EE 3CL4, §8 1 / 77 Tim Davidson Transfer functions Frequency EE3CL4: Response Plotting the Introduction to Linear Control Systems freq. resp. Mapping Section 8: Frequency Domain Techniques Contours Nyquist’s criterion Ex: servo, P control Tim Davidson Ex: unst., P control Ex: unst., PD contr. Ex: RHP Z, P contr. McMaster University Nyquist’s Stability Criterion as a Design Tool Winter 2020 Relative Stability Gain margin and Phase margin Relationship to transient response
EE 3CL4, §8 2 / 77 Outline Tim Davidson 1 Transfer functions Transfer functions Frequency Frequency Response 2 Response Plotting the Plotting the freq. resp. 3 freq. resp. Mapping 4 Mapping Contours Contours Nyquist’s Nyquist’s criterion 5 criterion Ex: servo, P control Ex: servo, P control Ex: unst., P control Ex: unst., PD contr. Ex: unst., P control Ex: RHP Z, P contr. Ex: unst., PD contr. Nyquist’s Stability Ex: RHP Z, P contr. Criterion as a Design Tool Relative Stability 6 Nyquist’s Stability Criterion as a Design Tool Gain margin and Phase margin Relative Stability Relationship to transient response Gain margin and Phase margin Relationship to transient response
EE 3CL4, §8 4 / 77 Transfer Functions: Tim Davidson A Quick Review Transfer • Consider a transfer function functions Frequency � i ( s + z i ) Response G ( s ) = K � j ( s + p j ) Plotting the freq. resp. Mapping • Zeros: − z i ; Poles: − p j Contours Nyquist’s • Note that s + z i = s − ( − z i ) , criterion Ex: servo, P control • This is the vector from − z i to s Ex: unst., P control Ex: unst., PD contr. • Magnitude: Ex: RHP Z, P contr. Nyquist’s � i | s + z i | j | s + p j | = | K | prod. dist’s from zeros to s Stability | G ( s ) | = | K | Criterion as a � prod. dist’s from poles to s Design Tool Relative Stability Gain margin and Phase margin • Phase: Relationship to transient response ∠ G ( s ) = ∠ K + sum angles from zeros to s − sum angles from poles to s
EE 3CL4, §8 6 / 77 Frequency Response Tim Davidson Transfer functions Frequency • For a stable, linear, time-invariant (LTI) system, the Response steady state response to a sinusoidal input is Plotting the freq. resp. a sinusoid of the same frequency but possibly different Mapping magnitude and different phase Contours Nyquist’s criterion • Sinusoids are the eigenfunctions of convolution Ex: servo, P control Ex: unst., P control Ex: unst., PD contr. • If input is A cos( ω 0 t + θ ) Ex: RHP Z, P contr. and steady-state output is B cos( ω 0 t + φ ) , Nyquist’s Stability then the complex number B / Ae j ( φ − θ ) Criterion as a Design Tool is called the frequency response of the system at Relative Stability Gain margin and frequency ω 0 . Phase margin Relationship to transient response
EE 3CL4, §8 7 / 77 Frequency Response, II Tim Davidson Transfer functions Frequency Response Plotting the freq. resp. Mapping • If a stable LTI system has a transfer function G ( s ) , Contours then the frequency response at ω 0 is G ( s ) | s = j ω 0 Nyquist’s criterion Ex: servo, P control Ex: unst., P control • What if the system is unstable? Ex: unst., PD contr. Ex: RHP Z, P contr. Nyquist’s Stability Criterion as a Design Tool Relative Stability Gain margin and Phase margin Relationship to transient response
EE 3CL4, §8 9 / 77 Plotting the frequency response Tim Davidson Transfer functions • For each ω , G ( j ω ) is a complex number. Frequency Response • How should we plot it? Plotting the freq. resp. • G ( j ω ) = � � e j ∠ G ( j ω ) � � G ( j ω ) Mapping Contours � versus ω , and ∠ G ( j ω ) versus ω � � � G ( j ω ) Plot Nyquist’s criterion Ex: servo, P control �� � • Plot 20 log 10 � � G ( j ω ) versus log 10 ( ω ) , and Ex: unst., P control � Ex: unst., PD contr. ∠ G ( j ω ) versus log 10 ( ω ) Ex: RHP Z, P contr. Nyquist’s Stability • G ( j ω ) = Re � � � � G ( j ω ) + j Im G ( j ω ) Criterion as a Design Tool � �� � � � Plot the curve Re G ( j ω ) , Im G ( j ω ) on an “ x – y ” plot Relative Stability Gain margin and Phase margin � e j ∠ G ( j ω ) as ω changes (polar plot) � � � G ( j ω ) Equiv. to curve Relationship to transient response
EE 3CL4, §8 10 / 77 Polar plot, example 1 Tim Davidson Transfer Let’s consider the example of an RC circuit functions Frequency Response Plotting the freq. resp. Mapping Contours Nyquist’s criterion Ex: servo, P control Ex: unst., P control • G ( s ) = V 2 ( s ) 1 V 1 ( s ) = Ex: unst., PD contr. 1 + sRC Ex: RHP Z, P contr. Nyquist’s 1 • G ( j ω ) = 1 + j ω/ω 1 , where ω 1 = 1 / ( RC ) . Stability Criterion as a Design Tool ω/ω 1 Relative Stability 1 • G ( j ω ) = 1 +( ω/ω 1 ) 2 − j Gain margin and 1 +( ω/ω 1 ) 2 Phase margin Relationship to transient response 1 +( ω/ω 1 ) 2 e − j atan( ω/ω 1 ) 1 • G ( j ω ) = √
EE 3CL4, §8 11 / 77 Polar plot, example 1 Tim Davidson ω/ω 1 1 • G ( j ω ) = 1 +( ω/ω 1 ) 2 − j 1 +( ω/ω 1 ) 2 Transfer functions 1 +( ω/ω 1 ) 2 e − j atan( ω/ω 1 ) 1 • G ( j ω ) = √ Frequency Response Plotting the freq. resp. Mapping Contours Nyquist’s criterion Ex: servo, P control Ex: unst., P control Ex: unst., PD contr. Ex: RHP Z, P contr. Nyquist’s Stability Criterion as a Design Tool Relative Stability Gain margin and Phase margin Relationship to transient response
EE 3CL4, §8 12 / 77 Polar plot, example 2 Tim Davidson K Consider G ( s ) = s ( s τ + 1 ) . Transfer functions • Poles at origin and s = − 1 /τ . Frequency • To use geometric insight to plot polar plot, Response Plotting the K /τ rewrite as G ( s ) = freq. resp. s ( s + 1 /τ ) Mapping Contours � = K /τ � � • Then � G ( j ω ) | j ω | | j ω + 1 /τ | Nyquist’s criterion and ∠ G ( j ω ) = − ∠ ( j ω ) − ∠ ( j ω + 1 /τ ) Ex: servo, P control Ex: unst., P control Ex: unst., PD contr. Ex: RHP Z, P contr. Nyquist’s Stability Criterion as a Design Tool Relative Stability Gain margin and Phase margin Relationship to transient response
EE 3CL4, §8 13 / 77 K /τ Polar plot, ex. 2, G ( s ) = Tim Davidson s ( s + 1 /τ ) Transfer functions Frequency Response Plotting the freq. resp. Mapping Contours Nyquist’s criterion Ex: servo, P control Ex: unst., P control • When ω → 0 + , | G ( j ω ) | → ∞ , ∠ G ( j ω ) → − 90 ◦ from below Ex: unst., PD contr. Ex: RHP Z, P contr. Tricky Nyquist’s Stability − K ω 2 τ ω K • To get a better feel, write G ( j ω ) = Criterion as a ω 2 + ω 4 τ 2 − j ω 2 + ω 4 τ 2 Design Tool Hence, as ω → 0 + , G ( j ω ) → − K τ − j ∞ Relative Stability Gain margin and Phase margin • As ω increases, distances from poles to j ω increase. Relationship to transient response Hence | G ( j ω ) | decreases • As ω increases, angle from pole at − 1 /τ increases. Hence ∠ G ( j ω ) becomes more negative
EE 3CL4, §8 14 / 77 K /τ Polar plot, ex. 2, G ( s ) = Tim Davidson s ( s + 1 /τ ) Transfer functions Frequency Response Plotting the freq. resp. Mapping Contours Nyquist’s criterion Ex: servo, P control Ex: unst., P control Ex: unst., PD contr. Ex: RHP Z, P contr. √ • When ω = 1 /τ , G ( j ω ) = ( K /τ ) / � � e − j ( 90 ◦ + 45 ◦ ) ( 1 /τ )( 2 /τ ) Nyquist’s √ Stability 2 ) e − j 135 ◦ i.e., G ( j ω ) | ω = 1 /τ = ( K τ/ Criterion as a Design Tool Relative Stability • As ω approaches + ∞ , both distances from poles get large. Gain margin and Phase margin Hence | G ( j ω ) | → 0 Relationship to transient response • As ω approaches + ∞ , angle from − 1 /τ approaches − 90 ◦ from below. Hence ∠ G ( j ω ) approaches − 180 ◦ from below
EE 3CL4, §8 15 / 77 K /τ Polar plot, ex. 2, G ( s ) = Tim Davidson s ( s + 1 /τ ) Transfer functions Frequency Response Plotting the freq. resp. Mapping Contours Nyquist’s criterion Ex: servo, P control Summary Ex: unst., P control Ex: unst., PD contr. • As ω → 0 + , G ( j ω ) → − K τ − j ∞ Ex: RHP Z, P contr. Nyquist’s Stability • As ω increases, Criterion as a | G ( j ω ) | decreases, ∠ G ( j ω ) becomes more negative Design Tool √ Relative Stability • When ω = 1 /τ , G ( j ω ) = ( K / 2 ) e − j 135 ◦ Gain margin and Phase margin Relationship to transient response • As ω approaches + ∞ , G ( j ω ) approaches zero from angle − 180 ◦
EE 3CL4, §8 16 / 77 K /τ Polar plot, ex. 2, G ( s ) = Tim Davidson s ( s + 1 /τ ) Transfer functions Frequency Response Plotting the freq. resp. Mapping Contours Nyquist’s criterion Ex: servo, P control Ex: unst., P control Ex: unst., PD contr. Ex: RHP Z, P contr. Nyquist’s Stability Criterion as a Design Tool Relative Stability Gain margin and Phase margin Relationship to transient response
Recommend
More recommend
