{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Chapter 3" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 状態空間モデルの記述" ] }, { "cell_type": "code", "execution_count": 1, "metadata": { "tags": [] }, "outputs": [], "source": [ "from control.matlab import ss, ssdata\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "状態空間モデルの記述は，ssを用います\n", "\n", "行列A, B, C, Dは，array_like data (リスト）か string（文字列）で記述します" ] }, { "cell_type": "code", "execution_count": 2, "metadata": { "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ ": sys[2]\n", "Inputs (1): ['u[0]']\n", "Outputs (1): ['y[0]']\n", "States (2): ['x[0]', 'x[1]']\n", "\n", "A = [[ 0. 1.]\n", " [-1. -1.]]\n", "\n", "B = [[0.]\n", " [1.]]\n", "\n", "C = [[1. 0.]]\n", "\n", "D = [[0.]]\n", "\n" ] } ], "source": [ "# array_like data\n", "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)\n" ] }, { "cell_type": "code", "execution_count": 3, "metadata": { "tags": [] }, "outputs": [], "source": [ "# string (ver.0.10.0から非対応)\n", "# 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": 4, "metadata": { "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ ": sys[3]\n", "Inputs (1): ['u[0]']\n", "Outputs (1): ['y[0]']\n", "States (3): ['x[0]', 'x[1]', 'x[2]']\n", "\n", "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", "\n", "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": 5, "metadata": { "tags": [] }, "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)\n" ] }, { "cell_type": "code", "execution_count": 6, "metadata": { "tags": [] }, "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)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## 伝達関数モデルの記述" ] }, { "cell_type": "code", "execution_count": 7, "metadata": {}, "outputs": [], "source": [ "from control.matlab import tf, tfdata\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "伝達関数モデルの記述は，tfを用います" ] }, { "cell_type": "code", "execution_count": 8, "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(0*s + 1)の係数\n", "Dp = [1, 2, 3] # 伝達関数の分母多項式(1*s^2 + 2*s + 3)の係数\n", "P = tf(Np, Dp)\n", "print('P(s)=', P)\n" ] }, { "cell_type": "code", "execution_count": 9, "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)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "ラプラス演算子を定義して記述することもできます" ] }, { "cell_type": "code", "execution_count": 10, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "P(s)= \n", " 1\n", "-------------\n", "s^2 + 2 s + 3\n", "\n" ] } ], "source": [ "s = tf('s')\n", "P = 1/(s**2 + 2*s + 3)\n", "print('P(s)=', P)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 練習問題" ] }, { "cell_type": "code", "execution_count": 11, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$$\\frac{s + 2}{s^3 + 5 s^2 + 3 s + 4}$$" ], "text/plain": [ "TransferFunction(array([1, 2]), array([1, 5, 3, 4]))" ] }, "execution_count": 11, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P = tf([1, 2], [1, 5, 3, 4])\n", "P\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "分母多項式の展開" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle s^{3} + 5 s^{2} + 8 s + 4$" ], "text/plain": [ " 3 2 \n", "s + 5⋅s + 8⋅s + 4" ] }, "execution_count": 12, "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)\n" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$$\\frac{s + 3}{s^3 + 5 s^2 + 8 s + 4}$$" ], "text/plain": [ "TransferFunction(array([1, 3]), array([1, 5, 8, 4]))" ] }, "execution_count": 13, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P = tf([1, 3],[1, 5, 8, 4])\n", "P\n" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$$\\frac{s + 3}{s^3 + 5 s^2 + 8 s + 4}$$" ], "text/plain": [ "TransferFunction(array([1, 3]), array([1, 5, 8, 4]))" ] }, "execution_count": 14, "metadata": {}, "output_type": "execute_result" } ], "source": [ "P1 = tf([1, 3], [0, 1]) # (s+3)/1\n", "P2 = tf([0, 1], [1, 1]) # 1/(s+1)\n", "P3 = tf([0, 1], [1, 2]) # 1/(s+2)\n", "P = P1 * P2 * P3**2\n", "P\n" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "data": { "text/latex": [ "$$\\frac{s + 3}{s^3 + 5 s^2 + 8 s + 4}$$" ], "text/plain": [ "TransferFunction(array([1, 3]), array([1, 5, 8, 4]))" ] }, "execution_count": 15, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = tf('s')\n", "P = (s+3)/((s+1)*(s+2)**2)\n", "P\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 分母・分子多項式の係数の抽出" ] }, { "cell_type": "code", "execution_count": 16, "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)\n" ] }, { "cell_type": "code", "execution_count": 17, "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)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## モデルの変換" ] }, { "cell_type": "code", "execution_count": 18, "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": [ "from control.matlab import tf, tf2ss, ss2tf\n", "\n", "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)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 可制御正準形" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "正準形への変換は，controlのcanonikcal_formを使います（control.matlabには関数がありません）" ] }, { "cell_type": "code", "execution_count": 19, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "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 ss, 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", "\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(Pr)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "等価変換の行列は T です" ] }, { "cell_type": "code", "execution_count": 20, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "[[-5.55111512e-16 -2.50000000e-01 1.00000000e+00]\n", " [ 2.50000000e-01 2.50000000e-01 -2.50000000e-01]\n", " [-8.33333333e-02 -0.00000000e+00 8.33333333e-02]]\n" ] } ], "source": [ "print(T)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 可観測正準形" ] }, { "cell_type": "code", "execution_count": 21, "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", "[[ 1.18423789e-16 2.00000000e+00 1.00000000e+00]\n", " [ 1.00000000e+01 3.00000000e+00 -1.00000000e+00]\n", " [ 1.50000000e+01 -2.10000000e+01 1.20000000e+01]]\n" ] } ], "source": [ "Pr, T = canonical_form(Pss, form='observable')\n", "print(Pr)\n", "print(T)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## ブロック線図の結合" ] }, { "cell_type": "code", "execution_count": 22, "metadata": { "tags": [] }, "outputs": [], "source": [ "from control.matlab import tf, ss, series, parallel, feedback\n" ] }, { "cell_type": "code", "execution_count": 23, "metadata": { "tags": [] }, "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)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 直列結合" ] }, { "cell_type": "code", "execution_count": 24, "metadata": { "tags": [] }, "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)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "分母分子の共通因子 s+1 が約分されません。\n", "\n", "この場合は，minreal を使います。minreal は不可制御，不可観測なモードの削除（最小実現）をするものです。" ] }, { "cell_type": "code", "execution_count": 25, "metadata": { "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "S= \n", " 1\n", "-----------\n", "s^2 + s + 1\n", "\n" ] } ], "source": [ "print('S=', S.minreal())\n" ] }, { "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": 26, "metadata": { "tags": [] }, "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))\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 並列結合" ] }, { "cell_type": "code", "execution_count": 27, "metadata": { "tags": [] }, "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)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### フィードバック結合" ] }, { "cell_type": "code", "execution_count": 28, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "S= \n", " s^3 + 2 s^2 + 2 s + 1\n", "-----------------------------\n", "s^4 + 3 s^3 + 5 s^2 + 5 s + 2\n", "\n", "S= \n", " s^2 + s + 1\n", "---------------------\n", "s^3 + 2 s^2 + 3 s + 2\n", "\n" ] } ], "source": [ "S = S1 / (1 + S1*S2)\n", "print('S=', S)\n", "S = feedback(S1, S2)\n", "print('S=', S)\n" ] }, { "cell_type": "code", "execution_count": 29, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "S= \n", " s^2 + s + 1\n", "---------------------\n", "s^3 + 2 s^2 + 3 s + 2\n", "\n" ] } ], "source": [ "print('S=', S.minreal())\n" ] }, { "cell_type": "code", "execution_count": 30, "metadata": { "tags": [] }, "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)\n" ] }, { "cell_type": "code", "execution_count": 31, "metadata": { "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "S= \n", " 1\n", "-----------\n", "s^2 + s + 2\n", "\n" ] } ], "source": [ "print('S=', S.minreal())\n" ] }, { "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": 32, "metadata": { "tags": [] }, "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())\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### 練習問題" ] }, { "cell_type": "code", "execution_count": 33, "metadata": { "tags": [] }, "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)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## プロパー性" ] }, { "cell_type": "code", "execution_count": 34, "metadata": { "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "1\n", "-\n", "1\n", "\n" ] } ], "source": [ "from control.matlab import tf, tf2ss\n", "\n", "S1 = tf([1, 1], [0, 1])\n", "S2 = tf([0, 1], [1, 1])\n", "\n", "S = series(S1, S2)\n", "print(S.minreal())\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "厳密にプロパー" ] }, { "cell_type": "code", "execution_count": 35, "metadata": { "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", " 1\n", "-----\n", "s + 1\n", "\n" ] }, { "data": { "text/latex": [ "$$\n", "\\left(\\begin{array}{rll|rll}\n", "-1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n", "\\hline\n", "1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&0\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n", "\\end{array}\\right)\n", "$$" ], "text/plain": [ "StateSpace(array([[-1.]]), array([[1.]]), array([[1.]]), array([[0.]]))" ] }, "execution_count": 35, "metadata": {}, "output_type": "execute_result" } ], "source": [ "print(S2)\n", "tf2ss(S2)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "インプロパー" ] }, { "cell_type": "code", "execution_count": 36, "metadata": { "tags": [] }, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "s + 1\n", "-----\n", " 1\n", "\n" ] } ], "source": [ "print(S1)\n", "# tf2ss(S1) # 状態方程式に変換できない\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "プロパー" ] }, { "cell_type": "code", "execution_count": 37, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "s + 2\n", "-----\n", "s + 1\n", "\n" ] }, { "data": { "text/latex": [ "$$\n", "\\left(\\begin{array}{rll|rll}\n", "-1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n", "\\hline\n", "1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}&1\\phantom{.}&\\hspace{-1em}&\\hspace{-1em}\\phantom{\\cdot}\\\\\n", "\\end{array}\\right)\n", "$$" ], "text/plain": [ "StateSpace(array([[-1.]]), array([[1.]]), array([[1.]]), array([[1.]]))" ] }, "execution_count": 37, "metadata": {}, "output_type": "execute_result" } ], "source": [ "S3 = tf([1, 2], [1, 1])\n", "print(S3)\n", "tf2ss(S3)\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## ラプラス変換" ] }, { "cell_type": "code", "execution_count": 38, "metadata": { "tags": [] }, "outputs": [], "source": [ "import sympy as sp\n" ] }, { "cell_type": "code", "execution_count": 39, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle \\left( \\frac{1}{s}, \\ 0, \\ \\text{True}\\right)$" ], "text/plain": [ "⎛1 ⎞\n", "⎜─, 0, True⎟\n", "⎝s ⎠" ] }, "execution_count": 39, "metadata": {}, "output_type": "execute_result" } ], "source": [ "s = sp.Symbol('s')\n", "t = sp.Symbol('t', positive=True)\n", "sp.init_printing()\n", "sp.laplace_transform(1, t, s)\n" ] }, { "cell_type": "code", "execution_count": 40, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle \\left( \\frac{1}{s^{2}}, \\ 0, \\ \\text{True}\\right)$" ], "text/plain": [ "⎛1 ⎞\n", "⎜──, 0, True⎟\n", "⎜ 2 ⎟\n", "⎝s ⎠" ] }, "execution_count": 40, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.laplace_transform(t, t, s)\n" ] }, { "cell_type": "code", "execution_count": 41, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\left( \\frac{1}{a + s}, \\ - a, \\ \\text{True}\\right)$" ], "text/plain": [ "⎛ 1 ⎞\n", "⎜─────, -a, True⎟\n", "⎝a + s ⎠" ] }, "execution_count": 41, "metadata": {}, "output_type": "execute_result" } ], "source": [ "a = sp.Symbol('a', real=True)\n", "sp.laplace_transform(sp.exp(-a*t), t, s)\n" ] }, { "cell_type": "code", "execution_count": 42, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\left( \\frac{w}{s^{2} + w^{2}}, \\ 0, \\ \\text{True}\\right)$" ], "text/plain": [ "⎛ w ⎞\n", "⎜───────, 0, True⎟\n", "⎜ 2 2 ⎟\n", "⎝s + w ⎠" ] }, "execution_count": 42, "metadata": {}, "output_type": "execute_result" } ], "source": [ "w = sp.Symbol('w', real=True)\n", "sp.laplace_transform(sp.sin(w*t), t, s)\n" ] }, { "cell_type": "code", "execution_count": 43, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\left( \\frac{s}{s^{2} + w^{2}}, \\ 0, \\ \\text{True}\\right)$" ], "text/plain": [ "⎛ s ⎞\n", "⎜───────, 0, True⎟\n", "⎜ 2 2 ⎟\n", "⎝s + w ⎠" ] }, "execution_count": 43, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.laplace_transform(sp.cos(w*t), t, s)\n" ] }, { "cell_type": "code", "execution_count": 44, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": 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"text/latex": [ "$\\displaystyle \\left( \\frac{w}{w^{2} + \\left(a + s\\right)^{2}}, \\ - a, \\ \\text{True}\\right)$" ], "text/plain": [ "⎛ w ⎞\n", "⎜─────────────, -a, True⎟\n", "⎜ 2 2 ⎟\n", "⎝w + (a + s) ⎠" ] }, "execution_count": 44, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.laplace_transform(sp.exp(-a*t)*sp.sin(w*t), t, s)\n" ] }, { "cell_type": "code", "execution_count": 45, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle \\left( \\frac{a + s}{w^{2} + \\left(a + s\\right)^{2}}, \\ - a, \\ \\text{True}\\right)$" ], "text/plain": [ "⎛ a + s ⎞\n", "⎜─────────────, -a, True⎟\n", "⎜ 2 2 ⎟\n", "⎝w + (a + s) ⎠" ] }, "execution_count": 45, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.laplace_transform(sp.exp(-a*t)*sp.cos(w*t), t, s)\n" ] }, { "cell_type": "markdown", "metadata": { "toc-hr-collapsed": true, "toc-nb-collapsed": true }, "source": [ "## 逆ラプラス変換" ] }, { "cell_type": "code", "execution_count": 46, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAZElEQVQYGWP8//8/Aww0NDQYAdmrgdgYyP4AEmcBMgSA9GwgfgfEJkCsBMRwAFIAUhkKEgGyy4AUyBQ4YIKzcDBGFUACBj0chKHhJQQLN0ZQZAGDGBRBIOACxKC4OQfE94B4NwDm+hiAOyllRAAAAABJRU5ErkJggg==", "text/latex": [ "$\\displaystyle 1$" ], "text/plain": [ "1" ] }, "execution_count": 46, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(1/s, s, t)\n" ] }, { "cell_type": "code", "execution_count": 47, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAcAAAANCAYAAABlyXS1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAnElEQVQYGW2PgQ3CMAwEEyZAjNBZ6AbdgTGSNWAF2AA2QHSEjIDYINxBU7UISy+/8/7YjrXW0CLnvIU/QA8vmyZMeU/uwNP6V+x5K7he/0SdFwUjppR8OADnyW9A5z24kKBpANBvbV7OdN4I5liK7cu1ON3nCddZgTSnrkCTy3xCQxNX8xAGOrom7iiKFh0K5DG6MsR5R3AG1ifzG7XmPqHm1k1JAAAAAElFTkSuQmCC", "text/latex": [ "$\\displaystyle t$" ], "text/plain": [ "t" ] }, "execution_count": 47, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(1/s**2, s, t)\n" ] }, { "cell_type": "code", "execution_count": 48, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle e^{- a t}$" ], "text/plain": [ " -a⋅t\n", "ℯ " ] }, "execution_count": 48, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(1/(s+a), s, t)\n" ] }, { "cell_type": "code", "execution_count": 49, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEIAAAAVCAYAAADy3zinAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADkElEQVRYCc2Y61EbQQyAzx4KgKSC2B3w6MB0AKSCQAdh8gv+ZaCDhA4CHUAHPDrAHYS4A/J9y97OPXznFzbWjNhdSStptZJuTfb6+pqtC56dnfVW4cs4O91sTeD8/Pw7rmyvyJ1etJfMdbyBScCmHjKP4Cnz35PkZ+Wj84A9e4yns+6dVx5bBn6Un2fajDAQm+DOvIab9uGIen80BUE++Azqw7sB+i5RdqJ+lU4VCITvkN1iPHHTO8MF+n616BzAMwgvLTLzsrSr/Wyq0pjXyjT7CO4/cKtJFp7ODhj7TTKL0LXP/i9TZcQihtr24oS9YdgmA8+MuJkgswhb+0cbasAhu7UGR2CoGcav0HdA0/IadLxjfciYMbrnCpT+BzTFPJiwB94jYx22wT5My64E7NMXy1Bf1L8NTR+CTubSj8HPoP0jNXDm+mXthzJm1Cd98yyerwra39+AqdILRp1KwDpXZMRU4lcjAesnFtKfGT+BOhsOzqgzj4w3YNuN7yJX6w/s0TmD7iEsi5Jv0EJzjXwvIwVCHuiZcvDwBlNbtaBD0/9BNwrsorS4GXLNwaZm5UF1NqUvc4MkeLNtoM0mve4zALku1xm6DfJ9WLzxq/u1eRv5ynt4dTRdiPt7ZoSRd2HTcpNKvMlJaY1Ygoc0K0+qwS1z3zLJG2sCD5UCHIWG+ObBhCPwZ5jxB7pB0mb15h/gNQVC+qYZIfg+8OA90Hqy7mopC31lgH0PpD/pdjUOPQSO0bJRplgWBs5HUh4olgFM/yawrEddNmksY/TV2Ac7LO0Px8yN8DLBTPQw4yCUFT6k22VelA1lAy0EJiqQluSlwfcM1cDIykGdL2aEgnbgBGw2yqakDWaZYFqGixhjpNQf8MkMKMo6r6b7OJpfv1JwKrbMiGE3Eu3CxWhLdl1VUJVRTkXzgjflp3YcBAdlRN/8oVS8WefJH3hmkHtSsCKtVFrwq2BbeOrwk1QFRv9vQcLv8y2KbKTbzP0khVRlNDjfQA1K96ZGoHQbl3RLS/lAR0d4e7AuQdR9zVh7NUJTj33K90PGutgLJEnL+5h29F+ZK/AFlFZ6Y7CuATrsHyfr8MTWkUMcKt52zeFlELBpsL3wfncZBmbU6VfKDPoI8Ge/9qf79blMD7kN09n693ZWBtGedkPJrUNGeHh7SF7vqwqG9lImfniPyE8db+iAcZYXbb59phEb/neq9DvoP/zfDjqp2RB6AAAAAElFTkSuQmCC", "text/latex": [ "$\\displaystyle \\sin{\\left(t w \\right)}$" ], "text/plain": [ "sin(t⋅w)" ] }, "execution_count": 49, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(w/(s**2+w**2), s, t)\n" ] }, { "cell_type": "code", "execution_count": 50, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle \\cos{\\left(t w \\right)}$" ], "text/plain": [ "cos(t⋅w)" ] }, "execution_count": 50, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(s/(s**2+w**2), s, t)\n" ] }, { "cell_type": "code", "execution_count": 51, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle e^{- a t} \\sin{\\left(t w \\right)}$" ], "text/plain": [ " -a⋅t \n", "ℯ ⋅sin(t⋅w)" ] }, "execution_count": 51, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(w/((s+a)**2+w**2), s, t)\n" ] }, { "cell_type": "code", "execution_count": 52, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle e^{- a t} \\cos{\\left(t w \\right)}$" ], "text/plain": [ " -a⋅t \n", "ℯ ⋅cos(t⋅w)" ] }, "execution_count": 52, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform((s+a)/((s+a)**2+w**2), s, t)\n" ] }, { "cell_type": "code", "execution_count": 53, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAgAAAAOCAYAAAASVl2WAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAZElEQVQYGWP8//8/Aww0NDQYAdmrgdgYyP4AEmcBMgSA9GwgfgfEJkCsBMRwAFIAUhkKEgGyy4AUyBQ4YIKzcDBGFUACBj0chKHhJQQLN0ZQZAGDGBRBIOACxKC4OQfE94B4NwDm+hiAOyllRAAAAABJRU5ErkJggg==", "text/latex": [ "$\\displaystyle 1$" ], "text/plain": [ "1" ] }, "execution_count": 53, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(1/s, s, t)\n" ] }, { "cell_type": "code", "execution_count": 54, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAcAAAANCAYAAABlyXS1AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAnElEQVQYGW2PgQ3CMAwEEyZAjNBZ6AbdgTGSNWAF2AA2QHSEjIDYINxBU7UISy+/8/7YjrXW0CLnvIU/QA8vmyZMeU/uwNP6V+x5K7he/0SdFwUjppR8OADnyW9A5z24kKBpANBvbV7OdN4I5liK7cu1ON3nCddZgTSnrkCTy3xCQxNX8xAGOrom7iiKFh0K5DG6MsR5R3AG1ifzG7XmPqHm1k1JAAAAAElFTkSuQmCC", "text/latex": [ "$\\displaystyle t$" ], "text/plain": [ "t" ] }, "execution_count": 54, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(1/s**2, s, t)\n" ] }, { "cell_type": "code", "execution_count": 55, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle e^{- a t}$" ], "text/plain": [ " -a⋅t\n", "ℯ " ] }, "execution_count": 55, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(1/(s+a), s, t)\n" ] }, { "cell_type": "code", "execution_count": 56, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle \\sin{\\left(t w \\right)}$" ], "text/plain": [ "sin(t⋅w)" ] }, "execution_count": 56, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(w/(s**2+w**2), s, t)\n" ] }, { "cell_type": "code", "execution_count": 57, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle \\cos{\\left(t w \\right)}$" ], "text/plain": [ "cos(t⋅w)" ] }, "execution_count": 57, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(s/(s**2+w**2), s, t)\n" ] }, { "cell_type": "code", "execution_count": 58, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle e^{- a t} \\sin{\\left(t w \\right)}$" ], "text/plain": [ " -a⋅t \n", "ℯ ⋅sin(t⋅w)" ] }, "execution_count": 58, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform(w/((s+a)**2+w**2), s, t)\n" ] }, { "cell_type": "code", "execution_count": 59, "metadata": { "tags": [] }, "outputs": [ { "data": { "image/png": "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", "text/latex": [ "$\\displaystyle e^{- a t} \\cos{\\left(t w \\right)}$" ], "text/plain": [ " -a⋅t \n", "ℯ ⋅cos(t⋅w)" ] }, "execution_count": 59, "metadata": {}, "output_type": "execute_result" } ], "source": [ "sp.inverse_laplace_transform((s+a)/((s+a)**2+w**2), s, t)\n" ] } ], "metadata": { "kernelspec": { "display_name": "Python 3 (ipykernel)", "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": "3.11.3" } }, "nbformat": 4, "nbformat_minor": 4 }