Douglas A. Lawrence, MemberSchool of EECSOhio University
Athens, Ohio 4S701
JoyH. Kelly, Senior Member
Sverdrup TechnologyEglin Air Force Base, FL 32542
Johnny H. Evers, Senior Member
Munitions DirectorateAir Force Research Laboratory
Eglin Air Force Base, FL 32542
AbstractThe application of a recently developed methodologyfor gain scheduled control system synthesis to thedesign of a pitch channel missile autopilot is described.A control signal interpolation strategy is used togenerate a family of linear autopilots from a collectionof point designs. Conditions are given under which anonlinear gain scheduled autopilot exists thatlinearizes to the family of linear autopilots and theinterpolation scheme plays an important role insatisfying these conditions. Nonlinear simulationresults indicate satisfactory performance over thespecified flight envelope.
1. Introduction
Gain scheduling is a commonly practiced methodfor missile autopilot design in which it is possible tomeet performance objectives over a wide flightenvelope and still take advantage of the wealth of toolsand experience for linear controller synthesis.Essentially, the objective is to construct an autopilotthat schedules an appropriate linear autopilot for everyflight condition in a specified range. This implicitlyimposes a linearisation requirement on the gainscheduled autopilot that is often overlooked.
Typically, linear autopilots are designed at adiscrete collection of operating points and theparameters of these linear point designs are then insome way interpolated as functions of a designated setof scheduling variables to yield a linear autopilot forany operating point The end result is a family oflinear autopilots parameterized by the schedulingvariables. It should be emphasized that the schedulingvariables are treated as frozen parameters as opposed totime-varying signals in this phase of the designprocess.
In the final implementation, the autopilotparameters are instantaneously adjusted based on
measurements of the scheduling variables as dictatedby the particular interpolation scheme. The result is again scheduled autopilot that has a linear parameter-varying (LPV) structure but is nonlinear in thescheduling variables. Moreover, the schedulingvariables act as time-varying input signals to the gainscheduled autopilot Consequently, when the gainscheduled autopilot is linearized about an operatingpoint, first-order terms corresponding to perturbationsin the scheduling variables arise that have nocounterpart in the linear autopilot family. Thus, thelinearized autopilot does not exactly match theinterpolated linear autopilot for that operating pointAny such mismatch constitutes a so-called hiddencoupling term that can potentially degrade and evendestabilize system performance. [8]
Recent investigations on gain scheduling from anonlinear systems perspective have considered moregeneral controller realizations to see if hidden couplingterms can be avoided. This can be accomplished usinga novel controller architecture proposed in [1] providedthat time-derivatives of certain measured signals areavailable. Technical conditions have been derived in[6] that allow the hidden coupling term issue to beanalyzed for any gain scheduled controller.
In this paper we show, in the context of alongitudinal missile autopilot design problem, that acontrol signal interpolation technique, in addition toother advantages discussed in [3], permits amodification to the implementation in [1] for whichtime-differentiated signals are not required. Our gainscheduled autopilot retains the popular LPV structureand yet is free of hidden coupling terms. It isimportant to note that this is achieved withoutimposing any constraints on the linear point designs.In fact, control signal interpolation affords additionalfreedom in this regard compared to other interpolationschemes.
The remainder of this paper is organized asfollows. The nonlinear missile model is presented in
the next section. The gain scheduled autopilot designis described in Section 3. Simulation results arediscussed in Section 4. Finally, concluding remarksare given in Section 5.
2. Missile DescriptionWe consider the hypothetical tail-controlled
missile model introduced in [8] and refined in [7]whose longitudinal behavior is characterized by thefollowing variables:
missile speed in feet/secangle-of-attack in radianspitch rate in radians/secelevator deflection in radianspitch attitude angle in radiansnormal acceleration in geesaltitude in feet
a.Q56r\zh
M mach numberq dynamic pressure in Ibs./ft. 2
The short-period longitudinal aerodynamics aredescribed by
sin(a)}fgcos(e-a)J+e
(1)
and the remaining longitudinal aerodynamics aredescribed byVm = — [CA cos(a)+CN sin(a)] - gsin(6 -a)
m(2)
(3)
A = FMsin(e-a)Normal acceleration is given by
r\,=—mgThe aerodynamic coefficients CA,CN, andcharacterize the aerodynamic axial force, normal force,and pitching moment, respectively, of the missile andare modeled by:
(4)
with polynomial coefficient values listed Table 1. Theatmospheric model that describes the altitudedependence of air density and the speed of sound isalso taken directly from [7].
The elevator control surface actuator is modeledby
(5)
where 8 c is the commanded elevator deflection inradians. Additional missile and actuator parametersare listed in Table 2.
Table 1. Aerodynamic Coefficient ParametersCA CN CM
AW=-3L023cn =-9.717
aa=- 0300 an = 19373 am = 40.440bm= -64.015cm= 2.922
dm =-11803em=-ni9
Table 2. Missile and Actuator ParametersSymbolSmgdIye.<»a
Descriptionsurface areamassaccel. due to gravityreference lengthpitch moment of inertiaactuator damping ratio
^jjcjttjatojja^ujrayh&eo^^^^
Value0.44 ft213.98 slug32.2 ft/s2
0.75 ft182.5 slug-ft20.7150rad/s
3. Gain Scheduled Autopilot DesignIn general terms, the objective is to design a gain
scheduled pitch channel autopilot such that theresulting closed-loop system exhibits asymptotictracking of piece-wise constant normal accelerationcommands with acceptable transient response over asuitable flight envelope. The gain schedulingmethodology of [6] applied to the autopilot designproblem at hand involves the following steps:
Step 1. Calculate the family of constant operatingpoints for the missile/actuator dynamics andparameterize by a designated set of schedulingvariables.
Step 2. For the corresponding family of linearizedmissile/actuator models, design a family oflinear autopilots to meet prescribed designgoals at each constant operating point
Step 3. Construct a gain scheduled autopilot thatlinearizes to the appropriate linear autopilot ateach constant operating point
Step 4. Check performance of the resulting nonlinearclosed-loop system.
For the first step in the design process we consideronly short-period equilibria and are interested inconstant solutions to the algebraic equations resultingfrom the constraints d = Q = 0:
0=-r—oo 1
T —— [CN°cos(a0)-Cx°sin(a°)}fgcos(eo-a0)
(6)
The corresponding constant normal acceleration valueintroduces the additional constraint
tlr° = —— Cw°+cos(e°). (7)mg
Since (6) and (7) constitute three independentequations in seven unknowns, there is a continuousfamily of solutions that can be parameterized in severalways. Here we parameterize by angle-of-attack, machnumber, altitude, and pitch attitude angle anddetermine pitch rate, elevator deflection, and normalacceleration as functions of these variables to obtain:
Q0 = Q°la.0,M°,h°,Q0]S0 = 8°[a0,A/0,A0,eo] (8)
In addition, the actuator dynamics yield8C° = 5°[a°, M°, A°,6°]. (9)
For the second step in the design process, we focuson the short period aerodynamics (1), the actuatordynamics (5), and normal acceleration (3), for which(8) and (9) characterize a parameterized family ofconstant operating points. For the associatedparameterized family of linearizations, we firstconsider a parameterized family of linear autopilots ofthe form:
where r\c denotes commanded normal acceleration.To implement the control signal interpolation
strategy over an entire flight envelope, thecorresponding three-dimensional scheduling variableset must be divided into an appropriate number of cellswith linear point designs constructed for the eightvertices of each cell. On the interior of a cell, thecontrol signals generated by the eight linear pointdesigns operating in parallel are to be interpolated.Finally, a procedure to switch between sets of linearpoint designs as scheduling variable trajectories crosscell boundaries must be devised. For simplicity we donot address this important issue here, but ratherconfine our attention to a single cell in the largerscheduling variable set defined by
Linear H2 point designs were constructed at the eightvertices using a synthesis model very similar to thatdescribed in [7]. On the interior of the cell, linearlyinterpolating the control signals generated by each ofthe point designs leads to a parameterized family oflinear autopilots (10) with
As an indication of the performance and stabilityrobustness achieved by the family of linear autopilotsover the entire cell, we consider the operating pointcorresponding to the center of the cell: 5 degrees angle-of-attack, mach 3.25, and 25k ft altitude. For theassociated linearized short-period aerodynamics,actuator dynamics, and linear autopilot, a closed-loopunit step response is given in Fig. 1 and a Bode plot ofthe open-loop return difference, indicating thefrequency dependent distance from the open-loopNyquist curve to the -1 point, is given in Fig 2. This
For the first step in the design process we consideronly short-period equilibria and are interested inconstant solutions to the algebraic equations resultingfrom the constraints d-O = 0:
sin(a°)}fg cos(9° -
(6)
The corresponding constant normal acceleration valueintroduces the additional constraint
5700
t\° = -— CN ° +cos(6°). (7)mg
Since (6) and (7) constitute three independentequations in seven unknowns, there is a continuousfamily of solutions that can be parameterized in severalways. Here we parameterize by angle-of-attack, machnumber, altitude, and pitch attitude angle anddetermine pitch rate, elevator deflection, and normalacceleration as functions of these variables to obtain:
Q° = Q°[a.0,M°,h°,Q0]S0 = 8°[a0,A/0,A0,eo] (8)
In addition, the actuator dynamics yield5c° = 8°[a.0,M0,h0,Q0]. (9)
For the second step in the design process, we focuson the short period aerodynamics (1), the actuatordynamics (5), and normal acceleration (3), for which(8) and (9) characterize a parameterized family ofconstant operating points. For the associatedparameterized family of linearizations, we firstconsider a parameterized family of linear autopilots ofthe form:
where r\c denotes commanded normal acceleration.To implement the control signal interpolation
strategy over an entire flight envelope, thecorresponding three-dimensional scheduling variableset must be divided into an appropriate number of cellswith linear point designs constructed for the eightvertices of each cell. On the interior of a cell, thecontrol signals generated by the eight linear pointdesigns operating in parallel are to be interpolated.Finally, a procedure to switch between sets of linearpoint designs as scheduling variable trajectories crosscell boundaries must be devised. For simplicity we donot address this important issue here, but ratherconfine our attention to a single cell in the largerscheduling variable set defined by
{@eR3 10 <. a £ 10°, 3.0 <>M <, 3.5, 20k £ h £ 30kft.Linear H2 point designs were constructed at the eightvertices using a synthesis model very similar to thatdescribed in [7]. On the interior of the cell, linearlyinterpolating the control signals generated by each ofthe point designs leads to a parameterized family oflinear autopilots (10) with
A(@) = A =
5(0) = B =
0
5,
— pg(0)Z)g]where (Ai,BitCt,£>,),/ = !,...,8 are state spacerealizations of the eight point designs and linearinterpolation of the individual control signals isachieved by the scalar-valued p, (0) 's. An importantfeature of this interpolation scheme as it relates to theconstruction of a nonlinear gain scheduled autopilot inthe next step of the design procedure is that the A andB matrices above are independent of 0.
As an indication of the performance and stabilityrobustness achieved by the family of linear autopilotsover the entire cell, we consider the operating pointcorresponding to the center of the cell: 5 degrees angle-of-attack, mach 3.25, and 25k ft altitude. For theassociated linearized short-period aerodynamics,actuator dynamics, and linear autopilot, a closed-loopunit step response is given in Fig. 1 and a Bode plot ofthe open-loop return difference, indicating thefrequency dependent distance from the open-loopNyquist curve to the -1 point, is given in Fig 2. This
situation essentially represents worst case linearperformance achieved over the cell and yet these plotsindicate acceptable transient response and stabilitymargin.
The third step in the design process is to constructa nonlinear gain scheduled autopilot of the form:
*/ = 1e - V:
xc = a(xc,x},T\,,Q,e) (11)8C = c(xc,Xj,r\z,Q,&)
that schedules the family of linear autopilots in thesense that the following two requirements are met.Requirement 1. At each constant operating point, thenonlinear autopilot generates the correct constantcontrol value. Specifically, there must exist functionsXj °(0), xc °(0) that satisfy (suppressing arguments)
Requirement 2. At each constant operating point, thenonlinear autopilot linearizes to the appropriate linearautopilot Specifically, the following partial derivativeidentities must hold.
da , n „dxf
da
dc(13)
dc
and80 < o—— (xc°,;50 C
dc , „(14)
The identities in (14) ensure that (11) will notintroduce hidden coupling terms since the family oflinear autopilots does not exhibit linear terms in thescheduling variables. The following theorem, adaptedfrom [6, Theorem 1], establishes necessary andsufficient conditions for the existence of a gainscheduled autopilot that meets these two requirements.
Proof. For necessity, suppose there is a nonlinearautopilot that satisfies Requirements 1 and 2. Thendifferentiating (12) with respect to 0 and substitutingthe identities in (13) and (14) yields (15). Forsufficiency, suppose there are functionsXj °(0), xc °(0) that satisfy (15) and consider
Since A and £ are independent of 0, the first identityin (15) gives
Axc°(@) = 0 (17)
which gives the first identity in (12). It is also clearthat the first two identities in (13) and the first identityin (14) are satisfied. Next, it is easy to see that thesecond identity in (12) and the last two identities in(13) hold for any choice of functions x, °(0), xc °(0)and it remains to check that the second identity in (14)is satisfied. Differentiating (16) with respect to 0 andevaluating at any constant operating point yields
which is zero due to the second identity in (15).ODDIt is worth emphasizing that since this particularinterpolation scheme leads to constant A and Bmatrices, the first identity in (15) is equivalent to (17).Consequently, for any choice of x} °(0), we can satisfythe first identity in (15) by taking
is the DC gain of the interpolated linear autopilot.Hence, the existence conditions reduce to a singleequation (20) which can be solved for dx;°/d& toyield a partial differential equation of the form
(22)V 'Unfortunately, these conditions are restrictive and
typically will not be satisfied. To circumvent thisdifficulty, an alternate state-space representation forthe parameterized family of linear autopilots will bederived having the same parameterized transferfunction as (10) for which a gain scheduled autopilotthat fully meets analogous linearization requirementscan be constructed. The construction makes use oflinearly equivalent integrator placements as in [1], butbecause of the control signal interpolation strategyemployed, time-derivatives of input signals to theautopilot are not required.
To begin, the parameterized transfer functionassociated with (10) is
"J- 0 0"(23)
which, from an input-output viewpoint, is equivalent to"/ 0 0"
In order for there to exist a gain scheduled autopilotmeeting appropriately modified versions ofRequirements 1 and 2, there must exist functionszc°(0) and Zj°(®) that satisfy
In the fourth step of this gain schedulingmethodology, nonlocal performance is normallyanalyzed under the assumption that all exogenoussignals assumed to be constant during Step 2 aresufficiently slowly-varying. [2] [5] Here, we haveimplicitly treated the state variables governed by (2) as
exogenous signals acting on the short-periodaerodynamics (1). However, since the dynamics of (1)and (2) are coupled, the usual slowly-varying analysisis not directly applicable. A modification to thisanalysis is described in [4] to address this coupling andconfirm the performance exhibited in the simulationresults of the next section.
4. Simulation ResultsA Simulink simulation was constructed for the
closed-loop system consisting of the missileaerodynamics (1) and (2), the actuator dynamics (5),and die gain scheduled autopilot (25). Simulationresults are presented here for an initial conditioncorresponding to the constant operating pointparameterized by:
angle-of-attack 0 degreesmach number 3.5altitude 25,000/eefpitch attitude 0 degrees
and a -40 g normal acceleration step command appliedat f=0. This scenario was selected to generate ascheduling variable profile that spans much of thepreviously specified cell in order to test the validity ofthe control signal interpolation scheme. Variousresponses are shown in Figs. 3 through 8. Fig. 3depicts the normal acceleration profile whose transientresponse is in good agreement with that of the closed-loop linearizations at the design points. Fig. 4indicates that angle-of-attack passes through most ofits 0 to 10 degree range, Fig. 7 indicates that machnumber passes through most of its 3.0 to 3.5 range,while Fig. 8 shows that altitude remains near themidpoint of its 20k to 30k ft. range.
5. Concluding RemarksIn practice, gain scheduled autopilots are often
implemented as LPV controllers based on interpolatedlinear point designs. Autopilots designed in thisfashion will typically fail to satisfy a basic linearizationrequirement resulting in the presence of hiddencoupling terms that may adversely affect performance.
In this paper, we have considered an approachcombining a control signal interpolation technique anda particular state-space realization that leads to anautopilot that retains an LPV structure but is free ofhidden coupling terms. Our approach has implicationsbeyond the scope of the missile autopilot designproblem considered here and may therefore be ofpractical interest
6. References[1] I. Kaminer, A. Pascoal, P. Khargonekar, C.
Thompson, "A Velocity Algorithm for theImplementation of Gain Scheduled Controllers,"Automatica, Vol. 31, No. 8, pp. 1185-1191, 1995.
[2] M. Kelemen, "A Stability Property," IEEETransactions on Automatic Control, Vol. AC-31,No. 8, pp. 766-768, 1986.
[3] J. H. Kelly and J. H. Evens, "An InterpolationStrategy for Scheduling Dynamic Compensators,"Proceedings of the 1997 AIAA Guidance,Navigation and Control Conference, New Orleans,Louisiana, 1997.
[4] D. A. Lawrence, "Tools for the Analysis andDesign of Gain Scheduled Missile Autopilots,"Final Report, 1997 AFOSR Summer ResearchExtension Program, December, 1997.
[5] D. A. Lawrence and W. J. Rugh, "On a StabilityTheorem for Nonlinear Systems with SlowlyVarying Inputs," IEEE Transactions on AutomaticControl, Vol. AC-35, No. 7, pp. 860-864, 1990.
[6] D. A. Lawrence and W. J. Rugh, "GainScheduling Dynamic Linear Controllers for aNonlinear Plant," Automatica, Vol. 31, No. 5, pp.381-390,1995.
[7] C. P. Mracek and J. R. Cloutier, "Full EnvelopeMissile Longitudinal Autopilot Design Using theState-Dependent Riccati Equation Method,"Proceedings of the 1997 AIAA Guidance,Navigation and Control Conference, New Orleans,Louisiana, 1997.
[8] R. A. Nichols, R. T. Reichert, and W. J. Rugh,"Gain Scheduling for //-Infinity Controllers: AFlight Control Example," IEEE Transactions onControl Systems Technology, Vol. 1, No. 2, pp.69-79, 1993.
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