Robust Flight Control Design for the HIRM based on Linear Quadratic Control

Aerosp. Sci. Technol.4 (2000) 423–438 2000 Éditions scientifiques et médicales Elsevier SAS. All rights reservedS1270-9638(00)01067-1/FLA

Robust Flight Control Design for the HIRM based on Linear QuadraticControl

Francesco Amatoa, Massimiliano Matteia, Stefano Scalab, Leopoldo Verdeb,*

a Dipartimento di Informatica e Sistemistica, Università degli Studi di Napoli Federico II, Via Claudio 21, Napoli, 80125, Italyb Centro Italiano Ricerche Aerospaziali, Via Maiorise, 81043 Capua (CE), Italy

Received 11 January 2000; revised 26 April 2000; accepted 7 June 2000

Abstract In this paper the application of Linear Quadratic Optimal Control based techniques to the HIRM (HighIncidence Research Model) Robust Flight Control design problem is considered. The structure of the proposedcontroller is basically a Proportional Integral action, both for the longitudinal and for the lateral-directionalparts, designed with optimality criteria in order to achieve the performance and robustness requirements.Moreover, the control scheme is completed by the compensation of some physical non linearities, somedemand shaping filters and a switching logic based on the angle of attack and on the Mach number. Most of thedesign requirements are easily satisfied; some others need a heavier tuning of the controller design parameters.A procedure to tune such parameters, formulated in terms of an optimization problem, is proposed. Simulationresults, from the HIRM automated evaluation procedure are also provided at the end of the paper, from whichit is shown that the proposed controller meets the requirements. 2000 Éditions scientifiques et médicalesElsevier SAS

aircraft control / robust control / proportional integral control / linear quadratic methods / parametricuncertainties / non linearity compensation

Nomenclature

DOF Degrees Of FreedomGARTEUR Group for Aeronautical Research

and Technology in EURopeHIRM High Incidence Research ModelLQ Linear QuadraticPI Proportional Integralg0 sea level gravity accelerationh altitudem aircraft massp, q, r body axis angular ratesθ,φ Euler anglesp0 sea level pressurepw velocity vector roll rateqref pitch rate reference signalq dynamic pressure

* Correspondence and reprints; E-mail: [email protected]

R universal gas constantRl rectangle within the operating en-

velopeT0 sea level temperatureV velocity vector airspeedo point in the operating envelopeO operating envelopeK(o) controller designed inoα angle of attackβ angle of sideslipγ vertical flight path angleλ rate of change of temperature with

altitudeρ air densityA, Bu, C system matricesx,u, y state, input and output of the lin-

earised system

424 F. Amato et al. / Aerosp. Sci. Technol. 4 (2000) 423–438

xi , e, yi state, input and output of the PIcontroller integrators

Aa , Ba , Ba2 matrices of the augmented systemKp ,Ki ,Ka PI controller matricesQa , R LQ weighting matricesqi , ri diagonal elements of the LQ weight-

ing matricesz vector of the diagonal elements of

the LQ weighting matricesJ LQ cost functionδCS, δCD symmetrical and differential ca-

nards deflectionδTS, δTD symmetrical and differential tailerons

deflectionδR rudder deflectionδTH throttle commandf , fi , ci , gi optimisation cost function, its ele-

ments and weights, its constraintsno, nc number of elements of cost func-

tion, number of constraintsPaerlong, Paerlat longitudinal and lateral-directional

aerodynamic uncertaintiesKi , Φi gain and phase offsets at the input

of the actuatorsF polytope of gain and phase offsetsVlat, ν hyper-rectangle of gain and phase

offsets, element ofVlatP,π ,Hw(π,ρ) uncertainty set, element of P, hyper-

rectangular neighbourhood ofπp(z,π) closed-loop characteristic polyno-

mial function of parametersz andπ

ρ radius of the greatest stable hyper-rectangle

Nic, Mag(ω), Ph(ω) Nichols plot and its parameterisedequation

d distance functionsL(t), U(t) Lower and upper bounds for angle

of sideslipt time

1. Introduction

In the last thirty years a big effort has been spent bycontrol researchers to develop robust control techniquesfor multivariable systems (Linear Quadratic OptimalControl,H∞ control,µ synthesis, etc.); to this end see,among others, the books [3,6,7].

Although these methodologies are very powerful de-sign techniques, the issue concerning their application toreal world plants is still a fertile field of research and de-bate between theoreticians and practitioners.

In this context, within the activities of the Groupfor Aeronautical Research and Technology in EURope(GARTEUR), Action Group FM(AG08) (see [1,2] formore details) has investigated the applicability of theabove-mentioned control techniques to the controller de-sign problem for the High Incidence Research Model

(HIRM) described in the GARTEUR report [8,9] (thisreport is also available at web page http://www.nlr.nl/public/hosted-sites/garteur/sum26.html).Universities, Re-search Centers and Industries from seven European coun-tries have been involved in this program, and to eachresearch team the task of applying a particular controlmethodology has been assigned.

In this regard a co-operation between the Dynamics,Control and Automation Department of the Centro Ital-iano Ricerche Aerospaziali (CIRA) and the System andControl Group of the Dipartimento di Informatica e Sis-temistica of the University of Naples Federico II (UNAP)has been established to apply the Linear Quadratic (LQ)based techniques to the design problem of the HIRM.

The work performed by this research team is describedin [1,2]; due to the limited time to our disposal and tothe objective difficulty of the performance specifications,many requirements were left partially or completelyunsatisfied. However, after the end of the project, thecollaboration between CIRA and UNAP to improve thedesign has continued.

First we recall that LQ based techniques, which arebasically oriented to the control of linear plants, cannotbe applied in a straightforward way to take into accountall the requirements given in [8,9]. Therefore, we haveproposed a design cycle based on the introduction ofsome auxiliary functions to achieve an optimal choice ofthe design parameters of the controller, namely the LQmatrix weightings.

The aim of this paper is to describe the whole projectconducted by CIRA and UNAP, together with the im-provements provided by the authors to the original de-sign. The improvements are:

(a) The first change in the controller structure is thedecoupling of the PI controller in a longitudinaland a lateral directional part; these have nowbeen designed separately (this allows us to reducethe controller order, with obvious benefits for thewhole control scheme).

(b) The second change concerns the method employedto control the angle of attack during roll manoeu-vres. In the first version of the control system thePI gains were gain scheduled on the pilot roll ratedemand, in such a way that integral control of theangle of attack error signal was added only duringthe roll manoeuvres to minimise deviations fromthe value of the angle of attack at the beginning ofthe manoeuvre. Further, the other gains of the PIcontroller changed according to the pilot roll ratedemand.In the current version a non-linear static compen-sation is used to modify the pilot demand to the PIcontroller, whose gains are no longer scheduled onpilot demand. Integral control of the angle of at-tack is no longer needed, since the compensationinverts the non linear equation of the angle of at-tack time derivative,α, to compute the demand to

F. Amato et al. / Aerosp. Sci. Technol. 4 (2000) 423–438 425

the PI controller in such a way that variations ofthe angle of attack are cancelled during the lateralmanoeuvre. The same compensation is also usedto implement carefree handling of angle of attack.

(c) The third change is the implementation of a func-tion to compensate gravity vector effects.

(d) The robust stability analysis of the closed loopsystem, which is now performed in a rigorous wayby applying the procedure proposed in [5,11].

The resulting characteristics of the proposed controlscheme are the following:• the controller is decoupled into a longitudinal and a

lateral-directional controller;• both controllers are basically Proportional plus In-

tegral (PI) control laws with matrix gains computedby means of the standard LQ technique;• the robustness and performance issues are addressed

by means of a design cycle which optimises asuitable cost function, over the possible weightingmatrices of the LQ design.

Moreover• shaping of the pilot demands and compensation of

physical non linearities are included, in order to takeinto account physical limitations of the non linearaircraft and variations of the aircraft dynamics in theoperating envelope;• the controller gains are scheduled, on the basis of

the Mach number and of the angle of attack, so as tocover the whole operating envelope of the aircraft.

The paper is organised as follows. The chosen controllerarchitecture is discussed in section 2; the gain schedulingconsiderations are detailed in section 3; the translation ofthe HIRM design criteria into an optimisation problem isprovided in section 4; section 5 is devoted to the descrip-tion of the design cycle, while in section 6 the numericalresults from the evaluation procedure are presented; theseresults show the improvement obtained with respect tothe original design. Finally some concluding remarks aregiven in section 7.

2. The controller architecture

The control system architecture proposed for the HIRMdesign problem is schematically shown infigure 1. Ittakes as inputs the four pilot demands and the measuredvariables and computes the required deflections of theavailable control surfaces and of the throttle. From a func-tional point of view, the controller can be divided into thefollowing main sub-components:

(i) a set of demand shaping filters;(ii) the computation of velocity vector roll rate, dy-

namic pressure, vertical flight path angle and airdensity from the measured variables;

(iii) state feedback Proportional-Integral action de-signed with LQ optimality criteria;

(iv) a compensation of the effects due to the physicalnon linearities (dynamic pressure, air density andgravity);

(v) a carefree handling function for angle of attackand load factor limitation.

2.1. Demand shaping filters

As usual in aeronautical applications, an important roleis played by the Demand Shaping filters (see for example[3,8]).

In our controller scheme, the filters have been used toachieve the following tasks:

(i) to avoid the saturation of the control commands;(ii) to allow the satisfaction of the Gibson criteria

and the specification on dropback according to therequirements given in [8,9].

Indeed, if the pilot gives a sudden demand on one of thevariables to be controlled, the control command couldexhibit peaks or a high rate of variation. When dealingwith a non linear plant, where saturations and rate limitersare acting, it may happen that the closed loop system stateis brought out of the stability region.

On the other hand, the requirements on the input-output behaviour of the plant expressed by the Gibsoncriteria can be accomplished by introducing linear filtersat the input of the plant with an adequate placement oftheir poles and zeros.

Two kind of filters have been considered in our con-troller:

(i) non linear filters which avoid the demand to be toohigh or too fast in correspondence with the criticalpoints of the flight envelope;

(ii) linear filters consisting of stable linear dynamicsystem whose location of poles and zeros allowsfor the satisfaction of the Gibson criteria.

2.2. The computation of velocity vector roll rate, airdensity, dynamic pressure and flight path angle

Since lateral stick deflection demands a velocity vectorroll rate,pw, which is not a measured variable, we needto find an efficient way to estimate or to compute thisvariable on the basis of the set of measured variables.

The expression ofpw in terms of the angular rates inthe body axesp, q andr and the angles of attack,α, andof sideslip,β , is:

pw = p cosα cosβ + q sinβ + r sinα cosβ.

Since thea andβ state variables are measured, we canuse the transformation above to obtain the actual valueof pw. Note that, when demanding the velocity vectorroll rate, α is kept almost constant andβ is kept tozero to perform the manoeuvre satisfactorily. This meansthat the matrix transformation turns out to be almost


Figure 1. The control system architecture.

constant; hence the non linearity introduced by this termis expected to be sufficiently mild so that it does notdeteriorate the closed loop system performance.

Computation of air density is made from the measuredaltitude according to the following expression, which isvalid in the HIRM altitude envelope (h < 20000 feet)

ρ = p0

R(T0+ λh)(T0+ λhT0

)− g0Rλ

.

The dynamic pressure is computed from the measuredairspeed and computed air density according to thefollowing equation, which is assumed to hold, given thelimited Mach envelope:

q = 0.5ρV 2.

The vertical flight path angle,γ , is computed using themeasures of the attitude angles and of the angles of attackand sideslip.

2.3. The PI feedback control action

The core of the proposed control scheme is a multi-variable PI feedback action. We used the LQ control tech-nique to compute the PI matrix gains following the guide-lines drawn in [3]. The PI gains design is done separatelyfor the longitudinal and the lateral-directional parts ofthe controller. The following development applies to bothaxes.

Let us consider the linearised model of the HIRMaircraft in the form

x =Ax +Buu,y = Cx,

in correspondence with a certain flight condition, andassume the complete measurability of the state variables.

To satisfy the HIRM requirements given in [8,9] wehave to synthesise a controller which allows for theregulation of four output variables: the velocity vectorair speed,V ; the pitch rate,q ; the velocity vector rollrate,pw; and the sideslip angle,β . The controls usedare the symmetrical and differential tailerons,δTS andδTD, the symmetrical and differential canards,δCS andδCD, the rudderδR, and the throttle commandδTH. Thestates, control inputs and controlled outputs are, for thelongitudinal control law design problem

x = [V α q ϑ]T,u= [δTS δCS δTH]T,y = [V q]T,

and, for the lateral directional design problem

x = [β p r ϕ]T,u= [δTD δCD δR]T,y = [β pw]T.

Let us now make reference to the closed loop schemeshown infigure 2, where we consider the linear plant tobe the linearised HIRM model. The state space realisationof the integrator is

xi = e,yi = xi,

wheree= r−y is the tracking error andr is the referencesignal.

We have the following closed loop system state spaceequation

xa =[A+BuKp BuKi

−C 0

]xa +

[0

I

]r, (1)


Figure 2. Closed loop scheme with PI controller.

where

xa =[x

xi

].

Equation (1) can be rewritten in a compact form as

xa = (Aa +BaKa)xa +Ba2r,

where

Aa =[A 0

−C 0

], Ba =

[Bu

0

], Ba2=

[0

I

],

are the state space matrices of the augmented system andKa = [Kp Ki ] is a state feedback gain acting on such asystem.

Obviously the design of a state feedback controlleron the augmented system allows us to compute, via anappropriate partitioning of the matrixKa , the propor-tional and integral gain matrices for the original model(A,Bu,C).

As is always the case with LQ based techniques, thekey point for the design is the choice of the weightingmatricesQ andR for the quadratic cost function relatedto the auxiliary system

J =∫ (xTa Qaxa + uTRu

)dt.

Indeed our main objective is to keep the tracking errore as low as possible, while maintaining the controlvariables within the prescribed ranges. This means thatin the quadratic cost function, defined for the system(Aa,Ba ), the last four states, which are the states ofthe integrators, should be emphasised by increasing therelativeQa terms.

When the choice of theR matrix is concerned, a goodtrade-off between performance and control activity must

be found; in any case we assume, to simplify the design,Qa andR to be diagonal matrices.

The PI controller structure proposed has four apprecia-ble properties:

(i) the low order of the compensator (only four linearstates coming from the integrators);

(ii) the simplicity in the computation of the gainmatrices (we use the solution of two standard LQproblems, each of order 6:4 from the aircraft and2 from the controller);

(iii) some intrinsic robustness properties guaranteedby the LQ control (see [3]);

(iv) finally the decoupling between the longitudinaland lateral directional controller simplifies thetuning of the controller gains, since two problemsof lower order have to be solved instead ofone problem of higher order. This is even moreappreciable if redesign of the controller gainshas to be done for some reason; indeed in thiscase it could be sufficient to repeat the tuning ofonly one of the controllers, longitudinal or lateraldirectional, whilst the full controller should betuned again, if it were not decoupled.

However there are several problems in using this tech-nique which forced us to add other components to thecontrol scheme:

(i) the technique requires the complete accessibilityof the state;

(ii) it is not possible to take into account the nonlinear nature of the plant;

(iii) there is no way to directly take into account all ofthe robustness and performance specifications asrequired by the HIRM design problem.

Concerning point (i), the complete accessibility of thesix DOF aircraft model states is a matter of fact; theproblem is the presence of the additional states of sensorsand actuators. By means of an extensive campaign ofsimulations on the complete linearised model of the


aircraft we could verify that the dynamics of sensors andactuators have a minor influence on the performance ofour controller. Hence we propose to neglect them in thesynthesis of the controller and to reintroduce them whenverifying the performance of the closed loop system.

To overcome problem (ii) we introduced subsystemsinto the controller actions: a non linear scaling of thecontrol commands (see section 2.4) and a switchinglogic between different sets of matrix gains scheduledwith respect to Mach number and angle of attack (seesection 3).

Finally, the way to overcome problem (iii) will bediscussed in section 4.

2.4. Compensation of physical non linearities

As usual when using a linear controller for a non linearplant, the stability and performance of the non linearclosed loop system is not guaranteed; indeed, there arestrong variations of the linearised models of the plantaround different operating conditions.

A possible way to avoid these problems is to compen-sate some of the non linearities of the model to make thelinearised model matrices as similar as possible in a largeregion of the operating envelope.

For aircraft, it is possible to take into account thenon linear effect of the airspeed and of the altitude onthe forces and moments and try to compensate it bymeans of a non linear scaling of some of the gainsof the controller. Indeed the aerodynamic forces andmoments generated by the control surfaces are linearlydependent on the dynamic pressureq = 0.5ρV 2. If thecontroller has been designed for the linearised modelaround one operating point, where the nominal dynamicpressure isq0= 0.5ρ0V

20 , when the controller is applied

in other operating conditions the control law is amplifiedor reduced depending on an increase or a reduction of thedynamic pressure.

In order to normalise the aerodynamic surface controlpower we have scaled the control commands at the outputof the PI action with a termq0/q.

A similar consideration applies to the thrust of theengines, whose nominal value is the sea level thrust. Theactual thrust is scaled with respect to the nominal onewith the ratio of the air densities at actual and sea levelaltitude; therefore we have scaled back the demandedthrust with the inverse ratio of the air densityρ0/ρactual.

Further compensations of physical non linearities im-plemented into the computation of the demanded throttleare: the inversion of the static non linearity between thethrottle and the demanded thrust, and the compensationof the gravity vector effect on the airspeed, through theaddition to the demanded thrust of a component given by

mg sinγ

cosα cosβ.

2.5. Control of angle of attack for lateralmanoeuvres and carefree handling

Direct regulation of the angle of attack of HIRM isrequired in two cases. First, during a lateral manoeuvre,since the pilot demands the velocity vector roll rate,which means that angle of attack should be kept constantduring this manoeuvre. Second, a carefree handlingfunction is desirable, in order to recover from situationswhere the pilot demands would bring the aircraft outof the prescribed angle of attack envelope. A unifiedsolution to these two requirements is provided in the formof a static non linear compensation of angle of attackvariations, computed via an inverse dynamics technique,which sets the reference command,qref, for the pitch axisPI controller.

The HIRM envelope is limited in the angle of attackto [−10◦, 30◦]. Further a proof of the protection againstviolating the given limits is required in the HIRM defin-ition document [8,9]. We have implemented this carefreehandling function in a controller subsystem which sets apitch rate reference to be followed by the ProportionalIntegral longitudinal controller.

In particular, given the above angle of attack envelope,the pitch rate reference is taken from the pitch ratedemand shaping filter ifα ∈ [−8◦,28◦]. If, instead,α >28◦ or α <−8◦, the pitch rate value to keepα constant,qref, is computed from the expression

0= α = qref− (p cosα + r sinα) tgβ

+ (anz + g cosθ cosϕ)cosα

V cosβ.

The signalqref has also been used to hold the angle ofattack constant during velocity vector roll manoeuvres.

3. Gain scheduling considerations

Let us denote byO = [0.15,0.5] × [−10◦,+30◦] ⊂R2 the operating envelope in the mach-angle of attackplane of the HIRM aircraft and consider the linearisedmodel of the HIRM around a pointo ∈O

x =A(o)x +Bu(o)u.

It is clear that the controller designed for a pointo, sayK(o), according to the procedure proposed in the previ-ous subsections, will work satisfactorily in a neighbour-hood of pointo, but its performance will degrade far frompoint o. Therefore, it is necessary to schedule the con-troller versus the operating envelopeO . More preciselylet us consider a representation ofO in the form

R1,R2, . . . ,Rr :r⋃l=1

Rl ⊇O


with Rl , l = 1, . . . , r, suitable rectangles belonging to theoperating envelope plane.

The problem is designing an output feedback gainscheduled controllerK(o) in the form

K(·) :o ∈Rl→Kl, l = 1, . . . , r

such that the overall closed loop system is asymptoticallystable for allo ∈Rl , l = 1, . . . , r.

Stability of the gain scheduled scheme is guaranteedunder the assumption that the aircraft reaches the steadystate regime before switching to another configuration.In other words, supposing that at the instantt∗ thereis a switching between regionRi and regionRj ofthe operating envelope, we assume that the aircraft willremain in regionRj for a sufficiently long time before anew transition will occur. Critical situations can only betaken into account via simulation tests.

4. The translation of HIRM design criteria into anoptimisation problem

One of the drawbacks of the LQ control technique isthat it is not possible to take directly into account all ofthe specifications on performance and robustness givenin the HIRM problem definition [8,9]. The designer hasto iterate on the choice of the weighting matrices tryingto achieve his objectives; this procedure to synthesise asuitable controller could take a very long time. In orderto save time we introduced some auxiliary functions,dependent on the design parameters, which allow us tocheck if the given requirements are satisfied.

We defined two kinds of auxiliary functions:• the functions denoted bygi translate the more

stringent specifications of the control problem; thesefunctions are positive if the given requirement is notsatisfied;• the functions denoted byfi translate less stringent

specifications; in particular the closerfi is to zerothe closer we are to the complete satisfaction of thespecification.

Let Qa = diag(q1, . . . , q6) and R = diag(r1, r2, r3) bethe weighting matrices for one of the two decoupled LQproblems described in section 2.3. The parameter designvector is defined as

z := [q1 . . . q6 r1 r2 r3 ]T.

The following optimisation problem can be posed:

minzf (z)

such that

gi(z) < 0, i = 1, . . . , nc,

where

f (z)=no∑i=1

cif2i (z)

and gi(z) are the functions to be minimised and theconstraints respectively,ci are weighting elements,nc isthe number of constraints andno is the number of termsin the cost function which results from the performanceand robustness requirements.

Remark1. – Note that the solution of the above opti-misation problem automatically finds a pair of optimalweighting matrices for the LQ problem. However thegoodness of the solution found depends on the choice ofthe weightsci of the optimisation functionf (z). Theseweights can be tuned, according to the designer experi-ence, to improve the performance of the designed con-troller.

The above formulation applies to both the longitudinaland lateral-directional design, the only difference beingin the number and definition of the functionsfi andgi . In the following we will describe some of thefiandgi functions utilised to translate part of the HIRMrequirements (see [1,2]).

4.1. Stability analysis under modelling errors andperturbations at the input of the plant

Since the design is performed on the linearised modelof the plant, in the synthesis phase we considered themodelling errors as uncertainties entering the system ma-trices of the aircraft. We used for the stability analysisa methodology and software tools developed by the au-thors in [11] and already applied to other aeronauticalproblems (see for example [5]), which allow us to checkthe stability of a linear time-invariant system dependingon uncertain parameters ranging in a hyper-rectangle. Inparticular, this analysis tool requires the computation ofthe characteristic polynomial of the system in correspon-dence with the vertices of the hyper-rectangle.

Modelling errors are introduced as follows. Considerone of the regionsRl defined in section 3 and consider theuncertain linearised model of the aircraft around a pointo ∈ Rl comprehensive of sensors, actuators and controllerstates in the form

x =A(o,p)x +B(o,p)u,y = C(o)x,

wherep is the vector of the model uncertainties whichbelongs to the hyper-rectangle Paer according to [8,9].However, when the number of parametric uncertaintiesis large, engineering judgement and experience can beused to select the most significant parameters againstwhich to test robustness; therefore Paer is a subset of theuncertainty set considered in [8,9]. Further, as already


Figure 3. Closed loop broken at the input of the actuators.

Figure 4. The polytope F for the evaluation of the stabilitytest.

Figure 5. The region to be avoided by the Nichols plots inthe single loop analysis.

said, during the design phase, the longitudinal and lateraldirectional problems can be checked separately in orderto lower the order of the robust stability analysis problemto be solved at each iteration; the coupled system can bechecked in a post-design analysis. Therefore, we denoteby Paerlongand Paerlatthe uncertainties of the longitudinaland lateral directional models respectively.

In a similar way, both for the longitudinal and lateraldirectional problem, we can take into account the HIRMrobustness requirements concerning the perturbations atthe input of the actuators (seefigure 3). For the sake ofsimplicity in the sequel of the section we shall refer onlyto the lateral-directional case.

The stability test required in Muir et al. [8,9] deals withthe simultaneous and independent gain,Ki , and phase,Φi , offsets at the input ofith actuator as shown infigure 3.The offsets must attain values in the polytope F depictedin figure 4. This gives us the possibility to define anauxiliary vector of uncertain parameters

v = [K1 Φ1 K2 Φ2 K3 Φ3 ] ∈ Vlat

such thatv ranges in the hyper-rectangleVlat, when theoffsets range in F.

In order to use the tools available for stability robust-ness analysis we approximated the phase offset with asecond order Pade approximation.


Let P=Paerlat×Vlat be the whole uncertainty set of thelateral-directional model. Letπ0= (pT

0 , vT0 )

T ∈ P be thevector of nominal parameters and

Hw(π0, ρ) :={π : ‖π − π0‖w∞ < ρ

}be an hyper-rectangular neighbourhood ofπ0 with radiusρ, where

‖x‖w∞ =maxiwi |xi|, wi > 0.

It is always possible to select a weighting vectorw0 suchthatHw0(π0,1)= P . By using the algorithm proposed in[11] it is possible to solve the following problem for thegiveno ∈Rl :

ρ(z)= supρ

such that

p(z,π) is stable, ∀π ∈Hw0(π0, ρ),

wherep(z,π) is the closed loop characteristic polyno-mial of the lateral-directional model in correspondenceof the controller parametersz and of the uncertain para-metersπ .

Checking the HIRM requirement on stability robust-ness to modelling errors and perturbations at the input of

the plant is therefore equivalent to check that the radiusof the greatest hyper-rectangle still stable,ρ, is greaterthan one.

This is readily translated into the constraint function

g1(z)= 1− ρ(z).

4.2. Single loop analysis: Nichols plots

In [8,9] a performance specification is given in termsof the single input-single output Nichols plot of thefrequency response between each actuator demand andthe corresponding error signal, obtained by breaking theloop at the point shown infigure 3while leaving the otherloops closed; these Nichols plots should avoid the regionsshown infigure 5.

Let us consider the parameterised equation of theNichols plot obtained by breaking theith channel of theloop in the frequency interval of interestω ∈ [ω1,ω2], sayNici ,

Magi =Magi (ω)

Phi =Phi (ω)

and let (Magc(τ ), Phc(τ )), τ ∈ [0,1], the parameterisedequation of the contour of the region to be avoided,denoted byD. We can define a relative distance betweenthe Nichols plot and the regionD (seefigure 6):

Figure 6. Definition of distance for single loop Nichols plot constraint.


d(D,Nici )

=

−minω,τ

√(Magc −Magi )2+ (Phc −Phi )2,

if Nici ∩D = ∅, ω ∈ [ω1,ω2], τ ∈ [0,1],minτ

√(Magc −Magi ($))2+ (Phc −Phi ($))2,

if Nici ∩D 6= ∅, τ ∈ [0,1]where$ = ($2 − $1)/2, and$1 and$2 > $1 arethe values ofω whereNici intersects the contour ofD.The requirement is equivalent to impose that, for a givenpoint o ∈ Rl , the relative distance between the Nyquistplot of the transfer function under consideration and theregion depicted infigure 5 be negative. Therefore wecan add the following constraints, one for each actuator,whose negativeness guarantees the satisfaction of thisperformance requirement

gi(z)= d(D,Nici ).

4.3. Physical considerations

In [8,9] other requirements are given which take intoaccount the physical behaviour of the plant. Because ofthe non linear nature of these specifications, it is notpossible to take any of them directly into account in theLQ controller synthesis.

On the other hand they can be considered in theformulation of a number of auxiliary functions, part ofthem computed on the basis of numerical simulationsperformed on the closed loop non linear system.

As an example, consider the two specifications givenon the sideslip time response; the former concerns thecoupling between roll and sideslip, while the latterregards the response to a sideslip demand.

Let β(t, z) be the sideslip time response of the closedloop non linear system to a roll rate demand in the timeinterval[0,10], obtained for a certain design parameterz.We introduce the following auxiliary function

f1(z)= maxt∈[0,10]β

2(t, z).

To satisfy the second requirement we make reference tothe region shown infigure 7, sayB. The contour of thisregion is defined by the following parametric equations

L= L(t) t ∈ [0,10],U =U(t),

which represent the lower and upper bounds ofB respec-tively.

Performing a ten seconds simulation of the non linearclosed loop system, we obtain the sideslip time responseβ(t, z). Let us define a time dependent distance function

Figure 7. The region for the evaluation of the sideslip timeresponse requirement.

in the form:

d(t, z)={

0, if L(t)6 β(t, z)6U(t),L(t)− β(t, z), if β(t, z)6 L(t),β(t, z)−U(t), if β(t, z)>U(t).

We introduce the following auxiliary function

f2(z)= maxt∈[0,10]

d(t, z).

Similar auxiliary functions have been defined on the othervariables (see [1,2]).

5. The design cycle

The controller design cycle is divided in seven steps,which are described below.Step 1Consider a pointo ∈O .Step 2Consider a rectangleR containingo; denote the fourvertices ofR by o(l), l = 1, . . . ,4.Step 3Solve the optimisation problem

minz,lf(l)(z)

such that

gi(l)(z) < 0, i = 1, . . . , nc, l = 1, . . . ,4,

wheref(l)(z) andgi(l)(z) are the optimisation functionand theith constraint relative to vertexo(l) respectively.Step 4If the problem in Step 3 admits a solutionand R ⊃ Ogoto Step 6;Else If the problem in Step 3 admits a solutionandR ⊂ O , consider a setR1 ⊃ R, setR = R1 and gotoStep 2;Else If the problem in Step 3 does not admit a solution,consider a setR1⊂R, setR =R1 andgotoStep 2.


Note that, as stated, the iterative algorithms composedof Steps 1–4 terminate only when we obtain a rectangleR

which covers the operating envelopeO . Clearly, anotherstopping condition is obtained when the size of therectangle obtained at the iterationk + 1 differs for lessthan a pre-specified toleranceε from the size of therectangle obtained at the iterationk.

In this procedure we implicitly assume that stabilitywith respect to the four verticeso(l), l = 1, . . . ,4, guar-antees stability for allo ∈ R; this is, obviously, not al-ways true, and an a posteriori check is required. Such atest can be performed either via a force gridding ofR orby consideringo as an uncertain parameter attaining val-ues inR and by applying again the algorithm proposedin [5,11].Step 5Repeat the Steps 1–4 until a set of rectangles, whoseunion cover the operating envelopeO , is obtained.Step 6Design the demand shaping filters.Step 7Test the performance of the complete closed loop nonlinear system by means of numerical simulations.

The numerical tools which actually support the abovedesign cycle are all the assessed tools to solve theLQ problems, and some numerical optimisation codes,problem oriented, developed to quicken each step of thedesign procedure. Among these, a SimulinkTM basedUser Interface for the Design and Analysis of the FlightControl System has been built, by which the designercan run MATLABTM functions developed for the HIRMproblem.

6. Simulation results

The HIRM design problem definition is complementedwith an Automated Evaluation Procedure [8,9] inMATLAB TM/SimulinkTM which has been developed andused in the GARTEUR project to ensure comparison ofthe results achieved by HIRM controllers designed bymeans of different control methods. In this section a setof figures which are the results of the HIRM AutomatedEvaluation Procedure are presented to show the perfor-mance and robustness of the proposed control scheme;the analysis of the following figures clearly shows thestrong improvements obtained with respect to the origi-nal design developed in [1,2].

Figures 8 to 11 show the results for the HIRM ro-bustness requirements. The Nichols plots do not en-ter into the exclusion zone, thus showing that the con-troller guarantees robustness with respect to these re-quirements.

From figures 12to 15 satisfaction of the Gibson re-quirements for both pitch and roll axis is demonstrated.In particular the Nichols plots (figures 12and13) corre-sponding to the frequency response between pitch stickforce and pitch attitude, computed for input amplitudesof 10% and 30% of the maximum (at Mach 0.4 and al-titude 10000 ft) and 1% and 10% of the maximum (atMach 0.24 and altitude 20000 ft), pass through the re-quired Level 1 region. The same considerations can berepeated for the roll.

Figures 16and17show the behaviour of the controlledaircraft in response to longitudinal stick step commandof maximum and minimum amplitudes at flight condi-tion Mach= 0.4,h= 10000 feet; infigure 16the normalacceleration and infigure 17the angle of attack, together

Figure 8. Nichols plot: nominal flight, Mach 0.4, altitude10000 ft. Figure 9. Eigenvalue plot: multivariable uncertainty, Mach 0.4,

altitude 10000 ft.


Figure 10. Nichols plot: parametric uncertainty, Mach 0.4,altitude 10000 ft.

Figure 11. Nichols plot: parametric uncertainty, Mach 0.24,altitude 20000 ft.

Figure 12.Gibson criterion – pitch: Mach 0.4, altitude 10000 ft,input amplitudes of 10% (continuous) and 30% (dashed) of themaximum.

Figure 13. Gibson criterion – pitch: Mach 0.24, altitude20000 ft, input amplitudes of 1% (continuous) and 10%(dashed).

Figure 14. Gibson criterion – roll: Mach 0.4, altitude 10000 ft,input amplitudes of 10% (continuous) and 30% (dashed).

Figure 15. Gibson criterion – roll: Mach 0.24, altitude20000 ft, input amplitudes of 1% (continuous) and 10%(dashed) of the maximum.


Figure 16. Behaviour of normal acceleration at Mach 0.4 andaltitude 10000 ft.

Figure 17.Behaviour of angle of attack at Mach 0.4 and altitude10000 ft.

Figure 18. Frequency response from pitch rate to taileron(continuous) and canard (dashed).

Figure 19. Frequency response from normal acceleration totaileron (continuous) and canard (dashed).

with their limits, are given. The plots show the satisfac-tion of the carefree handling requirements.

Figures 18and 19 show that the structural couplingrequirements are satisfied since the plotted frequencyresponses from pitch rate and normal acceleration totaileron and canard respectively are lower than the pre-scribed high frequency limits.

Figures 20to 22 are a subset of the assessment ma-noeuvres available in the HIRM Automated EvaluationProcedure. They are responses of the HIRM to step in-puts in the pilot demands for a flight condition withMach number 0.3 and altitude 5000 feet.Figure 20 isthe response to a pitch rate step demand of 5◦/sec fromt = 2 sec tot = 6 sec. Infigure 21the performance ofthe controller when the pilot demands a 360◦ roll loop

around the velocity vector for the same flight condition isshown; the angle of attack changes by less than 1◦ duringthe whole manoeuvre, thus showing the effectiveness ofthe proposed controller; the pitch rate follows closely thereference signal generated to keepα constant.Figure 22is the response to an airspeed step demand of 50 m/secfrom t = 2 sec tot = 40 sec.

7. Concluding remarks

In this paper we have considered the application of theLQ optimal control based techniques to the HIRM ro-bust flight control problem. In particular the controlleris decoupled into a longitudinal and a lateral directionalcontroller; both controllers are basically PI control laws


Figure 20.Assessment manoeuvre: response to a pitch rate step demand of 5 deg/sec fromt = 2 to t = 6 sec.

Figure 21.Assessment manoeuvre: response to a 360 deg loop around the velocity vector fromt = 2 to t = 7 sec.


Figure 22.Assessment manoeuvre: response to an airspeed step demand of 50 m/sec fromt = 2 to t = 40 sec.

whose matrix gains are chosen according to the standardLQ technique. Since LQ techniques are oriented to thecontrol of certain linear plants, they cannot be applied ina straightforward way to take into account both the ro-bustness and the performance requirements, which haveoften to be evaluated on the whole non linear system. Todeal with this problem, the robustness and performanceissues have been translated into some auxiliary functionswhich define the cost function and the constraints of asuitable optimisation problem which is part of the con-troller Design Cycle; the variables of such problem arethe entries of the weighting matrices of the LQ problems.

To evaluate the suitability of the proposed controlscheme and of the related design cycle in problemssuch as the HIRM design control problem, we wouldlike to summarise the aspects which, in our opinion,can be considered as the main advantages and the maindrawbacks of the method.

Of course we have the following advantages:(i) the controller has a low order and the scheme is

quite simple to implement, since it is essentially aPI controller; this kind of structure also simplifiesthe understanding of its physical action on theaircraft;

(ii) a thorough theoretical background is not requiredto apply the proposed method; furthermore, the

conditions that the augmented plant should satisfyare very mild and can be easily checked;

(iii) since the LQ method is very well assessed in thecontrol literature, the software to be used either isalready available in the main control packages orquite easy to implement;

(iv) most of the design requirements are easily satis-fied; some other requirements can be satisfied bymeans of tuning the controller parameters;

(v) re-design of flight control laws with this methodshould be straightforward, provided that the phys-ical problem is not significantly changed.

On the other hand, the main drawbacks are:(i) the design is partially based on a trial and error

procedure (the adjustment of the weightsci ofthe optimisation functionf ) which does notguarantee a short design time; however, basedon our experience, the procedure does not needmany iterations. In principle, one could enlargethe design vectorz to also include the weightsci and therefore arrive to a completely automatedprocedure. This approach could render very heavythe computational burden of the optimisationproblem if the number of functionsfi is large,and therefore it should be avoided in such a case;


(ii) a complete simulation campaign on the non linearclosed loop system is needed to validate thecontroller in each flight condition; on the otherhand it is certainly true that this last point is alsocommon to any practical flight control systemdesign (as stressed for example in the AGARDreport [10]) especially if linear plant oriented.

References

[1] Amato F., Mattei M., Scala S., Verde L., HIRM designchallenge presentation document: the Linear Quadraticmethods approach, GARTEUR/TP-088-26, 1997.

[2] Amato F., Mattei M., Scala S., Verde L., HIRM design viaLQ methods, in: Magni, Bennani, Terlouw (Eds.), RobustFlight Control, Springer Verlag, 1998.

[3] Anderson B.D.O., Moore J.B., Linear Optimal Control,Prentice Hall, 1992.

[4] Blight R.L., Dailey J.D., Gangsaas D., Practical controllaw design for aircraft using multivariable techniques, Int.J. Control (1994).

[5] Cavallo, De Maria G., Verde L., Robust Flight ControlSystems: a parameter space design, J. Guid. ControlDynam. 15 (5) (1992) 1207–1215.

[6] Dorato P. (Ed.), Robust Control, IEEE Press, New York,1987.

[7] Dorato P., Yedavalli R.K., Recent Advances in RobustControl, IEEE Press, New York, 1990.

[8] Muir E.A. et al., Robust flight control design challengeproblem formulation and manual: The High IncidenceResearch Model (HIRM), GARTEUR/TP-088-4, 1997;see also

[9] Muir E.A. et al., The HIRM design challenge problemdescription, in: Magni, Bennani, Terlouw (Eds.), RobustFlight Control, Springer Verlag, 1998.

[10] Handling qualities of unstable highly augmented aircraft,AGARD Advisory Report No. 279, AGARD AR-279,1991.

[11] Verde L., Controllo Robusto di Aeromobili: Metodi Para-metrici per l’analisi e la sintesi, PhD Thesis, Universitàdegli Studi di Napoli, February 1992.

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Robust Flight Control Design for the HIRM based on Linear Quadratic Control