Differential Equations

# Linearity let y1t and y2 denote the output of a

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

Ask a homework question - tutors are online