Regional pole assignment with eigenstructure robustness

Abstract

The paper provides a computational procedure for a type ofrobust regional pole assignment problem. It allows closed-loop poles to be settled at certain perturbation insensitive locations within some prespecified regions in the complex plane. The novelty ofour approach lies in the versatility of the proposed algorithm which provides a rich set of constrained subregions applicable for the assignment of individual or subsets of closedloop poles. in contrast to other conventional regional pole assignment methods. The algorithm is based on a gradient flow formulation on a differentiable potentialfunction which provides II minimizing solution for the Frobenius condition number of the closedloop eigenvector matrix. A numerical example is employed to illustrate the technique.The improvement on the eigenstructure robustness is compared with different kinds of constrained region.

Robust eigenstructure assignment with minimum subspace separation

“…with w being the normalized eigenvector corresponding to the maximum eigenvalue of P: The proof of the above gradient formulas can be obtained in a similar way as described in the robust regional pole assignment framework developed by Lam and Tam [54].…”

Closed-form solutions to the matrix equation AX − EXF = BY with F in companion form

“…where row N i p−1 i=0 is given by (13). The vectors yi, i ∈ I [1, p] can also be obtained via the same procedure.…”

“…The linear matrix equation (3) is closely related with many problems in conventional linear control systems theory, such as pole/eigenstructure assignment design [1−5] , Luenberger-type observer design [6−9] , robust fault detection [10−12] , regional pole assignment [13] , robust partial pole-placement [14] , constraint control [15] and so on, and has been investigated by several researchers [5, 16−20] . When dealing with eigenstructure assignment, observer design and model reference control for descriptor linear systems, the more general linear matrix equation (1), with E being usually singular, is encountered.…”

Robust matrix 𝒟U‐stability analysis

“…Nominal transient performances might be obtained by assigning closed-loop state matrix eigenvalues that strongly influence settling time, damping ratio and so on. Nevertheless, rather than to require most of the flexibility offered by the multivariable control law by assigning a strict spectrum, it may be more discerning to locate the desired eigenvalues in some appropriate areas of tolerance [1,2]. Furthermore, a strict placement seems utopian while the model is just an approximation of the real system behaviour.…”

“…Then a point is raised: how to get both required performances and robustness against uncertainty while computing a pole placement? To this crucial question, some authors answer by reaching some satisfying nominal performances through a pole placement while improving an index of closed-loop system robustness owing to the degrees of freedom provided by eigenvectors and sets of tolerance [1,2]. Such works require to define analytical robustness criteria in the sense of eigenvalue location.…”