place_poles(A, B, poles, method='YT', rtol=0.001, maxiter=30)
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.
The Tits and Yang (YT), paper is an update of the original Kautsky et al. (KNV) paper . 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 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.
is available as a technical report on the following URL: https://hdl.handle.net/1903/5598
State-space representation of linear system AX + BU
.
Desired real poles and/or complex conjugates poles. Complex poles are only supported with method="YT"
(default).
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.
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 :None:None:`rtol` the algorithm stops. Default is 1e-3.
Maximum number of iterations to compute the gain matrix. Default is 30.
full_state_feedback is composed of:
gain_matrix
gain_matrix
computed_poles
computed_poles
requested_poles
requested_poles
X
X
rtol
rtol
nb_iter
nb_iter
Compute K such that eigenvalues (A - dot(B, K))=poles.
A simple example demonstrating real pole placement using both KNV and YT algorithms. This is example number 1 from section 4 of the reference KNV publication ():
>>> from scipy import signal
... import matplotlib.pyplot as plt
>>> A = np.array([[ 1.380, -0.2077, 6.715, -5.676 ],
... [-0.5814, -4.290, 0, 0.6750 ],
... [ 1.067, 4.273, -6.654, 5.893 ],
... [ 0.0480, 4.273, 1.343, -2.104 ]])
... B = np.array([[ 0, 5.679 ],
... [ 1.136, 1.136 ],
... [ 0, 0, ],
... [-3.146, 0 ]])
... P = np.array([-0.2, -0.5, -5.0566, -8.6659])
Now compute K with KNV method 0, with the default YT method and with the YT method while forcing 100 iterations of the algorithm and print some results after each call.
>>> fsf1 = signal.place_poles(A, B, P, method='KNV0')
... fsf1.gain_matrix array([[ 0.20071427, -0.96665799, 0.24066128, -0.10279785], [ 0.50587268, 0.57779091, 0.51795763, -0.41991442]])
>>> fsf2 = signal.place_poles(A, B, P) # uses YT method
... fsf2.computed_poles array([-8.6659, -5.0566, -0.5 , -0.2 ])
>>> fsf3 = signal.place_poles(A, B, P, rtol=-1, maxiter=100)
... fsf3.X array([[ 0.52072442+0.j, -0.08409372+0.j, -0.56847937+0.j, 0.74823657+0.j], [-0.04977751+0.j, -0.80872954+0.j, 0.13566234+0.j, -0.29322906+0.j], [-0.82266932+0.j, -0.19168026+0.j, -0.56348322+0.j, -0.43815060+0.j], [ 0.22267347+0.j, 0.54967577+0.j, -0.58387806+0.j, -0.40271926+0.j]])
The absolute value of the determinant of X is a good indicator to check the robustness of the results, both 'KNV0'
and 'YT'
aim at maximizing it. Below a comparison of the robustness of the results above:
>>> abs(np.linalg.det(fsf1.X)) < abs(np.linalg.det(fsf2.X)) True
>>> abs(np.linalg.det(fsf2.X)) < abs(np.linalg.det(fsf3.X)) True
Now a simple example for complex poles:
>>> A = np.array([[ 0, 7/3., 0, 0 ],
... [ 0, 0, 0, 7/9. ],
... [ 0, 0, 0, 0 ],
... [ 0, 0, 0, 0 ]])
... B = np.array([[ 0, 0 ],
... [ 0, 0 ],
... [ 1, 0 ],
... [ 0, 1 ]])
... P = np.array([-3, -1, -2-1j, -2+1j]) / 3.
... fsf = signal.place_poles(A, B, P, method='YT')
We can plot the desired and computed poles in the complex plane:
>>> t = np.linspace(0, 2*np.pi, 401)See :
... plt.plot(np.cos(t), np.sin(t), 'k--') # unit circle
... plt.plot(fsf.requested_poles.real, fsf.requested_poles.imag,
... 'wo', label='Desired')
... plt.plot(fsf.computed_poles.real, fsf.computed_poles.imag, 'bx',
... label='Placed')
... plt.grid()
... plt.axis('image')
... plt.axis([-1.1, 1.1, -1.1, 1.1])
... plt.legend(bbox_to_anchor=(1.05, 1), loc=2, numpoints=1)
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.signal._ltisys.place_poles
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