The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze

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Rank ................................................................................................................ 25 Bases and null spaces ...................................................................................... 26 Roots and stability ........................................................................................... 27 Special constant matrices related to polynomials ........................................... 29 Divisors and multiples.................................................................................................. 30 Scalar divisor and multiple ............................................................................. 30 Division ........................................................................................................... 31 Division with remainder ................................................................................. 31 Greatest common divisor ................................................................................ 31 Coprimeness .................................................................................................... 31 Least common multiple ................................................................................... 32 Matrix divisors and multiples ......................................................................... 33 Matrix division ................................................................................................ 33 Matrix division with remainder ...................................................................... 33 Greatest common left divisor .......................................................................... 34 Least common right multiple .......................................................................... 35 Dual concepts .................................................................................................. 36 Transposition and conjugation ................................................................................... 36 Transposition ................................................................................................... 36 Complex coefficients ...................................................................................... 36 Conjugated transposition ................................................................................ 37 Reduced and canonical forms ..................................................................................... 38 Row degrees .................................................................................................... 38 Row and column reduced matrices ................................................................. 39 Row reduced form ........................................................................................... 39 Triangular and staircase form ......................................................................... 39 Another triangular form .................................................................................. 40 Hermite form ................................................................................................... 40 Echelon form ................................................................................................... 41 Smith form ...................................................................................................... 41 Invariant polynomials ..................................................................................... 42 Polynomial matrix equations ...................................................................................... 42 Diophantine equations .................................................................................... 42 Bézout equations ............................................................................................. 44
Matrix polynomial equations .......................................................................... 44 One-sided equations ........................................................................................ 45 Two-sided equations ....................................................................................... 45 Factorizations ............................................................................................................... 46 Symmetric polynomial matrices ..................................................................... 47 Zeros of symmetric matrices ........................................................................... 47 Spectral factorization ...................................................................................... 50 Non-symmetric factorization .......................................................................... 52 Matrix pencil routines.................................................................................................. 53 Transformation to Kronecker canonical form ................................................. 54 Transformation to Clements form ................................................................... 55 Pencil Lyapunov equations ............................................................................. 55 3 Discrete-time and two-sided polynomial matrices ........................................ 57 Introduction .................................................................................................................. 57 Basic operations ............................................................................................................ 57 Two-sided polynomials ................................................................................... 57 Leading and trailing degrees and coefficients ................................................ 57 Derivatives and integrals ................................................................................. 58 Roots and stability ........................................................................................... 59 Norms .............................................................................................................. 59 Conjugations ................................................................................................................. 59 Conjugate transpose ........................................................................................ 60 Symmetric two-sided polynomial matrices................................................... 61 Zeros of symmetric two-sided polynomial matrices ....................................... 62 Matrix equations with discrete-time and two-sided polynomials ............................ 65 Symmetric bilateral matrix equations ............................................................. 65 Non-symmetric equation ................................................................................. 66 Discrete-time spectral factorization ................................................................ 66 Resampling ................................................................................................................... 68 Sampling period .............................................................................................. 68 Resampling of polynomials in z -1 .................................................................... 69 Resampling and phase ................................................................................... 69 Resampling of two-sided polynomials ............................................................ 70
Page 2 and 3: PolyX, Ltd E-mail: info@polyx.com S
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 46 and 47: Factorizations returns X = Y = 0.25
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
Page 96 and 97:
Conversion to the Control System To
Page 98 and 99:
y1 0 y2 0 Continuous-time model. Gt
Page 100 and 101:
Conversion from the Control System
Page 102 and 103:
Conversion to the Symbolic Math Too
Page 104 and 105:
Conversion from the Symbolic Math T
Page 106 and 107:
class(ans) ans = mdf 106
Page 108 and 109:
A abcd · 65 addition · 10, 22, 57
Page 110:
Q quick start · 8 quotient · 2 R
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The Polynomial Toolbox for MATLAB - DCE FEL ČVUT v Praze

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