Uso do calculo simbólico na Matlab

Funções importantes para o uso do cálculo simbólico

• vpa(x) Variable-precision arithmetic: evaluate each element of the symbolic input x to at least d significant digits, where d is the value of the digits function. The default value of digits is 32.
• digits(d) Change variable precision used: sets the precision used by vpa to d significant decimal digits. The default is 32 digits.
• double(s) Convert symbolic values to MATLAB double precision: converts the symbolic value s to double precision. Converting symbolic values to double precision is useful when a MATLAB® function does not accept symbolic values.
• x = sym('x') creates symbolic variable x
• syms x Create symbolic variables and functions: Create symbolic variable x.
• p = poly2sym(c) Create symbolic polynomial from vector of coefficients: creates the symbolic polynomial expression p from the vector of coefficients c.
• S = solve(eqn,var) Equations and systems solver.
• subs(s,new) Symbolic substitution.
• g = matlabFunction(f) Convert symbolic expression to function handle or file.
• simplify(S) Algebraic simplification
• expand(S) Symbolic expansion of polynomials and elementary functions
• F = factor(x) Factorization
• numden(A) Extract numerator and denominator
• partfrac(expr,var) Partial fraction decomposition
• latex(S) LaTeX form of symbolic expression

Algumas dicas sobre o cálculo simbólico

• Choose Symbolic or Numeric Arithmetic
• Perform Symbolic Computations
• Create Symbolic Numbers, Variables, and Expressions
• Create Symbolic Functions

syms x y
f = sin(x)^2 + cos(y)^2;
diff(f)

ans =
2*cos(x)*sin(x)

• Extract Numerators and Denominators of Rational Expressions
• Generate MATLAB Functions from Symbolic Expressions
• Use subs to Evaluate Expressions and Functions
• Formula Rearrangement and Rewriting

Exemplo aplicado a substituição de variáveis

Definindo uma função de transferência $H(s)={\frac {s^{2}+1}{3\,s^{2}+4\,s+6}}$ como uma função simbólica

 syms s
 H(s) = (s^2 + 1)/(3*s^2 + 4*s + 6)

H(s) =
(s^2 + 1)/(3*s^2 + 4*s + 6)

Imprimindo H(s) em formato LaTeX.

 latex(H(s))

ans =
\frac{s^2 + 1}{3\, s^2 + 4\, s + 6}

Substituindo a variável s pela f(z) = 2*(z-1)/(z+1);

syms z
Hz(z) = subs(H,2*(z-1)/(z+1))

Hz(z) =
((2*z - 2)^2/(z + 1)^2 + 1)/((3*(2*z - 2)^2)/(z + 1)^2 + (4*(2*z - 2))/(z + 1) + 6)

Obtem-se a expressão: $H(z)={\frac {{\frac {{\left(2\,z-2\right)}^{2}}{{\left(z+1\right)}^{2}}}+1}{{\frac {3\,{\left(2\,z-2\right)}^{2}}{{\left(z+1\right)}^{2}}}+{\frac {4\,\left(2\,z-2\right)}{z+1}}+6}}$

Agrupando os termos comuns em potências de z

Hz2 = collect(Hz)

Hz2 =
(5*z^2 - 6*z + 5)/(26*z^2 - 12*z + 10)

Obtem-se a expressão: $H(z)={\frac {5\,z^{2}-6\,z+5}{26\,z^{2}-12\,z+10}}$