Frequency Domain Analysis and Design

제 7 장 Frequency Domain Analysis and Design

7.1 Summary

7.2 Bode Diagram Method

7.3 Frequency Response Analysis

7.4 앞섬 뒤짐보상기 Design법

7.5 요점정리

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- 박노해, 길 잃은 날의 지혜

7.1 Summary

- Frequency Domain Control System Design and Analysis using Bode Diagram

1) Bode diagram : Express frequency domain response with two graphs, one for the magnitude response and the other for phase response. Horizontal axis is frequency expressed logarithmic scale. Vertical axis of magnitude response is expressed in dB, vertical axis of the phase response is degree expressed in linear scale.

2) Bode Diagram for the systems in series can be obtained by adding the diagrams for the sub-systems => Easy. Bode diagram for the simple system can be draw by hand.

3) Cemtool has the commands that easily draw Bode Diagram for general transfer functions => Bode Diagram of complicated system can be obtained using a computer.

4) Control system designed in the Frequency Domain guarantees the performances of the control system in spite of small error in the design process or variables =>

Possible to develop a more robust control system compared to time domain design.

5) Stability margin : The relative indices that express closed loop stability with respect to uncertainties of the corresponding open loop system (relative stability).

- Gain Margin : The Maximum margin variation that maintains closed loop stability when the phase is fixed.

- Phase Margin : Maximum phase variation that maintains closed loop stability when margin is fixed.

- Desirable stability margin : Gain Margin ≥ 6 [dB], Phase Margin = 30∼60[◦]

6) 앞섬(뒤짐) 보상기 : 대상시스템의 위상을 적절히 앞서도록(뒤지도록) 보상하는 장치로서 각각 1차의 전달함수로 표시된다. 이 보상기들은 따로 쓰이거나 또는 주파수역을 달리하 여 함께 쓰이기도 하며, 함께 사용할 경우에는 2차 전달함수로 표시된다.

7) 앞섬보상기 : 시스템의 상승시간을 빠르게 하고 대역폭을 증가시키는 등 과도응답특성을 개선시킨다.

뒤짐보상기 : 시스템의 정상상태 오차를 줄이는 등 정상상태 응답특성을 개선시키지만 상승시간이나 정착시간이 느려진다.

앞섬/뒤짐보상기 : 과도응답과 정상상태 응답특성을 함께 개선시키기 위해 쓰인다.

7.2 Bode Diagram Method

Bode Diagram(Bode diagram) : 1942 H. W. Bode. Express frequency domain response with two graphs, one for the magnitude response and the other for phase response. Horizontal axis is frequency expressed logarithmic scale. Vertical axis of magnitude response is expressed in dB, vertical axis of the phase response is degree expressed in linear scale.

Benefits of Bode Diagram :

1) Bode Diagram for the systems in series can be obtained by adding the diagrams for the sub-systems => Easy. Bode diagram for the simple system can be draw

by hand.

2) Frequency is expressed with logarithmic scale => Frequency response can be expressed in very wide frequency range.

3) Control system designed in the Frequency Domain guarantees the performances of the control system in spite of small error in the design variables => Possible to develop a more robust control system compared to time domain design.

4) Bode Diagram is frequency response of the open loop transfer function but the characteristics of closed loop system can be found from it.

<Remarks>

Performance indices :

- Band-pass gain G_B :

- Cutoff frequency ω_c : Frequency at which magnitude is 1/ 2 of GB or decrease by 3[dB].

- Bandwidth ω_B : Frequency range that passes through the system.

- Resonant frequency ω_r : Frequency at which resonance takes place.

- Resonant peak M_r : Maximum gain at resonant frequency

- Stability margin : Relative indices for stability. Rages for Gain and Phase that guarantee stability. GM: gain margin, PM: phase margin

|G(jw)|_dB

w_c w w_r M_r

G_B G_B-3

<Fig. 4.7> Frequency Domain Performance Indices

7.2.1 Bode Diagram Plot

- Basic factors of the transfer function : `Constant', `1^st order factor', `2^nd order factor'

1) ConstantG(j^ω) =K case :

Magnitude is 20 log |K|[dB], Phase is 0[◦] forK> 0 , 180 [◦] for K< 0..

<Fig. 7.1> Bode Diagram for the Constant Factor

2) 1st order factorG(j^ω) =1/(j^ωT) :

Magnitude isG_dB=-20 logωT[dB] => Decrease - 20[dB] as frequency increases 10 times. ωT= 1 =>G_dB= 0 [dB]. G(j^ω) =-j/(ωT) => Phaseφ(ω)=-90[◦].

Bode Diagram for G(j^ω)=j^ωT is symmetric with respect to horizontal axis compared to <Fig. 7.2>.

<Fig. 7.2> Bode Diagram for 1st order factor : G(j^ω) =1/j^ωT

3) 1st order factor G(j^ω) =1/(j^ωT+1) :

Magnitude is G_dB=-20 log ω²T ²+1 =-10 log (ω²T ²+1)[dB],G_dB=-3[dB] for

ωT= 1, G_dB≈0 [dB] forωT≤0.1 . Cutoff frequency and Bandwidth are ω_c=ω_B = 1/T, Band-pass gain is G_B= 0[dB].G_dB≈-20 logωT[dB] whenωT≥10 => Decreases - 20 [dB] when ω increases 10 times. Phaseφ(ω) =-tan ^{- 1}ωT => φ(ω_c)=-45 [◦]

and φ(ω)≈0 [◦] for ωT≤0.1 , φ(ω)≈-90[◦] forωT≥10 . Bode Diagram can be expressed as <Fig. 7.3>.

Bode Diagram for G(j^ω) =j^ωT+1 is symmetric transform of <Fig. 7.3> with respect to horizontal axis.

<Fig. 7.3> Bode Diagram of 1st order factor G(j^ω) =1/(j^ωT+1) 4) 2nd order factor G(j^ω) =1/[ (j^ω/ω_n)²+2ζ(j^ω/ω_n)+1] :

Magnitude is |G(j^ω)| = 1/ [1-(ω/ω_n)²]²+[2ζ(ω/ω_n)]². Resonance take places at the frequency that minimize denominator of |G(j^ω)| => Resonant frequency and Resonant peak are as follows.

ω_r = ω_n 1-2ζ², 0≤ζ≤0.707 M_r = |G(j^ω_r)|= 1

2ζ 1-ζ²

(7.1)

Resonance take places for the damping ratioζ< 0.707 . Damping ratio ζ ↓=> _ converges to ω_n and M_r ↑. Cutoff frequency and Bandwidth when the Gain is 1/ 2 are as follows..

ω_c = ω_B = ω_n

(

1-2ζ²+ 4ζ⁴-4ζ²+2

)

^{1/ 2 (7.2)}

Phase is φ(ω) =-tan ^{- 1} 2ζω/ω_n

1-(ω/ω_n)² , φ(ω)=-90[◦]forω=ω_n,φ(ω)≈0 forω= 0.1ω_n, φ(ω)≈-180 [◦] forω= 10ω_n. Bode Diagram for 2nd order factor is as <Fig. 7.4>.

Bode Diagram for G(j^ω) = (j^ω/ω_n)²+2ζ(j^ω/ω_n)+1 is symmetric transform of <Fig.

7.4> with respect to horizontal axis.

<Fig. 7.4> Bode Diagram of 2nd order factor : G(j^ω) =1/[ (j^ω/ω_n)²+2ζ(j^ω/ω_n)+1]

7.2.2 Using Computer Software CemTool Commands :

[mag,phase] = bode(num,den,w); (7.3)

Here, num, den are coefficients of the numerator and denominator in the system transfer function, w are frequencies where frequency responses are calculated.

[예] Define 100 frequencies in the range of 10 ^{- 2}~ 10³[rad/sec] :

w = logspace(-2,3,100); (7.4)

<주> ① If the range is not defined : use w=logspace(-1,1,50).

② Phase is expressed in [◦], 'mag' is expressed in absolute value =>

'20*log(mag)' should be used to express in [dB].

Plotting Bode Diagram :

① Auto. frequency range logspace(-1,1,50) : bode(num,den);

② Specific frequency range : bode(num,den,w);

<주> Plot the Bode Diagram but mag. and phase are not stored.

Ex..3 : Plot the Bode Diagram for the following system using CemTool.

G(s) = 25 s²+4s+25 Frequency range id 0.01∼100[rad/sec].

Sol. :

num = 25;

den = [1 4 25];

w = logspace(-2,2,100);

[mag,pha] = bode(num,den,w);

subplot(2,1,1);

semilogx(w,20*log(mag));

title("G(s)의 Bode Diagram");

xtitle("주파수 [rad/sec]");

ytitle("크기 [dB]");

subplot(2,1,2);

semilogx(w,phase);

xtitle("주파수 [rad/sec]");

ytitle("위상 [degree]");

Result : Bode Diagram given in <Fig. 7.7>.

□

<주> Defining the location and title of the Fig. : CEMTool>>num=25; den=[1 4 25];

CEMTool>>w=logspace(-2,2,100);

CEMTool>>bode(num,den,w);

Result : Same as <Fig. 7.7> but title of the Fig. does not appear and sign for the horizontal and vertical axis expressed in English.

<Fig. 7.7> 예제7.3의 Bode Diagram

7.3 Frequency Response Characteristics Analysis

7.3.1 Definition of Stability Margin

Absolute stability : Expression of stable or not.

Relative stability :

- Expression of the degree of stability.

- Stability margin, expressed by the gain margin and phase margin.

- Transfer Function for the system with uncertainty :

G_T(s)=G(s)Ke ^-jθ (7.5)

Def. 7.1 : For the unit feedback system in <Fig. 7.8>, Gain Margin is the range of K that guarantee closed loop system stability with θ= 0. Phase margin is the range of

θ that guarantee closed loop system stability withK= 1.

□

r + G(s) -

e y

불확실성 Ke^-jq

불확정 플랜트 C(s)=1

<Fig. 7.8> 안정성여유를 정의하는 단위되먹임시스템

<주> ① In Def. 7.1, controller is assumed asC(s) = 1 , ifC(s)≠1 , this Def. can be applied to transfer function G(s)C(s).

② In Def. 7.1, it is assumed that gain and phase do not change simultaneously.

Actually they can change simultaneously and in that case stability margin decreases generally. But in this case, stability margin is hard to calculate =>

Stability margin is calculated as in the Def. 7.1.

Stability margin in the Nyquist diagram :

G(jw)

실수축 허수축

-1

PM

1/GM

<Fig. 7.9> Stability margin in the Nyquist diagram

   log_  _  

    ∠_   _    (7.6) Here,G(j^ω) is frequency response for given open loop transfer function, Phase crossover frequencyω_pc is the frequency at which phase response of the open loop transfer function crosses -180[◦], gain crossover frequencyω_gc is the frequency at which gain response of the open loop transfer function crosses 0 dB.

Stability margin in the Bode diagram :

|G(jw)|

G(jw) 0 dB

_o

-180 PM

GM log w

log w w_gc

w_pc

<Fig. 7.10> Stability margin in the Bode diagram

<주> ① For the stable open loop system, Gain margin and phase margin should be greater than 0 for the unit feedback closed loop system to be stable.

Margins ↑ => Relative stability ↑.

② Stability margins ↑ => Relative stability ↑ & Time response characteristics speed↓. Desirable stability margins considering stability and performance are phase margin 30∼ 60 [◦] gain margin ≥ 6 dB.

③ To improve relative stability, controller is added to the unit feedback system in <Fig. 7.8>. Gain margin is easy to control => Generally, the controller is designed to adjust phase margin of the system.

④ Stability margin in <Fig. 7.9>, <Fig. 7.10> can be applied to the stable open loop system, necessary and sufficient condition for stability is that both gain margin and phase margin are greater than 0.

7.3.2 Calculation of Stability Margins

Ex. 7.4 : For the system with following open loop transfer function, plot system Bode Diagram and calculate gain margin and phase margin from it.

G(s)H(s) = 10

s(1+0.02s)(1+0.2s)

Sol. : CemTool commands for the Bode Diagram of the open loop transfer function :

num = 10;

den = conv([1 0],conv([0.02 1],[0.2 1]));

bode(num,den);

Result : <Fig. 7.11>. Here gain and phase margin can be found using 'trace' function in CemTool.. In <Fig. 7.11>, gain crossover frequency is about 6.2[rad/sec], phase angle for this frequency is about -148.5[◦] => phase margin is 180+(-148.5)=31.5[◦].

<Fig. 7.11> 예제7.4의 위상여유 구하기

Bode Diagram in <Fig. 7.11> is plotted in the automatic frequency range (0.1-10 rad/s).

But the phase crossover frequency is out of this range and gain margin can not be obtained from <Fig. 7.11> => Redraw the bode diagram in a different frequency range.

CEMTool>>w=logspace(1,2);

CEMTool>>bode(num,den,w);

Redrawn bode diagram : <Fig. 7.12> In this figure phase crossover frequency is about 15.3[rad/sec] and corresponding gain is about -14.2[dB] => Gain margin is about 14.2[dB].

□

<Fig. 7.12> 예제7.4의 이득여유 구하기

CemTool Commands :

[mag,phase]=bode(num,den,w);

[Gm,Pm,Wpc,Wgc]=margin(mag,phase,w);

Here, Gm and Pm are gain margin and phase margin. Wpc and Wgc are phase and gain crossover frequency. Gm is expressed as absolute value of gain margin =>

Calculated one more time to express in dB.

Ex. 7.5 : Obtain gain margin, phase margin, phase and gain crossover frequency for the system in Ex. 7.4 using ‘margin' command.

풀이 : CemTool commands

num=10; den=conv([1 0],conv([0.02 1],[0.2 1]));

w=logspace(0,2,100);

[mag,phase]=bode(num,den,w);

[Gm,Pm,Wpc,Wgc]=margin(mag,phase,w);

Gm=20*log(Gm); // 이득여유를 [dB]로 바꿈 Results :

Gm = 14.8080[dB], Pm = 31.7147[◦], Wpc = 15.8114[rad/sec], Wgc = 6.2179[rad/sec]

These numerical values are quite accurate and very similar to the results of Ex. 7.4.

Small discrepancies are due to the error in the reading the graphs.

□

7.3.3 Relation with Time Domain Characteristics Constants

Frequency Domain and Time Domain Characteristics Constants have specific relations.

=> Should be understood as the basic knowledge for the controller design.

In most controller design, performance target is given as the Time Domain Characteristics Constants => The relations should be used to obtain Frequency Domain Characteristics Constants from Time Domain Characteristics Constants =>

Controller can be designed in the frequency domain.

The relations can not be clearly expressed for the general system but can be expressed in the standard 1st and 2nd order system.

r + G(s) -

y

<Fig. 7.13> Unit Feedback System

1^st Order System

Open Loop Transfer Function : Unit feedback system in <Fig. 7.13>

G(s) = 1

T_cs^, T_c> 0 Stability Margin :

- Gain Margin GM = ∞ [dB]

- Phase Margin PM = 90 [◦]

- Phase crossover frequency ω_pc는 없음.

- Gain crossover frequency ω_gc = 1

T_c ^[rad/sec]

Closed Loop System : Standard 1st order system with time constantT_c T(s) = 1

T_cs+1

Frequency Response Characteristics of the Closed Loop System : - Band-pass gain G_B = 0 [dB]

- Bandwidth ω_B = 1

T_c ^[rad/sec]

- Cutoff frequency ω_c = ω_B = 1

T_c ^[rad/sec]

- Resonant frequencyω_r and Resonant peakM_rare can not be defined.

<주> In the frequency response characteristics of the standard 1st order system, Gain crossover frequency, Bandwidth and Gain crossover frequency are identical as inverse of time constant. Other characteristics have no relation with time domain characteristics.

2^nd Order System

Open loop transfer function :

G(s) = ω²_n

s(s+2ζ ω_n) , ζ,ω_n≥ 0 (7.7) Closed loop system : Standard 2nd Order System

T(s) = ω²_n s²+2ζ ω_ns+ω²_n Stability Margin : (익힘문제 7.4 참조)

-Gain Margin GM = ∞ [dB] (7.8)

-Phase Margin PM = tan ^{- 1} 2ζ

1+4ζ⁴-2ζ² [◦]

-Phase Crossover Frequency ω_pc = ∞ [rad/sec]

-Gain Crossover Frequency ω_gc = ω_n 1+4ζ²-2ζ²[rad/sec]

<주> Gain Margin와 Phase Crossover Frequency are all ∞ and independent of system coefficients. Phase margin and gain crossover frequency are dependent only on ζ.

Phase Margin vs. damping ratio curve <Fig. 7.14> : Expression of Eq. (7.8) PM ≈

{

^{10060+50 (ζ, ζ}-0.6), 0.6 ≤^{0 ≤ζ≤0.6}ζ≤0.8

70+25 (ζ-0.8), 0.8 ≤ζ≤1.2 (7.9)

<Fig. 7.14> Phase margin vs. damping ratio curve

Max. overshoot M_p and Resonant peakM_r for the closed loop system : Function of the damping ratio ζ that can be obtained from Eq. (5.10) and (7.1).

M_p = e ^{-π ζ/ 1-ζ2}, 0≤ζ< 1 M_r = 1

2ζ 1-ζ², 0≤ζ≤0.707

In this equations,M_p and M_r all monotonically decrease with respect to ζ => M_p ↑→

M_r ↑. Analytical relation betweenM_p and M_r without ζ is hard to obtain => Obtain ζ corresponding to given M_p using <Fig. 7.15> then calculate M_r from the ζ. For example, if the performance target is overshoot ≤ 10% (M_p≤ 0.1 ), then corresponding damping ratio ζ≥ 0.59 from <Fig. 7.15> and M_r≤ 1.05 .

<Fig. 7.15> Overshoot vs. damping ratio curve

Standard 2nd order system with constant damping ratio ζ= 0.5 , Rising time, resonant frequency and bandwidth can be obtained as follows using Eq. (5.8), (7.1), (7.2).

t_r = 1.8ω_n , ζ= 0.5 ω_r = ω_n 1-2ζ² = ω_n

2

|

ζ= 0.5

ω_B = ω_n

(

1 - 2ζ²+ 4ζ⁴- 4ζ²+2

)

^{1/ 2} = 1.27ω_n

|

_{ζ= 0.5}

<주> For the constant damping ratio, resonant frequenct and bandwidth are proportional to each other and inversely proportional to rising time. So, rising time ↓ =>

resonant frequency and bandwidth ↑.