MIT16_30F10_assn06

MIT16_30F10_assn06 - 16.30/31 Prof J P How and Prof E...

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± ± ± 16.30/31 October 29, 2010 Prof. J. P. How and Prof. E. Frazzoli Due: November 5, 2010 T.A. B. Luders 16.30/31 Homework Assignment #6 Goals: More on LQR; LQ servo; DOFB compensators; intro to LQ robustness 1. Consider the system x ˙ = ( A + Δ I 2 ) x + Bu, 1 5 1 ² ³ A = , B = , C = 1 0 . y = C x , 2 3 0 The Δ I 2 term corresponds to possible uncertainty in knowledge of the plant dynamics, where I 2 is the identity matrix. (You may use Matlab throughout this problem.) In parts (a)-(c), we consider the nominal dynamics, i.e. Δ = 0. (a) Design an LQR controller using Q = I 2 , R = 0 . 01, and implement the controller to perform reference tracking via u = r K x . Describe the tracking performance of this controller to a step input, and give the resulting steady-state error. ¯ ¯ (b) Suppose we modify the control law to take the form u = Nr K x , where N is a ¯ scalar. How should N be chosen to minimize steady-state error? Verify that this results in good tracking performance for a
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MIT16_30F10_assn06 - 16.30/31 Prof J P How and Prof E...

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