A MODEL-FOLLOWING NEURO-ADAPTIVEAPPROACH FOR ROBUST CONTROL OF HIGH

PERFORMANCE AIRCRAFTS

Radhakant Padhi∗, P. Narayana Rao∗∗,Siddharth Goyal ∗∗∗, Abha Tripathi ∗∗∗∗

∗Asst. Professor, Dept. of Aerospace Eng., Indian Institute ofScience (IISc), Bangalore, India.

∗∗Former Graduate Student, Dept. of Aerospace Eng., IISc,Bangalore, India.

∗∗∗Former Summer Trainee, Dept. of Aerospace Eng., IISc,Bangalore, India.

∗∗∗∗Former Project Assistant, Dept. of Aerospace Eng., IISc,Bangalore, India.

Abstract: Based on dynamic inversion, a relatively straightforward approach is presentedin this paper for nonlinear flight control design of high performance aircrafts, whichdoes not require the normal and lateral acceleration commands to be first transferredto body rates before computing the required control inputs. This leads to substantialimprovement of the tracking response. Promising results are obtained from six degree-of-freedom simulation studies of F-16 aircraft, which are found to be superior as comparedto an existing approach (which is also based on dynamic inversion). The new approachhas two potential benefits, namely reduced oscillatory response (including elimination ofnon-minimum phase behavior) and reduced control magnitude. Next, a model-followingneuron-adaptive design is augmented the nominal design in order to assure robustperformance in the presence of parameter inaccuracies in the model. Note that in theapproach the model update takes place adaptively online and hence it is philosophicallysimilar to indirect adaptive control. However, unlike a typical indirect adaptive controlapproach, there is no need to update the individual parameters explicitly. Instead theinaccuracy in the system output dynamics is captured directly and then used in modifyingthe control. This leads to faster adaptation, which helps in stabilizing the unstable plantquicker. The robustness study from a large number of simulations shows that the adaptivedesign has good amount of robustness with respect to the expected parameter inaccuraciesin the model.

Keywords: Dynamic inversion, neuro-adaptive design, aircraft control, longitudinalmaneuver, lateral maneuver, pilot command tracking, autopilot design

1. INTRODUCTION

To enhance the maneuverability and increase the lift-to-drag ratio, high performance aircrafts are often de-signed to be naturally unstable. Such aircrafts are alsooften required to operate at the edge of the flight enve-

lope for combat superiority, which is supposed to be aswide as possible. Besides, the flight dynamics of suchaircrafts are also inherently highly nonlinear (they arevery often forced to fly in high angle of attack regime)and the aerodynamics involved is also quite complex.Because of these reasons, the flight control design for

high-performance aircrafts is a very challenging task.Traditional control system design techniques basedon classical control theory are becoming increasinglyinadequate to meet the stringent performance require-ments with assured stability and there is a strong needto use modern control design methods. Many mod-ern control design techniques have been attempted tocome up with effective flight control design, the taskof which is essentially to track the pilot commandswhile simultaneously assuring stability and robustnessof the overall plant. Techniques such asH∞ robustcontrol design [Lin et al., 1997], sliding mode control[Schumacher, Cottrill & Yeh, 1999], model-referencecontrol [Bodson, 2002], adaptive control [Ying et al.,2004], intelligent control designs like neuro-control[Willis, 1999], fuzzy-logic control [Won et al., 1999]etc. have been attempted in the literature.

Gain scheduled control design, where a number ofgains are first designed at various expected operatingpoints in the flight envelope which are then interpo-lated based on some physically meaningful schedulingvariable(s) is perhaps a very intuitive basic designapproach that has found wide acceptability in industry.This philosophy has also been used for control designof high performance aircraft [Lee & Spillman, 1997,Shin, Balas & Kaya, 2001]. However, potential disad-vantages of gain scheduling control design include along tuning process, no theoretical guarantee of stabil-ity for the interpolated gain (which is a major concernfor unstable plants like unstable aircrafts) etc.

Out of the many advanced techniques that have ap-peared in the literature, a promising approach is basedon the idea of optimal sliding mode [Schumacher,Cottrill & Yeh, 1999], where a two-loop structureshas been designed for tracking pilot commands (whichare assumed to be angle of attack, roll angle com-mands while assuring near-zero side slip angle). Inthis approach, an optimal sliding surface is gener-ated using a state dependent Riccati equation (SDRE)based optimal controller. This is followed by a slidingcontroller in the inner loop using Lyapunov theory.The performance of this controller has been evaluatedusing the six degree-of-freedom nonlinear model of F-16 aircraft. Note that SDRE approach can only assuresuboptimality and local stability [Cloutier, 1997], andhence such an approach is inadequate for the widerange of flight conditions.

A popular technique, which serves as a ‘universal gainscheduling controller’ (and hence avoids the tediousgain scheduling process), is dynamic inversion [Ennset al., 1994, Kim & Calise, 1997, Kaneshige, Bull &Totah, 2000, Khalil, 2002]. This technique is essen-tially based on the technique of feedback lineariza-tion. It leads to a number of potential advantages;namely asymptotic (rather exponential) stability of theerror dynamics thereby leading to perfect tracking,a simple closed form expression for the controller(hence no computational concerns), preserving many

of the benefits of the PID design etc. However, as thedynamic inversion is rather sensitive to the issue ofparameter inaccuracy and modeling errors, there is astrong need of augmenting this technique with someother robust/adaptive techniques, to make it useful inpractice. A potential approach in this regard is the ideaof online dynamic function approximation taking thehelp of evolving methods like ‘neuro-adaptive tech-nique’ [Kim & Calise, 1997]. The main philosophythat is exploited heavily in system theory applicationsis that neural networks have the universal functionapproximation property, which helps a controller toadapt to plants having unmodelled dynamics and time-varying parameters.

First, assuming perfect knowledge about the plant,this paper proposes a new relatively straightforwardapproach based on dynamic inversion which is appliedfor flight control design and the results are comparedwith an existing approach which is also based on dy-namic inversion [Menon, 1993]. The innovations ofthe current approach as compared to the existing ap-proach include (i) elimination of the necessity of trans-ferring the normal and lateral acceleration commandsto equivalent body rate commands before computingthe control surface defections (which is usually donein practice), (ii) reduction of tuning parameters in thecontrol design process, (iii) the assumption that dou-ble derivative of velocity vector components in bodyX and Y directions to be is zero, which is a morerealistic assumption as compared to assuming theirsingle derivatives to be zero and (iv) elimination ofthe assumption that double derivative of desired Eulerangles are zero. Note that the new approach leads toreduction of control magnitude and reduced oscilla-tory response. It also substantially reduces nonmin-imum phase behaviour of the closed loop response.The proposed method is applied for longitudinal andlateral control of F-16 aircraft. Combined longitudinaland lateral control (velocity vector roll) has also beenexperimented in the numerical simulation studies. Thepilot commands assumed in longitudinal mode arenormal acceleration and forward velocity of the air-craft. In lateral mode, roll rate and the desired velocityare used as pilot commands. In the approach, the fastdynamics corresponding to states i.e. body rates whichare controlled by the inputs (i.e. aileron, elevator andrudder deflection) are updated for every time step∆t,while slow dynamics corresponding to state total ve-locity which is controlled by thrust is updated afterevery five time steps. Promising results are obtainedfrom the fully six-degree-of-freedom (6-DOF) non-linear simulation, which are found to be superior ascompared to an existing approach [Menon, 1993].

Unfortunately, even though the dynamic inversiontechnique has evolved as a promising tool for non-linear control design substituting the extensive gainscheduling approach, there are a few important is-sues with respect to the technique as well. Becauseof modeling error and parameter inaccuracies, inver-

sion of the model does not lead to exact cancelationof the nonlinearities. Because of this, the techniquebecomes sensitive to the parameter inaccuracies andunmodelled dynamics, and hence, there is a need toaugment this technique with adaptive/robust controldesign tools. This problem of sensitivity to parameterinaccuracies (both aerodynamic and inertia parameterinaccuracies) is addressed next by augmenting the dy-namic inversion technique with a neuro-adaptive de-sign approach [Padhi, 2007], the basic philosophy ofwhich has been recently proposed by the first authorof this paper along with his earlier co-workers. Thisadaptive control design is carried out in two steps: (i)synthesis of a set of neural networks which capturematched unmodelled (neglected) dynamics or modeluncertainties because of parametric variations and (ii)synthesis of a controller that drives the state of theactual plant to that of a desired nominal model. Theneural network weight update rule is derived usingLyapunov theory, which guarantees both stability ofthe error dynamics (in a practical stability sense) andboundedness of the weights of the neural networks.Note that even though this technique has been usedalong with dynamic inversion technique, the basic phi-losophy and procedure is independent of the techniqueused to design the nominal controller, and hence canbe used in conjunction with any known control designtechnique. An interested reader can see the referencefor details of this approach. However, for complete-ness of this paper, the necessary steps are given in asubsequent section of this paper (including all designparameters). Note that the proposed approach in thispaper is slightly different from the one proposed in[Padhi, 2007]. This is because here we are particularlyinterested in the output robustness (i.e. performancerobustness), where as in the reference cited the pri-mary focus was to enhance stability (i.e. robust stabil-ity). In the aircraft problem discussed in this paper, arobustness study from large number of simulations hasbeen carried out by randomly varying the parameters(this has been done due to lack of precise mathemat-ical tools for rigorous mathematical analysis). Thisstudy clearly shows that the control design has goodamount of robustness with respect to expected levelsof parameter inaccuracies in the model. Note that inthe approach the model update takes place adaptivelyonline and hence it is philosophically similar to in-direct adaptive control. However, unlike a typical in-direct adaptive control approach, there is no need toupdate the individual parameters explicitly. Instead theinaccuracy in the system output dynamics is captureddirectly and then used in modifying the control. Thisleads to faster adaptation, which helps in stabilizingthe unstable plant quicker.

2. NONLINEAR SIX-DOF AIRCRAFTDYNAMICS

2.1 Equations of motion

Assuming the airplane to be a rigid body and earthto be flat, the complete set of Six-DOF equations ofmotion in the body frame are given by the followingset of differential equations [Roskam, 1995].

U = RV−QW−gsinΘ+(FAx +T)/m (1)

V = PW−RU+gcosΘsinΦ+FAy/m (2)

W = QU−PV +gcosΘcosΦ+FAz/m (3)

P = c1QR+c2PQ+c3LA +c4NA (4)

Q = c5PR+c6(R2−P2)+c7MA (5)

R= c8PQ−c2QR+c4LA +c9NA (6)

Φ = P+QsinΦ tanΘ+RcosΦ tanΘ (7)

Θ = QcosΦ−RsinΦ (8)

Ψ = (QsinΦ+RcosΦ)secΘ (9)

xE

yE

zE

=

cosΨ −sinΨ 0sinΨ cosΨ 0

0 0 1

cosΘ 0 sinΘ0 1 0

−sinΘ 0 cosΘ

1 0 00 cosΦ −sinΦ0 sinΦ cosΦ

UVW

(10)

where

c1

c2

c3

c4

c8

c9

, 1

(IXXIZZ− I2XZ)

IZZ(IYY− IZZ)− I2XZ

IXZ(IZZ + IXX− IYY)IZZ

IXZ

I2XZ + IXX(IXX− IYY)

IXX

[c5 c6 c7

]T , 1IYY

[(IZZ− IXX) IXZ 1

]T

In the above equationsU,V,W are the velocity com-ponents andP,Q,R are the roll, pitch and yaw ratesrespectively about the body-fixed axes.Φ,Θ,Ψ arethe Euler angles andxE, yE, zE are the coordinatesof ground fixed inertial frame [Roskam, 1995]. Notethath =−zE whereh is the height of the aircraft fromground.FAx,FAy,FAz are the aerodynamic componentsof external forces andT is thrust acting along thebodyX-axis (it is assumed that thrust passes throughCG and produces no moment component). Similarly,LA,MA,NA are the aerodynamic components of the air-plane.IXX, IYY, IZZ, IXZ represent the moment of iner-tias of the airplane in the body frame(note that the air-craft is assumed to be symmetric about theXZ planeand hence,IXY = IXZ = 0). mandg represent mass andacceleration due to gravity respectively (both assumedconstants).

2.2 Aerodynamic forces and moments

The aerodynamic forces and moments along and aboutthe bodyX,Y,Z directions are given by [Nguyen et al.,1979].

[FAX FAY FAZ

]T = qS[

CXt CYt CZt

]T(11)

[LA MA NA

]T = qS[

bCLt cCMt bCNt

]T(12)

whereq is the dynamic pressure andS is wing plat-form area. The non-dimensional aerodynamic forcecoefficientsCXt ,CYt ,CZt and the moment coefficientsCLt ,CMt ,CNt are expressed as multivariate nonlinearfunctions [Morelli, 1998] and are given by

CXt = Cx(α,δe)+Cxq(α)q (13)

CYt = Cy(β ,δa,δr)+Cyp(α)p+Cyr (α)r (14)

CZt = Cz(α,β ,δe)+Czq(α)q (15)

CLt = Cl (α,β )+Clp(α)p+Clr (α)r+Clδa

(α,β )δa +Clδr(α,β )δr

(16)

CMt = Cm(α,δe)+Cmq(α)q+CZt (xcgre f −xcg) (17)

CNt = Cn(α,β )+Cnp(α)p+Cnr (α)r+Cnδa

(α,β )δa

+Cnδr(α,β )δr −CYt (xcgre f −xcg)(

cb)

(18)

wherep, pb/2V, q, qc/2V, r , rb/2V andxcgre f ,xcg.

The following physical data of F-16 used was obtainedfrom [Nguyen et al., 1979].m = 637.14 slugs, S=300 f t2, b = 30 f t, c = 11.32f t, IXX = 9496slugs−f t2, IYY = 55814slugs− f t2, IZZ = 63100slugs−f t2, IXZ = 982 slugs− f t2. All of the actuators aremodeled as first order lags with limits on deflectionand rates [Nguyen et al., 1979]. The thrust actuator hasunity gain and rate limit of 10000lb/sec. The elevator,aileron and rudder each has a gain of (1/0.0495)sec−1

and rate limits of±60 deg/sec, ±80 deg/sec, ±120deg/secrespectively.

In this paper the velocity vector roll maneuver is alsoconsidered. During a velocity vector roll, the aircraftrotates about an axis aligned with its velocity vector.In this case it is convenient to use the wind frame, inwhich the equation of motion of the aircraft are writtenas [Wang & Stengel, 2005]

VT = (Fwx/m)−gsinγ (19)

α = Q− (Qw/cosβ )−Pcosαtanβ −Rsinαtanβ(20)

β = Rw +Psinα−Rcosα (21)

Pw = Pcosαcosβ +(Q− α)sinβ +Rsinαcosβ (22)

Qw =−(Fwz/mVT)− (g/VT)cosγcosφ (23)

Rw = (Fwy/mVT)+(g/VT)cosγsinφ (24)

ϕ = Pw +Qwsinϕ tanγ +Rwcosϕ tanγ (25)

γ = Qwcosϕ−Rwsinϕ (26)

ψ = (Qw sinϕ +Rwcosϕ)secγ (27)

whereVT ,α ,β are the total velocity, angle of attackand side slip angle respectively andPw,Qw,Rw are the

roll, pitch and yaw rates respectively about the windaxes andα is the angle of attack.ϕ,γ,ψ are the wind-axis Euler angles andFwx is the wind axis total force.

2.3 Equations in compact notation

First, equations (1)-(6) are written in a structured formas follows.

XV = fV(X)+[

gV(X) dV][

UA

σT

](28)

XR = fR(X)+gR(X)UA (29)

where,X , [VT α β P Q RΦ Θ h]T , XV , [U V W]T ,XR , [P Q R]T , UA , [δa δe δr ]

T , σT , T/Tmax, Uc ,[UT

A σT]T

.

Note thatΨ, xE, yE are not considered as part of thestate equation, as they are not coupled with the otherequations (due to the flat earth assumption). Otherterms in Equations (28) - (29) are defined as follows.

fV1 ,

RV−QW−gsinΘPW−RU+gcosΘsinΦQU−PV +gcosΘcosΦ

fV2 , qSm

Cx(α)+Cxq(α)qCy(β )+Cyp(α)p+Cyr (α)r

Cz(α ,β )+Cxq(α)q

gV(X) , qSm

0 Cxδe0

Cyδa0 Cyδr

0 Czδe0

dV ,[

Tmax/m 0 0]T

fR(X) = ( fR1 + fR2 fR3 fR5)gR(X) = fR2 fR3 fR4

fR1 ,

c1QR+c2PQc5PR+c6(R2−P2)

c8PQ−c2QR

fR2 , qS

c3 0 c4

0 c7 0c4 0 c9

fR3 ,

b 0 00 c 00 0 b

fR4 ,

Clδa0 Clδr

0 Cmδe0

Cnδa0 Cnδr

fR5 ,

Cl (α,β )+Clp(α)p+Clr (α)rCm(α)+Cmq(α)q

Cn(α,β )+Cnp(α)p+Cnr (α)r

The longitudinal accelerationnx, lateral accelerationny and normal accelerationnz are defined as

nx , (Fx/m) = QW−RV+gsinΘ+U (30)

ny , (Fy/m) = UR−WP−gsinΦcosΘ+V (31)

nz ,−(Fz/m) = UQ−VP+gcosΦcosΘ−W (32)

Alternately, these quantities can also be written as

nx = fnx +gnxUA (33)

ny = fny +gnyUA (34)

nz = fnz +gnzUA (35)

where the termsfnx, fny, fnz,gnx,gny,gnz are appropri-ately defined.

Also note that from equations (28) and (29) one canwrite

VT = fVT (X)+[

gVT (X) dVT

][UA

σT

](36)

P = fP(X)+gP(X)UA (37)

Q = fQ(X)+gQ(X)UA (38)

where fVT , (1/VT)XTV fV gVT , (1/VT)XT

V gV

dVT , (U/VT)dV

fP , fR(1, :) gP , gR(1, :) fQ , fR(2, :) gQ , gR(2, :)

Similarly, in wind axis frame, normal acceleration(nwz) can be written as

nwz = fnwz +gnwzUA (39)

where fnwz = (−sinα fnx +cosα fnz)gnwz = (−sinαgnx +cosαgnz)

3. FLIGHT CONTROL DESIGN

3.1 Nominal control design

Following the philosophies presented in literature[Kaneshige, Bull & Totah, 2000, Kim & Calise, 1997,Menon, 1993], the objective is to design a controllersuch that the roll angleΦ → Φ∗, normal accelerationnz → n∗z, lateral accelerationny → n∗y and total ve-locity VT → V∗

T whereΦ∗,n∗z,V∗T are the commanded

values from the pilot.n∗y = 0 is preselected in thedesign process (this assumes turn co-ordination). Notethat Φ∗ = 0 is also preselected in longitudinal ma-neuvers. Here we would like to point out that in anavailable literature [Menon, 1993]n∗z,n∗y are typicallyreplaced by proportional and integral error terms in theCommand Augmentation System (an outer loop). Inthis procedure, the required assumptions involved areV = W = 0 and[Φ∗ Θ∗ Ψ∗]T = 0. Note that since anintegral feedback is used, it may lead to“control wind-up”, and hence, it is advisable to have an associatedwind-up prevention logic (which is not mentioned inthe reference). In this paper, however, the need for anintermediate command transformation and the needto introduce any integral feedback for the errors inacceleration commands are eliminated. Moreover, itis assumed thatV = W = 0, which is a more realisticassumption as compared to assumingV =W = 0. Fur-thermore, the additional assumption[Φ∗ Θ∗ Ψ∗]T = 0is also not necessary. The mathematical details of thisnew procedure is outlined below in fair detail. First,new variablesaz,a∗z anday,a∗y are defined as

az , nz+W, a∗z , n∗z +W (40)

ay , ny−V, a∗y , n∗y−V (41)

The new method relies on the key observation that([nz ny]T → [n∗z n∗y]T)⇔ ([az ay]T → [a∗z a∗y]T); thisis because of the one-to-one correspondence between

them. From Equations (31)-(32) and (40)-(41) it canbe seen that

az = UQ−VP+gcosΦcosΘ (42)

ay = UR−WP−gsinΦcosΘ (43)

Taking derivatives of both sides with respect to timeand using Equations (7)-(9) and (28)-(29), the follow-ing equations are obtained,

az = faz(X)+[

gaz(X) daz

][UA

σT

](44)

ay = fay(X)+[

gay(X) day

][UA

σT

](45)

The symbols used in Equations (44)-(45) are definedbelow

[ faz(X) fay(X)]T , A1 fR(X)+B1 fV(X)+C1 (46)

[gaz(X) gay(X)]T , A1gR(X)+B1gV(X) (47)

[daz day] , B1dV (48)

whereA1 ,[ −V U 0−W 0 U

]; B1 ,

[Q −P 0R 0 −P

]

C1 , g

[ −cosΘsinΦ −cosΦsinΘ−cosΘcosΦ sinΘsinΦ

][ΦΘ

]

3.1.1. Longitudinal maneuver In the longitudinalmaneuver case, goal is to assure that[Φ nz ny]T →[Φ∗ = 0 n∗z n∗y = 0]T . However, from Equation (7) it isobserved that the controlUA does not appear in theΦequation. So, we first convert theΦ∗ command to thecommand in roll rateP∗. For doing this, we define theerrorΦ , (Φ−Φ∗) and enforce a stable desired errordynamics as follows

˙Φ+(1/τΦ)Φ = 0 (49)

where τΦ > 0 is the time constant of the error dy-namics. By substituting forΦ from Equation (7) andassumingΦ∗ to be constant, the desired roll rate (de-noted asP∗) is given by the following expression

P∗ =−(QsinΦ+RcosΦ) tanΘ− 1τΦ

(Φ−Φ∗) (50)

where τΦ > 0 is the desired time constant. Next,definingXT , [P nz ny]T , X∗T , [P∗ n∗z n∗y = 0]T

and XT , (XT −X∗T), we aim to design a controllersuch that the following stable linear error dynamics isenforced.

˙XT +KXT = 0 (51)

where the gain matrixK is selected to be a positivedefinite matrix. A relatively easier way to select thegain matrixK is to choose theith diagonal element tobe 1/τi , whereτi > 0 is the desired time constant ofthe ith channel of the error dynamics. The gainK isselected as

K = diag(1/τP,1/τnz,1/τny) (52)

where τP, τnz, τny are the time constants for rollrate, normal acceleration and lateral error dynamicsrespectively. With the assumptionV = W = 0, fromEquation (40), it is clear that[nz ny] = [az ay]. FromEquations (37) and (44)-(45), it follows that

fP +gPUA

faz +gazUA

fay +gayUA

− X∗T +

K

Pfnz +gnzUA

fny +gnyUA

−

P∗

n∗zn∗y

= 0

(53)

Carrying out necessary algebra, the expression for thecontroller can be written as

UA = A−1U bU (54)

where AU , [gTP gT

azgT

ay]T + K[0T gT

nzgT

ny]T , bU ,

−[ fP faz fay]T −K[(P−P∗) ( fnz−n∗z) ( fny−n∗y)]T .

Similarly, definingVT , (VT −V∗T ) (note thatVT is a

slower variable), the following error dynamics is alsoenforced

˙VT +kVTVT = 0 (55)

where the gain matrixkVT = 1/τVT > 0 (whereτVT istime constant for velocity error dynamics) is selectedto be a positive definite matrix. By solving Equation(36) and expression for thrust can be expressed as

σT =−d−1VT

cVT (56)

wherecVT , {( fVT +gVTUA)−V∗T +kVT (VT −V∗

T )}Note that the control vector[δa δe δr ] is updated afterevery time stepdt while σT is updated after every fivetime steps5dt (since it a is slow variable). An imple-mentation schematic of the controller in longitudinalmaneuver is given in Figure 1.

Command Transformation

Control Computation Aircraft

Compute n nz y

Compute

βαPQR

V

h

φ

αβ

θ

T

q_

δδδσ T

a

e

r

P *

zn*

TV*

yn* = 0

φ* = 0

V hT

q_

q_

Fig. 1. Implementation of new control design in longitudinal mode

3.1.2. Lateral maneuver During lateral maneuvers,the objectives are to driveP → P∗, ny → n∗y, h →h∗ and VT → V∗

T . Note that the appropriaten∗z (suchthat h → h∗) is automatically computed in this pro-cess. However, from Equation (10) it is observed thatthe controlUA does not appear in theh equation andhence,UA cannot be computed directly from this goal.Because of this, a command transfer loop is intro-duced for converting the height commandh∗ to pitchangle commandΘ∗, and subsequently, to the pitch ratecommandQ∗. In this process, first an error expressionis defined ash , (h− h∗) and a stable height-errordynamics is enforced as

˙h+(1/τh)h = 0 (57)

where τh is the desired time constant for this errordynamics. By substituting forh from Equation (10),this can be expanded as

[U sinΘ−V cosΘsinΦ−WcosΘcosΦ]−h∗+(1/τh)(h−h∗) = 0

(58)

The variable Θ is solved (denoted asΘ∗) fromEquation (58) with a nonlinear equation solver (e.g.Newton-Raphson technique [Gupta, 1995]). Similarly,a stable first-order error dynamics is enforced for thepitch angle dynamics by definingΘ = (Θ−Θ∗) andthen enforcing the following dynamics.

˙Θ+(1/τΘ)Θ = 0 (59)

whereτΘ > 0 is the desired time constant. Substitutingfor Θ equation from Equation (8) and assumingΘ∗to be constant at each instant of time (quasi-steadyassumption), an expression forQ (and denote it asQ∗)can be obtained as

Q∗ = (1/cosΦ)[RsinΦ− (1/τΘ)(Θ−Θ∗)] (60)

The pitch rateQ∗ is assumed to be quasi-steady (heldconstant at each instant of time). Next, defineXT ,[P Q ny] , X∗T , [P∗ Q∗ n∗y = 0] andXT , (XT−X∗T).The objective now is to synthesize a controller suchthat Equation (51) is satisfied. In this case, the gainmatrix is selected to be of the form

K = diag(1/τP,1/τQ,1/τny) (61)

Following the steps outlined before and carrying outthe necessary algebra, an expression for control canbe written in the following form.

UA = A−1U bU (62)

where,

AU , [gTP gT

Q (gTay

+(1/τny)gTny

)]T

bU ,−[ fP fQ fay]T−K[(P−P∗) (Q−Q∗) ( fny−n∗y)]

T

σT can be calculated by using the same expression asused in Section 3.1.1. Note that the command trans-formation fromh∗ to Q∗ can be considered as an outerloop, whereas the subsequent control computation canbe interpreted as an inner loop. Due to the quasi-steady

assumptions, one should guaranteeτh > τΘ > τQ, sothat the inner-loop dynamics is faster than the outer-loop dynamics. An implementation schematic of thecontroller in longitudinal maneuver is given in Figure2.

Command Transformation

Control Computation Aircraft

Compute

βαPQR

V

h

φ

αβ

θ

T

q_

δδδσ T

a

e

r

P *

TV*

yn* = 0

φ*

V hT

Compute n y

* h

*θ

*Q

Optional

q_

q_

Command Transformation

Command Transformation

Fig. 2. Implementation of new control design in lateral mode

3.1.3. Combined longitudinal and lateral maneuverThe significance of this maneuver is obvious, for

it allows the pilot to quickly slew and point the air-craft’s nose using a presumably “fast” roll maneuver,without pulling g’s and turning. In this case, goal isXT →X∗T andVT →V∗

T , whereXT , [P nz ny]T , X∗T ,[P∗ n∗z n∗y = 0]T . HereP∗w command is given aboutvelocity vector and furtherP∗ is calculated fromP∗w.Using Equations (20), (22) and (23),P∗ can be ex-pressed as

P∗ = fP∗ +gP∗UA (63)

where

fP∗ ,

(1/cosα(cosβ + tanβsinβ ))×{P∗w−Rsinα(sinβ tanβ +cosβ )−tanβ (( fnwz/VT)+(g/VT)cosγcosΦ)

}

gP∗ , (−tanβ cosα(cosβ + tanβsinβ ))(gnwz/VT).

Now, a controller is designed such that the stable errordynamics Eq. (51) is satisfied and the gain matrixK isselected to be a positive definite matrix which is takenas same as in Eq. (52). With the assumptionV = W =0, from Eq. (40), it is clear that[nz ny] = [az ay].Hence from Eqs. (37), ( 44) and (45), one can write

fP +gPUA

faz +gazUA

fay +gayUA

−X∗T

+K

Pfnz +gnzUA

fny +gnyUA

−

fP∗ +gP∗UA

n∗zn∗y

= 0

(64)

Simplifying Eq.(64) and carrying out the necessary al-gebra, the expression for the controller can be derivedas

UA = A−1U bU (65)

whereAU , [gTP gT

azgT

ay]T + K[−gT

P∗ gTnz

gTny

]T bU ,−[ fP faz fay]

T −K[(P− fP∗) ( fnz − n∗z) ( fny − n∗y)]TσT can be calculated with the same expression as inSection 3.1.1. An implementation schematic of thecontroller in combined mode is given in Figure 3.

Command Transformation

Control Computation Aircraft

Compute n nz y

Compute

βαPQR

V

h

φ

αβ

θ

T

q_

δδδσ T

a

e

r

P *

zn*

TV*

yn* = 0

V hT

q_

q_

P *w

Fig. 3. Implementation of new control design in combined mode

Note that in the dynamic inversion design, the dynam-ics of the output tracking error is always guaranteedto be globally asymptotically stable (because of theenforcement of stable linear dynamics). But in generalthere arises the issue of stability of internal dynam-ics. Even though its difficult to show the stability ofinternal dynamics analytically using the complicatednonlinear six-DOF equation, from our numerous sim-ulation studies we have observed that the design ap-proach proposed here does not lead to the instabilityof internal dynamics.

3.2 Neuro-adaptive control design

3.2.1. Generic theory In this approach, as wepointed out earlier the first aim is to come up witha nominal controller, which will meet the goals forthe nominal model. The class of nonlinear system isfocused on, which can be represented by the followingequation

Xd = fd(Xd)+Gd(Xd)Ud (66)

Yd = hd(Xd) (67)

whereXd ∈Rn andUd ∈Rm, Yd ∈Rp are the state andcontrol variables of the nominal system respectively.The objective here is to design a controllerUd so thatYd→Yd, whereYd(t) is the commanded signal, whichis assumed to be bounded and smooth. The nominalcontrollerUd has been designed using dynamic inver-sion technique. Equation (68) may not truly represent

the actual plant because of the presence of uncertain-ties in the model. Using the chain rule of derivative,the expression forY can be derived as:

Yd = fYd(Xd)+GYd(Xd)Ud (68)

wherefYd , [∂h/∂Xd] fd(Xd) & GYd , [∂h/∂Xd]Gd(Xd).Now the actual plant output is represented as

Y = fY(X)+GY(X)U +dY(X) (69)

dY(X) is an unknown function that arises due to pa-rameter uncertainties and unknown disturbances. ThecontrollerU needs to be designed online such that thestates of the actual plant follow the respective statesof the nominal model. In other words, the goal is toensure thatY→Yd as t→∞. To achieve this, the ideafollowed here is to first capture the unknown functiondY(X), which is accomplished through a neural net-work approximation. For this purpose, an intermedi-ate step is needed, which is to define anapproximatesystemas follows

Ya = fY(X)+GY(X)U + dY(X)+Ka(Y−Ya)Ya(0) = Y(0)

(70)

whereKa is selected as a positive definite gain matrix.A relatively easy way of doing this is to selectKa asa diagonal matrix with theith element beingkai > 0.Even though the selection ofkai > 0 ∀ i = 1, . . . ,nsatisfies the need of theka being positive definite,it is desirable to choosekai > 1.5 because it leadsto a smaller bound in the tracking error (this willbecome clear towards the end of this section). Notethat whenever a function is approximated and only theapproximate function is kept in the dynamics, then themodified equation no more represents the true dynam-ics (because of the function approximation error). Thisis the primary reason to introduce theYa dynamics.The approach followed here for ensuringY → Yd in-volves two steps:(i) Y →Ya and(ii) Ya →Yd, whichare discussed next. A pictorial representation of thesesteps is shown in the Fig.4.

Step1 Capturing dY(X) and ensuringY→Ya. To

Ya

Y Yd

Approximate tracking error

Actual tracking error Nominal tracking error

Fig. 4. Philosophy of model following approach

capture the unknown function, first writedY(X) ,[dY1(X) ... dYn(X)

]wheredYi (X),i = 1,2, ....n is the

ith component of thedY(X). EachdY(X) is approxi-mated asdY(X) in a separate linear-in-the-weight neu-ral network. We assume that the unknown function

dY(X) can be represented by the basis function vectorφ(X)

dYi (X) = WiTφi(X) (71)

whereWi is the weight vector of theith neural net-work andφi(X) is its basis function vector for eachchannel. This neural network function approximationis depicted in Fig. 5.

di

φih1

φihi

Wi1

Win

(X1,X2) = WiTφ^

i^

Y1

Y1n

Ya1

Yan

Fig. 5. Linear-in-weight neural network

At this point, it needs to be mentioned that eventhough generic radial basis functions can be used forthis purpose. It is probably wiser to incorporate someprior knowledge about the system and judiciously se-lect the basis functions, which will lead to faster learn-ing of the unknown function. Note that the combina-tion of n sub-networks can be interpreted to constitutea single neural network that representsdY(X). Theidea of havingn neural networks forn independentchannels is to facilitate simpler mathematical analysis.More important, it leads to faster training becauseof reduced computational complexity, as none of theweights are linked to more than one output function.The next task is to update the weights of the neuralnetwork (i.e. to train them). Towards this end, theerror between the actual state and the correspondingapproximate state is defined as

eai,Yi −Yai (72)

From Eq. (69) and Eq. (70), the equations for theith

channeleai is written as

eai , dYi (X)− dYi(X)−kai ieai

= WTiφi(X)+ εi −kaieai

(73)

whereW , (Wi −W) is the error between the idealweight and actual weight of the neural network.Next,define a series of Lyapunov function candidatesLi , i = 1,2, ...n such that

Li =eai pieai

2+

WTi γi

−1Wi

2(74)

wherepi > 0 andγi > 0. Taking the time derivative ofboth sides of Eq. (74), using the fact that˙W = − ˙Wi

(SinceWi is a constant) and on substituting foreaifrom Eq. (74)

Li = eai pi eai +WTi γi

−1 ˙Wi

= eai pi(WTi φi(X)+ εi −kai eai)+WT

i γi−1 ˙Wi

(75)

Note that our objective is to come up with a meaning-ful condition that will ensureLi < 0 which will ensurethe stability of the error dynamics (of tracking error aswell as weight error). However, the expression forLi

containsWi (which is unknown), and hence, nothingcan be concluded about the sign ofLi . To get rid ofthis difficulty, force the term multiplying it to zeroand obtain the following weight update rule (trainingalgorithm) for theith neural network.

˙Wi = γieai piφi(X)−σiγiWi (76)

whereγi can be interpreted as a learning rate for theith network (its numerical value essentially dictates therate of capturing the unknown function)dYi (X). Notethat Eq. (76) is the weight update (learning) rule forWi . Select the initial condition asWi(0) = 0. This iscompatible with the fact that ifdYi (X) = 0 (i.e. there isno error in the model), then automaticallyˆdYi (X) = 0.

From the previous discussion we know that˙Wi =− ˙Wi ,therefore Eq. (75) becomes

Li = eai pi(WTi φi(X)+ εi −kaieai)−WT

i γi−1 ˙Wi (77)

Substitutingeai from Eq. (73) and˙Wi from Eq. (76) inEq. (77), we get

Li = eai piεi −kai e2ai

pi +σiWTi Wi (78)

HoweverWTi Wi , the last term from Eq. (78) can be

derived as follows

WTi Wi =

22

(WT

i Wi)

=12

(2WT

i Wi −2WTi Wi

)(79)

Further expanding2WiTWi , the first term from Eq.(79)

becomes

2WiTWi = Wi

TWi +WiTWi

= WiT (

Wi +Wi)+

(Wi −Wi

)TWi

= WiT (

Wi −Wi)+Wi

TWi +WTi Wi

=−WiTWi +Wi

TWi +WTi Wi

(80)

Using Eq. (80), Eq. (79) can be expressed as

WiTWi =

12

(−Wi

TWi +Wi

TWi

+WTi Wi −Wi

TWi −WiTWi

)

≤ 12

(−‖Wi‖2−‖Wi‖2 +‖Wi‖2)(81)

Therefore the last term in Eq. (78) satisfies the follow-ing inequality

σiWiTWi ≤−1

2σi‖Wi‖2− 1

2σi‖Wi‖2 +

12

σi‖Wi‖2

(82)

Equation (78) can now be rewritten as

Li ≤eai piεi −e2

aipikai −

12

σi‖Wi‖2

−12

σi‖Wi‖2 +12

σi‖Wi‖2

≤e2

aipi

2+

ε2i pi

2−e2

aipikai −

12

σi‖Wi‖2

−12

σi‖Wi‖2 +12

σi‖Wi‖2

≤e2

aipi

2+

ε2i pi

2− 1

2σi‖Wi‖2

−12

σi‖Wi‖2 +12

σi‖Wi‖2

(83)

In the expression 83,Li < 0 is only possible ifeai2 > βi

or |eai |>√

2βi where

βi ,[

ε2i pi2 − 1

2σi‖Wi‖2− 12σi‖Wi‖2 + 1

2σi‖Wi‖2].

Thus, it is evident that by selecting a smallσi andsufficiently good set of basis functions, the approx-imation errorεi will be reduced, which will help inkeeping the error bound small.

Step2EnsuringYa→Yd and Computation ofU

As pointed out earlier, while ensuringY→Ya and cap-turing the unknown functiondY(X) as a functional ap-proximationdY(X), it is simultaneously ensured thatYa→Yd as t→∞. To achieve this objective, the con-troller U is designed such that the following stableerror dynamics is satisfied

(Ya−Yd)+Kg(Ya−Yd) = 0 (84)

where Kg is chosen to be a positive definite gainmatrix. A relatively easy way of choosing the gainmatrix is to haveKg = diag(1/τ1 . . .1/τn) , whereτi

can be interpreted as the desired time constant for theith channel of the error dynamics in Eq. (84). From Eq.(68) and Eq. (70) and carrying out necessary algebra,the adaptive control is obtained as

U =−G−1Yd

fY + dY(X)+Ka(Y−Ya)−fY(Xd)−GY(Xd)Ud+K(Ya−Yd)

(85)

Note that even though we have used this techniquein conjunction with dynamic inversion, the genericneuro-adaptive control design presented in this sectioncan be implemented with ’any’ baseline controller tomake it robust with respect to parametric and model-ing inaccuracies.

3.2.2. Problem specific equationsFor longitudinalas well as lateral modes, first a nominal controllerwas designed using dynamic inversion technique asdescribed earlier. Then randomness was introducedby assuming Gaussian distributions around the nom-inal parameter values (which were considered as themean value of the distribution). The uncertain pa-rameters (mass, moment of inertia and aerodynamic

coefficients) are varied at random using Gaussian dis-tribution from 1% to 3%. A 3σ spread was assumedto cater to this distribution of the data and numberswere generated from this distribution at random. Notethat even though the system dynamics was simulatedwith these random values, this information was notused in the control design (i.e. in the control designonly nominal values of the parameters were used).Next a neuro-adaptive controller was designed usingthe technique described earlier.

Longitudinal maneuver In longitudinal mode theoutput vector is

Y(X) = [φ nz ny VT ]T (86)

But as the control does not appear in theφ equation,a command transformation is done and the outputdynamics is written as follows

Y(X) = [P az ay VT ]T (87)

The first three output dynamics are dealt together andthe fourth one, which ultimately gives the solutionfor the thrust (which is a slow variable) is handledseparately.

In this problem, the error equation is given by

Pa

aza

aya

VTa

−

Pd

azd

ayd

VTd

+K

Pa−Pd

nza−nzd

nya−nyd

VTa−VTd

= 0 (88)

Substituting the relevant expressions, 88 can be writ-ten as

fPa

faza

faya

fVTa

+

gPa 0gaza

daza

gayadaya

gVTadVT

[UAa

σTa

]+

K

Pa−Pd

nza−nzd

nya−nyd

VTa−VTd

+

dP

daz

day

dVT

+Ka

P−Pa

nz−nza

ny−nya

VT −VTa

= 0

(89)

From, 89 the control can be obtained as below

[UAa

σTa

]=

gPa ogaza

daza

gayadaya

gVTadVTa

−1

+

fPd

fazd

fayd

fVTd

+

gPd ogazd

dazd

gayddayd

gVTddVTd

[UAd

σTd

]−

dP

daz

day

dVT

−

fPa

faza

faya

fVTa

−K

Pa−Pd

nza−nzd

nya−nyd

VTa−VTd

−Ka

P−Pd

nz−nzd

ny−nyd

VT −VTd

Lateral maneuver In lateral mode the output vectoris

Y(X) = [φ h ny VT ]T (90)

But as the control does not appear in theφ equation,a command transformation is done and the outputdynamics is written as follows

Y(X) = [P Q ay VT ]T (91)

The first three output dynamics are dealt together andthe fourth one, which ultimately gives the solutionfor the thrust (which is a slow variable) is handledseparately.In this problem, the error equation is given by

Pa

Qa

aya

VTa

−

Pd

Qd

ayd

VTd

+K

Pa−Pd

Qa−Qd

nya−nyd

VTa−VTd

= 0 (92)

Substituting the relevant expressions, 92 can be writ-ten as

fPa

fQa

faya

fVTa

+

gPa 0gQa dQa

gayadaya

gVTadVT

[UAa

σTa

]

+K

Pa−Pd

Qa−Qd

nya−nyd

VTa−VTd

+

dP

dQ

dayˆdVT

+Ka

P−Pa

Q−Qa

ny−nya

VT −VTa

= 0

(93)From , 93 the control can be obtained as below

[UAa

σTa

]=

gPa 0gQa dQa

gayadaya

gVTadVTa

−1

+

fPd

fQd

fayd

fVTd

+

gPd 0gQd dQd

gayddayd

gVTddVTd

[UAd

σTd

]−

dP

dQ

dayˆdVT

−

fPa

fQa

faya

fVTa

−K

Pa−Pd

Qa−Qd

nya−nyd

VTa−VTd

−Ka

P−Pd

Q−Qd

ny−nyd

VT −VTd

A necessity in the process of neural network training isto choose appropriate basis functions. Trying to retainthe generic nature of this problem, the basis functionsare chosen as Gaussian functions. The basis functionvector in each channel is chosen as follows

ϕi(X) =

e− 1

2

((yi−yia)

2

σ21

)

e− 1

2

((yi−yia)

2

σ22

)

e− 1

2

((yi−yia)

2

σ23

)

For better approximation, we have selected threeGaussian basis functions about each of the mean val-

ues asσ1 = 0.1 σ2 = 1 σ3 = 10. Note that the selec-tion of Gaussian functions as the basis functions is inaccordance with the universal function approximationtheory of neural networks. The initial conditions of theneural network weights were assumed to be zero.

3.3 Actuator dynamics

Actuators for the control surface and engine are mod-eled as first order lags with limits on their minimumand maximum values as well as their rates. The mag-nitude and rate limits, as well as time constants ofthe actuators used in the nonlinear aircraft model areshown in Table 3.

4. NUMERICAL EXPERIMENTS

4.1 Numerical data selection

The low fidelity aerodynamic model data, as well asthe Moment-of-Inertia data, for the F-16 aircraft isused in our simulation study [Nguyen et al., 1979].A fourth-order Runge-Kutta technique [Gupta, 1995]with fixed step size of 50 msec is used for numericalintegration, which was motivated from [Keviczky &Balas, 2006]

4.1.1. Trim condition The trim condition for steadylevel flight is calculated by minimizing the followingcost function(J) with specified initial velocity andaltitude [Russell, 2003].

J = 5h2 +10Φ2 +10Θ2 +10Ψ2

+2VT2 +10α2 +10β 2 +10P2 +10Q2 +10R2

The cost function is minimized by the Matlab’s func-tion fminsearch, which finds values for the free pa-rameters by using an iterative technique. The trimcondition values (as found by minimizing(J) ) atspecified velocityVT0 = 580( f t/sec) and altitudeh0 =10,000f t are given in Table 1.

4.1.2. Selection of control design parameters

Nominal control design After some trial and errortuning, the values selected for the time constants (de-sign parameters) are given in Table 2. Note that thesome of these time constant values are not required inthe longitudinal mode whereasτnz value is not neces-sary in the lateral mode. Also note that the numericalvalues of the design parameters are kept same in both

the modes, which facilitates easier implementation(especially in the combined longitudinal and lateralmode).

In order to compare the performance of the new con-trol design method proposed here with an existingversion [Menon, 1993] (see appendix for a summaryof it), gain values ofk1 = k3 = 1, k2 = k4 = 30were selected for the command augmentation system.Similarly in the attitude orientation system, parametervalues ofkvi = 2ζiωni , kpi = ω2

niwith ζ1 = 1.5, ζ2 =

0.9, ζ3 = 0.9 andω1 = 2, ω2 = 5, ω3 = 5 rad/sec(i = 1,2,3) were selected for each of the attitude angleerror dynamic channels and the time constants forthis case as well. It is important to point out that theexisting technique need eleven design parameters inthe longitudinal case and twelve design parameters inthe lateral case. In the new approach presented in thisstudy, however only five design parameters are neededfor the longitudinal mode and seven parameters arerequired for the lateral case. This significantly lessnumber of design parameters with better performanceis clearly a potential advantage of the new approach.

Adaptive control design The learning rate for thelongitudinal and lateral mode is60except for the totalvelocity in the lateral mode which is40. The scalarselected for longitudinal mode isPp = 0.0001, Pnz =0.05, Pny = 0.001, PVT = 0.005. And for the lateralmode it isPp = Pq = Pny = 0.0001, PVT = 0.05. Thegains and for the longitudinal mode are given asK =diag(5, 5, 5, 4), andKa = diag(0.05, 0.05, 0.05, 1).The gains selected for the lateral mode areK =diag(5, 4, 4, 4), andKa = diag(0.05, 0.8, 0.05, 0.05).

4.2 Analysis of results

4.2.1. Nominal control design In the representativenumerical results presented here, the goal is to trackthe reference commands for 90 sec in longitudinalcase and 60 sec in lateral case. Within this time slot,the command is altered to reflect possible real-lifescenarios. In all plots (Figures 6 - 17), the solid linesrepresent the results from the new approach presentedin this report, whereas the dashed lines represent theresults from the existing approach [Menon, 1993].

Longitudinal maneuver In Figures 6 - 9, simulationresults for a longitudinal maneuver are shown, start-ing from the trim condition. The upper value ofn∗zassumed for low-fidelity model of F-16 [Keviczky &Balas, 2006] is 2.0g. The following sequence of com-mand signals is input:[Φ∗ n∗y V∗

T ] = [0 0 VT0]throughout the maneuver,n∗z = 0.9965g for t = (0−1)sec,2.0g for t = (1−15) sec,0.5g for t = (15−65)sec,1.0g for t = (65−90) sec.

In Figure 6, it is clear that the goal of normal accel-eration tracking is met for both approaches. However,

Table 1.Trim Condition Values

Parameter α0 β0 Φ0 Θ0 Ψ0 δa0 δe0 δr0 σT0

Trim Value 1.497 deg 0 deg 1.497 deg 0 deg 0 deg 0 deg -1.81 deg 0 deg 0.09

Table 2.Time Constant Values (N/R: Not Required)

Time constant (sec) τP τnz τny τVT τΦ τΘ τQ τh

Longitudinal 0.3 1.5 2 3 1 N/R N/R N/RLateral 0.3 N/R 2 3 1 0.3 0.1 5

Table 3.Engine and Control surface actuator’s description

Engine(throttle) Elevator Aileron rudderupper level limit 19000lb +25deg +21.5deg +30deglower level limit 1000lb -25deg -21.5deg -30degupper rate limit +10000lb/s +60deg/sec +80deg/sec +120deg/seclower rate limit -10000lb/s -60deg/sec -80deg/sec -120deg/sectime constant 1 0.0495 0.0495 0.0495

the new approach offers several improvements. First,the transient oscillations have much smaller overshootand the frequency of oscillation is lesser (which isalso evident from the normal acceleration and pitchrate history in Figures 6 and 8). This leads to betterhandling quality of the airplane. From Figure 6, it canalso be observed that even though the normal accel-eration is eventually tracked in the existing approach[Menon, 1993] successfully, the closed loop shows anon-minimum phase behavior (i.e. the initial responseis in the opposite direction with respect to the com-mand before recovering back).

0 20 40 60 800

0.5

1

1.5

2

2.5

Time (Sec)

n z(g)

0 20 40 60 80−1

−0.5

0

0.5

1

Time (Sec)

n y(g)

0 20 40 60 80−1

−0.5

0

0.5

1

Time (Sec)

P (

deg/

sec)

0 20 40 60 80550

600

650

700

750

800

Time (Sec)

VT (

ft/s)

Fig. 6. Roll angle, Normal acceleration, Lateral acceleration andTotal velocity in longitudinal maneuver

From Figure 7, it is evident that the control surfacedeflections are approximately zero in both the ap-proaches (i.e aileron and rudder deflections). How-ever, the final elevator deflection requirement is less inthe new approach. Moreover, the existing approach ex-hibits oscillations of relatively higher frequency in theelevator. The main difference here is that, in existingmethod thrust saturation starts very early as comparedto this new method (which is shown in Figure 7) and itgoes below the lower permissible limit of thrust (takenas 1000 lb [Nguyen et al., 1979]). After saturation, thevelocity deviates from its desired goal, but in the newmethod it is able to recover and the desired goal isachieved leading to better tracking.

0 20 40 60 800

20

40

60

80

100

Time (Sec)T

hrus

t (%

)

0 20 40 60 80−1

−0.5

0

0.5

1

Time (Sec)

Aile

ron

defle

ctio

n (d

eg)

0 20 40 60 80−3

−2

−1

0

1

2

Time (Sec)

Ele

vato

r de

flect

ion

(deg

)0 20 40 60 80

−1

−0.5

0

0.5

1

Time (Sec)

Rud

der

defle

ctio

n (d

eg)

Fig. 7.Aileron, Elevator and Rudder deflections and Thrust levelin longitudinal maneuver

0 20 40 60 80−1

−0.5

0

0.5

1

Time (Sec)

P (

deg/

sec)

0 20 40 60 80−10

−5

0

5

Time (Sec)

Q (

deg/

sec)

0 20 40 60 80−1

−0.5

0

0.5

1

Time (Sec)

R (

deg/

sec)

0 20 40 60 80−1

−0.5

0

0.5

1

Time (Sec)

Φ (

deg)

0 20 40 60 80−20

0

20

40

60

Time (Sec)

Θ (

deg)

0 20 40 60 80−1

−0.5

0

0.5

1

Time (Sec)

Ψ (

deg)

Fig. 8. Roll, Pitch, Yaw rate and Euler angles in longitudinalmaneuver

The aerodynamic and state variables are also pre-sented in Figures 8 and 9. It is clear that all thenon-tracked state variables remain within reasonablevalues throughout the maneuver (i.e. the internal dy-namics remains stable). Note that even though onlyone set of representative results are presented here,simulation studies for a large number of cases led tosimilar advantages and did not show instability in anyof the cases.

0 20 40 60 80−1

0

1

2

3

4

5

6

Time (Sec)

α (d

eg)

0 20 40 60 80−1

−0.5

0

0.5

1

Time (Sec)

β (d

eg)

0 20 40 60 80150

200

250

300

350

400

Time(sec)

Dyn

amic

Pre

ssur

e (lb

/ft2 )

0 20 40 60 800.5

0.55

0.6

0.65

0.7

0.75

Time(sec)

M

Fig. 9.Aerodynamic variables in longitudinal maneuver

Lateral maneuver Simulation results for a lateralmaneuver from the trim condition are presented inFigures 10 - 13. The sequence of command signalsapplied consists ofP∗ = −10 deg/sec for t = (0−7) sec, P∗ = 10 deg/sec for t = (7− 14) sec andP∗ = 0 deg/secfor t = (14−60) sec. Throughout themaneuver, it was assumed thatV∗

T =VT0, h∗ = h0 (theinitial condition values). However, the pilot can essen-tially select any other reasonable values. From Figure11, in case of new approach, it is clearly shown thataileron, elevator and rudder deflections are relativelylesser at 7 sec. Besides, in the existing approach theelevator and rudder deflection histories show relativelyhigh magnitude and high frequency transient oscil-lations, which should preferably be avoided. Thesetrends are absent in the performance of this new ap-proach. From the graph, it is clear that thrust requiredis also less (around half) as compared to existingmethod, which is again an advantage. Note that thelateral accelerations in both approaches remain closeto zero (which is also evident in small side-slip an-gle in Figure 13), which was a requirement for themaneuvers. However, in the new approach both side-slip angles remain more closer to zero and oscillationsare also relatively lesser. In other words, it leads tobetter turn-coordination. Pitch rate also more magni-tude with more oscillations, which is shown in Figure12. All other state and aerodynamic variables are alsoplotted in Figures 12 and 13, they also shows highfrequency transient oscillations with more magnitudeas compared to this new approach.

In the second case,P∗ command is generated usingbank angle command as discussed before. In this case,the sequence of command signals applied consisted ofΦ∗ = −40 degfor t = (0−10) sec,Φ∗ = 40 degfort = (10− 20) sec andΦ∗ = 0 deg for t = (20− 60)sec. Throughout the maneuver it was assumed thatV∗

T = VT◦ , h∗ = h◦ (the initial condition values).

Trajectories of the tracked states and associated con-trols are presented in Figures 14 - 17. The responseplots for the tracked variables (as shown in Figure14) do not show much of the difference, except that

0 20 40 60−20

−10

0

10

20

Time (Sec)

P (

deg/

sec)

0 20 40 600.99

0.995

1

1.005

1.01x 10

4

Time (Sec)

Alt

(ft)

0 20 40 60−0.1

−0.05

0

0.05

0.1

Time (Sec)

n y(g)

0 20 40 60570

575

580

585

590

Time (Sec)

VT (

ft/s)

Fig. 10.Roll rate, height, Lateral acceleration and Total velocityin the lateral maneuver

0 20 40 600

20

40

60

80

Time (Sec)T

hrus

t (%

)

0 20 40 60−4

−2

0

2

Time (Sec)

Aile

ron

defle

ctio

n (d

eg)

0 20 40 60−5

0

5

10

Time (Sec)

Ele

vato

r de

flect

ion

(deg

)

0 20 40 60−4

−2

0

2

4

Time (Sec)

Rud

der

defle

ctio

n (d

eg)

Fig. 11.Aileron, Elevator and Rudder deflections and Thrust levelin lateral maneuver

0 20 40 60−20

0

20

Time (Sec)

P (

deg/

sec)

0 20 40 60−20

0

20

Time (Sec)

Q (

deg/

sec)

0 20 40 60−5

0

5

Time (Sec)

R (

deg/

sec)

0 20 40 60−100

0

100

Time (Sec)

Φ (

deg)

0 20 40 600

1

2

3

Time (Sec)

Θ (

deg)

0 20 40 60−50

0

50

Time (Sec)

Ψ (

deg)

Fig. 12. Roll, Pitch and Yaw rate and Euler angles in lateralmaneuver

the ny deviation is more (about 5 times) in the exist-ing method. However, in Figure 15 it is shown thatthe magnitudes of the aileron and rudder controllers(main controllers for lateral case) are much lower inthe new method whenever the bank angle commandis changed. From Figure 16, it is evident that there issudden increase in the roll rate to about80 (deg/sec)in the existing approach, which is much higher ascompared to the new technique proposed. From a largenumber of simulation studies it was found that theuntracked states remain within reasonable limits.

0 20 40 600

2

4

6

8

10

Time (Sec)

α (d

eg)

0 20 40 60−0.2

0

0.2

0.4

0.6

Time (Sec)

β (d

eg)

0 20 40 60294

296

298

300

302

Time(sec)

Dyn

amic

Pre

ssur

e (lb

/ft2 )

0 20 40 600.536

0.538

0.54

0.542

0.544

Time(sec)

M

Fig. 13.Aerodynamic variables in lateral maneuver

0 20 40 60−50

0

50

Time (Sec)

Φ (

deg)

0 20 40 600.99

0.995

1

1.005

1.01x 10

4

Time (Sec)

Alt

(ft)

0 20 40 60−0.1

−0.05

0

0.05

0.1

Time (Sec)

n y(g)

0 20 40 60570

575

580

585

590

Time (Sec)

VT (

ft/s)

Fig. 14.Roll angle, height, Lateral acceleration and Total velocityin lateral maneuver

0 20 40 607

8

9

10

11

12

Time (Sec)

Thr

ust (

%)

0 20 40 60−20

−10

0

10

Time (Sec)

Aile

ron

defle

ctio

n (d

eg)

0 20 40 60−3

−2

−1

0

Time (Sec)

Ele

vato

r de

flect

ion

(deg

)

0 20 40 60−5

0

5

10

Time (Sec)

Rud

der

defle

ctio

n (d

eg)

Fig. 15.Aileron, Elevator and Rudder deflections and Thrust levelin lateral maneuver

Combined longitudinal and lateral maneuverInmany recent literature [Wang & Stengel, 2005, Muir,1998, Pachter, 1996] velocity vector roll is performedat constant angle of attack, which is predominantly alateral maneuver but coupled with some longitudinalcomponent as well. This is done to get a faster re-sponse from the aircraft. We have also performed suchan exercise, by simultaneously giving normal accel-eration command (instead of maintaining a constantangle of attack) with the velocity vector roll command.The command set also includes a velocity command

0 20 40 60−100

0

100

Time (Sec)

P (

deg/

sec)

0 20 40 60−5

0

5

Time (Sec)

Q (

deg/

sec)

0 20 40 60−5

0

5

Time (Sec)

R (

deg/

sec)

0 20 40 60−50

0

50

Time (Sec)

Φ (

deg)

0 20 40 600

1

2

3

Time (Sec)

Θ (

deg)

0 20 40 60−40

−20

0

20

Time (Sec)

Ψ (

deg)

Fig. 16. Roll, Pitch and Yaw rate and Euler angles in lateralmaneuver

0 20 40 601

1.5

2

2.5

3

Time (Sec)

α (d

eg)

0 20 40 60−0.4

−0.2

0

0.2

0.4

Time (Sec)

β (d

eg)

0 20 40 60294

295

296

297

Time(sec)

Dyn

amic

Pre

ssur

e (lb

/ft2 )

0 20 40 600.536

0.537

0.538

0.539

0.54

Time(sec)M

Fig. 17.Aerodynamic variables in lateral maneuver

and the lateral acceleration command (which is zero).Even though this is a difficult task in general, wewish to point out that the approach proposed hereworks well for this case as well. Simulation resultsfor combined longitudinal and lateral maneuver arepresented in Figures 18 - 21. In this maneuver the pi-lot commands given are normal acceleration, velocityvector roll rate, lateral acceleration and total velocity.Note that promising results have also been obtainedwith constant angle of attack maneuvers along withvelocity vector roll (in lieu of normal accelerationcommand). However, those results are not presentedhere to contain the length of the paper.

0 20 40 60−15

−10

−5

0

5

10

15

Time (Sec)

Pw

(de

g/se

c)

0 20 40 60−0.1

−0.05

0

0.05

0.1

Time (Sec)

n y (g)

0 20 40 600

0.5

1

1.5

2

2.5

Time (Sec)

n z (g)

0 20 40 60570

575

580

585

590

Time (Sec)

VT (

ft/s)

Fig. 18.Roll rate, Lateral acceleration, Normal acceleration andTotal velocity in combined maneuver

0 20 40 600

10

20

30

40

50

60

Time (Sec)

Thr

ust (

%)

0 20 40 60−1.5

−1

−0.5

0

0.5

1

Time (Sec)

Aile

ron

(deg

)

0 20 40 60−2.2

−2

−1.8

−1.6

−1.4

Time (Sec)

Ele

vato

r (d

eg)

0 20 40 60−0.6

−0.4

−0.2

0

0.2

Time (Sec)

Rud

der

(deg

)

Fig. 19.Aileron, Elevator and Rudder deflections and Thrust levelin combined maneuver

0 10 20 30 40 50 60−100

−50

0

50

Time (Sec)

Φ (

deg)

0 10 20 30 40 50 600

10

20

30

Time (Sec)

Θ (

deg)

0 10 20 30 40 50 60−80

−60

−40

−20

0

Time (Sec)

Ψ (

deg)

Fig. 20.Euler angles in combined maneuver

4.2.2. Neuro-Adaptive control design In this sec-tion, we present a set of representative results whichshows that the adaptive control performs much betterover the nominal control.

In the longitudinal mode, the pilot commands given(for which results are given here) are roll angle,normal and lateral acceleration and total velocity[φ , nz, ny, VT ]T . The necessary control histories andthe associated tracking performance are given in Fig-

0 20 40 60−1

0

1

2

3

4

5

6

Time (Sec)

α (d

eg)

0 20 40 60−0.2

−0.15

−0.1

−0.05

0

0.05

Time (Sec)

β (d

eg)

0 20 40 60250

260

270

280

290

300

Time (sec)

Dyn

amic

Pre

ssur

e (lb

/ft2 )

0 20 40 600.538

0.54

0.542

0.544

0.546

0.548

0.55

0.552

Time (sec)

M

Fig. 21.Aerodynamic variables in combined maneuver

ures 22 and 23 respectively. Note that there are threeplots in each of these figures, namely (i) the nominalplot (in which the nominal control is applied to thenominal plant), (ii) the actual plot (which representthe scenario in which the nominal control formulais applied to the actual plant with the actual statefeedback) and (iii) the adaptive case, in which theadaptive control is applied to the actual plant. It isobvious from Figure 23 that the actual case leads tothe control saturation in thrust (which opens the loop)and hence leads to failure in the tracking performance(see Figure 22). However the adaptive case does notlead to the control saturation and, as seen in Figure22, the tracking performance is much better. In fact,the performance with adaptive control is so close tothe nominal case that it feels as if there is no failure!

Similar results have been obtained in the lateral modeas well. In this case, the pilot commands assumed areroll rate, lateral acceleration, height and total velocity[P, ny, h, VT ]T . In our simulation studies we havenoticed that the rudder deflection and the thrust goout of bounds for the actual case. However in theadaptive case these are well within the limits. In thetracking performance, the lateral acceleration and thetotal velocity goes away from the desired values forthe actual case, whereas in the adaptive case theyclosely follow the desired values. Note that to containthe length we have not included the plots in this paper.

4.2.3. Robustness study for parameter inaccuracyEven though the results presented in Section is quiteencouraging, to have a better idea of the robustness en-hancement in performance, it is necessary to carry outfurther studies to infer about the robustness enhance-ment of the adaptive controller. To best of our knowl-edge, however, no systematic mathematical analysistool is available for robustness analysis of nonlinearcontrol designs. Hence, we have followed a proba-bilistic analysis approach from large number of sim-ulation studies as an alternative (which can be inter-preted as a Monte-Carlo simulation study).

0 20 40 60 800

2

4

6

Time (Sec)

nz(g

)

0 20 40 60 80−1

−0.5

0

0.5

1

Time (Sec)

ny(g

)

0 20 40 60 80−1

−0.5

0

0.5

1

Time (Sec)

P (

deg

/sec

)

0 20 40 60 80400

600

800

1000

1200

1400

Time (Sec)

VT (

ft/s

)

NominalActualAdaptive

Fig. 22.Outputs in longitudinal maneuver

0 20 40 60 800

20

40

60

80

100

Time (Sec)

Th

rust

(%

)

0 20 40 60 80−1

−0.5

0

0.5

1

Time (Sec)

Aile

ron

def

lect

ion

(d

eg)

0 20 40 60 80−2.5

−2

−1.5

−1

Time (Sec)

Ele

vato

r d

efle

ctio

n (

deg

)

0 20 40 60 80−1

−0.5

0

0.5

1

Time (Sec)

Ru

dd

er d

efle

ctio

n (

deg

)

NominalActualAdaptive

Fig. 23.Aileron, Elevator and Rudder deflections and Thrust levelin longitudinal maneuver

In this study, first we perturbed the aerodynamic forceand moment coefficients as well as the inertia pa-rameters (i.e. mass and moment of inertia values) byvarious percentages of their nominal values. Then, weselected random numbers for each of the parametervalues from within this bound using gaussian distri-bution, where mean value (µ) is taken as nominalvalue of the parameter and the standard deviation (σ )is taken as one-third of the maximum allowed pertur-bation in that parameter. Note that the controller wasmade ignorant of this perturbation.

In each simulation study, the aim was to declare it aseither a ‘success’ or a ‘failure’ and then compute aprobability of success from a large number of simula-tions. We decided to put a set of criteria for declaringa case as success, provided the maximum overshootand steady state errors of the performance outputs arewithin the following limits: (i) nz is within 20% ofits commanded value (which is sufficiently large toexpose the weakness of the nominal design), (ii)VT iswithin 1% of its commanded value (1 % is taken sinceVT itself is a high value), (iii)−5 deg< Φ < 5 degand (iv) −0.05g < ny < 0.05g. Note that absolutenumerical bounds are necessary forΦ and ny sincetheir nominal values are zeros. A simulation run wasconsidered as a success only if ‘all’ of the above con-ditions are satisfied. Similarly, in the lateral case with

bank angle command option, we decided that the fol-lowing bounds to be met for the maximum overshootand steady state errors to declare a simulation run asa success: (i)Φ is within 10%, (ii) VT is within 1%,(iii) h is within 1 % and (iv)−0.05g < ny < 0.05g.Note that the percentage values are with respect to thecorresponding commanded values. In the lateral casewith roll rate command, however, the first conditionis replaced with the condition that the overshoot andsteady state error forP should remain within 10%. Inthe case of combined longitudinal and lateral maneu-ver case, we combined the criteria of longitudinal andlateral cases and decided that the following boundsshould be met for the maximum overshoot and steadystate errors to declare a simulation run as a success: (i)nz is within 20%, (ii)Pw is within 10%, (iii)h is within1% and (iv)−0.05g < ny < 0.05g.

Next, we clubbed the aerodynamic coefficients intoone group and the inertia parameters (mass and mo-ment of inertias) into another. Then we put variouscombinations of their possible percentage errors atdiscrete values and for each such selected combina-tion, ran 50 simulation case. From this exercise, wecalculated the percentage of success and the resultsobtained are summarized in Table 4. Note that the twolateral cases have not been reported separately sincethe results obtained for them were found to be samefor all perturbation cases.

From Table 4, the followings are evident:

• The nominal controller does not have sufficientrobustness. Even for small 2% perturbationsin aerodynamic parameters, there is robustnessdegradation. In fact, there is substantial amountof degradation for 5% and more perturbation ofparameter values.

• With the application of adaptive control, for purelongitudinal and lateral modes the success was100 % (there was no failure at all), even with10% perturbation of parameter values. This obvi-ously indicates substantial amount of robustnessenhancement.

• The adaptive controller also leads to enhance-ment of robustness in combined lateral and longi-tudinal maneuvers. The success rate is not 100 %in this case (primarily because this is a punishingmaneuver). However, with 10% perturbation ofparameter values, the success rate observed was84 %, which is quite high.

5. CONCLUSION

A new relatively straightforward approach based ondynamic inversion technique is presented in this paperfor nonlinear flight control design of high performanceaircrafts. This approach does not require the normaland lateral acceleration commands to be first trans-

Table 4.Robustness in Various Modes

Aerodynamic Coefficient Perturbation 1 % 1 % 2 % 2 % 5 % 5 % 10 % 10 %Inertia Parameter Perturbation 5 % 10 % 5 % 10 % 5 % 10 % 5 % 10 %Nominal Success 100 % 100 % 96 % 92 % 76 % 70 % 48 % 40 %Adaptive Success(Longitudinal Mode) 100 % 100 % 100 % 100 % 100 % 100 % 100 % 100 %Adaptive Success(Lateral Mode) 100 % 100 % 100 % 100 % 100 % 100 % 100 % 100 %Adaptive Success(Combined Mode) 100 % 100 % 100 % 100 % 100 % 100 % 94 % 84 %

ferred to body rates before computing the requiredcontrol inputs. This leads to substantial improve-ment of the tracking response, which is demonstratedfrom the six degree-of-freedom simulation studies ofF-16 aircraft in longitudinal, lateral and combinedlongitudinal-lateral maneuvers. The new approach hastwo potential benefits, namely reduced oscillatory re-sponse (including elimination of non-minimum phasebehavior) and reduced control magnitude. A model-following neuro-adaptive design has also been aug-mented the nominal design in order to assure robustperformance in the presence of parameter inaccura-cies in the model. The robustness study from a largenumber of simulations shows that the adaptive designenhances the robustness of the nominal design sub-stantially.

Even though the results are quite promising, we wishto point out that an important possible direction offuture research would be to test it in the presence ofexternal noise (like wind guest, for example), sensornoise etc. To address this issue explicitly, a state esti-mator (say an extended Kalman filter) will be neces-sary in the loop. This may also open up the issue of‘data fusion’ for better usage of the data from multiplesensors. Another possible direction of future researchwould be to do a careful rigorous comparison study ofthe adaptive control design proposed in this paper withone/more of the existing neuro-adaptive design ideas.

Acknowledgement

This research was supported by Defense Research andDevelopment Organisation (DRDO)-India, under thecontract CSSP/DRDO582 and CSSP/DRDO585.

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