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Symbolic solution to expansion of natural frequencies method for determining index_1.png

Written by Nasser M. Abbasi, based on lecture given by Professor Barmish, ECE 717, University of Wisconsin, Madison
11/1/2014

The method of finding index_2.png using expansion of natural frequencies is first applied to a 2×2 matrix to illustrate the symbolic method. Then a general purpose function is written at the end using the symbolic method and applied to much larger problems for illustrations. This is a standalone function which can be called to determine index_3.png for matrices A which can have repeated eigenvalues.  The result is verified by comparing the output to Mathematica buildin function MatrixExp[]

Example 1 from lecture done using symbolics

Define A

index_4.gif

index_5.png

Find eigenvalues of A

index_6.png

Set the eigenvalue multiplers

index_7.png

Apply index_8.png algorithm using expansion of natural frequencies

index_9.gif

index_10.png

Automatic generation of  lists of unknowns Y(k,i) and list of symbols to b[i] to use for solving Ax=b

index_11.gif

index_12.png

index_13.png

Take repeated derivatives of index_14.png to generate the set of equations to solve for Y(k,i)

index_15.png

index_16.png

The problem can be viewed as Ax=b where x=Y(k,i) is the unknowns and b are the symbolic names for the matrix powers.

index_17.gif

index_18.png

Now solve Ax=b

index_19.png

index_20.png

Replace symbols b(i) on the right of the above, by the actual numerical values of the matrix powers

index_21.gif

index_22.png

Now apply these numerical values to the solution found earlier in order to obtain numerical values for Y(k,i)

index_23.gif

index_24.png

Now since Y(k,i) are now found,  evaluate index_25.png found above using the above solution

index_26.png

index_27.png

Compare to Mathematica buildin function

index_28.png

index_29.png

Function to automatically calculate index_30.png using symbolic natural frequencies method

index_31.png

Example 2 from lecture solved using the above function

index_32.gif

λ multiplier
3 1
2 3
1 2

index_33.png

index_34.png

index_35.png

index_36.png

index_37.png

index_38.png

index_39.png

Compare to buildin function to verify

index_40.png

index_41.png

index_42.png

index_43.png

Verification Method, suggested by Professor Barmish. We Differentiate Matrix index_44.png found above and see it is the same as index_45.png for t=0

index_46.gif

k index_47.png index_48.png
0 index_49.png index_50.png
1 index_51.png index_52.png
2 index_53.png index_54.png
3 index_55.png index_56.png
4 index_57.png index_58.png
5 index_59.png index_60.png
Created with the Wolfram Language