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BUILT-IN MATHEMATICA SYMBOL
LQEstimatorGains[ssm, {w, v}]
gives the optimal estimator gain matrix for the StateSpaceModel ssm, with process and measurement noise covariance matrices w and v.
LQEstimatorGains[ssm, {w, v, h}]
includes the cross-covariance matrix h.
LQEstimatorGains[{ssm, sensors}, {...}]
specifies sensors as the noisy measurements of ssm.
LQEstimatorGains[{ssm, sensors, dinputs}, {...}]
specifies dinputs as the deterministic inputs of ssm.
Details and Options
- The standard state-space model ssm can be given as StateSpaceModel[{a, b, c, d}] in either continuous time or discrete time:
-
continuous-time system 
discrete-time system - The descriptor state-space model ssm can be given as StateSpaceModel[{a, b, c, d, e}] in either continuous time or discrete time:
-
continuous-time system 
discrete-time system - The input
can include the process noise 
, as well as deterministic inputs 
. - The argument dinputs is a list of integers specifying the positions of
in 
. - The output
consists of the noisy measurements 
as well as other outputs. - The argument sensors is a list of integers specifying the positions of
in 
. - LQEstimatorGains[ssm, {...}] is equivalent to LQEstimatorGains[{ssm, All, None}, {...}].
- The noisy measurements are modeled as
, where 
and 
are the submatrices of 
and 
associated with 
, and 
is the noise. - The process and measurement noises are assumed to be white and Gaussian:
-
, 
process noise 
, 
measurement noise - The cross-covariance between the process and measurement noises is given by
. - If omitted, h is assumed to be a zero matrix.
- The estimator with the optimal gain
minimizes 
, where 
is the estimated state vector. - For continuous-time systems, the optimal gain is computed as
, where 
is the solution of the continuous algebraic Riccati equation 
. The matrix 
is the submatrix of 
associated with the process noise. - For discrete-time systems, the optimal gain is computed as
, where 
is the solution of the discrete Riccati equation 
. - The optimal estimator is asymptotically stable if
is nonsingular, the pair 
is detectable, and 
is stabilizable for any 
.
New in 8 | Last modified in 9
