constructs the feedback regulator for the StateSpaceModel ssm with estimator and feedback gain matrices l and κ, respectively.


uses only sensors as the measured outputs of ssm.


specifies finputs as the feedback inputs of ssm.


specifies einputs as the exogenous deterministic inputs.

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
  • EstimatorRegulator 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 EstimatorRegulator.
  • The input can include stochastic inputs , feedback inputs , and exogenous deterministic inputs .
  • The arguments finputs and einputs are lists of integers specifying the positions of and 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 .
  • EstimatorRegulator[ssm,{}] is equivalent to EstimatorRegulator[{ssm,All,All,None},{}].
  • The estimator gains l can be computed using EstimatorGains, LQEstimatorGains, or DiscreteLQEstimatorGains.
  • The feedback gains κ can be computed using StateFeedbackGains, LQRegulatorGains, or DiscreteLQRegulatorGains.
  • Block diagram of the system with its regulator:


open allclose all

Basic Examples  (1)

Construct a state-feedback regulator from known estimator and state-feedback gains:

Scope  (5)

The regulator for a SISO system with estimator gain L and regulator gain K:

Its transfer function:

The closed-loop system:

The regulator for a descriptor state-space model has the same descriptor matrix:

A state-feedback regulator for a system with two sensor outputs, one feedback input, one exogenous deterministic input, and one stochastic input:

An observer-based optimal regulator for a deterministic continuous-time system:

The transfer function of the estimator-regulator:

Compute the estimator-regulator for a NonlinearStateSpaceModel:

Applications  (3)

A state-feedback regulator for the discrete-time model of a satellite's attitude control system:

A regulator for a simple pendulum:

An LQ regulator for a SISO system:

Properties & Relations  (2)

The closed-loop poles are the poles of the state-feedback and estimator subsystems:

The poles of the closed-loop system with state feedback matrix k:

The poles of the estimator with gain matrix l:

These make up the closed-loop poles for the system with the estimator-regulator:

Construct an LQG regulator using optimal estimator gains and state-feedback gains:

LQGRegulator gives the same result:

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


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


@misc{reference.wolfram_2020_estimatorregulator, author="Wolfram Research", title="{EstimatorRegulator}", year="2014", howpublished="\url{}", note=[Accessed: 03-March-2021 ]}


@online{reference.wolfram_2020_estimatorregulator, organization={Wolfram Research}, title={EstimatorRegulator}, year={2014}, url={}, note=[Accessed: 03-March-2021 ]}


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


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