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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 closed-loop 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 times-scales are very short compared to the system dynamics Impossible to predict w ( ) for &gt; 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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lec12 - MIT OpenCourseWare http://ocw.mit.edu 16.323...

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