#### The H∞-optimization in locally convex spaces

Journal of Applied Mathematics and Stochastic Analysis
THE H?-OPTIMIZATION IN LOCALLY CONVEX SPACES1
LI-XIN MA 0 1
0 CHUAN-GAN HU Nankai University Department of Mathematics Tianjin 300071 , P.R. China
1 Dezhou University Department of Mathematics Dezhou 253023 , Shandong , P.R. China
In this paper, the ordinary L?-control theory is extended to locally convex spaces through the form of a parameter. The algorithms of computing the infimal model-matching error and the infimal controller are presented in a locally convex space. Two examples with the form of a parameter are enumerated for computing the infimal model-matching error and the infimal controller.
1. Introduction
? J ? ? ? 0? ? ??
The subset of Z L? consisting of all real-rational functions of = and 0 is denoted by Z VL?.
Let ? J4? ? ? 0? ? ? oe ?4?0?, 0 ? ?8, 4 oe "? #. In Z L?, the order may be defined as
follows:
Definition 1.2: For any J"? J# ? Z L?, we call ?" ? ?# if ?"?0? ? ?#?0?, 0 ? ?8.
Definition 1.3: We call J ?=? 0? to be strong proper if J ?=? 0? ? Z VL? and
sup0??8 ? J ??? 0? ? ? ?, strictly strong proper if J ??? 0? ? !. We call J ?=? 0? to be
stable if J ?=? 0? ? Z VL? and J ?=? 0? has no poles in the closed right half-plane e?=? ? !
(for any fixed 0 in ?8?.
If J ?=? 0? is real-rational about = in e?=? ? !, then J ?=? 0? ? Z VL? if and only if J is
strong proper and stable (for any fixed B ? ?8?.
Similarly, we define the transfer function matrix
K?B? 0? oe ? XX"$??==?? 00?? X#?!=? 0? ?? O?=? 0? oe ? U?=? 0??
Then the model-matching problem is ? X"? ? ? 0? ? X#? ? ? 0?U? ? ? 0?X$? ? ? 0? ? ? oe
minimum in the sense of Definition 1.2, where X4?4 oe "? #? $? ? Z VL?, and O represents a
controller in Z VL?.
We shall give the algorithms of computing the model-matching error ? and the optimal
controller U in the form of the parameter case.
2. The Minimal Realization
Definition 2.1: A scalar-valued parameter function X ?=? 0? in Z VL? is inner if
X ? ? =? 0?X ?=? 0? oe "
?#?"?
and outer if it has no zeros in e?=? ? !. The zeros of an inner-function all lie in e?=? ? !,
the number of its zeros is called degree.
Theorem 2.1: If scalar-valued parameter function X ?=? 0? ? Z VL?, then
?3? there is a factorization X ?=? 0? oe X3?=? 0?X9?=? 0?, where X3 is an inner
function, X9 is an outer function;
?33? ? X3?4=? 0? ? oe " for any fixed 0 ? ?8;
?333? if X ?4=? 0? ? ! for all = in ?!? ?? and any fixed 0 in ?8, then X9?"?=? 0? exists and
X9?"?=? 0? ? Z VL?.
Proof: ?3? Let X3?=? 0? be the product of all factors of the form ?+?0? ? =???+?0? ? =?,
here +?0? ranges over all zeros of X ?=? 0? in e?=? ? !, counting multiplicities, being a
polynomial of 0, and define X9 oe X ?=? 0??X3?=? 0?. Then X3?=? 0? and X9?=? 0? are inner and
outer respectively, and X ?=? 0? oe X3?=? 0?X9?=? 0?.
?33? From ?3? we derive X3? ? =? 0?X3?=? 0? oe ". Particularly, if = oe 4=, then
X3? ? 4=? 0?X3?4=? 0? oe ". Thus X3?4=? 0?X3?4=? 0? oe ", i.e. ? X3?4=? 0? ? oe ".
?333? If X ??? 0? ? ! and X ?4=? 0? has no zeros on the imaginary axis for any fixed
0 ? ?8, then so is X9?=? 0?. Consequently, X9?"?=? 0? ? Z VL?? Q.E.D.
Suppose that
O?-? oe -8 ? O"-8?" ? ? ? O8 oe !
?#?#?
is the characteristic equation of a matrix E?0?, and that -"? -#? ?? -8 are the characteristic
roots of E?0?, where - is a polynomial of 0.
The matrix
?
G ?0? oe ??
!
!
?
? ? O8
"
!
?
? O8?"
!
"
?
? O8?#
has rank 8, denoted by ?E?0?? G ?0??.
Definition 2.4: Given an < ? 7 matrix K?=? 0? whose elements are rational functions of =
(for any fixed 0 in ?8?, we wish to find matrices E?0?, F?0? and G ?0? depending on 0,
having dimensions 8 ? 8, 8 ? 7 and < ? 8, respectively, such that
K?=? 0? oe G ?0??=M8 ? E?0???"F?0?
where M8 is the unit matrix of order 8. ?E?0?? F?0?? G ?0?? !? is termed a realization of K?=? 0?
of order 8. All the above realizations will include matrices K?=? 0? having the least
dimensions be called the minimal realizations.
? ? ?
Theorem 2.2: If ?E?0?? F?0?? G ?0?? !? is c.c. or c.o., then so is ?E ?0?? F ?0?? G ?0?? !??,
? ? ?
where E ?0?, F ?0? and G ?0? are of algebraical equivalence via a square matrix,
respectively.
Proof: Using the algebraic relation of equivalence among matrices, there is a square
matrix T ?0? such that
? ? ?
E ?0? oe T ?0?E?0?T ?"?0?F ?0? oe T ?0?F?0?G ?0? oe G ?0?T ?"?0??
?#?$?
The
?#?%?
?#?&?
?#?'?
?#?(?
? ? ? ? 8?"
rank?F ?0? E ?0?F ?0??E
be the monic least common denominator of all element <34?=? 0?, and let
<?=? 0?V?=? 0? oe =;?"V!?0? ? =;?#V"?0? ? ? ? V;?"?0??
?#?)?
where V3?0? is a constant < ? 7 matrix depending on a parameter 0. Then a realization of
V?=? 0? is
?
E?0? oe ??
!
!
M7
!
!
M7
? ? 1;?0?
? 1;?"?0?
Finally, combining (2.11) and the expression for G ?0? in (2.9), we have
G ?0??=M7 ? E?0???"F?0?
oe ?V;?"?0? V;?#?0??V!?0???M7 =M7?=;?"M7??<?=? 0?
oe V?=? 0??
?#?*?
?#?"!?
?#?""?
It follows that
using the series of ?/E?0?>?.
Consider the product
3. Lyapunov Equations
The Lyapunov equations are
Define the two controllability and observability gramians:
E?0?P-?0? ? P-?0?EX ?0? oe F?0?FX ?0?
EX ?0?P9?0? ? P9?0?E?0? oe G X ?0?G ?0??
?
P-?0? oe ' /?E?0?>F?0?FX ?0?/?EX ?0?>.>
!
?$?"?
?$?#?
?
P9?0? oe ' /?EX ?0?>G X ?0?G ?0?/?E?0?>.>?
!
Definition 3.1: A matrix E?0? is said to be antistable if all the eigenvalues of E?0? are in
e?=? ? !.
Theorem 3.1: If E?0? is antistable, then P-?0? and P9?0? are the unique solutions of
?$?"? and ?$?#?, respectively.
Proof: From the definition of P-?0? we derive
E?0?P-?0? ? P-?0?EX ?0?
?
oe ' ?E?0?/?E?0?>F?0?FX ?0?/?EX ?0?> ? /?E?0?>F?0?FX ?0?/?EX ?0?>EX ?0??.>
!
?
oe ' .?/?E?0?>F?0?FX ?0?/?EX ?0?>?
!
oe F?0?FX ?0? ? lim ?/?E?0?>F?0?FX ?0?/?EX ?0?>??
> ? ?
lim ?/?E?0?>F?0?FX ?0?/?EX ?0?>? oe !?
> ? ?
Since E?0? is antistable,
Then P-?0? is the solution of (3.1).
Proof the uniqueness of P-?0? is as follows. If E?0? and F?0? are 8 ? 8 and 7 ? 7 matrices, having characteristic roots -3?0?, ?3?0? and vectors A3?0?, C3?0?, respectively, then
?E?0? ? F?0???A3?0? ? C3?0??
oe E?0?A3?0? ? F?0?C3?0?
oe -3?0?A3?0? ? ?4?0?C4?0?
oe -3?0??4?0?A3?0? ? C4?0??
So, the characteristic roots of E?0? ? F?0? are
-3?0??4?0?, 3 oe "? #? ?? 8 and 4 oe "? #? ?? 7?
E?0?\?0? ? \?0?F?0? oe G ?0??
be a matrix equation, where \?0? and G ?0? are 8 ? 7 matrices.
If E?0? is an 8 ? 8 matrix and \?0? is an 8 ? 7 matrix, then the matrix equation
E?0?\?0? oe G ?0? can be written as the form
?E?0? ? M7?\?0? oe G ?0??
\?0? oe ?\""? ?? \"7? ?? \8"? ?? \87?
?$?$?
?$?%?
is the column 78-vector formed from the roots of \?0? taken in order.
Similarly \?0?F?0? oe G ?0? can be written as
Using (3.4) and (3.6), equation (3.3) can be written as the form
?M8 ? FX ?0??\?0? oe G ?0??
?E?0? ? M7 ? M8 ? FX ?0??\?0? oe G ?0??
Let H?0? oe E?0? ? M7 ? M8 ? FX ?0?. Thus
H?0?\?0? oe G ?0??
The solution of (3.8) is unique if and only if the 78 ? 78 matrix H?0? is nonsingular. To
find the condition for this to hold, consider
?M8 ? %E?0?? ? ?M7 ? %FX ?0?? oe M8 ? M7 ? %H?0? ? %#E?0? ? F?0?
which has characteristic roots
?M ? %-3?0???" ? %?4?0?? oe " ? %?-3?0? ? ?4?0?? ? %#-3?0??4?0??
It follows by comparing terms in % that H?0? has characteristic roots -3?0? ? ?4?0?,
3 oe "? #? ?? 8 and 4 oe "? #? ?? 7. Hence, H?0? is nonsingular if and only if there are no
characteristic roots of E?0? and F?0? such that -3?0? ? ?4?0? oe !, and this is the condition
for the solution \?0? of matrix equation (3.3) to be unique. Because E?0? is antistable, the
characteristic roots -3?0? ? -3?0? of E?0? ? EX ?0? are not zero. Consequently, the solution
of the Lyapunov equation (3.1) is unique.
P9?0? is the unique solution of (3.2) with similar proof above. Q.E.D.
4. Infimal Model-Matching Error
Define and So and
0 ?=? 0? oe ?E?0?? A?0?? G ?0?? !??
1?=? 0? oe ? ? EX ?0?? -?"?0?P9?0?A?0?? FX ?0?? !?
\?=? 0? oe V?=? 0? ? -?0?0 ?=? 0??1?=? 0??
0 ?=? 0? oe G ?0??=M ? E?0???"A?0? ? Z VL#?
1?=? 0? oe FX ?0??=M ? EX ?0???"-?"?0?P9?0?A?0? ? Z VL#?
Definition 4.1: Let ??0? denote the infimal model-matching error:
??0? oe inf? ? X"? ? ? 0? ? X#? ? ? 0?U? ? ? 0?X$? ? ? 0? ? ?? U ? Z VL???
?$?'?
?$?(?
?$?)?
?%?"?
?%?#?
A matrix U in Z VL? satisfying ??0? oe ? X"? ? ? 0? ? X#? ? ? 0?U? ? ? 0?X$? ? ? 0? ? ? is called
optimal.
If X3?=? 0? are scalar-valued, then there is no need for both X#?=? 0? and X$?=? 0?. So we
may as well suppose X$?=? 0? oe ". It is also assumed that X#?"?=? 0? ? Z VL? to avoid the
trivial instance of the problem.
Returning to the model-matching problem, bringing in an inner-outer factorization of
X#?=? 0?? X#?=? 0? oe X#3?=? 0?X#9?=? 0?, we have
? X"? ? ? 0? ? X#? ? ? 0?U? ? ? 0? ? ?
oe ? X#3? ? ? 0??X#?3 "? ? ? 0?X"? ? ? 0? ? X#9? ? ? 0?U? ? ? 0?? ? ?
oe ? X#?3 "? ? ? 0?X"? ? ? 0? ? X#9? ? ? 0?U? ? ? 0? ? ?
oe ? V? ? ? 0? ? \? ? ? 0? ? ??
?%?%?
Theorem 4.1: The infimal in ?%?$? is achieved if X#?=? 0? has no zeros on the extended
imaginary axis. In this case, the optimal U?=? 0? is determined by the following property?
X"?=? 0? ? X#?=? 0?U?=? 0? is a scalar multiple of an inner function of degree less than the
number of zeros of X#?=? 0? in e?=? ? ! ?for any fixed 0 in ?8?.
Proof: Suppose F" oe Z L V? and F oe Z VP?, then from Proposition A 2.2 in [7], the
infimum in (4.4) is achieved.
?
Assume \ ?=? 0? oe X"?=? 0? ? X#?=? 0?U?=? 0?, then using (4.4) we have
?
? \ ? ? ? 0? ? ? oe ? \? ? ? 0?X#?3 "? ? ? 0? ? ??
?
Consequently, the minimization of \ ?=? 0? can be accomplished by minimizing
? ?
\ ?=? 0?X#?3 "?=? 0? and multiplying the result by X#?3 "?=? 0?. Now as \ ?=? 0?X#?3 "?=? 0? is
analytic in e?=? ? !, except for the poles of X#?3 "?=? 0?, which are ,4, 4 oe "? ?? <
?
(depending on 0?. So \ ?=? 0?X#?3 "?=? 0? must have a continuation to the entire plane,
?
with poles at ,4 and ? ,4. Therefore, \ ?=? 0?X#?3 "?=? 0? is rational, and has the form
?
\ ?=? 0?X#?3 "?=? 0? oe G ?0? ##4738oeoe""??==??,-43??00??????==??-,34??00????
?%?&?
where 7 ? <, e?-3?0?? ? !, G ?0? ? !.
As X#3?=? 0? is an inner function, obviously rational, so \?=? 0? is also rational.
? ? ?
Since ? \ ?4=? 0? ? oe ? a.e., \ ?=? 0? is inner and the zeros of \ ?=? 0? must be among -3 in
(4.5). Q.E.D.
Definition 4.2: The Z P: space, " ? : ? ?, will be viewed as :th power integrable
functions about = and 0. When : oe ?, Z P? is the space of essentially bounded functions
(for any fixed 0 in ?8?.
Definition 4.3: The Z VP: space, " ? : ? ?, will be viewed as a subset of Z P:, which
consists of all real-variational functions of = and 0.
Definition 4.4: Let J ?=? 0? ? Z P? and 1?=? 0? ? Z P#. Then the operator
AJ ?=?0?? AJ ?B?0?1?=? 0? oe J ?=? 0?1?=? 0?
is called the Laurent operator.
A related operator is AJ ?=?0? ? Z L#, the restriction of AJ ?=?0? to Z L#, which maps Z L# to
Z P# where J ?=? 0? ? Z P?.
For J ?=? 0? in Z P?, the Hankel operator with symbol J ?=? 0?, denoted by >J ?=?0?, maps
Z L# to Z L#? and is defined as
>J ?=?0?? oe #"AJ ?=?0? ? Z L#?
where Z P# oe Z L# ? Z L#?, and C" is the projection from Z P# onto Z L#?.
Using a similar method to the classical methods we have the following conclusion:
Theorem 4.2: There exists a closest Z VL?-function \?=? 0? to a given Z VP?-function
V?=? 0?, and ? V? ? ? 0? ? \? ? ? 0? ? ? oe ? >V???0? ? .
From Section 3, a factor V?=? 0? can be written as V"?=? 0? ? V#?=? 0? with V"?=? 0?
strictly proper and analytic in e?=? ? ! and V#?=? 0? in Z VL?. Then V"?=? 0? has the
minimal state-space realization
V"?=? 0? oe ?E?0?? F?0?? G ?0?? !??
And from Section 3, with @?0? oe -?"?0?P9?0?A?0? we derive
and
P-?0?@?0? oe -?0?A?0?
P9?0?A?0? oe -?0?@?0??
Theorem 4.3: The infimal model-matching error ??0? equals ? >V? ? ? 0? ? and the
unique optimal \ equals V?=? 0? ? -?0?0 ?=? 0??1?=? 0??
Proof: From Theorem 4.2, we derive that there is a function \?=? 0? in Z L? such that
It is claimed that every \?=? 0? in Z L? satisfying (4.8) also satisfies
But (4.9) has a unique solution \?=? 0? oe V?=? 0? ? -?0?0 ?=? 0??1?=? 0?. We know that
? V? ? ? 0? ? \? ? ? 0? ? ? oe ? >V???0? ? ?
V?=? 0? ? \?=? 0?1?=? 0? oe >V?=? 0?1?=? 0??
>V?=?0?1?=? 0? oe -?0?0 ?=? 0?
holds. In fact, add and subtract =P-?0? on the left-hand side in (3.1) to get
? ?=M ? E?0??P-?0? ? P-?0??=M ? EX ?0?? oe F?0?FX ?0??
Now pre-multiply by G ?0??=M ? E?0???" and post-multiply by ?=M ? EX ?0???"@?0? to get
? G ?0?P-?0??=M ? EX ?0??@?0? ? G ?0??=M ? E?0???"P-?0?@?0?
oe G ?0??=M ? E?0???"F?0?FX ?0??=M ? EX ?0???"@?0??
The first function on the left-hand side belongs to Z L#; from (4.1) and (4.6), the second
function equals -?0?0 ?=? 0?; and from (4.2) and (4.6), the function on the right-hand side
equals V"?=? 0?1?=? 0?. Project both sides of (4.12) onto Z VL#? to get
-?0?0 ?=? 0? oe #"V"?=? 0?1?=? 0? oe >V"?=?0?1?=? 0??
?%?'?
?%?(?
?%?)?
?%?*?
?%?"!?
?%?""?
But >V"?=?0? oe >V?=?0?? hence (4.10) holds. It follows that (4.10) and Theorem 4.2 imply
??0? oe -?0?. There is \?=? 0? oe V?=? 0? ? ??0?0 ?=? 0??1?=? 0?. Set ??0? oe -?0? and
U?=? 0? oe X#?"?=? 0?\?=? 0??
?%?"#?
Since X A#9?=? 0? and X#?9"?=? 0? ? Z VL?, (4.12) sets up the one-to-one correspondence
between functions U?=? 0? in Z VL? and functions \?=? 0? in Z VL?. The optimal \?=? 0?
yields the optimal U?=? 0? via (4.9).
5. Steps of Computation
From Section 2 through Section 4, we derive that in the form of parameter valued case, the
steps in the design procedure on the L?-optimization in locally convex spaces are as follows:
Step 1: Do an inner-outer factorization
Step 2: Define
and find a minimal realization
Step 3: Solve the equations
X#?=? 0? oe X#3?=? 0?X#9?=? 0??
V?=? 0? oe X#?3 "?=? 0?X"?=? 0?
V?=? 0? oe ?+?0?? F?0?? G ?0?? !? ? ?a function in Z VL???
E?0?P-?0? ? P-?0?EX ?0? oe F?0?FX ?0?
EX ?0?P9?0? ? P9?0?E?0? oe G X ?0?G ?0??
and
and
Step 4: Find the maximum eigenvalue -# of P-?0?P9?0? and a corresponding eigenvector
A?0?.
Step 5: Define
0 ?=? 0? oe ?E?0?? A?0?? G ?0?? !?
1?=? 0? oe ? ? EX ?0?? -?"?0?P9?0?A?0?? FX ?0?? !?
\?=? 0? oe V?=? 0? ? -?0? 01??==??00?? ?
Step 6: Set ??0? oe -?0? and U?=? 0? oe X#?9"?=? 0?\?=? 0??
For a single-input and single-output design in the form of parameter valued case, we have
a similar to ordinary computing method.
Example 1:
T ?=? 0? oe ??= ? "??= ? #?????= ? "??=# ? = ? " ? 0#?? ? Z VL?, =" oe 0.01, % oe 0.1.
? T?=? 0? oe R?=? 0??Q?=? 0??
P- oe ? ?(#%?)"?#("( ?$'%?)%?*(( ?, P9 oe ? ""??$# ""??%$ ??
E oe ? !" #! ?? F oe ? ?"##%??%"*#" ?? G oe c " " d?
respectively.
Step (3)
P- oe ? ??#%?)??"#"?#????"?#""#"??"#%"*?? ???")#??"$#??????%"*#?" ?%*%?*? ?, P9 oe ?""??#$ ""??$% ??
0.0044
P-P9 oe ? 0.0031
? 0.0025 "
? 0.0017 ?? - oe 0.05113, A oe ? ? 0.7209 ??
Step (5) 0 ?=? oe ?0.2791= ? 1.2791????= ? "??= ? #???
1?=? oe -?"? ? 0.0141= ? 0.0657)???= ? "??= ? #???
\?=? oe 6.15??= ? "??= ? #??= ? !?%?????"!= ? "?#?= ? %?''???
Step 5: U"?=? 0? oe ? 6.15??= ? 0.4??=# ? = ? " ? 0#?????= ? "?#?= ? %?''???
Step 6:
0.615?=?!?%??=?"??=#?=?"?0#?
O?=? 0? oe ?=%?6.145=$?12.54=#?13.53=?0.0232).
Note O?=? 0? ? VL?, but O?=? 0? ? Z VL?.
Example 2:
Step 1:
Step 2:
Step 3:
Step 4: When 5 oe ",
T ?=? 0? oe ??="?!"=???="#???==??#"??00##?? ? Z VL?, =" oe 0.01,% oe 0.15?
? T ?=? 0? oe R ?=? 0??Q ?=? 0??
R ?=? 0? oe ? T ?=? 0?, Q ?=? 0? oe " oe \?=? 0?, ] ?=? 0? oe !?
[ ?=? 0? oe ?= ? "???"!= ? "?.
Step (2) V?=? 0? oe ? ??= ? "??= ? # ? 0#?????"!= ? "??= ? # ? 0#???
then E?0?? F?0? and G ?0? in the minimal realization of V?=? 0? are E?0? oe
# ? 0#, F?0? oe ? #??# ? 0#??$ ? 0#????"!0# ? #"? and G ?0? oe ", respectively.
Step (3)
P-?0? oe #?#?0#??$?0#?# "
?"!0#?#"?# , P9?0? oe #?#?0#? ?
Step (4) P-?0?P9?0? oe ?# ? 0#?#??"!0# ? #"?#?
then
Step (5)
\?=? 0? oe ?=??=#??#0?#?0?#"?!?=*?=?"?*?0"#!?0#"?)?#"? ?
??0? oe "!$0?#?0##" , U?=? 0? oe ? ?"*!?==?#?"?=??"!"0?# ?0##?"? .
*?=#?=?"?0#?
U"?=? 0? oe ? ?"!=?"??"!0#?#"??=?"? ?
*?=#?=?"?0#??=?"?
O?=? 0? oe ?"!=?"???"!0#?#"?=#??""!0#?##?=??"*0#?$*?? .
Note O?=? 0? ? VL?, but O?=? 0? ? Z VL?.
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