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Problem 1. Consider the closed-loop system below:
Using Nyquist stability criterion, find the acceptable range for gain K (positive or negative) for
the closed-loop system to remain stable.
Problem 2. Consider the closed-loop system below:
(a) Plot the Nyquist diagram of the loop transfer function with K = 1.
(b) Use the Nyquist stability criterion, to find out the range of gain K (positive or negative) such that
the closed-loop system will remain stable.
(c) For those values of K that make the closed-loop system unstable, use the Nyquist stability criterion
to determine the number of closed-loop poles in RHP.
(d) Verify your results by roughly sketching the root loci for both K > 0 and K < 0.
Problem 3. Consider the feedback system shown below:
These homework problems are compiled using the different textbooks listed on the course syllabus
1
ECE141 – Principles of Feedback Control Homework 6 2
Assume:
H(s) = 1, and, G(s) = 5000(s + 2)(s + 3)
s
3(s + 20)(s + 30)
(a) Plot the Nyquist diagram of the loop transfer function L(s) = KH(s)G(s) where K = 1 and
H(s) = 1.
(b) How many times does the Nyquist diagram encircle the critical point -1? Is the closed-loop system
stable? (Remember again that nyquist function in MATLAB does not properly handle open-loop
poles on the jω axis. You may instead want to use lnyquist1(num,den) function posted on
CCLE.)
(c) Using the Nyquist stability criterion, find the upward gain margin and the downward gain margin.
(d) Confirm your result in Part (c) by plotting root-locus and finding the gain values associated with
the jω axis crossings. (You can use rlocus and rlocfind functions).
(e) Plot the Bode diagram of the loop transfer function, and again confirm the gain margin values
you obtained in Part (c). Also find out the phase margin. (You can use bode function, and then
right-click on your Bode diagram, and select the Characteristics -> All Stability Margins
submenu to show all the crossover frequencies and the associated stability margins.)
Problem 4. Consider the feedback system shown below:
(a) Determine the value of the gain Kp in order to achieve 45◦ phase margin.
(b) For the same gain you found in Part (a) (i.e., for PM=45◦
), what would be the approximate
damping ratio of the closed-loop system? What would be the associated approximate percentage
overshoot (P.O.)?
(c) For the same gain you found in Part (a), plot the actual step response of the closed-loop system,
and find out the actual P.O. and the damping ratio. How do the actual values compare with what
you had predicted approximately based on the 45◦ phase margin?
(d) For the gain you found in Part (a), plot the Nyquist diagram of the loop transfer function. On the
same figure, plot a unit circle centered at the origin. And by showing on the figure, confirm that
the system does indeed have 45◦ phase margin. (You can use lnyquist1 and plot unit circle
functions which are posted on CCLE).
Problem 5. Conditional Stability: The closed-loop system below shows the model of a feedback
amplifier where the output saturation is also modeled within the loop:
ECE141 – Principles of Feedback Control Homework 6 3
(a) Assume the output of G(s) remains smaller than 10 µVolts. Therefore, we can ignore the saturation
nonlinearity and replace it with a constant gain K = 105
. Hand sketch the asymptotes for the
Bode magnitude and phase plots of the loop transfer function L(s) = KG(s). Then use bode
function in MATLAB to verify your plots.
(b) How many times does the phase curve cross -180 degrees? This is a common characteristics of
conditionally stable systems where the large slope (i.e., high roll-off) of the magnitude curve results
in a phase cruve that drops below −180◦ over a range of frequencies before it recovers and goes
back up such that we will still get positive phase margin at the gain crossover frequency.
(c) We know that larger loop gain at low frequencies is desirable to achieve lower sensitivity for the
closed-loop system. At the same time, we know that the gain curve should not have very large
slope at the gain crossover frequency, or else we will end up with poor phase margin. This is
one reason why conditionally stable systems may sometimes be desirable, in that they can give us
larger loop gain and thus lower sensitivity while still maintaining a small slope at the gain crossover
frequency and thus a decent phase margin. To see that, consider the following alternative loop
transfer function:
L2(s) = 5 × 105
(2.5 × 103s + 1)(10−3s + 1)2
(1)
Use bode function in MATLAB and plot the Bode diagrams of both L(s) and L2(s) on the same
figure. How do the two Bode diagrams compare? Which one will lead to lower sensitivity at low
frequencies for the closed-loop system? Please discuss.
(d) Use margin function in MATLAB to find and compare the gain margins and the phase margins
of L(s) and L2(s). Remember that the gain margin returned by margin function is in linear scale.
So the gain margin in dB will be 20 log(gm) where gm is returned by the margin function.
(e) Notice that in case of the conditionally stable L(s), the gain margin returned by margin function
is smaller than 1.0. That is because the margin function has only returned the downward gain
margin. Find out the upward gain margin for L(s). To do that, right-click on the Bode diagram
you plotted in Part (c), and select the Characteristics -> All Stability Margins submenu
to show all the crossover frequencies and the associated stability margins.
(f) To better appreciate why L(s) is conditionally stable, and thus, unlike L2(s), has both a nonzero
downward gain margin and a finite upward gain margin, use nyquist function in MATLAB to
plot the Nyquist diagrams for both L(s) and L2(s) on the same figure. Then to see the behavior
around the critical point -1, open a new figure (using figure command) and plot the two Nyquist
diagrams again, but this time set your axis limits using the command axis([-10 1 -2 2]). How
ECE141 – Principles of Feedback Control Homework 6 4
many times does each Nyquist diagram encircle the -1 point? Please discuss the differences between
the Nyquist diagrams of L(s) and L2(s).
(g) [Optional, 5 pt. Extra Credit] Now let’s look at why the saturation block, which is quite
common in feedback amplifiers, can be problematic for the conditionally stable L(s). Recall that
nonlinear systems do not have transfer functions. But the transfer characteristics of some nonlinear
elements may sometimes be represented by the so-called Describing Functions, which essentially
look at the fundamental frequency component at the output in the presence of a sinusoidal input,
while ignoring all other harmonics that could be generated due to the nonlinearity. Now, for the
saturation element, the Describing Function can be obtained as:
GD(E) = 2 × 105
π
sin−1
10−5
E
+
10−5
E
r
1 −
10−10
E2
, ]0
◦
(2)
where E is the amplitude of the input sinusoid. Obtain the value of GD assuming that the input
voltage of the saturation block increases to E = 92.5µVolts. The effective loop gain will thus be
reduced by the amount equal to GD
105 . Compare this value with the downward gain margin you had
obtained for L(s) in Part (d). Then replace the nonlinear saturation block with the constant gain
equal to GD, and use the feedback and pole functions in MATLAB to obtain the closed-loop pole
locations. As you shall notice, the closed-loop system will move into instability. And that is the
challenge with the conditionally stable systems where the closed-loop system can become unstable
due to a nonzero downward gain margin, and the effective loop gain reduction that may happen
due to the nonlinearities in the loop.
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