AIAA-98-48 Direct Adaptive Reconfigurable Control of a ailless Fighter Aircraft A.J. Calise, S. Lee and M. Sharma Georgia Institute of echnology School of Aerospace Engineering Atlanta, GA 333 Abstract his paper describes a neural network based direct adaptive control approach to the problem of reconfigurable flight control. A ailless fighter aircraft configuration with multiple and redundant control actuation devices is used to illustrate the level to which handling qualities can be maintained in the presence of large scale failures in the actuation channels. Of significance here is the speed with which recovery and maintenance of handling qualities can take place. he main advantage lies in eliminating the need for parameter identification during the recovery phase, and limiting the potential need for parameter identification to the problem of control allocation following the failure. A second by-product of this work is that the need for an accurate aerodynamic data base for purposes of flight control design can be significantly reduced. Moreover, the need for etensive off-line analysis, in-flight tuning and validation of gain schedules, and contingency coding necessary to handle a large set of possible failure modes is substantially reduced. hus the overall approach may also be viewed as a direct path to substantially reducing the cost associated with the development of new aircraft. Introduction raditional methods of flight control design typically imply a large amount of apriori data, and involve the development of a large array of gain schedules. While this approach has proved highly successful in the past, it represents an ever increasing burdensome and costly task when viewed in the contet of reconfigurable flight control design. he goal of control reconfiguration is to maintain handling qualities in the presence of a large universe of damage and failure modes. In contrast to robust control design methods of the past, the emphasis in control reconfiguration involves a combination of on-line parameter identification control redesign and/or adaptation for a degraded mode of flight. raditional approaches to flight control reconfiguration can entail four major and separate problems: ) Failure detection ) Failure isolation and characterization 3) System identification of the degraded system, and 4) Flight control reconfiguration to accommodate the degraded sensor / actuator / airframe configuration. hese are generally treated in the literature as isolated problems because each problem has fundamentally different objectives. hese problems become increasing more complicated when one attempts to accommodate a single failure among multiple potential sources of failures, and when the characterization of the failure is different for each source. Failures may occur in sensor suites, in actuators or possibly in the airframe as a result of battle damage. Also failures may be partial (such as partial loss or deformation of a surface area, increase in gyro drift, reduced pressure or voltage in an actuator, etc.) or they may be total. Even in the case of an isolated total failure, the nature of the failure can have important implications on how the system Professor, AIAA Fellow, Phone: (44) 894-745, Fa: 4-76, Email: anthony.calise@ae.gatech.edu. Graduate Research Assistants
should be reconfigured. For eample, a control surface may be frozen due to surface damage, or it may be free due to loss of torque from the actuator. he problem is further compounded by the fact that flight control reconfiguration implies that the control system gains must be redesigned in real time. he complications here are immense since this process entails gain scheduling, and it requires a reasonably accurate knowledge of the low frequency dynamics of the aircraft. Another disadvantage of the traditional approach outlined above is that the main problems identified must in general be solved sequentially. he design of each subsystem requires a tradeoff between generality and the speed and accuracy of the underlying algorithms. Also, these problems increase in compleity. For eample, failure detection is easier and can be performed more reliably than failure isolation, and so on. Consequently, the state of the art in these problem areas has a decreasing level of maturity as we progress down the list. here eists no reliable approach to flight control reconfiguration in response to multiple potential sources of large scale asymmetric damage to the airframe using present day linear control theory. At the opposite end of the scale, failure detection and isolation of sensor failures, and state estimation in the presence of degraded sensor data is a mature field with numerous successful implementations,. hus the most difficult challenge is to design a fault tolerant control system that can accommodate asymmetric failures in actuation or in the airframe. Recent eamples of approaches that employ parameter identification and on-line control redesign may be found in Ref s 3 and 4. Reference 3 places emphasis on problems associated with identification of time-varying parameters, and singularities that can occur due to insufficient ecitation, and uses a receeding horizon optimal linear regulator approach for on-line control redesign. his approach has been matured to the level of flight tests on a VISA/F-6 aircraft. his paper eamines the application of a neural network based direct adaptive control approach, with the goal of eliminating the need for parameter identification for purposes of stabilizing the aircraft and maintaining handling qualities in the immediate time period following damage to the airframe or failure in one or more of the actuator channels. he approach is taken from Ref. 5, and entails augmenting an eisting flight control system architecture with a neural network that provides online adaptation. he eisting architecture is based on the method of feedback linearization, and consists of the baseline control law as described in Ref. 6. While it is recognized that failure identification and isolation remain as an important aspect of the overall problem, the goal in this work is to relegate its use to areas such as control reallocation through adaptive modifications to the Control Effector Manager 6 (CEM). he results presented here are limited to the use of a fied effector manager, and therefore represent a worst case performance level that does not make any use of failure identification and/or online parameter identification. he outline for this paper is as follows. We first describe the manner in which the neural network (NN) based adaptive controller of Ref. 5 was adapted for application to the tailless aircraft configuration of Ref. 6. Emphasis is placed on first and second order forms that were used to match the corresponding response models employed to define the handling quality criteria for each control channel. his is followed by a summary of several eample simulation results. Conclusion based on these results are given at the end. NN Based Adaptive Flight Control Architecture he main purpose of the adaptive controller is to compensate for dynamic inversion error which would eist in case of actuator failures and/or airframe. he baseline controller of Ref. 6 computes effective control surface deflections in the roll, pitch and yaw channels. hese effective deflections at then distributed to the real control actuators by the CEM. Each channel has to satisfy handling quality criteria specified in the form of first and second order transfer functions. Below we outline the adaptation of the basic approach of Ref. 5 so that it become compatible with these criteria. Roll Channel: In the roll channel, the specification in Ref. 6 calls for first order response to a command in roll-rate about the aircraft stability ais. he roll rate dynamics can be epressed in the form p& s = f (, δ ) () where p s is the roll rate in stability ais, δ = [δ p,δ q,δ r ] is the vector of effective controls (one for each channel) and is the state vector. Handling qualities in the roll channel are defined by the first order transfer function psf ( s) = () p ( s) τ s + sc r
or p& sf = ( psc psf ) (3) τ r where τ r is time constant, p sf is command filter output of roll rate in stability ais, and p sc is the pilot command. Eq. s () and (3) represent a response model which generates desired output roll rate and its time derivative. he role of the response model is identical to that of a reference model in model following adaptive control (MRAC). Figure illustrates the roll channel implementation incorporating the response model of Eq. (), the baseline inversion logic ( f ˆ ) and the neural network. he resulting pseudo control input to the baseline inversion logic is u ~ p ( t) = K p p ( t) + p& sf ( t) uˆ adp ( t) (4) where u p (t) : input into inverse dynamics with adaptation (modified pseudo control) ˆ ( t ) : adaptation signal (NN output) u adp p& sf (t) : filtered roll acceleration For a linearly parameterized network, the adaptation input is defined by uˆ adp = N j= pj pj p wˆ ξ = wˆ ξ (5) where ξ p are the basis functions of the network, and N wˆ p R is its weighting vector. he update law for the weights can be derived as a special case of the more general case outlined in the Appendi w & ˆ p = γ ~ pξ p p (6) where γ p > is the roll channel learning rate, and the superscript ë~í denotes the roll rate error signal in the linear portion of the feedback loop. o insure boundedness of the weighting vector a standard e- modification was also incorporated as follows: w & ˆ p = γ p ( ~ pξ ~ p + η p p w p ) (7) where η p > is e-modification factor. p p sc Response Model p cf - p s &p cf u p p $δ Control K ~ p f $ Effector p - Manager $u ad Roll Channel NN u p q $δ $ δ r δ p δ q δ r Figure. Structure of the Roll Channel Adaptive Controller Pitch and Yaw Channels: Handling qualities in the pitch and yaw channels are specified in second order form. he transfer functions are described with side-slip angle ( β ) and angle of attack (α ) as output variables. he architecture for these channels is similar to the roll channel architecture, and the derivation of the network weight adaptation law in second order form can be found in Ref. 5, which uses a deadzone. A derivation which avoids the use of a deadzone is given in the Appendi. he desired response model has the second order form cf ωn = (8) s + ζω s + ω c n n where is used to denote α and β. he network weights are adapted according to the following equation with ~ =, cf ( e Pbξ η e w ) w i = γ i i i i + i i i (9) where e = ~ &~ () i K K Pi = [ ] [ ] b = () di p i K + K p i di K + K p i p i Kp i + K di K p i () 3
where the gains K p i and K d i are defined by choosing a natural frequency and damping ration for the error transient which has the same form of Eq. (8): K p i = ωd (3) Kd i = ζωd he subscript i in above representations denotes α or β in case of pitch or yaw channel respectively. Figure illustrates the structure of the yaw channel controller, and the pitch channel has a similar architecture. Neural Network Architecture: he architecture of the roll and yaw channel neural networks chosen for this application is illustrated in Figure 3. he pitch channel has the same structure, but uses onlyα, q, and σ ( u q ) as inputs to the basis functions. Note that in both cases, the network input u i depends on the network output u ˆ adi. his implies that a fied point solution must eist for u ˆ adi. As discussed in Ref. 5, eistence is insured by inserting the sigmoidal activation function σ ( u i ). he total number of weights employed in the pitch, roll, and yaw channels are 8, 7, and 7 respectively. he selection of inputs to the neural network is critical in any application, and it depends on the functional form the inversion error. Inversion error eists even in normal flight due to errors in the aerodynamic table, and due to approimations used in deriving the inversion function. In general, the inversion error is much more severe when a failure or damage occurs. he inputs into each channel were selected to supply sufficient information to allow the output of the network to nearly cancel any inversion error that might arise in either situation. Note that in general, inversion error depends on the pseudo control vector. In our implementation, it is assumed that this function dependence etends to only the pseudo control for the local channel. In general, one could feed all the pseudo control variables as inputs to all three networks. However, the fied point condition pertains only to the local pseudo control variable. && β cf $ δ $ p δ q βc Response Model β cf & β cf - ~ β ~ &β K p K d - u $ r δ Control f $ r Effector Manager $u ad δ p δ q δ r β β & Pitch Channel NN u r Figure. Structure of the Yaw Channel Adaptive Controller. 4
α α α ξi ( α, β, p, r, σ ) Π $w i ξ i β p r β p r M Π M ξ in Π $w in Π $w i $w in Σ $u adi σ( u i ) σ ξ ( α, β, p, r, σ ) in σ Figure 3. Structure of Neural Network Numerical Eample he model used for the numerical eample is the tailess fighter aircraft as described in Ref. 6. he maneuver is generated by the following sequence of commands: ) zero commands for. seconds )..one cycle square wave roll rate command with a period of 5. seconds. 3) one cycle square wave alpha command with a period 5. seconds 4) zero commands for 3 seconds he maneuver begins in trimmed flight at h = ft, Mach =.6. Numerous failures described in Ref. 6 have been investigated at this flight condition, but only the failure in which the left aftbody flap (ABF) is locked at 3 degrees at. sec is shown here. his failure mainly affects the lateral motion, since it produces an unepected yaw moment. Figure 4 illustrates time histories of selected states of the unfailed (nominal) case, the failed case with adaptation, and the failed case without adaptation. Note that in the unfailed case, the roll rate command results in nearly inverted flight (roll angle = 8 o ) at approimately 3.3 seconds into the maneuver. he roll maneuver is performed with near zero sideslip, and at constant alpha. he effect of ABF fail can be easily found in time history of β, which has to be maintained near zero through the flight. In the failed case, the longitudinal and lateral state responses with adaptation are very similar to the unfailed case. However, the responses without adaptation ehibit large errors in sideslip, and in roll rate during the alpha command portion of the maneuver. he alpha response is also highly underdamped in the case without adaptation. Figure 5 illustrates the time history of some of the important control effectors. In the unfailed case, the control histories with and without adaptation are essentially identical. he effect of the failure is mainly seen in the symmetric and differential ABF profiles. In general, in the failed cases, the case with adaptation results in reasonable control levels, with significantly fewer oscillations in the control histories in comparison to the failed case without adaptation. Figure 6 illustrate several weight histories in each channel for the failed case. All weights histories are well behaved bounded values. By comparison, in the unfailed case, the weight remained nearly zero. he most interesting result in this figure is fact that the e-modification results in a strict boundary on bias weight in the roll channel. Note flat bottom at - in the roll_wt plot. his result can be deduced by eamining the form in Eq. (7). he e-modification factor was chosen as η p =.5. For the bias term, ξ =, and the roll rate error is positive. So the epression becomes zero when w bias =-. 5
3 Normal w/ NN w/o NN 8 6 alpha (deg) beta (deg) 4 - - - -4 5 Pitch (deg) - - -3 Ps (deg/sec) -5-4 - 68 66 Airspeed (ft/sec) 64 6 phi (deg) 6-58 - Figure 4. Longitudinal and Lateral States 6
3 Sym EF (deg) - Normal - w/ NN w/o NN -3 5 Yaw VC (deg) 5-5 - 3 4 Sym ABF (deg) 3 Diff ABF (deg) - - -3 3 Pitch VC (deg) - Diff Canard (deg) 5-5 - - - -3 - Figure 5 Control Effectors 7
4 pitch_wt - -4 3 roll_wt - - -3 - yaw_wt -4-6 -8 - Figure 6 ime History of Selected Weights 8
Conclusion his paper shows that model reference adaptive control using linear in the parameter neural networks can be used as one approach to control reconfiguration. he proposed adaptive control, as a byproduct, can also compensate the inversion error that can occur in nominal flight conditions as well. Appendi For the stability analysis, one can separately treat each degree of freedom, and therefore we drop the subscript i in what follows. he error dynamics of second order system in Fig. can be defined as e& = Ae + b[ uˆ ad ] (A) where A = K p K d (A) = f (,, & δ ˆ) f (,, & δ ) (A3) Eq. (A3) shows the definition of inversion error and * ˆ represents optimized inversion error with the neural networks structures in Eq. (5). he difference * between ˆ and is bounded as ˆ * ε (A4) Under the eistence assumption of a fied point, estimation error is defined as ˆ* ~ uˆ ad = w ( t)ξ (A5) where * w ~ ( t) = wˆ ( t) w (A6) An equivalent epression for Eq. (A) with Eq. (A5) is e& = Ae + bw ~ ξ + b[ ˆ * ] (A7) For the i th degree of freedom, define the candidate Lyapunov function w~ w~ L = e Pe + (A8) where γ >. For K P > and K D >, A in Eq (A7) is stable, and for all Q > the solution of A P + PA = Q (A9) is unique and positive definite. Differentiating Eq (A8), substituting Eq (A7), and using Eq (A9), gives * ( ˆ ) L& = e Qe + e Pb ~ ~ & + W e Pb + W γ (A) For the adaptation in Eq (9) without e-modification term, and setting Q= I, Eq (A) reduces to L& = e e + e Pb( ˆ * ) (A) e + e Pb Using e Pe λ it follows that ( P) ( P) e e Pe L& + ε e Pe λ λ which is strictly negative when e Pe 3 { λ ( )} ( P) (A) (A3) > ε P (A4) his is sufficient to show, via the LaSalle- Yoshizawa theorem (see for eample, Reference 7), that e(t) and w(t) remain bounded. Furthermore, if ε=, (no NN approimation error), then lim e( t) =. t he adaptation law in Eq (9) is improved by the addition of the e-modification. his helps to further contain parameter growth, and to improve robustness to unmodelled dynamics. References. Willsky, A.,S., "A Survey of Several Failure Detection Methods," Automatica, Vol., No. 6, 976.. Massoumnia, M.A., Verghese, G.C., and Willsky, A.S., "Failure Detection and Identifica-tion, " IEEE rans. on Auto. Control, Vol. AC-34, No. 3, 989. 3. Monaco, J., Ward, D., Barron, R. and Bird, R., "Implementation and Flight est Assessment of an Adaptive Reconfigurable Flight Control System," AIAA-97-3738, Guidance, Navigation and Control Conference, August 997. 9
4. Chandler, P., Pachter, M. and Mears, M., "System Identification for Adaptive and Reconfigurable Control," AIAA Journal of Guidance, Control, and Dynamics, Vol. 8, pp. 56-54, 995. 5 Kim, Byoung S. and Calise, Anthony J., Nonlinear Flight Control Using Neural Netorks, AIAA Journal of Guidance, Control, and Dynamics, Vol., pp. 6-33, 997. 6. Wise, K.A., Brinker, J.S., "Reconfigurable / Damage Adaptive Flight Control for ailless Fighter Aircraft," AIAA Guidance, Navigation, and Control Conference, 998 (submitted for presentation ). 7. McFarland, Michael B., Adaptive Nonlinear Control of Missiles Using Neural Networks, Ph.D. hesis, School of Aerospace Engineering, Georgia Inst. Of echnology, Atlanta, GA, July, 997.