Nonlinear Feedback linearisation

7/28/2019 Nonlinear Feedback linearisation

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A feedback linearizationcontrol the nonlinear 5-Do

flywheel suspended by PMbiased HMB

Authors: Wen Tong, Fang Jiancheng

Presen

Arpit Ch

Digvija


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Contents Introduction


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Intro AMBs(Active magnetic bearings) instead of normal ball-bea

Advantage: High Rotation speed

No contact character

Nearly zero friction

Long life span

No auxiliary lubricant system

Types:

EMB(Electro-magnetic bearing)

PM(permanent magnet) biased HMB(Hybrid magnetic bearing)


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EMB vs PM biased HMB

In EMB, suspending forces for rotor are generated completecurrents.

Whereas in HMBs PM provide the biased flux and current gthe regulating force to realize the stable suspension of the r

Power consumption is reduced greatly in the application of

biased HMB, because the biased flux is proved by the permamagnet without any external power consumption as opposebiased flux is generated by biased current.


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Need of controller

Effective controller is required for Magnetic suspensionflywheel(MSFW) because of the the instability of the open-lAMB caused by the negative displacement stiffness of themagnetic force .

The dynamic model of the PM biased HMB is nonlinear andcomplicated, and its nonlinearity is caused by the superposi

PM flux and EM flux and the magnetic flux coupling in differdirections.

It is simplified by assuming that permanent magnet in HMB equivalent to biased current in the EMB, so an equivalent EMmodel describing the HMB system can be obtained


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Need of exact linearization

Based on the equivalent model, using the Taylor series expa

the neighborhood of the operating point(or equilibrium poiapproximate linear model of the nonlinear HMB system canobtained for the controller design.

This method has an advantage that the control model is verand the controller can be realized simply.

For the application of the HMB with high restriction on powand accuracy(such as MSFW), the controller design must bethe nonlinear model of system.


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Sketch of MSFW

Diagram 1:


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Model development For convenience, there are three coordinate frames introdu

system, as shown in Figure. OXYZ is the stator-fixed frame, o

rotor-fixed frame and oabc is the intermediate frame.


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Five DOFs According to the design of flywheel(MSFW), The rotation alon

direction is controlled by the driven motor and independent of tThe translation z is controlled by the axial HMB and can be regardecoupled with other four coupled axial DOFs[x, y, , ].

The four axial DOFs of the rotor could be calculated from the foupicked up by the displacement sensors[xma, yma, xmb, ymb] as showEq.(1):

x = xma+ xmb

= (xma- xmb)/L

y = yma+ ymb

= (yma- ymb)/L

L is the distance between the distance between the two displacesensors located at the two ends the stator of the MSFW, as showfollowing figure


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MSFW description

The MSFW includes four sets of radial HMBs.

As shown in following figures, each set consist of two seriesconnected coils placed diagonally opposite to generate restforces.


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rotor


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Take the HMB in X direction for example. It is assumed that the control cur

sent to the opposite connected coils in X+ and X-. Control currents flowing

two coils in X+ and X- are iax and iax, respectively. fax is the resultant restor

acting on the rotor, which is the difference between the two magnetic forc

in X+ and X- directions.

fax = fax+ - fax-= [(pax++ iax++ iax-ax+)2 - (pax- - iax- - iax+ax-)

2 ]/ (20A)

Where,

pax+, pax- are PM flux in air-gap of X+ and X-,

iax+

, iax+ax-

are EM flux in X+ and X- generated by the control current iax

,

iax-, iax-ax+ are EM flux in X+ and X- generated by the control current -iax.

permeability of the vacuum and A is the area of one pole. The other three

forces fay, fbx and fby would be obtained using the same methods described


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Rotor Dynamics

ma and mb are distances from HMBs to the origin of the OXYZ.

fx, fy, px, and py are the disturbances caused by the residual static and dynamic

imbalances.

The state variables are chosen as [x y ]. The generalized magnetic forces actin

on the system are [fx py fy px].


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Dynamic model of the nonlinear sys the MSFW is a 4-input, 4-output, 8-order nonlinear system.

The dynamics model is consisted of three parts. The first is sequation f(x) which is regarded as the uncontrollable permamagnetic force and the couple caused by the gyroscopic efferotor. The second is the controllable magnetic force gi (x)uithe input part of the system. The inputs of the system are seu=[iax ibx iay iby]. The last one is the disturbances caused by thdynamic imbalances.


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where m is themassoftheroto

the moments ofinertialofthe

directions.


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The input functions of the nonlinear system are described in detail as follows:

The output functions for the selected outputs:

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Nonlinear Feedback linearisation