Automation Systems (lecture/laboratory) - exam preparation exercises, pt.1, 2017
collected: Jakub Mo˙zaryn, PhD, Eng June 16, 2017
Exercise 1
For a given differential equation of the system
T12d2y dt + T2
dy
dt + y = ku (1)
determine the transfer function.
Exercise 2
For a given transfer function of the system
G(s) = s2+ 1
s3+ 10s2+ 2s + 1 (2)
determine the euqation of the motion.
Exercise 3
For a given differential equation of the system d2y
dt2 + 4dy
dt + y = 2du
dt + 3u (3)
determine the transfer function and static characteristics.
Exercise 4
For a given differential equation of the system
Ad2y
dt2 + Bdydt + Cy = 5(Tdx
dt + x) − x3
x = x1− Ddx2 dt
(4)
draw block diagram of the system and static characteristics.
Determine the answer y(t) to the step function x(t) = 1(t)xst of the system from Fig. 1.
Figure 1: Block diagram - Ex. 5
Exercise 6
Determine the answer y(t) to the impulse function x(t) = δ(t) of the system from Fig. 2.
Figure 2: Block diagram - Ex. 6
Exercise 7
There is given block diagram of the system (Fig. 3). Determine the equation of motion and sketch its static characteristics.
Figure 3: Block diagram - Ex. 7
Exercise 8
Determine analytically, wether in the system described with trasfer function (5) there will be dumped oscillations or not.
G(s) = 10
4s2+ 4s + 1 (5)
Exercise 9
Calculate trasfer functions of the systems described with the following equations
f (t) = eαtcos(ωt) (6)
f (t) = teαt (7)
f (t) = αt3e−βt (8)
Exercise 10
Determine equations of motion of systems described with following transfer functons:
F (s) = a s2+ a2
a
s2− a2 (10)
F (s) = 1 s3
a
s − a2 (11)
Exercise 11
Determine equations of motion of systems described with following transfer functons:
G(s) = s2+ 1
s(s + 1)(s − 2) (12)
G(s) = s2+ 2s + 1
s2(s + 3)3 (13)
G(s) = 1
s3− s (14)
G(s) = s
(s + 1)2(s + 2)4 (15)
Exercise 12
In each case there is given trasfer function. Determine the dynamical char- acteristics for a given excitation u(t), sketch the excitation and answer of the system.
G(s) = 1 s(s + 1) u(t) = 1(t)
(16)
G(s) = s2
(s + 1)(2s + 1)(s + 4) u(t) = 3 · 1(t)
(17)
G(s) = 1 s(s + 1) u(t) = δ(t)
(18)
G(s) = s2 (s − 1)2(s + 1) u(t) = t
(19)
Exercise 13
Determine the transfer function of the system described with block diagram given in Fig. 4.
Figure 4: Block diagram - Ex. 13
Exercise 14
In Table 1 there are given results of the measurements determined for the sys- tem given in Fig. 5. Sketch the Nyquist plot G(jω), and Bode plots L(ω), φ(ω)
Figure 5: Block diagram - Ex. 14
ω 0 0.01 0.02 0.05 0.1 0.2 0.5 1
A1 2 2 2 2 2 2 2 2
A2 ∞ 20 16 12 8 4 2 1
φ -90◦ -100◦ -120◦ -150◦ -180 ◦ -210 ◦ -240◦ -270◦
Table 1: Values of the aplitudes and phase shifts, for the input and output signals of the system, for different frequencies - Ex. 14
Setch Bode plots for the system desctibed by block driagram given in Fig. 6.
Figure 6: Block diagram - Ex. 15
Exercise 16
Determine values of T , for wchich system described with transfer function (20) has aperiodical answer to step function.
G(s) = 2
4s2+ T s + 1 (20)
Exercise 17
Determine aplitude and frequency of the settled input signal of the system described with transfer function (21)
G(s) = 2s
2s + 1 (21)
while its input is excited with a signal (22)
x(t) = 5 sin(0.1t) (22)
Exercise 18
Determine the transfer fucntion of the system described with block diagram given in Fig. 7.
Figure 7: Block diagram - Ex. 18
