LQEstimatorGains
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 - LQEstimatorGains also accepts nonlinear systems specified by AffineStateSpaceModel and NonlinearStateSpaceModel.
- For nonlinear systems, the operating values of state and input variables are taken into consideration, and the gains are computed based on the approximate Taylor linearization.
- 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. - LQEstimatorGains supports a Method option. The following explicit settings can be given:
-
"CurrentEstimator" constructs the current estimator "PredictionEstimator" constructs the prediction estimator - The current estimate is based on measurements up to the current instant.
- The prediction estimate is based on measurements up to the previous instant.
- For continuous-time systems, the current and prediction estimators are the same. 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
of the current estimator is computed as 
, where 
is the solution of the discrete Riccati equation 
. - The optimal gain
of the prediction estimator for a discrete-time system is computed as 
. - The optimal estimator is asymptotically stable if
is nonsingular, the pair 
is detectable, and 
is stabilizable for any 
.
Examples
open allclose allBasic Examples (3)
Scope (7)
Determine the optimal estimator gains of a continuous-time system:
The gains for a discrete-time system with nonzero cross-covariance:
The Kalman gains for a continuous-time system with cross-correlated noises:
Use the first output as the measurement:
Use the second output as the measurement:
The Kalman gains for a system in which the last four inputs are stochastic disturbances:
Estimator gain for a system with two deterministic inputs and two stochastic inputs:
The poles of the Kalman estimator:
Find the optimal gains for a descriptor state-space model:
The gains for an AffineStateSpaceModel:
Applications (1)
Properties & Relations (4)
Compute the Kalman estimator gains using the underlying Riccati equation:
LQEstimatorGains gives the same result:
The gains for a discrete-time system can be computed using DiscreteRiccatiSolve:
LQEstimatorGains gives the same result:
It is equivalent to the conjugate transpose of optimal regulator gains of the dual system:
