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 "prop".

Details 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

Examples

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Basic Examples  (1)

Exactly linearize a system using feedback and nonlinear transformations:

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

Simulate the resulting closed-loop system:

Scope  (21)

Basic Uses  (5)

Obtain the linearized system:

Design a controller using the linear system:

The closed-loop system:

Simulate the response to a unit step input:

Obtain the residual system:

Compute the eigenvalues of the residual system:

Explicitly specify the new state and feedback variables:

The results in terms of the specified variables:

Obtain a property directly:

Obtain multiple properties directly:

Transfomed System Properties  (1)

The transformed system has two subsystems:

The entire transformed system:

It can also be assembled from lsys and rsys:

Controller and Estimator Properties  (5)

Find the nonlinear state feedback controller :

The controller designed using the exactly linearized system:

Obtain the controller for the original system:

Find a nonlinear state estimator:

Compute the estimator gains for the exactly linearized system:

Obtain the estimator for the original system:

Plot the estimated state trajectories:

Compare the actual and estimated state trajectories:

Find the controlled closed-loop system:

The feedback gains computed using the linearized system:

Obtain the closed-loop system:

Simulate the closed-loop system:

The closed-loop system using a dynamic controller:

The linear system:

The linear controller and estimator design:

The controller for the original system:

The closed-loop system:

The full controller for the original system:

The feedback gains computed using the linearized system:

The full controller for the original system:

The inputs to the full controller are the reference inputs and the state feedback:

Plot the control input to the system:

Compensator Properties  (5)

The feedback compensator:

The compensated system consists of independent single-input, single-output loops:

The first output is influenced only by the first input:

And the second output is influenced only by the second input:

Other variants of the feedback compensator:

The systems model of the inverse feedback compensator:

The inverse feedback compensator as a list of transformation rules:

The decoupling matrix:

The matrix must be invertible for any control design to be valid:

Obtain the decoupling matrix from the inverse feedback transformation:

If possible, a precompensator is computed to make the decoupling matrix nonsingular:

The precompensator in series with the system results in a well-defined vector of relative orders:

The decoupling matrix of just the original system is singular:

If possible, for scalar systems a postcompensator is computed:

With the postcompensator, the system is exactly feedback linearizable:

The post compensator is a combination of the outputs of the system:

Different outputs result in different linear systems:

Zero Dynamics  (5)

The zero dynamics system:

It can also be obtained from the residual system by deleting all its inputs:

The zero dynamics manifold:

Compute it from the inverse state transformation:

A system with no residual dynamics rsys:

Since there are no residual dynamics, linear control design can be used:

Simulate a step response for the closed-loop system:

A system with stable residual dynamics:

The residual dynamics are stable, so feedback-design-based linear methods are valid:

Use linear feedback design, and find the feedback law:

Simulate the system:

A system with unstable residual dynamics rsys:

The eigenvalues have a positive real part, so the rsys is unstable:

For such nonminimum phase systems, the designs based on linear techniques are not valid:

The designed feedback is unable to stabilize the intrinsically ill-behaved system:

Options  (2)

Method  (2)

For linear systems, an identity transformation and feedback are used by default:

The Burnovsky form moves the hidden modes to the residual system:

For nonlinear systems, the result is in Burnovsky form by default:

It may be possible to decouple the input from the residual dynamics:

Applications  (8)

Electromechanical Systems  (2)

Design a controller that stabilizes a magnetic levitation system using exact linearization, and compare with a design based on approximate linearization:

The affine model can be obtained directly from the governing equations:

The system without feedback is unstable, here with initial value {0.2,0.,0.1}:

It is completely feedback linearizable, since it has no residual dynamics:

The controller design based on the linear system:

The closed-loop system using original state variables:

Simulate the system with initial value {0.3,0.,0.31305}:

Design based on the approximately linearized system:

The closed-loop system with the controller based on linearization:

Simulation of the system with controller based on approximate linearization:

The design based on feedback linearization has a better response:

Find a stabilizing controller for a two-wheeled inverted pendulum (e.g. Segway) using voltages to the two DC wheel motors as inputs:

The AffineStateSpaceModel of the system with states {θ,θ',ψ,ψ',ϕ,ϕ'}:

Feedback linearize the system:

The zero dynamics have purely oscillatory behavior:

Design a controller using the linearized subsystem:

Compute the state response of the closed-loop system:

The plots show that the oscillations are in the pendulum's pitch:

The two wheels are at rest in steady state:

And so is the yaw motion of the pendulum:

Mechanical Systems  (2)

Design a vibration controller for a flexible structure, and compute the control effort expended. A two-mode model of a flexible structure with no damping: »

The two modes:

The model is completely feedback linearizable:

Design a controller using the linear system to suppress the vibrations:

The modes of the closed-loop system:

The control effort:

Design a controller to suppress the oscillations in an axial flow compressor. A model of the compressor with throttle as input: »

Simulations show the presence of surge-like oscillations:

Feedback linearize the system:

Design a controller to suppress the oscillations:

The closed-loop system:

Simulations show that the oscillations have been suppressed:

Chemical Systems  (2)

Design a controller to improve an isothermal continuous stirred-tank reactor process: »

The affine system with states and inputs :

The system is completely feedback linearizable:

Design a controller based on the linearized system:

The closed-loop system:

The response of the closed-loop system from a non-equilibrium initial condition:

The response of the uncontrolled system is far worse:

Assume that the inputs to the system are random and normally distributed:

The response of the controlled system:

The response of the open-loop system:

Again, the response of the controlled system is better:

Design a controller for a room cooling system: »

The air and water flow are the inputs and are the states:

Feedback linearize the system:

The zero dynamics are stable:

Compute a controller using the linearized system:

The closed-loop system:

The temperature in the room from various initial conditions as a function of time in hours:

The outside temperature is not present in the closed-loop system, as it is offset by the feedback:

The control inputs to the system can be determined using the full controller:

The outside air temperature on a warm day:

The temperature plot:

The control inputs, which are the air and water flows:

Plot the control inputs:

Electrical Systems  (2)

Design a controller for better speed response in an induction motor subject to varying loads: »

The model of the motor with inputs :

Simulations show the effect of torque on the angular velocity:

The model with an integrator in the q axis:

Feedback linearize the system:

Design a state feedback controller for the linear system:

The closed-loop system:

Simulate the closed-loop system for various torque values:

Design a controller that regulates quantities in a wind energy conversion system: »

Its feedback linearization:

The zero dynamics are stable:

The linear system consists of a double integrator and three single integrators:

A controller that gives desired closed-loop transfer functions in each linearized and decoupled loop:

The closed-loop system also has the same linear characteristics plus the zero dynamics:

The response of the system to a unit step applied to each input in turn:

The first input affects only the first output:

And the closed-loop behaves as the designed linear system:

This is true of the three other decoupled SISO loops as well:

Properties & Relations  (9)

The linear system lsys is given in Burnovsky form:

The Burnovsky form consists of chains of integrators:

The length of integrator chains is determined by the relative orders:

The Burnovsky form is controllable and observable:

The order of the linear system is the sum of the relative orders:

Since the order of the linear system lsys is less than 4, there are residual dynamics:

The modes of the zero dynamics of a linear system are its transmission zeros:

The problem of zeroing the output:

The trajectories of the zero dynamics:

The inverse state transformation:

Compute the input that gives zero output:

The output is essentially zero for the nonzero input:

The linear system lsys together with the residual system rsys gives the transformed system:

The transformed system tsys has lsys and rsys in parallel:

Obtain it directly from the linearization data object:

Obtain the transformed system from the original system using feedback and coordinate transformation:

It can also be obtained by first applying the transformation and then feedback:

The original system can be reacquired from the transformed system:

The transformed system:

The feedback transformation as a static system:

Apply the feedback followed by the state transform to tsys:

Apply the state transform followed by the feedback to tsys:

Both are equivalent to asys:

The zero dynamics are obtained from the residual dynamics by deleting all the inputs:

Possible Issues  (1)

A nonidentity feedback causes a mismatch between the specified and actual estimator poles:

This may cause the estimator to be unstable:

Adjust the pole specifications to obtain a stable estimator:

Introduced in 2014
 (10.0)