Introduction to State-Space Models

Topic #5 16.30/31 Feedback Control Systems State-Space Systems • What are state-space models? • Why should we use them? • How are they related to the transfer functions used in classical control design and how do we develop a statespace model? • What are the basic properties of a state-space model, and how do we analyze these? Fall 2010 16.30/31 5–2 SS Introduction • State space model: a representation of the dynamics of an N th order system as a first order differential equation in an N -vector, which is called the state. • Convert the N th order differential equation that governs the dy­ namics into N first-order differential equations • Classic example: second order mass-spring system mp̈ + cṗ + kp = F • Let x1 = p, then x2 = ṗ = ẋ1, and ẋ2 = p̈ = (F − cṗ − kp)/m = (F − cx2 − kx1)/m � � � �� � � � ṗ 0 1 p 0 ⇒ = + u p̈ −k/m −c/m ṗ 1/m • Let u = F and introduce the state � � � � x1 p x= = ⇒ ẋ = Ax + Bu x2 ṗ • If the measured output of the system is the position, then we have that � � � � � � p � � x1 y=p= 1 0 = 1 0 = Cx ṗ x2 September 21, 2010 Fall 2010 16.30/31 5–3 • Most general continuous-time linear dynamical system has form ẋ(t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) + D(t)u(t) where: • t ∈ R denotes time • x(t) ∈ Rn is the state (vector) • u(t) ∈ Rm is the input or control • y(t) ∈ Rp is the output • A(t) ∈ Rn×n is the dynamics matrix • B(t) ∈ Rn×m is the input matrix • C(t) ∈ Rp×n is the output or sensor matrix • D(t) ∈ Rp×m is the feedthrough matrix • Note that the plant dynamics can be time-varying. • Also note that this is a multi-input / multi-output (MIMO) system. • We will typically deal with the time-invariant case ⇒ Linear Time-Invariant (LTI) state dynamics ẋ(t) = Ax(t) + Bu(t) y(t) = Cx(t) + Du(t) so that now A, B, C, D are constant and do not depend on t. September 21, 2010 Fall 2010 16.30/31 5–4 Basic Definitions • Linearity – What is a linear dynamical system? A system G is linear with respect to its inputs and output u(t) → G(s) → y(t) iff superposition holds: G(α1u1 + α2u2) = α1Gu1 + α2Gu2 So if y1 is the response of G to u1 (y1 = Gu1), and y2 is the response of G to u2 (y2 = Gu2), then the response to α1u1 + α2u2 is α1y1 + α2y2 • A system is said to be time-invariant if the relationship between the input and output is independent of time. So if the response to u(t) is y(t), then the response to u(t − t0) is y(t − t0) • Example: the system ẋ(t) = 3x(t) + u(t) y(t) = x(t) is LTI, but ẋ(t) = 3t x(t) + u(t) y(t) = x(t) is not. • A matrix of second system is a function of absolute time, so re­ sponse to u(t) will differ from response to u(t − 1). September 21, 2010 Fall 2010 16.30/31 5–5 • x(t) is called the state of the system at t because: • Future output depends only on current state and future input • Future output depends on past input only through current state • State summarizes effect of past inputs on future output – like the memory of the system • Example: Rechargeable flashlight – the state is the current state of charge of the battery. If you know that state, then you do not need to know how that level of charge was achieved (assuming a perfect battery) to predict the future performance of the flashlight. • But to consider all nonlinear effects, you might also need to know how many cycles the battery has gone through • Key point is that you might expect a given linear model to ac­ curately model the charge depletion behavior for a given number of cycles, but that model would typically change with the number cycles September 21, 2010 Fall 2010 16.30/31 5–6 Creating State-Space Models • Most easily created from N th order differential equations that describe the dynamics • This was the case done before. • Only issue is which set of states to use – there are many choices. • Can be developed from transfer function model as well. • Much more on this later • Problem is that we have restricted ourselves here to linear state space models, and almost all systems are nonlinear in real-life. • Can develop linear models from nonlinear system dynamics September 21, 2010 Fall 2010 16.30/31 5–7 Equilibrium Points • Often have a nonlinear set of dynamics given by ẋ = f (x, u) where x is once gain the state vector, u is the vector of inputs, and f (·, ·) is a nonlinear vector function that describes the dynamics • First step is to define the point about which the linearization will be performed. • Typically about equilibrium points – a point for which if the system starts there it will remain there for all future time. • Characterized by setting the state derivative to zero: ẋ = f (x, u) = 0 • Result is an algebraic set of equations that must be solved for both xe and ue • Note that ẋe = 0 and u̇e = 0 by definition • Typically think of these nominal conditions xe, ue as “set points” or “operating points” for the nonlinear system. • Example – pendulum dynamics: θ¨ + rθ̇ + gl sin θ = 0 can be written in state space form as � � � � ẋ1 x2 = ẋ2 −rx2 − gl sin x1 • Setting f (x, u) = 0 yields x2 = 0 and x2 = − rlg sin x1, which implies that x1 = θ = {0, π} September 21, 2010 Fall 2010 16.30/31 5–8 Linearization • Typically assume that the system is operating about some nominal state solution xe (possibly requires a nominal input ue) • Then write the actual state as x(t) = xe + δx(t) and the actual inputs as u(t) = ue + δu(t) • The “δ” is included to denote the fact that we expect the variations about the nominal to be “small” • Can then develop the linearized equations by using the Taylor series expansion of f (·, ·) about xe and ue. • Recall the vector equation ẋ = f (x, u), each equation of which ẋi = fi(x, u) can be expanded as d (xei + δxi) = fi(xe + δx, ue + δu) dt � � ∂fi �� ∂fi �� ≈ fi(xe, ue) + δx + δu ∂x �0 ∂u �0 where � � ∂fi ∂fi ∂fi = ··· ∂x ∂xn ∂x1 and ·|0 means that we should evaluate the function at the nominal values of xe and ue. • The meaning of “small” deviations now clear – the variations in δx and δu must be small enough that we can ignore the higher order terms in the Taylor expansion of f (x, u). September 21, 2010 Fall 2010 • Since 16.30/31 5–9 d dt xe i = fi(xe, ue), we thus have that � � d ∂fi �� ∂fi �� (δxi) ≈ δx + δu dt ∂x �0 ∂u �0 • Combining for all n state equations, gives (note that we also set “≈” → “=”) that � ⎤ � ⎤ ⎡ ⎡ � ∂f1 � ∂f1 �� ⎢ ∂x � ⎥ ⎢ ∂u � ⎥ ⎢ ⎥ ⎢ 0 � ⎥ �0 ⎥ ⎢ ⎢ ⎥ ⎢ ∂f2 �� ⎥ ⎢ ∂f2 �� ⎥ ⎢ ⎥ ⎢ ⎥ d ⎢ ∂x �0 ⎥ ⎢ ∂u �0 ⎥ δx = ⎢ ⎥ δx + ⎢ ⎥ δu ⎢ ... ⎥ ⎢ ... ⎥ dt ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ � ⎥ � ⎥ ⎢ ⎢ ⎣ ∂fn � ⎦ ⎣ ∂fn � ⎦ � � � ∂x 0 ∂u �0 = A(t)δx + B(t)δu where ⎡ ⎢ ⎢ ⎢ A(t) ≡ ⎢ ⎢ ⎢ ⎣ September 21, 2010 ∂f1 ∂x1 ∂f2 ∂x1 ∂fn ∂x1 ∂f1 ∂x2 ∂f2 ∂x2 ∂fn ∂x2 ··· ··· ... ··· ∂f1 ∂xn ∂f2 ∂xn ∂fn ∂xn ⎤ ⎡ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎢ ⎢ ⎢ and B(t) ≡ ⎢ ⎢ ⎢ ⎣ 0 ∂f1 ∂u1 ∂f2 ∂u1 ∂fn ∂u1 ∂f1 ∂u2 ∂f2 ∂u2 ∂fn ∂u2 ··· ··· ... ··· ∂f1 ∂um ∂f2 ∂um ∂fn ∂um ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ 0 Fall 2010 16.30/31 5–10 • Similarly, if the nonlinear measurement equation is y = g(x, u) and y(t) = ye + δy, then � ⎤ � ⎤ ⎡ ⎡ � ∂g1 � ∂g1 �� ⎢ ∂u � ⎥ ⎢ ∂x � ⎥ ⎢ . 0⎥ ⎢ . 0⎥ ⎥ ⎢ . . ⎥ δy = ⎢ . � ⎥ δx + ⎢ ⎢ . � ⎥ δu ⎣ ∂gp � ⎦ ⎣ ∂gp � ⎦ � � � ∂x 0 ∂u �0 = C(t)δx + D(t)δu • Typically drop the “δ” as they are rather cumbersome, and (abusing notation) we write the state equations as: ẋ(t) = A(t)x(t) + B(t)u(t) y(t) = C(t)x(t) + D(t)u(t) which is of the same form as the previous linear models • If the system is operating around just one set point then the partial fractions in the expressions for A–D are all constant → LTI lin­ earized model. September 21, 2010 Fall 2010 16.30/31 5–11 Stability of LTI Systems • Consider a solution xs(t) to a differential equation for a given initial condition xs(t0). • Solution is stable if other solutions xb(t0) that start near xs(t0) stay close to xs(t) ∀ t ⇒ stable in sense of Lyapunov (SSL). • If other solutions are SSL, but the xb(t) do not converge to xs(t) ⇒ solution is neutrally stable. • If other solutions are SSL and xb(t) → x(t) as t → ∞ ⇒ solution is asymptotically stable. • A solution xs(t) is unstable if it is not stable. • Note that a linear (autonomous) system ẋ = Ax has an equilibrium point at xe = 0 • This equilibrium point is stable if and only if all of the eigenvalues of A satisfy Rλi(A) ≤ 0 and every eigenvalue with Rλi(A) = 0 has a Jordan block of order one.1 • Thus the stability test for a linear system is the familiar one of determining if Rλi(A) ≤ 0 • Somewhat surprisingly perhaps, we can also infer stability of the orig­ inal nonlinear from the analysis of the linearized system model 1 more on Jordan blocks on 6–??, but this basically means that these eigenvalues are not repeated. September 21, 2010 Fall 2010 16.30/31 5–12 • Lyapunov’s indirect method2 Let xe = 0 be an equilibrium point for the nonlinear autonomous system ẋ(t) = f (x(t)) where f is continuously differentiable in a neighborhood of xe. As­ sume � ∂f �� A= ∂x �xe Then: • The origin is an asymptotically stable equilibrium point for the nonlinear system if Rλi(A) < 0 ∀ i • The origin is unstable if Rλi(A) > 0 for any i • Note that this doesn’t say anything about the stability of the nonlinear system if the linear system is neutrally stable. • A very powerful result that is the basis of all linear control theory. 2 Much more on Lyapunov methods later too. September 21, 2010 Fall 2010 16.30/31 5–13 Linearization Example • Example: simple spring. With a mass at the end of a linear spring (rate k) we have the dynamics mẍ = −kx but with a “leaf spring” as is used on car suspensions, we have a nonlinear spring – the more it deflects, the stiffer it gets. Good model now is mẍ = −k1x − k2x3 which is a “cubic spring”. Fig. 1: Leaf spring from http://en.wikipedia.org/wiki/Image:Leafs1.jpg • Restoring force depends on deflection x in a nonlinear way. Fig. 2: Response to linear k = 1 and nonlinear (k1 = k, k2 = −2) springs (code at the end) September 21, 2010 Fall 2010 16.30/31 5–14 • Consider the nonlinear spring with (set m = 1) ÿ = −k1y − k2y 3 gives us the nonlinear model (x1 = y and x2 = ẏ) � � � � d y ẏ = ⇒ ẋ = f (x) −k1y − k2y 3 dt ẏ • Find the equilibrium points and then make a state space model • For the equilibrium points, we must solve � � ẏ f (x) = =0 −k1y − k2y 3 which gives ẏe = 0 and k1ye + k2(ye)3 = 0 � • Second condition corresponds to ye = 0 or ye = ± −k1/k2, which is only real if k1 and k2 are opposite signs. • For the state space model, ⎡ ∂f1 ∂f1 ⎢ ∂x1 ∂x2 A=⎢ ⎣ ∂f2 ∂f2 ∂x1 ∂x2 ⎤ ⎡ ⎥ ⎥ ⎣ ⎦ = 0 −k1 − 3k2(y)2 0 ⎦ 0 0 ⎡ = ⎣ 0 1 −k1 − 3k2(ye)2 0 and the linearized model is δ˙x = Aδx September 21, 2010 1 ⎤ ⎤ ⎦ Fall 2010 16.30/31 5–15 • For the equilibrium point ye = 0, ẏe = 0 ⎡ A0 = ⎣ 0 1 −k1 0 ⎤ ⎦ which are the standard dynamics of a system with just a linear spring of stiffness k1 • Stable motion about y = 0 if k1 > 0 • Assume that k1 = −1,√k2 = 1/2, then we should get an equilibrium point at ẏ = 0, y = ± 2, and since k1 + k2(ye)2 = 0 then ⎡ ⎤ 0 1 ⎦ A1 = ⎣ −2 0 which are the dynamics of a stable oscillator about the equilibrium point Fig. 3: Nonlinear response (k1 = −1, k2 = 0.5). The figure on the right shows the oscillation about the equilibrium point. September 21, 2010 Fall 2010 16.30/31 5–16 Code: Nonlinear System Sim 1 2 3 function test=nlplant(ft); global k1 k2 x0 = [−1 2]/10; 4 5 6 k1=1;k2=0; [T,x]=ode23('plant', [0 20], x0); % linear 7 8 9 k2=−2; [T1,x1]=ode23('plant', [0 20], x0); %nonlinear 10 11 12 13 14 15 16 17 18 19 20 21 22 23 figure(1);clf; subplot(211) plot(T,x(:,1),T1,x1(:,1),'−−'); legend('Linear','Nonlinear') ylabel('X','FontSize',ft) xlabel('Time','FontSize',ft) subplot(212) plot(T,x(:,2),T1,x1(:,2),'−−'); legend('Linear','Nonlinear') ylabel('V','FontSize',ft) xlabel('Time','FontSize',ft) text(4,0.3,['k 2=',num2str(k2)],'FontSize',ft) return 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 % use the following to cll the function above close all set(0, 'DefaultAxesFontSize', 12, 'DefaultAxesFontWeight','demi') set(0, 'DefaultTextFontSize', 12, 'DefaultTextFontWeight','demi') set(0,'DefaultAxesFontName','arial') set(0,'DefaultAxesFontSize',12) set(0,'DefaultTextFontName','arial') set(gcf,'DefaultLineLineWidth',2); set(gcf,'DefaultlineMarkerSize',10) global k1 k2 nlplant(14) print −f1 −dpng −r300 nlplant.png k1=−1;k2=0.5; % call plant.m x0 = [sqrt(−k1/k2) .25]; [T,x]=ode23('plant', [0:.001:32], x0); figure(1);subplot(211);plot(T,x(:,1));ylabel('y');xlabel('Time');grid subplot(212);plot(T,x(:,2));ylabel('dy/dt');xlabel('Time');grid figure(2);plot(x(:,1),x(:,2));grid hold on;plot(x0(1),0,'rx','MarkerSize',20);hold off; xlabel('y');ylabel('dy/dt') axis([1.2 1.7 −.25 .25]);axis('square') 47 48 49 1 2 3 4 5 6 print −f1 −dpng −r300 nlplant2.png print −f2 −dpng −r300 nlplant3.png function [xdot] = plant(t,x); % plant.m global k1 k2 xdot(1) = x(2); xdot(2) = −k1*x(1)−k2*(x(1))ˆ3; xdot = xdot'; September 21, 2010 Fall 2010 16.30/31 5A–1 Linearization Example: Aircraft Dynamics • The basic dynamics are: �˙ I F� = m�v̇cI and T� = H 1 � F = �v̇cB + BI ω � × �vc m �˙ B + BI ω � ⇒ T� = H � ×H ⇒ Transport Thm. q u X Y v p • Basic assumptions are: w 1. Earth is an inertial reference frame 2. A/C is a rigid body 3. Body frame B fixed to the aircraft (�i, �j, �k) • Instantaneous mapping of �vc and BI ω � = P�i + Q�j + R�k ⎡ ⎤ P ⇒ BI ωB = ⎣ Q ⎦ R BI r Z Image by MIT OpenCourseWare. ω � into the body frame: �vc = U�i + V �j + W �k ⎡ ⎤ U ⇒ (vc)B = ⎣ V ⎦ W • If x and z axes in plane of symmetry, can show that Ixy = Iyz = 0, but value of Ixz depends on specific body frame selected. • Instantaneous mapping of angular momentum � = Hx�i + Hy�j + Hz�k H into the body frame given by ⎡ ⎤ ⎡ ⎤⎡ ⎤ Hx Ixx 0 Ixz P HB = ⎣ Hy ⎦ = ⎣ 0 Iyy 0 ⎦ ⎣ Q ⎦ Hz Ixz 0 Izz R September 21, 2010 Fall 2010 16.30/31 5A–2 • The overall equations of motion are then: 1 � F = �v̇cB + BI ω � × �vc m ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ U̇ X 0 −R Q U 1 ⇒ ⎣ Y ⎦ = ⎣ V̇ ⎦ + ⎣ R 0 −P ⎦ ⎣ V ⎦ m −Q P 0 W Z Ẇ ⎡ ⎤ U̇ + QW − RV = ⎣ V̇ + RU − P W ⎦ Ẇ + P V − QU �˙ B + T� = H BI � ω � ×H ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤ IxxṖ + Ixz Ṙ L 0 −R Q Ixx 0 Ixz P ⎦+⎣ R ⇒⎣M ⎦ = ⎣ 0 −P ⎦ ⎣ 0 Iyy 0 ⎦ ⎣ Q ⎦ Iyy Q̇ N −Q P 0 Ixz 0 Izz R Izz Ṙ + Ixz Ṗ ⎡ ⎤ IxxṖ + Ixz Ṙ +QR(Izz − Iyy ) + P QIxz = ⎣ Iyy Q̇ +P R(Ixx − Izz ) + (R2 − P 2)Ixz ⎦ Izz Ṙ + Ixz Ṗ +P Q(Iyy − Ixx) − QRIxz • Equations are very nonlinear and complicated, and we have not even said where F� and T� come from ⇒ need to linearize to develop analytic results • Assume that the aircraft is flying in an equilibrium condition and we will linearize the equations about this nominal flight condition. September 21, 2010 Fall 2010 16.30/31 5A–3 Linearization • Can linearize about various steady state conditions of flight. • For steady state flight conditions must have F� = F�aero + F�gravity + F�thrust = 0 and T� = 0 ∗ So for equilibrium condition, forces balance on the aircraft L = W and T = D • Also assume that Ṗ = Q̇ = Ṙ = U̇ = V̇ = Ẇ = 0 • Impose additional constraints that depend on flight condition: ∗ Steady wings-level flight → Φ = Φ˙ = Θ˙ = Ψ˙ = 0 • Define the trim angular rates and velocities ⎡ ⎤ ⎡ ⎤ P Uo BI o o ωB = ⎣ Q ⎦ (vc)B = ⎣ 0 ⎦ R 0 which are associated with the flight condition. In fact, these define the type of equilibrium motion that we linearize about. Note: • W0 = 0 since we are using the stability axes, and • V0 = 0 because we are assuming symmetric flight September 21, 2010 Fall 2010 16.30/31 5A–4 • Proceed with linearization of the dynamics for various flight conditions Nominal Velocity Perturbed Velocity ⇒ ⇒ Perturbed Acceleration Velocities U0 , W0 = 0, V0 = 0, U = U0 + u W =w V =v ⇒ ⇒ ⇒ U̇ = u̇ Ẇ = ẇ V̇ = v̇ Angular Rates P0 = 0, Q0 = 0, R0 = 0, P =p Q=q R=r ⇒ ⇒ ⇒ Ṗ = ṗ Q̇ = q̇ Ṙ = ṙ Angles Θ0 , Φ0 = 0, Ψ0 = 0, Θ = Θ0 + θ Φ=φ Ψ=ψ ⇒ ⇒ ⇒ Θ̇ = θ̇ Φ̇ = φ̇ Ψ˙ = ψ̇ September 21, 2010 Fall 2010 16.30/31 5A–5 • Linearization for symmetric flight U = U0 + u, V0 = W0 = 0, P0 = Q0 = R0 = 0. Note that the forces and moments are also perturbed. 1 [X0 + ΔX] = U̇ + QW − RV ≈ u̇ + qw − rv ≈ u̇ m 1 [Y0 + ΔY ] = V̇ + RU − P W m ≈ v̇ + r(U0 + u) − pw ≈ v̇ + rU0 1 ˙ + P V − QU ≈ ẇ + pv − q(U0 + u) [Z0 + ΔZ] = W m ≈ ẇ − qU0 ⎡ ⎤ ⎡ ⎤ ΔX u̇ 1 ⇒ ⎣ ΔY ⎦ = ⎣ v̇ + rU0 ⎦ m ΔZ ẇ − qU0 1 2 3 • Attitude motion: ⎡ ⎤ ⎡ ⎤ IxxṖ + Ixz Ṙ +QR(Izz − Iyy ) + P QIxz L ⎣M ⎦ = ⎣ Iyy Q̇ +P R(Ixx − Izz ) + (R2 − P 2)Ixz ⎦ N Izz Ṙ + Ixz Ṗ +P Q(Iyy − Ixx) − QRIxz ⎡ ⎤ ⎡ ⎤ ΔL Ixxṗ + Ixz ṙ ⎦ ⇒ ⎣ ΔM ⎦ = ⎣ Iyy q̇ ΔN Izz ṙ + Ixz ṗ September 21, 2010 4 5 6 Fall 2010 16.30/31 5A–6 • To understand equations in detail, and the resulting impact on the vehicle dynamics, we must investigate terms ΔX . . . ΔN . • We must also address the left-hand side (F� , T� ) • Net forces and moments must be zero in equilibrium condition. • Aerodynamic and Gravity forces are a function of equilibrium con­ dition AND the perturbations about this equilibrium. • Predict the changes to the aerodynamic forces and moments using a first order expansion in the key flight parameters ∂X ∂X ∂X ∂X ∂X g ΔX = ΔU + ΔW + ΔẆ + ΔΘ + . . . + ΔΘ + ΔX c ∂U ∂W ∂Θ ∂Θ ∂Ẇ g ∂X ∂X ∂X ∂X ∂X = u+ w+ ẇ + θ + ... + θ + ΔX c ∂U ∂W ∂Θ ∂Θ ∂Ẇ • ∂X ∂U called stability derivative – evaluated at eq. condition. • Clearly approximation since ignores lags in aerodynamics forces (as­ sumes that forces only function of instantaneous values) September 21, 2010 Fall 2010 16.30/31 5A–7 Stability Derivatives • First proposed by Bryan (1911) – has proven to be a very effective way to analyze the aircraft flight mechanics – well supported by numerous flight test comparisons. • The forces and torques acting on the aircraft are very complex nonlin­ ear functions of the flight equilibrium condition and the perturbations from equilibrium. • Linearized expansion can involve many terms u, u, ˙ u, ¨ . . . , w, w, ˙ w, ¨ ... • Typically only retain a few terms to capture the dominant effects. • Dominant behavior most easily discussed in terms of the: • Symm. variables: U , W , Q & forces/torques: X, Z, and M • Asymm. variables: V , P , R & forces/torques: Y , L, and N • Observation – for truly symmetric flight Y , L, and N will be exactly zero for any value of U , W , Q ⇒ Derivatives of asymmetric forces/torques with respect to the sym­ metric motion variables are zero. • Further (convenient) assumptions: 1. Derivatives of symmetric forces/torques with respect to the asym­ metric motion variables are small and can be neglected. 2. We can neglect derivatives with respect to the derivatives of the motion variables, but keep ∂Z/∂ ẇ and Mẇ ≡ ∂M/∂ ẇ (aero­ dynamic lag involved in forming new pressure distribution on the wing in response to the perturbed angle of attack) 3. ∂X/∂q is negligibly small. September 21, 2010 Fall 2010 16.30/31 5A–8 ∂()/∂() X u • v 0 w • p 0 q ≈0 r 0 Y 0 • 0 • 0 • Z • 0 • 0 • 0 L 0 • 0 • 0 • M • 0 • 0 • 0 N 0 • 0 • 0 • • Note that we must also find the perturbation gravity and thrust forces and moments � � ∂X g �� ∂Z g �� = −mg cos Θ0 = −mg sin Θ0 ∂Θ �0 ∂Θ �0 • Aerodynamic summary: � ∂X � � ∂X � 1A ΔX = ∂U 0 u + ∂W 0 w ⇒ ΔX ∼ u, αx ≈ w/U0 2A ΔY ∼ β ≈ v/U0, p, r 3A ΔZ ∼ u, αx ≈ w/U0, α̇x ≈ ẇ/U0, q 4A ΔL ∼ β ≈ v/U0, p, r 5A ΔM ∼ u, αx ≈ w/U0, α̇x ≈ ẇ/U0, q 6A ΔN ∼ β ≈ v/U0, p, r September 21, 2010 Fall 2010 16.30/31 5A–9 • Result is that, with these force, torque approximations, equations 1, 3, 5 decouple from 2, 4, 6 • 1, 3, 5 are the longitudinal dynamics in u, w, and q ⎡ ⎤ ⎡ ⎤ ΔX mu̇ ⎣ ΔZ ⎦ = ⎣ m(ẇ − qU0) ⎦ ΔM Iyy q̇ ⎡ � ∂X � ∂U 0 u + � ∂X � ∂W 0 w + � ∂X g � c θ + ΔX ⎢ � � � � ⎢� � ⎢ ∂Z u + � ∂Z � w + ∂Z ẇ + ∂Z q + � ∂Z g � θ + ΔZ c ≈ ⎢ ∂U 0 ∂Q 0 ∂Θ 0 ∂W 0 ∂Ẇ 0 ⎢ � � � � ⎣� � � ∂M � ∂M ∂M ẇ + ∂M q + ΔM c ∂Q ∂U 0 u + ∂W 0 w + ∂Ẇ ∂Θ 0 0 0 • 2, 4, 6 are the lateral dynamics in v, p, and r ⎡ ⎤ ⎡ ⎤ ΔY m(v̇ + rU0) ⎣ ΔL ⎦ = ⎣ Ixxṗ + Ixz ṙ ⎦ ΔN Izz ṙ + Ixz ṗ ⎡� ∂Y ∂V 0 v � � ∂Y � � ∂Y � + ∂P 0 p + ∂R 0 r + +ΔY ⎢ ⎢ � ∂L � � ∂L � � ∂L � c ≈ ⎢ v + p + ∂P 0 ∂R 0 r + ΔL ⎢ ∂V 0 ⎣� � � ∂N � � ∂N � ∂N c ∂V 0 v + ∂P 0 p + ∂R 0 r + ΔN September 21, 2010 ⎤ c ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦