{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Example 4: Minimum Frobenius Norm Static Output Feedback"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The example system has been taken from \\[1, Example 4.2\\]. The eigenvalue placement with minimum Frobenium norm is discussed in \\[2, Example 4\\]. The open-loop system has eigenvlaues at $\\pm1$, $\\pm2$. We want to place two eigenvalues at -3 and -4, i.e., we carry out a partial eigenvalue assignment.\n",
"\n",
"1. Yannakoudakis, A. G. (2016): *The static output feedback from the invariant point of view*. IMA Journal of Mathematical Control and Information, 33(3), 639-668. DOI: <https://doi.org/10.1093/imamci/dnu057>\n",
"2. 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": 37,
"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}, s, l_{0}, l_{1}, l_{2}, q]</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\Bold{Q}[k_{11}, k_{12}, k_{21}, k_{22}, s, l_{0}, l_{1}, l_{2}, q]$$"
],
"text/plain": [
"Multivariate Polynomial Ring in k11, k12, k21, k22, s, l0, l1, l2, q over Rational Field"
]
},
"execution_count": 37,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"%display latex\n",
"R.<k11,k12,k21,k22,s,l0,l1,l2,q> = PolynomialRing(QQ, order='lex')\n",
"R"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"State space system"
]
},
{
"cell_type": "code",
"execution_count": 38,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\left(\\begin{array}{rrrr}\n",
"-1 & 0 & 0 & 0 \\\\\n",
"0 & -2 & 0 & 0 \\\\\n",
"0 & 0 & 1 & 0 \\\\\n",
"0 & 0 & 0 & 2\n",
"\\end{array}\\right), \\left(\\begin{array}{rr}\n",
"1 & 0 \\\\\n",
"1 & 0 \\\\\n",
"1 & 1 \\\\\n",
"1 & 0\n",
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
"1 & 1 & 1 & 1 \\\\\n",
"0 & 0 & 0 & 1\n",
"\\end{array}\\right)\\right)</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\left(\\begin{array}{rrrr}\n",
"-1 & 0 & 0 & 0 \\\\\n",
"0 & -2 & 0 & 0 \\\\\n",
"0 & 0 & 1 & 0 \\\\\n",
"0 & 0 & 0 & 2\n",
"\\end{array}\\right), \\left(\\begin{array}{rr}\n",
"1 & 0 \\\\\n",
"1 & 0 \\\\\n",
"1 & 1 \\\\\n",
"1 & 0\n",
"\\end{array}\\right), \\left(\\begin{array}{rrrr}\n",
"1 & 1 & 1 & 1 \\\\\n",
"0 & 0 & 0 & 1\n",
"\\end{array}\\right)\\right)$$"
],
"text/plain": [
"(\n",
"[-1 0 0 0] [1 0] \n",
"[ 0 -2 0 0] [1 0] \n",
"[ 0 0 1 0] [1 1] [1 1 1 1]\n",
"[ 0 0 0 2], [1 0], [0 0 0 1]\n",
")"
]
},
"execution_count": 38,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = matrix(R,[[-1,0,0,0],\n",
" [0,-2,0,0],\n",
" [0,0,1,0],\n",
" [0,0,0,2]])\n",
"B = matrix(R,[[1,0],[1,0],[1,1],[1,0]])\n",
"C = matrix(R,[[1,1,1,1],[0,0,0,1]])\n",
"(A,B,C)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Symbolic feedback matrix"
]
},
{
"cell_type": "code",
"execution_count": 39,
"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",
"\\end{array}\\right)</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n",
"k_{11} & k_{12} \\\\\n",
"k_{21} & k_{22}\n",
"\\end{array}\\right)$$"
],
"text/plain": [
"[k11 k12]\n",
"[k21 k22]"
]
},
"execution_count": 39,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"K = matrix(R,[[k11,k12],[k21,k22]])\n",
"K"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Closed-loop characteristic polynomial"
]
},
{
"cell_type": "code",
"execution_count": 41,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}k_{12} k_{21} s^{2} - k_{11} k_{22} s^{2} + 4 k_{11} s^{3} + k_{12} s^{3} + k_{21} s^{3} + s^{4} + 3 k_{12} k_{21} s - 3 k_{11} k_{22} s + 2 k_{12} s^{2} + k_{21} s^{2} + 2 k_{12} k_{21} - 2 k_{11} k_{22} - 10 k_{11} s - k_{12} s - 4 k_{21} s - 5 s^{2} - 2 k_{12} - 4 k_{21} + 4</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}k_{12} k_{21} s^{2} - k_{11} k_{22} s^{2} + 4 k_{11} s^{3} + k_{12} s^{3} + k_{21} s^{3} + s^{4} + 3 k_{12} k_{21} s - 3 k_{11} k_{22} s + 2 k_{12} s^{2} + k_{21} s^{2} + 2 k_{12} k_{21} - 2 k_{11} k_{22} - 10 k_{11} s - k_{12} s - 4 k_{21} s - 5 s^{2} - 2 k_{12} - 4 k_{21} + 4$$"
],
"text/plain": [
"k12*k21*s^2 - k11*k22*s^2 + 4*k11*s^3 + k12*s^3 + k21*s^3 + s^4 + 3*k12*k21*s - 3*k11*k22*s + 2*k12*s^2 + k21*s^2 + 2*k12*k21 - 2*k11*k22 - 10*k11*s - k12*s - 4*k21*s - 5*s^2 - 2*k12 - 4*k21 + 4"
]
},
"execution_count": 41,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"CP = det(s*matrix.identity(4)-(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": 45,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 k_{12} k_{21} - 2 k_{11} k_{22} - 78 k_{11} - 8 k_{12} - 10 k_{21} + 40, 6 k_{12} k_{21} - 6 k_{11} k_{22} - 216 k_{11} - 30 k_{12} - 36 k_{21} + 180\\right]</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[2 k_{12} k_{21} - 2 k_{11} k_{22} - 78 k_{11} - 8 k_{12} - 10 k_{21} + 40, 6 k_{12} k_{21} - 6 k_{11} k_{22} - 216 k_{11} - 30 k_{12} - 36 k_{21} + 180\\right]$$"
],
"text/plain": [
"[2*k12*k21 - 2*k11*k22 - 78*k11 - 8*k12 - 10*k21 + 40,\n",
" 6*k12*k21 - 6*k11*k22 - 216*k11 - 30*k12 - 36*k21 + 180]"
]
},
"execution_count": 45,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"q1,r1 = CP.quo_rem(s+3)\n",
"q2,r2 = CP.quo_rem(s+4)\n",
"[r1,r2]"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Lagrangian function"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html><script type=\"math/tex; mode=display\">\\newcommand{\\Bold}[1]{\\mathbf{#1}}k_{11}^{2} l_{0} - 2 k_{11} k_{22} l_{1} - 6 k_{11} k_{22} l_{2} - 78 k_{11} l_{1} - 216 k_{11} l_{2} + k_{12}^{2} l_{0} + 2 k_{12} k_{21} l_{1} + 6 k_{12} k_{21} l_{2} - 8 k_{12} l_{1} - 30 k_{12} l_{2} + k_{21}^{2} l_{0} - 10 k_{21} l_{1} - 36 k_{21} l_{2} + k_{22}^{2} l_{0} - l_{0} q + 40 l_{1} + 180 l_{2} + q</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}k_{11}^{2} l_{0} - 2 k_{11} k_{22} l_{1} - 6 k_{11} k_{22} l_{2} - 78 k_{11} l_{1} - 216 k_{11} l_{2} + k_{12}^{2} l_{0} + 2 k_{12} k_{21} l_{1} + 6 k_{12} k_{21} l_{2} - 8 k_{12} l_{1} - 30 k_{12} l_{2} + k_{21}^{2} l_{0} - 10 k_{21} l_{1} - 36 k_{21} l_{2} + k_{22}^{2} l_{0} - l_{0} q + 40 l_{1} + 180 l_{2} + q$$"
],
"text/plain": [
"k11^2*l0 - 2*k11*k22*l1 - 6*k11*k22*l2 - 78*k11*l1 - 216*k11*l2 + k12^2*l0 + 2*k12*k21*l1 + 6*k12*k21*l2 - 8*k12*l1 - 30*k12*l2 + k21^2*l0 - 10*k21*l1 - 36*k21*l2 + k22^2*l0 - l0*q + 40*l1 + 180*l2 + q"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"L = q + l0*(k11^2+k12^2+k21^2+k22^2-q) + l1*r1 + l2*r2\n",
"L"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Neccessary optimility condition and associated polynomial ideal"
]
},
{
"cell_type": "code",
"execution_count": 47,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"0 : k11 : 2*k11*l0 - 2*k22*l1 - 6*k22*l2 - 78*l1 - 216*l2\n",
"1 : k12 : 2*k12*l0 + 2*k21*l1 + 6*k21*l2 - 8*l1 - 30*l2\n",
"2 : k21 : 2*k12*l1 + 6*k12*l2 + 2*k21*l0 - 10*l1 - 36*l2\n",
"3 : k22 : -2*k11*l1 - 6*k11*l2 + 2*k22*l0\n",
"4 : l0 : k11^2 + k12^2 + k21^2 + k22^2 - q\n",
"5 : l1 : -2*k11*k22 - 78*k11 + 2*k12*k21 - 8*k12 - 10*k21 + 40\n",
"6 : l2 : -6*k11*k22 - 216*k11 + 6*k12*k21 - 30*k12 - 36*k21 + 180\n",
"7 : 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} - 2 k_{22} l_{1} - 6 k_{22} l_{2} - 78 l_{1} - 216 l_{2}, 2 k_{12} l_{0} + 2 k_{21} l_{1} + 6 k_{21} l_{2} - 8 l_{1} - 30 l_{2}, 2 k_{12} l_{1} + 6 k_{12} l_{2} + 2 k_{21} l_{0} - 10 l_{1} - 36 l_{2}, -2 k_{11} l_{1} - 6 k_{11} l_{2} + 2 k_{22} l_{0}, k_{11}^{2} + k_{12}^{2} + k_{21}^{2} + k_{22}^{2} - q, -2 k_{11} k_{22} - 78 k_{11} + 2 k_{12} k_{21} - 8 k_{12} - 10 k_{21} + 40, -6 k_{11} k_{22} - 216 k_{11} + 6 k_{12} k_{21} - 30 k_{12} - 36 k_{21} + 180, -l_{0} + 1\\right)\\Bold{Q}[k_{11}, k_{12}, k_{21}, k_{22}, s, l_{0}, l_{1}, l_{2}, q]</script></html>"
],
"text/latex": [
"$$\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(2 k_{11} l_{0} - 2 k_{22} l_{1} - 6 k_{22} l_{2} - 78 l_{1} - 216 l_{2}, 2 k_{12} l_{0} + 2 k_{21} l_{1} + 6 k_{21} l_{2} - 8 l_{1} - 30 l_{2}, 2 k_{12} l_{1} + 6 k_{12} l_{2} + 2 k_{21} l_{0} - 10 l_{1} - 36 l_{2}, -2 k_{11} l_{1} - 6 k_{11} l_{2} + 2 k_{22} l_{0}, k_{11}^{2} + k_{12}^{2} + k_{21}^{2} + k_{22}^{2} - q, -2 k_{11} k_{22} - 78 k_{11} + 2 k_{12} k_{21} - 8 k_{12} - 10 k_{21} + 40, -6 k_{11} k_{22} - 216 k_{11} + 6 k_{12} k_{21} - 30 k_{12} - 36 k_{21} + 180, -l_{0} + 1\\right)\\Bold{Q}[k_{11}, k_{12}, k_{21}, k_{22}, s, l_{0}, l_{1}, l_{2}, q]$$"
],
"text/plain": [
"Ideal (2*k11*l0 - 2*k22*l1 - 6*k22*l2 - 78*l1 - 216*l2, 2*k12*l0 + 2*k21*l1 + 6*k21*l2 - 8*l1 - 30*l2, 2*k12*l1 + 6*k12*l2 + 2*k21*l0 - 10*l1 - 36*l2, -2*k11*l1 - 6*k11*l2 + 2*k22*l0, k11^2 + k12^2 + k21^2 + k22^2 - q, -2*k11*k22 - 78*k11 + 2*k12*k21 - 8*k12 - 10*k21 + 40, -6*k11*k22 - 216*k11 + 6*k12*k21 - 30*k12 - 36*k21 + 180, -l0 + 1) of Multivariate Polynomial Ring in k11, k12, k21, k22, s, l0, l1, l2, q over Rational Field"
]
},
"execution_count": 47,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"vars = [k11,k12,k21,k22,l0,l1,l2,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": 48,
"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": 48,
"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": 49,
"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": 49,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"J = I.change_ring(PolynomialRing(QQ, 'k11,k12,k21,k22,l0,l1,l2,q'))\n",
"J.dimension()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Algebraic variety, computation of possible solutions w.r.t. q"
]
},
{
"cell_type": "code",
"execution_count": 50,
"metadata": {},
"outputs": [
{
"data": {
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