交直流马达控制_CHAPTER 3

Figure 3-10 KP-versus-KD parameter plane for the attitude control system with a PD controller.
Frequency-Domain Design
Figure 3-11 Bode plot of G ( s ) =
K P = R2 R1 , K D = Rd C d
Figure 3-4(b) Three-op-amp circuit.
¾ 3-2-1 Time-Domain Interpretation of PD Control
Figure 3-5 Waveforms of y(t), e(t), and de(t)/dt, showing the effect of derivative control. (a) Unit-step response. (b) Error signal. (c) Time rate of change of the error signal.
Figure 3-9 Unit-step response of the attitude control system with and without PD controller.
Hale Waihona Puke Table 3-1 Attributes of the Unit-Step Responses of the System in Example 10-1 with PD Control KD 0 0.0005 0.00177 0.0025 tr (sec) 0.00125 0.0076 0.00199 0.00103 ts (sec) 0.0151 0.0076 0.0049 0.0013 Maximum Overshoot (%) 52.2 25.7 4.2 0.7
¾ 3-3-2 Frequency-Domain Interpretation and Design of PI Control
G c (s) = K + K I K I [1 + ( K P / K = S S
I
)S ]
P
The Bode plot of GC ( jω ) is shown in Fig 3-18
¡E GC ( jω ) at ω = ∞ is 20 log10 K P dB ,which represents an attenuations if the value of K P is less than 1. ¡E The phase of GC ( jω ) is always negative , which is detrimental to stability. Thus
KP = R2 1 , KI = R1 Ri Ci
The forward-path transfer function of the compensated system is ωn 2 ( K P s + K I ) G ( s ) = Gc ( s )GP ( s ) = 2 s ( s + 2δωn ) Clearly, the PI controller are 1. Adding a zero at s = − KI KP to the forward-path transfer function.
The PI controller is
Gc ( s ) = K P + KI s
The transfer function of the three-op-amp circuit in Fig 3-16(b) is E ( s ) R2 1 Gc ( s ) = o = + Ein ( s ) R1 Ri Ci s
1. Improving damping and reducing maximum overshoot. 2. Reducing rise time and settling time. 3. Increases BW. 4. Improves GM, PM, and Mr. 5. Possibly accentuate noise at higher frequencies. 6. Possibly require a relatively large capacitor in circuit implementation.
2. Adding a pole at s=0 to the forward-path transfer function. This means that the system type is increased by 1 to a type 2 system. Figure 3-17 Pole-zero configuration of a PI controller.
¾ 3-2-2 Frequency-Domain Interpretation of PD Control
Gc ( s) = K P + K D s = K P (1 + Figure 3-6 Bode diagram of 1 + KD s) KP
KD s , KP =1 KP
¾
3-2-3 Summary of Effects of PD Control
Settling time ts ≤ 0.005 sec
Time-Domain Design To satisfy the specified steady-state error, K should be set at 181.17.
However, with the value of K, the damping ratio of the system is 0.2. With the PD controller and K=181.17. The forward-path transfer function of the system becomes θ y ( s) 815.265( K P + K D s) G ( s) = = θ e ( s) s ( s + 361.2) The closed-loop transfer function is θ y ( s) 815.265( K P + K D s) = 2 θ e ( s) s + (361.2 + 815265K D ) s + 815261K P The ramp-error constant K v = lim sG ( s ) =
Chapter 3 Design of Control System
¡±3 -1 Introduction
Control system design involves the following three steps : 1. Determine what the system should do and how to do it (design specifications). 2. Determine the controller or compensator configuration relative to how it is connected to the controlled process. 3. Determine the parameter values of the controller to achieve the design goals. Figure 3-1 Controlled process
¾ 3-1-1 Design Specifications
Often include specifications about : ¡E relative stability (phase margin, BW, gain margin) ¡E steady-state accuracy (error) ¡E transient-response (rise-time, overshoot, settling time) ¡E sensitivity to parameter variations (robustness)
815265(1 + K D s ) . s ( s + 361.2)
Table 3-2 Frequency-Domain Characteristics of the System in Example 10-1 with PD Control KD 0 0.0005 0.00177 0.0025
s →∞
815265K P = 2257.1K P 361.2
ess =
1 0.000443 = KP Kv
Let K P = 1 which is acceptable from the steady-state error, The damping ratio
δ=
361.2 + 815265K D = 0.2 + 451.46 K D 1805.84 δ = 1 , K D = 0.001772
ts (sec) 0.0151 0.0076 0.0049 0.0013
Maximum Overshoot (%) 52.2 25.7 4.2 0.7
¡± 3-3 Design with the PI Controller
Figure 3-15 Control system with PI controller
EX 3-1 The forward-path transfer function of the system is given 4500 K G ( s) = s ( s + 361.2)
Specifications : Steady-state error due to unit-ramp input ≤ 0.000443 Maximum overshoot ≤ 5 percent Rise time tr ≤ 0.005 sec