Signals and Systems, 2/E by Simon Haykin and Barry Van Veen Copyright © 2003 John Wiley & Sons....

Signals and Systems, 2/E by Simon Haykin and Barry Van VeenCopyright © 2003 John Wiley & Sons. Inc. All rights reserved.

Figure 9.1 (p. 664)Two different structures for the realization of an accumulator. (a) Feed-forward structure of infinite order. (b) First-order recusive structure.


Figure 9.2 (p. 665)(a) Simple parallel RC circuit driven by current source i(t). (b) Block diagram of the RC circuit, formulated in terms of two variables: the current i1(t) through the capacitor C and the voltage v(t) across c. This figure clearly displays the presence of a feedback loop, even though there is no physical evidence of feedback in the RC circuit itself.


Figure 9.3 (p. 666)Block diagram representations of single-loop feedback system: (a) time-domain representation and (b) s-domain representation.


Figure 9.4 (p. 667)s-domain representation of a scheme for measuring the return difference F(s), which is defined as the difference between the unit signal applied to the plant and the return signal G(s)H(s).


Figure 9.5 (p. 667)s-domain representation of a scheme for measuring the loop transfer function L(s).


Figure 9.6 (p. 669)Block diagram of single-loop feedback amplifier.


Figure 9.7 (p. 670)Block diagram of a single-loop feedback system that includes a disturbance inside the loop.


Figure 9.8 (p. 671)Feedback system for Problem 9.2.


Figure 9.9 (p. 674)(a) Conventional symbol for operational amplifier. (b) Operational amplifier with input and output voltages.


Figure 9.10 (p. 674)Operational amplifier embedded in a single-loop feedback circuit.


Figure 9.11 (p. 675)Ideal model for the feedback circuit of Fig. 9.10.


Figure 9.12 (p. 676)Reformulation of the feedback circuit of Fig. 9.10 so that it corresponds to the basic feedback system of Fig. 9.3(b).


Figure 9.13 (p. 677)Operational amplifier circuit used as an integrator in Example 9.2


Figure 9.14 (p. 678)Operational amplifier circuit for Example 9.3.


Figure 9.15 (p. 679)(a) Block diagram of open-loop control system and (b) configuration for calculation of the error signal e(t).


Figure 9.16 (p. 680)Control system with unity feedback.


Figure 9.17 (p. 681)Reformulation of the feedback control systems of Fig. 9.16.


Figure 9.18 (p. 681)A pair of equivalent block diagrams used to change fig. 9.16 into the equivalent form shown in Fig. 9.17.


Figure 9.19 (p. 683)Transient response of first-order system, plotted against the normalized time t/, where is the time constant of the system.


Figure 9.20 (p. 684)Transient response of the second-order system with T(0) = 1 and n = 1 for three different values of the damping ratio : overdamped ( = 2), critically damped ( = 1), and underdamped ( = 0.1)


Figure 9.21 (p. 686)Effect of feedback, with increasing K, on the locations of the poles of a first-order system.


Figure 9.22 (p. 687)Effect of feedback, with increasing K, on the locations of the poles of a second-order system. The loop transfer function has poles at s = 0 and s = –1/.


Figure 9.23 (p. 691)RC audio oscillataor.


Figure 9.24 (p. 694)Illustrating the angle criterion of Eq. (9.67) and the magnitude criterion of Eq. (9.68) for the loop transfer function .

)/ - )(1/ - 1 (

)/ - (1 )(

d*sdss

csKsL


Figure 9.25 (p. 697)Diagram showing the intersection point (i.e., the centroid of the root locus) of the three asymptotes for the feedback system of Example 9.8.


Figure 9.26 (p. 698)Root locus of third-order feedback system with loop transfer function

.3) 2)( 1)( (

6 )(

sss

KsL


Figure 9.27 (p. 698)Unity-feedback system for Example 9.9.


Figure 9.28 (p. 700) Root locus of closed-loop control system with loop transfer function

12) 4)( - (

2) (0.5 )(

sss

sKsL


Figure 9.29 (p. 701)(a) Contour C traversed in counterclockwise direction in s-plane. (b) and (c) Two-possible ways in which contour C is mapped onto F-plane.


Figure 9.30 (p. 702)Illustration of the definition of encirclement. As point s1 traverses contour C in the s-plane in the counterclockwise direction, as shown in (a), point A is encircled by contour only once and point B is encircled twice, both in the clockwise direction in the F-plane, as shown in (b).


Figure 9.31 (p. 703)Diagram for Problem 9.14.


Figure 9.32 (p. 704)Nyquist contour.


Figure 9.33 (p. 705)Nyquist diagrams representing (a) a stable system and (b) an unstable system.


Figure 9.34 (p. 706)Nyquist diagram for three-stage feedback amplifier with loop frequency response with K = 6.

3) 2)( 1)( (

6 )(

jjj

KjL


Figure 9.35 (p. 708)Straight-line approximation to gain component of Bode diagram for open-loop response for K = 6.

3) 2)( 1)( (

6 )(

jjj

KjL


Figure 9.36 (p. 708)(a) Exact gain response and (b) phase response for open-loop response for K = 6.

3) 2)( 1)( (

6 )(

jjj

KjL


Figure 9.37 (p. 709)Illustration of the definitions of (a) gain margin and gain crossover frequency and (b) phase margin and phase crossover frequency.


Figure 9.38 (p. 711)Nyquist diagram illustrating the notion of conditional stability.


Figure 9.39 (p. 712)Block diagram of sampled-data feedback control system, which includes both discrete-time and continuous-time components.


Figure 9.40 (p. 713)Impulse response of zero-order hold.


Figure 9.41 (p. 713)Model of sampled-data feedback control system shown in Fig. 9.39.


Figure 9.42 (p. 715)Illustration of Property 2 of the Laplace transform of a sampled signal. (a) Pole-zero map of A(s). (b) Pole-zero map ofwhere s is the sampling frequency.

),(1

)( sks

jk - sAT

sA


Figure 9.43 (p. 716)Block diagram of sampled-data system obtained by reformulating the model of Fig. 9.41


Figure 9.44 (p. 720)Root locus of sampled-data system with loop transfer function

)21

1)( - (

21

)( -z z

KzzL


Figure 9.45 (p. 723)Root locus diagram illustrating the application of MATLAB command r l o c f i n d.


Figure 9.46 (p. 726)Harold Black’s original feedback diagram and equation.

0) (for - 1

Input

Output de

μβ

μ


Figure P9.22 (p. 727)


Figure P9.23 (p. 727)


Figure P9.24 (p. 727)


Figure P9.25 (p. 728)


Figure P9.26 (p. 728)


Figure P9.42 (p. 729)


Figure P9.43 (p. 729)


Figure P9.44 (p. 729)


Figure P9.46 (p. 730)


Figure P9.47 (p. 731)


Figure P9.48 (p. 731)


Figure P9.49 (p. 731)


Figure P9.50 (p. 731)


Figure P9.51 (p. 731)


Figure P9.53 (p. 732)


Figure P9.54 (p. 732)


Figure P9.55 (p. 733)


Figure P9.56 (p. 733)

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