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.
- 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
.
Examples
open allclose allBasic Examples (3)
Scope (7)
Applications (1)
Properties & Relations (4)
Possible Issues (2)
See Also
DiscreteLQEstimatorGains KalmanEstimator RiccatiSolve DiscreteRiccatiSolve EstimatorGains ObservableModelQ LQGRegulator StateOutputEstimator EstimatorRegulator
Related Guides
Introduced in 2010
(8.0)
| Updated in 2014 (10.0)