[American Institute of Aeronautics and Astronautics AIAA Guidance, Navigation, and Control...

Autonomous Missile Avoidance using Nonlinear Model

Predictive Control

Dr. Leena Singh

Sr. Member Technical Staff

The Charles Stark Draper Laboratory

Cambridge, MA 02420

[email protected]

In this paper, we present a new way to formulate an autonomous missile avoiding feed-

back control law, and successfully enable a UAV to autonomously avoid an oncoming track-

ing missile. We used nonlinear Model Predictive Control to pose and solve a finite-horizon

H∞-like control problem to synthesize the optimal, aggressive flight control strategy. The

key result of this paper is the formalization of the optimization cost function for missile-

avoidance - in our algorithm, the optimizer maximizes the line-of-sight rate between aircraft

and missile. This research takes advantage of the knowledge that most tracking missiles

use proportional-navigation feedback control; Line-of-sight rate is central to a tracking mis-

sile’s strategy to intercept its target. The algorithm predicts the behaviour of the missile

in response to any bounded-horizon control action that the aircraft might employ and thus

identifies the correct instant to employ a suitable high-G evasive maneuver that it synthe-

sizes via a control profile. To formulate this optimization problem, we recognized that the

key limitation in the missile system is the rotational limits on its seeker’s platform - the mis-

sile itself is considerably more agile and faster than any aircraft. Conversely, if the seeker

loses sight of its target for an instant, it cannot re-aqcuire the target, the lock is broken,

and the target is out of harms way. Therefore, for the missile to lose its target, the aircraft

must maneuver in such a way that the missile’s seeker gimbals are exercised beyond their

performance range. Our optimization-based feedback controller quantitatively computes

the correct, missile-avoiding maneuver to make and initiates it at the correct flash-point.

Qualitatively, the behaviour of the aircraft using this autonomous missile-avoidance logic

is similar to succesful pilotted escapes - this is a welcome validation of our work which

also serves to mathematically formalize why the human tactics work. The paper includes

results for interception experiments for a UAV against two distinct types of missiles - a

faster missile and a slower missile. The faster missile is characterized by a higher feed-

back gain on the proportional navigation feedback control law, and the slower missile by

the smaller feedback gain. Proportional navigation feedback gains are typically limited to

a pre-specified, known interval. We exercised our missile avoiding controls at both ends

of the known spectrum. Simulation results indicated that the MPC controller needed to

run at a faster rate and recompute more frequently against the aggressive missile than it

did against the slower missile. We have also simulated these experiments with different

interception angles between aircraft and missile; this paper presents results for the more

strategic interception situation in which the aircraft has to maneuver very dynamically to

avoid the missile. This paper also addresses mathematical modifications with control basis

functions used to compute the nonlinear controls in a robust, but computationally efficient

way.

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American Institute of Aeronautics and Astronautics

AIAA Guidance, Navigation, and Control Conference and Exhibit16 - 19 August 2004, Providence, Rhode Island

AIAA 2004-4910

Copyright © 2004 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I. Introduction

In recent years, there has been renewed interest in unmanned aircraft that can engage in tactical militarymissions in which the aircraft must autonomously evade threats. Such tactical missions require that theunmanned aircraft must autonomously assess threats, analyze mission requirements, and manage aggressive,agile flight operations. Fielded unmanned aircraft such as Predator are presently only capable of safe, high-altitude, flight in near-trim regimes. In this paper, we present a novel problem - i.e., to autonomouslyevade a tracking missile that has locked onto an aircraft - and succesfully present a formal control-theoreticapproach to autonomously compute aircraft evasive maneuvers with the associated controls, to dodge themissile. Missiles are considerably more agile and maneuverable than aircraft and have higher physicaldynamics limits; hence conventional wisdom considers it futile to try to “out-run” a missile. However, weknow from observing fighter pilots that precision maneuvers at close range can saturate the target seekeron the missile and thus force the missile to lose the aircraft. In this paper, we have used this informationto formulate our missile evasion strategy. This research uses finite, receding horizon optimal control, viz.Model Predictive Control, to synthesize aggressive flight operations and control. An important contributionof this paper is the structure of the optimization index that reliably enables missile-avoidance.

Model Predictive Control (MPC), which builds upon classical optimal control theory, explicitly considerssystem constraints (input, output, state, or environment) as it synthesizes control sequences that minimizethe stated objective function; its solution is the (locally) optimal sequence of controls in a finite futuretime horizon. Solving an optimization problem online - even a finite horizon optimization problem - iscomputationally burdensome; consequently, MPC has only been significantly used to control systems withlarge time-constants such as those found in process control applications and in chemical engineering. Recentlyit is being popularly investigated as an alternative to existing control design methods, due to recent accessto faster, cheaper computers, and efficient numerical algorithms for solving optimization problems.1–3 ModelPredictive Control enables real-time onboard optimal control because it is a repeating, finite-horizon optimalcontrol strategy - it decomposes the complete optimal control problem to mission-end into a sequence of muchshorter term, finite receding horizon optimal control sub-problems. At each point in time, the optimizercomputes the optimal sequence of controls within some short prediction horizon based upon reducing themission scope to that within this small prediction horizon. The first control of this optimal control sequenceis commanded. In the next time cycle, the optimizer resolves the optimal control for a control trajectorybased upon another small view of the mission with the time window slid forward by one time interval.Model Predictive Control for aircraft guidance and control applications has been previously addressed andwell developed4, 5 for applications with constraints.

At each call, MPC’s optimizer outputs the admissible, optimal sequence of controls of length equal tothis finite control horizon, such that the control sequence minimizes the objective function and satisfiesconstraints within the prediction horizon. Of the control sequence produced in this window, we apply onlythe control associated with time t = 0 (i.e. tnow). In our practice, two noteworthy points are: (1) we define anMPC computation rate fMPC in contrast to the true control rate fc and that the first k = fc

fMP Ccontrols are

applied as feed-forward commands to an inner-loop controller and (2) the residual part of previous optimalcontrol solution is used as the initial guess to seed the optimization in the next cycle.

In this paper, we introduce an MPC-based, strategic feedback control algorithm that autonomouslycomputes the aircraft controls that will avoid the missile; the algorithm succesfully avoids the missile eventhough the aircraft is slower and less agile than the missile. The aircraft dynamics are nonlinear as are theclosed loop missile dynamics. Our control-theoretic avoidance algorithm considers the capabilities of thecomplete missile system: because the dynamics of the sensor-based tracking system on the missile is bandlimited to the rates that the seeker’s gimbal can maintain, we have identified that an onboard controllermust attempt to maximize predicted lines-of-sight rates within the finite future horizon. We posed an H∞

optimization problem that seeks to maximize the maximum achievable line-of-sight rate between the aircraftand missile trajectories; this is the quantity that forms the error signal in the missile’s feedback control law.A similar cost function was previously explored6 for a multi-objective optimization control problem.

Section II introduces some background about pilotted fighter aircrafts in dog-fights and motivates the

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choice of this algorithm. Section A summarizes the missile’s proportional navigation control law. SectionIII describes our system model-based finite-horizon optimization. Section IV presents simulation resultsshowing a missile tracking a aircraft and the aircraft’s evasive maneuvers in dodging the system.

II. Pilot Operation Assesment and Background

Experienced fighter pilots choose from a suite of efficient and often-effective evasive maneuvers whenthey find that they have engaged a missile. These maneuvers are (1) to wait until the missile is in veryclose range and then initiate a sharp turn (dive, pullup, left or right pullouts) so that the missile’s sensoris unable to track the aircraft (i.e. saturate missile’s sensor gimbals); the missile sensor “loses” sight ofthe aircraft and the aircraft wins the game. Other tactics popularly employed when the missile is still at adistance include (1) entering a steep dive so that the missile’s seeker has to identify the aircraft against thefeature-rich background in ground-fall, and (2) dodging behind terrain feature such as a hilltop so that themissile “loses” line-of-sight with the aircraft. All these tactics indicate that the aircraft pilot really evadesthe missile’s feedback sensor - the seeker and/or image tracker; she does not attempt to out-run the missile.

Traditionally missiles use proportional navigation in their tracking control systems. Proportional naviga-tion computes the line of sight rate between the aircraft and the missile from the distance (range) betweenthe two vehicles, their relative speeds, and the estimated time to nearest intercept or “t-go”. When line-of-sight rate is zero, the missile and aircraft are on a perfect collision course and if both maintain theirrespective speeds, the missile will need no further corrections to intercept the aircraft. When this rate ishigh, the missile has to turn quite sharply to intercept the aircraft. Missile seeker platforms are typicallybandlimited to about 30o−45o/s tracking capability;7 if the aircraft makes rapid maneuvers that exceed theseeker’s tracking ability (at 30o − 45o/s), then the target (aircraft) moves beyond the seeker’s field of viewand the missile loses the target. Because line-of-sight rate increases as the range to target decreases, clearly,cross-range motion in proximity to the missile force greater line-of-sight rates than the same maneuversfurther away.

In this research, we show how we exploit this tactic to formulate our autonomous missile-avoiding aircraftguidance and control strategy.

A. Missile Dynamics and Proportional Navigation Control

The missile is modelled as a pure double integrator in position: Xmiss = N c where N c is the accelerationcontrol command output from the proportional navigation control law, filtered through a second order lowpass filter. This approach was derived from existing literature7 and is considered an acceptable way tomodel a missile in a scenario with an aircraft because any lags due to missile dynamics are insignificant whencompared with the typical aircraft dynamic modes. Effectively, we say that the realized acceleration of themissile is identically equal to the commanded (control) with no delays and lags between the missile autopilotcommands. Proportional navigation control - the missile’s feedback tracking control algorithm - is coarselysummarized below.

• Compute range: relative position vector to aircraft. Compute relative velocity of aircraft

• Compute closing speed : component of relative velocity along range

• Determine tgo: range separation divided by closing speed

• Find the Zero Effort Miss (ZEM) vector: minimum separation of trajectories at present headings andvelocities of aircraft and self (missile)

• Find ZEM⊥: ZEM perpendicular to line-of-sight vector

• Evaluate λ: line-of-sight rate vector from ZEM⊥, range, and tgo

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• Compute proportional-navigation feedback control: Nc = KN ∗ λ, where KN is a feedback gain (navi-gation ratio) and varies between 2 and 4.

Missile

Target

RangeV

λ

m

VΤ

Vclosing

Figure 1. Key elements in the pro-nav algorithm

III. Model Predictive Control for Aircraft Control

Typically, in a Model Predictive Control (MPC) problem, the control designer formulates a discrete time,optimal regulation or tracking problem via an objective function which consists of weighted state trackingerror and control effort terms to be minimized. Just as in classical optimal control, the optimal solution is thecontrol action that minimizes this objective function. In receding horizon control, this function only penalizespredicted error and control effort terms in some time horizon of N future terms, t = Th = NTs, (where Ts isthe discrete time system’s sampling time) and not to t∞ or mission completion as in LQR. At each time tl,MPC’s optimizer then computes the bounded time sequence of N+1 aircraft controls, [u∗

k], k = l : l + N , inthe interval T = [tl, ..tl+N ], that minimizes the cost:

Jl =∑l+N

k=l εTk+1Qεk+1 + uT

k Ruk (1)

J∗l = min

uk

Jl (2)

where εk is the tracking error relative to some reference trajectory, at time tk. Q and R are the state errorand control weighting terms that indicate the relative importance of keeping the corresponding state variableerror small. The optimization is subject to a dynamic model constraint x = F (x, u) which establishes theexplicit relationship between control uk, the independent optimization variable, and state (or tracking error)εk, εk+1 . . .; the dependent optimization variable. Only the first controls u∗

l , of this optimal control solutionsequence is ever applied to the plant. The system responds to this control signal, its new states are recordedand the above optimization is repeated in the new Th second long interval that now extends from absolutetime [(l + 1) · Ts : (l + 1 + N) · Ts].

Typically, we choose the forward control horizon, Th so that the inequality constraints beyond Th do notsignificantly affect the control profile at the present time, ie. ul. Presently, there are no formal analyticalways to ensure that a horizon length is “long enough” for general classes of nonlinear dynamics. Formalmethods for selecting bounds on horizon length that guarantee stability of the optimal solution are beinginvestigated for unconstrained MPC. In this paper, we assessed different horizons k = 0 : N and selected a

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10 second interval within which the optimal control solution was robust to further changes in Th and did notchange appreciably for longer Th horizons.

In our missile-avoidance guidance and control algorithm, we wish to optimally determine the (predicted)aircraft trajectory that maximizes the maximum (predicted) line-of-sight rate achievable in the optimizationinterval between missile and aircraft based upon current positions and velocities of aircraft and missile.In essence, in this research, we pose a discrete time, nonlinear H∞ game-theory problem that computesthe feedback control to maximize the maximum achievable line-of-sight rate within the prediction horizon.Therefore, we define the optimal performance index Jl at time tl as:

J∗l (u∗, Th) = min

u∗

(

min∀k∈{l:l+N}

(−λ2k)

)

(3)

N = Th/Ts. This is a good choice for the optimization cost function, because we know that if the in-stantaneous los-rate (λ) exceeds the missile sensor’s tracking capability, its feedback control system cannotcompute the feedback error signal; it cannot form the tracking control; and that loss of track is sufficient tolose the target. LOS-R is a complex nonlinear measure derived from relative position and velocity states ofthe aircraft and missile. Therefore, unlike many MPC applications, this problem is a nonlinear worst-caseoptimization problem and cannot be written in the familiar finite-horizon LQR-like form as a sequence oftracking error terms. This optimization problem is subject to the aircraft’s dynamic model as a differentialconstraint. Our aircraft dynamic model is presented in Sec. A as derived from.8

To synthesize the optimal control profile, we predict both the aircraft’s trajectory produced by an admis-sible, candidate control sequence as well as the missile’s response to that candidate aircraft trajectory. Thealgorithm predicts the missile’s trajectory from a missile dynamic model and an onboard implementationof the proportional navigation feedback control (as part of the aircraft’s anti-missile autopilot logic). Thenonlinear optimizer repeats this process posing candidate control sequences and evaluating the cost functionassociated with that candidate control. The aircraft control sequence that produces the highest achievableLOS-R becomes the optimal solution for this MPC cycle. The controls are applied for one iteration and thesearch then repeats based upon the new positions and velocities of aircraft and missile.

A. Aircraft Dynamics

The following nonlinear dynamic model is used to model the aircraft’s behaviour.

V ac = −ωac × V ac − KdV1.5ac − RI

ac · g + F (4)

ωac = J−1in (ωac × Jinωac) + T (5)

XI = RacI V ac (6)

The last equation is the kinematic transformation from aircraft body frame velocities to inertial. In addition,we can augment the vehicle dynamics with an inner stability augmentation system (SAS) that stabilizes theaircraft’s attitude dynamics.9 V ac is the aircraft’s velocity vector in the aircraft frame, XI is the positionof the aircraft in a ground-based inertial frame, and R

BA is the rotation matrix from a B frame of reference

to an A frame of reference. The N inputs to the dynamic model are a simplified pseudo-control vector:U = [F , T ] ∈ <p; these inputs are computed by the optimizing control system and augmented by the innerloop SAS. In ongoing research, we will deploy the SAS-augmented aircraft dynamics and relax the modelcomplexity used in the optimizing prediction step. The augmented aircraft model decouples the nonlinearmodel into the separable lateral/longitudinal form; each dynamic block can further be written in quasi-linearform. This structure can then be exploited to simplify the optimization.

We imposed constraints on achievable aircraft actuation commands, aircraft speed, and actuation rates:

(ui)min ≤ ui(k) ≤ (ui)max i = {1, 6} (7)

Vmin ≤ ‖V ‖(i) ≤ Vmax (8)

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Aircraft forward and lateral thrust, and lift were set such that F /m not exceed 3.5G. We did not constrainthe missile dynamics; however, we do limit the missile’s seeker’s tracking rates to 45o/s and therefore, itsfeedback control 2-norm.

B. Input Control Bases

In this phase of the research, our prime objective was to verify our choice of the optimization cost functionfor missile-evasion as introduced in Sec. III and in Equation 3. We did not attempt to significantly mitigatethe computational overhead of solving a non-convex optimization problem with any linearizing assumptions.However, we reduce the dimensionality of the optimization problem by restricting the optimizer’s controldesign space. We introduce M discrete, scaleable control basis functions that are each Th seconds long andconstrain the optimizer to synthesize the optimal control commands by only scaling this comparatively smallnumber, M, of basis functions. With this approach we reduce the dimensions of the optimization problemfrom a (N · p = Th/Ts · p) dimensional problem to (M · p) dimensions where we select (M << Th/Ts) andp = dim(ui) is the number of inputs to the aircraft, Eqs. 4 and 5. In our formulation, the MPC control systemcomputes perturbations to a nominal control sequence. The nominal control sequence at any time ti is afeedforward signal equal to the unused portion of the previously computed optimal control trajectory. Thuswe reuse the results of the previous optimization iteration when we set up the next optimization problem.We constrain the control perturbations to M ∈ = scaleable control basis functions as shown in Figure 2; thisapproach is derived from the literature and is detailed in the reference.10 In general, the control bases canbe any real function - they do not have to be the saturating ramps shown in Figure 2. With this choice ofM bases in our Th second horizon, each resulting control (Fx, Fy, . . .) can form any M-dimensional polytopicfunction. Note that unlike the treatment in Singh2001,,10 where the authors used these linear perturbationalcontrol bases to both reduce optimization dimension and linearize the dynamic model around a nominaltrajectory, in this work, we only use them to reduce the optimization dimension in each cycle.

T

Basis_1Basis_2

Basis_M

HTime

Con

trol

bas

is fu

nctio

ns

1

Figure 2. Sequence of saturating ramps that form the control bases. Each ramp starts when the previous has

saturated. The optimizer computes the M scale functions that then determine the shape of the control during

this prediction interval.

The control is constructed by superposing the M basis ramps. Each control input (u1 or u2) is represented

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in terms of its bases as:

uk

uk+1

...

uk+n1

...

uk+2n1

...

uk+N

=

du 0 0 . . . 0

2du 0 0 0...

......

...

1 du 0 0...

......

...

1 1 du 0...

......

...

1 1 1 1 1

·

ν1

ν2

ν3

...

ν6

(9)

U = S· V (10)

where n1 is the length of each ramp, (n1Ts is therefore the time before the second control basis ramps up).Note also that in this representation U = SV only pertains to the perturbational component of the input ;the actual inputs applied will include the feedforward, nominal control U0 so that the control applied at thekth control channel will be Wk = U0(k) + S(k, :) · V ∗. The optimizer produces the optimal sequence of M · pscale factors, V ∗, of perturbatioal controls, in each optimization cycle. As mentioned earlier, each candidate

control sequence per optimization iteration at time ti : Wk =

[

F

T

]

k

is evaluated on the nonlinear aircraft

and missile dynamic models, Eq. 6 to compute the cost of using that control.Other functions have also been succesfully used to solve similar optimization and optimal aircraft control

problems with discrete basis functions; in particular, polynomial basis functions such as the Laguerre andLegendre polynomials were analyzed11 for aircraft guidance and control and in process control12 and highbandwidth robotics control applications.13 Analysis11 has demonstrated that Laguerre polynomials areespecially useful as control basis functions compared with other polynomials such as Legendre and Hermitepolynomials for applications in aircraft guidance and control as they effectively span the controllable sub-space with fewer basis elements. In this research, we were able to further reduce the computational complexityusing half the number of the Laguerre polynomial bases (5) than the saturating ramps, with little comparativereduction in performance. Reducing the number of basis functions used in the optimal control synthesisreduces optimization time and therefore improves the numerical efficiency of the optimization problem. TheLaguerre control basis functions which are scaled to produce the optimal control perturbations are shown inFig. 3. All other aspects of the use of these as control basis functions remains consistent with the developmentabove.

An important distinction between applying the ramp and Laguerre polynomial bases as control basisfunctions is necessary. The saturating ramps inherently admit a nearly smooth (if the corners are filtered)function because all basis elements linearly transition to 1 starting at 0. The Laguerre basis functions incontrast have non-zero initial values; consequently, each time we superpose the scaled perturbations to formthe applied control vector, this “optimal” control vector will contain a sudden jump at the initial time. Inthis work, therefore, we filtered the output of the optimizer before applying the controls to the aircraft toensure that there are no step actuations.

C. Nonlinear Control Optimization

Given feedback about the aircraft and missile’s present positions and velocities, the optimizer compares thecosts of various candidate control sequences by computing the vector of scale factors, V, to the control basisfunctions according to Eq. 10. We add the scaled basis functions to the nominal control trajectory overthe prediction horizon to produce the Th second candidate control sequence. Presently, we use MATLAB’sfmincon() nonlinear optimizer and provide the nonlinear, model-based cost evaluation function routine.The routine predicts the aircraft’s response to the Th second long candidate control sequence supplied by the

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0 20 40 60 80 100 120 140 160 180 2000

0.2

0.4

0.6

0.8

1

4th order Laguerre Polynomials Functions

Figure 3. Fourth order Laguerre Polynomials are used as an alternative set of control bases. The optimizer

computes the optimal scale factors to these 5 polynomial functions.

optimizer, the expected response of the missile to that candidate aircraft trajectory, and returns expectedcost (maximum line-of-sight rate) induced by that control sequence in the prediction interval. The optimizerevaluates multiple feasible control perturbations and selects those perturbations that produce the largest(predicted) LOS-R in the prediction horizon. Note that Th need not extend to cover the entire time-horizonto intercept, tgo. In the early stages, the predicted control trajectories returned by the optimizer will actuallynot evoke much maneuvering by the aircraft. Figures obtained from example simulation runs in the nextsection demonstrate this.

Initially, the nominal control trajectory is in steady level trim flight; in subsequent iterations of the MPCmissile avoidance control, this nominal control trajectory becomes the residual control sequence computedin previous iterations and correctly time indexed, it constitutes the new feedforward control sequence.

The control hierarchy and MPC’s operations described above is briefly summarized in Figure 4. Theguidance logic calls the optimizer with the present states of the aircraft and missile. The optimizer searchsfor the best control sequence that it predicts will induce the largest instantaneous line-of-sight rate in the10second prediction horizon. It applies the first 0.25 seconds worth of the controls. 0.25 seconds later, thesearch for the optimal control sequence starts again based upon the new aircraft and missile staes.

Solving a nonconvex optimization problem with nonlinear dynamic constraints is computationally cum-bersome. This primary goal of this research, however, was to verify the choice of optimization cost functionfor autonomous missile avoidance. We recognize the computational complexity inherent in this formulation;our approach to discretize the control sequence during the control computation helped to ameliorate theproblem somewhat. We are investigating this problem further.

IV. Results

We defined and solved the problem as presented above. We used the following key figures in our tests. Weset the missile and aircraft’s control rates and our simulation rate at 20Hz. The aircraft’s MPC update rates(rate at which the optimizer runs and new controls are computed) were 4Hz, and constrained the aircraftcontrols as described previously. We used a 10 second prediction horizon, the control prediction horizon wasalso 10 seconds. We discretized the available controls using 10 saturating ramp bases per control channel(forward thrust, lateral thrust, lift, etc.) and constrained the aircraft to 3.5G thrust limits. The number ofbasis functions is determined analytically; the using more than 10 ramps produces no significant improvement

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Optimization Engine

Simulate missileResponse Traj.

LO

S_R

U*

U*(1)U*(2:N)

U*(1)

Apply

Nom

inal

FF

cont

rol s

eque

nce

MPC Optimizer

A/C Guidance

Aircraft

Missile

Aircraft Inner & servos

trajectorySimulate AC

Compute LOS−R

controlscandidate

Pred. A/C pos/vel

Pred. missilepos/vel.

Missile pos, vel

Feed

back

A/C pos, vel

A/C pos, vel

Missile pos/vel

Figure 4. Flow-of-Control in the MPC-based autonomous feedback control algorithm.

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in the solution and only impedes the optimization process as it increases the optimization dimension; usingfewer than 8 bases, the optimizer is unable to find an escape solution that also satisfies the control constraints,in some scenarios. Consequently, we have settled on 10 ramp bases for this aircraft and missile system totrade off stabilizing and convergent control synthesis against computational dimension. The MPC updaterate was also derived empirically by establishing the slowest update at which the aircraft autopilot couldstill drive the aircraft to avoid the missile.

In all experiments, the missile approaches the aircraft from behind and below the target aircraft - thiscorresponds to a difficult avoidance configuration. We assume that the missile can auto-destruct within a10m proximity of the target which will destroy the aircraft before a true intercept. Therefore, we designaterange R < 10m separation between aircraft and missile is tantamount to collision.

We pose and approach the control computation problem in the following way. Initially, we assume thatthe aircraft is in steady level trim flight when the missile locks onto it and the aircraft autonomous missioncontrol system detects the lock. The missile avoidance logic then initiates. We assume that the aircraft alsohas feedback sensors and estimators that record and feeds back the missile’s coordinates and flight speeds.

When the aircraft detects the missile, this algorithm takes over the guidance and control operations ofthe aircraft. Figures to follow summarize the results. It is notable that this max(max(LOS-R)) strategygenerates the same qualitative behaviour as the skilled fighter pilot. The aircraft essentially maintains avery steady track, even veering into the path of the missile. In the last few seconds before the steady-statetime-to-collision when tgo is small, the aircraft maneuvers sharply away from the missile. The line-of-sightrates escalate rapidly and the aircraft is off-screen from the missile’s seeker. The missile loses the aircraft. Inour preliminary experiments, we attempted the missile avoidance simulation experiments with the missile indifferent approach tangents to the aircraft - the most challenging avoidance problem occurs when the missileapproachs parallel to and behind the aircraft. This is therefore our experimental scenario for which we arereporting results.

Missile systems avail of one important customization depending upon their own thrustor maneuveringcapability. The feedback gain or navigation ratio in the missile’s proportional navigation feedback controllaw typically varies from 2 to about 4,7 - gains higher than this range destabilize the pro-nav feedbackcontroller. We present simulation results with 2 different choice of pro-nav gains: KN = 2.5 and KN = 3.5.With the smaller nav ratio, the minimum miss distance is about 90m. However, with the larger nav ratio, theminimum miss distance fell to nearly half that at about 48m. We assumed in this research that the aircraft’spredictor knows the proportional navigation function’s feedback gain. Indeed, in a future step, we intendto include a gain estimator in the aircraft’s logic that will estimate this gain factor early in the dog-fight.At that point, the avoidance logic can proceed exactly as we have described above. We first present resultsobtained by using the saturating ramp basis functions to synthesize the optimal perturbational control.

Figures 5.a-b show the first example dog-fight with the less reactive (lower navigation ratio) missile. Thetrajectory shows that the aircraft stays the course until close to the missile. It then initiates the high-Gturn maneuver in the plane perpendicular to the missile’s velocity vector. Figure 5.b shows a close-up of theengagement. Figure 6.a shows the range plots - the aircraft comes within 90m of the missile when it engagesthe evasive maneuver. The last figure plots the acceleration profiles in the maneuver. The reader will noticethat the aircraft maintains steady flight until the missile is close.

The second set of figures present results using the more agile (higher navigation ratio) missile. Themissile still starts below the aircraft, but this is a much faster responding vehicle. The reader will note thatthe aircraft’s optimal strategy is somewhat different in this case - the aircraft actively approaches the missilebefore it deploys the evasive turning maneuver. The miss distance was less than 50 meters in this case.These plots are shown in Figures 7.a-b and 8.a-b.

Finally, we present the results obtained using Laguerre polynomials as the control basis functions. For thismissile avoidance scenario, we present results obtained with an intermediate navigation ratio at K = 2.75.The miss distance (200m) obtained in this case is noticeably larger than with the saturating ramp functions.Note that the polynomials admit improved performance (miss distances) with fewer basis elements comparedwith the ramps. The computational effort is therefore also noticeably reduced. We filtered the computedaircraft control with a corner frequency at 5Hz: UFF + ULag where ULag is the optimally scaled Laguerre

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Figure 5. Engagement Scenario with missile with low navigation ratio

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st A

ngle

(b) Aircraft acceleration commands in engagement.

Figure 6. Engagement Scenario with missile with low navigation ratio. Missile approachs to within 90m of

aircraft before rolling into sharp lateral turn out of range of the missile.

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(b) Close-up of trajectories near minimum separationpoint

Figure 7. Engagement Scenario with the more agile missile and high navigation ratio

77 78 79 80 81 82 830

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Figure 8. Engagement Scenario with the more agile missile with high navigation ratio. Missile approaches

to within less than 50m of aircraft. Aircraft relies on high gravity assisted turn into the missile to maximize

LOS-R.

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function that spans the 10second control prediction horizon. The instantaneous line of sight rates reachesa maximum value of nearly 80 deg /sec when the aircraft makes its avoidance maneuver. Figure 9b showsthe aircraft and missile configuration at the center of the maneuver. The aircraft rapidly veers across themissile’s path from above it; this rapid turn in the near field produces the rapid rise in the line-of-sights andsaturates the missile’s seeker with instantaneous rates at nearly 80 deg /sec when the aircraft and the missileare at the coordinates marked by the cyan squares in the plot. At this juncture the missile is about 300mfrom the aircraft and loses track. The minimum separation distance is 200m and occurs after the missile haslost track of the aircraft, but continues along its last vector.

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(a) Aircraft (blue) and Missile (green) Trajectories inflight

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(b) Close-up of trajectories near minimum separationpoint

Figure 9. Engagement Scenario with low nav ratio missile and laguerre polynomial control basis functions.

Coordinates at minimum separation and maximum line-of-sight rate are labelled.

Additionally, we have run this scenario with the more reactive missile with pro-nav gain = 3.5 and usingthe same Laguerre Bases to synthesize the NMPC-based controls. As with the saturating ramps, the aircraftavoids the missile, but the miss distances are smaller at 160m and the escape line-of-sight rate is also lowerat 60 deg /sec. Both results, however, are correspondingly higher than those produced with the saturatingramps.

Performance with the laguerre functions is noticeably better; however, in each case, the aircraft is ableto avoid the missile with this NMPC tactic that attempts to maximize the maximum instantaneous line ofsight rate between the aircraft and the missile in any prediction interval.

V. Conclusion

In this paper, we introduced a new way to mathematically formulate and successfully solve the missileavoidance problem. This research uses Model Predictive Control to synthesize aggressive flight operationsand control for this missile-evasion application. An important contribution of this paper is the structure ofthe optimization cost function that reliably enables missile-avoidance. We showed in this paper that we cansuccessfully evade missiles that use proportional-navigation feedback control (indeed, most tracking missilesuse this feedback control law), we predict the behaviour of the missile in response to any control action thatthe aircraft might employ and thus identify the correct instant to employ a high-G evasive maneuver. Theimportant realization in this work was that the key limitation in the missile system is the constraints of its

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0 20 40 60 80 100 120 1400

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(a) Close up of Range separation between missile andaircraft: Range plot. Rmin = 200m.

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(b) Aircraft LOS-R during engagement

Figure 10. Engagement Scenario with the low nav ratio missile annd laguerre polynomial control basis func-

tions. Missile approaches to nearly 200m of the aircraft. Aircraft relies on high gravity assisted turn into the

missile to maximize LOS-R.

seeker platform - the missile itself is considerably more agile and faster than the aircraft. Therefore, for themissile to lose its target, the aircraft must maneuver in such a way that the seeker gimbals are exercisedbeyond their performance range to saturation at which point the seeker can no longer track the targetaircraft. Our optimization-based feedback controller quantitatively computes the correct, missile-avoidingmaneuver to make and initiates it at the correct flash-point. Qualitatively, this tactic is similar to thesuccesful pilot-inspired one. However, because the pilot relies upon intuition and training, he has to “feel”the correct maneuver initiation-time - our algorithm explicitly computes the right intensity of turn to makeand the correct time to initiate it.

Future research will include methods to estimate the navigation ratio of the pro-nav guided missile,and will relax the complexity of the nonlinear model used in the optimization loop and thus set up acomputationally tractable problem. The present solution relies on MATLAB and its fmincon() optimizationfunction which is extremely slow.

References

1D.Q.Mayne, J.B.Rawlings, C.V.Rao, and P.O.M.Scokaert, “Constrained model predictive control: Stability and optimal-ity,” in Automatica, vol. 36, pp. 789–814, 2000.

2C.E.Garcia, D.M.Prett, and M.Morari, “Model predictive control: Theory and practice - a survey,” in Automatica, vol. 25,1989.

3S.Boyd, C.Crusius, and A. Hansson, “Control applications of nonlinear convex programming,” in Journal of Process

Control, no. 5–6 in 8, pp. 313–324, 1998.4J. M. Maciejowski, Predictive Control with Constraints. Pearson Education POD; 1st edition, 2002.5E.F.Camacho, C. Bordons, and M. Johnson, Model Predictive Control. Springer Verlag, 1999.6H.A.Hindi, B.Hassibi, and S.P.Boyd, “Multiobjective h2/h∞-optimal control via finite dimensional q-parametrization

and linear matrix inequalities,” in American Control Conference, June 1998.7P.Zarchan, Tactical and Strategic Missile Guidance, 2nd Ed. Washington, DC: AIAA Progress in Astronautics and

Aeronautics, Vol 157, 1994.8B. Etkin, Dynamics of Flight - Stability & Control 3e. John Wiley and Sons Ltd, March 20, 1996.

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9S. Heise and J. M. Maciejowski, “Model predictive control of a supermaneuverable aircraft,” in AIAA Guidance Navigation

and Control, 1996.10L.Singh and J.Fuller, “Trajectory generation for a uav in urban terrain, using nonlinear mpc,” in American Control

Conference, June 2001.11T. Lapp, “Guidance and control using model predictive control for low altitude rt terrain following flight,” in M.S.Thesis,

MIT, Dept. of Aero and Astro, 2004.12M. Huzmezan and G. Dumont, “Direct adaptive predictive control using subspace identification in laguerre domain in

the presence of constraints,” in IEEE Mediterranean Controls Conference, Patras, Greece, July 2000.13S. Jung and J. Wen, “Nonlinear model predictive control for the swing-up of a rotary inverted pendulum,” in To appear,

ASME J. on Dynamics, Measurements, and Control, 2004.

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