Linearized Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018

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1 Linearized Equaions of Moion Rober Sengel, Aircraf Fligh Dynamics MAE 331, 2018 Learning Objecives Develop linear equaions o describe small perurbaional moions Apply o aircraf dynamic equaions Reading: Fligh Dynamics , , , , Copyrigh 2018 by Rober Sengel. All righs reserved. For educaional use only. hp:// hp:// 1 How Is Sysem Response Calculaed? Linear and nonlinear, ime-varying and imeinvarian dynamic models Numerical inegraion ime domain ) Linear, ime-invarian LTI) dynamic models Numerical inegraion ime domain ) Sae ransiion ime domain ) Transfer funcions frequency domain ) 2 1

2 Inegraion Algorihms Exac x T ) = x 0) + f x Recangular Euler) Inegraion x k ) = x k 1 ) + δ x k 1, k ) x k 1 ) + f [ x k 1 ),u k 1 ),w k 1 )]δ δ = k k 1 Trapezoidal modified Euler) Inegraion ~MATLAB s ode23) x k ) x k 1 ) + 1 [ 2 δ x + δ x 1 2 ] where δ x 1 = f [ x k 1 ),u k 1 ),w k 1 )]δ δ x 2 = f [ x k 1 ) + δ x 1,u k ),w k )]δ T 0 ),u ),w ) d See MATLAB manual for descripions of ode45 and ode15s 3 Numerical Inegraion: MATLAB Ordinary Differenial Equaion Solvers * Explici Runge-Kua Algorihm Adams-Bashforh-Moulon Algorihm Numerical Differeniaion Formula Modified Rosenbrock Mehod Trapezoidal Rule Trapezoidal Rule w/back Differeniaion * hp:// Shampine, L. F. and M. W. Reichel, "The MATLAB ODE Suie," SIAM Journal on Scienific Compuing, Vol. 18, 1997, pp

3 Nominal and Acual Trajecories Nominal or reference) rajecory and conrol hisory { x N ), u N ),w N )} for in [ o, f ] x : dynamic sae u : conrol inpu w : disurbance inpu x N T ) = x N 0 ) + f x N Acual rajecory perurbed by Small iniial condiion variaion, Δx o o) Small conrol variaion, Δu) T 0 ),u N ),w N ) d { x), u),w) } for in [ o, f ] { } = x N ) + Δx), u N ) + Δu),w N ) + Δw) x T ) = x 0) + f x T 0 ),u ),w ) d 5 Boh Pahs Saisfy he Dynamic Equaions Dynamic models for he acual and he nominal problems are he same x N ) = f[x N ),u N ),w N )], x N o x) = f[x),u),w)], ) given ) given x o Δx o ) = x o ) x N o ) Differences in iniial condiion and forcing... Δu) = u) u N ) Δw) = w) w N ) ) in * + o, f, -... perurb rae of change and he sae x) = x N ) + Δx) x) = x N ) + Δx) in o, f 6 3

4 Approximae Neighboring Trajecory as Linear Perurbaion o Nominal Trajecory x) = x N ) + Δx) = f[x N ) + Δx), u N ) + Δu), w N ) + Δw),] New rajecory ~ nominal pah + linear perurbaion x) = x N ) + Δx) New rajecory [change in noaion] f[x N ),u N ),w N ),]+ f x Δx) + f u Δu) + f w Δw) 7 Perurbaion Dynamics Approximaed by Linearized Equaion Solve for he nominal and perurbaion rajecories separaely Nominal Equaion x N ) = f[x N ),u N ),w N ),], x N o ) given Perurbaion Equaion Δx) f Δx) + f Δu) + f Δw) x x=x N ) u=u N ) u x=x N ) u=u w=w N ) N ) w x=x N ) u=u w=w N ) N ) w=w N ) F)Δx) + G)Δu) + L)Δw), Δx o dim x dim u ) = dim Δx) = n 1 ) = dim Δu) = m 1 ) = dim Δw) = s 1 dim w ) given 8 4

5 Jacobian Marices Express Soluion Sensiiviy o Small Perurbaions Sabiliy marix, F, is square F) = f x x=x N ) u=u N ) w=w N ) dimf) = n n = x 1 x n f n f n x 1 x n x=x N ) u=u N ) w=w N ) 9 Sensiiviy o Conrol Perurbaions, G G) = f = u x=x N ) u=u N ) w=w N ) dimg) = n m u 1 u m f n u 1 f n u m x=x N ) u=u N ) w=w N ) 10 5

6 Sensiiviy o Disurbance Perurbaions, G L) = f = w x=x N ) u=u N ) w=w N ) diml) = n s w 1 w s f n w 1 f n w s x=x N ) u=u N ) w=w N ) 11 Scalar Example of Nominal and Perurbaion Equaions Acual Nonlinear Sysem x ) = x ) + 2x 2 ) + 3u ) + 4w 3 ) Nominal Nonlinear Sysem x N ) = x N ) + 2x 2 N ) + 3u N ) + 4w 3 N ) Linear Perurbaion Sysem 2 Δx ) = Δx ) + 4x N )Δx ) + 3Δu ) +12w N Time-Varying Parameer Time-Varying Parameer )Δw ) 12 6

7 Comparison of Damped Linear and Nonlinear Sysems Linear Spring x 1 ) = x 2 ) x 2 ) = 10x 1 ) x 2 ) Displacemen Rae of Change Linear Spring Force vs. Displacemen Spring Damper Linear plus Siffening Cubic Spring x 1 ) = x 2 ) Cubic Spring Force vs. Displacemen x 2 ) = 10x 1 ) 10x 1 3 ) x 2 ) Linear plus Weakening Cubic Spring x 1 ) = x 2 ) Cubic Spring Force vs. Displacemen x 2 ) = 10x 1 ) + 0.8x 1 3 ) x 2 ) 13 MATLAB Simulaion of Linear and Nonlinear Dynamic Sysems MATLAB Main Scrip Nonlinear and Linear Examples clear span = [0 10]; xo = [0, 10]; [1,x1 = ode23nonlin,span,xo); xo = [0, 1]; [2,x2] = ode23nonlin,span,xo); xo = [0, 10]; [3,x3] = ode23lin,span,xo); xo = [0, 1]; [4,x4] = ode23lin,span,xo); subplo2,1,1) plo1,x1:,1),k,2,x2:,1),b,3,x3:,1),r,4,x4:,1),g) ylabelposiion), grid subplo2,1,2) plo1,x1:,2),k,2,x2:,2),b,3,x3:,2),r,4,x4:,2),g) xlabeltime), ylabelrae), grid Linear Sysem x 1 ) = x 2 ) x 2 ) = 10x 1 ) x 2 ) funcion xdo = Lin,x) Linear Ordinary Differenial Equaion x1) = Posiion x2) = Rae xdo = [x2) -10*x1) - x2)]; Nonlinear Sysem x 1 ) = x 2 ) x 2 ) = 10x 1 ) + 0.8x 3 1 ) x 2 ) funcion xdo = NonLin,x) Nonlinear Ordinary Differenial Equaion x1) = Posiion x2) = Rae xdo = [x2) -10*x1) + 0.8*x1)^3 - x2)]; 14 7

8 Linear and Siffening Cubic Springs: Small and Large Iniial Condiions Cubic erm Large Rae Small Rae Linear and nonlinear responses are indisinguishable wih small iniial condiion 15 Linear and Weakening Cubic Springs: Small and Large Iniial Condiions Cubic erm Large Rae Small Rae 16 8

9 Linear, Time-Varying LTV) Approximaion of Perurbaion Dynamics 17 Siffening Linear-Cubic Spring Example Nonlinear, ime-invarian NTI) equaion x 1 ) = f 1 = x 2 ) x 2 ) = f 2 = 10x 1 ) 10x 1 3 ) x 2 ) " Inegrae nonlinear equaions o produce nominal pah x 1N 0) x 2N 0) f 0 " f 1N f 2N d " x 1N ) x 2N ) Analyical evaluaion of parial derivaives in " 0, f x 1 = 0; = 10 30x 2 1N ); x 1 x 2 = 1 x 2 = 1 u = 0; w = 0 u = 0; w =

10 Nominal NTI) and Perurbaion LTV) Dynamic Equaions Nonlinear, ime-invarian NTI) nominal equaion x N ) = f[x N )], x N 0) given x 1N ) = x 2N ) x 2N ) = 10x 1N ) 10x 1N 3 ) x 2N ) " x 1N 0) x 2N 0) Example = 0 9 " Perurbaions approximaed by linear, ime-varying LTV) equaion Δx 1 ) Δx 2 ) Δx) = F)Δx), Δx0) given = x 2 1N ) ) 1 Δx 1 ) Δx 2 ) " Example Δx 1 0) Δx 2 0) = " Comparison of Approximae and Exac Soluions x 2N 0) = 9 Δx 2 0) = 1 x 2N ) + Δx 2 ) = 10 x 2 ) = 10 x N ) Δx) x N )+ Δx) x) x N ) Δ x) x N ) + Δ x) x) 20 10

11 Suppose Nominal Iniial Condiion is Zero Nominal soluion remains a equilibrium x N ) = f[x N )], x N 0) = 0, x N ) = 0 in [ 0, ] Perurbaion equaion is linear and ime-invarian LTI) " Δ x 1 ) " Δ x 2 ) = 0 1 " Δx 1 ) " ) 1 Δx 2 ) 21 Separaion of he Equaions of Moion ino Longiudinal and Laeral- Direcional Ses 22 11

12 Rigid-Body Equaions of Moion Rae of change of Translaional Velociy u = X / m gsinθ + rv qw v = Y / m + gsinφ cosθ ru + pw w = Z / m + gcosφ cosθ + qu pv Rae of change of Translaional Posiion x I = cosθ cosψ )u + cosφsinψ + sinφsinθ cosψ )v + sinφsinψ + cosφsinθ cosψ )w y I = cosθ sinψ )u + cosφcosψ + sinφsinθsinψ )v + sinφcosψ + cosφsinθsinψ )w z I = sinθ )u + sinφ cosθ)v + cosφ cosθ)w Rae of change of Angular Velociy I xy = I yz = 0) ) I I I 2 xx zz xz ) zz zz yy xx xx yy p = I zz L + I xz N I xz I yy I xx I zz ) p + " I xz ) r}q + I I I q = " M I xx I zz ) pr I xz p 2 r 2 ) I yy r = I xz L + I xx N I xz I yy I xx I zz )r + " I xz ) p}q + I I I ) I I I 2 xx zz xz ) Rae of change of Angular Posiion φ = p + qsinφ + r cosφ)anθ θ = q cosφ r sinφ ψ = qsinφ + r cosφ)secθ " Sae Vecor x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 u v w x y z = p q r φ θ " ψ 23 " Inermingled Axes x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 u v w x y z = p q r φ θ " ψ Reorder he Sae Vecor Firs six elemens of he sae are longiudinal variables Second six elemens of he sae are laeraldirecional variables x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 new Pariioned Axes = x Lon x La Dir = u w x z q θ v y p ψ 24 r φ 12

13 Longiudinal Equaions of Moion Dynamics of velociy, posiion, angular rae, and angle primarily in he verical plane u = X / m gsinθ + rv qw " x 1 = f 1 w = Z / m + gcosφ cosθ + qu pv " x 2 = f 2 x I = cosθ cosψ )u + cosφ sinψ + sinφ sinθ cosψ )v + sinφ sinψ + cosφ sinθ cosψ )w " x 3 = f 3 z I = sinθ )u + sinφ cosθ )v + cosφ cosθ )w " x 4 = f 4 ) pr I xz p 2 r 2 ) q = M I xx I zz I yy " x 5 = f 5 θ = qcosφ rsinφ " x 6 = f 6 x Lon ) = f[x Lon ),u Lon ),w Lon )] 25 Laeral-Direcional Equaions of Moion Dynamics of velociy, posiion, angular rae, and angle primarily ou of he verical plane v = Y / m + gsinφ cosθ ru + pw " x 7 = f 7 y I = cosθ sinψ )u + cosφ cosψ + sinφ sinθ sinψ )v + sinφ cosψ + cosφ sinθ sinψ )w " x 8 = f 8 ) I I I 2 xx zz xz ) I I I 2 xx zz xz p = I zz L + I xz N { I xz I yy I xx I zz ) p + I 2 xz + I zz I zz I yy ) r }q r = I xz L + I xx N { I xz I yy I xx I zz )r + I 2 xz + I xx I xx I yy ) p }q φ = p + qsinφ + r cosφ)anθ " x 11 = f 11 ψ = qsinφ + r cosφ)secθ " x 12 = f 12 ) " x 9 = f 9 ) " x 10 = f 10 x LD ) = f[x LD ),u LD ),w LD )] 26 13

14 Sensiiviy o Small Moions 12 x 12) sabiliy marix for he enire sysem F) = f... 1 x 1 x f x 1 x 12 Four 6 x 6) blocks disinguish longiudinal and laeral-direcional effecs Effecs of longiudinal perurbaions on longiudinal moion Effecs of laeral-direcional perurbaions on longiudinal moion F = Effecs of longiudinal perurbaions on laeral-direcional moion F Lon La Dir F Lon Lon F La Dir F La Dir Effecs of laeral-direcional perurbaions on laeral-direcional moion 27 Sensiiviy o Small Conrol Inpus G) = 12 x 6) conrol marix for he enire sysem δ E δt δ F δ A δ R δ SF δ E δt δ F δ A δ R δ SF δ E 2 δt 2 δ F 2 δ A 2 δ R 2 δ SF Four 6 x 3) blocks disinguish longiudinal and laeraldirecional conrol effecs Effecs of longiudinal conrols on longiudinal moion G = Effecs of longiudinal conrols on laeral-direcional moion G Lon La Dir G Lon Effecs of laeral-direcional conrols on longiudinal moion Lon G La Dir G La Dir Effecs of laeral-direcional conrols on laeral-direcional moion 28 14

15 Sensiiviy o Small Disurbance Inpus u w ) w w ) q w ) w) = v w ) " p w ) r w ) Disurbance inpu vecor and perurbaion Four 6 x 3) blocks disinguish longiudinal and laeral-direcional effecs Effecs of longiudinal disurbances on longiudinal moion L = L Lon La Dir L Lon Effecs of longiudinal disurbances on laeral-direcional moion Axial wind, m / s Normal wind, m / s Piching wind shear, deg / s or rad / s Laeral wind, m / s Rolling wind shear, deg / s or rad / s Yawing wind shear, deg / s or rad / s Effecs of laeral-direcional disurbances on longiudinal moion Lon L La Dir L La Dir " Δu w ) Δw w ) Δq w ) Δw) = Δv w ) Δp w ) Δr w ) Effecs of laeral-direcional disurbances on laeral-direcional moion 29 Decoupling Approximaion for Small Perurbaions from Seady, Level Fligh 30 15

16 Resric he Nominal Fligh Pah o he Verical Plane Nominal laeral-direcional moions are zero x La DirN = 0 x La DirN = 0 Nominal longiudinal equaions reduce o x Lon ) = f Lon x Lon u N = X / m gsinθ N q N w N w N = Z / m + gcosθ N + q N u N x IN = cosθ N )u N + sinθ N )w N z IN = sinθ N )u N + cosθ N )w N q N = M I yy ),u Lon ) Nominal Sae Vecor x Lon x La Dir ) ) Nominal u N ) w N ) x N ) z N ) q N ) = θ N ) θ N = q N 31 Δx Lon Δx La Dir Resric he Nominal Fligh Pah o he Verical Plane However, laeral-direcional perurbaions may no be zero Δx La Dir 0 Δx La Dir 0 ) ) = F ) 0 Lon 0 F La Dir ) + G ) 0 Lon 0 G La Dir ) [See Secion 4.1, Fligh Dynamics, for F and G definiions] Δx Lon Δx La Dir ) ) ) ) Δu Lon Δu La Dir Perurbaion Sae Vecor Δx Lon Δx La Dir ) ) ) ) ) ) ) ) ) ) ) ) Δu Δw Δx Δz Δq ) ) = Δθ Δv Δy Δp Δr Δφ Δψ 32 16

17 Nominal Reference) Moion Plus Perurbaion x N ) = f[x N ),u N )] Δx) = F)Δx) + G)Δu) 33 Resric he Nominal Fligh Pah o Seady, Level Fligh Longiudinal ODEs replaced by Algebraic Equaions for Equilibrium Trim) 0 = X / m gsinθ N q N w N 0 = Z / m + gcosθ N + q N u N V N = cosθ N )u N + sinθ N )w N 0 = sinθ N )u N + cosθ N )w N 0 = M I yy 0 = q N Δx ) = F )Δx ) + G )Δu Trimmed Sae Vecor excep x N) is consan u w x z q θ Trim u Trim w Trim V = N 0 ) z N 0 θ Trim Perurbaion Model: F and G are consan ) = FΔx ) + GΔu ) Longiudinal and laeral-direcional perurbaions are essenially decoupled Δx Lon ) Δx La Dir ) = F Lon 0 Δx Lon ) 0 F La Dir Δx La Dir ) + G Lon 0 Δu Lon ) 0 G La Dir Δu La Dir ) 34 17

18 Small Longiudinal and Laeral- Direcional Perurbaion Effecs Assume he airplane is symmeric and is nominal pah is seady, level fligh Small longiudinal and laeral-direcional perurbaions are approximaely uncoupled from each oher 12 x 12) sysem is block diagonal consan, i.e., linear, ime-invarian LTI) DECOUPLED ino wo separae 6 x 6) sysems F = F Lon 0 0 F La Dir G = G Lon 0 0 G La Dir L = L Lon 0 0 L La Dir 35 6 x 6) LTI Longiudinal Perurbaion Model Dynamic Equaion Δ x Lon ) = F Lon Δx Lon ) + G Lon Δu Lon ) + L Lon Δw Lon ) " Δx 1 Δx 2 Δx 3 Δx Lon = Δx 4 Δx 5 Δx 6 Sae Vecor Lon = " Δu Δw Δx Δz Δq Δθ Conrol Vecor Δu Lon = ΔδT ΔδE ΔδF Disurbance Vecor " Δw Lon = Δu wind Δw wind Δq wind 36 18

19 LTI Longiudinal Response o Iniial Pich Rae 37 6 x 6) LTI Laeral-Direcional Perurbaion Model Dynamic Equaion Δ x La Dir ) = F La Dir Δx La Dir ) + G La Dir Δu La Dir ) + L La Dir Δw La Dir ) Sae Vecor Δx 1 Δx 2 Δx 3 Δx La Dir = Δx 4 Δx 5 Δx 6 La Dir = Δv Δy Δp Δr Δφ Δψ Conrol Vecor Δu La Dir = Δδ A Δδ R ΔδSF ) ) ) Disurbance Vecor " Δw Lon = Δv wind Δp wind Δr wind 38 19

20 LTI Laeral-Direcional Response o Iniial Yaw Rae 39 Nex Time: Longiudinal Dynamics Reading: Fligh Dynamics , Airplane Sabiliy and Conrol Chaper 7 Learning Objecives 6 h -order -> 4 h -order -> hybrid equaions Dynamic sabiliy derivaives Long-period phugoid) mode Shor-period mode 40 20

21 Supplemenal Maerial 41 How Do We Calculae he Parial Derivaives? F) = f x x=x N ) u=u N ) w=w N ) G) = f u x=x N ) u=u N ) w=w N ) L) = f w x=x N ) u=u N ) w=w N ) Analyically Symbolic evaluaion of analyical models of F, G, and L Numerically Firs cenral differences in fx,u,w) 42 21

22 Numerical Esimaion of he Jacobian Marix, F) x 1 ) x 1 ) f 1 f 2 x 1 + Δx 1 ) x 2 x n x 1 + Δx 1 ) x 2 x n f 1 x 1 Δx 1 ) x 2 x n x=xn ) u=u N ) w=w N ) 2Δx 1 ; f 2 x 1 Δx 1 ) x 2 x n x=xn ) u=u N ) w=w N ) 2Δx 1 ; x 2 ) x 2 ) f 1 f 2 x 1 x 1 x 2 + Δx 2 x f 2 Δx 2 1 x n x n ) ) 2Δx 2 ) x 1 x 1 x 2 + Δx 2 x f 2 Δx 2 2 x n x n 2Δx 2 Coninue for all n x n elemens of F) ) x=xn ) u=u N ) w=w N ) x=xn ) u=u N ) w=w N ) 43 Numerical Esimaion of he Jacobian Marix, G) u 1 ) u 1 ) f 1 f 2 u 1 + Δu 1 ) u 2 u m u 1 + Δu 1 ) u 2 u m f 1 u 1 Δu 1 ) u 2 u m x=xn ) u=u N ) w=w N ) 2Δu 1 ; f 2 u 1 Δu 1 ) u 2 u m x=xn ) u=u N ) w=w N ) 2Δu 1 ; u 2 ) u 2 ) f 1 f 2 u 1 u 1 u 2 + Δu 2 u f 2 Δu 2 1 u m u m ) ) 2Δu 2 ) u 1 u 1 u 2 + Δu 2 u f 2 Δu 2 2 u m u m 2Δu 2 Coninue for all n x m elemens of G) ) x=xn ) u=u N ) w=w N ) x=xn ) u=u N ) w=w N ) 44 22

23 w 1 w 1 ) ) f 1 f 2 Numerical Esimaion of he Jacobian Marix, L) w 1 + Δw 1 ) w 2 w s w 1 + Δw 1 ) w 2 w s f 1 w 1 Δw 1 ) w 2 w s x=xn ) u=u N ) w=w N ) 2Δw 1 ; f 2 w 1 Δw 1 ) w 2 w s x=xn ) u=u N ) w=w N ) 2Δw 1 ; w 2 ) w 2 ) f 1 f 2 w 1 w 1 w 2 + Δw 2 w f 2 Δw 2 1 w s w s ) ) 2Δw 2 ) w 1 w 1 w 2 + Δw 2 w f 2 Δw 2 2 w s w s 2Δw 2 ) Coninue for all n x s elemens of L) x=xn ) u=u N ) w=w N ) x=xn ) u=u N ) w=w N ) 45 23