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Example 4: Minimum Frobenius Norm Static Output Feedback¶

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.

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
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 [37]:

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

Out[37]:

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

State space system

In [38]:

A = matrix(R,[[-1,0,0,0],
        [0,-2,0,0],
        [0,0,1,0],
        [0,0,0,2]])
B = matrix(R,[[1,0],[1,0],[1,1],[1,0]])
C = matrix(R,[[1,1,1,1],[0,0,0,1]])
(A,B,C)

Out[38]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\left(\begin{array}{rrrr} -1 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 1 & 0 \\ 1 & 1 \\ 1 & 0 \end{array}\right), \left(\begin{array}{rrrr} 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{array}\right)\right)$

Symbolic feedback matrix

In [39]:

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

Out[39]:

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

Closed-loop characteristic polynomial

In [41]:

CP = det(s*matrix.identity(4)-(A-B*K*C))
CP

Out[41]:

$\newcommand{\Bold}[1]{\mathbf{#1}}k_{12} k_{21} s^{2} - k_{11} k_{22} s^{2} + 4 k_{11} s^{3} + k_{12} s^{3} + k_{21} s^{3} + s^{4} + 3 k_{12} k_{21} s - 3 k_{11} k_{22} s + 2 k_{12} s^{2} + k_{21} s^{2} + 2 k_{12} k_{21} - 2 k_{11} k_{22} - 10 k_{11} s - k_{12} s - 4 k_{21} s - 5 s^{2} - 2 k_{12} - 4 k_{21} + 4$

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

In [45]:

q1,r1 = CP.quo_rem(s+3)
q2,r2 = CP.quo_rem(s+4)
[r1,r2]

Out[45]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[2 k_{12} k_{21} - 2 k_{11} k_{22} - 78 k_{11} - 8 k_{12} - 10 k_{21} + 40, 6 k_{12} k_{21} - 6 k_{11} k_{22} - 216 k_{11} - 30 k_{12} - 36 k_{21} + 180\right]$

Lagrangian function

In [46]:

L = q + l0*(k11^2+k12^2+k21^2+k22^2-q) + l1*r1 + l2*r2
L

Out[46]:

$\newcommand{\Bold}[1]{\mathbf{#1}}k_{11}^{2} l_{0} - 2 k_{11} k_{22} l_{1} - 6 k_{11} k_{22} l_{2} - 78 k_{11} l_{1} - 216 k_{11} l_{2} + k_{12}^{2} l_{0} + 2 k_{12} k_{21} l_{1} + 6 k_{12} k_{21} l_{2} - 8 k_{12} l_{1} - 30 k_{12} l_{2} + k_{21}^{2} l_{0} - 10 k_{21} l_{1} - 36 k_{21} l_{2} + k_{22}^{2} l_{0} - l_{0} q + 40 l_{1} + 180 l_{2} + q$

Neccessary optimility condition and associated polynomial ideal

In [47]:

vars = [k11,k12,k21,k22,l0,l1,l2,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 - 2*k22*l1 - 6*k22*l2 - 78*l1 - 216*l2
1  :  k12  :  2*k12*l0 + 2*k21*l1 + 6*k21*l2 - 8*l1 - 30*l2
2  :  k21  :  2*k12*l1 + 6*k12*l2 + 2*k21*l0 - 10*l1 - 36*l2
3  :  k22  :  -2*k11*l1 - 6*k11*l2 + 2*k22*l0
4  :  l0  :  k11^2 + k12^2 + k21^2 + k22^2 - q
5  :  l1  :  -2*k11*k22 - 78*k11 + 2*k12*k21 - 8*k12 - 10*k21 + 40
6  :  l2  :  -6*k11*k22 - 216*k11 + 6*k12*k21 - 30*k12 - 36*k21 + 180
7  :  q  :  -l0 + 1

Out[47]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(2 k_{11} l_{0} - 2 k_{22} l_{1} - 6 k_{22} l_{2} - 78 l_{1} - 216 l_{2}, 2 k_{12} l_{0} + 2 k_{21} l_{1} + 6 k_{21} l_{2} - 8 l_{1} - 30 l_{2}, 2 k_{12} l_{1} + 6 k_{12} l_{2} + 2 k_{21} l_{0} - 10 l_{1} - 36 l_{2}, -2 k_{11} l_{1} - 6 k_{11} l_{2} + 2 k_{22} l_{0}, k_{11}^{2} + k_{12}^{2} + k_{21}^{2} + k_{22}^{2} - q, -2 k_{11} k_{22} - 78 k_{11} + 2 k_{12} k_{21} - 8 k_{12} - 10 k_{21} + 40, -6 k_{11} k_{22} - 216 k_{11} + 6 k_{12} k_{21} - 30 k_{12} - 36 k_{21} + 180, -l_{0} + 1\right)\Bold{Q}[k_{11}, k_{12}, k_{21}, k_{22}, s, l_{0}, l_{1}, l_{2}, q]$

Dimension of the ideal

In [48]:

I.dimension()

Out[48]:

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

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

In [49]:

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

Out[49]:

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

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

In [50]:

lsg = J.variety(QQbar)
lsg

Out[50]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[\verb|{q:|\phantom{\verb!x!}\verb|49.86765685705741?,|\phantom{\verb!x!}\verb|l2:|\phantom{\verb!x!}\verb|1.724779377670393?,|\phantom{\verb!x!}\verb|l1:|\phantom{\verb!x!}\verb|-4.776564836819256?,|\phantom{\verb!x!}\verb|l0:|\phantom{\verb!x!}\verb|1,|\phantom{\verb!x!}\verb|k22:|\phantom{\verb!x!}\verb|-0.004657285406960964?,|\phantom{\verb!x!}\verb|k21:|\phantom{\verb!x!}\verb|5.312689537081823?,|\phantom{\verb!x!}\verb|k12:|\phantom{\verb!x!}\verb|4.652185288969493?,|\phantom{\verb!x!}\verb|k11:|\phantom{\verb!x!}\verb|-0.011708391316228208?}|, \verb|{q:|\phantom{\verb!x!}\verb|999.2471217743665?,|\phantom{\verb!x!}\verb|l2:|\phantom{\verb!x!}\verb|-7.002551934616844?,|\phantom{\verb!x!}\verb|l1:|\phantom{\verb!x!}\verb|17.53660754400994?,|\phantom{\verb!x!}\verb|l0:|\phantom{\verb!x!}\verb|1,|\phantom{\verb!x!}\verb|k22:|\phantom{\verb!x!}\verb|-22.72982463317064?,|\phantom{\verb!x!}\verb|k21:|\phantom{\verb!x!}\verb|14.43443587369492?,|\phantom{\verb!x!}\verb|k12:|\phantom{\verb!x!}\verb|15.21077467795653?,|\phantom{\verb!x!}\verb|k11:|\phantom{\verb!x!}\verb|6.548403517217146?}|, \verb|{q:|\phantom{\verb!x!}\verb|1561.763567399865?,|\phantom{\verb!x!}\verb|l2:|\phantom{\verb!x!}\verb|-0.01707246399143216?,|\phantom{\verb!x!}\verb|l1:|\phantom{\verb!x!}\verb|108.04659597236198?,|\phantom{\verb!x!}\verb|l0:|\phantom{\verb!x!}\verb|1,|\phantom{\verb!x!}\verb|k22:|\phantom{\verb!x!}\verb|-39.00476708252040?,|\phantom{\verb!x!}\verb|k21:|\phantom{\verb!x!}\verb|3.953570842581141?,|\phantom{\verb!x!}\verb|k12:|\phantom{\verb!x!}\verb|4.962917040643787?,|\phantom{\verb!x!}\verb|k11:|\phantom{\verb!x!}\verb|-0.3611707055916909?}|, \verb|{q:|\phantom{\verb!x!}\verb|-4.916404336627678?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|18.52750021398926?*I,|\phantom{\verb!x!}\verb|l2:|\phantom{\verb!x!}\verb|4.339721358523375?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|0.2633215598930059?*I,|\phantom{\verb!x!}\verb|l1:|\phantom{\verb!x!}\verb|-12.01723568378310?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|0.7361405887374441?*I,|\phantom{\verb!x!}\verb|l0:|\phantom{\verb!x!}\verb|1,|\phantom{\verb!x!}\verb|k22:|\phantom{\verb!x!}\verb|2.503462143622678?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|0.3220443233695608?*I,|\phantom{\verb!x!}\verb|k21:|\phantom{\verb!x!}\verb|7.930611453339347?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|9.557597773794300?*I,|\phantom{\verb!x!}\verb|k12:|\phantom{\verb!x!}\verb|9.59540187514585?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|8.99762547548652?*I,|\phantom{\verb!x!}\verb|k11:|\phantom{\verb!x!}\verb|2.508671109495064?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|0.1866574327692599?*I}|, \verb|{q:|\phantom{\verb!x!}\verb|-4.916404336627678?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|18.52750021398926?*I,|\phantom{\verb!x!}\verb|l2:|\phantom{\verb!x!}\verb|4.339721358523375?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|0.2633215598930059?*I,|\phantom{\verb!x!}\verb|l1:|\phantom{\verb!x!}\verb|-12.01723568378310?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|0.7361405887374441?*I,|\phantom{\verb!x!}\verb|l0:|\phantom{\verb!x!}\verb|1,|\phantom{\verb!x!}\verb|k22:|\phantom{\verb!x!}\verb|2.503462143622678?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|0.3220443233695608?*I,|\phantom{\verb!x!}\verb|k21:|\phantom{\verb!x!}\verb|7.930611453339347?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|9.557597773794300?*I,|\phantom{\verb!x!}\verb|k12:|\phantom{\verb!x!}\verb|9.59540187514585?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|8.99762547548652?*I,|\phantom{\verb!x!}\verb|k11:|\phantom{\verb!x!}\verb|2.508671109495064?|\phantom{\verb!x!}\verb|-|\phantom{\verb!x!}\verb|0.1866574327692599?*I}|, \verb|{q:|\phantom{\verb!x!}\verb|1357.768400840521?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|0.?e-177*I,|\phantom{\verb!x!}\verb|l2:|\phantom{\verb!x!}\verb|-32.12818743969861?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|0.?e-176*I,|\phantom{\verb!x!}\verb|l1:|\phantom{\verb!x!}\verb|0.4586019187827692?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|0.?e-175*I,|\phantom{\verb!x!}\verb|l0:|\phantom{\verb!x!}\verb|1,|\phantom{\verb!x!}\verb|k22:|\phantom{\verb!x!}\verb|-35.98956877727162?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|0.?e-176*I,|\phantom{\verb!x!}\verb|k21:|\phantom{\verb!x!}\verb|5.067929623658367?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|0.?e-175*I,|\phantom{\verb!x!}\verb|k12:|\phantom{\verb!x!}\verb|6.057612470277891?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|0.?e-174*I,|\phantom{\verb!x!}\verb|k11:|\phantom{\verb!x!}\verb|0.3751806979787524?|\phantom{\verb!x!}\verb|+|\phantom{\verb!x!}\verb|0.?e-176*I}|\right]$

Selection of the minimum solution q

In [51]:

lx = lsg[0]
lx

Out[51]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\verb|{q:|\phantom{\verb!x!}\verb|49.86765685705741?,|\phantom{\verb!x!}\verb|l2:|\phantom{\verb!x!}\verb|1.724779377670393?,|\phantom{\verb!x!}\verb|l1:|\phantom{\verb!x!}\verb|-4.776564836819256?,|\phantom{\verb!x!}\verb|l0:|\phantom{\verb!x!}\verb|1,|\phantom{\verb!x!}\verb|k22:|\phantom{\verb!x!}\verb|-0.004657285406960964?,|\phantom{\verb!x!}\verb|k21:|\phantom{\verb!x!}\verb|5.312689537081823?,|\phantom{\verb!x!}\verb|k12:|\phantom{\verb!x!}\verb|4.652185288969493?,|\phantom{\verb!x!}\verb|k11:|\phantom{\verb!x!}\verb|-0.011708391316228208?}|$

Numerical feedback gain matrix

In [52]:

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

Out[52]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left(\begin{array}{rr} -0.0117083913162282 & 4.65218528896949 \\ 5.31268953708182 & -0.00465728540696096 \end{array}\right)$

Frobenius norm of the computed feedback matrix

In [53]:

K0.norm('frob')

Out[53]:

$\newcommand{\Bold}[1]{\mathbf{#1}}7.061703537890657$

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

In [54]:

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

Out[54]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[-4.000000000000001?, -3.000000000000001?, -1.930622507449263?, -0.9874187533371403?\right]$

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

In [55]:

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

Out[55]:

$\newcommand{\Bold}[1]{\mathbf{#1}}\left[-4.0000000000000036, -0.9874187533371406, -3.0, -1.9306225074492622\right]$