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Unformatted text preview: 2.160 System Identification, Estimation, and Learning Lecture Notes No. 8 March 6, 2006 4.9 Extended Kalman Filter Process Model ) w u y v + _ y ˆ In many practical problems, the process dynamics are nonlinear. Dynamics Kalman Gain & Covariance Update (Linearized If the process is nonlinear but smooth, its linearized approximation may be used for the process model. Extension to nonlinear system using linearization Consider a nonlinear, continuous system ( , , ( x ¡ = t u x f ) + t w ) … n − dim (87) , y = t x h ( ( ) + t v ) … l − dim (88) f (.), and h(.) : known but nonlinear, differentiable functions u : input (deterministic forcing term; assumed zero) w , v : uncorrelated process and measurement noises ( [ ( E [ t w )] = 0 , t v E )] = 0 T T T ( ) 0 t ≠ s v t v E ( s ) ] = ( ) E [ w t w ( s ) ] = [ ( ) 0 t ≠ s E [ v t w ( s ) ] = 0 Q t = s Q t = s The original Kalman filter is not applicable to this class of nonlinear systems. 1 Approach Linearize the nonlinear system around the state that is either: 1) Predetermined…linearized Kalman filter e.g. a trajectory to track. or 2) Estimated in realtime using online measurement …Extended Kalman Filter 4.9.1 Linearized Kalman Filter Actual Absence of Process Noise: w = 0 State jectory j x ∆ , Discrepancy Process track of Tra Nominal Tra ectory Desired Tra Feedback control to keep the desired trajectory j. Time t x * ( ) = a nominal trajectory in the state space satisfying the noiseless state equation: * t x ( * ( ) , ¡ ( ) = t t x f ) (89) absence of process noise ( Consider derivation ∆ t x ) ( * ( ( t x ) = t x ) ∆ + t x ) ( 9 ) ¡ ( ¡ * ( ¡ ( t x ) = t x ) ∆ + t x ) ( 9 1 ) Taylor expansion * t x f ( , ( ) = x f ∆ + t x ) , (92) * ∂ f ≅ t x f )+ * ∆ x x = x ( , ∂ x where 2 ∂ f 1 ∂ f 1 " ∂ f 1 ∂ x 1 ∂ x 2 ∂ x n ∂ f 2 % # (93) ∂ f n × n * = ∂ x 1 ∈ R Jacobian x = x ∂ x # % ∂ f ∂ f n n … ∂ x 1 ∂ x n * x = x Combining (91) and (92)...
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This note was uploaded on 02/27/2012 for the course MECHANICAL 2.160 taught by Professor Harryasada during the Spring '06 term at MIT.
 Spring '06
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