{
"cells": [
 {
		   "cell_type": "markdown",
	   "metadata": {},
	   "source": [
       "# Chapter 10: Stability"
	   ]
	},
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.02_01: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"s = poly(0, 's');\n",
"p=poly([12],'s','coeff');\n",
"q=poly([0 2 4 1],'s','coeff');\n",
"G=p/q;\n",
"H=poly([0.5],'s','coeff');\n",
"//characteristic equation is 1+G(s)H(s)=0\n",
"y=1+G*H\n",
"r=numer(y)\n",
"disp('=0',r,'characteristics equation is')   "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.02_02: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"s = poly(0, 's');\n",
"p=poly([7],'s','coeff');\n",
"q=poly([ 2 3 1],'s','coeff');\n",
"G=p/q;\n",
"H=poly([0 1],'s','coeff');\n",
"//characteristic equation is 1+G(s)H(s)=0\n",
"y=1+G*H\n",
"r=numer(y)\n",
"disp('=0',r,'characteristics equation is')   "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.02_03: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"s = poly(0, 's');\n",
"G=syslin('c',2/(s^2+2*s))\n",
"H=syslin('c',1/s);\n",
"//characteristic equation is 1+G(s)H(s)=0\n",
"y=1+G*H\n",
"r=numer(y)\n",
"disp('=0',r,'characteristics equation is')   "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.03: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"s=%s;\n",
"m=s^3+5*s^2+10*s+3;\n",
"r=coeff(m)\n",
"n=length(r);\n",
"routh=[r([4,2]);r([3,1])];\n",
"routh=[routh;-det(routh)/routh(2,1),0];\n",
"t=routh(2:3,1:2); //extracting the square sub block of routh matrix\n",
"routh=[routh;-det(t)/t(2,1),0]\n",
"c=0;\n",
"for i=1:n\n",
"if (routh(i,1)< 0)\n",
"c=c+1;\n",
"end \n",
"end\n",
"if(c>=1)\n",
"printf('system is unstable')\n",
"else ('system is stable')\n",
"end\n",
"  "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.04: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"s=%s;\n",
"m=s^3+2*s^2+3*s+10;\n",
"r=coeff(m)\n",
"n=length(r);\n",
"routh=[r([4,2]);r([3,1])];\n",
"routh=[routh;-det(routh)/routh(2,1),0];\n",
"t=routh(2:3,1:2); //extracting the square sub block of routh matrix\n",
"routh=[routh;-det(t)/t(2,1),0]\n",
"c=0;\n",
"for i=1:n\n",
"if (routh(i,1)<0)\n",
"c=c+1;\n",
"end \n",
"end\n",
"if(c>=1)\n",
"printf('system is unstable')\n",
"else ('system is stable')\n",
"end\n",
"  "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.05_01: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"ieee(2)\n",
"s=%s;\n",
"m=s^4+6*s^3+21*s^2+36*s+20\n",
"r=coeff(m)\n",
"n=length(r);\n",
"routh=[r([5,3,1]);r([4,2]),0]\n",
"routh=[routh;-det(routh(1:2,1:2))/routh(2,1),-det(routh(1:2,2:3))/routh(2,2),0];\n",
"routh=[routh;-det(routh(2:3,1:2))/routh(3,1),-det(routh(2:3,2:3))/routh(3,2),0];\n",
"routh=[routh;-det(routh(3:4,1:2))/routh(4,1),0,0];\n",
"disp(routh,'routh=')\n",
"c=0;\n",
"for i=1:n\n",
"if (routh(i,1)<0)\n",
"c=c+1;\n",
"end \n",
"end\n",
"if(c>=1)\n",
"printf('system is unstable')\n",
"else ('system is stable')\n",
"end\n",
"  \n",
"  "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.05_02: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"s=%s;\n",
"m=s^5+6*s^4+3*s^3+2*s^2+s+1\n",
"r=coeff(m)\n",
"n=length(r)\n",
"routh=[r([6,4,2]);r([5,3,1])]\n",
"routh=[routh;-det(routh(1:2,1:2))/routh(2,1),-det(routh(1:2,2:3))/routh(2,2),0]\n",
"routh=[routh;-det(routh(2:3,1:2))/routh(3,1),-det(routh(2:3,2:3))/routh(3,2),0]\n",
"routh=[routh;-det(routh(3:4,1:2))/routh(4,1),-det(routh(3:4,2:3))/routh(4,2),0]\n",
"routh=[routh;-det(routh(4:5,1:2))/routh(5,1),0,0]\n",
"c=0;\n",
"for i=1:n\n",
"if (routh(i,1)<0)\n",
"c=c+1;\n",
"end \n",
"end\n",
"if(c>=1)\n",
"printf('system is unstable')\n",
"else ('system is stable')\n",
"end\n",
"  "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.06: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"ieee(2)\n",
"s=%s;\n",
"m=s^5+2*s^4+4*s^3+8*s^2+3*s+1\n",
"r=coeff(m);  //Extracts the coefficient of the polynomial\n",
"n=length(r); \n",
"routh=[r([6,4,2]);r([5,3,1])]\n",
"syms eps;\n",
"routh=[routh;eps,-det(routh(1:2,2:3))/routh(2,2),0];\n",
"routh=[routh;-det(routh(2:3,1:2))/routh(3,1),-det(routh(2:3,2:3))/routh(3,2),0];\n",
"routh=[routh;-det(routh(3:4,1:2))/routh(4,1),-det(routh(3:4,2:3))/routh(4,2),0];\n",
"routh=[routh;-det(routh(4:5,1:2))/routh(5,1),0,0];\n",
"disp(routh,'routh=') \n",
"//To check the stability\n",
"routh(4,1)=8-limit(5/eps,eps,0); //Putting the value of eps=0 in routh(4,1)\n",
"disp(routh(4,1),'routh(4,1)=')\n",
"routh(5,1)= limit(routh(5,1),eps,0); //Putting the value of eps=0 in routh(5,1)\n",
"disp(routh(5,1),'routh(5,1)=')\n",
"routh\n",
"printf('There are two sign changes of first column,hence the system is unstable \n')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.07: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"s=%s;\n",
"m=s^5+2*s^4+2*s^3+4*s^2+4*s+8\n",
"routh=routh_t(m) //This Function generates the Routh table\n",
"c=0;\n",
"for i=1:n\n",
"if (routh(i,1)<0)\n",
"c=c+1;\n",
" end \n",
"  end\n",
"  if(c>=1)\n",
"    printf('system is unstable')\n",
"  else ('system is stable')\n",
"  end\n",
"  \n",
"  "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.08: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"s=%s;\n",
"m=s^4+5*s^3+2*s^2+3*s+2\n",
"r=coeff(m)\n",
"n=length(r);\n",
"routh=[r([5,3,1]);r([4,2]),0]\n",
"routh=[routh;-det(routh(1:2,1:2))/routh(2,1),-det(routh(1:2,2:3))/routh(2,2),0];\n",
"routh=[routh;-det(routh(2:3,1:2))/routh(3,1),-det(routh(2:3,2:3))/routh(3,2),0];\n",
"routh=[routh;-det(routh(3:4,1:2))/routh(4,1),0,0]\n",
"  c=0;\n",
" for i=1:n\n",
"    if (routh(i,1)<0)\n",
"      c=c+1;\n",
"    end \n",
"  end\n",
"  if(c>=1)\n",
"    printf('system is unstable')\n",
"  else ('system is stable')\n",
"  end\n",
"  \n",
"  "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.09: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"ieee(2);\n",
"s=%s;\n",
"m=s^5+s^4+3*s^3+3*s^2+4*s+8\n",
"r=coeff(m);  //Extracts the coefficient of the polynomial\n",
"n=length(r); \n",
"routh=[r([6,4,2]);r([5,3,1])]\n",
"syms eps;\n",
"routh=[routh;eps,-det(routh(1:2,2:3))/routh(2,2),0];\n",
"routh=[routh;-det(routh(2:3,1:2))/routh(3,1),-det(routh(2:3,2:3))/routh(3,2),0];\n",
"routh=[routh;-det(routh(3:4,1:2))/routh(4,1),-det(routh(3:4,2:3))/routh(4,2),0];\n",
"routh=[routh;-det(routh(4:5,1:2))/routh(5,1),0,0];\n",
"disp(routh,'routh=') \n",
"//To check the stability\n",
"routh(4,1)=limit(routh(4,1),eps,0); //Putting the value of eps=0 in routh(4,1)\n",
"disp(routh(4,1),'routh(4,1)=')\n",
"routh(5,1)= limit(routh(5,1),eps,0); //Putting the value of eps=0 in routh(5,1)\n",
"disp(routh(5,1),'routh(5,1)=')\n",
"routh\n",
"printf('There are two sign changes of first column,hence the system is unstable \n')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.10: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"ieee(2);\n",
"syms  s k;\n",
"m=s^4+4*s^3+7*s^2+6*s+k;\n",
"cof_a_0 = coeffs(m,'s',0);\n",
"cof_a_1 = coeffs(m,'s',1);\n",
"cof_a_2 = coeffs(m,'s',2);\n",
"cof_a_3 = coeffs(m,'s',3);\n",
"cof_a_4 = coeffs(m,'s',4);\n",
"    \n",
"r=[cof_a_0 cof_a_1 cof_a_2 cof_a_3  cof_a_4] \n",
"\n",
"n=length(r);\n",
"routh=[r([5,3,1]);r([4,2]),0]\n",
"routh=[routh;-det(routh(1:2,1:2))/routh(2,1),-det(routh(1:2,2:3))/routh(2,2),0];\n",
"routh=[routh;-det(routh(2:3,1:2))/routh(3,1),-det(routh(2:3,2:3))/routh(3,2),0];\n",
"routh=[routh;-det(routh(3:4,1:2))/routh(4,1),0,0];\n",
"disp(routh,'routh=')\n",
"\n",
" "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.11: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"ieee(2);\n",
"syms  s k;\n",
"m=(24/100)*s^3+s^2+s+k;\n",
"cof_a_0 = coeffs(m,'s',0);\n",
"cof_a_1 = coeffs(m,'s',1);\n",
"cof_a_2 = coeffs(m,'s',2);\n",
"cof_a_3 = coeffs(m,'s',3);\n",
"r=[cof_a_0 cof_a_1 cof_a_2 cof_a_3] \n",
"n=length(r);\n",
"routh=[r([4,2]);r([3,1])]\n",
"routh=[routh;-det(routh)/routh(2,1),0]\n",
"t=routh(2:3,1:2); //extracting the square sub block of routh matrix\n",
"routh=[routh;-det(t)/routh(3,1),0]\n",
"disp(routh,'routh=');\n",
"routh(3,1)=0  //For marginaly stable system\n",
"k=1/0.24;\n",
"disp(k,'K(marginal)=')\n",
"disp('=0',(s^2)+k,'auxillary equation')\n",
"s=sqrt(-k);\n",
"disp(s,'Frequency of oscillation(in rad/sec)=')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.12: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"ieee(2);\n",
"syms p K s;\n",
"m=s^3+(p*s^2)+(K+3)*s+(2*(K+1))\n",
"cof_a_0 = coeffs(m,'s',0);\n",
"cof_a_1 = coeffs(m,'s',1);\n",
"cof_a_2 = coeffs(m,'s',2);\n",
"cof_a_3 = coeffs(m,'s',3);\n",
"r=[cof_a_0 cof_a_1 cof_a_2 cof_a_3] \n",
"n=length(r);\n",
"routh=[r([4,2]);r([3,1])];\n",
"routh=[routh;-det(routh)/routh(2,1),0];\n",
"t=routh(2:3,1:2); //extracting the square sub block of routh matrix\n",
"routh=[routh;-det(t)/routh(3,1),0];\n",
"disp(routh,'routh=')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.13: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"ieee(2);\n",
"s=%s;\n",
"m=2*s^4+(4*s^2)+1\n",
"routh=routh_t(m)    //Generates the Routh Table\n",
"roots(m)            //Gives the Roots of the Polynomial(m)\n",
"disp(0,'the number of roots lying in the left half plane of s-plane=')\n",
"disp(0,'the number of roots lying in the right half plane of s-plane=')\n",
"disp(4,'the number of roots lying on the imaginary axis=')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.14: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"ieee(2);\n",
"syms  s k;\n",
"m=s^4+6*s^3+10*s^2+8*s+k;\n",
"cof_a_0 = coeffs(m,'s',0);\n",
"cof_a_1 = coeffs(m,'s',1);\n",
"cof_a_2 = coeffs(m,'s',2);\n",
"cof_a_3 = coeffs(m,'s',3);\n",
"cof_a_4 = coeffs(m,'s',4);\n",
"r=[cof_a_0 cof_a_1 cof_a_2 cof_a_3  cof_a_4] \n",
"n=length(r);\n",
"routh=[r([5,3,1]);r([4,2]),0]\n",
"routh=[routh;-det(routh(1:2,1:2))/routh(2,1),-det(routh(1:2,2:3))/routh(2,2),0];\n",
"routh=[routh;-det(routh(2:3,1:2))/routh(3,1),-det(routh(2:3,2:3))/routh(3,2),0];\n",
"routh=[routh;-det(routh(3:4,1:2))/routh(4,1),0,0];\n",
"disp(routh,'routh=')\n",
"\n",
" "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.15: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"ieee(2);\n",
"syms s T;\n",
"m=s^2+(2-T)*s+1\n",
"cof_a_0 = coeffs(m,'s',0);\n",
"cof_a_1 = coeffs(m,'s',1);\n",
"cof_a_2 = coeffs(m,'s',2);\n",
"r=[cof_a_0 cof_a_1 cof_a_2] \n",
"n=length(r);\n",
"routh=[r([3,1]);r(2),0];\n",
"routh=[routh;-det(routh)/routh(2,1),0];\n",
"disp(routh,'routh=')"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 10.16: equatio.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//equation//\n",
"ieee(2)\n",
"s=%s;\n",
"m=s^6+2*s^5+7*s^4+10*s^3+14*s^2+8*s+8\n",
"routh=routh_t(m);\n",
"disp(routh,'routh=')\n",
" c=0;\n",
" for i=1:n\n",
"    if (routh(i,1)<0)\n",
"      c=c+1;\n",
"    end \n",
"  end\n",
"  if(c>=1)\n",
"    printf('system is unstable')\n",
"  else ('system is stable')\n",
"  end\n",
"  \n",
"  "
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.08: value.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//value//\n",
"s=%s;\n",
"H=syslin('c',(s+2)/((s+1)*s*(s+4)));\n",
"evans(H,100)\n",
"printf('From the graph we observed that, \n a)The no of loci ending at inf is 2 \n b)Three loci will start from s= 0,-1 & -4,\n c)One loci will end at -2 & remaining two will end at inf') \n",
""
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 6.0: coefficient.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"//coefficient//\n",
"ieee(2);\n",
"s=%s;\n",
"m=s^5+2*s^4+4*s^3+8*s^2+3*s+1\n",
"r=coeff(m);  //Extracts the coefficient of the polynomial\n",
"n=length(r); \n",
"routh=[r([6,4,2]);r([5,3,1])]\n",
"syms eps;\n",
"routh=[routh;eps,-det(routh(1:2,2:3))/routh(2,2),0];\n",
"routh=[routh;-det(routh(2:3,1:2))/routh(3,1),-det(routh(2:3,2:3))/routh(3,2),0];\n",
"routh=[routh;-det(routh(3:4,1:2))/routh(4,1),-det(routh(3:4,2:3))/routh(4,2),0];\n",
"routh=[routh;-det(routh(4:5,1:2))/routh(5,1),0,0];\n",
"disp(routh,'routh=') \n",
"//To check the stability\n",
"routh(4,1)=8-limit(5/eps,eps,0); //Putting the value of eps=0 in routh(4,1)\n",
"disp(routh(4,1),'routh(4,1)=')\n",
"routh(5,1)= limit(routh(5,1),eps,0); //Putting the value of eps=0 in routh(5,1)\n",
"disp(routh(5,1),'routh(5,1)=')\n",
"routh\n",
"printf('There are two sign changes of first column,hence the system is unstable \n')"
   ]
   }
],
"metadata": {
		  "kernelspec": {
		   "display_name": "Scilab",
		   "language": "scilab",
		   "name": "scilab"
		  },
		  "language_info": {
		   "file_extension": ".sce",
		   "help_links": [
			{
			 "text": "MetaKernel Magics",
			 "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
			}
		   ],
		   "mimetype": "text/x-octave",
		   "name": "scilab",
		   "version": "0.7.1"
		  }
		 },
		 "nbformat": 4,
		 "nbformat_minor": 0
}