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
2 Polynomial matrices Introduction In this chapter we review in a tutorial style many of the functions and operations defined for polynomials and polynomial matrices. This exposition continues in Chapter 3 Discrete-time and two-sided polynomial matrices. Chapter 5 is devoted to next objects, Polynomial matrix fractions. Functions and operations for linear timeinvariant systems defined by polynomial matrix fractions are discussed in Chapter 6, LTI systems. Chapter 6, Control system design covers the applications of polynomial matrices in control system design. More detailed information is available in Chapter 12, Objects, properties, formats and in the manual pages included in the companion volume Commands. Polynomials and polynomial matrices Polynomials are mathematical expressions of the form 2 n P s P0 Ps 1 P2 s Pn s where P0 , P1 , P2 , , P n are real or complex numbers, coefficients, and s is the indeterminate variable. Matrix polynomials are of the same form but with P0 , P1 , P2 , , P n numerical matrices, coefficients matrices. Alternatively and usually, these objects can be viewed as polynomial matrices, i.e. matrices whose individual entries are polynomials. These two concepts will be used interchangeably. For system and control engineers, s may be treated as the derivative operator d dt acting on continuous-time signals xt . It can be also considered as the Laplace transform complex variable. Entering polynomial matrices Polynomials and polynomial matrices are most easily entered using one of the indeterminate variables s, p that are recognized by the Polynomial Toolbox, combined with the usual MATLAB conventions for entering matrices. Thus, typing P = [ 1+s 2*s^2 2+s^3 4 ]
The pol command The Polynomial Matrix Editor The default indeterminate variable defines the matrix 2 1s 2s Ps () 3 2s4 and returns P = 1 + s 2s^2 2 + s^3 4 For other available indeterminate variables such as z, q, z^–1 or d see Chapter 3, Discrete-time and two-sided polynomial matrices. Polynomials and polynomial matrices may also be entered in terms of their coefficients or coefficient matrices. For this purpose the pol command is available. Typing P0 = [1 2;3 4]; P1 = [3 4;5 1]; P2 = [1 0;0 1]; P = pol([P0 P1 P2],2,'s') for instance, defines the polynomial matrix P = according to 1 + 3s + s^2 2 + 4s 3 + 5s 4 + s + s^2 P( s) P0 P1s P2 s 2 More complicated polynomial matrices may be entered and edited with the help of the Polynomial Matrix Editor (see Chapter 4). Note that if any of the default indeterminates is redefined as a variable then it is no longer available as an indeterminate. Typing s = 1; P = 1+s^2 results in
- 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 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
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2 <strong>Polynomial</strong> matrices<br />
Introduction<br />
In this chapter we review in a tutorial style many of the functions and operations<br />
defined <strong>for</strong> polynomials and polynomial matrices. This exposition continues in<br />
Chapter 3 Discrete-time and two-sided polynomial matrices. Chapter 5 is devoted to<br />
next objects, <strong>Polynomial</strong> matrix fractions. Functions and operations <strong>for</strong> linear timeinvariant<br />
systems defined by polynomial matrix fractions are discussed in Chapter 6,<br />
LTI systems. Chapter 6, Control system design covers the applications of polynomial<br />
matrices in control system design.<br />
More detailed in<strong>for</strong>mation is available in Chapter 12, Objects, properties, <strong>for</strong>mats and<br />
in the manual pages included in the companion volume Commands.<br />
<strong>Polynomial</strong>s and polynomial matrices<br />
<strong>Polynomial</strong>s are mathematical expressions of the <strong>for</strong>m<br />
<br />
2<br />
n<br />
P s P0 Ps 1 P2 s Pn s<br />
where P0 , P1 , P2 , , P n are real or complex numbers, coefficients, and s is the<br />
indeterminate variable.<br />
Matrix polynomials are of the same <strong>for</strong>m but with P0 , P1 , P2 , , P n numerical matrices,<br />
coefficients matrices. Alternatively and usually, these objects can be viewed as<br />
polynomial matrices, i.e. matrices whose individual entries are polynomials. <strong>The</strong>se<br />
two concepts will be used interchangeably.<br />
For system and control engineers, s may be treated as the derivative operator d dt<br />
acting on continuous-time signals xt . It can be also considered as the Laplace<br />
trans<strong>for</strong>m complex variable.<br />
Entering polynomial matrices<br />
<strong>Polynomial</strong>s and polynomial matrices are most easily entered using one of the<br />
indeterminate variables s, p that are recognized by the <strong>Polynomial</strong> <strong>Toolbox</strong>, combined<br />
with the usual <strong>MATLAB</strong> conventions <strong>for</strong> entering matrices. Thus, typing<br />
P = [ 1+s 2*s^2<br />
2+s^3 4 ]
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