Differential Equations

ansn an1sn1 a1s1 a0y bmsm bm1sm1

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Unformatted text preview: )vo Equivalently, dis d2vo dvo ⇒ 36 + 3is = 144 + 48 + vo dt dt dt The transfer function is vo 36s + 3 ⇒ = is 144s2 + 48s + 1 Transfer Functions Eqn (1) in operator notation is . . . (ansn + an−1sn−1 + . . . + a1s1 + a0)y = (bmsm + bm−1sm−1 + . . . + b1s1 + b0)u ￿m bi si y i = ￿n=0 i u i=0 ais = G (s ) This is called the system’s transfer function. Since y = G(s)u, the output is the transfer function times the input. System’s Perspective u ✲ G (s ) y ✲ Think of u as the cause, y as the effect. u is the forcing function, y is the response u is the system’s input, y is its output Transfer Function Terminology A transfer function is said to be . . . rational if it’s the ratio of two polynomials proper if m ≤ n strictly proper if m < n improper if m ≥ n strictly improper if m > n biproper if m = n A complex number z is a zero of the transfer function if G(z ) = 0. Zeros are the roots of the numerator polynomial. A complex number z is a pole of the transfer function if G(z ) = ∞. Poles are the roots of the denominator polynomial. Forced Response Suppose that u(t) = Aept, where A and p are given complex numbers. Signals of this form are called complex exponentials. To solve eqn (1), assume a solution of the form y (t) = Bept. Then, du = pAept = pu, dt d2u = p2u, dt2 dk u = pk u dtk and similarly for y , dk u = pk u dtk etc. Substituting into n m ￿ dk y ￿ dk u ak k = bk k dt dt k =0 k =0 gives n ￿ ak pk Bept = k =0 m ￿ bk pk Aept k =0 n m ￿ ￿ = Bept ak pk = Aept bk p k k =0 k =0 ￿m bk pk k ⇒ B = A ￿n=0 ak pk k =0 = AG(p) y = Bept = G(p)Aept = G(p)u Theorem 1 If u(t) = Aept, then one solution of eqn (1) is y (t) = G(p)u(t). Proof As above. This solution is called the forced response. If the input is a complex exponential, the output has a complex exponential exponential of the same frequency. Since y = Bept = G(p)Aept = G(p)u yB ⇒ = = G (p ) uA the transfer function represents the gain of the system. The gain varies with th...
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This note was uploaded on 01/22/2014 for the course ENG 282IN taught by Professor Haines during the Spring '12 term at Pima CC.

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