![]() We can see that when then the function becomes zero, and when the function value approaches infinity (for real p_). The constant is referred to as a pole of the function. The constant is referred to as a zero of the function. The last expression, can be written as follows: Case 1: Distinct Poles- (Cover-up method)įirst, let us explain what are the poles and zeros of a function. In the sequel, we present a systematic and easy to use approach for computing partial fraction expansion. Finally, code line 5 is used to compute the expansion.Īlthough the above-explained approach might work for more complicated functions, it becomes tedious and time-consuming if we have many terms. Once we have that variable, with code line 3 we can define a symbolic expression for our function. The code line “syms s” is used to define a symbolic variable. Namely, the following code computes the solution: How to verify this result? We can simply substitute these values of and in ( 2) and see if we can get the original expression, given by ( 1). Using this logic, from the last equation, we obtain the following systme of equations: These polynomials are equal if their coefficients multiplying the terms with the same power of, equal. The expressions on both sides of the last equations are polynomials in. These expressions are equal if the following equation is satisfied ![]() We want to find the constants and, such that ( 4) and ( 1) are equal. Namely, from the last equation, we have:īy multiplying the expressions on the right-hand side of the last expression, we obtain: First, we are going to use a high-school level approach for solving this problem. The expansion ( 2) is called the partial fraction expansion. ![]() Where and are constants that need to be determined. So our goal is to try to represent this function as ![]() We want to expand the function in partial fractions.įor example, say that we have a (transfer) function: Once we have such simpler terms, in the case of control systems, we can compute the inverse Laplace transform of simpler terms to obtain the system response in the time domain. Given a rational function or a transfer function in the case of control systems, we want to represent such a function as a sum of simpler terms. Often, in practice, we are faced with the following problem. ![]()
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