MATLAB Toolboxes - Air Transportation Systems Laboratory

Introduction to MATLAB 

MATLAB Toolboxes 

Computer Applications in CEE 

Drs. Trani and Rakha 

Civil and Environmental Engineering 

Virginia Polytechnic Institute and State University 

Spring 2000 

Virginia Polytechnic Institute and State University 1 of 31

Sample Toolboxes 

• MATLAB has been extended over the years to respond to 

the needs of various users 

• Several toolboxes exist to add to the power of the original 

language. For example: 

- Simulink 

- Fuzzy Logic 

- Neural Networks 

- Optimization 

- Controls 

- C/C++ compiler library 

- Real-time workshop 


Simulink 

• Simulink is a powerful toolbox to solve systems of 

differential equations 

• Simulink has applications in Systems Theory, Control, 

Economics, Transportation, etc. 

• The Simulink approach is to represent systems of ODE 

using block diagram nomenclature 

• Simulink provides seamless integration with MATLAB. 

In fact, Simulink can call any MATLAB function 

• Simulink interfaces with other MATLAB toolboxes such 

as Neural Network, Fuzy Logic, and Optimization 

routines 


Simulink Building Blocks 

• Simulink has a series of libraries to construct models 

• Libraries have object blocks that encapsulate code and 

behaviors 

• Connectors between blocks establish causality and flow 

of information in the model 


Simulink Interface 

• The main application of Simulink is to model continuous 

systems 

• Perhaps systems that can be described using ordinary 

differential equations 


Typical Simulink Libraries 

Shown are some typical Simulink libraries (Macintosh) 


Sample Simulink Library (Windows and UNIX) 

The new Simulink interface in Windows/UNIX uses 

standard OOP interfaces (Visual C++, Visual Basic) 


Simulink Example 

The following example illustrates the use of Simulink to 

solve a two vehicle car-following problem. This problem 

has been studied for the past 40 years in traffic flow 

theory 

ẋ˙ ( t + τ) fc = k( ẋ() t lc – ẋ t ) 

k 

() fc 

where: is a gain constant of the response process 

ẋ˙ ( t + τ) fc 

is the acceleration of the following vehicle 

ẋ() t lc 

ẋ() t fc 

is the speed of the leading vehicle 

is the speed of the following vehicle 


Set-Up of the Problem 

• Assume the velocity profile of the leading vehicle is 

known (we “drive” this car to test the response of the 

following vehicle) 

• Initially, assume no time lags in the acceleration response 

of the following vehicle ( ) 

τ = 0 

• Test an emergency braking maneuver executed by the 

first vehicle at 3 m/s 2 

• Test a new scenario with a deterministic time lag 

response time of 0.75 seconds 

• Verify that both cars do not collide 


Simulink Representation of the Car-Following 

Problem 

Car Following Problem 

Mux 

Mux1 

Speed Profiles 

1 

Xdot 

1 

X 

s 

Integrator 

s 

Integrator1 

Mux 

Mux 

Distance Profiles 

Speed LTU 

k (Xdot_ltu - Xdot_ftu) 

2 

Constant 

Xdot_ltu - Xdot_ftu 

1 

s 

Integrator2 


Car-Following Model Output 

Velocity profiles for leading and trailing vehicles 

Speed (m/s) 

Time (s) 


Car-Following Model Output 

The time space diagram below illustrates an emergency 

braking maneuver for the leading vehicle 

Space (m) 

Time (s) 


Car-Following Model with Delay 

A pure transport delay block is added to the original 

model simulating the transport lag dynamics of a manmachine 

system 

Mux 

Mux1 

Speed Profiles 

Xdot 

1 

1 

X 

s 

Integrator 

s 

Integrator1 

Mux 

Mux 

Distance Profiles 

Transport 

Delay 

Speed LTU 

k (Xdot_ltu - Xdot_ftu) 

1 

Constant 

Xdot_ltu - Xdot_ftu 

1 

s 

Integrator2 


Output of the Car-Following Model with Delay 

• The output suggests that a collision occurs with 

=0.75s. 

Space (m) 

τ 

Time (s) 


Brief on C/C++ Compiler 

• Converts MATLAB M-Files to C and C++ code 

• Using this toolbox all graphics and math code can be 

converted to develop standalone applications 

• Typically requires MATLAB compiler and the MATLAB 

C/C++ library 

• New features in version 5.3 also convert cell and 

structure arrays 

• My limited experience with this toolbox suggest 

substantial gains in speed if no vector operations have 

been implemented in the code 

• Vector operations show very little improvement when the 

code is compiled 


Neural Network Toolbox 

P 

Input 

P 

RxQ 

1 

W1 

Z1xR 

b1 

n1 

Z1xQ 

F1 

a1 

Z1xQ 

W2 

Z2xZ1 

1 b2 Z2xQ 1 

F2 , a2 W3 

F3 

n2 Z2xQ 

n3 

Z3xZ2 

b3 

Z3xQ 

a3 

Z3xQ 

Z1x1 

Z2X1 

Z3x1 

• Provides 100+ functions to simplify the training, 

generalization and implementation of artificial neural 

networks 

• The functional implementation of the ANN toolbox is 

just through a series of MATLAB functions 

• All ANN functions are execued from the command line 


Relevant Functions Provided 

• Layer initialization functions (Nguyen-Widrow) 

• Learning functions (gradients, perceptron, Widrow-Hoff) 

• Analysis functions (max. learning rate, error surface) 

• Line search functions (Golden section, backtraking) 

• Network functions (competitive, feed-forward wth 

backpropagation) 

• Performance functions (mean absolute error, SEE) 

• Training functions (BFGS, Bayesion regularization, 

Fletchel-Powell, Lavenberg-Marquardt, etc.) 

• Transfer functions (log signmoid, tangent sigmoid, etc.) 


Problem: 

Example of ANN Using MATLAB 

• To estimate aircraft fuel consumption based on aircraft 

performance parameters 

• Implementation on fast-time simulation models 

(SIMMOD, RAMS, etc.) 

Solution 

• Prototype model using MATLAB’s neural Network 

Toolbox 

• Verify the accuracy of the model with the current stateof-the 

art 


Sample Data Presented in Flight Manual 


Modeling Approach 

Training Data Set 

Learning Module 

Data Normalization 

Import Data 

Input learning pairs 

Forward 

Propagation 

Compute 

Error 

Initialize connection 

Weights 

Change and 

Update weights 

Configuration 

Parameters 

eg - maximum allowable error 

me- maximum allowable iterations 

YES 

NO 

Error< 

eg 

Number 

of cycles > me 

YES 

End of learning 

NO 


Selected ANN Topology 

1 

w 1,1 

sum 

b 1 1 

sum 

n 1 1 

f1 

f1 

a 1 1 

w 2 

1,1 

sum 

sum 

2 

b 

1 

n 2 1 

f2 

f2 

a 2 1 

Target 

sum 

f1 

sum 

f2 

Altitude 

Mach 

sum 

f1 

sum 

f2 

sum 

n 3 1 

f3 

Fuel burn 

Initial 

Weight 

sum 

f1 

sum 

f2 

b 3 1 

ISA Cond. 

sum 

f1 

sum 

f2 

1 

w 8,4 

sum 

f1 

sum 

f2 

a 2 8 

sum 

f1 

sum 

n 2 8 

f2 

n 1 8 

a 1 8 

w 2 8,8 

b 2 8 

Where f1 - log-sigmoid transfer function 

f2 - tansigmoid transfer function 

f3 - purelin transfer function 


ANN Data 

The data sets used to develop and train the ANN are 

shown below. Generalization of the ANN was conducted 

using another (random) data set 

Flight Phase 

Number of Testing Points 

Takeoff and Climbout Nor applicable (linear regression 

used instead) 

Climb to Cruise Altitude 

850 (Fuel) 

850 (Distance) 

Cruise 805 

Descent 

1210 (Fuel) 

140 (Distance) 


Summary of Results (Fokker F100) 

The results shown in the table illustrate the accuracy of 

the model 

Flight Phase 

Climb 

• Distance 

• Fuel 

Cruise Specific 

Air Range 

Mean 

Error (%) 

0.377 

1.026 

Standard 

Deviation (%) 

0.305 

0.190 

Null Hypothesis 

(t-test at α 0.01) 

= 

Accept 

Accept 

-0.034 0.334 Accept 


Flight Phase 

Descent 

• Distance 

• Fuel 

Mean 

Error (%) 

1.760 

1.423 

Standard 

Deviation (%) 

1.860 

1.177 

Null Hypothesis 

(t-test at α = 0.01) 

Accept 

Accept 


Sample Results (Climb Fuel) 

• The results of training an ANN with 3 layers are shown 

below 

6000 

5000 

* 

Actual Data 

Estimated Data 

Climb Fuel (lb) 

4000 

3000 

2000 

1000 

0 

0 5 10 15 20 25 30 35 40 

Pressure Altitude (kft) 


Absolute Error of ANN 

The errors of the ANN are very small as depicted in this 

graph for SFC 

160 

Frequency 

140 

120 

100 

80 

60 

40 

20 

Complete flight envelope 

0.60 < Mach < 0.75 

10,000 ft < altitude < 37,000 ft 

58,000 lb < weight < 98,000 lb 

805 data points 

0 

-1.5 -1 -0.5 0 0.5 1 1.5 

Specific Range Error (%) 


Application of the Model to Complete Flight Plans 

The ANN model developed has been applied to a generic 

flight trajectory generator with very accurate results 

Flight plan way-points 

Constant heading 

segments 

30 

Descent 

Climb 

Altitude (kft) 

20 

10 

0 

80 MIA 

85 

Longitude (deg) 

90 

95 

Pseudo-globe circle route 

100 26 

27 

26.5 

28 

27.5 

DFW 

29 

28.5 

Latitude (deg) 

29.5 


Complete Flight Plan Correlation 

Flight 

ROA a - 

MDW b 

MIA c -DFW d 

ROA-LGA e 

ATL f -MIA 

Cruise Flight 

Level (FL) 

Distance 

(nm) / 

Time (hr) 

a.ROA - Roanoke Regional Airport (Virginia) 

b.MDW - Midway Airport (Illinois) 

c.MIA - Miami International (Florida) 

d.DFW - Dallas-Forth Worth International (Texas) 

e.LGA - Laguardia Airport (New York) 

f. ATL - Atlanta Hartsfield International Airport (Georgia) 

Flight Manual 

Fuel Burn 

(lb) 

Neural Net 

Fuel Burn (lb) 

Percent 

Difference (%) 

280 448 / 1:08 6,457 6,546 1.37 

310 448 / 1:10 6,360 6,330 0.46 

310 972 / 2:24 11,851 11,865 0.12 

350 972 / 2:13 11,510 11,544 0.29 

290 352 / 0:57 5,298 5,260 0.71 

330 352 / 0:58 5,343 5,429 1.61 

290 518 / 1:20 6,990 7,047 0.80 

330 518 / 1:21 7,009 7,082 1.04 


Sample Source Code 

% Initialize Weights and Biasis 

nns = 8; % Number of Neurons in each layer 

nns2 = 8; 

%******************************** 

% For Climb Distance ** 

%******************************** 

[W31_cb_d,b31_cb_d,W32_cb_d,b32_cb_d,W33_cb_d, 

b33_cb_d ]=initff(P1_cbd,nns,'logsig',nns2 ... 

,'tansig',T1_cbd,'purelin'); 

% Taining of the neural networks using Lavenberg- 

Marquardt Alogrithm 


[W31_cb_d,b31_cb_d,W32_cb_d,b32_cb_d,W33_cb_ 

d,b33_cb_d ]= trainlm(W31_cb_d,b31_cb_d,'logsig' ... 

,W32_cb_d,b32_cb_d,'tansig',W33_cb_d,b33_cb_d,'pure 

lin',P_cbd,Ta_cbd,tp); 

%******************************** 

% For Climb Fuel ** 

%******************************** 

[W31_cb_f,b31_cb_f,W32_cb_f,b32_cb_f,W33_cb_f,b3 

3_cb_f ]=initff(P1_cbf,nns,'logsig',nns2,'tansig' ... 

,T1_cbf,'purelin'); 

% Taining of the neural networks using Lavenberg- 

Marquardt Alogrithm 

[W31_cb_f,b31_cb_f,W32_cb_f,b32_cb_f,W33_cb_f,b3 

3_cb_f ]= 

trainlm(W31_cb_f,b31_cb_f,'logsig',W32_cb_f ... 


,b32_cb_f,'tansig',W33_cb_f,b33_cb_f,'purelin',P_cbf, 

Ta_cbf,tp);

MATLAB Toolboxes - Air Transportation Systems Laboratory

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