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Example 3: Minimum Frobenius Norm Static Output Feedback¶

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.

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.
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
Röbenack, K.; Gerbet, D.: Minimum Norm Partial Eigenvalue Placement for Static Output Feedback Control.

International Conference on System Theory, Control and Computing (ICSTCC),
October 20-23, 2021, Iași, Romania

Polynomial ring

In [22]:

%display latex
R.<k11,k12,k21,k22,k31,k32,s,l0,l1,l2,l3,l4,l5,q> = PolynomialRing(QQ, order='lex')
R

Out[22]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\Bold{Q}[k_{11}, k_{12}, k_{21}, k_{22}, k_{31}, k_{32}, s, l_{0}, l_{1}, l_{2}, l_{3}, l_{4}, l_{5}, q]$

State space system

In [23]:

A = matrix(R,[[0,1,0,0,0],
        [0,0,1,0,0],
        [0,0,0,1,0],
        [0,0,0,0,1],
        [0,0,0,0,0]])
B = matrix(R,[[1,0,0],[1,0,0],[0,1,0],[0,1,0],[0,0,1]])
C = matrix(R,[[1,0,0,0,0],[0,1,0,0,0]])
(A,B,C)

Out[23]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left(\begin{array}{rrrrr} 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right), \left(\begin{array}{rrr} 1 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right), \left(\begin{array}{rrrrr} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \end{array}\right)\right)$

Symbolic feedback matrix

In [24]:

K = matrix(R,[[k11,k12],[k21,k22],[k31,k32]])
K

Out[24]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} k_{11} & k_{12} \\ k_{21} & k_{22} \\ k_{31} & k_{32} \end{array}\right)$

Closed-loop characteristic polynomial

In [25]:

CP = det(s*matrix.identity(5)-(A-B*K*C))
CP

Out[25]:

$\newcommand{\Bold}[1]{\mathbf{#1}}k_{11} s^{4} + k_{12} s^{4} + s^{5} - k_{12} k_{21} s^{2} + k_{11} k_{22} s^{2} + k_{11} s^{3} + k_{22} s^{3} - k_{12} k_{21} s + k_{11} k_{22} s + k_{21} s^{2} + k_{22} s^{2} - k_{12} k_{31} + k_{11} k_{32} + k_{21} s + k_{32} s + k_{31}$

Remainders of polynomial division for an eigenvalue placement at $-3,-4,-5,-2\pm2j$

In [26]:

q1,r1 = CP.quo_rem(s+5)
q2,r2 = q1.quo_rem(s+4)
q3,r3 = q2.quo_rem(s+3)
q4,r4 = q3.quo_rem(s^2 + 4*s + 8)
r4a = r4.coefficient({s:0})
r4b = r4.coefficient({s:1})

Lagrangian function

In [27]:

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
L

Out[27]:

$\newcommand{\Bold}[1]{\mathbf{#1}}k_{11}^{2} l_{0} + 20 k_{11} k_{22} l_{1} - 8 k_{11} k_{22} l_{2} + k_{11} k_{22} l_{3} + k_{11} k_{32} l_{1} + 500 k_{11} l_{1} - 308 k_{11} l_{2} + 85 k_{11} l_{3} - 11 k_{11} l_{4} + k_{11} l_{5} + k_{12}^{2} l_{0} - 20 k_{12} k_{21} l_{1} + 8 k_{12} k_{21} l_{2} - k_{12} k_{21} l_{3} - k_{12} k_{31} l_{1} + 625 k_{12} l_{1} - 369 k_{12} l_{2} + 97 k_{12} l_{3} - 12 k_{12} l_{4} + k_{12} l_{5} + k_{21}^{2} l_{0} + 20 k_{21} l_{1} - 8 k_{21} l_{2} + k_{21} l_{3} + k_{22}^{2} l_{0} - 100 k_{22} l_{1} + 52 k_{22} l_{2} - 11 k_{22} l_{3} + k_{22} l_{4} + k_{31}^{2} l_{0} + k_{31} l_{1} + k_{32}^{2} l_{0} - 5 k_{32} l_{1} + k_{32} l_{2} - l_{0} q - 3125 l_{1} + 2101 l_{2} - 660 l_{3} + 89 l_{4} - 16 l_{5} + q$

Neccessary optimility condition and associated polynomial ideal

In [28]:

vars = [k11,k12,k21,k22,k31,k32,l0,l1,l2,l3,l4,l5,q]
PLIST = []
for ii in range(len(vars)):
    print(ii," : ",vars[ii]," : ",diff(L,vars[ii]))
    PLIST.append(diff(L,vars[ii]))
I = Ideal(PLIST)
I

0  :  k11  :  2*k11*l0 + 20*k22*l1 - 8*k22*l2 + k22*l3 + k32*l1 + 500*l1 - 308*l2 + 85*l3 - 11*l4 + l5
1  :  k12  :  2*k12*l0 - 20*k21*l1 + 8*k21*l2 - k21*l3 - k31*l1 + 625*l1 - 369*l2 + 97*l3 - 12*l4 + l5
2  :  k21  :  -20*k12*l1 + 8*k12*l2 - k12*l3 + 2*k21*l0 + 20*l1 - 8*l2 + l3
3  :  k22  :  20*k11*l1 - 8*k11*l2 + k11*l3 + 2*k22*l0 - 100*l1 + 52*l2 - 11*l3 + l4
4  :  k31  :  -k12*l1 + 2*k31*l0 + l1
5  :  k32  :  k11*l1 + 2*k32*l0 - 5*l1 + l2
6  :  l0  :  k11^2 + k12^2 + k21^2 + k22^2 + k31^2 + k32^2 - q
7  :  l1  :  20*k11*k22 + k11*k32 + 500*k11 - 20*k12*k21 - k12*k31 + 625*k12 + 20*k21 - 100*k22 + k31 - 5*k32 - 3125
8  :  l2  :  -8*k11*k22 - 308*k11 + 8*k12*k21 - 369*k12 - 8*k21 + 52*k22 + k32 + 2101
9  :  l3  :  k11*k22 + 85*k11 - k12*k21 + 97*k12 + k21 - 11*k22 - 660
10  :  l4  :  -11*k11 - 12*k12 + k22 + 89
11  :  l5  :  k11 + k12 - 16
12  :  q  :  -l0 + 1

Out[28]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(2 k_{11} l_{0} + 20 k_{22} l_{1} - 8 k_{22} l_{2} + k_{22} l_{3} + k_{32} l_{1} + 500 l_{1} - 308 l_{2} + 85 l_{3} - 11 l_{4} + l_{5}, 2 k_{12} l_{0} - 20 k_{21} l_{1} + 8 k_{21} l_{2} - k_{21} l_{3} - k_{31} l_{1} + 625 l_{1} - 369 l_{2} + 97 l_{3} - 12 l_{4} + l_{5}, -20 k_{12} l_{1} + 8 k_{12} l_{2} - k_{12} l_{3} + 2 k_{21} l_{0} + 20 l_{1} - 8 l_{2} + l_{3}, 20 k_{11} l_{1} - 8 k_{11} l_{2} + k_{11} l_{3} + 2 k_{22} l_{0} - 100 l_{1} + 52 l_{2} - 11 l_{3} + l_{4}, -k_{12} l_{1} + 2 k_{31} l_{0} + l_{1}, k_{11} l_{1} + 2 k_{32} l_{0} - 5 l_{1} + l_{2}, k_{11}^{2} + k_{12}^{2} + k_{21}^{2} + k_{22}^{2} + k_{31}^{2} + k_{32}^{2} - q, 20 k_{11} k_{22} + k_{11} k_{32} + 500 k_{11} - 20 k_{12} k_{21} - k_{12} k_{31} + 625 k_{12} + 20 k_{21} - 100 k_{22} + k_{31} - 5 k_{32} - 3125, -8 k_{11} k_{22} - 308 k_{11} + 8 k_{12} k_{21} - 369 k_{12} - 8 k_{21} + 52 k_{22} + k_{32} + 2101, k_{11} k_{22} + 85 k_{11} - k_{12} k_{21} + 97 k_{12} + k_{21} - 11 k_{22} - 660, -11 k_{11} - 12 k_{12} + k_{22} + 89, k_{11} + k_{12} - 16, -l_{0} + 1\right)\Bold{Q}[k_{11}, k_{12}, k_{21}, k_{22}, k_{31}, k_{32}, s, l_{0}, l_{1}, l_{2}, l_{3}, l_{4}, l_{5}, q]$

Dimension of the ideal

In [29]:

I.dimension()

Out[29]:

$\newcommand{\Bold}[1]{\mathbf{#1}}1$

Change of the ring (without the variable s), dimension of the new ideal over the new ring

In [30]:

J = I.change_ring(PolynomialRing(QQ, 'k11,k12,k21,k22,k31,k32,l0,l1,l2,l3,l4,l5,q'))
J.dimension()

Out[30]:

$\newcommand{\Bold}[1]{\mathbf{#1}}0$

Algebraic variety, computation of possible solutions w.r.t. q

In [31]:

lsg = J.variety(QQbar)
lsg

Out[31]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\verb|{q:|\phantom{\verb!x!}\verb|149947.5561771188?,|\phantom{\verb!x!}\verb|l5:|\phantom{\verb!x!}\verb|-21194.45750708729?,|\phantom{\verb!x!}\verb|l4:|\phantom{\verb!x!}\verb|-27092.99027708953?,|\phantom{\verb!x!}\verb|l3:|\phantom{\verb!x!}\verb|-5957.462984593230?,|\phantom{\verb!x!}\verb|l2:|\phantom{\verb!x!}\verb|-737.9872520913827?,|\phantom{\verb!x!}\verb|l1:|\phantom{\verb!x!}\verb|2.648427531308446?,|\phantom{\verb!x!}\verb|l0:|\phantom{\verb!x!}\verb|1,|\phantom{\verb!x!}\verb|k32:|\phantom{\verb!x!}\verb|373.1040444944647?,|\phantom{\verb!x!}\verb|k31:|\phantom{\verb!x!}\verb|17.35255610531564?,|\phantom{\verb!x!}\verb|k22:|\phantom{\verb!x!}\verb|101.10404449446474?,|\phantom{\verb!x!}\verb|k21:|\phantom{\verb!x!}\verb|-3.907738998902481?,|\phantom{\verb!x!}\verb|k12:|\phantom{\verb!x!}\verb|14.104044494464742?,|\phantom{\verb!x!}\verb|k11:|\phantom{\verb!x!}\verb|1.895955505535258?}|, \verb|{q:|\phantom{\verb!x!}\verb|154095.7160695417?,|\phantom{\verb!x!}\verb|l5:|\phantom{\verb!x!}\verb|-19190.75511277376?,|\phantom{\verb!x!}\verb|l4:|\phantom{\verb!x!}\verb|-31766.99313915121?,|\phantom{\verb!x!}\verb|l3:|\phantom{\verb!x!}\verb|-7105.995278228797?,|\phantom{\verb!x!}\verb|l2:|\phantom{\verb!x!}\verb|-880.9931460774156?,|\phantom{\verb!x!}\verb|l1:|\phantom{\verb!x!}\verb|2.853303091171581?,|\phantom{\verb!x!}\verb|l0:|\phantom{\verb!x!}\verb|1,|\phantom{\verb!x!}\verb|k32:|\phantom{\verb!x!}\verb|204.7678011058971?,|\phantom{\verb!x!}\verb|k31:|\phantom{\verb!x!}\verb|-221.4622564769529?,|\phantom{\verb!x!}\verb|k22:|\phantom{\verb!x!}\verb|-67.23219889410296?,|\phantom{\verb!x!}\verb|k21:|\phantom{\verb!x!}\verb|76.3779508219885?,|\phantom{\verb!x!}\verb|k12:|\phantom{\verb!x!}\verb|-154.2321988941030?,|\phantom{\verb!x!}\verb|k11:|\phantom{\verb!x!}\verb|170.2321988941030?}|, \verb|{q:|\phantom{\verb!x!}\verb|225149.0338066715?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|155419.5856845946?*I,|\phantom{\verb!x!}\verb|l5:|\phantom{\verb!x!}\verb|-31011.72845957045?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|23105.21464151249?*I,|\phantom{\verb!x!}\verb|l4:|\phantom{\verb!x!}\verb|-47805.83748537921?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|21831.42312177592?*I,|\phantom{\verb!x!}\verb|l3:|\phantom{\verb!x!}\verb|-10603.30148707911?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|4693.838637620011?*I,|\phantom{\verb!x!}\verb|l2:|\phantom{\verb!x!}\verb|-1342.004818623881?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|549.3420250633582?*I,|\phantom{\verb!x!}\verb|l1:|\phantom{\verb!x!}\verb|-6.788613886025828?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|15.11280913409759?*I,|\phantom{\verb!x!}\verb|l0:|\phantom{\verb!x!}\verb|1,|\phantom{\verb!x!}\verb|k32:|\phantom{\verb!x!}\verb|360.1474105331525?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|45.56369481778983?*I,|\phantom{\verb!x!}\verb|k31:|\phantom{\verb!x!}\verb|-344.7980682089170?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|153.5432720434014?*I,|\phantom{\verb!x!}\verb|k22:|\phantom{\verb!x!}\verb|88.1474105331525?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|45.56369481778983?*I,|\phantom{\verb!x!}\verb|k21:|\phantom{\verb!x!}\verb|-72.07186170530965?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|68.91537574625540?*I,|\phantom{\verb!x!}\verb|k12:|\phantom{\verb!x!}\verb|1.147410533152441?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|45.56369481778983?*I,|\phantom{\verb!x!}\verb|k11:|\phantom{\verb!x!}\verb|14.85258946684756?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|45.56369481778983?*I}|, \verb|{q:|\phantom{\verb!x!}\verb|225149.0338066715?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|155419.5856845946?*I,|\phantom{\verb!x!}\verb|l5:|\phantom{\verb!x!}\verb|-31011.72845957045?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|23105.21464151249?*I,|\phantom{\verb!x!}\verb|l4:|\phantom{\verb!x!}\verb|-47805.83748537921?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|21831.42312177592?*I,|\phantom{\verb!x!}\verb|l3:|\phantom{\verb!x!}\verb|-10603.30148707911?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|4693.838637620011?*I,|\phantom{\verb!x!}\verb|l2:|\phantom{\verb!x!}\verb|-1342.004818623881?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|549.3420250633582?*I,|\phantom{\verb!x!}\verb|l1:|\phantom{\verb!x!}\verb|-6.788613886025828?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|15.11280913409759?*I,|\phantom{\verb!x!}\verb|l0:|\phantom{\verb!x!}\verb|1,|\phantom{\verb!x!}\verb|k32:|\phantom{\verb!x!}\verb|360.1474105331525?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|45.56369481778983?*I,|\phantom{\verb!x!}\verb|k31:|\phantom{\verb!x!}\verb|-344.7980682089170?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|153.5432720434014?*I,|\phantom{\verb!x!}\verb|k22:|\phantom{\verb!x!}\verb|88.1474105331525?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|45.56369481778983?*I,|\phantom{\verb!x!}\verb|k21:|\phantom{\verb!x!}\verb|-72.07186170530965?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|68.91537574625540?*I,|\phantom{\verb!x!}\verb|k12:|\phantom{\verb!x!}\verb|1.147410533152441?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|45.56369481778983?*I,|\phantom{\verb!x!}\verb|k11:|\phantom{\verb!x!}\verb|14.85258946684756?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|45.56369481778983?*I}|\right]$

Selection of the minimum solution q

In [32]:

lx = lsg[0]
lx

Out[32]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|{q:|\phantom{\verb!x!}\verb|149947.5561771188?,|\phantom{\verb!x!}\verb|l5:|\phantom{\verb!x!}\verb|-21194.45750708729?,|\phantom{\verb!x!}\verb|l4:|\phantom{\verb!x!}\verb|-27092.99027708953?,|\phantom{\verb!x!}\verb|l3:|\phantom{\verb!x!}\verb|-5957.462984593230?,|\phantom{\verb!x!}\verb|l2:|\phantom{\verb!x!}\verb|-737.9872520913827?,|\phantom{\verb!x!}\verb|l1:|\phantom{\verb!x!}\verb|2.648427531308446?,|\phantom{\verb!x!}\verb|l0:|\phantom{\verb!x!}\verb|1,|\phantom{\verb!x!}\verb|k32:|\phantom{\verb!x!}\verb|373.1040444944647?,|\phantom{\verb!x!}\verb|k31:|\phantom{\verb!x!}\verb|17.35255610531564?,|\phantom{\verb!x!}\verb|k22:|\phantom{\verb!x!}\verb|101.10404449446474?,|\phantom{\verb!x!}\verb|k21:|\phantom{\verb!x!}\verb|-3.907738998902481?,|\phantom{\verb!x!}\verb|k12:|\phantom{\verb!x!}\verb|14.104044494464742?,|\phantom{\verb!x!}\verb|k11:|\phantom{\verb!x!}\verb|1.895955505535258?}|$

Numerical feedback gain matrix

In [33]:

K0 = K.subs(k11=RR(lx['k11']),k12=RR(lx['k12']),
            k21=RR(lx['k21']),k22=RR(lx['k22']),
            k31=RR(lx['k31']),k32=RR(lx['k32']))
K0

Out[33]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} 1.89595550553526 & 14.1040444944647 \\ -3.90773899890248 & 101.104044494465 \\ 17.3525561053156 & 373.104044494465 \end{array}\right)$

Frobenius norm of the computed feedback matrix

In [34]:

K0.norm('frob')

Out[34]:

$\newcommand{\Bold}[1]{\mathbf{#1}}387.2306240176759$

Verification of the closed-loop eigenvalues (over the rational field)

In [35]:

A0 = matrix(QQ,A-B*K0*C)
A0.eigenvalues()

Out[35]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[-4.999999999999944?, -4.000000000000080?, -2.999999999999979?, -1.999999999999999? - 1.999999999999999? \sqrt{-1}, -1.999999999999999? + 1.999999999999999? \sqrt{-1}\right]$

Verification of the closed-loop eigenvalues (as floating point numbers)

In [36]:

A0 = matrix(RDF,A-B*K0*C)
A0.eigenvalues()

Out[36]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[-5.0000000000000435, -2.0000000000000067 + 2.0000000000000058i, -2.0000000000000067 - 2.0000000000000058i, -3.99999999999994, -2.9999999999999916\right]$