Feedback Linearization
Feedback linearization is an exact linearization process that computes state and feedback transformations to linearize a nonlinear system and allows for the design of nonlinear controllers using linear techniques. Compare controller designs based on exact and approximate linearizations for a magnetically levitated system.
The affine model can be obtained directly from the governing equations.
Out[2]= | ![](HTMLImages.en/feedback-linearization/O_23.png) |
It is completely feedback linearizable, since there are no residual dynamics.
Out[4]= | ![](HTMLImages.en/feedback-linearization/O_24.png) |
Compute the stabilizing feedback gains using the exactly linearized system.
Out[5]= | ![](HTMLImages.en/feedback-linearization/O_25.png) |
Simulate the closed-loop system for given initial conditions.
Out[7]= | ![](HTMLImages.en/feedback-linearization/O_26.png) |
Compute stabilizing feedback gains using the approximately linearized system.
Out[8]= | ![](HTMLImages.en/feedback-linearization/O_27.png) |
The design based on exact linearization has a better response.
Out[10]= | ![](HTMLImages.en/feedback-linearization/O_28.png) |
The nonlinear controller used in the exact linearization design.
Out[11]= | ![](HTMLImages.en/feedback-linearization/O_29.png) |
The control effort expended.
Out[13]= | ![](HTMLImages.en/feedback-linearization/O_30.png) |
The effort expended for the exact case is much lower than that of the approximate.
Out[15]= | ![](HTMLImages.en/feedback-linearization/O_31.png) |
An animation of the ball being levitated using the nonlinear controller.
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