Tartalomjegyzék

CCS 1. gyakorlat

TELJES MATLAB SCRIPT KIEGÉSZÍTŐ FÜGGVÉNYEKKEL

file:   d2017_09_14_gyak1.m
author: Peter Polcz <ppolcz@gmail.com>

Created on 2017. September 16.

Output:

┌d2017_09_14_gyak1
│   - Persistence for `d2017_09_14_gyak1` reused (inherited) [run ID: 4948, 2017.09.16. Saturday, 15:21:54]
│   - Script `d2017_09_14_gyak1` backuped

Szimbolikus számítások

Szimbolikus valtozok letrehozasa

syms t s real

Szimbolikus kifejezesek szorzatainak felbontasa

expand((t+1)^2)

Output:

ans =
t^2 + 2*t + 1

Szimbolikus fuggveny letrehozasa

f(t) = sin(3*t) + 2 + 3*s;

Szimbolikus kifejezes letrehozasa

g = 4*cos(3*t) + 2 + s^2;

Szimbolikus fuggveny kiertekelese adott pontban

f(1) % t <-- 1

Output:

ans =
3*s + sin(3) + 2

Szimbolikus kifejezes kiertekelese adott pontban

subs(g,t,1)

Output:

ans =
s^2 + 4*cos(3) + 2

Szimbolikus fuggveny/kifejezes derivalasa

diff(f)

Output:

ans(t) =
3*cos(3*t)

Szimulikus kifejezes hatarozatlan integralja egy valtozo szerint

int(f,t)

Output:

ans(t) =
2*t - cos(3*t)/3 + 3*s*t

Szimulikus kifejezes hatarozott integralja egy valtozo szerint. Pl. $\int_0^1 f(t,s) \mathrm{d}t$

int(f,t,0,1)

Output:

ans =
3*s - cos(3)/3 + 7/3

Szimbolikus fuggveny/kifejezes derivalasa s szerint

diff(f,s)
diff(g,s)

Output:

ans(t) =
3
ans =
2*s

Szimbolikus Laplace transzformáció és inverze

Paraméter

syms a real

f = sin(3*t) + 2 + exp(-4*t) + exp(-a*t)*sin(t);
F = laplace(f,t,s);
pretty(F)

Output:

  1           1           3     2
----- + ------------ + ------ + -
s + 4          2        2       s
        (a + s)  + 1   s  + 9

Inverz transzformáció

syms R C real
H(s) = s/(s*R + 1/C)
h(t) = ilaplace(H,s,t)
pretty(h)

Output:

H(s) =
s/(R*s + 1/C)
h(t) =
dirac(t)/R - exp(-t/(C*R))/(C*R^2)
              /    t  \
           exp| - --- |
dirac(t)      \   C R /
-------- - ------------
    R             2
               C R

Parciális törtekre bontás szimbolikusan

Egy nagyon elvetemült függvény

H = (s^2 + 5*s + 7) / (s^7 + s^2 + 2*s + 1);
pretty(H)
help sym/partfrac
H_partfrac = partfrac(H,s,'factormode','complex')
H_vpa = vpa(H_partfrac,2)
disp('H(s) = ')
pretty(H_vpa)

Output:

    2
   s  + 5 s + 7
-----------------
 7    2
s  + s  + 2 s + 1

 PARTFRAC Partial fraction decomposition.
    PARTFRAC(F, X), where F is a rational function in X, 
    returns the partial fraction decomposition of F. 
    The denominator is factored over the rationals.
    If x is not given then it is determined using SYMVAR. 
    PARTFRAC(F, X, 'FACTORMODE', MODE)
    uses the factorization mode MODE to factor the denominator.
    Possible factorization modes are 'rational'(default), 'real', 'complex', 
    and 'full'. See FACTOR for details.
    In full mode, the result can also be a symbolic sum ranging over the roots of the denominator.
 
    Examples:
       partfrac(1/(x^2 + x), x)
       returns 1/x - 1/(x+1)
 
       partfrac(1/(x^2 + x + 1), x, 'FactorMode', 'full')
       returns - (3^(1/2)*1i)/(3*(x - (3^(1/2)*1i)/2 + 1/2)) + (3^(1/2)*1i)/(3*(x + (3^(1/2)*1i)/2 + 1/2))
 
    See also SYM/FACTOR, SYM/SIMPLIFYFRACTION.

H_partfrac =
(- 0.32270280751035776003772966614116 + 1.413924711271345340490620232093i)/(s + 0.6999571751005907727532107980535 + 0.48593229028651428604616692679117i) + (- 0.32270280751035776003772966614116 - 1.413924711271345340490620232093i)/(s + 0.6999571751005907727532107980535 - 0.48593229028651428604616692679117i) + (- 0.59384831282075522573117593742214 + 0.19579475391242410405854456578991i)/(s - 1.0741448302011052863629514808799 + 0.6346129136683528778466053725399i) + (- 0.45160745977862704320065693696877 - 0.49644356498450036858787474108365i)/(s + 0.021527797673323922542404047107598 - 1.119850355377281321873827505792i) + (2.7363171602194800579391250810641 + 6.3129380067080851518680765608996e-40i)/(s + 0.70531971485438118213467327143752) + (- 0.59384831282075522573117593742214 - 0.19579475391242410405854456578991i)/(s - 1.0741448302011052863629514808799 - 0.6346129136683528778466053725399i) + (- 0.45160745977862704320065693696877 + 0.49644356498450036858787474108365i)/(s + 0.021527797673323922542404047107598 + 1.119850355377281321873827505792i)
H_vpa =
(- 0.32 + 1.4i)/(s + 0.7 + 0.49i) + (- 0.32 - 1.4i)/(s + 0.7 - 0.49i) + (- 0.59 + 0.2i)/(s - 1.1 + 0.63i) + (- 0.45 - 0.5i)/(s + 0.022 - 1.1i) + (2.7 + 6.3e-40i)/(s + 0.71) + (- 0.59 - 0.2i)/(s - 1.1 - 0.63i) + (- 0.45 + 0.5i)/(s + 0.022 + 1.1i)
H(s) = 
 - 0.32 + 1.4i     - 0.32 - 1.4i     - 0.59 + 0.2i      - 0.45 - 0.5i
--------------- + --------------- + --------------- + ----------------
s + 0.7 + 0.49i   s + 0.7 - 0.49i   s - 1.1 + 0.63i   s + 0.022 - 1.1i

                 -40
     2.7 + 6.3 10       - 0.59 - 0.2i      - 0.45 + 0.5i
   + --------------- + --------------- + ----------------
         s + 0.71      s - 1.1 - 0.63i   s + 0.022 + 1.1i

Egy szép függvény parciális törtekre bontása

H1 = 1 / (s^2 + 3*s + 2);
H2 = partfrac(H1,s);
pretty(H1 == H2)

Output:

      1           1       1
------------ == ----- - -----
 2              s + 1   s + 2
s  + 3 s + 2

Komplex parciális törtekre bontás

H1 = 1 / (s^2 + 2*s + 2);
H2 = partfrac(H1,s,'factormode','complex');
pretty(H1 == H2)

Output:

      1                0.5i             0.5i
------------ == - -------------- + --------------
 2                s + 1.0 - 1.0i   s + 1.0 + 1.0i
s  + 2 s + 2

Szimbolikus parciális törtekre bontás - összehasonlítás

Factormode: rational [default]

H1 = (s^2 + 1) / expand((s^2 + 3*s + 2) * (s+4) * (s^2 + 2*s + 2));
H2 = partfrac(H1,s,'factormode','rational');
pretty(H1 == H2)

Output:

                 2
                s  + 1                       2           5           17
------------------------------------- == --------- - --------- + ----------
 5      4       3       2                3 (s + 1)   4 (s + 2)   60 (s + 4)
s  + 9 s  + 30 s  + 50 s  + 44 s + 16

       3 s    1
       --- - --
        10   10
   + ------------
      2
     s  + 2 s + 2

Factormode: complex

H2 = sym(partfrac(H1,s,'factormode','complex'));
pretty(H1 == H2)

Output:

                 2
                s  + 1                     0.15 + 0.2i
------------------------------------- == --------------
 5      4       3       2                s + 1.0 - 1.0i
s  + 9 s  + 30 s  + 50 s  + 44 s + 16

     0.66666666666666666666666666666667     1.25
   + ---------------------------------- - -------
                   s + 1.0                s + 2.0

     0.28333333333333333333333333333333     0.15 - 0.2i
   + ---------------------------------- + --------------
                   s + 4.0                s + 1.0 + 1.0i

Factormode: full

H2 = sym(partfrac(H1,s,'factormode','full'));
pretty(H1 == H2)

Output:

                 2
                s  + 1                       2           5           17
------------------------------------- == --------- - --------- + ----------
 5      4       3       2                3 (s + 1)   4 (s + 2)   60 (s + 4)
s  + 9 s  + 30 s  + 50 s  + 44 s + 16

       3   1        3   1
      -- + -i      -- - -i
      20   5       20   5
   + --------- + ----------
     s + 1 - i   s + 1 + 1i

Numerikus parciális törtekre bontás

Alakítsuk át a szimbolikus kifejezést numerikussá. Számláló nevező szétválasztása:

[num,den] = numden(H1)

Output:

num =
s^2 + 1
den =
s^5 + 9*s^4 + 30*s^3 + 50*s^2 + 44*s + 16

Polinomok nimerikus reprezentációja.

$$\frac{B(s)}{A(s)} = \frac{r_1}{s - p_1} + \frac{r_2}{s - p_2} + ... + \frac{r_n}{s - p_n} + K(s) $$

B = sym2poly(num)
A = sym2poly(den)

Output:

B =
     1     0     1
A =
     1     9    30    50    44    16

Numerikus parciális törtekre bontás:

[r,p,k] = residue(B,A)

Output:

r =
   0.2833 + 0.0000i
  -1.2500 + 0.0000i
   0.1500 + 0.2000i
   0.1500 - 0.2000i
   0.6667 + 0.0000i
p =
  -4.0000 + 0.0000i
  -2.0000 + 0.0000i
  -1.0000 + 1.0000i
  -1.0000 - 1.0000i
  -1.0000 + 0.0000i
k =
     []

End of the script.

Output:

└ 2.636 [sec]

Published with MATLAB® R2015a, and sanitarized by BeautifulSoup (Python script using BSoup is written by Polcz).