SLYY211 E-book

10.3 Transient response

The numerator and denominator of a transfer function, $G (s)$ can be factorized to express the transfer function itself in terms of poles and zeros,

Equation 97.

y (s) = A \frac{(s - z_{1}) (s - z_{2}) . . . (s - z_{m})}{(s - p_{1}) (s - p_{2}) . . . (s - p_{n})} u (s)

This rational function yields q terms, state vectors, through partial fraction expansion,

Equation 98.

y (s) = \frac{ε_{1}}{s - r_{1}} + \frac{ε_{2}}{s - r_{2}} + . . . + \frac{ε_{q}}{s - r_{q}}

The time response is a sum of exponential terms, where each index is a denominator root,

Equation 99.

y (t) = ε_{1} e^{r_{1} t} + ε_{2} e^{r_{2} t} + . . . + ε_{n} e^{r_{n} t} + ε_{n + 1} e^{r_{n + 1} t} + . . . + ε_{q} e^{r_{q} t}

Table 10-2 Response portions.

Response Type	Representation	Value	Description
Transient Response	$y_{c} (t)$	$ε_{1} e^{r_{1} t} + ε_{2} e^{r_{2} t} + . . . + ε_{n} e^{r_{n} t}$	The n terms in y(t) with roots originating from comprise the transient response
Steady State Response	$y_{p} (t)$	$ε_{n + 1} e^{r_{n + 1} t} + . . . + ε_{q} e^{r_{q} t}$	The a-n terms originating from u(s) comprise the steady state response