Chapter 9, 9OSCILLATIONS IN NONLINEAR SAMPLED-DATA SYSTEMS Video Solutions, Multiple-Input Describing Functions and Nonlinear System Design

Problem 1

Consider the sampled ideal relay shown in Fig. 9-1, with $x(t)=A \sin \left(\omega t+45^{\circ}\right)$. The $t$ scale has one of the sampling points at its origin. Sketch the waveforms of $x(t), y(t)$, and $y^*(t)$ for several cycles of $x(t)$, in the case $\omega / \omega,=1 / \pi$. Also indicate on a frequency scale the locations of the harmonic components of $y^*(t)$. What do you conclude about the applicability of describing function theory?
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05:27

Problem 2

Repeat Prob, 9-1 with $\omega / \omega_s=\frac{2}{7}$.

Khoobchandra Agrawal

Numerade Educator

05:27

Problem 3

Repeat Prob. 9-1 with $\omega / \omega,=\frac{1}{4}$.

Khoobchandra Agrawal

Numerade Educator

01:13

Problem 4

(a) For $T,=1 \mathrm{sec}, K D=10, \tau=3 \mathrm{sec}$, find the possible limit cycle modes of the system of Fig. 9-2 and the amplitude of each at $c(t)$.
(b) What is the general relation between $T$, and $\tau$ which will ensure that there can be no very large amplitude limit cycle?
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Raushan Kumar

Numerade Educator

Problem 5

What limit cycle modes might the system of Fig. 9-3 display with no compensation, $G(s)=1$ ? What is the maximum possible average offset at the output?
Design $G(s)$ so that this system has only the 1,1 limit cycle mode. What now is the maximum possible average offset at the output? Suggest an additional compensation which will reduce this offset by a factor of 10 .
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Problem 6

Find the maximum permissible relay hysteresis $\delta$ such that the system of Fig. 9.4 will display no limit cycle mode of lower frequency than the 3,3 mode.
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Problem 7

Calculate the $z$-transform describing function for the sampled relay with dead zone for each of the modes shown in Fig. 9.2-8. By comparison with Eqs. (9.2-40), verify in these cases the general relation
$$
N^{\bullet}(A, \varphi)=T_{\mathrm{s}} N(A, \varphi) \quad \omega<\frac{1}{2} \omega_{\mathrm{s}}
$$

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Problem 8

Use the sampled describing function method as presented in Sec. 9.2 to find the range of $\tau / T$, for which the 2,2 limit cycle mode is possible in the system of Fig. 9-5. Interpret your result in terms of the graphical construction suggested for two-level relay systems in Sec. 9.1.
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00:19

Problem 9

Solve Prob. 9-8 using the $z$-transform describing function method.

Amy Jiang

Numerade Educator

Problem 10

For the system of Fig. 9-5, derive the conditions which define the 1, 1 limit cycle mode, using both the sampled and $z$-transform describing function methods. Note that in this case, where $\omega=\frac{1}{2} \omega$, the $z$-transform describing function method gives a necessary condition, but not sufficient conditions to define the limit cycle.

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Problem 11

The system of Fig. 9-6 uses a unit-sensitivity digital lead compensator to stabilize an inertia plant. Use the sampled describing function method to determine the range of $\tau / T$, which makes the 4,4 limit cycle mode impossible.

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Problem 12

Solve Prob. 9-11, using the $z$-transform describing function method.

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01:16

Problem 13

What is the minimum permissible value of dead zone, $\delta$, which guarantees that the system of Fig. 9-7 will not limit-cycle in the absence of input?
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Harshita Goel

Numerade Educator

01:54

Problem 14

If $\delta=1$, in the system of Fig. 9-7, what limit cycle modes are possible? Suggest a compensation which will eliminate all but the 1,1 mode.

James Kiss

Numerade Educator

Problem 15

Determine the range of $k$ for which a limit cycle of period $T=4 T$, and form $a$, $0,-a, 0$ would be stable in the system of Fig. 9-8.
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02:10

Problem 16

Use the difference-equation method to find the exact range of $\tau / T$, which makes the 4, 4 limit cycle mode impossible in the system of Fig. 9-6. Compare this result with those given by the describing functions in Probs. 9-11 and 9-12.

Vipender Yadav

Numerade Educator

05:16

Problem 17

Repeat Prob. 9-16, using the transform method.

Rory Naguib

Numerade Educator

Problem 18

What is the maximum gain $K$ that the pulse-width-modulated system of Fig. 9-9 can tolerate without exhibiting any of the limit cycle modes for which the describing function is given in Appendix F? Suggest a compensation which will permit this gain to be doubled without a limit cycle.
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