{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Chapter 3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using ControlSystems"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 伝達関数モデルの記述"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = tf([0, 1], [1, 2, 3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = tf(\"s\")\n",
    "P = 1/(s^2+2s+3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 練習問題"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = tf([1, 2], [1, 5, 3, 4])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "分母多項式の展開"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Symbolics\n",
    "@variables s\n",
    "expand( (s+1)*(s+2)^2 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = tf([1, 3],[1, 5, 8, 4])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "s = tf(\"s\")\n",
    "P = 1/(s+1)/(s+2)^2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 分母・分子多項式の係数の抽出"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "den(P)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "num(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 状態空間モデルの記述"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "A = [0 1; -1 -1]\n",
    "B = [0; 1]\n",
    "C = [1 0]\n",
    "D = 0\n",
    "P = ss(A, B, C, D)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 練習問題"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A,B,C,D行列の抽出"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "println(\"A=\", P.A)\n",
    "println(\"B=\", P.B)\n",
    "println(\"C=\", P.C)\n",
    "println(\"D=\", P.D)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sysA, sysB, sysC, sysD = ssdata(P)\n",
    "println(\"A=\", sysA)\n",
    "println(\"B=\", sysB)\n",
    "println(\"C=\", sysC)\n",
    "println(\"D=\", sysD)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## ブロック線図の結合"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "S1 = tf( [0, 1], [1, 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "S2 = tf( [1, 1], [1, 1, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 直列結合"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "S = S2 * S1;\n",
    "println(S)\n",
    "\n",
    "S = series(S1, S2);\n",
    "println(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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "minreal(S)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 並列結合"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "S = S1 + S2;\n",
    "println(S)\n",
    "\n",
    "S = parallel(S1, S2)\n",
    "println(S)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### フィードバック結合"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "S = S1*S2 / (1 + S1*S2)\n",
    "println(S)\n",
    "\n",
    "S = feedback(S1*S2, 1)\n",
    "println(S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "minreal(S)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 練習問題"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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",
    "S12 = feedback(S1, S2)\n",
    "S123 = series(S12, S3)\n",
    "S = feedback(S123, S4)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "toc-hr-collapsed": true,
    "toc-nb-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": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "P = tf( [0, 1], [1, 1, 1])\n",
    "\n",
    "Pss = ss(P) # 伝達関数モデルから状態空間モデルへの変換\n",
    "println(Pss)\n",
    "Ptf = tf(Pss) # 状態空間モデルから伝達関数モデルへの変換\n",
    "println(Ptf)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### プロパー性"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "S1 = tf([1, 1], [0, 1])\n",
    "S2 = tf([0, 1], [1, 1])\n",
    "\n",
    "S = series(S1, S2)\n",
    "minreal(S)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(S2)\n",
    "ss(S2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(S1)\n",
    "ss(S1)"
   ]
  }
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