{"id":2799,"date":"2011-01-17T11:36:39","date_gmt":"2011-01-17T10:36:39","guid":{"rendered":"http:\/\/www.walkingrandomly.com\/?p=2799"},"modified":"2011-01-17T11:36:39","modified_gmt":"2011-01-17T10:36:39","slug":"zero-pole-gain-models-in-mathematica-8","status":"publish","type":"post","link":"https:\/\/walkingrandomly.com\/?p=2799","title":{"rendered":"Zero-Pole-Gain models in Mathematica 8"},"content":{"rendered":"

MATLAB’s control system toolbox has functions for converting transfer functions<\/a> to either state space representations<\/a> or zero-pole-gain models as follows<\/p>\n

%Create a simple transfer function in MATLAB\r\nnum=[2 3];\r\nden = [1 4 0 5];\r\nmytf=tf(num,den);\r\n\r\n%convert this to State Space form\r\ntf2ss(num,den)\r\nans =\r\n\r\n0   1   0\r\n0   0   1\r\n-5  -0  -4\r\n\r\n%Convert to zero-pole-gain form\r\n[z,p,k]=tf2zp(num,den)\r\nz = -1.5000\r\np =\r\n\r\n-4.27375 + 0.00000i\r\n0.13687 + 1.07294i\r\n0.13687 - 1.07294i\r\n\r\nk =  2<\/pre>\n
Now that Mathematica 8 is available I wondered what the Mathematica equivalent to the above would look like.  The conversion to a StateSpace model is easy:<\/p>\n
mytf=TransferFunctionModel[(2 s + 3)\/(s^3 + 4 s^2 + 5), s]\r\nStateSpaceModel[mytf]<\/pre>\n
\"Transfer<\/p>\n
\"State<\/p>\n
but I couldn’t find an equivalent to MATLAB’s tf2zp<\/a> function.<\/p>\n
I can get the poles and zeros easily enough (as an aside it’s nice that Mathematica can do this exactly)<\/p>\n
TransferFunctionPoles[mytf]\r\nTransferFunctionZeros[mytf]<\/pre>\n
\"State
\nbut what about gain?\u00a0 There is no TransferFunctionGain[] function as you might expect. There is a TransferFunctionFactor[] function but that doesn’t do what I want either.<\/p>\n
It turns out that there is a function that will do what I want but it is hidden from the user a little in an internal function<\/p>\n
Control`ZeroPoleGainModel[mytf]\r\n\r\nControl`ZeroPoleGainModel[{{{{-(3\/2)}}}, {1\/\r\n     3 (-4 - 16 (2\/(263 - 3 Sqrt[5865]))^(\r\n        1\/3) - (1\/2 (263 - 3 Sqrt[5865]))^(1\/3)), -(4\/3) +\r\n     8\/3 (1 + I Sqrt[3]) (2\/(263 - 3 Sqrt[5865]))^(1\/3) +\r\n     1\/6 (1 - I Sqrt[3]) (1\/2 (263 - 3 Sqrt[5865]))^(1\/3), -(4\/3) +\r\n     8\/3 (1 - I Sqrt[3]) (2\/(263 - 3 Sqrt[5865]))^(1\/3) +\r\n     1\/6 (1 + I Sqrt[3]) (1\/2 (263 - 3 Sqrt[5865]))^(1\/3)}, {{2}}}, s]<\/pre>\n
That’s not as user friendly as it could be though.  How about this?<\/p>\n
tf2zpk[x_] := Module[{z, p, k,zpkmodel},\r\n  zpkmodel = Control`ZeroPoleGainModel[x];\r\n  z = zpkmodel[[1, 1, 1, 1]];\r\n  p = zpkmodel[[1, 2]];\r\n  k = zpkmodel[[1, 3, 1]];\r\n  {z, p, k}\r\n  ]<\/pre>\n
Now we can get very similar behavior to MATLAB:<\/p>\n
In[10]:= tf2zpk[mytf] \/\/ N\r\n\r\nOut[10]= {{-1.5}, {-4.27375, 0.136874 + 1.07294 I,\r\n  0.136874 - 1.07294 I}, {2.}}<\/pre>\n
Hope this helps someone out there.\u00a0 Thanks to my contact at Wolfram who told me about the Control`ZeroPoleGainModel function.<\/p>\n
Other posts you may be interested in<\/strong><\/p>\n