ME360Partial Fraction Expansions PDF

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ME 3600 Control Systems Partial Fraction Expansions o Linear ordinary differential equations (ODE's) may be solved using Laplace transforms. There are three steps in the solution process. 1. The Laplace transform is used to convert the differential equation into an algebraic equation. 2. The algebraic equation is manipulated to solve for the Laplace transform of the solution. The concept of partial fractions is used (as necessary) to separate the solution into its elementary components. 3. The inverse Laplace transform is taken of each of the elementary components to find the solution in the time domain. o Given a single differential equation for the variable x(t ) , application of the Laplace transform method gives an algebraic equation for X ( s) , the Laplace transform of x(t ) . This algebraic equation is then solved to find X ( s) . In general, X ( s) will be a ratio of two polynomials in the complex variable s . o If the form of this ratio is found in the Laplace transform table, then the inverse transform immediately provides the solution x(t ) . o However, if the form of X ( s) is more complicated than those found in the table, then the concept of partial fractions is used to separate the solution into its elementary components (all of which are found in the table). The sum of the inverse Laplace transforms of the elementary components provides the solution x(t ) . o As mentioned above, when we apply the Laplace transform method to a single differential equation, we find the Laplace transform of the solution in the form X (s ) 

Q(s ) numerator polynomial in s  P(s ) denominator polynomial in s

o The form of the partial fraction expansion of X ( s) depends on the roots of the characteristic equation (denominator polynomial). We will consider only the three cases described below. (Note that in each case we are assuming that the denominator polynomial is of higher order than the numerator polynomial)

Kamman – ME 3600 – page: 1/3

Case 1: Characteristic equation has real, unequal roots only When the characteristic equation has N real, unequal roots, then X ( s) can be written as X (s ) 

Q(s )   s  s1  s  s2 s  sN 

In this case, the partial fraction expansion may be written as X (s ) 

K1 K2 KN   s  s1  s  s2  s  s N 

where the constants Ki (i  1,, N ) are determined using the equation K i   s  si  X ( s) s s  i  si  s1  si  s2 

Q (s i )

Case 2: Characteristic equation has complex roots only When the characteristic equation has M pairs of complex-conjugate roots, we can write X ( s) in the form X (s ) 

Q(s) (s 2  2ns  n2 )1 (s 2  2ns   n2 )2

 ns  n2 )M



In this case, the partial fraction expansion is of the form X (s ) 

A1 s  B1 (s 2  2 n s  n2 )1



A2 s  B2 (s 2  2 n s  2n )2

 

AM s  BM ( s 2  2 n s  2n ) M



The Ai and Bi (i  1, may be determined by clearing fractions and solving the simultaneous equations that result from equating the coefficients of like powers of s.

Kamman – ME 3600 – page: 2/3

Case 3: Characteristic equation has real, unequal and complex roots When the characteristic equation has N real, unequal roots and M pairs of complexconjugate roots, we can write X(s) in the form X (s ) 

Q (s )



s  s1 s  s 2 

In this case, the partial fraction expansion is of the form X (s ) 

K1 K2 KN      s  s1  s  s2  s  s N  A1 s  B1 2

(s  2n s  2n )1



A2 s  B2 2

( s  2 n s  2n ) 2

 

AM s  BM 2

( s  2 n s  2n ) M



The Ki (i  1, may be determined as in Case 1, and once the K i are known, the Ai and may be determined by clearing fractions and equating the coefficients of Bi (i  1, like powers of s as in Case 2.

Notes: 1. The method of clearing fractions can be used exclusively to solve for the coefficients in the partial fraction expansions. However, the above method for determining the coefficients for the real roots is more convenient in that they may be determined one at a time. 2. Partial fraction expansions can be defined for systems with repeated real roots as well, but they will not be covered here.

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