Pole assignment problem

Let be a commutative ring (cf. Commutative ring) and let be a pair of matrices of sizes and , respectively, with coefficients in . The pole assignment problem asks the following. Given , does there exist an -matrix , called a feedback matrix, such that the characteristic polynomial of is precisely ? The pair is then called a pole assignable pair of matrices. The terminology derives from the "interpretation" of as (the essential data of) a discrete-time time-invariant linear control system:

(a1)

where , , or also, when or , a continuous-time time-invariant linear control system:

(a2)

where , .

In both cases, state feedback (see Automatic control theory), , changes the pair to .

The transfer function of a system (a1) or (a2) with output is equal to

(a3)

and therefore the terminology "pole assignment" is used.

The pair is a coefficient assignable pair of matrices if for all there is an -matrix such that has characteristic polynomial .

The pair is completely reachable, reachable, completely controllable, or controllable if the columns of the -reachability matrix

(a4)

span all of . 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 , see again Automatic control theory.

A cyclic vector for an -matrix is a vector such that is a basis for , i.e., such that is completely reachable. Now consider the following properties for a pair of matrices :

a) there exist a matrix and a vector such that is cyclic for ;

b) is coefficient assignable;

c) is pole assignable;

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

A ring 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 have the coefficient assignment property?" J. Pure Appl. Algebra , 112 (1996) pp. 239–246