Frequency-Domain Analysis of Control System 2022-2-3
2022-2-3 1 Frequency-Domain Analysis of Control System
Frequency Response Frequency response is the anal ysis of the response of systemswhen subjected to a sinusoidal change in input When a linear system is subjected to a sinusoidal input its ultimate response is also a sustained sinusoidal wave, with the same frequency The figure below compares the output response of a system (solid line)with a sinusoidal input ( dashed line )disturbing the system 2022-2-3
2022-2-3 2 Frequency Response Frequency response is the analysis of the response of systemswhen subjected to a sinusoidal change in input. When a linear system is subjected to a sinusoidal input, its ultimate response is also a sustained sinusoidal wave, with the same frequency. The figure below compares the output response of a system (solid line) with a sinusoidal input (dashed line) disturbing the system
Phase shift 0.5 10 20 Tim A m plitude r atio =b/a This particular graph shows the response of the system: G(S)=3 5s+i to the sinusoidal input of: u= sin(t). The figure shows that the response of the system lags the input changes slightly this lag is known as the phase shift. The ratio of the amplitudes of the input and output sinusoids is known as the amplitude ratio Both the magnitude and the phase shift of a system will change ith the frequency of the input into the system 2022-2-3 3
2022-2-3 3 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 0 1 0 2 0 3 0 T im e O u t p u t P h a s e S h ift A B A m p l i t u d e R a t i o = B / A This particular graph shows the response of the system: to the sinusoidal input of: u = sin (t). The figure shows that the response of the system lags the input changes slightly, this lag is known as the phase shift. The ratio of the amplitudes of the input and output sinusoids is known as the amplitude ratio. Both the magnitude and the phase shift of a system will change with the frequency of the input into the system. G s s ( ) 3 5 1
If the general system represented by the following transfer function: G(3)=P(s)(where @(s)and P(s)are polynomials is terms of s)is subjected to a sinusoidal input of frequency a then the amplitude ratio of the resulting response will be given by: AR= modulus of@)=G(o), where GGo)is found of replacing s in g(byo. Similarly the phase shift will be given by: phase shift=zG(o). For a given sinusoidal input of a sin(ot+9), the output will be a sinusoid of: aG(jo)sin(ot+9+2G(o) Example l: Determine the phase angle and ar of the system S G(s)=-,Replace s by jo G()= JO 0-j Jo 0Jo 2022-2-3 4
2022-2-3 4 If the general system represented by the following transfer function: (where Q(s) and P(s) are polynomials is terms of s) is subjected to a sinusoidal input of frequency w, then the amplitude ratio of the resulting response will be given by: , where G(jw) is found of replacing s in G(s) by jw. Similarly the phase shift will be given by: . For a given sinusoidal input of , the output will be a sinusoid of: . G s Q s P s ( ) ( ) ( ) AR modulus of G(jw )= G(jw) phase shift G(jw) a sinwt aG(jw)sinwt G(jw) Example 1: Determine the phase angle and AR of the system 1/s. G s s s j G j j j j j j ( ) , ( ) 1 1 1 0 1 Replace by w, w w w w w w
Given a complex number of the form: y =a+jb, the modulus and argument of it are as follows √a2+b2,∠y=tani Therefore AR=G(jO) tan ∞=-90°orp=-tan- Example 2: Determine the phase angle and ar of the system 4 G(S)= when subjected to a +1 sinusoidal input r=2sin (3t+600) 2022-2-3 5
2022-2-3 5 Given a complex number of the form: y = a+jb, the modulus and argument of it are as follows: y a b y b a 2 2 1 , tan Therefore, AR G j o ( ) , tan tan w w 1 90 1 1 or Example 2: Determine the phase angle and AR of the system when subjected to a sinusoidal input R=2sin(3t+60 0). G s s ( ) 4 1