Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Introduction to MATLAB IISteve GuJan 25, 2007Outline•Matrix Operation–Matrix functions–Element-wise operations•Dynamic Systems–Classification–2nd1st Order Equations•Introduction to SimulinkPart I•Matrix Operation–Matrix functions–Element-wise operationsMatrix Functions•Many mathematical functions work on the individual entries of a matrix or vector•For example, abs(M) takes the absolute value of each entry of M:>> abs( [1.2 -0.3 -0.5; 0.4 -0.7 0.2] ) 1.2 0.3 0.5 0.4 0.7 0.2•Other such functions are sin, cos, sqrt, etc.Matrix Functions•Many functions work on the columns of matrices•Example, the max function:–max(v) finds the maximum entry of a vector–max(M) returns a row vector containing the maximum entry of each column of a matrix–max(max((M)) returns the maximum entry a matrixMatrix Functions>> max( [3 5 2 -3] ) 5>> M = [3 8 2 -1; 5 3 7 3; 6 -10 5 2; 9 3 4 3] 3 8 2 -1 5 3 7 3 6 -10 5 2 9 3 4 3 >> max( M ) % maximum in each column 9 8 7 3>> max( M’ ) % maximum in each row 8 7 6 9Similar Functions•Similar functions include:–max min sum –mean sort>> M = [1 2 3; 4 5 6; 7 8 9] 1 2 3 4 5 6 7 8 9>>sum( M ) 12 15 18>> min( M ) 1 2 3Similar Functions•Some functions work on the entire column:–sort>> M = rand( 3, 6 ) 0.21116 0.61004 0.77120 0.74940 0.55795 0.20212 0.55104 0.43918 0.06933 0.09868 0.82398 0.24698 0.19782 0.94107 0.20374 0.41644 0.91337 0.96385>> sort( M ) 0.19782 0.43918 0.06933 0.09868 0.55795 0.20212 0.21116 0.61004 0.20374 0.41644 0.82398 0.24698 0.55104 0.94107 0.77120 0.74940 0.91337 0.96385Matrix Functions•some mathematical functions work on the entire matrix•For example, det(M) takes the determinant of matrix M:>> det( [1 2;3 4] )>>ans-2•Other such functions are eig, norm, cond, etc.Element-wise Products•How have heard repeatedly that if A = (ai,j) and B = (bi,j) are matrices, then AB means matrix multiplication:•Often though, the simple product is useful:22222121121211112221121122211211bababababbbbaaaa222212212122112122121211211211112221121122211211bababababababababbbbaaaaElement-wise Products•Matlab allows you to do this with the .* operator:>> A = [1 2 3; 4 5 6] 1 2 3 4 5 6>> B = [1 2 4; 8 16 32] 1 2 4 8 16 32>> A .* B 1 4 12 32 80 192Element-wise Powers•You also learned that if A = (ai,j) then An means repeated matrix multiplication:•Again, sometimes you just want to raise each entry to an exponent:222221212211222211211aaaaaaaa222122121221121221212112112211222211211aaaaaaaaaaaaaaaaaaElement-wise Powers•Matlab allows you to do this with the .^ operator:>> A = [1 2 3; 4 5 6] 1 2 3 4 5 6>> A.^2 1 4 9 16 25 36>> A.^6 1 32 243 1024 3125 7776Dot Product•Note that matrix multiplication is a series of dot products:>> A = [1 2; 3 4];>> B = [2 3; 5 7];>> A*B 12 17 26 37>> [A(1,:)*B(:,1) A(1,:)*B(:,2) A(2,:)*B(:,1) A(2,:)*B(:,2)]; 12 17 26 37Part II•Dynamic Systems–Classification–2nd1st Order EquationsDynamic Systems•What we consider is Linear, Deterministic, Stationary, Discrete Dynamic Systems:Last Week…The equation for the motion: Remark: Second Order Difference Equation2nd1st Order Equations2nd1st Order EquationsAlso, we have: G=0, H=[1,0]Therefore, we’ve reduced 1 second order equation to a system of 2 first order equationsPart III•Introduction to SimulinkStarting Simulink1. Start MATLAB2. Click the Simulink icon on MATLAB toolbar;Enter the simulink command at the MATLAB prompt3. Starting Simulink displays the Simulink Library BrowserSimulink Library BrowserDisplays a tree-structured view of the Simulink block librariesSignal GeneratorGenerate various waveformsParameters and Dialog BoxSimulation Parameters•Set the simulation parameters by choosing Simulation Parameters from the Simulation menuStarting the Simulation•Pull down the Simulation menu and choose the Start command (or Ctrl+T)Ctrl+TBlocksetsBlocksets are specialized collections of Simulink blocks built for solving particular classes of problemsA Working ExampleResults & DemoThe End•Thank