Control System Toolbox™

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

and a vector p of desired self-conjugate closed-loop pole locations, place computes a gain matrix K such that the state feedback places the closed-loop poles at the locations p. In other words, the eigenvalues of match the entries of p (up to the ordering).

K = place(A,B,p) computes a feedback gain matrix K that achieves the desired closed-loop pole locations p, assuming all the inputs of the plant are control inputs. The length of p must match the row size of A. place 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 or .

[K,prec,message] = place(A,B,p) also returns prec, an estimate of how closely the eigenvalues of match the specified locations p (prec 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, message contains a warning message.

You can also use place for estimator gain selection by transposing the A matrix and substituting C' for B.

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

Example

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 at p = [-1 -1.23 -5.0] by

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

Algorithm

place uses the algorithm of [1] which, for multi-input systems, optimizes the choice of eigenvectors for a robust solution. We recommend place rather than acker even for single-input systems.

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. and N.K. Nichols, "Robust Pole Assignment in Linear State Feedback," Int. J. Control, 41 (1985), pp. 1129-1155.