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Multiple Model Adaptive Evasion Against a Homing
Missile
Robert Fonod∗and Tal Shima†
Technion - Israel Institute of Technology, 32000 Haifa, Israel
Multiple model adaptive evasion strategies from an incoming homing missile are pre-
sented. The problem is formulated in the context of an evading target aircraft having
imperfect information on the relative state and on the employed guidance law and guid-
ance parameters of the missile. The missile’s guidance strategy is assumed to belong in a
finite set of linear guidance laws and to be fixed throughout the engagement. Arbitrary-
order linear missile and target dynamics, bounded target control, and nonlinear kinematics
are also assumed. The filter used to identify the missile’s guidance strategy is a nonlinear
adaptation of the multiple model adaptive estimator, in which each model represents a pos-
sible guidance law and corresponding guidance parameters of the attacking missile. Specific
limiting cases are carefully analyzed in which the attacking missile uses proportional nav-
igation, augmented proportional navigation, or optimal guidance law. Matched optimal
evasions from these specific cases are also derived and fit into the framework of the multi-
ple model adaptive control approach. For the limiting cases, an alternative reduced-order
approach is proposed to save computational resources. Considering noisy bearing-only
measurements, the performance of the proposed evasion concepts is compared through a
Monte Carlo simulation campaign to the scenario when the target has full knowledge about
the attacking missile and the relative state.
Nomenclature
[0] = matrix of zeros with indicated dimension
a(·)= lateral acceleration, m/s2
amax
(·)= maximal maneuvering capability, m/s2
A,B,C= state-space model of the linearized evasion problem
A(·),B(·),C(·),d(·)= state-space model of the entity’s dynamics
f= nonlinear equations of motion
H= measurement matrix
J(·)= cost function
k(·)= parameter used for linearization
N0= navigation gain
p= number of possible regimes/filters
P= state covariance matrix
r= range, m
R= measurement covariance matrix
sMT = switching function
S= covariance matrix of the innovation sequence
t,tgo,tf= time, time-to-go, and final time, respectively, s
Ts= sampling period
∗Postdoctoral Fellow, Department of Aerospace Engineering, robert.fonod@technion.ac.il.
†Associate Professor, Department of Aerospace Engineering, tal.shima@technion.ac.il. Associate Fellow AIAA.
1
V(·)= speed, m/s
v= measurement noise vector
u(·)= acceleration command, m/s2
x= state vector used for estimation
y= state vector of the linearized evasion problem
y(·)= entity’s internal dynamics state vector
zk= measurement vector at time tk
z1:k= measurement history up to time tk
X−O−Y= Cartesian reference frame
Z= zero-effort miss, m
α= ratio between the weight on the control effort and the miss distance
ξ= target-missile relative displacement normal to the initial LOS, m
γ(·)= flight-path angle, rad
Λj=jth regime conditioned likelihood function
µj= probability that the jth regime is correct
¯
µ= vector of posterior probabilities representing PN, APN, and OGL
νj= innovation sequence of the jth regime
σ2
(·)= measurement noise variance
θ= normalized time-to-go
λ= angle between the line of sight and the XIaxis, rad
%, ¯%= design parameters of the reduced-order approach
τ(·)= first-order time constant, s
Φ= transition matrix
N= Gaussian distribution
U= set of possible missile guidance laws
(˜
·)= approximated value
(ˆ
·)= estimated value
Subscripts
0= initial values
k= step of the discrete time tk
r, λ = along and normal to the line of sight
T, M, E = target, missile, and evasion, respectively
Superscripts
j=jth regime/filter
⊥= normal to the initial line of sight
∗= optimal solution
I. Introduction
Guidance laws for intercepting a moving target, such as aircraft in this study, have traditionally been
developed for one-on-one engagements. Usually, optimal control theory is applied, and perfect information
and linearized kinematics are assumed [1]. For example, proportional navigation (PN) is the optimal guidance
law for a scenario between an attacking missile with ideal dynamics and a non-maneuvering target [2]. The
same is true for augmented proportional navigation (APN), if the target performs a constant maneuver [3].
Taking into account first-order missile dynamics the well-known optimal guidance law (OGL) was obtained
by Cottrell [4]. In [5], this work was extended to arbitrary known target maneuvers. All these guidance
laws are linear and have the same general form of an effective navigation gain N0(constant or time-varying)
multiplied by a zero-effort-miss term and divided by time-to-go squared.
In order to respond to the threat from homing missiles employing such guidance laws, a significant effort
2
was made to extend the protection capabilities of targeted aircrafts. Among the systems developed for this
purpose are electronic countermeasures (e.g., jammers) and various kinds of decoys (chaff or flares). Besides
these means of protection, an aircraft can perform an evasive maneuver, which can be either arbitrary or
optimally adjusted against the incoming missile. In case of random maneuvers, two types are suggested:
random telegraph [6, 7] and periodic sine or square wave maneuver with a random phase, in a frequency
that is matched to the interceptor’s navigation gain and time constant [8, 9].
It is also possible to develop an optimal evasion strategy using optimal control theory tools, however
it requires information on the missile’s future behavior, i.e., its guidance law and guidance parameters.
Practically, it implies that the pilot is alerted that a missile of known type has been launched against his
aircraft. A case study where such a problem was formulated as a one-sided optimal control problem against
a PN-guided missile was presented in [10–12]. In these works some simplifying assumptions were applied,
such as two dimensional analysis and constant pursuer and evader speeds with bounded maneuverabilities.
Nonlinear engagement kinematics along with assumptions on first order missile dynamics were considered for
the problem formulation in [11]. A numerical solution was presented over a set of various initial engagements
conditions. In [12] a linearization around the collision course was made assuming ideal missile dynamics.
This study was extended in [10] to a nonlinear model. The resulting optimal evasion strategies obtained in
the above works were found to have a bang-bang structure, i.e. applying maximum available acceleration
normal to the line of sight (LOS) for a given period of time.
Although there has been substantial work in the literature on target evasion, most of the research con-
centrated on PN guided interceptor missiles, which leaves the strategies against other missile guidance laws
lacking. In a recent work of [13], optimal evasion strategies for a target aircraft from a homing missile em-
ploying a linear guidance law was derived. The problem was analyzed for arbitrary-order linear missile and
target dynamics, bounded target controls, and assuming perfect information. The underlying assumption in
this solution is that the missile’s guidance strategy is exactly known to the target. Furthermore, the lack of
appropriate guidance law and guidance parameter identification solutions greatly reduces the utility of the
available target-evasion strategies.
Motivated to enhance aircraft survivability in situations where the relative state and the missile strategy
are not exactly known, a multiple model adaptive evasion strategy is proposed which greatly relaxes the
assumptions made in [13]. It is assumed that the missile is chasing the target using one of a finite set of
linear guidance laws and guidance parameters, which is referred to as “regime” or “mode”. Each such regime
generates different missile acceleration commands and will therefore result in different missile trajectories.
The active regime has to be identified first. In this paper, an extended Kalman filter (EKF) based multiple
model adaptive estimator (MMAE) approach is used to identify the missile’s active guidance strategy. Similar
approach was used in [14] to design a cooperative multiple model guidance for an aircraft defending missile.
In the MMAE approach, each model in this scheme represents a possible guidance law and corresponding
guidance parameters of the attacking missile. The idea is to run a bank of filters in parallel, with each filter
matching a different possible regime. The estimation is then a weighted sum of the state estimates from
each individual filter in the bank, and the weights represent the probability of each regime being correct
based on the measurement history. The regime probabilities at each time step are updated based on Bayesian
inference, using the previous time step’s regime probabilities and the regime-conditioned likelihood of the new
measurement. For each considered regime, an optimal target evasion law is paired. Each optimal evasion
law is derived based on a linearized model, but implemented in the nonlinear setting. The final optimal
target maneuver is computed in a multiple model adaptive control (MMAC) framework using the posterior
probabilities of each regime being correct. In the MMAC approach, the estimation of each elementary filter is
fed into a “controller” (matched optimal evasion strategy in our case) matched to the filter’s specific regime
and the total scheme control command is then determined by the minimum mean square error (MMSE)
criterion or maximum a posteriori probability (MAP) criterion. Additionally, a set of “classical” missile
guidance laws such as PN, APN, and OGL, as well as the corresponding optimal evasion strategies from
these laws are presented. For these classical guidance guidance laws, an alternative reduced-order approach
is proposed to reduce the computational burden of the full-order MMAE. In this reduced-order approach,
the number of MMAE regimes is reduced to maximum three. Each regime represents one of the classical
guidance laws, i.e., PN, APN, or OGL. The corresponding guidance parameter is treated as an unknown
parameter that has to be estimated.
From observability point of view, if only LOS angle measurements are available, the quality of the
estimation of the relative state depends upon the intercept trajectory, which is reflected by the system
3
observability. For example, proportional navigation attempts to null the LOS rate and consequently range
and range-rate are not observable [15]. So, in case of bearing-only measurements, the range can be hardly
reconstructed from LOS measurements. This, in turn, might result in poor evasion performance. The
bearing-only target tracking problem with application to missile guidance has been largely studied in the
past [16–18]. Contrary, there exist only scarce open literature on guidance law and guidance parameter
identification using bearing-only measurements. Nevertheless, identifying a highly maneuverable missile in
the presence of missile maneuver uncertainty and noisy measurements is still a challenging problem and is
limited by the estimation performance. A solution to improve range observability is to maneuver away from
the collision triangle [19], this causes the LOS to rotate which then will give some insights on the relative
range. These principles are also incorporated into the proposed evasion strategy.
The remainder of this paper is organized as follows. The next section presents the mathematical models
of the missile-target engagement. The optimal target evasion strategy is presented in Sec. III, followed by
the derivation of a multiple model adaptive control based evasion strategy in Sec. IV. A comprehensive
performance analysis of the proposed approach is presented in Sec. VI, followed by concluding remarks.
II. Mathematical Models
This section presents the full nonlinear kinematics and dynamics equations of the missile-target evasion
problem, which will serve for analysis. Then, linearized equations, used for the derivation of the optimal
target evasion strategy, are presented. Measurement models and considered assumptions are also discussed.
A. Nonlinear Kinematics and Dynamics
The studied problem consists of two entities: an evading target aircraft and an attacking missile. Next, for
brevity, the target aircraft is referred as target and the attacking missile as missile. The engagement will be
analyzed in two-dimensions. In Figure 1 a schematic view of the planar point mass missile-target engagement
geometry is shown, where XI-OI-YIrepresents a Cartesian inertial reference frame. The missile and target
related variables are denoted by the subscripts M and T, respectively. The speed, acceleration, and flight-
path angles are denoted by V,a, and γ, respectively; the range between the missile and target is r, and λ
is the angle between the LOS and XIaxis.
OI
YI
XI
Missile
(M)
Target
(T)
λ
γM
γT
VT
VM
aT
aM
a⊥
M
a⊥
T
r
Figure 1: Planar missile-target engagement geometry
Neglecting the gravitational force, the engagement kinematics, expressed in a polar coordinate system
(r, λ)attached to the missile, is:
˙r=Vr,(1)
˙
λ=Vλ/r, (2)
4
where the relative velocities along and perpendicular to the LOS are
Vr=−VMcos(γM−λ)−VTcos(γT+λ),(3)
Vλ=−VMsin(γM−λ) + VTsin(γT+λ).(4)
The running time is denoted as t. The endgame initiates at t= 0 with ˙r(t= 0) <0and terminates at t=tf,
where
tf= arg
t
inf{r(t) ˙r(t)=0}, t > 0,(5)
allows to define the time-to-go by
tgo ,tf−t. (6)
At t=tf, the missile-target separation r(tf)is minimal and is often referred to as “miss distance”.
During the endgame, the missile and the target are assumed to move at a constant speed and to perform
lateral maneuvers only. Moreover, arbitrary-order linear missile and target dynamics are assumed, i.e.,
˙
yi=Aiyi+Biui
ai=Ciyi+diui
˙γi=ai/Vi
, i ={M, T },(7)
where yi∈Rniis the internal state vector of the ith entity’s dynamics, aiand uiare the ith entity’s
acceleration and acceleration command, respectively. It is also assumed that the target’s maneuver capability
is limited to |uT| ≤ amax
T. In Eq. (7), the term Ciyiis denoted as aiS and represents, if it exists, the part
of the acceleration with dynamics (for example, an angle of attack generating lift). The second part of the
acceleration, i.e. diui, represents the direct lift, which can be obtained immediately from deflection of the
steering mechanism such as the canard or tail (neglecting servo dynamics).
The missile and target accelerations normal to the LOS, routinely used in guidance logic, are denoted by
a⊥
Mand a⊥
T, respectively, satisfying
a⊥
M=aMcos(γM−λ),(8a)
a⊥
T=aTcos(γT+λ).(8b)
Note that in Eq. (7), and in the remainder of this paper, bold-italic is used to represent matrix or vector.
B. Linearized Equations of Motion
If during the endgame the missile and target deviations from the collision triangle are small, then the
linearization around the initial LOS is justified [1]. In Fig. 2, the linearized planar geometry and the
corresponding kinematics variables are depicted. The X-axis, aligned with the LOS used for linearization,
is denoted as LOS0. The relative displacement between the target and missile normal to this direction is ξ.
Under linearization assumption, a⊥
Mand a⊥
Tare approximated by
a⊥
M≈kMaM, kM= cos(γM0−λ0),(9a)
a⊥
T≈kTaT, kT= cos(γT0+λ0),(9b)
where the subscript ”0” denotes the initial value around which linearization has been performed. It is
assumed that the missile is launched in a colison course, i.e., that |γM0−λ0|< π/2and |γT0+λ0|< π/2.
During the endgame, the missile and target are assumed to move at a constant speed. Thus, once a
collision triangle is reached and maintained, the speed Vris constant and the interception time tfcan be
assumed fixed and approximated by
˜
tf≈ −r0/Vr(10)
Let us define the state vector yof the linearized missile-target evasion problem as
y=hy1y2yT
MyT
TiT,hξ˙
ξyT
MyT
TiT
.
5
OI
YI
XI
Missile
(M)
Target
(T)
λ
γM
γT
VT
VM
aT
aM
Y
X
λ0
y
a⊥
M
a⊥
T
λ0
LOS0
r
reference
Predicted collision point
Figure 2: Linearized planar missile-target engagement geometry
Then, the missile-target equations of relative motion normal to LOS0can be expressed as
˙y1=y2
˙y2=kTaT−kMaM
˙
yM=AMyM+BMuM
˙
yT=ATyT+BTuT
(11)
Using Eqs. (7) and (9), the above equations can be rewritten into a matrix form as
˙
y=Ay +Bu⊥
T+Cu⊥
M,(12)
A=
0 1 [0]1×nM[0]1×nT
0 0 −kMCMkTCT
[0]nM×1[0]nM×1AM[0]nM×nT
[0]nT×1[0]nT×1[0]nT×nMAT
,B=
0
dT
[0]nM×1
k−1
TBT
,C=
0
−dM
k−1
MBM
[0]nT×1
,
where [0] denotes a matrix of zeros with a given dimension, u⊥
Mand u⊥
Tbeing, respectively, the missile and
target acceleration commands normal to LOS0. Note that the command u⊥
i, i ∈ {M, T }is related to ui
analogously as a⊥
iis related to ai, see Eqs. (8) and (9).
C. Measurement Model
The target is assumed to be equipped with an electro-optic seeker and/or a radar. Thus, one may measure:
a)both rand λ, or b)only λ, i.e., bearing-only measurement. The discrete-time measurement vector
zk∈Rnzis assumed to be acquired at a given sampling time Tm
sand being corrupted by a zero-mean
mutually independent white Gaussian noise sequence vk∈Rnz. The measurement model, when all possible
measurements are available, is
zk=Hxk+vk="rk
λk#+vk,(13)
where
vk∼ N ([0]nz×1,R),R=diag(σ2
r, σ2
λ),
xkis the relevant state vector (being defined later) at discrete time tk, and His the appropriate measurement
matrix. If rkis not available, the appropriate row is eliminated from Hand the appropriate row and column
are eliminated from R, respectively.
6
III. Optimal Target Evasion
In this section, the nonlinear implementation of the optimal target evasion strategy from a missile em-
ploying a linear guidance law is presented.
A. Missile Guidance Law
In this paper, a family of linear guidance laws, commonly derived under the assumption of linear kinematics,
perfect information, and unbounded controls, is considered [1]. This common practice resulted in a class
of guidance laws, which all have the same linear form as a function of the missile-target linear state yand
eventually the target’s control u⊥
T, i.e.,
u⊥
M=K(tgo)y+KuT(tg o)u⊥
T,(14)
where
K(tgo) = hK1K2KMKTi.
Note that Eq. (14) represents a wide variety of linear guidance laws. In the next two sections, a set of
“classical” missile guidance laws such as PN, APN, and OGL as well as the corresponding optimal evasion
strategies from these laws will be presented.
Remark 1. In Eq. (14), it is assumed that the missile’s current controller may be dependent on the target’s
current controller. Actually, if the missile uses an optimal-control based guidance law, then the underlying
assumption in its derivation was that the target’s controller is known not just at the current time but also
from the current time until the end of the scenario [13].
B. Optimal Evasion Problem - Linear Setting
Using the equations of motion (EOM) of the linearized missile-target engagement, see Eq. (12), together
with the general form of the linear missile guidance law, see Eq. (14), the EOM of the one-sided evasion
problem are obtained
˙
y=AE(tgo)y+BE(tg o)u⊥
T,(15)
where
AE(tgo) =
0 1 [0]1×nM[0]1×nT
−dMK1−dMK2−kMCM−dMKMkTCT−dMKT
k−1
MBMK1k−1
MBMK2AM+k−1
MBMKMk−1
MBMKT
[0]nT×1[0]nT×1[0]nT×nMAT
,
BE(tgo) = h0dT−dMKuTk−1
MBT
MKuTk−1
TBT
TiT
.
Based on Eq. (15), the optimal target evasion strategy from Eq. (14) has been derived in [13] and is briefly
recalled in the following theorem.
Theorem 1.The optimal target evasion strategy from a homing missile employing a linear guidance law of
the form of Eq. (14), maximizing the following cost function
JE=y2
1(tf)/2(16)
subject to the EOM of of Eq. (15) and under the constraint that the target’s control |uT| ≤ amax
Tis bounded,
is given by
u⊥∗
T=a⊥max
Tsign(sMT )sign(ZM T ), ZMT (0) 6= 0,(17)
where sMT is the switching function and ZM T is the well-known zero-effort-miss (ZEM) distance given by,
respectively,
sMT =DEΦE(tf, t)BE(tg o),(18)
ZMT =DEΦE(tf, t)y,(19)
7
and DEbeing a constant vector
DE=h1 0 [0]1×nM[0]1×nTi.(20)
The transition matrix ΦE(tf, t), associated with the homogenous solution of Eq. (15), having the structure
ΦE(tf, t) = ΦE(tgo) =
Φ11 Φ12 Φ1MΦ1T
Φ21 Φ22 Φ2MΦ2T
Φ31 Φ32 Φ3MΦ3T
Φ41 Φ42 Φ4MΦ4T
,(21)
satisfies ˙
ΦE(tf, t) = −ΦE(tf, t)AE(tgo),ΦE(tf, tf) = I.(22)
In Eq. (17), a⊥max
Tis a projection of the target’s maximum lateral acceleration amax
Tin the direction per-
pendicular to LOS0, and is given by: a⊥max
T=kTamax
T.
Proof. The proof of Theorem 1 can be found in [13].
Remark 2. If the engagement is initialized in the singular region, i.e., ZM T (0) = 0, or if ZMT (t)=0, t > 0,
then it was suggested in [13] that the optimal evasion strategy u⊥∗
Tshould be chosen as either a⊥max
Tor
−a⊥max
T. In fact, if the missile employs OGL homing strategy with α= 0 (for more details about α, see the
discussion in Sec. IVB), then the target has no chance to escape and thus ZM T (t)=0for any t.
It was also shown in [13] that if the optimal evasion strategy u⊥∗
Tis used within the linear setting, the
optimal ZEM dynamics, i.e., ˙
Z∗
MT , is governed by
˙
Z∗
MT =sign(ZM T )|sMT |a⊥max
T.(23)
Consequently, |ZM T |is a monotonically increasing function of time, satisfying ∀t∈[0, tf]
sign(ZMT (t)) = sign(ZM T (tf)), ZMT (0) 6= 0.(24)
Defining m=Rtf
0|sMT |a⊥max
Tdt, the expected (under the assumption of linearity) miss distance is obtained
as follows
Miss(exp)=|y1(tf)|=|ZMT (tf)|=|ZM T (0)|+m. (25)
The consequence of Eq. (24) is that ZMT does not change its sign during the engagement. Therefore, if
ZMT (0) 6= 0 is known, the only parameter needed for the computation of the optimal target control u⊥∗
Tis
the time-to-go tgo that is used to evaluate the sign of switch function sMT (tgo ). However, this is only valid
if the linearization assumptions hold.
Remark 3. The optimal target evasion strategy of Eq. (17) was derived under the assumption of unbounded
missile control command u⊥
M. It was shown in [12] that if the missile acceleration a⊥
Mor u⊥
Mare bounded,
then in order to exploit the missile’s saturation the target maneuver switches should occur earlier in the
engagement.
C. Nonlinear Implementation
Based on Theorem 1, the target’s optimal evasion strategy is a closed-loop guidance law in which the target
needs to compute sMT (tg o)and ZM T (t). These variables are in general functions of the time-to-go tgo and
the linear state y. To implement such evasion strategy within the nonlinear setting, u⊥∗
Tmust be projected
in the direction normal to the target’s velocity vector, denoted as u∗
T. Assuming small deviations from the
collision triangle leads to kT/cos(γT+λ)≈1, hence u∗
Tcan be approximated as
u∗
T=u⊥∗
T
cos(γT+λ)≈amax
Tsign(sMT )sign(ZM T ), ZMT (0) 6= 0.(26)
Once a collision triangle is reached and maintained, the speed Vris constant and the approximation of
the interception time ˜
tf, given by Eq. (10), can be assumed fixed throughout the engagement. Therefore,
the relevant components of ΦE(˜
tf, t), needed to determine sMT (tg o), can be precomputed off-line.
8
Note that ZMT , given by Eq. (19), is a function of the linear state y. The displacement ξnormal to
LOS0can be expressed at any time by
ξ=rsin(˜
λ),(27)
where ˜
λ=λ−λ0. Differentiating Eq. (27) with respect to time, yields
˙
ξ=Vrsin(˜
λ) + Vλcos(˜
λ).(28)
Using the above expressions, Eqs. (27) and (28), for the computation of ZMT replaces the dependency on ξ
and ˙
ξby the kinematics variables Vr,Vλ,r, and λ. Moreover, assuming small deviations from the collision
triangle, thus λ−λ0≈0, the relative displacement ξcan be assumed zero and the time derivative ˙
ξcan be
approximated as ˙
ξ=Vλ.
Remark 4. As outlined previously, |ZMT |is monotonically increasing in time only if the linearity assump-
tions hold. If these assumptions are violated or if uncertain states are used to compute ZMT , then ZM T might
change its sign during the evasion. Such phenomena might result in unnecessary switches of u∗
T, which con-
sequently might lead to undesired reduction in the resulting miss distance. This phenomena is obviously not
appreciated and a workaround will be discussed in Sec. VD.
Remark 5. It is obvious that if the collision triangle changes its geometry throughout the engagement
(requiring relinearization), the transition matrix ΦE(tg o), used to determine the optimal switches sMT (tgo)
and the ZEM distance ZMT (t), is not necessary the optimal transition matrix. Thus, one may consider to
recompute ΦE(tgo )every time when a significant change in the linearized collision geometry is observed.
IV. MMAE for Missile Identification
The underlying assumption of Theorem 1 is that the missile’s guidance strategy is exactly known to the
target. In this section, based on Magill’s pioneering work [20], a multiple model online identification scheme
is proposed to identify the employed missile guidance law and all the parameters (relative state and time-
to-go) required for the proper implementation of the evasion strategy given in Sec. V. The set of “classical”
missile guidance laws, used to validate the proposed evasion strategy in Sec. VI, is also presented. Finally,
an alternative reduced-order approach is proposed to save computational resources.
A. Multiple Model Adaptive Estimation
In the multiple model estimation approach, it is assumed that the system operates in one of a finite number
of models. The operating model is often called as mode or regime of the system. If the regime that the
system obeys is fixed, that is, no switching from one mode to another occurs during the estimation process,
then the static multiple model estimator, also known as the MMAE approach, is considered [20, 21]. While
the model that is in effect stays fixed, each model has its own dynamics, so the overall estimator is dynamic.
Let’s assume that the guidance strategy uMof the missile, being fixed throughout the engagement, is
one of ppossible ones (the system operates in one of pregimes)
uM∈ U ={u1
M, . . . , up
M}.(29)
Each guidance strategy uj
Mis characterized by a set of five parameters {Kj
1, Kj
2,Kj
M,Kj
T, Kj
uT}. These
parameters exclusively define the jth regime and can be functions of tgo. Each such regime will generate
different missile acceleration commands uM, defined by the linear guidance law of Eq. (14), and therefore will
result in different missile trajectories. If the target-related parameters such as yT,γT, and VTare assumed
to be known to a very high accuracy (via some navigation system), then, based on the constant missile speed
assumption, the jth regime dynamics is governed by the following set of nonlinear equations
˙r=Vr
˙
λ=Vλ/r
˙
yM=AMyM+BMuj
M
˙γM= (CMyM+dMuj
M)/VM
˙
VM= 0
,(30)
9
where Vrand Vλare given by Eqs. (3) and (4), respectively, and uj
Mis the missile’s acceleration command
obeying the jth regime defined using Eqs. (14),(27),(28) and (8a) as
uj
M=Kj
1rsin(˜
λ) + Kj
2Vrsin(˜
λ) + Vλcos(˜
λ)+Kj
MyM+Kj
TyT+Kj
uTu⊥
T
cos(γM−λ).(31)
The discrete-time version of Eq. (30), used for MMAE design, can be compactly rewritten as
xk=fj
k−1(xk−1, u⊥
T),(32)
where xk,[r, λ, yT
M, γM, VM]Tis the state vector at time tk=kTs, Ts>0used for estimation, fj
k−1is a
vector function derived by integrating of Eq. (30) from tk−1to tk, and jis the particular regime.
Remark 6. In the above model, an assumption has been made that the parameters of the missile dynamics,
i.e., AM,BM,CM, and dM, are known exactly. If this is not the case, the missile dynamics can be
approximated by e.g., a first-order strictly proper dynamics, i.e., AM=−1/τM,BM= 1/τM,CM= 1,
dM= 0, and the uncertainty on τMcan be treated in two different ways: a) several values of τMcan be
represented as different regimes and the problem can be addressed in the same way as the guidance law
uncertainty is addressed, or b) τMcan be added as a constant state in Eq. (30) similarly as VM.
As the engagement model in Eq. (30) is nonlinear, a mode-matched EKF is used to calculate the state
estimate, ˆ
xj, and the associated regime probability, µj, assuming the jth regime being correct, see [14] for
more details. Note that other nonlinear filtering techniques, such as particle filter [22] or unscented Kalman
filter [23], can be utilized for this purpose as well. The main idea is to design and run in parallel a bank of
pfilters, each matching a different regime j. Then, the regime probabilities at each time step are calculated
based on Bayesian inference, using the previous time step’s regime probabilities (weights) and the mode-
conditioned likelihood of the new measurement. Starting with the initial probability µj
0that uj
Mis correct
(missile employs uj
M), i.e.,
Prob{uj
M|Z0}=µj
0,∀j∈ {1, . . . , p},(33)
where Z0is some information known a priori, Pp
j=1 µj
0= 1 since the correct law is among the assumed
ppossible ones, then using Bayes’ rule, the posterior probability, given the measurement data z1:k,
{zi;i= 1, . . . , k}up to time k µj
k, is given by the recursion [21]
µj
k,Prob{uj
M|z1:k}=p(zk|z1:k−1, uj
M)Prob{uj
M|z1:k−1}
p(zk|z1:k−1).(34)
Using the initial probabilities of Eq. (33) and applying the total probability of Eq. (34) results in the following
weight update formula for time k
µj
k=Λj
kµj
k−1
Pnr
i=1 Λi
kµi
k−1
,∀j∈ {1, . . . , p},(35)
where Λj
k,p(zk|z1:k−1, uj
M)is the jth regime-conditioned likelihood function calculated using the jth filter
innovations process statistics. Under the linear-Gaussian assumptions, Λj
kis Gaussian and given by
Λj
k=p(νj
k) = N(νj
k; [0]nz×1,Sj
k),(36)
where νj
kand Sk
kare the innovation and its covariance from the jth mode-matched filter. It is obvious that
µj
k≥0and that Pp
i=1 µi
k= 1. In a nonlinear and/or non-Gaussian setting, Gaussian likelihood functions
are used, although they are clearly approximations [21].
Remark 7. The fixed missile guidance law assumption, i.e., that the missile does not switch between guidance
laws throughout the engagement, is a very realistic assumption for the endgame. If this assumption does not
hold, a dynamic multiple model estimator can be derived based on the interacting multiple model (IMM)
approach [18, 21]. The IMM allows transitions between regimes, but the probabilities of these transitions are
needed to be known.
10
B. MMAE for Classical Missile Guidance Laws
Among the large family of linear guidance laws, special attention is given here to the most well-known
guidance laws of PN [2], APN [3], and OGL [4]. These guidance laws are more widely known in the following
nonlinear form: (rather than in the form of Eq. (31))
ui
M=N0
i
Zi
t2
go cos(γM−λ), i ∈ {PN,APN,OGL},(37)
where N0is the effective navigation gain and Zis the missile’s ZEM distance. The expression for the ZEM
distance is different for each guidance law, as it is dependent on the model used and assumptions made
regarding the future target maneuvers.
Remark 8. In the above context, the ZEM represents the missile-target separation normal to LOS0at the
final time tf, i.e., y1(tf) = ξ(tf), which will be obtained if the attacking missile does not apply any control
from the current time onward and the target aircraft continues employing the expected maneuver strategy
that is known to the missile.
•PN-Guided Missile
Under the assumptions of ideal missile dynamics and no target maneuver (i.e., u⊥
T(t) = 0,∀t≥0), the
obtained missile guidance law guaranteeing zero miss distance is PN with
ZP N =−Vrt2
go ˙
λ, (38)
where ˙
λis given by Eq. (2). If N0
P N = 3, this guidance law minimizes the control effort.
•APN-Guided Missile
Extending the results to the case in which the target is assumed to perform a constant maneuver (i.e.,
u⊥
T(t) = const.,∀t≥0), APN was obtained with
ZAP N =ZP N +aTcos(γT+λ)t2
go/2.(39)
•OGL-Guided Missile
Additionally assuming that the missile’s closed-loop acceleration dynamics can be approximated by a first-
order strictly proper transfer with a time constant τM, OGL was obtained with
ZOGL =ZAP N −τ2
Mδ(θ)aMS cos(γM−λ),(40)
where the normalized time-to-go θand the function δ(θ)being
θ=tgo/τM,(41)
δ(θ) = e−θ+θ−1.(42)
The navigation gains of PN and APN are constant, whereas that of OGL is time-varying, i.e.,
N0
OGL(θ) = 6θ2δ(θ)
3+6θ−6θ2+ 2θ3−3e−2θ−12θe−θ+ 6α/τ3
M
,(43)
where αrepresents the ratio between the weights on the control effort (integral of the acceleration command
squared) and the miss distance in the quadratic cost function JOGL used in the OGL formulation:
JOGL =b
2y2
1(tf) + 1
2
tf
Z
0
(uMcos(γM0−λ0))2dt, b ,1/α. (44)
Note, letting α→0yields to a perfect intercept requirement. If the missile has a first-order strictly
proper dynamics and if the missile will be provided perfect (or even relatively precise) measurements on
11
the relative state, then the target has no chance to escape, no matter what maneuver it applies or what
the initial conditions are. This is because OGL was designed taking into account such missile dynamics
and assuming unbounded control [13]. In such a case, the aircraft survivability can be enhanced only by
denying the missile such “perfect” information or by using a defender missile to intercept the incoming thread
[13, 14, 24]. Therefore, the case when α= 0 is out of scope of this paper.
In order to fit the classical guidance laws, introduced earlier, into the MMAE scheme of Sec. IVA, it is
required to define a set Ucof possible regimes, as in Eq. (29), as follows
Uc=nuP N
MN0
P N ∈¯
N0
P N , uAP N
MN0
AP N ∈¯
N0
AP N , uOGL
M(α∈¯
α)o,(45)
where ¯
N0
P N ={N0(1)
P N , . . . , N 0(p1)
P N },¯
N0
AP N ={N0(1)
AP N , . . . , N 0(p2)
AP N },¯
α={α(1), . . . , α(p3)}
represent the sets of considered navigation gains for PN and APN and the set of αparameters for OGL.
Overall there are p=P3
i=1 piregimes. For each regime it is necessary to construct the mode-matched EKF
and run all the pfilters in parallel.
C. Reduced Order MMAE for Classical Missile Guidance Laws
It is obvious that if the total number of considered regimes pis too large, it might pose some severe com-
putational challenges. Identification of the above presented classical missile guidance laws requires the
identification of the ZEM (either ZP N ,ZAP N , or ZOGL ) and the navigation gain (either N0
P N ,N0
AP N , or
N0
OGL). With regards to the navigation gain, in the case of PN and APN the requirement is to identify
its constant value, whereas for OGL the requirement is to identify α. This special structure of the classical
guidance laws, see Eq. (37), allows to reduce the computational burden of the MMAE algorithm by reducing
the total number of regimes to p= 3. Thus, a reduced set Ur
cof possible regimes is defined as follows
Ur
c=uP N
M(N0
P N ), uAP N
M(N0
AP N ), uOGL
M(α),(46)
where the guidance parameters N0
P N ,N0
AP N , and αare treated as unknown system parameters that need
to be estimated. This can be easily incorporated into the existing MMAE scheme of Sec. IVA by adding,
respectively, N0
P N ,N0
AP N , and αas a constant state (similarly as the constant missile speed VM, i.e., ˙
VM= 0)
into the EOM given by Eq. (30) for each regime. For more details on parameter estimation see e.g., [21].
V. Multiple Model Adaptive Evasion
In this section, the proposed MMAE scheme of Sec. IV and the optimal target evasion strategy of Sec. III
are combined in a MMAC (multiple model adaptive control) configuration, followed by the derivation of
specific optimal target evasions from the classical missile guidance laws. Finally, the MMAC adaptation to
the reduced-order problem of Sec. IVC is given and some implementation issues are discussed.
A. MMAC-based Target Evasion
In the MMAC approach [25], the state estimate of each elementary filter is fed into the “controller” which
is paired with the filter’s specific regime. In our case, the paired “controller” corresponds to the optimal
evasion strategy matched to the filter’s particular regime. This framework fits our missile-target evasion
problem as each regime jin the MMAE approach of Sec. IVA corresponds to a known missile strategy uj
M,
thus an optimal evasion strategy u∗j
T, matching uj
M, can be easily derived based on developments presented
in Sec. III. Finally, the total target control command uTis determined in the MMAC sense by one of
the following approaches: a)MMSE - minimum mean square error, or b)MAP - maximum a posteriori
probability.
In the MMSE approach, uTis a weighted average of controls from each filter-matched controller in the
bank. The weighting is based on filter-matched controller’s posterior probabilities µj
k, j = 1, . . . , p. The
target acceleration command at time step kcan therefore be calculated as
u(mmse)
T=
p
X
j=1
µj
ku∗j
T,(47)
12
where
u∗j
T=amax
Tsign(sj
MT )sign(Zj
MT ),(48)
sj
MT and Zj
MT represent, respectively, the switching function of Eq. (18) and the ZEM distance of Eq. (19)
of the jth regime, both evaluated using the jth regime-conditioned state estimate ˆ
xj
k|kat time step k.
In the MAP criterion sense, uTis determined as the control associated with the maximum a posteriori
probability, i.e.,
u(map)
T=u∗j
T, j = argmax
i∈{1,...,p}
(µi
k).(49)
Note that unlike the MMAE approach, the MMAC approach is heuristic (an approximation), but seems
to yield good performance when a controller can be matched to each possible regime [14, 26].
B. Optimal Evasion from Classical Missile Guidance Laws
Here, specific optimal target evasion strategies, matched against the classical missile guidance laws of PN,
APN and OGL, are derived. The derivation can be extended to other missile guidance laws using the same
formulation and similar derivation steps, unless they comply with the linear form of Eq. (14).
First, uMof Eq. (37) need to be expressed in the general linear form u⊥
Mof Eq. (14) which was used in
the derivation of the optimal evasion strategy in Theorem 1. Assuming small deviations from the collision
triangle, the displacement ξnormal to LOS0can be approximated by
ξ≈r(λ−λ0)(50)
Differentiating Eq. (50) with respect to time yields
ξ+˙
ξtgo =−Vrt2
go ˙
λ. (51)
It can be seen that the right hand side of Eq. (51) is actually identical with the expression for ZPN in
Eq. (38). Armed with the above expression, K(tgo)=[K1K2KMKT]and KuTcan be easily defined
for PN, APN, and OGL-guided missiles as follows:
•Evasion from PN-Guided Missile
K1=N0
P N
t2
go , K2=N0
P N
tgo ,KM= [0]1×nM,KT= [0]1×nT, KuT= 0.
•Evasion from APN-Guided Missile
K1=N0
AP N
t2
go , K2=N0
AP N
tgo ,KM= [0]1×nM,KT=kT
N0
AP N CT
2, KuT=N0
AP N dT
2.
•Evasion from OGL-Guided Missile
K1=N0
OGL(θ)
t2
go , K2=N0
OGL(θ)
tgo , KuT=N0
OGL(θ)dT
2,
KM=−kM
N0
OGL(θ)δ(θ)CM
θ2,KT=kT
N0
OGL(θ)CT
2.
Now, for all uj
M∈ Uc(or uj
M∈ Ur
c, if the guidance parameter N0or αis known/identified), the optimal
switching function sj
MT and the ZEM distance function Zj
MT can be determined by solving the following set
of differential equations in reverse time:
dΦ11
dtgo
=Φ1Mk−1
MBMK1−Φ12dMK1,Φ11 (0) = 1
dΦ12
dtgo
=Φ1Mk−1
MBMK2+ Φ11 −Φ12dMK2,Φ12 (0) = 0
dΦ1M
dtgo
=Φ1M(AM+k−1
MBMKM)−Φ12(kMCM+dMKM),Φ1M(0) = [0]1×nM
dΦ1T
dtgo
=Φ1Mk−1
MBMKT+Φ1TAT+ Φ12(kTCT−dMKT),Φ1T(0) = [0]1×nT.
(52)
13
It can be noticed that in the above cases K2=K1tgo and that the first two equations of Eq. (52) are
independent from the last one. This allows the first two solutions of Eq. (52) being related as
Φ12 = Φ11tgo ,(53)
and thus reduce the number of equations needed to be solved in Eq. (52) to three. For all three classical
guidance laws, the expression for ZMT has the same form and is given by
ZMT =−Φ11 Vrt2
go ˙
λ+Φ1MyM+Φ1TyT.(54)
The switching function given in Eq. (18), for arbitrary order dynamics, yields
sMT = Φ12 (dT−dMKuT) + Φ1Mk−1
MBMKuT+Φ1Tk−1
TBT.(55)
Finally, for each classical guidance law uj
M∈ Uc, an optimal (matched) evasion strategy u∗j
Tof the form of
Eq. (48) can be obtained using Eqs. (54) and (55), together with the corresponding solution Φj
Eto Eq. (52)
matching the particular regime. It is important to note that sM T is purely a function of tgo . Thus, more
accurate tgo estimation yields to more precise timing of the required switches.
Remark 9. The set of equations in Eq. (52) can be solved numerically, see [27] for more details. For special
cases in which it is assumed that the target has ideal dynamics and the missile has first-order strictly proper
dynamics, closed-form solutions have been derived in [12, 13] for PN and APN-guided missile with integer
valued navigation constant N0.
C. Target Evasion Based on Reduced-order MMAE
It is obvious that the reduced-order MMAE approach proposed in Sec. IVC cannot be used within the
MMAC setting as the three regimes defined in Eq. (46) are functions of the unknown parameters N0
P N ,
N0
AP N , and α. Thus, a matched (optimal) evasion u∗j
Tagainst any uj
M∈ Ur
ccan be only computed when the
corresponding parameter (N0
P N ,N0
AP N , or α) has been identified. This leads to the following reduced-order
evasion strategy:
u(red)
T=
%amax
Tif µj
k6= 1,∀j∈ {1. . . p}OR tgo ≥20 ×τmax
M,
¯%amax
Tsign(sj
MT )otherwise,
(56)
where sj
MT is the switching function of Eq. (55) corresponding to the identified guidance law, i.e., PN, APN,
or OGL, computed as in Sec. VB, using the respective estimated parameter N0
P N ,N0
AP N , or α, taken at
time step when the “if” condition in Eq. (56) was first time violated. The design parameter %can be chosen
arbitrarily as ±1and defines the target’s initial maneuver direction, τmax
Mstands for the largest time constant
of the missile dynamics, and ¯%is defined as
¯%=%sign(sj
MT (tg o >20 ×τmax
M)) (57)
to ensure that, if the guidance law jhas been identified early enough, the resulting evasion strategy will
follow the switches dictated entirely by sj
MT . The condition 20 ×τmax
Mis enforced in order to allow extra
time to the estimation scheme to converge to the correct parameter N0
P N ,N0
AP N , or α. This is reasonable
as the target switches occur at the very end of the engagement.
Note that any target maneuver direction switch (switches) occurring earlier than ∼20 ×τmax
Mhas
a negligible effect on the resulting miss distance, but it might have a severe impact on the estimation
performance, especially when bearing-only measurements are considered.
D. Implementation Issues
For practical implementation of the target’s evasion strategy (as well as missile’s guidance law), the time-
to-go tgo, defined in Eq. (6), is commonly approximated as
˜
tgo ≈ −r/Vr, Vr<0.(58)
14
It has to be noted that observability in engagements with bearing-only measurements is a crucial factor
in any guidance loop [19, 28]. If collision course conditions hold, i.e., the missile and target stay on the LOS,
the range cannot be reconstructed from bearing-only measurements. A poor range estimation might result
in inappropriate timing of the optimal switches dictated by sMT and thus in a poor evasion performance.
The only way how to improve ˜
tgo accuracy is to improve range observability. Maneuvering away from the
collision triangle, i.e., forcing the collision triangle to rotate, can improve the performance of the estimation
process because, by altering the line of sight, the bearing measurement will return some insights on the
relative range [19].
As outlined in Remark 4, ZM T (t)might cause chattering of u∗
T. This chattering might happen when
ZMT (0) = 0 and uncertain states are used to compute ZM T (practical implementation). Such unwanted
chattering of uT(non-optimal bang-bang maneuvers) leads to a non-rotating collision triangle, thus poor
range estimate and reduction of the achievable miss distance. A workaround to solve this problem is to use
a dead-zone-like function applied on the sign of ZMT (t). By this, if
|ZMT (t)|< , (59)
i.e., the ZEM distance is smaller than some prescribed value , the target will not change the direction of its
maneuver command, unless the sign of sMT (tgo)is changed or |ZMT (t)| ≥ . Note that in the linear setting,
|ZMT (t)|is a monotonically increasing function of time, see Eq. (24). Thus, by letting → ∞, it is actually
assumed that the monotonically increasing property also holds for the nonlinear engagement. In that case,
the nonlinear evasion strategy of Eq. (26) reduces to
u∗
T≈amax
Tsign(sMT )sign(ZM T (0)), ZMT (0) 6= 0.(60)
If ZMT (0) = 0, the evasion strategy for u∗
T(0) can be chosen arbitrarily as either umax
Tor −umax
T, and
sign(ZMT (0)) ,sign(u∗
T(0)) should be assumed fixed in Eq. (60) throughout the engagement.
Let us now consider the above discussions within the MMAC configuration. Each “controller” in the
bank has its own switching function sj
MT entirely defined by Φj
E, pre-computed off-line using the uj
Mbeing
correct assumption. Each sj
MT is evaluated based on its own tgo estimate, which is calculated using Eq. (58)
and ˆ
xj
k|k. To avoid chattering when Zj
MT (0) ≈0for all j∈ {1, . . . , p}(this can happen when all the
filters in the bank are initialized with the same initial guess. Then, after the first measurement update,
Zj
MT (0) = Zi
MT (0),∀i, j ∈ {1, . . . , p}since z0is the same for all filters), u∗j
Tshould be initialized with the
same value (umax
Tor −umax
T) for all j. By this, uTcomputed using MMSE or MAP approach will not be
affected by the initial transitions (convergence) of the probabilities µj
k. The probabilities will not play any
role until the first switch occurs in any sj
MT . Overall, this implementation ensures that the resulting target
command uT, computed by Eq. (47) or Eq. (49), will have a bang-bang structure and also it ensures that,
in order to enhance observability, the best option is to maneuver away from the collision triangle.
The reduced-order approach, given in Sec. VC, is designed such that it is not affected by initial chattering.
However the estimation problem is more challenging as there is an additional uncertainty on the guidance
parameters, i.e., on N0
P N ,N0
AP N , and α, respectively. Therefore, one must ensure that the estimated
guidance parameters, along with the estimated state and tgo approximation, have reasonable accuracy. A
way to reduce noise in the guidance parameters is to apply a moving average on them.
VI. Performance Analysis
In this section, the set of “classical” missile guidance laws, presented in Sec. IVB, is used to demonstrate
the performance of the proposed evasion concepts throughout numerical simulations. First, the simulation
environment together with the interception scenario are presented, followed by a sample run analysis. Then,
using Monte Carlo (MC) simulations, the estimation performance is analyzed in open loop and the miss
distance is evaluated in closed loop.
A. Simulation Environment and Scenario
The planar nonlinear kinematics, missile and target dynamics, presented in Sec. A, are used for analysis.
All engagements are initiated at a horizontal separation of 5 km between the missile and the target, thus
r0= 5000 m and λ0= 0 rad. Both missile and target have constant speed. The target’s speed is VT= 300
15
m/s and the missile’s speed is VM= 600 m/s. For the analysis, it is assumed that the missile and the target
have first-order strictly proper dynamics with time constants τM= 0.2s and τT= 0.5s. Hence, matrices
in Eq. (7) degenerate to Ai=−1/τi,Bi= 1/τi,Ci= 1, and di= 0,i∈ {M, T }. The target’s maneuver
capability is limited to amax
T= 10 g. No saturation is applied on the missile acceleration command uM.
The EOM of the missile-target engagement are solved using a fourth-order Runge-Kutta (4RK) algorithm.
The stopping criterion for the simulation is when Vrchanges its sign (i.e., when Vr≥0). To ensure precise
miss distance evaluation, high resolution integration is performed when the range is at range less than
r < 25∆t(VM+VT), where ∆t > 0is the nominal integration step. All studied engagements start with the
missile and the target on a perfect collision triangle. Given the target’s initial flight path angle γT0, the
missile’s initial flight path angle γM0is determined such that the initial collision conditions hold, i.e.,
VMsin(γM0−λ0)−VTsin(γT0+λ0)=0.(61)
The missile is assumed to have perfect information on the relative state, its own and target’s parameters,
respectively, and being guided towards the target using one of the following guidance laws: PN, APN or OGL,
with navigation gains ¯
N0
i∈ {3,3.5,4,4.5,5}for i={PN,APN}and weights ¯
α∈ {0.0001,0.001}for OGL.
The MMAE regimes of Eq. (45) are matched exactly to the above missile guidance laws and parameters, i.e.,
the navigation gains of PN and APN are represented by five regimes each, whereas for OGL two regimes are
considered. Thus, in total, p= 12 EKFs are required to be run in parallel. In the case of the reduced-order
MMAE, only p= 3 filters are designed and run, matching the above guidance laws of PN, APN and OGL,
however without good knowledge about the respective guidance parameters. The prior probability of each
guidance law is 1/3and the initial probability of each regime is equal within the respective guidance law,
i.e., µi
0= 1/15, i = 1,...,10 and µ11
0=µ12
0= 1/6.
The estimation problem formulated in Sec. IV address two possible sensor choice combinations, see
Sec. IIC. However, in our simulations, only the more difficult case is considered in which the target has
bearing-only measurements. The states needed for the proposed evasion strategy employment are estimated
at a frequency of 200 Hz (i.e., Ts= 1/200 s). The nonlinear EOM of Eq. (32) are propagated using the
4RK algorithm. The measurements are acquired at a sampling frequency of 50 Hz (i.e., Tm
s= 1/50 s).
The simulated measurement noises are with σλ= 1 mrad. The tuning parameters of the EKFs have been
chosen by numerical simulations. A blind range of 50 m is assumed. If r < 50, no new measurements are
acquired and the EKFs work in open loop, i.e., perform only the time propagation step at a given rate Ts. All
filters in the bank are initialized with the same initial conditions sampled from a Gaussian distribution, i.e.,
ˆ
x0|0∼ N (x0,P0|0),where x0is the true state vector and P0|0=diag 502,(3π/180)2,(1g)2,(3π/180)2,502
is the initial covariance matrix of the filters. For the reduced-order MMSE approach, the covariance matrix
P0|0is augmented for each EKF with an additional diagonal element of the value of 52,52, and 0.0082,
corresponding to the uncertainty on the guidance parameter N0
P N ,N0
AP N , and α, respectively. After the
filters are initialized, they run recursively on their own estimates. Their likelihood functions are used to
update the mode probabilities. The latest mode probabilities are used to compute the target evasion evasion
command.
B. Sample Run
A sample run of a target evasion from a PN-guided missile with N0
P N = 4 is considered here. The considered
initial flight path angle of the target is γT0=π/12 rad. The initial flight path angle of the missile γM0
satisfies Eq. (61).
Figure 3 shows the planar trajectories of the missile and target in the simulated sample run. Particularly,
the target assumes perfect information, i.e., uses true states and its evasion strategy is exactly matched to
the active missile guidance law and parameters. Also, the time-to-go required by the evasion law is computed
as in Eq. (58), where range and range-rate are obtained from the true relative position and relative velocity.
The resulting miss distance in this sample scenario is approximately 0.51 m. It can be seen that the applied
evasion strategy also forces the collision triangle to rotate with time.
Based on the same sample run, Figs. 4-5 present, among others, the posterior probabilities ¯
µk=
[µP N
k, µAP N
k, µOGL
k]of each guidance law being correct as a function of time for different MMAE approaches.
Figure 4 shows the behavior of ¯
µkand the probability µj
k, j = 1,...,12 of each guidance law parameter
(regime) being correct for the full-order MMAE approach. In this case, the PN guidance law has been
identified as the missile’s guidance law in approximately 1 sec, and after approximately 2.5 sec its navigation
gain has also been correctly identified.
16
X (m)
0 1000 2000 3000 4000 5000
Y (m)
-200
0
200
400
600
800
1000
1200
1400
1600
1800
2000
1s
1s
2s
2s
3s
3s
4s
4s
5s
5s
6s
6s
7s
7s
8s
8s 9s
9s
Target
Missile
Miss Distance = 0.51118 (m)
Figure 3: Planar engagement trajectories of a missile-target engagement. Missile uses PN with N0
P N = 4.
Target MMAC with MMSE.
Figure 5 depicts the posterior probabilities ¯
µkof each guidance law being correct and the estimated
guidance parameters, N0
P N ,N0
AP N , and α, using the reduced-order MMAE. It can be seen that the correct
guidance law takes a bit longer to be identified in this sample run, i.e., approximately 2 sec. However, the
correct guidance parameter N0
P N = 4 is identified after approximately 3 sec. A 1 sec moving average window
is used to mitigate the impact of the noise estimate. Note that as soon as the probability of a particular
filter reaches 0, it is turned off, thus allowing additional computational savings.
Figure 4: Full-order MMAE approach’s sample regime probabilities. Missile uses PN with N0
P N = 4.
Figure 6 presents the acceleration profiles of the target for different scenarios. In this figure, the ac-
celeration aT(solid line) is driven by the target’s acceleration command uT(dashed line). The top frame
shows the optimal sequence and timing of the target switches against a missile employing PN guidance law
with N0
P N = 4. This evasion is obtained when the target assumes perfect information. The other three
frames below represent the proposed MMAE-based approaches: MMAC with MMSE, MMAC with MAP,
and the reduced-order approach of Sec. VC. It can be seen that while both MMAC approaches perform
almost identically to the perfect information (optimal) case, the reduced-order approach executes the first
switch with some chattering. This occurs, as will be shown in the next subsection, due to less accurate
17
Figure 5: Reduced-order MMAE approach’s sample regime probabilities and guidance parameter estimates.
Missile uses PN with N0
P N = 4.
time-to-go estimation. It can be also noticed that the optimal switches occur approximately 1.5 sec before
the end of the engagement (tgo ≈1.5). This confirms the proposed bound 20 ×τM= 4 sec in Eq. (56) for
the reduced-order approach.
012345678910
-20
0
20 Matched evasion (perfect information)
u∗
T/aT(g)
012345678910
-20
0
20 MMAC with MMSE
ummse
T/aT(g)
012345678910
-20
0
20 MMAC with MAP
umap
T/aT(g)
Time (sec)
012345678910
-20
0
20 Reduced order approach
ured
T/aT(g)
Figure 6: Target acceleration and acceleration command for different scenarios. Target evasion against n
PN guided missile with N0
P N = 4.
C. Monte Carlo Study - Open Loop
To demonstrate and compare the estimation performance of the full-order and the reduced-order MMAE
approach, a 500-run MC simulation campaign was performed considering bearing-only measurements. To
allow a fair comparison between the two approaches, perfect information is used again to compute the optimal
target evasion strategy. This allows to consider, in each MC run, the same measurement and target controls
sequences for both MMAE approaches. However, in each MC run, the filters are initiated with different
random initial guess, see Sec. VI.A, and different noise seeds are used to generate the measurement noises.
The same engagement scenario is considered here as in the previous subsection, i.e., the missile uses PN
18
guidance with N0
P N = 4. The target’s initial angle is γT0=π/12.
Figures 7-11 present the obtained state estimation results for both approaches. The errors shown are
computed using the blended state estimates, ˆ
xk|k, and blended covariances, Pk|k, computed using the mode-
conditioned state estimates ˆ
xj
k|kand error covariances Pj
k|kas follows
ˆ
xk|k=
p
X
i=1
µj
kˆ
xj
k|k,
Pk|k=
p
X
i=1
µj
kPj
k|k+ (ˆ
xj
k|k−ˆ
xk|k)(ˆ
xj
k|k−ˆ
xk|k)T.
Additionally to the estimated states, the time-to-go approximation errors (function of the estimated states)
are also shown in Fig. 12. In all the figures, the dash-dotted line stands for the mean of the estimation
errors, the solid line is the error from a sample run, the thick solid line is the actual standard deviation
(actual 1σ) of the estimation errors, the dotted line is the sample run’s 1σestimation error bound (±σf ilter )
predicted by the filter, and the vertical dashed line indicates the beginning of the blind range of the sensor.
Note that the sample run (sample error and ±σfilter) shown in Figs. 7-12 corresponds to the same sample
run as presented in the previous “Sample Run” subsection in Figs. 4-5.
Time (sec)
0123456789
Range error (m)
-200
-150
-100
-50
0
50
100
150
200
sample err.
expected 1σ(±σfilter)
mean err.
actual 1σ
blind range (50m)
(a) Full-order MMAE with 12 regimes
Time (sec)
0123456789
Range error (m)
-200
-150
-100
-50
0
50
100
150
200
sample err.
expected 1σ(±σfilter)
mean err.
actual 1σ
blind range (50m)
(b) Reduced-order MMAE with 3 regimes
Figure 7: Range estimation error performance
The obtained results show relatively fast convergence of all of the estimated states but the range. Es-
pecially note the small error in estimating the missile’s acceleration, which is in general known to be hard
to estimate. The hangoff errors that can be seen in most of the cases at the beginning of the engagement
are due to the initial transitions (convergence) of the weights when the correct regime has still not been
identified and some filters might be already diverging. However, when the filter is matched, i.e., the regime
probabilities converged, the estimates becomes very good. It can be also observed that the standard devia-
tions of the errors (actual 1σ) are consistent with those predicted by the filter. The results also suggest that
the reduced-order approach has slightly worse estimation performance than the full-order approach. This is
obvious since it deals with additional uncertainty on the missile guidance parameters.
Despite the poor range estimate, the time-to-go approximation yields relatively good performance, see
Fig. 12. This is especially important as the time-to-go is primarily used in the proposed evasion concept
and, as it was discussed earlier, its accuracy is considered to have significant effect on the closed loop evasion
performance (resulting miss distance). This will be studied in the next subsection. It should be noted that
measurements with Tm
s= 1/25 s and σλ= 2 mrad were also considered. The obtained results did not differ
significantly from those presented in this section. Thus, we have omitted these results for brevity of the
presentation.
19
Time (sec)
0123456789
LOS angle error (mrad)
-1.5
-1
-0.5
0
0.5
1
1.5
sample err.
expected 1σ(±σfilter)
mean err.
actual 1σ
blind range (50m)
(a) Full-order MMAE with 12 regimes
Time (sec)
0123456789
LOS angle error (mrad)
-1.5
-1
-0.5
0
0.5
1
1.5
sample err.
expected 1σ(±σfilter)
mean err.
actual 1σ
blind range (50m)
(b) Reduced-order MMAE with 3 regimes
Figure 8: LOS angle estimation error performance
Time (sec)
0123456789
Acceleration error (g)
-10
-8
-6
-4
-2
0
2
4
6
8
10
sample err.
expected 1σ(±σfilter)
mean err.
actual 1σ
blind range (50m)
(a) Full-order MMAE with 12 regimes
Time (sec)
0123456789
Acceleration error (g)
-10
-8
-6
-4
-2
0
2
4
6
8
10
sample err.
expected 1σ(±σfilter)
mean err.
actual 1σ
blind range (50m)
(b) Reduced-order MMAE with 3 regimes
Figure 9: Missile acceleration estimation error performance.
D. Monte Carlo Study - Closed Loop
Here, the effect of the “estimator in the loop” on the closed-loop performance of the proposed evasion
strategies is evaluated and compared with: i)the case when the target performs optimal evasion matched to
the missile’s active guidance strategy (perfect information case), and ii)the case when the target, throughout
the entire engagement, applies maximal acceleration command to one side (blind evasion).
The analysis is based on a set of 500 MC runs. The performance is compared in terms of the achieved miss
distance. All engagements start with the missile and the target being on the collision triangle. For each run,
the active missile guidance strategy, uj
M, was randomly selected among the 12 possible regimes, presented
in Sec. VI.A, and obeying its initial probability µj
0. The considered engagement scenario is symmetric with
respect to the X-axis. Thus, only positive values of γT0are considered. These values are uniformly drawn
from the interval [0, π/6] rad.
Figure 13 presents the obtained miss distances by means of the cumulative distribution functions (CDFs).
The expected miss distance Missexp, defined in Sec. III.B, is also considered for illustration. The results
suggest that the achieved evasion performances of the two full-order approaches, i.e., MMAC with MMSE
20
Time (sec)
0123456789
Flight path angle error (rad)
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
sample err.
expected 1σ(±σfilter)
mean err.
actual 1σ
blind range (50m)
(a) Full-order MMAE with 12 regimes
Time (sec)
0123456789
Flight path angle error (rad)
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
0.1
sample err.
expected 1σ(±σfilter)
mean err.
actual 1σ
blind range (50m)
(b) Reduced-order MMAE with 3 regimes
Figure 10: Missile flight path angle estimation error performance
Time (sec)
0123456789
Missile speed error (m/s)
-60
-40
-20
0
20
40
60
sample err.
expected 1σ(±σfilter)
mean err.
actual 1σ
blind range (50m)
(a) Full-order MMAE with 12 regimes
Time (sec)
0123456789
Missile speed error (m/s)
-60
-40
-20
0
20
40
60
sample err.
expected 1σ(±σfilter)
mean err.
actual 1σ
blind range (50m)
(b) Reduced-order MMAE with 3 regimes
Figure 11: Missile velocity estimation error performance
and MMAC with MAP, are relatively close to the evasion performance of the deterministic scenario when
the target has perfect information. It can be also seen that MMSE and MAP blending yield very similar
performance. The evasion performance slightly deteriorates when the reduced-order approach is considered.
However, it still performs much better than the blind evasion to one side. Note that the above presented
results correspond to bearing-only measurements. It is obvious that the evasion performance could be further
improved by using an additional measurement of the relative range, yielding to more accurate time-to-go
estimate.
VII. Conclusion
Multiple model adaptive evasion strategies for a target aircraft from a homing missile employing a linear
guidance law have been proposed. The performance of these schemes has been analyzed through extensive
Monte Carlo simulations. It was shown that the proposed approaches allow fast enough identification of
the employed missile guidance strategy and thus enabling the target to apply, early enough, the optimal
21
Time (sec)
0123456789
Time-to-go approximation error (s)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
sample err.
mean err.
actual 1σ
blind range (50m)
(a) Full-order MMAE with 12 regimes
Time (sec)
0123456789
Time-to-go approximation error (s)
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
sample err.
mean err.
actual 1σ
blind range (50m)
(b) Reduced-order MMAE with 3 regimes
Figure 12: Time-to-go approximation error performance
Miss distance (m)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
Empirical miss distance‘s CDF
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Expected miss distance (Missexp )
Optimal (matched) evasion (u∗
T)
MMAC with MMSE (ummse
T)
MMAC with MAP (umap
T)
Reduced order approach (ured
T)
Maneuvering one side (uT=amax
T)
Figure 13: Miss distance CDF based on 500 Monte Carlo runs.
evasion maneuvers matched to the missile’s active guidance. Moreover, the proposed evasion methods force
the collision triangle to rotate, thus helping to enhance observability, especially that of the range which is
critical when the target aircraft acquires LOS angle measurements only. From evasion perspective, for the
missile and target having first-order linear dynamics, it was shown that the resulting evasion performance,
achieved by the full-order approach (imperfect information), is very close to the best performance achievable
(perfect information). This comparison thus indicates that, for targets having limited maneuver capability
and carrying sensors that provide bearing-only measurements, the degradation in avoidance capability from
a homing missile may not be as serious as it could have been expected.
Acknowledgments
This effort was sponsored by the U.S. Air Force Office of Scientific Research, Air Force Materiel Command,
under grant number FA9550-15-1-0429. The U.S. Government is authorized to reproduce and distribute
reprints for Governmental purpose notwithstanding any copyright notation thereon.
22
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