Bài giảng Control system design - Chapter VIII: Frequency Response Methods - Nguyễn Công Phương

Nguyễn Công Phương CONTROL SYSTEM DESIGN Frequency Response Methods Contents I. Introduction II. Mathematical Models of Systems III. State Variable Models IV. Feedback Control System Characteristics V. The Performance of Feedback Control Systems VI. The Stability of Linear Feedback Systems VII. The Root Locus Method VIII.Frequency Response Methods IX. Stability in the Frequency Domain X. The Design of Feedback Control Systems XI. The Design of State Variable Feedback Systems XII. Robust Control Systems XIII.Digital Control Systems sites.google.com/site/ncpdhbkhn 2 Frequency Response Methods 1. Introduction 2. Frequency Response Plots 3. Performance Specifications in the Frequency Domain 4. Frequency Response Methods Using Control System Software sites.google.com/site/ncpdhbkhn 3 Introduction (1) • The frequency response of a system: the steady – state response of the system to a sinusoidal input signal. • The sinusoid is a unique input signal, and the resulting output signal for a linear system, as well as signals throughout the system, is sinusoidal in the steady state. • It differs from the input waveform only in amplitude and phase angle. sites.google.com/site/ncpdhbkhn 4 Introduction (2) =ms() = ms () T( s ) n q( s ) + ∏(s p i ) i=1 Aω rt()= A sinω t → Rs () = s2+ ω 2 k kα s + β →=Ys() RsTs ()() =1 ++ ... n + + +2 + ω 2 sp1 spsn −− − αs + β  →ytke( ) =p1 t ++ ... kepn t + L 1   1 n s2+ω 2  α+ β  = = −1 s ysteady− state ( t ) lim yt ( ) lim L   t→∞ t →∞ s2+ω 2  =ATj(ω )sin( ωφφ t + ), =∠ Tj ( ω ) sites.google.com/site/ncpdhbkhn 5 Frequency Response Methods 1. Introduction 2. Frequency Response Plots 3. Performance Specifications in the Frequency Domain 4. Frequency Response Methods Using Control System Software sites.google.com/site/ncpdhbkhn 6 Frequency Response Plots (1) Gj(ω )= Gs () = Re[( Gj ω )] + Im[( Gj ω )] s= j ω =R()ω + jX () ω =Gj()ω ejφ( ω ) = Gj ()[()] ω ∠ φ ω =RX2(ω ) + 2 () ω ∠ {tan[− 1 XR ( ω )/ ( ω )]} sites.google.com/site/ncpdhbkhn 7 Frequency Response Plots (2) Ex. 1 1 1V ( s ) + + VsV()= () s = Is () = . 1 R 2 capacitor Cs Cs 1 + R V( s ) V( s ) Cs 1 2 V( s ) 1 C →G( s ) =2 = − − + V1( s ) RCs 1 1 Positive ω →Gj(ω ) = Gs () = 0.5 s= j ω ω + Negative ω j( RC ) 1 0.4 1 1 0.3 =, ω = j(ω / ω )+ 1 1 RC 0.2 1 0.1 ) ω 0 X( 1ω / ω -0.1 = − j 1 +ωω2 + ωω 2 -0.2 1(/)1 1(/) 1 -0.3 -0.4 1 −1 = ∠tan ( − ω / ω ) -0.5 + ω ω 2 1 1 ( /1 ) 0 0.2 0.4 0.6 0.8 1 R( ω) sites.google.com/site/ncpdhbkhn 8 Frequency Response Plots (3) Semilog plots of the magnitude (in decibels) and phase (in degrees) of a transfer function versus frequency T= 20log T T = T φ →  dB 10 φ = = φ φ φ TTTT123... (T 1 1)(T 2 2)(T 3 3 )... = () φ+ φ + φ + T1 T 2 T 3 ... 1 2 3 ... 20logT= 20log T + 20log T + 20log T + ... →  10 10 1 10 2 10 3 φ= φ + φ + φ +  1 2 3 ... sites.google.com/site/ncpdhbkhn 9 Frequency Response Plots (4) 2  ± jω  j2 ζω j ω  K()1 j ω 1 + 1 +1 +  ...   ω ω  z1 k  k  T(ω ) = 2  jω  j2 ζω j ω  1+   1 +2 +    ... p ω ω  1  n n   K : gain 1 1 1 : poleat theorigin :simplepole 2 : quadraticpole jω jω j2ζ ω j ω  1+ 1+2 + p ω ω  1 n n  2 jω j2ζ ω j ω  jω : zeroat theorigin 1+ :simplezero 1+1 + :quadraticzero z ω ω  1 k k  sites.google.com/site/ncpdhbkhn 10 Frequency Response Plots (5) T= 20log K T(ω ) =K →  dB 10 φ = 0 φ TdB 20log K 10 0 0.1 1 10 100 ω 0.1 1 10 100 ω sites.google.com/site/ncpdhbkhn 11 Frequency Response Plots (6) = − ω 1 TdB 20log 10 T(ω ) = →  jω φ = − 90 o TdB 20 φ 0 0o 0.1 1 10 ω 0.1 1 10 ω −20 −90 o sites.google.com/site/ncpdhbkhn 12 Frequency Response Plots (7) = ω TdB 20log 10 T(ω ) =j ω →  φ = 90 o φ o TdB 90 20 0 0o 0.1 1 10 ω 0.1 1 10 ω −20 sites.google.com/site/ncpdhbkhn 13 Frequency Response Plots (8)  ω = − + j TdB 20log10 1 1  p1 T(ω ) = →  jω +  − ω  1 φ =tan 1  −  p1   p1  TdB 5 0 -5 -10 -15 -20 ω -25 0.1 p1 p1 10 p1 sites.google.com/site/ncpdhbkhn 14 Frequency Response Plots (9)  ω = − + j TdB 20log10 1 1  p1 T(ω ) = →  jω +  − ω  1 φ =tan 1  −  p1   p1  φ 0.1 p1 p1 10 p1 100 p1 0 ω -10 -20 -30 -40 -50 -60 -70 -80 -90 -100 sites.google.com/site/ncpdhbkhn 15 Frequency Response Plots (10)  ω = + j TdB 20log10 1 jω  z1 T(ω )= 1 + →  z ω  1 φ = −1  tan    z1  TdB 25 20 15 10 5 0 ω -5 0.1 z1 z1 10 z1 sites.google.com/site/ncpdhbkhn 16 Frequency Response Plots (11)  ω = + j TdB 20log10 1 jω  z1 T(ω )= 1 + →  z ω  1 φ = −1  tan   φ  z1  100 90 80 70 60 50 40 30 20 10 ω 0 0.1 z1 z1 10 z1 100 z1 sites.google.com/site/ncpdhbkhn 17 Frequency Response Plots (12)  2 j2ζ ω j ω  T =−20log 1 +2 + dB 10 ω ω  1  n n  T(ω ) = →  ζ ω ω  2 j2 2 j  − j2ζ ω / ω  1+ +   φ = − tan 1 2 n  ω ω  − ω2 ω 2  n n  1 / n  20 ζ = TdB 2 0.05 10 ζ 2 = 0.2 ζ = 0 2 0.4 -10 ζ 2 = 0.707 ζ =1.5 -20 2 -30 ω -40 ω ω ω ω 0.1 n n 10 n 100 n sites.google.com/site/ncpdhbkhn 18 Frequency Response Plots (13)  2 j2ζ ω j ω  T =−20log 1 +2 + dB 10 ω ω  1  n n  T(ω ) = →  ζ ω ω  2 j2 2 j  − j2ζ ω / ω  1+ +   φ = − tan 1 2 n  ω ω  − ω2 ω 2  n n  1 / n  ω ω ω ω φ 0.1 n n 10 n 100 n 0 ω -20 ζ = -40 2 1.5 -60 ζ = -80 2 0.707 -100 ζ = 2 0.4 -120 ζ = -140 2 0.2 -160 ζ = 2 0.05 -180 sites.google.com/site/ncpdhbkhn 19 Frequency Response Plots (14)  2 j2ζ ω j ω  T =20log 1 +2 + 2 dB 10 ω ω  j2ζ ω j ω   n n  T(ω )= 1 +2 +  →  ω ω  n n  − j2ζ ω / ω  φ = tan 1 2 n  − ω2 ω 2   1 / n  40 TdB ζ = 30 2 1.5 ζ = 20 2 0.707 10 ζ = 2 0.4 0 -10 ζ = ζ = 2 0.05 2 0.2 ω -20 ω ω ω ω 0.1 n n 10 n 100 n sites.google.com/site/ncpdhbkhn 20 Frequency Response Plots (15)  2 j2ζ ω j ω  T =20log 1 +2 + 2 dB 10 ω ω  j2ζ ω j ω   n n  T(ω )= 1 +2 +  →  ω ω  n n  − j2ζ ω / ω  φ = tan 1 2 n  − ω2 ω 2   1 / n  180 φ ζ 2 = 0.05 160 ζ 2 = 0.2 140 ζ 2 = 0.4 120 100 80 ζ 2 = 0.707 60 ζ = 2 1.5 40 20 ω 0 ω ω ω ω 0.1 n n 10 n 100 n sites.google.com/site/ncpdhbkhn 21 Frequency Response Plots (16) 1 N N jω  K (jω ) N 1+  (jω ) z  20N dB/decade 20log 10 K 1 ω 20N dB/decade −20N dB/decade 1 ω z ω ω 90 N o 90 N o ω 0o 0o ω ω −90 N o z z 10 z ω 10 sites.google.com/site/ncpdhbkhn 22 Frequency Response Plots (17) 1 1 N 2 N 2 N ζ ω ω   ζ ω ω   ω  j2 1 j j2 2 j + j 1+ +    1+ +    1  ω ω  ω ω  p  n n   k k   ω k p 40N dB/decade ω ω −20N dB/decade −40N dB/decade ω n ω o 180 N ω k ω ω p 10 k 10 k 10 p 10 p 0o ω o ω 0 0o ω ω 10 ω ω − o − o n n n 180 N 90 N 10 sites.google.com/site/ncpdhbkhn 23 Frequency Response Plots (18) Ex. 2 500 jω 10 jω T(ω ) = = (jω+ 5)( j ω + 10) (1+jω /5)(1 + j ω /10) 10 jω T(ω ) = (1+jω /5)(1 + j ω /10) 10 jω − − = 90o− tan 1 (ω /5) − tan 1 ( ω /10) 1+jω /51 + j ω /10  1 1 T=+20log 10 20log j ω + 20log + 20log  dB 10 10 101+jω /5 10 1 + j ω /10 →   o− 11 − 1  1  φ =90 + tan + tan    ω/5  ω /10  sites.google.com/site/ncpdhbkhn 24 Frequency Response Plots (19) Ex. 2 500 jω 10 jω T(ω ) = = (jω+ 5)( j ω + 10) (1+jω /5)(1 + j ω /10) 1 1 T=+20log 10 20log j ω + 20log + 20log dB 10 10 101+jω /5 10 1 + j ω /10 40 TdB 30 20log10 10 K 20 20log K 10 10 ω 0 ω -10 -20 0.1 1 5 10 20 100 200 500 sites.google.com/site/ncpdhbkhn 25 Frequency Response Plots (20) Ex. 2 500 jω 10 jω T(ω ) = = (jω+ 5)( j ω + 10) (1+jω /5)(1 + j ω /10) 1 1 T=+20log 10 20log j ω + 20log + 20log dB 10 10 101+jω /5 10 1 + j ω /10 40 TdB 30 20log10 10 (jω ) N 20 10 ω 20N dB/decade 20log 10 j ω 0 1 ω -10 -20 0.1 sites.google.com/site/ncpdhbkhn1 5 10 20 100 20026 500 Frequency Response Plots (21) Ex. 2 500 jω 10 jω T(ω ) = = (jω+ 5)( j ω + 10) (1+jω /5)(1 + j ω /10) 1 1 T=+20log 10 20log j ω + 20log + 20log dB 10 10 101+jω /5 10 1 + j ω /10 40 TdB 30 20log10 10 1 20 N jω  1 +  p  10 ω 20log 10 j ω p 0 ω −20N dB/decade 1 -10 20log 10 1+ jω / 5 -20 0.1 sites.google.com/site/ncpdhbkhn1 5 10 20 100 20027 500 Frequency Response Plots (22) Ex. 2 500 jω 10 jω T(ω ) = = (jω+ 5)( j ω + 10) (1+jω /5)(1 + j ω /10) 1 1 T=+20log 10 20log j ω + 20log + 20log dB 10 10 101+jω /5 10 1 + j ω /10 40 TdB 30 20log10 10 1 20 N jω  1 +  p  10 ω 20log 10 j ω p 0 ω 1 1 20log −20N dB/decade -10 20log 10 + ω 10 1+ jω / 5 1j /10 -20 0.1 sites.google.com/site/ncpdhbkhn1 5 10 20 100 20028 500 Frequency Response Plots (23) Ex. 2 500 jω 10 jω T(ω ) = = (jω+ 5)( j ω + 10) (1+jω /5)(1 + j ω /10) 1 1 T=+20log 10 20log j ω + 20log + 20log dB 10 10 101+jω /5 10 1 + j ω /10 40 Slope = 0 + 20 – 20 TdB 30 20log10 10 20 10 ω 20log 10 j 0 ω 1 1 20log -10 20log 10 1+ jω /10 10 1+ jω / 5 -20 0.1 1 sites.google.com/site/ncpdhbkhn5 10 20 100 200 500 29 Frequency Response Plots (24) Ex. 2 500 jω 10 jω T(ω ) = = (jω+ 5)( j ω + 10) (1+jω /5)(1 + j ω /10) o− 11 − 1  1  φ =90 + tan + tan   ω/5  ω /10  φ 90 o 90 ω N (j ) 60 40 20 ω 0 90 No -20 -40 -60 ω -90 0.1 0.5 1 5 10 20 50 100 200 500 sites.google.com/site/ncpdhbkhn 30 Frequency Response Plots (25) Ex. 2 500 jω 10 jω T(ω ) = = (jω+ 5)( j ω + 10) (1+jω /5)(1 + j ω /10) o− 11 − 1  1  φ =90 + tan + tan   ω/5  ω /10  φ 90 o 90 1 N jω  60 1+  p  40 20 ω 0 p -20 10 p 10 p − 1  -40 tan 1 0o ω ω  -60 /5  o −90 N -90 0.1 0.5 1 5 10 20 50 100 200 500 sites.google.com/site/ncpdhbkhn 31 Frequency Response Plots (26) Ex. 2 500 jω 10 jω T(ω ) = = (jω+ 5)( j ω + 10) (1+jω /5)(1 + j ω /10) o− 11 − 1  1  φ =90 + tan + tan   ω/5  ω /10  φ 90 o 90 1 N jω  60 1+  p  40 20 ω 0 p -20   10 p 10 p −1 1 − 1  tan   -40 tan 1 ω /10  0o ω ω  -60 /5  o −90 N -90 0.1 0.5 1 5 10 20 50 100 200 500 sites.google.com/site/ncpdhbkhn 32 Frequency Response Plots (27) Ex. 2 500 jω 10 jω T(ω ) = = (jω+ 5)( j ω + 10) (1+jω /5)(1 + j ω /10) o− 11 − 1  1  φ =90 + tan + tan   ω/5  ω /10  φ 90 o 90 60 40 20 0 ω -20 −1 1  − 1  tan   -40 tan 1 ω /10  ω  -60 /5  -90 0.1 0.5 1 sites.google.com/site/ncpdhbkhn5 10 20 50 100 200 500 33 Frequency Response Plots (28) Ex. 3 1000(jω + 20) 8(1+ jω / 20) T(ω ) = = ωω+2 ω 2 + ω +  ωω+2 + ω + ω 2  jj( 5) ( j ) 40 j 100  jj(1 /5) 1 j 4/10 ( j /10)  40 TdB 20 20log10 8 ω 0 -20 K -40 20log 10 K -60 ω -80 -100 0.1 1 5 10 20 50 100 sites.google.com/site/ncpdhbkhn 34 Frequency Response Plots (29) Ex. 3 1000(jω + 20) 8(1+ jω / 20) T(ω ) = = ωω+2 ω 2 + ω +  ωω+2 + ω + ω 2  jj( 5) ( j ) 40 j 100  jj(1 /5) 1 j 4/10 ( j /10)  40 jω TdB 20log 1 + 10 20 20 20log10 8 ω 0 -20 jω  N 1 +  z  -40 20N dB/decade -60 z ω -80 -100 0.1 1 5 10 20 50 100 sites.google.com/site/ncpdhbkhn 35 Frequency Response Plots (30) Ex. 3 1000(jω + 20) 8(1+ jω / 20) T(ω ) = = ωω+2 ω 2 + ω +  ωω+2 + ω + ω 2  jj( 5) ( j ) 40 j 100  jj(1 /5) 1 j 4/10 ( j /10)  40 jω TdB 20log 1 + 10 20 20 20log10 8 ω 0 1 20log 10 -20 jω 1 (jω ) N -40 1 ω -60 −20N dB/decade -80 -100 0.1 1 5 10 20 50 100 sites.google.com/site/ncpdhbkhn 36 Frequency Response Plots (31) Ex. 3 1000(jω + 20) 8(1+ jω / 20) T(ω ) = = ωω+2 ω 2 + ω +  ωω+2 + ω + ω 2  jj( 5) ( j ) 40 j 100  jj(1 /5) 1 j 4/10 ( j /10)  40 ω T 1 + j dB 20log 20log10 1 10 (1+ jω / 5) 2 20 20 20log10 8 ω 0 1 20log 10 -20 jω 1 N jω  1+  p  -40 p -60 ω −20N dB/decade -80 -100 0.1 1 5 10 20 50 100 sites.google.com/site/ncpdhbkhn 37 Frequency Response Plots (32) Ex. 3 1000(jω + 20) 8(1+ jω / 20) T(ω ) = = ωω+2 ω 2 + ω +  ωω+2 + ω + ω 2  jj( 5) ( j ) 40 j 100  jj(1 /5) 1 j 4/10 ( j /10)  40 ω T 1 + j dB 20log 20log10 1 10 (1+ jω / 5) 2 20 20 20log10 8 ω 0 1 20log 1 10 ω N -20 j ζ ω ω  2  +j2 2 + j  1 ω ω  k k   -40 ωk ω -60 −40N dB/decade -80 1 20log 10 1+jω 4 /10 + ( j ω /10) 2 -100 0.1 1 5 10 20 50 100 sites.google.com/site/ncpdhbkhn 38 Frequency Response Plots (33) Ex. 3 1000(jω + 20) 8(1+ jω / 20) T(ω ) = = ωω+2 ω 2 + ω +  ωω+2 + ω + ω 2  jj( 5) ( j ) 40 j 100  jj(1 /5) 1 j 4/10 ( j /10)  40 TdB 20 ω 0 -20 -40 -60 -80 -100 0.1 Frequency1 Response - 5 10 20 50 100 39 sites.google.com/site/ncpdhbkhn Frequency Response Plots (34) Ex. 3 1000(jω + 20) 8(1+ jω / 20) T(ω ) = = ωω+2 ω 2 + ω +  ωω+2 + ω + ω 2  jj( 5) ( j ) 40 j 100  jj(1 /5) 1 j 4/10 ( j /10)  φ 90 50 ω 0 -50 N jω  1 +  -90 z  o 90 N -180 -200 0o z z 10 z ω 10 -270 -300 -360 0.1 0.2 0.5 sites.google.com/site/ncpdhbkhn1 2 10 50 100 200 40 500 Frequency Response Plots (35) Ex. 3 1000(jω + 20) 8(1+ jω / 20) T(ω ) = = ωω+2 ω 2 + ω +  ωω+2 + ω + ω 2  jj( 5) ( j ) 40 j 100  jj(1 /5) 1 j 4/10 ( j /10)  φ 90 50 ω 0 -50 1 -90 (jω ) N -180 ω -200 −90 No -270 -300 -360 0.1 0.2 0.5 sites.google.com/site/ncpdhbkhn1 2 10 50 100 200 41 500 Frequency Response Plots (36) Ex. 3 1000(jω + 20) 8(1+ jω / 20) T(ω ) = = ωω+2 ω 2 + ω +  ωω+2 + ω + ω 2  jj( 5) ( j ) 40 j 100  jj(1 /5) 1 j 4/10 ( j /10)  φ 90 50 ω 0 -50 1 N -90 jω  1+  p  p -180 p 10 p 10 -200 0o ω − o 90 N -270 -300 -360 0.1 0.2 0.5 sites.google.com/site/ncpdhbkhn1 2 10 50 100 200 42 500 Frequency Response Plots (37) Ex. 3 1000(jω + 20) 8(1+ jω / 20) T(ω ) = = ωω+2 ω 2 + ω +  ωω+2 + ω + ω 2  jj( 5) ( j ) 40 j 100  jj(1 /5) 1 j 4/10 ( j /10)  φ 90 50 ω 0 -50 1 N -90 ζ ω ω  2  +j2 2 + j  1 ω ω  k k   ωk ω ω 10 k 10 k -180 -200 0 o ω −180 N o -270 -300 -360 0.1 0.2 0.5 sites.google.com/site/ncpdhbkhn1 2 10 50 100 200 43 500 Frequency Response Plots (38) Ex. 4 Find the transfer function from the Bode plot? 40 Slope = 0 H dB ω N (j ) 30 20 20N dB/decade 10 1 ω jω 0 ω -10 -20 0.1 1 5 10 20 100 200 500 sites.google.com/site/ncpdhbkhn 44 Frequency Response Plots (39) Ex. 4 Find the transfer function from the Bode plot? 40 Slope = 0 H dB K 30 20log10 10 20 20log 10 K 10 ω jω 0 ω -10 -20 0.1 1 5 10 20 100 200 500 sites.google.com/site/ncpdhbkhn 45 Frequency Response Plots (40) Ex. 4 Find the transfer function from the Bode plot? 40 H Slope = 0 1 dB N jω  30 20log10 10 1 +  p  p 20 ω −20N dB/decade 10 jω 0 ω 1 -10 20log 10 1+ jω / 5 -20 0.1 1 5 10 20 100 200 500 sites.google.com/site/ncpdhbkhn 46 Frequency Response Plots (41) Ex. 4 10 jω Find the transfer function from the Bode plot? T(ω ) = (1+jω /5)(1 + j ω /10) 40 H Slope = 0 1 dB N jω  30 20log10 10 1+  p  p 20 ω −20N dB/decade 10 1 ω 20log 10 j 1+ jω /10 0 ω 1 -10 20log 10 1+ jω / 5 -20 0.1 1 5 10 20 100 200 500 sites.google.com/site/ncpdhbkhn 47 Frequency Response Methods 1. Introduction 2. Frequency Response Plots 3. Performance Specifications in the Frequency Domain 4. Frequency Response Methods Using Control System Software sites.google.com/site/ncpdhbkhn 48 Performance Specifications in the Frequency Domain (1) ω= ω − ζζ2 < resonant n 1 2 , 0.707 −1 =ω =− ζ ζ2 ζ < Mpω G( j r )( 2 12) , 0.707 20 ζ = TdB 2 0.05 10 ζ 2 = 0.2 ζ = 0 2 0.4 1 G(ω ) = 2 -10 j2ζ ω j ω  ζ = 0.707 1+2 +   2 ω ω ζ = n n  -20 2 1.5 -30 ω -40 ω ω ω ω 0.1 n n 10 n 100 n sites.google.com/site/ncpdhbkhn 49 Performance Specifications in the Frequency Domain (2) −1 =ω =− ζ ζ2 ζ < Mpω G( j r )( 2 1 2) , 0.707 5.5 5 4.5 4 3.5 ω p M 3 2.5 2 1.5 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ζ −ζω = +n t ω + θ y( t ) 1 Be cos(1 t ) Resonant magnitudes should be relatively small: Mpω < 1.5, for example sites.google.com/site/ncpdhbkhn 50 Frequency Response Methods 1. Introduction 2. Frequency Response Plots 3. Performance Specifications in the Frequency Domain 4. Frequency Response Methods Using Control System Software sites.google.com/site/ncpdhbkhn 51 Frequency Response Methods Ex. Using Control System Software 500 s T( s ) = (s+ 5)( s + 10) sites.google.com/site/ncpdhbkhn 52