FeedbackLinearize

FeedbackLinearize[asys]
input-output linearizes the AffineStateSpaceModel asys by state transformation and feedback.

FeedbackLinearize[asys,{z,v}]
specifies the new states z and the new control inputs v.

FeedbackLinearize[asys,{z,v},"prop"]
computes the property .

Details and OptionsDetails and Options

  • FeedbackLinearize is also known as exact linearization.
  • FeedbackLinearize will construct a linear system lsys from a nonlinear system asys in such a way that you can use linear control design techniques for the linear system lsys to control the nonlinear system asys.
  • FeedbackLinearize returns a LinearizingTransformationData object that can be used to extract the properties needed for analysis and design based on feedback linearization.
  • The transformed system tsys consists of a linear system lsys and possibly a residual system rsys with internal dynamics that need to be stable, but is otherwise not observable.
  • Properties related to the transformed system include:
  • "LinearSystem"systems model lsys
    "ResidualSystem"systems model rsys
    "TransformedSystem"systems model tsys
  • By designing a stabilizing controller cs for lsys, the resulting closed-loop system will be stable, provided that the residual system rsys is stable.
  • In order to deploy the controller for the original nonlinear system asys, you need to transform the controller cs to use the original variables.
  • Properties related to transforming the controller and estimator to original coordinates:
  • {"OriginalSystemController",cs}controller cs in original coordinates
    {"OriginalSystemEstimator",es}estimator for and
    {"ClosedLoopSystem",cs}closed-loop system in original coordinates
    {"OriginalSystemFullController",cs}systems model of cs in original coordinates
  • Further detailed properties of feedback linearization are also available and can be used to deploy alternative simulations and implementations of controllers, estimators, etc.
  • The system asys is connected with a feedback compensator, precompensator, and postcompensator to give a modified system , where is the modified input, is the state vector that consists of with possible additional compensator states, and is the modified output.
  • The feedback compensator is essentially a transformation between and given by , where is the decoupling matrix.
  • Compensator properties include:
  • "FeedbackCompensator"systems model from to
    "InverseFeedbackCompensator"systems model from to
    "InverseFeedbackTransformation"list of rules
    "DecouplingMatrix"matrix
    "PreCompensator"systems model from to
    "PostCompensator"systems model from to
  • To get an explicitly linear system lsys and a possible residual system rsys, you need to perform a state transformation .
  • Properties related to state transformation and zero dynamics include:
  • "InverseStateTransformation"list of rules
    "ZeroDynamicsSystem"systems model
    "ZeroDynamicsManifold"manifold on which the rsys state evolves
  • FeedbackLinearize takes a Method option with the following settings:
  • Automaticautomatically determine method (default)
    "Identity"apply identity feedback with identity transformation
    "Burnovsky"return lsys in Burnovsky form

ExamplesExamplesopen allclose all

Basic Examples  (1)Basic Examples  (1)

Exactly linearize a system using feedback and nonlinear transformations:

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Use the resulting linear system to design the closed-loop behavior of the system:

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Simulate the resulting closed-loop system:

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Introduced in 2014
(10.0)