ITERATIVE CONTROL LAW ADJUSTMENT FOR AIRCRAFT LOADALLEVIATION PURPOSE

P. Mouyon∗, C. Cumer∗, Y. Losser∗∗ONERA/DCSD, B.P. 4025, F31055 Toulouse, France

e-mail: mouyon@cert.fr, cumer@cert.fr, losser@cert.fr

Abstract: The adjustment of fixed structure con-trollers becomes a major issue in the development ofaircraft flight control systems. TheH2 adjustmentproblem we deal with here is indeed a key point when-ever the structural aircraft behavior in the presence ofturbulent wind must be improved. Mechanical loadsundergone by the aircraft express as theH2 norm ofa given transfer function. Our objective is to adjustselected gains of the current control law in order to re-duce thisH2 norm while minimizing all other changesin the closed loop behavior.

The paper presents the development of a Lyapunovapproach based on the classical Lyapunov computa-tion of H2 norms. An analytic sensitivity analysisis carried out that leads to an efficient gradient likesearch. The application of this new methodology en-lights the balance to be done between bending mo-ments of the wing and the tail, and also the particularrole of some closed-loop gains.

Keywords: Load alleviation.H2 problems. Controllaw optimization. Adjustment tools. Aircraft controllaws.

IntroductionThis paper presents some methodological aspects

of H2 adjustment and preliminary application resultsto the mechanical load alleviation problem for a largeflexible aircraft in the presence of turbulent wind. Theins and outs of the adjustment problem are describedin the first section. The methodological point of viewis then developed in sections II and III. And the lastsection describes the application results.

Our objective is to tune selected gains of a con-troller in order to reduce theH2 norm of a given trans-fer function while minimizing all other changes in theclosed loop behavior. TheH2 norm is the mechanicalload criterium, and the constraints illustrate the wishto keep invariant the current closed loop behavior with

This work is supported by SPAé.

respect to other classical flight dynamic criteria. Thesatisfaction of this later point simply results here fromthe fact that we develop an iterative small gain cor-rection methods. At each step the adjustment can bestopped if unexpected changes appear. A more accu-rate processing of this constraint would be interestingbut is out of the scope of this paper.

In the second section we develop a sensitivity anal-ysis of H2 criterium based on Lyapunov theory. Weshow that the sensitivity with respect to controllergains expresses as a function of the matrix solutionsof a set of coupled Lyapunov equations. This analyticexpression of the sensitivity is then used in section IIIwhere anH2 adjustment procedure is proposed.

The application results described in the last sectionwere obtained when applying the above mentionedH2

adjustment procedure to an aircraft load alleviationproblem. A large flexible aircraft model is considered.It is well representative of the structural behavior upto a few Hz (thus including most important flexiblemodes). The control law involves both active con-trol of the lower frequency flexible modes and passivecontrol of higher frequency flexible modes. A fewgains related to the main loops are to be adjusted. Thesensitivity analysis and the adjustment procedure en-light the antoganism between bending moments of thewing and the tail, but also the particular role of somefeedback loops.

The adjustment problemThe aircraft control law design is seldom a direct

operation. Generally multiple stages are required toobtain a satisfactory result. When the design condi-tions are modified, to start again the work since thebeginning is not necessarily the most effective way.Thus the need for a retuning of the control laws mayarise all along the development of an aircraft, eachtime the design model or the specifications evolve.

Obviously the availability of efficient adjustmenttools is of particular importance in the end of theproject where delays may have highly negative eco-

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nomic consequences on the overall project. At thisstage the structure of the controller is frozen, andthe adjustment procedures only intend to adapt somecontroller parameters so that every control law re-quirements are satisfied. Thus the development of ad-justment tools dedicated to fixed structure controllersbecomes a major issue in the development of flightcontrol systems.

Among various objectives, adjustment ofH2 normsis often required because a lot of design objectivesexpress asH2 criteria. Adjustment tools allow tocompensate for differences between the aircraft de-sign model and the actual aircraft model that includesall the more recent knowledge of the aircraft behav-ior. This later model often includes a lot of modesneglected at the design step, but may also differ be-cause of new available flight test identification results.

Another need forH2 adjustment tools results fromthe fact that a lot ofH2 constraints are just roughlytaken into account for when designing the control law,or even not at all. For example an accurate evaluationof the aircraft behavior in the presence of turbulencewind in terms of servomechanisms fatigue, mechani-cal loads undergone by the aircraft structure or passen-gers comfort may lead to the adjustment of the controllaw gains.

As regards methodological issues, one must notethat fixed structure design problems are known to begenerally non convex. Existence and uniqueness ofstabilizing controllers of a given order or structure isstill an open question.1 Here we bypass this question,assuming an initial feasible point has been yet found,so that we just have to preserve stability.

Further more the use of anH2 sensitivity tool inorder to tune a given control law may be seen as a rem-iniscence of2,3 where more complex mixedH2/H∞design problems are studied. Indeed, even if our ob-jective is not to find anH2 optimal solution, iterativetechniques dedicated to the minimization ofH2 normsprovide a good basis for the development ofH2 adjust-ment procedures. Related works may also be foundin4,5,6 where homotopic and descent algorithms arediscussed.

Direct synthesis approaches such the one describedin7 is out off the scope of the present paper, since wefollow here a multiple steps design approach.

H2 norm sensitivity analysisWe consider anM− ∆ representation of a closed

system model as depicted on figure (1). The matrix

PerturbationModel System

ParametrizedController

Structured Adjustable Matrix Gain

K=

w e

u y

Fig. 1 M−∆ representation of a closed system

∆ = K involves the controller gains. Some of themare to be adjust in order to decrease theH2 norm ofthe transferG from w to e. Such a structured gain isoften calleddecentralized static output feedbackandcaptures a large class of controller architectures.8

The state space representation writes as:

x = (A+BKCy)x+Bwwe = (Ce+DeuKCy)x

(1)

TheA matrix includes the open loop system dynamic,the controller dynamic and the perturbation model dy-namic. Thusu and y on figure (1) are not the truesystem input and output, but fictitious signals that areequivalent to control and measurement signals for theM−∆ representation.

We denote byJ the squaredH2 norm of G: J =‖G‖2

2. The classical computation ofJ within the Lya-punov approach is first recalled.

J = trace{(Ce+DeuKCy)X(Ce+DeuKCy)T}

(2)

whereX satisfies the Lyapunov equation

(A+BKCy)X +X(A+BKCy)T +BwBTw = 0 (3)

Within the stochastic framework, ifw is a white noiseprocess with power spectral density equal to 1, thenthe definite non negative matrixX is the covariance ofthe state variablex. Since all closed loop matrices arelinear functions of the matrix gainK it is always pos-sible to shift them towards zero. Thus we can assumethat the initial value ofK is zero.

The sensitivity matrix ofJ with respect toK is amatrix denotedS= ∂J/∂K with the same dimensionsasK. It depends onK. Its value at the initial pointK =K0 = 0 is given by the following lemma. However,note that this lemma only applies to non zero terms ofS. SinceK is a structured gain, ifKi, j is fixed, thenSi, j = 0.

Lemma 0.1 The general term of the sensitivity matrixS is given by:

Si, j = tr{CeW CT

e +DiCj X CTe +CeX (DiCj)T}

where X≥ 0 and W satisfy the set of Lyapunov equa-tions:

AX+XAT +BwBTw = 0

AW+WAT +BiCj X +X (BiCj)T = 0(4)

and where Bi (resp. Di) stands for the ith column of B(resp. Deu) and Cj stands for the jth row of Cy.

The dimension ofK is m× p. We denote byei,m theith vector of the natural basis ofR m. We consider avariation ofK with norm ρ and which is zero every-where except at rowi column j. This means thatKwrites:

K = ρei,meTj,p (5)

We have:Bi = Buei,m, Cj = eTj,pCy, andDi = Deuei,m.

Then, from equation (2), J expresses as a function ofρ:

(A+ρBiCj)X +X (A+ρBiCj)T +BwBTw = 0

J(ρ) = tr{(Ce+ρDiCj)X (Ce+ρDiCj)T

}The derivative ofJ with respect to this parameterρ =Ki j is the general term of the sensitivity matrixSi j =dJ/dρ:

Si j = tr{

DiCj X (Ce+ρDiCj)T

+(Ce+ρDiCj)W (Ce+ρDiCj)T

+(Ce+ρDiCj)X (DiCj)T}

whereW is the matrix defined byW = dX/dρ.In order to evaluate this later matrix we compute

the derivative of the Lyapunov equation (3) satisfiedparX. Its derivative w.r.tρ writes as:

BiCj X +(A+ρBiCj)W+W (A+ρBiCj)T +X (BiCj)T = 0

Finally takingρ = 0 yields to the expected lemma re-sult.

An H2 adjustment procedureThe behavior ofJ about the initial pointK0 = 0 is

described at the first order by:

J = J0 + tr{

ST(K−K0)}

(6)

whereJ0 is the value whenK = 0. Our objective is toreduceJ but without any other important change in the

closed loop behavior. This constraint led us not to usea gradient search algorithm but a descent algorithmalong the direction of maximal sensitivity.

Let us consider a Singular Value Decomposition ofS : S= UΣVT . We can write:K = K0 +URVT . Thenwe have :

J−J0 = tr{VΣTUTURVT

}= tr

{VΣTRVT

}= tr

{VTVΣTR

}= tr

{ΣTR

}= tr

{RΣT

}= ∑i Rii σi

Let us remark that‖K −K0‖2 = ‖URVT‖2 = ‖R‖2.Thus if this norm is fixed equal toρ, the maximal de-crease ofJ is achieved with

K = K0−ρσ1

u1vT1 (7)

whereu1 andv1 are the left and right singular vectorsassociated to the smallest singular valueσ1 of S.

The first order variation ofJ is J = J0−ρ. With :ρ = αJ0, the stepα is just equal to the expectedrelative decrease ofJ. A typical value isα = 10%.The proposedH2 adjustment procedure is then thefollowing.

Procedure(i) Compute the current closed loop model in order toshift K towards0.(ii) Compute dK∗ = u1vT

1 /σ1, the direction of varia-tion for K maximizing the sensitivity of S about K= 0.(iii) Search for ρ over the range[0, 0.1× J0] so thatthe gain variation−ρdK∗ yields to a smaller J value,while keeping the closed loop stable.

Before each step of the algorithm, the previouslycomputedK is included in the system closed loopmodel. Then the sensitivity is estimated, and aKvariation is proposed along the maximal sensitivitydirection. A one dimensional search algorithm maybe used to solve step(iii) . Furthermore, in order toensure that the iterations stay inside the stability do-main, a logarithmic barrier functional may be used inthe spirit of interior point methods. However this isa theoretical solution, and simpler and more intuitiveprocesses may be as efficient.

We propose to just test severalα values of a prede-fined grid and jump to the best point. Such a simplealgorithm obviously requires a stability monitoringprocess. For example in case of instability we cansimply reduce theα value (or theα grid step) untilstability is recovered.

From a computational point of vue, note that thenumber of Lyapunov equations to be solved is equal

to the number of free parameters inK. Furthermore,each Lyapunov equation involves the sameA matrix.Classical Lyapunov solvers use a schur factorizationfollowed by a Gauss pivoting method applied to solvea set of linear triangular equations. Consequently thecomputational cost of the schur factorization needsnot to be repeated. As regards the SVD, it must bepointed out that it applies to a matrix whose dimensionis only equal to that ofK. Then, even with high di-mensional state space models, the overall convergencespeed may be kept reasonable. Futhermore, since theadjustment procedure only involves the largest singu-lar value economic dedicated routines may be used.

Retuning dedicated to load alleviationThe used model is representative of the vertical ba-

havior of a large four engine civil aircraft, with con-ventional control surfaces. It is a linear aero-elasticmodel based on a modal description. The overall air-craft model with its control law involves more than250 states. The mechanical load outputs we dealt withhere are:1. the vertical bending moment at the wing root,2. the vertical bending moment at the tail root,3. the torsion moment at a forward fuselage point,4. the torsion moment at a rear fuselage point.This preliminary study uses a Dryden turbulent windmodel. The Lyapunov analysis introduced above al-lows us to compute the mechanical load power with-out carrying on any simulation.

The current control law is based on five measure-ment outputs: the pitch rateq, and the normal ac-celeration at four points (crew station, forward andrear fuselage, left and right external engines). Thefive components of the input vector correspond to rud-der and aileron deflections. The controller involvesfive selected gains to be adjust in order to lower themechanical loads. Gains 1 to 3 correspond to theclassical control of pitch rate and normal accelerationresponses. Gains 4 and 5 are intended to reduce thevibration level by aileron and elevator actuation re-spectively.

Single output minimization

For each of the four mechanical load outputs an ad-justment of the current control law has been carriedout that is intented to minimize the power of the outputunder consideration. The Lyapunov analysis of theseresults are summerized in the following table (1):

Moment Adjustment case1 2 3 4

Bending Wing root -10 34 46 10Bending Tail root 20 -60 500 1398Torsion Forward fuselage 9 78 -40 48Torsion Rear fuselage -2 66 -4 -64

Table 1 Load power variations (%)

This table brings to the fore the balance between thetwo bending moments. If the bending moment at thewing root is reduced by 10 percent (first column),the bending moment at the tail root is increased by20 %. On the other hand (secund column) if thislater moment is minimized down to 60 %, then theformer increases by 34 %. As regards the torsionfuselage moments, the table depicts the fact that theminimization of these moments may result in a verylarge increase of the bending moments (columns 3 and4). Thus, in order to acheive a reasonable adjustment,these two criteria must be completed with wing andtail bending terms.

Sensitivity analysis

We now focus on the sensitity analysis. The overallvariation of the 5 control gains, for each the mini-mization objective, is described in table (2). The ta-ble gives the multiplication coefficients to be appliedto control gains. It clearly appears that the sensitiv-ity is maximal with respect to gain number 1 to 3.Undoubtly such a minimization of the structural mo-ments will thus induce an importante degradation ofthe flight dynamic response. The optimization pro-cedure must be improved by adding a more efficientway to take into account for invariant constraints onthe flight dynamic response.

Gain Adjustment casenumber 1 2 3 4

1 0.0 1.0 1.0 1.02 2.6 -0.2 -0.3 1.03 1.0 -0.4 2.5 4.84 1.0 0.6 1.0 1.05 1.0 1.0 1.0 1.0

Table 2 Multiplicative variations of control gains

A first attempt to keep unchanged the flight dy-namic behavior, is to fix gains number 1 to 3. Theresults described hereafter were obtained under thishypothesis.

case 5:When the adjustment is restricted to gain4, the procedure is only able to reduce moment

number 2. This means that gain number 4 is wellfitted to control the bending tail root. The multi-plicative variation of the gain number 4 is equalto−0.3, which yields to[20,−11, 8, 22] % mo-ment variations.

case 6: As regards the adjustment of gain 5, itallows to minimize all moments but best resultsare obtained when the bending wing root is min-imized (−37 %). Nevertheless this is achieved atthe expense of an unacceptable increase of othermoments.

Multi-output criteria

We now study some multi-output criteria. If aglobal criterium involving all (normalized) momentsis minimized w.r.t the five gains, the only changedgain is the path control gain 1 which is multiplied by0.3. This leads to the case 7 results in table (3) and(4). The trade off between flight dynamic responseand structural moment level is obviously enlighted.

Moment Adjustment case7 8 9 10

Bending Wing root -5 -2 20 3Bending Tail root 1 -52 605 -6Torsion Forward fuselage -3 26 -36 0Torsion Rear fuselage -5 16 -34 2

Table 3 Load power variations (%) for global criteria

A criterium involving both bending moments may bealso reduced (case 8). These both moments decrease,but both torsion moments increase. Once again flightdynamic control gains are lowered. The path controlgain 1 is even decreased down to zero. Indeed it is aquite obvious and intuitive fact that bending momentsmay be reduced by relaxing the control of the aircraftvertical path.

A global reduction of torsion moments may be alsofound (case 9). In that case the path control gain num-ber 1 is kept unchanged. Aileron control (gain 4) isstrongly changed.

If we come back to an adjustment restricted to gain4 and 5, a minimization is obviously more difficultto carry out. The minimization of the global bendingmoment criterium results in a small decrease of thebending tail (case 10). And a global reduction of tor-sion moments was not found to be able w.r.t these twogains.

Gain Adjustment casenumber 7 8 9 10

1 0.3 0.0 1.0 1.02 1.0 0.2 0.3 1.03 1.0 -0.1 3.2 1.04 1.0 1.0 0.4 0.65 1.0 1.0 1.0 1.0

Table 4 Multiplicative variations of control gains forglobal criteria

Efficiency and reliabilityWe fisrt give here some indications about how the

adjustment algorithm behave when optimization pa-rameters evolve. Then some results about the CPUtime cost are presented.

Optimization parameters

The procedure is mainly governed by two choices:the descent direction and the one-dimensional opti-mization method within this direction. Free param-eters are theα-range, i.e. the maximalα value, andtheα-grid (number of points, values).

The previously used α-grid is equal to[−1,−0.7,−0.3] × 10 %. The α-range is thusequal to 10 %. If the range is reduced from 10% down to 5 % the results of the single outputminimizations are changed as described in tables (5)and (6).

Moment Adjustment case1 2 3 4

Bending Wing root -11 26 25 211Bending Tail root 39 -61 243 12615Torsion Forward fuselage 17 73 -33 997Torsion Rear fuselage 3 59 -7 -88

Table 5 Load power variations (%) with reduced α-range

Gain Adjustment casenumber 1 2 3 4

1 0.0 0.5 1.0 1.02 3.6 -0.8 -0.1 0.53 1.0 -0.5 2.1 6.64 1.0 1.0 1.0 1.05 1.0 1.0 1.0 1.0

Table 6 Multiplicative variations of control gains withreducedα-range

These results are very similar to those yet describedin tables (1) and (2) for the bending moments mini-mizations. As regards torsion moments, even if theacheived minimal values are significantly different,the evolution of the four moments is coherent with thepreviously obtained results.

Equivalent results are still obtained when theα-range is reduced to 1 %. In that case the optimalcriterium values are[−9.8,−61,−41,−90] %. How-ever the number of required iterations increases dras-tically (see table 7).

Adjustment caseα-range 1 2 3 4

10 % 6 13 8 115 % 6 25 11 471 % 14 104 59 236

exponential 8 16 10 35conditional 7 33 14 51

Table 7 Number of iterations w.r.t the α-range

A practicle way to limit the number of iteration is touse a variable range. In this spirit, we test an expo-nential reduction of theα-range from 10 % down to3 % over a 60 iterations horizon (see table 7, line 4).Another idea is to reduce the range only when the firstorder approximation seems not to be valid. For ex-ample we test a conditional reduction of theα-range(from 10 % down to 1 %) whenever the current cri-terium reduction is less than 80 % of the expectedρ-value (see table 7, line 5).

Another advantage of the range reduction is thatthis might allow to recover the local converge prop-erty of gradient optimization methods that is knownto exist when some special kinds of decreasing steplength are used (not satisfied within our numerical ex-periments).

As regards theα-grid, we tried to refine the grid.Results of tables (8) and (9) where obtained with anα-grid equal to[−1,−0.9,−0.8, . . . − 0.2,−0.1] × 10%.

Moment Adjustment case1 2 3 4

Bending Wing root -10 36 83 137Bending Tail root 20 -60 906 8416Torsion Forward fuselage 9 79 -48 634Torsion Rear fuselage -2 68 8 -86

Table 8 Load power variations (%) with refined α-grid

Once again, the results of the first two adjustmentcases are very similar to the initial ones. Cases 3 and4 are significantly better from the optimal acheivedvalue, but the degradation of the other moments is ob-viously greater.

Gain Adjustment casenumber 1 2 3 4

1 0.0 1.0 1.0 1.02 2.6 -0.3 -0.5 0.53 1.0 -0.4 2.9 6.44 1.0 0.6 0.1 1.05 1.0 1.0 4.4 1.0

Table 9 Multiplicative variations of control gains withrefined α-grid

The main drawback of theα-grid refinement is theincrease in the computation. Nevertheles we foundthat it is worth to refine the grid only in the zone ofsmall values. Furthermore the refinement may be usedconditionnaly to the deviation of the local criteriumbehaviour from the first order expected approxima-tion. The procedure becomes then quite similar to theone with anα-range conditional adaptation, and theresults are almost the same.

All theses numerical experiments yield to resultsequivalent to those of tables (1) and (2). So, from ourpoint of vue, the benefit of such refinements is not ob-vious. The approach is quite robust w.r.t the involvedparameters.

Computation cost

An important characteristic of models involved inthe study of mechanical loads undergone by the air-craft structure is the high dimension. Even if thisstudy is only carried out here with the aim of check-ing the impact of control laws on the mechanical loadsand adjust the laws so that critical levels are not ex-ceeded, the number of states to be considered is quiteimportant.

As regards the computation cost, we have first de-velopped a simple Matlab routine without any specialcare dedicated to the computation cost. The averagestep duration is less than 1.4 s on a SPARC station IV,when a 25 dimensional model is used. In order to in-crease the size of the models to which the procedureapplies a more adapted Matlab routine has then beendevelopped. It relies on the fact already pointed outthat the schur factorization involved by classical Lya-punov solvers needs not to be repeat within each iter-ation (because it applies to the sameA matrix). Such

a routine yields to an average step duration which isequal to 0.3 s.

With this new algorithm, numerical experimentsshow that the average CPU time duration of each it-eration increases linearly when the size of the aircraftstate space model is greater than about 130 (figures2).Above this limit each iteration increases of about 0.1sper state. These results were obtained with reduced

20 40 60 80 100 120 140 160 180 200 2200

2

4

6

8

10

12

14

Model dimension

Ave

rage

CP

U ti

me

Fig. 2 One iteration average CPU time v.s model order

state space models computed from a unique initialaircraft structural model. For that purpose a reduc-tion procedure dedicated to aircraft structural modelreduction has been developed.

It is then possible and interesting to compareacheived performances. Expected performances aredepicted on figure (3).

20 40 60 80 100 120 140 160 180 200 2200

10

20

30

40

50

60

70

80

90

Model dimension

Load

pow

er d

ecre

ase

(%)

Case 1Case 2Case 3Case 4

Fig. 3 Algorithm robustness

Since the performance criterium is computed withrespect to the current reduced model we expect to

found a decreasing curve (better results might beacheived with low dimensional systems). Howeverthe figure seems to show that the acheived perfor-mances are quite sensitive with respect to the modelsize. Perhaps some more care must be paid to the tun-ing of the reduction parameters. Furthermore it willbe intersting to compare the acheived performanceswith repect to the same full order model. This will bepresented in a forthcoming paper.

ConclusionWe propose to use a gradient-like algorithm to solve

the iterativeH2 adjustment problem. Numerical com-putations of the sensitivity function are detailled andthe adjustment algorithm is detailed. Such a tool isof great interest within the framework of autopilotdesign: it allows more flexibility within the designprocess, and can be used to tune fixed structure con-trollers.

Preliminary results related the adjustment of agiven control law in order to reduce mechanical loadswere presented. They show that the physical balancebetween all outputs and also the intuitive tuning of thegains are well recovered by the algorithm. However abetter way to deal with constraints (invariance of flightdynamic criteria) must be found. This could be doneby considering mixed adjustment problems (H2/H2 orH2/H∞).

From a theoretical point of view, the main draw-back of the proposed iterative approach is that stabilitymust be checked at each iteration. An efficient em-pirical grid-search is proposed but a barrier penaltyfunction could also be introduced in order to preservestability.

Other tools may be used to solve the same prob-lem. For example an interesting way of research maybe found in9,10 where an approach based on the LMIcomputation ofH2 norms is developped. Within thisframework an iterated component wise optimizationalgorithm is proposed which is linked to the work in11

and the K-iteration approach described in.12 This ap-proach ensures stability preservation but is more timeconsuming.

Nevertheless it is worth to be pointed out that theapproach developped here is able to deal with high di-mensional aircraft models (up to a few hundreds) thatare commonly used for aircraft load analysis. Indeed,fast and accurate solvers are now available for the in-volved Lyapunov equations. Even if some numericalimprovments are still required, this may be consideredhas a decisive advantage.

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