fundamentals of engineering supplied-reference handbook - Ventech!

3. Transient responses are obtained by using Laplace
Transforms or computer solutions with numerical
integration.
Common Compensator/Controller forms are
⎛ 1 ⎞
PID Controller GC(s) = K
⎜
⎜1
+ + TDs
⎟
⎝ TI
s ⎠
Lag or Lead Compensator GC(s) =
depending on the ratio of T1/T2.
Routh Test
For the characteristic equation
⎛ 1+
sT ⎞
⎜ 1 K
⎟
⎝1
+ sT2
⎠
ans n + an–1s n–1 + an–2s n–2 + … + a0 = 0
the coefficients are arranged into the first two rows of an
array. Additional rows are computed. The array and
coefficient computations are defined by:
where
an an–2 a n–4 … … …
a n–1 a n–3 a n–5 … … …
b1 b2 b3 … … …
c1 c2 c3 … … …
a
b =
b
1
2
…
a
=
a − ana
a
n−1
n−2
n−1
a − ana
a
n−1
n−4
n−1
n−3
n−5
a
c =
1
c
2
a
=
b − a
b
n−3
1
n−5
1
1
b − a
b
1
n−1
b
n−1
The necessary and sufficient conditions for all the roots of
the equation to have negative real parts is that all the
elements in the first column be of the same sign and
nonzero.
First-Order Control System Models
The transfer function model for a first-order system is
Ys () K
= , where
Rs () τ s+
1
K = steady state gain,
τ = time constant
The step response of a first-order system to a step input of
magnitude M is
−t/ τ −t/ τ
0
yt () = ye + KM(1−
e )
In the chemical process industry, y0 is typically taken to be
zero and y(t) is referred to as a deviation variable.
2
b
3
89
MEASUREMENT AND CONTROLS (continued)
For systems with time delay (dead time or transport lag) θ
the transfer function is
−θs
Ys () Ke
=
Rs () τ s+
1
The step response for t ≥ θ to a step of magnitude M is
( t ) / ( t )/
( )
− −θ τ − −θ τ
yt () = ⎡ye 0 + KM1−e ⎤ut
( −θ)
, where
⎣ ⎦
u(t) is the unit step function.
Second-Order Control-System Models
One standard second-order control-system model is
( )
( )
2
ωn
2 2
+ 2ζω
n +ω n
Y s K
=
Rs s s
K = steady state gain,
ζ = the damping ratio,
, where
ωn = the undamped natural (ζ = 0) frequency,
2
ωd = ωn
1− ζ , the damped natural frequency,
and
2
ω =ω 1−2ζ , the damped resonant frequency.
r n
If the damping ratio ζ is less than unity, the system is said to
be underdamped; if ζ is equal to unity, it is said to be
critically damped; and if ζ is greater than unity, the system
is said to be overdamped.
For a unit step input to a normalized underdamped secondorder
control system, the time required to reach a peak value
tp and the value of that peak Mp are given by
t
p
M
= π ⎜
⎛ωn 1−
ζ
⎝
p
= 1+
e
−πζ
2
2
1−ζ
⎟
⎞
⎠
For an underdamped second-order system, the logarithmic
decrement is
δ =
1 ⎛ x
ln⎜
m ⎜
⎝ xk
k
+m
⎞
⎟
=
⎠
2πζ
1−
ζ
where xk and xk+m are the amplitudes of oscillation at cycles
k and k + m, respectively. The period of oscillation τ is
related to ωd by
ωd τ = 2π
The time required for the output of a second-order system to
settle to within 2% of its final value is defined to be
Ts
=
ζω
4
n
2