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# hw6 - EML 4312 Fall 2009 State Space Homework 1 Consider...

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EML 4312 Fall 2009 State Space Homework 1. Consider the following block diagram K G p + Input Output Block diagram. where G p = s 2 + 3 s 3 + s 2 + 5 s + 1 . Express the transfer function in terms of a state space realization. CHECK your Answer using methods from class. Sol: Using the Reachable cannonical form gives A = 0 1 0 0 0 1 1 5 1 B = 0 0 1 C = £ 3 0 1 ¤ check H = £ 3 0 1 ¤ s 1 0 0 s 1 1 5 s + 1 1 0 0 1 = s 2 + 3 s 3 + s 2 + 5 s + 1 1

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2. Are the following state space systems controllable, observable, or both? a) A = 1 1 0 1 ¸ B = 1 0 ¸ C = £ 0 1 ¤ Sol: C ( A, B ) = £ B AB ¤ = 1 1 0 0 ¸ det 1 1 0 0 ¸ = 0 Not controllable O ( C, A ) = £ C CA ¤ = 0 1 0 1 ¸ det 0 1 0 1 ¸ = 0 Not Observable b) A = a 0 0 b ¸ B = 0 1 ¸ C = £ 0 1 ¤ C ( A, B ) = £ B AB ¤ = 0 0 1 b ¸ det 0 0 1 b ¸ = 0 Not controllable. O ( C, A ) = £ C CA ¤ = 0 1 0 b ¸ det 0 1 0 b ¸ = 0 Not Observable. c) A = a 1 0 b ¸ B = 0 1 ¸ C = £ 1 1 ¤ C ( A, B ) = £ B AB ¤ = 0 1 1 b ¸ det 0 1 1 b ¸ = 1 2
Controllable.

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