In the previous section, we showed how solutions to the heat equation can be produced. These solutions use half of the available degrees of freedom to so that the left-most and right-most agent pass through preset waypoints. In this section, we show how this method can be adapted to construct solutions to Burgers’ equation.
4.2.1
Optimization of the trajectories
It is possible to turn a solutionθto the heat equation constructed in the previous sections into a solutionφto Burgers’ equation. This is done thanks to Hopf-Cole transformation that was recalled and proved intheorem 2.1.3:
φ=−2µθx
θ (4.9)
Creating an appropriate solution is thus a two-sided problem. On the one hand, the solution to the heat equationθhas to be of strictly constant sign. We may see in the sample controls depicted infigure 4.1that adding waypoints create some “overshoot” between these waypoints. This can lead the solution to cross the origin. On the other hand, the solutions to Burgers’ equation obtained from the Hopf-Cole transformation should follow a determined set of conditions. This two-sided problem is solved by means of optimization.
Formally, we may define our goals as follows. First, we want the solution to the heat equation to be strictly positive. We define c, the minimum of φ(x, t) on the unity square (e.g. A = {(x, t), 0 ¶ x, t ¶ 1}). Thanks to the minimum principle (see e.g. [Logan, 2008, part. 7.2.1, p. 353]), this minimum is located on the border of the unity square (e.g. A− Awhere Ais the interior of A: A={(x, t), 0 < x, t < 1}). This minimum can be found as:
c = minn inf
0¶x¶1f0(x), inf0¶x¶1f1(x), inf0¶t¶1φ0(t), inf0¶t¶1φ1(t)
o
, (4.10)
Second, if the first condition is verified, i.e. if c > 0, we may evaluate the solution to Burgers’ equation using the Hopf-Cole transformation. This solution u(x, t) on A is evaluated using a finite difference scheme. Doing this, we want the function u(x, t) to meet specific requirements. These requirements can depend on the objectives of the system. In the case of trajectories in space, these objectives may encompass collision avoidance. In the following, we will minimize the lengths of the paths of the leader and the anchor. This length is evaluated as:
〈u〉(x) = Z 1
0
|ut(x, s)|ds − |u(x, 1) − u(x, 0)|. (4.11)
The overall cost function is then defined as:
〈f0, f1〉 =
¨
c if c ¶ 0
A convenient way to find extrema of certain functions is to use Particle Swarms Optimization (short PSO, see subsection 1.1.2 – Methods based on Particle Swarm Optimization). An advantage of PSO over other optimization frameworks, is that it does not need derivations of the cost function. Therefore, this process is adapted to optimization in high dimension or with noisy data. A simple description of the PSO algorithm can be given in two steps as follows:
Initialization: A set of n particles – called the swarm – is created with random positions. These
positions are uniformly distributed over the solution space [xmin, xmax]n. The speeds of
the particles are randomly initialized. This initialization is done with value uniformly distributed over [−|xmax− xmin|, |xmax− xmin|]n. The local best position of each particle is initialized to the particle’s initial position while the global best particle is searched among them.
Evolution: At each step of the evolution, the speed of each particle is updated based on its
current speed and position and on the knowledge of the best local and global particles. The particles are then moved according to the new speed and evaluated.
It is believed that the swarm will converge to the global optimum if it exists or, if not, to a local optimum. However this convergence depends on the various parameters that can be chosen and the swarm can, in some cases, diverge or oscillate. The choice of these parameters has thus to be performed with care whereas with a certain freedom.
Inspired by the way the coefficients for the state representations f0 and f1 were found, we search for a way to find these coefficients using the Hopf-Cole transformation. We may write, for each of the agent of index xk,:
u(tj, xk)
−2µ fj(xk) =f
′
j(xk) (4.13)
We search for an appropriate matrix Mjso that:
pj1=Mjpj0, (4.14)
Where pjicorresponds either to pior qiin the preceding. We write Uj the diagonal matrix of the scaled positions (−u(xjk, tj)/2µ)0¶k<Nof the N agents at time tj. We write Ajthe matrix of generic term ((xk)2l/(2l)!)0≤k<N,0≤l<Nand Bjthe matrix of generic term ((xk)2l+1/(2l + 1)!)0≤k<N,0≤l<N. D is the derivation matrix :
D = 0 1 0 . . . 0 ... ... · · · · · · · · · · · · · · · · · 0 1 0 · · · · · 0 (4.15)
Equation (4.13)can be written as:
Uj(Ajpj0+Bjpj1) =BjDpj0+Ajpj1, (4.16)
(UjBj− Aj)pj1= (BjD− UjAj)pj0, (4.17)
During the optimization process, the particles are chosen in the N-dimensional space of pj0 (or 2N-dimensional space of pj0 and pj+10 as will be explained in the following). The odd coefficient are then computed based on the positions of the agents and on the particles. Algorithm 1.1
is then adapted and applied to this system. As a result, it outputs a possibly optimal set of coefficients for the state representation f0 and f1. This is adapted as explained in the following subsections.
4.2.2
Leaders and followers
In our framework, we distinguish two types of agents: leaders and followers. The agents introduced in the previous section for which a start and an end state can be imposed are leaders. Their positions are determined and chosen at all the desired transition states. As a consequence of the adopted formalism, a function fi inequation (3.166)represents a formation of N leaders.
In addition to these leaders, another class of agents is introduced: followers. This class is inspired by the agents used in [Meurer and Krsti´c,2011]. Their positions in the start and end states are a result of the position of the leaders. The number of followers is not limited by the representation chosen inequation (3.166). These followers are described by indices chosen between the indices of the leading agents. This choice can represent a specific topology.
The status of leaders and followers can be changed between successive deployments and the number of leaders is not fixed. A system of three leaders and two followers during a deployment can be turned in any system with two to five leaders in the following deployment. However, changing the organization of the system impacts the complexity and the efficiency of the optimization problem.
4.2.3
Transition between successive steps
Let there be three successive states of a formation of N leaders and M followers at time t0, t1
and t2. These three formations are described by sets of N ordered positions s0, s1 and s2. The first transition from s0to s1is obtained by optimizing the functions f0and f1 of the form given inequation (3.166)on page78representing respectively s0 and s1.
To create the transition from state s1 to state s2, two approaches are considered. The first solution consists in evaluating only the best possible representation f2of the state s2 by keeping the representation f1of the state s1. This reduces the size of the optimization space by a factor
two and is compatible with the use of followers. Since the representation of the state is kept, the position of the followers is coherent. However, the resulting solution is not the best possible solution and can result in longer trajectories.
The second solution consists in evaluating again the representation f1and f2 of the states s1 and s2. This solution leads to better optimized trajectories at the cost of a wider optimization space. Moreover, as the representation of the start state changes, the starting positions of the followers is not coherent with their ending position at the previous step. An error is induced that has to be corrected by the closed-loop controllers of the followers.
0 2 4 6 8 10 101 102 t 〈u 〉 (0 ) + 〈u 〉 (1 ) optimizing f1 keeping f1
Figure 4.3 – Time evolution of the cost for a team of four leaders (average on 50 runs with fixed points,γ=1.65,µ=1.0).
The difference in optimization time and efficiency between these two solutions is represented infigure 4.3for a team of four leaders. In our configuration, the optimization of both the start and the final representations (depicted in light blue, dashed) gives better results than optimizing the only final representation (depicted in dark blue), provided the optimizer has enough time. In the case where a solution is needed in the shortest possible time, keeping the representation f1
is the best solution. Advantages and drawbacks of these two methods are gathered intable 4.1
keeping fi optimizing fi resolution speed + -
optimized - +
adapted to followers + -
Table 4.1 – Comparison of the two suggested methods for state representation optimization.
These methods are illustrated infigure 4.4on the next page where the final state of the second deployment is the start state of the first deployment (s2=s0). Three possible return methods are presented. The first deployment from s0to s1is drawn in light blue. As it is the first deployment,
both the start and end formation representations have been optimized. This results in short and smooth paths.
For the way back, the first possibility, drawn in dark blueinfigure 4.4on the facing page, has been obtained by optimizing again both the start and end representations. This results in equally short and smooth paths whereas different from the direct way. This is a sign of the nonlinearity of Burgers’ equation as the trajectories between two points is not the same depending on the sense of travel. The second possibility, drawn in light green, was obtained by keeping the representation of the final formation of the previous deployment as the representation of the new start formation and optimizing the representation of the new final formation. This results in a path longer than in the previous case but was obtained in notably less time. A last
0 0.2 0.4 0.6 0.8 1 −0.2 0.1 0.4 t, 1− t u (0, t) ,u (1, t) s0→ s1 s1→ s0optimizing f1 s1→ s0keeping f1 s1→ s0with f1↔ f0
Figure 4.4 – Three return possibilities (γ=1.65,µ=1.0).
possibility, that is not ensured to work, is given as the dark green dashed line. This is obtained by only exchanging the role of the previous start and final formation representation. Due to the definitions of the controls inequations (3.167)and(3.168)(and notably due to the choice of powers of t− 1 instead of powers of 1 − t), the systems is not invariant by the change of variable t← 1 − t. In the case where the resulting solution to the heat equation is strictly positive, it is still possible to compute a solution to Burgers’ equation that matches the given start and end formation. However, this solution undergoes no optimization process and the resulting solution, as in the case depicted infigure 4.4, is not optimal at all.