Consider the following system A11 0 2-422 = QA21 A22 [7]10-6 (S) where 21 R...

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Consider the following system A11 0 2-422 = QA21 A22 [7]10-6 (S) where 21 € R¹¹ and 12 € R2 are the states, y R2 is the output, and a is the positive parameter. (a) Determine if the matrix A is normal. (b) Find the expression for the resolvent of the above matrix A and use the inverse Laplace transform to determine et Hint: The resulting state transition matrix should be partitioned conformably with the partition of the matrix A and its components should be expressed in terms of A11, A21, A22, and a. You may find the following expression useful: (c) For A11 = A 0 A-1 -D-¹CA-¹ D-1 [ ĉ ¦ ] = [ -1 C [ -1 -2 -5 A22 = -1, A21 = [12] find the right and the the left eigenvectors of the matrix A. Your result should be expressed in terms of the parameter a and the left eigenvectors should be normalized to satisfy bi-orthogonality condition. (d) For the system given in part (c), determine the expression for the system's output arising from the initial conditions in 21 and 22. You should use the results obtained in part (c) here. (e) Sketch the output components determined in part (d). How do your results change if a is in- creased? Explain your observations. Consider the following system A11 0 2-422 = QA21 A22 [7]10-6 (S) where 21 € R¹¹ and 12 € R2 are the states, y R2 is the output, and a is the positive parameter. (a) Determine if the matrix A is normal. (b) Find the expression for the resolvent of the above matrix A and use the inverse Laplace transform to determine et Hint: The resulting state transition matrix should be partitioned conformably with the partition of the matrix A and its components should be expressed in terms of A11, A21, A22, and a. You may find the following expression useful: (c) For A11 = A 0 A-1 -D-¹CA-¹ D-1 [ ĉ ¦ ] = [ -1 C [ -1 -2 -5 A22 = -1, A21 = [12] find the right and the the left eigenvectors of the matrix A. Your result should be expressed in terms of the parameter a and the left eigenvectors should be normalized to satisfy bi-orthogonality condition. (d) For the system given in part (c), determine the expression for the system's output arising from the initial conditions in 21 and 22. You should use the results obtained in part (c) here. (e) Sketch the output components determined in part (d). How do your results change if a is in- creased? Explain your observations.