Output feedback invariants

Abstract

The paper is concerned with the problem of determining a complete set of invariants for output feedback. Using tools from geometric invariant theory it is shown that there exists a quasi-projective variety whose points parameterize the output feedback orbits in a unique way. If the McMillan degree n ≥ mp, the product of number of inputs and number of outputs, then it is shown that in the closure of every feedback orbit there is exactly one nondegenerate system. * The main results of this paper were announced

On Equivalence of Linear Control Systems and Its Usage to Verification of the Adequacy of Different Models for a Real Dynamic Process

“…Except for the classification questions of linear control systems, invariant theory is widely used in the problem of output feed-back design both for linear and bilinear control systems (see, for examples, papers [2] - [6], [38], [39], [41,42], and [50]). Here it should be said that in article [5] it was succeeded to get the constructive solution of output feed-back design problem for system (2.1), (2.2) in the case mp > n.…”

Some classification problems for control systems and the similarity problem for sets of square matrices

“…Numerous papers (see the bibliography in [12,13]) deal with the investigation of the similarity action on L(n, m) and on the space L(n, m, p) of n-dimensional m-input p-output systems; nevertheless, there are unsolved problems [for example, the description of the set of regular points in L(n, m, p)] in this situation as well. The action of the complete output feedback group in the context of invariant theory was considered in [12][13][14].…”

Some classification problems for control systems and the similarity problem for sets of square matrices

“…For the first time, the classification problem for linear control systems for the transformations (2.3) was stated in [9], and later it was considered in [1-4, 10, 11]. The action of the complete output feedback group in the context of invariant theory was considered in [12][13][14]. Numerous papers (see the bibliography in [12,13]) deal with the investigation of the similarity action on L(n, m) and on the space L(n, m, p) of n-dimensional m-input p-output systems; nevertheless, there are unsolved problems [for example, the description of the set of regular points in L(n, m, p)] in this situation as well.…”