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MIT16_30F10_rec10

MIT16_30F10_rec10 - 16.30/31 Fall 2010 Recitation 10...

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16.30/31, Fall 2010 Recitation # 10 Brandon Luders November 15, 2010 In this recitation, we explore the Linear Quadratic Estimator (LQE) problem. 1 Linear Quadratic Estimator 1.1 Motivation In Topic 14, when discussing estimators, we pointed out that there is a duality between the regulator problem (choosing a feedback K to set the closed-loop poles of A BK ) and the estimator problem (choosing a feedback L to set the closed-loop poles of A LC ). We have also talked at length about designing an optimal feedback K using the Linear Quadratic Regulator (LQR) problem, whose parameters are the state matrix A , input matrix B , state weighting matrix Q , and input weighting matrix R . Thus, one might expect to be able to identify an “optimal” (in some sense) estimator using the dual quantities: A T , C T , and some matrices Q ˜ and R ˜ . This is easily done in Matlab: K = lqr(A,B,Q,R) L = (lqr(A’,C’,Q,R))’ But, what exactly is this “optimal” estimator? In what sense is it optimal? And how are the Q ˜ and R ˜ matrices supposed to be selected? This is the Linear Quadratic Estimator (LQE) problem, and is discussed below. (Notes adapted from Prof. How’s 16.323 materials.) You already knew how to design LQE estimators from Topic 14; the purpose of this recitation is to provide context, aiding in the selection of Q ˜ and R ˜ matrices. 1.2 Deriving LQE Consider the standard state space model, now augmented with multiple sources of noise: x ˙ = A x + B u + w , (1) y = C x + v . (2) Here w is a process noise , modeling uncertainty in the system model (e.g., wind disturbance), while v is a sensing noise , modeling uncertainty in the measurements (e.g., a poor rate gyro).
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