Robust Control of Rotor-Bearing Systems Using H-infinity Design

Copyright © IFAC 12th Triennial World Congress. Sydney. Australia. 1993

ROBUST CONTROL OF ROTOR-BEARING SYSTEMS USING H-INFINITY DESIGN C.R. Burrows, P.S. Keough and C. Mu School of Mechanical Engineering, University ofBath, Claver/on Down. Ba/h BA7 2AY, UK

Abstract. This paper describes a procedure for designing an H-infinity (H_) controller for a rotor-bearing system. The system consists of a rotor. supported by journal bearings. together with a magnetic actuator which is used to apply the control forces. The frequency characteristics of the actuator and modelling uncertainty at higher frequencies are accounted for. A computer simulation shows that rotor vibration can be significantly reduced by using the H_ control strategy. The system robustness is also demonstrated in a comparison with pole placement strategies. In particular, the H_ strategy does not suffer from modelling uncertainty spillover and is effective in stabilising the system at high running speeds, the destabilising influence being caused by journal bearing oil whirl.

Key Words. H_ control; rotor vibration; robust control; rotating machinery; stability

approach to compensate for broad band excitation is to use eigenvalue assignment (Stanway and Burrows, 1981), but the difficulties of implementing full state feedback control are not trivial, even for a rigid rotor (Bleuler and Schweitzer, 1983). In cases where bending modes occur, model reduction techniques must be used when designing the pole-placement controller and this may introduce problems due to spillover (Salm and Schweitzer, 1984).

1. INTRODUCTION A central issue in the design of high speed machines is that of limiting rotor vibrations. This problem has a long history starting with Rankine's attempt to explain the violent vibrations which occur at 'critical speeds' (Rankine, 1869). In the 1920's, the demand for higher machine speeds resulted in rotors being operated in the super critical region and this required a more fundamental understanding of rotor dynamics and in particular rotor stability. It was recognized that the bearings had a significant influence on rotor vibrations, and oil-film bearings were known to be the cause of instability under certain operational conditions. Various bearing designs were proposed to prevent oil-whirl (Newkirk and Taylor, 1925) and this has remained an active research topic (Keogh et

Burrows et al. (1993) have shown that state feedback can be used to suppress the critical speeds of a flexible rotor without encountering problems due to spillover, but the approach was dependent upon having a good model for the system. This is frequently not the case due to the inherent difficulty of accurately modelling the dynamic characteristics of oil-film bearings. Thus there is a need for establishing robust algorithms capable of controlling the vibration of flexible rotors in the presence of model uncertainty or truncation effects. This paper examines the ability of an ll_ controller to fulfil these requirements.

al.(1990». The inherent limitations in using passive bearing elements to control the vibration of flexible rotors was one of the major factors which led to the study of magnetic bearings as machine elements (Schweitzer, 1975; Schweitzer and Lange, 1976). This in turn focused attention on the need to develop effective strategies for controlling the various modes of vibration excited during normal machine operation. One approach, effective in reducing synchronous vibrations, is to use an open-loop adaptive strategy (Burrows and Sahinkaya, 1983), which is inherently stable and effective in compensating for parameter change. A more general

NOTATION

Ac' Bc, Cc, Dc - controller state space matrices A.. - reduced order system matrix in modal space A - full order system matrix A w •• B w•• CwIt DO'. - state space weighting matrices A ..1• B w1 ' C..1• D..1 - state space weighting matrices BI - control force distribution matrix B•• - reduced order disturbance modal space matrix B 2m - reduced order control modal space matrix

621

BI BI

where 1.1 denotes the Euclidean norm and cr(G(s))Z is the maximum eigenvalue of (G(sflG(s). Thus, if s ::: jO), for a frequency response analysis, then the spectral norm depends on the frequency of vibration. To obtain a measure of GUO)) over a complete frequency range, the H_ norm is defined as

full order disturbance input matrix full order control input matrix Cl.. - reduced order control modal space matrix Cl.. - reduced order measurement modal space matrix in modal space Cl - full order control output matrix Cl - full order measurement output matrix Ch - controlled displacement output matrix Ch - controlled velocity output matrix Cb - measured displacement output matrix Cl> - measured velocity output matrix d - disturbance vector d.. - disturbance vector due to uncertain high frequency modes D - damping matrix D I .. - feedforward matrix from disturbance to controlled modal output D I .. - feedforward matrix from disturbance to measured modal output Gf - disturbance distribution matrix G - gyroscopic matrix G1J(s) - transfer function matrices K - stiffness matrix M - mass matrix P... - retained modal state vector Pr - high order modal state vector q - generalized coordinate vector of rotor T.(s) - controller transfer function matrix s - Laplace transform variable u - control input vector x - state vector x wll x w2' X. - weighting and controller state vectors Z - controlled output vector Zl - weighted control force vector Z2 - weighted controlled output vector cr - singular value X - observed output vector Q - rotor running speed 0) - rotor vibration frequency -

-

101 : -

sup o(GUO))) : sup 0)

e

j

18 lz 18ilz 0

(3)

which has also been related to notional input and output signal energies,

le,l, • [ fe,'(I)e,(t)dt le.l,

= [

fe:
r r

provided they exist and the poles of G(s) lie strictly in the left half plane, i.e. the system is stable. Thus, the H_ norm also provides an upper bound on the output to input energy ratio over all possible finite energy input signals. The basic state space representation required of any linear system for H _ control design is i : Ax + Bid + Blu

Z : Clx + Dud + D 12 u

(4)

X : C 2 x + Dzld + D 22 u Here, d is regarded as an input due both to modelling uncertainty and physical disturbances. The two output equations represent measured states X together with states to be controlled z. Normally the measured states are a subset of the states to be controlled. The transfer function relationship between inputs and outputs must be of the form

2. H-infinity CONTROL DESIGN (5)

This section will detail the application of the H_ control design technique to a rotor-bearing system. Initially, a preliminary statement of terminology is given.

where

2.1. Definitions and Objectives Consider a transfer function matrix G(s) to relate a vector input 8i to a vector output 80 by 80

:

Consider the controller to be based on the measured states only with u ::: T.(s)X giving

(1)

G(s)8 j

(6)

The spectral norm of G(s) is defined by IIG(s)ll s

:

sup x,tO

I G(s)xl hi :

o(G(s))

and the objective in the H_ design is to find the controller T.(s) such that

(2)

622

IG n + G 12 T c ( / -GzzTyl Gz1 )L

(9)

is minimised under the constraint that the system is stable, i.e. to reduce the signal energy transfer from the input to the control states.

z = C1x , X = Czx the modelled system may have around 100 generalised coordinates or, equivalently, around 200 states. It is therefore difficult to design a controller based directly on (9). To overcome this problem, the system eigenvalues and eigenvectors are solved for and the system states of (9) are transformed into modal coordinates by

2.2. Rotor-Bearing System Modelling The first stage in any application must involve system modelling. Fig. 1 shows a symmetric, flexible shaft with a disc mounted at either end. Two journal bearings are used to support the rotor and a magnetic actuator is located between the two journal bearings.

x

1

'-

Fig.I.

-,- -2

4

l

-

JOOl1lal boarioa

, •

-~

7

0-4- t- Magnetic bearing

0.13 dia

--

(10)

Vp

where V is the eigenvector matrix. If the eigenvalues are ordered in ascending frequency and the modal vector is partitioned into low and high frequency states by

2.112

-

=

• -+---

0.1 ilia

9

~

Jownal bearing

'-

then the modal form of (9) is

Layout of rotor-bearing system showing nodes (dimensions in m)

(11)

x = Cz",P",

The use of finite element techniques, based upon Timoshenko beam theory, enables the equation of motion for lateral vibration to be established in the form Mij + (D +QG)tj + Kq

= Hfu

+

C2r Pr

Thus, when high frequency modal states Pr are ignored, the representation of the reduced model is

+ Gfd (7)

(12)

where q is the vector of generalised coordinates, U is the control vector, and d is the vector of physical disturbances e.g. unbalance forcing.

X = Cz",P", + Dz",d", where d", is a notional disturbance accounting for the missing high frequency modes.

The states to be controlled may be defined from (8a)

2.3. Control Design For Rotor-Bearing System

and the measured states defined from X.

= Cz.q ,

X.

= CzA

Initially, two weighting functions are introduced which recognise the capability of the magnetic actuator to deliver force, together with possible spillover effects arising from unmodelled modes. These are designated by the transfer function matrices W1(s) and WzCs). Typically, W1(s) would reflect the frequency response of the magnetic actuator and filter out inputs from the unmodelled modes. The weighting Wis) may be chosen to assign importance to frequency ranges, e.g. around critical speed values. The transfer functions operate on the control force and control states according to

(8b)

Since displacement transducers are the likely means of measurement, velocities need not be included in the measured states, but they are retained for generality. The state space representation follows from

X ' [:]

,

z =[::]

X=

[~:]

and is of the form

(13)

The equivalent state space representation of these weightings will take the general forms 623

(14a)

(17)

It is to be noted that the controller matrices depend on running speed. Also, for validation of robustness properties, the controller should be applied to the full order model.

and (14b) the augmented system from (12), (14a), and (14b) is of the fonn

3. COMPUTER SIMULATION AND RESULTS For the rotor-bearing system configuration shown in Fig.l, two journal bearings are positioned at stations 2 and 8 and the magnetic actuator is located at station 4. The transducers are at stations I, 4, and 9 in horizontal and vertical directions to supply measured lateral displacements. The computer simulation was based upon software which wa<; written so that the system modelling, modal analysis, system reduction, selection of the weighting functions, Connulation of equations (15), H_ controIler design, and the dynamic response of the uncontrolled and controIled system, could be performed sequentially. The reduced order model was initially based on retention of the four lowest frequency modes from the full order system model.

(15)

This system now has the same form that of equation (4). The controller transfer function is based on the measured states (16)

u '" Tc(s) X

Critical speeds occur when the system natural frequencies, which are running speed dependent, are synchronous with the running speed. By checking for small real parts of the eigenvalues (low damping levels), two critical speeds at 308 rad/sec and 366 rad/sec are important. The corresponding modes of vibration involve significant rotor flexure. There are also lower critical speeds involving mainly rigid body motions of the rotor, but the journal bearings ensure that these are well damped. Therefore, the controIler was designed only at the two flexure critical speeds.

and is designed to mInImIse the H_ nonn of the transfer function between

and

~]

With appropriate choice of weighting transfer functions the control force may be constrained within the limits of the magnetic actuator capability and the influence from the neglected high frequency modes minimised. The system is shown in block diagram form in Fig.2.

The choice of the weighting matrix W1(s) was based around a first order model of the magnetic actuator. The break point frequency of the inverse of W1(s) was selected at 76 rad/sec, which is that of a magnetic actuator which has been used in experiments on rotor control [10]. It is to be noted that this weighting function does not usually depend upon the rotor running speed. Furthermore, with this choice, the high frequency signal energy trdnsferred from disturbance to controller output is constrained so that the high frequency unmodelled modes have minimum influence on the controller. The weighting Wis) was chosen to be a constant so that unifonn vibration reduction over the frequency range of the modes in the reduced order model was considered important.

z

Controller

Fig.2.

Representation of controlled system The maximum singular value of the controIler transfer function matrix

Finally, the controller (16) may be represented in state variable fonn by

624

2 ;..:.x.=..lO:::..-4

E

20

---:-_ _----.

•

" 11

-- Uncontrolled Controlled

1.5

"

,"",

I ll)

] .....

o 10- 1

~

10 2

10 5

Frequency _ Rad/Sec

:: f

"

11

1

<

I I 1I

I

I'

I1

'I

,

" "

I

"

I

,

"

" , ," ,,

0.5

,

OL-----==----------'

o

Fig.3.

Largest singular value of H_ controller for .Q = 308 rad/sec

500

Rotating speed _ Rad/Sec a. left disc location

(18)

corresponding to the design at 308 rad/sec, is shown in Fig.3. It is seen that the controller has low pass filter characteristics so that high frequency disturbances contained in the system measured states X have reduced influence on the system. This characteristic would also enhance the ability of the controller to overcome measurement errors, such as those caused by surface roughness on the shaft. For the design at 360 rad/sec, a similar result was obtained, but with slightly increased levels. This reflects the increase in unbalance force disturbance with running speed.

4 5 ;..:.x.=..1O:::..-_

I
.g

.....a

-

0 ..

~

!'

-- Uncontrolled Controlled

4

E

•

3 2

" " " "' I " ''

, ,

,

I

I

'

:

'

I

I,~

,

"

,

\I

" ,, '\

,-

, ,,

1

OL-----===-----------'

o

500

Rotating speed _ Rad/Sec

Ideally, the controller should be designed at every running speed, but in cases where the resonant frequencies associated with the critical speeds do not vary greatly with running speed, it is sufficient to use only the designs at those critical speeds. Thus, in the present case, only two designs are required and computer storage requirements are reduced. The control was the applied corresponding to the 308 rad/sec design for running speeds below 330 rad/sec and corresponding to the 366 rad/sec design for running speeds above 330 rad/sec.

b. actuator location

2.5 ;..:.x~10~-_4 E

2

____,.___--_,

-- Uncontrolled Controlled

I
.g .a ....

To evaluate rotor synchronous response, it is supposed that an unbalance mass is situated on the right disc. This will cause an elliptical orbit at each rotor station and the major axes, i.e. maximum amplitudes of the vibration, are shown in FigA, both for the controlled and uncontrolled systems. FigAa shows the vibration level at the left disc position, FigAb at the magnetic actuator position, and FigAc at the right disc position. The controlled vibration is reduced significantly around the two critical speeds, while the control influence is reduced away from the critical speeds. It is seen that the better control results are obtained at the left disc and actuator positions compared with the right disc position. This

I

1.5 1

0.5 0'---=::...----------' 500 o Rotating speed _ Rad/Sec c. right disc location

Fig.4.

625

Unbalance responses for uncontrolled and H_ controlled rotor

Table 1 Open and close-loop eigenvalues with controller designed at Q = 308 rad/sec

Running speed

308

366

660

700

Uncontrolled

Pole placement - constant feedback

Pole placement - dynamic feedback

H_infmity control

-41 ± 257i • -50 ± 274i • -2.0 ± 308i • -3.6 ± 361i -73 ± 609i -139 ± 1055i

-1.2 ± 245i -40 ± 257i -55 ± 272i +5.5 ± 434i -76 ± 612i -138 ± 1054i

-41 ± 257i -47 ± 274i -16 ± 325i +15 ± 350i -73 ± 610i -139 ± 1055i

-41 ± 257i -51 ± 276i -35 ± 312i -7.2 ± 363i -73 ± 609i -139 ± 1055i

-32 ± 268i -37 ± 286i -2.6 ± 303i -3.9 ± 367i -83 ± 594i -131 ± 1071i

-2.4 ± 245i -31 ± 268i -41 ± 284i +7.3 ± 436i -86 ± 596i -130 ± 107li

-32 ± 268i -34 ± 285i -14 ± 322i +12 ± 355i -83 ± 594i -131 ± 107li

-32 ± 268i -34 ± 286i -40 ± 310i -7.7 ± 369i -83.5 ± 594i -131 ± 107li

-1.5 ± 276i -2.1 ± 310i +2.6 ± 330i -2.9 ± 396i -100 ± 508i -496 ± 583i

-8.5 ± 234i -2.0 ± 310i -4.7 ± 329i +17± 450i -103 ± 510i -509 ± 602i

-4.2 ± 286i -2.2 ± 311i +3.8 ± 337i +5.6 ± 386i -100 ± 508i -493 ± 583i

-29 ± 27li -2.1 ± 310i -0.95 ± 331i -6.5 ± 398i -100 ± 508i -495 ± 583i

-1.3 ± 273i +1.4 ± 316i +6.9 ± 335i -2.4 ± 399i -99 ± 498i -460 ± 568i

-9.1 ± 233i +1.4 ± 316i -0.5 ± 334i +18 ± 452i -101 ± 499i -472 ± 586i

-3.8 ± 284i +1.2 ± 316i +8.3 ± 340i +5.8 ± 390i -99 ± 498i -459 ± 568i

-28 ± 267i +1.2 ± 316i +4.0 ± 336i -5.9 ± 402i -99 ± 498i -460 ± 568i

• modes on which controller design is based

is due to the asymmetry in the forcing at the magnetic actuator and unbalance locations. There are also speed ranges away for the critical speeds where the controlled response is marginally higher than the uncontrolled response, indicating the influence of designing the controller using the reduced order model. However, these levels are not significant for rotor vibration.

upon different reduced order models which retained various modes from the full order model. For the design at running speed 308 rad/sec, the following case was considered: a) three lowest frequency modes retained. For the design at the running speed 366 rad/sec, the following cases were considered:

A speed independent H~ controller was also applied to the system for comparison. The controller designed at 308 rad/sec was used throughout the speed range.

b) three lowest frequency modes retained; c) four lowest frequency modes retained. For comparison, a pole placement design, with and without a first order model of the magnetic actuator, was also evaluated for cases a), b), and c). These designs were applied to the full order system over a running speed range and the system eigenvalues evaluated. The results for case a) are shown in Table 1. It is seen that the uncontrolled system is unstable, due to oil whirl, for running speeds at 660 rad/sec and above. The H~ control strategy is seen to significantly enhance the stability of the retained full order modes without degrading the omitted modes. Similar results

It was found that a marginal degradation of controlled response occurred, mainly around the second critical speed. This gives the possibility of implementing the control in a simpler way in practice, but this did rely on the close proximity of the two critical speeds.

The assessment of the robustness of the H~ control strategy may be made by examining system eigenvalues. This was evaluated for H~ designs based

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characteristics of circular arc journal bearings. I.MechE. seminar on Developments in Plain Bearings for the 90' s, 45-54. Newkirk, BL & Taylor, H.D. (1925). Shaft whirling due to oil action in journal bearings. General Elec.Rev., 28, 559-568. Rankine, WJ. (1869). On the centrifugal force of rotating shafts. Engineer 27, p249. Salm, J. & Schweitzer, G., (1984). Modelling and control of a flexible rotor with magnetic bearings. Proc.3rd I.Mech.E./ntl.ConfVibs in Rotating Machinery, York, 553-561. Schweitzer, G. (1975). Stabilization of self excited rotor vibrations by an active damper. In Dynamics of rotors (ed. F. I. Niordson), 472-493, Berlin: Springer-Verlag Schweitzer, G. & Lange, R, (1976). Characteristics of a magnetic rotor bearing for active vibration control. Proc.l.Mech.E.l st Intl.ConfVibs in Rotating Machinery, Cambridge, 301-306. Stanway, R & Burrows, C.R (1981). Stabilization of a flexible rotor by eigenvalue assignment. lEE ConfControl and its Application, 102-105.

were found for cases b) and c). The pole placement strategies were implemented to shift the reduced order model poles to those achieved by the f{~ strategy. However, when applied to the full order system, severe spillover problems are encountered in the unmodelled modes and the pole shifting of the retained modes is reduced. Inclusion of the dynamic model of the actuator does improve this problem, but spillover problems are avoided only if the desired pole shifting is reduced.

4. CONCLUSIONS A computer simulation has been performed to indicate the procedure of designing an f{~ controller for a rotor-bearing system. The frequency characteristics of a magnetic actuator were included in the control design and account was also taken of unmodelled modes. Rotor unbalance response was attenuated significantly by using the f{~ controller. Furthermore, a speed independent controller had almost the same effect on the vibration reduction as the case of speed dependent controller. This is of importance in real time control, but the validity of this result did depend on the close proximity of two critical speeds. Although the control design was based upon a reduced order system, it caused no instability problems when applied to the full order system, thus demonstrating the robustness of the H ~ control strategy. This point will be considered in future experimental work using a rig on which the simulation results were based.

ACKNOWLEDGMENT This work forms part of a programme supported by the Science and Engineering Research Council.

5. REFERENCES Bleuler, H. & Schweitzer, G. (1983). Dynamics of a magnetically suspended rotor with decentralized control. In lASTED Applied Dynamics and Control Symposium, Copenhagen. Burrows, C.R., Keogh, P.S. & Tasaltin, R. (1993). Closed-loop vibration control of flexible rotor - an experimental study. Accepted for publication in Proc.I.Mech.E.Part C. Burrows, C.R & Sahinkaya, M.N. (1983). Vibration control of multi-mode rotor bearing systems. Proc.R.SocLond.A386,77-94. Keogh, P.S. Wale, G.D. & Walton, M.H. (1990). Influence of downstream grooving on dynamic

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