Continuous Control Systems: a review

SYSTEM
y(t) = H(s)e^st


Figure 1.2. Response of a dynamic system to periodic inputs

From Equation 1.1.6 one gets


^dy(t)  s H (s) _e^st
dt

(1.1.7)

and by introducing Equation 1.1.7 in Equation 1.1.1, while bearing in mind that

u(t) = e^st, one obtains

(s  ¹ ) H (s)e ^st  ^G e^st

(1.1.8)

T T


¹ e^st is an eigenfunction of the system because its functional properties are preserved when passing through the system (only the amplitude and the phase are modified).

H(s), which gives the gain and phase deviation introduced by the system of Equation 1.1.1 at different complex frequencies, is known as the transfer function. The transfer function H(s) is a function of only the complex variable s. It represents the ratio between the system output and input when the input is e^st. From Equation 1.1.8, it turns out that, for the system described by Equation 1.1.1, the transfer function is

H (s) 
G

1 s T

(1.1.9)

The transfer function H(s) generally appears as a ratio of two polynomials in s (H(s)=B(s)/A(s)). The roots of the numerator polynomial (B(s)) define the “zeros” of the transfer function and the roots of the denominator polynomial (A(s)) define the “poles” of the transfer function. The “zeros” correspond to those complex frequencies for which the system gain is null and the “poles” correspond to those complex frequencies for which the system gain is infinite.

Note that the transfer function H(s) can also be obtained by two other techniques:

• 
  Replacing p by s in Equation 1.1.2 and algebraic computation of the y/u
  

ratio.

• 
  Using the Laplace transform (Ogata 1990).
  

The use of the representation of dynamic models in the form of transfer functions presents a certain number of advantages for the analysis and synthesis of closed-loop control systems. In particular the concatenation of dynamic models described by transfer functions is extremely easy.

3.