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Unformatted text preview: 16.31 Handout #7 Prof. J. P. How November 30, 2007 T.A. TBD Due: Dec 7, 2007 16.31 Homework Assignment #7 1. LQR solutions . Consider a model of a first order unstable plant with the state equation x ( t ) = ax ( t ) au ( t ) a > and the cost functional J = x 2 ( t ) + u 2 ( t ) dt > (a) Analytically determine the steady-state LQR gain K as a function of and a (b) Analytically determine the location of the closed-loop poles using this gain for < < . (c) Draw a Nyquist plot of the loop transfer function T ( s ) = K ( sI A ) 1 B u for three very different values of and discuss how the shape changes as a function of . Confirm that your plot of the LTF will never penetrate the unit circle centered at-1. 2. Performance Limits . Let G c be a stabilizing controller for the plant G ( s ) G ( s ) = ( s ) ( s )( s + ) for > 0, > 0, and 0. Suppose that | S (j ) | < 1, [ c , c ], where S ( s ) = (1 +...
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This note was uploaded on 11/07/2011 for the course AERO 16.31 taught by Professor Jonathanhow during the Fall '07 term at MIT.
- Fall '07