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
Factorizations returns X = Y = 0.25 - 0.5s 0.5 0.25 + 0.5s -0.5 -0.25 - 0.5s 0.5 0.75 + 0.5s -0.5 Various other scalar and matrix polynomial equations may be solved by directly applying appropriate solvers programmed in the Polynomial Toolbox, such as the equation A( s) X ( s) X ( s) A ( s) B( s) Table 2 lists all available polynomial matrix equation solvers. Table 2. Equation solvers Equation Name of the routine AX B axb AXB C axbc AX BY C axbyc A X X A B axxab A X Y A B axyab AX YB C axybc XA AX B xaaxb XA B xab XA YB C xaybc To indicate the polynomial matrix equation is unsolvable, the solver returns matrices of correct sizes but full of NaNs. X0 = axb(A,B) X0 = NaN NaN NaN NaN 46
Symmetric polynomial matrices Zeros of symmetric matrices Besides linear equations special quadratic equations in scalar and matrix polynomials are encountered in various engineering fields. The equations typically contain a symmetric polynomial matrix on the right hand side. A square polynomial matrix M which is equal to its conjugated transpose M' is called (para-Hermitian) symmetric. In the scalar case with real coefficients, 2 4 2 3 4 M s s s M s only even coefficients are nonzero. In the scalar case with complex coefficients, 2 3 4 2 3 3 4 M s is s is s M s even coefficients are real while odd ones are imaginary. In the matrix case or 2 1 0 1 3203 M s s s s M s 1 2 10 2 4 3 0 2 3 2 2i 1 i i 1 i M s s M s 1 i 2 2i 1 i 7i even coefficients are (Hermitian) symmetric, odd coefficients are (Hermitian) antisymmetric. M s as a function of complex variable, we can set s to various points s i of the complex plane. Thus we obtain numerical matrices Ms i that are (Hermitian) symmetric and hence they have only real eigenvalues. If all eigenvalues of Ms i are positive for all s i on the imaginary axis then Ms is called positive definite. Such a polynomial matrix, of course, has no zeros on the imaginary axis. If all the eigenvalues are nonnegative then Ms is called nonnegative definite. Yet another important class consists of matrices with constant number of positive and negative eigenvalues of Ms i over the whole all imaginary axis. Such Ms is called indefinite (in matrix sense). All the above cases can be factorized. Should the number of positive and negative eigenvalues change as s i ranges the imaginary axis, however, the matrix Ms is indefinite in the scalar sense and cannot be factorized in any symmetric sense. Treating Nonnegative definite symmetric polynomial matrix Ms with real coefficients has its zeros distributed symmetrically with respect to the imaginary axis. Thus if it has a zero s i , it generally has a quadruple of zeros si , si, si , si , all of the same multiplicity. If s i is real, then it is only a couple si, si. If s i is imaginary, then it is also a couple si, s i but it must be of even multiplicity. Finally, if si 0 then it is a singlet of even multiplicity. zpplot(4+s^4), grid 47
- Page 2 and 3: PolyX, Ltd E-mail: info@polyx.com S
- Page 4 and 5: Rank ..............................
- Page 6 and 7: Resampling of polynomials in z ....
- Page 8 and 9: 1 Quick Start Initialization Help E
- Page 10 and 11: Typing the name of the matrix P now
- Page 12 and 13: Concatenation and working with subm
- Page 14 and 15: ans = 1 0 0 1 s^2 s^3 1 - s 1 The c
- Page 16 and 17: 2 Polynomial matrices Introduction
- Page 18 and 19: Changing the default indeterminate
- Page 20 and 21: Degrees and leading coefficients R
- Page 22 and 23: 3 + 8s integral(F) ans = 2s + 1.5s^
- Page 24 and 25: isunimod(U) ans = 1 Also the adjoin
- Page 26 and 27: Bases and null spaces ans = 1 confi
- Page 28 and 29: If P( s) is square then its roots a
- Page 30 and 31: and simply type or or hurwitz(p) an
- Page 32 and 33: Least common multiple If the only c
- Page 34 and 35: Greatest common left divisor 0 -3 +
- Page 36 and 37: Dual concepts M = lrm(A,B) M = 0.32
- Page 38 and 39: 0 1 s^2 -s^3 -s 0 Reduced and canon
- Page 40 and 41: Another triangular form Hermite for
- Page 42 and 43: Invariant polynomials The entries a
- Page 44 and 45: Bézout equations Matrix polynomial
- Page 48 and 49: zpplot(1-s^2), grid zpplot(1+2*s^2+
- Page 50 and 51: Spectral factorization zpplot(1+2*i
- Page 52 and 53: Non-symmetric factorization or 1 0
- Page 54 and 55: Transformation to Kronecker canonic
- Page 56 and 57: common roots (including roots at in
- Page 58 and 59: Derivatives and integrals ans = ans
- Page 60 and 61: Conjugate transpose The operation o
- Page 62 and 63: Zeros of symmetric twosided polynom
- Page 64 and 65: For polynomial matrices with comple
- Page 66 and 67: Non-symmetric equation Discrete-tim
- Page 68 and 69: Resampling Sampling period Discrete
- Page 70 and 71: Resampling of two-sided polynomials
- Page 72 and 73: 4 The Polynomial Matrix Editor Intr
- Page 74 and 75: Matrix Pad window Matrix Pad button
- Page 76 and 77: 5 Polynomial matrix fractions Intro
- Page 78 and 79: The sdf command Comparison of fract
- Page 80 and 81: Reverse managed by switching the Po
- Page 82 and 83: The mdf command -1 + s 0.5s ------
- Page 84 and 85: Leftdenominatorfraction The ldf com
- Page 86 and 87: Gmdf==Grdf ans = Grdf==Gldf ans = 1
- Page 88 and 89: Entrywise and matrix division ans =
- Page 90 and 91: Coefficients and coefficient matric
- Page 92 and 93: Transfer function matrices or f2 =
- Page 94 and 95: 0 and then just types F=mdf(A,B,C,D
Factorizations<br />
returns<br />
X =<br />
Y =<br />
0.25 - 0.5s 0.5<br />
0.25 + 0.5s -0.5<br />
-0.25 - 0.5s 0.5<br />
0.75 + 0.5s -0.5<br />
Various other scalar and matrix polynomial equations may be solved by directly<br />
applying appropriate solvers programmed in the <strong>Polynomial</strong> <strong>Toolbox</strong>, such as the<br />
equation<br />
<br />
A( s) X ( s) X ( s) A ( s) B( s)<br />
Table 2 lists all available polynomial matrix equation solvers.<br />
Table 2. Equation solvers<br />
Equation Name of the routine<br />
AX B<br />
axb<br />
AXB C<br />
axbc<br />
AX BY C<br />
axbyc<br />
<br />
A X X A B<br />
axxab<br />
<br />
A X Y A B<br />
axyab<br />
AX YB C<br />
axybc<br />
<br />
XA AX B<br />
xaaxb<br />
XA B<br />
xab<br />
XA YB C<br />
xaybc<br />
To indicate the polynomial matrix equation is unsolvable, the solver returns matrices<br />
of correct sizes but full of NaNs.<br />
X0 = axb(A,B)<br />
X0 =<br />
NaN NaN<br />
NaN NaN<br />
46
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