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STANKOVIC
´
et al.: DECENTRALIZED OVERLAPPING CONTROL OF A PLATOON OF VEHICLES 819
have the appealing block-diagonal structure from (7), which
immediately allows the application of the sequential design
methodology proposed below. For example, for
this
matrix would have the form
where . Notice that the structure in ((7) is ob-
tained by an adequate choice of
in (34), satisfying (35) (see
also [7], [8], and [16]).
After expansion, the subsystems in
represented by (7) ap-
pear as disjoint. Our immediate aim is then to design the state
feedback controller
for by local design of state feedback
controllers
for the subsystems (the gain are
assumed to be the same for all the subsystems for simplicity).
We shall apply a methodology based on LQ optimization with
respect to the local performance indexes attached to the sub-
systems. Connections between local and global performance in-
dices in the context of the inclusion of performance indexes for
(i.e., for ) and are discussed from the general standpoint
in [6], [16], and [17]. However, in the context of the desired per-
formance and the assumed information structure in
, we shall
define the local performance indexes directly, starting from the
basic platoon control requirements presented in Section II.
As the vehicle dynamics represents formally a part of each
subsystem, we shall describe the details of the proposed control
strategy sequentially, starting from the leading vehicle.
B. Leading Vehicle Control Law
The leading vehicle is supplied with the reference command
and uses its own state vector for control design. Formally
speaking, according to (1) and (4), it is represented by the
following model:
(8)
where
. The optimal state feedback control law
will be found from the condition for the minimum of the perfor-
mance index
(9)
where
is a time-varying reference supplied to the
first vehicle, which is known entirely in advance, and
and are the corresponding weighting matrices. Ac-
cording to [1], the resulting LQ optimal tracking control law is
given by
(10)
The obtained control law is, in general, only suboptimal, since
the feedforward matrix is constant [1]. It is not, however, far
from the optimum, having in mind characteristic forms of the
reference signals within platoons.Apriori choice of the criterion
weights provides different tracking properties.
Notice that the steady-state error with respect to a
constant velocity reference always reduces to zero, i.e.,
and , having
in mind that
is singular [1]. Namely,if diag ,
the only nonzero term in the two-dimensional vector
in
(10) is
(for a vector ,
denotes its th component). The condition implies
(since and, having in mind that
, we obtain finally
(for a matrix , denotes its th element). The last equa-
tion coincides, however, with the scalar equation corresponding
to the (1,1)th element in the matrix Riccati equation for
in
(10).
C. Subsystem State Feedback
Control design for the subsystems defined by (6) can, in gen-
eral, be based on the LQ optimal tracking methodology, pro-
viding the entire input vector
as a function of
the subsystem state vector
. However, such a solution is not in
accordance with the assumed platoon controller structure, since
the control signal
in the st vehicle would become
in such a way a function of
, and , i.e., of the state vector
of the following,
th vehicle. Having this in mind, we shall pro-
pose in this section a different approach, based on sequential op-
timization, following the idea exposed in [12], [17] in relation
with the systems possessing the general lower block triangular
(LBT) structure. Namely, as the subsystem model (6) possesses
the LBT structure with two main diagonal blocks, we shall at-
tach to it two criteria
(11)
(12)
where
,
is the complete set of reference commands corresponding to all
the components of
, while , , and
are appropriately defined weighting matrices.
The sequential design procedure starts from the top of the hi-
erarchy by finding the optimal input
minimizing the crite-
rion
for the model extracted from
(6) (see general aspects of treating LBT structures in [12], [16],
and [17]). The solution to this optimization problem coincides
with the control law for the leading vehicle defined by (10) with
and [the leading vehicle model represents
formally a part of (6)], i.e.,
(13)
where
and are defined by (10). After implementing (13)
in (6), one obtains
(14)