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
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- 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:
-
Automatic automatically determine method (default) "Identity" apply identity feedback with identity transformation "Burnovsky" return lsys in Burnovsky form
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Examples
open allclose allBasic Examples (1)
Scope (21)
Basic Uses (5)
Transfomed System Properties (1)
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 controller and estimator design:
The controller for the original system:
The full controller for the original system:
The feedback gains computed using the linearized system:
The full controller for the original system:
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The inputs to the full controller are the reference inputs and the state feedback:
Compensator Properties (5)
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 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:
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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:
Zero Dynamics (5)
It can also be obtained from the residual system by deleting all its inputs:
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:
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:
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Options (2)
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:
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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:
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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:
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 model is completely feedback linearizable:
Design a controller using the linear system to suppress the vibrations:
The modes of the closed-loop system:
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:
Simulations show that the oscillations have been suppressed:
Chemical Systems (2)
Design a controller to improve an isothermal continuous stirred-tank reactor process: »
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The affine system with states and inputs
:
The system is completely feedback linearizable:
Design a controller based on the linearized 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: »
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The air and water flow are the inputs and are the states:
Feedback linearize the system:
Compute a controller using the linearized 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:
Electrical Systems (2)
Design a controller for better speed response in an induction motor subject to varying loads: »
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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:
Simulate the closed-loop system for various torque values:
Design a controller that regulates quantities in a wind energy conversion system: »
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 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:
The zero dynamics are obtained from the residual dynamics by deleting all the inputs:
Text
Wolfram Research (2014), FeedbackLinearize, Wolfram Language function, https://reference.wolfram.com/language/ref/FeedbackLinearize.html.
CMS
Wolfram Language. 2014. "FeedbackLinearize." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/FeedbackLinearize.html.
APA
Wolfram Language. (2014). FeedbackLinearize. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/FeedbackLinearize.html