alternative approach to aeroservoelastic design and clearance

Alternative approach to aeroservoelastic design and clearance

unsteady aerodynam'cs

R . Tay1 o r R.W. Pratt 6.D.Caldwell

fl ight control aeroservodynamic system

Indexing terms: Aeroservoelastic design, Servo-hydraulic attenuation

Abstract: The in-flight interaction between an aircraft's flight control system (FCS), structural dynamics and unsteady aerodynamics has been of concern to aircraft designers for many years. Frequently, due to a limited understanding of the array of complex issues involved, a very conservative approach is essential to achieve a FCS which will obtain flight clearance. The paper examines the impact of the nonlinear nature of the servo-hydraulic actuation system on the aeroservoelastic interaction. A method of predicting the occurrence of a limit cycling condition in such a system is presented, with comparisons made to experimental results. In addition, the paper demonstrates how an application of actuator performance limits can be used to predict a maximum level of structural response. Finally, the possibility of reducing the current clearance requirements from a consideration of this maximum structural response is discussed.

1 Introduction

The in-flight interaction between the aerodynamics and structural dynamics of an aircraft has been of importance to aircraft designers for many years. Aeroelastic- ity, as such an interaction is known, manifests itself in various forms, such as static divergence and more com- monly flutter. Static divergence of an aircraft lifting or control surface is rarely encountered, but has been known to occur [l]. Such a static divergence is nor- mally one of the characteristics of forward swept wings. It is only recently that advances in materials science and structural design have made the use of a forward swept wing as in the X-29 aircraft a viable proposition.

The more common occurrence of an aeroelastic interaction is in the form of flutter which, as the name suggests, is an oscillatory structural response brought about by an interaction between the structural dynam- 0 IEE, 1996 IEE Proceedings online no 19960091 Paper first received 9th June 1995 and in revised form 11th October 1995 R Taylor and R W Pratt are with the Department of Aeronautical and Automotive Engineering and Transport Studies, Loughborough Univer- sity, LE11 3TU, UK B D Caldwell is with British Aerospace PLC, Military Aircraft Division, Warton Aerodrome, Lancashire, PR4 IAX, UK

ics and unsteady aerodynamics. The occurrence of a flutter condition within the aircraft flight-envelope is generally to be avoided. During the initial flight-envelope expansion of new aircraft, strict test procedures must be adhered to in order to prevent, where possible, the dramatic and potentially destructive consequences of flutter. Naturally, if an in-flight flutter condition is detected, structural modifications or limitation of the flight envelope is necessary.

Recently, the inclusion of automatic flight control systems with powered control surfaces has given rise to a new class of interactions known as aeroservoelasticity [2-41. As the name suggests, this is an interaction between the unsteady aerodynamics, structural dynamics and flight control system of the aircraft as demonstrated by the interaction triangle of Fig. 1 [3].

dynamics

Fig. 1 Interaction triangle

In the case of flutter, the energy needed to sustain the structural oscillation is provided entirely by the unsteady aerodynamics. In an aeroservoelastic interaction however, the flight control system (FCS) can provide some excitation of the structural response. This is possible because the aircraft motion sensors detect the structural vibrations in addition to the rigid body motion of the aircraft. As a result, provided that the frequency of the structural motion is within the bandwidth of the FCS, the aircraft control surfaces will be moved in an attempt to control the structural response. This motion of the control surfaces may further excite the structure through either inertial excitation or further unsteady aerodynamic loading. If further excitation of the structure does result, then an unstable structural oscillation will occur.

Whereas in the case of flutter, interactions only occur in-flight as a result of the influence of the aerodynamics, aeroservoelastic interactions may occur when the aircraft is both on the ground and in flight.

As the drive to produce more efficient structural designs increases, the occurrence of structural modes of frequency well within the bandwidth of the FCS will increase. In fact, for large civil aircraft, the frequency

1 IEE Proc-Control Theory Appl., Vol. 143, No. I , January 1996

of some structural modes can be of the order of 1-2Hz. In addition, the use of high-gain FCS and higher bandwidth control surface actuators increases the bandwidth of the FCS, making this overlap between the structural modes and FCS more likely [3, 5, 61.

The increasing use of digital flight control systems introduces a further aspect to the problem of predicting and preventing aeroservoelastic interactions. The digital nature of the control system can result in further excitation of a structural mode [7], or the aliasing of a high-frequency structural mode to within the bandwidth of the FCS [SI.

In accounting for the digital nature of the control system, suitable anti-aliasing filters may be included in the feedback path to prevent such an effect. Impor- tantly however, this is at the cost of additional phase lag at rigid-body frequencies.

At present, the FCS designer must ensure that the stability margins of the structural modes satisfy definite requirements [9] in order to clear an aircraft for flight. In order to achieve these clearance requirements, it is usual practice to include notch or low-pass filters [lo] within the FCS which are designed to attenuate the feedback of the structural response to acceptable levels. However, the inclusion of such filters results in the addition of undesirable phase lags to the system.

The current requirements and procedures for clearance of the aircraft are restricted by the understanding of the aeroservoelastic problem. To improve on the clearance requirements currently employed, a greater level of understanding concerning how each system component effects the aeroservoelastic interaction must be achieved. Recent work has concentrated on the modelling of the unsteady aerodynamic effects [l I] and on the development of software packages for the design and optimisation of the necessary feedback filtering [12, 131.

As a vital part of the flight control system, and hence the aeroservoelastic closed loop, the role of the actuation system within the problem of aeroservoelasticity has been largely assumed. Most current studies of a typical system involve the use of low order linear actuator models although some studies now involve the use of higher order linear models [14]. In addition, some studies involving the use of nonlinear actuation system models have been completed [15-171. It has been shown [17, 181, however, that the presence of a typical aeroservoelastic feedback signal can result in changes to the actuators frequency response at FCS frequencies as a result of the nonlinear nature of the actuation system. Such a change in actuator performance could have serious consequences in terms of stability of the rigid body dynamics.

Clearly, there exist many uncertainties within the modelling of a typical aeroservoelastic system. In particular, unreliable modelling of the aircraft structure and system nonlinearities will have a significant effect on the flight control system design process. One possible solution of such a problem involves the use of multivariable robust control techniques such as H,, whereby the controller is designed to provide satisfactory performance in the presence of modelling uncertainty [19]. To apply such techniques, however, a better understanding of the aeroservoelastic process needs to be obtained [20].

This paper examines the consequences of nonlinear actuation on the aeroservoelastic problem. The per-

2

main main actuation main control valve ram -system -actuation + ''Ive T t i I I I ' I i

Fig.2 Actuation system block diagram

2 Nonlinear actuation system modelling

The basic components of a typical aircraft servo- hydraulic actuation system are shown in Fig. 2. In this case, the system comprises of four main blocks, namely actuator control system, main valve actuation, the main valve block itself and the main ram, where the input to the system is in the form of main ram demand signals from the flight control system and the output is in the form of main ram position. It is possible to obtain equations to describe the dynamics of each of these blocks in turn. The modelling of actuation systems is a large topic in itself, and the interested reader should refer to more specialised documents [21, 221 for a more in depth treatment.

Each of the four actuation system components con- tain considerable nonlinear elements, such as saturation nonlinearities representing current limits of the electro- hydraulic servo valves. In addition, valve travel limits, valve backlash, valve friction, fluid compressibility, hysteresis, valve port profiles and port flow equations add more nonlinearities to the model.

In this case, the actuation system model used is based on the Jaguar fly-by-wire (FBW) taileron actuator, since this component is available for experimental verification of the results. This actuator was designed spe- cifically for a digital fly-by-wire aircraft and is representative of actuators found on aircraft currently in service.

The nonlinear model used in the analysis is derived from earlier work [23, 241 in which a typical actuator was tested by the manufacturer [25]. A SIMULINK model of the nonlinear actuator was used for all analy- ses and, naturally, certain assumptions had to be made for the nonlinear model to operate efficiently within this environment. The model contained a large number of the nonlinearities inherent in the actuator, including the valve port flow equations, valve travel limits, valve port shaping and software rate limiting. However, the model does not include factors such as ram loading, hysteresis, friction and damping.

3 Performance boundary for nonlinear actuation system

Until recently, any analysis of an aeroservoelastic system involved only linear actuation system models. One consequence of this would be that the amplitude of any unstable structural mode would be unbounded, with

IEE Proc.-Control Theory Appl., Vol. 143, No. 1, January 1996

obvious catastrophic results. In reality, servo-hydraulic actuation systems are nonlinear in nature, there being limits to servo valve displacement, main valve displacement and main ram travel in particular. These limits can be interpreted as constraints on the main ram acceleration, velocity and displacement respectively. For any given actuator demand frequency, there will exist a demand amplitude at which the actuator performance is being limited in terms of these constraints. For example, for a low-frequency demand signal, the actuator will be limited only by the ram travel limit. Once this limit has been exceeded any increase in demand amplitude will not be met due to the physical limit on the ram extension. For a higher frequency demand signal, a point exists where the main valve becomes fully open, equating to the main ram travel- ling at its maximum rate. Thus ram displacement would be limited by this maximum rate and the time period of the demand signal.

- a,-21 ' ' ' ' ' ' ' ' ' 1 e o 002 0 0 4 006 008 010 c

tirne,s Fig.3 1 5 m m input signal (1) Rate limiter input signal (11) Rate limiter output signal

Rate limiter iput/output time response characteristics for 50Hz,

1 0" 1 o1 1 OL frequency, Hz

Fig.4 Jaguar FEW taileron actuator performance limit for nonlinear model

In addition to these performance limits which are determined by the hardware, it is common practice to incorporate rate limits in the software within the FCS so as to protect the actuator from potentially damaging demands. The effect of such rate limiting on actuator performance can be demonstrated by examining the input and output time responses of such a rate limiter in isolation as shown in Fig. 3. From the Figure, any subsequent increase in the amplitude of the input signal to the rate limiter will not result in an increase in the amplitude of the output waveform. It is this output signal from the rate limiter that forms the demand signal for the actuator. Consequently, the rate-limiting in the software is responsible in part for limiting the performance of the actuator.

The performance limit for a taileron actuator on the FBW Jaguar is shown in Fig. 4. This Figure represents output amplitudes obtained from a nonlinear actuation system model for increasing demand amplitude. From the Figure it is clear that the responses for increasing input amplitudes converge on a single boundary representing the performance limit for the actuator.

k 10' c 5 ,} . , , . . . . . . . . ..\

" E -2 e 10

1 n-3 I " 100 1 0' 1 o2

frequency, Hz

Fig. 5 (i) Main ram travel limit alone (ii) As for (i), but including servo valve travel limit (iii) As for (ii), but including main valve travle limit (iv) As for (iii), but including software rate limiting

Performance limit of actuation system model

The contribution of the differing hardware and software limits to this performance boundary can be demonstrated as shown in Fig. 5. These results represent the performance boundary for the nonlinear actuation system model as each of the hardware and software limits are introduced into the model.

From the above results, the major cause of the performance limit of the actuation system is the software rate limiting of the input signal. This is to be expected as the purpose of the software rate limiting is to prevent main valve saturation. Considering the boundary for the case of main ram travel limit alone, the results show that there is no change in the boundary with input frequency. As the servo valve and main valve limits are introduced, the results demonstrate that the performance limit is lowered from the simple travel limit case, as would be expected.

1 0" 10' 10' frequency. Hz

Fig. 6 Jaguar FB W taileron actuator performance limit

Experimental single-input frequency response tests on a representative actuation system have confirmed the results obtained from simulation as shown in Fig. 6, where the output amplitudes for the actuator main ram have been obtained for increasing demand frequency and amplitude. In addition, the performance boundary as predicted from simulation is included in Fig. 6. It can be seen that there is good correlation between the experimental and simulated results.

3 IEE Puoc.-Control Theory Appl., Vol. 143, No. I , January 1996

4 iction of limit cycles

The nonlinear nature of the actuation system can be seen to play a vital role in the propagation of aeroservoelastic signals around the aircraft closed-loop. The existence of a performance boundary to the actuator as shown above, and the nonlinear nature of the actuation system in general, indicates that an unstable aeroservoelastic interaction would result in a limit cycling condition.

x(t) c(t) G,(jw,E ) G,(Jw) Gp(jw)

rate Limit actuator load '1'

Fig.7 Bloclc diagram of the actuator test rig

ie r t ia

Fig. 8 Schematic of test rig

The existence of limit cycles in a nonlinear system can be predicted from the characteristic equation of a system with the nonlinear elements replaced by their describing functions [26]. Consider the system as shown in Fig. 7. Such a system is representative of the test rig available for experimental verification of the results. The test rig consists of a Jaguar FBW taileron actuator driving a load made up of a pivoted mass-spring system as shown in Fig. 8. The LVDT shown in Fig. 8 was set so as to measure the deflection of the spring, as signified by the signal 6(t) in Fig. 7. This signal was then combined with the actuator ram deflection, x( t ) within the rig controller before feedback to the actuator input signal. The two gains in the feedback path, K, and K2, were included so as to vary the characteristics of the system in order to generate a suitable structural interaction. Although the rig itself is simple in comparison with an entire aircraft system, it exhibits many characteristics of the full system. The nonlinear actuator is driving a load which exhibits a structural mode. This structural mode is sensed by the control system and fed back to the actuator input as in the aircraft.

To predict the existence of limit cycles within the system, it is necessary to replace the nonlinearities within the system by their describing functions. To simplify the analysis, the only nonlinearity that will be consid- ered is the software rate limit function. As a result, the actuator itself is assumed to behave linearly down- stream of the rate limiter. As discussed earlier, the pur-

pose of the rate limiter within the control software is to prevent saturation of the actuator main valve and as such, the rate limiter is the main nonlinearity which limits the performance of the actuator.

Considering the block diagram for the test rig as shown in Fig. 7, it is possible to derive the characteristic equation of the system to give

(1) 1 + Gn(~m,E){Gi(~w)K2 - G I ( J ~ ) K I K ~

f Gi(aw)Gz(~w)KiK2} = 0 The solution of the characteristic equation is the limit cycle condition, which can be predicted from a re- arrangement of eqn. 1 giving

Provided that the describing function for the rate limiter can be derived, and that the linear components within the system can be adequately modelled, it will be possible to predict the existence of limit cycling conditions within the system.

I

Tc w w T t, JL -

time,s Fig.9 Input/output characteristics of rate limit function

To derive a describing function for the rate limit, consider the inputloutput characteristics of such a rate limiter as shown in Fig. 9. In this case, the characteristics are shown after a length of time sufficient for a steady relationship to be achieved. In addition it is assumed that the input signal is a pure sinusoid and exceeds the rate limit sufficiently such that the output waveform is triangular.

From Fig. 9, the amplitude of the triangular output waveform can be derived as

y = - 7i.P

2w ( 3 ) where p is the maximum rate as shown in Fig. 9. Fou- rier analysis of such a triangular waveform, of amplitude Y, reveals that the amplitude of the fundamental is

4P YI, = - W T

(4)

with an infinite number of other harmonics. Neglecting these higher order harmonics, the gain of the rate limiter can be expressed as

(5) for an input sinusoid of the form as shown in Fig. 9. The generation of higher harmonic components in the actuator output signal, along with the possible generation of low-frequency subharmonic components has been discussed in detail in [17].

To derive the phase change of the rate limiter, con-

IEE Proc -Control The0i.y Appl, Vol 143, No 1, January 1996 4

sider the input/output relationship of Fig. 9 once again. From the Figure, it can be seen that the phase lag between the two signals can be represented by the time delay, z. To obtain an expression for this time delay, it is necessary to locate the time at which the input signal is equal to the output signal, such that

;.P E sin wt, = - 2w

Considering that tm occurs after t = d 2 0 , the above equation can be solved for tm such that

7 r 1 t , = - - -asin (&) w w Therefore, the time delay, z, can be expressed as

r = W 1 { 5 - asin (&) }

(7)

Finally, since the phase lag between the input and output signals can be expressed as

Therefore the describing function of the rate limiter under the assumptions applied earlier is

iG,(jw, E ) = - r w (9)

LG,( jw, E ) = - - - asin - {; (22d)} (11) The above expressions allow the prediction of the

existence of the limit cycling condition from the solution of the characteristic equation for the system as given in eqn. 2. This is provided that the linear elements of the system can be accurately modelled. These two linear elements consist of a linearisation of the actuator dynamics, which can be represented by a fifth order transfer function [27], and a linear model of the load. The load can be represented by a second order transfer function matched to frequency response data from the rig itself giving

5221

G 2 ( S ) = s2 + 4.3s + 5221 One method of solution of the resulting characteristic

equation is to plot both sides of eqn. 2 and find the intersection of the two loci. Unfortunately, the describing function for the rate limiter is both frequency and input amplitude dependent, resulting in an infinite number of loci. A solution to the characteristic equation can be found iteratively. The solution for gain val- ues of K1 = 5 and K2 = 0.1 is shown in Fig. 10.

2 01 I

rate limiter (wl=ll 1 Hz)

solution point

E=5 8mm

-2 0 -1.5 -1 0 -05 0 0 5 real

Solution of the characteristic equation for kl = 5 0, k2 = 0 I

W1:ll 1 HZ

-2 0

Fig. 10

The locus representing the linear elements of the characteristic equation is plotted for varying frequency.

IEE Proc -Control Theory A p p l , Vol 143, No 1, January 1996

The second locus representing the rate limiter element of the characteristic equation is plotted for a single value of frequency, ml, and varying error signal amplitudes, E.

Since the two loci intersect at a frequency of m1 on the loci representing the left hand side of eqn. 2, this represents the solution of the characteristic equation. In the case of this simple system, there is only a single solution of the characteristic equation. If the load was of much higher order, with many structural modes for example, there may well be several solutions of the resulting characteristic equation. As a result, the frequency of the limit cycle within the simple system as shown in Fig. 7 can be predicted as being 11.1 Hz with an amplitude, E, of 5.8mm.

Time domain simulations of the system indeed resulted in a limit cycle condition arising. From the simulation, the frequency of the limit cycle was 11.1 Hz with an amplitude of E = 6.8mm. The error between the predicted amplitude and the amplitude seen in the simulation is due to the differences between the linear actuator model used in the prediction and the true nonlinear actuator model used in the time domain simulation.

-101 " " " ' 1 ' 9 5 9 6 9 7 9 8 9 9 100

Comparison of simulated and experimental limit cycle time,s

Fig. 11 (I) Simulation results (11) Experimental results

The results from the simulation and experimentation are compared in Fig. 11. It can be seen that the simulation provides a good match with the amplitude and frequency of the limit cycle obtained from the rig test.

It should be noted, however, that in the case of Fig. 11, both simulation and experimental results included the presence of a low-frequency demand signal. This low-frequency signal is the excitation signal used to initiate the limit cycle on the experimental rig.

To conclude, application of limit cycle prediction techniques to the example system has shown to provide a good estimate of the limit cycle frequency and amplitude. Time domain simulation of the system has also shown that the limit cycle can be accurately predicted from time domain modelling.

5 Application of limit cycle prediction to aircraft systems

The above example has demonstrated how it is possible to obtain a good approximation of the frequency and amplitude of a limit cycle present in a representative aeroservoelastic system. In reality, the aircraft system is significantly inore complex than the example used here, and thought must be given to an application of such prediction techniques in the case the aircraft. Inevita- bly, there is a high degree of uncertainty present in the

5

modelling of the flexible aircraft [IO]. Although ground vibration tests provide reliable measurements of the open-loop gain of the aircraft system, there exists a large degree of uncertainty in the phase response of the system. This is due in part to uncertainties in the modelling of the unsteady aerodynamics and also in the phase relationships between the many possible signal paths that exist within a typical flight control system. At present, clearance procedures allow for this uncertainty in the phase by neglecting its influence on the stability of the system and by assuming in-phase addition of all the signal paths [lo].

If the phase response of the system cannot be relied upon, then the use of the above limit cycle prediction techniques is limited.

Consider the characteristic equation of the example system as given in eqn. 2. If no phase information is available, then the solution of the characteristic equation becomes one of gains only such that

The effect of this is that where as in the earlier case there was a single solution, there now becomes an infinite number of possible solutions. The actual solution that exists in reality is dependent on the phase relationship.

Rearranging eqn. 13, and applying eqn. 10 gives

Considering the earlier result of eqn. 4, the above result represents the maximum output of the rate limiter at any given frequency multiplied by the open-loop gain of the linear elements of the system. Consider now that

where X(w) is the maximum output of the linear actuator and rate limiter combination at any given frequency. It is, therefore, possible to apply the actuator performance limit of Fig. 4 to the prediction of the limit cycle amplitudes by substituting the performance envelope for X(w) in eqn. 14. The importance of this is that it enables the amplitudes of the limit cycles to be predicted from the performance limit of the actuator and the gain response of the remaining linear elements of the system. In addition, the linearisation of the actuator is no longer necessary.

E lo2 E

-2 10" - a C l I

frequency, Hz Fig. 12 Potential limit cycle amplitudes for test system

Applying the result of eqn. 14 to the simple system currently under consideration, and using the results of Fig. 4 enables a specification of the possible limit cycle

6

amplitudes as shown in Fig. 12. Here the amplitude of an 11.1Hz limit cycle can be seen. This amplitude matches exactly that found in the time domain simulation of the system.

h 4 actuator inboard flaperon

I L ' I

I I

I I f e e d m t filterinq

Fig. 13 Aircraft system block diagram

This method of applying the actuator performance limits to the linear system can be applied to the aircraft system in order to specify a maximum amplitude to the limit cycles that may occur in that case. Consider the aircraft system as shown in Fig. 13.

In this case, which is typical of modem aircraft of canard-delta configuration, there are three surfaces which are used by the flight control system to control the longitudinal dynamics of the aircraft. In calculating the stability of the structural modes it is common practice to assume an in-phase addition of the contributions from the three control surfaces. This is as a result of the uncertainty of the phase of the flexible aircraft once again.

frequency, Hz Fig. 14 craft

Maximum sensed structural response envelope for example air-

,,, 004 m . ;- 002

c U 0

x -002

VI -0OL

c

P r

Q

c

0 2 4 6 8 tirne,s

Fig. 15 Pitch rate time response for example aircraft system

Applying the same assumption to the prediction of the limit cycle amplitudes, it is possible to specify the maximum limit cycle amplitude at any point around the aircraft system closed loop. For example, applying the actuator performance limits as given in Fig. 4, to a model of such a flexible aircraft results in an envelope

IEE Proc -Control Theory Appl, Vol 143, No I , January 1996

of maximum structural response as sensed by the motion sensor unit. Such a maximum level of response envelope for a generic combat aircraft is shown in Fig. 14.

This maximum structural response envelope thus specifies a maximum level to the amplitude of the limit cycling condition arising as a result of the presence of an unstable structural mode.

To demonstrate this, a time response of aircraft pitch rate for the above generic aircraft system with no structural mode filters is included as Fig. 15 for a typical pilot demand input.

In this case, an unstable structural mode exists at a frequency of approximately 16Hz, resulting in a limit cycling condition of amplitude 0.047radsh at the aircraft motion sensor. This level of response for such a structural mode frequency is below the maximum predicted in Fig. 14. The fact that the actual response amplitude is less than that predicted in Fig. 14 is sim- ply that the derivation of the maximum response envelope assumed in-phase addition of the three signal paths, which would be rarely the case.

6 Alternative approach to aeroservoelastic design and clearance

To use the maximum level of the structural response to our benefit, consider a situation where it is possible to derive an upper limit of this structural response to ensure satisfactory control of the rigid body aircraft. This could be compared with the unfiltered maximum response as shown in Fig. 14 resulting in a specification for the attenuation requirements for the structural mode filters. Such an approach would therefore ensure that satisfactory control of the rigid body aircraft would be maintained at all times.

Consideration of the effects of structural signals on actuator performance [17, 181 has revealed that the software rate limiter can introduce significant phase lag into the system as a result of the presence of a high-frequency structural mode signal. Examining the response of such a rate limiter in more detail, it can be shown that a certain level of rate limiting might be acceptable in terms of rigid body control. For example, the phase response of a rate limiter subjected to a l.0mm 3Hz demand signal in the presence of structural signals of varying amplitude and frequency is shown in Fig. 16.

O r

L a - 6 0 1 6

0 2 4 6 8 10

Phase response of rate limit in the presence of 1 Omm 3Hz structural signal amplitude, mm of ram demand

Fig.16 iemand signal and structural signals of varying amplitude and frequency

From the Figure, the effect of a structural signal on the performance of the actuator under these particular input conditions can be quantified. For example, a structural signal of frequency 7.1Hz and amplitude 6 n m demand will induce approximately 10” of phase lag nto the system as a result of the rate limit being

exceeded. The application of such results is that it enables a maximum allowable level of structural noise at a particular frequency to be set. For example, suppose that it was decided that a phase lag of up to 10” could be tolerated. From Fig. 16, this would enable specification of a maximum boundary for the structural noise amplitude at the actuator input. Once such a specification had been obtained, comparison with the maximum level of response as shown in Fig. 14 would give a requirement for the attenuation requirements of the structural mode filters.

One problem with this approach however, is that such limit cycle conditions will have a very serious effect on other elements of the aircraft system. Possible consequences of limit cycles existing within the aircraft closed loop are increased structural fatigue, wear of actuator components such as valve seals, and a large level of electrical power dissipation within the flight control system. Naturally, the existence of such limit cycles is to be discouraged even if they do not pose a threat to the satisfactory control of the rigid body aircraft.

A compromise may be adopted however. Such a compromise would be that the filters are designed to give nominal stability of the structural modes such that limit cycle conditions would not arise. At present, these filters are designed to give a maximum open-loop gain of -9dB for the structural modes [lo]. If the attenuation was reduced from this level to -ldB for example, then there would be a significant saving in the phase lag introduced into the system by these filters. One consequence of this action would be that any error in the modelling of the system could result in a limit cycling condition arising.

Consider a situation where an error in the modelling has indeed resulted in an in-flight limit cycle condition arising. Such a condition would be more likely at high aircraft incidence where the FCS gains are highest. Pro- vided that such a possibility has been investigated in terms of the limit cycle amplitude and its effect on the actuator performance, rigid body stability will be maintained. The aircraft incidence could then be safely reduced, whereupon the limit cycle would dissipate as a result of the reduction in FCS gain.

If such an in-flight interaction were encountered, then it would be possible to correct the flexible aircraft model accordingly, redesigning the structural mode filters so as to maintain the -1 dB maximum open-loop gain.

7 Conclusions

The consequences of the nonlinear nature of a typical servohydraulic actuation system on the aeroservoelastic phenomena has been investigated. In the case of an unstable structural mode, it has been shown that the resulting limit cycling condition can be predicted. In the presence of phase uncertainty however, this prediction is limited to that of a prediction of the amplitude of any resultant limit cycle. Such amplitude prediction can be used to advantage however in that in enables investigation of the effects of such a signal on actuator performance in response to rigid body stabilisation signals. This could, in turn, lead to a reduction in the attenuation requirements of the structural mode filters resulting in a significant saving in the inherent phase lags of such filters at rigid body frequencies.

‘EE Proc-Control Theory Appl., Vol. 143, No. I , January 1996 I

The authors gratefully acknowledge the support for the project provided by British Aerospace plc, Military Aircraft Division, Warton, England.

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17 TAYLOR, R , PRATT, R W , and CALDWELL, B D ‘The effect of actuator nonlineanties on aeroservoelasticity’ Paper AIAA-94-3628, AIAA conference on Guidance, navigation and control, Scottsdale, 1-3 August 1994

‘Actua- tor performance changes in the presence of aeroservoelastic feedback signals’ Paper AIAA-95-3 180, AIAA conference on Guidance, navigation and control, Baltimore, 7-9 August 1995

19 BRESLIN, S G., and GRIMBLE, M.J . ‘The control of an ASTOVL (advanced short take-off and vertical landing) aircraft using H, methods’ IEE colloquium on Multivariable methods for flight control applications, London, May 1994, Digest 19941132

20 TAYLOR, R , PRATT, R W , and CALDWELL, B D. ‘An overview of the effects of aeroservoelasticity on the design of flight control systems’ IEE colloquium on Multivariable methods forflight control applications, London, May 1994, Digest 19941132

21 MERRIT, H E ‘Hydraulic control systems’, (J Wiley & Sons, New York, 1967)

22 GUILLON, M: ‘Hydraulic servo systems’, (Butterworths, Lon- don, 1969)

23 STIRLING, B ‘Jaguar FBW actuation system mathematical model’ British Aerospace PLC company report, AEI399 (unclassified)

24 BECK, J.. ‘Jaguar fly-by-wire taileron actuation system model matched to frequency response and impedance’ British Aerospace PLC company report, AEI485 FCSDGR 210 (unclassified)

25 BECK, J . ‘Jaguar FBW taileron actuation system tests at Dowty Boulton Paul’ British Aerospace PLC company report AEi486 FCSDGR 21 1 (unclassified)

26 NAGRATH, 1.J , and GOPAL, M.: ‘Control systems engineering’ (J Wiley & Sons, New York, 1982)

27 BECK, J ‘Jaguar fly-by-wire linear taileron actuatin system model for frequency response and impedance’ British Aerospace PLC company report, AEl504 FCSDGR 245 (unclassified)

16 MAHARAJ, D Y , ROSENBERG, G , and COWLING, D A

18 TAYLOR, R , PRATT, R W , and CALDWELL, B D

IEE Proc.-Control Theory Appl., Vol 143, No. I , January 1996

alternative approach to aeroservoelastic design and clearance

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