1 Chapter 9: Design via Root Locus
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1 1 Figure 9.1 a. Sample root locus, showing possible design point via gain adjustment (A) and desired design point that cannot be met via simple gain adjustment (B); b. responses from poles at A and B
2 2 Figure 9.2 Compensation techniques: a. cascade; b. feedback
3 3 Figure 9.3 Pole at A is: a. on the root locus without compensator; b. not on the root locus with compensator pole added; (figure continues)
4 4 Figure 9.3 (continued) c. approximately on the root locus with compensator pole and zero added
5 5 Figure 9.4 Closed-loop system for Example 9.1: a. before compensation; b. after ideal integral compensation
6 6 Figure 9.5 Root locus for uncompensated system of Figure 9.4(a)
7 7 Figure 9.6 Root locus for compensated system of Figure 9.4(b)
8 8 Figure 9.7 Ideal integral compensated system response and the uncompensated system response of Example 9.1
9 9 Figure 9.8 PI controller
10 10 Figure 9.9 a. Type 1 uncompensated system; b. Type 1 compensated system; c. compensator pole-zero plot
11 11 Figure 9.10 Root locus: a. before lag compensation; b. after lag compensation
12 12 Figure 9.11 Compensated system for Example 9.2
13 13 Figure 9.12 Root locus for compensated system of Figure 9.11
14 14 Table 9.1 Predicted characteristics of uncompensated and lag-compensated systems for Example 9.2
15 15 Figure 9.13 Step responses of uncompensated and lag-compensated systems for Example 9.2
16 16 Figure 9.14 Step responses of the system for Example 9.2 using different lag compensators
17 17 Figure 9.15 Using ideal derivative compensation: a. uncompensated; b. compensator zero at 2; (figure continues)
18 18 Figure 9.15 (continued) c. compensator zero at 3; d. compensator zero at 4
19 19 Figure 9.16 Uncompensated system and ideal derivative compensation solutions from Table 9.2
20 20 Table 9.2 Predicted characteristics for the systems of Figure 9.15
21 21 Figure 9.17 Feedback control system for Example 9.3
22 22 Figure 9.18 Root locus for uncompensated system shown in Figure 9.17
23 23 Table 9.3 Uncompensated and compensated system characteristics for Example 9.3
24 24 Figure 9.19 Compensated dominant pole superimposed over the uncompensated root locus for Example 9.3
25 25 Figure 9.20 Evaluating the location of the compensating zero for Example 9.3
26 26 Figure 9.21 Root locus for the compensated system of Example 9.3
27 27 Figure 9.22 Uncompensated and compensated system step responses of Example 9.3
28 28 Figure 9.23 PD controller
29 29 Figure 9.24 Geometry of lead compensation
30 30 Figure 9.25 Three of the infinite possible lead compensator solutions
31 31 Figure 9.26 Lead compensator design, showing evaluation of uncompensated and compensated dominant poles for Example 9.4
32 32 Table 9.4 Comparison of lead compensation designs for Example 9.4
33 33 Figure 9.27 s-plane picture used to calculate the location of the compensator pole for Example 9.4
34 34 Figure 9.28 Compensated system root locus
35 35 Figure 9.29 Uncompensated system and lead compensation responses for Example 9.4
36 36 Figure 9.30 PID controller
37 37 Figure 9.31 Uncompensated feedback control system for Example 9.5
38 38 Figure 9.32 Root locus for the uncompensated system of Example 9.5
39 39 Table 9.5 Predicted characteristics of uncompensated, PD-, and PID- compensated systems of Example 9.5
40 40 Figure 9.33 Calculating the PD compensator zero for Example 9.5
41 41 Figure 9.34 Root locus for PD-compensated system of Example 9.5
42 42 Figure 9.35 Step responses for uncompensated, PD-compensated, and PID-compensated systems of Example 9.5
43 43 Figure 9.36 Root locus for PIDcompensated system of Example 9.5
44 44 Figure 9.37 Uncompensated system for Example 9.6
45 45 Figure 9.38 Root locus for uncompensated system of Example 9.6
46 46 Table 9.6 Predicted characteristics of uncompensated, lead-compensated, and lag-leadcompensated systems of Example 9.6
47 47 Figure 9.39 Evaluating the compensator pole for Example 9.6
48 48 Figure 9.40 Root locus for lead-compensated system of Example 9.6
49 49 Figure 9.41 Root locus for lag-leadcompensated system of Example 9.6
50 50 Figure 9.42 Improvement in step response for lag-leadcompensated system of Example 9.6
51 51 Figure 9.43 Improvement in ramp response error for the system of Example 9.6: a. leadcompensated; b. lag-leadcompensated
52 52 Figure 9.44 a. Root locus before cascading notch filter; b. typical closed-loop step response before cascading notch filter; (figure continues)
53 53 Figure 9.44 (continued) c. pole-zero plot of a notch filter; d. root locus after cascading notch filter; e. closed-loop step response after cascading notch filter.
54 54 Table 9.7 Types of cascade compensators (slide 1 of 2) (continued next slide)
55 55 Table 9.7 Types of cascade compensators (slide 2 of 2)
56 56 Figure 9.45 Generic control system with feedback compensation
57 57 Figure 9.46 A position control system that uses a tachometer as a differentiator in the feedback path. Can you see the similarity between this system and the schematic on the front end papers? Photo by Mark E. Van Dusen.
58 58 Figure 9.47 a. Transfer function of a tachometer; b. tachometer feedback compensation
59 59 Figure 9.48 Equivalent block diagram of Figure 9.45
60 60 Figure 9.49 a. System for Example 9.7; b. system with rate feedback compensation; c. equivalent compensated system; d. equivalent compensated system, showing unity feedback
61 29 Chapter 13: Digital Control Systems Figure a. Digital control system showing the digital computer performing compensation; b. continuous system used for design; c. transformed digital system Nise/Control Systems Engineering, 3/e
62 30 Chapter 13: Digital Control Systems Figure Closed-loop response for the compensated system of Example showing effect of three different sampling frequencies Nise/Control Systems Engineering, 3/e
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