = Z,,
I='1 ....
n,
and
]]5,]l~=fl,,
t=l,
..,n.
In addition, we have
z,(k)r,(~- k) +
k=0
E Z(k)r,(~-- k
t~l
> E k=N+,
~ e,(k)r,(57i
/t = N +
k=0~,(t,)r,(~7- k) [
-,
t=l
~-
n
1
N
fl,l~,(~7-k)l - E ~ Iz,(k)r,(~7-k)l
k=N+l
t/
t=l
k=0
max/N I
->Z
fl, I r , ( d ) I - 2~
d=0
t=l
fi, l r , ( N k~0
k) I
98
M Kharnmash, J B Pearson / Robust dtsturbance rejectton
(
~l
>-- ~ f l ,
i~1
n
IIe, l l j
2nfl,
-
IV
l=
(N+l)fl,
= ~/3,[Ie, lIj-e.
t=l
Hence, at follows that
IIP~(R~5, + - . . + R . 5 . ) I I . ~ >- Iln~ [l~,fl~ + " • + I [ R . I I _ ~ & - ~ ,
whxch completes the proof
[]
Lemma 5.3. Let T and S be m £~'~-~<q I f (v, w ) ~ P N ( E q ) × P N ( I q ) C ( q
× 8 q , then gwen any e>O,
fll >- II v I1~, and f12 ~-- II W 1[~ , there exist an mteger N > N and sequences (~, ~ ) ~ P ~ ( d ~ ) × P ~ ( g ~ )
sattsfylng
P~=v,
PN~'=w,
II~ll,,=Ba,
II)~/),~=B2,
l) P ~ ( S 6 + T~)]1 ~ >_ max { )lS, I)fl~ + Jl T, lift2) - e _
l <~t<~q
Proof. I m m e d m t e from L e m m a 5.2_
[]
Proposition 5.4. Let T and S both be t n ~ - i
sequences 4 ~-~C~'q and x ~ 8q such that
[Ix[I
×q
wtth T sattsfymg IIT [I ~ < 1• Then gwen e > O, there extst
II S, I1,,,
1 [ 4 1 1 ~ > _ _ em a x l _' l [ T , _i,
=1,
and
Ile,~ll~lle,(ax+Z~)ll~,
n = O , 1, 2,
Proof. Let e > 0 be given. We m a y assume the given e satisfies 0 < e < max, II S, [I _~, since proving the
proposition for such an e imphes ~ts vahdity for a larger one. We start by constructing two sequences of
truncated elements of {p'~, namely (x 1, x 2,
) and (~1, ~2, -- ), satisfying the following propertms
x, ~ P~,, ( { g ) ,
~,~PN,(lq)
P~x,+ l = x , ,
P,',4,+a=4,,
IlP,4,1l~ < l l P , ( a x , +
where N,+, > N,,
[Ix,[l~=l,
T4,)[l~,
n = O , 1,
lle~,ll~=X,-1,
.,
where
2M = max { II co, I[ 0~- e },
4 0 = 0,
t
X~=max{llT,[l~,Xk_l+
IIS, l [ ~ , - e } ,
k>~2
t
The construction of these sequences goes as follows" Define (x 0, ( 0 ) - = (0, 0 ) ~ P _ l ( g q ) × P l ( g q ) Next apply L e m m a 5.3 to (x0, 40) with fll = 1 and f12 = X0 to obtain an integer, N~ > 0, and (x a, 4a)
P~¢,(I~) X P ~ , ( ? ~ ) . N o w apply the same lemma to thas new pmr with fll = 1 and f12 = ~ to get an integer
N 2 > N 1 and a another pmr (x z, 42) ~ PN~(gq) × PN~(g~) Repeating this procedure mdefimtely yields the
two desired sequences It can be checked by mduct~on that these two sequences in fact satxsfy the
properties stated above,
N o w x and 4 can be defined as the pomtwlse hrrut of these two sequences, 1.e. let
x(k)=
limx,(k),
4(k)=
hm4,(k)
N o t e that since Vj >_t P ~ x , = PN, Xs and PN,~, = PN,~j, the hrmts as defined above exist Furthermore, we
have that
Piv x = X , ,
PN,~ = ~,,
l = O, 1,
. .
M Khammash, J B Pearson / Robust &sturbance rejectton
99
Tins implies that
lIe~ll~-<lle~(Sx+T~)]l~,
n=0,1,2,
.
The fact that for each ~, PNX = X, where x, ~ C~ with II x, II ~ -- 1 implies that x ~ Cq and
[[ x If ~ = 1. To show that ~ ~ t ~ , it suffices to show that hm, ~ ~2,, < oo in which case II @ If ~ = hm, ~ , ) , , .
To this goal, consider the sequence of real numbers m question, {~,},~1 which has been recursively
defined above According to L e m m a 5.1, this sequence converges, and its hmit is exactly equal to
max, {( II S, II ~ - 0 / ( 1 - It T, II ~,)} This completes that proof. []
Proposition 5.5. L e t x = { x ( i ) } L 0 and y = { y ( t ) } L o
be m ~ q . I f
II P,,x II ~o -< II e , y II ~, n = 0, 1, 2 . . . .
then there exists A E ~ . ~
q such that II A II -< l and A y = x.
Proof. The proposition holds trivially if y = 0: just pick A itself to be zero. So assume y 4= 0. To prove the
proposition we will construct a A that has the desired properties. W e start b y ~dentffylng a subset of the
y 0 ) ' s , call it y ( q ) , Y0z), - winch, depending on y, m a y or m a y not be fimte. This subset m a y be
defined b y l n d u c t m n in the following manner: Let q be the smallest integer such that y ( q ) =~ 0 On the
other hand, given Y 0 , ) let i,+~ be the smallest integer greater that t~ such that I Y(Z.+l) I ~ -> I Y(Z,) I~Using the x ( t ) ' s and y ( i , ) ' s we are now ready to construct A through specifying its pulse response matrix.
So let
Ao,o
A'=
Ao,1
A0,2
fltl,0
AI,1
ml 2
"'"
A2,o
A2~
A:,2
..
where
Aq,q .-~- [ X ( l l ) / Y ( l l ) ] ,
A q + l , q "..~- [X(/1 + 1 ) / y ( 1 1 ) ] . . . .
A,z,t,-= [x(12)/y(12)],
A
=
[x(,,)/y(,,)]
A,a+I Q -~- [ x ( / 2 q- 1 ) / y ( 1 2 ) ]
.
.
.
,
A 2_1, q := I x ( / 2 _
, k
1)/y(ll)],
1.,: .= [ x ( f 3 - 1 ) / y 0 2 ) ] '
.
and 0 otherwise
Notice that each row of any of these matrices has at most one nonzero element, which, by the choice of
the y ( i , ) ' s , will have its absolute value less than or equal to one.
A will have the form
/~11,11
Atl+l,i 1
A / 2 - I ,I l
A=
0
0
/~12 12
At 2 +1 12
A t~-l,t2
0
A,, ,~
M Khammash, J B Pearson / Robust disturbance rejectton
100
f r o m which it is easy to see that A is c a u s a l a n d t h a t A x = y . F u r t h e r m o r e , it follows f r o m the r e m a r k s
l m m e d x a t e l y f o l l o w i n g the d e f l n m o n of A t h a t
I1~11 = supll(&a
-)11-<1
,.a,,2
1
T i n s c o m p l e t e s the proof.
[]
4.1. [I T II ~e < 1 xs b o t h n e c e s s a r y a n d s u f f i c m n t for the reverse m the s t a t e m e n t of the
t h e o r e m to exist for all A c a u s a l a n d satzsfymg II AII < 1 [1]
T h e t h e o r e m h o l d s t n v m l l y if either T or S is zero So a s s u m e that they are b o t h n o n z e r o Let x ~ g q
b e a r b i t r a r y a n d d e f i n e v = (I + TA)- ~Sx_ It follows t h a t
Proof of Theorem
Sx=(I+
rA)y
S , x = y , + T,Ay,
l <t<q
[[S,l[~,[Ix[l~>-Ily, l[~-
[[ T` I[ J II AY I[ ~ >- [ [ Y , [ [ ~ - [I T, I [ ~ ¢ [ l Y [ I ~ ,
l_<t_<q
B u t for s o m e ~ = rn, it h o l d s t h a t ][ y,. I[ ~ = 11Y I[ ~- T h e r e f o r e
IIS., II ~,11 x II ~ >--II y ) l ~ ( 1 -
liT,,, II ~,)
whJch Implies t h a t
-
l[ Y II ~
[I Sm I[ ~
II St I[
<
< max
I [ x l [ ~ - 1 - 11T,,, I[~
l_<,_<q 1 -- lIT, I[a¢
W e have s h o w n t h a t for a n y c a u s a l A with II A [I < 1, m a x I <, < q II S, I1~ / ( 1 - II T, II ~ ) b o u n d s f r o m a b o v e
[ 1 ( I + Tzl)-~SII, a n d to p r o v e the t h e o r e m it is e n o u g h to show t h a t g i v e n e > 0, there exists a c a u s a l k
with II z~ II -< 1 such t h a t
IIS, I ) ~
l~<,_<q 1 - liT, [I .z.'
II(l+ Zz~)-~S[I > m a x
So let e > 0 b e given By P r o p o s m o n 5 4, there exist s e q u e n c e s ~ a n d x in ~
{Ixl[~ = 1,
II~ll~ >
max
a<_,<_q 1 -
II s, fl ~,
liT, I1~
such t h a t
e,
and
IIP.@II~IIP.(Sx+T~)LI~,
n=0,1,2,
.
So d e f i n e y = Sx + T~ It follows that
IlYl{,~ = IfSx+ Tfl[,~ > [l~l[ ~ >
I[S,l[~,
max
l_<,_<q 1 - liT, I1~,
M o r e o v e r , b y P r o p o s i t i o n 5.5 there exists a c a u s a l A w i t h l[ a II < 1 such that A y = --4- T h e r e f o r e we c a n
write
y=Sx-TAy
~
(l+TA)y=Sx
~
y=(I+TA)
~Sx.
Since II Y l[ ,~ > max1 <,_< q I[ S, II ~,/(1 - I[ 7], II ~,) - e a n d II x I[ ~ = 1, we h a v e t h a t
I 1 ( I + Ta )- ~S II >-- m a x
l_~,_<q 1 -
)1 S, II ~,
liT, II•
e,
[]
M Khammash, J B Pearson / Robust disturbance rejection
101
6. Conclusion
It has been shown how stability robustness a n d p e r f o r m a n c e r o b u s t n e s s c a n be i n c o r p o r a t e d together in
one design procedure when the p e r f o r m a n c e is m e a s u r e d b y the n o r m of the sensitivity function. A n
expression for the worst case n o r m of tins f u n c t i o n has b e e n derived a n d it has b e e n shown how it can be
m i n i m i z e d subject to robust stablhty constraints. T h e next step will be to investigate how sirmlar results
can be o b t a i n e d when loop functions other that the sensitivity f u n c t i o n are of interest. Finally, it should be
m e n t i o n e d that even though the p e r t u r b a t i o n s considered here were multlplicatlve p e r t u r b a t i o n s , the
s i t u a u o n is almost identical when additive p e r t u r b a t i o n s are assumed.
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