place

Pole placement design

Syntax

K = place(A,B,p) [K,prec,message] = place(A,B,p)

Description

Given the single- or multi-input system

$\dot{x} = A x + B u$

and a vectorpof desired self-conjugate closed-loop pole locations,placecomputes a gain matrixKsuch that the state feedbacku= –Kxplaces the closed-loop poles at the locationsp. In other words, the eigenvalues ofA–BKmatch the entries ofp(up to the ordering).

K = place(A,B,p)places the desired closed-loop polespby computing a state-feedback gain matrixK. All the inputs of the plant are assumed to be control inputs. The length ofpmust match the row size ofA.placeworks 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 inAorB.

[K,prec,message] = place(A,B,p)returnsprec, an estimate of how closely the eigenvalues ofA–BKmatch the specified locationsp(precmeasures 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,messagecontains a warning message.

You can also useplacefor estimator gain selection by transposing theAmatrix and substitutingC'forB.

l = place(A',C',p).'

Examples

Pole Placement Design

Consider a state-space system(a,b,c,d)with two inputs, three outputs, and three states. You can compute the feedback gain matrix needed to place the closed-loop poles atp = [-1 -1.23 -5.0]by

p = [-1 -1.23 -5.0]; K = place(a,b,p)

Algorithms

placeuses 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.

版本历史

Introduced before R2006a