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Flight control system design considering rate saturation
Aerospace
Science
and Technology,
1998,
no. 4, 265-215
Flight control system design considering
rate saturation Holger Duda*
DLR, Institute for flight mechanics, German Aerospace Center, 38022 Braunschweig, Germany (Received
Duda H., Aerospace
Science
and
Technology,
4 September
1997, accepted 12 November
1998, no. 4, 265-275.
In this paper the design process of modem flight control systems is discussed with respect to pilot-inthe-loop oscillations (PIO) due to rate saturation. At first the PI0 phenomenon is characterised briefly. Well-known design criteria with respect to flight control system stability and handling qualities based on linear aircraft models are presented. A new design criterion considering the nonlinear effects of rate limiting is described. The criterion is based on the ‘open loop onset point’ (OLOP) of the rate limiter in a Nichols chart. The background for the development, and the verification of the OLOP-criterion are presented briefly. In order to demonstrate the applicability of the OLOP-criterion in the design process a simple example is examined. The flight control system parameters of a basically unstable aircraft are optimised with respect to the linear design criteria and the OLOP-criterion. Two alternative designs were discussed, a ‘nominal design’ and a ‘low gain design’ with reduced feedback loop gain. The low gain design is favourable with respect to rate limiting in the feedback loop of the flight control system, which is proved by nonlinear simulations in the time domain. Finally, the utilisation and the design of a rate limitation in the forward path of the flight control system are discussed. 0 Elsevier, Paris
Abstract
flying qualities / flight control system / aircraft-pilot nonlinear system / rate limiting Zusammenfassung
coupling / pilot-in-the-loop
* Correspondence
oscillations /
Auslegung von Flugsteuerungssystemen bei begrenzter Stellgeschwindigkeit. In diesem Papier wird der AuslegungsprozeB modemer Flugsteuerungssysteme im Hinblick auf Flugzeug-Pilot-Kopplungen (Pilot-in-the-Loop Oscillations, kurz PIO) bei Stellgeschwindigkeitsbegrenzungen diskutiert. Zunachst wird das PIO-PhSinomen kurz charakterisiert. Die bekannten Auslegungskriterien fur Reglerstabilitat und gute Flugeigenschaften basierend auf linearen Flugzeugmodellen werden vorgestellt. Ein neues Auslegungskriterium zur Bertlcksichtigung nichtlinearer Effekte bei Stellgeschwindigkeitsbegrenzungen wird beschrieben. Das Kriterium basiert auf dem Aktivierungspunkt der Stellgeschwindigkeitsbegrenzung (Open Loop Onset Point, kurz OLOP) im Nicholsdiagramm. Der EntwicklungsprozeB und die Verifikation des OLOP-Kriteriums werden kurz dargestellt. Zur Demonstration der Anwendbarkeit des OLOPKriteriums im EntwurfsprozeB neuer Flugsteuerungssysteme wird ein vereinfachtes Beispielflugzeug betrachtet. Die Parameter des Flugsteuerungssystems fur ein instabiles Basisflugzeug werden unter Berticksichtigung der linearen Auslegungskriterien und des OLOP-Kriteriums optimiert. Zwei alternative Varianten werden untersucht, eine Auslegung mit hoher und eine mit reduzierter Rtickftlhrverstarkung. Die Auslegung mit reduzierter Rtickftihrverstarkung zeigt sich als vorteilhaft im Hinblick auf Stellgeschwindigkeitsbegrenzungen, wie durch nichtlineare Simulationen im Zeitbereich gezeigt wird. AbschlieBend wird die Auslegung einer Stellgeschwindigkeitsbegrenzung im Vorwartszweig der Flugsteuerung diskutiert. 0 Elsevier, Paris Flugeigenschaften / Flugsteuerungssystem I Flugzeug-Pilot-Kopplung Stellgeschwindigkeitsbegrenzung
Aerospace
1997)
and reprints
Science and Technology,
1270-9638,
98/04/O
Elsevier,
Paris
I nichtlineares System I
266
H. Duda
Nomenclature angle of attack amplitude elevator deflection pitch stick force frequency at - 180” phase angle frequency response elevator per stick force phase angle crossover phase angle actuator transfer function prefilter transfer function aircraft transfer functions aircraft gain (control sensitivity) flight control system feedback gains Hight control system forward path gain flight control system feedback loop gain pilot gain describing function elevator per stick force rate limiting element describing function vertical load factor phase rate at - 180” phase angle pitch rate commanded pitch rate maximum rate maximum rate in the feedback loop maximum rate in the forward path Laplace variable pitch angle input amplitude of the rate limiter frequency crossover frequency closed loop onset frequency onset frequency short period frequency short period damping
1. Introduction The high demands on performance and handling qualities of modern aircraft within a large flight envelope require the implementation of complex electronic flight control systems. Especially for highly manoeuvrable military aircraft, this control technology has opened completely new opportunities
for optimisation. In the latest military developments basically unstable aircraft designs were utilised, such as the US American F-22, the Swedish JAS39 or the European EF2000. A basically unstable design can provide benefits in flight performance up to . 18 % [17]. Also in the civil aircraft branch the use of modern electronic flight control systems provides a large potential for improvements. The development of the Airbus A320 can be regarded as a milestone in this process; a pure electronic link was used for the ailerons and elevators for the first time in an operational civil transport aircraft [2]. The use of this fly-by-wire technology allowed a stronger influence on the flight characteristics, for instance the aircraft behaviour is less dependent on the flight condition and centre of gravity location. It is even possible to achieve a commonality of flying characteristics within a certain aircraft family, which is a profitable element considering pilot training and certification aspects [7 ]. Besides the economic benefits due to better flight performance, the use of electronic flight control systems contributes to increased flight safety; improved handling qualities and flight envelope protection functions strongly reduce pilot workload; fault detection and isolation algorithms are used in combination with highly sophisticated reconfiguration concepts to cope with system failures, such as the loss of a control surface. However, in spite of all these safety improvement a significant handling qualities problem came up again with the introduction of electronic flight control systems: pilotin-the-loop oscillations or pilot-induced oscillations (PIO). PI0 can be considered as a closed loop destabilisation of the aircraft-pilot loop and it occurs often under situations when the pilot proves to be unable to adapt himself to a sudden change of the vehicle dynamics during a high demanding flying task (the trigger) [I, 141. PI0 are not pilot failures, but result from a failure in the design process. A significant correlation was found between PI0 incidents reported during the complete history of aviation and rate saturation in flight control systems. Nearly all catastrophic PI0 cases were associated with rate saturation, such as the crashes of the YF-22 in 1992 and two JAS39 prototypes in 1989 and 1993. In the past, most PI0 cases occurred within the fighter aircraft branch, but also civil airliners with electronic flight control systems have demonstrated PI0 problems [ 131. In spite of the very strong correlation between PI0 susceptibility and rate limiting, no design criteria are established, which address nonlinear rate saturation effects. Until recently several design criteria were available in the frequency and time domains, but they were only validated on the basis of linear aircraft models. Although the time domain criteria can cope
Flight control system design considering Auslegung van Flugsteuerungssystemen
rate saturation/
with nonlinear systems, they were unable to explain the rate saturation effects sufficiently. In the time domain one can only recognise that an additional time delay due to rate limiting occurs. But a method to predict the magnitude of this additional time delay in the design process has been lacking. Frequency domain criteria provide a better understanding of the dynamic behaviour of the system, but they are limited to linear systems. Therefore, a new criterion is required, which combines the advantages of ‘frequency domain thinking’ with the possibility of treating nonlinear systems. This deficiency in the design requirements was the background for starting a research project at DLR. On the basis of a closed loop system describing function analyses a new design criterion with respect to rate saturation in flight control systems was developed [3-51. In this criterion the ‘open loop onset point’ (OLOP) of rate limitation is assessed in a Nichols chart. Within the framework of a German/Swedish cooperation between DLR and FFA the OLOP-criterion was validated on the basis of new experimental data [6, 81. In this paper the OLOP-criterion is presented and its application potential to flight control system design is discussed on the basis of a representative example aircraft.
“PI0 are sustained or uncontrollable oscillations resulting from the efforts of the pilot to control the aircraft.” Several different types of oscillations forced by the pilot are known, which differ in frequency and amplitude. In the roll axis high frequency low amplitude oscillations are known, called ratcheting [141. Such oscillations basically occur at frequencies above 10 rads’ and are caused by interactions between the cockpit controls and the pilot’s neuromuscular system. They are characterised by pilot comments about some potential PI0 problems, but will not result in a catastrophe. The corresponding phenomenon in the pitch axis is called bobble, which is mainly caused by structural modes. Lower frequency high amplitude oscillations represent the common meaning of real PIO. For the understanding and analysis a further classification was recently introduced [ 111. Category I PI0
Essentially linear aircraft-pilot oscillations Category II PI0 Quasi-linear aircraft-pilot oscillations with significant rate or position limiting Category III PI0 Nonlinear aircraft-pilot oscillations with transitions 3. Linear
2. Pilot-in-the-loop
oscillations
267
bei begrenzter Stellgeschwindigkeit
(PIO)
The PI0 problem has been known for years, while the introduction of full-authority, digital fly-by-wire flight control systems has increased the potential for PIO. For the understanding and analysis of PI0 three main elements have to be considered [ 121: - the pilot, - the aircraft, and - the trigger. The weakest point in the analysis of PI0 are pilot models, since the pilot behaviour is highly nonlinear and very complex. However, pilot models are available for special pilot behavioural patterns [ 121. The aircraft is represented in this scenario by the linear aircraft and flight control system dynamics. The trigger can have different forms, for example a nonlinear effect in the flight control system or a transition in the pilot behavioural pattern, but it always causes a sudden change in the closed loop dynamics of the aircraftpilot system. This may lead to a strong misadaptation of the pilot, but it has to be stressed again that PI0 is not a result of a pilot failure. The following definition can be found in the US American military specifications [ 11, 161. 1998. no. 4
design criteria
(Category
I PIO)
The main effects causing Category I PI0 are well known as excessive lags due to time delays and various digital filter dynamics in the flight control system. These effects lead to a ‘high frequency phase rolloff in the frequency response. The established frequency domain PI0 criteria such as the ‘phase rate criterion’ [ 181 are well suited for predicting Category I problems. Several flight control system design criteria are established based on linear systems. One can roughly distinguish between the following two groups of design criteria: - flight control system stability - handling qualities requirements. The first group of design criteria concerns the stability of the aircraft system (in this context, ‘aircraft’ means the basic aircraft including flight control system). Figure 1 shows the typical boundaries for phase and amplitude margin in a Nichols chart [2]. The open loop aircraft frequency response has to be checked against this criterion. For example in the pitch axis the loop h,as to be opened at the elevator actuator. The second group of design criteria concerns the handling qualities of the piloted aircraft. Most common handling qualities requirements are collected in the US American military specifications [ 15, 161. The requirements are graded regarding the aircraft class and the flight phase category, such as the following:
268
H. Duda
gradient for nIL,/a. for Category A flight phases [ 151. These boundaries can be interpreted by the ‘control anticipation parameter’ (CAP), which is defined as the ratio of initial pitch acceleration to quasi-stationary vertical load factor. The requirements for the short period damping are summarised in table I [ 1.51.
20 E * a x
10
5
critical point, 0
Table
I. Short
period
-10
Category Level
-20 -150
-100
chase angle [deg] Figure chart.
1. Boundaries
Category Category Category
damping
for phase
and amplitude
margin
in a Nichols
A
Non-terminal flight phases, precision tracking required, e.g. air-to-air combat B Non-terminal flight phases, without precision tracking, e.g. cruise C Terminal flight phases, accurate flight path control, e.g. approach and landing
The levels of handling follows:
qualities
are defined as
inlnx
I
0.35
1.3
0.30
2.0
2
0.25
2.0
0.20
2.0
3
0.15
-
0.15
The main cause for Category I PI0 - the high frequency phase rolloff - is not sufficiently addressed by the short period criteria. Therefore, special high order system criteria were developed, such as the ‘phase rate criterion’ [18]. Figure 3 shows the calculation of the phase rate parameter PRlso for the frequency response of the pitch angle 0 by pitch stick force F,,. Additionally, the boundaries Phase angle over frequency
frequency
requirement
(Category
(O/F,,)
frequency a) Determination
0
0.5
1O” b) Phase
period
B
c Llllll
“-
Figure 2. Short phases).
Category
c map
In the pitch axis the short period dynamics (frequency and damping) are relevant for the aircraft manoeuvrability. Figure 2 presents the requirement for the short period frequency depending on the
n,la [ghad]
A, C
&in
Level 1 Satisfactory without improvements Level 2 Adequate to accomplish the mission Level 3 Controllable, excessive pilot workload
10
requirements
A flight Figure
3. Phase rate
criterion
f180
of phase
1
rate PR18c
f, 80 WI
rate criterion
1.5
boundaries
2
Flight control system design considering Auslegung von Flugsteuerungssystemen
rate saturation/
for the phase rate PR1sO and the frequency flsO at - 180” phase angle are presented. Physically, the phase rate parameter has the following significance: a too high phase rate means that with a small increase in frequency a strong additional phase delay occurs, which is always a problem in a closed loop system. It is highly correlated with the additional time delay in the complete system.
4. The OLOP-criterion
(Category
269
bei begrenzter Stellgeschwindigkeit
II PIO)
The nonlinear effects of rate saturation in the flight control system are not considered within the criteria presented above, since they are only valid for linear aircraft models. Figure 4 shows two typical positions, where rate limiters are installed as software elements in the flight control systems of highly augmented aircraft. The limiters in the feedback loop of the flight control system have the task to protect the actuators against overload. Otherwise the oil supply of the actuation system could break down. The limiters in the forward path of the flight control system are installed to protect the system against high input rates by the pilot, preventing a saturation of the feedback loop rate limiters. This protection is particularly important for unstable highly augmented aircraft.
limiting
situation [5]
The onset frequency of the rate limiter is the ratio of the maximum rate R and the current input amplitude ~,l<, of the rate limiter
For the calculation of the describing function of a rate limited closed loop system a special method was developed. The application of this method to a highly augmented aircraft with a rate limitation in the feedback loop is presented in$gure 5. In that case the closed loop system describing function is characterised by a jump phenomenon after rate limiting onset, which can be recognised in a Nichols chart as a phase jump. In the presented example offigure 5, the phase jump leads to a dramatic loss of phase and amplitude margin indicating the potential for an instability of the closed loop system. This instability was verified by a nonlinear &n&lation in time domain [3].
Open Loop Frequency Response q/q,
,/---
,,, pilot
15c
[- e]
5.1 radisec
: rate limiter
4 >** Figure Hight
4. Typical implementation control systems.
of software
rate limiters
‘, :
in modern
-10 -
To properly analyse closed loop systems with rate limiting, nonlinear system theory has to be applied. In contrast to linear systems, where a complete system theory is available, nonlinear systems can be analysed only for special cases. Several methods for treating nonlinear systems are known for this purpose, such as the theory of Lyapunow, the Popow criterion, the phase plane method, the numeric simulation in time domain, and the describing function technique [9, lo]. The latter is rated as the most promising tool to analyse PI0 due to rate limiting in flight control systems. A nonlinear system can be represented by a quasilinear system using the describing function and the remnant. The describing function can be interpreted as the quasi-linear frequency response for sinusoidal inputs. The remnant is a high frequency signal, which can be neglected if the linear part of the system has low pass character. The describing function was calculated using a Fourier series for the fully developed rate
-15 -20 -300
_,,A
,’ I’
,’ /’
/’
,’ .---- describing function linear open loop onset l point (OLOP) _I
-250
Figure 5. Jump phenomena augmented aircraft.
-150 -100 phase angle [deg]
-200
due
to
rate
limiting
onset
-50
in highly
This describing function analysis was the background for the development of a new design criterion based on the ‘open loop onset point’ (OLOP) of the rate limiter in the Nichols chart. In$gure 5, the OLOP can be identified as the point where the phase jump starts. The OLOP is defined as the frequency response value of the open loop aircraft system at the closed loop onset frequency cZ,,,,~,~~. The latter is the frequency where the rate limiter is activated for the first time for maximum pilot input amplitude (worst case). A
270
simplified method was derived to calculate the OLOP without utilising the describing function technique, but extended to the analysis of PIO: I. Definition
of a simple (high) gain pilot model,
2. Calculation of the ‘linear’ closed loop frequency response from the stick input to the input of the rate limiter (in the pitch axis F$,q), 3. Determination of the closed loop onset frequency ^ wonset considering stick and control surface limits. 4. Calculation of the relevant open loop frequency response and separation into amplitude A (w) and phase angle @ (w),
15 5. 2. $10 .=3 -5 E a5 0 -
-10
5. OL()P = [@(4,nset)rA (&,set)]. The definition of the pilot model is now treated more in detail. The pilot model has to be adjusted to the linear aircraft dynamics, which means that the pilot has adapted himself to an aircraft behaviour without rate saturation. It is recommended that simple gain pilot models be used since the pilot usually reacts as a simple gain during fully developed PI0 (‘synchronous precognitive behaviour’) [ 121. The gain is adjusted based upon the linear crossover phase angle of the open loop aircraft-pilot system @,. A gain spectrum from Qc = 110” (low pilot gain) up to Q’,, = -160” (high pilot gain) should be used. This gain spectrum can provide the following results: if an aircraft is rated as Category II PI0 prone by an OLOP based on a low gain pilot model, PI0 is very likely; if an aircraft is rated as Category II PI0 free based on a high gain pilot model, PI0 is very unlikely after rate limiting onset. The OLOP-criterion was verified by evaluation of several different configurations from roll axis databases with rate limiters in the forward path and feedback loop of the flight control systems [5]. Aircraft models from US American flight test programs were used (LATHOS: lateral high order system; F-18 inflight simulation; YF-16 first flight). Figure 6 presents the correlation between the OLOP-criterion and the Category II PI0 potential based upon nonlinear simulations with pilot models. It appears that the OLOP-criterion is applicable to both forward path and feedback loop rate limiters using the same PI0 boundary. But the feedback limiters provide a much stronger Category II PI0 potential, especially for high feedback loop gains. Category II PI0 due to rate limiters in the forward path is mainly possible for very high pilot gains or extremely low maximum rates. More details on this subject are presented in reference [5]. New experimental results utilising ground based simulations indicate the validity of the OLOP-criterion to predict Category II PI0 [S].
0 LATHOS, no PI0 @ LATHOS, PI0 possible . LATHOS, PI0 0 F-18, no PI0 0 F-18, PI0 possible + F-18, PI0 a YF-16, no PI0 A YF-16, PI0 possible / A YF-16, PI0
-5
-15
Figure
-180 6. Verification
-160
-140
-120 -100 phase angle [deg]
of the OLOP-criterion
5. Flight control
[S]
system design
The design of new flight control systems is usually performed on the basis of linear aircraft models at several reference points within the entire flight envelope. The nonlinear effects of rate limiting are commonly investigated in a second step, since the rate limitation is not installed due to an optimisation of the flight control system, but due to safety aspects. The importance of considering the rate saturation effects as early as possible is discussed in the following. Modern fighter aircraft are designed basically unstable with a time to double amplitude well under I s. They are completely uncontrollable by the pilot without the flight control system, but the augmented aircraft must have Level 1 handling qualities. In order to achieve this, control power is required, which has classically been achieved with increased control surface sizing. This is in direct conflict with the performance of the aircraft resulting in smaller control surfaces that must move very rapidly. This explains the occurrence of several PI0 cases caused by rate limiting of modern highly augmented aircraft. It is obvious that the design process has to be a compromise between maximum aircraft performance and flight safety regarding flying qualities problems due to rate saturation.
5.1. A simple design example The OLOP-criterion can be utilised within the design process of new flight control systems to optimise the parameters with respect to performance and flight safety. This will be demonstrated by a simple design example, which covers several aspects
Flight control system design considering Auslegung van Flugsteuerungssystemen
rate saturation/
271
bei begrenzter Stellgeschwindigkeit
basic aircraft:
actuator:
prefilter:
with (5; 0) = (s2 + 2 @IS + 02), (a) = (s + 4 Figure
7. Structure
of
a simple
design
example
of the design process with respect to the rate saturation problem. In practice, additional topics have to be taken into account. Figure 7 shows the structure of the simple design example including typical parameters to be optimised. An unstable second order aircraft model (only short period mode) is considered. The gain K,, represents the aircraft control authority. A second order linear actuator model is used. A software rate limitation of n% = 100” .s-i and an amplitude limitation of 30” are assumed for the elevator. The maximum stick force is limited to 80 N. These parameters are prescribed by hardware requirements. The following flight control system parameters can be optimised with respect to the requirements. The feedback gains K, and K, have to be adjusted in order to attain the required short period characteristics (frequency and damping). The feedback loop gain K% can be adjusted in order to define the bandwidth of the system. The feedforward gain KR is responsible for the steady state gain (sensitivity) of the augmented aircraft. The prefilter time constants have to be optimised with respect to the phase rate criterion. The presented values were obtained from a manual optimisation and will not be changed any more in the following examination. The maximum rate of the limiter in the forward path RR has to be defined, so that the feedback loop rate limiters can not be saturated anymore. 5.2. Linear
design
In the ‘following design process a gradient n, /o = 10 g . rad-i is assumed. According to jigure 2 and table I the following values for short period frequency and damping are required for Category A flight phases to achieve Level 1: Frequency: 1.7 rad . s-r 5 wSp 5 6 rad . s-l IYYX. no 4
Damping:
6, 2 0.35
A manual optimisation provided the feedback gains for unity feedback loop gain (Kfb = 1): K, = 0.6s
K,, = 2.5
This leads to the following short period characteristics: W sp = 5.9 rad . s-l es,, = 0.7” With Ke = KK = K,, = 1 this configuration is referred to as the ‘nominal design’. It is obvious that this design has a large potential for feedback loop gain reduction, since the short period frequency is at the upper limit and the damping is much higher than the required value. The effects of feedback loop gain reduction are presented in jipre 8a with respect to the linear design criteria. Regarding the short period frequency and damping requirements a feedback loop gain reduction to KB, = 0.2 would be possible, while still staying within the requirements (figure 8~). But, this would lead to a very poor design with respect to the phase rate criterion (Level 3 not fulfilled) (figure 8b). In order to achieve a Level 1 design a feedback loop gain of about Ka = 0.68 is required to have a sufficient margin to the Level 2 boundary. This configuration is referred in the following as the ‘low gain design’. The loop stability criterion shows that in both cases (Ke = 1 and Kfb = 0.68) sufficient phase and amplitude margin are available (figure 8~). 5.3. Considering
rate limitation
Based upon the linear design criteria two alternatives were presented above, which meet the linear design requirements. The next step is to consider the effects of feedback loop rate limitation (assuming no forward path rate limitation at this stage, Rff = x). For the application of the OLOP-criterion, at first the closed loop onset frequency ij,,,,t has to be determined. For
272
H. Duda
60 3 s 40 3 "a E m 20
’
frequency [radkec] 4.g S6
lo
a) Closed loop onset frequency -I1
I
-4
-2
O
Re(s)
I 2
iii 91
a) Short period frequency and damping
I
t s 5 (
9 5.6
Kfb = 0.68
-1
-180
-140 -100 phase angle [deg]
b) OLOP-criterion (flight control system loop) Figure 9. Evaluation of the OLOP-criterion dashed: nominal design).
b) Phase rate criterion
(solid:
low gain design:
limiter is activated first time, when the frequency response curve crosses a straight line with a slope of -20 dB.decade-’ crossing zero dB in the maximum rate (lOO”.s-’ in this case). Additionally, the deflection limit of 30” is presented in jigure 9n, which does not cross the frequency response curves of either configurations (nominal and low gain). The following closed loop onset frequencies were obtained: Nominal (Kn, = 1): ij,,,,,,,+ z 4.1) rad . s-’ Low gain (Kn, = 0.68): LjOnSI,+ Z 5.6: ratl ‘s-l. -180
-140 -100 phase angle [deg] c) Flight control system loop stability
Figure 8. Evaluation of feedback loop gain IT+),.
this purpose the pilot input (stick limiter (elevator amplitude of 80
linear
design
criteria
depending
on
the
linear frequency response from the force F,,) to the input of the rate deflection 5,) for maximum input N is utilised @gure 9~). The rate
Figure 9b presents the OLOP-parameters of the two configurations. For the nominal design the OLOPparameter is located in the critical area, which means a destabilisation of the closed loop aircraft system after rate limiting onset is predicted. For the low gain design the OLOP-criterion predicts no destabilisation after rate limiting onset. These predictions are examined by nonlinear simulations in time domain. A 3-2-l-l signal is used with maximum input amplitude of 80 N. The rate limited elevator deflection and the pitch rate are
Flight control system design considering rate saturation/ Auslegung van Flugsteuerungssystemen bei begrenzter Stellgeschwindigkeit
273
Stick Force 100 50 _ E 2
o--so-100
.. Elevator
Deflection
40 -
Pitch Rate ,Uk,
I
j 1*;~~-r,,-~~~~ ,1 -18; -140 -100 phase angle [cieg]
Figure
11. OLOP-criterion @, = -150
rate limiter,
for the aircraft-pilot loop, feedback loop deg (solid: low gain design; dashed: nominal
design). time [set]
Figure 10. Nonlinear simulation limitation in the feedback loop nominal design).
in time domain, effect (solid: low gain design;
of rate dashed:
presented in figure 10. The time histories show that rate saturation occurs in both cases, but only for the nominal configuration this leads to an instability of the closed loop system. The low gain configuration remains stable during rate saturation. This means that rate saturation in flight control systems can be harmless even for designs with basically unstable aircraft under certain conditions (OLOP-criterion is fulfilled)! This investigation shows that the low gain design is favourable although it is slightly worse with respect to the phase rate criterion than the high gain design. In practice during the flight control system design process optimum search algorithms are utilised to minimise a cost function depending on the design criteria. Within this optimisation process the linear design criteria and the nonlinear OLOP-criterion drive the flight control system parameters in opposite directions; considering only the linear design criteria would lead to a system with a high feedback loop gain, but stability problems may occur after rate limiting onset. Therefore, it is highly recommended to apply the OLOP-criterion within the early design process in order to find the best compromise between the linear performance and flight safety after rate limiting onset. One issue not addressed up to now is the pilot-inthe-loop situation. For this task the OLOP-criterion can be applied to the aircraft-pilot loop considering a pure gain pilot model based upon the linear crossover phase angle of the open loop aircraft-pilot loop QC [5]. In the design process a high gain pilot should be assumed in order to characterise a worst case behaviour. In this case a crossover phase angle gC = -150” is assumed, 1998, no. 4
Table II. Required gain design
measures
to provide
Measure
flight
safety
for
Drawbacks
Reduction of the forward gain to Kff = 0.65
path
Strong reduction in performance (agility)
Increasing the maximum to Rfh = 150Cieg.s-1
rate
Increased to faster
Increasing the (control surface K,, = 1.6 Rate limitation path
the high
aircraft gain sensitivity) to in the forward
aircraft
cost and weight actuators
due
Increased control surface size (stronger actuators, reduced performance) How to design rate?
the maximum
which is a fairly high gain. Figure 11 presents the OLOP-parameters of the two configurations. The low gain design is predicted to be PI0 free after rate limiting onset, therefore, it can be considered as the optimum with respect to the design criteria used. If the high gain design is required due to other circumstances, the following measures are possible in order to provide flight safety with respect to rate saturation in the feedback loop (table ZZ). The presented values for K,,, Rn and R.a were based on OLOP-analyses. Each measure has obvious drawbacks, except the implementation of a rate limitation in the forward path. Therefore, this option is examined more in detail. 5.4. Rate limitation
in the forward
path
The rate limiter in the forward path should be designed so that no rate limiting in the feedback loop is possible due to high input rates by the pilot. For this
274
H. Duda
task the describing function technique can be utilised. Figure 12 presents the linear frequency response from the pilot input (pitch stick force Fclg) to the input of the rate limiter (commanded elevator deflection h,) for maximum input amplitude (80 N) for the high gain design (the same as presented in jigur~ 9~). Additionally, the describing function considering the rate limitation in the forward path is plotted. In this case a maximum rate of Rtf RZ292 N SK’ is required to make sure that the feedback loop rate limiter straight line with -20 dB.decade-’ is not crossed. This means that for sinusoidal input signals it is impossible for the pilot to saturate the feedback loop limiters, which was the design aim of the forward path rate limiter.
Stick Force 100 50. z
;
O-
‘\
i
-I : ‘1 I’! ‘1 , 11
\ /
.ti
\
,
\
-50 -100~
Elevator
Deflection
40
-40 ’
Pitch Rate
2OOr
-60 g 8 40 a “a 5 20
-200 ’ 0
~o”setv+f) 0
Figure
4
6
i 10
8
time [set] &de
gain
2
.‘.I
1
12. Definition design.
Rff
nOo”*et(hb)
Figure
es
2,7 4,9 10 frequency [radkec]
of the maximum % 292 N 5-I
rate in the forward
13. Nonlinear
the forward
path.
path
simulation c 292 N
(8,
in time domain with rate limiter S- ’ ), high gain design.
in
high
In order to prove these results a nonlinear simulation in time domain is carried out. Again, the 3-2-l-l signal with maximum input amplitude of 80 N is applied figure 13). The rate limited stick force and the pitch rate are presented. Unlike the simulation without rate limitation in the forward path presented in Jigure IO, no instability occurs for the high gain design anymore for the same input signal. The advantages of rate limitation in the forward path were significant in this example, but it has to be taken into account that now PI0 can occur due to this rate limitation. To assess the PI0 potential due to rate saturation in the forward path, the ‘OLOPcriterion’ is applied to the aircraft-pilot loop. Figure 14 presents the OLOP-criterion on the basis of a high gain pilot model (@jr = -150’). The OLOP-parameter is located very near to the boundary in the dangerous area indicating some PI0 potential (figure 14). This result has to be interpreted carefully. A Category II PI0 might occur only for very high pilot gains and maximum input amplitude. The pilot has still the possibility to break out of this PI0 by reducing his gain (aggressiveness). This pilot gain reduction would not help anymore after a saturation of the feedback loop rate limiters (if no rate limiter would have been installed in the forward path), as it is demonstrated in figure 10. In that case the aircraft becomes explosively unstable. This means that rate saturation in the forward path has to be rated much less dangerous than rate saturation in the feedback loop.
@, = -150 deg
-IO-180
-140
-100 phase angle [deg]
Figure
14. OLOP-criterion
path rate - 1w.
limitation
(Rff
z
for the aircraft-pilot 292 N S-’ ). high
loop with forward gain design, @, =
The forward path rate limitation has minor adverse effects, but it helps to avoid an aircraft instability due to high pilot input rates. For a specific aircraft it has to be decided if priority is focused on maximum linear system performance for small to medium input amplitudes. In that case the high gain design with rate limitation in the forward path is preferable, although a minor PI0 potential is predicted for full stick deflections. Otherwise, if priority is focused on maximum flight safety for all input amplitudes the low gain design should be favoured. Also the potential for the other measures presented in table II should be examined for a specific design.
Flight control system design considering Auslegung van Flugsteuerungssystemen
rate saturation/
bei begrenzter Stellgeschwindigkeit
The OLOP-criterion provides significant support to find the optimum design with respect to all parameters early in the design process. This will definitely save time and money, since the effort for changing flight control system parameters is lowest in the beginning of the design process.
6. Conclusions The flight control system design process of an basically unstable aircraft with respect to pilot-in-theloop oscillations (PIO) and rate saturation effects was discussed. The applicability of the new design criterion (‘open loop onset point’, OLOP) to predict flying qualities or stability problems due to rate saturation was demonstrated. Additionally, typical linear design criteria were utilised. Considering only linear design criteria for the optimisation of the flight control system parameters may lead to a design with stability problems after rate limiting onset. Therefore, it is highly recommended to consider the OLOP-criterion early in the design process. The OLOP-criterion is a very valuable tool in order to optimise the feedback loop and forward path gains of the flight control system, the required maximum rates of the actuators, and the required control power of the basic aircraft. It was proved that the implementation of a rate limiter in the forward path of the flight control system definitely has favourable effects on flight safety. For the definition of the maximum rate in the forward path the describing function technique can be utilised. However, there may be some adverse effects, which can be examined with the OLOP-criterion.
References [l] Ashkenas I.L., Jex H.R., McRuer D.T., Pilot-induced oscillations: their cause and analysis, NORAIR Report NOR-64-143, 1964. 12) Brockhaus R., Flugregelung, Springer Verlag, Berlin, Heidelberg, New York, 1994. 131Duda H., Effects of rate limiting elements in flight control systems - a new PIO-criterion, AIAA paper 95-3304.
1995.
275
[4] Duda H., Prediction of pilot-in-the-loop oscillations due to rate saturation, Journal of Guidance, Navigation, and Control 20 (3) 1997. [5] Duda H., Flying qualities criteria considering rate limiting, DLR-FB 97-15, 1997 (in German). [6] Duda H., Minutes on a Workshop on Pilot-in-the-Loop Oscillations, DLR, Braunschweig. June 12-l 3. 1997, DLR IB 1 I l-97/25, 1997. [7] Duda H., Bouwer G. Bauschat J.M., Hahn K.-U., Autopilot design based on the model following control approach, in: Magni J.-F., Bennani S., Terlouw J. (Eds.), Robust Flight Control - A Design Challenge, Lecture Notes in Control and lnformation Sciences, Springer Verlag, London, 1997. [8] Duda H., Hovmark G., Forssell L., Prediction of Category II aircraft-pilot couplings - new experimental results, AlAA Paper 97-3499, 1997. [9] Fiillinger O., Nichtlineare Regelungen, Bd. I & 2, 3. Auflage, R. Oldenbourg Verlag, Miinchen, Wien, 198 I, 1982.
[IO] Graham D., McRuer D., Analysis of Nonlinear Control Systems, John Wiley & Sons, Inc., New York, London, 1961.
[ 1I] Klyde D.H., McRuer D.T., Myers T.T., Unified pilot-induced oscillation theory, TR- I3 13- I, Systems Technology Inc., Hawthorne, 1995. [ 121 McRuer D., Pilot-induced oscillations and human dynamic behaviour, NASA CR 4683, 1995. [ 131 McRuer D. et al., Aviation Safety and Pilot control. Understanding and Preventing Unfavorable Pilot-Vehicle Interactions, ASEB National Research Council, National Academy Press, Washington D.C., 1997. 1141 Mitchell D.G., Hoh R.H., Aponso B.L., Klyde D.H., The measurement and prediction of pilot-in-the-loop oscillations, AIAA Paper 94-3670, 1994. [ 151 N.N., Military specification; flying qualities of piloted airplanes, MIL-F-8785 C, Wright Paterson AFB, 1980. 1161 N.N., Military standard; flying qualities of piloted aircraft, MIL-STD-1797 A, Wright Paterson AFB. 1990.
1171Wedekind G., Mangold P., Integration of aerodynamic, performance, stability and control requirements into the design process of modern unstable fighter aircraft configurations, AGARD-LS- 153, Paper 2, 1987. [18] Wiinnenberg H. (Ed.), Handling qualities of unstable highly augmented aircraft, AGARD-AR-279, I99 I.
Science
and Technology,
1998,
no. 4, 265-215
Flight control system design considering
rate saturation Holger Duda*
DLR, Institute for flight mechanics, German Aerospace Center, 38022 Braunschweig, Germany (Received
Duda H., Aerospace
Science
and
Technology,
4 September
1997, accepted 12 November
1998, no. 4, 265-275.
In this paper the design process of modem flight control systems is discussed with respect to pilot-inthe-loop oscillations (PIO) due to rate saturation. At first the PI0 phenomenon is characterised briefly. Well-known design criteria with respect to flight control system stability and handling qualities based on linear aircraft models are presented. A new design criterion considering the nonlinear effects of rate limiting is described. The criterion is based on the ‘open loop onset point’ (OLOP) of the rate limiter in a Nichols chart. The background for the development, and the verification of the OLOP-criterion are presented briefly. In order to demonstrate the applicability of the OLOP-criterion in the design process a simple example is examined. The flight control system parameters of a basically unstable aircraft are optimised with respect to the linear design criteria and the OLOP-criterion. Two alternative designs were discussed, a ‘nominal design’ and a ‘low gain design’ with reduced feedback loop gain. The low gain design is favourable with respect to rate limiting in the feedback loop of the flight control system, which is proved by nonlinear simulations in the time domain. Finally, the utilisation and the design of a rate limitation in the forward path of the flight control system are discussed. 0 Elsevier, Paris
Abstract
flying qualities / flight control system / aircraft-pilot nonlinear system / rate limiting Zusammenfassung
coupling / pilot-in-the-loop
* Correspondence
oscillations /
Auslegung von Flugsteuerungssystemen bei begrenzter Stellgeschwindigkeit. In diesem Papier wird der AuslegungsprozeB modemer Flugsteuerungssysteme im Hinblick auf Flugzeug-Pilot-Kopplungen (Pilot-in-the-Loop Oscillations, kurz PIO) bei Stellgeschwindigkeitsbegrenzungen diskutiert. Zunachst wird das PIO-PhSinomen kurz charakterisiert. Die bekannten Auslegungskriterien fur Reglerstabilitat und gute Flugeigenschaften basierend auf linearen Flugzeugmodellen werden vorgestellt. Ein neues Auslegungskriterium zur Bertlcksichtigung nichtlinearer Effekte bei Stellgeschwindigkeitsbegrenzungen wird beschrieben. Das Kriterium basiert auf dem Aktivierungspunkt der Stellgeschwindigkeitsbegrenzung (Open Loop Onset Point, kurz OLOP) im Nicholsdiagramm. Der EntwicklungsprozeB und die Verifikation des OLOP-Kriteriums werden kurz dargestellt. Zur Demonstration der Anwendbarkeit des OLOPKriteriums im EntwurfsprozeB neuer Flugsteuerungssysteme wird ein vereinfachtes Beispielflugzeug betrachtet. Die Parameter des Flugsteuerungssystems fur ein instabiles Basisflugzeug werden unter Berticksichtigung der linearen Auslegungskriterien und des OLOP-Kriteriums optimiert. Zwei alternative Varianten werden untersucht, eine Auslegung mit hoher und eine mit reduzierter Rtickftlhrverstarkung. Die Auslegung mit reduzierter Rtickftihrverstarkung zeigt sich als vorteilhaft im Hinblick auf Stellgeschwindigkeitsbegrenzungen, wie durch nichtlineare Simulationen im Zeitbereich gezeigt wird. AbschlieBend wird die Auslegung einer Stellgeschwindigkeitsbegrenzung im Vorwartszweig der Flugsteuerung diskutiert. 0 Elsevier, Paris Flugeigenschaften / Flugsteuerungssystem I Flugzeug-Pilot-Kopplung Stellgeschwindigkeitsbegrenzung
Aerospace
1997)
and reprints
Science and Technology,
1270-9638,
98/04/O
Elsevier,
Paris
I nichtlineares System I
266
H. Duda
Nomenclature angle of attack amplitude elevator deflection pitch stick force frequency at - 180” phase angle frequency response elevator per stick force phase angle crossover phase angle actuator transfer function prefilter transfer function aircraft transfer functions aircraft gain (control sensitivity) flight control system feedback gains Hight control system forward path gain flight control system feedback loop gain pilot gain describing function elevator per stick force rate limiting element describing function vertical load factor phase rate at - 180” phase angle pitch rate commanded pitch rate maximum rate maximum rate in the feedback loop maximum rate in the forward path Laplace variable pitch angle input amplitude of the rate limiter frequency crossover frequency closed loop onset frequency onset frequency short period frequency short period damping
1. Introduction The high demands on performance and handling qualities of modern aircraft within a large flight envelope require the implementation of complex electronic flight control systems. Especially for highly manoeuvrable military aircraft, this control technology has opened completely new opportunities
for optimisation. In the latest military developments basically unstable aircraft designs were utilised, such as the US American F-22, the Swedish JAS39 or the European EF2000. A basically unstable design can provide benefits in flight performance up to . 18 % [17]. Also in the civil aircraft branch the use of modern electronic flight control systems provides a large potential for improvements. The development of the Airbus A320 can be regarded as a milestone in this process; a pure electronic link was used for the ailerons and elevators for the first time in an operational civil transport aircraft [2]. The use of this fly-by-wire technology allowed a stronger influence on the flight characteristics, for instance the aircraft behaviour is less dependent on the flight condition and centre of gravity location. It is even possible to achieve a commonality of flying characteristics within a certain aircraft family, which is a profitable element considering pilot training and certification aspects [7 ]. Besides the economic benefits due to better flight performance, the use of electronic flight control systems contributes to increased flight safety; improved handling qualities and flight envelope protection functions strongly reduce pilot workload; fault detection and isolation algorithms are used in combination with highly sophisticated reconfiguration concepts to cope with system failures, such as the loss of a control surface. However, in spite of all these safety improvement a significant handling qualities problem came up again with the introduction of electronic flight control systems: pilotin-the-loop oscillations or pilot-induced oscillations (PIO). PI0 can be considered as a closed loop destabilisation of the aircraft-pilot loop and it occurs often under situations when the pilot proves to be unable to adapt himself to a sudden change of the vehicle dynamics during a high demanding flying task (the trigger) [I, 141. PI0 are not pilot failures, but result from a failure in the design process. A significant correlation was found between PI0 incidents reported during the complete history of aviation and rate saturation in flight control systems. Nearly all catastrophic PI0 cases were associated with rate saturation, such as the crashes of the YF-22 in 1992 and two JAS39 prototypes in 1989 and 1993. In the past, most PI0 cases occurred within the fighter aircraft branch, but also civil airliners with electronic flight control systems have demonstrated PI0 problems [ 131. In spite of the very strong correlation between PI0 susceptibility and rate limiting, no design criteria are established, which address nonlinear rate saturation effects. Until recently several design criteria were available in the frequency and time domains, but they were only validated on the basis of linear aircraft models. Although the time domain criteria can cope
Flight control system design considering Auslegung van Flugsteuerungssystemen
rate saturation/
with nonlinear systems, they were unable to explain the rate saturation effects sufficiently. In the time domain one can only recognise that an additional time delay due to rate limiting occurs. But a method to predict the magnitude of this additional time delay in the design process has been lacking. Frequency domain criteria provide a better understanding of the dynamic behaviour of the system, but they are limited to linear systems. Therefore, a new criterion is required, which combines the advantages of ‘frequency domain thinking’ with the possibility of treating nonlinear systems. This deficiency in the design requirements was the background for starting a research project at DLR. On the basis of a closed loop system describing function analyses a new design criterion with respect to rate saturation in flight control systems was developed [3-51. In this criterion the ‘open loop onset point’ (OLOP) of rate limitation is assessed in a Nichols chart. Within the framework of a German/Swedish cooperation between DLR and FFA the OLOP-criterion was validated on the basis of new experimental data [6, 81. In this paper the OLOP-criterion is presented and its application potential to flight control system design is discussed on the basis of a representative example aircraft.
“PI0 are sustained or uncontrollable oscillations resulting from the efforts of the pilot to control the aircraft.” Several different types of oscillations forced by the pilot are known, which differ in frequency and amplitude. In the roll axis high frequency low amplitude oscillations are known, called ratcheting [141. Such oscillations basically occur at frequencies above 10 rads’ and are caused by interactions between the cockpit controls and the pilot’s neuromuscular system. They are characterised by pilot comments about some potential PI0 problems, but will not result in a catastrophe. The corresponding phenomenon in the pitch axis is called bobble, which is mainly caused by structural modes. Lower frequency high amplitude oscillations represent the common meaning of real PIO. For the understanding and analysis a further classification was recently introduced [ 111. Category I PI0
Essentially linear aircraft-pilot oscillations Category II PI0 Quasi-linear aircraft-pilot oscillations with significant rate or position limiting Category III PI0 Nonlinear aircraft-pilot oscillations with transitions 3. Linear
2. Pilot-in-the-loop
oscillations
267
bei begrenzter Stellgeschwindigkeit
(PIO)
The PI0 problem has been known for years, while the introduction of full-authority, digital fly-by-wire flight control systems has increased the potential for PIO. For the understanding and analysis of PI0 three main elements have to be considered [ 121: - the pilot, - the aircraft, and - the trigger. The weakest point in the analysis of PI0 are pilot models, since the pilot behaviour is highly nonlinear and very complex. However, pilot models are available for special pilot behavioural patterns [ 121. The aircraft is represented in this scenario by the linear aircraft and flight control system dynamics. The trigger can have different forms, for example a nonlinear effect in the flight control system or a transition in the pilot behavioural pattern, but it always causes a sudden change in the closed loop dynamics of the aircraftpilot system. This may lead to a strong misadaptation of the pilot, but it has to be stressed again that PI0 is not a result of a pilot failure. The following definition can be found in the US American military specifications [ 11, 161. 1998. no. 4
design criteria
(Category
I PIO)
The main effects causing Category I PI0 are well known as excessive lags due to time delays and various digital filter dynamics in the flight control system. These effects lead to a ‘high frequency phase rolloff in the frequency response. The established frequency domain PI0 criteria such as the ‘phase rate criterion’ [ 181 are well suited for predicting Category I problems. Several flight control system design criteria are established based on linear systems. One can roughly distinguish between the following two groups of design criteria: - flight control system stability - handling qualities requirements. The first group of design criteria concerns the stability of the aircraft system (in this context, ‘aircraft’ means the basic aircraft including flight control system). Figure 1 shows the typical boundaries for phase and amplitude margin in a Nichols chart [2]. The open loop aircraft frequency response has to be checked against this criterion. For example in the pitch axis the loop h,as to be opened at the elevator actuator. The second group of design criteria concerns the handling qualities of the piloted aircraft. Most common handling qualities requirements are collected in the US American military specifications [ 15, 161. The requirements are graded regarding the aircraft class and the flight phase category, such as the following:
268
H. Duda
gradient for nIL,/a. for Category A flight phases [ 151. These boundaries can be interpreted by the ‘control anticipation parameter’ (CAP), which is defined as the ratio of initial pitch acceleration to quasi-stationary vertical load factor. The requirements for the short period damping are summarised in table I [ 1.51.
20 E * a x
10
5
critical point, 0
Table
I. Short
period
-10
Category Level
-20 -150
-100
chase angle [deg] Figure chart.
1. Boundaries
Category Category Category
damping
for phase
and amplitude
margin
in a Nichols
A
Non-terminal flight phases, precision tracking required, e.g. air-to-air combat B Non-terminal flight phases, without precision tracking, e.g. cruise C Terminal flight phases, accurate flight path control, e.g. approach and landing
The levels of handling follows:
qualities
are defined as
inlnx
I
0.35
1.3
0.30
2.0
2
0.25
2.0
0.20
2.0
3
0.15
-
0.15
The main cause for Category I PI0 - the high frequency phase rolloff - is not sufficiently addressed by the short period criteria. Therefore, special high order system criteria were developed, such as the ‘phase rate criterion’ [18]. Figure 3 shows the calculation of the phase rate parameter PRlso for the frequency response of the pitch angle 0 by pitch stick force F,,. Additionally, the boundaries Phase angle over frequency
frequency
requirement
(Category
(O/F,,)
frequency a) Determination
0
0.5
1O” b) Phase
period
B
c Llllll
“-
Figure 2. Short phases).
Category
c map
In the pitch axis the short period dynamics (frequency and damping) are relevant for the aircraft manoeuvrability. Figure 2 presents the requirement for the short period frequency depending on the
n,la [ghad]
A, C
&in
Level 1 Satisfactory without improvements Level 2 Adequate to accomplish the mission Level 3 Controllable, excessive pilot workload
10
requirements
A flight Figure
3. Phase rate
criterion
f180
of phase
1
rate PR18c
f, 80 WI
rate criterion
1.5
boundaries
2
Flight control system design considering Auslegung von Flugsteuerungssystemen
rate saturation/
for the phase rate PR1sO and the frequency flsO at - 180” phase angle are presented. Physically, the phase rate parameter has the following significance: a too high phase rate means that with a small increase in frequency a strong additional phase delay occurs, which is always a problem in a closed loop system. It is highly correlated with the additional time delay in the complete system.
4. The OLOP-criterion
(Category
269
bei begrenzter Stellgeschwindigkeit
II PIO)
The nonlinear effects of rate saturation in the flight control system are not considered within the criteria presented above, since they are only valid for linear aircraft models. Figure 4 shows two typical positions, where rate limiters are installed as software elements in the flight control systems of highly augmented aircraft. The limiters in the feedback loop of the flight control system have the task to protect the actuators against overload. Otherwise the oil supply of the actuation system could break down. The limiters in the forward path of the flight control system are installed to protect the system against high input rates by the pilot, preventing a saturation of the feedback loop rate limiters. This protection is particularly important for unstable highly augmented aircraft.
limiting
situation [5]
The onset frequency of the rate limiter is the ratio of the maximum rate R and the current input amplitude ~,l<, of the rate limiter
For the calculation of the describing function of a rate limited closed loop system a special method was developed. The application of this method to a highly augmented aircraft with a rate limitation in the feedback loop is presented in$gure 5. In that case the closed loop system describing function is characterised by a jump phenomenon after rate limiting onset, which can be recognised in a Nichols chart as a phase jump. In the presented example offigure 5, the phase jump leads to a dramatic loss of phase and amplitude margin indicating the potential for an instability of the closed loop system. This instability was verified by a nonlinear &n&lation in time domain [3].
Open Loop Frequency Response q/q,
,/---
,,, pilot
15c
[- e]
5.1 radisec
: rate limiter
4 >** Figure Hight
4. Typical implementation control systems.
of software
rate limiters
‘, :
in modern
-10 -
To properly analyse closed loop systems with rate limiting, nonlinear system theory has to be applied. In contrast to linear systems, where a complete system theory is available, nonlinear systems can be analysed only for special cases. Several methods for treating nonlinear systems are known for this purpose, such as the theory of Lyapunow, the Popow criterion, the phase plane method, the numeric simulation in time domain, and the describing function technique [9, lo]. The latter is rated as the most promising tool to analyse PI0 due to rate limiting in flight control systems. A nonlinear system can be represented by a quasilinear system using the describing function and the remnant. The describing function can be interpreted as the quasi-linear frequency response for sinusoidal inputs. The remnant is a high frequency signal, which can be neglected if the linear part of the system has low pass character. The describing function was calculated using a Fourier series for the fully developed rate
-15 -20 -300
_,,A
,’ I’
,’ /’
/’
,’ .---- describing function linear open loop onset l point (OLOP) _I
-250
Figure 5. Jump phenomena augmented aircraft.
-150 -100 phase angle [deg]
-200
due
to
rate
limiting
onset
-50
in highly
This describing function analysis was the background for the development of a new design criterion based on the ‘open loop onset point’ (OLOP) of the rate limiter in the Nichols chart. In$gure 5, the OLOP can be identified as the point where the phase jump starts. The OLOP is defined as the frequency response value of the open loop aircraft system at the closed loop onset frequency cZ,,,,~,~~. The latter is the frequency where the rate limiter is activated for the first time for maximum pilot input amplitude (worst case). A
270
simplified method was derived to calculate the OLOP without utilising the describing function technique, but extended to the analysis of PIO: I. Definition
of a simple (high) gain pilot model,
2. Calculation of the ‘linear’ closed loop frequency response from the stick input to the input of the rate limiter (in the pitch axis F$,q), 3. Determination of the closed loop onset frequency ^ wonset considering stick and control surface limits. 4. Calculation of the relevant open loop frequency response and separation into amplitude A (w) and phase angle @ (w),
15 5. 2. $10 .=3 -5 E a5 0 -
-10
5. OL()P = [@(4,nset)rA (&,set)]. The definition of the pilot model is now treated more in detail. The pilot model has to be adjusted to the linear aircraft dynamics, which means that the pilot has adapted himself to an aircraft behaviour without rate saturation. It is recommended that simple gain pilot models be used since the pilot usually reacts as a simple gain during fully developed PI0 (‘synchronous precognitive behaviour’) [ 121. The gain is adjusted based upon the linear crossover phase angle of the open loop aircraft-pilot system @,. A gain spectrum from Qc = 110” (low pilot gain) up to Q’,, = -160” (high pilot gain) should be used. This gain spectrum can provide the following results: if an aircraft is rated as Category II PI0 prone by an OLOP based on a low gain pilot model, PI0 is very likely; if an aircraft is rated as Category II PI0 free based on a high gain pilot model, PI0 is very unlikely after rate limiting onset. The OLOP-criterion was verified by evaluation of several different configurations from roll axis databases with rate limiters in the forward path and feedback loop of the flight control systems [5]. Aircraft models from US American flight test programs were used (LATHOS: lateral high order system; F-18 inflight simulation; YF-16 first flight). Figure 6 presents the correlation between the OLOP-criterion and the Category II PI0 potential based upon nonlinear simulations with pilot models. It appears that the OLOP-criterion is applicable to both forward path and feedback loop rate limiters using the same PI0 boundary. But the feedback limiters provide a much stronger Category II PI0 potential, especially for high feedback loop gains. Category II PI0 due to rate limiters in the forward path is mainly possible for very high pilot gains or extremely low maximum rates. More details on this subject are presented in reference [5]. New experimental results utilising ground based simulations indicate the validity of the OLOP-criterion to predict Category II PI0 [S].
0 LATHOS, no PI0 @ LATHOS, PI0 possible . LATHOS, PI0 0 F-18, no PI0 0 F-18, PI0 possible + F-18, PI0 a YF-16, no PI0 A YF-16, PI0 possible / A YF-16, PI0
-5
-15
Figure
-180 6. Verification
-160
-140
-120 -100 phase angle [deg]
of the OLOP-criterion
5. Flight control
[S]
system design
The design of new flight control systems is usually performed on the basis of linear aircraft models at several reference points within the entire flight envelope. The nonlinear effects of rate limiting are commonly investigated in a second step, since the rate limitation is not installed due to an optimisation of the flight control system, but due to safety aspects. The importance of considering the rate saturation effects as early as possible is discussed in the following. Modern fighter aircraft are designed basically unstable with a time to double amplitude well under I s. They are completely uncontrollable by the pilot without the flight control system, but the augmented aircraft must have Level 1 handling qualities. In order to achieve this, control power is required, which has classically been achieved with increased control surface sizing. This is in direct conflict with the performance of the aircraft resulting in smaller control surfaces that must move very rapidly. This explains the occurrence of several PI0 cases caused by rate limiting of modern highly augmented aircraft. It is obvious that the design process has to be a compromise between maximum aircraft performance and flight safety regarding flying qualities problems due to rate saturation.
5.1. A simple design example The OLOP-criterion can be utilised within the design process of new flight control systems to optimise the parameters with respect to performance and flight safety. This will be demonstrated by a simple design example, which covers several aspects
Flight control system design considering Auslegung van Flugsteuerungssystemen
rate saturation/
271
bei begrenzter Stellgeschwindigkeit
basic aircraft:
actuator:
prefilter:
with (5; 0) = (s2 + 2 @IS + 02), (a) = (s + 4 Figure
7. Structure
of
a simple
design
example
of the design process with respect to the rate saturation problem. In practice, additional topics have to be taken into account. Figure 7 shows the structure of the simple design example including typical parameters to be optimised. An unstable second order aircraft model (only short period mode) is considered. The gain K,, represents the aircraft control authority. A second order linear actuator model is used. A software rate limitation of n% = 100” .s-i and an amplitude limitation of 30” are assumed for the elevator. The maximum stick force is limited to 80 N. These parameters are prescribed by hardware requirements. The following flight control system parameters can be optimised with respect to the requirements. The feedback gains K, and K, have to be adjusted in order to attain the required short period characteristics (frequency and damping). The feedback loop gain K% can be adjusted in order to define the bandwidth of the system. The feedforward gain KR is responsible for the steady state gain (sensitivity) of the augmented aircraft. The prefilter time constants have to be optimised with respect to the phase rate criterion. The presented values were obtained from a manual optimisation and will not be changed any more in the following examination. The maximum rate of the limiter in the forward path RR has to be defined, so that the feedback loop rate limiters can not be saturated anymore. 5.2. Linear
design
In the ‘following design process a gradient n, /o = 10 g . rad-i is assumed. According to jigure 2 and table I the following values for short period frequency and damping are required for Category A flight phases to achieve Level 1: Frequency: 1.7 rad . s-r 5 wSp 5 6 rad . s-l IYYX. no 4
Damping:
6, 2 0.35
A manual optimisation provided the feedback gains for unity feedback loop gain (Kfb = 1): K, = 0.6s
K,, = 2.5
This leads to the following short period characteristics: W sp = 5.9 rad . s-l es,, = 0.7” With Ke = KK = K,, = 1 this configuration is referred to as the ‘nominal design’. It is obvious that this design has a large potential for feedback loop gain reduction, since the short period frequency is at the upper limit and the damping is much higher than the required value. The effects of feedback loop gain reduction are presented in jipre 8a with respect to the linear design criteria. Regarding the short period frequency and damping requirements a feedback loop gain reduction to KB, = 0.2 would be possible, while still staying within the requirements (figure 8~). But, this would lead to a very poor design with respect to the phase rate criterion (Level 3 not fulfilled) (figure 8b). In order to achieve a Level 1 design a feedback loop gain of about Ka = 0.68 is required to have a sufficient margin to the Level 2 boundary. This configuration is referred in the following as the ‘low gain design’. The loop stability criterion shows that in both cases (Ke = 1 and Kfb = 0.68) sufficient phase and amplitude margin are available (figure 8~). 5.3. Considering
rate limitation
Based upon the linear design criteria two alternatives were presented above, which meet the linear design requirements. The next step is to consider the effects of feedback loop rate limitation (assuming no forward path rate limitation at this stage, Rff = x). For the application of the OLOP-criterion, at first the closed loop onset frequency ij,,,,t has to be determined. For
272
H. Duda
60 3 s 40 3 "a E m 20
’
frequency [radkec] 4.g S6
lo
a) Closed loop onset frequency -I1
I
-4
-2
O
Re(s)
I 2
iii 91
a) Short period frequency and damping
I
t s 5 (
9 5.6
Kfb = 0.68
-1
-180
-140 -100 phase angle [deg]
b) OLOP-criterion (flight control system loop) Figure 9. Evaluation of the OLOP-criterion dashed: nominal design).
b) Phase rate criterion
(solid:
low gain design:
limiter is activated first time, when the frequency response curve crosses a straight line with a slope of -20 dB.decade-’ crossing zero dB in the maximum rate (lOO”.s-’ in this case). Additionally, the deflection limit of 30” is presented in jigure 9n, which does not cross the frequency response curves of either configurations (nominal and low gain). The following closed loop onset frequencies were obtained: Nominal (Kn, = 1): ij,,,,,,,+ z 4.1) rad . s-’ Low gain (Kn, = 0.68): LjOnSI,+ Z 5.6: ratl ‘s-l. -180
-140 -100 phase angle [deg] c) Flight control system loop stability
Figure 8. Evaluation of feedback loop gain IT+),.
this purpose the pilot input (stick limiter (elevator amplitude of 80
linear
design
criteria
depending
on
the
linear frequency response from the force F,,) to the input of the rate deflection 5,) for maximum input N is utilised @gure 9~). The rate
Figure 9b presents the OLOP-parameters of the two configurations. For the nominal design the OLOPparameter is located in the critical area, which means a destabilisation of the closed loop aircraft system after rate limiting onset is predicted. For the low gain design the OLOP-criterion predicts no destabilisation after rate limiting onset. These predictions are examined by nonlinear simulations in time domain. A 3-2-l-l signal is used with maximum input amplitude of 80 N. The rate limited elevator deflection and the pitch rate are
Flight control system design considering rate saturation/ Auslegung van Flugsteuerungssystemen bei begrenzter Stellgeschwindigkeit
273
Stick Force 100 50 _ E 2
o--so-100
.. Elevator
Deflection
40 -
Pitch Rate ,Uk,
I
j 1*;~~-r,,-~~~~ ,1 -18; -140 -100 phase angle [cieg]
Figure
11. OLOP-criterion @, = -150
rate limiter,
for the aircraft-pilot loop, feedback loop deg (solid: low gain design; dashed: nominal
design). time [set]
Figure 10. Nonlinear simulation limitation in the feedback loop nominal design).
in time domain, effect (solid: low gain design;
of rate dashed:
presented in figure 10. The time histories show that rate saturation occurs in both cases, but only for the nominal configuration this leads to an instability of the closed loop system. The low gain configuration remains stable during rate saturation. This means that rate saturation in flight control systems can be harmless even for designs with basically unstable aircraft under certain conditions (OLOP-criterion is fulfilled)! This investigation shows that the low gain design is favourable although it is slightly worse with respect to the phase rate criterion than the high gain design. In practice during the flight control system design process optimum search algorithms are utilised to minimise a cost function depending on the design criteria. Within this optimisation process the linear design criteria and the nonlinear OLOP-criterion drive the flight control system parameters in opposite directions; considering only the linear design criteria would lead to a system with a high feedback loop gain, but stability problems may occur after rate limiting onset. Therefore, it is highly recommended to apply the OLOP-criterion within the early design process in order to find the best compromise between the linear performance and flight safety after rate limiting onset. One issue not addressed up to now is the pilot-inthe-loop situation. For this task the OLOP-criterion can be applied to the aircraft-pilot loop considering a pure gain pilot model based upon the linear crossover phase angle of the open loop aircraft-pilot loop QC [5]. In the design process a high gain pilot should be assumed in order to characterise a worst case behaviour. In this case a crossover phase angle gC = -150” is assumed, 1998, no. 4
Table II. Required gain design
measures
to provide
Measure
flight
safety
for
Drawbacks
Reduction of the forward gain to Kff = 0.65
path
Strong reduction in performance (agility)
Increasing the maximum to Rfh = 150Cieg.s-1
rate
Increased to faster
Increasing the (control surface K,, = 1.6 Rate limitation path
the high
aircraft gain sensitivity) to in the forward
aircraft
cost and weight actuators
due
Increased control surface size (stronger actuators, reduced performance) How to design rate?
the maximum
which is a fairly high gain. Figure 11 presents the OLOP-parameters of the two configurations. The low gain design is predicted to be PI0 free after rate limiting onset, therefore, it can be considered as the optimum with respect to the design criteria used. If the high gain design is required due to other circumstances, the following measures are possible in order to provide flight safety with respect to rate saturation in the feedback loop (table ZZ). The presented values for K,,, Rn and R.a were based on OLOP-analyses. Each measure has obvious drawbacks, except the implementation of a rate limitation in the forward path. Therefore, this option is examined more in detail. 5.4. Rate limitation
in the forward
path
The rate limiter in the forward path should be designed so that no rate limiting in the feedback loop is possible due to high input rates by the pilot. For this
274
H. Duda
task the describing function technique can be utilised. Figure 12 presents the linear frequency response from the pilot input (pitch stick force Fclg) to the input of the rate limiter (commanded elevator deflection h,) for maximum input amplitude (80 N) for the high gain design (the same as presented in jigur~ 9~). Additionally, the describing function considering the rate limitation in the forward path is plotted. In this case a maximum rate of Rtf RZ292 N SK’ is required to make sure that the feedback loop rate limiter straight line with -20 dB.decade-’ is not crossed. This means that for sinusoidal input signals it is impossible for the pilot to saturate the feedback loop limiters, which was the design aim of the forward path rate limiter.
Stick Force 100 50. z
;
O-
‘\
i
-I : ‘1 I’! ‘1 , 11
\ /
.ti
\
,
\
-50 -100~
Elevator
Deflection
40
-40 ’
Pitch Rate
2OOr
-60 g 8 40 a “a 5 20
-200 ’ 0
~o”setv+f) 0
Figure
4
6
i 10
8
time [set] &de
gain
2
.‘.I
1
12. Definition design.
Rff
nOo”*et(hb)
Figure
es
2,7 4,9 10 frequency [radkec]
of the maximum % 292 N 5-I
rate in the forward
13. Nonlinear
the forward
path.
path
simulation c 292 N
(8,
in time domain with rate limiter S- ’ ), high gain design.
in
high
In order to prove these results a nonlinear simulation in time domain is carried out. Again, the 3-2-l-l signal with maximum input amplitude of 80 N is applied figure 13). The rate limited stick force and the pitch rate are presented. Unlike the simulation without rate limitation in the forward path presented in Jigure IO, no instability occurs for the high gain design anymore for the same input signal. The advantages of rate limitation in the forward path were significant in this example, but it has to be taken into account that now PI0 can occur due to this rate limitation. To assess the PI0 potential due to rate saturation in the forward path, the ‘OLOPcriterion’ is applied to the aircraft-pilot loop. Figure 14 presents the OLOP-criterion on the basis of a high gain pilot model (@jr = -150’). The OLOP-parameter is located very near to the boundary in the dangerous area indicating some PI0 potential (figure 14). This result has to be interpreted carefully. A Category II PI0 might occur only for very high pilot gains and maximum input amplitude. The pilot has still the possibility to break out of this PI0 by reducing his gain (aggressiveness). This pilot gain reduction would not help anymore after a saturation of the feedback loop rate limiters (if no rate limiter would have been installed in the forward path), as it is demonstrated in figure 10. In that case the aircraft becomes explosively unstable. This means that rate saturation in the forward path has to be rated much less dangerous than rate saturation in the feedback loop.
@, = -150 deg
-IO-180
-140
-100 phase angle [deg]
Figure
14. OLOP-criterion
path rate - 1w.
limitation
(Rff
z
for the aircraft-pilot 292 N S-’ ). high
loop with forward gain design, @, =
The forward path rate limitation has minor adverse effects, but it helps to avoid an aircraft instability due to high pilot input rates. For a specific aircraft it has to be decided if priority is focused on maximum linear system performance for small to medium input amplitudes. In that case the high gain design with rate limitation in the forward path is preferable, although a minor PI0 potential is predicted for full stick deflections. Otherwise, if priority is focused on maximum flight safety for all input amplitudes the low gain design should be favoured. Also the potential for the other measures presented in table II should be examined for a specific design.
Flight control system design considering Auslegung van Flugsteuerungssystemen
rate saturation/
bei begrenzter Stellgeschwindigkeit
The OLOP-criterion provides significant support to find the optimum design with respect to all parameters early in the design process. This will definitely save time and money, since the effort for changing flight control system parameters is lowest in the beginning of the design process.
6. Conclusions The flight control system design process of an basically unstable aircraft with respect to pilot-in-theloop oscillations (PIO) and rate saturation effects was discussed. The applicability of the new design criterion (‘open loop onset point’, OLOP) to predict flying qualities or stability problems due to rate saturation was demonstrated. Additionally, typical linear design criteria were utilised. Considering only linear design criteria for the optimisation of the flight control system parameters may lead to a design with stability problems after rate limiting onset. Therefore, it is highly recommended to consider the OLOP-criterion early in the design process. The OLOP-criterion is a very valuable tool in order to optimise the feedback loop and forward path gains of the flight control system, the required maximum rates of the actuators, and the required control power of the basic aircraft. It was proved that the implementation of a rate limiter in the forward path of the flight control system definitely has favourable effects on flight safety. For the definition of the maximum rate in the forward path the describing function technique can be utilised. However, there may be some adverse effects, which can be examined with the OLOP-criterion.
References [l] Ashkenas I.L., Jex H.R., McRuer D.T., Pilot-induced oscillations: their cause and analysis, NORAIR Report NOR-64-143, 1964. 12) Brockhaus R., Flugregelung, Springer Verlag, Berlin, Heidelberg, New York, 1994. 131Duda H., Effects of rate limiting elements in flight control systems - a new PIO-criterion, AIAA paper 95-3304.
1995.
275
[4] Duda H., Prediction of pilot-in-the-loop oscillations due to rate saturation, Journal of Guidance, Navigation, and Control 20 (3) 1997. [5] Duda H., Flying qualities criteria considering rate limiting, DLR-FB 97-15, 1997 (in German). [6] Duda H., Minutes on a Workshop on Pilot-in-the-Loop Oscillations, DLR, Braunschweig. June 12-l 3. 1997, DLR IB 1 I l-97/25, 1997. [7] Duda H., Bouwer G. Bauschat J.M., Hahn K.-U., Autopilot design based on the model following control approach, in: Magni J.-F., Bennani S., Terlouw J. (Eds.), Robust Flight Control - A Design Challenge, Lecture Notes in Control and lnformation Sciences, Springer Verlag, London, 1997. [8] Duda H., Hovmark G., Forssell L., Prediction of Category II aircraft-pilot couplings - new experimental results, AlAA Paper 97-3499, 1997. [9] Fiillinger O., Nichtlineare Regelungen, Bd. I & 2, 3. Auflage, R. Oldenbourg Verlag, Miinchen, Wien, 198 I, 1982.
[IO] Graham D., McRuer D., Analysis of Nonlinear Control Systems, John Wiley & Sons, Inc., New York, London, 1961.
[ 1I] Klyde D.H., McRuer D.T., Myers T.T., Unified pilot-induced oscillation theory, TR- I3 13- I, Systems Technology Inc., Hawthorne, 1995. [ 121 McRuer D., Pilot-induced oscillations and human dynamic behaviour, NASA CR 4683, 1995. [ 131 McRuer D. et al., Aviation Safety and Pilot control. Understanding and Preventing Unfavorable Pilot-Vehicle Interactions, ASEB National Research Council, National Academy Press, Washington D.C., 1997. 1141 Mitchell D.G., Hoh R.H., Aponso B.L., Klyde D.H., The measurement and prediction of pilot-in-the-loop oscillations, AIAA Paper 94-3670, 1994. [ 151 N.N., Military specification; flying qualities of piloted airplanes, MIL-F-8785 C, Wright Paterson AFB, 1980. 1161 N.N., Military standard; flying qualities of piloted aircraft, MIL-STD-1797 A, Wright Paterson AFB. 1990.
1171Wedekind G., Mangold P., Integration of aerodynamic, performance, stability and control requirements into the design process of modern unstable fighter aircraft configurations, AGARD-LS- 153, Paper 2, 1987. [18] Wiinnenberg H. (Ed.), Handling qualities of unstable highly augmented aircraft, AGARD-AR-279, I99 I.