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Unformatted text preview: EE221A Linear System Theory Problem Set 8 Professor C. Tomlin Department of Electrical Engineering and Computer Sciences, UC Berkeley Fall 2007 Issued 11/27; Due 12/6 Problem 1: State vs. Output Feedback. Consider the plant described by: X = AX + Bu (1) y = CX (2) where A = bracketleftbigg 1 7- 4 bracketrightbigg , B = bracketleftbigg 1 2 bracketrightbigg , C = [1 3] (3) Find the closed loop characteristic equation if the feedback is: ( a ) u =- [ f 1 f 2 ] X , and ( b ) u =- ky . Problem 2: Controllability and Observability. u(s) (a) s+a u(s) x (s) (= y(s)) 1 K 2 s+a K 2 2 x (s) C 2 + 1 s+a K 1 1 x (s) C 1 + y(s) (b) Figure 1: Figure for Problem 2. Consider the systems shown in Figures 1 ( a ) and ( b ). Is system ( a ), with state variable x 1 as shown, controllable and observable? For what conditions on a i ,K i , and C i is system ( b ), with state variables x 1 and x 2 as shown, controllable and observable? By referring to the definitions of controllability and observability, explain these results.observable?...
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