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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 16.323 Principles of Optimal Control Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 16.323 Lecture 12 Stochastic Optimal Control Kwaknernaak and Sivan Chapter 3.6, 5 Bryson Chapter 14 Stengel Chapter 5 Spr 2008 16.323 121 Stochastic Optimal Control Goal: design optimal compensators for systems with incomplete and noisy measurements Consider this first simplified step: assume that we have noisy system with perfect measurement of the state. System dynamics: x ( t ) = A ( t ) x ( t ) + B u ( t ) u ( t ) + B w ( t ) w ( t ) Assume that w ( t ) is a white Gaussian noise 20 N (0 ,R ww ) The initial conditions are random variables too, with E [ x ( t )] = , and E [ x ( t ) x T ( t )] = X Assume that a perfect measure of x ( t ) is available for feedback. Given the noise in the system, need to modify our cost functions from before consider the average response of the closedloop system 1 1 t f J s = E x T ( t f ) P t f x ( t f ) + ( x T ( t ) R xx ( t ) x ( t ) + u T ( t ) R uu ( t ) u ( t )) dt 2 2 t Average over all possible realizations of the disturbances. Key observation: since w ( t ) is white, then by definition, the corre lation timesscales are very short compared to the system dynamics Impossible to predict w ( ) for > t , even with perfect knowledge of the state for t Furthermore, by definition, the system state x ( t ) encapsulates all past information about the system Then the optimal controller for this case is identical to the deter ministic one considered before. 20 16.322 Notes June 18, 2008 Spr 2008 16.323 122 Spectral Factorization Had the process noise w ( t ) had color (i.e., not white), then we need to include a shaping filter that captures the spectral content (i.e., temporal correlation) of the noise ( s ) Previous picture: system is y = G ( s ) w 1 , with white noise input y w 1 G ( s ) New picture: system is y = G ( s ) w 2 , with shaped noise input y w 2 G ( s ) Account for the spectral content using a shaping filter H ( s ) , so that the picture now is of a system y = G ( s ) H ( s ) w 1 , with a white noise input G ( s ) H ( s ) w 1 w 2 y Then must design filter H ( s ) so that the output is a noise w 2 that has the frequency content that we need How design H ( s ) ? Spectral Factorization design a stable mini mum phase linear transfer function that replicates the desired spectrum of w 2 . Basis of approach: If e 2 = H ( s ) e 1 and e 1 is white, then the spec trum of e 2 is given by e 2 ( j ) = H ( j ) H ( j ) e 1 ( j ) where e 1 ( j ) = 1 because it is white....
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 Spring '08
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