problemset5_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 Five Solution C. Caramanis Due: Wednesday, October 21, 2009. This problem set focuses on the new concepts introduced in the last two classes: reachability and state feedback. As usual, we also work in some exercise with important concepts from linear algebra. 1. Reachable Canonical Form: Consider the system from above: A = 14 3 26 1 2 52 ,B = 0 0 1 (a) What is the reachable canonical form, ˜ A , for the matrix A ? (b) In class we showed that ˜ A and A are related by the relationship: ˜ A = TAT 1 ,forsome invertible matrix T . 1 We showed in class that this kind of transformation preserves eigenvalues, and therefore also determinants. Use Matlab (you can compute by hand, if you wish) to verify that ˜ A and A havethesamee igenva lues ,andthesamedeterm inant (equal to the product of the eigenvalues). (c) Follow the procedure we outlined in class (and also outlined in the text) to compute the invertible matrix T that can be used to transform from A to ˜ A . Check your answer to see that this indeed has worked. Solution : (a) The characteristic polynomial of A is given by p A ( λ )= | λI A | = λ 3 7 λ 2 3 λ +9 This gives the reachable canonical form ˜ A = 73 9 10 0 01 0 , ˜ B = 1 0 0 (b) The characteristic polynomial of ˜ A is given by p ˜ A ( λ | λI ˜ A | = λ 3 7 λ 2 3 λ Since p ˜ A ( λ p A ( λ ), A and ˜ A have the same set of eigenvalues. Their determinant values are also same as determinant of a matrix is equal to the product of its eigenvalues. 1 This is known as a similarity transformation . 1
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( c )A c co rd ingtoEqua t ion(6 . 8) of the textbook, and also as we derived in class, T = ˜ W r W 1 r ,where ˜ W r is the reachability matrix for ( ˜ A, ˜ B ). This gives W r = 03 1 0 1 2 12 3 , ˜ W r = 175 2 01 7 00 1 T = ˜ W r W 1 r = 39 7 152 7 1 5 7 22 7 0 1 7 3 7 0 T can also be found by hand by solving ˜ B = TB and ˜ AT = TA simultaneously. 2. Show that the change of coordinates we are so freely using is not bogus: Consider the system: ˙ x = Ax + Bu. Let T be some invertible matrix, and consider the change of coordinates z = Tx . Write down the dynamics for the system in the z coordinates. Show that if the original system is reachable, then so is the new one, and then con- versely, if the new system is reachable, then the original one must be reachable. There- fore, you are showing that reachability is a physical property of the system and is not representation-dependent.
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problemset5_Soln - The University of Texas at Austin...

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