cupyx.scipy.signal.place_poles#

cupyx.scipy.signal.place_poles(A, B, poles, method='YT', rtol=0.001, maxiter=30)[source]#

Compute K such that eigenvalues (A - dot(B, K))=poles.

K is the gain matrix such as the plant described by the linear system AX+BU will have its closed-loop poles, i.e the eigenvalues A - B*K, as close as possible to those asked for in poles.

SISO, MISO and MIMO systems are supported.

Parameters:

A (ndarray) – State-space representation of linear system AX + BU.
B (ndarray) – State-space representation of linear system AX + BU.
poles (array_like) – Desired real poles and/or complex conjugates poles. Complex poles are only supported with method="YT" (default).
method ({'YT', 'KNV0'}, optional) –
Which method to choose to find the gain matrix K. One of:
- ’YT’: Yang Tits
- ’KNV0’: Kautsky, Nichols, Van Dooren update method 0
See References and Notes for details on the algorithms.
rtol (float, optional) – After each iteration the determinant of the eigenvectors of A - B*K is compared to its previous value, when the relative error between these two values becomes lower than rtol the algorithm stops. Default is 1e-3.
maxiter (int, optional) – Maximum number of iterations to compute the gain matrix. Default is 30.

Returns:

full_state_feedback –

full_state_feedback is composed of:

gain_matrix1-D ndarray: The closed loop matrix K such as the eigenvalues of A-BK are as close as possible to the requested poles.
computed_poles1-D ndarray: The poles corresponding to A-BK sorted as first the real poles in increasing order, then the complex congugates in lexicographic order.
requested_poles1-D ndarray: The poles the algorithm was asked to place sorted as above, they may differ from what was achieved.
X2-D ndarray: The transfer matrix such as X * diag(poles) = (A - B*K)*X (see Notes)
rtolfloat: The relative tolerance achieved on det(X) (see Notes). rtol will be NaN if it is possible to solve the system diag(poles) = (A - B*K), or 0 when the optimization algorithms can’t do anything i.e when B.shape[1] == 1.
nb_iterint: The number of iterations performed before converging. nb_iter will be NaN if it is possible to solve the system diag(poles) = (A - B*K), or 0 when the optimization algorithms can’t do anything i.e when B.shape[1] == 1.

Return type:

Bunch object

Notes

The Tits and Yang (YT), [2] paper is an update of the original Kautsky et al. (KNV) paper [1]. KNV relies on rank-1 updates to find the transfer matrix X such that X * diag(poles) = (A - B*K)*X, whereas YT uses rank-2 updates. This yields on average more robust solutions (see [2] pp 21-22), furthermore the YT algorithm supports complex poles whereas KNV does not in its original version. Only update method 0 proposed by KNV has been implemented here, hence the name 'KNV0'.

KNV extended to complex poles is used in Matlab’s place function, YT is distributed under a non-free licence by Slicot under the name robpole. It is unclear and undocumented how KNV0 has been extended to complex poles (Tits and Yang claim on page 14 of their paper that their method can not be used to extend KNV to complex poles), therefore only YT supports them in this implementation.

As the solution to the problem of pole placement is not unique for MIMO systems, both methods start with a tentative transfer matrix which is altered in various way to increase its determinant. Both methods have been proven to converge to a stable solution, however depending on the way the initial transfer matrix is chosen they will converge to different solutions and therefore there is absolutely no guarantee that using 'KNV0' will yield results similar to Matlab’s or any other implementation of these algorithms.

Using the default method 'YT' should be fine in most cases; 'KNV0' is only provided because it is needed by 'YT' in some specific cases. Furthermore 'YT' gives on average more robust results than 'KNV0' when abs(det(X)) is used as a robustness indicator.

[2] is available as a technical report on the following URL: https://hdl.handle.net/1903/5598