Linearized Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 216 Learning Objectives Develop linear equations to describe small perturbational motions Apply to aircraft dynamic equations Reading: Flight Dynamics 234-242, 255-266, 274-297, 321-325, 329-33 Copyright 216 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/mae331.html http://www.princeton.edu/~stengel/flightdynamics.html 1 Assignment 6 due: End of day, Nov. 22, 216 Code the aerodynamic, thrust, and inertial properties of the HondaJet in AeroModel.m MATLAB function for FLIGHTver2.m 2
Review Questions Describe the functions of airplane control systems: Elevator/Stabilator Ailerons/Elevons Rudder Spoilers How can you control the yaw dynamics of a flying wing airplane? What are compensating ailerons? Why aren t all control surfaces all moving? How are the mechanical dynamics of control systems modeled? What is a horn balance, and how does it work? What are control tabs, and how do they work? 3 How Is System Response Calculated? Linear and nonlinear, time-varying and timeinvariant dynamic models Numerical integration time domain Linear, time-invariant LTI dynamic models Numerical integration time domain State transition time domain Transfer functions frequency domain 4
Integration Algorithms Exact x T = x + f x t Rectangular Euler Integration xt k = xt k1 + xt k1,t k xt k1 + f [ xt k1,ut k1,wt k1 ]t t = t k t k1 T,u t,w t dt Trapezoidal modified Euler Integration ~MATLAB s ode23 xt k xt k1 + 1 [ 2 x + x 1 2 ] where x 1 = f [ xt k1,ut k1,wt k1 ]t = f [ xt k1 + x 1,ut k,wt k ]t See MATLAB manual for descriptions of ode45 and ode15s 5 Numerical Integration: MATLAB Ordinary Differential Equation Solvers * Explicit Runge-Kutta Algorithm Adams-Bashforth-Moulton Algorithm Numerical Differentiation Formula Modified Rosenbrock Method Trapezoidal Rule Trapezoidal Rule w/back Differentiation * http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/ref/ode23.html. Shampine, L. F. and M. W. Reichelt, The MATLAB ODE Suite, SIAM Journal on Scientific Computing, Vol. 18, 1997, pp 6 1-22.
Nominal and Actual Trajectories Nominal or reference trajectory and control history { x N t, u N t,w N t} for t in [t o,t f ] x : dynamic state u : control input w : disturbance input Actual trajectory perturbed by Small initial condition variation, x o t o Small control variation, ut { xt, ut,wt } for t in [t o,t f ] = x N t + xt, u N t + ut,w N t + wt { } 7 Both Paths Satisfy the Dynamic Equations Dynamic models for the actual and the nominal problems are the same x N t = f[x N t,u N t,w N t], x N t o xt = f[xt,ut,wt], given given x t o Differences in initial condition and forcing... xt o = xt o x N t o ut = ut u N t wt = wt w N t in * + t o,t f, -... perturb rate of change and the state xt = x N t + xt xt = x N t + xt in * t o,t f +, 8
Approximate Neighboring Trajectory as a Linear Perturbation to the Nominal Trajectory x N t = f[x N t,u N t,w N t,t] xt = x N t + xt = f[x N t + xt,u N t + ut,w N t + wt,t] Approximate the new trajectory as the sum of the nominal path plus a linear perturbation xt = x N t + xt f[x N t,u N t,w N t,t]+ f x xt + f u ut + f w wt 9 Linearized Equation Approximates Perturbation Dynamics Solve for the nominal and perturbation trajectories separately Nominal Equation x N t = f[x N t,u N t,w N t,t], x N t o given Perturbation Equation dimx = n 1 dimu = m 1 dimw = s 1 xt f xt x x=x N t + f ut u=u N t u w=w N t x=x N t + f wt u=u N t w w=w N t x=x N t u=u N t w=w N t Ftxt+ Gtut+ Ltwt, x t o given dimx = n 1 dimu = m 1 dimw = s 1 1
Jacobian Matrices Express Solution Sensitivity to Small Perturbations Stability matrix, F, is square Ft = f x x=x N t u=u N t w=w N t dimf = n n = f 1 f 1 x 1 x n f n f n x 1 x n x=x N t u=u N t w=w N t 11 Sensitivity to Control Perturbations, G Gt = f = u x=x N t u=u N t w=w N t dimg = n m f 1 f 1 u 1 u m f n f n u 1 u m x=x N t u=u N t w=w N t 12
Sensitivity to Disturbance Perturbations, G Lt = f = w x=x N t u=u N t w=w N t diml = n s f 1 f 1 w 1 w s f n f n w 1 w s x=x N t u=u N t w=w N t 13 Scalar Example of Nominal and Perturbation Equations Actual System x t = x t + 2 t + 3u t + 4w 3 t Nominal System x N t = x N t + 2 N t + 3u N t + 4w 3 N t Perturbation System 2 x t = x t + 4x N tx t + 3u t +12w N tw t 14
Comparison of Damped Linear and Nonlinear Systems Linear Spring x 1 t = t t = 1x 1 t t Displacement Rate of Change Linear Spring Force vs. Displacement Spring Damper Linear plus Stiffening Cubic Spring x 1 t = t t = 1x 1 t 1x 3 1 t t Linear plus Weakening Cubic Spring x 1 t = t t = 1x 1 t +.8x 3 1 t t Cubic Spring Force vs. Displacement Cubic Spring Force vs. Displacement 15 MATLAB Simulation of Linear and Nonlinear Dynamic Systems MATLAB Main Script Nonlinear and Linear Examples clear tspan = [ 1]; xo = [, 1]; [t1,x1 = ode23nonlin,tspan,xo; xo = [, 1]; [t2,x2] = ode23nonlin,tspan,xo; xo = [, 1]; [t3,x3] = ode23lin,tspan,xo; xo = [, 1]; [t4,x4] = ode23lin,tspan,xo; subplot2,1,1 plott1,x1:,1,k,t2,x2:,1,b,t3,x3:,1,r,t4,x4:,1,g ylabelposition, grid subplot2,1,2 plott1,x1:,2,k,t2,x2:,2,b,t3,x3:,2,r,t4,x4:,2,g xlabeltime, ylabelrate, grid Linear System x 1 t = t x 2 t = 1x 1 t t function xdot = Lint,x Linear Ordinary Differential Equation x1 = Position x2 = Rate xdot = [x2-1*x1 - x2]; Nonlinear System x 1 t = t x 2 t = 1x 1 t +.8x 3 1 t t function xdot = NonLint,x Nonlinear Ordinary Differential Equation x1 = Position x2 = Rate xdot = [x2-1*x1 +.8*x1^3 - x2]; 16
Linear and Stiffening Cubic Springs: Small and Large Initial Conditions Cubic term Linear and nonlinear responses are indistinguishable with small initial condition 17 Linear and Weakening Cubic Springs: Small and Large Initial Conditions Cubic term 18
Linear, Time-Varying LTV Approximation of Perturbation Dynamics 19 Stiffening Linear-Cubic Spring Example x 1N N Nonlinear, time-invariant NTI equation x 1 t = f 1 = t t = f 2 = 1x 1 t 1x 1 3 t t Integrate equations to produce nominal path t f f 1N f 2N dt x 1N t N t Analytical evaluation of partial derivatives in,t f f 1 x 1 = ; f 1 = 1 f 2 = 1 3 1N t; x 1 f 2 = 1 f 1 u = ; f 1 w = f 2 u = ; f 2 w = 2
Nominal NTI and Perturbation LTV Dynamic Equations Nonlinear, time-invariant NTI nominal equation x N t = f[x N t], x N given x 1N t = N t N t = 1x 1N t 1x 1N 3 t N t x 1N N Example = 9 Perturbations approximated by linear, time-varying LTV equation x 1 t t xt = Ftxt, x given = 1 1 + 3 1N t 1 x 1 t t Example x 1 = 1 21 Comparison of Approximate and Exact Solutions Initial Conditions N = 9 =1 N t+ t =1 t =1 x N t xt x N t+ xt xt x N t xt x N t + xt xt 22
Suppose Nominal Initial Condition is Zero Nominal solution remains at equilibrium x N t = f[x N t], x N =, x N t = in [, ] Perturbation equation is linear and time-invariant LTI x 1 t t = 1 1 3 1 x 1 t t 23 Separation of the Equations of Motion into Longitudinal and Lateral- Directional Sets 24
Grumman F9F Rate of change of Translational Velocity Rate of change of Translational Position Rate of change of Angular Velocity I xy = I yz = Rate of change of Angular Position Rigid-Body Equations of Motion Scalar Notation u = X / m gsin + rv qw v = Y / m + gsin cos ru + pw w = Z / m + g cos cos + qu pv x I = cos cos u + cos sin + sin sin cos v + sin sin + cos sin cos w y I = cos sin u + cos cos + sin sin sin v + sin cos + cos sin sin w z I = sin u + sin cos v + cos cos w 2 I xx I zz I 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 pr I xz p 2 r 2 q = M I xx I zz I yy 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 = p + qsin + r cos tan = qcos r sin = qsin + r cos sec 2 I xx I zz I xz State Vector x 1 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 1 x 11 x 12 u v w x y z = p q r 25 Reorder the State Vector x 1 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 1 x 11 x 12 u v w x y z = p q r First six elements of the state are longitudinal variables Second six elements of the state are lateraldirectional variables x 1 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 1 x 11 x 12 new = x Lon x LatDir = u w x z q v y p r * 26
Longitudinal Equations of Motion Dynamics of velocity, position, angular rate, and angle primarily in the vertical plane u = X / m gsin + rv qw x 1 = f 1 w = Z / m + gcos cos + qu pv = f 2 x I = cos cos u + cos sin + sin sin cos v + sin sin + cos sin cos w x 3 = f 3 z I = sin u + sin cos v + cos cos w x 4 = f 4 pr I xz p 2 r 2 q = M I xx I zz I yy x 5 = f 5 = qcos rsin x 6 = f 6 x Lon t = f[x Lon t,u Lon t,w Lon t] 27 Lateral-Directional Equations of Motion Dynamics of velocity, position, angular rate, and angle primarily out of the vertical plane v = Y / m + gsin cos ru + pw x 7 = f 7 y I = cos sin u + cos cos + sin sin sin v + sin cos + cos sin sin w x 8 = f 8 I 2 xxi zz I 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 = p + qsin + r cos tan x 11 = f 11 = qsin + r cos sec x 12 = f 12 x 9 = f 9 x 1 = f 1 x LD t = f[x LD t,u LD t,w LD t] 28
Sensitivity to Small Motions 12 x 12 stability matrix for the entire system Ft = f 1 f... 1 x 1 x 12......... f 12 f... 12 x 1 x 12 Four 6 x 6 blocks distinguish longitudinal and lateral-directional effects Effects of longitudinal perturbations on longitudinal motion Effects of lateral-directional perturbations on longitudinal motion F = Effects of longitudinal perturbations on lateral-directional motion F Lon LatDir F Lon Lon F LatDir F LatDir Effects of lateral-directional perturbations on lateral-directional motion 29 Sensitivity to Small Control Inputs Gt = 12 x 6 control matrix for the entire system f 1 E f 1 T f 1 F f 1 A f 1 R f 1 SF f 2 E f 2 T f 2 F f 2 A f 2 R f 2 SF.................. f 12 E f 12 T f 12 F f 12 A f 12 R f 12 SF Four 6 x 3 blocks distinguish longitudinal and lateraldirectional control effects Effects of longitudinal controls on longitudinal motion G = Effects of longitudinal controls on lateral-directional motion G Lon LatDir G Lon Effects of lateral-directional controls on longitudinal motion Lon G LatDir G LatDir Effects of lateral-directional controls on lateral-directional motion 3
Sensitivity to Small Disturbance Inputs Disturbance input vector and perturbation wt = u w t w w t q w t v w t p w t r w t Axial wind, m / s Normal wind, m / s Pitching wind shear, deg / s or rad / s Lateral wind, m / s Rolling wind shear, deg / s or rad / s Yawing wind shear, deg / s or rad / s wt = u w t w w t q w t v w t p w t r w t Four 6 x 3 blocks distinguish longitudinal and lateral-directional effects Effects of longitudinal disturbances on longitudinal motion L = L Lon LatDir L Lon Effects of longitudinal disturbances on lateral-directional motion Effects of lateral-directional disturbances on longitudinal motion Lon L LatDir L LatDir Effects of lateral-directional disturbances on lateral-directional motion 31 Decoupling Approximation for Small Perturbations from Steady, Level Flight 32
Restrict the Nominal Flight Path to the Vertical Plane Nominal lateral-directional motions are zero x Lat DirN = x Lat DirN = Nominal longitudinal equations reduce to u N = X / m gsin N q N w N w N = Z / m + gcos N + q N u N x IN = cos N u N + sin N w N z IN = sin N u N + cos N w N q N = M I yy N = q N x 1 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 1 x 11 x 12 Nominal State Vector N = x Lon x LatDir = N 33 u N w N x N z N q N N Restrict the Nominal Flight Path to Steady, Level Flight Specify nominal airspeed V N and altitude h N = z N Calculate conditions for trimmed equilibrium flight See Flight Dynamics and FLIGHT program for a solution method Trimmed State Vector is constant = X / m gsin N q N w N = Z / m + gcos N + q N u N V N = cos N u N + sin N w N = sin N u N + cos N w N = M I yy = q N u w x z q Trim = u Trim w Trim V N t t z N Trim 34
Small Longitudinal and Lateral- Directional Perturbation Effects Assume the airplane is symmetric and its nominal path is steady, level flight Small longitudinal and lateral-directional perturbations are approximately uncoupled from each other 12 x 12 system is block diagonal constant, i.e., linear, time-invariant LTI decoupled into two separate 6 x 6 systems F = F Lon F LatDir G = G Lon G LatDir L = L Lon L LatDir 35 6 x 6 LTI Longitudinal Perturbation Model Dynamic Equation x Lon t = F Lon x Lon t + G Lon u Lon t + L Lon w Lon t x Lon = State Vector x 1 x 3 x 4 x 5 x 6 Lon = u w x z q Control Vector u Lon = T E F Disturbance Vector w Lon = u wind w wind q wind 36
LTI Longitudinal Response to Initial Pitch Rate 37 6 x 6 LTI Lateral-Directional Perturbation Model Dynamic Equation x Lat Dir t = F Lat Dir x Lat Dir t + G Lat Dir u Lat Dir t + L Lat Dir w Lat Dir t x Lat Dir = State Vector x 1 x 3 x 4 x 5 x 6 Lat Dir v y p = r * Control Vector u Lat Dir = A R SF Disturbance Vector w Lon = v wind p wind r wind 38
LTI Lateral-Directional Response to Initial Yaw Rate 39 Next Time: Longitudinal Dynamics Reading: Flight Dynamics 452-464, 482-486 Airplane Stability and Control Chapter 7 Learning Objectives 6 th -order -> 4 th -order -> hybrid equations Dynamic stability derivatives Long-period phugoid mode Short-period mode 4
Supplemental Material 41 How Do We Calculate the Partial Derivatives? Ft = f x x=x N t u=u N t w=w N t Gt = f u x=x N t u=u N t w=w N t Lt = f w x=x N t u=u N t w=w N t Numerically First differences in fx,u,w Analytically Symbolic evaluation of analytical models of F, G, and L 42
Numerical Estimation of the Jacobian Matrix, Ft f 1 t x 1 f 2 t x 1 f 1 f 2 x 1 + x 1 x n x 1 + x 1 x n * f 1 x 1 * x 1 x n x=xn t u=u N t w=w N t 2x 1 ; * f 2 x 1 * x 1 x n x=xn t u=u N t w=w N t 2x 1 ; f 1 t f 2 t f 1 f 2 x 1 + x n x 1 + x n * f 1 2 * f 2 2 Continue for all n x n elements of Ft x 1 * x n x 1 * x n x=xn t u=u N t w=w N t x=xn t u=u N t w=w N t 43 Numerical Estimation of the Jacobian Matrix, Gt f 1 t u 1 f 2 t u 1 f 1 f 2 u 1 + u 1 u 2 u m u 1 + u 1 u 2 u m * f 1 u 1 * u 1 u 2 u m x=xn t u=u N t w=w N t 2u 1 ; * f 2 u 1 * u 1 u 2 u m x=xn t u=u N t w=w N t 2u 1 ; f 1 t u 2 f 2 t u 2 f 1 f 2 u 1 u 2 + u 2 u m u 1 u 2 + u 2 u m * f 1 2u 2 * f 2 2u 2 Continue for all n x m elements of Gt u 1 u 2 * u 2 u m u 1 u 2 * u 2 u m x=xn t u=u N t w=w N t x=xn t u=u N t w=w N t 44
f 1 t w 1 f 2 t w 1 f 1 f 2 Numerical Estimation of the Jacobian Matrix, Lt w 1 + w 1 w 2 w s w 1 + w 1 w 2 w s * f 1 w 1 * w 1 w 2 w s x=xn t u=u N t w=w N t 2w 1 ; * f 2 w 1 * w 1 w 2 w s x=xn t u=u N t w=w N t 2w 1 ; f 1 t w 2 f 2 t w 2 f 1 f 2 w 1 w 2 + w 2 w s w 1 w 2 + w 2 w s * f 1 2w 2 * f 2 Continue for all n x s elements of Lt 2w 2 w 1 w 2 * w 2 w s w 1 w 2 * w 2 w s 45 x=xn t u=u N t w=w N t x=xn t u=u N t w=w N t