The H∞-optimization in locally convex spaces

International Journal of Stochastic Analysis, Aug 2018

In this paper, the ordinary H∞-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.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

http://downloads.hindawi.com/journals/ijsa/2002/590258.pdf

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?. [1] [2] [3] [4] Advances in ns Research Hindawi Publishing Corporation ht p:/ www.hindawi.com Advances in Hindawi Publishing Corporation ht p:/ www.hindawi.com bability and Statistics Hindawi Publishing Corporation ht p:/ www.hindawi.com Hindawi Publishing Corporation ht p:/ www.hindawi.com Hindawi Publishing Corporation ht p:/ www.hindawi.com The Scientiifc World Journal Hindawi Publishing Corporation ht p:/ www.hindawi.com International Journal of Combinatorics Hindawi Publishing Corporation ht p:/ www.hindawi.com Submit your manuscr ipts Journal of Mathematics Mathematical Pro blems gineering Discrete Nature and Society International Journal of Mathematics and Mathematical Sciences Journal of Discrete Mathematics ht p:/ w w.hindawi.com Journal of Francis , B.A. , A Course in L ?-Control Theory , Springer-Verlag, Berlin, Heidelberg, New York 1987 . Francis , B.A. and Zames , G. , On L ? -optimal sensitivity theory for SISO feedback systems , IEEE Trans. Auto . Contr. AC- 29 ( 1984 ), 9 - 16 . Francis , B.A. and Doyle , J.C. , Linear control theory with an L?-optimality criterion , SIAM J. Control and Optim . 25 ( 1987 ), 815 - 844 . Keulen , B.V. , L ? -control with measurement-feedback for linear infinite-dimensional systems , J. of Math. Sys., Estim. and Contr . 3 ( 1993 ), 373 - 411 . Nehari , Z. , On bounded bilinear forms , Ann. of Math. 65 ( 1957 ), 153 - 162 . Petrushev , P.P. and Popov , V.A. , Rational Approximation of Real Function , Cambridge University Press, New York, New Rochelle, Melbourne, Sydney 1987 . Zames , G. and Francis , B.A. , Feedback, minimax sensitivity, and optimal robustness , IEEE Trans. Auto . Contr. AC- 28 ( 1983 ), 585 - 601 . Volume 2014 Volume 2014 Volume 2014 Hindawi Publishing Corporation ht p:/ www .hindawi.com


This is a preview of a remote PDF: http://downloads.hindawi.com/journals/ijsa/2002/590258.pdf

Chuan-Gan Hu, Li-Xin Ma. The H∞-optimization in locally convex spaces, International Journal of Stochastic Analysis, DOI: 10.1155/S1048953302000102