Second and Higher Order Systems

Abstract

In this section, we shall obtain the response of a typical second-order control system to a step input. In terms of damping ratio and natural frequency , the system shown in figure 1 , and the closed loop transfer function ///// given by the equation 1 [ 222 [ [ [ 1 This form is called the standard form of the second-order system. The dynamic behavior of the second-order system can then be description in terms of two parameters and. We shall now solve for the response of the system shown in figure 1, to a unit-step input. We shall consider three different cases: the underdamped 0 0 0 0 1 , critically damped 1, and overdamped 1 1) Underdamped Case : In this case, the closed-loop poles are complex conjugates and lie in the left-half s plane. The ///// can be written as 2 Where 1 1 1 , the frequency is called damped natural frequency. For a unit step-input, can be written 222 3 By apply the partial fraction expansion and the inverse Laplace transform for equation 3, the response can give by