Control Engineering Problems with Solutions
Mathematical Models and Block Diagrams
2.3 Problems
Problem 2.1
Find the poles and zeros of the transfer function
stable. Check your result using Matlab.
V V
and determine if it is
V V V V V
[0, -2, -4, -0.50.8
Control Engineering Problems with Solutions
Mathematical Models and Block Diagrams
0
0
1 , T 0 , D = 0 .]
B= C =
0
1
0
2
Problem 2.5
Determine the transfer function for Figure P2.3 if G1 ( s ) =
Find its poles and zeros.
[
s +1
2
1
, G2 ( s
Control Engineering Problems with Solutions
Feedback Loop Stability
The percentage overshoot for the closed loop step response is 100 exp(]S / 1 ] 2 ) . The results are
shown plotted against in Figure 5.21.
100
90
% Overshoot
80
70
60
50
Phase margin degr
Control Engineering Problems with Solutions
Transfer Functions and their Time Domain Responses
Then expand the numerator and compare the coefficients in the powers of s, thus for example thes
cubed term which is zero on the left hand side (lhs) and the co
Control Engineering Problems with Solutions
Feedback Loop Stability
> w1=logspace(-2,-0.05,500);
> w2=logspace(0.05,2,500);
> w=[w1 w2];
> nyquist(g,w)
The problem with this plot is that the plot routine joins together the points obtained after the low
fr
Control Engineering Problems with Solutions
Frequency Responses and their Plottin
Z 2 2 jZ 16
2
which has a real part of 2
and therefore tends to -2 as the
jZ ( jZ 1)( jZ 1)
+1
frequency tends to zero. The Nyquist plot is shown in Figure 4.8.
By calcula
Control Engineering Problems with Solutions
Feedback Loop Stability
The negative real axis is cut at approximately -1.14 and -0.574. For K = 1 the encirclements N = 0 as the
small one near the origin is in the counterclockwise direction and the one at inf
Control Engineering Problems with Solutions
Frequency Responses and their Plottin
Problem 4.4
Sketch the asymptotic Bode diagram for G ( s )
0.5(1 4 s ) / s (1 s s 2 )(1 0.5s )and estimate the
phase shift when the gain is unity. Check your result using Ma
Control Engineering Problems with Solutions
Feedback Loop Stability
5 Feedback Loop Stability
5.1 Introduction
The problems in this chapter are concerned with the determination of the stability and relative stability
of a closed loop feedback system. Thi
Control Engineering Problems with Solutions
Feedback Loop Stability
14
5
Nyquist Diagram
x 10
4
3
Imaginary Axis
2
1
0
-1
-2
-3
-4
-5
-14
-12
-10
-8
-6
-4
-2
Real Axis
0
2
14
x 10
Figure 5.10 Nyquist plot for G(s) of example 5.4(a).
One can generate a fre
Control Engineering Problems with Solutions
Feedback Loop Stability
Nyquist Diagram
1
0.8
B
0.6
Imaginary Axis
0.4
0.2
A
0
-0.2
-0.4
-0.6
-0.8
C
-1
-3
-2
-1
0
1
2
3
Real Axis
Figure 5.12 Nyquist plot for two frequency vectors for example 5.4(a).
The true
Control Engineering Problems with Solutions
State Space Models and Transformation
or
nTm
( J L n 2 J m )TL ( FL n 2 Fm )TL
Taking the first equation and denoting the total inertia and friction referred to the motor shaft
by J and F, that is J
J m J L / n
Control Engineering Problems with Solutions
Frequency Responses and their Plottin
Nyquist Diagram
15
10
Imaginary Axis
5
0
-5
-10
-15
-5
0
10
5
15
20
Real Axis
Figure 4.4 Nyquist plot for example 4 with lossy integrator.
Example 4.5
Find for the transfer
Control Engineering Problems with Solutions
Mathematical Models and Block Diagrams
Example 2.6
Enter the transfer * V
L into Simulink using transfer function blocks, a state space
V V
representation, and using integrators?
The simplest approach is to us
Control Engineering Problems with Solutions
State Space Models and Transformation
which on cancelling the s+4 factor gives 3/(s+1). Alternatively for the following commands the reduced
model is obtained.
> Gm=minreal(G)
1 state removed.
a=
x1
x1 -1
b=
Control Engineering Problems with Solutions
5 4
, B
1 0
b) A
1
, C
1
5 1 1
c) A 4 1 2 , B
4 1 0
15 16 8
17 10 , B
15
16 17 8
d) A
1
State Space Models and Transformation
2 , D
2
3,C
1
0.
4
1
1 , C
1
1 2 , D = 0
1
0 1 , D = 0
1
s
Control Engineering Problems with Solutions
Frequency Responses and their Plottin
1) A second order numerator term with a natural frequency of 4 and damping ratio of 0.25
with an asymptotic approximation of an increasing gain at 12dB/octave from = 4 and
Control Engineering Problems with Solutions
Feedback Loop Stability
Problem 5.5
For the following transfer functions obtain the range of positive values of K for which the system will
be stable by the Routh-Hurwitz criterion and use of the Nyquist diagram
Control Engineering Problems with Solutions
Frequency Responses and their Plottin
Nyquist Diagram
25
20
15
10
System: G
Real: -0.133
Imag: -1
Frequency (rad/sec): 16.6
Imaginary Axis
5
0
System: G
Real: 31.6
Imag: -7.88
Frequency (rad/sec): 3.99
-5
-10
-1
Control Engineering Problems with Solutions
Feedback Loop Stability
If the gain K is reduced to less than 0.2, however, there will be no encirclements of the (-1, 0) point, so
that N = 0 and Z = 2. Thus the system is unstable for K > 0.2.
c) For this case
Control Engineering Problems with Solutions
d) In this case G ( s)
Feedback Loop Stability
10 K ( s 1) 2
and since the transfer function contains a triple integration it
s 3 ( s 2 1.2 s 16)
starts at low frequencies at infinity with a phase of -270 and fi
Control Engineering Problems with Solutions
State Space Models and Transformation
To obtain the diagonal form using Matlab the command is canon(G,modal) and because of the various
alternative solutions the representation returned by Matlab may differ acco
Control Engineering Problems with Solutions
s 5
(c) For this system sI A 4
4
1 1
s 1 2
1
s
State Space Models and Transformation
3
2
which gives the characteristic equation s 6 s 11s 6
0,
which has eigenvalues of -1, -2 and -3. The corresponding eigenve
Control Engineering Problems with Solutions
Feedback Loop Stability
Nyquist Diagram
5
4
3
Imaginary Axis
2
1
0
-1
-2
-3
-4
-5
-2
0
4
2
6
10
8
Real Axis
Figure 5.15 Nyquist plot for Example 5.4(c) with a leaky integrator.
Excellent Economics and Business p
Control Engineering Problems with Solutions
State Space Models and Transformation
a=
x1
x2
x1 -4 0
x2 0
-1
b=
u1
x1
0
x2 -1.414
c=
x1
x2
y1 -1.455 -2.121
d=
u1
y1
0
From which it can be seen that the transfer function is 3/(s+1
Control Engineering Problems with Solutions
State Space Models and Transformation
6 State Space Models and
Transformations
6.1 Introduction
The purpose of the examples in this chapter is to cover basic aspects of state space modelling, transfer
functions
Control Engineering Problems with Solutions
State Space Models and Transformation
(ii) Expanding in partial fractions gives G ( s )
2 / 3 3 / 2 11 / 6
so that one diagonal representation
s 1 s 2 s 4
is A/
0
1 0
0 2 0 , B/
0
0 4
1
1 , C/
1
2 / 3
3
Control Engineering Problems with Solutions
Transfer Functions and their Time Domain Responses
Problem 3.6
What is the maximum overshoot for the step response of Problem 3.5. Check the result using Matlab
[20.8%, or .208 per unitcfw_ e / 2 ]
Problem 3.7
F
Control Engineering Problems with Solutions
Feedback Loop Stability
It can be seen from Figure 5.6 that to satisfy this condition the frequency response of the forward
loop transfer function G(s) must not have a negative real part greater than 0.5, as the
Control Engineering Problems with Solutions
Feedback Loop Stability
Problem 5.9
Determine from the Nyquist plot and the use of M circles the maximum allowable value of K for the
system of Problem 5.8 to have a closed loop frequency response with a maximum