Study Guide
For the Following Block Diagram Find the State Space Representation
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University:
Kentucky State University -
Course:
PHY 212 | General Physics II Academic year:
2020
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Views:
114
Pages:
69
Author:
strofamay58
EE440
Selected problems #4
For the following block diagram find the state space representation
Suppose
Consider a negative feebdback control system with G ( s ) Gc ( s )
the output of the closed loop control system is given as
s 8
s
Y ( s)
R( s )
D( s) , where Y the output, R the input, and D the
( s 4)(s 2)
s4
1
disturbance. Calculate the value of the steady state error assume that R ( s )
,
s
1
D(s) 2 .
s
1
, and H ( s ) 1 .
s ( s 3)
Design PD-Controller ( Gc (s) k p k d s ) such that the closed loop poles are at 1 j
For a unity feedback control system with G ( s)
ks p
s2
2
. answer the following three
questions
a. Determine The sensitivity of the steady state Error signal to variation in p (for k=1,p=10
and s=1)
b. The system is not a standard second order (zero in the numerator) but can be
approximated to second order with settling time 4 second specify the condition of
approximation
c. Design k and p such that the unit step has no oscillation
The state space model of a single input single output (SISO) system is given by
xt Axt Bu t and yt Cxt , answer the following 4 questions
a. If the system is stable, find the steady state error due to unit step input is
b. If the state matrix A is a 22 matrix with A11 0, A12 1, A21 a 2, A22 a 2 4a 3 ,
where a is a real positive number, find the range of values of a for asymptotic stability
c.
If A=
d.
If A=
and C 1 0, find the system response for a unit step input
,B
, B
and C 1 0, then for critically damped unit step
response with settling time =1sec, the values of k1, k2 are
A single input single output (SISO) system has the following state space representation.
Answer the following five sub-questions
1 0
0
x u
x
3 4 1
y 1 0 x
a. Compute the system transfer function.
b. Find the present over shoot and the peak time in the unit step response
c. A feedback controller u(t)=-k1x1-k2x2+r(t) is designed to improve the performance of the
system, r(t) is the reference signal. Design k1 and k2 in order to have a percentage
overshoot of 4.3% and a settling time of 2 seconds.
d. Can you obtain also a peek time Tp=2.1sec without perturbing the values obtained in (c).
Given the block diagram of a control system as shown below
Where G ( s )
k1 ( s 1)
s 3 ( s 2)
and H ( s )
k2 s
. Answer the following four sub-questions
s a. Find the system type number, and the closed loop transfer function
b. Determine the position, velocity and acceleration error constants (kp, kv, ka ) and the
steady state errors due to unit step, ramp and acceleration input, respectively.
c. Determine the closed loop system sensitivity to k1 and if s=1, k1=1, =2 and k2=10.
d. If the amplifier gain k1 is not subject to any undesired changes while k2 is of bad quality.
Hence, it is subject to serious changes. Would you keep the configuration as it is in the
figure above or would you exchange k1 & k2 locations? Explain your answer (No more
than one line answer).
A unity feedback control system with transfer function G( s)
k
s(s p)
has the following
performance indices settling (ts) =4 sec. and rise time( tr)=1 sec. Answer the following 4
sub-questions
a. What are the values of k and p of the system?
b. Does the system satisfy the performance settling time t p 1.6 sec. & percent over shoot
POS 12.4 %? Explain your answer
c. What is the system steady state error to a ramp input?
d. What is the value of p if k is adjusted to 2 and the system exhibits critical damping.
For the closed loop system shown below, if a PI controller
answer the following six questions
, then
a. The sensitivity of the closed loop TF with respect to changes in K1, when K2=1 and s =1
b. The conditions on k1 &k2 for which the closed loop system is asymptotically stable are
c. The values of k1 &k2 such that the steady state error due to unit step input is < 0.1 and
that due to unit ramp unit is equal to 0.01 are
d. If K2=0 & K1>0, then the values of the position, velocity and acceleration error constants
( , &
) for the system are;
e. What will happen to the settling time, if k1 is increased while k2 is set to zero
f. What will happen to the system percentage overshoot will If k1 is increased while k2 is set
to zero EE 440
summer semester 2014-2015
Selected problems
Q1. A multi-tank level control system is shown in Fig. Q1, if each tank can be modeled
as a simple first order system such that:
𝑑𝑄𝑜𝑢𝑡
+ 𝑄𝑜𝑢𝑡 = 𝑄𝑖𝑛
𝑑𝑡
where R is the outlet resistance and A is the area of the tank. Answer the following
questions
a) Develop the transfer function of each tank, and assemble them into a block diagram,
showing the flows Qc, Q1, Q2, Q3, and Qd, on the diagram.
RA
b) Now assume a flow controller of the form:
Qc = Kc e = Kc (Qr − Q3 )
where Qr is the reference flow and Kc is the controller gain. Draw the resulting block
diagram for the entire closed-loop system
c) Now assume that R1 = R2 = R3 = 1,and A1 = 3, A2 = 2, A3 = 1. Find the closed loop
𝑄
transfer function T(s) = 𝑄3 for this system (i.e., the system with feedback).
𝑟
Assume Qd is neglected.
d) Then, find the ordenary differential equation that govern the relation between Qr
and Q3
Fig.Q1 Q2. For the system shown in Fig.Q2., using Mason’s gain formula, find the output Y2(s)
and Y1(s)
H1
G1
G4
G2
G3
H2
Fig. Q2.
Q3. For the following control system,shown in Fig.Q3. Answer the following
a) Compute the closed loop transfer function Y(s)/R(s).
b) Compute the controller gain K such the unit step response is critically damped.
c) If K =2, find the unite step response.
Fig. Q3.
Q4. Consider the feedback control system in Fig. Q4. Define the tracking error as E(s) =
R(s) Y(s). Determine a suitable H(s) such that the tracking error is zero for any input
R(s) in the absence of a disturbance input (that is, when T d(s) = 0).
Fig. Q4. Q5. In a nuclear power generating plant, heat from a reactor is used to generate steam for turbines. The
rate of the fission reaction determines the amount of heat generated, and this rate is controlled by rods
inserted into the radioactive core. The rods regulate the flow of neutrons. If the rods are lowered into the
core, the rate of fission will diminish; if the rods are raised, the fission rate will increase. By automatically
controlling the position of the rods, the amount of heat generated by the reactor can be regulated. Draw a
functional block diagram for the nuclear reactor control system shown in Figure Q5. Show all blocks and
signals.
Fig. Q5
Q6. Given the following differential equation, solve for y(t) if all initial conditions are zero. Use the
Laplace transform
Q7. Find the inverse Laplace transform of Problem : Find the Closed loop Transfer Function of the following Figs. Selected problem #2
The block diagram below represents an automobile speed control. The operator sets the
desired speed Vd(s), which is compared to the actual speed of the automobile V(s), in order
to develop the error E(s). Answer the following 5 questions(Q1-Q5)
Q1. The closed-loop system time constant in second of the automobile if Gc(s)=1.375, is
A. 30
B. 2
C. 4
D. 15
E. None
Q2. For proportional control, Gc(s)= Kp=1, what is the steady-state error in automobile
speed
when the desired speed is Vd 1?
A. 0.2
B. 1/3
C. 0
D. 3
E. None
𝐾
Q3 For proportional integral control, Gc(s)=𝐾𝑝 + 𝑆𝐼, the values of (Kp , KI) such that the
unit step response is critically damped with closed loop time constant equals to 2
A. (2, 2)
B. (7,1.875)
C. (1.875,2)
D. (7,2)
E. None
0.1
Q4. For integral control, Gc ( s)
, the steady state error in speed due to unit ramp
s
input is
A. 0
B. 0.2
C. 0.833
D. 5
E.
None
Q5. For proportional control, Gc ( s ) 0.125 , the closed loop dc gain is
A. 2
B. 0.2
C. 1
D. 4
A
E. None
Q6. Which one of the following figures represents the impulse response of the following
pole s 1 j 3
B
C
Q7. The impulse response at which 0 < 1 is
D
E A
B
C
D
E
If the unit step response of a system is y (t ) 1 e 2t sin 3.464 t 60 u (t ) . Answer the
following three questions (Q6-Q7)
Q6. The system is
(a)underdamped (b)overdamped (c) critically damped
(d) undamped
Q7. The system has closed loop poles at
(a) 2 j 3.464 (b) 2,2 (c) 2 j1.732
(d) no enough information
Y ( s)
a
, for an input r (t ) 5u (t ) , if the output has a steady-state value
R( s ) s b
of 10 and y (t 2) 5 . Answer the following three questions(Q9-Q11)
Q9. The time constant of the system in second is
(a) 0.347 (b) 0.694
(c) 1 (d) 2.886
Q10. The value of a is
(a) 0.347 (b) 0.694
(c) 1 (d) 2.886
Q11. The value of b is
(a) 0.347 (b) 0.694
(c) 1 (d) 2.886
Let T ( s)
For the Following Block Diagram Find the State Space Representation
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