Computation of supremal (A,B)-invariant and controllability subspaces

Robust Eigenstructure Assignment in Geometric Control Theory

“…First, we show that the set of parameter matrices K such that rank X K < r has Lebesgue measure zero. From [13 …”

“…Except for stabilizability and detectability subspaces, which require eigenspace computations, the traditional algorithms employed to compute the aforementioned subspaces are based on monotonic sequences of subspaces that converge in a finite number of steps (typically not greater than the system order) to the desired subspace. An alternative approach was taken by Moore and Laub in [13], who proposed an algorithm for the computation of the largest output-nulling reachability subspace that employs the Rosenbrock system matrix pencil. That paper made a number of restrictive assumptions, and perhaps for this reason the methods in [13] have been given only marginal attention.…”

“…An alternative approach was taken by Moore and Laub in [13], who proposed an algorithm for the computation of the largest output-nulling reachability subspace that employs the Rosenbrock system matrix pencil. That paper made a number of restrictive assumptions, and perhaps for this reason the methods in [13] have been given only marginal attention. An approach via the special coordinate basis (SCB), which avoided the restrictive assumptions of [13], was given in [5], [6], [7].…”

“…That paper made a number of restrictive assumptions, and perhaps for this reason the methods in [13] have been given only marginal attention. An approach via the special coordinate basis (SCB), which avoided the restrictive assumptions of [13], was given in [5], [6], [7].…”

“…Similarly, the SCB method of [5] was incorporated into the computation of the friends in the MATLAB Linsyskit toolbox; 2 the atea.m routine is used for computing the friends and is described in [10]. However, it has been acknowledged by many authors [13], [8] that a major drawback of the applicability of the geometric approach is the lack of algorithms for the computation of friends that deliver a robust closed-loop eigenstructure, in which the closed-loop eigenvalues are rendered insensitive to perturbations in the state matrices. The classical methods for the computation of friends do not attempt to address the robustness aspects of the problem and leave unexploited all the degrees of freedom in the computation of the friend.…”