A Hybrid, PDE-ODE Controller for Intercepting an Intelligent, Well-informed Target in a Stationary, Cluttered Environment

Abstract

A new class of intelligent controllers that can semantically embed an agent in a spatial context constraining its behavior in a goal-oriented manner was suggested in [1,11]. A controller of such a class can guide an agent in a stationary unknown environment to a fixed target zone along an obstacle-free trajectory. Here, an extension is suggested that would enable the interception of an intelligent target that is maneuvering to evade capture amidst stationary clutter (i.e. the target zone is moving). This is achieved by forcing the differential properties of the potential field used to induce the control action to satisfy the wave equation. Theoretical developments, as well as, proofs of the ability of the modified control to intercept the target along an obstacle-free trajectory are supplied. Simulation results are also provided.

Full text

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A Hybrid, PDE-ODE Controller for Intercepting an
Intelligent, Well-informed Target in a Stationary,
Cluttered Environment
Ahmad A. Masoud
Electrical Engineering Department, KFUPM, P.O. Box 287, Dhahran 31261, Saudi Arabia, e-mail: masoud@kfupm.edu.sa
Abstract - A new class of intelligent controllers that can
semantically embed an agent in a spatial context constraining
its behavior in a goal-oriented manner was suggested in [1,11].
A controller of such a class can guide an agent in a stationary
unknown environment to a fixed target zone along an obstacle-
free trajectory. Here, an extension is suggested that would
enable the interception of an intelligent target that is
maneuvering to evade capture amidst stationary clutter (i.e. the
target zone is moving). This is achieved by forcing the
differential properties of the potential field used to induce the
control action to satisfy the wave equation. Theoretical
developments, as well as, proofs of the ability of the modified
control to intercept the target along an obstacle-free trajectory
are supplied. Simulation results are also provided.
I. Introduction
The survival of a specie is dependent, among other things, on its
ability to develop its navigation competence to successfully
handle a prey-hunter situation. The developed capabilities are
subjective in nature, that is: the structure of the capture scheme
is built around cues that are meaningful to the hunter and related
to aspects of the prey’s personality. In a situation where an
intelligent prey is being hunted, the prey may be aware of the
hunter. Even more, it may be aware of what the hunter expects
from it. In such a situation, the prey may initiate a deception
scheme that is masked by its expected behavior with the goal of
engaging in an interactive message exchange with the hunter.
The structure of this exchange is based on the model which the
prey has for the hunter’s decision making process. Its aim is to
out-maneuver its pursuer and evade capture. Through this
exchange, the prey may even acquire a soft control of the hunter
that could reverse the role of each. Here, the nesting of action
release, or equivalently message exchange, has the form of I
KNOW that IT KNOWS that I KNOW etc. [2]. The deeper the
nesting is, the more likely that the actions released by the party
concerned drive it to the desired conclusion. Such subjectivity
seems inherent in the structure of these kind of problems. This
has convinced the majority of researchers in the area of the need
to use evolutionary techniques (e.g. neural networks, genetic
algorithms, etc.) to “absorb” the personality profile of the target
in order to derive a successful capture scheme. While profiling
may work most of the time, the situation is dramatically different
for the case of active intelligent targets. These targets are
capable of adjusting their trajectories to enhance their chance of
survival by extending their domain of awareness to include that
of their pursuers. To the best of this author’s knowledge, this
class of problems was not addressed in the literature [3].
This paper hypothesizes the existence of objective techniques for
target interception whose success is directly tied to the potential
of achieving a desired conclusion instead of being dependant on
the psychology of the prey. Testing the above hypothesis is
carried out with the help of a newly introduced class of
intelligent motion controllers [1,11].Up-till-now such
controllers are implemented using harmonic potential fields
(HPFs). HPFs possess many useful properties that make them
excellent tools for navigation [13]. Most notably, is that an HPF
is also a Morse function (see appendix). Despite the efficiency
of HPF-based methods in tackling environments with
sophisticated geometry, they can only engage simple, stationary
targets (a sitting duck). In this paper a modified version of the
control that utilizes the wave potential is suggested for engaging
active, intelligent targets in a provably-correct manner.
This paper is organized as follows: in section II the problem is
formulated. Section III provides a background on PDE-ODE
controllers. Sections IV and V presents the PDE and the ODE
components needed for constructing the control. Section VI
contains motion analysis. Simulation results and conclusions are
in sections VII and VIII respectively.
II. Problem Formulation
Designing a method for intercepting a moving target can be a
challenging task, especially if an active intelligent target is
engaged. The target may attempt to intelligently maneuver to
evade capture using full knowledge of its surroundings which
may be as complex as a maze. It may also be well-informed
about the movements of its pursuer. Here, planning the
movements of the pursuer in a manner that can cope with the
above situation is carried out using the nonlinear, dynamical
system:
•
=
( ( ( ),
The above system is required to make:
1
xg V x t xtx
p
∈
( ), )) ( )0.
ΓΩ
2
and x(t) 0 S œ t ,
where S is an admissible subset of the N-dimensional state
space, ' is the boundary of the non-admissible subset of state
space (O, '= MO), xp (t) is the trajectory of the target ( xp (t) 0 S
œ t ), x(t) 0 RN, g: R 6RN, V: RN×RN×RM 6 R, where '0 RM,
M#N-1. The above implies the followings:
1- the target may be intelligent,
2- the target may have an accurate model of its environment, as

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well as information about the movements of its pursuer,
3- no psychological profile of the target, its tendencies, and
habits or, for that matter, a statistical model of the target’s
behavior are needed ,
4- the pursuer has a good model of the environment, and full
information about the movements of the target.
The target is assumed to have limited power so that it cannot
instantly change its position or orientation (i.e. xp(t)0CL, L$2).
III A Background
The controller of interest (figure-1) functions to convert the goal,
constraints on behavior, and the available representation of an
environment into a sequence of actions {u0,..,uL}. These actions must
yield a corresponding sequence of states {X0,...,XL} so that the final
state XL is the goal state of the agent, and all the transient states satisfy
the constraints on behavior. The action sequence is called a plan which
is a member of a field of plans (action field) that densely covers state
space so that regardless of the starting point, a plan always exists to
move the agent to its goal.

X g V
•
= ( )
Figure-1: A PDE-ODE Controller.
The information needed for synthesizing the action field from a PDE-
ODE Controller is generated by the synergetic interaction of a massive
number of differential systems (micro-agents), figure-2. A micro-
agent has a structure that is identical to the agent being
controlled, with the exception that its state is stagnant and
immobilized to an a priori known location in state space.
Figure-2: A collective of micro-agents.
The actions of the micro-agent collective are emulated using a
potential field that is operated on by a vector partial differential
operator. The synergetic interaction needed to convert the
individual actions of the micro-agents into a guidance-capable
group action is achieved using the two steps:
1- the individual micro-agents are informationally coupled
(figure-3). This is achieved using the proper partial differential
governing relation,
2- the process of morphogenesis [4], which is responsible for
guiding the group structure of the action field, is activated by
factoring the influence of the agent’s environment in the
behavior generation process. This influence is factored-in using
boundary control action. The overall control structure is induced
by solving the boundary value problem (BVP) that is
constructed using the partial differential governing relation from
step-1, and the boundary control action.

Figure-3: An informationally-coupled micro-agent.
The above mode of behavior synthesis is in conformity with the
artificial life approach to behavior generation [5]. Figure-4
shows the evolution of the action field for an HPF-based
controller.
Figure-4: Evolution of action structure
IV. The PDE Component
Scalar potential fields that describe changes in both space and
time are suitable for synthesizing action fields that can be used
for tracking moving targets. The partial differential relation that
may be used to govern the differential properties of such
surfaces is the wave equation (WE):
1
V
a
∂
where a is a positive constant, L is the gradient operator, L2 is
Laplace operator. The reason the WE is chosen over other
spatio-temporal governing relations (e.g the diffusion equation)
is mainly due to the nature of its solution. From the method of
separation of variables, the solution of a field that is dependent
on both space and time (V(x,t)), and is governed by the WE may
be written in the form:

V x t R x T t( , ) ( ) ( )
=
where R is the position-only dependent component of the
solution, and T is time-only dependent component. Therefore,
the WE may be placed in the form:
3
∇=
2
2
2
2
V
t
∂
4
5

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The only way equation 5 can be satisfied is for the position and
time dependent terms to be equal to the same constant which is,
for convenience, chosen equal to -82 :
6
As a result, T(t), and R(x) may be computed by solving the
following Helmholtz equations (HEs):

∇+
R
λ
∂
∂
t
is well-known that the fundamental solution of an N-D HE
provides N orthogonal basis functions capable of representing an
arbitrary scalar function that is defined on that space. Therefore,
the above set of equations yields N+1 orthogonal basis enough
to represent (using the generalized Fourier series expansion) any
piecewise continuous function in both time and space.
N-D HE 7
=
22
0R
1-D HE It
λ
a
2
2
2
0
T
T
+=
()
A. BVP-1, The Dirichlet Case:
The generating BVP is (Figure-5): solve
8
subject to V(x,xp(t),')=C for x0', and V(xp(t),xp(t),')=0,
where C is a positive constant, and xp(t) is the target’s
trajectory.
Figure-5: Boundary Conditions, the Dirichlet case.
B. BVP-2, The Homogeneous Neumann Case:
The generating BVP is: solve
9
subject to V(x(0),xp(t),')=C, MV(x,xp(t),')/Mn=0 for all x0', and
V(xp(t),xp(t),')=0, where n is a unit vector normal to '.
Existence and uniqueness of the solution of the above BVPs
were proven in [6,7].
V. The ODE System
The interception trajectory is generated using the first order
nonlinear differential system:
10
It can be shown that the above ODE system is capable of
satisfying condition 2. Unfortunately, the system encounters a
singularity when x(t)=xp(t). To remedy this problem, the
condition in 2 is relaxed to:
11
where 1>>D>0. A singularity-free ODE system that can
accomplish the above is:

where 12
and 0(x) is a monotonically increasing function that satisfies the
followings: 0(0) = , , 0(D) = D D>,>0,
13
A form for 0(x) that satisfies the above conditions is:
14
It ought to be noticed that the partial, implicit dependence of
MV/Mt on dxp/dt does not imply that the velocity of the target is
needed for its computation. Any numerical procedure for the
solution of the time dependent potential computes MV/Mt using
the a finite difference approximation:
V x t xtV x t
dt
Constructing the above approximation requires only that the
time-dependant position of the target be estimated.
. 15
dt x ),t dt
pp
( ( ), ( ), )( ((), )
ΓΓ−−−
A. The solution for the ODE System in 12 Exists
The nonlinear ODE system in 12 satisfies the global Lipschitz
condition [8]. In other words, for every time interval J0[0,4), ›
the constants kJ< 4, and hJ< 4, so that
16
and
œ t 0 [0,J], and œ x,y 0 Rn . This means that the solution of 12
does exist, and is unique. This in turns imply:
1- x is differentiable almost everywhere (i.e. dx/dt exists),
2- the relation:
holds for t where dx/dt exists,

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3- x(t) satisfies: 17
VI. Motion Analysis
In this section, the ability of the suggested PDE-ODE system to
enforce the convergence and avoidance constraints in 2 is
examined. The proof is based on Liapunov second method [9].
A key element of the proof is showing that the wave potential
in section IV is a Liapunov function candidate (LFC).
A. The Wave Potential is an LFC
It is well-known that the solution of the WE is analytic. This
satisfies the requirement that a LFC be differentiable, or at least
continuous. It can also be shown that surfaces with differential
properties that are governed by the linear, elliptic, WE partial
differential operator satisfy the maximum Principle (i.e they are
free of local extreme, where minima or maxima of such
functions can only occur on the boundary of the space on which
V is defined (')), [10]. This in turn leads to the satisfaction of
the second condition required by an LFC, that is:
1- V(x(ti) , xp (ti), ') = 0 at, and only at x=xp, 18
2- V(x(ti), xp (ti), ') > 0 x0S, œ ti 0 t.

B. Liapunov’s Direct Method
A point xp is considered to be an equilibrium point of the system
in 1 if
g(xp (t),xp(t), ') = 0 œt $0 . 19

This equilibrium point is considered to be globally,
asymptotically stable (i.e. x(t)6xp(t) as t64 ) if › a LFCh:
1- dV(x(ti), xp(ti),')/dt =0 at and only at x=xp, 20
2- dV(x(ti), xp(ti),')/dt < 0 x0S, œ ti 0 t.
C. Convergence Analysis
The time derivative of the wave potential is:
. 21
V x t x( ( ),t V x t x( ( ),tx
V x t x ( ( ),
∂
t
t
pp
t
p
••
= ∇+
( ), )( ), )
( ), )
ΓΓ
Γ
∂
Substituting the value of dx/dt from 10, we have:
22
Since, from the maximum principle, V contains no local extrema
in S (note that ', and xp ó S) where, by design, the minimum
is placed at xp and the maximum at ', the time derivative of the
wave potential satisfies the conditions in 20. Therefore, V is a
valid Liapunov Function. Hence, global asymptotic convergence
to xp from anywhere in S is guaranteed.
In a similar way, it can be shown that for the singularity-free,
dynamical system in 12 dV/dt < 0 œ x0S-BD , where
23
This implies that: 24
D.1. Avoidance Analysis (the Dirichlet Case)
Let '* be a thin region surrounding '. Proving that motion will
not be steered towards ' inside '* is sufficient to prove that the
forbidden regions will not be entered.
Assuming that x is initially inside '*, let xn be the distance
between x and ':

xx
n
=
n
where n denotes a unit vector normal to '. Since n is not a
function of time, the time derivative of xn is equal to:
, 25
t
26
Since the state is initially assumed to be outside ' (xn > 0),
proving that a measure of the length of xn is always non-
decreasing is sufficient to prove that the state will never enter
the forbidden regions.
Let Va be a measure of xn:
27
The time derivative of Va may be computed as:
28
Since the value of V on ' is constrained to a constant C,
29
Since x is assumed to be very close to ' (x0'*), the following
approximation may be constructed:
30
Also, from the maximum principle, we have
31
Therefore 32
Using the above results it can be seen that the value of the time
derivative of the Liapunov function inside '* is equal to
V
n
∂
Therefore, the guidance field will always push x away from '*,
steering it away from '. In other words, the forbidden regions
will be avoided.
33
Vx
n
•
≈ −>
20
∂
D.2. Avoidance Analysis (the Neumann Case)
In the Neumann case MV/Mn is set to zero at '. Substituting this
in 28, we have:
•
= 0
In other words, motion cannot proceed towards '; hence the
forbidden regions will not be entered.
x0' 34
V

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VII Simulation Results
In this section, the tracking and region avoidance capabilities of
the wave potential approach are tested. Three types of partial
differential relations are used to govern the point properties of
the potential field:
1-The Lpalace Equation

applied in a quasi stationary manner with the ODE system:

, 35
∇=
2
0V
. 36
xV
•
= −∇
2-The Diffusion Equation 37
the same ODE system in 36 is used.
3- The suggested Wave Equation approach.
The results from the three approaches are compared for a
stationary, linearly moving, and slowly moving targets . It was
observed that all techniques exhibit equivalent capabilities in
terms of converging to the target and avoiding the forbidden
regions. However, the disparity between the performances of the
above techniques greatly widened when the linear motion of the
target was augmented with rapid, sinusoidal oscillation. Figure-6
shows the response of the quasi stationary Laplace approach. As
can be seen, the rapid fluctuations which the target
superimposed along its escape path confused the tracker leaving
it undecided whether to exit the corridor from the right side or
the left side (figure-7). The total failure of the quasi-stationary
Laplace strategy is a consequence of its total disregard to the
temporal dependence of events and its reliance on spatial
information only in guiding the tracker to the target. The
absence of the dimension of time from the strategy of the
interceptor opens a “hatch” for the target through which this
neglected dimension is exploited not only
Figure-6: Response of the tracker, Lapalace.
Figure-7: x,y components of trajectory vs time, Lapalce.
to evade capture by the interceptor, but even to totally paralyze
it. Here, the target is aware that the interceptor is tracking it
along the minimum distance path with no regard to where the
target is moving to next, or how fast it is going there. Therefore,
once the target observes a move of the tracker, it quickly
repositions itself so that the distance of the interceptor to the
new location of the target is shorter if the interceptor moves
along a direction that is opposite to the one it has. By
continuously repeating this relatively simple maneuver, the
target is able to bring the motion of the interceptor to a standstill
along the x-axis trapping it in a limit cycle along the y-axis
(figure-7). Although the ODE system for the Diffusion strategy
uses only spatial information to lay an interception trajectory to
the target, temporal information is indirectly utilized in encoding
the target and environment information in the potential field. For
this case (figure-8), despite the ability of the tracker to proceed
in the general direction of the target, it fails to keep up with its
rapid fluctuating movements.
Figure-8: Response of the tracker, Diffusion .
As for the Wave Equation strategy, the interceptor was able to
closely follow the target (figure-9).
Figure-9: Response of the tracker, Wave.
VIII Conclusion
This paper focuses on a special class of pursuit-evasion
problems concerned with intercepting a well-informed, active,
intelligent target that is maneuvering to escape capture in a
known, stationary, cluttered environment. The paper examines
the possibility of the existence of a tracking method that is not
dependent on the psychological profile of the prey or the manner
it may utilize the knowledge it has to evade capture. It seems
that the suggested wave potential-based controller has the ability
to exhibit such objectivity in goal-oriented, action synthesis.
There are still many questions that need to be answered about
the behavior of the suggested approach. For example, the ability
of the tracker to always intercept the target regardless of the

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amount of information which the target has, or the efficiency of
its utilization needs to be carefully examined. Another question
is concerned with the effect of placing dynamical constraints on
the tracker’s ability to intercept a target. Another issue has to do
with the effect of delays caused by the potential field synthesis
process on the interception ability of the pursuer. These are
some of the important questions which future investigations of
this approach should carefully answer.
Acknowledgment
The author thanks KFUPM for its support of this work.
Appendix
A. Definition: Let V(X) be a smooth ( at least twice
differentiable), scalar function (V(X): RN 6 R). A point Xo is
called a critical point of V if the gradient vanishes at that point
(LV(Xo)=0); otherwise, Xo is regular. A critical point is Morse,
if its Hessian matrix (H(Xo)) is nonsingular. V(X) is Morse if all
its critical points are Morse [12].
B. Proposition: If V(X) is a harmonic function defined in an N-
dimensional space (RN) on an open set S, then the Hessian
matrix at every critical point of V is nonsingular, i.e. V is Morse.
Proof: There are two properties of harmonic functions that are
used in the proof:
1- a harmonic function (V(X)) defined on an open set S contains
no maxima or minima, local or global in S. An extrema of V(X)
can only occur at the boundary of S,
2- if V(X) is constant in any open subset of S, then it is constant
for all S.
Other properties of harmonic functions may be found in [13].
Let Xo be a critical point of V(X) inside S. Since no maxima or
minima of V exist inside S, Xo has to be a saddle point. Let
V(X) be represented in the neighborhood of Xo using a second
order Taylor series expansion:
V(X)V(Xo)V(Xo) (X Xo)
= + ∇−
*X-X0*<<1. 38
1
2(X
Xo) H(Xo)(XXo)
T
T
+
−−
Since Xo is a critical point of V, we have:
V V(X)-V(Xo)
=
*X-X0*<<1. 39
1
2(X
Xo) H(Xo)(XXo)
'
T
=
−−
Notice that adding or subtracting a constant from a harmonic
function yields another harmonic function , i.e. V` is also
harmonic. Using eigenvalue decomposition:

1
2(X
..
00.
⎣

⎡
⎢
⎢
⎢
⎢
00.
40
VXo) U
00
.
⋅
0
0
.
0
U(XXo)
'TT
1
2
N
=−
⎡
⎢
⎢
⎢
⎢
⎤
⎥
⎥
⎥
⎥
⎦
⎤
⎥
⎥
⎥
⎥
−
λ
λ
λ
=
⋅
⎣⎦
=
=∑
i 1
1
2
00
.
0
0
.
0
..
T
1
2
N
ξ
λ
λ
λ
ξλ ξ
i i
1
2
N
2
where U is an orthonormal matrix of eigenvectors, 8’s are the
eigenvalues of H(Xo), and >=[>1 >2 ..>N]T = U(X-Xo). Since V`
is harmonic, it cannot be zero on any open subset S; otherwise,
it will be zero for all S, which is not the case. This can only be
true if and only if all the 8i’s are nonzero. In other words, the
Hessian of V at a critical point Xo is nonsingular. This makes
the harmonic function V also a Morse function.
It ought to be mentioned that a navigation function defined in
[14] is a special case of a harmonic potential field. According to
[14] a navigation function must satisfy the following properties:
1- it is smooth (at lest C2),
2- it contains only one minimum located at the target point,
3- it is a Morse function,
4- it is maximal and constant on '.
A harmonic function (V) is C4 and Morse. Harmonic functions
are extrema-free in S. Their maxima and minima can only
happen at the boundary of S. In the harmonic approach ' and
the target point (XT) are treated as the bounary of S. Through
applying the appropriate boundary conditions, the minimum of
V is forced to occur on XT. Also by the application of the
Drichlet boundary conditions, the value of V is forced to be
maximal and constant at '. The Drichlet condition (constant
potential on the boundary) is one of many settings used in
constructing a harmonic potential that may be used for
navigation.
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