Signal Processing - November 2016 - 77

the coordination region (4f). The problem captures the basic
requirements: safety through (4e), liveness through (4f), and
performance through the objective function. Problem 1 can
be conveniently reformulated in a discrete time domain by
discretizing the systems dynamics (4b). Furthermore, to
solve a finite dimensional problem, receding horizon control (RHC) or model predictive control (MPC) schemes can
be used [7], where a finite time optimal control problem is
solved every sampling time instant. In particular, as explained
in Figure 3, RHC seeks future input and state trajectories at
every sampling time instant over a finite time horizon, so
as to minimize the cost function, subject to the constraints.
The first element of the computed control input sequence is
applied to the system, and, at the next time step, the problem
is formulated and solved over a shifted time horizon. This
RHC approach also allows us to account for the future, but by
only committing the control action for the current time, we
are able cope with limited disturbances (e.g., due to imperfect
sensing or communication). This is important, as we will see
in the next section.

Challenges in solving the coordination problem
Although finite dimensional, solving problem 1 in a receding horizon framework is extremely challenging, not only
from the control perspective, but also due to imperfect communications as well as uncertainties induced by the sensors.
While all these challenges are interrelated, we break them
down as follows.

Control challenges
The main control-related challenges involve, first, the ability to
compute good, feasible control actions, and, second, the ability
to guarantee that the closed-loop system has certain desired
properties. Regarding the former, note that the mathematical
problem of finding the actions of N vehicles that allow them
to pass the coordination zone without colliding is inherently
a combinatorial problem. For a given initial configuration, a
multitude of feasible temporal crossing orders (i.e., different
orders in which one vehicle passes a coordination zone before
another) might exist, and the optimal ordering can only be
found by a structured exploration of the different alternatives.
It is therefore no surprise that even the problem of finding a
feasible solution to (4) is NP-hard in general [8]. Exact solutions to the OCP are therefore intractable for relevant problem sizes, and either heuristics or approximations must be
employed. Regarding the properties of closed-loop control,
there are several challenges. For instance, given the severity
of constraint violations in the OCP, any controller needs to
ensure persistent feasibility. If satisfied, this property ensures
that any action taken does not put the system in a state from
which no feasible actions exists, i.e., that no vehicle is ever
put in a state from which a collision is inevitable. The closedloop controller must also ensure stability, e.g., to make certain
that the crossing order does not change every time the solution
is recomputed. Additionally, the aforementioned issues are
linked, as the computational challenges of the mathematical

Prediction Horizon
Output Reference
Measured Output
Predicted Output
Previous Inputs

Optimal Input at k
Optimal
Input at k + 1

Past

k k+1

Future

k+N

Figure 3. An illustration of an RCH scheme. The sketch depicts how, in
an RCH algorithm, a present decision is made based on the current state
of the system and its predicted future behavior.

coordination problem, for instance, might promote distributed solutions. In that case, the closed-loop controller needs
to guarantee the aforementioned properties while the solution
is obtained iteratively and possibly asynchronously over the
wireless vehicular network.

Communications challenges
Irrespective of how the OCP is solved, information exchange
is required between the involved entities (e.g., vehicles and
possibly dedicated infrastructure). First and foremost, this
includes the information necessary to formulate the OCP,
e.g., the models of vehicle dynamics, road geometry, state
measurements, and static and dynamic map information. In
addition, it also includes information required to solve it,
e.g., internal messaging in a distributed, iterative algorithm.
Communication between entities will be greatly affected by
the impairments associated with wireless channels, including the inherent randomness and correlation of the channel, interference due to simultaneous transmissions, and a
limited communication range. In combination with limited
communication resources (bandwidth, power), this results
in packet drops and random latencies in packet arrivals. For
automobile applications, it was pointed out that the current
standards for V2V and V2I communication cannot ensure
time-critical message dissemination in dense scenarios [9].
In general, it is desired to keep the communication load low,
as wireless channel congestion is envisioned to be one of the
major challenges related to vehicular networks [10], [11].
Overall, the communication subsystem forms a bottleneck
for the OCP, related both to its formulation and the means by
which it is solved.

Sensing challenges
The vehicles' own perception of their current locations and
the positions of surrounding vehicles is fundamentally uncertain. Both are based on observations from sensors
such as cameras, radar, lidar, the global navigation satellite system (GNSS), and inertial navigation sensors, which
deliver observations that are corrupted by noise and clutter

IEEE SIgnal ProcESSIng MagazInE

November 2016

Table of Contents for the Digital Edition of Signal Processing - November 2016