# hw12s - MEEN 651 Control System Design Homework 12 LQR LQG...

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MEEN 651 Control System Design Homework 12: LQR, LQG, and LTR Solutions Assigned: Tuesday, 18 Nov. 2008 Due: Tuesday, 25 Nov. 2008, 5:00 pm Problem 1) Problem 7.57

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Problem 2) Consider the diagrams provided below, where G 1 (s) and G 2 (s) are stable, completely controllable, and completely observable. For each configuration, determine if the composite system is always: a) Stable b) Completely Controllable c) Stabilizeable Please provide a proof or counter-example to justify your answer. Parallel Connection Series Connection Feedback Connection Parallel connection: Stable. If both subsystems are stable, then the composite system is also stable. Completely controllable. Simple proof using Hautus Rosenbrock criteria. Stabilizable. Clearly, since it is completely controllable. Series connection: Stable. If both subsystems are stable, then the composite system is also stable. Not always completely controllable. Counterexample is a simple pole-zero cancellation. Stabilizeable. Although some modes may not be controllable, all modes are stable. Feedback connection: Not always stable. Simple counterexample with K/(s+1) under positive feedback. Not always completely controllable. Counterexample is simple pole-zero cancellation. Not always stabilizable.
Problems 3) – 5) are concerned with the linearized and normalized dynamics of the magnetically suspended ball, described by: ݔሶ ሺݐሻൌݔ ሺݐሻ ሺݐሻ ൅ ݑሺݐሻ Problem 3) Assume a quadratic performance index of: ܬൌන ሾݔ ൅2ݔ ݔ ൅ݔ ൅ݑ ሿ݀ݐ a) Show that the solution to the Algebraic Ricatti Equation associated with this system and performance index is given by: ܲൌ൤ ߚߚ , with ߚ൐0 . What is the value of ߚ ?

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## This note was uploaded on 09/20/2009 for the course MEEN 651 taught by Professor Staff during the Fall '08 term at Texas A&M.

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hw12s - MEEN 651 Control System Design Homework 12 LQR LQG...

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