SOLVED CH-5120: Modern Control Theory Assignment 1

 Problem 1: Change of basis (2 + 2 points) Part-1.1: Change of basis for a vector Let � = # � �&, where a and b are the last two digits of your roll number. Thus, if this is CH17B987, then � = # 8 7 &. (If your roll number ends in 00, then use a = 1 and b = 1). 1. Express this vector x in terms of new basis, �! = # 1 1 & and �" = # 1 −1 &. Part-1.2: Change of bases for linear transformation Recall that the problem of blending of two streams was a three-input-two-output problem. The inputs were flowrates �!, �", �#$% and the outputs were ℎ, �&. The gain matrix is given by: � = # 4 2 4 0.5 1 0 & 2. How will this matrix change if the domain space is expressed in terms of the following bases: �! = 6 1 0 0 7, �" = 6 1 1 0 7, �' = 6 0 1 −1 7 and the co-domain space is expressed in terms of �! = # 1 1 &, �" = # 1 −1 & Problem 2: Linearly Dependence (1 + 3 points) Let �!, �", �' ∈ ℝ( be linearly independent vectors in n-dimensional space. 3. If the above three vectors are linearly independent, what is/are the possible value(s) of n? (a) n = 1 (b) n = 2 (c) n = 3 (d) n = 4 (e) n = 5 4. Consider the three vectors � = �! + �", � = �! + �', � = �" + �'. Are the vectors u, v, w linearly independent? Prove this. Problem 3: Null and Image Spaces (4 points) Consider a matrix L = # � � 4 0.5 1 0 &, where a and b are the last two digits of your roll number. If your roll number ends in 00, use a = 1 and b = 1. 5. Using definition, determine null space and image space of L. Using MATLAB: Not Graded, For Practice Only Also confirm the same using SVD (please use MATLAB for SVD).   Problem 4: Eigenvalue Decomposition and Matrix Exponent (1 + 1 + 1 + 1 points) Consider a matrix: � = # 1 0 � � &, where a and b are the last two digits of your roll number. Thus, if this is CH17B987, then � = # 1 0 8 7 &. (If your roll number ends in 00, use a = 1 and b = 1). 6. Obtain the characteristic equation and hence compute the eigenvalues of B. 7. Substitute B in its characteristic equation and thus verify Cayley Hamilton Theorem. 8. Perform eigenvalue decomposition for the matrix B 9. Using eigenvalue decomposition, compute matrix exponent �& Using MATLAB: Not Graded, For Practice Only Use MATLAB and compute the matrix exponent of B. Problem 5: Jordan Decomposition (2 + 2 points) 10. Find the eigenvalues of the matrix, � = # 1 1 −1 3 &. Since eigenvalues are repeated, compute eigenvector and generalized eigenvector 11. Representing � = �Λ�)! in Jordan canonical form, compute the matrix exponent Hints for Problems 4 and 5 • Please see the discussion about effect of similarity transform on exponent • Consider the following rules exp F# � 0 0 � &H = I �* 0 0 �*J, exp F# � � 0 � &H = I �* ��* 0 �* J