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MIT16_30F10_assn04

MIT16_30F10_assn04 - 16.30/31 Prof J P How and Prof E...

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16.30/31 October 15, 2010 Prof. J. P. How and Prof. E. Frazzoli Due: October 22, 2010 T.A. B. Luders 16.30/31 Homework Assignment #4 Goals: Modal analysis, transfer matrices, controllability and observability (part 1), linear system theory 1. Consider the system with two states, and the state-space model matrices given by: A = 6 1 , B = 1 , C = 1 0 , 5 0 K where K R is a parameter to be specified. (a) Find the transfer function G ( s ) for the system. Discuss the structure of G ( s ) for various values of K . (b) Form the observability matrix for the system. Is the system observable for all values of K ? (c) Form the controllability matrix for the system. Is the system controllable for all values of K ? (d) Compare your observations in parts (b) and (c) with those in part (a). 2. Given the transfer function from input u ( t ) to output y ( t ), Y ( s ) s 2 4 s + 3 = U ( s ) ( s 2 + 6 s + 8)( s 2 + 25) (a) Develop a state space model for this transfer function, in the standard form x ˙ = Ax + Bu y = Cx + Du.

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MIT16_30F10_assn04 - 16.30/31 Prof J P How and Prof E...

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