feedback linearization presented by : shubham bhat (eces-817)
TRANSCRIPT
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Feedback Linearization
Presented by : Shubham Bhat
(ECES-817)
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Feedback Linearization- Single Input case
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Feedback Linearization- Single Input case
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Feedback Linearization- Contd.
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Feedback Linearization- Contd.
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Problem Statement
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Input Output Linearization-Contd.
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Input Output Linearization-Contd.
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Internal Dynamics
•If we need to differentiate the output r times to generate an explicit relationship between output y and input u, the system is said to have a relative degree r.
•The system order is n. If r<= n, there is an part of the system dynamics which has been rendered “unobservable”. This part is called the internal dynamics, because it cannot be seen from the external input-output relationship.
•If the internal dynamics is stable, our tracking control design has been solved. Otherwise the tracking controller is meaningless.
•Therefore, the effectiveness of this control design, based on reduced-order model, hinges upon the stability of the internal dynamics.
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Internal Dynamics- Contd
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Internal Dynamics in Linear Systems
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•Extending the notion of zeros to nonlinear systems is not a trivial proposition.
•For nonlinear systems, the stability of the internal dynamicsmay depend on the specific control input.
•The zero-dynamics is defined to be the internal dynamicsof the system when the system output is kept at zero by the input.
•A nonlinear system whose zero dynamics is asymptotically stable is an asymptotically minimum phase system.
•Zero-Dynamics is an intrinsic feature of a nonlinear system, which does not depend on the choice of control law or the desired trajectories.
Extension of Internal Dynamics to Zero Dynamics
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•Lie derivative and Lie bracket
•Diffeomorphism
•Frobenius Theorem
•Input-State Linearization
•Examples
•The zero dynamics with examples
•Input-Output Linearization with examples
•Opto-Mechanical System Example
Mathematical Tools
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Lie Derivatives
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Properties of Lie Brackets
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Diffeomorphisms and State transformations
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Example
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Frobenius Theorem- Completely Integrable
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Frobenius theorem
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Input-State Linearization
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Conditions for Input-State Linearization
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How to perform input-state Linearization
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Consider a mechanism given by the dynamics which represents a single link flexible joint robot.Its equations of motion is derived as
Because nonlinearities ( due to gravitational torques) appear in the first equation, While the control input u enters only in the second equation, there is no easy wayto design a large range controller.
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Example system
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Checking controllability and involuvity conditions.
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It has rank 4 for k>0 and IJ> infinity. Furthermore, since the above vector fields are constant, they form an involutive set.Therefore the system is input-state linearizable.
Example system- Contd.
![Page 42: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/42.jpg)
Let us find out the state-transformation z = z(x) and the inputtransformation so that input-state linearization is achieved.vxxu )()(
)(cos
)(sin
.,
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Example system - Contd.
![Page 43: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/43.jpg)
Accordingly, the input transformation is
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))((
exp
)/()(
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xI
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Example system- Contd.
![Page 44: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/44.jpg)
Finally, note thatThe above input-state linearization is actually global, because the diffeomorphism z(x) and the input transformation are well defined everywhere.Specifically, the inverse of the state transformation is
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Example system- Contd.
![Page 45: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/45.jpg)
Input-Output Linearization of SISO systems
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Generating a linear input-output relation
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![Page 47: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/47.jpg)
Normal Forms
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Zero Dynamics
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![Page 50: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/50.jpg)
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Zero Dynamics- Contd.
![Page 51: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/51.jpg)
Local Asymptotic Stabilization
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Example System
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![Page 53: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/53.jpg)
Global Asymptotic Stability
Zero Dynamics only guarantees local stability of a control system based on input-output linearization.
Most practically important problems are of global stabilization problems.
An approach to global asymptotic stabilization based on partial feedback linearization is to simply consider the control problem as a standard lyapunov controller problem, but simplified by the fact that putting the systems in normal form makes part of the dynamics linear.
The basic idea, after putting the system in normal form, is to view as the “input” to the internal dynamics, and as the “output”.
![Page 54: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/54.jpg)
Steps for Global Asymptotic Stability
•The first step is to find a “ control law” which stabilizes the internal dynamics.
•An associated Lyapunov function demonstrating the stabilizing property.
•To get back to the original global control problem.
•Define a Lyapunov function candidate V appropriately as a modified version of
•Choose control input v so that V be a Lyapunov function for the whole closed-loop dynamics.
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0V
0V
![Page 55: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/55.jpg)
Local Tracking Control
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Inverse Dynamics
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![Page 59: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/59.jpg)
Application of Feedback Linearization to Opto-Mechanics
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(cos)))((tansin2
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For the double slit aperture, the irradiance at any point in space is given as:
= wavelength = 630 nmk = wave number associated with the wavelength a = center-to-center separation = 32 umb = width of the slit = 18 umz = distance of propagation =1000 um
![Page 60: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/60.jpg)
Plant Model
-
+ )1(
1
sMotor Dynamics Plant Model
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)(1)(sin 2 assumeAXcY
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Plant Model
![Page 61: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/61.jpg)
Input-State Linearization
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Pole-Placement loop21 xz z
Plant Model
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+ )1(
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sMotor Dynamics Plant Model
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![Page 63: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/63.jpg)
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Input-Output Linearization
![Page 64: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/64.jpg)
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![Page 65: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/65.jpg)
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![Page 66: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/66.jpg)
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![Page 67: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/67.jpg)
Conclusion
Control design based on input-output linearization can be made in 3 steps:
•Differentiate the output y until the input u appears
•Choose u to cancel the nonlinearities and guarantee tracking convergence
•Study the stability of the internal dynamics
If the relative degree associated with the input-output linearization is the same as the order of the system, the nonlinear system is fully linearized.
If the relative degree is smaller than the system order, then the nonlinear system is partially linearized and stability of internal dynamics has to be checked.
![Page 68: Feedback Linearization Presented by : Shubham Bhat (ECES-817)](https://reader035.vdocuments.site/reader035/viewer/2022062217/56649f275503460f94c3eadd/html5/thumbnails/68.jpg)
Homework Problems
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