MIT16_30F10_rec07

MIT16_30F10_rec07 - 16.30/31, Fall 2010 - Recitation # 7...

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± ± ± ± ± ± 16.30/31, Fall 2010 Recitation # 7 Brandon Luders October 25, 2010 In this recitation, we consider controllability and observability, as well as the assumptions on the LQR formulation and how to solve it by hand. 1 Controllability and Observability Consider the following state space model: x ˙ = Ax + Bu y = Cx 0 1 0 0 2 1 A = , B = , C = . 8 6 1 1 4 2 We are interested in determining whether this model is controllable and/or observable. First, let’s compute the transfer function matrix; since there are two inputs and two outputs, we expect the transfer function matrix to be a 2 × 2 matrix. G ( s ) = C ( sI A ) 1 B 2 1 ± s 1 ± 1 0 0 ± = 4 2 8 s + 6 1 1 ±² ±³ ± 2 1 1 s + 6 1 0 0 = s ( s + 6) + 8 s 1 1 4 2 ± 8 ± ± 1 2 1 s + 6 1 0 0 = s 2 + 6 s + 8 4 2 8 s 1 1 ± ± 1 2 1 1 1 = ( s + 2)( s + 4) 4 2 s s 1 s + 2 s + 2 = ( s + 2)( s + 4) 2( s + 2) 2( s + 2) 1 1 1 = . s + 4 2 2 Note the pole-zero cancellation that took place at s = 2; we should expect this to show up as a loss of either controllability or observability. First, let’s compute the controllability matrix M c : 0 0 1 1 M c = [ B AB ] = 1 1 6 6 . For the system to be controllable, M c must have full rank. Keep in mind that an a × b matrix M (where a = b ) is considered to have full rank if rank( M ) = min { a, b } . So all we need is for M c to have rank 2 - which it does. Thus the system is controllable.
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± ± ² ± ± ² ± Next, let’s compute the observability matrix M o : ± 2 1 = C = 4 2 . M o CA 8 4 16 8 For the system to be observable, M o must have rank 2. Yet we can see that all rows of M o are multiples of each other, and thus rank( M o ) = 1 < 2. Thus this system is not observable. We know that a pole-zero cancellation occurred at s = 2, so let’s investigate the con- trollability and observability of the mode it corresponds to. We can ±nd the left eigenvector w T corresponding to s = 2 by solving the system of equations w T ( sI A ) = 0.
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MIT16_30F10_rec07 - 16.30/31, Fall 2010 - Recitation # 7...

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