homework6-2010

# homework6-2010 - ECE 750 linear systems Autumn 2010...

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ECE 750 linear systems Autumn 2010 Excercise set # 6 (no credit): Observability, Detectability and Output-Feedback Design This homework set will not be graded or collected. Students can use it in preparation for the ﬁnal exam. Solutions will be posted on Monday, December 6. Problem 1. Given the linear system ˙ x = Ax + Bu y = Cx (1) where x R n , u R m and y R p , prove that there exists a dynamic output-feedback controller of the form ˙ ξ = + Gy u = Hξ , ξ R ν (2) such that the closed-loop system is internally stable only if ( A,B ) is stabilizable and ( C,A ) is detectable. Problem 2 For a linear system of the form (1), prove that controllability and stabilizability are invariant under regular state feedback, and observability and detectability are invariant under regular output-injection. Namely, prove the following: 1. If ( A,B ) is controllable/stabilizable, then for any K R m × n the pair ( A + BK,B ) is controllable/stabilizable. 2. If (

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## homework6-2010 - ECE 750 linear systems Autumn 2010...

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