gives the state feedback gains for the system specification sspec that minimizes an output cost function with weights wts.


gives the value of the property "prop".

Details and Options

  • LQOutputRegulatorGains is also known as linear quadratic output regulator, linear quadratic output controller or optimal controller.
  • LQOutputRegulatorGains is typically used to stabilize a system or improve its performance.
  • The controller is typically given by a state feedback , where is the computed gain matrix.
  • The inputs u of sys consist of feedback inputs uf and possibly other inputs ue.
  • The outputs y of sys consist of regulated outputs yr and possibly other outputs.
  • The system specification sspec is the system sys together with the uf and yr specifications.
  • LQOutputRegulatorGains minimizes a quadratic cost function with weights q, r and p of the regulated outputs yr and feedback inputs uf of a linear system sys:
  • continuous-time system
    discrete-time system
  • LQ design works for linear systems as specified by StateSpaceModel:
  • continuous-time system
    discrete-time system
  • The resulting feedback gain matrix is then computed as kappa=TemplateBox[{{r, _, 1}}, Inverse].x_1:
  • continuous-time system
    discrete-time system
  • The matrix xr is the solution of the Riccati equation:
  • a.x_r+x_r.a-x_1.TemplateBox[{{r, _, 1}}, Inverse].x_1+c_r.q.c_r=0continuous-time algebraic Riccati equation
    a.x_r.a-x_r-x_1.TemplateBox[{{r, _, 1}}, Inverse].x_1+c_r.q.c_r=0discrete-time algebraic Riccati equation
  • The submatrices bf, cr and drf correspond to the feedback inputs uf and regulated outputs yr.
  • 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 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
    "RegulatedOutputs"Allthe regulated outputs yr
  • 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
  • For nonlinear systems such as AffineStateSpaceModel, NonlinearStateSpaceModel and SystemModel, the system will be linearized around its stored operating point.
  • LQOutputRegulatorGains[,"Data"] returns a SystemsModelControllerData object cd that can be used to extract additional properties using the form cd["prop"].
  • LQOutputRegulatorGains[,"prop"] can be used to directly give the value of cd["prop"].
  • Possible values for properties "prop" include:
  • "ClosedLoopPoles"poles of the linearized "ClosedLoopSystem"
    "ClosedLoopSystem"system csys with ue and as input and y as output
    {"ClosedLoopSystem", cspec}detailed control over the form of the closed-loop system
    "ControllerModel"model cm with and x as input and uf as output
    "Design"type of controller design
    "DesignModel"model used for the design
    "FeedbackGains"gain matrix κ or its equivalent
    "FeedbackGainsModel"model gm with x as input and as output
    "FeedbackInputs"inputs uf of sys used for feedback
    "InputModel"input model sys
    "InputCount"number of inputs u of sys
    "OpenLoopPoles"poles of "DesignModel"
    "OutputCount"number of outputs y of sys
    "RegulatedOutputs"outputs yr of sys that are regulated
    "SamplingPeriod"sampling period of sys
    "StateCount"number of states x of sys
  • The diagram of the feedback gains model gm, controller model cm, and closed-loop system csys.

  • Possible keys for cspec include:
  • "InputModel"input model in csys
    "Merge"whether to merge csys
    "ModelName"name of csys


open allclose all

Basic Examples  (2)

A set of optimal output-weighted state feedback gains for a continuous-time system:

LQ output regulator gains for a discrete-time system:

Scope  (27)

Basic Uses  (8)

Compute the state feedback gain of a system with equal weighting for the output and input:

The closed-loop system:

Compute the gain for an unstable system:

The gain stabilizes the unstable system:

Compute the state feedback gains for a multiple-output system:

The dimensions of the result correspond to the number of inputs and the system's order:

Compute the gains for a system with 3 outputs and 2 inputs:

Reverse the weights of the feedback inputs:

Typically, the feedback input with the bigger weight has the smaller norm:

Compute the gains when the cost function contains cross-coupling of the outputs and feedback inputs:

Choose the feedback outputs for multiple-output systems:

Choose the first output:

Choose the second output:

Choose the feedback inputs for multiple-input systems:

Choose the first input:

Choose the second input:

Compute the gains for a nonlinear system:

The controller is returned as a vector and takes operating points into consideration:

The controller for the approximate linear system:

Plant Models  (6)

Properties  (10)

LQOutputRegulatorGains returns the feedback gains by default:

In general, the feedback is affine in the states:

It is of the form κ0+κ1.x, where κ0 and κ1 are constants:

The systems model of the feedback gains:

An affine systems model of the feedback gains:

The closed-loop system:

The poles of the linearized closed-loop system:

The model used to compute the feedback gains:

The gains of the design model and input model:

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 one disturbance and one feedback input:

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  (2)

Construct an output-weighted state-feedback gain matrix for an aircraft model:

Plot the closed-loop output response:

Plot the control signals:

A two-loop circuit modeled as a descriptor system:

The circuit has a large magnitude response at certain frequencies:

Find an optimal closed-loop system for unity cost matrices:

The closed-loop response is attenuated:

Properties & Relations  (3)

Equivalent output regulator gains can be computed using LQRegulatorGains:

LQOutputRegulatorGains gives the same result:

Compute the LQ output regulator gains by solving the underlying Riccati equation:

LQOutputRegulatorGains gives the same result:

Compute the gains for a discrete-time system using DiscreteRiccatiSolve:

LQOutputRegulatorGains gives the same result:

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


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


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


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


@misc{reference.wolfram_2024_lqoutputregulatorgains, author="Wolfram Research", title="{LQOutputRegulatorGains}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/LQOutputRegulatorGains.html}", note=[Accessed: 27-May-2024 ]}


@online{reference.wolfram_2024_lqoutputregulatorgains, organization={Wolfram Research}, title={LQOutputRegulatorGains}, year={2021}, url={https://reference.wolfram.com/language/ref/LQOutputRegulatorGains.html}, note=[Accessed: 27-May-2024 ]}