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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p1101901.png" /> be a commutative ring (cf. [[Commutative ring|Commutative ring]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p1101902.png" /> be a pair of matrices of sizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p1101903.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p1101904.png" />, respectively, with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p1101905.png" />. The pole assignment problem asks the following. Given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p1101906.png" />, does there exist an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p1101907.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p1101908.png" />, called a feedback matrix, such that the [[Characteristic polynomial|characteristic polynomial]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p1101909.png" /> is precisely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019010.png" />? The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019011.png" /> is then called a pole assignable pair of matrices. The terminology derives from the  "interpretation"  of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019012.png" /> as (the essential data of) a discrete-time time-invariant linear control system:
+
Let $R$ be a [[commutative ring]] and let $(A,B)$ be a pair of matrices of sizes $(n \times n)$ and $(n \times m)$, respectively, with coefficients in $R$. The pole assignment problem asks the following. Given $r_1,\ldots,r_n$, does there exist an $(m \times n)$-matrix $F$, called a feedback matrix, such that the [[characteristic polynomial]] of $A+BF$ is precisely $(X-r_1)\cdots(X - r_n)$? The pair $(A,B)$ is then called a pole assignable pair of matrices. The terminology derives from the  "interpretation"  of $(A,B)$ as (the essential data of) a discrete-time time-invariant linear control system:
 +
\begin{equation}\label{eq:a1}
 +
x(t+1) = Ax(t) + Bu(t)
 +
\end{equation}
 +
where $x(t) \in R^n$, $u(t) \in R^m$, or also, when $R = \mathbf{R}$ or $\mathbf{C}$, a continuous-time time-invariant linear control system:
 +
\begin{equation}\label{eq:a2}
 +
\dot x(t)  = Ax(t) + Bu(t)
 +
\end{equation}
 +
where $x(t) \in R^n$, $u(t) \in R^m$.
  
<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/p/p110/p110190/p11019013.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
In both cases, state feedback (see [[Automatic control theory]]), $u \mapsto u + Fx$, changes the pair $(A,B)$ to $(A+BF,B)$.
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019015.png" />, or also, when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019016.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019017.png" />, a continuous-time time-invariant linear control system:
 
 
 
<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/p/p110/p110190/p11019018.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
 
 
 
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019020.png" />.
 
 
 
In both cases, state feedback (see [[Automatic control theory|Automatic control theory]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019021.png" />, changes the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019022.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019023.png" />.
 
 
 
The [[Transfer function|transfer function]] of a system (a1) or (a2) with output <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019024.png" /> is equal to
 
 
 
<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/p/p110/p110190/p11019025.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>
 
  
 +
The [[transfer function]] of a system \eqref{eq:a1} or \eqref{eq:a2} with output $y(t) = C x(t)$ is equal to
 +
\begin{equation}\label{eq:a3}
 +
T(s) = C(sI-A)^{-1}B
 +
\end{equation}
 
and therefore the terminology  "pole assignment"  is used.
 
and therefore the terminology  "pole assignment"  is used.
  
The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019026.png" /> is a coefficient assignable pair of matrices if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019027.png" /> there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019028.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019029.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019030.png" /> has characteristic polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019031.png" />.
+
The pair $(A,B)$ is a ''coefficient assignable'' pair of matrices if for all $a_1,\ldots,a_n \in R$ there is an $(m\times n)$-matrix $F$ such that $A+BF$ has characteristic polynomial $X^n + a_1X^{n-1} + \cdots + a_n$.
 
 
The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019032.png" /> is completely reachable, reachable, completely controllable, or controllable if the columns of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019033.png" />-reachability matrix
 
  
<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/p/p110/p110190/p11019034.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>
+
The pair $(A,B)$ is ''completely reachable'', ''reachable'', ''completely controllable'', or ''controllable'' if the columns of the $(n\times nm)$-reachability matrix
 +
\begin{equation}\label{eq:a4}
 +
(B,AB,\ldots,A^{n-1}B)
 +
\end{equation}
 +
span all of $R^n$. All four mentioned choices of terminology are used in the literature. The reachability matrix \eqref{eq:a4} is also called the controllability matrix. This terminology also derives from the  "interpretations" \eqref{eq:a1} and \eqref{eq:a2} of a pair $(A,B)$, see again [[Automatic control theory]].
  
span all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019035.png" />. All four mentioned choices of terminology are used in the literature. The reachability matrix (a4) is also called the controllability matrix. This terminology also derives from the "interpretations"  (a1) and (a2) of a pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019036.png" />, see again [[Automatic control theory|Automatic control theory]].
+
A [[cyclic vector]] for an $(n\times n)$-matrix $M$ is a vector $v\in R^n$ such that $(v,MV,\ldots,M^{n-1}v)$ is a basis for $R^n$, i.e., such that $(M,v)$ is completely reachable. Now consider the following properties for a pair of matrices $(A,B)$:
  
A [[cyclic vector]] for an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019037.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019038.png" /> is a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019039.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019040.png" /> is a basis for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019041.png" />, i.e., such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019042.png" /> is completely reachable. Now consider the following properties for a pair of matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019043.png" />:
+
a) there exist a matrix $F$ and a vector $w \in R^m$ such that $Bw$ is cyclic for $A+BF$;
  
a) there exist a matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019044.png" /> and a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019045.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019046.png" /> is cyclic for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019047.png" />;
+
b) $(A,B)$ is coefficient assignable;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019048.png" /> is coefficient assignable;
+
c) $(A,B)$ is pole assignable;
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019049.png" /> is pole assignable;
+
d) $(A,B)$ is completely reachable.  
  
d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019050.png" /> is completely reachable. Over a [[Field|field]] these conditions are equivalent and, in general, a)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019051.png" />b)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019052.png" />c)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019053.png" />d). In control theory, the implication d)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019054.png" />a) for a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019055.png" /> is called the Heyman lemma, and the implication d)<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019056.png" />c) for a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019057.png" /> is termed the pole shifting theorem.
+
Over a [[field]] these conditions are equivalent and, in general, a)$\Rightarrow$b)$\Rightarrow$c)$\Rightarrow$d). In control theory, the implication d)$\Rightarrow$a) for a field $R$ is called the Heyman lemma, and the implication d)$\Rightarrow$c) for a field $R$ is termed the pole shifting theorem.
  
A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019058.png" /> is said to have the FC-property (respectively, the CA-property or the PA-property) if for that ring d) implies a) (respectively, d) implies b), or d) implies c)). Such a ring is also called, respectively, an FC-ring, a CA-ring or a PA-ring. As noted, each field is an FC-ring (and hence a CA-ring and a PA-ring). Each Dedekind domain (cf. also [[Dedekind ring|Dedekind ring]]) is a PA-ring. The ring of polynomials in one indeterminate over an [[Algebraically closed field|algebraically closed field]] is a CA-ring, but the ring of polynomials in two or more indeterminates over any field is not a PA-ring (and hence not a CA-ring) [[#References|[a4]]].
+
A ring $R$ is said to have the FC-property (respectively, the CA-property or the PA-property) if for that ring d) implies a) (respectively, d) implies b), or d) implies c)). Such a ring is also called, respectively, an FC-ring, a CA-ring or a PA-ring. As noted, each field is an FC-ring (and hence a CA-ring and a PA-ring). Each Dedekind domain (cf. also [[Dedekind ring]]) is a PA-ring. The ring of polynomials in one indeterminate over an [[algebraically closed field]] is a CA-ring, but the ring of polynomials in two or more indeterminates over any field is not a PA-ring (and hence not a CA-ring) [[#References|[a4]]].
  
 
For a variety of related results, see [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a5]]].
 
For a variety of related results, see [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a5]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W. Brewer,  J.W. Bunce,  F.S. van Vleck,  "Linear systems over commutative rings" , M. Dekker  (1986)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Brewer,  D. Katz,  W. Ullery,  "Pole assignability in polynomial rings, power series rings, and Prüfer domains"  ''J. Algebra'' , '''106'''  (1987)  pp. 265–286</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Bumby,  E.D. Soutrey,  H.J. Sussmann,  W. Vasconcelos,  "Remarks on the pole-shifting theorem over rings"  ''J. Pure Appl. Algebra'' , '''20'''  (1981)  pp. 113–127</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Tannenbaum,  "Polynomial rings over arbitrary fields in two or more variables are not pole assignable"  ''Syst. Control Lett.'' , '''2'''  (1982)  pp. 222–224</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Brewer,  T. Ford,  L. Kingler,  W. Schmale,  "When does the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p110/p110190/p11019059.png" /> have the coefficient assignment property?"  ''J. Pure Appl. Algebra'' , '''112'''  (1996)  pp. 239–246</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top">  J.W. Brewer,  J.W. Bunce,  F.S. van Vleck,  "Linear systems over commutative rings" , M. Dekker  (1986)</TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Brewer,  D. Katz,  W. Ullery,  "Pole assignability in polynomial rings, power series rings, and Prüfer domains"  ''J. Algebra'' , '''106'''  (1987)  pp. 265–286</TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Bumby,  E.D. Soutrey,  H.J. Sussmann,  W. Vasconcelos,  "Remarks on the pole-shifting theorem over rings"  ''J. Pure Appl. Algebra'' , '''20'''  (1981)  pp. 113–127</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top">  A. Tannenbaum,  "Polynomial rings over arbitrary fields in two or more variables are not pole assignable"  ''Syst. Control Lett.'' , '''2'''  (1982)  pp. 222–224</TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Brewer,  T. Ford,  L. Kingler,  W. Schmale,  "When does the ring $K[g]$ have the coefficient assignment property?"  ''J. Pure Appl. Algebra'' , '''112'''  (1996)  pp. 239–246</TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Latest revision as of 20:38, 12 November 2017

Let $R$ be a commutative ring and let $(A,B)$ be a pair of matrices of sizes $(n \times n)$ and $(n \times m)$, respectively, with coefficients in $R$. The pole assignment problem asks the following. Given $r_1,\ldots,r_n$, does there exist an $(m \times n)$-matrix $F$, called a feedback matrix, such that the characteristic polynomial of $A+BF$ is precisely $(X-r_1)\cdots(X - r_n)$? The pair $(A,B)$ is then called a pole assignable pair of matrices. The terminology derives from the "interpretation" of $(A,B)$ as (the essential data of) a discrete-time time-invariant linear control system: \begin{equation}\label{eq:a1} x(t+1) = Ax(t) + Bu(t) \end{equation} where $x(t) \in R^n$, $u(t) \in R^m$, or also, when $R = \mathbf{R}$ or $\mathbf{C}$, a continuous-time time-invariant linear control system: \begin{equation}\label{eq:a2} \dot x(t) = Ax(t) + Bu(t) \end{equation} where $x(t) \in R^n$, $u(t) \in R^m$.

In both cases, state feedback (see Automatic control theory), $u \mapsto u + Fx$, changes the pair $(A,B)$ to $(A+BF,B)$.

The transfer function of a system \eqref{eq:a1} or \eqref{eq:a2} with output $y(t) = C x(t)$ is equal to \begin{equation}\label{eq:a3} T(s) = C(sI-A)^{-1}B \end{equation} and therefore the terminology "pole assignment" is used.

The pair $(A,B)$ is a coefficient assignable pair of matrices if for all $a_1,\ldots,a_n \in R$ there is an $(m\times n)$-matrix $F$ such that $A+BF$ has characteristic polynomial $X^n + a_1X^{n-1} + \cdots + a_n$.

The pair $(A,B)$ is completely reachable, reachable, completely controllable, or controllable if the columns of the $(n\times nm)$-reachability matrix \begin{equation}\label{eq:a4} (B,AB,\ldots,A^{n-1}B) \end{equation} span all of $R^n$. All four mentioned choices of terminology are used in the literature. The reachability matrix \eqref{eq:a4} is also called the controllability matrix. This terminology also derives from the "interpretations" \eqref{eq:a1} and \eqref{eq:a2} of a pair $(A,B)$, see again Automatic control theory.

A cyclic vector for an $(n\times n)$-matrix $M$ is a vector $v\in R^n$ such that $(v,MV,\ldots,M^{n-1}v)$ is a basis for $R^n$, i.e., such that $(M,v)$ is completely reachable. Now consider the following properties for a pair of matrices $(A,B)$:

a) there exist a matrix $F$ and a vector $w \in R^m$ such that $Bw$ is cyclic for $A+BF$;

b) $(A,B)$ is coefficient assignable;

c) $(A,B)$ is pole assignable;

d) $(A,B)$ is completely reachable.

Over a field these conditions are equivalent and, in general, a)$\Rightarrow$b)$\Rightarrow$c)$\Rightarrow$d). In control theory, the implication d)$\Rightarrow$a) for a field $R$ is called the Heyman lemma, and the implication d)$\Rightarrow$c) for a field $R$ is termed the pole shifting theorem.

A ring $R$ is said to have the FC-property (respectively, the CA-property or the PA-property) if for that ring d) implies a) (respectively, d) implies b), or d) implies c)). Such a ring is also called, respectively, an FC-ring, a CA-ring or a PA-ring. As noted, each field is an FC-ring (and hence a CA-ring and a PA-ring). Each Dedekind domain (cf. also Dedekind ring) is a PA-ring. The ring of polynomials in one indeterminate over an algebraically closed field is a CA-ring, but the ring of polynomials in two or more indeterminates over any field is not a PA-ring (and hence not a CA-ring) [a4].

For a variety of related results, see [a1], [a2], [a3], [a5].

References

[a1] J.W. Brewer, J.W. Bunce, F.S. van Vleck, "Linear systems over commutative rings" , M. Dekker (1986)
[a2] J. Brewer, D. Katz, W. Ullery, "Pole assignability in polynomial rings, power series rings, and Prüfer domains" J. Algebra , 106 (1987) pp. 265–286
[a3] R. Bumby, E.D. Soutrey, H.J. Sussmann, W. Vasconcelos, "Remarks on the pole-shifting theorem over rings" J. Pure Appl. Algebra , 20 (1981) pp. 113–127
[a4] A. Tannenbaum, "Polynomial rings over arbitrary fields in two or more variables are not pole assignable" Syst. Control Lett. , 2 (1982) pp. 222–224
[a5] J. Brewer, T. Ford, L. Kingler, W. Schmale, "When does the ring $K[g]$ have the coefficient assignment property?" J. Pure Appl. Algebra , 112 (1996) pp. 239–246
How to Cite This Entry:
Pole assignment problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pole_assignment_problem&oldid=42284
This article was adapted from an original article by M. Hazewinkel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article