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DiscreteLQEstimatorGains[ssm,{w,v},τ]

gives the optimal discrete-time estimator gain matrix with sampling period τ for the continuous-time StateSpaceModel ssm, with process and measurement noise covariance matrices w and v.

DiscreteLQEstimatorGains[{ssm,sensors},{w,v},τ]

specifies sensors as the noisy measurements of ssm.

DiscreteLQEstimatorGains[{ssm,sensors,dinputs},{w,v},τ]

specifies dinputs as the deterministic inputs of ssm.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Properties & Relations  
Possible Issues  
See Also
Related Guides
History
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DiscreteLQEstimatorGains

DiscreteLQEstimatorGains[ssm,{w,v},τ]

gives the optimal discrete-time estimator gain matrix with sampling period τ for the continuous-time StateSpaceModel ssm, with process and measurement noise covariance matrices w and v.

DiscreteLQEstimatorGains[{ssm,sensors},{w,v},τ]

specifies sensors as the noisy measurements of ssm.

DiscreteLQEstimatorGains[{ssm,sensors,dinputs},{w,v},τ]

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}], where a, b, c, and d represent the state, input, output, and transmission matrices of the continuous-time system .
  • The descriptor continuous-time state-space model ssm defined by can be given as StateSpaceModel[{a,b,c,d,e}].
  • 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 .
  • DiscreteLQEstimatorGains[ssm,{…},τ] is equivalent to DiscreteLQEstimatorGains[{ssm, All,None},{…},τ].
  • 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 estimator with the optimal gain minimizes , where is the estimated state vector.
  • DiscreteLQEstimatorGains computes the estimator gains based on the discrete equivalent of the noise matrices.
  • The state-space model ssm is discretized using the zero-order hold method.

Examples

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Basic Examples  (1)

Compute the discrete LQ estimator gains for a continuous-time state-space model:

Wolfram Language code: DiscreteLQEstimatorGains[StateSpaceModel[{{{0, 1}, {0, -5}}, {{0, 0}, {1, 0.1}}, {{1, 0}}, {{0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {(| | | | -- | - | | 10 | 0 | | 0 | 1 |), {{1}}}, 0.1]

Scope  (3)

Compute the discrete-time Kalman gains for a state-space model:

Wolfram Language code: ssm = StateSpaceModel[{{{-1.7, 50, 160}, {0.22, -1.4, -32}, {0, 0, -12}}, {{-272, 0.02, 0.1}, {0, -0.0035, 0.004}, {14, 0, 0}}, {{1, 0, 0}, {0, 1, 0}}, {{0, 0, 0}, {0, 0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: DiscreteLQEstimatorGains[ssm, {(| | | | | ----- | ----- | ----- | | 0.001 | 0 | 0 | | 0 | 0.001 | 0 | | 0 | 0 | 0.001 |), (| | | | - | - | | 1 | 0 | | 0 | 1 |)}, 0.5]//MatrixForm

The gains based on the measurement of just the second output:

Wolfram Language code: DiscreteLQEstimatorGains[{ssm, 2}, {(| | | | | ----- | ----- | ----- | | 0.001 | 0 | 0 | | 0 | 0.001 | 0 | | 0 | 0 | 0.001 |), (⁠1⁠)}, 0.5]//MatrixForm

The gains for a system in which all inputs except the first are stochastic:

Wolfram Language code: ssm = StateSpaceModel[{{{0, 0, 1, 0}, {0, 0, 0, 1}, {-29.4, 19.6, 0, 0}, {14.7, -14.7, 0, 0}}, {{0, 1, 0, 0, 0}, {0, 0, 1, 0, 0}, {0, 0, 0, 1, 0}, {0.125, 0, 0, 0, 1}}, {{1, 0, 0, 0}, {0, 0, 1, 0}}, {{0, 0, 0, 0, 0}, {0, 0, 0, 0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: DiscreteLQEstimatorGains[{ssm, All, {1}}, {10^6(⁠| | | | | | - | - | - | - | | 1 | 0 | 0 | 0 | | 0 | 1 | 0 | 0 | | 0 | 0 | 1 | 0 | | 0 | 0 | 0 | 1 |⁠), (⁠| | | | - | - | | 1 | 0 | | 0 | 1 |⁠)}, 0.2]

Find the optimal gains for a descriptor state-space model:

Wolfram Language code: DiscreteLQEstimatorGains[StateSpaceModel[{{{-0.3, 0.65, 0}, {0, -1, 1}, {0.25, -0.5, -0.6}}, {{-1, 0}, {0.5, 0}, {0.7, -0.6}}, {{1, 0, 0}, {0, -1, -1}}, {{0, 0}, {0, 0}}, {{3, 0, 0}, {0, 5, 0}, {0, 0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {(⁠| | | | -- | -- | | 1. | 0. | | 0 | 2. |⁠), (⁠| | | | -- | -- | | 5. | .1 | | .1 | 3. |⁠)}, .1]

Properties & Relations  (1)

Find estimator gains using DiscreteLQEstimatorGains:

Wolfram Language code: ssm = StateSpaceModel[{{{0, 1, 0}, {-1, -1, 0}, {0, 0, -1}}, {{0, 0}, {1, 0}, {0, 1}}, {{2, 1, 1}}, {{0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None];
Wolfram Language code: Subscript[l, 1] = DiscreteLQEstimatorGains[{ssm, All, 1}, {(⁠16⁠), (⁠1⁠)}, 0.5]//Chop

Create a discrete-time Kalman estimator with the gains and a discretized model:

Wolfram Language code: kalman1 = StateOutputEstimator[{ToDiscreteTimeModel[ssm, 0.5, Method -> "ZeroOrderHold"], All, 1}, Subscript[l, 1]];

This is different from that obtained by discretizing a continuous-time estimator:

Wolfram Language code: kalman2 = ToDiscreteTimeModel[KalmanEstimator[{ssm, All, 1}, {(⁠16⁠), (⁠1⁠)}], 0.5, Method -> "ZeroOrderHold"];

Response of the first estimator in the presence of process and measurement noises:

Wolfram Language code: pnoise = Interpolation[Thread[{Range[0, 20], RandomReal[NormalDistribution[0, 4], {21}]}], t];
Wolfram Language code: mnoise = Interpolation[Thread[{Range[0, 20], RandomReal[NormalDistribution[0, 1], {21}]}], t];
Wolfram Language code: noisyout = OutputResponse[ssm, {UnitStep[t], pnoise}, {t, 20}] + mnoise;
Wolfram Language code: res1 = OutputResponse[kalman1, {UnitStep[t], noisyout[[1]]}, {t, 20}];

Response of the discretized Kalman estimator:

Wolfram Language code: res2 = OutputResponse[kalman2, {UnitStep[t], noisyout[[1]]}, {t, 20}];

The responses are different:

Wolfram Language code: Show[{ListPlot[res1, Joined -> True], ListPlot[res2, PlotStyle -> Dashed, Joined -> True]}, PlotRange -> All]

Possible Issues  (1)

The system must be detectable:

Wolfram Language code: DiscreteLQEstimatorGains[StateSpaceModel[{{{-1, 0, 0}, {0, 2, 0}, {0, 0, 3}}, {{1, 1}, {-1, 0}, {0, 1}}, {{1, 0, 0}, {0, 0, 1}}, {{0, 0}, {0, 0}}}, SamplingPeriod -> None, SystemsModelLabels -> None], {(⁠| | | | - | - | | 1 | 0 | | 0 | 1 |⁠), (⁠| | | | - | - | | 1 | 0 | | 0 | 1 |⁠)}, 0.1]
Wolfram Research (2010), DiscreteLQEstimatorGains, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteLQEstimatorGains.html (updated 2012).

Text

Wolfram Research (2010), DiscreteLQEstimatorGains, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteLQEstimatorGains.html (updated 2012).

CMS

Wolfram Language. 2010. "DiscreteLQEstimatorGains." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2012. https://reference.wolfram.com/language/ref/DiscreteLQEstimatorGains.html.

APA

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

BibTeX

@misc{reference.wolfram_2026_discretelqestimatorgains, author="Wolfram Research", title="{DiscreteLQEstimatorGains}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteLQEstimatorGains.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_discretelqestimatorgains, organization={Wolfram Research}, title={DiscreteLQEstimatorGains}, year={2012}, url={https://reference.wolfram.com/language/ref/DiscreteLQEstimatorGains.html}, note=[Accessed: 13-June-2026]}