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PREr)lC'iION OF PILDT-AIRCRAFT SPABlLlTY The matter 4s of partir:ular concern when the subject is BOLXDARIES AND PERKNUANCE CONTOURS a skilled pilot and the task is controlling a maneuvering aircraft. for tbe ssccess of the mission 8nd the pilot's om safety are strong motivating factors. During rapid lureuvering.
the aircraft's dynamic cnaracteriatica can c h ~ g e nrkedly in a Robert F. Stengel and John R. Broussard matter of seconds. and the pilot n y be called up00 to . I * e The Analytic Sciences Corporation changes in his control strategy jus+. to mintain stability. much S I X Jacob Way less carry out his mission. More olten than not. the pilot Readlng. Massachusetts 01867 who performs such maneuvers has mastered the I.ecess8ry proce- dural adaptation and executes it with precisi~n. On rare occasions. even the skilled pilot m8y get into trouble. adapt- ing his control strategy to suit poorly chosen criteria, or perhaps not adapting t all. In high-perforp.~ce aircraft.
this apparent lapse cru cause a pilot-induced "deprture."
i.e.. a loss-of-control incident rblch, if n o ' corrected i~.e- diately, can lead to a spin 8nd possible loss o f the aircraft.
Control-theoretic pilot models can p r w i d e important new The problers for study are not only hor to Podel the pilot's discretionary behavior in departure-prone mneuvering treks.
lnsights regarding the stability and performance character- but how to relate such a model to the lore frequant, near- istics of the piiot-aircraft system. Optimal-control pilot optimal behavior of the skilled pilot.
models can be formed for a wide range of flight conditions, sug- Festing that the h u m n pilot can maintain stability if he adapts The approach taken in this paper is to define a se- his control strategy ta the aircraft'b changing dynamics. Of quence of optimal-control pilot m d e l s which correspbni, to the partico;ar concern is the effect of sub-optimal pilot adaptation as an aircraft transitions from low LO high angle-of-attack dur- aircraft's changing dynamics 8s it perforpq a nOPin8l Plleilver " ,- u ing rapid maneuvering. as the changes in aircraft stability and and to examine the effects of pilot-aircraft a & l miclrmrtcb on control response can be extreme. This paper examines the effects closed-loop stability and statistical tracking error. The
maneuver -- a "wind-up turn" -- begins at lor angle of attack
of optimal and slrb-optimal effort during a typical high-gm maneuver. and it introduces the concept of min~mum-control effort (ao) and proceeds to a high ao. As the maneuver progresses.
there is a dramatic variation in the optimal piloting strategy.
(YCE) adaptation. Limited experimental results tend to support including, in soma cases. a change in sign of tbe pilot's st*- t5e MCE adaptation concept.
bilizing commands to the aircraft. Prom the ,mtset, it is IhTRDDUCTION clear that sufficient mislrnteh could lead to closed-loop inst8- bility. but the rationale for large sismatcb remains to be de- termined.
A hywthesis for sismatcb is found in the mini-con- Since Tustin first llkened the command and response trol-effolt m d e l of pilot adaptation, w&icb. simply stated.
patterns of an:i-aircraft gunners to rudimentary control sys- suggests t+at the pilot selects not the optimal strategy but tems (Ref. 1). the intriguing notion that control-theoretic m r t h e ~ t l ~ a l m d e l s can characterize the numan operator has the one w h ~ c h minimizes the variance of stick and pedal motions (in the mlslotched case). With the ainiaum-control-effort (MKS) been carried tu a higl. state of developrnt. Frequency-dmin mrdels have proven capable o f capturing fundamental aspects o f pilot model, closed-loop stability can be rintaincd, but the the human operator's behavior in a straightfornard and logical margin for cr.-or is reduced, by comparison to the optimal stra- tegy. Where tLr pilot has a chciee of control outputs. (e-g..
fashion (as in Refs. 2 and 3). whlle time-domain models have denanstrated that a well-nut:vated sub?ect can. in fact. behave stick alane. 2eall alone. or colbination of the two), tbe pilot model also predicts the point during tbe maneuver at like an optimal coutrol systecn in various single- and multi- which tbe pilo. may choose to transition from one c - d mode axls trackisg tasks (Refs. 4 to 6 ) . Nevertheless. a number o f to another to maintain stability with miniaril effort.
perplexing problcms remain In the study of wha' might be crlled the pilot's "discretionary conlrol behavior." i.e.. given that The YCE pilot model has yet to be vnlidrted by exhaus- the subject is physically and emotionally capable of p e r f o m - tjve experimentation. but there is remarkable agreement between ing a task in an optimal manner. why might that subject choose to do otherwise?
* P r e s e n t e r n e 1 3 : h h n u a l Conference o? Manual Control.
Cambridge. Uassachusetts, June 15-17. 1977.
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. d u c u P L c 4 1 M U U ) & O ~ * d L I ~ C 0 w a * L U G * ti- ma 1 u 0 3 d - 0 1 r . - B x h a Q Z ~ B M ~ UU U U C C l d C h c da4 A d h C R d M D E M 01 Q L 01 B R 1 U 01 C u d g r L . S h 3 d u d d U > @ u $ s h k : 4 P J b a u II- o P B M I ? WCIC-Id 6 4 E P 3 Il- s 3 . H 0 2 ..3 - 0 - B 0 I O - B . U 1 UQ.3 CI 1 0 1 c * c W01&R*M e - t . a - o m a * QJ 0 d V B * L ' L * tation is itself a difficult task. The pilot may choose t . o adopt Tab e 3 response patterns which are more consistent and, therefore. sub- Control ;aim of the Ad pted Pilot Yodel Using optimal with respect to the criteria used to generate the pilot Lateral Stick and Foot Pedals model. Furthermore, even if the pilot chooses to adapt optimally in a dynamic maneuver, there is likely to be a lag between the AIRCRAW aircraft's actual flight condition and the pilot's adaptation point. Consequently, it is instructive to examine cases in which ATTACK, a6/ar, a6/ap, ad/a~p, the aircraft's dynamics and the control stratcgy adopted by the deu/deg/ deg/deg/ degldeg pilot are mismatched. In the examples which follow, it is assumed CONTROL
- that the pilot formulates an optimal control strategy for an assumed
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angle of attack,np , which may or may not be the same as the Stick aircraft's angle of attack, aA, during the maneuver.
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Figure 2 illustrates the boundaries between pilot- aircraft stability and instability for independent variations of
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aA and aP (during the wind-up turn). When the pilot model output is lateral stick el-mmand alone (Fig. 2s). mismatch introduces 24.6 +O. 160 regions af Dutch ruil and spiral mode instability which are approxi- mately symmetric about the line of perfect adaptation. There is Pedals 1.02 +O. 0342 a "stzbility neck" in the vicinity of aA=18 deg, where ap must be 8.72 +O.0137 very close to mA to maintain stability. Not only is adaptation in
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t!ds region potentially difficult (as indicated by Table 2). it is ~ r u c i a l that it be nearly optimal to prevent loss of control. If the pilot model uses foot pedals as well as lateral stick (Fig. 2b).
stability margins are substantially increased. .ud there are no unstable regions for ap greater than a&. If the pilot model uses foot pedals alone for control, the entire region is stable; in other words, a mismatch of as much as 30 deg between ad and ap Table 4 causes no instability if the pilot model does not use lateral-stick Control Gains of the Adapted Pilot Model Using motions for control.
Foot Pedals Alone AIRCRAFT SIDE YAW ROLL ROLL YAW ANGLE OF VELD2ITY RATE, RATE, ANGLE, ANGLE.
ATTACS , aalav, aalar, aa/ap, aa/a+ a a ~ a b , degldeg deg/deg
deg I deg/fps deg/deg/ deg/degl
sec sec mat* U, P U a, . . 1 . ( . w ....HIUI".III, a) Lateral Stick Alone b) Lateral Stick and Foot Pedals Figure 2: Effects of Pilot Model Adaptation on Maneuvering Plight Stability The "separation theorem" of stochastic, linear-optimal control gains are mismatched. The results which follow demonstrate control (Ref. 10) is only partially applicable to the optimal- the effects of sub-optimal control strategy on piloting perforarace; control pilot model. The stability results presented here depend the effects of sub-optir~al estimation remain to be determfned.
on the pilot's control strategy, but not on his estimation law; therefor-, control results are separated from estimation results, Figure 3 presents contours of equal rms values under the but the converse is not t ~ . u r The pilot model estimntor gains asswnption that the pilot uses lateral stick motions alone for (K) and eigenvalues depend on the control strategy as a consequence control. These contours would scale up or down as the turbulence of signal-dependent neuronotor and observation noise, but the level changed, so absolute values are less important than relative pilot model control gains ( C ) and "closed-loop" aircraft eigen- values in evaluating the figure.
values do not depend on K. Thus, the stability boundaries of Fig. 2 apply as long as the pilot model is atle to make a stable estimate of the aircraft's state. "Vertigo" Is an example of a circumstance in which the pilot estimator dons not converge.
In the present case, the pilot model experiences estimator diver- gence with dual control at high aA (Fig. 2b) because the signal- dependent neuromotor noise is free to grow without bound in the estimator solution (Ref. 9); however, if the neuromotor noise level is bounded (standard deviation of 1 in for stick motion and 0 . 7 in for pedal motion), a stable estimator solution is obtained for all ap.
PERFORKANCX CONTOURS In addition to predicting stability boundaries for vary- ing flight conditions, the pilot-aircraft model (Fig. 1) can be used to predict the statistics ot tracking errors and control usage within the stable regions. (Outside the stable regions. these statistics grow without bound). The method applied here is covariance analysis (Ref. 11). in which the aircraft is assumed to be driven by a gaussian disturbance (turbulence with an rms level of 1.52 rnisec (5 fps) and a correlation time of 0.3 sec), and the pilot-aircraft model is used to compute,the rms values of state and control perturbations which result Since the covariances of system variables are used to define the pilot model estimator gains, the same equations can be used to evaluate the statistical performance of the pilot. In order to use these equations, the pilot model estimator is assumed to be adapted at each flight
condition (defined by a,,); hence. when ap + l x A , only the pilot model
X, is defined as the expected value of
h he state covariance matrix
the products of the states,'i.e., X = E ( ~ & A ~ T ) . and the rlns values of the states are the square roots of the diagonal elements of X.
The control covariance matrix, U, and the associated rms values are similarly defined. The covariance propagation equations are detailed in Ref. 9.
Figure 3: Performance Contours for control with Lateral Stick Alone The contours for roll rate m s ( a ), side velocity rma (0 .
equivalent to sideslip angle timeg forward kelocity) , andv' stick motion rms (06 ) define surfaces of rms values in much 1 a t
---
the same way as a topographical map displays hills and valleys.
It 1s apparent that the valleys in Fig. 3 do not lie along the line of perfect adaptation, nor do they overlay each other with complete regularity. There are two reasons for this. Although the control gains are meant to minimize a weighted sum of state and control covariances, this is no guarantee that the covariances of individual components will be minimized at the same time.
There is a tradeoff between tracking error and control usage, so it is likely that decreases in one component all1 be accompanied by increases in another. The second reason is that the partial breakdown in the separation theorem leads to the possibility that alternate control strategies could yield lower values of control cost than the separately optimal control law.
does have tho effect of ap- Controlling to minimize o6 lat proximately minimizing o although increased control activity is P' required rn the region of the stability neck (Fig. 3). YaintainiIIg Perfornmnce Contour8 for Control with Uteral Stick Figure 4 perfect adaptation lor aA = 17 to 20 deg requires four times the and Foot Pedals W c .
In addition to sub- rms control motion that is used at low aA.
w stantial gain variation with flight condition (Table 2 ) . perfect adaptation also leads to increased control effort, again suggesting that the pilot may choose to change his control mode or to adapt in sub-optimal fashion.
Performance contours for a pilot model using both stick and pedals demonstrate that the addition of pedal control has little effect on a (except at low a), but it does reduce oV P (as might be expected) and increase stability margins (Fig. 4 ) .
to high RrIgles of The pilot model can retain low r~ and ov P attack with low control effort by fixing up in the vicinity of 10 deg (Fig. 5). a decidely sub-optimal policy by the criteria used for pilot model computation. Suboptimal or not, this adaptation approach could have definite appeal to the human pilot, for it provides perceptably low tracking error and control effort with ~inirnal adaptation; however, if the pilot wishes to fly to a6=30 deg and beyond. he must adopt a marked change in control strategy tc avoid spiral mode instability and estjmator divergence.
Reference 9 presents additional results for pedal-alone control which demonstrate that stability and low control effort can be maintained with the stick centered for all aA considered. Control Coage Contoura for Control with Lateral Figure 5 Stick and Foot Pedal.
MINIMUM-CONTROL-EFFORT PILOT Y l The combination of reduced control effort and reduced variation in control strategy suggests a hypothesis for minimum- conrrol-effort (MCE) pilot adaptation, which also can predict at what point the pilot is likely to switch his contrb; mode. Figure 6 illustrates two MCE adaptation patterns for the wind-up turn, with the heavy line tracing the corresponding aA-up relationship.
For aA below 17 deg, there is no significant lateral control re- duction associated with using the pedals as well as the stick, and the MCE model is "content" to use stick alone. The MC'E model is slightly overadapted at low aA and slightly underadapted at qA=12 dey; hence, the net amount of adaptation is lower than that implied by fully optimal control.
As aA increases, the MCE strategy is headed for a stability boundary; the pilot can avoid the boundary by adopting a more nearly optimal strategy, but this requires substantially increased control effort. As alternatives, he can either blend in foot pedal comnand (coordinated adaptation) or use the pedals alone (stick-centered adaptation) for lateral-directional control. The advantage of the first approach is that relatively good maneuveriug precision can be maintained with both controls without requiring counter-intuitive control style; however, the coordinated use of both controls is a difficult task. Judging from Table 4, the use of pedals alone may be a more easily learned task which results in modest increases in o and uv.
P Experimental results indicate that the MCE pilot model hypothesis does, in fact, describe a realistic pattern of pilot adaptation. Figure 7 is a partial time history of a wind-up turn maneuver in which a trained pilot is flying a ground-based simulation of the Subject aircraft. The aerodynamic model of the aircraft is the same as that used in the linear analysis, although the nonlinear, time-varying equations of motien drive the simulator. Below 18-deg angle of attack, the pllot controls with stick alone. As aA increases beyond 10 deg, stick motions and sideslip excursions build up. At o A = 18 deg, the pilot begins to use the rudder pedals actively, while his use of the stick is substantially diminished. This result tends to confirm Figure 6 Prediction of Pilot Behavior with Minimum- the MCE pilot model, elthough further validation is warranted.
Control-Eftort (MCE) Adaptation Model This r?rk was conducted under Contract No. X00014-754-0432 for the Office of Navel Research.
1. Tustin. A.. "The Nature of the Operator's Response in Manual Journal Control and Its Implications for-controller ~ e s i ~ n . " of IEE, Vol. 94, 1947, pp. 190-202.
2. McRuer, D.T., and Jex, H.R., "A Review of Quasi-Linear Pilot Models," IEEE Transactions on Human Factors in Electronics, Vol. HFE-8. No. 3, Sept. 1967, pp. 231-249.
3. McRuer, D.T., "Development of Pilot-in-the-Loop Analysis," Journal of Airsraft, Vol. 10, No. 9, Sept. 1973, pp. 515-524- 4. Obermayer, R.I., and Yuckler, F.A. .,,"Modern rantrol Syatem Theory and Human Control Functiona~ NASA CR-256, July 1965.
5. Kleinman, D.L., Baron, S., and Levison, W.H., "An Optimal Control Model of Human Response." Automatica, Voi. 8 , No. 3 , Figure 7 Results of Manned Simulation May 1070, pp. 357-383.
Kleinman, D.L., and Baron, S., "A Control Theoretic Model 6.
CONCLUSION for Piloted Approach to Landing, Autoautica, Val. 0 . No. 3, May 1973, pp. 339-347.
Whether or not a pilot experiences difficulties in maneuvering flight depends upon how he adapts his control strat- 7. Stengel, R.F., Taylor, J.H. Broussnrd, J.R., and Berry. P.W., egy to changing flight conditions. Stability boundaries plotted "High Angle of Attack Stability and Control," ONR-CR215-237-1, as functions of the aircraft's actual rr and the 3 assumed oy the April. 1976.
pilot model in forming a control strategy illustrate that the pilot's adapration must be very nearly optimal to maintain sta- 8. Broussard, J.R., and Stengel, R.F., "Stability of the Pilqt- bility in certain flight conditions. Consideration of statisticai Aircraft System in Maneuvering Flight," Proceedings of the tracking error and control usage within stable boundaries leads 12th Annual Conference on Manual Control, May 1976, pp. 778-791.
to the concept of minirnum-control-effort (MCE) adaptation in the pilot model. The KCE model provides a rationale tcr non-optimal Stengel, R.F., Broussard, J.R., Berry, P.U., and Taylor, J.H., 9.
adaptation which accounts for fundamental changes in the control "Modern Methods of Aircraft Stability and Control Analysis, modes selected by the pilot, such as the decision to use stick ONR-CR215-237-2, Yay 1977.
and pedals in a coordinated fashion rather than stick alone.
10. Tse, E., "On the Optimal Control of Stoch~etic Linsar Systems," IEEE Transactions on Automatic Control, Vol. AC-16, NO. 6, Dec. 1971, pp. 776-785.
1 1 Gelb, A,, ed., Applied Optima: Estimation, M.I.T. Press, Cambridge, 1974.