problem9

problem9 - EE221A Linear System Theory Problem Set 9...

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Unformatted text preview: EE221A Linear System Theory Problem Set 9 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2010 Issued 11/23; Due 12/2 Problem 1: How does state feedback affect observability? Consider the following system: bracketleftbigg ˙ x 1 ˙ x 2 bracketrightbigg = bracketleftbigg- 2 1 1 bracketrightbiggbracketleftbigg x 1 x 2 bracketrightbigg + bracketleftbigg 1 bracketrightbigg u (1) y = [1 2] bracketleftbigg x 1 x 2 bracketrightbigg (2) and assume that you would like to design a feedback controller of the form u =- Fx + r , where x = [ x 1 x 2 ] T and r is a reference input signal. (a) Show that the system is observable. (b) Show that there exists a state feedback gain matrix F = [ f 1 f 2 ] such that the closed loop system resulting from setting u =- Fx + r is not observable. (c) Now, compute a matrix F of the form F = [1 f 2 ] such that the closed loop system (as in part (b)) is not observable....
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