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

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Basic Examples  (3)

The Kalman gain matrix for a continuous-time system:

The gains for a discrete-time system:

The gains for an unobservable system:

Although unobservable, the system is detectable:

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:

Assemble the estimator:

Compute the actual and estimated responses:

Plot the responses:

Applications  (1)

Compute the Kalman gains that smooth the response of a stochastic system:

The response of the system to a sine input with process and measurement noise:

Filtered response :

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:

Find the optimal estimator:

It is equivalent to the conjugate transpose of optimal regulator gains of the dual system:

The dual relationship for a discrete-time system:

Possible Issues  (2)

The measurement noise covariance matrix must be positive definite:

Optimal estimator gains can be computed for an unobservable system only if it is detectable:

The last mode is unstable and unobservable:

Wolfram Research (2010), LQEstimatorGains, Wolfram Language function, https://reference.wolfram.com/language/ref/LQEstimatorGains.html (updated 2014).

Text

Wolfram Research (2010), LQEstimatorGains, Wolfram Language function, https://reference.wolfram.com/language/ref/LQEstimatorGains.html (updated 2014).

CMS

Wolfram Language. 2010. "LQEstimatorGains." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014. https://reference.wolfram.com/language/ref/LQEstimatorGains.html.

APA

Wolfram Language. (2010). LQEstimatorGains. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/LQEstimatorGains.html

BibTeX

@misc{reference.wolfram_2022_lqestimatorgains, author="Wolfram Research", title="{LQEstimatorGains}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/LQEstimatorGains.html}", note=[Accessed: 09-June-2023 ]}

BibLaTeX

@online{reference.wolfram_2022_lqestimatorgains, organization={Wolfram Research}, title={LQEstimatorGains}, year={2014}, url={https://reference.wolfram.com/language/ref/LQEstimatorGains.html}, note=[Accessed: 09-June-2023 ]}