Method of resolving a problem of optimal control

Abstract

A method for determining, using a calculation unit, optimal control u of a vehicle engine for a problem of optimal control, having a state function x, initial state constraints xo, final state constraints xƒ, and a cost function ƒ0, involving determining, using the calculation unit, a two-level optimal control problem, and resolving said optimal control problem, said solution being determined by implementing a resolution method using Pontryagin's maximum principle.

Description

Method for solving an optimal control problem The present invention relates to a method for solving an optimal control problem. It further relates to a computing unit configured to implement said method. State of the prior art Methods are known for solving an optimal control problem such as minimizing fuel consumption. of a hybrid electric vehicle (HEV), for the English Hybrid Electric Vehicle, over a given or predicted time cycle A common cycle used is the WLTC cycle (for Worldwide Harmonized Light Vehicles Test Cycle), lasting 1800 seconds, which is divided into four parts: a low speed zone lasting 589 seconds, a moderate speed zone lasting 433 seconds, a high speed zone lasting 455 seconds and finally a last one at "very" high speed which lasts 323 seconds and is the most energy-intensive. The control system is of dimension 2: the action levers, also called degrees of freedom, are the distribution of the torque between the thermal engine and the electric motor (defined by the selection of the thermal torque ), and gear ratio control (selection of ). State of charge (for English State of Charge) of the battery must generally be fixed, as for standard homologation cycles requiring a maintenance scenario of the (or to “iso-SOC”). The optimal control problem can be to determine: with: the dynamics of the battery state of charge: constraints on the evolution of the state of charge: constraints on the commands, which can be seen as belonging to an admissible domain (which is not further developed here): constraints on the initial state and the final state of charge: The state of the art proposes several methods for solving optimal control problems, including: an equivalent consumption minimization strategy (ECMS). This is an online method, but suboptimal. direct methods: these methods require determining a discretization in time and solving an optimization problem in very high dimensions (these methods are based on the Karush-Kunh-Tucker (KKT) conditions) dynamic programming methods: these methods require determining a discretization in time and exhaustively exploring the set of trajectory possibilities in state and using the Bellman principle (and exploring all possibilities). reinforcement learning methods: these methods require determining a discretization in time and control and using the Bellman principle (learning the Q function iteratively).indirect methods: transform the optimal control problem (OCP) into a boundary value problem (BVP) using the Pontryagin Maximum Principle (PMP).

[0006] Indirect methods are preferred because they provide a process that provides a much faster solution than dynamic programming methods, and they are much more accurate than other methods. However, indirect methods implementing the Pontryagin maximum principle require the resolution of a low-dimensional optimization problem. This is usually determined by a "shooting method" based on iterative simulations of an internal model. But in the case of an application to cycles with a relatively long horizon, this method has some drawbacks: a possibly poor convergence or even a divergence of the associated optimization problem, too great sensitivity to initial values, a possibly long computation time, due to internal simulations over long horizons, difficulties in taking into account state constraints (in the case of using HEVs: min-max limitations of the SOC)

[0007] The purpose of this description is to propose a method which overcomes at least one of the aforementioned drawbacks. To this end, according to a first aspect of the invention, a method for determining an optimal control by a calculation unit is proposed. of a vehicle motorization for an optimal control problem, presenting a state function , initial state constraints , final state constraints , a cost function , which is written: with : the state, of dimension n, control, dimension , the cost function, the state dynamics function, : the initial state constraints, : the final state constraints According to the invention, the method comprises a determination by the calculation unit of a two-level optimal control problem, and a resolution of said two-level optimal control problem, said two-level optimal control problem being written: with : said solution being determined by implementing a resolution method using the simple shooting method of the Pontryagin maximum principle. The single shot method is so named in contrast to the multiple shot method. Advantageously, prior to a resolution of said optimal two-level control problem, the optimal passage costs , from an initial state to a final state are replaced by estimated passage costs predetermined before solving the optimal control problem by the computing unit, the two-level optimal control problem becoming two independent problems, namely an optimization problem And optimal ordering problems According to one embodiment, the resolution steps And ( are carried out online. Preferably, the method further comprises a step of determining the estimated passage costs. predetermined from a set comprising initial states , final states and an optimal cost of passage . For example, each of the optimal passage costs from an initial state , to a final state is determined by solving the intermediate optimal control problem : According to one possibility, the intermediate optimal control problem is solved by implementing a solution method using the Pontryagin maximum principle, preferably by a shooting method. According to a second aspect of the invention, a calculation unit is proposed for determining an optimal control of a vehicle engine, said calculation unit being configured to implement a method according to the first aspect of the invention, or one or more of its improvements. Description of figures Other advantages and particularities of the invention will appear on reading the detailed description of implementations and embodiments which are in no way limiting, with regard to the appended drawings in which: represents, in a principle diagram of an exemplary embodiment of a method according to the invention. Description of embodiments The embodiments described below being in no way limiting, it will be possible in particular to consider variants of the invention comprising only a selection of the characteristics described, subsequently isolated from the other characteristics described, if this selection of characteristics is sufficient to confer a technical advantage or to differentiate the invention compared to the state of the prior art. This selection comprises at least one characteristic, preferably functional without structural details, or with only a part of the structural details if this part only is sufficient to confer a technical advantage or to differentiate the invention compared to the state of the prior art. In the figures, an element appearing in several figures retains the same reference. General description The invention addresses a method for solving an optimal control problem which can be written in its general form: with: the state, of dimension n, control, dimension , the cost function, the state dynamics function possibly state constraints: , possibly order constraints: : the initial state constraints : the final state constraintsTransition to an equivalent two-level problem said One idea that is the basis of the invention is to transform the optimal control problem in an equivalent two-level problem, with one level, said , presenting an optimal control problem, and another level, said , comprising a plurality of optimal control problems, from intermediate times on the interval fixed in advance. This transformation can be written: With The division into intermediate time can, for example, be carried out according to a predetermined period. The transformation leads to solving iteratively optimal control problems on shorter intervals and then to introduce the resolutions of these shorter problems in the part to solve the associated optimization problem. Approximation of the problem and resolution This resolution of the problem includes: an offline step, including: a step of creating (E1) a database including, for each of the time intervals , , THE being fixed beforehand, and for each of the initial states and each of the final states , an optimal passage cost ,then for each of the time intervals a learning step (E2) of cost function from said optimal passage costs , each of the cost functions being a learning of points .an online step, including: a resolution (E3) of the problem ,a resolution (E4) of the problem . Creating the database To determine the optimal passage cost from an initial state in time to a final state in time , we solve the optimal control problem ( to determine an optimal intermediate control trajectory, and the intermediate passage cost associated with said optimal intermediate control trajectory is determined. For example, if the number of possible initial states at time is 11 and the final state number at time is also 11, and that the data creation strategy is an exhaustive exploration of possible combinations, the database ultimately contains 121 optimal costs associated with a transition from an initial state to a final state. There are techniques, known to those skilled in the art, for only having to explore the possible final states. There are other strategies where all possible combinations are not exhaustively calculated, even without defining a regular grid (other sampling methods) (example: Latin hypercube, random draws, etc.) Optimal control problems ( being independent, it is possible to solve them in parallel. Determination of cost functions

[0020] Determination of cost functions includes learning about the different databases previously created. Several methods can be implemented to determine the functions and in particular: multilinear interpolation methods, which nevertheless require regularity of the database, approximation methods using polynomial functions, neural networks. Resolution of the problem The problem of is replaced by the resolution of the problem following in which we seek the optimal intermediate states to minimize the sum of the estimated optimal costs: Several methods can be implemented to find the optimal intermediate states for each of the times :dichotomy method (only if dimension is 1: ), gradient descent, evolutionary algorithm to avoid encountering a local minimum, brute force, dynamic programming. Solving the problem Solving the problem includes, for each of the intervals , a determination of the optimal intermediate control solution of the following optimal intermediate control problem, using the Pontryagin maximum principle firing method: Application The present application proposes in particular an application to the resolution of an optimal control problem such as that of minimizing fuel consumption. of a HEV hybrid electric vehicle. The optimal control problem can then be written: with state dynamics: state constraints: control constraints: my initial and final constraints: We note that the problem is a very special case of the previous problem. It is sufficient to apply the method previously presented to solve it. Of course, the invention is not limited to the examples which have just been described and numerous adjustments can be made to these examples without departing from the scope of the invention. For example, it is envisaged to use the exposed resolution method to determine optimal controls for the power distribution of hydrogen vehicles. In addition, the various features, forms, variations and embodiments of the invention may be combined with each other in various combinations to the extent that they are not incompatible or mutually exclusive.

Claims

Method for determining an optimal order by a calculation unit of a vehicle motorization for an optimal control problem (OCP), presenting a state function , initial state constraints , final state constraints , a cost function , which is written: with : the state, of dimension n, control, dimension , the cost function, the state dynamics function, : the initial state constraints, : the final state constraints comprising a determination by the calculation unit of an optimal control problem (B0CP) at two levels (Macro, Micro), and a resolution of said optimal control problem, said optimal control problem at two levels being written: with : being the cost of moving from one state to another to a state , said solution being determined by implementing a resolution method using the Pontryagin maximum principle. Method according to the preceding claim, in which prior to a resolution of said optimal two-level control problem, the optimal passage costs , from an initial state to a final state are replaced by estimated passage costs predetermined before solving the optimal control problem by the computing unit, the two-level optimal control problem (BOCP) becoming two independent problems, namely an optimization problem and N+1 optimal ordering problems (E3, E4): Method according to one of the two preceding claims, in which the steps of solving the problems And are carried out online. Method according to one of the two preceding claims, when dependent on claim 2, further comprising a step of determining (E2) the estimated passage costs. predetermined from a set comprising initial states , final states and an optimal cost of passage . Method according to the preceding claim, in which each of the optimal passage costs from an initial state , to a final state is determined (E1) by solving the intermediate optimal control problem: Method according to the preceding claim, in which the intermediate optimal control problem is solved by implementing a resolution method using the Pontryagin maximum principle. Calculation unit for determining an optimal order of a vehicle engine by implementing a method according to any one of the preceding claims.

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