16.322 Stochastic Estimation and Control, Fall 2004 Prof vander Velde New estimate =Old estimate Gain(Measurement residual) This is the form of the modern recursive estimator. If this is formulated directly the case where several parameters are being estimated, the(02+o2)is a matrix inversion. However, if only one scalar measurement is being processed, the inversion is a scalar reciprocal 02 0. 000- 01=020202+02) =(1-KO k≡G(a2+o Extensions to this simple problem Several estimated parameters instead of one o No conceptual difficulty o Get vector and matrix operations instead of scal Dynamic parameters instead of static o No real difficulty if they obey a set of linear differential equations veral simultaneous measurements instead of one o No difficulty if they are linearly related to x Biased measurement noise o Estimate the bias Correlated measurement noise o Estimate the noise(different form of filter)or work with independent measurement differences Non-normal noises o Makes maximum likelihood difficult o Requires full distribution rather than mean and variance Nonlinear system constraints or measurements o Makes things very difficult o Requires more information than mean and variance Statistics in State Space Formulation In the state space formulation we depend on the concept of the shaping filter exclusively. Even if the input statistics are time varying, we suppose the input to be generated by passing white noise through a suitable time-varying shaping filter. In the non-stationary case it is not clear how to generate the shaping system directl Page 5 of 816.322 Stochastic Estimation and Control, Fall 2004 Prof. Vander Velde Page 5 of 8 New estimate = Old estimate + Gain(Measurement residual) This is the form of the modern recursive estimator. If this is formulated directly in the case where several parameters are being estimated, the ( ) 1 2 2 σ σ 0 z − + is a matrix inversion. However, if only one scalar measurement is being processed, the inversion is a scalar reciprocal. ( ) ( ) ( ) 2 2 0 2 2 2 22 10 0 1 2 22 2 2 10 0 2 0 1 22 2 0 0 111 1 z z z z z z k k σ σ σ σ σ σσ σ σσ σ σ σ σσ σ − − + =+= = + = − ≡ + Extensions to this simple problem: • Several estimated parameters instead of one o No conceptual difficulty o Get vector and matrix operations instead of scalar • Dynamic parameters instead of static o No real difficulty if they obey a set of linear differential equations • Several simultaneous measurements instead of one o No difficulty if they are linearly related to x • Biased measurement noise o Estimate the bias • Correlated measurement noise o Estimate the noise (different form of filter) or work with independent measurement differences • Non-normal noises o Makes maximum likelihood difficult o Requires full distribution rather than mean and variance • Nonlinear system constraints or measurements o Makes things very difficult o Requires more information than mean and variance Statistics in State Space Formulation In the state space formulation we depend on the concept of the shaping filter exclusively. Even if the input statistics are time varying, we suppose the input to be generated by passing white noise through a suitable time-varying shaping filter. In the non-stationary case it is not clear how to generate the shaping system directly