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ecen314-lec3 - Lecture 3 Systems Described by Linear...

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Lecture 3: Systems Described by Linear Constant-Coefficient Differential/Difference Equations Tie Liu September 5, 2011 The input-output relationship of a system can be either explicit or implicit . An impor- tant example for an implicit description of a system is that described by a linear constant- coefficient differential/difference equation (LCCDE). Consider an RC circuit as shown below: + - x ( t ) y ( t ) R C + - We have i ( t ) = C dy ( t ) dt (1) Ri ( t ) + y ( t ) = x ( t ) (2) Plugging (1) into (2), we have RC dy ( t ) dt + y ( t ) = x ( t ) (3) To be concrete, let RC = 2 so the input-output relationship of the system is described by the LCCDE 2 dy ( t ) dt + y ( t ) = x ( t ) (4) 1
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Suppose that the input x ( t ) = e t u ( t ) where u ( t ) is the CT unit step function defined as u ( t ) = 1 , t > 0 1 / 2 , t = 0 0 , t < 0 (we do not really care about the value of u ( t ) at the particular point t = 0; it can be any finite number.) To find the corresponding output y ( t ), one needs to solve the LCCDE (4). The complete solution to a LCCDE consists of the sum of a particular solution y p ( t ) and the homogeneous solution y h
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