The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze
The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze
Spectral factorization zpplot(1+2*i*s-s^2),grid One of quadratic equations in scalar and matrix polynomials frequently encountered is the polynomial spectral factorization A( s) X ( s) JX ( s) and the spectral co-factorization A( s) X ( s) JX ( s) . In either case, the given polynomial matrix A( s) satisfies A( s) A ( s) (we say it is para-Hermitian symmetric) and the unknown X ( s) is to be stable. The case of A( s) positive definite on the stability boundary results in J I . Spectral factorization with J = I is the main tool to design LQ and LQG controllers as well as Kalman filters. On the other hand, if A( s) is indefinite in matrix sense then J diag l+1, + 1, , + 1, 1, 1, 1q . This is the famous J -spectral factorization problem, which is an important tool for robust control and filter design based on H 50
norms. The Polynomial Toolbox provides two macros called spf and spcof to handle spectral factorization and co-factorization, respectively. By way of illustration consider the para-Hermitian matrix A = 34 - 56s^2 -13 - 22s + 60s^2 36 + 67s -13 + 22s + 60s^2 46 - 1e+002s^2 -42 - 26s + 38s^2 36 - 67s -42 + 26s + 38s^2 59 - 42s^2 Its spectral factorization follows by typing [X,J] = spf(A) X = J = 2.1 + 0.42s 5.2 + 0.39s -2 + 0.35s -5.5 + 4s 4.3 + 0.64s -7.4 - 5.5s 0.16 + 6.3s -0.31 - 10s -0.86 + 3.5s 1 0 0 0 1 0 0 0 1 while the spectral co-factorization of A is computed via [Xcof,J] = spcof(A) Xcof = J = 2.7 + 0.42s 4.8 + 4s 2 + 6.3s -1.6 + 0.39s 0.93 + 0.64s -6.5 - 10s 4.3 + 0.35s 2.7 - 5.5s 5.8 + 3.5s 1 0 0 0 1 0 0 0 1 The resulting J reveals that the given matrix A is positive-definite on the imaginary axis. On the other hand, the following matrix is indefinite B = 5 -6 - 18s -8 -6 + 18s -41 + 81s^2 -22 - 18s -8 -22 + 18s -13 Its spectral factorization follows as [Xf,J] = spf(B) Xf = J = 3.3 2.9 - 0.92s -1.6 1.8 6.6 + 0.44s 3.4 1.6 2.5 + 9s -2 51
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- Page 42 and 43: Invariant polynomials The entries a
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Spectral<br />
factorization<br />
zpplot(1+2*i*s-s^2),grid<br />
One of quadratic equations in scalar and matrix polynomials frequently encountered<br />
is the polynomial spectral factorization<br />
A( s) X (<br />
s) JX ( s)<br />
and the spectral co-factorization<br />
A( s) X ( s) JX (<br />
s)<br />
.<br />
In either case, the given polynomial matrix A( s)<br />
satisfies A( s) A ( s)<br />
<br />
(we say it is<br />
para-Hermitian symmetric) and the unknown X ( s)<br />
is to be stable. <strong>The</strong> case of A( s)<br />
positive definite on the stability boundary results in J I .<br />
Spectral factorization with J = I is the main tool to design LQ and LQG controllers as<br />
well as Kalman filters. On the other hand, if A( s)<br />
is indefinite in matrix sense then<br />
J diag l+1, + 1, , + 1, 1, 1, 1q<br />
. This is the famous J -spectral factorization<br />
problem, which is an important tool <strong>for</strong> robust control and filter design based on H <br />
50
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