EstimatorRegulator

EstimatorRegulator[sspec,{l,κ}]

gives the output feedback controller with estimator and regulator gains l and κ for the system specification sspec.

EstimatorRegulator[,"prop"]

gives the value of the property "prop".

Details and Options

  • EstimatorRegulator is also known as observer controller or estimator controller.
  • EstimatorRegulator is used to assemble a control system consisting of an estimator and regulator with the given estimator and regulator gains.
  • The optimal controller is typically given by , where are estimates of the states of sys.
  • The inputs u of sys consist of feedback inputs uf and possibly other deterministic inputs ue and stochastic inputs uw.
  • The outputs y of sys consist of measured outputs ym and possibly other outputs.
  • The system specification sspec is the system sys together with the uf, ue and ym specifications.
  • The system specification sspec can have the following forms:
  • StateSpaceModel[]linear control input and linear state
    AffineStateSpaceModel[]linear control input and nonlinear state
    NonlinearStateSpaceModel[]nonlinear control input and nonlinear state
    SystemModel[]general system model
    <||>detailed system specification given as an Association
  • The detailed system specification can have the following keys:
  • "InputModel"sysany one of the models
    "FeedbackInputs"Allthe feedback inputs uf
    "ExogenousInputs"Nonethe exogenous inputs ue
    "MeasuredOutputs"Allthe measured outputs ym
  • The inputs and outputs can have the following forms:
  • {num1,,numn}numbered inputs or outputs numi used by StateSpaceModel, AffineStateSpaceModel and NonlinearStateSpaceModel
    {name1,,namen}named inputs or outputs namei used by SystemModel
    Alluses all inputs or outputs
    Noneuses none of the inputs or outputs
  • The estimator gains l can be computed using EstimatorGains, LQEstimatorGains or DiscreteLQEstimatorGains.
  • The feedback gains κ can be computed using StateFeedbackGains, LQRegulatorGains, LQOutputRegulatorGains or DiscreteLQRegulatorGains.
  • EstimatorRegulator[,"Data"] returns a SystemsModelControllerData object cd that can be used to extract additional properties using the form cd["prop"].
  • EstimatorRegulator[,"prop"] can be used to directly get the value of cd["prop"].
  • Possible values for properties "prop" include:
  • "ClosedLoopPoles"poles of "ClosedLoopSystem"
    "ClosedLoopSystem"system csys with uw, ue, as input and y as output
    {"ClosedLoopSystem",cspec}detailed control over the form of the closed-loop system
    "ControllerModel"model cm with , ue, ym as input and uf as output
    "Design"type of controller design
    "DesignModel"model used for the design
    "EstimatorGains"gain matrix
    "EstimatorRegulatorModel"model erm with uf, ue, ym as input and as output
    "ExogenousInputs"deterministic and non-feedback inputs ue of sys
    "FeedbackGains"gain matrix κ or its equivalent
    "FeedbackGainsModel"model gm with as input and as output
    "FeedbackInputs"inputs uf of sys used for feedback
    "InputModel"input model sys
    "InputsCount"number of inputs u of sys
    "MeasuredOutputs"measured outputs ym of sys
    "OpenLoopPoles"poles of the Taylor linearized sys
    "OutputsCount"number of outputs y of sys
    "SamplingPeriod"sampling period of sys
    "StateEstimatorModel"model sem with uf, ue, ym as input and as output
    "StatesCount"number of states x of sys
  • Possible keys for cspec include:
  • "InputModel"input model in csys
    "Merge"whether to merge csys
    "ModelName"name of csys
  • The diagram of the feedback gains model or regulator model gm, state-estimator model sem, estimator-regulator model erm, controller model cm, and closed-loop system csys.

Examples

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Basic Examples  (1)

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

Scope  (24)

Basic Uses  (7)

A system with one feedback input and one measurement:

The feedback input and measurement are inputs to the estimator-regulator:

A system with one noisy input as well:

Noisy inputs are not inputs to the estimator-regulator:

A system with one feedback and one exogenous input:

The exogenous input is an input to the estimator-regulator:

A system where only some of the outputs are measured:

Only the feedback input and measured output are fed to the estimator-regulator:

Compute the gains using pole placement:

Assemble the estimator-regulator:

Compute the gains optimally:

Assemble the estimator-regulator:

Compute the feedback gains optimally:

Use pole placement to compute the estimator gains:

Assemble the estimator-regulator:

Plant Models  (5)

A continuous-time StateSpaceModel:

The controller for a set of gains:

The closed-loop system:

Compute it automatically:

The poles of the closed-loop system:

A descriptor StateSpaceModel:

The controller has the same descriptor matrix:

Its closed-loop system:

And poles:

A discrete-time StateSpaceModel:

The controller for a set of gains:

Its closed-loop system:

And poles:

An AffineStateSpaceModel with a set of gains:

Its controller:

The closed-loop system:

Its poles are that of the Taylor linearized model:

A NonlinearStateSpaceModel with a set or gains:

Its controller:

The closed-loop system:

Its poles are that of the Taylor linearized system:

Properties  (9)

By default, EstimatorRegulator returns the controller comprising the estimator and regulator:

The estimator model:

The feedback gains model:

The closed-loop system:

The closed-loop poles:

The estimator-regulator feedback model:

In this model, the feedback input is fed back directly:

Assemble the estimator and regulator with feedback to get the same result as before:

The closed-loop system differs from the computed one only in the input matrix:

The design method:

Properties related to the input model and gains:

Get the controller data object:

The list of available properties:

The value of a specific property:

Closed-Loop System  (3)

Compute the closed-loop system based on a linear design:

Assemble the closed-loop system with the nonlinear model:

Compare the response of the two systems:

Assemble the merged closed loop of a plant with an EstimatorRegulator:

The unmerged closed-loop system:

When merged, it gives the same result as before:

Explicitly specify the merged closed-loop system:

Create a closed-loop system with a desired name:

The closed-loop system has the specified name:

The name can be directly used to specify the closed-loop model in other functions:

The simulation result:

Applications  (3)

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

A regulator for a simple pendulum:

Design a regulator:

Design an estimator:

Assemble the estimator-regulator:

An LQ regulator for a system with one feedback input and one measured output:

Design an optimal estimator:

Design an optimal regulator:

Assemble the estimator-regulator:

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, https://reference.wolfram.com/language/ref/EstimatorRegulator.html (updated 2021).

Text

Wolfram Research (2010), EstimatorRegulator, Wolfram Language function, https://reference.wolfram.com/language/ref/EstimatorRegulator.html (updated 2021).

BibTeX

@misc{reference.wolfram_2021_estimatorregulator, author="Wolfram Research", title="{EstimatorRegulator}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/EstimatorRegulator.html}", note=[Accessed: 19-September-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_estimatorregulator, organization={Wolfram Research}, title={EstimatorRegulator}, year={2021}, url={https://reference.wolfram.com/language/ref/EstimatorRegulator.html}, note=[Accessed: 19-September-2021 ]}

CMS

Wolfram Language. 2010. "EstimatorRegulator." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2021. https://reference.wolfram.com/language/ref/EstimatorRegulator.html.

APA

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