Transient Response Of Higher-order Systems

Transient response of higher-order systems

A pair of complex-conjugate poles yields a second-order term in s. Since the factored form of the higher-order characteristic equation consists of first and second-order terms, Equation above can be rewritten as

 

 

where q 2r = n. If the closed-loop poles are distinct, Equation above can be expanded into partial fractions as follows:

 

 

From this last equation, we see that the response of a higher-order system is composed of a number of terms involving the simple functions found in the responses of first- and second-order systems. The unit-step response e(t), the inverse Laplace transform of C(s), is then

 

 

Thus the response curve of a stable higher-order system is the sum of a number of exponential curves and damped sinusoidal curves. If all closed-loop poles lie in the left-half s plane, then the exponential terms and the damped exponential terms in Equation above will approach zero as time t increases. The steady-state output is then c(∞) = a.

Let us assume that the system considered is a stable one. Then the closed-loop poles that are located far from the jω axis have large negative real parts. The exponential terms that correspond to these poles decay very rapidly to zero. (Note that the horizontal distance from a closed-loop pole to the jω axis determines the settling time of transients due to that pole. The smaller the distance is, the longer the settling time.)

Remember that the type of transient response is determined by the closed-loop poles, while the shape of the transient response is primarily determined by the closed loop zeros. The poles of the input R(s) yield the steady-state response terms in the solution, while the poles of C(s)/R(s) enter into the exponential transient-response terms and/or damped sinusoidal transient-response terms. The zeros of C(s)/R(s) do not affect the exponents in the  exponential terms, but they do affect the magnitudes and signs of the residues.