Time Response of Linear, Time-Invariant (LTI) Systems!

Robert Stengel, Aircraft Flight Dynamics!MAE 331, 2016

•! Methods of time-domain analysis–! Continuous- and discrete-time models–! Transient response to initial conditions and inputs–! Steady-state (equilibrium) response–! Phase-plane plots–! Response to sinusoidal input

Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html

http://www.princeton.edu/~stengel/FlightDynamics.html

Reading:!Flight Dynamics!298-313, 338-342!

Airplane Stability and Control!Sections 11.1-11.12!

1

Learning Objectives

Stability Augmentation!Chapter 20, Airplane Stability and

Control, Abzug and Larrabee!•! What are the principal subject and scope of the

chapter?!•! What technical ideas are needed to understand

the chapter?!•! During what time period did the events covered

in the chapter take place?!•! What are the three main "takeaway" points or

conclusions from the reading?!•! What are the three most surprising or

remarkable facts that you found in the reading?!2

Review Questions!!! Why can we normally reduce the 6th-order lateral-

directional equations to 4th-order without compromising accuracy?!

!! What are the lateral-directional modes of motion?!!! What variables are in the stability-axis 4th-order

equations?!!! Why are the stability-axis equations useful?!!! Why are most airplanes symmetric about the body-

axis vertical plane?!!! What is a “vortical wind”?!!! Under what circumstances can the 4th-order set be

replaced by two 2nd-order sets?!!! What benefits accrue from the separation, and what

are the limitations?! 3

Assignment #7"due: End of day, Dec 9, 2016!

•! Simulated flight test!•! Assess the aircraft's response at various

flight conditions using the AeroModel.m and DataTable.m files created in the last assignment !

•! 3-member teams!4

2nd-Order Short-Period (LTI) Model

•! What can we do with it?–! Integrate equations to obtain time histories of initial

condition, control, and disturbance effects–! Examine steady-state conditions–! Identify effects of parameter variations–! Determine modes of motion–! Define frequency response

Gain insights about system dynamics

5

! !q t( )! !" t( )

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'((=

Mq M"

1) Lq VN*+,

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(((

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(((!0E t( )

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(((!"wind t( )

Linear, Time-Invariant (LTI) System Model

Dynamic equation (ordinary differential equation)

!!x(t) = F!x(t)+G!u(t)+L!w(t), !x(to ) given

State and output dimensions need not be the same

dim !x(t)[ ] = n "1( )dim !y(t)[ ] = r "1( ) 6

!y(t) = Hx!x(t)+Hu!u(t)+Hw!w(t)Output equation (algebraic transformation)

System Response to Inputs and Initial Conditions

•! Solution of the linear, time-invariant (LTI) dynamic model

!!x(t) = F!x(t) +G!u(t) + L!w(t), !x(to ) given

!x(t) = !x(to ) + F!x(" ) +G!u(" ) + L!w(" )[ ]to

t

# d"

•! ... has two parts–! Unforced (homogeneous) response to initial conditions–! Forced response to control and disturbance inputs

7

Response to Initial Conditions!

8

Unforced Response to Initial Conditions

The state transition matrix, !!, propagates the state from to to t by a single multiplication

!x(t) = !x(to )+ F!x(" )[ ]d"to

t

# = eF t$to( )!x(to ) = %% t $ to( )!x(to )

eF t!to( ) =Matrix Exponential

= I+ F t ! to( ) + 12!F t ! to( )"# $%

2+ 13!F t ! to( )"# $%

3+ ...

= && t ! to( ) = State TransitionMatrix

Neglecting forcing functions

9

Initial-Condition Response via State Transition

!! = I + F "t( ) + 12!F "t( )#$ %&

2+13!F "t( )#$ %&

3+ ...

!x(t1)="" t1 # to( )!x(to )!x(t2 )="" t2 # t1( )!x(t1)!x(t3)="" t3 # t2( )!x(t2 )!

If (tk+1 – tk) = !t = constant, state transition matrix is

constant

!x(t1)="" #t( )!x(to )=""!x(to )!x(t2 )=""!x(t1)=""

2!x(to )!x(t3)=""!x(t2 )=""

3!x(to )…

Incremental propagation of ""x

Propagation is exact10

Discrete-Time Dynamic Model

!x(tk+1)= !x(tk )+ F!x(" )+G!u(" )+L!w(" )[ ]d"tk

tk+1

#

!x(tk+1) = "" # t( )!x(tk )+"" # t( ) e$F % $tk( )&' ()d%tk

tk+1

* G!u(tk )+L!w(tk )[ ]= ""!x(tk )+ ++!u(tk )+ ,,!w(tk )

Response to continuous controls and disturbances

Response to piecewise-constant controls and disturbances

With piecewise-constant inputs, control and disturbance effects taken outside the integral

Discrete-time model of continuous system = Sampled-data model

11

Sampled-Data Control- and Disturbance-Effect Matrices

!! = eF" t # I( )F#1G

= I# 12!F" t + 1

3!F2" t 2 # 1

4!F3" t 3 + ...$

%&'()G" t

!x(tk ) = ""!x(tk#1)+ $$!u(tk#1)+ %%!w(tk#1)

As !!t becomes very small

!! " t#0$ #$$ I+ F" t( )%% " t#0$ #$$ G" t&& " t#0$ #$$ L" t 12

!! = eF" t # I( )F#1L

= I# 12!F" t + 1

3!F2" t 2 # 1

4!F3" t 3 + ...$

%&'() L" t

Discrete-Time Response to Inputs

!x(t1)=""!x(to )+##!u(to )+$$!w(to )!x(t2 )=""!x(t1)+##!u(t1)+$$!w(t1)!x(t3)=""!x(t2 )+##!u(t2 )+$$!w(t2 )!

Propagation of ""x, with constant !!, ##, and $$

!t = tk+1 " tk

13

Continuous- and Discrete-Time Short-Period System Matrices

•! !!t = 0.1 s

•! !!t = 0.5 s

F = !1.2794 !7.98561 !1.2709

"

#$

%

&'

G = !9.0690

"

#$

%

&'

L = !7.9856!1.2709

"

#$

%

&'

!! = 0.845 "0.6940.0869 0.846

#

$%

&

'(

)) = "0.84"0.0414

#

$%

&

'(

** = "0.694"0.154

#

$%

&

'(

!! = 0.0823 "1.4750.185 0.0839

#

$%

&

'(

)) = "2.492"0.643

#

$%

&

'(

** = "1.475"0.916

#

$%

&

'(

•! Continuous-time ( analog ) system

•! Sampled-data ( digital ) system

!!t has a large effect on the digital model

!t = tk+1 " tk

!! = 0.987 "0.0790.01 0.987

#

$%

&

'(

)) = "0.09"0.0004

#

$%

&

'(

** = "0.079"0.013

#

$%

&

'(

•! !t = 0.01 s

14

Continuous- and Discrete-Time Short-Period Models

! !q t( )! !" t( )

#

$%%

&

'((= )1.3 )8

1 )1.3#

$%

&

'(

!q t( )!" t( )

#

$%%

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'((+ )9.1

0#

$%

&

'(!*E t( )

Note individual acceleration and difference sensitivities to state and control perturbations

!qk+1!" k+1

#

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&

'((= 0.85 )0.7

0.09 0.85#

$%

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'(

!qk!" k

#

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'((+ )0.84

)0.04#

$%

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'(!*Ek

Differential Equations Produce State Rates of Change

Difference Equations Produce State Increments

Learjet 23MN = 0.3, hN = 3,050 mVN = 98.4 m/s

!t = 0.1sec

15

!!x t( ) = F!x t( ) +G!u t( )!y t( ) = Hx!x t( ) +Hu!u t( )

Initial-Condition Response

Doubling the initial condition doubles the output

!!x1!!x2

"

#$$

%

&''= (1.2794 (7.9856

1 (1.2709"

#$

%

&'

!x1!x2

"

#$$

%

&''+ (9.069

0"

#$

%

&'!)E

!y1!y2

"

#$$

%

&''= 1 0

0 1"

#$

%

&'

!x1!x2

"

#$$

%

&''+ 0

0"

#$

%

&'!)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

16

Commercial Aircraft of the 1940s•! Pre-WWII designs, reciprocating engines•! Development enhanced by military transport and bomber versions

–! Douglas DC-4 (adopted as C-54)–! Boeing Stratoliner 377 (from B-29, C-97)–! Lockheed Constellation 749 (from C-69)

!!iiss""rriiccaall FFaacc""iiddss

17

Commercial Propeller-Driven Aircraft of the 1950s•! Reciprocating and turboprop engines•! Douglas DC-6, DC-7, Lockheed Starliner 1649, Vickers

Viscount, Bristol Britannia, Lockheed Electra 188

18

Bristol Brabazon“Jumbo Turboprop”

Brabazon Committee study for a post-WWII jet-powered mailplane with small passenger compartment

deHavilland Vampire, 1943

19

deHavilland Swallow, 1946

Commercial Jets of the 1950s•! Low-bypass ratio turbojet

engines•! deHavilland DH 106 Comet

(1951)–! 1st commercial jet transport–! engines buried in wings–! early takeoff accidents

•! Boeing 707 (1957)–! derived from USAF KC-135–! engines on pylons below

wings–! largest aircraft of its time

•! Sud-Aviation Caravelle (1959)–! 1st aircraft with twin aft-

mounted engines

deHavilland Comet

Boeing 707

Sud-Aviation Caravelle

20

Superposition of Linear Responses!

21

Step Response

•! Stability, speed of response, and damping are independent of the initial condition or input

Doubling the input doubles the output

!!x1!!x2

"

#$$

%

&''= (1.2794 (7.9856

1 (1.2709"

#$

%

&'

!x1!x2

"

#$$

%

&''+ (9.069

0"

#$

%

&'!)E

!y1!y2

"

#$$

%

&''= 1 0

0 1"

#$

%

&'

!x1!x2

"

#$$

%

&''+ 0

0"

#$

%

&'!)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

Step Input

!"E t( ) = 0, t < 0#1, t $ 0

%&'

('

22

Superposition of Linear Step Responses

Stability, speed of response, and damping are independent of the initial condition or input

!!x1!!x2

"

#$$

%

&''= (1.2794 (7.9856

1 (1.2709"

#$

%

&'

!x1!x2

"

#$$

%

&''+ (9.069

0"

#$

%

&'!)E

!y1!y2

"

#$$

%

&''= 1 0

0 1"

#$

%

&'

!x1!x2

"

#$$

%

&''+ 0

0"

#$

%

&'!)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

23

2nd-Order Comparison: Continuous- and Discrete-Time LTI

Longitudinal Models

Short Period

Phugoid

! !V t( )! !" t( )

#

$%%

&

'(() *0.02 *9.8

0.02 0#

$%

&

'(

!V t( )!" t( )

#

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'((+ 4.7

0#

$%

&

'(!+T t( )

! !q t( )! !" t( )

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'((= )1.3 )8

1 )1.3#

$%

&

'(

!q t( )!" t( )

#

$%%

&

'((+ )9.1

0#

$%

&

'(!*E t( )

!qk+1!" k+1

#

$%%

&

'((= 0.85 )0.7

0.09 0.85#

$%

&

'(

!qk!" k

#

$%%

&

'((+ )0.84

)0.04#

$%

&

'(!*Ek

!Vk+1!" k+1

#

$%%

&

'((= 1 )0.98

0.002 1#

$%

&

'(

!Vk!" k

#

$%%

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'((+ 0.47

0.0005#

$%

&

'(!*Tk

Differential Equations Produce State Rates of Change

Difference Equations Produce State Increments

!t = 0.1sec

24

Short Period

Phugoid

Equilibrium Response!

25

Equilibrium Response

!!x(t) = F!x(t) +G!u(t) + L!w(t)

0 = F!x(t) +G!u(t) + L!w(t)

!x* = "F"1 G!u *+L!w *( )

Dynamic equation

At equilibrium, the state is unchanging

Constant values denoted by (.)*

26

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 =f11 f12f21 f22

!

"##

$

%&&; G =

g1g2

!

"##

$

%&&; L =

l1l2

!

"##

$

%&&

!x1 *!x2 *

"

#$$

%

&''= (

f22 ( f12( f21 f11

"

#$$

%

&''

f11 f22 ( f12 f21( )g1g2

)

*

++

,

-

.

.!u *+l1l2

)

*

++

,

-

.

.!w*"

#

$$

%

&

''

2nd-order example

sI! F = " s( ) = s2 + f11 + f22( )s + f11 f22 ! f12 f21( )= s ! #1( ) s ! #2( ) = 0

Re #i( ) < 0

System Matrices

Equilibrium Response with Constant Inputs

Requirement for Stability

27

Equilibrium Response ofApproximate Phugoid Model

!xP* = "FP"1 GP!uP *+LP!wP *( )

!V *

!" *#

$%%

&

'((= )

0 VNLV

)1g

VNDV

gLV

#

$

%%%%%

&

'

(((((

T*TL*TVN

#

$

%%%

&

'

(((!*T * +

DV

)LVVN

#

$

%%%

&

'

(((!VW

*

+

,--

.--

/

0--

1--

Equilibrium state with constant thrust and wind perturbations

28

Equilibrium Response ofApproximate Phugoid Model

!V * = "L#TLV

!#T * +!VW*

!$ * =1gT#T + L#T

DV

LV

%

&'

(

)*!#T *

With L!!T ~ 0, steady-state velocity perturbation depends only on the horizontal wind

Constant thrust perturbation produces steady climb rateCorresponding dynamic response

to thrust step, with L!!T = 0

Steady horizontal wind affects velocity but not flight path angle

29

Equilibrium Response ofApproximate Short-Period Model

!xSP* = "FSP"1 GSP!uSP *+LSP!wSP *( )

!q*

!" *

#

$%%

&

'((= )

L"VN

M"

1 )Mq

#

$

%%%

&

'

(((

L"VN

Mq + M"

*

+,-

./

M0E

)L0EVN

#

$

%%%

&

'

(((!0E* )

M"

)L"VN

#

$

%%%

&

'

(((!"W

*

1

233

433

5

633

733

Equilibrium state with constant elevator and wind perturbations

30

Equilibrium Response ofApproximate Short-Period Model

Steady pitch rate and angle of attack response to elevator perturbation are not zero

Steady vertical wind affects steady-state angle of attack but not pitch rate

!q* = "

L#VN

M$E%

&'(

)*

L#VN

Mq + M#

%

&'(

)*

!$E*

!# * = "M$E( )

L#VN

Mq + M#

%

&'(

)*

!$E + !#W*

with L!E = 0

Dynamic response to elevator step with L!!E = 0

31

Phase Plane Plots!

32

A 2nd-Order Dynamic Model

!!x1!!x2

"

#$$

%

&''=

0 1() n

2 (2*) n

"

#$$

%

&''

!x1!x2

"

#$$

%

&''+ 1 (1

0 2"

#$

%

&'

!u1!u2

"

#$$

%

&''

33

!x1 t( ) : Displacement (or Position)!x2 t( ) : Rate of change of Position

! n : Natural frequency, rad/s" : Damping ratio, -

State ( Phase ) Plane Plots

Cross-plot of one component against another

Time is 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

!!x1!!x2

"

#$$

%

&''(

0 1)*n

2 )2+*n

"

#$$

%

&''

!x1!x2

"

#$$

%

&''+ 1 )1

0 2"

#$

%

&'

!u1!u2

"

#$$

%

&''

34

Dynamic Stability Changes the State-Plane Spiral

Damping ratio = 0.1 Damping ratio = 0.3 Damping ratio = –0.1

35

Tactical Aircraft Maneuverability!Chapter 10, Airplane Stability and

Control, Abzug and Larrabee!•! What are the principal subject and scope of the

chapter?!•! What technical ideas are needed to understand

the chapter?!•! During what time period did the events covered

in the chapter take place?!•! What are the three main "takeaway" points or

conclusions from the reading?!•! What are the three most surprising or

remarkable facts that you found in the reading?!36

Scalar Frequency Response!

37

Speed Control of Direct-Current Motor

u(t) = Ce(t)wheree(t) = yc (t) ! y(t)

Control Law (C = Control Gain)

Angular Rate

38

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, yc(t), rad/s, is constant–! Control gain, C, scales command-following

error to motor input voltage39

Model of Dynamics and Speed Control

Dynamic equation

y(t) = 1J

u(t)dt0

t

! =CJ

e(t)dt0

t

! =CJ

yc (t) " y(t)[ ]dt0

t

!

dy(t)dt

=u(t)J

=Ce(t)J

=CJyc (t) ! y(t)[ ], y 0( ) given

Integral of the equation, with y(0) = 0

Direct integration of yc(t)Negative feedback of y(t)

40

Step Response of Speed Controller

y(t)= yc 1! e!CJ

"

#$

%

&'t(

)**

+

,--= yc 1! e

.t() +,= yc 1! e! t /(

)*+,-

•! where! "" = –C/J = eigenvalue or root of the system (rad/s)! ## = J/C = time constant of the response (sec)

•! Solution of the integral, with step command

yc t( ) =0, t < 01, t ! 0

"#$

%$

41

Angle Control of a DC Motor

Closed-loop dynamic equation, with y(t) = I2 x(t)

u(t) = c1 yc (t) " y1(t)[ ]" c2y2(t)

!x1(t)!x2 (t)

!

"##

$

%&&=

0 1'c1 / J 'c2 / J

!

"##

$

%&&

x1(t)x2 (t)

!

"##

$

%&&+

0c1 / J

!

"##

$

%&&yc

Control law with angle and angular rate feedback

!n = c1 J ; " = c2 J( ) 2!n42

c1 /J = 1 c2 /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

!n = c1 J ; " = c2 J( ) 2!n

43

Angle Response to a Sinusoidal Angle Command

Amplitude Ratio (AR) =ypeakyCpeak

Phase Angle !( ) = "360#t peakPeriod

, deg

•! Output wave lags behind the input wave

•! Input and output amplitudes different

yC t( ) = yCpeaksin!t

44

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°•! Lag of angle output

oscillation, compared to input, is small

yc (t) = sin t / 6.28( ), deg !n = 1 rad / s" = 0.707

45

At Higher Input Frequency, Phase Angle Lag Increases

yc (t) = sin t( ), deg

46

At Even Higher Frequency, Amplitude Ratio Decreases and

Phase Lag Increasesyc (t) = sin 6.28t( ), deg

47

Angle and Rate Response of a DC Motor over Wide Input-Frequency Range !!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)

Very low damping

Moderate damping

High damping

48

Input Frequencies (previous slides)

Next Time:!Transfer Functions and

Frequency Response!Reading:!

Flight Dynamics!342-357!

49

•! Frequency domain view of initial condition response•! Response of dynamic systems to sinusoidal inputs•! Transfer functions•! Bode plots

Learning Objectives

Supplemental Material!

50

Example: Aerodynamic Angle, Linear Velocity, and Angular Rate Perturbations

Learjet 23MN = 0.3, hN = 3,050 m

VN = 98.4 m/s

!" ! !wVN!" = 1°# !w = 0.01745 $ 98.4 = 1.7m s

!% ! !vVN!% = 1°# !v = 0.01745 $ 98.4 = 1.7m s

!p = 1° / s

!wwingtip = !p b2"# $% = 0.01745 & 5.25 = 0.09m s

!q = 1° / s!wnose = !q xnose ' xcm[ ] = 0.01745 & 6.4 = 0.11m s

!r = 1° / s!vnose = !r xnose ' xcm[ ] = 0.01745 & 6.4 = 0.11m s

Aerodynamic angle and linear velocity perturbations

Angular rate and linear velocity perturbations

51

Continuous- and Discrete-Time Dutch-Roll Models

!!r t( )! !" t( )

#

$

%%

&

'

(() *0.11 1.9

*1 *0.16#

$%

&

'(

!r t( )!" t( )

#

$%%

&

'((+ *1.1

0#

$%

&

'(!+R t( )

!rk+1!"k+1

#

$%%

&

'(() 0.98 0.19

*0.1 0.97#

$%

&

'(

!rk!"k

#

$%%

&

'((+ *0.11

0.01#

$%

&

'(!+Rk

Differential Equations Produce State Rates of Change

Difference Equations Produce State Increments

52

!t = 0.1sec

Continuous- and Discrete-Time Roll-Spiral Models

!!p t( )! !" t( )

#

$%%

&

'(() *1.2 0

1 0#

$%

&

'(

!p t( )!" t( )

#

$%%

&

'((+ 2.3

0#

$%

&

'(!+A t( )

!pk+1!"k+1

#

$%%

&

'(() 0.89 0

0.09 1#

$%

&

'(

!pk!"k

#

$%%

&

'((+ 0.24

*0.01#

$%

&

'(!+Ak

Differential Equations Produce State Rates of Change

Difference Equations Produce State Increments

53

!t = 0.1sec

4th- Order Comparison: Continuous- and Discrete-Time Longitudinal Models

Phugoid and Short Period

! !V t( )! !" t( )! !q t( )! !# t( )

$

%

&&&&&&

'

(

))))))

=

*0.02 *9.8 0 00.02 0 0 1.30 0 *1.3 *8

*0.02 0 1 *1.3

$

%

&&&&

'

(

))))

!V t( )!" t( )!q t( )!# t( )

$

%

&&&&&&

'

(

))))))

+

4.7 00 00 *9.10 0

$

%

&&&&

'

(

))))

!+T t( )!+E t( )

$

%&&

'

())

!Vk+1!" k+1

!qk+1!# k+1

$

%

&&&&&

'

(

)))))

=

1 *0.98 *0.002 *0.060.002 1 0.006 0.120.0001 0 0.84 *0.69*0.002 0.0001 0.09 0.84

$

%

&&&&

'

(

))))

!Vk!" k

!qk!# k

$

%

&&&&&

'

(

)))))

+

0.47 0.00050.0005 *0.0020 *0.840 *0.04

$

%

&&&&

'

(

))))

!+Tk!+Ek

$

%&&

'

())

Differential Equations Produce State Rates of Change

Difference Equations Produce State Increments !t = 0.1sec

54