Eecec 425 CH 7 Compensator Design Using Frequency Response Method PDF

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CH 7: Compensator Design Using Frequency Response Methods The primary objective is to present procedures for the design and compensation of single-input-single-output linear time-invariant control systems. Compensation is the modification of the system dynamics to satisfy the given specifications. The approach to the control system design and compensation used in this section is the frequency response method. Before discussing the compensations based on the frequency response methods, we will see the relationship between the closed-loop transient and the open-loop frequency responses and then the closed-loop transient and the frequency responses.

Relationship between Closed-Loop Transient and Open-Loop Frequency Responses Consider a unity feedback a 2nd-order system shown below: R(s)

n2 s (s  2wn )

Y(s)

We will see the relationship between the closed-loop transient and the closed -loop frequency responses. a) Relationship between Damping Ratio  (Percent Overshoot-PO) and Phase Margin (PM) The open-loop and the closed-loop transfer functions of the 2nd order system are given by

GOL (s) 

 2n s( s  2wn )

and

GCL (s) 

 2n Y (s)  2 R(s) s  2 wn s   2n

respectively. In order to evaluate the phase margin, we first find the frequency for which GOL( j)  1

1

Hence, GOL ( j  ) 

 2n



  2  j 2wn 

 2n  4  4 2 n2 2

1

The frequency satisfying the last equation is called the gain crossover frequency,  gc ,and is

 gc   n 2  2  1  4

4

The phase angle of GOL ( j ) at this gain crossover frequency,  gc is   2 2  1  4 4    gc       90 o  tan 1  GOL ( j gc )  90 o  tan 1   2    2 n    Then the phase margin is   2 2  1  4    gc   PM  180 o   G OL ( j gc )  90 o  tan 1   90 o  tan 1  2   2n   2 or PM  tan 1  2 2  1  4  4

4

    

The last equation shows the relationship between the phase margin (PM) and damping ratio, , (PO). This relationship will enable us to evaluate the percent overshoot (PO) from the phase margin (PM) found from the open-loop frequency response. The equation of the phase margin (PM) is plotted in Figure 1 (a). The plots of the phase margin (PM) versus the percent overshoot (PO) and the percent overshoot (PO) versus damping ratio are shown in Figure 1(b) and (c), respectively. Phase Margin (PM) vs Percent Overshoot (PO) (%)

Phase Margin (PM) vs Damping Ratio

80

90

75 80

70

Phase Margin (PM) in (degrees)

Phase Margin (PM) in (degrees)

65 70 60 50 40 30 20

60 55 50 45 40 35 30 25 20 15 10

10

5 0

0

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8 Damping Ratio

(a)

2

2.2 2.4 2.6 2.8

3

0

0

5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 Percent Overshoot (PO) (%)

(b) 2

Percent Overshoot (PO) vs Damping Ratio 100 90

Percent Overshoot (PO) (%)

80 70 60 50 40 30 20 10 0

0

0.1

0.2

0.3

0.4 0.5 0.6 Damping Ratio

0.7

0.8

0.9

1

(c) Figure 1. (a) Phase margin (PM) versus damping ratio and (b) phase margin (PM) versus percent overshoot (PO) and (c) percent overshoot (PO) versus damping ratio. As they are seen from Figure 1 (a) and (b), the phase margin (PM) increases as damping ratio ( ) increases and the phase margin (PM) decreases as percent overshoot (PO) increases. b) Relationship between Damping Ratio  (Percent Overshoot-PO) and ClosedLoop Frequency Response Consider a unity feedback a 2nd-order system shown above. The magnitude closed-loop frequency response is GCL ( j) 

 2n n2   2  j 2 wn  n2 ( n2   2 ) 2  4 2  n2  2

The magnitude plot of GCL( j) is shown in Figure 2.

Figure 2. The magnitude plot of GCL ( j ) .

3

The maximum value of GCL( j) occurs at the frequency  p ( r ) where

p  r  n 1  2  2 Since GCL( j) exhibits a peak at the frequency  p it is called the peak frequency,  p , or since the system resonates at the same frequency, this frequency is also called the resonant frequency,  r . GCL( j) has a peak at the peak frequency,  p (the resonant frequency,  r ) the maximum value of

GCL ( j ) is given by M p  GCL ( j p )  GCL ( j r ) 

1 2 1  

2

where M p  GCL ( j  p )  GCL ( j r ) is defined as the resonant peak magnitude. The plot of GCL ( j ) versus  and the percent overshoot (PO) are shown in Figure 3. |Gcl(jw)| vs Percent Overshoot (PO) (%)

5

5

4

4 |Gcl(jw)|

|Gcl(jw)|

|Gcl(jw)| vs Damping Ratio

3

2

1 0.1

3

2

0.2

0.3

0.4 0.5 Damping Ratio

0.6

0.7

1 0

10

20

30 40 50 Percent Overshoot (PO) (%)

60

70

80

(a) (b) Figure 3. (a) the plot of GCL ( j ) versus  and (b) the plot of GCL( j) versus the percent overshoot (PO). Note that GCL( j) increases as  decreases and there will not be a peak at 1 . For a given  , the larger value of  p (  r ) frequencies above zero if   2 the larger  n , and the faster the transient response for the system. c) Relationship between Setting Time, Peak Time, Rise Time and Closed-Loop Frequency Response. Another relationship between the frequency response and time response is between the speed of time response (as measured by t s , t p and t r ) and the bandwidth of the closed loop frequency response which is defined here to be

4

that frequency, BW , at which the magnitude response curve is

1 (or 3db 2

down from its value at zero frequency). The bandwidth of closed-loop system is found by setting GCL ( j) 

GCL ( j) 

1 2

, i.e.

 n2 n2 1   2 2   j 2 wn   n 2 ( n2   2 )2  4 2  n2  2

The result is BW  n (1  2 2 )  4

4

 4

2

2

Figure 4(a) and (b) show plots of BW /  n versus damping ratio  and percent overshoot (PO), respectively. BW/wn vs Damping Ratio 1.6

BW/wn vs Percent Overshoot (PO) (%) 1.6

1.4

1.5

1.2

1.4 1.3 1.2

BW/wn

BW/wn

1 0.8 0.6

1.1 1 0.9

0.4

0.8

0.2 0 0

0.7

0.2 0.4 0.6 0.8

1

1.2 1.4 1.6 1.8 Damping Ratio

2

2.2 2.4 2.6 2.8

3

0

10

20

30 40 50 60 70 Percent Overshoot (PO) (%)

(a)

80

90

100

(b)

Figure 4. Plots of (a) BW /  n versus damping ratio  and (b) BW /  n versus percent overshoot (PO). BW is directly proportional to  n ,that is, BW increases and decreases with  n . As it is seen from Figure 4, for a given  n , BW gets smaller as  gets larger and BW gets larger as PO gets larger.

To relate BW to settling time, we substitute  n 

4 into the last equation,  ts

and obtain BW 

4  ts

(1 2 2 )  4 4  4 2  2

5

Similarly, the relationship between BW and t p is obtained as

BW 

 t p 1 

(1 2 2 ) 

2

To relate BW to rise time, we substitute n 

4 4  4 2  2

1  0.4167  2.917 2 into the tr

BW equation and obtain 1  0.4167  2.917 2 BW  (1  2 2 )  4 4  4  2  2 tr The relationship between BW normalized by t s , t p and t r , and damping ratio

are shown in Figure 5. BW *tp vs Damping Ratio 7

BW*ts vs Damping Ratio

120

6.5

100

6

BW*tp

BW*ts

140

80

5.5

60

5 40

4.5

20

0

4 0

0.1

0.2

0.3

0.4 0.5 0.6 Damping Rat io

0.7

0.8

0.9

1

0

0.1

0.2

0.3

(a)

0.4 0.5 0.6 Damping Rat io

0.7

0.8

0.9

1

(b) BW*tr vs Damping Ratio

2.3 2.2 2.1

BW*tr

2 1.9 1.8 1.7 1.6 1.5

0

0.1

0.2

0.3

0.4 0.5 0.6 Damping Rat io

0.7

0.8

0.9

1

(c) Figure 5. Normalized bandwidth (BW) versus damping ratio for (a) settling time, (b) peak time and (c) rise time. The relationship between  n normalized by t r and damping ratio is shown in Figure 6.

6

wn*tr vs Damping Ratio 4

3.5

3

wn*tr

2.5

2

1.5

1

0.5 0

0.1

0.2

0.3

0.4 0.5 0.6 Damping Ratio

0.7

0.8

0.9

1

Figure 6. The relationship between  n normalized by t r and damping ratio. For a given  , both t s and t r get smaller but BW gets larger and PO remains the same as  n gets larger. Hence the system has a faster response. On the other hand, for a given  n , t s , PO and BW get smaller but t r gets larger as

 gets larger. BW and t r are inversely proportional to each other. Therefore, increasing  n increases BW and decreases t r and increasing  decreases BW and increases t r . As a result, in the design of control systems using the frequency response methods, we try to reshape the frequency response of the open-loop transfer function to meet both the phase margin requirement (percent overshoot) and the bandwidth requirement (settling and/or rise time(s)) such that the desired percent overshoot is achieved by meeting the requirement on the phase margin and the desired t s and/or t r can be achieved

by adjusting BW with  n while keeping PM constant (unchanged). However, it is not easy to achieve the desired t s and/or t r by adjusting BW with  n while keeping PM constant (unchanged) since the relationships between these terms are very complicated. In other words, there is no easy way to relate all requirements prior to reshaping task. Thus, the reshaping of the frequency response of the open-loop transfer function can lead to several trials until all transient requirements are met. Improving Transient Response Frequency response methods are not as intuitive as the root-locus. With root locus, we can identify a specific point as having a desired transient response characteristic. We can then design cascade compensation to operate at that point and meet the transient response specifications. In the case of the design of control systems using the frequency response methods, we try to reshape the frequency response of the open-loop transfer function to meet both the phase margin requirement (percent overshoot) and the bandwidth requirement (settling and/or rise time(s)) by several trials until all transient requirements are met since there is no easy way to relate all requirements prior to reshaping task.

7

Improving Transient Response An advantage of using frequency response design methods is to meet both all the transient response and the steady-state error requirements at the same time. Compensation based on Frequency Response Methods In this section, we will see the following frequency response methods to improve both the steady-state error and transient response. 1. Lead Compensation 2. Lag Compensation 3. Lag-Lead Compensation However, these design methods can be easily extended to the designs of PD, PI and PID controllers. 1. Lead Compensation Consider the cascade lead compensation shown in Figure 7.

R(s)

E(s)

Lead Compensator Gc(s)

Gai n

Plant

K

Gp(s)

C(s)

Figure 7.The cascade lead compensation. The transfer function of the lead compensator is

Gc ( s) 

U ( s) s z  KC E (s ) sp

where z, p and KC are the zero, pole and the gain of the lead compensator, respectively. The pole is located to the left of the zero. Thus, p  z ,  z   p and

c  z   p  0 . Therefore, it is called the “lead compensator”. The gain of the lead compensator, KC , is chosen such that it does not affect the steady-state error but the lead compensator improves the transient response when the lead compensator is inserted into the system. Thus,

8

s z z lim Gc ( s)  Gc (0)  lim KC  KC  1 s 0 s  p s 0 p or

KC 

p 1 z

In fact, KC is the dc gain of the compensator. The pole-zero diagram of the lead compensator is shown in Figure 8.

Figure 8. The pole-zero diagram of the lead compensator.

Electrical Lead Network The electrical lead compensator network is shown in Figure 9. This network consists 1 of a passive network and an amplifier whose gain is .



Figure 9. The electrical lead compensator network. The transfer function of the lead compensator is

Gc ( s) 

Vo ( s) s z  KC Vi( s) s p 9

1 1 and p  are the zero, pole and the gain of the lead compensator, T T R2  1 and T  R1 C . Thus, the transfer function of the respectively. Here,   R1  R2 lead compensator in terms of  and T is

where z 

V ( s) 1 G c (s )  o  V i (s ) 

1 T 1 s T s

Active Electrical Lead Network

The active electrical lead compensator is shown in Figure 10. The transfer function of the lead compensator is

Gc ( s) 

Vo ( s) s z  KC Vi( s) s p

p 1 RC 1 1 1 1 , p   and K C    1 1  1 are the zero, pole T R1C1 T R 2C 2 z  R2 C 2 1 1 z and the gain of the lead compensator, respectively. Note that p  R2 C2 R1 C1 or R1C1  R2 C2 . Thus, the transfer function of the lead compensator in terms of  and T is where z 

1 V ( s) 1 T G c (s )  o  V i (s )   1 s T s

10

Figure 10. The active electrical lead compensator network. Steady-State Error of Lead Compensator

Note that the steady-state error of the lead compensator is

1 1 1 T  1 T 1 lim Gc ( s)  Gc (0)  lim 1  1 s 0  s 0 s T T s

Thus, it does not affect the steady-state error but the lead compensator improves the transient response when the lead compensator is inserted into the system. Lead Compensator Frequency Response In order to find the phase angle of the lead compensator,  c , the following transfer function of the lead compensator must be evaluated at j .

1 1 T Gc ( s )   s 1 T s

Thus,

11

1 1 T Gc ( j )   j  1 T j 

Then, the phase angle of the lead compensator,  c , is

c  Gc ( j )  tan 1(T )  tan  1(T ) The frequency at which the maximum value of the phase occurs is obtained by d differentiating  c with respect to  and setting c  0 , i.e., d

dc T T 0   2 d  1  (T ) 1  ( T ) 2 The solution of the last equation is 1  max  T  Note that,  max is equal to the geometric mean of two corner frequencies, e.i.,

1 1 1  T T T  In other words, the maximum phase angle occurs halfway between the pole and the zero frequencies on the logarithmic frequency scale. This Then, substituting 1  max  into Gc ( j ) , we have T 

 max  z p 

1 1 j 1 jmax  1 1 (   j1 1 (   j1)(1  j   T Gc ( j max )      j  1 (1   )  (1  j  )  j  1 max T

Gc ( j max ) 

1 (2   j (1  ) (1   ) 

Then, the maximum value of the phase angle at  max is determined from the triangle in Figure 11.

12

Figure 11. The maximum phase shift,  max . Thus, the maximum phase shift,  max , is  1     1  

 max  sin 1 

The maximum phase shift,  max , versus  is shown in Figure 12. Maximum Phase Angle vs Alpha 70

Maximum Phase Angle in (degrees)

60

50

40

30

20

10

0

0

0.1

0.2

0.3

0.4

0.5 Alpha

0.6

0.7

0.8

0.9

1

Figure 12. The maximum phase shift,  max , versus  . Or the value of the  is found from



1  sin(max ) 1  sin(max )

The  versus maximum phase shift,  max , is shown in Figure 13.

13

Alpha vs Maximum Phase Angle in(degrees) 1 0.9 0.8 0.7

Alpha

0.6 0.5 0.4 0.3 0.2 0.1 0

0

10

20 30 40 50 Maximum Phase Angle in (degrees)

60

70

Figure 13. The  versus maximum phase shift,  max . The magnitude of the lead compensator at  max is

Gc ( j max ) 

1



The frequency response of the lead compensator is shown in Figure 14.

14

Figure 14. The frequency response of the lead compensator. As it is seen from the frequency response of the lead compensator, the phase diagram is raised at  max and the magnitude plot is raised at higher frequencies. Thus, the phase angle of the compensated system at  max is raised by  max and the magnitude plot of the compensated system at  max is raised by

20 log

1



1



1 and it is also raised at higher frequencies by



(or

(or

20 log

1



).

Hence, the lead compensator increases the bandwidth by increasing the gain crossover frequency and also increases the phase margin of the compensated system. In time domain, the lead compensator lowers the percent overshoot (increases the damping ratio) and may lower the settling time and/or the rise time. The effect of the lead compensation on the system is shown in Figure 15.

15

16

17

18

Figure 15. The effect of the lead compensation on the system. One advantage of the frequency response method over the root locus is that we can first implement a steady-state error requirement and then design a transient response. The specification of transient response with the constraint of a steadystate error is easier to implement with the frequency response technique than the root-locus method. As a result, first, the requirement on the steady-state error is met by adjusting gain. After meeting the steady-state error requirement, the requirement on the transient response is satisfied by inserting the lead compensator into the system since the lead compensator will not affect the steady-state requirement (

lim Gc ( s)  Gc (0) ...