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 ) ( 2 1 2 2 2 1 1 1 a a s T s T a s T s T a s G C lead lag
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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.
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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:
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Section 9-
9-3
Example 9-7-1 (cont.)
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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
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.
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Section 9-
9-6
8, p. 577
Section 9-
9-7
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Section 9-
9-8
Second-Order Active Filter
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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
•
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n
Section 9-
9-10
Example 9-8-1
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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.
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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
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
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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.
• 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
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Section 9-
9-26
Example 9-9-1 (cont.)
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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
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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.
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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
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a = 0.1T = 100
Section 9-
9-30
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Section 9-
9-31
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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:
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Section 9-
9-33
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Section 9-
9-34
Example 9-10-1 (cont.)
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Section 9-
9-35
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Section 9-
9-36
Example 9-10-1 (cont.)
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Section 9-
9-37
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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:
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Section 9-
9-39
Example 9-10-2 (cont.)
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Section 9-
9-40
Example 9-10-2 (cont.)
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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
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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:
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Section 9-
9-43
Example 9-10-3 (cont.)
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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.
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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.