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Unformatted text preview: 20 KALMAN FILTER 20.1 Introduction In the previous section, we derived the linear quadratic regulator as an optimal solution for the fullstate feedback control problem. The inherent assumption was that each state was known perfectly. In real applications, the measurements are subject to disturbances, and may not allow reconstruction of all the states. This state estimation is the task of a modelbased estimator having the form: x + Bu + H ( y C x = A x ) (241) The vector x represents the state estimate, whose evolution is governed by the nominal A and B matrices of the plant, and a correction term with the estimator gain matrix H . H operates on the estimation error mapped to the plant output y , since C = y . Given x statistical properties of real plant disturbances and sensor noise, the Kalman Filter designs an optimal H . +  y x + + + 1/s A H C x y Bu 20.2 Problem Statement We consider the statespace plant model given by: x = Ax + Bu + W 1 (242) y = Cx + W 2 . There are n states, m inputs, and l outputs, so that A has dimension n n , B is n m , and C is l n . The plant subject to two random input signals, W 1 and W 2 . W 1 represents 20.3 Step 1: An Equation for 99 disturbances to the plant, since it drives x directly; W 2 denotes sensor noise, which corrupts the measurement y . An important assumption of the Kalman Filter is that W 1 and W 2 are each vectors of unbiased, independent white noise, and that all the n + l channels are uncorrelated. Hence, if E ( ) denotes the expected value, E ( W 1 ( t 1 ) W 1 ( t 2 ) T ) = V 1 ( t 1 t 2 ) (243) E ( W 2 ( t 1 ) W 2 ( t 2 ) T ) = V 2 ( t 1 t 2 ) (244) E ( W 1 ( t ) W 2 ( t ) T ) = 0 n l . (245) Here ( t ) represents the impulse (or delta) function. V 1 is an n n diagonal matrix of intensities, and V 2 is an l l diagonal matrix of intensities. The estimation error may be defined as e = x x . It can then be verified that x + Bu + H ( y C e = [ Ax + Bu + W 1 ] [ A x )] (246) = ( A HC ) e + ( W 1 HW 2 ) . The eigenvalues of the matrix A HC thus determine the stability properties of the es timation error dynamics. The second term above, W 1 + HW 2 is considered an external input. The Kalman filter design provides H that minimizes the scalar cost function J = E ( e T W e ) , (247) where W is an unspecified symmetric, positive definite weighting matrix. A related matrix, the symmetric error covariance , is defined as = E ( ee T ) . (248) There are two main steps for deriving the optimal gain H ....
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.154 taught by Professor Michaeltriantafyllou during the Fall '04 term at MIT.
 Fall '04
 MichaelTriantafyllou

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