Transfer Functions and Frequency Response
 Robert Stengel, Aircraft Flight Dynamics
 MAE 331, 2012" •  Frequency domain view of initial condition response" •  Response of dynamic systems to sinusoidal inputs" •  Transfer functions" •  Bode plots" Copyright 2012 by Robert Stengel. All rights reserved. For educational use only.! http://www.princeton.edu/~stengel/MAE331.html ! http://www.princeton.edu/~stengel/FlightDynamics.html ! Laplace Transform of Initial Condition Response Laplace Transform of a Dynamic System " Δ  x(t ) = F Δx(t ) + G Δu(t) + LΔw(t ) •  System equation! •  Laplace transform of system equation! sΔx(s) − Δx(0) = F Δx(s ) + GΔ u(s) + LΔw(s) dim(Δx) = (n × 1) dim(Δu ) = (m × 1) dim(Δw) = (s ×1) Laplace Transform of a Dynamic System " •  Rearrange Laplace transform of dynamic equation! sΔx(s) − FΔ x(s) = Δx(0) + GΔ u(s) + LΔw(s) sI − F [ ] Δx(s) = Δx(0) + GΔ u(s) + LΔw(s) Δx(s) = sI − F [ ] −1 Δx(0) + G Δu(s) + LΔw(s) [ ] Initial ! Condition! Control/Disturbance ! Input! 4 th -Order Initial Condition Response " •  Longitudinal dynamic model (time domain)! Δx(s) = sI − F [ ] −1 Δx(0) Δ  V (t) Δ  γ (t) Δ  q(t) Δ  α (t) $ % & & & & & ' ( ) ) ) ) ) = −D V −g −D q −D α L V V N 0 L q V N L α V N M V 0 M q M α − L V V N 0 1 − L α V N $ % & & & & & & & ' ( ) ) ) ) ) ) ) ΔV(t) Δ γ (t) Δq(t) Δ α (t) $ % & & & & & ' ( ) ) ) ) ) , ΔV(0) Δ γ (0) Δq(0) Δ α (0) $ % & & & & & ' ( ) ) ) ) ) given •  Longitudinal model (frequency domain)! ΔV(s) Δ γ (s) Δq(s) Δ α (s) $ % & & & & & ' ( ) ) ) ) ) = sI − F Lon [ ] −1 ΔV(0) Δ γ (0) Δq(0) Δ α (0) $ % & & & & & ' ( ) ) ) ) ) Elements of the Characteristic Matrix Inverse" sI − F Lon ≡ Δ Lon (s) = s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 Adj sI − F Lon ( ) = n V V (s) n γ V (s) n q V (s) n α V (s) n V γ (s) n γ γ (s) n q γ (s) n α γ (s) n V q (s) n γ q (s) n q q (s) n α q (s) n V α (s) n V α (s) n V α (s) n V α (s) $ % & & & & & & ' ( ) ) ) ) ) ) sI − F Lon [ ] −1 = Adj sI − F Lon ( ) sI − F Lon = C T s ( ) Δ Lon (s) (4 × 4) 1×1 ( ) •  Denominator is scalar! •  Numerator is an (n x n) matrix of polynomials! (sI – F) –1 Distributes and Shapes the Effects of Initial Conditions" sI −F Lon [ ] −1 = n V V (s) n γ V (s) n q V (s) n α V (s) n V γ (s) n γ γ (s) n q γ (s) n α γ (s) n V q (s) n γ q (s) n q q (s) n α q (s) n V α (s) n V α (s) n V α (s) n V α (s) $ % & & & & & & ' ( ) ) ) ) ) ) s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0  A Lon s ( ) (4 × 4) 1×1 ( ) •  Denominator determines the modes of motion" •  Numerator distributes each element of the initial condition to each element of the state! Δx(s) = Adj sI −F Lon ( ) sI − F Lon Δx(0) = A Lon s ( ) Δx(0) 4 ×1 ( ) Relationship of (sI – F) –1 to State Transition Matrix, (t,0)" •  Initial condition response! Δx(s) = sI − F [ ] −1 Δx(0) = A s ( ) Δx(0) Δx(t ) = Φ t, 0 ( ) Δx(0) Time ! Domain! Frequency ! Domain! •  Δx(s) is the Laplace transform of Δx(t)! Δx(s) = A s ( ) Δx(0) = L Δx(t ) [ ] = L Φ t,0 ( ) Δx(0) # $ % & = L Φ t,0 ( ) # $ % & Δx(0) Relationship of (sI – F) –1 to State Transition Matrix, (t,0)" sI − F [ ] −1 = A s ( ) = L Φ t, 0 ( ) # $ % & = Laplace transform of the state transition matrix •  Therefore,! Initial Condition Response of a Single State Element " A s ( )  sI − F [ ] −1 •  Typical (ij th ) element of A(s)! a ij (s) = k ij n ij (s) Δ(s) = k ij s q + b q−1 s q−1 + +b 1 s + b 0 ( ) s n + a n−1 s n−1 + + a 1 s + a 0 ( ) = k ij s − z 1 ( ) ij s − z 2 ( ) ij s − z q ( ) ij s − λ 1 ( ) s − λ 2 ( ) s − λ n ( ) Initial Condition Response of a Single State Element " p i (s) = k i1 s q 1 + +b 0 ( ) 1 Δx 1 (0)++ k in s q n + +b 0 ( ) n Δx n (0)  k p i s q max + + b 0 ( ) •  All terms have the same denominator polynomial" •  Terms sum to produce a single numerator polynomial! Δx i (s) = a i1 s ( ) Δx 1 (0)+ a i2 s ( ) Δx 1 (0)++ a in s ( ) Δx n (0)  p i s ( ) Δ s ( ) •  Initial condition response of Δx i (s)! Real, scalar" Partial Fraction Expansion of the Initial Condition Response" •  Scalar response can be expressed with n parts, each containing a single mode! Δx i (s) = p i s ( ) Δ s ( ) = d 1 s − λ 1 ( ) + d 2 s − λ 2 ( ) + d n s − λ n ( ) $ % & & ' ( ) ) i , i =1,n where, for each i, the (possibly complex) coefficients are d j = s − λ j ( ) p i s ( ) Δ s ( ) s= λ j , j = 1,n Partial Fraction Expansion of the Initial Condition Response" •  Time response is the inverse Laplace transform! Δx i (t) = L −1 Δx i (s) [ ] = L −1 d 1 s − λ 1 ( ) + d 2 s − λ 2 ( ) +  d n s − λ n ( ) $ % & ' ( ) i = d 1 e λ 1 t + d 2 e λ 2 t +  + d n e λ n t ( ) i , i = 1, n Each element’s time response contains every mode of the system (although some coefficients may be zero)" Scalar and Matrix Transfer Functions Response to a Control Input" •  Neglect initial condition" •  State response to control" sΔx(s) = FΔx(s)+ GΔu(s)+ Δx(0), Δx(0)  0 Δx(s) = sI − F [ ] −1 G Δu(s) •  Output response to control" Δy(s) = H x Δx(s)+H u Δu(s) = H x sI − F [ ] −1 GΔu(s)+ H u Δu(s) = H x sI − F [ ] −1 G + H u { } Δu(s) Transfer Function Matrix" •  Frequency-domain effect of all inputs on all outputs" •  Assume control effects do not appear directly in the output: H u = 0" •  Transfer function matrix! H (s) = H x sI − F [ ] −1 G  H x A s ( ) G r × n ( ) n × n ( ) n × m ( ) = r × m ( ) First-Order Transfer Function " y s ( ) u s ( ) = H (s) = h s − f [ ] −1 g = hg s − f ( ) (n = m = r = 1) •  Scalar transfer function (= first-order lag)!  x t ( ) = fx t ( ) + gu t ( ) y t ( ) = hx t ( ) •  Scalar dynamic system! Second-Order Transfer Function " H(s) = H x A s ( ) G = h 11 h 12 h 21 h 22 ! " # # $ % & & adj s − f 11 ( ) − f 12 − f 21 s − f 22 ! " # # $ % & & det s − f 11 ( ) − f 12 − f 21 s − f 22 ( ) ( ) * * + , - - g 11 g 12 g 21 f 22 ! " # # $ % & & (n = m = r = 2) •  Second-order transfer function matrix! r × n ( ) n × n ( ) n × m ( ) = r × m ( ) = 2 × 2 ( )  x t ( ) =  x 1 t ( )  x 2 t ( ) ! " # # $ % & & = f 11 f 12 f 21 f 22 ! " # # $ % & & x 1 t ( ) x 2 t ( ) ! " # # $ % & & + g 11 g 12 g 21 f 22 ! " # # $ % & & u 1 t ( ) u 2 t ( ) ! " # # $ % & & y t ( ) = y 1 t ( ) y 2 t ( ) ! " # # $ % & & = h 11 h 12 h 21 h 22 ! " # # $ % & & x 1 t ( ) x 2 t ( ) ! " # # $ % & & •  Second-order dynamic system! Longitudinal Transfer Function Matrix " •  With H x = I, and assuming" –  Elevator produces only a pitching moment" –  Throttle affects only the rate of change of velocity" –  Flaps produce only lift! H Lon (s) = H x Lon sI − F Lon [ ] −1 G Lon = H x Lon A Lon s ( ) G Lon = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 " # $ $ $ $ % & ' ' ' ' n V V (s) n γ V (s) n q V (s) n α V (s) n V γ (s) n γ γ (s) n q γ (s) n α γ (s) n V q (s) n γ q (s) n q q (s) n α q (s) n V α (s) n γ α (s) n q α (s) n α α (s) " # $ $ $ $ $ $ % & ' ' ' ' ' ' 0 T δ T 0 0 0 L δ F / V N M δ E 0 0 0 0 −L δ F / V N " # $ $ $ $ $ % & ' ' ' ' ' Δ Lon s ( ) Longitudinal Transfer Function Matrix " H Lon (s) = n δ E V (s) n δ T V (s) n δ F V (s) n δ E γ (s) n δ T γ (s) n δ F γ (s) n δ E q (s) n δ T q (s) n δ F q (s) n δ E α (s) n δ T α (s) n δ F α (s) $ % & & & & & & ' ( ) ) ) ) ) ) s 2 + 2 ζ P ω nP s + ω n P 2 ( ) s 2 + 2 ζ SP ω n SP s + ω n SP 2 ( ) •  There are 4 outputs and 3 inputs! Douglas AD-1 Skyraider! Longitudinal Transfer Function Matrix " ΔV(s) Δ γ (s) Δq(s) Δ α (s) $ % & & & & & ' ( ) ) ) ) ) = H Lon (s) Δ δ E(s) Δ δ T (s) Δ δ F(s) $ % & & & ' ( ) ) ) •  Input-output relationship! Scalar Transfer Function from Δu j to Δy i " H ij (s) = k ij n ij (s) Δ(s) = k ij s q + b q−1 s q−1 + + b 1 s + b 0 ( ) s n + a n−1 s n−1 + + a 1 s + a 0 ( ) # zeros = q! # poles = n" •  Just one element of the matrix, H(s)" •  Denominator polynomial contains n roots" •  Each numerator term is a polynomial with q zeros, where q varies from term to term and ≤ n – 1 ! = k ij s − z 1 ( ) ij s − z 2 ( ) ij s − z q ( ) ij s − λ 1 ( ) s − λ 2 ( ) s − λ n ( ) Scalar Frequency Response Function" H ij (j ω ) = k ij j ω − z 1 ( ) ij j ω − z 2 ( ) ij j ω − z q ( ) ij j ω − λ 1 ( ) j ω − λ 2 ( ) j ω − λ n ( ) • Substitute: s = j ω ! • Frequency response is a complex function of input frequency, ω " –  Real and imaginary parts, or" –  ** Amplitude ratio and phase angle ** ! = a( ω )+ jb( ω ) → AR( ω ) e j φ ( ω ) Transfer Function Matrix for Short-Period Approximation " •  Transfer Function Matrix (with H x = I, H u = 0)" H SP (s) = I 2 A SP s ( ) G SP = s − M q ( ) −M α − 1− L q V N # $ % & ' ( s + L α V N ( ) ) * + + + + , - . . . . -1 M δ E −L δ E V N ) * + + + , - . . . Δ  x SP = Δ  q Δ  α # $ % % & ' ( ( ≈ M q M α 1 − L q V N + , - . / 0 − L α V N # $ % % % & ' ( ( ( Δq Δ α # $ % % & ' ( ( + M δ E −L δ E V N # $ % % % & ' ( ( ( Δ δ E •  Dynamic Equation" Transfer Function Matrix for Short-Period Approximation " •  Transfer Function Matrix (with H x = I, H u = 0)" H SP (s ) = A SP s ( ) G SP = s + L α V N ( ) M α 1− L q V N # $ % & ' ( s − M q ( ) ) * + + + + , - . . . . M δ E −L δ E V N ) * + + + , - . . . s − M q ( ) s + L α V N ( ) − M α 1− L q V N # $ % & ' ( Transfer Function Matrix for Short-Period Approximation " H SP (s) = M δ E s + L α V N ( ) − L δ E M α V N $ % & ' ( ) M δ E 1− L q V N * + , - . / − L δ E V N ( ) s − M q ( ) $ % & ' ( ) $ % & & & & & ' ( ) ) ) ) ) s 2 + −M q + L α V N ( ) s − M α 1− L q V N * + , - . / + M q L α V N $ % & ' ( ) = M δ E s + L α V N − L δ E M α V N M δ E ( ) $ % & ' ( ) − L δ E V N ( ) s + V N M δ E L δ E 1− L q V N * + , - . / − M q $ % & ' ( ) 0 1 2 3 4 5 6 3 $ % & & & & & ' ( ) ) ) ) ) Δ SP s ( ) Transfer Function Matrix for Short-Period Approximation " H SP (s)  k q n δ E q (s) k α n δ E α (s) # $ % % & ' ( ( s 2 + 2 ζ SP ω n SP s + ω n SP 2 = Δq(s) Δ δ E(s) Δ α (s) Δ δ E(s) # $ % % % % % & ' ( ( ( ( ( dim = 2 x 1" Scalar Transfer Functions for Short- Period Approximation " Δq(s) Δ δ E(s) = M δ E s + L α V N − L δ E M α V N M δ E ( ) % & ' ( ) * s 2 + −M q + L α V N ( ) s − M α 1 − L q V N + , - . / 0 + M q L α V N % & ' ( ) * = k q s − z q ( ) s 2 + 2 ζ SP ω n SP s + ω n SP 2 Δq(s) Δ α (s) # $ % % & ' ( ( = Δq(s) Δ δ E(s) Δ α (s) Δ δ E(s) # $ % % % % & ' ( ( ( ( Δ δ E(s) Δ α (s) Δ δ E(s) = − L δ E V N ( ) s + V N M δ E L δ E 1 − L q V N % & ' ( ) * − M q + , - . / 0 1 2 3 4 3 5 6 3 7 3 s 2 + −M q + L α V N ( ) s − M α 1 − L q V N % & ' ( ) * + M q L α V N + , - . / 0 = k α s − z α ( ) s 2 + 2 ζ SP ω n SP s + ω n SP 2 •  Pitch Rate Transfer Function" •  Angle of Attack Transfer Function" Short-Period Frequency Response (s = j  ) Expressed as Amplitude Ratio and Phase Angle" Pitch-rate frequency response" Angle-of-attack frequency response" Δq( j ω ) Δ δ E( j ω ) = k q j ω − z q ( ) − ω 2 + 2 ζ SP ω n SP j ω + ω n SP 2 = AR q ( ω ) e j φ q ( ω ) Δ α ( j ω ) Δ δ E( j ω ) = k α j ω − z α ( ) − ω 2 + 2 ζ SP ω n SP j ω + ω n SP 2 = AR α ( ω ) e j φ α ( ω ) Bode Plot (Frequency Response of a Scalar Transfer Function)  Angle and Rate Response of a DC Motor over Wide Input- Frequency Range " !  Long-term response of a dynamic system to sinusoidal inputs over a range of frequencies" !  Determine experimentally or " !  from the Bode plot of the dynamic system! Very low damping! Moderate damping! High damping! Bode Plot Portrays Response to Sinusoidal Control Input" •  Express amplitude ratio in decibels " € AR(dB) = 20log 10 AR original units ( ) [ ] 20 dB = factor of 10! € Δq( j ω ) Δ δ E( j ω ) = k q j ω − z q ( ) − ω 2 + 2 ζ SP ω n SP j ω + ω n SP 2 = AR q ( ω ) e j φ q ( ω ) •  Plot AR(dB) vs. log 10 ( ω input )" •  Plot phase angle, ϕ (deg) vs. log 10 ( ω input )" •  Asymptotes form “skeleton” of response amplitude ratio" •  Asymptotes change at poles and zeros" Products in original units are sums in decibels! # zeros = 1! # poles = 2" Constant Gain Bode Plot" € H( j ω ) = 1 € H( j ω ) = 10 € H( j ω ) = 100 y t ( ) = hu t ( ) Slope = 0dB / dec, Amplitude Ratio = constant Phase Angle = 0° Integrator Bode Plot" € H( j ω ) = 1 j ω € H( j ω ) = 10 j ω y t ( ) = h u t ( ) dt 0 t ∫ Slope = −20dB / dec Phase Angle = −90° Differentiator Bode Plot" H ( j ω ) = j ω € H( j ω ) =10 j ω y t ( ) = h du t ( ) dt Slope = +20dB / dec Phase Angle = +90° Sign Change" H ( j ω ) = − h j ω y t ( ) = −h u t ( ) dt 0 t ∫ H ( j ω ) = − j ω y t ( ) = −h du t ( ) dt Slope = −20dB / dec Phase Angle = +90° Slope = +20dB / dec Phase Angle = −90° Integral! Derivative! Multiple Integrators and Differentiators" H ( j ω ) = h j ω ( ) 2 y t ( ) = h d 2 u t ( ) dt 2 H ( j ω ) = h j ω ( ) 2 y t ( ) = h u t ( ) dt 2 0 t ∫ 0 t ∫ Slope = −40dB / dec Phase Angle = −180° Slope = +40dB / dec Phase Angle = +180° Double Integral! Double Derivative! Constant Gain, Integrator, and Differentiator Bode Plots Form the Asymptotes for More Complex Transfer Functions" +20 " dB/dec" +40 " dB/dec" 0 " dB/dec" +20 " dB/dec" –20 " dB/dec" Bode Plots of First-Order Lags" H red ( j ω ) = 10 j ω +10 ( ) H blue ( j ω ) = 100 j ω +10 ( ) H green ( j ω ) = 100 j ω +100 ( ) Bode Plot Asymptotes, Departures, and Phase Angles for First-Order Lags" •  General shape of amplitude ratio governed by asymptotes" •  Slope of asymptotes changes by multiples of ±20 dB/dec at poles or zeros" •  Actual AR departs from asymptotes" •  Phase angle of a real, negative pole" –  When ω = 0, ϕ = 0°" –  When ω = λ , ϕ =–45°" –  When ω -> ∞, ϕ -> –90°" •  AR asymptotes of a real pole" –  When ω = 0, slope = 0 dB/ dec" –  When ω ≥ λ , slope = –20 dB/ dec" [...]... http://www.mathworks.com/matlabcentral/fileexchange/10183-bode-plot-with-asymptotes" 2nd-Order Pitch Rate Frequency Response" bode.m" asymp.m" First- and Second-Order Departures from Amplitude Ratio Asymptotes " Frequency Response AR Departures in the Vicinity of Poles " •  Difference between actual amplitude ratio (dB) and asymptote = departure (dB)" •  Results for multiple roots are additive" •  from McRuer, Ashkenas, and Graham, Aircraft Dynamics and Automatic Control, Princeton... opposite sign" First- and SecondOrder Phase Angles " Phase Angle Variations in the Vicinity of Poles" •  Results for multiple roots are additive" •  from McRuer, Ashkenas, and Graham, Aircraft Dynamics and Automatic Control, Princeton University Press, 1973" •  LHP zero variations have opposite sign" •  RHP zeros have same sign" Next Time: Control Devices and Systems Reading Flight Dynamics, 214-234 Virtual... jω ) + 2 ( 0.1) (100 ) ( jω ) + 100 2 & '$ ' $ #( # zeros = 0! # poles = 4" € •  Resonant peaks and large phase shifts at each natural frequency" •  Additive AR slope shifts at each natural frequency" Left-Half-Plane Transfer Function Zero " •  Zeros are numerator singularities " H (jω ) = Right-Half-Plane Transfer Function Zero " k ( jω − z1 ) ( jω − z2 ) ( jω − λ1 )( jω − λ2 ) ( jω − λn ) H ( jω )... 4th-Order Transfer Function with 2nd-Order Zero " Second-Order Transfer Function Zero " H( jω ) = ( jω − z)( jω − z* ) = € [( jω ) 2 + 2(0.1)(100)( jω ) + 100 2 ] • Complex pair of zeros produces an amplitude ratio notch at its natural frequency " H( jω ) = € [( jω ) [( jω ) 2 2 + 2(0.1)(10)( jω ) + 10 2 2 + 2(0.05)(1)( jω ) + 1 ][( jω ) 2 ] + 2(0.1)(100)( jω ) + 100 2 ] Elevator-toNormal-Velocity Frequency. .. ( jω ) +10 2 2 H green ( jω ) = 10 3 ( jω ) + 2 (0.1) (10) ( jω ) +10 2 2 Effect of Damping Ratio ! Effects of Gain and Natural Frequency ! H blue ( jω ) = 10 2 ( jω ) + 2 (0.4 ) (10) ( jω ) +10 2 2 H red ( jω ) = 10 2 ( jω ) + 2 (0.707) (10) ( jω ) +10 2 2 Amplitude Ratio Asymptotes and Departures of Second-Order Bode Plots (No Zeros) " •  AR asymptotes of a pair of complex poles " –  When ω = 0,... 2 ] + 2(0.1)(100)( jω ) + 100 2 ] Elevator-toNormal-Velocity Frequency Response" ( ) 2 M δ E s 2 + 2ζω n s + ω n Approx Ph ( s − z3 ) Δw(s) n w (s) = δE ≈ 2 2 2 Δδ E(s) Δ Lon (s) s + 2ζω n s + ω n s 2 + 2ζω n s + ω n SP Ph ( ) ( ) 0 dB/dec! 0 dB/dec! +40 dB/dec! • (n – q) = 1" • Complex zero almost (but not quite) cancels phugoid response" –40 dB/dec! Phugoid" Short " Period" –20 dB/dec! ... Press, 1973" •  LHP zero variations have opposite sign" •  RHP zeros have same sign" Next Time: Control Devices and Systems Reading Flight Dynamics, 214-234 Virtual Textbook, Part 16 Bode Plots of 1st- and 2nd-Order Lags " Supplementary Material H red ( jω ) = H blue ( jω ) = 10 ( jω + 10) 100 2 ( jω ) 2 + 2(0.1)(100)( jω ) + 100 2 € Bode Plots of 3rd-Order Lags " & # 10 &# 100 2 ( H blue ( jω ) = % . Transfer Functions and Frequency Response Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012" •  Frequency domain view of initial condition response& quot; •  Response of. Rate Transfer Function" •  Angle of Attack Transfer Function" Short-Period Frequency Response (s = j  ) Expressed as Amplitude Ratio and Phase Angle" Pitch-rate frequency response& quot; Angle-of-attack. e j φ α ( ω ) Bode Plot (Frequency Response of a Scalar Transfer Function)  Angle and Rate Response of a DC Motor over Wide Input- Frequency Range " !  Long-term response of a dynamic