Differential Equations

Linearity let y1t and y2 denote the output of a

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Unformatted text preview: e value of p. Aept✲ G (s ) G(p)Aept ✲ A special case of the theorem is p = 1. If the input is constant, u(t) = A, one solution is that the output is the constant y (t) = G(0)A = G(0)u. So G(0) is the D.C. gain of the system. Linearity Let y1(t) and y2 denote the output of a system resulting from the inputs u1(t) and u2(t) respectively. u1(t)✲ u2(t)✲ G (s ) y1(t) ✲ G (s ) y2(t) ✲ The system is said to be Linear if the input u3(t) = c1u1(t) + c2u2(t) results in the output y3(t) = c1y1(t) + c2y2(t) for all constants c1 and c2, and all inputs u1(t) and u2. c1u1(t) + c2u2(t) ✲ G (s ) c1y1(t) + c2y2(t) ✲ Example - Amplifier or Gain u(t) ✲ K y (t) ✲ This system is linear since y3(t) = K (c1u1(t) + c2u2(t)) = c1 (c1u1(t)) + c2 (Ku2(t)) = c1y1(t) + c2y2(t) Example Consider the system y (t) = (u(t))2 i.e. the output is the square of the input. y3(t) = (c1u1(t) + c2u2(t))2 y3(t) = c2u2(t) + c2u2(t) 11 22 ￿= c1y1(t) + c2y2(t) = c1u2(t) + c2u2(t) 1 2 So this system is non-linear. Example - The Integrator u(t) ✲ ￿ y (t) = y (t) ✲ ￿ u(t)dt dy (t) = u(t) dt 1 Transfer function is s This system is linear because ￿ y3(t) = (c1u1(t) + c2u2(t)) dt ￿￿ ￿ ￿￿ ￿ = c1 u1(t)dt + c2 u2(t)dt = c1y1(t) + c2y2(t) Example - The Differentiator u(t) ✲ d dt y (t) ✲ du(t) y (t) = dt Transfer function is s This system is linear because d y3(t) = (c1u1(t) + c2u2(t)) dt ￿ ￿ ￿ ￿ du1(t) du2(t) = c1 + c2 dt dt = c1y1(t) + c2y2(t) Theorem 2 Systems described by n m ￿ dk y ￿ dk u ak k = bk k dt dt k =0 (1) k =0 are linear. Proof Suppose that the inputs u1 and u2 give rise to the outputs y1(t) and y2. This means that n m ￿ dk y1 ￿ dk u1 ak k = bk k dt dt k =0 n ￿ k =0 dk y2 ak k = dt k =0 m ￿ k =0 dk u2 bk k dt (2) (3) Multiply (2) by c1 and (3) by c2, and add. ⇒ = n ￿ k =0 m ￿ k =0 n dk y1 ￿ dk y2 ak k + ak k dt dt dk u1 bk k + dt k =0 m ￿ k =0 dk u2 bk k dt dk Using the fact that k is linear gives dt n ￿ dk ⇒ ak k ( c 1 y 1 + c...
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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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