This is a post on serendipity, because what this is about is actually neither control systems, Z<\/em>-, or Laplace transforms but a numbers game. Bare with me.<\/p>\n\n\n\n Many systems can be modelled using a rational function of the form<\/p>\n\n\n You can easily put this into Nice. Let’s generalise this a little bit:<\/p>\n\n\n\n By playing with the values for Then there is the bilinear transform called Tustin’s method that approximately maps the function G<\/em> from the Laplace domain into the Z<\/em> domain.<\/p>\n\n\n Let’s do this in Sweet. Now, I want to have the representation in two polynomials in z<\/em> (numerator and denominator) to extract the coefficients (because that’s what I wanted to hack into code):<\/p>\n\n\n\n This gives you a nice, codeable result:<\/p>\n\n\n\n … but my eye spotted a pattern here. Do you see it? It’s easier to see in matrix representation, and I’ll only look at Here is the resulting matrix for N = M = 3 and for N = M = 5:<\/p>\n\n\n Here are the resulting matrices for N = M = 4:<\/p>\n\n\n Here are the resulting matrices for N = M = 6<\/p>\n\n\n The odd values for N and M produce matrices with all non-zero values, so I’ll look at those first.<\/p>\n\n\n\n What I find stunning is the fact that there are so many patterns and yet I fail to think of a pattern that creates the whole matrix. For example, the first column clearly follows the binomial distribution:<\/p>\n\n\n where k<\/em> is the row number starting with 0.<\/p>\n\n\n\n The first and last rows of the matrix are power series of 2 and -2 respectively:<\/p>\n\n\n where l<\/em> is the column number starting at 0.<\/p>\n\n\n\n Every column has a different sign pattern starting at all positive<\/em> on column 0 and alternating, starting with +<\/em> on the last column.<\/p>\n\n\n\n Ignoring the signs, the columns are related like this<\/p>\n\n\n\n Maybe this follows from the power series mentioned earlier. But looking at the coefficients in each column the binomial pattern is lost as you move to the inside.<\/p>\n\n\n\n And last but not least wild zeros appear when you look at the even values for N and M:<\/p>\n\n\n\n Apparently, in the middle row as well as the middle column, every other value is zero.<\/p>\n\n\n\n All right, I’m absolutely sure mathematicians and control systems theoreticians have looked at this back in the day and this is all solved and done. Still, I found it interesting to look at.<\/p>\n\n\n\n If you want to create those values yourself, you can use this little script, and modify Oh, and of course you can change Recently, I was working on digital control systems, namely PID controllers. But this proved to become a weird pattern generator.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[1],"tags":[30,31,29],"class_list":["post-175","post","type-post","status-publish","format-standard","hentry","category-uncategorized","tag-mathematics","tag-numbers","tag-sympy"],"_links":{"self":[{"href":"https:\/\/bowfinger.de\/blog\/wp-json\/wp\/v2\/posts\/175","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/bowfinger.de\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/bowfinger.de\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/bowfinger.de\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/bowfinger.de\/blog\/wp-json\/wp\/v2\/comments?post=175"}],"version-history":[{"count":52,"href":"https:\/\/bowfinger.de\/blog\/wp-json\/wp\/v2\/posts\/175\/revisions"}],"predecessor-version":[{"id":233,"href":"https:\/\/bowfinger.de\/blog\/wp-json\/wp\/v2\/posts\/175\/revisions\/233"}],"wp:attachment":[{"href":"https:\/\/bowfinger.de\/blog\/wp-json\/wp\/v2\/media?parent=175"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/bowfinger.de\/blog\/wp-json\/wp\/v2\/categories?post=175"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/bowfinger.de\/blog\/wp-json\/wp\/v2\/tags?post=175"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}![\"](\"https:\/\/bowfinger.de\/blog\/wp-content\/ql-cache\/quicklatex.com-0d1b4c7b91f7cc326bd87fab8fc9cb57_l3.png\" "\\"Rendered")
<\/p>\n\n\n\nsympy<\/code> and play around with it.<\/p>\n\n\n\nimport sympy as sy\n\ns, z, Ts = sy.symbols(\"s, z, T_s\")\n\na0 = sy.S.One\na1, a2 = sy.symbols(\"alpha_1, alpha_2\")\nb0, b1, b2 = sy.symbols(\"beta_0, beta_1, beta_2\")\n\nG = (b0 * s**2 + b1 * s + b2) \/ (a0 * s**2 + a1 * s + a2)\nG # use `print(sy.latex(G))` to get the LaTeX representation<\/code><\/pre>\n\n\n![\"](\"https:\/\/bowfinger.de\/blog\/wp-content\/ql-cache\/quicklatex.com-9c972f6fcf78ff27f4f7e7825a131d56_l3.png\" "\\"Rendered")
<\/p>\n\n\n\nimport sympy as sy\n\ns, z, Ts = sy.symbols(\"s, z, T_s\")\n\nN, M = 2, 2\n\na = sy.symbols(f\"alpha_1:{N+1}\")\na = (sy.S.One, *a)\nb = sy.symbols(f\"beta_0:{M+1}\")\n\nG = sy.Add(*[b[i] * s**i for i in range(M+1)]) \/ sy.Add(*[a[i] * s**i for i in range(N+1)])<\/code><\/pre>\n\n\n\nN<\/code> and M<\/code> we can now generate any order of rational function we wish. Again, not the thing this article is about.<\/p>\n\n\n\n![\"](\"https:\/\/bowfinger.de\/blog\/wp-content\/ql-cache\/quicklatex.com-bef9bdc094adebe0f86ae6bcd34849fb_l3.png\" "\\"Rendered")
<\/p>\n\n\n\nsympy<\/code>:<\/p>\n\n\n\ntustin = 2\/Ts * (z-1)\/(z+1)\n\nG.subs({s: tustin}).simplify()<\/code><\/pre>\n\n\n![\"](\"https:\/\/bowfinger.de\/blog\/wp-content\/ql-cache\/quicklatex.com-ee1a1f8d9dae033c13bc580b6ebdb54d_l3.png\" "\\"Rendered")
<\/p>\n\n\n\nnum, den = G.as_numer_denom()\nnum = sy.Poly(num, z)\nden = sy.Poly(den, z)\n\nnum.coeffs()\nden.coeffs()<\/code><\/pre>\n\n\n\nnum: [ T_s**2*beta_0 + 2*T_s*beta_1 + 4*beta_2,\n 2*T_s**2*beta_0 - 8*beta_2,\n T_s**2*beta_0 - 2*T_s*beta_1 + 4*beta_2]\nden: [ T_s**2 + 2*T_s*alpha_1 + 4*alpha_2,\n 2*T_s**2 - 8*alpha_2,\n T_s**2 - 2*T_s*alpha_1 + 4*alpha_2]<\/pre>\n\n\n\n
num<\/code> (i.e. the expression with \u03b2i<\/sub><\/em> in it, because I set \u02510<\/sub> to 1):<\/p>\n\n\n![\"](\"https:\/\/bowfinger.de\/blog\/wp-content\/ql-cache\/quicklatex.com-0111e93f51a33f323518a06ce9b77417_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"](\"https:\/\/bowfinger.de\/blog\/wp-content\/ql-cache\/quicklatex.com-1dcb0a9e54b893690c2472b5a7a7f433_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"](\"https:\/\/bowfinger.de\/blog\/wp-content\/ql-cache\/quicklatex.com-d20fc9b003b94abf81b6ecbcf9ceeb43_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"](\"https:\/\/bowfinger.de\/blog\/wp-content\/ql-cache\/quicklatex.com-820ba4837214deab8e2d618e1f703e23_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"\begin{pmatrix}N \\ k\end{pmatrix}\"](\"https:\/\/bowfinger.de\/blog\/wp-content\/ql-cache\/quicklatex.com-a7684aab495933bff8c72f4c31be9ad0_l3.png\" "\\"Rendered")
<\/p>\n\n\n\n![\"](\"https:\/\/bowfinger.de\/blog\/wp-content\/ql-cache\/quicklatex.com-708eda5fdbd59193ec017ea25de109ae_l3.png\" "\\"Rendered")
<\/p>\n\n\n\nN<\/td> #0 \u21b7 #N-1<\/td> #1 \u21b7 #N-2<\/td> #2 \u21b7 #N-3<\/td> #3 \u21b7 #N-4<\/td><\/tr> 3<\/td> \u0fbe 8<\/td> \u0fbe 2<\/td> <\/td> <\/td><\/tr> 5<\/td> \u0fbe 32<\/td> \u0fbe 8<\/td> \u0fbe 2<\/td> <\/td><\/tr> 7<\/td> \u0fbe 128<\/td> \u0fbe 32<\/td> \u0fbe 8<\/td> \u0fbe 2<\/td><\/tr><\/tbody><\/table> N<\/code> and M<\/code>. But beware, at N = 10 the script runs for 10 seconds. <\/p>\n\n\n\nimport sympy as sy\n\ns, z, Ts = sy.symbols(\"s, z, T_s\")\n\nN, M = 3, 3 # <-- Modify here\n\na = sy.symbols(f\"alpha_1:{N+1}\")\na = (sy.S.One, *a)\nb = sy.symbols(f\"beta_0:{M+1}\")\n\nG = sy.Add(*[b[i] * s**i for i in range(M+1)]) \/ sy.Add(*[a[i] * s**i for i in range(N+1)])\n\ntustin = 2\/Ts * (z-1)\/(z+1)\nG = G.subs({s: tustin}).simplify()\n\nnum, den = G.as_numer_denom()\nnum = sy.Poly(num, z)\nden = sy.Poly(den, z)\n\nfor l in num.coeffs():\n p = sy.Poly(l, Ts)\n for i, B in enumerate(p.all_coeffs()):\n try:\n coeff, _ = B.as_two_terms()\n except:\n coeff = 0 if B is sy.S.Zero else 1\n print(f\"{repr(coeff):6s} & \", end=\"\")\n print()<\/code><\/pre>\n\n\n\nN<\/code> and M<\/code> independently from each other, but I could not be bothered to look at this as well.<\/p>\n","protected":false},"excerpt":{"rendered":"