# place

Pole placement design

## Description

Pole placement is a method of calculating the optimum gain matrix used to assign closed-loop poles to specified locations, thereby ensuring system stability. Closed-loop pole locations have a direct impact on time response characteristics such as rise time, settling time, and transient oscillations. For more information, see Pole Placement.

From the figure, consider a linear dynamic system in state-space form

$$\dot{x}=Ax+Bu$$

$$y=Cx+Du$$

For a given vector `p`

of desired self-conjugate
closed-loop pole locations, `place`

computes a gain matrix
`K`

such that the state feedback *u* =
–*Kx* places the poles at the locations `p`

. In other
words, the eigenvalues of *A* – *BK* will match the entries
of `p`

(up to the ordering).

places the desired closed-loop poles `K`

= place(`A`

,`B`

,`p`

)`p`

by computing a state-feedback gain
matrix `K`

. All the inputs of the plant are assumed to be control inputs.
`place`

also 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*.

`[`

also returns `K`

,`prec`

] = place(`A`

,`B`

,`p`

)`prec`

, an accuracy estimate of how closely the eigenvalues of
*A* – *BK* match the specified locations
`p`

(`prec`

measures the number of accurate decimal
digits in the actual closed-loop poles). A warning is issued if some nonzero closed-loop
pole is more than 10% off from the desired location.

## Examples

## Input Arguments

## Output Arguments

## Tips

You can use

`place`

for estimator gain selection by transposing the`A`

matrix and substituting`C'`

for matrix`B`

as follows, as shown in Pole Placement Observer Design. You can use the resultant estimator gain for state estimator workflows using`estim`

.

## 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.

## Version History

**Introduced before R2006a**