Transcript Automatic Control Systems

Automatic Control Systems
P43 2-6 (a)(e)
Find the inverse Laplace transforms of the
following functions. Perform partial-fraction
expansion on G(s) first ,then use the Laplace
transform table.
(a)
1
Gs  
ss  2s  3
(e)
1
G s  
s  13
P70 3-8(b)
Find the transfer functions Y2/Y1 and Y1/Y2 of the
SFGs shown in Fig.3P-8
Fig.3P-8
P311 7-13
For the control systems shown in Fig.7P-7, find the values of K and Kt
so that the maximum overshoot of the output is approximately 4.3
percent and the rise time tr is approximately 0.2 sec. Simulate the
systems with any time-response simulation program to check the
accuracy of your solutions.
Fig.7P-7
P229 6-9
The block diagram of a motor-control system with
tachometer feedback is shown in Fig.6P-9.Find the range
of the tachometer constant Kt so that the system is
asymptotically stable.
Fig.6P-9
P230 6-12(a)
The conventional Routh-Hurwitz criterion gives
information only on the location of the zeros of a
polynomial F(s) with respect to the left half and right half
of the s-place. Devise a linear transformation s=f (p,α) ,
where p is a complex variable, so that the Routh- Hurwitz
criterion can be applied to determine whether F(s) has
zeros to the right of the line s=-α,where αis a positive real
number. Apply the transformation to the following
characteristic equations to determine how many roots are
to the right of the line s=-1 in the s-plane.
(a)
F s   s 2  5s  3  0
P309 7-3(c)
Determine the step, ramp, and parabolic error
constants of the following unity-feedback control
systems. The forward-path transfer functions are
given.
(c)
1
Gs  
s1  0.1s 1  0.5s 
P309 7-5 (a) (d)
The following transfer functions are given for a
single-loop nonunity-feedback control system. Find
the steady-state errors due to a unit-step input, a
unit-ramp input, and a parabolic input, t 2 2us t 
(a)
G s  
1
s2  s  2
1
H S  
S 1
(d)
Gs  
1
s 2 s  12
H s   5s  2
P347 8-2(a)
For the loop transfer functions that follows, find
the angle of departure or arrival of the root loci at
the designated pole or zero.
(a)
G s H s  
Ks
s  1 s 2  1


Angle of arrival (K<0) and angle of departure (K>0)at
s=j.
P347
8-5(h)
Construct root-locus diagram for each of the
following control systems for which the poles and
zeros of G(s)H(s) are given. The characteristic
equation is obtained by equating the numerator of
1+ G(s)H(s) to zero.
(h) Poles at 0, 0, -8, -8; no finite zeros.
P424
9-9(a) (b)
The loop transfer functions L(s) of single –feedback-loop systems
are given below. Sketch the Nyquist plot of L(jω) for ω=0 to ω=∞.
Determine the stability of the closed-loop system. If the system is
unstable, find the number of poles of the closed-loop transfer
function that are in the right-half s-plane. Solve for the intersect of
L(jω)on the negative real axis of the L(jω)-plane analytically. You may
construct the Nyquist plot of L(jω)using any computer program.
(a)
(b)
Ls  
20
s1  0.1s 1  0.5s 
10
Ls  
s1  0.1s 1  0.5s 
P428 9-26(c)
The forward-path transfer functions of unity-feedback
control systems are given in the following. Plot the Bode
diagram of G(jω)/K and do the following: (1) Find the
value of K so that the gain margin of the system is 20 dB.
(2) Find the value of K so that the phase margin of the
system is 45o.
(c )
K
G(s) 
(s  3)3