Pole placement design
K = place(A,B,p)
[K,prec,message] = place(A,B,p)
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 u =
–Kx places the closed-loop poles at the
locations p. In other words, the eigenvalues of A – BK match
the entries of p (up to the ordering).
K = place(A,B,p) places the desired closed-loop poles p by computing a state-feedback gain matrix K. All the inputs of the plant are assumed to be 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 A or B.
[K,prec,message] = place(A,B,p) returns prec,
an 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).
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).'
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 at p =
[-1 -1.23 -5.0] by
p = [-1 -1.23 -5.0]; K = place(a,b,p)
place 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.
[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.