At the end of the lecture, I should be able to: Identify various control modes to achieve the desired level of performance. Predict behavior of systems with derivative controller
Identify various control modes to achieve the desired level of performance. • Proportional mode (P) • Derivative mode (D) • Integral mode (I)
• Combination of modes (PD, PI and PID)
Derivative Control, GC(s)=Kds To prevent systems becoming too oscillatory as
Kp is increased, derivative (differential) control action may be incorporated into the controller (P+D control). The controller transfer function is then of the form:
GC (s) K p sKd Signal proportional to the error
or
K GC ( s) K p 1 s d K p
Signal proportional to the derivative of the error
How does it work?? Derivative action works by anticipating errors. If the
rate at which the error signal is changing is high then a large overshoot will probably occur. Output c(t)
Error e(t)
1. Error is large and +ve the rate of change of the error (derivative) is -ve 1 the signal fed to the system will be reduced
1
Time
Time
How does it work?? Derivative action works by anticipating errors. If the
rate at which the error signal is changing is high then a large overshoot will probably occur. Output c(t)
2
2. Error is -ve
Error e(t)
the derivative remains -ve
the signal fed to the system is now increased pushing the system back to its desired position Time
2
Time
How does it work?? Derivative action works by anticipating errors. If the
rate at which the error signal is changing is high then a large overshoot will probably occur. Output c(t)
3
Error e(t)
3. Error is still -ve the derivative becomes +ve the signal fed to the system is decreased
Time
Time
3
Effect of (P+D) control on transient and steady state response Example 18: Examine the effects of (P+D) control on the performance of the system below:
+ -
GC (s) K p sKd
1 G( s) 2 5s 6 s 1
H ( s) 1
Effect of (P+D) control on transient and steady state response Method: 1) Calculate the overall transfer function
GC ( s)G( s) G' ( s) 1 H ( s)GC ( s)G( s)
K p sK d
+
G(s)
H(s)
G( s) G' ( s) 1 H ( s)G( s)
5s 6 s 1 K p sK d 1 2 5s 6 s 1 K p sK d =0 2 5s (6 K d ) s ( K p 1) 2
Characteristic Equation
Effect of (P+D) control on transient and steady state response Method: 2) For various values of Kd, vary Kp, find the roots of C.E.
5s 2 (6 K d ) s ( K p 1) 0 s1, 2
b b 2 4ac 2a
lim[ e(t )] lim[ sE ( s )]
3) For each value of Kd and Kp calculate the steady state error to a unit step input
t
s 0
1 lim s 0 s R( s ) 1 H ( s )GC ( s )G ( s ) 1 1 1 lim s 0 s K p s K d 1 K p s 1 2 5 s 6 s 1
Effect of (P+D) control on transient and steady state response Method: 4) Tabulate results Kp 0.5
Kd=0.25
Kd=0.5
Kd=0.9
S1
S2
S1
S2
S1
-0.34
-0.89
-0.32
-0.95
-0.30
S2
e(t)
-0.99 0.667
1
0.500
2
0.333
4
0.200
6
0.143
8
0.111
10
0.091
Effect of (P+D) control on transient and steady state response Summary: As you fill out the table, you should find that: In general increasing Kd adds damping to system making it less oscillatory but does not reduce the steady state error.
Increasing Kd is comparable to increasing the damping term C
Increasing Kd tends to amplify any high frequency
noise which can be damaging. Pure derivative action is not physically possible.
In practice there will always be some delay, though often delay is relatively small in comparison with the system being controlled.
