This paper investigates the application of velocity-based gain-scheduling

techniques to a demanding, highly nonlinear, missile control design task. Scheduling on

instantaneous incidence (a rapidly varying quantity) is well known to lead to

considerable difficulties with conventional gain-scheduling methods. Moreover, the

missile dynamics do not have well-defined relative degree at operating points where the

lateral velocity is zero and thus conventional feedback linearisation methods cannot be

applied to design a controller

Leith, Douglas J.

Leithead, W.E.

Tsourdos, A.

White, B.A.

English

MURAL - Maynooth University Research Archive Library

VELOCITY-BASED GAIN-SCHEDULED LATERAL AUTO-PILOT FOR AN AGILE MISSILED.J.Leith*, A.Tsourdos†, B.A.White†, W.E.Leithead**Dept. of Electronic & Electrical Engineering, University of Strathclyde, U.K.†Dept. of Aerospace, Power & Sensors, Cranfield University-RMCS, U.K.Abstract:  This paper investigates the application of velocity-based gain-schedulingtechniques to a demanding, highly nonlinear, missile control design task.  Scheduling oninstantaneous incidence (a rapidly varying quantity) is well known to lead toconsiderable difficulties with conventional gain-scheduling methods.  Moreover, themissile dynamics do not have well-defined relative degree at operating points where thelateral velocity is zero and thus conventional feedback linearisation methods cannot beapplied to design a controller.1. INTRODUCTIONGain-scheduling control is widely employed inflight control applications, where high performance hasto be achieved over a broad operating envelope.  In theclassical gain-scheduling design approach, a non-linearcontroller is constructed by continuously interpolating,in some manner, between the members of a family oflinear controllers.  Each linear controller is, typically,associated with a specific equilibrium operating pointof the missile and is designed to ensure that, locally tothe equilibrium operating point, the performancerequirements are met.  By employing a series expansionlinearisation which, locally to the equilibrium operatingpoint, has similar dynamics to the aircraft, lineartechniques may be used to resolve this local designtask.  Continuity is, therefore, maintained withestablished linear design techniques for which aconsiderable body of experience has been accumulated.While this traditional gain-scheduling approach isextremely successful in most flight control applications(McLean 1990),  the trend is towards vehicleconfigurations where the conventional gain-schedulingconditions may not always be satisfied.  Gain-scheduledcontrollers are traditionally designed on the basis of thedynamics relative to a family of trim conditionsassuming that the airspeed is slowly varying.  However,during aggressive manoeuvring the vehicle may be farfrom equilibrium with rapidly varying airspeed.   Inaddition, the requirement to operate at high angles ofattack can necessitate scheduling on rapidly varyingquantities such as the instantaneous incidence angle.  Itshould be noted that scheduling on instantaneousincidence is well known to be problematical and isalmost always avoided in classical schedulingarrangements.  Specifically, in the example consideredin this paper, a conventional gain-scheduling designapproach fails to lead to a stabilising controller.Faced with these kind of issues (which are alsorelevant in many other applications), an recent years anumber of alternative approaches have been proposedwhich attempt to extend gain-scheduling methodsincluding those based on local model networks andTakagi-Sugeno fuzzy models (see, for example,Johansen & Murray-Smith 1997, Hunt & Johansen1997).  However, the latter typically include off-equilibrium information at the cost of moving tononlinear (especially affine) formulations and so losethe continuity with linear methods which is one of theprinciple advantages of classical gain-scheduling.Moreover, the requirement for some form of slowvariation condition is often retained either directly orindirectly via assumptions implicit in the formulation(Leith & Leithead 1999a).  The velocity-based analysisand design framework, recently proposed in Leith &Leithead (1998a,b), associates a linear system withevery operating point of a nonlinear system, not just theequilibrium operating points.  This approach therebyrelaxes the restriction to near equilibrium operationwhile maintaining the continuity with linear methods asrequired.  Moreover, it can be shown that the velocity-based approach does not inherently involve any slowvariation requirement (Leith & Leithead 1999b).Velocity-based techniques provide a useful bridgebetween classical control approaches and AI methodssuch as those based on local model networks andTakagi-Sugeno fuzzy models, generalising the seriesexpansion linearisation which forms the basis of theformer while retaining the blended multiple modelstructure of the latter.  The aim of this paper is toinvestigate the application of velocity-based gain-scheduling techniques to a demanding, highlynonlinear, missile control design task.2. MODELLING FOR CONTROLThe missile lateral dynamics are (White et al. 1998,Tsourdos et al. 1998)  | | | | | |  | | | |  | | | |r b v b r b v b v b v b v b uv a r a a v v a v a uv U r a a v v a v a uy o               4 5 122 3 6 73 1 2 4 51 2 4 5           (1)where r is the yaw rate (rad/s), v the lateral velocity(m/s), y the lateral acceleration (m/s2), u the fin angle(rad) and Uo is the nominal forward speed (m/s).  (Notethat the missile dynamics do not have well-definedrelative degree at operating points where the lateralvelocity is zero and thus conventional feedbacklinearisation methods cannot be applied to design acontroller).  Differentiating, the corresponding velocity-based formulation is obtained  | | | | sgn( ) sgn( )| |    | | sgn( ) | |    | | sgn( ) | |  w b v b w b v b v b b r v b u v wb v b uw a w a a v a u v w a v a ua a v a u v w a v a ur r vv r vy v                4 5 122 3 4 66 73 1 2 4 4 51 2 4 4 53 222                                            (2)with   ,  r w v wr v  .  The velocity-based linearisationassociated with an operating point is obtained by simply“freezing” (2) at the relevant operating point and thecollection of such linearisations forms the velocity-based linearisation family.   The solution to thevelocity-based linearisation associated with a particularoperating point locally approximates the solution to thenonlinear dynamics, (1), and the solutions to thevelocity-based linearisation family can be piecedtogether to recover the global solution to (1).  Thevelocity-based framework therefore allows continuitywith established linear methods to be maintained evenin the nonlinear context.Since there are a continuum of operating points, thevelocity-based linearisation family associated with anonlinear system has, of course, infinitely manymembers.  It is therefore attractive to determine a(perhaps approximate) finite parameterisation of thefamily on which to base control designs.  Thisrequirement leads naturally to consideration of ablended multiple model representation of thelinearisation family whereby the linearisations at asmall number of representative operating points areblended together/interpolated between to produce anapproximation to the exact linearisation family.  Such arepresentation is closely related to neuro-fuzzymodelling approaches including Local Model Networksand Takagi-Sugeno fuzzy methods .  However, incontrast to the latter methods, the velocity-basedrepresentation (Leith & Leithead 1999a) Uses genuinely linear local models (not affine). The dynamics are directly related to the localmodels: the solution to a velocity-based blendedmultiple model system, locally to a specificoperating point, is described by the solution to thelinear system obtained by “freezing” the blendedmultiple model system at the relevant operatingpoint.  The resulting frozen system is simply aweighted linear combination of the local models. A still stronger property is that the solution to theblended multiple model system, locally to aspecific operating point, is approximated by theweighted linear combination of the solutions to thelocal models.In the present application,  the velocity-basedlinearisation family associated with the missile lateraldynamics is approximated by the blended family  ( )  ( ) wwvwwv urvrv	 	 A B(3)with A A B B( ) (| | ) , ( ) (| | )v vi i    w v w vii ii1515(4)where A A1 2338 0 621000 324285 5 701000 1188       . ..,. ..A A A3 4 5232 15371000 20 52125 48 471000 37 79018 99 921000 55 07         . ..,. ..,. ..B BB B B1              5 5 0 2 82 8 3 0 05 1 6 6 82 6 5 7 24 8 3 0 82 4 8 4 44 1 5 8 82 1 3 8 83 4 8 6 81 7 9 3 223 4 5..,....,..,..The coefficients in (2) which depend on yaw rate, r, andfin angle, u, are neglected so that the coefficients in (3)vary only with lateral velocity.  For simplicity, standardtriangular membership functions, wi, centred on lateralvelocities of 0 m/s, 10 m/s, 20 m/s, 40 m/s and 60 m/sare used with overlap occurring only between theweighting functions associated with neighbouringcentres.   This scheme corresponds to straightforwardlinear interpolation between the local models.  Despitethe very strongly nonlinear nature of the dynamics, onlyfive local models are sufficient to capture the dynamicsthrough the entire operating envelope up to around60g (the accuracy of the approximation can beassessed by comparing the transfer functions of thelinearisations associated with (2) and (3) over a rangeof operating points but plots are omitted here owing tospace considerations).   Note that provided (3) is asufficiently accurate approximation to the exact familyand/or the controller is sufficiently robust, a controllerwhich achieves the performance objectives with theseapproximate dynamics is also guaranteed to achievesatisfactory performance when used with the exactplant dynamics, (1).3. VELOCITY-BASED CONTROL DESIGNThe requirement is to achieve a closed-loopacceleration response with rise time of around 0.1seconds with less that 20% overshoot.  Thisrequirement is to be satisfied over the full of operatingenvelope of 50g.  The control design consideredadopts an inner/outer loop type of structure common inflight control applications.  A lateral velocity innerloop is designed first and then enclosed within anacceleration outer loop.  The inner loop is designed tohave sufficiently higher bandwidth than the outer loopthat the design of the two loops can be effectivelydecoupled.  One immediate advantage of this approachis that the dynamics from fin angle to lateral velocityare minimum-phase (the dynamics from fin to lateralacceleration are non-minimum-phase) and so an inversecan be determined directly without the need to use aminimum-phase approximation.Inner-loopThe structure of the inner controller is shown in Fig. 1.Design of  G1  and  A 1   The linear dynamics,  A 1 , arean approximate pole-zero inverse of the actuatordynamics while  G1  is approximately inverts thenonlinear dynamics associated with velocity-basedlinearisation family, (3).    G1  is designed using thevelocity-based inversion approach of Leith & Leithead(1999b).    Velocity-based linearisation families areassociated with the plant and the controller.  Considerselecting the controller velocity-based linearisationssuch that at every operating point the closed-loopcombination of the plant and controller linearisationshas the approximately the same transfer function.  Theclosed-loop linearisation family then has similartransfer function at every operating point.  This is,however, not sufficient to ensure that the controllerapproximately inverts the plant dynamics.  The transferfunction only specifies the realisation of a linearisationto within a linear state transformation.  It is, therefore,also necessary to ensure that the state-space realisationsof the members of the controller linearisation family aresuitably compatible with one another.  (When the rateof variation of a nonlinear system is sufficiently slow, itcan be shown that the controller becomes insensitive tothe choice of realisation.  However, in the presentexample the plant dynamics are rapidly varying and thecontroller dynamics are highly sensitive to the choice ofrealisation.  Indeed, it is not difficult to find choices ofrealisation which lead to controllers that fail to anunstable closed-loop).  In the present example, anappropriate realisation of approximate velocity-basedinverse is  ( ) ( )    ( ) ( )  w A w Bu C w Di   i ii i iv vv v(5)whereAi(v)=A(v)-B(v)D-1(v)C(v),Bi(v) = -B(v)D-1(v)Ci(v)=D-1(v)C(v),   Di(v)=D-1(v)with C D( ) , ( )v vc d  1 .  Note that the values ofd and c used when deriving the approximate inverseare design parameters which determine the accuracy ofthe inversion achieved.  Analogously to linear pole-zeroinversion of systems with relative degree greater thanzero, the approximate inverse contains high frequencypoles to ensure that it is realisable.  Roughly speaking,d determines the frequency of the poles and cinfluences the damping.  Since the direct coupling termis zero in (3) (the relative degree is greater than zero) ,the inversion error tends to zero as d and c tend tozero.  In the present example, values of 5 and 0.2,respectively are found to achieve an adequate degree ofaccuracy.Design of C  The approximate inverse, G-1, isaugmented with controller dynamics, C, consistingsimply of integral action.  The gain, K, of the integratoris selected such that the inner loop bandwidth is around130 rad/s, see figure 2.Controller Realisation The velocity-based inverse, (5),cannot be directly implemented owing to the derivativeaction at the input (shown in figure 3a).  However,owing to the integral action in the controller dynamics,C, the controller structure in figure 3a may beequivalently formulated as in figure 3b and is nowrealisable.  Note that a direct link is maintained betweenthe gains in the controller implementation and the gainsin the original (unrealisable) design.Outer-loopThe lateral acceleration is related to lateral velocityand fin angle by  | | sgn( ) | |  y va a v a u v w a v a u    1 2 4 4 52    (6)Owing to the relatively high bandwidth of the innerloop, the lateral velocity can be considered effectivelyequal to the lateral velocity demand input to the innerloop controller.  An outer loop controller with thesimple structure shown in figure 4 is thereforesufficient.  The nonlinear gain, gnl includes a factor121 2a a v | |   to compensate for the nonlinear gainrelating acceleration to velocity in (6).  An additionalfactor is included to compensate for the effect of thedirect coupling term in (6) at low frequencies, namely  (| | )gg w vnl i ii 14(7)with g1=1, g2=2, g3=5, g4=5 and triangular weightingfunctions, wi centred on lateral velocities of 1 m/s,30m/s, 50 m/s and 60 m/s.  The overall nonlinearcontrol gain, gnl, is therefore g n la a v1 22 | |  .  Thenonlinear gain is augmented with controller dynamicsconsisting of integral action.  The gain of the integratoris selected such that the outer loop bandwidth is around20 rad/s, see figure 5.Controller Realisation The outer loop controller alsoincludes integral action and so can be realised directlyas discussed previously.4. PERFORMANCETypical step responses of the controlled system areshown in Figure 6.  It can be seen that the performancerequirements are satisfied over the operating envelope(50g).  Typical stability margins and open-loopcrossover frequencies of the velocity-basedlinearisations of the inner and outer control loops areInner looplateralvelocity(m/s)gainmargin(dB)phasemargin(deg)cross-overfrequency(rad/s)0 9.35 72.7 130.820 9.23 72.9 128.650 9.60 73.5 123.9Outer looplateralvelocity(m/s)gainmargin(dB)phasemargin(deg)cross-overfrequency(rad/s)0 12.3 57.3 17.320 14.80 55.6 18.750 14.28 55.3 18.7The stability margins vary due to the approximatenature of the particular inverse dynamics used andowing to the inaccurate inversion that results fromusing a scheduled approximation to the exact plantdynamics.5. SUMMARYThe application of velocity-based gain-schedulingtechniques to a demanding, highly nonlinear, missilecontrol design task is investigated.  Scheduling oninstantaneous incidence (a rapidly varying quantity) iswell known to lead to considerable difficulties withconventional gain-scheduling methods.  Moreover, themissile dynamics do not have well-defined relativedegree at operating points where the lateral velocity iszero and thus conventional feedback linearisationmethods cannot be applied to design a controller.   It isdemonstrated that the velocity-based framework can,however, be used to successfully design a gain-scheduled controller which achieves the performancerequirements while maintaining the continuity withlinear methods which is the principle advantage ofclassical gain-scheduling approaches.ACKNOWLEDGEMENTD.J. Leith acknowledges the generous support providedby the Royal Society for the work presented.REFERENCESHUNT, K.J.,JOHANSEN,T.A., 1997, Design and analysis ofgain-scheduled control using local controller networks.International Journal of Control, 66, 619-651.JOHANSEN, T.A., MURRAY-SMITH,R., 1997, Theoperating regime approach to nonlinear modelling andControl.  Multiple Model Approaches to Modelling & Control(Ed. Murray-Smith, R., Johansen, T.A.) (Taylor & Francis,London).LEITH, D.J., LEITHEAD, W.E., 1998a, Gain-Scheduled &Nonlinear Systems: Dynamic Analysis by Velocity-BasedLinearisation Families.  International Journal of Control, 70,289-317. 1998b, Gain-Scheduled Controller Design: AnAnalytic Framework Directly Incorporating Non-EquilibriumPlant Dynamics. International Journal of Control, 70, 249-269.1999a, Analytic Framework for Blended Multiple ModelSystems Using Linear Local Models. International Journal ofControl, 72, 605-619. 1999b, Input-Output Linearisation byVelocity-Based Gain-Scheduling.  International Journal ofControl, 72, 229-246.MCLEAN, D.,  1990, Automatic Flight Control Systems.Prentice-Hall, London, 1990.Tsourdos, A., Blumel,A. White, B.A., 1998, TrajectoryControl of a Non-linear Homing Missile. Proceeding of the14th IFAC Symposium on Automatic Control inAerospace,pp.118-123, Aug. 1998, Korea.White, B.A., Tsourdos, A., BlumelA., 1998, LateralAcceleration Control Design of a Non-linear Homing Missile.Proceeding of the 4th IFAC Nonlinear ControlSystems Design Symposium, pp.707-712, July 1998, TheNetherlands.u G 1  A 1 AC Gdesiredvelocityactuatorvcontrollerdynamicsapproximateplantinverseapproximateactuatorinverse-Figure 1 Structure of inner-loop controller10010110210310410-1010-5100105open-loop tfs of innerl100101102103104-400-300-200-1000Figure 2  Transfer functions of open-loop velocity-based linearisations of inner-loop at lateral velocities of 0 m/s, 20m/s and 50 m/s (corresponding roughly to lateral accelerations of 0g, 20g and 50g).  The uniform nature of thedynamics is clearly evident.  ( ) ( )    ( ) ( )  w A w Bu C w Di   i ii i iv vv v A 1Kcontrollerdynamics, Capproximateplantinverseapproximateactuatorinversed/dt u  u i(a)  ( ) ( )    ( ) ( )  w A w Bu C w Di   i ii i iv vv v A 1K u  u i(b)Figure 3 Alternative formulations of controller vgnlinner-loopCdesiredacceleration y vdcontrollerdynamicsnonlineargain-plantFigure 4 Structure of outer loop controller10010110210310410-1010-5100105open-loop tfs of outer loop100101102103104-800-600-400-2000Figure 5  Transfer functions of open-loop velocity-based linearisations of outer-loop at lateral velocitiesof 0 m/s, 20 m/s and 50 m/s (corresponding roughly to0g, 20g and 50g).  It can be seen that the dynamics arerelatively uniform at frequencies below around 60rad/s; that is, at frequencies up to and above the cross-over frequency of the outer-loop.y (m/s2)0 0.2 0.4 0.6-100200500                                     time (s)Figure 6 Step responses with velocity-based gain-scheduled controller

Velocity-Based Gain-Scheduled Lateral Auto-Pilot For An Agile Missile

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Velocity-Based Gain-Scheduled Lateral Auto-Pilot For An Agile Missile

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