1
Introduction to Automatic Control
Systems
1.1
INTRODUCTION
An automatic control system is a combination of components that act together in
such a way that the overall system behaves automatically in
Mathematical Background
37
7 The Initial Value Theorem
This theorem refers to the behavior of the function f t as t ! 0 and, for this reason,
is called the initial value theorem. This theorem is given
Automatic Control Systems
13
of pressure, the corresponding pressure on the surface A is indeed very big. Finally,
knowing that a decrease or increase in the distance qt corresponds to a decrease or
i
Mathematical Background
49
2 Matrix Multiplication
Consider the matrices A aij and B bij of dimensions n m and m p, respectively. Then, their product AB is the n p matrix C cij , whose elements cij ar
Mathematical Background
(b) Lf f tg F s
61
s2
ss2 4
5. Find the inverse Laplace transforms of the functions
(a)
1
s 1s 2s 3
(b)
1
s as b2
(c)
1
s 2 2s 9
(d)
s4
s 4s 8
(e)
1
s s 4
(f)
s1
s s 2 4
(g)
s
Mathematical Models of Systems
2.
109
is the determinant of the signal-ow graph, which is given by
1 L1 L2 L3
3.
4.
where
a. L1 is the gain of every loop and L1 is the sum of the gains of all the
l
Mathematical Models of Systems
97
Hence, the transfer function H s of the equivalent open-loop system is
H s
Gs
1 GsF s
3:10-3
Special Case
If the feedback-path transfer function F s is unity, i.e.,
Mathematical Models of Systems
85
Since the network is a SISO system, the above state equations have the same form as
that of Eqs (3.7-15), with the exception that in this case all values are scalar.
Mathematical Models of Systems
Figure 3.4
73
A two-loop network.
1t
1t
R 1 i1 t
i t dt
i t dt vt
C 01
C 02
t
1
di2 1 t
i t dt R2 i2 t L
i t dt 0
C 01
dt C 0 2
with initial conditions vc 0 V0 and iL