feedback bhat

Feedback LinearizationPresented by : Shubham Bhat

(ECES-817)

Feedback Linearization- Single Input case

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Input Output Linearization-Contd.

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.

1

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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

•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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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- Contd.

Let us find out the state-transformation z = z(x) and the inputtransformation so that input-state linearization is achieved.vxxu )()(

)(cos

)(sin

.,

0000

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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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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”.

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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Input-State Linearization

.,

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Pole-Placement loop21 xz z

Plant Model

-+ )1(

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Motor Dynamics Plant Model

U(x,v)X2 Y

11zKv -

0

Input-State Linearization- Block diagram

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)sin()sin()(sin)sin(

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Input-Output Linearization

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Zero Dynamics

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.

Homework Problems

2

122

211

1

222

112

21

2

)1(

xyuxxx

uxkxxfordynamicszerotheofstabilityglobalCheck

xyuxxaxxx

xxforcontrolleroutputinputlinearaDesign

feedback bhat

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