Time Response of Linear, Time-Invariant Systems 
 Robert Stengel, Aircraft Flight Dynamics
 MAE 331, 2012" •  Time-domain analysis" –  Transient response to initial conditions and inputs" –  Steady-state (equilibrium) response" –  Continuous- and discrete-time models" –  Phase-plane plots" –  Response to sinusoidal input" 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 ! Linear, Time-Invariant (LTI) Longitudinal Model" 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) $ % & & & & & ' ( ) ) ) ) ) + 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 δ E(t) Δ δ T (t ) Δ δ F(t) $ % & & & ' ( ) ) ) •  Steady, level flight" •  Simplified control effects " •  Neglect disturbance effects " •  What can we do with it?" –  Integrate equations to obtain time histories of initial condition, control, and disturbance effects" –  Determine modes of motion" –  Examine steady-state conditions" –  Identify effects of parameter variations" –  Define frequency response " Gain insights about system dynamics! Linear, Time-Invariant System Model" •  General model contains" –  Dynamic equation (ordinary differential equation)" –  Output equation (algebraic transformation) " € Δ ˙ x (t) = FΔx(t) + GΔu(t) + LΔw(t), Δx(t o ) given Δy(t) = H x Δx(t) + H u Δu(t) + H w Δw(t) •  State and output dimensions need not be the same" dim Δx(t ) [ ] = n × 1 ( ) dim Δy(t ) [ ] = r × 1 ( ) System Response to Inputs and Initial Conditions" •  Solution of the linear, time-invariant (LTI) dynamic model " Δ  x(t) = FΔx(t) + GΔu(t) + LΔw(t), Δx(t o ) given Δx(t) = Δx(t o ) + FΔx( τ ) + GΔu( τ ) + LΔw( τ ) [ ] t o t ∫ d τ •  has two parts" –  Unforced (homogeneous) response to initial conditions" –  Forced response to control and disturbance inputs" Response to Initial Conditions Unforced Response to Initial Conditions" •  The state transition matrix, Φ, propagates the state from t o to t by a single multiplication" Δx(t ) = Δx(t o )+ FΔx( τ ) [ ] d τ t o t ∫ = e F t−t o ( ) Δx(t o ) = Φ t − t o ( ) Δx(t o ) e F t −t o ( ) = Matrix Exponential = I + F t − t o ( ) + 1 2! F t − t o ( ) " # $ % 2 + 1 3! F t − t o ( ) " # $ % 3 + = Φ t − t o ( ) = State Transition Matrix •  Neglecting forcing functions" Initial-Condition Response via State Transition" Φ = I + F δ t ( ) + 1 2! F δ t ( ) # $ % & 2 + 1 3! F δ t ( ) # $ % & 3 + Δx(t 1 ) = Φ t 1 − t o ( ) Δx(t o ) Δx(t 2 ) = Φ t 2 − t 1 ( ) Δx(t 1 ) Δx(t 3 ) = Φ t 3 − t 2 ( ) Δx(t 2 )  •  If (t k+1 – t k ) = Δ t = constant, state transition matrix is constant" Δx(t 1 ) = Φ δ t ( ) Δx(t o ) = ΦΔx(t o ) Δx(t 2 ) = ΦΔx(t 1 ) = Φ 2 Δx(t o ) Δx(t 3 ) = ΦΔx(t 2 ) = Φ 3 Δx(t o ) … •  Incremental propagation of Δx" •  Propagation is exact" Discrete-Time Dynamic Model" Δx(t k+1 ) = Δx(t k )+ FΔx( τ )+ GΔu( τ )+ LΔw( τ ) [ ] d τ t k t k+1 ∫ Δx(t k+1 ) = Φ δ t ( ) Δx(t k )+ Φ δ t ( ) e −F τ −t k ( ) & ' ( ) d τ t k t k+1 ∫ GΔu(t k )+ LΔw(t k ) [ ] = ΦΔx(t k )+ ΓΔu(t k )+ ΛΔw(t k ) •  Response to continuous controls and disturbances" •  Response to piecewise-constant controls and disturbances" Ordinary Difference Equation! •  With piecewise-constant inputs, control and disturbance effects taken outside the integral" •  Discrete-time model = Sampled-data model" Sampled-Data Control- and Disturbance-Effect Matrices" Γ = e F δ t − I ( ) F −1 G = I − 1 2! F δ t + 1 3! F 2 δ t 2 − 1 4! F 3 δ t 3 + $ % & ' ( ) G δ t Λ = e F δ t − I ( ) F −1 L = I − 1 2! F δ t + 1 3! F 2 δ t 2 − 1 4! F 3 δ t 3 + $ % & ' ( ) L δ t Δx(t k ) = ΦΔx(t k −1 ) + ΓΔu(t k −1 ) + ΛΔw(t k −1 ) •  As δ t becomes very small" Φ δ t →0 $ →$$ I + F δ t ( ) Γ δ t →0 $ →$$ G δ t Λ δ t →0 $ →$$ L δ t Discrete-Time Response to Inputs" Δx(t 1 ) = ΦΔx(t o )+ ΓΔu(t o )+ ΛΔw(t o ) Δx(t 2 ) = ΦΔx(t 1 )+ ΓΔu(t 1 )+ ΛΔw(t 1 ) Δx(t 3 ) = ΦΔx(t 2 )+ ΓΔu(t 2 )+ ΛΔw(t 2 )  •  Propagation of Δx, with constant Φ, Γ, and Λ" δ t = t k +1 − t k Continuous- and Discrete-Time Short-Period System Matrices" •  δ t = 0.1 s" •  δ t = 0.5 s" F = −1.2794 −7.9856 1 −1.2709 " # $ % & ' G = −9.069 0 " # $ % & ' L = −7.9856 −1.2709 " # $ % & ' Φ = 0.845 −0.694 0.0869 0.846 # $ % & ' ( Γ = −0.84 −0.0414 # $ % & ' ( Λ = −0.694 −0.154 # $ % & ' ( Φ = 0.0823 −1.475 0.185 0.0839 # $ % & ' ( Γ = −2.492 −0.643 # $ % & ' ( Λ = −1.475 −0.916 # $ % & ' ( •  Continuous-time (analog) system" •  Sampled-data (digital) system" δ t has a large effect on the digital model" δ t = t k +1 − t k Φ = 0.987 −0.079 0.01 0.987 # $ % & ' ( Γ = −0.09 −0.0004 # $ % & ' ( Λ = −0.079 −0.013 # $ % & ' ( •  δt = 0.01 s" Example: Continuous- and Discrete-Time Models" Δ  q Δ  α # $ % % & ' ( ( = −1.3 −8 1 −1.3 # $ % & ' ( Δq Δ α # $ % % & ' ( ( + −9.1 0 # $ % & ' ( Δ δ E • Note individual acceleration and difference sensitivities to state and control perturbations" Short Period" Δq k +1 Δ α k +1 # $ % % & ' ( ( = 0.85 −0.7 0.09 0.85 # $ % & ' ( Δq k Δ α k # $ % % & ' ( ( + −0.84 −0.04 # $ % & ' ( Δ δ E k Differential Equations Produce State Rates of Change" Difference Equations Produce State Increments" Learjet 23! M N = 0.3, h N = 3,050 m" V N = 98.4 m/s" δ t = 0.1sec Example: Continuous- and Discrete-Time Models" Δ  p Δ  φ # $ % % & ' ( ( ≈ −1.2 0 1 0 # $ % & ' ( Δp Δ φ # $ % % & ' ( ( + 2.3 0 # $ % & ' ( Δ δ A Roll-Spiral" Δp k +1 Δ φ k +1 # $ % % & ' ( ( ≈ 0.89 0 0.09 1 # $ % & ' ( Δp k Δ φ k # $ % % & ' ( ( + 0.24 −0.01 # $ % & ' ( Δ δ A k Differential Equations Produce State Rates of Change" Difference Equations Produce State Increments" Example: Continuous- and Discrete-Time Models" Δ  r Δ  β # $ % % & ' ( ( ≈ −0.11 1.9 −1 −0.16 # $ % & ' ( Δr Δ β # $ % % & ' ( ( + −1.1 0 # $ % & ' ( Δ δ R Dutch Roll" Δr k +1 Δ β k +1 # $ % % & ' ( ( ≈ 0.98 0.19 −0.1 0.97 # $ % & ' ( Δr k Δ β k # $ % % & ' ( ( + −0.11 0.01 # $ % & ' ( Δ δ R k Differential Equations Produce State Rates of Change" Difference Equations Produce State Increments" Initial-Condition Response" •  Doubling the initial condition doubles the output" Δ  x 1 Δ  x 2 " # $ $ % & ' ' = −1.2794 −7.9856 1 −1.2709 " # $ % & ' Δx 1 Δx 2 " # $ $ % & ' ' + −9.069 0 " # $ % & ' Δ δ E Δy 1 Δy 2 " # $ $ % & ' ' = 1 0 0 1 " # $ % & ' Δx 1 Δx 2 " # $ $ % & ' ' + 0 0 " # $ % & ' Δ δ E % Short-Period Linear Model - Initial Condition! ! F = [-1.2794 -7.9856;1. -1.2709];! G = [-9.069;0];! Hx = [1 0;0 1];! sys = ss(F, G, Hx,0);! ! xo = [1;0];! [y1,t1,x1] = initial(sys, xo);! ! xo = [2;0];! [y2,t2,x2] = initial(sys, xo);! plot(t1,y1,t2,y2), grid! ! figure! xo = [0;1];! initial(sys, xo), grid! Angle of Attack Initial Condition" Pitch Rate Initial Condition" Phase Plane Plots State (Phase) Plane Plots" •  Cross-plot of one component against another" •  Time or frequency not shown explicitly" % 2nd-Order Model - Initial Condition Response! ! clear! z = 0.1; % Damping ratio! wn = 6.28; % Natural frequency, rad/s! F = [0 1;-wn^2 -2*z*wn];! G = [1 -1;0 2];! Hx = [1 0;0 1];! sys = ss(F, G, Hx,0);! t = [0:0.01:10];! xo = [1;0];! [y1,t1,x1] = initial(sys, xo, t);! ! plot(t1,y1)! grid on! ! figure! plot(y1(:,1),y1(:,2))! grid on! Δ  x 1 Δ  x 2 " # $ $ % & ' ' ≈ 0 1 − ω n 2 −2 ζω n " # $ $ % & ' ' Δx 1 Δx 2 " # $ $ % & ' ' + 1 −1 0 2 " # $ % & ' Δu 1 Δu 2 " # $ $ % & ' ' Dynamic Stability Changes the State-Plane Spiral" •  Damping ratio = 0.1" •  Damping ratio = 0.3" •  Damping ratio = –0.1" Superposition of Linear Responses Step Response" •  Stability, speed of response, and damping are independent of the initial condition or input" •  Doubling the input doubles the output" Δ  x 1 Δ  x 2 " # $ $ % & ' ' = −1.2794 −7.9856 1 −1.2709 " # $ % & ' Δx 1 Δx 2 " # $ $ % & ' ' + −9.069 0 " # $ % & ' Δ δ E Δy 1 Δy 2 " # $ $ % & ' ' = 1 0 0 1 " # $ % & ' Δx 1 Δx 2 " # $ $ % & ' ' + 0 0 " # $ % & ' Δ δ E % Short-Period Linear Model - Step! ! F = [-1.2794 -7.9856;1. -1.2709];! G = [-9.069;0];! Hx = [1 0;0 1];! sys = ss(F, -G, Hx,0); % (-1)*Step! sys2 = ss(F, -2*G, Hx,0); % (-1)*Step! ! % Step response! step(sys, sys2), grid! Δ δ E t ( ) = 0, t < 0 −1, t ≥ 0 % & ' ( ' Superposition of Linear Responses" •  Stability, speed of response, and damping are independent of the initial condition or input" Δ  x 1 Δ  x 2 " # $ $ % & ' ' = −1.2794 −7.9856 1 −1.2709 " # $ % & ' Δx 1 Δx 2 " # $ $ % & ' ' + −9.069 0 " # $ % & ' Δ δ E Δy 1 Δy 2 " # $ $ % & ' ' = 1 0 0 1 " # $ % & ' Δx 1 Δx 2 " # $ $ % & ' ' + 0 0 " # $ % & ' Δ δ E % Short-Period Linear Model - Superposition! ! F = [-1.2794 -7.9856;1. -1.2709];! G = [-9.069;0];! Hx = [1 0;0 1];! sys = ss(F, -G, Hx,0); % (-1)*Step! ! xo = [1; 0];! t = [0:0.2:20];! u = ones(1,length(t));! ! [y1,t1,x1] = lsim(sys,u,t,xo);! [y2,t2,x2] = lsim(sys,u,t);! ! u != zeros(1,length(t));! [y3,t3,x3] = lsim(sys,u,t,xo);! ! plot(t1,y1,t2,y2,t3,y3), grid! 2 nd -Order Comparison: Continuous- and Discrete- Time LTI Longitudinal Models" Short ! Period" Phugoid" Δ  V Δ  γ # $ % % & ' ( ( ≈ −0.02 −9.8 0.02 0 # $ % & ' ( ΔV Δ γ # $ % % & ' ( ( + 4.7 0 # $ % & ' ( Δ δ T Δ  q Δ  α # $ % % & ' ( ( = −1.3 −8 1 −1.3 # $ % & ' ( Δq Δ α # $ % % & ' ( ( + −9.1 0 # $ % & ' ( Δ δ E Δq k +1 Δ α k +1 # $ % % & ' ( ( = 0.85 −0.7 0.09 0.85 # $ % & ' ( Δq k Δ α k # $ % % & ' ( ( + −0.84 −0.04 # $ % & ' ( Δ δ E k ΔV k +1 Δ γ k +1 # $ % % & ' ( ( = 1 −0.98 0.002 1 # $ % & ' ( ΔV k Δ γ k # $ % % & ' ( ( + 0.47 0.0005 # $ % & ' ( Δ δ T k Learjet 23! M N = 0.3, h N = 3,050 m" V N = 98.4 m/s" Differential Equations Produce State Rates of Change" Difference Equations Produce State Increments" δ t = 0.1sec 4 th - Order Comparison: Continuous- and Discrete-Time Longitudinal Models" Phugoid and Short Period" Δ  V Δ  γ Δ  q Δ  α $ % & & & & & ' ( ) ) ) ) ) = −0.02 −9.8 0 0 0.02 0 0 1.3 0 0 −1.3 −8 −0.002 0 1 −1.3 $ % & & & & ' ( ) ) ) ) ΔV Δ γ Δq Δ α $ % & & & & ' ( ) ) ) ) + 4.7 0 0 0 0 −9.1 0 0 $ % & & & & ' ( ) ) ) ) Δ δ T Δ δ E $ % & ' ( ) ΔV k+1 Δ γ k+1 Δq k+1 Δ α k+1 $ % & & & & & ' ( ) ) ) ) ) = 1 −0.98 −0.002 −0.06 0.002 1 0.006 0.12 0.0001 0 0.84 −0.69 −0.0002 0.0001 0.09 0.84 $ % & & & & ' ( ) ) ) ) ΔV k Δ γ k Δq k Δ α k $ % & & & & & ' ( ) ) ) ) ) + 0.47 0.0005 0.0005 −0.002 0 −0.84 0 −0.04 $ % & & & & ' ( ) ) ) ) Δ δ T k Δ δ E k $ % & & ' ( ) ) Learjet 23! M N = 0.3, h N = 3,050 m" V N = 98.4 m/s" Differential Equations Produce State Rates of Change" Difference Equations Produce State Increments" δ t = 0.1sec Equilibrium Response Equilibrium Response" Δ  x(t ) = FΔx(t ) + GΔu(t) + LΔw(t ) 0 = FΔx(t ) + GΔu(t ) + LΔw(t ) Δx* = −F −1 GΔu * +LΔw * ( ) •  Dynamic equation" •  At equilibrium, the state is unchanging" •  Constant values denoted by (.)*" Steady-State Condition" •  If the system is also stable, an equilibrium point is a steady-state point, i.e.," –  Small disturbances decay to the equilibrium condition" F = f 11 f 12 f 21 f 22 ! " # # $ % & & ; G = g 1 g 2 ! " # # $ % & & ; L = l 1 l 2 ! " # # $ % & & Δx 1 * Δx 2 * " # $ $ % & ' ' = − f 22 − f 12 − f 21 f 11 " # $ $ % & ' ' f 11 f 22 − f 12 f 21 ( ) g 1 g 2 ) * + + , - . . Δu *+ l 1 l 2 ) * + + , - . . Δw * " # $ $ % & ' ' 2 nd -order example" sI − F = Δ s ( ) = s 2 + f 11 + f 22 ( ) s + f 11 f 22 − f 12 f 21 ( ) = s − λ 1 ( ) s − λ 2 ( ) = 0 Re λ i ( ) < 0 System Matrices" Equilibrium " Response with Constant Inputs" Requirement for Stability" Equilibrium Response of" Approximate Phugoid Model" Δx P * = −F P −1 G P Δu P * +L P Δw P * ( ) ΔV * Δ γ * # $ % % & ' ( ( = − 0 V N L V −1 g V N D V gL V # $ % % % % % & ' ( ( ( ( ( T δ T L δ T V N # $ % % % & ' ( ( ( Δ δ T * + D V −L V V N # $ % % % & ' ( ( ( ΔV W * + , - - . - - / 0 - - 1 - - •  Equilibrium state with constant thrust and wind perturbations" Steady-State Response of" Approximate Phugoid Model" ΔV * = − L δ T L V Δ δ T * + ΔV W * Δ γ * = 1 g T δ T + L δ T D V L V % & ' ( ) * Δ δ T * •  With L δ T ~ 0, i.e., no lift produced directly by thrust, steady-state velocity depends only on the horizontal wind" •  Constant thrust produces steady climb rate" •  Corresponding dynamic response to thrust step, with L δ T = 0" Steady horizontal wind affects velocity but not flight path angle! Equilibrium Response of" Approximate Short-Period Model" Δx SP * = −F SP −1 G SP Δu SP * +L SP Δw SP * ( ) Δq * Δ α * # $ % % & ' ( ( = − L α V N M α 1 −M q # $ % % % & ' ( ( ( L α V N M q + M α * + , - . / M δ E − L δ E V N # $ % % % & ' ( ( ( Δ δ E * − M α −L α V N # $ % % % & ' ( ( ( Δ α W * 1 2 3 3 4 3 3 5 6 3 3 7 3 3 •  Equilibrium state with constant elevator and wind perturbations" Steady-State Response of" Approximate Short-Period Model" •  Steady pitch rate and angle of attack response to elevator are not zero" •  Steady vertical wind affects steady-state angle of attack but not pitch rate" Δq * = − L α V N M δ E % & ' ( ) * L α V N M q + M α % & ' ( ) * Δ δ E * Δ α * = − M δ E ( ) L α V N M q + M α % & ' ( ) * Δ δ E + Δ α W * with L δ E = 0" Dynamic response to elevator step with L δ E = 0! Scalar Frequency Response Speed Control of Direct-Current Motor" u(t) = C e(t) where e(t) = y c (t) − y(t) •  Control Law (C = Control Gain)" Angular Rate" Characteristics of the Motor" •  Simplified Dynamic Model" –  Rotary inertia, J, is the sum of motor and load inertias" –  Internal damping neglected" –  Output speed, y(t), rad/s, is an integral of the control input, u(t)! –  Motor control torque is proportional to u(t) " –  Desired speed, y c (t), rad/s, is constant" –  Control gain, C, scales command-following error to motor input voltage" Model of Dynamics and Speed Control" •  Dynamic equation" y(t) = 1 J u(t)dt 0 t ∫ = C J e(t)dt 0 t ∫ = C J y c (t) − y(t) [ ] dt 0 t ∫ dy(t) dt = u(t) J = Ce(t) J = C J y c (t) − y(t) [ ] , y 0 ( ) given •  Integral of the equation, with y(0) = 0" • Direct integration of y c (t)" • Negative feedback of y(t)" Step Response of Speed Controller" y(t) = y c 1− e − C J " # $ % & ' t ( ) * * + , - - = y c 1− e λ t ( ) + , = y c 1− e − t τ ( ) * + , - •  where"   λ = –C/J = eigenvalue or root of the system (rad/s)"   τ = J/C = time constant of the response (sec)" Step input : y C (t) = 0, t < 0 1, t ≥ 0 " # $ % $ •  Solution of the integral, with step command" y c t ( ) = 0, t < 0 1, t ≥ 0 " # $ % $ Angle Control of a DC Motor" •  Closed-loop dynamic equation, with y(t) = I 2 x(t)! € u(t) = c 1 y c (t) − y 1 (t) [ ] − c 2 y 2 (t)  x 1 (t)  x 2 (t) ! " # # $ % & & = 0 1 −c 1 / J −c 2 / J ! " # # $ % & & x 1 (t) x 2 (t) ! " # # $ % & & + 0 c 1 / J ! " # # $ % & & y c •  Control law with angle and angular rate feedback! ω n = c 1 J ; ζ = c 2 J ( ) 2 ω n c 1 /J = 1 " c 2 /J = 0, 1.414, 2.828" % Step Response of Damped " Angle Control" " F1 = [0 1;-1 0];" G1 = [0;1];" " F1a = [0 1;-1 -1.414];" F1b = [0 1;-1 -2.828];" " Hx = [1 0;0 1];" " Sys1 = ss(F1,G1,Hx,0);" Sys2 = ss(F1a,G1,Hx,0);" Sys3 = ss(F1b,G1,Hx,0);" " step(Sys1,Sys2,Sys3)" Step Response of Angle Controller, with Angle and Rate Feedback" •  Single natural frequency, three damping ratios! ω n = c 1 J ; ζ = c 2 J ( ) 2 ω n Angle Response to a Sinusoidal Angle Command" € Amplitude Ratio (AR) = y peak y C peak Phase Angle = −360 Δt peak Period , deg •  Output wave lags behind the input wave" •  Input and output amplitudes different! y C t ( ) = y C peak sin ω t Effect of Input Frequency on Output Amplitude and Phase Angle" •  With low input frequency, input and output amplitudes are about the same" •  Rate oscillation leads angle oscillation by ~90 deg" •  Lag of angle output oscillation, compared to input, is small" y c (t) = sin t / 6.28 ( ) , deg ω n = 1 rad / s ζ = 0.707 At Higher Input Frequency, Phase Angle Lag Increases" y c (t) = sin t ( ) , deg [...]... deg Angle and Rate Response of a DC Motor over Wide InputFrequency Range " Very low damping! Moderate damping! High damping! !  Long-term response of a dynamic system to sinusoidal inputs over a range of frequencies" !  Determine experimentally from time response or " !  Compute the Bode plot of the system s transfer functions (TBD)! Next Time: Root Locus Analysis Reading Flight Dynamics, 357-361, . Time Response of Linear, Time- Invariant Systems 
 Robert Stengel, Aircraft Flight Dynamics MAE 331, 2012" •  Time- domain analysis& quot; –  Transient response to initial. •  Damping ratio = –0.1" Superposition of Linear Responses Step Response& quot; •  Stability, speed of response, and damping are independent of the initial condition or input" • . time response or " !  Compute the Bode plot of the systems transfer functions (TBD)! Very low damping! Moderate damping! High damping! Next Time:  Root Locus Analysis  Reading Flight