AIAA Guidance, Navigation, andControl Conference and Exhibit14-17 August 2000 Denver, CO

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AOO-37027AIAA-2000-3970

ROBUSTNESS OF A NONLINEAR MISSILE AUTOPILOTDESIGNED USING DYNAMIC INVERSION

Michael B. McFarland* and Shaheen M. Hoque^Raytheon Electronic Systems

Tucson, Arizona, USA

Abstract

An inner-loop/outer-lqop dynamic inversion controlarchitecture with output redefinition in the inner-loop isemployed to synthesize a full-envelope nonlinearautopilot for an air-to-air missile. The autopilot trackspitch and yaw acceleration commands while regulatingbody roll rate to zero. Essential design parameters arescheduled with flight condition for optimalperformance. The autopilot has been subjected to avariety of robustness tests. Linear robustness analysiswas based on linearized models computed locallythroughout the flight envelope and includeddetermination of classical gain and phase margins aswell as vector margins associated with simultaneousgain and phase variations. Nonlinear robustnessanalysis was based on single-run and Monte Carlosimulations and included determination of maximumallowable gain variations and input delays as well assensitivity to variations in aerodynamic parameters.

1. Introduction

Dynamic inversion is a simple methodology fornonlinear feedback control when the open-loop plant isfeedback-linearizable without the need for a statetransformation. Ref. 1 provides an excellentintroduction to the basics of this nonlinear controltechnique. Many real systems, however, do not meetthe restrictive assumptions that must be satisfied beforedynamic inversion can be applied. Non-minimum-phasesystems, for example,' are not suitable for dynamicinversion. Because tail-controlled missiles are non-minimum-phase when acceleration is selected as thecontrolled output, dynamic inversion should not beapplied directly to these systems. Most investigatorsavoid this problem by selecting the aerodynamic angle-

Senior Engineer. Senior Member AIAA.^ Senior Engineer.

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of-attack and sideslip angles or body-axis pitch and yawrates for control. Either of these is a special case of thetechnique known as output redefinition. A particularlyinteresting treatment of output redefinition for missileapplications may be found in Ref. 2.

This paper follows the approach described inRefs. 3 and 4 for application to strike munitions. In thisprevious work, an inner-loop/outer-loop structure wasused to transform the acceleration tracking problem intoa new inner-loop tracking problem. The inner-loopoutput was selected to be a combination of aerodynamicangles and body rates, rather than simply one or theother. Since performance specifications for munitionsvary over the flight envelope, important autopilotparameters are scheduled with dynamic pressure. Basedon the results obtained in Refs. 3 and 4, this architectureis capable of achieving desired performance.

As with most nonlinear control schemes, therobustness of dynamic inversion autopilots is frequentlysuspect. In Refs. 5 and 6, relatively simple dynamicinversion controllers were shown to have robustnessproperties inferior to both modern and classical lineardesigns in many cases. To date, however, the literaturedoes not contain documentation of a thoroughrobustness analysis of the type of dynamic inversionautopilot described in Refs. 3 and 4. Because of theclassical design of the outer-loop, and the presence ofintegral action, it is expected that this more complicateddynamic inversion architecture will possess desirablerobustness properties.

This paper begins with an overview of thenonlinear dynamic inversion autopilot structuredescribed in Refs. 3 and 4. In order to confirm previousperformance results obtained for air-to-groundmunitions, performance results based on nonlinearsimulations of a specific air-to-air missile are presented.Linearized analysis is then used to determine classicalgain and phase margins as well as vector margins in thepitch, yaw, and roll autopilot channels. Nonlinearsimulation analysis is also used to validate thelinearized results. Monte Carlo simulation studiesdemonstrate the control system's robustness to

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parameter errors in the nonlinear aerodynamic model.Finally, concluding remarks are presented.

2. Nonlinear Missile Dynamics

The generic missile diagram presented in Figure 1illustrates the variables and coordinate frames typicallyinvolved in missile autopilot design. The (XB, JB, ZB)axes are fixed to and rotate with the missile body. Notethat ZB lies in the plane of two of the missile's tail fins.The missile's translational and rotational velocitieshave components (u, v, w) and (p, q, r), respectively,along the body-axes. The magnitude of the translationalvelocity is VM- The angle fa is of special interest inmissile autopilot design.

<XT = COS-1(u/VM)

(4)

With simplifying assumptions, it can be shown that thesix-degree-of-freedom missile dynamics are governedby the following differential equations:

a = q — (p cos a + r sin a)cos a tan ft(az cos a — a x sincrjcosor

-r + (p cos ft + q sin /?)cos ft tan a(dycosft-citsmft^osft (5)

= tan~'(v/ w) (1)

It represents the orientation of the "maneuver-plane"defined by the (XM, ZM) axes. In order to take fulladvantage of the physical symmetries of a missileairframe, we define the sideslip angle similar to a asfollows:

(2)

This is in contrast to the usual definition ofaerodynamic sideslip angle:

(3)

=P-(<lsin <t>A + r cos <!>A )cot ar+ (ay cos <I>A - az sin <j>A }/V sin ar

(6)

where Eq. (5) describes the motion of the velocityvector relative to the missile body, and Eq. (6)describes the free rotation of the body. In Eq. (5), ax, ay,and az denote body-axis components of the missileacceleration. For compactness of notation, Eq. (5) maybe re-written as:

The angles or and 07-are defined as usual. x = T(x)(0 + af

* C

(7)

XB> XM

Figure 1: Missile Variables and Coordinate Axes

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where x =

a

<t>A

(0 = (8)

and we have introduced

T(x) =-cos2 a tan/?cos2/? tan a sin ft cos [I tan a

-sm a cos a tan/-1

- sin <I>A cot aT - cos cot aT

(9)

and df =

(az cos (x-ax sin or)cos a

(ay cos ft - ax sin /?)cos(10)

In order to develop a nonlinear dynamic inversionautopilot that employs available tabular data, theaerodynamic moments in Eq. (6) will be approximatedlinearly:

(U)M =M =M0+MSp6p

Eq. (6) may thus be rewritten in affme form:

i = f(x,co,S}*=F(x,(a)+B-SD\ (12)

V7«where F(x,co) = \M0/Iyy + &-/„//„, W[ (13)

and fl =

f±

0xx

0

(14)

Because the missile's acceleration dynamics arenon-minimum-phase, a blend of aerodynamic angles(a, ft) and body rates (q, r) is used in the inner-loop.

This output redefinition, described in detail in Ref. 2,allows the placement of open-loop zeros.Differentiating Eq. (15) yields

yT=[a fi r]KT

which may be re-written as follows:

(16)

(17)

Eq. (17) represents the inner-loop dynamics after outputredefinition. These dynamics are minimum-phase withzeros placed for dynamic inversion. To achievedecoupling, K\i and K2i in Eq. (17) are assigned zerovalues in this development.

3. Nonlinear Dynamic Inversion Autopilot

A typical missile autopilot accepts externalcommands in pitch and yaw acceleration, and regulatesbody-axis roll rate to zero. This is often described asSkid-To-Turn (STT) control. What follows is a briefreview of the dynamic inversion autopilot architectureof Ref. 3 as applied to an air-to-air missile. The readeris encouraged to consult Ref. 3 for more detaileddiscussion.

Nonlinear Autopilot StructureFigure 1 shows a diagram of the dynamic inversion

autopilot structure considered here. It is a two-loopdesign with acceleration regulated in the outer-loop.This approach avoids the problem of directly invertingnon-minimum-phase dynamics. The outer-loop controlsthe non-minimum-phase pitch and yaw accelerations,with dynamic inversion used in the inner-loop governedbyEq.(17).

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Figure 1: Nonlinear Autopilot Architecture

Figure 1 shows that the inner-loop has an explicitmodel following architecture, where a command filterrepresents the model. In the current development, as inRef. 3, first-order filters are used in each channel.

c j-—^ = diagyc

1 i = 1,2,3 (18)

These filters may also be used to generate the timederivatives of the commands, which are typically usedin model following control as described in Ref. 7.

In the outer-loop, proportional and integral gainsare chosen for each acceleration channel, while the rollrate command is passed directly through to the inner-loop. This is because the roll rate dynamics of the open-loop plant are intrinsically first-order and minimum-phase, making them suitable for dynamic inversionwithout additional manipulation. The outputs of theouter-loop P+I control subsystem become commandsthat drive the inner-loop. In the case of theaccelerations, this means that

yc = KP(ac -a}+K,^(ac-a)dr (19)

The outputs of Eq. (19) are fed into the inner-loop,where P+I control is again employed.

(20)

Although Kyi has been included here for completeness,integral action is not generally necessary in the inner-loop. Accordingly, Ky! will be selected equal to zero inthe subsequent application.

Since the missile has no air data sensors, a and /?can not be measured directly. Therefore, measurementof the inner-loop variable y is not possible. Instead, anestimate of y is constructed based on inertial estimatesof a and /? as indicated in Figure 1. The details of thisestimation process, however, are beyond the scope ofthe current discussion. It is sufficient for our purposesto assume that accurate estimates of a and ft areavailable based on measured inertial data. With thisexception, the dynamic inversion autopilot employs thesame feedback measurements as a classical design.

The dynamic inversion block in Figure 1 has as itsinputs all available feedback variables, including notonly accelerometer and rate gyro measurements butalso inertial velocities (u, v, w), time-varying missilemass (m), altitude (h), Mach number (A/), and theaforementioned estimates of a and /3. This blockimplements the nonlinear inverse of Eq. (17),represented mathematically by Eq. (21) below.

1 0 00 Kn K120 K2l K22

B(X)\ u-x) T22(x) T23x

CD-

h1 0 00 Kn KnU K. 21 "- 22

F(X)\ (21)

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Gain Selection Based on Linearized AnalysisAs explained in Ref. 3, the parameters in the inner-

and outer-loop linear controllers, as well as thecommand filters, may be selected based on a linearizedanalysis in which the closed-loop dynamics areapproximately second-order. To guarantee time-scale-separation between the desired response associated withthe command filter and the tracking errors in the inner-loop, the inner-loop proportional gain, KyP, is firstchosen to be three times faster than the command filterpole, so that in each channel

KyP=diag{3/Ti} (22)

where ris the time constant of the command filter.If the dynamic inversion is performed accurately,

then the closed-loop dynamics of the inner-loop matchthe model dynamics as desired. When this is the case,the inner-loop dynamics may be approximated asidentically equal to the filter dynamics and the outer-loop control may be designed using classical methods.Consider first the longitudinal channel. Approximatingthe acceleration az linearly as follows:

(23)

we proceed by introducing the following simplifyingdesign approximations:.

-l/Za»Ku/Vm

(24)(25)

Eq. (24) neglects the non-minimum-phase zero in thepitch response, Eq. (25) assumes that KU is sufficientlysmall. These approximations result in the followingloop transfer function from commanded to achieved at:

Cf,\= (26)

This expression can be further simplified by choosing

K,=l/Kn (27)

resulting in an exact pole-zero cancellation. The outer-loop proportional gain and the filter time constant Tmay then be selected by specifying a natural frequencyand damping ratio for the closed-loop system asfollows:

(28)(29)KP=-Kncon/2&a

The design equations are similar in the yawchannel. The roll channel implementation is simpler,since the transfer function from fin deflection to rollrate is first-order and contains no finite zeros.

Outer-Loop Performance SchedulingScheduling of desired performance is necessary

since the missile is not capable of achieving a uniformresponse at all flight conditions of interest. Typically, a63% rise time criterion is used. The rise time criterionis converted to a natural frequency specification tofacilitate the use of the outer-loop design equations ofRef. 3. Given a natural frequency, the 63% rise timecan be approximated by

trise = l/con (30)

In the air-to-air missile application considered here, theouter-loop natural frequency (ty, in each channel wasscheduled as a function of Mach number and altitude.The optimal value of a), was selected by increasing thevalue until one of the performance constraints limitingovershoot, undershoot, and cross-channel couplingwere violated.

Nonlinear Autopilot PerformancePerformance of the nonlinear dynamic inversion

autopilot was demonstrated using 6-DOF simulationanalysis. An available gain-scheduled classicalautopilot design was used for benchmarking purposes.This is not an especially fair comparison, since thegain-schedule is substantially simpler than the dynamicinversion. The comparison does, however, show thatincreasing the computational complexity of thecontroller and introducing tabular aerodynamic dataresults in quantifiable performance improvements.

Figures 2-4 show step responses for maximumachievable acceleration commands at an aerodynamicroll angle of fa = 0° using both the gain-schedule andthe dynamic inversion autopilot. In this case, the rollattitude of the missile body is such that the normalacceleration lies in the plane of one pair of the missile'stail fins, and not in the plane of the external mountinghooks (which are located between two adjacent fins).This is therefore an asymmetric flight condition withundesirable coupling and limited control authority.While the gain-schedule has a relatively low g-limit atthis flight condition, the dynamic inversion autopilot isable to achieve the same acceleration level as at thesymmetric fa = -45° condition (when the normalacceleration is directed between two fins and throughthe hooks). Other step response results obtainedthroughout the flight envelope are similar, althoughthere are some design points with only slight increasesin g-limits.

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J

2.5

2

1.5

1

0.5

0

-0 5

_ _ L

"//•

v

.UV_ _ _

f r-H

_ _ _

•*"•«==xss

— — Dynamic Inversion..... Gain Schedule

=====

_ _ _

=====

IL _ _ 1 _ _.

1

111

I

11

t0 0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2

1 (sec)

Figure 2: Nonlinear Control Increases G-Limits

t (sso)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8t(sec)

Figure 4: Control Activity Comparison

As described above, 63% rise time is theperformance measure for this missile autopilot design.For comparison, step commands of magnitude equal tothe g-limit of the gain scheduled design were given asinputs to both autopilots and responses were simulatedthroughout the flight envelope. Table 1 shows the 63%rise time of the dynamic inversion autopilot, normalizedby the corresponding rise time of the gain-scheduledcontroller These results indicate that the performance ofthe nonlinear autopilot at least equals that of the gain-scheduled linear autopilot at nearly all design points.

Figure 3: Nonlinear Control Decreases Coupling

Table 1: Normalized Rise Time of Dynamic Inversion Autopilot

OkftlOkft20kft30kft40kftSOkft70kft

0.931.041.030.720.58

--

1.011.010.930.730.65

--

0.920.761.000.880.951.02

-

0.690.760.820.840.810.75

-

-0.620.810.790.730.760.73

-0.620.730.700.670.750.71

---

0.650.650.520.44

4. Robustness Analysis

Robustness of the dynamic inversion autopilot wasanalyzed using two separate methods. First, linear gainand phase margins, vector margins, and thecorresponding crossover frequencies were computed.Second, parameter variation studies were conducted totest for robustness to Uncertainties in the aerodynamicparameters. Since acceleration feedback is available, noaerodynamic force models are required. Only tabularaerodynamic moment data are used in the dynamicinversion of Eq. (21).

Linearized AnalysisTo compute linear gain and phase margins, vector

margins, and the corresponding crossover frequencies,the non-linear dynamic inversion autopilot was trimmedand linearized numerically. The plant, however, waslinearized analytically using the technique described inRef. 8. This process guarantees that "perfect" feedbacklinearization does not occur when forming the closed-loop. Linear frequency responses were computed andused to determine crossover frequencies, gain margins,and phase margins in the classical, single-loop sense.

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Vector gain and phase margins were also computed.Vector margins are based on the point of closestapproach of the Nyquist plot to the critical point in thecomplex plane, and correspond to the worst-casecombination of simultaneous gain and phase variations.Bode and Nyquist plots were also examined manuallyfor selected design points to validate the computationalapproach. This analysis was completed for variousflight conditions defined by four independent variables:M, h, 0A, and Of. The flight envelope was representedby the following sets of independent variable values:

M 6 {0.8,1.0,1.2,1.6,2.0,3.0,5.0}h (kft]e {0,10,20,30,40,50,70}<t>A e {0°,-45°}aT e {2°,60,12°,200}

(21)

The values of the classical and vector gain andphase margins at selected flight conditions arepresented in Tables 2, 3 and 4 for the roll, pitch, andyaw autopilot channels, respectively.

Table 2: Roll Channel Stability Margins _______________

Mach Alt PhiA AlphaT

(kft) (deg) (deg)0.8 0.0 0.0 2.00.8 0.0 0.0 6.00.8 0.0 0.0 12.00.8 0.0 0.0 ' 20.00.8 0.0 -45.0 2.00.8 0.0 -45.0 6.00.8 0.0 -45.0 12.00.8 0.0 -45.0 20.02.0 20.0 0.0 2.02.0 20.0 0.0 6.02.0 20.0 0.0 12.02.0 20.0 -45.0 2.02.0 20.0 -45.0 6.02.0 20.0 -45.0' 12.02.0 20.0 -45.0 20.05.0 70.0 0.0 2.05.0 70.0 0.0 6.05.0 70.0 0.0 12.05.0 70.0 0.0 20.05.0 70.0 -45.0 2.05.0 70.0 -45.0 6.05.0 70.0 -45.0 12.05.0 70.0 -45.0 20.0

ClassicalLower GM

(dB)-69.41-56.17-12.79-6.09-68.32-55.17-9.82-23.97-69.80-32.03-5.34-37.25-35.40-38.26-15.13-73.60-1 1 .65-3.78-3.87-71 .55-56.58-9.25-4.41

ClassicalUpper GM

(dB)16.7316.6216.3810.2216.7316.5515.7215.5512.6412.5711.9512.6312.4412.088.4616.1815.8013.3310.2316.1215.4612.589.82

ClassicalPM

(deg)73.4772.3767.8956.0673.3372.2162.8466.9265.5165.0151.1065.3663.5362.6449.2172.2466.4646.9732.9372.1268.5549.4533.34

VectorLower GM

(dB)-5.16-5.15-5.12-4.24-5.16-5.14-5.04-5.03-4.67-4.66-4.51-4.67-4.63-4.58-4.18-5.11-5.06-3.77-3.85-5.10-5.03-4.58-3.89

VectorUpperGM

(dB)14.5314.4014.098.6114.5314.3413.3813.3710.8110.749.8910.8010.5910.288.3814.0213.566.807.1013.9613.2910.327.23

VectorPM

(deg)47.9247.7547.3136.6547.9247.6646.2646.2541.7041.5739.7541.6841.2640.6036.0547.2146.5331.5132.4247.1346.1340.6832.82

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Table 3: Pitch Channel Stability Margins

Mach Alt PhiA AlphaT

(kft) (deg) (deg)0.8 0.0 0.0 2.00.8 0.0 0.0 6.00.8 0.0 0.0 12.00.8 0.0 0.0 20.00.8 0.0 -45.0 2.00.8 0.0 -45.0 6.00.8 0.0 -45.0, 12.00.8 0.0 -45.0 20.02.0 20.0 0.0 2.02.0 20.0 0.0 6.02.0 20.0 0.0 12.02.0 20.0 -45.0 2.02.0 20.0 -45.0 6.02.0 20.0 -45.0 12.02.0 20.0 -45.0 20.05.0 70.0 0.0 2.05.0 70.0 0.0 . 6.05.0 70.0 0.0 12.05.0 70.0 0.0 20.05.0 70.0 -45.0 2.05.0 70.0 -45.0 6.05.0 70.0 -45.0 12.05.0 70.0 -45.0 20.0

ClassicalLower GM

(dB)-Inf-Inf-Inf

-9.27-170.22-172.03-163.49-106.40

-Inf-Inf

-11.48-172.94-187.65-188.60-148.26-12.20-16.53

-Inf-Inf

-10.70-14.68-163.61-160.31

ClassicalUpper GM

(dB)18.2518.8718.6716.3915.5817.6719.5819.9415.8516.2815.5014.6315.7616.7517.3418.1719.2323.0713.9119.2619.6820.1620.36

ClassicalPM

(deg)64.8668.0665.5157.1962.3466.5975.8172.2960.9363.7351.8559.8864.3063.6768.0760.8662.2771.4160.3359.3163.9269.6472.73

VectorLower GM

(dB)-5.27-5.33-5.30-5.08-5.04-5.25-5.40-5.40-5.02-5.08-4.93-4.90-5.03-5.13-5.19-5.25-5.33-5.48-4.78-5.33-5.37-5.41-5.43

VectorUpperGM

(dB)15.6416.2616.0113.7313.4515.3517.1417.2313.2813.7312.5512.3313.3614.1614.8015.3516.2718.2911.5216.2516.7517.3517.65

VectorPM

(deg)49.3450.0649.7846.7946.3748.9951.0051.0946.1146.7944.9344.5646.2447.4248.2948.9950.0752.0943.0950.0550.5951.2151.50

Table 4: Yaw Channel Stability Margins?lliKt5oTliliBsWs ,

Mach Alt PhiA. AlphaT

(kft) (deg) (deg)0.8 0.0 0.0 2.00.8 0.0 0.0 6.00.8 0.0 0.0 12.00.8 0.0 0.0 20.00.8 0.0 -45.0 2.00.8 0.0 -45.0 6.00.8 0.0 -45.0 12.00.8 0.0 -45.0' 20.02.0 20.0 0.0 2.02.0 20.0 0.0 6.02.0 20.0 0.0 12.02.0 20.0 -45.0 2.02.0 20.0 -45.0 6.02.0 20.0 -45.0 12.02.0 20.0 -45.0 20.05.0 70.0 0.0 2.05.0 70.0 0.0 6.05.0 70.0 0.0 • 12.05.0 70.0 0.0 20.05.0 70.0 -45.0 2.05.0 70.0 -45.0 6.05.0 70.0 -45.0 12.05.0 70.0 -45.0 20.0

ClassicalLower GM

(dB)-Inf-Inf

-20.05-9.05

-Inf-Inf-Inf-Inf-Inf-Inf

-9.79-Inf-Inf-Inf

-19.09-12.56-23.00

-Inf-Inf

-20.24-Inf

-18.86-8.85

ClassicalUpper GM

(dB)18.1318.4518.4417.8819.1219.8720.7623.3815.9916.1915.8116.2216.9418.6620.5218.2619.4223.5313.1318.6620.0112.195.61

ClassicalPM

(deg)64.0363.8763.0061.4070.1472.1066.9270.6162.5663.6953.2463.8168.2467.1859.7662.8963.6771.2661.7063.9066.7354.2432.73

VectorLower GM

(dB)-5.26-5.29-5.28-5.20-5.35-5.39-5.41-5.47-5.05-5.07-4.98-5.07-5.14-5.19-4.99-5.26-5.34-5.48-4.71-5.29-5.35-4.57-3.31

VectorUpperGM

(dB)15.5415.7915.7514.8616.5217.1317.2818.2713.4713.6712.9313.6914.3114.7813.0415.5116.4818.3411.0815.9016.5910.245.43

VectorPM

(deg)49.2249.5249.4848.3750.3450.9951.1452.0846.4146.7145.5546.7347.6248.2645.7449.1850.3052.1442.2449.6550.4240.5026.87

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The results in Tables 2, 3 and 4 show that thedynamic inversion autopilot has adequate linear gain,phase and vector margins for a wide range of flightconditions. Initially, poor vector gain and phasemargins are observed for <j)A - 0°, aT = 20°, M = 0.8, andh - 0 kft. Robustness at these conditions was thenimproved simply by reducing the gain of the outer-loop.The relatively low yaw channel margins observed atM = 5, 04 = 0°, 0^=20°, and /z = 70kft suggest thepossible need for further retuning, but this is an unusualflight condition and therefore did not merit significanttuning effort.

Nonlinear AnalysisThe linearized robustness results obtained above

have been verified at selected flight conditions usingnonlinear simulations. Gain variations and delays wereintroduced at the plant input, and each was increasedindependently until limit cycle behavior was observed.These limit cycle phenomena occurred at gain anddelay levels approximately 80% lower than thosepredicted by the classical gain and phase margins. Thisdifference may be accounted for by the fact that theclassical margins are based on linearized models. Thatanalysis, which we shall not describe in further detail,serves not only to confirm the classical margin results

but also to validate the numerical and analyticallinearization algorithms used to obtain the linearmodels.

Additional nonlinear robustness studies wereconducted by allowing important aerodynamicparameters to vary by ±25%. The parameters includedin this study were the tabular aerodynamic momentsand control derivatives: LQ, M0, N0, L&, N&, M^,, L^,,and Nfy. First, each of these parameters wasindividually varied by the maximum amount of ±25%.Next, all of the parameters were allowed to varyrandomly within the prescribed uncertainty range withuniform distribution. A total of 500 maximum-g stepresponses were simulated for each flight condition ateach of the two uncertainty levels. The effects ofparameter variation were observed by monitoringsteady-state acceleration errors and body-axis roll rates.Only stability was considered in this analysis, sincegood performance could not be expected under suchconditions. Table 5 summarizes parameter variationstudy results for five specific flight conditions. Notethat the M = 5, <j>A - 0°, and h = 70 kft case is unstablefor some random parameter variations. This illustratesthe importance of properly limiting the outer-loopcontrol bandwidth to preserve stability robustness.

Table 5: Selected Parameter Variation Results for 25% Uncertaintyf iaSipf ;": liftei Siiiii :• :iA • ' - ; r : aT " IligWiSiS PSI sSSHi

IK :{kft)K* E' egjtf ; M^W-iK^ i. .;:.:..:.....:.:.,.&..¥. ,.-.-. .-;. -iV.. -:.-. .V ™. V!XB^ -. . -: . -. -K :»,„•,,,.,. *fJ,:-,.-:.,,:,vf-,:^.i.^,.. 'f-.-S..., , .35fe££ ST.V.vA;'™ ir; iS&fcfsMt. ^SAf,f.f,Af,f,fy^^:f....y.-^..s;.-;A.-s;;},fS^^S. K^Z^,:K,:s:,}:Z*X,:y;:Zi;*X^»*:?f,':v**j> &immm.S*=*iifflSS3$i;E.: ^S^JsS-W-S™ ::»•::-:::; 438^

12345

0.8•1.22.03.05.0

0.050.020.00.070.0

0.00.00.00.00.0

1.01.01.01.01.0

500500500500500

500500500500420

100.0100.0100.0100.084.0

5. Conclusions

A full-envelope nonlinear acceleration autopilot foran air-to-air missile has been developed using thedynamic inversion architecture of Ref. 3, whichfeatures an inner-loop/outer-loop structure with outputredefinition in the inner-loop. Based on 63% rise time,this autopilot compares favorably to an existing gain-scheduled controller. In addition, significant increasesin achievable accelerations were observed using thenonlinear autopilot. These performance improvementsare, however, expected since the nonlinear autopilotincorporates the full • aerodynamic database of themissile and is therefore more complicated than thesimple gain-scheduled classical design.

An extensive robustness analysis was performed onthe dynamic inversion autopilot in order to compare itsrobustness with that of the gain scheduled design,which was constrained to have 6 dB upper gain marginand 30 deg phase margin at all design points. Classicalgain and phase margins for the nonlinear autopilotdesign were computed based on local linearizations ofboth the nonlinear plant and the nonlinear controllerthroughout the flight envelope. These classical marginswere validated by using a nonlinear simulation todetermine maximum allowable gain variations andinput delays. Vector gain and phase margins were alsocomputed based on these linearized models. Althoughthe classical margins were adequate for all designpoints, the vector margins revealed deficiencies inrobustness at a few design points. These issues may be

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addressed by re-tuning the autopilot design parameters.Additional nonlinear simulation studies included MonteCarlo analysis of sensitivity to variations in modeledaerodynamic parameters. The dynamic inversionautopilot exhibited robustness for variations of up to25% in the aerodynamic moment coefficients andcontrol derivatives. Based on these results, thenonlinear dynamic inversion architecture studied hereachieves robustness levels comparable to or better thanthat of a classical gain-scheduled design whilesimultaneously improving performance.

References

1. Snell, S.A., Enns, D. F., and Garrard, W. L.,"Nonlinear Inversion Flight Control for aSupermaneuverable Aircraft," AIAA Journal ofGuidance, Control and Dynamics, Vol. 15, No. 4,pp. 570-577, 1992.

2. Ryu, J. H., Park, C. S., and Tahk, M. J., "PlantInversion Control of Tail-Controlled Missiles,"Proceedings of the AIAA Guidance, Navigation,and Control Conference, pp. 1691-1696, 1997.

3. Sharma, M., Calise, A. J., and Corban, J. E.,"Application of an Adaptive Autopilot Design to aFamily of Guided Munitions," Submitted to theAIAA Guidance, Navigation, and ControlConference, Denver, 2000.

4. Calise, A. J., Sharma M., and Corban, J. E., "AnAdaptive Autopilot Design for Guided Munitions,"AIAA Journal of Guidance, Dynamics, andControl, to appear.

5. Drinker, J. S., and Wise, K. A., "Stability andFlying Qualities Robustness of a DynamicInversion Aircraft Control Law," AIAA Journal ofGuidance, Control, and Dynamics, Vol. 19, No. 6,pp. 1270-1277, 1996.

6. Schumacher, C., and Khargonekar, P., "AComparison of Missile Autopilot Designs Using H-Infinity Control With Gain Scheduling andDynamic Inversion," Proceedings of the AmericanControl Conference, pp. 2759-2763,1997.

7. Durham, W., "Dynamic Inversion and Model-Following Control," AIAA-96-3690, AIAAGuidance, Navigation, and Control Conference,San Diego, 1996.

8. Bar-On, J. R., and Adams, R. J., "Linearization of aSix-Degree-of-Freedom Missile for AutopilotAnalysis," AIAA Journal of Guidance, Control,and Dynamics, Vol. 21, No. 1, pp. 184-187, 1998.

10American Institute of Aeronautics and Astronautics