# LQOutputRegulatorGains

LQOutputRegulatorGains[sspec,wts]

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

LQOutputRegulatorGains[,"prop"]

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 :
•  continuous-time system discrete-time system
• The matrix xr is the solution of the Riccati equation:
•  continuous-time algebraic Riccati equation discrete-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" sys any one of the models "FeedbackInputs" All the feedback inputs uf "RegulatedOutputs" All the 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 All uses 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

# Examples

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)

Continuous-time StateSpaceModel:

Discrete-time StateSpaceModel:

Descriptor StateSpaceModel:

### 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).

#### Text

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

#### CMS

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

#### APA

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

#### BibTeX

@misc{reference.wolfram_2024_lqoutputregulatorgains, author="Wolfram Research", title="{LQOutputRegulatorGains}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/LQOutputRegulatorGains.html}", note=[Accessed: 10-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_lqoutputregulatorgains, organization={Wolfram Research}, title={LQOutputRegulatorGains}, year={2021}, url={https://reference.wolfram.com/language/ref/LQOutputRegulatorGains.html}, note=[Accessed: 10-September-2024 ]}