LQGRegulator

LQGRegulator[sspec,cvs,wts]

gives the optimal output feedback regulator for the stochastic system specification sspec with noise covariance matrices cvs that minimizes a cost function with weights wts.

LQGRegulator[…,"prop"]

gives the value of the property "prop".

Details and Options

  • LQGRegulator is also known as linear quadratic Gaussian regulator or stochastic linear regulator.
  • The optimal controller is typically used to stabilize a system or improve its performance in the presence of noisy inputs and measurements.
  • The optimal controller is typically given by , where is the computed gain matrix and is the state estimate.
  • 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.
  • LQGRegulator minimizes a quadratic cost function with weights q, r and p of the states x and feedback inputs uf of a linear stochastic system sys:
  • continuous-time system
    discrete-time system
  • The weights wts can have the following forms:
  • {q,r}cost function with no cross-coupling
    {q,r,p}cost function with cross-coupling matrix p
  • The regulator gains are computed using LQRegulatorGains.
  • The stochastic inputs uw and the measurement noise yv are considered to be white and Gaussian:
  • , process noise
    , measurement noise
    cross covariance
  • The covariance matrices cvs can have the following forms:
  • {w,v}covariance matrices and zero cross covariance
    {w,v,h}covariance and cross covariance matrices
  • The estimator gains are computed using LQEstimatorGains.
  • LQG design works for linear systems as specified by StateSpaceModel:
  • continuous-time system
    discrete-time system
  • The measurements ym are modeled as , where and are the submatrices of and associated with , and is the noise.
  • 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
  • LQGRegulator[…,"Data"] returns a SystemsModelControllerData object cd that can be used to extract additional properties using the form cd["prop"].
  • LQGRegulator[…,"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
    "EstimatorGains"gain matrix ℓ
    "EstimatorGainsDesignModel"model used for the estimator gains design
    "EstimatorRegulatorFeedbackModel"model with ue, ym as input and as output
    "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
    "FeedbackGainsDesignModel"model used for the feedback gains design
    "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 an LQG regulator for a system with one measured output and one feedback input:

Scope  (29)

Basic Uses  (4)

A system with one feedback input and one noisy measurement:

The input covariance matrix is empty because there are no noisy inputs:

A set of weights for the states and feedback input:

Compute the regulator:

Compute the regulator from a full system specification:

A system with one feedback input, one noisy input and one noisy measurement:

A set of covariance matrices and weights:

Compute the regulator:

A system with an exogenous deterministic input as well:

A set of covariance matrices and weights:

Compute the regulator:

A general stochastic system:

A set of covariance matrices and weights:

Compute the regulator:

Plant Models  (7)

Continuous-time StateSpaceModel:

Discrete-time StateSpaceModel:

Descriptor StateSpaceModel:

AffineStateSpaceModel:

NonlinearStateSpaceModel:

SystemModel:

A SystemModel with a noisy input:

The system specification:

A regulator design:

Properties  (15)

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

The feedback gains:

The feedback gains model:

The model used to compute the feedback gains:

The estimator gains:

The model of the state estimator:

The model used to compute the estimator gains:

For nonlinear systems, the model used to design the estimator is the nonlinear state-space model:

The model to design the feedback gains is linear:

It is essentially the Taylor linearized state-space model:

The controller is the estimator and feedback gains in series:

The computed controller is the same:

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:

Get the controller data object:

The list of available properties:

The value of a specific property:

Closed-Loop System  (3)

Assemble the closed-loop system for a nonlinear plant model:

The closed-loop system with a linearized model:

Compare the response of the two systems:

Assemble the merged closed-loop of a plant with an LQGRegulator:

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 in other functions:

Applications  (2)

A state-space model with one feedback input, one noisy input and one measurement:

Design an LQG controller:

Obtain the closed-loop system:

Simulate its response:

A descriptor system with one feedback input, one noisy input,\ and two noisy measurements:

The covariance matrices:

Create Gaussian noise sequences:

Interpolate the sequences to get noise signals:

The open-loop response is slow to settle:

Design an LQG regulator:

The closed-loop system:

The closed-loop system has a faster settling time:

Properties & Relations  (2)

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

The closed-loop poles:

The poles of the optimal state feedback subsystem:

The poles of the estimator:

The values are the same:

Construct an LQG regulator from the Kalman estimator and the optimal state-feedback gains model:

The Kalman estimator:

The feedback gains model:

Create the estimator-regulator manually:

LQGRegulator gives the same result:

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

Text

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

BibTeX

@misc{reference.wolfram_2021_lqgregulator, author="Wolfram Research", title="{LQGRegulator}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/LQGRegulator.html}", note=[Accessed: 22-September-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_lqgregulator, organization={Wolfram Research}, title={LQGRegulator}, year={2021}, url={https://reference.wolfram.com/language/ref/LQGRegulator.html}, note=[Accessed: 22-September-2021 ]}

CMS

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

APA

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