Neural network application to aircraft control system design

NASA Technical Memorandum 105151

,e

Neural Network Application to

Aircraft Control System Design

Terry Troudet

Sverdrup Technology, Inc.

Lewis Research Center Group

Brook Park, Ohio

and

Sanjay Garg and Walter C. Merrill

Lewis Research Center

Cleveland, Ohio

Prepared for the

Guidance; Navigat[o6 arid-Contr6i _Conference

sponsored by the American Institute of Aeronautics and Astronautics

New Orleans, Louisiana, August 12-14, 1991

(NASA-TM-105151) NEURAL NETWORK APPLICATION Nqi-27167 - Tn AIRCqAFT CQmTRnL SYSTEM DESIGN (NASA) 2O p CSCL OIC Unc1 as

G)/O_ 0026555

NEURAL NE'I_'ORK APPLICATION TO AIRCRAFT CONTROL SYSTEM DESIGN Terry Troudet Sverdrup Technology, Inc.

Lewis Research Center Group Brook Park, Ohio 44142 Sanjay Garg and Walter C. Merrill National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135 Abstract. The feasibility of using artificial neu- flight. Towards this objective, results are presented ral networks as control systems for modern, complex from a preliminary study of neurocontrol design for aerospace vehicles is investigated via an example air- an integrated aifframe/propubion model of a mod- craftcontrol design study. The problem considered ern fighter aircraft for the pilotedlongitudinal land- isthat of designinga controller for an integratedair- ing task. To gain insight in the characteristics of frame/propulsion longitudinaldynamics model of a the neurocontroller, linear analysis tools are applied to linearisedrepresentationsof the neurocontroller modern fighter aircraft to provide independent con- trolof pitch rate ancl airspeedresponses to pilot com- and to a baseline Hoe based controller. Closed loop mand inputs. An explicit model-following controller system performance and robustness of the neurocon- troller are evaluated and discussedin relation to the using H_ control design techniques is first designed to gain insightinto the control problem as well as Hoe based controller.

to provide a baselineforevaluation of the neurocon- The paper is organized as follows. The vehicle troller. Using the model of the desireddynamics as a model and the desired closed-loop dynamics are first command generator,a multilayerfeedforward neural discussed, and an explicit model-f31lowing H_ based network is _rainedto control the vehicle model within control design ispresented. The architecture used to the physical limitationsof the actuator dynamics. train the neurocontrolleris then presented and the This isachieved by minimizing an objectivefunction resultsof the neurocontrol]er are evaluated. A per- which is a weighted sum of trackingerrorsand con- formance and robustness analysisis then presented trol input commands and rates.To gain insight in the for the neurocontroller and the H.c based controller.

neurocontrol,linearised representations of the neuro- 2 Vehicle Model. The vehicle model con- controller are analyzed along a commanded trajec- sists of an integratedairframe and propulsion system tory. Linear robustness analysis tools are then ap- state-space representationfor a modern fighter air- pliedto the linearised neurocontroller models and to craft powered by a two-spool turbofan engine and the baseline B_ based contrdller. Future areas of equipped with a two-dimensional thrust-vectoring researchare identified to enhance the practical appli- and reversingnozzle.

cability of neural networks to _qight control design.

The flight condition used in thisapplication is rep- resentativeof the STOL (Short Take-offand Land- I Introduction. In the past few years,there ing) approach-to-landing task, with an airspeed of has been an increasinginterestin the control com- V0 = 120 Knots, a flight path angle of 70 = -3 deg, munity to exploitthe promise of axtificial neural net- and a pitch attitude of 8o = 7 deg. The linearised works to solve difficult control problems. However, dynamics of the vehicle model are of the form most of the neuralnetwork applications to control de- sign that have appeared in the literature [1,2], either t = As + a_., _ = cs; (I) dealt with robotic systems, or with control problems that are mainly of academic interest such as the in- where the statevector is verted pendulum problem. Only more recently have neural networks been applied to the control design = [u, w, Q, 8, h, N2, N25, P6, T41B] T , (2) of more complex problems, e.g. manufacturing pro- with cess[3]. The objectiveof thispaper isto investigate the applicability of neural networks as controllers for u = aircraft body axis forward velocity (ft/sec) aerospace vehicles with special emphasis on piloted w = aircraft body axis ver_ica_ velocity (ft/sec) Table 1: Desired Response Transfer Q = aircraft pitch rate (rads/sec) Funetlons.

8 = pitch angle (rads) Notation: .{ k(1/r)/[_,; w,] = h = altitude(it)

k(s + 1/¢)/(s 2 + +

N_ = engine fan speed (rpm) N_5 = core compressor speed (rpm) P6 = engine mixing plane pressure (psia) T,UB = engine high pressureturbine blade --R_ = 0--R_ = _ temperature (OR), _/sr.L ' OSEL and the control input vector is Actuator models were also used in control design and evaluation. The fuel flow actuator was modeUed _., = [WF, 6TV]r; (3) as I0 50 with GwF(s) -- _ + 10"8+ 50' (6) WF = engine main burner fuel flow rate (lbm/hr) with a maximum fuel flow rate IW.FI_ = 6TV = nozsie thrust vectoring angle (deg).

lO, O001bm/hr, and a rate limit IWF],,_ = The vehicle outputs to be controlled are 20, O001bra/hr/s. Note that the fuel flow here cor- responds to the perturbation from the trim value for

= IV, Q]r , (4)

the linear model. In this study, the value [WF_,._ is therefore chosen such that the total fuel flow limit where V is the aircraft velocity in ft/sec, and Q is will not be exceeded when a perturbation of a magni- the pitch rate in deg/s. The system matrices A, B, tude of WF,_, is commanded. The thrust vectoring and C are available in Ref.[4]. The open-loop vehicle actuator is modelled as eigenvaiues are: Gsrv(,)= --" (7) )q = 0.07, A_,3 = -0.09 ± j0.23, _'t = 1.06, $+ 15' Airfraxne modes As = -1.47 with a maximum thrust vector angle [6TV]m = = 10deg, and a rate limit16TVI,,,,= = 20deg/s.

and As a result, nonlir_earigiea appear in the control de- sign and evaluation in the form of actuatorsposition A6 = -1.40, A7 - -3.57 ,As - -6.96, and rate limits.

3 Hoo Control Design. Recent advances in A9 = -89.28 Propulsion modes.

H_ control theory [6]and computational algorithms Note that the airframe is statically unstable with a to solve for H_ optimal control laws [7 i have en- abled the applicationof thistheory to practical com- highly unstabie pitch mode. Open loop analysis also indicated a strong coupling in the response of the plex muhivariable control design problems. Many ex- controlled outputs f to control inputs fla. ample applicationsof Hoo based control designs for The control design objectiveis to design a control aerospace vehicles have appeared in recent literature [8-I0]. Prior to applying a neural network approach system that providesdecoupled command trackingof to control design for the example vehicleunder study, velocityand pitch rate from pilot control inputs with an H= baaed control law was obtained as a baseline aircraft responses compatible with Level I handling for the performance and robustness analysisof the qualities requirements [5].The desiredresponse dy- neurocontroller.

namics are selected to be of the form Within the framework of H_ optimization, the

$,. = + B. SEZ, = (5) control design problem for this example study was

formulated as the model-following problem shown in with gsEL = [VSZL, QsEL] z where VsE_ is the pilot Fig.l. The three transfer functions that are of in- velocity command in ft/s and QsEL is the pilot longi- terest for such a problem are the sensitivity func- tion S(s), the complementary sensitixdty function tudinai stick deflection in inches, and _c = [Vc, Qc] T, T(s), and the control transmission function C(s).

where the subscript "c" refers to the ideal response These representthe transfer functionsfrom the refer- in V and Q with units of ft/s and deg/s respectively.

ence commands to tracking errors,controlled vari- The system matrices A,_, B,_ and C,, are the state- ables, and commanded control inputs respectively, space representation of the ideal response transfer i.e. _(,) = S(,)_(_), _(_) = T(s)_c(s) and _c(s) = functionslisted in Table i.

actuator model, 5th order ideal response model, and C(s)G(s). In order to be able to influenceboth the ist order Ws and WT for the two controlled vari- low-frequency and high-frequency propertiesof the closed-loop system, itisdesirableto finda controller ables.The resulting 21st order H_ optimal controller obtained using the solution algorithm of Ref.[6] was K(s) which minimizes a weighted norm of a combi- nation of these three transferfunctions,i.e.: reduced to 13th order by residualising the high or- der modes. The maximum eigenvalue of the reduced order controller is [Aim= "- 6.83rads/sec, which im- plies that the controller can be implemented digitally

[ ]

Wc(j,,, )

with reasonable sampling rates. With this reduced-

(8)

order controller, the performance results in terms of The weighting functions Ws(ju_), WT(juJ) and closed-loop response, control effort and control rate Wc (ju_) are the "knobs" used by the control designer requirements, are shown in Figs.2and 3 fortwo cases to "tune" the controller K(s) such that the design of pilot command inputs: (i) V$_L = -20ft/s for objectives are met. For instance, choosing Ws to t > 0, Qs_zL = 0.5in for 0 < _ < 3sec and QsEz = 0in be large at low frequencies ensures good command for t > 3see; (2) VSEL = 20ft/s for t > 0 and QSEL trackingperformance, and choosing Wr to be largeat same as for command input case 1. From Fig.2,we high frequencies ensures robustness to high frequency note that forthe pilotcommand input in case I the unmodelled dynamics. Wc is chosen to ensure that velocity response obtained with the controller isquite control actuationbandwidths, as wellas rateand de- close to the ideal response, and the control input com- flection limits, are not exceeded in the control design.

mands and rates are reasonable. For the pilot com- For the aircraftexample, the integrated design mand input in case 2, the pitch rate response isquite model, P(s), in Fig.1 consisted of the vehicle model similarto that for case I; however, the velocityre- (i)and the actuatormodels (6)and (7).The idealre- sponse is degraded from the ideal response. Case 2 sponse model, R(s), in Fig.1 consisted of the desired is demanding in that the pilot is commanding the _..A__ model dynamics (5) with a high pass filter (,+0.i)on aircraft to pitch up as well as accelerate to a higher the pilotpitch rate command. This high pass filter velocity. As seen in Fig.3, the maximum fuel flow is added to reflect the factthat pitch rate cannot be rateis commanded by the controller foran extended commanded in steady-state.The outputs _ and the period of time in order to track the ideal response.

errors_ were scaled by theirapproximate maximum Note that the closed-loopsystem remains stable in values to be commanded by the pilotwith V ° = 20 the presence of the actuator limits, and the aircraft ft/secand Qo = 3 deg/sec. The sensitivity weights response tracksthe ideal response inthe steady-state.

Ws and the complementary sensitivity weights I_ 4 Neurocontrol Design. Although the were chosen as listed in Table 2.

strength of neural networks liesin their ability to handle nonlinearities in the controlled dynamics, the Table 2: Weights for /-/:_ Control Design.

control design fora linear aircraft model isbeing con- Controlled W s WT sidered in this paper to gain insightinto the neu- Variable ral network characteristics by using linear analysis tools.As discussed earlier, nonlinearities of concern V 3:_.50s._- i000 _t35.01,-_i 0.0022,,.t- 1 forpractical control design, such as actuator position and rate limits, are included in the design criteria.

Q ¢.z0,-_ _000 0.044, 67.02, _- 1 0.00044* -r ], The architecture fortrainingthe neurocontroller is This choice of H's and WT was based on the per- shown in detailin Fig.4. For each pilot selected tra- formance and robustness arguments discussed earlier.

jectory _SEL(t), a commanded trajectory _c(t) isgen- The weights Wc consisted of the controlcommands erated from (5). Prior to training,the commanded and ratesweighted by the inverse of actuator position variables it(t) are discretized and scaled to _(tt) us- and rate limitsfor WF and 5TV listed earlier. Note ing the same scaling as for the H_ design. Like- that the combination of tracking errors _ and aircraft wise, the dynamics of the actuators and of the vehi- clemodel are discretized and scaled after normaliz- outputs _ is used as a controller input instead of and ideal response, _, to avoid control saturation ing the control input vector by itsmaximum value due to large pilotinputs and undue amplification of (!WFI,r,a, , [_2_r[,_az). As for the Hoc design, the inadvertentpilot command noise. trsckir_g error at time t_ is the error between the The Hoc control design plant as discussed above scaled vehicle output vector and itsdesired scaled is of 21st order consisting of the 9th order aircraft valueat the same time t#,, i.e. _,(tt)= ,_(_k)--_*(tk).

model. 2nd order H'F actuator model, istorder _TV However, because ofthe time-discretization ofthe ac- corresponding input to the neural network. Other

tuator dynamics and vehicle model dynamics within

alternatives would be to low-pass filter the integral the trainingloop,a commanded control input vector error itself, or to remove the scalingfactor I/tt from generated at time _k by the neurocontroller willonly the time-averaged error as learningtakes place. Be- affect the aircraft output at time t,+_. Consequently, cause oftheirpotential to improve steady-state track- the trackingerrorat time tk+2 definesthe magnitudes ing, these latter approaches should be considered in of the weights increments at time _k- Said in another future neurocontroldesigns.)In Fig.4,the symbol A way, due to the time-discretization of the dynamics, represents a latch that isclocked every 6¢ seconds to the internalrepresentation of the neurocontroller has update the inputs to the neurocontroller, the actua- to be updated at time tt on the basis of information tors and the vehicle model. A network configuration which will be only available at a latertime _k+_. To of 15 neurons in the first hidden layer, and 10 neurons be consistent with the time-discretized design,knowl- in the second hidden layer,ischosen for the neuro- edge of the anticipated commanded vehicleouput at controller. Each neuron of the neurocontrollerhas time th+2, _(_t+2), is explicitly provided to the neu- the activation function: ral network at time _k dumng _eaining by means of the commanded error_,(tk) _(t_+2)-£°(_k). This = _anh(-); (9) procedure ensures that the proper action will be com- manded by the neurocontroller at time _k to achieve which limits its output y to the interval[-i,+i] for the desired tracking at time tk+_ during training.

any input signal z. For a given set of weights of When operatingthe trainedneural network in closed- the neural network, the two output neurons yieldthe loop however, the tracking error _z(tk) willbe used normalized commanded control input vector as input to the neurocontr011er instead of the com- manded error _,(tk)which is not available in the real . WFc 6TVc simulation because itrequiresknowledge of f'_tu_ pi- e'_(z,) = [IWFI_.,' liT-'_._ ] (I0) lot command inputs. This means that the trained which is applied to the scaled actuators. After a small neural network willbe trackingthe exact commanded time-interval 6_ = gt+l - tt, the actuators yield the trajectorywith a two-step time delay during simula- normalized actuator control output vector _(_t_.l) tion evaluation. Since the neurocontroller operates in as ¢ efined by (6) and (7). The normalized actuator the continuous time domain, thistwo-step time de- control output vector _2[(tt+l) is subsequently ap- lay should not adverselyaffect performance in closed- plied as input to the scaled vehicle model over the loop evaluation. That such isthe case was confirmed time-interval [tt.._, £t+_], and changes the state vec- by the closed-loop evaluation results to be presented tor of the vehicle model from £(tt+l) to $(tt+_). In later.

order to maximize the tracking performance while As shown in Fig.4, the two commanded control in- minimizing the costs associated with high control ef- puts are calculated by a two hidden-layer feedforward fort and high control rate requirements, the neural neuralnetwork with eightinput units(or four pairsof network is trained to minimize an objective function fan-out units associatedto the Q and V variables), that includes tracking errors, control effort and con- and two neurons in the output layer. These pairs trol rate requirements consist of the scaled output vector £J(_t); the com- manded error _z(_) between the scaled vehicleout- 1 T ,* put vector at time _k and its desiredscaled value at J(_*) 5( _, (Z*+_)-,-e,(_) + time tk+2; the discretetime-derivative of the track-

ing error, _,(tk); and the time-averageof the track- ) + )

(n)

ing error, i/tkf_o_(t)dt. As in the Hoc design, the where [,(_t+_) is the error between the scaled com- motivation behind using the combination of _(tt) manded vector _ (_t+_) and the scaled vehicle output and _,(it)as inputs to the neurocontroller, instead of _'(_t+_). The matrices _, _ and _ are 2x2 diago- £"(_k)and _ (_t+_), is to allow the neural network to reconstruct the command without direct feedforward nal matrices whose coefficients can be adapted so as ofthe command. The role of the error rates _,(it)is to modify the characteristics of the neurocontroller to provide the neural network with lead information, in order to achieve a practicalperformance/control- and the time-averaged error feedback i/tkf_o'[, (t)d_ effort trade-off. Expression (11) isof the same form is to minimize the steady-state tracking error for step as the objective function used in Ref.[11]to design a neurocontrollerfor the same airframe/propulsion command inputs. (The motivation behind scalingthe system, but without s.imulating the actuator dynam- f_[,(t)dt intoits time-average was to integral error #0 ics within the trainingloop. In Ref.[11_, itwas found improve backpropagation learning by bounding the neural network, a weight increment is given by that training the neural network to minimize only the tracking error led to high control effort and high con- 6wj,(p+X):_,_ = ao_,pA_,(p+x) (12) trol rate requirements. When the actuator dynamics were included in the closed-loop evaluation, this re- where a is the steepest descent coefficient, oi,p is sulted in a highly oscillatory pitch rate response and the output of the i th neuron of the pth layer, and a limit cycle behavior in velocity/fuel-flow response.

A_,(p+x ) is the effective error at the output of the However, a satisfactory trade-off between tracking j_^ neuron of the (p + 1)th layer. The effective errors performance and control effort could be achieved with Ak,(p+_ ) in the (p + 2) t_ layer are backpropagated to finite values of A and _. Since the bandwidth limiting effect of the actuators is now explicitly taken into ac- the (p + 1) th hidden layer to give the effective errors count within the training loop, much improvement in in the (p+ 1) th layer, as performance/control-effort trade-off is expected from the minimization of (11). L_,(p+x) = .fl(z_,(v+x)) x Sj,(p+x) The backpropagation algorithm [12] was used to with find the set of weights of the neurocontroller which minimize the objective function (11) over the set of pilot input commands. In order to backpropagate (11), a single layer feedforward neural network (per- and where p(z_,{p+x)) is the value of the derivative of ceptron) was used in place of the vehicle model in the the neural activation function for an input za,(p.x ) of training architecture of Fig.4. This neural network the jta neuron of the (p ÷ 1) ta layer. In the output emulator had 11 input units (corresponding to the layer, the effective errors A_._r are the gradients of two normalized actuator control outputs and to the the objective function (11) nine state variables of the vehicle model), and g lin- en, output neurons (corresponding to the nine state (14) variables of the vehicle model). Likewise, two feedfor- ' Oo.L _ ward neural networks were used to emulate the dis- cretized dynamics of the actuators. The second-order Whenever the neural activation is not differentiable dynamics of the fuel flow actuator were simulated by over the range of_]l possible neuron input values (as a three-layer network of linear and linear-thresholding is the case for the linear-thresholding neurons used for neurons. As shown in Fig.5, constraining fuel flow ef- emulating the actuators),/_ should be constructed to fort and fuel flow rate requirements is achieved by preserve the characteristics of a monotonous contin- thresholding the linear neurons of the two last lay- uous function. For example, the linear-thresholding ers. The first-order dynamics of the thrust vectoring activation function which is defined as actuator were simulated by the two-layer neural net- f,,,(') = • _fI_I <-I, work shown in Fig.6. Constraining the effort and rate requirements of the thrust vectoring actuator is .f,,,(_) = i _f• >__ I, achieved by means of linear-thresholding neurons.

The layers of an (N _- 1)-layer neural network can f,,,,(_) = -1 _f• <_-1. (IS) be labeled by an index p from 0 to N, p = 0 de- is clearly not differentiable over I-oo, +oo]. Since noting the input layer. Layer p has u(p) elements fu_ is piecewise differentiable, it would seem a-priori consisting of [u(p) - 1] neurons and one unit that is natural to define rum _ ruM(z) - 1 if fz I < 1, and permanently _on" and used to define the thresholds ruM(z) = 0 if Izl > I. With this definition of rum of the neurons of the (p + 1) th layer. With symmet- however, any time a neuron input zo would take a ric activation functions of the type (9), the threshold value outside of [-1, +1] during training, the neuron of a neuron is defined as the value of its input signal output would remain trapped to 1, if z0 > 1, or -1, if above which its output is positive, and below which z0 <: -1. For such neuron input values, the weights its output is negative. During training, the thresh- of the incoming connectionswould remain frozen, and olds are updated with backpropagation in a manner thiswould bLas the learning.In order to permit the similar to the updating of the weights [12].

neurons fullaccessto the output state space during The weight connecting the i th neuron of the pth training, ]_h_ isthus defined as layer to the jr^ neuron of the (p+ 1) th layer is denoted as wj,(_+x):i,p. The threshold of the jth fUM(z,,p) = i if lz,,_l _< I or if z,.p.S,,p < O, neuron of the (p+ 1) _u layer thus corresponds to /,,_,(_,._) = 0 o_h_._e. (16) w_,(p.1):_(p),_. For a single feedforward pass of the trajectories.

which willensure that the weights be properly in- 5 Neurocontrol Performance. The eval- cremented during training. S_,p which appears in uation architecture of the neurocontroller in closed- (16) is definedin (13). The serial arrangement of the neurocontroller, the neuro-emulator of the actuators, loop is shown in Figure 8. The neurocontroller was and the neuro-emulator of the vehicle model, consti- tested on step pitch rate input commands, different from the doublets used in training. The input com- tutes a largerneural network through which the ob- mands chosen to illustrate the neurocontrol perfor- jectivefunction (11), J(tk), can be backpropagated mance were defined by the step pitch rate command through time [2]using Eqs.(13)-(16). The connec- tions between neurocontrollers and neuro-emulators QsgL(t) = 0.5in for t _< 3see, Qs_L(t) = 0 for > 3see; applied simultaneously with one of the fol- which were used as backpropagating channels are in- lowing classes of step velocity commands: VsE_(t > dicated in Fig.7 over a period of three time-steps 6_, 0) = -20ft/sec (case 1); VSZL(¢ > 0) = 20ft/sec and the weights increments are calculated using (12).

(case 2).

The commanded trajectoriesused to train the When training the neural network without giving neural network were generated as follows. The pi- any consideration to the cost associated with large lot selectedpitch rate was a doublet centered at a control efforts and large control rates, i.e. _ = _ - {} time tc between 2.5s and 5s, with the characteris- in Eq.(11), the neurocontroller learns very satisfac- tics: QSF.L(t) = Qo for $ < to; QSzL(t) = -Qo torily to track the commanded outputs. However, for 2to >_ t > to; QSEL(_) = 0 for $ > 2_c- Note the fuel fiow is quite irregular, and both control in- that QszL corresponds to pilot longitudinal stick de- put commands generated by the neurocontroller ride flection with units in inches. The pilot selected air- the actuator rate limits. A study of the trade-off frame velocity was a step function characterized by between tracking performance and control effort re- VszL(t) = 0 for Z < 0 and VSEL(t) = Vo for t > 0.

quirement was conducted by training the neural net- The maximum intensities IQol and IVol of the ran- work with _ and _ of the form A = diag[AwF, A6TV] domly selected input commands were bounded by and _ = diag[/_wy,/_6rv], with the same training Q,_= = 0.5 in and V,_,, = 20 ft/s. This maxi- characteristics and the same matrix elements of mum value of QSEL corresponds to a maximum pitch used earlier. As in Ref.[11], the tracking error is found rate command of about 3 deg/sec. Random sets to actually decrease for small increases in values of _, of input trajectories were generated from uniform distributions of Qo, tc and V0 over [-Q,,_,, Q,,4z], and _.

The results from this trade-off study are shown in [2.5s, 5s] and i-V,,a=, V,_a,] respectively. The com- manded variables Q,(t) and V_($) were filtered from Figs.9-10 for cases 1 and 2 with the choice of param- QsEL(t) and V'.gL(¢) over a period of 12s with a eters _ = diag[pv,pQ] = d/ag[2000,20], A = 0.0I, time-step /_t = 0.02s. These types of commanded -- 0.1. The pitch rate response follows the com- trajectories represent typical pilot command inputs. manded trajectory very smoothly, in spite of the thrust vectoring requirement ETV reaching the ac- Training was performed in two phases. In the tuator rate limit at the initiation and end of the gross-tuning phase of the training_ a set of 4000 com- command. However, within the proposed training manded trajectories was randomly generated, and the scheme, any attempt to lower the rate of thrust vec- synaptic weights were updated at every time tk = k/St toring by increasing/_6rv resulted in a loss of track- after backpropagating J(tk)through the neural net- ing performance. In case 1, neurocontrol is very sat- work. This was done once for each trajectory of the isfactory both in pitch rate and velocity response. In training data set with a steepest-descent coefficient case 2, neurocontrol tracking is still very satisfactory a = 0.001. In the fine-tuning phase of the train- in pitch rate response, but is slightly less satisfactory ing, the synaptic weights were updated following a in velocity response owing to the physically demand- moving-window scheme: at every time tk, the weights ing effort of increasing simultaneously aircraft speed were incremented after backpropagating through the and pitch angle.

neural network the time-integral of the objective In order to estimate the effect of providing the function calculated over n_ sampled points or dur- neurocontroller with lead information during train- ing a period of r_.6t seconds, i.e. _x J(tk+i). As ing, the above process was repeated without feed- the width of the moving window was progressively in- ing the discrete time-derivatives of the tracking er- creased to cover an entire commanded trajectory, i.e.

ror, i.e. $,(tk), to the neural network during train- r_ = 12sec/O.O2sec = 600, the steepest descent co- ing. Without constraining control efforts and rates efficient a was progressively reduced from the initial (A = _ = 0), the tracking performance deteriorated value of 0.001 to 0.0001. In total, the neurocontroller significantly with the appearance of some ringing in was trained with approximately 10,000 commanded dynamics, parameter changes due to change in flight

the pitch rate response and a limit cycle behavior in

conditions and the margin of errorassociatedwith es- the velocity/fuel-flow response. The fuel flow require- ment and fuel flow rate were both much more oscilla- tirnating model parameters based on analytical tools and experimental data. A classic specification for ro- tory than when lead information was provided to the bustness,alsoused in the military specifications for neurocontroller during training. The fuel flow rate oscillated between the maximum and minimum rate design of flight control systems [5], is that of stability limit during and beyond the 12 sec training period. A margins, specifically gain and phase margin [14]. The toolsto determine these margins are fairly well devel- more oscillatory, behavior was also noted for the con- oped for linearsystems - classical Bode analysisfor trol effort and rate of the thrust vectoring. However, the situation improved significantly when constraints single-input single-outputsystems [14]and modern on control efforts and rates were applied during train- singularvalue and structuredsingular value analysis for multi-input multi-output systems [15, 16]. For ing. In this case, a satisfactory trade-off between nonlinearsystems, one way to determine robustness performance and control-effort was reached for val- ues of A and _ in the vicinity of AwF -- AsTv = 0102, is to conduct Monte Carlo type simulations using all /_wF "- 0.2 and tzs_v = 1.0. The results showed possible combinations of modelling uncertainties that a similar velocity/fuel-flow response with and with- can be expected. Another approach is to linearise the out lead information, but showed a noticeable degra- closed-loop system at various pointsalong a given tra- dation in the pitch-rate/thrust-vectoring response in jectoryand then apply the linearanalysistools.The latterapproach is lesstime consuming and provides comparison to the situation where lead information more insight into the characteristics of the nonlinear was provided to the neurocontroller. This degrada- tion in tracking performance resulted from the large system. Furthermore, this latter approach allows to value of the pitch rate constraint _sTv (one order perform a similaranalysisfor the linear Hoo based reduced order controller and the nonlinearneurocon- of magnitude larger than before), which was needed to decrease the tracking overshoots. In summary, troller, forsmall perturbations along a given trajec- lead information enabled the neurocontroller to over- tory.

come ringing and limit cycle behavior while increas- Since the vehicle model used in this analysis is ing tracking performance. Thus, within the present linear, only linear small perturbation models of the scheme of neural computation, any dynamic char- neurocontroller at different points along a given tra- acteristics required to achieve desirable performance jectory are needed to perform the type of robust- had to be incorporated into the neural network with ness analysis discussed earlier. Considering the closed an appropriate choice of inputs. An extension of the loop system response with the neurocontroller for the present neura_ architecture to generate such dynamic case 2 command inputs, corresponding to the results characteristics could be a feedfoeward neural network presented in Fig.10, the linear neurocontroller models with intermediat; feedback inputs, i.e. a recurrent were generated at times t = 0.5, 2, 4, 6, 8 and 10 sees.

neural archhecture as a dynamic neurocontroller.

The first three points in time correspond to tran- 6 Analysis of the Controllers. From sient control activity whereas the last three represent a comparison of the closed:loop response for the steady-state type command tracking with monoton- two command cases with the H_: based reduced or- ically decreasing tracking error. Note that the neu- der controller (Figs.2 and 3) and the neurocontroller rocontroller as shown in Fig.8 consists of 4 sets of (Figs.9 and 10] it is evident that the neurocontroller scaled (normalised) inputs: the time-averaged errors provides improved command tracking although at the 1/tfto_(t)dt, the error rates _ (t), the errors _ (t) and expense of increased control rate activity, both for the controlled outputs _'(t). The scaling, the time- 6TV and WF. Also the pitch vectoring control re- averaged error and derivative action were embedded quirements are higher and the fuel flow activity ex- within the neurocontroller during the linearisation hibits oscillatoR' behavior for the neurocontroller.

process to find a control structure consistent with the structureof the Hoe based controller which has Note that the results presented so fax have been with the nominal vehicle model used for control de- only the errors (@) and the controlled outputs (_,) as the inputs. The frequency response Bode plots of sign. Since thismodel is only a simplified version the lineaxized neurocontroller models were obtained of the vehicledynamics, an important criterion for design of conzrollers for flight vehiclesis that of ro- to gain insightinto the characteristics of the control bustness. Robustness is defined here as maintaining action. Bode gain plotsforthe thrust vectoring angle performance and stability in the presence of uncer- (6TV) response to all the inputs to the controller lin- earized at t -- 0.bsecare shown in Fig.11.The Bode tainties associatedwith the modelling process.Mod- ellinguncertainties are due to neglected high order gain plots forthe B'_ based controller are shown in troller and the neurocontroller is the compensation Fig.12. An example variationin the neurocontroller from the measurements of the controlled plant out- characteristics with the change in magnitude of the puts (V and Q) to the control inputs (WF and 6TV).

inputs to the controller along the trajectory isshown As mentioned earlier, this compensation is a %on- in Fig.13 interms of the Bode gain plotsforpitch rate stant" (varying with input magnitude) gain from error (eo) to thrust vectoring angle (6TV) response.

the controller inputs to outputs for the linearised Fig.13 shows that the neurocontroller gains decrease neurocontroller. However, as seen from Fig.12, the with time. This type of behavior was exhibited by all B'oo based controller has dynamics associated with the other input/output Bode plots of the linearised this part of the control compensation and also has neurocontroller models. So in effect, the neurocon- higher compensation gains than the linearisedneu- troller can be thought of as a set of linear controllers rocontroller (Fig.t1).The controller structure used with the controller parameters being a strong func- for the B'oo and the neurocontrol design is consis- tion of the magnitude and direction(relative magni- tent with the classical approach of flight control de- tude) of the inputs to the controller. Note that since sign wherein an inner loop compensation (i ---* _) the H_ based controller is linear,itsdynamics are is designed firstto provide stability augmentation independent of the magnitudes of the controller in- and place the augmented plant dynamics within the puts.

handling qualities specifications; and then the outer From Fig.ll we note that the neurocontroller ex- loop compensation (8 ---* fi)is designed to provide hibits PID (Proportional + Integral + Derivative) decoupled command tracking to reduce pilot work- control type behavior from the error inputs (ev and load. The significance of the difference between the eQ) to the thrust vectoring angle (_TV) output. This Hoo based controller and neurocontroller "inner loop" was also the case for the ev and eQ to WF response, compensation was studiedfurtherby considering fail- and was true all along the trajectory as shown par- ures in the outer compensation loops, i.efailurein tially (for eQ input) by the plots in Fig.13. This dy- the error sensors. Eigenvaiue analysis showed that namic behavior of the neurocontroller for the error the closed-loopsystem with Hoc based controller will inputs is directly due to allowing feedback of the in- remain stable for failures in any or both of the error tegral and derivative errors. Since no such dynamics sensor loops whereas the closed-loopsystem with the were added to feedback of V and Q to the neurocon- neurocontrollerlinearized at t - 0.05 sec was unsta- troller, the neurocontroller exhibits only proportional bleforfailure in either or both of the errorloops. The type behavior from these inputs.

response of the closed-loopsystem for case 2 com- Comparing Figs.ll and 12, we first note that the mands and failure in the eQ loop is shown in Fig.14 magnitude of the ev and eQ to 6TV response is much for both the H_¢ based controller and the nonlinear lower fJr the Hoc based controller compared to the neurocontroller. The H_ based controller still tracks particular linearized neurocontrolter models. This the velocity command and provides stableresponse in was also true for the error in!_uts to WF response.

pitch ratewhereas the neurocontroller givesa highly This result is a further confirmation that the con- unstable response. So the Hoo based controller is trol effort and control rate requirements to track a using the plant measurements (_) in a manner con- given set of commands will be higher for the neuro- sistent with the cl_ical idea of providing inner loop controller. Although the dynamic behavior of the Hoc plant augmentation. How to formulate the neurocon- based controller is more complex than the neurocon- troldesign problem such that the resulting controller troller, some integral and derivative action is evident exploits the plant measurement information to pro- in the eq to _TV response. The integral action was vide inner loop stability augmentation is an area of built into the Hoc based controller through the choice future research.

of the sensitivity weighting, however, unlike for the neurocontrol design the error rate information was Stability margin analysiswas performed forthe lin- earized neurocontrollermodels and the Hoo based not explicitly provided in the Ho_ controller. The Ho_ controller to quantify robustness of the control de- control synthesis procedure is such that it naturally builds in the amount of lead (error rate) information signs. Among the linearised neurocontroller models, into the controller that is necessary to meet the con- stability margins were worst for the one linearized around t - 0.05 sec,so only those resultsare dis- trol design objectives specified through the weighted cussed here. Structured singular value analysis [17] quantities. As evident from Figs:]]_ i2, the Ho¢ showed that the H= based controller has guaranteed based controller provides leadat a lower frequency in multivariable gain margins of-3.7 to 6.6 dB (gain fac- the eQ to 6TV response as compared to the linearized neurocontroller. tor of 0.(35 to 2.1) and phase margins of--.30 deg for simultaneous loop gain or phase changes at the plant Another differencebetween the H_ based con- output(V andQ)and margins of-3.8 to 7.2 dB and ismostly subject to the generalization ability of the backpropagation algorithm (in the present context, ±32.5 deg at the plant input (WF and 6TV). For the linearized neurocontroller, these multivariable mar- generalizing means providing 8table control for "off- gins were only -0.6to 0.6 dB and :i:3.4 deg for loop nominal" _ehicle model dynamics _ha_ were no_ _ed gain variations at the plant output, and -0.9 to 1.1 d_ring _ining). Because backpropagation is known dB and ±6.6 deg at the plant input. The low stabil- in general to have a limited ability to generalize [18], the robustness of the neurocontroller as trained in itymargins with the neurocontroller are indicative of poor robustnessin that the closed loop system might section4 could have been expected to be quite lim- ited.

be unstable for small uncertaintiesin the plant dy- namics. Since the multivariablemargins can some- Within the neural architecture of Fig.4,one possi- times be conservative, the stability robustness of the ble approach to enhance the robustness of the neu- closed-loopsystem was further evaluated using the rocontroller may be to include allmodelling uncer- more classical approach of Ubreaking" one loop at s tainties in the trainingdata set.Another possibility time, i.e. one loop open and other loops closed. This might be to modify the objectivefunction(11)used to one-loop-at-a-time analysisconfirmed the poor stabil- trainthe neurocontroller to reflect some of the char- ity margins of the neurocontroller.The closed-loop acterkstics of the functional (8) which isminimised in response of the system with the Hoc based controller the H_ based control design.

and the nonlinearneurocontroller for an added delay 7 Conclusions. The applicability of neu- of ra = 0.05 sec in the two control channels (WF ral networks for flight control design was analyzed and _TV') isshown in Fig.15. This value of r_ cor- through the process of designing a model-following responds to a phase loss of 8 deg at a frequency of 3 neurocontroller for the example of an integratedair- rads/sec,which is the frequency that corresponds to frame/propulsion model of a modern fighter aircraft the guaranteed multivariablephase margin of 6.6 deg for the piloted longitudinallanding task. For this for the linearized neurocontroller, and itisquite rep- two control inputs - two control outputs example, resentative ofthe kinds of time delays to be expected the control design problem was set up as the task of in practical implementation of complex Right control following the trajectories generated from a model of designs. From Fig.15 we note that the Ho_ based the desired vehicle response dynamics to pilot com- controlshows very little degradation in trackingper- mand inputs. The neurocontroller was trained by formance in the presence of time delay,whereas the simulating the non-linear dynamics of the actuators neurocontrollerexhibitslimit cycle behavior in the includingpositionand rate limits. The choice of the pitch controlled variable. A factor that may con- objective function and itsminimization over entire tributeto thislack of robustness isthe factthat the commanded trajectories were found to be critical to neuro-command rides the thrust vectoring rate limit the neurocontrol design. A satisfactory trade-off be- during initial and final transients.In contrast, the tween tracking performance and control effort could neuro-command is well below the fuel £ow rate limit, be achieved by an appropriate selection of the weights which results in robust velocitytrackingin the pres- of the objectivefunction.

ence of time-delay. Improving phase robustness char- The neurocontroller shows better performance acteristics of neurocontrollers and investigating their than a baseline H_ based controller designed forthe gain robustnesscharacteristics are areas that warrant same command trackingproblem. HoweveL the neu- furtherstudy.

rocontroller commands larger controlrates than the H= based controller, speciallyfor thrust vectoring In the neurocontroldesign,the weights of the neu- ral network (the in_.ernal _presen_s_Wn of the neu- where the neuro-cornmand rides the thrustvectoring rocontroller) were chosen to minimize the objective ratelh'nit during initial and final transients. The pos- sibility of improving the practicality of the proposed function (11) over an exhaustive set of pilot input commands to the nominal vehicle model by using neurocontrol design methodology, to prevent neuro- the backpropagation algorithm. No information on commands from riding actuator rate limits without signii_cant degradation of tracking performance, is modelling uncertainties and no constraint on "off'- currently being investigatedin light of the results nominal" actuator dynamics were provided to the from the minimization of the H_ based control de- neural network during training. Withou_ any con- straint other than control effort and rate limits, the sign.

trainedneuralnetwork learned to control the nominal To gain further insight into the neurocontroller characteristics, linearized small perturbation repre- vehicle model as e_ciently as possible(and within the sentations of the ne_rocontrollerwere obtained at resolution ofbackpropagation). Consequently,the ro- bustness ofthe neurocontroller as trainedin section4 different time points along a trajectorycorrespond- ing to a demanding set of tracking commands. A lin- B.A., "State-Sp_e 5olution_ to Standard H= and Hm Control Problems", IEEE T_n_. on A=tom_tic Coat.el, Vol.34, No.S, ear analysis of these linearised neurocontroller models Aug. 1989. pp.&_l-847.

and the H_ based controller showed some differences [7] "MATEIXx Robust Control Module", Intesr_ed Systems in the controUer characteristics. The major difference Inc., $sn_s Clara, CA, Dec. 1989.

between the two controllers is that the H_ based [8] Kammez, I., Kha_onekar, P.P., and Robcl, G., "Design of Locs_z_ Captm,_ an¢t T.rs_ Modes _or a Lateral Autop_lot controller is a "fixed" dynamic controller whose dy- using Hm Symth¢_', IBEB Co_t. S_. Ma_., VoL1O, No.4, namics are "automatically" determined through the pp. 13-_I, Ju_e 1990.

synthesis procedure such that the specified criterion is [9] Reichar_, R., "Applic:a_em of Boo Control to Missile Au- top;lot Design", _ P_4_m" 89.-3560, Guidance, Nav_on met in the best possible manner, whereas the neuro- and Control Conf., Boston, MA, Aug. 1989.

controller is an input-output mapping which is highly [10] G_rs, $., Mattem, D.L., BrishL M.M., and Ousts, P. J., dependent on the magnitude and direction of the in- "Ho_ Based Integrated Fli_/Propu_on Control Dem_ for • puts and _uy desired dynamic characteristics have to STOVL _t in _r_ition Fli_at", _ Paper 96-3335, be built into the neurocontroller by appropriate selec- Guldancc, NaviKation and CoaWol Conf., Portland, OR. Aug.

1990.

tion of inputs. For instance, both the H_ based con- [llJ _'oudet, T., Gm_, $., Mattmm, D.L., and Mcrr_ W.

troller and the neurocontroller have lead characteris- C., "Towards Pr_-ti¢_1 Control D_isn U_ Neural Compu- tics (rate feedback) from the tracking error measure- tation', _.nt. Oohat Con[. on Nem'_ Network*, Seattle, WA, July 1991.

ments to the control commands; however, the lead [12] R_, D. E., MeCI_, J. L., et al.: Pa_lel D_ characteristic was a result of the synthesis procedure tributcd Proce_,mg. E='plor_tso_ in the Micro_tr_ctur_ of Cog- for the H_ based controller which used only errors nition, Volume 1: Fnndatson_., _ Press, Cs=_bridS_, MA, 1986.

inputs, whereas for the neurocontroLler this lead char- [13] Jordan, M. I.: Ge_tes"/c Constraints On Under#_e_d acteristic could be obtained only by providing error Trajectories, Int. Joh_ Conf. on Neural Networks, Vol.1, rate as explicit inputs (measurements). Developing p.217, Wuhlns_on D.C., June 1989.

neurocontroi design methodologies that can synthe- [14] Ogata, K.: ._o_er_ Control Enj_neermg, Pr_mtice Ha]] size the dynamics needed by the neurocontroller to Inc., 1970.

[IS] Leh_o,,_k;, N. A,: Pmc_ca/ Rob_tn¢_ Me_,_e_ in achieve the desired performance is an area of future M_lti-_armble Contrvl S_atem Analy_i_, Ph.D. Theais (LIDS- research. A possible approach may lie in the use of TH-1093), M2.T., Cambridge, MA, M•y 1981.

recurrent neura_ architectures.

[16] Doyle, J. C.: $trscturr.d Unczrt=inq_ in Control S_t_n De_ifn, Proeeedinp of the 24th Conf. on Deeimon and Con- Linear stab_ty robustness analysis tools were ap- trol, Ft. Laudcrdal¢, FL, Dee. 1985, pp. 260-265.

plied to the iinearized neurocontroller models and to [17] Apkariam, P. ]:L: Structured $t•bilit_ _ob_tn¢_ Ira- the baseline H_ based controller. These analysis provzmznt b_ Eiger_pace A_*tgnment Techmq_u: A Hybrid tools showed that the neurocontroller will have very Met_odolo_, Jourmd of Guidance, Control an d Dynami¢_ {18] Troude_, T., and Merrill, W. C.: Neuromo_-ph_¢ Lca_'ntng poor stabiihy margins as compared to the H_ based o/ Contm_ou. Val_ed Meppin_ ]_om Nei.e Corr_pte_ Data, controller. The poor phase margins for the neuro- I_EE Trans. on New.t1 Networks, Vol.2, No.2, March 1991.

controller were confirmed in simulation wherein time delays of 0.05 see in both control channels resulted in a limit cycle pitch response with the neurocontroller, while there was little performance degradation with the//_ based controller. Since the issue of robus_ heSS is criticsi to practical implernentation of flight Weighted: control systems, a future area of research is to de- coneol velop methodologies for the synthesis of" robust neu- rocontrollers, and tools to analyze their robustness.

References.

[1] Sp¢©ial $e_t_an on Neural Networks _or Control Sylten_, IEEE Cont. S_. Ma4_., Vol.9, No.3, pp. =S-59, April 1989.

[2] $peeia_ I,-" or, N¢l, rul Net_oorkJ ,, ¢on¢,.oi $_Jtem,, IEEE Cont. Sys. Mag., Vol.10, No -_, pp 3-87, April 1990.

[3] White, D., and 5ofge, D., "Neural Network Based Proce_ Opthnizatlon anti Control", tg_ _'EEE Conference o_ Dec_- ston croci Coarro_ Honolulu, HI, Dee. 1990.

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Vol.14, No.l, J_a.-Feb. 1991.

[5] _fiiitar_ 5pooh, cation . Fl_ng Oua_tie_ of Piloted Air- planes", MIL.F-$T8_C, USAF, WPAFB, OH, Nov. 1980.

!6] Doyle, J.C., Glover, K., Klm_one.k_, P.P., and Praneis, Rgure 1 .---Block dmgl"l_ for I-L. con_ol design.

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Report Documentation Page

Nstlen_ W end _L O_Aoc_mSonNo. 3. eUc_aC_ No.

NASA TM-105151 5. P.q_rt _ 4. 1"me md 8ebeee Neural Network Application to Aircraft Control System Design e. p.dmn_ o,wtu,t_ c4x_ =. p,_m_no OmmUm_n F_od No.

7. _e_{s) E-6435 Terry Troudet, Sanjay Garg, and Walter C. Merrill 10. WoHL Unit No.

505 - 62 - 50 D.

p.dun_ o_._=uo. Nm_, .rid/CaSe= 11. C_trld or Gnvl Ho.

National Aeronautics and Space Administration Lewis Research Center Cleveland, Ohio 44135- 3191 13. Type Of Repod and Pwiod Covered Technical Memorandum 12. $ponsodng Agen,c'y N_ Ind Addmn National Aeronautics and Space Administration 14. ,_,o,_ Apr¢t Co_ Washington, D.C. 20546-0001 16.

St_dmmntaty Notes Prepared for the Guidance, Navigation and Control Conference sponsored by the American Institute of Aeronautics and Astronautics, New Orleans, Louisiana, August 12-14, 1991. Terry Troudet, Sverdrup Technology, Inc., Lewis Re- search Center Group, 2001 Aerospace Parkway, Brook Park, Ohio 44142; Sanjay Garg and Walter C. Merrill, NASA Lewis Research Center. Responsible person, Terry Troudet, (216) 433 -8524.

16. Absh'sct The feasibility of using artificial neural networks as control systems for modem, complex aerospace vehicles is investi- gated via an example aircraft control design study. The problem considered is that of designing a controller for an integrated airframe/propulsion longitudinal dynamics model of a modem fighter aircraft to provide independent control of pitch rate and airspeed responses to pilot command inputs. An explicit model-following controller using H. control design techniques is first designed to gain insight into the control problem as well as to provide a baseline for evaluation of the neurocontroller. Using the model of the desired dynamics as a command generator, a multilayer feedforward neural network is trained to control the vehicle model within the physical limitations of the actuator dynamics. This is achieved by minimizing an objective function which is a weighted sum of tracking errors and control input commands and rates. To gain insight in the neurocontrol, linearized representations of the non-linear neurocontrolier are analyzed along a commanded trajectory. Linear robustness analysis tools are then applied to the linearized neurocontroller models and to the baseline I_ based controller. Future areas of research ate identified to enhance the practical applicability of neural networks to flight control design.

18. I:_ Stm_m'ner'd 17. KoyWords (.%q_gested byAuthor(s)) Unclassified - Unlimited H-infinity control; Neural networks; Neurocontrol; Performance/control trade-off Subject Category 08 19. Security Classif. (ot _ report) 20. Security Classif. (of Ibis page) 21. No. ot pages A03 Unclassified Unclassified 20 I NASA FOgM l&_l OCT *For sale bythe NationalTechnicalInformafonService,Springfield, Virginia 22161 PRECEDING PAGE BLANK NOT FILMED