Section: New Results

Optimal Control and Averaging in Aerospace Engineering

Chance-constrained optimal control problems in aerospace

Participants : Jean-Baptiste Caillau, Max Cerf [Airbus Safran Launchers] , Achille Sassi [ENSTA Paristech] , Emmanuel Trélat [Univ. Paris VI] , Hasnaa Zidani [ENSTA Paristech] .

The aim is to minimize the fuel mass of the last stage of a three-stage launcher. Since the design parameters of the spacecraft are not exactly known prior to the launch, uncertainties have to be taken into account. Although these parameters are supposed to be uniformly distributed on fixed ranges, it is not desirable to use "worst-case" robust optimization as the problem may not even be feasible for some values of the parameters due to very strong sensitivities. The idea is to frame instead a stochastic optimization problem where these parameters are independent stochastic variables. The original constraint becomes a stochastic variable, and one only asks that the desired target is reached with some given probability. A key issue in solving this chance constrained problem is to approximate the probability density function of the constraint. Contrary to Monte-Carlo methods that require a large number of runs, kernel density estimation  [68] has the strong advantage to permit to build an estimator with just a few constraint evaluations. This approach allows to treat efficiently uncertainties on several design parameters of the launcher, including the specific impulse and index of the third stage and using a simple affine discretization of the control (pitch angle). In [19], we use the Kernel Density Estimation method to approximate the probability density function of a random variable with unknown distribution, from a relatively small sample, and we show how this technique can be applied and implemented for a class of problems including the Goddard problem (with bang-bang or bang-singular-bang controls) and the trajectory optimization of an Ariane 5-like launcher. This work has been done in collaboration with Airbus Safran Launchers at Les Mureaux.

An involved question in chance constrained optimization is the existence and computation of the derivative of the stochastic constraint with respect to deterministic parameter. This shall be investigated in the light of new results in the Gaussian case [78]. Using a single deterministic control to reach a given target (or a given level of performance) with some fixed probability when the parameters of the system are randomly distributed is very similar to issues of ensemble controllability addressed in the recent work [26]. One expects some insight from the comparison of the two viewpoints.

Metric approximation of minimum time control systems

Participants : Jean-Baptiste Caillau, Lamberto Dell'Elce, Jean-Baptiste Pomet, Jérémy Rouot.

Slow-fast affine control systems with one fast angle are considered in this work [20]. An approximation based on standard averaging of the extremal is defined. When the drift of the original system is small enough, this approximation is metric, and minimum time trajectories of the original system converge towards geodesics of a Finsler metric. The asymmetry of the metric accounts for the presence of the drift on the slow part of the original dynamics. The example of the $J_2$ effect in the two-body case in space mechanics is examined. A critical ratio between the $J_2$ drift and the thrust level of the engine is defined in terms of the averaged metric. The qualitative behaviour of the minimum time for the real system is analyzed thanks to this ratio. Work in progress aims at dealing with multiphase averaging for systems driven by several fast angles.

Approximation by filtering in optimal control and applications

Participants : Jean-Baptiste Caillau, Thierry Dargent [Thales Alenia Space] , Florentina Nicolau [Univ. Cergy-Pontoise] .

Minimum time control of slow-fast systems is considered in this analysis [8]. In the case of only one fast angle, averaging techniques are available for such systems. The approach introduced in [57] and [34] is recalled, then extended to time-dependent systems by means of a suitable filtering operator. The process relies upon approximating the dynamics by means of sliding windows. The size of these windows is an additional parameter that provides intermediate approximations between averaging over the whole fast angle period and the original dynamics. The motivation is that averaging over an entire period may not provide a good enough approximation to initialize a convergent numerical resolution of the original system; considering a continuous set of intermediate approximations (filtering over windows of size varying from the period to zero) may ensure convergence. The method is illustrated on problems coming from space mechanics and has been implemented as an addition to the industrial code T3D of Thales Alenia Space.

Higher order averaging

Participants : Jean-Baptiste Pomet, Thierry Dargent [Thales Alenia Space] , Florentina Nicolau [Univ. Cergy-Pontoise] .

A further step in defining a suitable approximation of slow-fast oscillating controlled systems is to go beyond the $O\left(\epsilon \right)$ uniform error provided by simple averaging. An original approach has been proposed in [58] and demonstrated numerically; it consists in correcting the boundary values of the slow averaged variables to ensure an $O\left({\epsilon }^2\right)$ average error, without the difficulties of classical second order averaging  [73] (that leads to an $O\left({\epsilon }^2\right)$ uniform error, that we do not need), and allows an $O\left(\epsilon \right)$ approximation of the angle. It is proved in [9] that it is indeed possible, at least for initial value problems, to compute order one corrections of the initial slow variables to guarantee such an error. From the numerical side, this process is a key to be able to initialize shooting methods on the non-averaged system by averaged solutions when using a model with full perturbations in orbit transfer.