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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
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
Section 9-
9-3
Example 9-7-1 (cont.)
7, p. 575
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
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
Section 9-
9-6
8, p. 577
Section 9-
9-7
8, p. 578
Section 9-
9-8
Second-Order Active Filter
8, p. 579
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
Section 9-
9-10
Example 9-8-1
8, p. 581
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
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
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
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
Section 9-
9-15
Example 9-8-1: Phase-Lag Controller
Second-Stage Phase-Lag Controller Design• Phase-lag controller:
8, p. 584
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
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
Section 9-
9-18
Example 9-8-1: Unit-Step Responses
8, p. 585
Section 9-
9-19
Example 9-8-1: Freq.-Domain Design
8, p. 586
attenuation = 44.86 dB
g
Section 9-
9-20
Example 9-8-1: Notch Controller
• PM = 13.7°
• Mr = 3.92
0435.0z
612.7p
8, p. 586
Section 9-
9-21
Example 9-8-1 (cont.)
8, p. 587
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
Section 9-
9-23
Example 9-8-1 (cont.)
8, p. 588
Section 9-
9-24
9-9 Forward andFeedforward Controllers
9, p. 588
Forward compensation:
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
Section 9-
9-26
Example 9-9-1 (cont.)
9, p. 590
Forward controller
Feedforward controller
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
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
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
Section 9-
9-30
10, p. 592
Section 9-
9-31
10, p. 593
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
Section 9-
9-33
10, p. 595
Section 9-
9-34
Example 9-10-1 (cont.)
10, p. 596
Section 9-
9-35
10, p. 596
Section 9-
9-36
Example 9-10-1 (cont.)
10, p. 597
Section 9-
9-37
10, p. 597
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
Section 9-
9-39
Example 9-10-2 (cont.)
10, p. 598
Section 9-
9-40
Example 9-10-2 (cont.)
10, p. 599
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
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
Section 9-
9-43
Example 9-10-3 (cont.)
10, p. 600
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
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
Section 9-
9-46
Example 9-11-1Second-order sun-seeker system:
11, p. 603
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
Section 9-
9-48
9-12 A Hydraulic Control System
12, p. 605
two-stage electro-hydraulic valve
double-acting single rod linear actuator
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
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
Section 9-
9-51
Orifice Equation• The classic orifice equation for the fluid flow:
– The discharge coefficient:
12, p. 607
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
Section 9-
9-53
Rectangular valve-port geometry
For the rectangular geometry:For the open-center valve:
12, p. 609
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
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
Section 9-
9-56
Transfer Function of Two-Stage Valve
12, p. 611
Section 9-
9-57
Modeling the Hydraulic System
12, p. 612
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
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
Section 9-
9-60
Transfer Func. of Translational System
12, p. 615
rfaq
mc
cf KKKK
AT
sTK
,
)1(
1
Section 9-
9-61
Applications: Rotational System
12, p. 616
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
Section 9-
9-63
Applications: Variable Load
12, p. 617
Section 9-
9-64
9-13 Control DesignP Control:
13, p. 617
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
Section 9-
9-66
P Control: root loci & responses
13, p. 619
Section 9-
9-67
PD Control
• Closed transfer function:
PD control
13, p. 621
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
Section 9-
9-69
PD Control: root loci & root contour• Characteristic equation (KD = 0):
13, p. 622
Section 9-
9-70
PD Control: responses
13, p. 623
Section 9-
9-71
PI Control
13, p. 626
PI control
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
Section 9-
9-73
PI Control: root loci & responses
13, p. 627
Section 9-
9-74
PI Control: attributes
PI
13, p. 627
Section 9-
9-75
PID Control
• Transfer function:
13, p. 628
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
Section 9-
9-77
PID Control: attributes
13, p. 629
Section 9-
9-78
PID Control: responses
13, p. 630