Digital Control Systems [MEE 4003] - Kckong.info

2.2. LAPLACE TRANSFORMS AND TRANSFER FUNCTIONS 31
[Example 2-10] Suppose that you adjusted the desired temperature of a room by−4 ◦ C
at t = 0. By controlling an air-conditioner, the room temperature was changed by
y(t) = −4 + 4e −3t . What is the transfer function of the room equipped with the airconditioner
The transfer function is defined by the relationship between the input signal and
the output signal, i.e.
G(s) = Y(s)
U(s) = L{−4+4e−3t }
L{−4}
= −4 s + 4
s+3
− 4 s
= 3
s+3
whereG(s) is the transfer function of the room equipped with the air-conditioner.
[Example 2-11] Suppose that a linear continuous system follows the equation of motion
as
ÿ +2ζω 0 ẏ +ω 2 0 y = K 0u(t)
where the initial conditions are all zeros.
The transfer function is obtained by taking the Laplace transform of the both
sides of the equation, i.e.
L{ÿ +2ζω 0 ẏ +ω 2 0y} = L{K 0 u(t)}
s 2 Y(s)+2ζω 0 sY(s)+ω 2 0 Y(s) = K 0U(s)
Thus, rearranging the equation above, the transfer function is obtained:
G(s) = Y(s)
U(s) = K 0
s 2 +2ζω 0 s+ω 2 0
2.2.4 Relationship between state space models and transfer functions
Conversion from state space to transfer function
Consider a state space model:
ẋ = Fx+Gu
y = Hx+Ju
Digital Control Systems, Sogang University
Kyoungchul Kong