### Greedy vs Centralized MPC for platooning

Have you seen a commercial of a train of cars traveling on a highway and keeping the same inter-vehicle distances and speeds? Yes! You think it is fun, comfortable and makes our trips easier and not nerve-racking. Guess what? It is more advantageous than what you think. Believe it or not, it can be a significant contribution to save the planet from the increasing atmospheric concentration of carbon dioxide and a great deal for the economy as well! However, the latest control strategies proposed that varying the distances and speeds at hilly terrains could even save more fuel. Curious about how the control design will look like? Then you are in the right place.

#### Introduction

One of the things that has been always drawing my attention is the automated

vehicular control strategies and how they could reshape the transport sector

dramatically. One of the methods that many automotive manufacturers have

been recently developing is what is called platooning. A platoon is a convoy of

trucks that maintains fixed inter-vehicular distances, as shown in the Figure 1,

and usually applied on highways.

Figure 1: Trucks Platoon

The advantages go beyond the driver’s convenience and comfort. Having a lead

truck with a large frontal-area would reduce the air drag force acting on the

succeeding vehicles. Therefore, the required torque to drive the trucks at cer-

tain speed will be decreased which lead to less fuel consumption. That means,

of course, less CO2 emissions and lower financial burdens.

However, in a single-vehicle level, there is another approach that has been inves-

tigated for a better fuel economy. This approach utilizes the future topography

information in order to optimize the speed and the gear for a vehicle travelling

in a hilly terrain by exploiting the vehicles’ potential and kinetic energies stor-

ages. In this approach the velocity will vary along the road depending on the

road gradient. The look-ahead strategy could be seen as a contradiction to the

platooning approach in which vehicles maintain almost the same speed along

the road.

#### HOW TO HANDLE IT?

A combination between these approaches could be implemented using the model

predictive control (MPC) scheme. Since there are many process constraints,

such as inter-vehicular distances, engine maximum torque, road velocity limits,

etc. MPC is a perfect candidate to handle these constraints especially that in

many cases the system will be operating close to the limits. The control design

could be handled in two approaches, the centralized control design and the

decoupled control design. In the centralized controller, as shown in the Figure

2, all the vehicles’ private data such as mass, engine specs, etc. in addition to

their states such as velocity and time headway are sent to the central predictive

controller via vehicle to vehicle communication, could be in one of the trucks

probably the lead vehicle or even in a cloud. One of the methods used for optimal

control is the convex quadratic programming problem (CQPP) in which every

local minimum is a global minimum. The problem is as follows

$$ min\,z = f_0(x) \\

f_i(x) \leq 0 \\

Ax = b $$

Where f0,f1,f2, …, fm, is the objective function, and the inequality constraints

are convex functions. However, the equality constraints are affine functions.

In the platoon case, some convexification is needed in order to get CQPP. Hense,

the problem is solved and the optimal speed and time headway references are

sent back to the vehicles’ local controllers. This approach optimizes the fuel

consumption for the whole platoon rather than individual vehicles in which the

group interest comes first. One of the drawbacks of this approach is that in order

to solve the problem you need to handle huge matrices since all the vehicles info

is handled at once. In other words, this approach is rather computationally

expensive.

Figure 2: Centralized adaptive cruise control

The decoupled architecture, as depicted in the Figure 3, could be a solution for

the computation capacity issues. Instead of handling the quadratic program-

ming (QP) problem for the whole platoon, each vehicle considers itself, which is

why called greedy. The problem starts to be solved from the leading vehicle and

goes backwards. Each vehicle solves the QP, considering the gaps in front of the

vehicle and the road topography, and sends states to the succeeding vehicles.

The pros of this approach are that trucks need not to share their private data

and the matrices sizes are much smaller. So the computation time is less than in

the greedy control strategy but the solution is not as optimal as the centralized

controller.

Figure 3: Greedy approach

#### CHALLENGES

As it is mentioned above, formulating a convex quadratic programing problem

is used to get the fuel-saving velocities. Since the vehicle dynamics are quite

nonlinear, linear approximations are needed, therefore, finding an appropriate

velocity reference is essential, assuming that the vehicle will be driven close

to the reference. Finding such reference should consider many factors such as

maximum traction force along the road, road limits and the cruise speed set by

the driver. One of the other challenges is gear optimization which could be solved

using dynamic programming. The complexity of dynamic programing problem

increases exponentially with the rise of the vehicles number, as a result, the

problem become computationally demanding, therefore, it is not very reliable

for the real-time implementation.