{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The Tits and Yang (YT), " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" }, { "__type": "Text", "__tag": 4046, "value": " paper is an update of the original Kautsky et al. (KNV) paper " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "1" }, { "__type": "Text", "__tag": 4046, "value": ". KNV relies on rank-1 updates to find the transfer matrix X such that " }, { "__type": "InlineCode", "__tag": 4051, "value": "X * diag(poles) = (A - B*K)*X" }, { "__type": "Text", "__tag": 4046, "value": ", whereas YT uses rank-2 updates. This yields on average more robust solutions (see " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" }, { "__type": "Text", "__tag": 4046, "value": " 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 " }, { "__type": "InlineCode", "__tag": 4051, "value": "'KNV0'" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "KNV extended to complex poles is used in Matlab's " }, { "__type": "InlineCode", "__tag": 4051, "value": "place" }, { "__type": "Text", "__tag": 4046, "value": " function, YT is distributed under a non-free licence by Slicot under the name " }, { "__type": "InlineCode", "__tag": 4051, "value": "robpole" }, { "__type": "Text", "__tag": 4046, "value": ". 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." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "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. 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Default is 30." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "K is the gain matrix such as the plant described by the linear system " }, { "__type": "InlineCode", "__tag": 4051, "value": "AX+BU" }, { "__type": "Text", "__tag": 4046, "value": " will have its closed-loop poles, i.e the eigenvalues " }, { "__type": "InlineCode", "__tag": 4051, "value": "A - B*K" }, { "__type": "Text", "__tag": 4046, "value": ", as close as possible to those asked for in poles." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "SISO, MISO and MIMO systems are supported." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/signal/_ltisys.py", "item_line": 2990, "item_type": "function", "aliases": [ "scipy.signal.place_poles" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "A simple example demonstrating real pole placement using both KNV and YT\nalgorithms. This is example number 1 from section 4 of the reference KNV\npublication ([1]_):\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy import signal\nimport matplotlib.pyplot as plt\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "A = np.array([[ 1.380, -0.2077, 6.715, -5.676 ],\n [-0.5814, -4.290, 0, 0.6750 ],\n [ 1.067, 4.273, -6.654, 5.893 ],\n [ 0.0480, 4.273, 1.343, -2.104 ]])\nB = np.array([[ 0, 5.679 ],\n [ 1.136, 1.136 ],\n [ 0, 0, ],\n [-3.146, 0 ]])\nP = np.array([-0.2, -0.5, -5.0566, -8.6659])\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nNow compute K with KNV method 0, with the default YT method and with the YT\nmethod while forcing 100 iterations of the algorithm and print some results\nafter each call.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "fsf1 = signal.place_poles(A, B, P, method='KNV0')\nfsf1.gain_matrix\n", "execution_status": "success", "out": "array([[ 0.20071427, -0.96665799, 0.24066128, -0.10279785],\n [ 0.50587268, 0.57779091, 0.51795763, -0.41991442]])\n" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "fsf2 = signal.place_poles(A, B, P) # uses YT method\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "fsf2.computed_poles\n", "execution_status": "failure", "out": "array([-8.6659+0.j, -5.0566+0.j, -0.5 +0.j, -0.2 +0.j])\n" }, { "__type": "Text", "__tag": 4046, "value": "\n" }, { "__type": "Code", "__tag": 4050, "value": "fsf3 = signal.place_poles(A, B, P, rtol=-1, maxiter=100)\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "fsf3.X\n", "execution_status": "failure", "out": "array([[ 0.52072442+0.j, -0.08409372+0.j, -0.56847937+0.j,\n 0.74823657+0.j],\n [-0.04977751+0.j, -0.80872954+0.j, 0.13566234+0.j,\n -0.29322906+0.j],\n [-0.82266932+0.j, -0.19168026+0.j, -0.56348322+0.j,\n -0.4381506 +0.j],\n [ 0.22267347+0.j, 0.54967577+0.j, -0.58387806+0.j,\n -0.40271926+0.j]])\n" }, { "__type": "Text", "__tag": 4046, "value": "\nThe absolute value of the determinant of X is a good indicator to check the\nrobustness of the results, both ``'KNV0'`` and ``'YT'`` aim at maximizing\nit. Below a comparison of the robustness of the results above:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "abs(np.linalg.det(fsf1.X)) < abs(np.linalg.det(fsf2.X))\nabs(np.linalg.det(fsf2.X)) < abs(np.linalg.det(fsf3.X))\n", "execution_status": "failure", "out": "np.True_\n" }, { "__type": "Text", "__tag": 4046, "value": "\nNow a simple example for complex poles:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "A = np.array([[ 0, 7/3., 0, 0 ],\n [ 0, 0, 0, 7/9. ],\n [ 0, 0, 0, 0 ],\n [ 0, 0, 0, 0 ]])\nB = np.array([[ 0, 0 ],\n [ 0, 0 ],\n [ 1, 0 ],\n [ 0, 1 ]])\nP = np.array([-3, -1, -2-1j, -2+1j]) / 3.\nfsf = signal.place_poles(A, B, P, method='YT')\n", "execution_status": "success", "out": "" }, { "__type": "Text", "__tag": 4046, "value": "\nWe can plot the desired and computed poles in the complex plane:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "t = np.linspace(0, 2*np.pi, 401)\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "plt.plot(np.cos(t), np.sin(t), 'k--') # unit circle\nplt.plot(fsf.requested_poles.real, fsf.requested_poles.imag,\n 'wo', label='Desired')\nplt.plot(fsf.computed_poles.real, fsf.computed_poles.imag, 'bx',\n label='Placed')\n", "execution_status": "failure", "out": "[]\n" }, { "__type": "Code", "__tag": 4050, "value": "plt.grid()\n", "execution_status": "success", "out": "" }, { "__type": "Code", "__tag": 4050, "value": "plt.axis('image')\nplt.axis([-1.1, 1.1, -1.1, 1.1])\nplt.legend(bbox_to_anchor=(1.05, 1), loc=2, numpoints=1)\n", "execution_status": "failure", "out": "\n" } ], "title": [], "level": 0, "target": null }, "see_also": [], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "A", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "B", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "poles", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "method", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "YT" }, { "__type": "SigParam", "__tag": 4030, "name": "rtol", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0.001" }, { "__type": "SigParam", "__tag": 4030, "name": "maxiter", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "30" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "place_poles" }, "references": [ ".. [1] J. Kautsky, N.K. Nichols and P. van Dooren, \"Robust pole assignment", " in linear state feedback\", International Journal of Control, Vol. 41", " pp. 1129-1155, 1985.", ".. [2] A.L. Tits and Y. Yang, \"Globally convergent algorithms for robust", " pole assignment by state feedback\", IEEE Transactions on Automatic", " Control, Vol. 41, pp. 1432-1452, 1996." ], "qa": "scipy.signal._ltisys:place_poles", "arbitrary": [], "local_refs": [ "A", "B", "full_state_feedback", "maxiter", "method", "poles", "rtol" ] }