线性控制课件 module 21 08-4-27-english_new

K p s Ki C 3 2 R s s K p s K i
Kp=0 s will disappear from the denominator system is unstable (Routh’s criterion) Conclusion: PI control is able to eliminate steady state error but has little effect to transient response.
＋
＋
o
R
＋ －
K p Kd s
＋ ＋
1 Js 2 cs
C
Fig.21.11 Secondorder PD control system.
The transfer function of the controller:
o K Kd s i
p
For second order system ( damping ratio is meaningless for first order system)
Ki Kp 1
If the pole close to imaginary axis, the damping ratio will be small, and it cause an oscillatory response. Note: I control is not used alone.

For second order system:
K p s Ki C 3 2 R s s K p s K i C s 3 2 T s s K p s K i i
R is unit step input: T is unit step disturbance: Draw the root locus using:
o Ki Kp Kd s i s
o 1 K p (1 Td s) i Ti s
Trial-and-Error Method
(1)Select Kp firstly to achieve the transient response desired while setting Ti and Td to zero (2)Then Ti is adjusted to satisfy steady state error requirements. This will degrade the transient response due to the inclusion of a closed loop zero. (3)Select a suitable Td to restored the transient response.
For second order system.
C K 2 R s s K C 1 2 T s s K
R is unit step input Css t 1 T is unit step disturbance. 1
Css t K
If K is made large, the disturbance approach 0. But the large K should cause an oscillatory response. Conclusion: P control is of limited success in trying to obtain good performance in terms of steady state error, disturbance rejection and transient response.
Css t 1
Compare it to standard second order system.
2 s 2 2n s n Js 2 c K d s K p
c Kd 2n J 2 n K p / J
The above obtained Kp can satisfy damping ratio. But a zero is exist, and it must be taken into account in determining the transient response.
Proportional-Plus-Derivative Control (PD)

PD control can influence the transient response by changing the damping ratio.
i
Kp
Kd s
Fig.21.10 PD controller T
The transfer function of the controller:
Ki o K p i s o K p s Ki i s
K p s Ki C 2 R s s 1 K p K i C s 2 T s s 1 K p K i
Controller e K×e
K
Fig.21.2 Proportional controller
T R ＋ ＋
＋ －
-
K
1 1 s
C
Fig.21.3 First-order type 1 system with disturbance.
For first order system.
R is unit step input: T is unit step disturbance:
The
output is proportional to the input plus the integral of the input.
i
Kp
＋ ＋
o
Ki s
Fig.21.6 PI controller T R
＋ －
K P s Ki s
＋ ＋
1 1 s
Fig.21.7 PI control of first- order sysrem.
C
Kp
＋ ＋
1 Js c
1 s
Kr
Fig. 21.12 Derivative control using a tachometer
Kd
Ktach
Fig.21.13 Implementation of tachometer gain K r

The derivative control is never used alone but is always with a proportional component. The reason is to avoid noise. The differentiator can magnify noise.
Linear Control Systems Engineering 线性控制系统工程
Module 21
Multimode Controllers (多模控制器)
21 Multimode Controllers
• Multimode control design:
change the type of the system to meet desired performance requirements.
Css t 1 Css t 0
GH s K p s Ki s 2 1 s
To make the system stable, the zero must closer to the imaginary axis than system pole. (otherwise the asymptote may be at the right of imaginary axis).
C K / 1 K R 1 s / 1 K C 1 / 1 K T 1 s / 1 K
K Css t 1 K 1 Css t 1 K
If K is made large, the output for input approach 1 and for disturbance approach 0. But the K is finite.
For first order system:
R is unit step input: T is unit step disturbance:
Css t 1 Css t 0
The syom 0 to 1, due to the controller.
Fig.21.16 Quarter-decay response.

Ziegler and Nichols first method The method only applies if the open loop step response of the plat has no overshoot. To obtain maximum slope P and the time L
• Mode control – Design the controller in the loop.
R ＋ － Controller Plant C
Feedback transducer
Fif.21.1 Definition of controller
Proportional Control

The output is proportional to the input.
T R
＋ －
K p s Ki s
＋ ＋
1 s (1 s )
C
Fig.21.8 PI control of secend-order sistem. Im
×
﹥ ﹤
Re
1
Ki K p
Fig.21.9 Root locus of PI-controlled second-order system