# why laplace transform is used in control system I. Find the solution y(0) of the differential equation by using Laplace transform and inverse Laplace transform (tables). Hint: take Laplace transform of both sides, solve for the output in s- domain, use partial fraction expansion if need be, and use inverse Laplace transform table to find the solution in time-domain. y-3y + 2y = 4 y(0)=1. )(0) = 0 (10 pts) 2. The Laplace transform of a dynamical system is given by this expression: r(s) s(s+4) (s+8) 2. 1 Find the system's response in time domain. Hint: find the inverse Laplace transform. (10pts) 2. 2 Find the differential equation that expresses the dynamical behavior of the system. (1Opts)
Applying the Laplace Transform to a First Order system. There are a number of steps involved in solving a first order differential equation with Laplace transforms. Many different systems can be modeled in this manner. For example, filling a tank with water, heating a tank, charging a capacitor, measuring temperature with a thermometer are all examples of first order systems that can be modeled to give a time constant and a static gain. Step 2 - Transform the model to the s domain. where K is the time constant. Step 3 - Decide on what type of input, e. g. a step input. Having converted the first order differential equation to the frequency domain and manipulated it algebraically to give the transfer function we can now think about the next step which is to apply a step input to the system and determine the output. One familiar input to a first order system is the step change or step input. A step change from 0 to 1 is equivalent to a function that is equal to 0 for time 0, and is equal to 1 for time 0.

The Laplace transform of such a function is 1/s. If the step input is not unity but some other value, a, then the Laplace transform is a/s. in the Laplace transformed equation above with a/s. This is the equation that describes the output in the s domain. Step 4 - Transform back to the time domain. Checking the Laplace transform tables shows an entry for a term that is of the form b/s(s+b). This is similar to the equation above with b = 1/. This gives the output response in the time domain for a step change of the input of magnitude a.

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