LIDS-P-1545
April 1986
An Interpolation Problem Associated With Hf Optimal
Design In Systems With Distributed Input Lags
by
Gilead Tadmor!
Abstract
A variety of Hf optimal design problems reduce to interpolation of
compressed multiplication operators, f(s) ->rk(w(s)f(s)), where w(s) is a
given rational function and the subspace K is of the form K = H2 00(s)H 2 .
Here we consider 0(s) = (1-ea-S)/(s-a), which stands for a distributed delay
in a system's input. The interpolation scheme we develop, adapts to a
broader class of distributed lags, namely, those determined by transfer
functions of the form B(e S)/b(s), where B(z) and b(s) are polynomials and
b(s) = 0 implies B(e- s ) = 0.
Interpolation, HP
Key Words.
eigenvalues, Time domain analysis.
spaces,
Distributed
delays,
Maximum
/Dr. Gilead Tadmor, Massachusetts Institute of Technology, Laboratory for
Information and Decision Systems, Room 35-312, 77 Massachusetts Ave.,
Cambridge, MA 02139 (617)253-6173.
2
0. Notations
H2 (resp. Ha) is the space of L2 (resp. L,)
functions on the imaginary
axis which admit analytic continuation in the right half plane.
source on HP spaces is [7].
A good
By abuse of notations we shall not distinguish
between a function in H2 and its inverse Lapace transform in L2 [0,-] (e.g.
by "^" or avdz).
presentation.
Hopefully, this will simplify,
rather than obscure, the
A star will denote the adjoint of an operator (e.g. T*).
3
1. Introduction
A variety of H a design problems (see e.g. the surveys [6,8]) give rise
to "interpolation" problems of the following form:
rational function.
(For simplicity we assume throughout this note that w(s)
Let 0(s) be an inner function in H a and
is both stable and minimum phase.)
let K be the subspace K =: H280(s)H 2 .
multiplication by w(s).
interpolating
Consider now the compression to K of
That is, the operator
T:f(s) -4nK(w(s)f(s))
An
Let w(s) be a proper
for f(s)eK
function for
T is q(s)
e HF having
these
two
properties.
(i)
11TIh
(ii)
Tf(s) = TK(T(s)f(s)).
(= the operator norm of T) = IVl¥Iq
,
Note that, necessarily, if ¥(s)
(1.1)
interpolates T, it is of the form
w(s) + 0(s)h(s) for some h(s) in Ha.
Our setup is slightly, but not significantly, different:
Instead of an
inner function 0(s), we have the H a function (1-e'-S)/(s-o), with Rea>O.
One easily observes that the outer part of this function is He-invertible;
hence, our subspace K =: H2 0(1-ea- S)/(s-a)H2 is the same as that obtained by
substituting (1-ea-S)/(s-a) by its inner part. The advantage of the present
setup is that it facilitates the time domain analysis, as most of ours will
be.
The treatment of the present particular examples is motivated by a
larger class of functions 0(s); namely those of the form 0(s) = B(e- s)/b (s),
where both B(z) and b(s) are polynomials and b(s) = 0 implies B(e-S ) = 0.
4
Functions 0(s) in this class correspond to delay operations formed (in the
time domain) by cascades of elementary operators of those next three types (i)
f(t) -)f(t-1)
(ii)
f(t) -)f(t) - eaf(t-1)
(iii)
f(t) - fealf(t-v)dz.
Operators
of the
first type were treated
in
[1,2,3]
[4,5].
and
Inspired by those works, the present author developed the theory in [10], to
suit combination of operators of both types (i) and (ii) (i.e., the case
b(s) a
1),
solution.
and suggested also a simple computational scheme
for
the
Our aim here is to adapt those theory and scheme to suit non
constant b(s).
For simplicity, the analysis is done explicitly only for the
elementary building stone of type (iii).
Using the methodology of [10], the
reader will then be able to obtain the more general result.
For future use, let us give w(s) two specific forms. The first is
P
w(s)
1+
i Bi.
+s
i=1
Since w(s) is stable, Repi , i=1,...,p, all belong to the open right half
plane. The number q = lim w(s) could be scaled to be either 0 or 1. The
s-)300
trivial case w(s)
i
a
is excluded.
The second form is
W(s) =-
(s)
dTs)
5
where n(s) and d(s) are coprime polynomials.
Merely for simplicity in notations, we assume that all our system's
parameters (i.e., a,
A,
¥i and pi) are real numbers.
For the same reason,
we also treat only the case where AI- -.,ip are distinct.
can be easily dispensed.
Both assumptions
6
2. An Outline of the Solution
The underlying idea behind the solution (following [1,2,3] and [4,5])
is this next observation of Sarason [9, Proposition 5.1]:
Lemma 2.1 (Sarason).
If the operator T has a maximal function, say
f(s), then the unique interpolating function for T is ¥(s) =: Tf(s)/f(s).
Furthermore, that q(s) is a constant multiple of an inner function.
Thereby, one is lead to a search for a maximal function for T, or
equivalently, for T*T. Since (it is easy to see) T*T is the sum of R2 I (I,
being the identity operator) and a compact, infinite dimensional operator,
the spectrum of T*T consists of countably many eigenvalues, each of a finite
multiplicity, and of their only accumulation point, which is q2 .
follows from Weyl's Lemma (see e.g. Ill, p.32 5 Theorem 1]).
This
Our main effort
will go thereby into the search for a maximal eigenvalue and eigenfunction
for T*T.
In some cases, however, a maximal eigenvalue does not exist.
may yet occur that w(s) itself is an interpolating function.
Then it
(Note that by
definition, w(s) satisfies condition (l.1.ii), but mostly (1.1.i) fails and
IlW(S) I
>
I ITI].)
Here are sufficient conditions for either of the two
cases.
Proposition 2.1.
If
lw(jo) I<] H]
The spectral point t 2 is not an eigenvalue of T*T.
for all weR, then a maximal eigenvalue does not exist, yet
w(s) is an interpolating function for T. If, on the other hand, there exist
00>0 such that Iw(jw)I >
HI
eigenvalues are larger than
exist.
for all weR with ItI>wo0
q2.
then countably many
In particular, a maximal eigenvalue does
7
This
statement
sumnerizes Propositions
3.1
and
3.2
Observation 3.4 in [10], where one can also find the proof.
and part
of
(The present
generalization requires no significant changes.)
Henceforth
we
work
eigenfunctions of T*T.
on
characterization
of
eigenvalues
and
The analysis will be carried out in the time domain.
So our first task is to find out the exact form of the subspace K and of the
operators T and T*, following an inversed Laplace transform.
This is done
in the following section 3. In section 4 we then construct (in terms of the
number
a
and
({(X2 ):X2 >0}.
the
function w(s))
a parameterized
family
of matrices
It has the property that X2 is an eigenvalue if and only if
det G(X2 ) = 0. Given an eigenvalue X2 , a corresponding eigenfunction is
then obtained from a right annihilating vector for W(X2 ).
One thus finds the solution to the interpolation problem after a search
for the maximal X2 which solves det D(X2) = 0. This next simple observation
(see [10, Observation 3.4]) helps in limiting the domain of that search.
Proposition 2.2.
the interval (q2,
The maximal eigenvalue of T*T (if it exists) lies in
cIw(s)II2].
8
3. The Time Domain Setup.
A function fl in K
= (1-ea-S)/(s-a)H2 has this next time domain
representation
f1 (t)
=-
eag(t-),
for t-O
where g(t) belongs to L 2 [0,w) and is understood as the zero function for
negative values of the argument.
Thereby, a function fo belongs to K if and
only if for each g(t) in L 2 [0,~) the following holds
0= I
<f0 (t),
=
J <f
0
0
eacg(t-T)dv> dt
eaIf 0 (t+,)dT, g(t)>dt.
The subspace K is characterized, therefore, by the requirement
f(t+)d =
for all t>O.
(3.1)
Differentiation of (3.1) yields a difference equation
eaf0 (t+l) - f0 (t) = 0
Observation 3.1.
for all t>0.
A function fo(t)
(3.2)
in L 2 [O,-) belongs to K (i.e.,
9
satisfies Eq. (3.1) for all t0>)
if and only if Eq. (3.1) is satisfied for
t=0 and Eq. (3.2) holds for all positive t.
The proof is immediate.
The observation tells us that a function in K
is determined by its restriction to [0,1], and that restriction belongs to
the subspace of L2 10,1] which is orthogonal to eat.
We also obtain the
projection onto K along L 2 10,-) (denoted Kr
.)
Observation
3.2.
f0(t) = (nKf)(t).
(i)
fo(t)
Given
a
function
f(t)
in
L 2 (0,o),
set
Then
=
(1-e 2)
e-ia(f(t+i)
2ae
t eaf(T+i)d,)
i=0
for te[0,1), and
(ii)
f0(t+1) = e af (t)
for all t>1.
Proof.
A short, yet indirect proof goes as follows:
First one
observes that indeed, (i) and (ii) force fo(t) to satisfy the conditions in
Observtion 3.1.
Second, it easily follows that when f(t) belongs to K
(i.e., it satisfies Eqs. (3.1) and (3.2)) then the function fo(t) defined in
(i) is just f(t) itself.
A constructive proof which sheds light on the way
wK was obtained in the first place, can be found in [10, section 2].
Corollary 3.3.
Let X c L2 [0,1] be the subspace of functions which are
orthogonal to eat, with its inner product weighted by the scalar 1/(l-e(i.e. <fg>x = 1/(1-e- 2 )<fg>L [0 ,1].)
Then K is isometric to X.
2a ),
10
Proof.
Since a function in K is determined by its restriction to the
interval [0,1] (i.e., by its representative in X), one only has to check the
claim concerning the inner product, which follows from condition (3.2).
Now let us find the representation of T as an operator on X (which is
well defined, by the corollary):
become convolutions.
Under inverse Laplace transform, products
Hence the uncompressed operator of multiplication by
w(s) turns into
T¥i
f(t) -)g(t) = if(t) +
e
(3.3)
i=1
In order to obtain Tf(t) we have to project the right hand side of (3.3)
onto K. By Observation 3.2, this goes in two steps: First compute
(1-e-2a )
2
e-iag(t+i)
i=O
and then project the sum (as a function on [0,1]) onto the subspace X.
Since the domain of T is the subspace K, we have nK(Tqf) = if.
computation is thus needed only for terms of the form
Using Eq. (3.2), one gets
The
0eA(T-t)f(t)dt.
+i
e (T-t-i) f(z)dz
=
i-1
i-!
e(q-i)P-q CL
=
-eO
q=0=
qe(
-(+e-ia
eP(r-t)f(T)dr
fe'
f
)d
+ e-ia< eP(r-t)f(
O
+e
the other, a=P, being very
(Here we chose to treat only the case aif,
similar.)
eS(~-t)f(r)d~
The following summation yields
co
(1-e 2a) ~ e i a+e
i=0O
e-(a+)
(,-t-i)f(T)d, =
a (- +J
t)f(T)d
e
+
.
Now comes the projection onto X (i.e., -subtraction of
(3.4)
12
2A<k(), eat>e a
t
2al
where k(t) stands for the right hand side of (3.4)).
It yields (collecting all terms of Tf(t)) -
P
Tf(t) = if(t) +
¥
0p.
(r-t)
ii
e1
f( )dT
i=1
p
~
+
-(a+8i)
i(s-t)
e
e ¥i e-
i=1
f(e)dT
(3.5)
1-e
+ e2aei=1
i a.ii=1
1
e if(T)dT
i -a
e -e
(Since f(t)I[o,1] belongs to X we could drop terms involving fleatf(t)dt.)
As is born out from Observation 3.2 and Corollary 3.3,
one can compute
T* by first taking the L2 [O,1] adjoint of the operators described in Eq.
(3.4), and then project onto X.
In our example, these straightforward
computations yield a very simple form
p
T*f(t)= If(t) +
Yi
i=1
P (t-)
e
f(T)dT.
(3.6)
13
4. The Eigenvalue/Eigenfunction Problem
Here are system representations for the operators T and T*:
Let
xi(t) = -.ixi(t) + Tif(t)
p+ (t) = axp+1 (t)
p+1
g(t) = If(t) +
(4.1)
~ x i(t)
i=l
i+p+l(t) = 0iXi+p+l(t) - yig(t)
i=l,.
p
2p+1
xi (t ) ,
h(t) = ng(t) + ~
i=p+2
and impose the boundary constraints
xi(1) = eaxi(0)
P
xp+l
=
(0)
i-
xi(1) = 0
Observation 4.1.
i
-a
i
(4.2)
x (0)
1
i=p+2, ...,2p+1
Suppose f(t) belongs to X.
Then g(t) = Tf(t) and
14
h(t)
=
if
T*g(t)
and
only
if
f(t),
g(t),
h(t)
and
functions
xl(t),...,x2p+l(t) satisfy Eq. (4.1) and the boundary conditions (4.2).
Proof.
The "only if" direction is trivial.
(4.2) are met.
Then by simple algebra
-(a+fi )
xi(O) = e 1-e
for i=l,...,p.
Assume Eqs. (4.1) and
)
J
f ()dS
This insures, in turn, that g(t) = Tf(t) and, indeed, h(t) =
T*g(t).
The following nice and useful observation was made by Flammn in a
somewhat different setup [2,3], yet his ideas carry over to our case as
well.
Observation 4.2.
Suppose a positive number x2 is an eigenvalue of T*T,
and that f(t) e X is an associated eigenfunction.
the set of functions, each of the form
Let tl(t),...,42p(t) be
(t) = tqe t , where the numbers p is
a root of order (at least) q+1 of this next equation
X d(s)d(-s) - n(s)n(-s) = 0.
(Recall the notation w(s) = n(s)/d(s).)
(4.3)
Then f(t) is a linear combination
of the functions gi(t), i=1,...,2p and of eat.
Proof.
If f(t) is an eigenfunction of T*T then we can substitute
15
P+l
X2 f(t) for h(t), and
nf(t)
+ E xi(t) for g(t) in Eq. (4.1).
xi(t) and f(t) are analytic on [0,1].
Necessarily,
Taking the Laplace transforms of
these functions' analytic extensions to [0, ], the following equation is
obtained
'I'
(A -2 )f(s)
=
cT(sI -
0 :a :-1
0
,-]---- 81.
,
CT
+
(
)
B
X2p+l
f(s))
(4.4)
(Here CT is the (2 p+l)-raw vector (1,...,1).)
Further, direct computation
shows that the coefficient of f(s) on the right hand side of (4.4) is
w(s)w(-s)-n 2 .
f(s) =
Thus Eq. (4.4) is equivalent to
2
X2-w(s)w(-s)
[
])1
(45)
x.1(0)
l
1(
2Xp+l (0)
+
Finally, one notices that poles of the inverse matrix, other than at
s=a, cancel with poles of w(s) or with those of w(-s).
Thus, the poles of
f(s) belong to the set of zeros of Eq. (4.3), and the point a.
Now, given X2 and the question: is k2 an eigenvalue?, the observation
16
tells us that we can restrict our search to the 2p-dimensional subspace
l (t ) ,...,t)2p(X. }
V=:sp{eat,
since
if
f(t)
(Indeed, this subspace is 2p dimensional
v0 eat
+
viti(t)
E
belongs
to
X
then
vO = - 2a/(e 2 a-1) E <ti(I), eaU>.)
Let us thus consider the restriction of T to V. It ranges in the 3pdimensional subspace U =: sp(eat, tl(t),...,2p(t), e-I t,...,e pt) n X.
The
restriction
w =:
sp{eat,
of
T*
to
{l(t),...,42p(t)
'
U,
e-0
in
turn,
takes
) X.
t,...,epp t}
values
These
in
restricted
operators can be represented by 2px3p and 3px4p matrices, denoted
2p
P
1
1
12
2p
'
and . P
11
....... T1
2
31
In
these
representations,
32J
the
coefficients of (in that order)
of e±it,...,efPt.
2p,
3p
and
l1 (t),...-2p(t),
1
4p
vectors
describe
the
e-1t,...,ef-Pt and then
The coefficient of eat does not appear since it is always
determined by the others.
The following are important facts.
Observation 4.3.
2px2p identity.
Proof.
The matrix T22 is invertible.
Let us consider only that case where Eq. (4.3) has 2p distinct
roots pl,... ,2p-
exposition.)
The product T*1 T 1 is equal to X2I, where I is the
(The other case requires just a little more complicated
Then T1 and T*1 are these next diagonal matrices
17
TI
=
diag{w(jl),,...-,w(R2p)}
and
T1i
= diag{w(-1),...,w(-p)
(This follows from substituting
{i(t) for f(t) in Eq. (3.5).)
Since pi are
roots of Eq. (4.3), the first statement is true.
Similarly, one finds that T22 (the block which is responsible for the
transformation of the subspace sp{e- P t,...,ee-Pt} into itself, by T*) is
this next
T22
diaglw(P 1),...,w()p)
} .
By assumption, w(s) is minimum phase and P1 ...,fp are positive numbers.
Hence the second statement in the observation.
Theorem 4.4.
T1
and T*2.
From these facts we get
Fix X2 and the corresponding matrices T1, T2, Tl1, T22,
Then X2 is an eigenvalue of T*T if and only if this next
matrix
231TI + T32T2
is singular (i.e., det 0(X2 ) = 0).
If indeed, X2 is an eigenvalue and
18
V2=p
is a right annihilating vector for a(X2)
(i.e.,
C(X2 )v = 0),
then the
function
2p
f(t) =
S
2p
vii(t) - eat
i=1
t (.i(,)
2a
e2a-
ea>
(4.6)
-
e
i=1
is an associated eigenfunction for X2.
Proof.
Suppose
T*Tf(t) = X2 f(t).
0(X2 )v
=
0.
Then
following
the previous proof,
Suppose, on the other hand, that x2 is an eigenvalue and
f(t), a corresponding eigenfunction.
Observation 4.2 tells us that f(t) has
the representation (4.6), for some vector v.
Since the coefficients of
e-pat, i=l,...,p in T*Tf(t) are all zero, it follows that both the vector
T22T2 v and (T31 T 1 + T32T 2 )v must vanish.
Finally, since T22 is invertible,
the condition a(X2 )v = 0 is met.
In the theorem's statement we did not give the specific formulas for
the various components in 0(X2 ).
The interested reader will readily obtain
them from Eqs. (3.5) and (3.6)
The theorem gives rise to the main part in the interpolation scheme:
One computes det 0a( 2) for a decreasing sequence of X2 in (12,
I1w(s)112],
until he hits the highest solution of det 0(X2 ) = 0. This will be the
highest eigenvalue of T*T.
The corresponding eigenfunction's values in
19
[0,1] are then given by Eq. (4.6) (for v such that f(X2 )v = 0).
The values
of f(t) for t>1 are thereby determined via f(t+i) = e-iaf(t), which is Eq.
(3.2).
In order to find the interpolating function one has to compute also
Tf(t) (via Eq. 3.5) and the Laplace transforms of f(t) and Tf(t):
Observation 4.5.
Suppose A2 is the maximal eigenvalue for T*T, let
,1"',~t2p
be the roots of Eq. (4.3) (for simplicity we assume these are 2p
distinct numbers) and let v be a vector satisfying D(X2 )v
=
0. Then the
unique interpolating function, ¥(s), for T, is given by
2p
p
w(i)viti(s) +
. 2p
T(s) =: i=
-(P.+s)
zi
+ vo
Z e S-1
.
. ea-s-1
.
va.
viti()
+
-e
I
a-s
i=1
where the p-vector z is z = T 2 v, and the numbers zo and vo are obtained so
as to ensure that f(t) and Tf(t) be in X, i.e.
2p
vn
a
O
and
e2a-1
vi
e2a_ ! i=1
3
y0
i(t)eatdt
20
2a
2a--
Zo =
W(Fi)vit ti(t)eatdt
ziJ e
+
i dt.
i=1
The functions ti(s) are the Laplace transforms of 4i(t), i.e.
o
ti(s)
.1
e ts
t=-i ( t)dt
i-s)t
= [ ei
dt
'0
=
-e
ei - s
Ai- s
By Sarason's Lemma 2.1, above, q(s) = Tf(s)/f(s), where f(s) is
Proof.
the maximal eigenfunction.
By Eq. (3.2), the transform of a function f(t)
in K is
f(s) =
J
e-tSf(t)dt
e
i ( sa+ )
J
e-tsf(s)ds
i=0
_e-(a+s)
e-tf(s)ds .
So, it suffices to compute the transforms along [0,1].
The term l/1-e-( a+S )
appears in both Tf(s) and f(s), so it is canceled by devision.
The rest of
21
the claim follows by direct computation.
22
5. Concluding Remarks
Via
the
relatively
simple
example
of
0(s)
=
(1-ea- s )/(s- a) we
demonstrated the main steps in an interpolation scheme for a class of
transfer functions with distributed delays.
Key steps in the analysis were
the computation of the projection mK' hence of T and T*, in the time domain,
and characterization of maximal eigenvalues of T*T as zeros of Q(X2 ).
The
main difference of our case from the corresponding non distributed one
(i.e.,
0(s) = 1-ea S) is that here one has to look for the maximal
eigenfunction within the subspace X, instead of within the whole of L2[0,1].
The methods developed in [10],
bearing some corrections due to the
introduction of the counterpart of X, enable one to handle the more general
case of multiple distributed lags (i.e., 0(s) = B(e-s)/b(s), as described in
section 2).
Indeed, then life becomes more complicated with heavier
formulas and computations. This is due, in part, to the fact that the role
played in the present example, by some scalars (e.g., e-a ) is taken, in the
more general case, by matrices.
The unfortunate lack of cormutativity of
matrix multiplication does not help.
Acknowledgement
I wish to thank Professor Sanjoy K. Mitter and Dr. David S. Flamm for
interesting discussions and enlightening remarks concerning this work.
23
References
[11
D.S. Flanmmn and S.K. Mitter, "Progress on H a Optimal Sensitivity
for Delay Systems I: Minimum Phase Plant with Input Delay, 1
Pole/Zero Weighting Function", LIDS Technical Report P-1513, MIT,
Novenber 1985.
[2]
D.S. Flamm and S.K. Mitter, "Progress on H a Optimal Sensitivity
for Delay Systems II: Minimum Phase Plant with Input Delay, Genera
Rational Weighting Function", preprint, LIDS-MIT, January 1986.
[31
D.S. Flamm, Ph.D. Thesis, LIDS-MIT, to appear.
[4]
C. Foias, A. Tannenbaum, and G. Zames, "Weighted Sensitivity
Technical Report, Dept. of
Minimization for Delay Systems,"
Electrical Engineering, McGill University, September [1985].
[5]
C. Foias, A. Tannenbaum, and G. Zames, "On the H1-Optimal
Sensitivity Problem for Systems with Delays," Technical Report,
Dept. of Electrical Engineering, McGill University, December 1985.
[6]
B.A. Francis and J. Doyle, "Linear Control Theory with an H a
Optimality Criterion", Systems Control Group Report #8501,
University of Toronto, October 1985.
17]
J.B. Garnett, Bounded Analytical Functions, Academic Press, New
York, 1981.
[8]
J.W. Helton, "Worst Case Analysis in the Frequency Domain: The H a
Approach to Control", IEEE Trans. Auto Control, AC-30 (1985), pp.
1154-1170.
[91
D. Sarason, "Generalized Interpolation in Ho," Trans. AMS, 127
(1967), pp. 170-203.
[10]
G. Tadmor, "Hi Weighted Sensitivity Minimization for Systems with
Commensurate Input Lags", LIDS-Technical Report P-1546, MIT, March
1986.
[11]
K. Yosida, Functional Analysis, Springer, New York 1965.