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Unformatted text preview: EE635  Control System Theory Jitkomut Songsiri 6. Linear Quadratic Gaussian Control Output feedback The Kalman filter LQG/LQR 61 Output feedback Consider a linear system x = Ax + Bu, y = Cx A statefeedback controller has a form u ( t ) = Kx ( t ) which requires the availability of the process measurement when the state variables are not accessible, one can use u ( t ) = K x ( t ) where x ( t ) is an estimate of x ( t ) based on the output y Linear Quadratic Gaussian Control 62 Fullorder observers x cannot be fully measured and the goal is to estimate x based on y our approach is to replicate the process dynamic in x x = A x + Bu Define the state estimation error e = x x , we can see e = Ax A x = Ae if A is stable, then the error goes to zero asymptotically if A is unstable, e is unbounded and x grows further apart from x to avoid this problem, one can consider a correction term as x = A x + Bu + L ( y y ) , x (0) = 0 (feed y back to the estimator) Linear Quadratic Gaussian Control 63 L is a given matrix, called observer gain matrix the error dynamic now is e = Ax A x L ( Cx C x ) = ( A LC ) e, e (0) = x (0) the observer error goes to zero if L is chosen such that A LC is stable we can make e goes to zero fast if the eigenvalues of A LC can be arbitrarily assigned eigenvalues of A LC are same as those of ( A T C T L...
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 Spring '12
 K.Smith
 Kalman, Quadratic Gaussian Control

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