{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Example 3: Minimum Frobenius Norm Static Output Feedback"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The example system has been taken from \\[1, Example 3\\]. The eigenvalue placement with minimum Frobenium norm is discussed in \\[2, 3\\]. The poles should be placed at $-3,-4,-5,-2\\pm2j$, i.e., we want to have a conjugate complex pair.\n",
"\n",
"1. Lee, T. H.,; Wang, Q. G.; Koh, E. K. (1994). *An iterative algorithm for pole placement by output feedback*. IEEE Transactions on Automatic Control, 39(3), 565-568.\n",
"2. Röbenack, K.: Voßwinkell, R.: Franke, M. (2018). *On the eigenvalue placement by static output feedback via quantifier elimination*. In 26th Mediterranean Conference on Control and Automation (MED), pp. 133-138. DOI: <https://doi.org/10.1109/MED.2018.8442817>\n",
"3. Röbenack, K.; Gerbet, D.: *Minimum Norm Partial Eigenvalue Placement for Static Output Feedback Control*.\n",
"[International Conference on System Theory, Control and Computing (ICSTCC)](https://icstcc2021.ac.tuiasi.ro/), \n",
"October 20-23, 2021, Iași, Romania"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Polynomial ring"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Bold{Q}[k_{11}, k_{12}, k_{21}, k_{22}, k_{31}, k_{32}, s, l_{0}, l_{1}, l_{2}, l_{3}, l_{4}, l_{5}, q]</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Bold{Q}[k_{11}, k_{12}, k_{21}, k_{22}, k_{31}, k_{32}, s, l_{0}, l_{1}, l_{2}, l_{3}, l_{4}, l_{5}, q]$$"
],
"text/plain": [
"Multivariate Polynomial Ring in k11, k12, k21, k22, k31, k32, s, l0, l1, l2, l3, l4, l5, q over Rational Field"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"%display latex\n",
"R.<k11,k12,k21,k22,k31,k32,s,l0,l1,l2,l3,l4,l5,q> = PolynomialRing(QQ, order='lex')\n",
"R"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"State space system"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\left(\\begin{array}{rrrrr}\n",
"0 & 1 & 0 & 0 & 0 \\\\\n",
"0 & 0 & 1 & 0 & 0 \\\\\n",
"0 & 0 & 0 & 1 & 0 \\\\\n",
"0 & 0 & 0 & 0 & 1 \\\\\n",
"0 & 0 & 0 & 0 & 0\n",
"\\end{array}\\right), \\left(\\begin{array}{rrr}\n",
"1 & 0 & 0 \\\\\n",
"1 & 0 & 0 \\\\\n",
"0 & 1 & 0 \\\\\n",
"0 & 1 & 0 \\\\\n",
"0 & 0 & 1\n",
"\\end{array}\\right), \\left(\\begin{array}{rrrrr}\n",
"1 & 0 & 0 & 0 & 0 \\\\\n",
"0 & 1 & 0 & 0 & 0\n",
"\\end{array}\\right)\\right)</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\left(\\begin{array}{rrrrr}\n",
"0 & 1 & 0 & 0 & 0 \\\\\n",
"0 & 0 & 1 & 0 & 0 \\\\\n",
"0 & 0 & 0 & 1 & 0 \\\\\n",
"0 & 0 & 0 & 0 & 1 \\\\\n",
"0 & 0 & 0 & 0 & 0\n",
"\\end{array}\\right), \\left(\\begin{array}{rrr}\n",
"1 & 0 & 0 \\\\\n",
"1 & 0 & 0 \\\\\n",
"0 & 1 & 0 \\\\\n",
"0 & 1 & 0 \\\\\n",
"0 & 0 & 1\n",
"\\end{array}\\right), \\left(\\begin{array}{rrrrr}\n",
"1 & 0 & 0 & 0 & 0 \\\\\n",
"0 & 1 & 0 & 0 & 0\n",
"\\end{array}\\right)\\right)$$"
],
"text/plain": [
"(\n",
"[0 1 0 0 0] [1 0 0] \n",
"[0 0 1 0 0] [1 0 0] \n",
"[0 0 0 1 0] [0 1 0] \n",
"[0 0 0 0 1] [0 1 0] [1 0 0 0 0]\n",
"[0 0 0 0 0], [0 0 1], [0 1 0 0 0]\n",
")"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = matrix(R,[[0,1,0,0,0],\n",
" [0,0,1,0,0],\n",
" [0,0,0,1,0],\n",
" [0,0,0,0,1],\n",
" [0,0,0,0,0]])\n",
"B = matrix(R,[[1,0,0],[1,0,0],[0,1,0],[0,1,0],[0,0,1]])\n",
"C = matrix(R,[[1,0,0,0,0],[0,1,0,0,0]])\n",
"(A,B,C)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Symbolic feedback matrix"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n",
"k_{11} & k_{12} \\\\\n",
"k_{21} & k_{22} \\\\\n",
"k_{31} & k_{32}\n",
"\\end{array}\\right)</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n",
"k_{11} & k_{12} \\\\\n",
"k_{21} & k_{22} \\\\\n",
"k_{31} & k_{32}\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[k11 k12]\n",
"[k21 k22]\n",
"[k31 k32]"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"K = matrix(R,[[k11,k12],[k21,k22],[k31,k32]])\n",
"K"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Closed-loop characteristic polynomial"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}k_{11} s^{4} + k_{12} s^{4} + s^{5} - k_{12} k_{21} s^{2} + k_{11} k_{22} s^{2} + k_{11} s^{3} + k_{22} s^{3} - k_{12} k_{21} s + k_{11} k_{22} s + k_{21} s^{2} + k_{22} s^{2} - k_{12} k_{31} + k_{11} k_{32} + k_{21} s + k_{32} s + k_{31}</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}k_{11} s^{4} + k_{12} s^{4} + s^{5} - k_{12} k_{21} s^{2} + k_{11} k_{22} s^{2} + k_{11} s^{3} + k_{22} s^{3} - k_{12} k_{21} s + k_{11} k_{22} s + k_{21} s^{2} + k_{22} s^{2} - k_{12} k_{31} + k_{11} k_{32} + k_{21} s + k_{32} s + k_{31}$$"
],
"text/plain": [
"k11*s^4 + k12*s^4 + s^5 - k12*k21*s^2 + k11*k22*s^2 + k11*s^3 + k22*s^3 - k12*k21*s + k11*k22*s + k21*s^2 + k22*s^2 - k12*k31 + k11*k32 + k21*s + k32*s + k31"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"CP = det(s*matrix.identity(5)-(A-B*K*C))\n",
"CP"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Remainders of polynomial division for an eigenvalue placement at $-3,-4,-5,-2\\pm2j$"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [],
"source": [
"q1,r1 = CP.quo_rem(s+5)\n",
"q2,r2 = q1.quo_rem(s+4)\n",
"q3,r3 = q2.quo_rem(s+3)\n",
"q4,r4 = q3.quo_rem(s^2 + 4*s + 8)\n",
"r4a = r4.coefficient({s:0})\n",
"r4b = r4.coefficient({s:1})"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Lagrangian function"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}k_{11}^{2} l_{0} + 20 k_{11} k_{22} l_{1} - 8 k_{11} k_{22} l_{2} + k_{11} k_{22} l_{3} + k_{11} k_{32} l_{1} + 500 k_{11} l_{1} - 308 k_{11} l_{2} + 85 k_{11} l_{3} - 11 k_{11} l_{4} + k_{11} l_{5} + k_{12}^{2} l_{0} - 20 k_{12} k_{21} l_{1} + 8 k_{12} k_{21} l_{2} - k_{12} k_{21} l_{3} - k_{12} k_{31} l_{1} + 625 k_{12} l_{1} - 369 k_{12} l_{2} + 97 k_{12} l_{3} - 12 k_{12} l_{4} + k_{12} l_{5} + k_{21}^{2} l_{0} + 20 k_{21} l_{1} - 8 k_{21} l_{2} + k_{21} l_{3} + k_{22}^{2} l_{0} - 100 k_{22} l_{1} + 52 k_{22} l_{2} - 11 k_{22} l_{3} + k_{22} l_{4} + k_{31}^{2} l_{0} + k_{31} l_{1} + k_{32}^{2} l_{0} - 5 k_{32} l_{1} + k_{32} l_{2} - l_{0} q - 3125 l_{1} + 2101 l_{2} - 660 l_{3} + 89 l_{4} - 16 l_{5} + q</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}k_{11}^{2} l_{0} + 20 k_{11} k_{22} l_{1} - 8 k_{11} k_{22} l_{2} + k_{11} k_{22} l_{3} + k_{11} k_{32} l_{1} + 500 k_{11} l_{1} - 308 k_{11} l_{2} + 85 k_{11} l_{3} - 11 k_{11} l_{4} + k_{11} l_{5} + k_{12}^{2} l_{0} - 20 k_{12} k_{21} l_{1} + 8 k_{12} k_{21} l_{2} - k_{12} k_{21} l_{3} - k_{12} k_{31} l_{1} + 625 k_{12} l_{1} - 369 k_{12} l_{2} + 97 k_{12} l_{3} - 12 k_{12} l_{4} + k_{12} l_{5} + k_{21}^{2} l_{0} + 20 k_{21} l_{1} - 8 k_{21} l_{2} + k_{21} l_{3} + k_{22}^{2} l_{0} - 100 k_{22} l_{1} + 52 k_{22} l_{2} - 11 k_{22} l_{3} + k_{22} l_{4} + k_{31}^{2} l_{0} + k_{31} l_{1} + k_{32}^{2} l_{0} - 5 k_{32} l_{1} + k_{32} l_{2} - l_{0} q - 3125 l_{1} + 2101 l_{2} - 660 l_{3} + 89 l_{4} - 16 l_{5} + q$$"
],
"text/plain": [
"k11^2*l0 + 20*k11*k22*l1 - 8*k11*k22*l2 + k11*k22*l3 + k11*k32*l1 + 500*k11*l1 - 308*k11*l2 + 85*k11*l3 - 11*k11*l4 + k11*l5 + k12^2*l0 - 20*k12*k21*l1 + 8*k12*k21*l2 - k12*k21*l3 - k12*k31*l1 + 625*k12*l1 - 369*k12*l2 + 97*k12*l3 - 12*k12*l4 + k12*l5 + k21^2*l0 + 20*k21*l1 - 8*k21*l2 + k21*l3 + k22^2*l0 - 100*k22*l1 + 52*k22*l2 - 11*k22*l3 + k22*l4 + k31^2*l0 + k31*l1 + k32^2*l0 - 5*k32*l1 + k32*l2 - l0*q - 3125*l1 + 2101*l2 - 660*l3 + 89*l4 - 16*l5 + q"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L = q + l0*(k11^2+k12^2+k21^2+k22^2+k31^2+k32^2-q) + l1*r1 + l2*r2 + l3*r3 + l4*r4a + l5*r4b\n",
"L"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Neccessary optimility condition and associated polynomial ideal"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0 : k11 : 2*k11*l0 + 20*k22*l1 - 8*k22*l2 + k22*l3 + k32*l1 + 500*l1 - 308*l2 + 85*l3 - 11*l4 + l5\n",
"1 : k12 : 2*k12*l0 - 20*k21*l1 + 8*k21*l2 - k21*l3 - k31*l1 + 625*l1 - 369*l2 + 97*l3 - 12*l4 + l5\n",
"2 : k21 : -20*k12*l1 + 8*k12*l2 - k12*l3 + 2*k21*l0 + 20*l1 - 8*l2 + l3\n",
"3 : k22 : 20*k11*l1 - 8*k11*l2 + k11*l3 + 2*k22*l0 - 100*l1 + 52*l2 - 11*l3 + l4\n",
"4 : k31 : -k12*l1 + 2*k31*l0 + l1\n",
"5 : k32 : k11*l1 + 2*k32*l0 - 5*l1 + l2\n",
"6 : l0 : k11^2 + k12^2 + k21^2 + k22^2 + k31^2 + k32^2 - q\n",
"7 : l1 : 20*k11*k22 + k11*k32 + 500*k11 - 20*k12*k21 - k12*k31 + 625*k12 + 20*k21 - 100*k22 + k31 - 5*k32 - 3125\n",
"8 : l2 : -8*k11*k22 - 308*k11 + 8*k12*k21 - 369*k12 - 8*k21 + 52*k22 + k32 + 2101\n",
"9 : l3 : k11*k22 + 85*k11 - k12*k21 + 97*k12 + k21 - 11*k22 - 660\n",
"10 : l4 : -11*k11 - 12*k12 + k22 + 89\n",
"11 : l5 : k11 + k12 - 16\n",
"12 : q : -l0 + 1\n"
]
},
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(2 k_{11} l_{0} + 20 k_{22} l_{1} - 8 k_{22} l_{2} + k_{22} l_{3} + k_{32} l_{1} + 500 l_{1} - 308 l_{2} + 85 l_{3} - 11 l_{4} + l_{5}, 2 k_{12} l_{0} - 20 k_{21} l_{1} + 8 k_{21} l_{2} - k_{21} l_{3} - k_{31} l_{1} + 625 l_{1} - 369 l_{2} + 97 l_{3} - 12 l_{4} + l_{5}, -20 k_{12} l_{1} + 8 k_{12} l_{2} - k_{12} l_{3} + 2 k_{21} l_{0} + 20 l_{1} - 8 l_{2} + l_{3}, 20 k_{11} l_{1} - 8 k_{11} l_{2} + k_{11} l_{3} + 2 k_{22} l_{0} - 100 l_{1} + 52 l_{2} - 11 l_{3} + l_{4}, -k_{12} l_{1} + 2 k_{31} l_{0} + l_{1}, k_{11} l_{1} + 2 k_{32} l_{0} - 5 l_{1} + l_{2}, k_{11}^{2} + k_{12}^{2} + k_{21}^{2} + k_{22}^{2} + k_{31}^{2} + k_{32}^{2} - q, 20 k_{11} k_{22} + k_{11} k_{32} + 500 k_{11} - 20 k_{12} k_{21} - k_{12} k_{31} + 625 k_{12} + 20 k_{21} - 100 k_{22} + k_{31} - 5 k_{32} - 3125, -8 k_{11} k_{22} - 308 k_{11} + 8 k_{12} k_{21} - 369 k_{12} - 8 k_{21} + 52 k_{22} + k_{32} + 2101, k_{11} k_{22} + 85 k_{11} - k_{12} k_{21} + 97 k_{12} + k_{21} - 11 k_{22} - 660, -11 k_{11} - 12 k_{12} + k_{22} + 89, k_{11} + k_{12} - 16, -l_{0} + 1\\right)\\Bold{Q}[k_{11}, k_{12}, k_{21}, k_{22}, k_{31}, k_{32}, s, l_{0}, l_{1}, l_{2}, l_{3}, l_{4}, l_{5}, q]</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(2 k_{11} l_{0} + 20 k_{22} l_{1} - 8 k_{22} l_{2} + k_{22} l_{3} + k_{32} l_{1} + 500 l_{1} - 308 l_{2} + 85 l_{3} - 11 l_{4} + l_{5}, 2 k_{12} l_{0} - 20 k_{21} l_{1} + 8 k_{21} l_{2} - k_{21} l_{3} - k_{31} l_{1} + 625 l_{1} - 369 l_{2} + 97 l_{3} - 12 l_{4} + l_{5}, -20 k_{12} l_{1} + 8 k_{12} l_{2} - k_{12} l_{3} + 2 k_{21} l_{0} + 20 l_{1} - 8 l_{2} + l_{3}, 20 k_{11} l_{1} - 8 k_{11} l_{2} + k_{11} l_{3} + 2 k_{22} l_{0} - 100 l_{1} + 52 l_{2} - 11 l_{3} + l_{4}, -k_{12} l_{1} + 2 k_{31} l_{0} + l_{1}, k_{11} l_{1} + 2 k_{32} l_{0} - 5 l_{1} + l_{2}, k_{11}^{2} + k_{12}^{2} + k_{21}^{2} + k_{22}^{2} + k_{31}^{2} + k_{32}^{2} - q, 20 k_{11} k_{22} + k_{11} k_{32} + 500 k_{11} - 20 k_{12} k_{21} - k_{12} k_{31} + 625 k_{12} + 20 k_{21} - 100 k_{22} + k_{31} - 5 k_{32} - 3125, -8 k_{11} k_{22} - 308 k_{11} + 8 k_{12} k_{21} - 369 k_{12} - 8 k_{21} + 52 k_{22} + k_{32} + 2101, k_{11} k_{22} + 85 k_{11} - k_{12} k_{21} + 97 k_{12} + k_{21} - 11 k_{22} - 660, -11 k_{11} - 12 k_{12} + k_{22} + 89, k_{11} + k_{12} - 16, -l_{0} + 1\\right)\\Bold{Q}[k_{11}, k_{12}, k_{21}, k_{22}, k_{31}, k_{32}, s, l_{0}, l_{1}, l_{2}, l_{3}, l_{4}, l_{5}, q]$$"
],
"text/plain": [
"Ideal (2*k11*l0 + 20*k22*l1 - 8*k22*l2 + k22*l3 + k32*l1 + 500*l1 - 308*l2 + 85*l3 - 11*l4 + l5, 2*k12*l0 - 20*k21*l1 + 8*k21*l2 - k21*l3 - k31*l1 + 625*l1 - 369*l2 + 97*l3 - 12*l4 + l5, -20*k12*l1 + 8*k12*l2 - k12*l3 + 2*k21*l0 + 20*l1 - 8*l2 + l3, 20*k11*l1 - 8*k11*l2 + k11*l3 + 2*k22*l0 - 100*l1 + 52*l2 - 11*l3 + l4, -k12*l1 + 2*k31*l0 + l1, k11*l1 + 2*k32*l0 - 5*l1 + l2, k11^2 + k12^2 + k21^2 + k22^2 + k31^2 + k32^2 - q, 20*k11*k22 + k11*k32 + 500*k11 - 20*k12*k21 - k12*k31 + 625*k12 + 20*k21 - 100*k22 + k31 - 5*k32 - 3125, -8*k11*k22 - 308*k11 + 8*k12*k21 - 369*k12 - 8*k21 + 52*k22 + k32 + 2101, k11*k22 + 85*k11 - k12*k21 + 97*k12 + k21 - 11*k22 - 660, -11*k11 - 12*k12 + k22 + 89, k11 + k12 - 16, -l0 + 1) of Multivariate Polynomial Ring in k11, k12, k21, k22, k31, k32, s, l0, l1, l2, l3, l4, l5, q over Rational Field"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"vars = [k11,k12,k21,k22,k31,k32,l0,l1,l2,l3,l4,l5,q]\n",
"PLIST = []\n",
"for ii in range(len(vars)):\n",
" print(ii,\" : \",vars[ii],\" : \",diff(L,vars[ii]))\n",
" PLIST.append(diff(L,vars[ii]))\n",
"I = Ideal(PLIST)\n",
"I"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Dimension of the ideal"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}1</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}1$$"
],
"text/plain": [
"1"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"I.dimension()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Change of the ring (without the variable s), dimension of the new ideal over the new ring"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}0</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}0$$"
],
"text/plain": [
"0"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"J = I.change_ring(PolynomialRing(QQ, 'k11,k12,k21,k22,k31,k32,l0,l1,l2,l3,l4,l5,q'))\n",
"J.dimension()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Algebraic variety, computation of possible solutions w.r.t. q"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"data": {
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"[-3.90773899890248 101.104044494465]\n",
"[ 17.3525561053156 373.104044494465]"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"K0 = K.subs(k11=RR(lx['k11']),k12=RR(lx['k12']),\n",
" k21=RR(lx['k21']),k22=RR(lx['k22']),\n",
" k31=RR(lx['k31']),k32=RR(lx['k32']))\n",
"K0"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Frobenius norm of the computed feedback matrix"
]
},
{
"cell_type": "code",
"execution_count": 34,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}387.2306240176759</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}387.2306240176759$$"
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"text/plain": [
"387.2306240176759"
]
},
"execution_count": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"K0.norm('frob')"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Verification of the closed-loop eigenvalues (over the rational field)"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-4.999999999999944?, -4.000000000000080?, -2.999999999999979?, -1.999999999999999? - 1.999999999999999? \\sqrt{-1}, -1.999999999999999? + 1.999999999999999? \\sqrt{-1}\\right]</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-4.999999999999944?, -4.000000000000080?, -2.999999999999979?, -1.999999999999999? - 1.999999999999999? \\sqrt{-1}, -1.999999999999999? + 1.999999999999999? \\sqrt{-1}\\right]$$"
],
"text/plain": [
"[-4.999999999999944?, -4.000000000000080?, -2.999999999999979?, -1.999999999999999? - 1.999999999999999?*I, -1.999999999999999? + 1.999999999999999?*I]"
]
},
"execution_count": 35,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A0 = matrix(QQ,A-B*K0*C)\n",
"A0.eigenvalues()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Verification of the closed-loop eigenvalues (as floating point numbers)"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-5.0000000000000435, -2.0000000000000067 + 2.0000000000000058i, -2.0000000000000067 - 2.0000000000000058i, -3.99999999999994, -2.9999999999999916\\right]</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[-5.0000000000000435, -2.0000000000000067 + 2.0000000000000058i, -2.0000000000000067 - 2.0000000000000058i, -3.99999999999994, -2.9999999999999916\\right]$$"
],
"text/plain": [
"[-5.0000000000000435,\n",
" -2.0000000000000067 + 2.0000000000000058*I,\n",
" -2.0000000000000067 - 2.0000000000000058*I,\n",
" -3.99999999999994,\n",
" -2.9999999999999916]"
]
},
"execution_count": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A0 = matrix(RDF,A-B*K0*C)\n",
"A0.eigenvalues()"
]
}
],
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"display_name": "SageMath 9.3",
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