Section 9- 9-117 9-7 Design with Lead-Lag Controller Transfer function of a simple lead-lag (or...
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Transcript of Section 9- 9-117 9-7 Design with Lead-Lag Controller Transfer function of a simple lead-lag (or...
![Page 1: Section 9- 9-117 9-7 Design with Lead-Lag Controller Transfer function of a simple lead-lag (or lag-lead) controller: The phase-lead portion is used mainly.](https://reader036.fdocuments.us/reader036/viewer/2022062409/56649d4b5503460f94a28d61/html5/thumbnails/1.jpg)
Section 9-
9-1
9-7 Design with Lead-Lag Controller• Transfer function of a simple lead-lag (or lag-lead)
controller:
• The phase-lead portion is used mainly to achieve a shorter rise time and higher bandwidth, and the phase-lag portion is brought in to provide major damping of the system.
• Either phase-lead or phase-lag control can be designed first.
7, p. 574
)1,1(1
1
1
1)( 21
2
22
1
11
aasT
sTa
sT
sTasGC
lead lag
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Section 9-
9-2
Example 9-7-1: Sun-Seeker SystemExample 9-5-3: two-stage phase-lead controller design
Example 9-6-1: two-stage phase-lag controller design
• Phase-lead control:From Example 9-5-3 a1 = 70 and T1 = 0.00004
• Phase-lag control:
7, p. 575
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Section 9-
9-3
Example 9-7-1 (cont.)
7, p. 575
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Section 9-
9-4
9-8 Pole-Zero-Cancellation Design:Notch Filter
• The complex-conjugate poles, that are very close to the imaginary axis of the s-plane, usually cause the closed-loop system to be slightly damped or unstable.
Use a controller to cancel the undesired poles
• Inexact cancellation:
8, p. 576
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Section 9-
9-5
Inexact Pole-Zero Cancellation
• K1 is proportional to 11, which is a very smaller number. Similarly, K2 is also very small.
• Although the poles cannot be canceled precisely, the resulting transient-response terms will have insignificant amplitude, so unless the controller earmarked for cancellation are too far off target, the effect can be neglected for all practical purpose.
8, p. 577
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Section 9-
9-6
8, p. 577
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Section 9-
9-7
8, p. 578
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Section 9-
9-8
Second-Order Active Filter
8, p. 579
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Section 9-
9-9
Frequency-Domain Interpretation
“notch” at the resonant frequency n.
• Notch controller do not affect the high- and low-frequency properties of the system
•
8, p. 580
n
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Section 9-
9-10
Example 9-8-1
8, p. 581
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Section 9-
9-11
Example 9-8-1 (cont.)• Loop transfer function:
• Resonant frequency 1095 rad/sec
• The closed-loop system is unstable.
8, p. 582
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Section 9-
9-12
Pole-Zero-Cancellation Design with Notch Controller• Performance specifications:
– The steady-state speed of the load due to a unit-step input should have an error of not more than 1%
– Maximum overshoot of output speed 5%
– Rise time 0.5 sec
– Settling time 0.5 sec
• Notch controller:
to cancel the undesired poles 47.66 j1094
• The compensated system:
Example 9-8-1: Pole-Zero Cancellation
8, p. 582
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Section 9-
9-13
Example 9-8-1 (cont.)G(s): type-0 system:
• Step-error constant:
• Steady-state error:
• ess 1% KP 99
• Let n = 1200 rad/sec and p = 15,000– Maximum overshoot = 3.7%
– Rise time tr = 0.1879 sec
– Settling time ts = 0.256 sec
9910198.1
2
8
n
8, p. 583
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Section 9-
9-14
Example 9-8-1: Two Stage Design• Choose n = 1000 rad/sec and p = 10
the forward-path transfer function of the system with the notch controller:
maximum overshoot = 71.6%
• Introduce a phase-lag controller or a PI controller toEq. (9-167) to meet the design specification given.
8, p. 583
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Section 9-
9-15
Example 9-8-1: Phase-Lag Controller
Second-Stage Phase-Lag Controller Design• Phase-lag controller:
8, p. 584
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Section 9-
9-16
Example 9-8-1: PI Controller
Second-Stage PI Controller Design
• PI controller:
• Phase-lag controller (9-169)
• KP = 0.005 and KI/KP = 20 KI = 0.1maximum overshoot = 1%rise time tr = 0.1380 secsettling time ts = 0.1818 sec
8, p. 584
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Section 9-
9-17
Example 9-8-1: Pole-Zero Cancellation
Sensitivity due to Imperfect Pole-Zero Cancellation• Transfer function:
maximum overshoot = 0.4%rise time tr = 0.17 secsettling time ts = 0.2323 sec
8, p. 585
Notch Controller
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Section 9-
9-18
Example 9-8-1: Unit-Step Responses
8, p. 585
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Section 9-
9-19
Example 9-8-1: Freq.-Domain Design
8, p. 586
attenuation = 44.86 dB
g
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Section 9-
9-20
Example 9-8-1: Notch Controller
• PM = 13.7°
• Mr = 3.92
0435.0z
612.7p
8, p. 586
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Section 9-
9-21
Example 9-8-1 (cont.)
8, p. 587
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Section 9-
9-22
Example 9-8-1: Notch-PI ControllerDesired PM = 80°• New gain-crossover frequency:
(9.32)
(9.25)
8, p. 586
sec43 radg
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Section 9-
9-23
Example 9-8-1 (cont.)
8, p. 588
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Section 9-
9-24
9-9 Forward andFeedforward Controllers
9, p. 588
Forward compensation:
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Section 9-
9-25
Example 9-9-1Second-order sun-seeker with phase-lag control (Ex. 9-6-1):
Time-response attributes:
maximum overshoot = 2.5%, tr = 0.1637 sec, ts = 0.2020 sec
• Improve the rise time and the settling timewhile not appreciably increasing the overshoot add a PD controller Gcf(s) to the system (forward)
add a zero to the closed-loop transfer functionwhile not affecting the characteristic equation
maximum overshoot = 4.3%, tr = 0.1069, ts = 0.1313
9, p. 589
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Section 9-
9-26
Example 9-9-1 (cont.)
9, p. 590
Forward controller
Feedforward controller
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Section 9-
9-27
9-10 Design of Robust Control Systems• Control-system application:
1. the system must satisfy the damping and accuracy specifications.2. the control must yield performance that is robust (insensitive) to external disturbance and parameter variations
• d(t) = 0
• r(t) = 0
10, p. 590
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Section 9-
9-28
Sensitivity
• Disturbance suppression and robustness with respect to variations of K can be designed with the same control scheme.
10, p. 591
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Section 9-
9-29
Example 9-10-1Second-order sun-seeker with phase-lag control (Ex. 9-6-1)
• Phase-lag controller low-pass filterthe sensitivity of the closed-loop transfer function M(s) with respect to K is poor
10, p. 591
a = 0.1T = 100
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Section 9-
9-30
10, p. 592
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Section 9-
9-31
10, p. 593
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Section 9-
9-32
Example 9-10-1 (cont.)• Design strategy: place two zeros of the robust controller
near the desired close-loop poles
• According to the phase-lag-compensated system,s = 12.455 j9.624
• Transfer function of the controller:
• Transfer function of the system with the robust controller:
10, p. 594
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Section 9-
9-33
10, p. 595
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Section 9-
9-34
Example 9-10-1 (cont.)
10, p. 596
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Section 9-
9-35
10, p. 596
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Section 9-
9-36
Example 9-10-1 (cont.)
10, p. 597
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Section 9-
9-37
10, p. 597
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Section 9-
9-38
Example 9-10-2Third-order sun-seeker with phase-lag control (Ex. 9-6-2)
Phase-lag controller: a = 0.1 and T = 20 (Table 9-19) roots of characteristic equation: s = 187.73 j164.93
• Place the two zeros of the robust controller at
180 j166.13
• Forward controller:
10, p. 597
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Section 9-
9-39
Example 9-10-2 (cont.)
10, p. 598
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Section 9-
9-40
Example 9-10-2 (cont.)
10, p. 599
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Section 9-
9-41
Example 9-10-3Design a robust system that is insensitive to the variation of
the load inertia.
Performance specifications: 0.01 J 0.02
Ramp error constant Kv 200
Maximum overshoot 5%
Rise time tr 0.05 sec
Settling time ts 0.05 sec
10, p. 599
s = 50 j86.6s = 50 j50
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Section 9-
9-42
Example 9-10-3 (cont.)• Place the two zeros of the robust controller at
55 j45 • K = 1000 and J = 0.01:
K = 1000 and J = 0.02:
• Forward controller:
10, p. 600
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Section 9-
9-43
Example 9-10-3 (cont.)
10, p. 600
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Section 9-
9-44
9-11 Minor-Loop Feedback Control
Rate-Feedback or Tachometer-Feedback Control
• Transfer function:
• Characteristic equation:
• The effect of the tachometer feedback is the increasing of the damping of the system.
11, p. 601
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Section 9-
9-45
Steady-State Analysis• Forward-path transfer function:
type 1 system
• For a unit-ramp function input:tachometer feedback ess = (2+Ktn)/n
PD control ess = 2/n
• For a type 1 system, tachometer feedback decrease the ramp-error constant Kv but does not affect the step-error constant KP.
11, p. 602
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Section 9-
9-46
Example 9-11-1Second-order sun-seeker system:
11, p. 603
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Section 9-
9-47
Example 9-11-1• Characteristic equation:
• Kt = 0.02:maximum overshoot = 0tr = 0.04485 sects = 0.06061 sectmax = 0.4 sec
11, p. 604
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Section 9-
9-48
9-12 A Hydraulic Control System
12, p. 605
two-stage electro-hydraulic valve
double-acting single rod linear actuator
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Section 9-
9-49
Modeling Linear Actuator
• The applied force:f: the force efficiency of the
actuator
• Volumetric efficiency:
for an ideal case
12, p. 606
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Section 9-
9-50
Four-Way Electro-Hydraulic ValveTwo-stage control valve:
• The first stage is an electrically actuated hydraulic valve, which controls the displacement of the spool of the second stage of the valve.
• The second stage is a four-way spool valve, which controls the fluid flow and pressure into and out of ports A and B of the actuators.
At the nominal operating conditions:
12, p. 606
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Section 9-
9-51
Orifice Equation• The classic orifice equation for the fluid flow:
– The discharge coefficient:
12, p. 607
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Section 9-
9-52
Liberalized Flow Equations forFour-Way Valve
• Orifice equation:
• Taylor series:
Take x0 = 0 and
Kq: flow gain
Kc: pressure-flow coefficient
12, p. 608
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Section 9-
9-53
Rectangular valve-port geometry
For the rectangular geometry:For the open-center valve:
12, p. 609
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Section 9-
9-54
Equations for Four-Way Valve
• Volumetric flow rates into and out of the actuator:
for critical centered valves
mxx
12, p. 610
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Section 9-
9-55
Input Voltage & Main Spool Displacement
• Flow equation of pilot spool:(the control valve is critically centered)
• Assuming an incompressible fluid:
• Displacement of pilot spool:
12, p. 611
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Section 9-
9-56
Transfer Function of Two-Stage Valve
12, p. 611
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Section 9-
9-57
Modeling the Hydraulic System
12, p. 612
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Section 9-
9-58
Mathematical Equations• Force balance equation for an ideal linear actuator:
• Expressing pressure level PA and pressure level PB:
• Expressing the pressure difference two sides of linear
actuator:
• General equation:
12, p. 612
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Section 9-
9-59
Applications: Translational Motion• The voltage fed back from the actuator displacement z:
• The input voltage to the two-stage valve Verror:
• Desired input zdesired and desired voltage Vdesired:
General equation:
Transfer function of two-stage valve:
12, p. 614
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Section 9-
9-60
Transfer Func. of Translational System
12, p. 615
rfaq
mc
cf KKKK
AT
sTK
,
)1(
1
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Section 9-
9-61
Applications: Rotational System
12, p. 616
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Section 9-
9-62
Transfer Function of Rotational System• The translational displacement of the rod in terms of the
angular displacement:
• Main valve displacement:
• System transfer function:
12, p. 615
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Section 9-
9-63
Applications: Variable Load
12, p. 617
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Section 9-
9-64
9-13 Control DesignP Control:
13, p. 617
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Section 9-
9-65
P Control: transfer function• Simplified hydraulic system transfer function:
(neglecting the pole at 142350)
• Apply a P controller:
closed-loop transfer function:
13, p. 618
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Section 9-
9-66
P Control: root loci & responses
13, p. 619
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Section 9-
9-67
PD Control
• Closed transfer function:
PD control
13, p. 621
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Section 9-
9-68
PD Control: design• Steady-state error for a unit-ramp input:
ess 0.00061 KP 5
• Damping ratio for KP = 5:
= 1 KD = 0.0066
• Setting time:
ts 0.005 KD 0.0044
• Stability requirement: KP 0 and KD 0.00307
13, p. 622
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Section 9-
9-69
PD Control: root loci & root contour• Characteristic equation (KD = 0):
13, p. 622
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Section 9-
9-70
PD Control: responses
13, p. 623
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Section 9-
9-71
PI Control
13, p. 626
PI control
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Section 9-
9-72
PI Control: design• Steady-state error for a ramp input:
ess 0.2 KI 0.015
• Characteristic equation:
stable
• KI/KP = 5
13, p. 626
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Section 9-
9-73
PI Control: root loci & responses
13, p. 627
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Section 9-
9-74
PI Control: attributes
PI
13, p. 627
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Section 9-
9-75
PID Control
• Transfer function:
13, p. 628
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Section 9-
9-76
PID Control: design
• PD controller:
Table 9-28 KD1 = 0.0066, KP1 = 5
• PI controller:
KI2/KP2 = 5
13, p. 629
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Section 9-
9-77
PID Control: attributes
13, p. 629
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Section 9-
9-78
PID Control: responses
13, p. 630