Assignment of infinite zero orders in linear systems using state feedback

Abstract

The problem of modifying the infinite zero orders in linear multivariable systems by nonregular state feedback is revisited. An entirely different algebraic approach is presented that offers new, inspiring insights into the problem. The approach is based on the properties of the invariant factors of a product of two proper rational matrices. A complete and explicit solution to the problem is established for linear multivariable systems described by quadruples (A, B, C, D). The system is first brought to Morse normal form so that one can identify the structural invariants and construct a state-feedback realizable compensator that assigns the prespecified infinite zero orders, then one returns to the original coordinates. The solvability condition, necessary and sufficient, is stated by using the system’s structural invariants; the use of conjugate lists of invariants is avoided. In addition to determining all infinite zero orders that can be assigned, the most important result of the new approach is a characterization of the structural properties of all compensators achieving every single assignable list of infinite zero orders. It turns out that this is a combinatorial problem. The solution set has the structure of a lattice of lists of nonnegative integers with a given sum, partially ordered by majorization.