Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
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Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile

Presented By
Mr. Partha Saradhi Vankara
st
1 Semester, M.Tech, Roll No:10GNC08

Department of Electrical Engineering
College of Engineering, Trivandrum
Thiruvananthapuram-16
2010


Abstract
In this topic, an interceptor (missile) with bounded acceleration, pursues a target which
makes a sudden step maneuver. A multiple-model adaptive estimator-guidance law is pre-
sented for state estimation of the target. The estimators proposed for each target model
is placed in a filter bank. These estimators differ in the expected timing of the target ma-
neuver jump and feature a mechanism to efficiently identify such a jump. The certainty
equivalence principle seems to be invalid in the case of bounded missile acceleration. There-
fore the guidance gain for each target model, depends on the measurement noise level, target
maneuver statistics, and saturation limit. These gains can be computed a priori and stored
in look-up table to be used online, which are different for each target model in the bank.
The expressions for the zero-effort miss for all models are identical; however, since their
value is independently evaluated by each estimator in the bank, they are in general different.
Simulation results show significant improvement over the deterministic optimal guidance law
when jump times vary from 10 to 3 missile time constants before intercept. A sensitivity
analysis to various noise levels and expected target maneuvers emphasizes the robustness of
this scheme.


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Contents
1 Introduction 1
2 Problem Formulation 3
3 Single Element Estimator-Controller Formulation 6
3.1 Single Element Temporal Multiple-Model Estimator Equations . . . . . . . . 6
3.2 Describing Function Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Saturated Optimal Guidance Law built for a Single Temporal Multiple-Model
Element Estimator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.4 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.4.1 Homing Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4.2 Gains Sensitivity to Noise Level . . . . . . . . . . . . . . . . . . . . . . 12
3.4.3 Gains Sensitivity to Expected Target Maneuver Magnitude . . . . . . 13
4 Multiple-Model Adaptive Control Approach Solution 15
4.1 Multiple-Model Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Multiple-Model Adaptive Control Solution . . . . . . . . . . . . . . . . . . . . 16
4.3 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3.1 Comparison to the Bound . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.3.2 Measurement Noise Level Impact . . . . . . . . . . . . . . . . . . . . . 19
4.3.3 Expected Target Maneuver Magnitude Impact . . . . . . . . . . . . . 20
5 Conclusions 22
iii
List of Figures
2.1 Linearized endgame geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.1 Error function(erf) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2 Sample SOGL guidance gains . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Sample CDF for target maneuver at θjump=7 , σjump = 60[m/s2 ]
3.3 . . . . . . . . 11
The rms miss distance vs θjump , σjump = 60[m/s2 ] . . . . . . .
3.4 . . . . . . . . 12
3.5 N’ sensitivity to angular measurement noise level σφd . . . . . . . . . . . . . 13
3.6 N’ sensitivity to expected target maneuver magnitude σjump . . . . . . . . . 14
4.1 MMAC11F approach solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
The rms miss distance vs θjump ,σjump = 60 [m/s2 ]; Comparision of MMAC11F
4.2
with the performance bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
The rms miss distance vs θjump , σjump = 60 [m/s2 ].Target jumps ±σjump
4.3
Comparision of MMAC11F with the performance bound . . . . . . . . . . . . 19
MMAC11F:The rms miss distance vs θjump ,σjump = 60 [m/s2 ]; Target behaves
4.4
as its model. Comparision of noise levels. . . . . . . . . . . . . . . . . . . . . 19
The rms miss distance vs θjump , σjump = 60 [m/s2 ].Target jumps ±σjump ;
4.5
Comparision of noise levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
MMAC11F:The rms miss distance vs θjump , σjump = 40,60,90 [m/s2 ]. Target
4.6
jumps with σjump . Comparison of expected target maneuver effect. . . . . . . 20
The rms miss distance vs θjump , σjump = 40,60,90 [m/s2 ]; Target jumps
4.7
±σjump .Comparison of expected target maneuver effect . . . . . . . . . . . . 21
iv
Chapter 1
Introduction

The most popular and widely used guidance law is the PN (Proportional Navigation) guid-
ance law. PN issues a guidance acceleration command, which is jointly proportional to the
inertial angular rate of the LOS between the missile and target, and the missile-to-target
closing velocity. PN results in a zero miss distance, if ideal dynamics are assumed and if
there are no constraints on the acceleration. If missile and target speeds are constant, as-
suming ideal dynamics, for a nonmaneuvering target, both the target and the missile move
close to straight lines that meet each other at the intercept point. These lines together with
the LOS of missile and target are referred to as the Collision Triangle which is the basis for
a linearized version of the general guidance problem.
PN also has its drawbacks. For example, in realistic interception scenarios between a
pursuer with limited maneuver ability and a target with large maneuver capability, it results
in a significant miss distance. To reduce the miss distance and to relax the large acceleration
requirements, Modern guidance laws are formulated, which required additional information
such as the predicted time to intercept (or) the time to go tgo and the missile and target
accelerations. An important concept of modern guidance laws is the Zero-effort miss (ZEM).
ZEM is the miss distance that would result if the missile made no further corrective com-
mands, and the target followed the assumed acceleration model. The PN guidance law can
be expressed using the ZEM concept. The ZEM is constructed assuming ideal dynamics and
taking target’s acceleration to be zero. The Augmented Proportional Navigation guidance
law (APN) suggests an augmentation term compensating for constant target acceleration by
including its effect in the ZEM term. The direct effect of APN compared to PN is to reduce
missile acceleration at the end of the conflict at the expense of larger acceleration at the
initial time, and to reduce the total required maneuver.
For a maneuvering target, assuming ideal dynamics for the missile, using the optimal
control theory, it was proven that APN with navigation coefficient N’=3 is in fact an opti-
mal guidance law. This guidance law minimizes, a quadratic cost function, of the integral of
the square of the missile acceleration subject to ZEM constraint. Removing the ideal missile
dynamics assumption and replacing it with first-order dynamics, and using the same optimal
control tools, resulted in the birth of the Optimal Guidance Law (OGL). In real-world prob-
lems, sensor and system noise are the major influences on miss distance in a pursuer-evader
scenario. In realistic scenario, it can be assumed that the pursuer’s state is available for
use in the guidance law. However, the evader’s state is generally not known to the pursuer.
A well-designed evader guidance law will make the evading target as difficult to predict as
possible. Therefore, target state estimation is essential. The common linear state estimator
is the Kalman filter. For a nonlinear system or nonlinear observations, the extended Kalman
filter may be used. The EKF updates a linearization around the current state estimate.
When the assumed target maneuver and the observation noise statistics differ from the ac-
tual ones, the performance of the Kalman filter is degraded.
1
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
One way of overcoming this disadvantage is a method known as the Multiple-Model
Adaptive Estimation (MMAE)method [1]. Here, the target acceleration is assumed to follow
one of the several target models. Then processing the observations simultaneously, using all
possible models and finally using a weighted average of the resulting estimates, is in fact an
optimal estimator structure.The MMAE assumes that the system is in one of a finite number
of modes. Each filter’s estimate is weighted through the filter’s weighting coefficient to form
the full multiple-model filter’s estimate. The weighting coefficients are a function of the
measurements available up to the current time. The disadvantage of the MMAE scheme is
that, the estimator is required to contain a large number of filters, describing all the possible
system modes.
An approximation of the MMAE scheme is the Temporal Multiple-Model (TMM) esti-
mator, with the important addition of input estimation. For the assumed target model, the
target acceleration is represented as a white noise process. Pursuer (missile) acceleration
saturation is another major source of increased miss distance. Saturation brings another
important issue for the estimator-guidance scheme. The generally assumed certainty equiv-
alence principle (CEP) is not valid for a limited acceleration control missile.The certainty
equivalence principle states that the optimal control law for a stochastic control problem is
the optimal control law for the associated deterministic problem for which the known states
are replaced by the estimated ones. When the CEP holds, it is possible to design an estimator
and a control law independently of each other. Here the control law may be designed on the
basis of deterministic states. When the CEP is not valid, it was proved that the estimator
may be designed independently of the control problem, but the optimal controller depends on
the statistics of the estimated state. This principle is known as the one-way separation (or)
General Separation Theorem (GST). It can be used to suggest an approach for saturation
inclusion in the guidance law. This new linear guidance law named Saturated OGL (SOGL)
describes saturation using the random input describing function (RIDF) method and solves
the stochastic controller problem. This unique formulation results in a guidance law gain
that depends on the estimated state statistics. [2] provides the basis for the guidance law
used in the current work. A later work [3] proposed the optimal nonlinear solution for this
problem by numerically solving the stochastic optimization problem.
When certainty equivalence is not valid, a separate guidance law should be constructed
for each estimator, wherein the initial part of the scenario is devoted to the model identifica-
tion after which one filter-guidance law branch is selected. The suboptimal multiple-model
adaptive control (MMAC) scheme [4] can be used, which degenerates to an MMAE estimator
and the APN guidance law.The derivation of the guidance law proceeds in two stages. In
the first stage, it is assumed that the instance that the target issues a step command in its
acceleration is known. But the size of the target maneuver jump must be estimated. In the
second stage, it is assumed that the target performs a single step maneuver at an unknown
time near intercept. A suboptimal guidance law is derived using the MMAC formulation,
making this estimation-dependent guidance law adaptive to the target’s maneuver with a
gain, varying according to the current estimated state error statistics.The remainder of this
paper [5] is organized as follows. The next section depicts the problem formulation, the
model used, the assumptions, and the available measurements. Next is the solution for a
single estimator element matched to the new guidance law and a performance evaluation.
This is followed by an MMAC type solution, results, analysis and conclusions.
Department of Electrical Engineering, College of Engineering, Trivandrum 2
Chapter 2
Problem Formulation

The dynamic model of the problem is constructed using the following assumptions. 1) the
missile-target engagement occurs in one plane, 2) missile target are represented by a point
mass model with linear dynamics, 3) the relative endgame trajectory is linearized about the
initial LOS direction, 4) the missile and target speeds remain constant, 5)missile acceleration
is bounded, 6) missile and the target are modeled as first-order dynamic systems and 7)
missile acquires noisy measurement of some state-variables. Figure 2.1 shows the linearized
endgame geometry. Here X axis is aligned with the initial LOS direction, r is the range
between the missile and the target, φ is the angle between current LOS and initial LOS, y is
the relative displacement perpendicular to the initial LOS, aT and aM are the accelerations
of the target and missile normal to the initial LOS respectively.
Figure 2.1: Linearized endgame geometry
The proposed target model for the estimator is a first-order system, represented by a
time constant τ T and noise. The deterministic part of the target model is given as:
1
aT
= (2.1)
aTC 1 + s τT
where aTc is the target’s acceleration command. For a constant acceleration command, the
general target dynamics are described by:
y ̇T = υT (2.2)
υ ̇T = aT (2.3)
−1 1 1
a ̇T = aT + aTc + wT (2.4)
τT τT τT
aTc = 0
̇ (2.5)
3
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
where yT ,υ T , aT are the target’s position, velocity and acceleration respectively and wT is
the white noise process, representing the target acceleration uncertainties( small, possibly
random, changes as would result from wind gusts, turbulence, target flight control effects,
etc.). All are perpendicular to the initial LOS direction. The acceleration command satisfies
the following assumptions: 1) acceleration command is a single step function. 2) magnitude
of the step command is taken from a Gaussian distribution. The state used in this problem
consists of the relative position, velocity, missile acceleration, target acceleration and target
acceleration command, all being perpendicular to initial LOS direction.
aTc ]T
X = [y y aM ̇ aT (2.6)
The missile is modeled as a first-order dynamic system with a time constant τ M having the
following acceleration transfer function:
aM 1
(2.7)
=
u 1 + sτM
where ‘u’ is the maneuver command. Since constant closing speed uC is assumed, and the
initial range between the missile and target is ‘r0 ’ , the final intercept time tf is approximated
by:
tf = r0 /υc (2.8)
The time to go is defined as
tgo ∆ tf − t (2.9)
=
and the normalized time to go is
θ∆ tgo /τM (2.10)
=
The bounded control is described as a standard saturation function.
= UM if u > UM (2.11)
sat(u) = u if − UM ≤ u ≤ UM (2.12)
= −UM if u < −UM (2.13)
here UM = aMmax is the missile’s maximal maneuver capability. The equation of motion is
given as:
̇
X = Ax + bsat(u) + Gw; x(0) = x0 (2.14)
where


0 1 0 0 0
−1
0 0 1 0 



−1/τM
0 0 0 0
A = (2.15)



−1/τT
0 0 0 1/τT 

0 0 0 0 0
 
0
 0 
 
b =  1/τM  (2.16)


 0 
0
 
0 0
 0 0 
 
G =  1/τM 0 (2.17)

 
 0 1/τT 
0 0
Department of Electrical Engineering, College of Engineering, Trivandrum 4
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
and w = [wM ,wT ]T is a two dimensional white noise, and wM and wT are the white Gaussian
process representing the uncertainty in missile and target acceleration. The standard devia-
tion of the missile’s acceleration is taken to be small (0.1-1g) and represents the uncertainties
of the missile’s sensors. The continuous-time measurement taken by the pursuing missile is
given as:
z = φ + υ ≈ cx + υ (2.18)
where the measurement matrix ‘c’ is
c = [1/r 0 0 0 0] (2.19)
and ‘u’ is the angle measurement error modeled as a continuous-time white Gaussian noise
with spectral density V.
2
V = σφ (2.20)
2 2 2
(For discrete-time measurements taken every Ts seconds with variance σφd , σφ = σφd Ts ).
This range ‘r’ is assumed to be measured accurately. The cost function to be minimized is:
tf
T
(X T QX + Ru2 )dt]
J = E[Xf Sf Xf + (2.21)
0
where the subscript ‘f’ denotes the value of the final time. For this interception problem,
the following matrices are chosen
 
sf 0 0 0 0
 0 0 0 0 0 
 
Sf =  0 0 0 0 0 (2.22)

 
 0 0 0 0 0 
0 0 0 0 0


0 0 0 0 0
0 0 0 0 0 



Q = 0 0 0 0 0 (2.23)



0 0 0 0 0 

0 0 0 0 0
therefore the weighting coefficients ‘sf ’ and ‘R’ become the design parameters of the problem.
Department of Electrical Engineering, College of Engineering, Trivandrum 5
Chapter 3
Single Element

Estimator-Controller Formulation
The problem of estimating a sudden step maneuver during the intercept of time-to-go values
(equal to several missile time constants) is a very practical and difficult one. This is the
time span in which a large portion of the miss distance is formed. The problem becomes
more complicated when the missile’s acceleration is bounded, causing another growth in
the miss distance. Knowing these two elements’ contribution to the guidance performance
degradation, we make use of the special attributes of the TMM estimator, and the saturation
inclusion in the guidance law to deal with these two major problems in the guidance system.
In this section, a single TMM - type element estimator matched to a target acceleration
step, of known magnitude distribution, at a known time is used. Later, this estimator will
be combined with the SOGL algorithm to create the guidance law (using the one-way sepa-
ration). The single element TMM estimator is the optimal estimator for the assumed target
model, and the SOGL will be matched exactly to this optimal estimator. This combination
will yield the best available performance such a pair can achieve.
3.1 Single Element Temporal Multiple-Model Estimator Equa-
tions
The initial statistics of the estimated state vector x are
ˆ
E[x(0)] = x0
ˆ (3.1)
E[ˆ (0) xT (0)] = Pxx
0
(3.2)
x ˆˆ
the estimation error, denoted as (e), is defined as:
e=x−x ˆ (3.3)
Assuming that x0 , w(t) and υ(t) are independent and using Eqn.(2.14), we obtain the
ˆ
following continuous-time filter equations:
̇
x = Aˆ + bsat(u) + kf (z − cˆ)
ˆ x x (3.4)
kf = Pee cT V −1 (3.5)
Pee = APee + Pee AT − Pee cT V −1 cPee + GGT
̇ (3.6)
0
Pee (0) = Pee (3.7)
The special feature of the TMM estimator, that allows it to identify sudden target accelera-
tion jumps, lies in the estimator initialization. The initialization includes two operations: 1)
6
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
initializing the estimator’s state vector, 2) initializing the estimator’s error covariance matrix.
The state vector is initialized using the best available data at the time of the initialization,
and setting the fifth state member to aTc = 0. The error covariance matrix is initialized by
ˆ
taking the previous value of the 4 by 4 matrix, forcing p55 to the jump uncertainty value
σ2jump and setting remaining elements in the fifth row and column to zero. The setting of the
remaining elements in the fifth row and column to zero, is due to the fact that the (assumed)
new acceleration command is history independent, and hence uncorrelated to the other state
variables. It is shown schematically below
p− p− p− p−
 
0
11 12 13 14
p− p− p− p− 0
 
21 22 23 24
p− p− p− p−
 
+ 0
Pee = (3.8)

31 32 33 34
p− p− p− −
 
p44 0
 
41 42 43
2
0 0 0 0 σjump
where ( )− represents the values before the initialization, and ( )+ the values after ini-
tialization. This is, in fact, the optimal estimator when (1) the time of the jump in the
target’s acceleration command is known, (2) and the target’s acceleration step command is
uncorrelated with the previous commands.
3.2 Describing Function Approach
The missile acceleration saturation will be approximated by an RIDF as shown. Let φ (ζ) be
an odd single-input-single-output nonlinearity driven by a zero mean Gaussian process(ζ).
The approximation of φ (ζ) by a linear gain leads to the following minimization problem:
min
− Lζ]2
L E[φ (ζ) (3.9)
where ‘L’ is a RIDF (random input describing function). Let φ (ζ) be the saturation function
defined by Eq. (2.11),(2.12)and (2.13) where ζ = u. The associated RIDF in this case is
given by: √
L = erf (Um / 2σu ) (3.10)
where η
1 2
e−ψ dψ
erf (η) = √ (3.11)
π −η
2
u ≈ N (0, σu ) (3.12)
L and (1-L) represent the probability of not reaching and reaching saturation respectively.

Examine figure 3.1, the x-axis is the error function’s argument Um / 2σu as in Eq.(3.10) and
the y-axis is the RIDF’s equivalent gain L. σu represents the range of values of the control
signal in a closed-loop system (due to state uncertainties). The following is observed: 1) if
the standard deviation of the acceleration command approaches zero, the argument increases
and the gain ‘L’ approaches 1, meaning the saturation is inactive. 2) If the standard deviation
of the acceleration command rises as in stochastic case, the argument decreases and the gain
‘L’ drops to values lower than 1, inducing an attenuation factor, simulating the effect of
saturation in the closed-loop system. Using the RIDF concept, we will obtain the optimal
linear controller such that
u = kc xˆ (3.13)
Calculating the control signal variance yields
2 T
σu = kc Pxx kc (3.14)
ˆˆ
the calculation of the covariance matrix Pxx is based on the approximation of the saturation
ˆˆ
nonlinearity by the RIDF. Using Eqns. (2.14), (3.10) and (3.13)
̇
x = (A + Lbkc )ˆ + kf (z − cˆ)
ˆ x x (3.15)
Department of Electrical Engineering, College of Engineering, Trivandrum 7
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
Figure 3.1: Error function(erf)
is obtained. Because υ = z − cˆ is the innovation sequence,
x
E[υ(t)υ T (τ )] = V δ(t − τ ) (3.16)
we can write the state covariance differential equation as:
̇
Pxx= (A + Lbkc )Pxx + Pxx (A + Lbkc )T + kf V kf T
(3.17)
ˆˆ ˆˆ ˆˆ
0
Pxx (0) = Pxx (3.18)
ˆˆ ˆˆ
3.3 Saturated Optimal Guidance Law built for a Single Tem-
poral Multiple-Model Element Estimator
After the estimator has been formulated and the error covariance matrix is known for the
entire scenario duration, the optimal guidance law may be calculated using the one-way
separation principle. The cost function to be minimized is given by
tf tf
f
J = E[xT Sf xf + (xT Qx+Ru2 )dt] = tr[(Pxx +Pee )Sf ]+ f T
(tr[(Pxx +Pee )Q]+Rkc Pxx kc )dt
ˆˆ ˆˆ
f ˆˆ
0 0
(3.19)
where the minimization is performed with respect to kc , subject to Eq. (3.17) and (3.18).
Pee does not depend on kc (the one-way separation principle) and the problem to be solved
is
tf
min f T
tr(Pxx Sf ) + [tr(Pxx Q) + Rkc Pxx kc ]dt (3.20)
kc ˆˆ ˆˆ
ˆˆ
0
subject to Eq. (3.17) and (3.18), where
Um
L = erf (3.21)
T
2kc Pxx kc
ˆˆ
The Hamiltonian associated with this problem is
H = [tr(Pxx Q) + Rkc Pxx kc ] + tr [(A + Lbkc )Pxx + Pxx (A + Lbkc )T + kf V kf ]S + (3.22)
T T
ˆˆ ˆˆ ˆˆ ˆˆ
2
Um
T
λ kc Pxx kc −
ˆˆ 2[erf −1 (L)]2
,where the symmetric matrix S and the scalar λ are Lagrange multipliers. The optimal
solution is obtained by solving the following equations:
∂H
=0 (3.23)
∂kc
Department of Electrical Engineering, College of Engineering, Trivandrum 8
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
∂H
̇
S=− (3.24)
∂Pxxˆˆ
and the transversality condition
S(tf ) = Sf (3.25)
L is also an optimization variable under the constraint of Eq.(3.21); thus,
∂H
=0 (3.26)
∂L
Equation (3.23) leads to the relation
L T
kc = − b S (3.27)
R+λ
and substituting Eq. (3.27) into Eq. (3.21) yields
2 2
L Um
bT SPxx S T b = (3.28)
ˆˆ
2[erf −1 (L)]2
R+λ
Equations (3.24) and (3.25) are equivalent to the following differential equation:
L2
̇
S = −Q − AT S − SA + SbbT S (3.29)
R+λ
S(tf ) = Sf (3.30)
Equation (3.26) is used to derive an expression of λ in terms of L:
R
λ= (3.31)

π exp{[erf −1 (L)]2 }
−1
2 L erf −1 (L)
Equation (3.28) can be rewritten as
R+λ Um
bT SPxx Sb = √ (3.32)
ˆˆ
2erf −1 (L)
L
Substituting Eq. (3.31) into Eq. (3.32), we obtain
R
R+ √
π exp{[erf −1 (L)]2 }
−1
L Um
erf −1 (L)
2
bT SPxx Sb = √ (3.33)
ˆˆ
2erf −1 (L)
L
Equations (3.27),(3.29) and (3.30) induce the following guidance law structure:
NSOGL (tgo )
u = ac = ZEMSOGL (3.34)
t2
go
with the navigation gain N’ SOGL taken from Eq. (3.27)
NSOGL (tgo ) = kc (tgo )t2 [1 0 0 0 0]T (3.35)
go
and
1 2
ZEMSOGL = y + y tgo − aM τM (e−θ +θ −1)+ aT τT (e−θT +θT −1)− aˆ c τT (e−θT +θT −1− θT )
̇ ˆ 2 ˆ 2 2
ˆ ˆ T
2
(3.36)
θT = tgo /τT (3.37)
Department of Electrical Engineering, College of Engineering, Trivandrum 9
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
3.4 Performance Evaluation
To evaluate the potential of the newly proposed estimator-guidance-law pair a performance
evaluation was carried out using a simulation of the one-dimensional engagement (perpen-
dicular to the initial LOS). A discrete-time implementation was chosen, sampled over the
sampling period Ts . The scenario’s parameters are described in Table 3.1. In the scenario
described, the target produces an acceleration step at a predefined normalized time (denoted
as θjump ) which is varied in steps of 0.1s during the last second of the scenario. The tar-
get generates commands as assumed by the estimator’s model, that is, commands a step
in acceleration for which the amplitude is chosen from a Gaussian distribution with the
known σ jump . For each target jump time, a matched single TMM element estimator was
constructed, inducing the covariance change as in Eq. (3.8) exactly at the known jump
time. Then, the covariance of this single element TMM estimator is fed into the SOGL
algorithm, resulting in a matched guidance law gain for this specific element estimator. This
estimator-guidance law pair represents the best available match and leads to a performance
bound. Typical guidance gains are shown in Figure 3.2 as a function of the normalized time
to go θ for different values of target maneuver times. As expected, the effective navigation
gain N’ is influenced by the covariance increase of the filter. All N’ gains, including that of
OGL, converge asymptotically to 3, the constant gain of PN. When the filter initialization
occurs a long time before the intercept (θjump = 10, 7) the gain resembles that of the original
formulated SOGL. When the initialization is expected to happen exactly at the engagement
Figure 3.2: Sample SOGL guidance gains
termination (θjump = 0) the gain is identical to that of the classical OGL, as for this cases
a non-maneuvering target is expected. If the target jump is expected to happen close to the
terminal time (θjump = 5, 2) the covariance increase causes a substantial growth in the N’
Department of Electrical Engineering, College of Engineering, Trivandrum 10
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
values for θ ∈ [0.5, 2]. This illustrates the inherent feature of the SOGL to increase, early in
the engagement, the gain when uncertainties rise. In a full functional estimator, this increase
in the covariance represents the chance of a target maneuver, which means that an increase
in N’ would occur when the target performs a jump. This agrees with the common guidance
engineering practice of increasing the gain when the target maneuvers.
3.4.1 Homing Performance
, OGL and SOGL guidance laws were compared, all using the same target state estimates
from a single TMM estimator. Hence, the achieved improvement is due solely to the dif-
ferent guidance laws used. For each jump time 100 Monte Carlo runs were simulated and
the rms of the miss distance was calculated. Figure 3.3 shows a sample miss-distance (MD)
cumulative distribution function (CDF) for a target maneuvering at θ = 7. The superiority
of SOGL over OGL and PN is evident. For example, when using SOGL, in 90 percent of
the cases the miss distance was smaller than about 0.25m, whereas for OGL and PN it was
lower than about 0.5 and 2.5 m, respectively. We will use the RMS of the MD from the 100
runs, when comparing the different guidance laws. Here it is 0.29, 0.54, 1.39 m when using
SOGL, OGL, and PN respectively.
Figure 3.4 shows the rms MD obtained form 100 Monte Carlo runs for each target ma-
Figure 3.3: Sample CDF for target maneuver at θjump=7 , σjump = 60[m/s2 ]
neuver time. In Figure 3.4 a line representing PN performance is included to emphasize the
substantial difference between PN, OGL, and SOGL.
Figure 3.4: The rms miss distance vs θjump , σjump = 60[m/s2 ]
In all comparisons similar PN behavior was observed, and so, to show the comparable
Department of Electrical Engineering, College of Engineering, Trivandrum 11
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
difference between OGL and SOGL, only these two will be presented in the figures to follow.
An examination of Figure 3.4 shows typical behavior: little to no improvement occurs in the
vicinity of θjump = 10 Most of the improvement appears around θjump = 5, 6, 7 ( about 30
percent), and from θjump = 3 and on, OGL and SOGL performance coincide. θjump = 0
represents the case of a nonmaneuvering target. The behavior at θjump = 10 are explained
by the fact that both OGL and SOGL have enough time to close the ZEM and obtain a
similar miss distance.
Also, by examining Figure 3.2, it is obvious that SOGL’s N’ for target jumps in this
region resembles that of the ordinary OGL. The region of θjump = 5, 6, 7 is where most
improvement is observed. This is expected because this is the region when N’ of SOGL is
significantly different from that of the ordinary OGL. This is the region in which, if a target
maneuvered and the maneuver were properly identified, it would be wise for the pursuer to
apply as much acceleration command as possible to null the ZEM. SOGL’s higher gain does
just that. The region of θjump ∈ [0, 4] is characterized by the fact that both OGL and SOGL
(and from θjump = 2, PN) gains are high enough to saturate the acceleration command (as
N’ is divided by tgo), so that the performance of SOGL and OGL are similar. It is important
to understand that the extra information used in SOGL (the covariance enlargement) is of
no use to the conventional OGL and does not change its structure or behavior; thus, OGL
is the reasonable comparison basis for SOGL.
3.4.2 Gains Sensitivity to Noise Level
For a matched estimator SOGL, the only difference between OGL and SOGL lies in the dif-
ferent behavior of the guidance gain N’. This fact implies that, by exploring the behavior of
N’ with some varying parameters, we will gain better understanding of this guidance system
and its expected behavior. In this section, we explore the varying behavior of the guidance
gain N’ when the angular measurement noise level changes. The scenario parameters are de-
fined in Table 3.1 with the varying angular measurement noise as defined in table 3.2. For the
sake of clarity, and to deliver only the important details, the labels of the axes were removed
from the figure and are elaborated as follows: 1) the horizontal x axis is the normalized time
to go θ, 2) the vertical y axis is the guidance gain N’ and 3) each graph’s title represents
the corresponding θ in which a jump had occurred and for which the estimator-guidance-law
pair was constructed.
The gains shown in Figure 3.5 , which corresponds to target jump times of θ = 9,8,7,....,1,
are the gains for which most of the important phenomena may be observed: (1) As observed
before, a short time after a filter has been initialised, the gain increases. This increase in
N’ grows larger with the increase in noise level. The phenomenon is best observed in θjump
= 4, 5. This is the SOGL’s reaction to an increase in the estimation statistics uncertainties
due to larger measurement noise. (2) The increase in measurement noise causes the gains to
rise earlier in the last few missile time constants of the intercept.
Again, this represents the SOGL’s feature of increasing the gain when uncertainties rise
and saturating the missile’s acceleration command earlier in the scenario. The phenomenon
Department of Electrical Engineering, College of Engineering, Trivandrum 12
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
Figure 3.5: N’ sensitivity to angular measurement noise level σφd
is best observed in θjump = 7, 8, and 9. Both effects represent the physical and common
engineering practice of increasing he gain when uncertainties in the expected ZEM values
increase to react fast to a possible target maneuver. The high values of the navigation gain
serve to saturate the guidance command for relatively small values of ZEM. The value of
the guidance gain primarily influences the specific value of ZEM required to saturate the
guidance command.
3.4.3 Gains Sensitivity to Expected Target Maneuver Magnitude
In this section, we explore the varying behavior of the guidance gain N’ when the expected
target maneuver magnitude changes. The scenario parameters are defined in Table 3.1 with
the varying expected maneuver magnitude σjump as defined in Table 3.3. For the sake of
clarity, and to deliver only the important details, the labels of the axes were removed from
the figure and are elaborated as follows: 1) the horizontal x axis is the normalized time to
go θ, 2) the vertical y axis is the guidance gain N’ and 3) each graph’s title represents the
corresponding θ in which a jump had occurred and for which the estimator-guidance-law pair
was constructed. The gains shown in Figure 3.6 which corresponds to target jump times of θ
= 9, 8, 7,...,1, are the gains with which most of the important phenomena may be observed.
(1) A short time after a filter has been initialized, the gain increases. This increase in N’
grows larger with the increase in expected target maneuver magnitude. The phenomenon is
best observed in θjump = 6, 7, and 8. Again this is the SOGL’s reaction to an increase in
the estimation statistics uncertainties due to larger expected target maneuver magnitude.
(2) The increase in expected target maneuver magnitude causes the gains to rise earlier
in the last few missile time constants of the intercept. This represents the SOGL’s feature of
increasing the gain when uncertainties rise, and saturating the missile earlier in the scenario.
Department of Electrical Engineering, College of Engineering, Trivandrum 13
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
Figure 3.6: N’ sensitivity to expected target maneuver magnitude σjump
The phenomenon is best observed in θjump = 3, 4, and 5.
(3) It is observed that the increase in expected target maneuver magnitude does not
increase the gain proportionally, but rather causes an abrupt and powerful increase in N’.
this is due to the fact that SOGL identifies the expected target maneuver magnitude as
being a very large expected ZEM uncertainty, causing a large increase in N’ and inducing a
saturated pursuer acceleration command as early as possible.
These effects, again explain the common engineering practice of increasing the gain when
uncertainties in the expected ZEM values increase, to react fast to a possible target maneuver.
The high gains, in the last few missile time constants of the intercept yield a saturating
pursuer acceleration command.
Department of Electrical Engineering, College of Engineering, Trivandrum 14
Chapter 4
Multiple-Model Adaptive Control

Approach Solution
In this section, an MMAC scheme is used to construct a feasible implementation of an
estimator-guidance- law scheme. In this scheme, the time of the jump in the target’s ac-
celeration is no longer assumed to be known. To derive this solution, a brief review of
the multiple-model approach is presented, followed by an overview of the MMAC approach
principles and the specific suggested MMAC solution for the problem at hand.
4.1 Multiple-Model Approach
Our approach is based on the use of the MM approach, which is now introduced. In the basic
MM approach, it is assumed that the system obeys one of a finite number of models. The
filter is constructed using a Bayesian framework. Starting with previous probabilities of each
model being correct, the corresponding posterior probabilities are obtained. It is assumed
that the model the system obeys is fixed (referred to as the system mode being fixed) for
the entire estimation process and that it is one of the ‘r’ possible models (the system is in
one of ‘r’ possible modes). Mathematically formulating the above, the system mode M is
M ∈ {Mj }r j=1 (4.1)
The prior probability that Mj is correct (the system is in mode j) given Z0 is
p Mj |Z 0 = μj (0) ; j = 1, ..., r (4.2)
where Z0 is the prior information. It can be noted that
Σr μj (0) = 1 (4.3)
j=1
because the correct model is among the assumed r possible models. It is assumed that all
models are linear Gaussian.
Using Bayes formula, the posterior probability of model j being correct, given the mea-
surement data up to time k ( referred to as the weighting coefficients), is obtained recursively
as
p z (k) |Z k−1 , Mj μj (k − 1)
μj (k) = p Mj |Z k = r ; j = 1, ...., r (4.4)
Σj=1 p [z (k) |Z k−1 , Mi ] μi (k − 1)
starting the recursive calculation with Eq. (4.2). The probability appearing in the nominator
of Eq. (4.4) is referred to as the likelihood function of mode j at time k, which, under the
linear Gaussian assumptions, is given by the expression
15
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
Λj (k) ≡ p z (k) |Z k−1 , Mj = p[vj (k)] ≈ N [0, Sj (k)] (4.5)
where vj and Sj are the innovation and its covariance from the mode matched filter cor-
responding to mode j. A filter matched to each mode is constructed, yielding a mode-
conditioned state estimate and a mode-conditioned error covariance. The probability of
each mode being correct is obtained according to Eq. (4.4) based on its likelihood function
(4.5) relative to the other filters’ likelihood functions. This filter structure consists of r linear
filters. 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 probabili-
ties are used to combine the mode-conditioned estimate and covariance for that time. Under
these assumptions, the probability density function of the state is a Gaussian mixture with
r terms:
p[x(k)|Z k ] = Σr μj (k) N [ˆj (k|k), Pj (k|k)]
x (4.6)
j=1
The combination of the mode-conditioned estimates and covariance is carried out as follows:
x(k|k) = Σr μj (k) xj (k|k)
ˆ (4.7)
j=1
and
P (k|k) = Σr μj (k) P j (k|k) + [xj (k|k) − x(k|k)][xj (k|k) − x(k|k)]T (4.8)
j=1
Equations (4.7) and (4.8) are exact, under the following assumptions: 1) the correct model
is among the set of models considered, and 2) the same model has been in effect from the
initial time. This multiple-model formulation is usually referred to as the multiple-model
adaptive estimation scheme.
4.2 Multiple-Model Adaptive Control Solution
Multiple-model adaptive control is an approximate (suboptimal) scheme for combining an
MMAE- type estimator with an estimator-dependent controller. When the controller is
independent of the estimator, it is possible to match the combined output estimate of the
MMAE estimator with a single controller, but this is not the case for out matched estimator-
controller pair. In the MMAC scheme, each MMAE- type filter is combined with its matched
controller and the controllers outputs are weighted using the same weighting coefficients of
the MMAE estimator, to form an equivalent control signal for the system’s plant. Most
real-world targets will rarely switch their maneuver direction, type, or magnitude in the last
10 time constants of the interception. At most, a single new maneuver will start at these
last instances of the engagements. And any maneuver starting at that time, with a target
time constant as assumed here (0.3s), will appear mostly as a step or some part of it.
Therefore, the assumption of a single step occurring in these last 10 time constants is per-
fectly reasonable and will be used here.The combined TMM-SOGL algorithm does not yield
a significant improvement if initialized earlier than 10 missile time constants before the inter-
cept time. This is also evident from the almost unchanged N’ resulting from initializations
before the 10 missile time constants limit. Understanding that the important time span to
be dealt with is the last 10 missile time constants, a MMAC- type guidance system structure
is formed.The MMAC system (figure 4.1) is constructed from 11 filter-guidance law branches
(this system will be identified here as “MMAC11F”) matched to the 10 possible target jump
times of the span (θjump = 10, 9, 8,1) and another filter representing a jump at θjump = 0,
which is equivalent to a filter representing no target jump at all. Each of the component
models of the multiple-model architecture assumes a jump in the target acceleration com-
mand at a known time, but with an unknown magnitude. The jump magnitude is assumed to
be zero mean, Gaussian distributed, with a known σ jump . In this case, until a jump is iden-
tified, the expected value of the target acceleration command remains unchanged, regardless
Department of Electrical Engineering, College of Engineering, Trivandrum 16
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
Figure 4.1: MMAC11F approach solution
of the known different jump times that are assumed in each of the component models of the
multiple-model architecture. That is, a constant acceleration command is assumed for all
models. Therefore, all the ZEMSOGLi , have identical expressions:
1 2
ZEMSOGLi = yi +yi tgo −aMi τM (e−θ +θ−1)+aˆ i τT (e−θT +θT −1)−aTci τT (e−θT +θT −1− θT )
ˆ ˆ ˆ 2 2
ˆ 2
̇ T
2
(4.9)
Although for all MMAC- based guidance laws the ZEM structure is identical, for each dif-
ˆ ˆ ˆ
ferent value of i (different model), appropriate values of yi , yi , aMi , aˆ i , and aTci are used.
̇ ˆ
T
The MMAC - based control law is
11
Ni (tgo )
ac = μi zi (tgo )
ˆ (4.10)
t2
go
i=1
Where zi (tgo ) is the expectation if the zero-effort miss conditioned on the measurements
ˆ
until the current time and on the assumed time ti of the step in the target acceleration,
μi is the conditional probability of the target acceleration step occurring at time ti , and
Ni (tgo ) is the optimal guidance gain associated with a step in the target acceleration at
time ti of unknown (random) amplitude. Remark1: The stochastic target model used in the
current work specifies the random behavior of the target acceleration from the beginning
of the engagement to intercept time. Consequently, the assumed model dictates the zero-
effort-miss calculation. The model assumes that the target acceleration is the sum of the two
components: a first-order Gauss-Markov process and a single step in the target acceleration
command, occurring at an unknown (random) time. In particular, the ZEM calculation in
Eq.(4.9)shows that, at any instant, the future values of the target acceleration are the sum of
the present estimate of the target acceleration (aˆ i ) decaying with the assumed target time
T
constant for the Gauss - Markov component and a constant acceleration command (aTci ) ˆ
acting through the assumed target time constant.
Remark 2: If the SOGL is replaced by OGL, then Ni (tgo ) = N (tgo ) for all i. In this case,
Eq. (4.10) can be rewritten as
11
N (tgo )
N (tgo )
ac = μi zi (tgo ) =
ˆ ze (tgo )
ˆ (4.11)
2 t2
tgo go
i=1
Department of Electrical Engineering, College of Engineering, Trivandrum 17
Target Maneuver Adaptive Guidance Law for a Bounded Acceleration Missile
where
11
ze (tgo ) =
ˆ μi zi (tgo )
ˆ (4.12)
i=1
Remark 3: The guidance gains Ni (tgo ) of the different models in Eq. (4.10) can be computed
a priori. As is common in the missile guidance community, these gains can then be stored
in a lookup table to be used online.
4.3 Performance Evaluation
4.3.1 Comparison to the Bound
To compare the performance with that of the previous section , the same scenario will be
addressed and two types of behavior will be evaluated: 1) A target behaving according to
the assumed model, for example, commands a step in acceleration for which the amplitude
is chosen form a Gaussian distribution with the known σjump . 2) A target performing a step
with a magni
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