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 a prespecied desired
manner.
A close examination of th
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 by the relation
lim f t lim sF s
t!0
2:3-16
s!1
assumi
Automatic Control Systems
72.
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25
RE Kalman. Mathematical description of linear dynamic systems. SIAM J Control
1:152192, 1963.
RE Kalman. When is a linear control system optim
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
increase in the height yt, it is obvious that in this wa
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 are
given by
cij
m
X
aik bkj aT bj
i
k1
where aT is the
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)
s3
s 2s 1
(h)
8
s3 s2 s 2
(i)
1
s2 1s2 4s 8
(j)
s2 1
s2
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
loops of the graph.
b. L2 is the product of the gains of
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., if F s 1, then Eq. (3.10-3)
takes on the form
!
Gs
H s
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. This is
because the network is a rst-order system, and
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 0 I0 .
Example 3.4.3
Consider the mechanical system sho