{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Chapter 3" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "from control.matlab import *" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 伝達関数モデルの記述" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "P(s)= \n", " 1\n", "-------------\n", "s^2 + 2 s + 3\n", "\n" ] } ], "source": [ "Np = [0, 1] # 伝達関数の分子多項式の係数 (0*s + 1)\n", "Dp = [1, 2, 3] # 伝達関数の分母多項式の係数 (1*s^2 + 2*s + 3)\n", "P = tf(Np, Dp)\n", "print('P(s)=', P)" ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "P(s)= \n", " 1\n", "-------------\n", "s^2 + 2 s + 3\n", "\n" ] } ], "source": [ "P = tf([0, 1], [1, 2, 3])\n", "print('P(s)=', P)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 練習問題" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", " s + 2\n", "---------------------\n", "s^3 + 5 s^2 + 3 s + 4" ] }, "execution_count": 4, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P = tf([1, 2], [1, 5, 3, 4])\n", "P" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "分母多項式の展開" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAKUAAAAWCAYAAAChdVwBAAAABHNCSVQICAgIfAhkiAAABHhJREFUaIHt2luoVFUcx/GPF9LMwsrKiLCLXSSNtBAEKzUfMrtLN+gyRRZhmU9iBXV6KO0mZtBDdDGKHiq0CxSaIpRkN8IMi9LMIkrMTne8lJ4e/ns4wz4z58zs2edMI/OFzRnWWnvt/2+tvf/r/1/r0KLFfsJsbMAfybUOMxpqUe9wFz4WGn/GmxjTUItaVOQSXICTcDIewD84vZFG9QIrcKN4EcdiObbhsEYa1aJ62nFro43oZYZiLy5qtCH7M/1z6GMArhET9n4O/dXKVnRUuLbl/KyDxZi159xvtczASvyAndiCVzCxQfbUw3U65+nm0oqBdXQ6VsSSg/EnLsXndfRXD79jcZnyv3J+zmKsxwc591sND2EefsFr2IFRIpSaievxYgPsysKxeELMz9A8Oz5ADMpZWCAGKUsSUBBfy+SMdmxNrt7mEfwkNGehILvOESJs2IYjU3VTkn63ZLSrWgrqm6ci/bAK34gx7eIpyy3fZ2NZctMubMdHeDDVbg824xORpa7H3DoN7kuq1QmPCU90ntDc14wUc/WhsLOUNWKlOqLCvbXo7AvmYKpIIP8u1yD9Ut6Nd3EmVmOR2AYZhPN7eFh/sZQ3gkG4Vth/p/AeA7ppX4vOJUnfU/BFrlZXzybhBCZgeKruHBHrripzXz3z2RuMxkI8ntjVI0fhX7wnluY0pYOxUHyBx4nYcgH2YXoGQwvqX77LJTlbcG6Z9rXofFLsUU4VS2jxyhIHFdSnc64Y4+14Soz5y8L7rdR1Wa9FZzUU1Gf/QLGqfoUDk7I2PSQ6pwrv8rX4KtPsKPk9QgTVI0SSsUG8kCsyGlwPz4mB3yiWsRNwO27B2yIz/aykfS06b0v+rk61uV8MaF+yWHyAz2JWSflmLNV1Wa9FZ19wL8Zhktg5qIrh+E28uW/gKhyas2FbVd6+KXctreNZjyZ9LE+VN6vOecLzLRIf3hCMF46gAw+n2tejM2/7JyS2p21sU8ZT9ks1GoP7xGnNEJHxvYN78GmVgrpjLoalys4Q2xrP65pFrxfbH1kYJWKxdhyeqms2nZNFQrMcl6fqhghveLQ4YSvNwrPqzNP+gWIV2ys85e6SurbEvll4uht7EDHINBGzdAhXP6inmzJSkM9WQ5pDkn53ddOmWXQWvf4dFeqXJfUzK9TnobMgm/3DVO9xF1N583yPyOZWiXhtkgicv6/RoEZSPOXobv+uWXQWX6BK2z7F8nKxY7G8UTp345kKdeOF91wrEqB1pZXjcGKZm0bhV3wnnyPJchRk9yCnKf/PESPF0t0htkWKNKvOK3Uemx6TqpsusvKdOsOU3tBZkP+K1qab7HsObhCbqhtFJnc8Lk7qbxLC/29cgfki3vpWZN8nijPiwXhLLH1FmlXnq8LLTcOXOv9baTQuFLnBfHEESfPqROdL+Xrye4KY6MH4ES+JM9dNDbGuZ9bgFOEZJuIgkXGuxQvJ1VHSvll17hPJymxcjctE4tIuPrwlYq+ySLPqbNGiRYsWLVq0aNEiF/4DlDWBvTq1630AAAAASUVORK5CYII=\n", "text/latex": [ "$$s^{3} + 5 s^{2} + 8 s + 4$$" ], "text/plain": [ " 3 2 \n", "s + 5⋅s + 8⋅s + 4" ] }, "execution_count": 5, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import sympy as sp\n", "sp.init_printing()\n", "\n", "s = sp.Symbol('s')\n", "sp.expand( (s+1)*(s+2)**2, s)" ] }, { "cell_type": "code", "execution_count": 6, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", " s + 3\n", "---------------------\n", "s^3 + 5 s^2 + 8 s + 4" ] }, "execution_count": 6, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P = tf([1, 3],[1, 5, 8, 4])\n", "P" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", " s + 3\n", "---------------------\n", "s^3 + 5 s^2 + 8 s + 4" ] }, "execution_count": 7, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P1 = tf([1, 3], [0, 1])\n", "P2 = tf([0, 1], [1, 1])\n", "P3 = tf([0, 1], [1, 2])\n", "P = P1 * P2 * P3**2\n", "P" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 分母・分子多項式の係数の抽出" ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[array([1., 3.])]]\n", "[[array([1, 5, 8, 4])]]\n" ] } ], "source": [ "print(P.num)\n", "print(P.den)" ] }, { "cell_type": "code", "execution_count": 9, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[1. 3.]\n", "[1 5 8 4]\n" ] } ], "source": [ "[[numP]], [[denP]] = tfdata(P)\n", "print(numP)\n", "print(denP)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 状態空間モデルの記述" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A = [[ 0 1]\n", " [-1 -1]]\n", "\n", "B = [[0]\n", " [1]]\n", "\n", "C = [[1 0]]\n", "\n", "D = [[0]]\n", "\n" ] } ], "source": [ "A = '0 1; -1 -1'\n", "B = '0; 1'\n", "C = '1 0'\n", "D = '0'\n", "P = ss(A, B, C, D)\n", "print(P)" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A = [[ 0 1]\n", " [-1 -1]]\n", "\n", "B = [[0]\n", " [1]]\n", "\n", "C = [[1 0]]\n", "\n", "D = [[0]]\n", "\n" ] } ], "source": [ "A = [ [0, 1], [-1, -1] ]\n", "B = [ [0], [1] ]\n", "C = [ 1, 0 ]\n", "D = [ 0 ]\n", "P = ss(A, B, C, D)\n", "print(P)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 練習問題" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A = [[1 1 2]\n", " [2 1 1]\n", " [3 4 5]]\n", "\n", "B = [[2]\n", " [0]\n", " [1]]\n", "\n", "C = [[1 1 0]]\n", "\n", "D = [[0]]\n", "\n" ] } ], "source": [ "A = '1 1 2; 2 1 1; 3 4 5'\n", "B = '2; 0; 1'\n", "C = '1 1 0'\n", "D = '0'\n", "P = ss(A, B, C, D)\n", "print(P)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### A,B,C,D行列の抽出" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A= [[1 1 2]\n", " [2 1 1]\n", " [3 4 5]]\n", "B= [[2]\n", " [0]\n", " [1]]\n", "C= [[1 1 0]]\n", "D= [[0]]\n" ] } ], "source": [ "print('A=', P.A)\n", "print('B=', P.B)\n", "print('C=', P.C)\n", "print('D=', P.D)" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A= [[1 1 2]\n", " [2 1 1]\n", " [3 4 5]]\n", "B= [[2]\n", " [0]\n", " [1]]\n", "C= [[1 1 0]]\n", "D= [[0]]\n" ] } ], "source": [ "sysA, sysB, sysC, sysD = ssdata(P)\n", "print('A=', sysA)\n", "print('B=', sysB)\n", "print('C=', sysC)\n", "print('D=', sysD)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## ブロック線図の結合" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", " 1\n", "-----\n", "s + 1\n", "\n", "\n", " s + 1\n", "-----------\n", "s^2 + s + 1\n", "\n" ] } ], "source": [ "S1 = tf( [0, 1], [1, 1])\n", "S2 = tf( [1, 1], [1, 1, 1])\n", "print(S1)\n", "print(S2)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 直列結合" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "S= \n", " s + 1\n", "---------------------\n", "s^3 + 2 s^2 + 2 s + 1\n", "\n", "S= \n", " s + 1\n", "---------------------\n", "s^3 + 2 s^2 + 2 s + 1\n", "\n" ] } ], "source": [ "S = S2 * S1\n", "print('S=', S)\n", "\n", "S = series(S1, S2)\n", "print('S=', S)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "分母分子の共通因子 s+1 が約分されない\n", "この場合は，minreal を使う" ] }, { "cell_type": "code", "execution_count": 17, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "\n", " 1\n", "-----------\n", "s^2 + s + 1" ] }, "execution_count": 17, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S.minreal()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "あるいは，状態空間モデルに変換してから結合する" ] }, { "cell_type": "code", "execution_count": 18, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", " 1\n", "-----------\n", "s^2 + s + 1\n", "\n", "\n", " 1\n", "-----------\n", "s^2 + s + 1\n", "\n" ] } ], "source": [ "S1ss = ss(S1) # 状態空間モデルへの変換\n", "S2ss = ss(S2) # 状態空間モデルへの変換\n", "\n", "S = S1ss * S2ss\n", "print(tf(S))\n", "S = series(S1ss, S2ss)\n", "print(tf(S))" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 並列結合" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "S= \n", " 2 s^2 + 3 s + 2\n", "---------------------\n", "s^3 + 2 s^2 + 2 s + 1\n", "\n", "S= \n", " 2 s^2 + 3 s + 2\n", "---------------------\n", "s^3 + 2 s^2 + 2 s + 1\n", "\n" ] } ], "source": [ "S = S1 + S2\n", "print('S=', S)\n", "S = parallel(S1, S2)\n", "print('S=', S)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### フィードバック結合" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "S= \n", " s^4 + 3 s^3 + 4 s^2 + 3 s + 1\n", "-----------------------------------------------\n", "s^6 + 4 s^5 + 9 s^4 + 13 s^3 + 12 s^2 + 7 s + 2\n", "\n", "S= \n", " s + 1\n", "---------------------\n", "s^3 + 2 s^2 + 3 s + 2\n", "\n" ] } ], "source": [ "S = S1*S2 / (1 + S1*S2)\n", "print('S=', S)\n", "S = feedback(S1*S2, 1)\n", "print('S=', S)" ] }, { "cell_type": "code", "execution_count": 21, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "S= \n", " 1\n", "-----------\n", "s^2 + s + 2\n", "\n" ] } ], "source": [ "print('S=', S.minreal())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "ポジティブフィードバックの場合" ] }, { "cell_type": "code", "execution_count": 22, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", " 1\n", "-------\n", "s^2 + s\n", "\n" ] } ], "source": [ "S = feedback(S1*S2, 1, sign = 1)\n", "print(S.minreal())" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 練習問題" ] }, { "cell_type": "code", "execution_count": 23, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "S1= \n", " 1\n", "-----\n", "s + 1\n", "\n", "S2= \n", " 1\n", "-----\n", "s + 2\n", "\n", "S3= \n", "3 s + 1\n", "-------\n", " s\n", "\n", "S4= \n", "2 s\n", "---\n", " 1\n", "\n", "S= \n", " 3 s^2 + 7 s + 2\n", "--------------------\n", "7 s^3 + 17 s^2 + 7 s\n", "\n" ] } ], "source": [ "S1 = tf(1, [1, 1])\n", "S2 = tf(1, [1, 2])\n", "S3 = tf([3, 1], [1, 0])\n", "S4 = tf([2, 0], [0, 1])\n", "\n", "print('S1=', S1)\n", "print('S2=', S2)\n", "print('S3=', S3)\n", "print('S4=', S4)\n", "\n", "S12 = feedback(S1, S2)\n", "S123 = series(S12, S3)\n", "S = feedback(S123, S4)\n", "print('S=', S)" ] }, { "cell_type": "markdown", "metadata": { "toc-hr-collapsed": true }, "source": [ "## 補遺（実現問題とプロパー性）" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 実現問題" ] }, { "cell_type": "code", "execution_count": 24, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A = [[-1. -1.]\n", " [ 1. 0.]]\n", "\n", "B = [[-1.]\n", " [ 0.]]\n", "\n", "C = [[ 0. -1.]]\n", "\n", "D = [[0.]]\n", "\n", "\n", " 1\n", "-----------\n", "s^2 + s + 1\n", "\n" ] } ], "source": [ "P = tf( [0, 1], [1, 1, 1])\n", "\n", "Pss = tf2ss(P) # 伝達関数モデルから状態空間モデルへの変換\n", "print(Pss)\n", "Ptf = ss2tf(Pss) # 状態空間モデルから伝達関数モデルへの変換\n", "print(Ptf)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 可制御正準形" ] }, { "cell_type": "code", "execution_count": 25, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A = [[1 2 3]\n", " [3 2 1]\n", " [4 5 0]]\n", "\n", "B = [[1]\n", " [0]\n", " [1]]\n", "\n", "C = [[0 2 1]]\n", "\n", "D = [[0]]\n", "\n", "------------\n", "A = [[ 3. 21. 24.]\n", " [ 1. 0. 0.]\n", " [ 0. 1. 0.]]\n", "\n", "B = [[1.]\n", " [0.]\n", " [0.]]\n", "\n", "C = [[ 1. 9. 27.]]\n", "\n", "D = [[0]]\n", "\n" ] } ], "source": [ "from control import canonical_form\n", "\n", "A = '1 2 3; 3 2 1; 4 5 0'\n", "B = '1; 0; 1'\n", "C = '0 2 1'\n", "D = '0'\n", "Pss = ss(A, B, C, D)\n", "\n", "Pr, T = canonical_form(Pss, form='reachable')\n", "print(Pss)\n", "print('------------')\n", "print(Pr)" ] }, { "cell_type": "code", "execution_count": 26, "metadata": {}, "outputs": [], "source": [ "### 可観測正準形" ] }, { "cell_type": "code", "execution_count": 27, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "A = [[ 3. 1. 0.]\n", " [21. 0. 1.]\n", " [24. 0. 0.]]\n", "\n", "B = [[ 1.]\n", " [ 9.]\n", " [27.]]\n", "\n", "C = [[1. 0. 0.]]\n", "\n", "D = [[0]]\n", "\n" ] } ], "source": [ "Po, T = canonical_form(Pss, form='observable')\n", "print(Po)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### プロパー性" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "1\n", "-\n", "1\n", "\n" ] } ], "source": [ "S1 = tf([1, 1], [0, 1])\n", "S2 = tf([0, 1], [1, 1])\n", "\n", "S = series(S1, S2)\n", "print(S.minreal())" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", " 1\n", "-----\n", "s + 1\n", "\n" ] }, { "data": { "text/plain": [ "A = [[-1.]]\n", "\n", "B = [[1.]]\n", "\n", "C = [[1.]]\n", "\n", "D = [[0.]]" ] }, "execution_count": 29, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(S2)\n", "tf2ss(S2)" ] }, { "cell_type": "code", "execution_count": 30, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "s + 1\n", "-----\n", " 1\n", "\n" ] }, { "data": { "text/plain": [ "A = []\n", "\n", "B = []\n", "\n", "C = []\n", "\n", "D = [[1.]]" ] }, "execution_count": 30, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(S1)\n", "tf2ss(S1)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## ラプラス変換" ] }, { "cell_type": "code", "execution_count": 31, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left ( \\frac{1}{s}, \\quad -\\infty, \\quad 0 < \\Re{\\left(s\\right)}\\right )$$" ], "text/plain": [ "⎛1 ⎞\n", "⎜─, -∞, 0 < re(s)⎟\n", "⎝s ⎠" ] }, "execution_count": 31, "metadata": {}, "output_type": "execute_result" } ], "source": [ "import sympy as sp\n", "s = sp.Symbol('s')\n", "t = sp.Symbol('t', positive=True)\n", "sp.init_printing()\n", "sp.laplace_transform(1, t, s)" ] }, { "cell_type": "code", "execution_count": 32, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left ( \\frac{1}{s^{2}}, \\quad -\\infty, \\quad 0 < \\Re{\\left(s\\right)}\\right )$$" ], "text/plain": [ "⎛1 ⎞\n", "⎜──, -∞, 0 < re(s)⎟\n", "⎜ 2 ⎟\n", "⎝s ⎠" ] }, "execution_count": 32, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.laplace_transform(t, t, s)" ] }, { "cell_type": "code", "execution_count": 33, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left ( \\frac{1}{a + s}, \\quad -\\infty, \\quad 0 < \\Re{\\left(s\\right)} \\wedge \\left|{\\arg{\\left (a \\right )}}\\right| < \\frac{\\pi}{2}\\right )$$" ], "text/plain": [ "⎛ 1 π⎞\n", "⎜─────, -∞, 0 < re(s) ∧ │arg(a)│ < ─⎟\n", "⎝a + s 2⎠" ] }, "execution_count": 33, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = sp.Symbol('a', real=True)\n", "sp.laplace_transform(sp.exp(-a*t), t, s)" ] }, { "cell_type": "code", "execution_count": 34, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$\\left ( \\frac{w}{s^{2} + w^{2}}, \\quad -\\infty, \\quad 2 \\left|{\\arg{\\left (w \\right )}}\\right| = 0 \\wedge 0 < \\Re{\\left(s\\right)}\\right )$$" ], "text/plain": [ "⎛ w ⎞\n", "⎜───────, -∞, 2⋅│arg(w)│ = 0 ∧ 0 < re(s)⎟\n", "⎜ 2 2 ⎟\n", "⎝s + w ⎠" ] }, "execution_count": 34, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w = sp.Symbol('w', real=True)\n", "sp.laplace_transform(sp.sin(w*t), t, s)" ] }, { "cell_type": "code", "execution_count": 35, "metadata": {}, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$$\\left ( \\frac{s}{s^{2} + w^{2}}, \\quad -\\infty, \\quad 2 \\left|{\\arg{\\left (w \\right )}}\\right| = 0 \\wedge 0 < \\Re{\\left(s\\right)}\\right )$$" ], "text/plain": [ "⎛ s ⎞\n", "⎜───────, -∞, 2⋅│arg(w)│ = 0 ∧ 0 < re(s)⎟\n", "⎜ 2 2 ⎟\n", "⎝s + w ⎠" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.laplace_transform(sp.cos(w*t), t, s)" ] }, { "cell_type": "code", "execution_count": 36, "metadata": {}, "outputs": [ { "data": { "image/png": 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\n", "text/latex": [ "$$\\left ( \\frac{w}{w^{2} + \\left(a + s\\right)^{2}}, \\quad -\\infty, \\quad 2 \\left|{\\arg{\\left (w \\right )}}\\right| = 0 \\wedge \\left|{2 \\arg{\\left (a \\right )} + 2 \\arg{\\left (1 + \\frac{s}{a} \\right )}}\\right| < \\pi\\right )$$" ], "text/plain": [ "⎛ w │ ⎛ s⎞│ ⎞\n", "⎜─────────────, -∞, 2⋅│arg(w)│ = 0 ∧ │2⋅arg(a) + 2⋅arg⎜1 + ─⎟│ < π⎟\n", "⎜ 2 2 │ ⎝ a⎠│ ⎟\n", "⎝w + (a + s) ⎠" ] }, "execution_count": 36, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.laplace_transform(sp.exp(-a*t)*sp.sin(w*t), t, s)" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\left ( \\frac{a + s}{w^{2} + \\left(a + s\\right)^{2}}, \\quad -\\infty, \\quad 2 \\left|{\\arg{\\left (w \\right )}}\\right| = 0 \\wedge \\left|{2 \\arg{\\left (a \\right )} + 2 \\arg{\\left (1 + \\frac{s}{a} \\right )}}\\right| < \\pi\\right )$$" ], "text/plain": [ "⎛ a + s │ ⎛ s⎞│ ⎞\n", "⎜─────────────, -∞, 2⋅│arg(w)│ = 0 ∧ │2⋅arg(a) + 2⋅arg⎜1 + ─⎟│ < π⎟\n", "⎜ 2 2 │ ⎝ a⎠│ ⎟\n", "⎝w + (a + s) ⎠" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.laplace_transform(sp.exp(-a*t)*sp.cos(w*t), t, s)" ] }, { "cell_type": "markdown", "metadata": { "toc-hr-collapsed": true }, "source": [ "## 逆ラプラス変換" ] }, { "cell_type": "code", "execution_count": 38, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA0AAAASCAYAAACAa1QyAAAABHNCSVQICAgIfAhkiAAAAHZJREFUKJFjYKACCGFgYJjMwMBwmIGB4RMDA8N/BgaGJYQ0XYAq/MzAwHCdWE2ODAwMqgwMDIwMDAwOuDSxoPH3EzKVgYGBgYkYRaOaBlwTeuQGQDEDAwODBJS2ZGBgWABlv2FgYChBN6SBAZJ0cOEH5LiMzgAA6XoX52TB9a4AAAAASUVORK5CYII=\n", "text/latex": [ "$$1$$" ], "text/plain": [ "1" ] }, "execution_count": 38, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(1/s, s, t)" ] }, { "cell_type": "code", "execution_count": 39, "metadata": {}, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAsAAAASCAYAAACNdSR1AAAABHNCSVQICAgIfAhkiAAAAKpJREFUKJHF0D2KQjEYheEHFbUfLNyClQhW7mQQxOZWVu5hCpfhOqxsLAR/qpHZhhaKjRbmDkFuVLDwQPgC583JSXhTY1zwfW+UCuBOmKtXknc4JIL+NQnXF60+VCJ4hSkGWGAWefOi9CwkZUXmfa/8cetHfXMtcUbtGVjBEdsUENdoof6oQgy3w9y8An+FuU/B5WhfxRBdNNFDA7+pwyP84eT23z8p8IO6AiXQHpeF1u16AAAAAElFTkSuQmCC\n", "text/latex": [ "$$t$$" ], "text/plain": [ "t" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(1/s**2, s, t)" ] }, { "cell_type": "code", "execution_count": 40, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$e^{- a t}$$" ], "text/plain": [ " -a⋅t\n", "ℯ " ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(1/(s+a), s, t)" ] }, { "cell_type": "code", "execution_count": 41, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\sin{\\left (t w \\right )}$$" ], "text/plain": [ "sin(t⋅w)" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(w/(s**2+w**2), s, t)" ] }, { "cell_type": "code", "execution_count": 42, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$\\cos{\\left (t w \\right )}$$" ], "text/plain": [ "cos(t⋅w)" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(s/(s**2+w**2), s, t)" ] }, { "cell_type": "code", "execution_count": 43, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$e^{- a t} \\sin{\\left (t w \\right )}$$" ], "text/plain": [ " -a⋅t \n", "ℯ ⋅sin(t⋅w)" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(w/((s+a)**2+w**2), s, t)" ] }, { "cell_type": "code", "execution_count": 44, "metadata": {}, "outputs": [ { "data": { "image/png": "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\n", "text/latex": [ "$$e^{- a t} \\cos{\\left (t w \\right )}$$" ], "text/plain": [ " -a⋅t \n", "ℯ ⋅cos(t⋅w)" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform((s+a)/((s+a)**2+w**2), s, t)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "Python 3", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.6.8" } }, "nbformat": 4, "nbformat_minor": 2 }