1

Aircraft Response Due To Atmospheric Inputs (c.f. Nelson Chapter 6 p.212)

We begin this section by repeating the longitudinal and lateral small disturbance dynamical system equations, but with simplifications obtained by neglecting variables that have been described as negligible. Neglecting , the longitudinal state equation on p.4.12 of these notes is:

. (1)

By setting the product of inertia , the lateral equations on p.4.44 of these notes are:

. (2)

Note that (1) and (2) include the velocities and , respectively, as state variables. While we have, on occasion, reformulated (1) and (2) to have state variables and , in this chapter it is important that we retain the forms (1) and (2). The reason is that we desire to incorporate turbulence as a second input to these dynamical systems. Since turbulence is wind velocity, by having states variables that are all velocities makes this incorporation easy, as we now show.

The atmosphere is not a uniform medium. It includes spatial and temporal dynamics associated with wind. Recall that the forces and moments acting on a plane depend on its motion relative to the local atmosphere. To demonstrate how the above equations are modified due to wind, we can write the plane x-axis velocity disturbance in relation to the local x-axis gust velocity, , as: . Repeating this for all disturbance variables gives:

linear velocities: ; ;

rotational velocities: ; ; .

The above state equations then become:

(3)

2

. (4)

Both (3) and (4) have the structure of the following dynamical system:

.

In previous sections we ignored the control inputs. We will do the same here. In this case, the above form becomes:

. (5)

In both (3) and (4) the input, u, is a 3-D vector, and the output is a 4-D vector. Hence, in the jargon of modern control theory, these are said to be examples of a Multi-Input/Multi-Output (MIMO) system.

For a MIMO(3,4) system, each of the 3 inputs relates to each of the 4 outputs. Hence, there are a total of 12 scalar-valued relationships. The transfer function associated with (5) is obtained by taking its Laplace transform under zero initial conditions. Doing this, we obtain , or:

. (6)

The matrix defined in (6) is the transfer function matrix that relates the input, u, to the output x.

In-Class Question: What is the dimension of ? Answer: 4x3

The (j,k) element of is a scalar-values transfer function that relates the jth output to the kth input. For example, in (1) the transfer function between the input and the output is the (1,1) element of the transfer function matrix.

We are now in a position to compute the transfer function matrices associated with (1) and with (2) . We can do this simply by identifying the specific matrices associated with the matrices A and B in . To obtain the 12 scalar-valued transfer functions for each system by hand would be a very laborious task. It is conceivable that a symbolic code (e.g. Mathematica) could perform this task and yield nice expressions. We will not undertake either task at this time. The motivated student is encouraged to ‘go for it’!

A Closer Look At Scalar-Valued Transfer Functions

We have already seen the value of having a scalar-valued transfer function. It allows us to specify an input, u, and obtain the output, x, using very simple mathematics, a table of Laplace transforms, and Matlab commands. To motivate the discussion, we will consider the “PURE VERTICAL OR PLUNGING MOTION” of a plane, considered in Nelson, p.218. In this section a plane constrained to undergo only vertical motion in response to a gust of wind is addressed. The development begins with:

.

3

Expressing the aerodynamic force in the z-direction as (i.e. a small perturbation force), results in

(i.e. equilibrium), and writing w as (i.e. unaccelerated flight) gives:

.

Since the vertical force is a function of the angle of attack, , and its derivative, , this can be expressed as:

. (1)

The change in the angle of attack due to the combination of its motion and the vertical wind gust is:

. (2a)

This gives: . (2b)

. (3)

As mentioned before, one standard for of a first order model has the form , where the ‘input’ is

, the ‘output’ is , the system time constant is , and the system static gain is . putting (3) in this form gives:

. (4)

Hence, the system time constant is and the static gain is . The system transfer function is obtained

by taking the Laplace transform of (4) assuming zero initial conditions. It is:

. (5)

From (5), we can easily obtain the response for a variety of gust profiles. We now consider some progressively realistic profiles.

Case 1: A very idealized (and mathematically simple) gust profile.

Suppose that the gust profile, has the form of a short burst of duration, T, and strength, Wo. To obtain its Laplace

transform, , we need three items from a table of Laplace transforms:

Item 1: The unit step function, has the form Its Laplace transform is .

Item 2: For any function with Laplace transform , the time-shifted function has Laplace

transform .

4

Item 3: For any function with Laplace transform , the scaled function has Laplace transform

.

The idealized gust pulse can be written as: . Using the above items, its Laplace transform is:

. (6)

From (5) and (6), the Laplace transform of the response is:

. (7)

We can use a table of transform pairs to obtain the expression for . But it is instructive to express (7) as the sum of two components:

. (8)

The first term on the right side of (8) is simply the scaled step response of the system . The second term on the right is the same response, but time-delayed by an amount T.

EXAMPLE PROBLEM 6.1 of Nelson (p.224) As above, both Nelson and Etkin typically assume that . Then we

have . Recall that . Hence, . For a general aviation plane flying at

125 fps, we obtain With these numerical values, the system transfer function is:

.

From entry #4 in the table of Laplace transform pairs, the corresponding system impulse response is:

.

We will assume that the gust magnitude is , and that its duration is . From entry #7 of the table and equation (8) above, the system response to this gust is:

. (9)

Note that in (9) we cannot cancel the two 1’s, since the second 1 is multiplied by the time-delayed unit step function. Consequently for any time , (9) is only:

(10)

To plot the response (() using Matlab we have two options:

5

Option 1: Define an array of discrete time values that extend from 0 to at least We will choose the maximum time to be 5 seconds. The interval, , between samples should be small relative to . We will choose

Hence the array of computation times will include a total of 1+5/.01=501 numbers. This array is computed as: tvec = 0:500; tvec = tvec*.01. The response portion given by (10) is then simply: w = 15*(1 – exp(-1.43*tvec). Its time-delayed version begins after the 2/.01=200th time index. And so: ws = [zeroes(1,200) w]; ws = ws(1:501).

These commands are summarized as:

>> tvec=0:500; tvec=.01*tvec;>> w=15*(1-exp(-1.43*tvec));>> ws=[zeros(1,200) w]; ws=ws(1:501);>> dw=w - ws;>> plot(tvec,dw)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

Time (sec)

dw (f

t/sec

)

Response to a 2-second Gust

Figure 1. Plane vertical speed response to a 2-second idealized gust with magnitude 15 ft/sec.

6

FREQUENCY CONTENT ASSOCIATED WITH INPUTS, OUTPUTS & TRANSFER FUNCTIONS

The profile in Case 1 was mathematically convenient, and does provide some basic insight as to how a gust might affect the dynamics of an aircraft. But anyone who has stood in an open field on a windy day knows that gusts of wind, and the wind, in general, are more complicated.

If you watch a tree during a gust, you will find that it doesn’t simply bend in a static way. Rather, it sways. That’s because a tree is an underdamped system with a natural frequency. If you give it an initial condition by pulling it with a rope, and then let go, it will sway back and forth at its damped natural frequency. Now, the only way that a natural frequency can be excited is if the input has energy at that frequency. And so, returning to models for a wind gust, if the chosen model has no energy at the natural frequency of the plane dynamics, then the response to that gust model will not involve those dynamics. This could result in very misleading predictions to the plane’s response to a real gust.

The key word that was used repeatedly here is frequency. To this point, we have been concerned with the relation between a transfer function and time domain behavior (e.g. settling time and period of oscillation). And so, a natural question is:

QUESTION: How does a system transfer function, say, , convey frequency information about the system dynamics?

To begin to answer this question, we will now prove a most fundamentally important connection between the time-domain (t) and Laplace-domain (s) representations of a system with transfer function . To emphasize this relationship, we will not state it as a fact. Rather, we will state it as a theorem. The proof of this theorem is central to understanding it.

THEOREM. A system transfer function, say, , is mathematically identical to the Laplace transform of the system impulse response.

Before proving this theorem, let’s be clear about the difference between a transfer function, , and the Laplace domain expression for the system response to an input, say, . Suppose, for example, that the input is a force, with units of lbf and that the response is displacement, with units of ft. Then the units of are [ft / lbf], while the units of the response, are ft. And so to say that the Laplace transform of the response to a unit impulse, is equal to makes no sense. They have different units! The above theorem does not say they are equal. It says they have the same mathematical expression. We will use the force/displacement setting in the proof.

PROOF: For a unit impulse input, , the Laplace transform is . The Laplace transform of the response, , to any input, , is simply . Hence, for the unit impulse input, this becomes

. Recall, that the standard convention is to use a lower case for the time domain expression, and the upper case for its Laplace transform. And so, since we have . The time-domain function is the inverse Laplace transform of the system transfer function, . It is also the time-domain response to a unit-impulse input. Hence, is called the system impulse response function. In summary then, the system transfer function is the Laplace transform of the system impulse response function. □

EXAMPLE PROBLEM 6.1 continued: The impulse response of the system with transfer function

is .

7

This response is plotted against the response to a 2-second wind gust below.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

5

10

15

20

25

30

35

40

45

Time (sec)

w(t)

(ft/s

)

Response to a 2-second gust & to a an impulse gust with intensity = 30

Figure 2. Plane vertical speed response to a 2-second idealized gust with magnitude 15 ft/sec, and to an impulse with intensity equal to 30.

Discussion: The response to the 2-second pulse is well-behaved at , whereas the impulse response makes a very sharp jump. This sharp jump means that the frequency content of the impulse response will include much higher frequencies that the frequency content in the 2-second pulse. But this may not be at all obvious to those who are new to this topic. To quantify this, we now give the answer to the above QUESTION:

ANSWER: The frequency content associated with a system with transfer function is obtained by setting . The quantity is called the system Frequency Response Function (FRF).

It follows that the frequency content associated with the steady state response , to any input, , is simply . For completeness, we give the following definition.

Definition 1. Let be a function of time, and assume that . The function

is called the Laplace Transform of . For , this transform becomes

, and is called the Fourier Transform of .

We are now in a position to quantitatively describe the frequency content associated with the two responses in Figure 2 above.

The frequency content associated with the impulse gust (with intensity = 30) is simply:

. (10a)

The frequency content associated with the response to the 2-second gust is obtained directly from (7) as:

8

, (10b)

The frequency content associated with (10a) and (10b) is shown below.

10-2

10-1

100

101

102

-50

-40

-30

-20

-10

0

10

20

30

40

50

Frequency (Hz)

dB

Frequeny content associated with Impulse (blue) & 2-sec Pulse (green)

Figure 3. The frequency content associated with (10a) BLUE and (10b) GREEN. NOTE: (10a) is scaled) for easier visual comparison.

Discussion: The frequency content of the two inputs is similar up to about 0.1 Hz. At higher frequencies the frequency content of the response to the 2-second pulse drops significantly relative to that associated with the impulse.

THE FREQUENCY CONTENT IN A RANDOM WIND PROFILE

Again, imagine you are standing in a field during gusty wind conditions. Clearly, no two gusts will feel exactly the same. In fact, the wind, in general is not exactly the same from second to second. And so, a more realistic model for a gust would be one that has elements of randomness to it. At issue here is not the energy content in a gust. Every gust will have different energy content. What is of central concern now is the expected frequency content. More specifically, it is the expected energy at each frequency. Consider a function . Its frequency description is given by its Fourier Transform:

. This is well-defined if decays fast enough as . (e.g. a decaying exponential.) But wind

(esp. upper atmospheric wind) is always present. It does not simply go away after some period of time. Furthermore, it has randomness. In fact, wind is an example of a random process. In this section we will assume it is a random process that has the same general behavior at in time. We will refer to such a process as a (weakly) stationary random process.

Definition 2. Assume that is a stationary random process. Then the Fourier Transform of is defined as

0 20 40 60 80 100Time Index (k)

-3

-2

-1

0

1

2

3N(0,1) White noise simulation

9

Remark: The Fourier Transform operation converts a time-domain random function, , to a random function . It is simply a different way of viewing the random function (i.e. as a function of instead of as a function of t). It does not reduce the level of variability.

Since is a random function, we need to address its frequency content in relation to the expected energy at any frequency. This is defined as:

Definition 3. The expected energy at a frequency is defined as . In Nelson (p.227-228) the symbol used

for it is . In words, is called the power spectral density (psd) function associated with the

random process . The symbol denotes the expected value operation.

Example 1. White Noise: To embark on this journey we will first clarify the words white noise. The term noise has many meanings. As with many bands, what is music to one person may be noise to another. This suggests that noise is in the category of undesirables (or, as HRC might say, deplorables). In the study of signals and systems, the term signal relates to something we would like to send/receive. The term noise relates to something that distorts the signal quality. Again, noise is viewed as undesirable. One might, therefore, reasonably conclude that noise is undesirable. However, this is not necessarily the case. In the case of simulating wind turbulence, white noise is used as the input to the turbulence shaping filter (to be discussed presently). So in this case it is a desirable noise to achieve this goal.

The adjective white in relation to noise refers to the fact that, first, the noise is a stationary random process (i.e. a time-indexed collection of random variables). What makes this process white is the fact that its psd is flat throughout the analysis bandwidth.

Definition 4. Consider any function of time .Let denote the sampled version of . The quantity

T is the sampling period. Hence, the quantity is called the sampling frequency. The analysis bandwidth

associated with is the frequency interval where is called the Nyquist frequency.

We will now simulate a portion of a white noise process using Matlab. Specifically:

%PROGRAM NAME: whitenoise.mn=100;nvec=0:n-1;%Simulation of normal(mu=0,std=1) noiseu=normrnd(0,1,n,1);figure(1)plot(nvec,u,'*--')title('N(0,1) White noise simulation')xlabel('Time Index (k)')grid

Figure E1.1 Simulation of normal(0,1) white noise.

There are a number of things to note:

1. Each value was obtained using the normal random number generator. Hence, the value generated at time index k in no way influences the value that will be generated at index . (i.e. the process is white).

0 1 2 3 4 5 6 7Frequency (rad/sec)

0

0.5

1

1.5

2

2.5Absolute Value of the Fourier Transform of u

0 0.5 1 1.5 2 2.5 3 3.5Frequency (rad/sec)

0.975

0.98

0.985

0.99

0.995

1

1.005

1.01

1.015

1.02

1.025Approximation of White Noise PSD

-60

-40

-20

0

20

Magn

itude

(dB)

10-1 100 101 102-180

-135

-90

-45

0

Phas

e (de

g)

Bode Diagram

Frequency (rad/s)

System: HFrequency (rad/s): 49.9Magnitude (dB): -39.9

10

2. While the N(0,1) distribution (i.e. the standard ‘bell curve’) theoretically extends over , rarely will a value be generated outside of the interval . The probability of such an occurrence is 2*normcdf(-4,0,1)=6.3342e-05. In fact, the probability that a value outside of the interval is only 2*normcdf(-3,0,1)=0.0027. On the average, there would be only 2-3 values outside of this interval for every 1000 values generated. As seen in Figure E1.1, one value is close, but no values are outside of this interval in this simulation.

3. The horizontal axis includes only the time index. This is to be expected since the random number generator simply generates a sequence of random numbers. Time only enters the picture when you determine that is should. In Figure E1.1 we could relabel the x-axis as ‘Time (sec)’. This corresponds to the choice of the sampling period . In this case, the analysis BW is .

To see why the BW is , we compute the Fourier transform:

%Fourier Transform of u:U=n^-0.5*abs(fft(u));dw=2*pi/n;wvec=0:dw:2*pi-dw;figure(2)plot(wvec,U)title('Absolute value of the Fourier Transform of u')xlabel('Frequency (rad/sec)')grid

Figure E1.2 Absolute value of the FT of u.

The student having a discerning eye will see that is symmetric about the Nyquist frequency . Hence, there is no new information above this frequency. The information is limited to the analysis bandwidth .

The thoughtful student might also note that the plot in Figure E1.2 is anything but flat; suggesting that the noise is not

white. Ah! But this is where comes in [c.f. Definition 3]. The above plot is not the expected value

of anything. We will now average 104 such plots:

%Approximating the PSD via simulations:nsim=10^4;u=normrnd(0,1,n,nsim);U=(n^-0.5*abs(fft(u))).^2;Upsd=mean(U,2);wvec2=wvec(1:n/2);Upsd2=Upsd(1:n/2);figure(3)plot(wvec2,Upsd2)title('Approximation of White Noise PSD')xlabel('Frequency (rad/sec)')grid

Figure E1.3. Simulation-based .

From this figure we see that, indeed, white noise does have a flat psd across the analysis bandwidth. □

Example 2. Colored Noise: In this example we desire to simulate not white, but colored noise; i.e. noise that does not have a flat psd. Specifically, we

0 50 100 150 200 250 300Time (sec)

-15

-10

-5

0

5

10

15

20Colored Noise Simulation

whitecolored

11

desire that the psd should have the shape magnitude shown at right.H=tf(25,[1 2 25]);figure(10)bode(H)grid

Figure E2.1 Bode plot for .

From Definition 3, we have .

For continuous-time white noise with , we then have . This gives :

.

Hence, for continuous-time white noise with , the psd for is the Bode plot magnitude (in dB) for .

To determine our analysis BW we decide the frequency, beyond which we can basically assume . Using the data cursor, we identify this frequency at that where the amplitude is 40dB below the system static gain. Hence, we choose . It follows that the sampling frequency is , and so the sampling period is

. The colored noise simulation is:

% Simulation of Colored Noise:H=tf(25,[1 2 25]);wN=50; %Chosen Nyquist frequencydt=pi/wN; %Chosen sampling period%Compute desired colored noise variance:fun=@(w) 25^2*abs(((1i*w).^2 + 2*(1i*w) + 25)).^-2;varY=(1/pi)*integral(fun,0,wN); %Variance of YstdY=sqrt(varY)t=0:dt:300; t=t';n=length(t);stdU=1/sqrt(dt);u=normrnd(0,stdU,n,1);y=lsim(H,u,t);stdhatY=std(y) %Estimate of varYfigure(11)plot(t,u,'--')hold onplot(t,y,'r','LineWidth',2)title('Colored Noise Simulation')xlabel('Time (sec)')gridlegend('white','colored')

Figure E2.2 Colored noise simulation.

Important Notes:

Note 1: In the above code, the total area associated with the turbulence theoretical psd was computed as:

.

12

In word, the total area associated with the psd is the process variance, which is often also called the process total power.The approximate nature of the leftmost equality results from the fact that we are only integrating up to the Nyquist frequency. For a truly continuous-time process we would integrate to infinity.

Note 2: The psd is defined as where is the autocorrelation

function. This definition results in the Fourier Transform pair relation . For a process

with , this relation gives, in particular:

. (1)

The rightmost equality is due to the fact that (i.e. the psd is an even function of ). In words, half of

the total power lies at and the other half lies at . [Note that if, as is generally assumed, is a continuous function of , then The power associated with any single frequency is zero. This is because the area associated with a single point is zero.

Now, it is not the area associated with that equals the total power . Rather, the total power is that area scaled by . To see why this scale factor is important, recall that we have the relation

. (2)

Example 1 Let be a standard white noise process. Such a process, by definition, has . We claim that the

autocorrelation function is . To prove this, simply take the Fourier transform of :

. (3)

Hence, the white noise variance (i.e. total power) is . In other words, continuous time white noise does not

exist! Even so, it is a valuable mathematical concept. To this end, suppose that we want to model a process is the output of a transfer function having the input . Then clearly . Hence,

. (4)

Substituting (4) into (1) gives: . In words, the variance is the total area associated with

for scaled by . To illustrate the importance of this scale factor, consider the following Dryden model MIL-F-8785C for the forward speed turbulence psd:

. (5)

We will now apply (1): .

0 1000 2000 3000 4000 5000-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

13

Let . Then , and so

.

Clearly, something is wrong here. What is wrong is that the factor in the turbulence model (5) should not be there. This error becomes especially important in relation to simulating turbulence. The corresponding shaping filter is given as:

. (6)

The following code simulates turbulence according to (5):dt=tau/1000;tmax=5000;t=0:dt:tmax; t=t';n=length(t);stdU=1/sqrt(dt);u=normrnd(0,stdU,n,1);Lu=1750; u0=180; stdug=1;gs=stdug*sqrt(2*Lu/(pi*u0));tau=Lu/u0;H=tf(gs,[tau 1]);ug=lsim(H,u,t);varug=var(ug)plot(t,ug)

varug = 0.3117

Clearly, the sample variance is not close to the specified variance 1.0. In fact: 0.2917*pi=0.9792. In words, the shaping filter (6) is not the correct filter! It should NOT include the factor of .

CONCLUSION: While I am very skeptical of critiquing anything that has been long accepted as correct, I will confidently conclude that the turbulence spectra and associated shaping filters given in the Matlab turbulence toolbox are WRONG. Furthermore, the results of the errors associated with the factor can be consequential. The simulations generate turbulence whose variance is a factor of less than what they should be. Were one to simulate the forces on a wing or a building using such filters, one would end up underestimating them. This could have disastrous consequences in relation to subsequent designs. Finally, it would behoove Matlab to explicitly mention that any specified turbulence variance relates to the corresponding psd via the relation (1) the includes the factor of .

105 110 115 120 125 130 135 140 145 150Time (sec)

-8

-6

-4

-2

0

2

4

6

8

10Colored Noise Simulation

whitecolored

14

Note 3: The white noise input for the continuous-time process should be infinite. This follows from the fact that its psd is equal to 1.0 for all . Clearly, the area associated with this psd is infinite. Hence, continuous-time white noise does not, in fact, exist! Even so, discrete-time white noise does exist. We simulated it using the commands:

stdU=1/sqrt(dt); u=normrnd(0,stdU,n,1);

Notice that the variance of this white noise was chosen to be , where the sampling period was consistent with

our chosen Nyquist frequency. As , .

Note 3: We computed the value in order to be able to validate the simulation. We should expect the y-values to be generally within the range: . It is clear from the figure that they are, for the most part, in this range.

The zoomed figure at right shows that the colored noise has clear oscillations. The filter has a natural frequency and a damping ratio ; so that the damped natural frequency is

. Hence, the damped natural period is

. So on the average, we should see ~4 oscillations in any 5 second interval. This is seen in the figure. Figure E2.2 Colored noise simulation(zoomed).

I would suggest that you choose a variety of small values for in the colored noise simulation code. They will all result in the same variance for , as they should.

15

Wind PSD Models in Relation to the (u,v,w) Directions

The following are the von Karman psd models for the three directions (u,v,w) given by the authors on p.228:

(6.52)

(6.53)

(6.54)

The authors state on p.229 that: The psds are for spatial frequency. However, as the plane passes through frozen (i.e. spatial) turbulence, it senses temporal frequency. The relationship between the spatial and temporal frequency is given by

(6.55).

This suggests that the reader need only substitute (6.55) into any of (6.52-55) to obtain the temporal turbulence psds. As illustrated in the above examples, and via the models given in Matlab, this is NOT the case. Specifically, it does not account for the inclusion of the factor of that is in the temporal models. This is a pretty big factor! We note that the CORRECT psds associated with (6.52-55) should NOT include the factor . Nor should the associated shaping filters include the factor .

NOTE: In past notes I had stated that the variances in (6.52-55) were the driving white noise variances. They are NOT. In fact, they are the variances of the turbulence processes, themselves. As noted in the above example, they all resume that the continuous-time white noise has a psd . Practically, this translates into having only up to the

chosen Nyquist frequency . This choice results in the sampling interval . One can then show that setting

will result in for .

Connection to Matlab Simulink & to Military Specifications

Von Karman Wind Turbulence Model (Continuous): Generate continuous wind turbulence with Von Kármán velocity spectra

Library: Environment/Wind

Description: The Von Kármán Wind Turbulence Model (Continuous) block uses the Von Kármán spectral representation to add turbulence to the aerospace model by passing band-limited white noise through appropriate forming filters. This block implements the mathematical representation in the Military Specification MIL-F-8785C and Military Handbook MIL-HDBK-1797.

According to the military references, turbulence is a stochastic process defined by velocity spectra. For an aircraft flying at a speed V through a frozen turbulence field with a spatial frequency of Ω radians per meter, the circular frequency ω is calculated by multiplying V by Ω . The following table displays the component spectra functions: [See Matlab documentation] □