matlab_toolboxes

matlab_toolboxes - Introduction to MATLAB MATLAB Toolboxes...

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Unformatted text preview: 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 Virginia Polytechnic Institute and State University 2 of 31 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 Virginia Polytechnic Institute and State University 3 of 31 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 Virginia Polytechnic Institute and State University 4 of 31 Simulink Interface • The main application of Simulink is to model continuous systems • Perhaps systems that can be described using ordinary differential equations Virginia Polytechnic Institute and State University 5 of 31 Typical Simulink Libraries Shown are some typical Simulink libraries (Macintosh) Virginia Polytechnic Institute and State University 6 of 31 Sample Simulink Library (Windows and UNIX) The new Simulink interface in Windows/UNIX uses standard OOP interfaces (Visual C++, Visual Basic) Virginia Polytechnic Institute and State University 7 of 31 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 x˙( t + τ ) fc = k ( x ( t ) lc – x ( t ) fc ) ˙ ˙ ˙ where: k is a gain constant of the response process x˙( t + τ ) fc is the acceleration of the following vehicle ˙ x ( t ) lc is the speed of the leading vehicle ˙ x ( t ) fc is the speed of the following vehicle ˙ Virginia Polytechnic Institute and State University 8 of 31 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/s2 • Test a new scenario with a deterministic time lag response time of 0.75 seconds • Verify that both cars do not collide Virginia Polytechnic Institute and State University 9 of 31 Simulink Representation of the Car-Following Problem Car Following Problem Mux Speed Profiles Mux1 1 Xdot X 1 s Integrator s Integrator1 Mux Mux Distance Profiles Speed LTU 2 k (Xdot_ltu - Xdot_ftu) Xdot_ltu - Xdot_ftu Constant 1 s Integrator2 Virginia Polytechnic Institute and State University 10 of 31 Car-Following Model Output Velocity profiles for leading and trailing vehicles Speed (m/s) Time (s) Virginia Polytechnic Institute and State University 11 of 31 Car-Following Model Output The time space diagram below illustrates an emergency braking maneuver for the leading vehicle Space (m) Time (s) Virginia Polytechnic Institute and State University 12 of 31 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 Speed Profiles Mux1 Xdot X 1 1 s Integrator s Integrator1 Mux Mux Distance Profiles Transport Delay Speed LTU Xdot_ltu - Xdot_ftu k (Xdot_ltu - Xdot_ftu) 1 Constant 1 s Integrator2 Virginia Polytechnic Institute and State University 13 of 31 Output of the Car-Following Model with Delay • The output suggests that a collision occurs with τ =0.75s. Space (m) Time (s) Virginia Polytechnic Institute and State University 14 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 Virginia Polytechnic Institute and State University 15 of 31 Neural Network Toolbox Input a3 P P W1 RxQ Z1xR 1 a1 n1 W2 Z1xQ F2 n2 Z2xZ1 , a2 W3 Z2xQ Z3xZ2 F3 n3 Z3xQ Z1xQ F1 b1 Z1x1 1 b2 Z2xQ Z2X1 1 b3 Z3xQ 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 Virginia Polytechnic Institute and State University 16 of 31 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.) Virginia Polytechnic Institute and State University 17 of 31 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 MATLAB’s neural Network Toolbox • Verify the accuracy of the model with the current stateof-the art Virginia Polytechnic Institute and State University 18 of 31 Sample Data Presented in Flight Manual Virginia Polytechnic Institute and State University 19 of 31 Modeling Approach Training Data Set Learning Module Import Data Data Normalization Forward Propagation Input learning pairs Compute Error Initialize connection Weights Configuration Parameters Change and Update weights YES Error< eg NO Number of cycles > me eg - maximum allowable error me- maximum allowable iterations NO YES End of learning Virginia Polytechnic Institute and State University 20 of 31 Selected ANN Topology n 1 1 sum 1 b 1 w 1,1 f1 a 1 1 w 2 1,1 n 2 1 sum f2 2 b 1 f1 sum 1 f2 sum a sum f1 sum f1 sum 1 f2 sum Target 2 f2 Altitude Mach n sum f1 f2 Initial Weight sum ISA Cond. sum f1 sum sum f1 sum f2 1 Fuel burn f3 f2 1 w 8,4 sum 3 b n f1 sum n sum 1 8 a 1 8 w 2 8,8 b a 3 1 2 8 2 8 f2 2 8 Where f1 - log-sigmoid transfer function f2 - tansigmoid transfer function f3 - purelin transfer function Virginia Polytechnic Institute and State University 21 of 31 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 Takeoff and Climbout Climb to Cruise Altitude Cruise Descent Number of Testing Points Nor applicable (linear regression used instead) 850 (Fuel) 850 (Distance) 805 1210 (Fuel) 140 (Distance) Virginia Polytechnic Institute and State University 22 of 31 Summary of Results (Fokker F100) The results shown in the table illustrate the accuracy of the model Flight Phase Climb • Distance • Mean Error (%) Standard Deviation (%) Null Hypothesis (t-test at α = 0.01 ) 0.377 1.026 0.305 0.190 Accept Accept -0.034 0.334 Accept Fuel Cruise Specific Air Range Virginia Polytechnic Institute and State University 23 of 31 Flight Phase Descent • Distance • Mean Error (%) Standard Deviation (%) Null Hypothesis (t-test at α = 0.01 ) 1.760 1.423 1.860 1.177 Accept Accept Fuel Virginia Polytechnic Institute and State University 24 of 31 Sample Results (Climb Fuel) • The results of training an ANN with 3 layers are shown below 6000 Actual Data 5000 Climb Fuel (lb) * Estimated Data 4000 3000 2000 1000 0 0 5 10 15 20 25 30 35 40 Pressure Altitude (kft) Virginia Polytechnic Institute and State University 25 of 31 Absolute Error of ANN The errors of the ANN are very small as depicted in this graph for SFC 160 Complete flight envelope 140 0.60 < Mach < 0.75 Frequency 120 10,000 ft < altitude < 37,000 ft 58,000 lb < weight < 98,000 lb 100 805 data points 80 60 40 20 0 -1.5 -1 -0.5 0 0.5 Specific Range Error (%) Virginia Polytechnic Institute and State University 1 1.5 26 of 31 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 Constant heading segments Flight plan way-points Altitude (kft) 30 Climb Descent 20 DFW 10 29.5 Pseudo-globe circle route 0 80 MIA 28 27.5 85 90 Longitude (deg) 29 28.5 27 26.5 95 100 Latitude (deg) 26 Virginia Polytechnic Institute and State University 27 of 31 Complete Flight Plan Correlation Cruise Flight Level (FL) Distance (nm) / Time (hr) Flight Manual Fuel Burn (lb) Neural Net Fuel Burn (lb) Percent Difference (%) ROAaMDWb 280 448 / 1:08 6,457 6,546 1.37 310 448 / 1:10 6,360 6,330 0.46 MIAc-DFWd 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 Flight ROA-LGAe ATLf-MIA 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) Virginia Polytechnic Institute and State University 28 of 31 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 LavenbergMarquardt Alogrithm Virginia Polytechnic Institute and State University 29 of 31 [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 LavenbergMarquardt 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 ... Virginia Polytechnic Institute and State University 30 of 31 ,b32_cb_f,'tansig',W33_cb_f,b33_cb_f,'purelin',P_cbf, Ta_cbf,tp); Virginia Polytechnic Institute and State University 31 of 31 ...
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This note was uploaded on 10/16/2011 for the course MECHANICAL 000 taught by Professor R during the Fall '11 term at IUPUI.

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