## Generalized Assignment Problem Matlab For Loop

## Description

Given the single- or multi-input system

and a vector of desired self-conjugate closed-loop pole locations, computes a gain matrix such that the state feedback *u* = –*Kx* places the closed-loop poles at the locations . In other words, the eigenvalues of *A* – *BK* match the entries of (up to the ordering).

places the desired closed-loop poles by computing a state-feedback gain matrix . All the inputs of the plant are assumed to be control inputs. The length of must match the row size of . works for multi-input systems and is based on the algorithm from [1]. This algorithm uses the extra degrees of freedom to find a solution that minimizes the sensitivity of the closed-loop poles to perturbations in *A* or *B*.

returns , an estimate of how closely the eigenvalues of *A* – *BK* match the specified locations ( measures the number of accurate decimal digits in the actual closed-loop poles). If some nonzero closed-loop pole is more than 10% off from the desired location, contains a warning message.

You can also use for estimator gain selection by transposing the matrix and substituting for .

Consider a state-space system with two inputs, three outputs, and three states. You can compute the feedback gain matrix needed to place the closed-loop poles at by

## Algorithms

uses the algorithm of [1] which, for multi-input systems, optimizes the choice of eigenvectors for a robust solution.

In high-order problems, some choices of pole locations result in very large gains. The sensitivity problems attached with large gains suggest caution in the use of pole placement techniques. See [2] for results from numerical testing.

## References

[1] Kautsky, J., N.K. Nichols, and P. Van Dooren, "Robust Pole Assignment in Linear State Feedback," *International Journal of Control,* 41 (1985), pp. 1129-1155.

[2] Laub, A.J. and M. Wette, *Algorithms and Software for Pole Assignment and Observers*, UCRL-15646 Rev. 1, EE Dept., Univ. of Calif., Santa Barbara, CA, Sept. 1984.

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