problemset4_Soln - The University of Texas at Austin...

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The University of Texas at Austin Department of Electrical and Computer Engineering EE362K: Introduction to Automatic Control—Fall 2009 Problem Set Four Solutions C. Caramanis Not Due. This problem set should serve as a start for your review for the test. It covers the concepts introduced in Chapter 5. This includes homogeneous and particular solutions to LTI systems (CT and DT); The matrix exponential; Jordan Canonical Form and linear algebra; and stability of LTI systems. Also, while it also covers some of the earlier concepts discussed, it is not exhaustive. We had a lot of practice with linearization in class and in previous problem sets, so while that is an important concept, it is not emphasized in this one. It is de±nitely important for the midterm. 1. Consider the matrix: A = 1111 2 0 30 2 8 000 2 5 0004 7 0000 6 (a) If you can, write down the Jordan Canonical Form (JCF) of this matrix and explain which results you are using in order to arrive to your answer. If you cannot, explain why there is ambiguity. (b) Is the system ˙ x = Ax stable, unstable, or stable i.s.L., for A the above matrix? (c) Is the system x [ k +1]= Ax [ k ] stable, unstable, or stable i.s.L., for A the above matrix? Solution: (a) The matrix is upper triangular, and hence the eigenvalues are precisely the values on the diagonal. Thus the eigenvalues are: λ =1 , 3 , 0 , 4 , 6. Since these values are distinct, the JCF of the matrix must be composed of 1 × 1 blocks, i.e., it is diagonal, with the above values along the diagonal. (b) The continuous time system is unstable, since there is an eigenvalue (several, in fact) with strictly positive real part. (c) The discrete time system is also unstable, since there is an eigenvalue (several, in fact) strictly outside the unit circle. 2. Suppose A is a 7 × 7 symmetric matrix with characteristic polynomial: p A ( λ )=3 · λ · ( λ +2) 2 · ( λ +1) 3 · ( λ + 12) . (a) If you can, write down the Jordan Canonical Form (JCF) of this matrix and explain which results you are using in order to arrive to your answer. If you cannot, explain why there is ambiguity. 1
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(b) Is the system ˙ x = Ax stable, unstable, or stable i.s.L., for A the above matrix? (c) Is the system x [ k +1]= Ax [ k ] stable, unstable, or stable i.s.L., for A the above matrix? Solution : (a) The eigenvalues are not distinct. They are: λ =0 , 2 , 2 , 1 , 1 , 1 , 12. However, since the matrix is symmetric, we know it is diagonalizable, and hence its JCF must be the diagonal matrix with the values above in its diagonal. (b) The continuous time system is neutrally aka marginally stable. This is because there are no eigenvalues strictly in the right half plane, and the one with real part equal to zero is in a 1 × 1Jordanb lock . (c) The discrete time system is unstable, since there is an eigenvalue (several, in fact) strictly outside the unit circle.
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problemset4_Soln - The University of Texas at Austin...

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