Frequency Response Techniques

ผูเรียบเรียง ธเนศ เคารพาพงศ, อํานวย สิทธิเจริญชัย แกไข 28 ธันวาคม 2545 หนา 1 240-371 : Frequency Response Techniques

Chapter 10

Frequency Response Techniques

1


Introduction

1 Bode diagram2 Nyquist diagram3 Relation between Closed- and Open-Loop Frequency Responses4 Stability in frequency5 Example


Sinusoidal frequencyresponse:a. system;b. transfer function;c. input and outputwaveforms

Concept of Frequency Response


System with sinusoidal input

r t =Asin t

R s =A

s22


C s =G s A

s22 G s =

∏i=1

M

sz i

∏j =1

N

s p j

G j =

∏i=1

M

z i j

∏j =1

N

p j j

=∣G j ∣e j

=arctanℑ [G j ]

ℜ [G j ]


A

s22

G s =a 1

s j

a 2

s − j ...

=−A

2jG − j

a 2=[ A

s j ]s= j

a1=[ A

s− j ]s=− j

=A

2jG j


C s =−A

2j

1s j

G − j A

2j

1s− j

G j

G j =∣G j ∣ e j

G − j =∣G − j ∣e − j =∣G j ∣e − j

c t =A

2j∣G j ∣ − e − j t e − j

e j t e j

=A ∣G j ∣ e j t −e − j t

2j


c t =A∣G j ∣sint

จากสมการจะเห็นไดวาเอาตพุตของระบบเชิงเสน(Linear system) จะมีขนาด(Gain) และ มุมเฟสที่เปลี่ยนไปจากอินพุต


Bode Diagram

G s =K

sk

∏i=1

M

sz i

∏j =1

N

sp j

Bode Diagram 1. Gain diagram – Plot between gain (dB) and frequency on semi-log diagram. 2. Phase diagram – Plot between phase (degree or radian) and frequency on semi-log diagram.


Gain diagram

dB =20 log ∣G j ∣

∣G j ∣=K ∣ j z 1∣ ... ∣ j z M ∣

∣ j k ∣ ∣ j p 1∣ ... ∣ j p N ∣

dB = 20log K 20log z 12 2 ... 20log z M

2 2

− 20log k − 20log p 12 2− ...− 20log p N

2 2


∡G j =∡ j z 1 ...∡ j z M

−∡ j p 1− ...−∡ j p N

1=∡ s z 1=arctan j

z 1

M=∡ s z M =arctan j

z M

⋮ ⋮

Phase diagram


1=∡ s p 1=arctan j

p 1

N=∡ s p 1=arctan j

p N

⋮ ⋮


∡ G j = 1 . . . M− 1− . . .−

N

การพลอตนั้นจะทําอยูในชวงความถี่() ที่จะทําการศึกษา

d B = 20log K 20log z 12 2 ... 20log z M

2 2

− 2olog k − 20log p 12 2− 20log p N

2 2


Frequency response plots for G(s) =1/(s +2): separate magnitude and phase


Bode plots of (s + a) : a. magnitude plot b. phase plot.

Asymptotic Approximation


Asymptotic and actual normalized and scaled magnitude response of (s + a)


Asymptotic and actual normalized and scaled phase response of (s +a)


Normalized and scaled Bode plots fora. G(s) = s; b. G(s) = 1/s;


Normalized and scaled Bode plots forc. G(s) = (s + a); d. G(s) = 1/(s + a)


Example (Nise) Draw the Bode plot for the system shown below

G s =K s 3

s s 1 s 2


Bode log- magnitude plot : a. components; b. composite


Bode phase plota. components; b. composite


Bode asymptotesfor normalizedand scaled G(s) =

a. magnitude;b. phase


Normalized and scaledlog-magnitude response for


Scaled phase response for


Normalized and scaled log magnitude response for


Scaled phase response for


Bode magnitudeplot for G(s) =(s + 3)/[(s + 2)(s2 + 2s + 25)]:a. components;b. composite


Polar plot


G j =K z 1 j ...

j k p 1 j ...


Introduction to the Nyquist Criterion

The NyC relates the stability of a closed-loop system toopened-loop frequency response and open loop pole locationThus knowledge of the opened-loop system's frequency res-ponse yields information about the stability of the closed-loop system.


Closed- loop control system


Mapping contour A through function F(s) to contour B


Examples of contour mappingZero outside the contour


Pole outside the contour


Zero inside the contour


Pole inside the contour


Pole and zero inside the contour


Vector representation of mapping


Contour enclosingright half-plane todetermine stability


Nyquist Stability Criterion

If a contour A that encircles the entire right half-plane ismapped through G(s)H(s), then the number of closed-loop poles, Z, in the right half-plane equals the number of open-loop poles, P, that are in the right half plane minus the number of counterclockwise revolutions, N, around -1 of the mapping;that is, Z=P-N. The mapping is called the Nyquist diagram, orNyquist plot of G(s)H(s).


Mapping examples:a. contour does notenclose closed- looppoles;b. contour doesenclose closed- looppoles


Detouring around open-loop poles:a. poles on contour;b. detour right;c. detour left


a. Contour;b. Nyquist diagram


Stability via the Nyquist Diagram

If the Nyquist path in s-plane encircles Z zeros and P polesof 1+G(s)H(s) and does not pass through any poles or zerosof 1+G(s)H(s) as a representative point s moves in the clock-wise direction along the Nyquist path, then the correspondingcontour in the G(s)H(s)-plane encircles the -1+j0 point N=Z-Ptime in clockwise direction (Negative values of N imply counter-clockwise encirclements)


1. There is no encirclement of the -1+j0 point. This impiles that the system is stable if there are no poles of G(s)H(s) in the right-half s-plane, otherwise the system is unstable.

2. There is a counterclockwise encirclement or encirclements of the -1+j0 point. In this case, the system is stable if the number of counterclockwise encirclements is the same as the nmber of polse of G(s)H(s) in the right-half s-plane, otherwise the system is unstable.

3. There is a clockwise encirclement or encirclements of the -1+j0 point. In this case, the system is unstable.


DemonstratingNyquist stability:a. system;b. contour;c. Nyquist diagram


a. Contour ;b. Nyquist diagram


a. Contour and root locus of system thatis stable for small gain and unstablefor large gain;b. Nyquist diagram


a. Contour and root locus of system that is unstable for small gain and stable for large gain;b. Nyquist diagram


a. Portion of contour to be mapped for Example 10.7b. Nyquist diagram of mapping of positive imaginary axis


Nyquist diagramshowing gain andphase margins

Gain Margin and Phase Margin


Gain and phase margins on the Bode diagrams


Relation between Closed-Loop and Open-Loop Frequency Response

Constant M cycles and Constant N cycles

T s =G s

1G s

R(s) C(s)G(s)+

-

T j =G j

1G j

Close-loop transfer function and frequency response


Let G j =P jQ

T j =P jQ

P 12Q2

M2= ∣T

2 j ∣=

P jQ

P 12Q2

P M

2

M2−1

2

Q2=

M2

M2−12


Constant M circles


=arctan Q

P −arctan Q

P 1

tan A±B =tan A±tan B

1±tan A tan B

N = tan =

Q

P −

Q

P 1

1 Q

P

Q

P 1


N =Q

P2 P Q

2

P2 P Q

2

Q

N= 0

P 12

2

Q −1

2N 2

=N

21

4N2


Constant N circles


Nyquist diagram forExample 10.11 andconstant M and Ncircles


Nichols Chart


Typical unnormalizedand unscaled Bodelog-magnitude plotsshowing the value ofstatic error constants:a. Type 0;b. Type 1;c. Type 2

Steady-State Error Characteristics from Frequency Response

Frequency Response Techniques

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