constructs an estimator for the StateSpaceModel ssm, with estimator gain matrix l.


uses only sensors as the measurements of ssm.


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 in either a continuous-time or a discrete-time system:
  • 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
  • StateOutputEstimator also accepts nonlinear systems specified by AffineStateSpaceModel and NonlinearStateSpaceModel.
  • For nonlinear systems, the operating values of state and input variables are taken into consideration when constructing the estimator.
  • The inputs can include stochastic inputs and deterministic inputs .
  • The argument dinputs is a list of integers specifying the positions of in .
  • The outputs can include measurements and other outputs.
  • The argument sensors is a list of integers specifying the positions of in .
  • StateOutputEstimator[ssm,l] is equivalent to StateOutputEstimator[{ssm,All,All},l].
  • The estimator gains l can be computed using EstimatorGains, LQEstimatorGains or DiscreteLQEstimatorGains.
  • StateOutputEstimator[ssm,LQEstimatorGains[ssm,],] gives a Kalman estimator.
  • StateOutputEstimator[ssm,EstimatorGains[ssm,],] gives a Luenberger estimator.
  • StateOutputEstimator 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.
  • StateOutputEstimator gives an estimator with dynamics for continuous-time systems. The matrices with subscripts and are submatrices associated with the deterministic inputs and the sensors .
  • The prediction estimator of a discrete-time system has dynamics .
  • For discrete-time systems, StateOutputEstimator[,Method->"CurrentEstimator"] gives an estimator with dynamics , and the current state estimate is obtained from the current measurement as .
  • For discrete-time systems, the prediction gain and the current gain have the relationship .
  • Block diagram for the system with estimator:
  • The inputs to the estimator model are the deterministic inputs and the measurements .
  • The outputs of the estimator model consist of the estimated states and estimates of the measurements .


open allclose all

Basic Examples  (3)

The output and state estimator for a continuous-time system:

An estimator of a system with unity estimator gain and a sensor at the second output:

For a discrete-time system, StateOutputEstimator assembles a discretetime estimator:

Scope  (8)

A linear estimator for a system with one measured output and one deterministic input:

Specify that the input is stochastic:

The estimator for a system in which all the outputs are measured and all inputs are deterministic:

Only the first output is measured:

The first output is measured and the first input is stochastic:

All the outputs are measured and all inputs are stochastic:

An estimator for a descriptor state-space model:

An estimator for an AffineStateSpaceModel:

Compute a set of gains based on the linearized system:

Construct the estimator:

Compute the actual and estimated responses:

Plot the responses:

Options  (2)

Method  (2)

By default, the estimator is based on the current measurements:

The prediction estimator:

For continuous-time systems, the current and prediction estimates are equivalent:

Applications  (1)

An observer for a continuous-time system:

Simulate the system with input -t and from a random initial condition:

Compare each state and its estimate:

Compare the outputs:

Properties & Relations  (2)

StateOutputEstimator estimates the states and outputs of a system:

Extract the state estimator:

The output estimator:

Construct a Kalman estimator for a discrete-time system:

Use KalmanEstimator directly:

Wolfram Research (2010), StateOutputEstimator, Wolfram Language function, (updated 2014).


Wolfram Research (2010), StateOutputEstimator, Wolfram Language function, (updated 2014).


Wolfram Language. 2010. "StateOutputEstimator." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2014.


Wolfram Language. (2010). StateOutputEstimator. Wolfram Language & System Documentation Center. Retrieved from


@misc{reference.wolfram_2024_stateoutputestimator, author="Wolfram Research", title="{StateOutputEstimator}", year="2014", howpublished="\url{}", note=[Accessed: 25-June-2024 ]}


@online{reference.wolfram_2024_stateoutputestimator, organization={Wolfram Research}, title={StateOutputEstimator}, year={2014}, url={}, note=[Accessed: 25-June-2024 ]}