{
"cells": [
 {
		   "cell_type": "markdown",
	   "metadata": {},
	   "source": [
       "# Chapter 11: Minimum Variance Control"
	   ]
	},
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.10: Minimum_variance_control_for_nonminimum_phase_example.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Minimum variance control for nonminimum phase example of Example 11.6 on page 416.\n",
"// 11.10 \n",
"\n",
"exec('mv_nm.sci',-1);\n",
"exec('cl.sci',-1);\n",
"exec('zpowk.sci',-1);\n",
"exec('polmul.sci',-1);\n",
"exec('polsize.sci',-1);\n",
"exec('polsplit3.sci',-1);\n",
"exec('flip.sci',-1);\n",
"exec('xdync.sci',-1);\n",
"exec('rowjoin.sci',-1);\n",
"exec('left_prm.sci',-1);\n",
"exec('t1calc.sci',-1);\n",
"exec('indep.sci',-1);\n",
"exec('seshft.sci',-1);\n",
"exec('makezero.sci',-1);\n",
"exec('move_sci.sci',-1);\n",
"exec('colsplit.sci',-1);\n",
"exec('clcoef.sci',-1);\n",
"exec('cindep.sci',-1);\n",
"exec('poladd.sci',-1);\n",
"exec('tfvar.sci',-1);\n",
"exec('l2r.sci',-1);\n",
"exec('transp.sci',-1);\n",
"exec('tf.sci',-1);\n",
"exec('covar_m.sci',-1);\n",
"\n",
"A = convol([1 -1],[1 -0.7]); dA = 2;\n",
"B = [0.9 1]; dB = 1; k = 1;\n",
"C = [1 -0.7]; dC = 1; int1 = 0;\n",
"[Sc,dSc,Rc,dRc] = mv_nm(A,dA,B,dB,C,dC,k,int1);\n",
"[Nu,dNu,Du,dDu,Ny,dNy,Dy,dDy,yvar,uvar] = ...\n",
"cl(A,dA,B,dB,C,dC,k,Sc,dSc,Rc,dRc,int1);\n",
""
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.11: Minimum_variance_control_of_viscosity_control_problem.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Minimum variance control of viscosity control problem\n",
"// 11.11\n",
"\n",
"// Viscosity control problem of MacGregor\n",
"\n",
"exec('mv_nm.sci',-1);\n",
"exec('polmul.sci',-1);\n",
"exec('polsize.sci',-1);\n",
"exec('zpowk.sci',-1);\n",
"exec('polsplit3.sci',-1);\n",
"exec('flip.sci',-1);\n",
"exec('xdync.sci',-1);\n",
"exec('rowjoin.sci',-1);\n",
"exec('left_prm.sci',-1);\n",
"exec('t1calc.sci',-1);\n",
"exec('indep.sci',-1);\n",
"exec('seshft.sci',-1);\n",
"exec('makezero.sci',-1);\n",
"exec('move_sci.sci',-1);\n",
"exec('colsplit.sci',-1);\n",
"exec('clcoef.sci',-1);\n",
"exec('cindep.sci',-1);\n",
"exec('cl.sci',-1);\n",
"exec('poladd.sci',-1);\n",
"exec('tfvar.sci',-1);\n",
"exec('l2r.sci',-1);\n",
"exec('transp.sci',-1);\n",
"exec('tf.sci',-1);\n",
"exec('covar_m.sci',-1);\n",
"\n",
"A = [1 -0.44]; dA = 1; B = [0.51 1.21]; dB = 1;\n",
"C = [1 -0.44]; dC = 1; k = 1; int1 = 1;\n",
"[Sc,dSc,Rc,dRc] = mv_nm(A,dA,B,dB,C,dC,k,int1);\n",
"[Nu,dNu,Du,dDu,Ny,dNy,Dy,dDy,yvar,uvar] = ...\n",
"cl(A,dA,B,dB,C,dC,k,Sc,dSc,Rc,dRc,int1);"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.12: General_minimum_variance_controller_design.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// General minimum variance controller design, as given by Eq. 11.66 on page 421 and Eq. 11.70 on page 422.\n",
"// 11.12\n",
"\n",
"// function [Sc,dSc,Rc,dRc] = gmv(A,dA,B,dB,C,dC,k,rho,int)\n",
"// implements the generalized minimum variance controller\n",
"// if int>=1, integrated noise is assumed; otherwise,\n",
"// it is not integrated noise\n",
"\n",
"function [Sc,dSc,R,dR] = gmv(A,dA,B,dB,C,dC,k,rho,int1)\n",
"zk = zeros(1,k+1); zk(k+1) = 1;\n",
"if int1>=1, [A,dA] = polmul([1 -1],1,A,dA); end\n",
"[Fk,dFk,Ek,dEk] = xdync(zk,k,A,dA,C,dC);\n",
"[Gk,dGk] = polmul(Ek,dEk,B,dB);\n",
"alpha0 = Gk(1)/C(1); \n",
"Sc = alpha0 * Fk; dSc = dFk; \n",
"[R,dR] = poladd(alpha0*Gk,dGk,rho*C,dC);\n",
"endfunction;"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.13: GMVC_design_of_first_example_by_MacGregor.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// GMVC design of MacGregor's first example, as discussed in Example 11.9 on page 421. \n",
"// 11.13\n",
"\n",
"// MacGregor's first control problem by gmv\n",
"\n",
"exec('gmv.sci',-1);\n",
"exec('cl.sci',-1);\n",
"exec('xdync.sci',-1);\n",
"exec('rowjoin.sci',-1);\n",
"exec('polsize.sci',-1);\n",
"exec('left_prm.sci',-1);\n",
"exec('t1calc.sci',-1);\n",
"exec('indep.sci',-1);\n",
"exec('seshft.sci',-1);\n",
"exec('makezero.sci',-1);\n",
"exec('move_sci.sci',-1);\n",
"exec('colsplit.sci',-1);\n",
"exec('clcoef.sci',-1);\n",
"exec('cindep.sci',-1);\n",
"exec('polmul.sci',-1);\n",
"exec('zpowk.sci',-1);\n",
"exec('poladd.sci',-1);\n",
"exec('tfvar.sci',-1);\n",
"exec('l2r.sci',-1);\n",
"exec('transp.sci',-1);\n",
"exec('tf.sci',-1);\n",
"exec('covar_m.sci',-1);\n",
"\n",
"A = [1 -1.4 0.45]; dA = 2; C = [1 -0.5]; dC = 1;\n",
"B = 0.5*[1 -0.9]; dB = 1; k = 1; int1 = 0;\n",
"rho = 1;\n",
"[Sc,dSc,Rc,dRc] = gmv(A,dA,B,dB,C,dC,k,rho,int1);\n",
"[Nu,dNu,Du,dDu,Ny,dNy,Dy,dDy,yvar,uvar] = ...\n",
"        cl(A,dA,B,dB,C,dC,k,Sc,dSc,Rc,dRc,int1);"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.14: GMVC_design_of_viscosity_problem.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// GMVC design of viscosity problem, as described in Example 11.10 on page 423.\n",
"// 11.14\n",
"\n",
"// MacGregor's Viscosity control problem by gmv\n",
"\n",
"exec('gmv.sci',-1);\n",
"exec('cl.sci',-1);\n",
"exec('polmul.sci',-1);\n",
"exec('polsize.sci',-1);\n",
"exec('xdync.sci',-1);\n",
"exec('rowjoin.sci',-1);\n",
"exec('left_prm.sci',-1);\n",
"exec('t1calc.sci',-1);\n",
"exec('indep.sci',-1);\n",
"exec('seshft.sci',-1);\n",
"exec('makezero.sci',-1);\n",
"exec('move_sci.sci',-1);\n",
"exec('colsplit.sci',-1);\n",
"exec('clcoef.sci',-1);\n",
"exec('cindep.sci',-1);\n",
"exec('poladd.sci',-1);\n",
"exec('zpowk.sci',-1);\n",
"exec('tfvar.sci',-1);\n",
"exec('l2r.sci',-1);\n",
"exec('transp.sci',-1);\n",
"exec('tf.sci',-1);\n",
"exec('covar_m.sci',-1);\n",
"\n",
"A = [1 -0.44]; dA = 1; B = [0.51 1.21]; dB = 1;\n",
"C = [1 -0.44]; dC = 1; k = 1; int1 = 1;\n",
"rho = 1;\n",
"[Sc,dSc,R1,dR1] = gmv(A,dA,B,dB,C,dC,k,rho,int1);\n",
"[Nu,dNu,Du,dDu,Ny,dNy,Dy,dDy,yvar,uvar] = ...\n",
"        cl(A,dA,B,dB,C,dC,k,Sc,dSc,R1,dR1,int1);\n",
""
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.15: PID_tuning_through_GMVC_law.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// PID tuning through GMVC law, as discussed in Example 11.11.\n",
"// 11.15\n",
"\n",
"exec('gmvc_pid.sci',-1);\n",
"exec('zpowk.sci',-1);\n",
"exec('ch_pol.sci',-1);\n",
"exec('polmul.sci',-1);\n",
"exec('polsize.sci',-1);\n",
"exec('xdync.sci',-1);\n",
"exec('rowjoin.sci',-1);\n",
"exec('left_prm.sci',-1);\n",
"exec('t1calc.sci',-1);\n",
"exec('indep.sci',-1);\n",
"exec('seshft.sci',-1);\n",
"exec('makezero.sci',-1);\n",
"exec('move_sci.sci',-1);\n",
"exec('colsplit.sci',-1);\n",
"exec('clcoef.sci',-1);\n",
"exec('cindep.sci',-1);\n",
"exec('filtval.sci',-1);\n",
"exec('polyno.sci',-1);\n",
"\n",
"// GMVC PID tuning of example given by Miller et al.\n",
"// Model\n",
"A = [1 -1.95 0.935]; B = -0.015; k = 1; Ts = 1;\n",
"\n",
"// Transient specifications\n",
"N = 15; epsilon = 0.1;\n",
"T = ch_pol(N,epsilon);\n",
"\n",
"// Controller Design\n",
"[Kc,tau_i,tau_d,L] = gmvc_pid(A,B,k,T,Ts);\n",
"L1 = filtval(L,1);\n",
"zk = zpowk(k);\n",
"\n",
""
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.16: Value_of_polynomial_p_evaluated_at_x.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Value of polynomial p(x), evaluated at x\n",
"// 11.16\n",
"\n",
"// finds the value of a polynomial in powers of z^{-1}\n",
"// function Y = filtval(P,z)\n",
"\n",
"function Y = filtval(P,z)\n",
"N = length(P)-1;\n",
"Q = polyno(P,'x');\n",
"Y = horner(Q,z)/z^N; \n",
"endfunction;"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.17: PID_tuning_through_GMVC_law.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// PID tuning through GMVC law\n",
"// 11.17\n",
"\n",
"// function [Kc,tau_i,tau_d,L] = gmvc_pid(A,B,k,T,Ts)\n",
"// Determines p,i,d tuning parameters using GMVC\n",
"// Plant model: Integrated white noise\n",
"// A, B in discrete time form\n",
"\n",
"function [Kc,tau_i,tau_d,L] = gmvc_pid(A,B,k,T,Ts)\n",
"\n",
"dA = length(A)-1; dB = length(B)-1;\n",
"dT = length(T)-1;\n",
"if dA > 2,\n",
"   disp('degree of A cannot be more than 2')\n",
"   exit\n",
"elseif dB > 1,\n",
"   disp('degree of B cannot be more than 1')\n",
"   exit\n",
"elseif dT > 2,\n",
"   disp('degree of T cannot be more than 2')\n",
"   exit\n",
"end\n",
"delta = [1 -1]; ddelta = 1;\n",
"\n",
"[Adelta,dAdelta] = polmul(A,dA,delta,ddelta);\n",
"\n",
"[Q,dQ,P,dP] = ...\n",
"xdync(Adelta,dAdelta,B,dB,T,dT);\n",
"PAdelta = P(1)*Adelta;\n",
"\n",
"[zk,dzk] = zpowk(k);\n",
"[E,degE,F,degF] = ...\n",
"xdync(PAdelta,dAdelta,zk,dzk,P,dP);\n",
"nu = P(1)*E(1)*B(1);\n",
"Kc = -1/nu*(F(2)+2*F(3));\n",
"tau_i = -(F(2)+2*F(3))/(F(1)+F(2)+F(3))*Ts;\n",
"tau_d = -F(3)/(F(2)+2*F(3))*Ts;\n",
"L(1) = 1+Ts/tau_i+tau_d/Ts;\n",
"L(2) = -(1+2*tau_d/Ts);\n",
"L(3) = tau_d/Ts;\n",
"L = Kc * L';\n",
"endfunction;"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.1: Recursive_computation_of_Ej_and_Fj.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Recursive computation of Ej and Fj\n",
"// 11.1\n",
"\n",
"function [Fj,dFj,Ej,dEj] = recursion(A,dA,C,dC,j)\n",
"Fo = C; dFo = dC;\n",
"Eo = 1; dEo = 0;\n",
"A_z = A(2:dA+1); dA_z = dA-1;\n",
"zi = 1; dzi = 0;\n",
"for i = 1:j-1\n",
"    if (dFo == 0)\n",
"        Fn1 = 0;\n",
"    else\n",
"        Fn1 = Fo(2:(dFo+1));\n",
"    end\n",
"    dFn1 = max(dFo-1,0);\n",
"    Fn2 = -Fo(1)*A_z;  dFn2 = dA-1;\n",
"    [Fn,dFn] = poladd(Fn1,dFn1,Fn2,dFn2);\n",
"    zi = convol(zi,[0,1]); dzi = dzi + 1;\n",
"    En2 = Fn(1)*zi; dEn2 = dzi;\n",
"    [En,dEn] = poladd(Eo,dEo,En2,dEn2);\n",
"    Eo = En; Fo = Fn;\n",
"    dEo = dEn; dFo = dFn;\n",
"end\n",
"if (dFo == 0)\n",
"    Fn1 = 0;\n",
"else\n",
"Fn1 = Fo(2:(dFo+1));\n",
"end;\n",
"dFn1 = max(dFo-1,0);\n",
"Fn2 = -Fo(1)*A_z;  dFn2 = dA-1;\n",
"[Fn,dFn] = poladd(Fn1,dFn1,Fn2,dFn2);\n",
"Fj = Fn; dFj = dFn;\n",
"Ej = Eo; dEj = dEo;\n",
"endfunction;\n",
""
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.2: Recursive_computation_of_Ej_and_Fj_for_the_system_presented_in_Example.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Recursive computation of Ej and Fj for the system presented in Example 11.2 on page 408.\n",
"// 11.2\n",
"\n",
"exec('poladd.sci',-1);\n",
"exec('polsize.sci',-1);\n",
"exec('recursion.sci',-1);\n",
"\n",
"C = [1 0.5]; dC = 1;\n",
"A = [1 -0.6 -0.16]; dA = 2;\n",
"j = 2;\n",
"[Fj,dFj,Ej,dEj] = recursion(A,dA,C,dC,j)\n",
""
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.3: Solution_of_Aryabhatta_identity.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Solution of Aryabhatta's identity Eq. 11.8, as discussed in Example 11.3 on page 409.\n",
"// 11.3\n",
"\n",
"exec('xdync.sci',-1);\n",
"exec('rowjoin.sci',-1);\n",
"exec('polsize.sci',-1);\n",
"exec('left_prm.sci',-1);\n",
"exec('t1calc.sci',-1);\n",
"exec('indep.sci',-1);\n",
"exec('seshft.sci',-1);\n",
"exec('makezero.sci',-1);\n",
"exec('move_sci.sci',-1);\n",
"exec('colsplit.sci',-1);\n",
"exec('clcoef.sci',-1);\n",
"exec('cindep.sci',-1);\n",
"\n",
"C = [1 0.5]; dC = 1; j=2;\n",
"A = [1 -0.6 -0.16]; dA = 2;\n",
"zj = zeros(1,j+1); zj(j+1) = 1;\n",
"[Fj,dFj,Ej,dEj] = xdync(zj,j,A,dA,C,dC)"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.4: 1st_control_problem_by_MacGregor.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// MacGregor's first control problem, discussed in Example 11.4 on page 213.\n",
"// 11.4\n",
"\n",
"exec('mv.sci',-1);\n",
"exec('cl.sci',-1);\n",
"exec('cosfil_ip.sci',-1);\n",
"exec('zpowk.sci',-1);\n",
"exec('xdync.sci',-1);\n",
"exec('rowjoin.sci',-1);\n",
"exec('polsize.sci',-1);\n",
"exec('left_prm.sci',-1);\n",
"exec('t1calc.sci',-1);\n",
"exec('indep.sci',-1);\n",
"exec('seshft.sci',-1);\n",
"exec('makezero.sci',-1);\n",
"exec('move_sci.sci',-1);\n",
"exec('colsplit.sci',-1);\n",
"exec('clcoef.sci',-1);\n",
"exec('cindep.sci',-1);\n",
"exec('polmul.sci',-1);\n",
"exec('poladd.sci',-1);\n",
"exec('tfvar.sci',-1);\n",
"exec('l2r.sci',-1);\n",
"exec('transp.sci',-1);\n",
"exec('tf.sci',-1);\n",
"exec('covar_m.sci',-1);\n",
"exec('polyno.sci',-1);\n",
"\n",
"// MacGregor's first control problem\n",
"A = [1 -1.4 0.45]; dA = 2; C = [1 -0.5]; dC = 1;\n",
"B = 0.5*[1 -0.9]; dB = 1; k = 1; int1 = 0;\n",
"[Sc,dSc,Rc,dRc] = mv(A,dA,B,dB,C,dC,k,int1);\n",
"[Nu,dNu,Du,dDu,Ny,dNy,Dy,dDy,yvar,uvar] = ...\n",
"cl(A,dA,B,dB,C,dC,k,Sc,dSc,Rc,dRc,int1);\n",
"\n",
"// Simulation parameters for stb_disc.xcos\n",
"Tc = Sc; gamm = 1; [zk,dzk] = zpowk(k);\n",
"D = 1; N_var = 1; Ts = 1; st = 0; \n",
"t_init = 0; t_final = 1000;\n",
"\n",
"[Tcp1,Tcp2] = cosfil_ip(Tc,1); // Tc/1\n",
"[Rcp1,Rcp2] = cosfil_ip(1,Rc); // 1/Rc\n",
"[Scp1,Scp2] = cosfil_ip(Sc,1); // Sc/1\n",
"[Bp,Ap] = cosfil_ip(B,A); // B/A\n",
"[zkp1,zkp2] = cosfil_ip(zk,1); // zk/1\n",
"[Cp,Dp] = cosfil_ip(C,D); // C/D\n",
"\n",
""
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.5: Minimum_variance_control_law_design.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Minimum variance control law design, given by Eq. 11.40 on page 413. \n",
"// 11.5\n",
"\n",
"// function [S,dS,R,dR] = mv(A,dA,B,dB,C,dC,k,int)\n",
"// implements the minimum variance controller\n",
"// if int>=1, integrated noise is assumed; otherwise,\n",
"// it is not integrated noise\n",
"\n",
"function [S,dS,R,dR] = mv(A,dA,B,dB,C,dC,k,int1)\n",
"zk = zeros(1,k+1); zk(k+1) = 1;\n",
"if int1>=1, [A,dA] = polmul([1 -1],1,A,dA); end\n",
"[Fk,dFk,Ek,dEk] = xdync(zk,k,A,dA,C,dC);\n",
"\n",
"[Gk,dGk] = polmul(Ek,dEk,B,dB);\n",
"S = Fk; dS = dFk; R = Gk; dR = dGk;\n",
"endfunction;"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.6: Calculation_of_closed_loop_transfer_functions.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Calculation of closed loop transfer functions\n",
"// 11.6\n",
"\n",
"// function [Nu,dNu,Du,dDu,Ny,dNy,Dy,dDy,yvar,uvar] = ...\n",
"//    cl(A,dA,B,dB,C,dC,k,S,dS,R,dR,int)\n",
"// int>=1 means integrated noise and control law:\n",
"// delta u = - (S/R)y\n",
"// Evaluates the closed loop transfer function and \n",
"// variances of input and output\n",
"\n",
"function [Nu,dNu,Du,dDu,Ny,dNy,Dy,dDy,yvar,uvar] = ...\n",
"   cl(A,dA,B,dB,C,dC,k,S,dS,R,dR,int1)\n",
"[zk,dzk] = zpowk(k);\n",
"\n",
"[BS,dBS] = polmul(B,dB,S,dS);\n",
"[zBS,dzBS] = polmul(zk,dzk,BS,dBS);\n",
"[RA,dRA] = polmul(R,dR,A,dA);\n",
"if int1>=1, [RA,dRA] = polmul(RA,dRA,[1 -1],1); end\n",
"\n",
"[D,dD] = poladd(RA,dRA,zBS,dzBS);\n",
"\n",
"[Ny,dNy] = polmul(C,dC,R,dR);\n",
"[Nu,dNu] = polmul(C,dC,S,dS);\n",
"\n",
"[Nu,dNu,Du,dDu,uvar] = tfvar(Nu,dNu,D,dD);\n",
"[Ny,dNy,Dy,dDy,yvar] = tfvar(Ny,dNy,D,dD);\n",
"\n",
"endfunction;"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.7: Cancellation_of_common_factors_and_determination_of_covariance.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Cancellation of common factors and determination of covariance\n",
"// 11.7\n",
"\n",
"// function [N,dN,D,dD,yvar] = tfvar(N,dN,D,dD)\n",
"// N and D polynomials in z^{-1} form; discrete case\n",
"\n",
"function [N,dN,D,dD,yvar] = tfvar(N,dN,D,dD)\n",
"\n",
"[N,dN,D,dD] = l2r(N,dN,D,dD);\n",
"N = N/D(1); D = D/D(1);\n",
"LN = length(N); LD = length(D); \n",
"D1 = D;\n",
"if LD<LN, D1 = [D zeros(1,LN-LD)]; dD1 = dD+LN-LD; end\n",
"H = tf(N,D1,1);//TS=1 (sampling time) has been taken constant in tfvar\n",
"yvar = covar_m(H,1); \n",
"endfunction;"
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.8: Computing_sum_of_squares.sce"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Computing sum of squares, as presented in Example 11.5 on page 415.\n",
"// 11.8\n",
"\n",
"exec('tf.sci',-1);\n",
"exec('covar_m.sci',-1);\n",
"\n",
"Y = tf([1 0],[1 -0.9],-1);\n",
"covar_m(Y,1)\n",
""
   ]
   }
,
{
		   "cell_type": "markdown",
		   "metadata": {},
		   "source": [
			"## Example 11.9: Minimum_variance_control_for_nonminimum_phase_systems.sci"
		   ]
		  },
  {
"cell_type": "code",
	   "execution_count": null,
	   "metadata": {
	    "collapsed": true
	   },
	   "outputs": [],
"source": [
"// Minimum variance control for nonminimum phase systems\n",
"// 11.9\n",
"\n",
"// function [Sc,dSc,Rc,dRc] = mv_mv(A,dA,B,dB,C,dC,k,int)\n",
"// implements the minimum variance controller\n",
"// if int>=1, integrated noise is assumed; otherwise,\n",
"// it is not integrated noise\n",
"\n",
"function [Sc,dSc,Rc,dRc] = mv_nm(A,dA,B,dB,C,dC,k,int1)\n",
"if int1>=1, [A,dA] = polmul([1 -1],1,A,dA); end\n",
"[zk,dzk] = zpowk(k);\n",
"[Bzk,dBzk] = polmul(B,dB,zk,dzk);\n",
"[Bg,Bb] = polsplit3(B); Bbr = flip(Bb);\n",
"RHS = convol(C,convol(Bg,Bbr)); dRHS = length(RHS)-1;\n",
"[Sc,dSc,Rc,dRc] = xdync(Bzk,dBzk,A,dA,RHS,dRHS);\n",
"endfunction;"
   ]
   }
],
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			{
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