DiscreteLQRegulatorGains

DiscreteLQRegulatorGains[sspec,wts,τ]

gives the discrete-time state feedback gains with sampling period τ for the continuous-time system specification sspec that minimizes a cost function with weights wts.

DiscreteLQRegulatorGains[…,"prop"]

gives the value of the property "prop".

Details and Options

  • DiscreteLQRegulatorGains is also known as discrete linear quadratic regulator, discrete linear quadratic cost equivalent regulator or discrete linear quadratic emulated regulator.
  • DiscreteLQRegulatorGains is typically used to compute a digital implementation of a regulator that can stabilize a system or improve its performance.
  • The controller is typically given by , where is the computed gain matrix.
  • The inputs u to sys consist of feedback inputs uf and other inputs ue.
  • The system specification sspec is the system sys together with the uf specification.
  • DiscreteLQRegulatorGains computes the discrete-time controller for a linear continuous-time system sys that minimizes an approximated discrete-time equivalent of a continuous-time cost function.
  • The continuous-time cost function is given by .
  • The discrete-time cost function is approximated as sum_(k=0)^infty(x(k).phi.x(k)+TemplateBox[{{{u, _, f}, (, k, )}}, ConjugateTranspose].rho.u_f(k)+2 TemplateBox[{{x, (, k, )}}, ConjugateTranspose].pi.u_f(k)), with the following terms:
  • state weight matrix
    input weight matrix
    cross-coupling weight matrix
  • 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 sys can be given as StateSpaceModel[{a,b,…}], where a and b represent the state and input matrices in the continuous-time system .
  • The discrete-time design model dsys is a zero-order hold approximation , with the following terms:
  • state matrix
    input matrix
  • 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
  • The feedback inputs can have the following forms:
  • {num1,…,numn}numbered inputs numi used by StateSpaceModel, AffineStateSpaceModel and NonlinearStateSpaceModel
    {name1,…,namen}named inputs namei used by SystemModel
    Alluses all inputs
  • For nonlinear systems such as AffineStateSpaceModel, NonlinearStateSpaceModel and SystemModel, the system will be linearized around its stored operating point.
  • DiscreteLQRegulatorGains[…,"Data"] returns a SystemsModelControllerData object cd that can be used to extract additional properties using the form cd["prop"].
  • DiscreteLQRegulatorGains[…,"prop"] can be used to directly give the value of cd["prop"].
  • Possible values for properties "prop" include:
  • "Design"type of controller design
    "DesignModel"model used for the design
    "DiscreteTimeClosedLoopPoles"poles of "DiscreteTimeClosedLoopSystem"
    "DiscreteTimeClosedLoopSystem"dcsys with ue(k) and (k) as input and y(k) as output
    {"DiscreteTimeClosedLoopSystem",<|"Merge"m|>}whether to merge the discrete-time closed-loop system
    "DiscreteTimeControllerModel"dcm with (k) and x(k) as input and uf(k) as output
    "DiscreteTimeDesignModel"approximated discrete-time model dsys
    "DiscreteTimeOpenLoopPoles"poles of dsys
    "DiscreteTimeWeights"weights ϕ, ρ, π of the approximated cost
    "FeedbackGains"gain matrix κ or its equivalent
    "DiscreteTimeFeedbackGainsModel"dgm with x(k) as input and uf(k) as output
    "FeedbackInputs"inputs uf of sys used for feedback
    "InputModel"input model sys
    "InputsCount"number of inputs u of sys
    "OpenLoopPoles"poles of "DesignModel"
    "OutputsCount"number of outputs y of sys
    "SamplingPeriod"sampling period τ
    "StatesCount"number of states x of sys
  • The diagram of the discrete-time feedback gains model dgm, discrete-time controller model dcm, and discrete-time closed-loop system dcsys.

Examples

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Basic Examples  (1)

Compute a set of discrete-time regulator gains for a continuous-time system:

Scope  (23)

Basic Uses  (7)

Compute the discrete-time state feedback gain of a system:

The discrete-time approximation is stable:

The discrete-time closed-loop system is even more stable:

The gain for an unstable system:

The discrete-time approximation is also unstable:

The discrete-time closed-loop system is stable:

The gains for various sampling periods:

The discrete-time approximations:

The discrete-time closed-loop systems:

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

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

Compute the gains for a multiple-input system:

Reverse the weights of the feedback inputs:

A higher weight does not lead to a smaller norm:

Increase the weight even more to reduce the relative signal size:

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

Compute the gains for a nonlinear system:

The controller for the approximate linear system:

Properties  (11)

DiscreteLQRegulatorGains returns the discrete-time 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 discrete-time closed-loop system:

The poles of the linearized closed-loop system:

Increasing the weight of the states makes the system more stable:

Increasing the weight of the feedback inputs makes the system less stable:

The model used to compute the feedback gains:

The poles of the design model:

The discrete-time approximate model used to compute the feedback gains:

The poles of the approximate discrete-time model:

The approximate discrete-time weights:

The design method:

Properties related to the input model:

Get the controller data object:

Applications  (1)

Design an optimal controller by emulation:

The step response of the emulated system:

Properties & Relations  (1)

A continuous-time system:

Compute the weights for the emulated discrete-time system:

Compute the discrete LQ gains using the emulated system and corresponding weights:

DiscreteLQRegulatorGains directly gives the same result:

Possible Issues  (1)

It is not possible to compute an optimal regulator for a system that is not stabilizable:

The second mode is not stabilizable:

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

Text

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

BibTeX

@misc{reference.wolfram_2021_discretelqregulatorgains, author="Wolfram Research", title="{DiscreteLQRegulatorGains}", year="2021", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteLQRegulatorGains.html}", note=[Accessed: 02-August-2021 ]}

BibLaTeX

@online{reference.wolfram_2021_discretelqregulatorgains, organization={Wolfram Research}, title={DiscreteLQRegulatorGains}, year={2021}, url={https://reference.wolfram.com/language/ref/DiscreteLQRegulatorGains.html}, note=[Accessed: 02-August-2021 ]}

CMS

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

APA

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