Root Locus Analysis of Parameter Variations
 Robert Stengel, Aircraft Flight Dynamics
 MAE 331, 2012" •  Effects of system parameter variations on modes of motion" •  Root locus analysis" –  Evanss rules for construction" –  Application to longitudinal dynamic models" 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 ! Characteristic Equation: A Critical Component of the Response’s Laplace Transform " sI − F [ ] −1 = Adj sI − F ( ) sI − F = C T s ( ) sI − F (n × n) 1×1 ( ) •  Characteristic equation defines the modes of motion! sI − F = Δ(s) = s n + a n −1 s n −1 + + a 1 s + a 0 = s − λ 1 ( ) s − λ 2 ( ) ( ) s − λ n ( ) = 0 Δx(s) = sI − F [ ] −1 Δx( 0) + G Δu(s) + LΔw(s) [ ] •  Recall: s is a complex variable! s = σ + j ω Real Roots of the Dynamic System " Δ(s) = s − λ 1 ( ) s − λ 2 ( ) ( ) s − λ n ( ) = 0 •  Roots are solutions of the characteristic equation" λ i = µ i (Real number) x t ( ) = x 0 ( ) e µ t •  Real roots" –  are confined to the real axis" –  represent convergent or divergent time response" –  time constant, τ = –1/ λ = –1/ µ , sec # s Plane = σ + j ω ( ) Plane Complex Roots of the Dynamic System " € δ = cos −1 ζ •  Complex roots" –  occur only in complex-conjugate pairs" –  represent oscillatory modes" –  natural frequency, ω n , and damping ratio, ζ , as shown" s Plane = σ + j ω ( ) Plane λ 1 = µ 1 + j ν 1 = − ζω n + j ω n 1− ζ 2 – time constant = –1/μ = 1/ζω n " Stable" Unstable" – decay of exponential time-" response envelope" λ 2 = µ 2 + j ν 2 = µ 1 − j ν 1 = λ 1 * = − ζω n − j ω n 1− ζ 2 Complex Roots, Damping Ratio, and Damped Natural Frequency " s − λ 1 ( ) s − λ 1 * ( ) = s − µ 1 + j ν 1 ( ) $ % & ' s − µ 1 − j ν 1 ( ) $ % & ' = s 2 − µ 1 − j ν 1 ( ) + µ 1 + j ν 1 ( ) $ % & ' s + µ 1 − j ν 1 ( ) µ 1 + j ν 1 ( ) = s 2 − 2 µ 1 s + µ 1 2 + ν 1 2 ( )  s 2 + 2 ζω n s + ω n 2 µ 1 = − ζω n = −1 Time constant ν 1 = ω n 1− ζ 2  ω n damped = Damped natural frequency x 1 t ( ) = Ae − ζω n t sin ω n 1− ζ 2 t + ϕ % & ' ( x 2 t ( ) = Ae − ζω n t ω n 1− ζ 2 % & ' ( cos ω n 1− ζ 2 t + ϕ % & ' ( Identical exponentially decaying envelopes for both displacement and rate" Corresponding Second-Order Initial Condition Response" General form of response" Multi-Modal LTI Responses Superpose Individual Modal Responses" •  With distinct roots, (n = 4) for example, partial fraction expansion for each state element is (Flight Dynamics, p. 325) " Δx i s ( ) = d 1 i s − λ 1 ( ) + d 2 i s − λ 2 ( ) + d 2 i s − λ 3 ( ) + d 2 i s − λ 4 ( ) •  Corresponding 4 th -order time response is" Δx i t ( ) = d 1 i e λ 1 t + d 2 i e λ 2 t + d 3 i e λ 3 t + d 4 i e λ 4 t •  Details in next lecture" Evanss Rules for Root Locus Analysis Root Locus Example: " 4 th- Order Longitudinal Characteristic Equation" Δ Lon (s) = s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 = s 4 + D V + L α V N − M q ( ) s 3 + g − D α ( ) L V V N + D V L α V N − M q ( ) − M q L α V N − M α $ % & ' ( ) s 2 + M q D α − g ( ) L V V N − D V L α V N $ % & ' ( ) + D α M V − D V M α { } s + g M V L α V N − M α L V V N ( ) = 0 Δ Lon (s) = s 2 + 2 ζω n s + ω n 2 ( ) Ph s 2 + 2 ζω n s + ω n 2 ( ) SP •  Typically factors into oscillatory phugoid and short-period modes " € with L q = D q = 0 Root Locus Analysis of Parametric Effects on Aircraft Dynamics " •  Parametric variations alter eigenvalues of F" •  Graphical technique for finding the roots with a new parameter value" Locus: the set of all points whose location is determined by stated conditions" s Plane! Phugoid " Roots" Short Period" Root" Short Period" Root" Example: How do the roots vary when we change pitch-rate damping, M q ?" Δ Lon (s) = s 4 + D V + L α V N − M q ( ) s 3 + g − D α ( ) L V V N + D V L α V N − M q ( ) − M q L α V N − M α $ % & ' ( ) s 2 + M q D α − g ( ) L V V N − D V L α V N $ % & ' ( ) + D α M V − D V M α { } s + g M V L α V N − M α L V V N ( ) = 0 •  M q could be changed by" –  Variation in aircraft aerodynamic configuration" –  Effect of feedback control, i.e., control of pitching moment (via elevator) that is proportional to pitch rate" Effect of Parameter Variations on Root Location " •  Let root locus gain = k = a i (just a notation change) " –  Option 1: Vary k and calculate roots for each new value" –  Option 2: Apply Evanss Rules of Root Locus Construction" Walter R. Evans" Δ Lon (s) = s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 = s − λ 1 ( ) s − λ 2 ( ) s − λ 3 ( ) s − λ 4 ( ) = s − λ 1 ( ) s − λ 1 * ( ) s − λ 3 ( ) s − λ 3 * ( ) = s 2 + 2 ζ P ω n P s + ω n P 2 ( ) s 2 + 2 ζ SP ω n SP s + ω n SP 2 ( ) = 0 Effect of a 0 Variation on Longitudinal Root Location " •  Example: Root locus gain, k = a 0 ! where d(s) = s 4 + a 3 s 3 + a 2 s 2 + a 1 s = s − λ ' 1 ( ) s − λ ' 2 ( ) s − λ ' 3 ( ) s − λ ' 4 ( ) n(s) = 1 Δ Lon (s) = s 4 + a 3 s 3 + a 2 s 2 + a 1 s " # $ % + k [ ] ≡ d(s)+ kn(s) = s − λ 1 ( ) s − λ 2 ( ) s − λ 3 ( ) s − λ 4 ( ) = 0 d s ( ) : Polynomial in s n s ( ) : Polynomial in s •  Example: Root locus gain, k = a 1 ! where d(s) = s 4 + a 3 s 3 + a 2 s 2 + a 0 = s − λ ' 1 ( ) s − λ ' 2 ( ) s − λ ' 3 ( ) s − λ ' 4 ( ) n(s) = s Δ Lon (s) = s 4 + a 3 s 3 + a 2 s 2 + ks + a 0 ≡ d(s)+ kn(s) = s − λ 1 ( ) s − λ 2 ( ) s − λ 3 ( ) s − λ 4 ( ) = 0 Effect of a 1 Variation on Longitudinal Root Location " Three Equivalent Equations for Defining Roots " 1 + k n(s) d(s) = 0 k n(s) d(s) = −1 = (1)e − j π (rad ) = (1)e − j180(deg) d(s) + k n(s) = 0 Longitudinal Equation Example" •  Original 4 th -order polynomial! Δ Lon (s) = s 4 + 2.57s 3 + 9.68s 2 + 0.202s + 0.145 = s 2 + 2 0.0678 ( ) 0.124s + 0.124 ( ) 2 " # $ % s 2 + 2 0.411 ( ) 3.1s + 3.1 ( ) 2 " # $ % = 0 Δ Lon (s) = s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 = s − λ 1 ( ) s − λ 2 ( ) s − λ 3 ( ) s − λ 4 ( ) = s − λ 1 ( ) s − λ 1 * ( ) s − λ 3 ( ) s − λ 3 * ( ) = s 2 + 2 ζ P ω n P s + ω n P 2 ( ) s 2 + 2 ζ SP ω n SP s + ω n SP 2 ( ) = 0 •  Typical flight condition! Phugoid" Short Period" Example: Effect of a 0 Variation" Δ(s) = s 4 + a 3 s 3 + a 2 s 2 + a 1 s + a 0 = s 4 + a 3 s 3 + a 2 s 2 + a 1 s ( ) + k = s s 3 + a 3 s 2 + a 2 s + a 1 ( ) + k = s s + 0.21 ( ) s 2 + 2.55s +9.62 " # $ % + k k s s + 0.21 ( ) s 2 + 2.55s + 9.62 ! " # $ = −1 •  Example: k = a 0 ! •  Original 4 th -order polynomial! Δ Lon (s) = s 4 + 2.57s 3 + 9.68s 2 + 0.202s + 0.145 = 0 •  Rearrange:! ks s 2 − 0.00041s + 0.015 " # $ % s 2 + 2.57s + 9.67 " # $ % = −1 •  Example: k = a 1 ! Δ(s) = s 4 + a 3 s 3 + a 2 s 2 + a 1 s +a 0 = s 4 + a 3 s 3 + a 2 s 2 + ks + a 0 = s 4 + a 3 s 3 + a 2 s 2 + a 0 ( ) + ks = s 2 − 0.00041s + 0.015 # $ % & s 2 + 2.57 s + 9.67 # $ % & + ks Example: Effect of a 1 Variation" •  Rearrange:! The Root Locus Criterion" •  All points on the locus of roots must satisfy the equation k[n(s)/d(s)] = –1" •  Phase angle(–1) = ±180 deg" k = a 0 : k n(s) d(s) = k 1 s 4 + a 3 s 3 + a 2 s 2 + a 1 s = −1 k = a 1 : k n(s) d(s) = k s s 4 + a 3 s 3 + a 2 s 2 + a 0 = −1 = k s − 0 ( ) s 4 + a 3 s 3 + a 2 s 2 + a 0 = −1 •  Number of roots = 4" •  Number of zeros = 0" •  (n – q) = 4" •  Number of roots = 4" •  Number of zeros = 1" •  (n – q) = 3" •  Number of roots (or poles) of the denominator = n" •  Number of zeros of the numerator = q" Spirule" Origins of Roots (for k = 0)" Δ(s) = d(s) + kn (s) k → 0 # →## d(s) •  Origins of the roots are the Poles of d(s)" Destinations of Roots (for k -> ±∞) " •  q roots go to the zeros of n(s)" d(s)+ kn(s) k = d(s) k + n(s) k→∞ # →## n(s) = s − z 1 ( ) s − z 2 ( )  No zeros when k = a 0 " One zero at origin when k = a 1 " Destinations of Roots (for k -> ±∞) " d(s)+ kn(s) n(s) ! " # $ % & = d(s) n(s) + k ! " # $ % & k→±R→±∞ ) →))) s n s q + k ! " # $ % & → s n−q ( ) ± R →±∞ •  (n – q) roots go to infinite radius from the origin" R(+)" R(–)" s n−q ( ) = Re − j180° → ∞ or Re − j 360° → −∞ s = R 1 n−q ( ) e − j180° n−q ( ) → ∞ or R 1 n−q ( ) e − j 360° n−q ( ) → −∞ •  Asymptotes of the root loci are described by" •  Magnitudes of roots are the same for given k" •  Angles from the origin are different" Destinations of Roots (for k -> ±∞) " 4 roots to infinite radius" 3 roots to infinite radius" (n – q) Roots Approach Asymptotes as k –> ±∞" θ (rad) = π + 2m π n − q , m = 0,1, ,(n − q) − 1 θ (rad) = 2m π n − q , m = 0,1, ,(n − q) − 1 •  Asymptote angles for positive k" •  Asymptote angles for negative k" Origin of Asymptotes = Center of Gravity" "c.g." = σ λ i − σ z j j =1 q ∑ i =1 n ∑ n − q Root Locus on Real Axis" •  Locus on real axis" –  k > 0: Any segment with odd number of poles and zeros to the right" –  k < 0: Any segment with even number of poles and zeros to the right" First Example: Positive and Negative Variations of k = a 0 " k s s + 0.21 ( ) s 2 + 2.55s + 9.62 ! " # $ = −1 Second Example: Positive and Negative Variations of k = a 1 " ks s 2 − 0.00041s + 0.015 " # $ % s 2 + 2.57s + 9.67 " # $ % = −1 Summary of Root Locus Concepts" Origins " of Roots" Destinations " of Roots" Center " of Gravity" Locus on " Real Axis" Root Locus Analysis of Simplified Longitudinal Modes Approximate Phugoid Model " •  Second-order equation" Δ  x Ph = Δ  V Δ  γ # $ % % & ' ( ( ≈ −D V −g L V V N 0 # $ % % % & ' ( ( ( ΔV Δ γ # $ % % & ' ( ( + T δ T L δ T V N # $ % % % & ' ( ( ( Δ δ T •  Characteristic polynomial" sI − F Ph = det sI − F Ph ( ) ≡ Δ(s) = s 2 + D V s + gL V / V N = s 2 + 2 ζω n s + ω n 2 gL V / V N , D V •  Parameters" Δ(s) = s 2 + D V s ( ) + k = s s + D V ( ) + k k = gL V /V N " Effect of L V or 1/V N Variation on Approximate Phugoid Roots " •  Change in damped natural frequency" ω n damped  ω n 1− ζ 2 Effect of D V Variation on Approximate Phugoid Roots " k = D V " Δ(s) = s 2 + gL V / V N ( ) + ks = s + j gL V / V N ( ) s − j gL V / V N ( ) + ks •  Change in damping ratio" ζ Approximate Short-Period Model " •  Approximate Short-Period Equation (L q = 0)" •  Characteristic polynomial" •  Parameters" Δ  x SP = Δ  q Δ  α # $ % % & ' ( ( ≈ M q M α 1 − L α V N # $ % % % & ' ( ( ( Δq Δ α # $ % % & ' ( ( + M δ E −L δ E V N # $ % % % & ' ( ( ( Δ δ E Δ(s) = s 2 + L α V N − M q $ % & ' ( ) s − M α + M q L α V N $ % & ' ( ) = s 2 + 2 ζω n s + ω n 2 M α , M q , L α V N Effect of M α on Approximate Short-Period Roots " k = M α " Δ(s) = s 2 + L α V N − M q $ % & ' ( ) s − M q L α V N $ % & ' ( ) − k = 0 = s + L α V N $ % & ' ( ) s − M q ( ) − k = 0 •  Change in damped natural frequency" Effect of M q on Approximate Short-Period Roots" Δ(s) = s 2 + L α V N s − M α − k s + L α V N $ % & ' ( ) = s − L α 2V N + L α 2V N $ % & ' ( ) 2 + M α * + , , - . / / 0 1 2 3 2 4 5 2 6 2 s − L α 2V N − L α 2V N $ % & ' ( ) 2 + M α * + , , - . / / 0 1 2 3 2 4 5 2 6 2 − k s + L α V N $ % & ' ( ) = 0 k = M q " •  Change primarily in damping ratio" Effect of L ! /V N on Approximate Short-Period Roots" Δ(s) = s 2 − M q s − M α + k s − M q ( ) = s + M q 2 − M q 2 $ % & ' ( ) 2 + M α * + , , - . / / 0 1 2 3 2 4 5 2 6 2 s + M q 2 − M q 2 $ % & ' ( ) 2 + M α * + , , - . / / 0 1 2 3 2 4 5 2 6 2 + k s − M q ( ) = 0 k = L α /V N " •  Change primarily in damping ratio" How do the 4 th -order roots vary when we change pitch-rate damping, M q ?" Δ Lon (s) = s 4 + D V + L α V N ( ) s 3 + g − D α ( ) L V V N + D V L α V N ( ) − M α $ % & ' ( ) s 2 + D α M V − D V M α { } s + g M V L α V N − M α L V V N ( ) − M q s 3 − D V M q ( ) + M q L α V N $ % & ' ( ) s 2 + M q D α − g ( ) L V V N − D V L α V N $ % & ' ( ) s = 0 •  Identify M q terms in the characteristic polynomial" How do the 4 th -order roots vary when we change pitch-rate damping, M q ?" Δ Lon (s) = d s ( ) − M q s 3 + D V + L α V N $ % & ' ( ) s 2 − D α − g ( ) L V V N − D V L α V N $ % & ' ( ) s { } = d s ( ) − M q s s 2 + D V + L α V N $ % & ' ( ) s − D α − g ( ) L V V N − D V L α V N $ % & ' ( ) { } = d s ( ) + kn s ( ) = 0 •  Group M q terms in the characteristic polynomial" k n(s) d(s) = −1 How do the 4 th -order roots vary when we change pitch-rate damping, M q ?" −M q s s 2 + D V + L α V N ( ) s − D α − g ( ) L V V N − D V L α V N # $ % & ' ( { } s 4 + D V + L α V N ( ) s 3 + g − D α ( ) L V V N + D V L α V N ( ) − M α # $ % & ' ( s 2 + D α M V − D V M α { } s + g M V L α V N − M α L V V N ( ) ) * + + , + + - . + + / + + = −1 •  Factor terms that are multiplied by M q to find the 3 zeros" –  2 zeros near origin similar to approximate phugoid roots, effectively canceling M q effect on them " −M q s s − z 1 ( ) s − z 2 ( ) s 2 + 2 ζ P ω n P s + ω n P 2 ( ) s 2 + 2 ζ SP ω n SP s + ω n SP 2 ( ) = −1 s 2 − z 1 s ( )  s 2 + 2 ζ P ω n P s + ω n P 2 ( ) [...]... (s 2 2 − z1s ) ( s − z2 ) + 2ζ Pω nP s + ω 2 nP ) (s 2 + 2ζ SPω nSP s + ω 2 nSP  ) (s −M q ( s − z2 ) 2 2 + 2ζ SPω nSP s + ω nSP ) = −1 Next Time: Transfer Functions and Frequency Response Reading Flight Dynamics, 342-355 Virtual Textbook, Part 15 . Locus Analysis of Parameter Variations Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012" •  Effects of system parameter variations on modes of motion" •  Root locus analysis& quot; – . 0 Root Locus Analysis of Parametric Effects on Aircraft Dynamics " •  Parametric variations alter eigenvalues of F" •  Graphical technique for finding the roots with a new parameter. Variation in aircraft aerodynamic configuration" –  Effect of feedback control, i.e., control of pitching moment (via elevator) that is proportional to pitch rate" Effect of Parameter Variations