An engineering student, Mia, is analyzing a system transfer function H(s)=5s2−2(s+2)(s−1)2H(s) = \frac{5s^2 - 2}{(s + 2)(s - 1)^2}H(s)=(s+2)(s−1)25s2−2. She attempts to decompose it into partial fractions for an inverse Laplace transform calculation.
Her incorrect attempt is shown below.
Step 1: 5s2−2(s+2)(s−1)2≡Ps+2+Q(s−1)2\frac{5s^2 - 2}{(s + 2)(s - 1)^2} \equiv \frac{P}{s + 2} + \frac{Q}{(s - 1)^2}(s+2)(s−1)25s2−2≡s+2P+(s−1)2Q
Step 2: 5s2−2≡P(s−1)2+Q(s+2)5s^2 - 2 \equiv P(s - 1)^2 + Q(s + 2)5s2−2≡P(s−1)2+Q(s+2)
Step 3: Let s=−2⇒18=9P⇒P=2s = -2 \Rightarrow 18 = 9P \Rightarrow P = 2s=−2⇒18=9P⇒P=2 Let s=1⇒3=3Q⇒Q=1s = 1 \Rightarrow 3 = 3Q \Rightarrow Q = 1s=1⇒3=3Q⇒Q=1
Mia's Result: 5s2−2(s+2)(s−1)2≡2s+2+1(s−1)2\frac{5s^2 - 2}{(s + 2)(s - 1)^2} \equiv \frac{2}{s + 2} + \frac{1}{(s - 1)^2}(s+2)(s−1)25s2−2≡s+22+(s−1)21
(i) By using a counter-example, show that the result obtained by Mia cannot be correct.
(ii) Explain the conceptual mistake Mia made in Step 1.
Determine the correct partial fraction decomposition for H(s)H(s)H(s), with constant numerators.
