- A compromise value of p should be selected to obtain M1 and M2 large or small at the same time.. - For a fixed ωn, the less the value of ζ , the more oscillatory the time response; and
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CONTROL SYSTEM DESIGN
Lecture Notes The Performance of Feedback Control Systems
4 - To analyze & design a control system, we must define & measure its
performance
- Generally, for a given p, M1 is large and M2 is small, or vice versa
- A compromise value of p should be selected to obtain M1 and M2 large (or small)
at the same time
7 - R s( ) 1 = : the Laplace transform of the unit impulse
- R s( ) A
s
= : the Laplace transform of the step
- R s( ) A2
s
= : the Laplace transform of the ramp
- R s( ) 23A
s
= : the Laplace transform of the parabolic
9 - R(s) is a unit step input
-
2
2
n
ω
+ + : the standard form of a second – order transfer function
- y(t): the time response of the system
- The steady – state of the output y(t) is 1
10 - The response in the time domain of a second – order system to a unit step input
- For a fixed ωn, the less the value of ζ , the more oscillatory the time response; and the time response crosses the steady – state value (1) several times
11 - R(s) is a unit impulse input
- The steady – state of the output y(t) is 0
- For a fixed ωn, the less the value of ζ , the more oscillatory the time response; and the time response crosses the steady – state value (0) several times
13 - The swiftness of the response is measured by the rise time T r & the peak time T p
- The settling time is defined as the time required for the system to settle within a certain percentage delta of the input amplitude
14 - Percent overshoot & peak time should be small, but they can not be small at the same time
- When ζ varies from 0 to 1, percent overshoot decreases and peak time increases; these two criteria can not get minimum at the same time
15 - The less the value of ωn, the faster the time response goes to steady state
1,2 n n 1
s = − ζω ±jω − ζ : roots of the denominator of the transfer function T(s)
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- s 4
n
T
ζω
= : settling time, see page #13
- s1,2= − ± 1 j1: a pair of conjugate roots is selected, their real part satisfies the condition of settling time (less than 4 seconds)
23 - T1(s) has a negative real pole
- This real pole dampens the overshoot and increases the settling time
25
-
= =
= + + + : time response to a unit step
- The position of a pole (in the s – domain) determines properties of the time
response
27 - E(s): the tracking error
- K p: the position error constant
-
1
ss
p
A e
K
=
+ : the steady – state error is inversely proportional to the position error constant
28 - K v: the velocity error constant
- ss
v
A e
K
= : the steady – state error is inversely proportional to the velocity error constant
29 - K a: the acceleration error constant
- ss
a
A e
K
= : the steady – state error is inversely proportional to the acceleration error constant
30 - Number of Integrations: the order of s in 1
s
31 - e ss = 0: for a step input, the steady – state error is zero
-
2
ss
A e
K K
= : for a ramp input, the steady – state error is nonzero
33 - The larger the value of K2, the less the value of error
35 -
0
1 1
1 2lim 40
5
s→ s
+
+
: suppose that K1 = 2
39 - ISE is the accumulation of the square of the error
- In steady state, e(t) and e2(t) are zero, but ISE is nonzero
41 -
2
1 ( )
T s
= + + : transfer function of the system, obtained directly from the block diagram model
- ISE, ITAE, and ITSE obtain their own minima at different values of ζ
- For instance, if we want to use ITAE to evaluate the performance of the system,
we should select ζ = 0.75
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- If ITSE is applied, then ζ should be 0.60
42
- ( )
( )
k k k d
P
Y s
T s
∆
=
∆
: + The transfer function, see Mathematical Models of Systems
+ This is the transfer function of the system when the input is the disturbance
T d (s)
- The effect of the disturbance is minimized if the output Y(s) in ( )
( )
k k k d
P
Y s
T s
∆
=
∆
is minimized
-
nontouching nontouching
N
=
∆ = − + − + : the determinant, see Mathematical
Models of Systems
43 - P1 = 1: the unique path connecting the input T d (s) to the ouput Y(s)
44 - K1= 0.5; K K K1 2 p = 2.5: for instance
- T s d( ) 1
s
= : suppose that the disturbance is a step
3 3
0.1 0
dI
K dK
−
= − + = : to minimize I with respect to K3, the derivative of I with respect to K3 is set to 0
45 - Figure of two plots: applying the obtained K3, the steady state of the output Y(s) (the red curve) responsing to the disturbance T d (s) (the blue line) is zero, so the
effect of the disturbance is minimized
51 - The step responses of the two transfer function are nearly identical
- It can be observed that the term (s + 10) has nearly no effect on the response of
T1(s), hence it can be eliminated
52 - A higher – order system G H (s) is simplified to a lower – order system G L (s)
- k!: the factorial of the positive integer k
54 - By definition, 0! = 1
57 - The step responses of the two transfer function are nearly identical