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A branch of control theory where the performance of a dynamical system (cf. [[Automatic control theory|Automatic control theory]]) is appraised in terms of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h0460202.png" />-norm. The Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h0460204.png" /> (named after G.H. Hardy, cf. [[Hardy classes|Hardy classes]]) consists of all complex-valued functions of a complex variable which are analytic and of bounded modulus in the open right half-plane. The norm of such a function is the supremum modulus:
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<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/h/h046/h046020/h0460205.png" /></td> </tr></table>
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By a theorem of Fatou (cf. [[Fatou theorem|Fatou theorem]]), such a function has a boundary value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h0460206.png" /> for almost-all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h0460207.png" />, and, moreover,
+
A branch of control theory where the performance of a dynamical system (cf. [[Automatic control theory|Automatic control theory]]) is appraised in terms of the  $  H  ^  \infty  $-
 +
norm. The Banach space  $  H  ^  \infty  $(
 +
named after G.H. Hardy, cf. [[Hardy classes|Hardy classes]]) consists of all complex-valued functions of a complex variable which are analytic and of bounded modulus in the open right half-plane. The norm of such a function is the supremum modulus:
  
<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/h/h046/h046020/h0460208.png" /></td> </tr></table>
+
$$
 +
\| F \| _  \infty  = \sup _ { \mathop{\rm Re}  s > 0 }  | F( s) | .
 +
$$
  
The theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h0460209.png" /> control was initiated by G. Zames [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], who formulated a basic feedback problem as an optimization problem with an operator norm, in particular, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602010.png" />-norm. Relevant contemporaneous works are those of J.W. Helton [[#References|[a4]]] and A. Tannenbaum [[#References|[a5]]].
+
By a theorem of Fatou (cf. [[Fatou theorem|Fatou theorem]]), such a function has a boundary value  $  F( i \omega ) $
 +
for almost-all  $  \omega $,
 +
and, moreover,
 +
 
 +
$$
 +
\| F \| _  \infty  =  \mathop{\rm esssup} _  \omega  | F( i \omega ) | .
 +
$$
 +
 
 +
The theory of $  H  ^  \infty  $
 +
control was initiated by G. Zames [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], who formulated a basic feedback problem as an optimization problem with an operator norm, in particular, an $  H  ^  \infty  $-
 +
norm. Relevant contemporaneous works are those of J.W. Helton [[#References|[a4]]] and A. Tannenbaum [[#References|[a5]]].
  
 
The theory treats dynamical systems represented as integral operators of the form
 
The theory treats dynamical systems represented as integral operators of 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/h/h046/h046020/h04602011.png" /></td> </tr></table>
+
$$
 +
y( t)  = \int\limits _ { 0 } ^ { t }  g( t- \tau ) x( \tau )  d \tau .
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602012.png" /> is sufficiently regular to make the input-output mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602013.png" /> a bounded operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602014.png" />. Taking Laplace transforms gives <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602015.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602016.png" /> is called the [[Transfer function|transfer function]] of the system and it belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602017.png" /> because the integral operator is bounded. Moreover, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602018.png" />-norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602019.png" /> equals the norm of the integral operator, i.e.,
+
Here $  g $
 +
is sufficiently regular to make the input-output mapping $  x \mapsto y $
 +
a bounded operator on $  L _ {2} [ 0 , \infty ) $.  
 +
Taking Laplace transforms gives $  Y( s)= G( s) X( s) $.  
 +
The function $  G $
 +
is called the [[Transfer function|transfer function]] of the system and it belongs to $  H  ^  \infty  $
 +
because the integral operator is bounded. Moreover, the $  H  ^  \infty  $-
 +
norm of $  G $
 +
equals the norm of the integral operator, i.e.,
  
<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/h/h046/h046020/h04602020.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
\| G \| _  \infty  = \sup _ {\| x \| _ {2} \leq  1 } \
 +
\| y \| _ {2} .
 +
$$
  
There are two prototype problems giving rise to an optimality criterion with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602021.png" />-norm. The first is the problem of robust stability of the feedback system
+
There are two prototype problems giving rise to an optimality criterion with the $  H  ^  \infty  $-
 +
norm. The first is the problem of robust stability of the feedback system
  
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h046020a.gif" />
 
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/h046020a.gif" />
Line 23: Line 59:
 
Figure: h046020a
 
Figure: h046020a
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602023.png" /> are transfer functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602024.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602028.png" /> are Laplace transforms of signals; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602029.png" /> represents a  "plant" , the dynamical system which is to be controlled, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602030.png" /> represents the  "controller"  (cf. also [[Automatic control theory|Automatic control theory]]). The figure stands for the two equations
+
Here $  P $
 +
and $  C $
 +
are transfer functions in $  H  ^  \infty  $,  
 +
and $  X _ {1} $,  
 +
$  X _ {2} $,  
 +
$  Y _ {1} $,  
 +
$  Y _ {2} $
 +
are Laplace transforms of signals; $  P $
 +
represents a  "plant" , the dynamical system which is to be controlled, and $  C $
 +
represents the  "controller"  (cf. also [[Automatic control theory|Automatic control theory]]). The figure stands for the two equations
  
<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/h/h046/h046020/h04602031.png" /></td> </tr></table>
+
$$
 +
Y _ {1}  = X _ {1} + PY _ {2} ,\  Y _ {2}  = X _ {2} + CY _ {1} ,
 +
$$
  
 
which can be solved to give
 
which can be solved to give
  
<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/h/h046/h046020/h04602032.png" /></td> </tr></table>
+
$$
 +
\left [ \begin{array}{c}
 +
Y _ {1} \\
 +
Y _ {2}
 +
\end{array}
 +
\right ]  = \
 +
\left [
 +
\begin{array}{cc}
 +
 
 +
\frac{1 }{1- PC }
 +
  &
 +
\frac{P }{1- PC }
 +
  \\
 +
 
 +
\frac{C }{1- PC }
 +
  &
 +
\frac{1 }{1- PC }
 +
  \\
 +
\end{array}
 +
 
 +
\right ]
 +
\left [ \begin{array}{c}
 +
X _ {1} \\
 +
X _ {2}
 +
\end{array}
 +
\right ] .
 +
$$
  
Therefore, the input-output mapping for the feedback system has four transfer functions. The feedback system is said to be internally stable if these four transfer functions are all in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602033.png" />. A simple sufficient condition for this is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602034.png" />.
+
Therefore, the input-output mapping for the feedback system has four transfer functions. The feedback system is said to be internally stable if these four transfer functions are all in $  H  ^  \infty  $.  
 +
A simple sufficient condition for this is $  \| PC \| _  \infty  < 1 $.
  
Internal stability is robust if it is preserved under perturbation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602035.png" />. There are several possible notions of perturbation, typical of which is additive perturbation. So suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602036.png" /> is perturbed to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602037.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602038.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602039.png" />. About <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602040.png" /> it is assumed that only a bound on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602041.png" /> is known, namely,
+
Internal stability is robust if it is preserved under perturbation of $  P $.  
 +
There are several possible notions of perturbation, typical of which is additive perturbation. So suppose $  P $
 +
is perturbed to $  P+ \Delta P $,  
 +
with $  \Delta P $
 +
in $  H  ^  \infty  $.  
 +
About $  \Delta P $
 +
it is assumed that only a bound on $  | \Delta P( i \omega ) | $
 +
is known, namely,
  
<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/h/h046/h046020/h04602042.png" /></td> </tr></table>
+
$$
 +
| \Delta P( i \omega ) |  < | R( i \omega ) | ,\  \textrm{ a.a. } \
 +
\omega ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602043.png" />. J.C. Doyle and G. Stein [[#References|[a6]]] showed that internal stability is preserved under all such perturbations if and only if
+
where $  R \in H  ^  \infty  $.  
 +
J.C. Doyle and G. Stein [[#References|[a6]]] showed that internal stability is preserved under all such perturbations if and only 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/h/h046/h046020/h04602044.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2 }
 +
\| RC( 1- PC) ^ {- 1 } \| _  \infty  < 1 .
 +
$$
  
This leads to the robust stability design problem: Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602046.png" />, find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602047.png" /> so that the feedback system is internally stable and (a2) holds.
+
This leads to the robust stability design problem: Given $  P $
 +
and $  R $,  
 +
find $  C $
 +
so that the feedback system is internally stable and (a2) holds.
  
The second problem relates to the same feedback system. Suppose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602048.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602049.png" /> represents a disturbance signal, and the objective is to reduce the effect of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602050.png" /> on the output <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602051.png" />. The transfer function from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602052.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602053.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602054.png" />. Suppose, in addition, that the disturbance is not a fixed signal, but can be the output of another system with any input in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602055.png" /> of unit norm; let this latter system have transfer function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602056.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602057.png" />. Then, in view of (a1), the supremal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602058.png" />-norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602059.png" /> over all such disturbances equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602060.png" />. This leads to the disturbance attenuation problem: Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602061.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602062.png" />, find <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602063.png" /> to achieve internal stability and minimize <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602064.png" />.
+
The second problem relates to the same feedback system. Suppose $  X _ {2} = 0 $,  
 +
$  X _ {1} $
 +
represents a disturbance signal, and the objective is to reduce the effect of $  X _ {1} $
 +
on the output $  Y _ {1} $.  
 +
The transfer function from $  X _ {1} $
 +
to $  Y _ {1} $
 +
equals $  ( 1- PC) ^ {- 1 } $.  
 +
Suppose, in addition, that the disturbance is not a fixed signal, but can be the output of another system with any input in $  L _ {2} [ 0 , \infty ) $
 +
of unit norm; let this latter system have transfer function $  W $
 +
in $  H  ^  \infty  $.  
 +
Then, in view of (a1), the supremal $  L _ {2} [ 0 , \infty ) $-
 +
norm of $  y _ {1} $
 +
over all such disturbances equals $  \| W( 1- PC) ^ {- 1 } \| _  \infty  $.  
 +
This leads to the disturbance attenuation problem: Given $  P $
 +
and $  R $,  
 +
find $  C $
 +
to achieve internal stability and minimize $  \| W( 1- PC) ^ {- 1 } \| _  \infty  $.
  
The above two problems are special cases of the more general standard <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602066.png" /> control problem. It can be solved by reduction to the Nehari problem of approximating a function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602067.png" /> (bounded functions on the imaginary axis) by one in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602068.png" />. A summary of this theory is in [[#References|[a7]]], and a detailed treatment is in [[#References|[a8]]].
+
The above two problems are special cases of the more general standard $  H  ^  \infty  $
 +
control problem. It can be solved by reduction to the Nehari problem of approximating a function in $  L _  \infty  $(
 +
bounded functions on the imaginary axis) by one in $  H  ^  \infty  $.  
 +
A summary of this theory is in [[#References|[a7]]], and a detailed treatment is in [[#References|[a8]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Zames,  "Feedback and complexity, Special plenary lecture addendum" , ''IEEE Conf. Decision Control'' , IEEE  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Zames,  "Optimal sensitivity and feedback: weighted seminorms, approximate inverses, and plant invariant schemes" , ''Proc. Allerton Conf.'' , IEEE  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Zames,  "Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses"  ''IEEE Trans. Auto. Control'' , '''AC-26'''  (1981)  pp. 301–320</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.W. Helton,  "Operator theory and broadband matching" , ''Proc. Allerton Conf.'' , IEEE  (1979)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Tannenbaum,  "On the blending problem and parameter uncertainty in control theory"  ''Techn. Report Dept. Math. Weizmann Institute''  (1977)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.C. Doyle,  G. Stein,  "Multivariable feedback design: concepts for a classical modern synthesis"  ''IEEE Trans. Auto. Control'' , '''AC-26'''  (1981)  pp. 4–16</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  B.A. Francis,  J.C. Doyle,  "Linear control theory with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602069.png" /> optimality criterion"  ''SIAM J. Control and Opt.'' , '''25'''  (1987)  pp. 815–844</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  B.A. Francis,  "A course in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602070.png" /> control theory" , ''Lect. notes in control and inform. science'' , '''88''' , Springer  (1987)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G. Zames,  "Feedback and complexity, Special plenary lecture addendum" , ''IEEE Conf. Decision Control'' , IEEE  (1976)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Zames,  "Optimal sensitivity and feedback: weighted seminorms, approximate inverses, and plant invariant schemes" , ''Proc. Allerton Conf.'' , IEEE  (1979)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Zames,  "Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses"  ''IEEE Trans. Auto. Control'' , '''AC-26'''  (1981)  pp. 301–320</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.W. Helton,  "Operator theory and broadband matching" , ''Proc. Allerton Conf.'' , IEEE  (1979)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Tannenbaum,  "On the blending problem and parameter uncertainty in control theory"  ''Techn. Report Dept. Math. Weizmann Institute''  (1977)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  J.C. Doyle,  G. Stein,  "Multivariable feedback design: concepts for a classical modern synthesis"  ''IEEE Trans. Auto. Control'' , '''AC-26'''  (1981)  pp. 4–16</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  B.A. Francis,  J.C. Doyle,  "Linear control theory with an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602069.png" /> optimality criterion"  ''SIAM J. Control and Opt.'' , '''25'''  (1987)  pp. 815–844</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  B.A. Francis,  "A course in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046020/h04602070.png" /> control theory" , ''Lect. notes in control and inform. science'' , '''88''' , Springer  (1987)</TD></TR></table>

Revision as of 19:42, 5 June 2020


A branch of control theory where the performance of a dynamical system (cf. Automatic control theory) is appraised in terms of the $ H ^ \infty $- norm. The Banach space $ H ^ \infty $( named after G.H. Hardy, cf. Hardy classes) consists of all complex-valued functions of a complex variable which are analytic and of bounded modulus in the open right half-plane. The norm of such a function is the supremum modulus:

$$ \| F \| _ \infty = \sup _ { \mathop{\rm Re} s > 0 } | F( s) | . $$

By a theorem of Fatou (cf. Fatou theorem), such a function has a boundary value $ F( i \omega ) $ for almost-all $ \omega $, and, moreover,

$$ \| F \| _ \infty = \mathop{\rm esssup} _ \omega | F( i \omega ) | . $$

The theory of $ H ^ \infty $ control was initiated by G. Zames [a1], [a2], [a3], who formulated a basic feedback problem as an optimization problem with an operator norm, in particular, an $ H ^ \infty $- norm. Relevant contemporaneous works are those of J.W. Helton [a4] and A. Tannenbaum [a5].

The theory treats dynamical systems represented as integral operators of the form

$$ y( t) = \int\limits _ { 0 } ^ { t } g( t- \tau ) x( \tau ) d \tau . $$

Here $ g $ is sufficiently regular to make the input-output mapping $ x \mapsto y $ a bounded operator on $ L _ {2} [ 0 , \infty ) $. Taking Laplace transforms gives $ Y( s)= G( s) X( s) $. The function $ G $ is called the transfer function of the system and it belongs to $ H ^ \infty $ because the integral operator is bounded. Moreover, the $ H ^ \infty $- norm of $ G $ equals the norm of the integral operator, i.e.,

$$ \tag{a1 } \| G \| _ \infty = \sup _ {\| x \| _ {2} \leq 1 } \ \| y \| _ {2} . $$

There are two prototype problems giving rise to an optimality criterion with the $ H ^ \infty $- norm. The first is the problem of robust stability of the feedback system

Figure: h046020a

Here $ P $ and $ C $ are transfer functions in $ H ^ \infty $, and $ X _ {1} $, $ X _ {2} $, $ Y _ {1} $, $ Y _ {2} $ are Laplace transforms of signals; $ P $ represents a "plant" , the dynamical system which is to be controlled, and $ C $ represents the "controller" (cf. also Automatic control theory). The figure stands for the two equations

$$ Y _ {1} = X _ {1} + PY _ {2} ,\ Y _ {2} = X _ {2} + CY _ {1} , $$

which can be solved to give

$$ \left [ \begin{array}{c} Y _ {1} \\ Y _ {2} \end{array} \right ] = \ \left [ \begin{array}{cc} \frac{1 }{1- PC } & \frac{P }{1- PC } \\ \frac{C }{1- PC } & \frac{1 }{1- PC } \\ \end{array} \right ] \left [ \begin{array}{c} X _ {1} \\ X _ {2} \end{array} \right ] . $$

Therefore, the input-output mapping for the feedback system has four transfer functions. The feedback system is said to be internally stable if these four transfer functions are all in $ H ^ \infty $. A simple sufficient condition for this is $ \| PC \| _ \infty < 1 $.

Internal stability is robust if it is preserved under perturbation of $ P $. There are several possible notions of perturbation, typical of which is additive perturbation. So suppose $ P $ is perturbed to $ P+ \Delta P $, with $ \Delta P $ in $ H ^ \infty $. About $ \Delta P $ it is assumed that only a bound on $ | \Delta P( i \omega ) | $ is known, namely,

$$ | \Delta P( i \omega ) | < | R( i \omega ) | ,\ \textrm{ a.a. } \ \omega , $$

where $ R \in H ^ \infty $. J.C. Doyle and G. Stein [a6] showed that internal stability is preserved under all such perturbations if and only if

$$ \tag{a2 } \| RC( 1- PC) ^ {- 1 } \| _ \infty < 1 . $$

This leads to the robust stability design problem: Given $ P $ and $ R $, find $ C $ so that the feedback system is internally stable and (a2) holds.

The second problem relates to the same feedback system. Suppose $ X _ {2} = 0 $, $ X _ {1} $ represents a disturbance signal, and the objective is to reduce the effect of $ X _ {1} $ on the output $ Y _ {1} $. The transfer function from $ X _ {1} $ to $ Y _ {1} $ equals $ ( 1- PC) ^ {- 1 } $. Suppose, in addition, that the disturbance is not a fixed signal, but can be the output of another system with any input in $ L _ {2} [ 0 , \infty ) $ of unit norm; let this latter system have transfer function $ W $ in $ H ^ \infty $. Then, in view of (a1), the supremal $ L _ {2} [ 0 , \infty ) $- norm of $ y _ {1} $ over all such disturbances equals $ \| W( 1- PC) ^ {- 1 } \| _ \infty $. This leads to the disturbance attenuation problem: Given $ P $ and $ R $, find $ C $ to achieve internal stability and minimize $ \| W( 1- PC) ^ {- 1 } \| _ \infty $.

The above two problems are special cases of the more general standard $ H ^ \infty $ control problem. It can be solved by reduction to the Nehari problem of approximating a function in $ L _ \infty $( bounded functions on the imaginary axis) by one in $ H ^ \infty $. A summary of this theory is in [a7], and a detailed treatment is in [a8].

References

[a1] G. Zames, "Feedback and complexity, Special plenary lecture addendum" , IEEE Conf. Decision Control , IEEE (1976)
[a2] G. Zames, "Optimal sensitivity and feedback: weighted seminorms, approximate inverses, and plant invariant schemes" , Proc. Allerton Conf. , IEEE (1979)
[a3] G. Zames, "Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses" IEEE Trans. Auto. Control , AC-26 (1981) pp. 301–320
[a4] J.W. Helton, "Operator theory and broadband matching" , Proc. Allerton Conf. , IEEE (1979)
[a5] A. Tannenbaum, "On the blending problem and parameter uncertainty in control theory" Techn. Report Dept. Math. Weizmann Institute (1977)
[a6] J.C. Doyle, G. Stein, "Multivariable feedback design: concepts for a classical modern synthesis" IEEE Trans. Auto. Control , AC-26 (1981) pp. 4–16
[a7] B.A. Francis, J.C. Doyle, "Linear control theory with an optimality criterion" SIAM J. Control and Opt. , 25 (1987) pp. 815–844
[a8] B.A. Francis, "A course in control theory" , Lect. notes in control and inform. science , 88 , Springer (1987)
How to Cite This Entry:
H^infinity-control-theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=H%5Einfinity-control-theory&oldid=47156