matlab toolboxes

Virginia Polytechnic Institute and State University

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Introduction to MATLAB

MATLAB Toolboxes

Computer Applications in CEE

Drs. Trani and Rakha

Civil and Environmental EngineeringVirginia Polytechnic Institute and State University

Spring 2000


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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


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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


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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


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Simulink Interface

The main application of Simulink is to model continuous systems

Perhaps systems that can be described using ordinary differential equations


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Typical Simulink Libraries

Shown are some typical Simulink libraries (Macintosh)


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Sample Simulink Library (Windows and UNIX)

The new Simulink interface in Windows/UNIX uses standard OOP interfaces (Visual C++, Visual Basic)


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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

where: is a gain constant of the response process

is the acceleration of the following vehicle

is the speed of the leading vehicle

is the speed of the following vehicle

x t +( ) fc k x t( ) lc x t( ) fc( )=k

x t +( ) fcx t( ) lcx t( ) fc


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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 ( )

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

0=


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Simulink Representation of the Car-Following Problem

X

Car Following Problem

Xdot

k (Xdot_ltu - Xdot_ftu)Xdot_ltu - Xdot_ftu

Speed Profiles

Speed LTU

Mux

Mux1

Mux

Mux

s

1

Integrator2

s

1

Integrator1s

1

IntegratorDistance Profiles

2

Constant


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Car-Following Model Output

Velocity profiles for leading and trailing vehicles

Time (s)

Speed (m/s)


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Car-Following Model Output

The time space diagram below illustrates an emergency braking maneuver for the leading vehicle

Time (s)

Space (m)


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Car-Following Model with Delay

A pure transport delay block is added to the original model simulating the transport lag dynamics of a man-machine system

X

Xdot

k (Xdot_ltu - Xdot_ftu)Xdot_ltu - Xdot_ftu

TransportDelay

Speed Profiles

Speed LTU

Mux

Mux1

Mux

Mux

s

1

Integrator2

s

1

Integrator1s

1

IntegratorDistance Profiles

1Constant


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Output of the Car-Following Model with Delay

The output suggests that a collision occurs with =0.75s.

Time (s)

Space (m)

Virginia Polytechnic Institute and State University 15 of 31

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

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

P

PRxQ

Input

1

W1

b1

W2

b2

W3

b3

Z1xR

Z1x1

n1

Z1xQ

a1 Z1xQ

Z2xZ1

Z2X1

n2

Z2xQ

, a2

1 1

Z3xZ2

Z3x1

Z2xQ n3

Z3xQ

a3

Z3xQ F2 F3

F1


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.)


Example of ANN Using MATLAB

Problem:

To estimate aircraft fuel consumption based on aircraft performance parameters

Implementation on fast-time simulation models (SIMMOD, RAMS, etc.)

Solution

Prototype model using MATLABs neural Network Toolbox

Verify the accuracy of the model with the current state-of-the art


Sample Data Presented in Flight Manual


Modeling Approach

Data Normalization

Training Data Set

Input learning pairs

Initialize connectionWeights

Learning Module

Import Data

ForwardPropagation

ComputeError

Change and Update weights

ConfigurationParameters

Error< eg

Numberof cycles > me

End of learning

YES

NO

YES

NO

eg - maximum allowable errorme- maximum allowable iterations


Selected ANN Topology

sum f1

sum f1

sum f1

sum f1

sum f1

sum f1

sum f1

sum f1

sum f2

sum f2

sum f2

sum f2

sum f2

sum f2

sum f2

sum f2

sum f3

Target

Altitude

Mach

InitialWeight

ISA Cond.

w1,11

w 8,4

1

b1 1

a11

a 18

w2

1,1

w2

8,8

n1

1

n1

8

n2

1

b2

1

a2

1

b3

1

n3

1Fuel burn

b2

8

n2

8

a2

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 PointsTakeoff and Climbout Nor applicable (linear regres-

sion 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 PhaseMean

Error (%)Standard

Deviation (%)Null Hypothesis(t-test at )

Climb

Distance

Fuel

0.377 1.026

0.3050.190

AcceptAccept

Cruise Specific Air Range

-0.034 0.334 Accept

0.01=


Descent

Distance

Fuel

1.7601.423

1.8601.177

AcceptAccept

Flight PhaseMean

Error (%)Standard

Deviation (%)Null Hypothesis(t-test at ) 0.01=


Sample Results (Climb Fuel)

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

0 5 10 15 20 25 30 35 400

1000

2000

3000

4000

5000

6000

Clim

b Fu

el (lb

)

Pressure Altitude (kft)

*Actual DataEstimated Data


Absolute Error of ANN

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

-1.5 -1 -0.5 0 0.5 1 1.50

20

40

60

80

100

120

140

160

Frequ

ency

Complete flight envelope

805 data points

0.60 < Mach < 0.7510,000 ft < altitude < 37,000 ft58,000 lb < weight < 98,000 lb

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

8085

9095

100 2626.5

2727.5

2828.5

2929.5

0

10

20

30

Longitude (deg) Latitude (deg)

Alti

tude

(kft)

Flight plan way-points

DFW

MIA

Pseudo-globe circle route

Constant headingsegments

ClimbDescent


Complete Flight Plan Correlation

FlightCruise Flight Level (FL)

Distance (nm) /

Time (hr)

Flight Manual Fuel Burn

(lb)Neural Net

Fuel Burn (lb)Percent

Difference (%)

ROAa-MDWb

a.ROA - Roanoke Regional Airport (Virginia)b.MDW - Midway Airport (Illinois)

280 448 / 1:08 6,457 6,546 1.37310 448 / 1:10 6,360 6,330 0.46

MIAc-DFWd

c.MIA - Miami International (Florida)d.DFW - Dallas-Forth Worth International (Texas)

310 972 / 2:24 11,851 11,865 0.12350 972 / 2:13 11,510 11,544 0.29

ROA-LGAe

e.LGA - Laguardia Airport (New York)

290 352 / 0:57 5,298 5,260 0.71330 352 / 0:58 5,343 5,429 1.61

ATLf-MIA

f. ATL - Atlanta Hartsfield International Airport (Georgia)

290 518 / 1:20 6,990 7,047 0.80330 518 / 1:21 7,009 7,082 1.04


Sample Source Code

% Initialize Weights and Biasis

nns = 8; % Number of Neurons in each layernns2 = 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,'purelin',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,b33_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);

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