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Chapter: Control Systems - Time Response Analysis

Impulse Response of a First Order System

The impulse response of a system is an important response. The impulse response is the response to a unit impulse.

Impulse Response of A First Order System

         

o   The impulse response of a system is an important response. The impulse response is the response to a unit impulse.

 

o   The unit impulse has a Laplace transform of unity (1).That gives the unit impulse a unique stature. If a system has a unit impulse input, the output transform is G(s), where G(s) is the transfer function of the system. The unit impulse response is therefore the inverse transform of G(s), i.e. g(t), the time function you get by inverse transforming G(s). If you haven't begun to study Laplace transforms yet, you can just file these last statements away until you begin to learn about Laplace transforms. Still there is an important fact buried in all of this.

 

o   Knowing that the impulse response is the inverse transform of the transfer function of a system can be useful in identifying systems (getting system parameters from measured responses).

 

In this section we will examine the shapes/forms of several impulse responses. We will start with simple first order systems, and give you links to modules that discuss other, higher order responses.

 

A general first order system satisfies a differential equation with this general form

 

If the input, u(t), is a unit impulse, then for a short instant around t = 0 the input is infinite. Let us assume that the state, x(t), is initially zero, i.e. x(0) = 0. We will integrate both sides of the differential equation from a small time, , before t = 0, to a small time, after t = 0. We are just taking advantage of one of the properties of the unit impulse.

 

The right hand side of the equation is just Gdc since the impulse is assumed to be a unit impulse - one with unit area. Thus, we have:

 

We can also note that x(0) = 0, so the second integral on the right hand side is zero. In other words, what the impulse does is it produces a calculable change in the state, x(t), and this change occurs in a negligibly short time (the duration of the impulse) after t = 0 That leads us to a simple strategy for getting the impulse response. Calculate the new initial condition after the impulse passes. Solve the differential equation - with zero input - starting from the newly calculated initial condition.

 

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