Namespaces
Variants
Actions

Difference between revisions of "Frequency theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
 +
<!--
 +
f0417101.png
 +
$#A+1 = 47 n = 0
 +
$#C+1 = 47 : ~/encyclopedia/old_files/data/F041/F.0401710 Frequency theorem
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
 +
 +
{{TEX|auto}}
 +
{{TEX|done}}
 +
 
A theorem that states conditions for the solvability of the Lur'e equations in control theory:
 
A theorem that states conditions for the solvability of the Lur'e equations in control theory:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f0417101.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1 }
 +
P  ^ {*} H + H P + h h  ^ {*}  = G ,\ \
 +
H q - h \kappa  = g ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f0417102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f0417103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f0417104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f0417105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f0417106.png" /> are given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f0417107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f0417108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f0417109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171010.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171011.png" /> matrices respectively, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171013.png" /> are the required <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171015.png" /> matrices. The Lur'e equations have two other equivalent forms: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171016.png" />,
+
where $  P $,  
 +
$  G = G  ^ {*} $,  
 +
$  q $,  
 +
$  g $,  
 +
$  \kappa $
 +
are given $  n \times n $,  
 +
$  n \times n $,  
 +
$  n \times m $,  
 +
$  n \times m $,  
 +
and $  m \times m $
 +
matrices respectively, and $  H = H  ^ {*} $,  
 +
$  h $
 +
are the required $  n \times n $
 +
and $  n \times m $
 +
matrices. The Lur'e equations have two other equivalent forms: If $  \mathop{\rm det}  \kappa \neq 0 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2 }
 +
H Q _ {0} H + ( P _ {0}  ^ {*} H + H P _ {0} ) + G _ {0}  =  0 ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171019.png" />, and in the general case
+
where $  Q _ {0} = Q _ {0}  ^ {*} \geq  0 $,  
 +
$  G _ {0} = G _ {0}  ^ {*} $,  
 +
and in the general case
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3 }
 +
2  \mathop{\rm Re}  x  ^ {*} H ( P x + q \xi ) = {\mathcal G} ( x , \xi )
 +
- | h  ^ {*} x - \kappa \xi |  ^ {2} \ \
 +
( \forall x , \xi ) ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171021.png" /> is a given Hermitian form of two vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171023.png" />;
+
where $  {\mathcal G} ( x , \xi ) $
 +
is a given Hermitian form of two vectors $  x \in \mathbf C  ^ {n} $,  
 +
$  \xi \in \mathbf C  ^ {m} $;
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171024.png" /></td> </tr></table>
+
$$
 +
{\mathcal G} ( x , \xi )  = x  ^ {*} G x + 2
 +
\mathop{\rm Re}  ( x  ^ {*} g \xi ) + \xi  ^ {*} \Gamma \xi ,
 +
$$
  
Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171028.png" />.
+
Moreover, $  \Gamma = \kappa  ^ {*} \kappa \geq  0 $,  
 +
$  G _ {0} = g \Gamma  ^ {-} 1 g  ^ {*} - G $,  
 +
$  P _ {0} = P - g \Gamma g  ^ {*} $,  
 +
$  Q _ {0} = q \Gamma  ^ {-} 1 q  ^ {*} $.
  
Let the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171029.png" /> be controllable: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171030.png" />. Then the Lur'e equations reduce to the case where
+
Let the pair $  \{ P , q \} $
 +
be controllable: $  \mathop{\rm rank}  \| q , Pq \dots P  ^ {n-} 1 q \| = n $.  
 +
Then the Lur'e equations reduce to the case where
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171031.png" /></td> </tr></table>
+
$$
 +
=   \mathop{\rm diag}  [ \lambda _ {1} \dots \lambda _ {h} ] ,\ \
 +
\lambda _ {j} + \lambda _ {h} \neq 0 ,\  \lambda _ {j} \in \mathbf R .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171032.png" /> and all the matrices are real, the Lur'e equations in scalar notation take the form
+
If $  m = 1 $
 +
and all the matrices are real, the Lur'e equations in scalar notation take the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171033.png" /></td> </tr></table>
+
$$
 +
\sum _ { k= } 1 ^ { n }  q _ {k}
 +
\frac{h _ {j} h _ {k} }{\lambda _ {j} + \lambda _ {k} }
 +
- h _ {j} \sqrt \Gamma  = \gamma _ {j} ,\  j = 1 \dots n ;
 +
$$
  
here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171034.png" /> is the required vector.
+
here $  h = [ h _ {1} \dots h _ {n} ] $
 +
is the required vector.
  
 
The frequency theorem asserts that for the Lur'e equations to be solvable it is necessary and sufficient that
 
The frequency theorem asserts that for the Lur'e equations to be solvable it is necessary and sufficient that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171035.png" /></td> </tr></table>
+
$$
 +
{\mathcal G} [ ( i \omega I - P )  ^ {-} 1 q \xi , \xi ]  \geq  0
 +
$$
  
for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171038.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171039.png" /> is the identity matrix). The frequency theorem also formulates a procedure for determining the matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171041.png" /> and asserts that if
+
for all $  \xi \in \mathbf C  ^ {m} $,  
 +
$  \omega \in \mathbf R  ^ {1} $,  
 +
$  \mathop{\rm det}  \| i \omega I - P \| \neq 0 $(
 +
$  I $
 +
is the identity matrix). The frequency theorem also formulates a procedure for determining the matrices $  H $
 +
and $  h $
 +
and asserts that if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171042.png" /></td> </tr></table>
+
$$
 +
\mathop{\rm det}  \Gamma  \neq  0 ,\  \mathop{\rm det}  \| i \omega I -
 +
P \|  \neq  0 ,\  {\mathcal G} [ \| i \omega I - P \|  ^ {-} 1 q \xi ,\
 +
\xi ]  > 0
 +
$$
  
(for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171043.png" />, and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171044.png" />), then there exist (unique) matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171046.png" /> such that (except for the case of equation (3)) the following is true: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041710/f04171047.png" /> is a Hurwitz matrix (see [[#References|[3]]]).
+
(for all $  \xi \neq 0 $,  
 +
and all $  \omega $),  
 +
then there exist (unique) matrices $  H $
 +
and $  h $
 +
such that (except for the case of equation (3)) the following is true: $  P + q \kappa  ^ {-} 1 h  ^ {*} $
 +
is a Hurwitz matrix (see [[#References|[3]]]).
  
 
The Lur'e equations in the form (2) are also sometimes called the matrix algebraic Riccati equation. The frequency theorem is used when solving problems on absolute stability [[#References|[2]]], [[#References|[4]]]–[[#References|[6]]], control and adaptation (see, for example, [[#References|[7]]]–[[#References|[9]]]).
 
The Lur'e equations in the form (2) are also sometimes called the matrix algebraic Riccati equation. The frequency theorem is used when solving problems on absolute stability [[#References|[2]]], [[#References|[4]]]–[[#References|[6]]], control and adaptation (see, for example, [[#References|[7]]]–[[#References|[9]]]).
Line 41: Line 114:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Lur'e,  "Some non-linear problems of the theory of automatic control" , Moscow-Leningrad  (1951)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.M. Popov,  "Hyperstability of control systems" , Springer  (1973)  (Translated from Rumanian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Yakubovich,  "A frequency theorem in control theory"  ''Sib. Math. J.'' , '''14''' :  2  (1973)  pp. 265–289  ''Sibirsk. Mat. Zh.'' , '''14''' :  2  (1973)  pp. 384–420</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.K. Gelig,  G.A. Leonov,  V.A. Yakubovich,  "Stability of non-linear systems with a unique equilibrium state" , Moscow  (1978)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> , ''Methods for studing non-linear systems of automatic control'' , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D.D. Siljak,  "Nonlinear systems. Parameter analysis and design" , Wiley  (1969)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.N. Fomin,  A.L. Fradkov,  V.A. Yakubovich,  "Adaptive control of dynamic objects" , Moscow  (1981)  (In Russian)</TD></TR><TR><TD valign="top">[8a]</TD> <TD valign="top">  J.C. Willems,  "Almost invariant subspaces: an approach to high gain feedback design I. Almost controlled invariant subspaces"  ''IEEE Trans. Autom. Control'' , '''1'''  (1981)  pp. 235–252</TD></TR><TR><TD valign="top">[8b]</TD> <TD valign="top">  J.C. Willems,  "Almost invariant subspaces: an approach to high gain feedback design I. Almost conditionally invariant subspaces"  ''IEEE Trans. Autom. Control'' , '''5'''  (1982)  pp. 1071–1084</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  W. Coppel,  "Matrix quadratic equations"  ''Bull. Austr. Math. Soc.'' , '''10'''  (1974)  pp. 377–401</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.I. Lur'e,  "Some non-linear problems of the theory of automatic control" , Moscow-Leningrad  (1951)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.M. Popov,  "Hyperstability of control systems" , Springer  (1973)  (Translated from Rumanian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.A. Yakubovich,  "A frequency theorem in control theory"  ''Sib. Math. J.'' , '''14''' :  2  (1973)  pp. 265–289  ''Sibirsk. Mat. Zh.'' , '''14''' :  2  (1973)  pp. 384–420</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.K. Gelig,  G.A. Leonov,  V.A. Yakubovich,  "Stability of non-linear systems with a unique equilibrium state" , Moscow  (1978)  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> , ''Methods for studing non-linear systems of automatic control'' , Moscow  (1975)  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  D.D. Siljak,  "Nonlinear systems. Parameter analysis and design" , Wiley  (1969)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  V.N. Fomin,  A.L. Fradkov,  V.A. Yakubovich,  "Adaptive control of dynamic objects" , Moscow  (1981)  (In Russian)</TD></TR><TR><TD valign="top">[8a]</TD> <TD valign="top">  J.C. Willems,  "Almost invariant subspaces: an approach to high gain feedback design I. Almost controlled invariant subspaces"  ''IEEE Trans. Autom. Control'' , '''1'''  (1981)  pp. 235–252</TD></TR><TR><TD valign="top">[8b]</TD> <TD valign="top">  J.C. Willems,  "Almost invariant subspaces: an approach to high gain feedback design I. Almost conditionally invariant subspaces"  ''IEEE Trans. Autom. Control'' , '''5'''  (1982)  pp. 1071–1084</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  W. Coppel,  "Matrix quadratic equations"  ''Bull. Austr. Math. Soc.'' , '''10'''  (1974)  pp. 377–401</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Revision as of 19:40, 5 June 2020


A theorem that states conditions for the solvability of the Lur'e equations in control theory:

$$ \tag{1 } P ^ {*} H + H P + h h ^ {*} = G ,\ \ H q - h \kappa = g , $$

where $ P $, $ G = G ^ {*} $, $ q $, $ g $, $ \kappa $ are given $ n \times n $, $ n \times n $, $ n \times m $, $ n \times m $, and $ m \times m $ matrices respectively, and $ H = H ^ {*} $, $ h $ are the required $ n \times n $ and $ n \times m $ matrices. The Lur'e equations have two other equivalent forms: If $ \mathop{\rm det} \kappa \neq 0 $,

$$ \tag{2 } H Q _ {0} H + ( P _ {0} ^ {*} H + H P _ {0} ) + G _ {0} = 0 , $$

where $ Q _ {0} = Q _ {0} ^ {*} \geq 0 $, $ G _ {0} = G _ {0} ^ {*} $, and in the general case

$$ \tag{3 } 2 \mathop{\rm Re} x ^ {*} H ( P x + q \xi ) = {\mathcal G} ( x , \xi ) - | h ^ {*} x - \kappa \xi | ^ {2} \ \ ( \forall x , \xi ) , $$

where $ {\mathcal G} ( x , \xi ) $ is a given Hermitian form of two vectors $ x \in \mathbf C ^ {n} $, $ \xi \in \mathbf C ^ {m} $;

$$ {\mathcal G} ( x , \xi ) = x ^ {*} G x + 2 \mathop{\rm Re} ( x ^ {*} g \xi ) + \xi ^ {*} \Gamma \xi , $$

Moreover, $ \Gamma = \kappa ^ {*} \kappa \geq 0 $, $ G _ {0} = g \Gamma ^ {-} 1 g ^ {*} - G $, $ P _ {0} = P - g \Gamma g ^ {*} $, $ Q _ {0} = q \Gamma ^ {-} 1 q ^ {*} $.

Let the pair $ \{ P , q \} $ be controllable: $ \mathop{\rm rank} \| q , Pq \dots P ^ {n-} 1 q \| = n $. Then the Lur'e equations reduce to the case where

$$ P = \mathop{\rm diag} [ \lambda _ {1} \dots \lambda _ {h} ] ,\ \ \lambda _ {j} + \lambda _ {h} \neq 0 ,\ \lambda _ {j} \in \mathbf R . $$

If $ m = 1 $ and all the matrices are real, the Lur'e equations in scalar notation take the form

$$ \sum _ { k= } 1 ^ { n } q _ {k} \frac{h _ {j} h _ {k} }{\lambda _ {j} + \lambda _ {k} } - h _ {j} \sqrt \Gamma = \gamma _ {j} ,\ j = 1 \dots n ; $$

here $ h = [ h _ {1} \dots h _ {n} ] $ is the required vector.

The frequency theorem asserts that for the Lur'e equations to be solvable it is necessary and sufficient that

$$ {\mathcal G} [ ( i \omega I - P ) ^ {-} 1 q \xi , \xi ] \geq 0 $$

for all $ \xi \in \mathbf C ^ {m} $, $ \omega \in \mathbf R ^ {1} $, $ \mathop{\rm det} \| i \omega I - P \| \neq 0 $( $ I $ is the identity matrix). The frequency theorem also formulates a procedure for determining the matrices $ H $ and $ h $ and asserts that if

$$ \mathop{\rm det} \Gamma \neq 0 ,\ \mathop{\rm det} \| i \omega I - P \| \neq 0 ,\ {\mathcal G} [ \| i \omega I - P \| ^ {-} 1 q \xi ,\ \xi ] > 0 $$

(for all $ \xi \neq 0 $, and all $ \omega $), then there exist (unique) matrices $ H $ and $ h $ such that (except for the case of equation (3)) the following is true: $ P + q \kappa ^ {-} 1 h ^ {*} $ is a Hurwitz matrix (see [3]).

The Lur'e equations in the form (2) are also sometimes called the matrix algebraic Riccati equation. The frequency theorem is used when solving problems on absolute stability [2], [4][6], control and adaptation (see, for example, [7][9]).

References

[1] A.I. Lur'e, "Some non-linear problems of the theory of automatic control" , Moscow-Leningrad (1951) (In Russian)
[2] V.M. Popov, "Hyperstability of control systems" , Springer (1973) (Translated from Rumanian)
[3] V.A. Yakubovich, "A frequency theorem in control theory" Sib. Math. J. , 14 : 2 (1973) pp. 265–289 Sibirsk. Mat. Zh. , 14 : 2 (1973) pp. 384–420
[4] A.K. Gelig, G.A. Leonov, V.A. Yakubovich, "Stability of non-linear systems with a unique equilibrium state" , Moscow (1978) (In Russian)
[5] , Methods for studing non-linear systems of automatic control , Moscow (1975) (In Russian)
[6] D.D. Siljak, "Nonlinear systems. Parameter analysis and design" , Wiley (1969)
[7] V.N. Fomin, A.L. Fradkov, V.A. Yakubovich, "Adaptive control of dynamic objects" , Moscow (1981) (In Russian)
[8a] J.C. Willems, "Almost invariant subspaces: an approach to high gain feedback design I. Almost controlled invariant subspaces" IEEE Trans. Autom. Control , 1 (1981) pp. 235–252
[8b] J.C. Willems, "Almost invariant subspaces: an approach to high gain feedback design I. Almost conditionally invariant subspaces" IEEE Trans. Autom. Control , 5 (1982) pp. 1071–1084
[9] W. Coppel, "Matrix quadratic equations" Bull. Austr. Math. Soc. , 10 (1974) pp. 377–401

Comments

The frequency theorem is better known as the Kalman–Yakubovich lemma or Kalman–Yacubovich lemma.

References

[a1] R.E. Kalman, "Lyapunov functions for the problem of Lurie in automatic control" Proc. Nat. Acad. Soc. USA , 49 : 2 (1963) pp. 201–205
[a2] B.D.O. Anderson, S. Vongpanitlerd, "Network analysis and synthesis: a modern systems theory approach" , Prentice-Hall (1973)
How to Cite This Entry:
Frequency theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Frequency_theorem&oldid=18832
This article was adapted from an original article by G.A. Leonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article