Pole assignment problem

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