Page 58 - MATLAB an introduction with applications

MATLAB Basics ——— 43

Table 1.42
Linear algebra
colspace Basis for column space
det Determinant
diag Create or extract diagonals
eig Eigenvalues and eigenvectors
expm Matrix exponential
inv Matrix inverse
jordan Jordan canonical form
null Basis for null space
poly Characteristic polynomial
rank Matrix rank
rref Reduced row echelon form
svd Singular value decomposition
tril Lower triangle
triu Upper triangle

1.20 THE LAPLACE TRANSFORMS

The Laplace transformation method is an operational method that can be used to find the transforms of time
functions, the inverse Laplace transformation using the partial-fraction expansion of B(s)/A(s), where A(s) and
B(s) are polynomials in s. In this Chapter, we present the computational methods with MATLAB to obtain the
partial-fraction expansion of B(s)/A(s) and the zeros and poles of B(s)/A(s).
MATLAB can be used to obtain the partial-fraction expansion of the ratio of two polynomials, B(s)/A(s)
as follows:
n
... b n
( )
Bs = num = b (1)s + b (2)s n− 1 + + ( )
n
As den a (1)s + a (2)s n− 1 + + ( )
( )
... a n
where a(1) ≠ 0 and num and den are row vectors. The coefficients of the numerator and denominator of
B(s)/A(s) are specified by the num and den vectors.
Hence num = [b(1) b(2) … b(n)]
den = [a(1) a(2) … a(n)]
The MATLAB command
r, p, k = residue(num, den)
is used to determine the residues, poles and direct terms of a partial-fraction expansion of the ratio of two
polynomials B(s) and A(s) is then given by
()
Bs = ks r (1) + r (2) + ... + r ( ) n
() =
A () s s − p (1) s − p (2) s − p ( ) n
The MATLAB command [num, den] = residue(r, p, k) where r, p, k are the output from MATLAB converts the
partial fraction expansion back to the polynomial ratio B(s)/A(s).
The command printsys (num, den,‘s’) prints the num/den in terms of the ratio of polynomials in s.
The command ilaplace will find the inverse Laplace transform of a Laplace function.

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