Method for achieving optimal control of a system based on a predictive model integrating an adaptive observer

Abstract

The present invention relates to a method for determining at least one control setpoint ut of a system comprising, at a time t: - a set of input quantities comprising the control setpoint ut and modelled in the form of a vector: Ut, - a set of output quantities modelled in the form of a vector: Yt, and - a set of quantities representing the state of the system and modelled in the form of a vector: xt; the method being implemented in a computing unit and comprising: - a first step E1 of programming a set of linear differential equations relating the input quantities, the quantities representing the state of the system and the output quantities, and - a step E4 of determining the optimal control setpoint by means of the quadratic programming method.

Description

[0001]DESCRIPTION Method for optimal control of a system based on a predictive model integrating an adaptive observer Technical field of the invention The present invention relates to the field of industrial system control and more particularly the automated optimization of the control law of a system. The invention relates more specifically to a method for determining at least one control setpoint and an electronic card in which the method of the invention is implemented. Technical background Conventionally, the formulation of the control law of an industrial system is carried out in a modeling step upstream of the implementation within the system control unit. It is based on a model presenting modeling uncertainties, which leads to control that is not sufficiently robust for certain applications, for example when the control is applied to a fuel cell electric vehicle system.Building a reliable model requires taking into account changes in the system over time, particularly related to aging phenomena. In order to take these changes into account, it is known to carry out empirical recalibrations defined through tests on real systems and carried out on test benches. However, these recalibrations can be tedious and the test results may not be representative of all the systems in the same series. To overcome these drawbacks, there are solutions consisting of integrating into the system an algorithm for adapting the control law according to the real state of the system. However, very often, control laws are based on physical models represented by nonlinear equations, making adaptation and parameter identification techniques complicated or even impossible.Furthermore, solving nonlinear equations requires greater computing power which must be implemented in particularly complex and expensive computers. Summary of the invention The invention proposes a method for determining at least one control setpoint ut of a system comprising at a time t: - a set of input quantities comprising the control setpoint utet modeled in the form of a vector: Ut, - a set of output quantities modeled in the form of a vector: Yt, and - a set of quantities representing the state of the system and modeled in the form of a vector: xt; the method being implemented in a computing unit and comprising a first step E1 of programming a set of linear differential equations between the input quantities, the state quantities of the system and the output quantities in the form: [Math.13]. - where θ = {^^ ^ ^ ^ ^ ^, … ^ t , ^ , ^, ^^} are slow-changing parameters with a slower variation over time than Ut , Yt and xt and- where A(θt), B(θt), C(θt) and D(θt) are matrices containing the slow-changing parameters of 1, -1 or 0; the method being characterized in that it comprises: - a step E4 of determining the optimal control setpoint by means of the quadratic programming method over a time window [T0, T0+NΔT] comprising N samples using the following quadratic criterion to be optimized: [Math.14] - where xideal is the vector representing the target state of the system, - où ^^ ^ , ^ ^ ^ , and (^^^^^^ − ^^) ^ , are the transposed vectors of the vectors ^ ^ ,^^, and (^^^^^^ − ^^)., and- where Q, R and K are weighting matrices of the quadratic criterion to be optimized,- where N is a natural integer,the control instruction u tbeing then obtained for N samples to come from the present instant Tp, the instruction being expressed for each sample of the instant Tp+i =Tp+iΔT as a function of:A(θ Tp+1 ), B(θ Tp+1 ), C(θ Tp+1 ) and D(θ Tp+1 ), x Tp+1 , Q, R and K.. The method may further comprise:- a step E2, between steps E1 and E4, of determining the slowly evolving parameters θ = {^ ^ ^ ^, … , ^ t ^ , ^ ^ , ^ ^ ^^} thanks to the minimization of the gap between the measured values ​​of the output quantities (Yt)mes. and the values ​​calculated using the equation math.13 of the output quantities Yt:[Math.15] In step E2, the slowly evolving parameters θt = {^^ ^ , ^ ^ ^ , ^ ^ ^, … , ^ ^ ^} are obtained numerically after several iterations of solving equations 1 and minimizing the difference e in the time window [T0, T0+NΔT]. The method may include a new iteration of step E4 of determining the optimal control setpoint using the quadratic programming method when one of the slowly changing parameters The method may include a step E5 of sending to a control unit of the system the optimal control instruction for the following instant T p+1after each new iteration of step E4 of determining the optimal control setpoint. The method may further comprise:- a step E3 after step E2 or simultaneously with step E2, of adaptive observation comprising the numerical calculation of the state of the system xt at time t by numerical resolution of equations 1 taking into account the minimum deviation e at time t. The method may be implemented in a fuel cell system comprising:- a battery,- a fuel cell, and -Ibat is the intensity of the electric current in the battery,- IFC is the intensity of the electric current in the fuel cell,- f(IFC) is a function of IFC used in the calculation of VFC,^ ^^^ with [Math.17] ^ = ^ ^ ^^ ^ ^^ ^ où : - Vbat is the electrical voltage in the battery,- VFC is the electrical voltage in the fuel cell,- mH2 is the consumption of dihydrogen in the fuel cell.^^^ with [Math.18] ^ = ^^ ^^ ^^^^^^ where: - SoC is the state of charge of the battery, - mH2 is the consumption of dihydrogen in the fuel cell, - Vloss is the voltage related to electrical losses in the system. The matrices A, B, C and D can have the following form: 00^ θ50^ 0 The method can be implemented in a system having fouling effects, or friction forces. The invention also relates to an electronic card in which the method according to the invention is implemented. Brief description of the figures Other characteristics and advantages of the invention will appear during the reading of the detailed description which follows for the understanding of which reference will be made to the appended drawings in which: [Fig. 1] is a graph which represents a set of values ​​for a control setpoint of a system as a function of time, the values ​​being obtained by a method according to an embodiment of the invention; [Fig. 2] is a flowchart representing the sequence of steps of a method according to an embodiment of the invention; [Fig. 3] is a flowchart representing steps E2 and E3 of the method of figure 2; [Fig.4] is a flowchart representing the sequence of steps E2 to E5 of the method of Figure 2; [Fig. 5] is a schematic representation of the operation of the components of a fuel cell electric vehicle system to which the method of the invention can be applied; [Fig. 6] is a flowchart schematically representing the application of the method of the invention to the system of Figure 5. Detailed description of the invention In the following description, identical, similar or analogous elements will be designated by the same reference numbers. Figure 1 is a graph which represents a set of values ​​for a control setpoint ut of a system as a function of time. The values ​​are obtained by means of the method of the invention allowing the control of a system by predictive model. Values ​​were calculated for each time step k, k ranging from 0 to N in the example.The method of the invention makes it possible to calculate, for each future time step in a given time window, an optimal control setpoint uk. The calculation is carried out during a step E4 of the method which will be detailed later. The calculation implements a direct quadratic programming method known per se. Each iteration of step E4 makes it possible to calculate a set of optimal control setpoint values ​​uk for the future time steps in a given time window. The set of values ​​obtained after an iteration of step E4 constitutes a control trajectory. The graph in Figure 1 includes several control trajectories resulting from the different iterations of step E4. Each control trajectory is shown by a specific line of varying darkness. A new control trajectory is calculated, i.e. a new iteration of step E4 is carried out, only when a parameter θk of the system varies.θk represents, for example, a parameter that changes as a result of system aging. In the example, between steps k=2 and k=3, the θk parameters do not vary, i.e. θ3 = θ2. The control trajectory is therefore not recalculated at step k=3, but it is recalculated for the other steps. The dotted line represents the control setpoint uk actually sent to the system control unit. For each step k, the control setpoint u. kactually sent to the system control unit corresponds to the last control setpoint value calculated for the step k in question. Figure 1 shows that the method of the invention makes it possible to carry out the control of a system by predictive model by determining an optimal control setpoint adapted according to the actual behavior of the system. Figure 2 represents the steps of a method according to an embodiment of the invention in the form of a flowchart. The method of the invention is implemented in a calculation unit of a system comprising for example a generic processor or a microprocessor or an electronic circuit. The calculation unit may be an integral part of a system control unit or may comprise wired or radio communication means with a system control unit.Furthermore, the calculation unit in which the method is implemented comprises a memory capable of storing values ​​for certain quantities involved in the algorithm of the method. The method comprises the following steps: - a first step E1 of programming a set of linear differential equations between the input quantities, the state quantities of the system and the output quantities; - a second step E2 of determining the slowly evolving parameters; - a third step E3 of adaptive observation comprising the numerical calculation of the state of the system at time t; - a fourth step E4 of determining the optimal control setpoint by means of the quadratic programming method; - a fifth step E5 of sending to the system the optimal control setpoint for the following time. The first step E1 corresponds to the implementation of the system behavior model in the calculation unit of the system.The first step E1 involves programming a set of linear differential equations in the system's computing unit. Linear differential equations establish relationships between input quantities, system state quantities, and output quantities. The equations are in the form: [Math.1]. - where θt = {^^ ^ , ^ ^ ^ , ^ ^ ^, … , ^ ^ ^} are slow-changing parameters with a slower variation over time than Ut , Yt and xt and- where A(θt), B(θt), C(θt) and D(θt) are matrices containing the slow-changing parameters {^^ ^ , ^ ^ ^ , ^ ^ ^ ^ , … , ^^}, or 1, -1 or 0. The slowly changing parameters θt = {^^ ^ , ^ ^ ^ , ^ ^ ^ ^ , … , ^^} are parameters that do not necessarily have a physical meaning and that vary slowly over time, particularly as a function of aging and wear phenomena occurring in the system components. The second step E2 corresponds to the identification of system parameters. The second step E2 involves the determination of the slowly changing parameters θt = {^^ ^ , ^ ^ ^ , ^ ^ ^ ^ , … , ^^}. Step E2 is based on a reasoning which consists in considering that the measured values ​​of the output quantities (Yt)mes must be equal in theory to the values ​​of the output quantities Yt calculated using equation 1. Thanks to the introduction of the measured values ​​in the equations and the resolution of the equation system, it is possible to deduce unknowns of the system. In practice, the slowly evolving parameters θt are obtained thanks to the minimization of the difference e between: - the measured values ​​of the output quantities (Yt)mes. ; and - the values ​​calculated using equation 1 of the output quantities Yt. The difference e is expressed in the following form: [Math.2]^ = (^^)^^^. − ^^ . Obtaining a zero difference e is an indicator of good convergence of the identification algorithm towards the good values ​​of the parameters θt. This convergence is subject to conditions of persistence of excitation of the input quantities u tThis is guaranteed by adding to the optimal control setpoint uk a very low amplitude signal rich in frequency. The low amplitude makes it possible to have a neglected impact on the cost function J that we wish to optimize. The third step E3 corresponds to adaptive observation, i.e. an estimation of the state of the system based on a model adapted during the life of the system. The model is adapted, i.e. updated according to the evolution of the system behavior, in particular with aging. The third step E3 is carried out after step E2 in order to have at step E3 the slowly evolving parameters θt to calculate the values ​​characterizing the state of the system. Step E3 includes the numerical calculation of the state of the system xt at time t by numerical resolution of equations 1 taking into account the minimum deviation e at time t and the slowly evolving parameters θt obtained in step E2.Alternatively, steps E2 and E3 are performed simultaneously by a global resolution of the equations. The fourth step E4 corresponds to the expression of the optimal control. The fourth step E4 involves the determination of the optimal control setpoint u. t using the quadratic programming method. The quadratic programming method is applied to a time window [T0, T0+NΔT] comprising N samples using the following quadratic criterion to be optimized:[Math.3] - where xideal is the vector representing the target state of the system - où ^^ ^ , ^ ^ ^ , and (^^^^^^ − ^^) ^ , are the transposed vectors of the vectors ^ ^,^^, and (^^^^^^ − ^^)., and- where Q, R and K are weighting matrices of the quadratic criterion to be optimized,- where N is a natural integer. Each term of the quadratic criterion involves the product of a vector by its transposed vector, hence the name "quadratic". The direct quadratic programming method is well known in the literature and is mainly applied to obtain an optimal control setpoint for linear models over a time window of N observations. Also, the details of the calculations implemented in the quadratic programming method are not included in this description.By calculating the minimum of the quadratic criterion J, the control instruction ut is obtained for N samples to come from the present instant Tp, the instruction being expressed for each sample of the instant Tp+i=Tp+iΔT , i being between 0 and N, as a function of: A(θTp+1), B(θTp+1), C(θTp+1) and D(θTp+1), xTp+1, Q, R and K which are terms that can be expressed numerically, in particular thanks to step E2 which makes it possible to obtain θ. Tp+1 and step E3 which allows us to obtain x Tp+1 digitally. The fifth step E5 corresponds to the control of the system by predictive model. The fifth step E5 involves sending to a system control unit the optimal control setpoint u Tp+1 of the next moment T p+1. The sending of the optimal control setpoint is carried out after each new iteration of step E4 of determining the optimal control setpoint. The system control strategy by predictive model is robust insofar as it is constantly adapted to the actual operation of the system, taking into account in particular the evolution of the system's operation with aging. Steps E2 to E5 are carried out during the operation of the system. They can be implemented in an embedded manner, that is to say in a calculation unit and a control unit integrated in the system. Figure 3 represents in the form of a flowchart steps E2 and E3. Step E2 of identification of system parameters and step E3 of adaptive observation are two closely linked steps. In these steps E2 and E3, the state estimators xk+1 and parameters θk+1 produce at each time step k the predicted values ​​at the following time step k+1.Figure 3 shows that to perform the estimation at the next time step k+1, the adaptive observer needs as input data: - the values ​​of the input quantities Uk sent to the control unit or measured, - the difference e between the values ​​of the output quantities (Yk) measured and the values ​​of the output quantities Yk calculated, the difference being provided for example by a comparator 12, - the values ​​of the state quantities of the system xk and the parameters θk at the present time step k stored in a memory 10. The state of the system xk+1 and the parameters θk+1 estimated for the next instant k+1 are output data of steps E2 and E3 which will then be used in step E4 to update the optimal control setpoint uk+1. In step E2, the slowly changing parameters θt = {^^. ^ , ^ ^ ^ , ^ ^ ^, … , ^ ^ ^} can be obtained numerically after several iterations of solving equations 1 and minimizing the gap e in the time window [T0, T0+NΔT]. Indeed, a recursive calculation allows an asymptotic convergence towards the good values ​​of θ t. Similarly, in step E3, the values ​​of the state quantities of the system xt can be obtained numerically after several iterations of solving equations 1 and minimizing the difference e in the time window [T0, T0+NΔT]. Figure 4 represents the sequence of steps E2 to E5 in the form of a flowchart. The predictive model control strategy which is at the origin of this sequence of steps E2 to E5 aims to establish a link between the updated model and the optimal control setpoint in order to obtain a robust optimizer. Figure 4 shows in the form of a three-column table at which level each step is located.The steps are divided into the three levels as follows: - steps E2 and E3 are at the level of adaptive observation, i.e., the determination of the state of the system based on updated measurements, - step E4 is at the level of the optimization algorithm, i.e., the expression of optimal control, and - step E5 is at the level of system control by predictive model, i.e., at the level of command. After steps E2 and E3, the performance of step E4 of determining the optimal control setpoint is subject to the verification of a logical test 14 whose condition is as follows: θk+1≠ θk. ​​In other words, step E4 is triggered when one of the slowly changing parameters θt = {^^. ^ , ^ ^ ^ , ^ ^^ ^, … , ^^} varies. Step E5 of sending the optimal control setpoint to the system is automatically triggered at the end of step E4. Thus, if the condition θk+1 ≠ θk is not verified, the command corresponding to the setpoint uk+1 intended to be sent to pask+1, is not modified compared to the setpoint value uk+1 already in memory and the update of the command is deactivated as shown in step E6. Indeed, if the slowly changing parameters θt remain constant, then the update of the optimal control setpoint is deactivated to avoid unnecessary calculations since the control setpoint values ​​u tcalculated for the N samples from time Tp remain unchanged. Thus, the control unit takes into account at each time Tp+i =Tp+iΔT the last control setpoint uTp+i calculated for this time. A new iteration of step E5 can be planned at the end of the time window TP+NΔT if no update of the uoptimal control setpoint has been carried out since time TP. The predictive model control strategy uses an internal model adapted over the life of the system by means of an adaptive observation method. Figure 5 schematically represents the operation of the components of a fuel cell electric vehicle system 16 to which the method of the invention can be applied. In the example, the vehicle is a car or a truck. Alternatively, the vehicle can be any other type of vehicle such as a boat or an airplane.The system 16 comprises: - a junction box or junction box 18 for centralizing the electrical connections between different electrical devices; - a battery 20, - a fuel cell 22, - auxiliary components 24 of the fuel cell, and - an electric motor 26. The arrows represent the exchanges of electrical power between the components. The battery 20 is capable of supplying or storing electrical energy, which is represented by a two-way arrow pointing both towards the battery 20 to symbolize that it receives electrical energy to be stored in a storage mode or pointing towards the junction box 18 to symbolize that the battery 20 supplies electrical energy in a supply mode.The fuel cell 22 provides electrical energy through the oxidation on one electrode of a reducing fuel, for example dihydrogen, coupled with the reduction on the other electrode of an oxidant, such as oxygen from the air. The fuel cell 22 transmits the electrical energy produced to the junction box 18. The auxiliary components 24 of the fuel cell, for example, comprise a compressor and a DC-DC converter which is a power converter that converts a direct current (DC) source from a specified voltage level to another different voltage level. The auxiliary components 24 consume a portion of the electrical energy. The electric motor 26 consumes electrical energy to provide engine torque but it can also provide electrical energy when it operates as a generator, for example during deceleration. The power balance at the terminals of the junction box 18 is expressed by the equation below.[Math.4]. O ù :- PwFC is the power supplied by the fuel cell 22,- Pwbat is the power supplied by the battery 20 (positive sign in power supply mode and negative sign in storage mode),- Pwreq is the power supplied to the electric motor 26 (positive sign if the electric motor 26 consumes energy and negative sign when the electric motor 26 operates as a generator),- Pwaux is the power consumed by the auxiliary components 24.The system comprises:- a set of input quantities modeled in the form of a vector: Ut,- a set of output quantities modeled in the form of a vector: Yt, and- a set of quantities representing the state of the system and modeled in the form of a vector: xt.The input quantities, the output quantities and the quantities representing the state of the system are quantities that depend on time t.In the following writings, the index "t" indicating the time dependence is sometimes omitted for simplicity the scriptures.The input quantities and the output quantities are quantities that can be measured or deduced using known information about the system. The input quantities include: - the intensity of the supply current of the IFC fuel cell, which constitutes the variable to be controlled and regulated and for which we seek to calculate an optimal control setpoint ut, and - the power to be delivered to the electric motor Pwreq, which is not a controllable quantity but which is assumed to be known since the journey to be made by the vehicle is known. The intensity of the supply current of the IFC fuel cell determines the operating speed of the fuel cell. Thus, the greater the intensity of the supply current of the IFC fuel cell, the greater the quantity of electrical energy produced by the fuel cell and therefore the greater the instantaneous power delivered by the fuel cell PwFC.Therefore, the greater the intensity of the supply current of the IFC fuel cell, the greater the consumption of dihydrogen to provide the power Pw. FC The power to be delivered to the electric motor Pwreq is linked to the powers Pw FC , Pw bat and Pw aux by the power balance at the terminals of the junction box 18 expressed above. Therefore, since Pw req can be deduced from the knowledge of the route to be taken by the vehicle, it is possible to introduce into the input variables the quantities involved in the calculation of PwFC, Pwbat and Pwaux, namely Ibat and IFC. Figure 6 is a flowchart representing the application of the method of the invention to the system 16 of Figure 5. The input quantities are modeled by the vector U below: [Math.5] - Ibat is the intensity of the electric current in the battery,- IFC is the intensity of the electric current in the fuel cell,- f(IFC) is a function of IFC involved in the calculation of VFC. More precisely, it is the voltage current characterization when the fuel cell system is in new condition (without aging or Vloss=0). The output quantities include:- the electric voltage in the battery Vbat,- the electric voltage in the fuel cell VFC,- the consumption of dihydrogen in the fuel cell mH2. The output quantities are modeled by the vector Y below:[Math.6] ^ ^ ^^^ ^ = ^ ^^ ^ ^^ ^ où :- Vbat is the electrical voltage in the battery,- VFC is the electrical voltage in the fuel cell,- mH2 is the consumption of hydrogen in the fuel cell.The quantities representing the state of the system include:- the state of charge of the battery SoC,- the consumption of hydrogen in the fuel cell mH2,- the voltage linked to electrical losses in the system Vloss which characterizes the aging state of the fuel cell. The quantities representing the state of the system are modeled by the vectorx below:[Math.7] ^^^ ^ = ^ ^ ^^ ^^^^^^ where: - SoC is the state of charge of the battery, - mH2 is the consumption of dihydrogen in the fuel cell, - Vloss is the voltage related to electrical losses in the system. In the system of figures 5 and 6, equations 1 translate as follows: [Math.8] ẋ = A(θ)x + B(θ)U x = A (θ )x ( )^Y^ = C(θ)x + D(θ)U discrete form → ^ ^^^ ^ ^ ^ + B^ θ^ U^Y^^ = C^(θ^)x^ + D^(θ^)U^ Matrices A, B, C and D have the following form: [Math. 9] 1 0 0 ^ = ^ 0 1 04^ 0 0 θ [Math.10] The slow-changing parameters are related to the aging of the battery 20 and the fuel cell 22. Indeed, the behavior of the battery 20 and that of the fuel cell 22 vary as they are used. Other equations make it possible to translate the operating constraints of the system 16. Thus, the relationships between current and voltage within the battery 20 are translated. Similarly, the relationships between current and voltage within the fuel cell 22 are translated. In the quadratic criterion J to be optimized, that is to say to minimize, the first term involving x t reflects the aging of the system 16 and the second term involving U treflects the consumption, in particular of hydrogen. Thus, minimizing the quadratic criterion J, leads to an optimization, that is to say a minimization, of both the aging of the system and its consumption. The algorithm makes it possible to calculate, through the intensity of the electric current IFC in the fuel cell 22, the energy to be supplied by the fuel cell 22 so as to achieve several objectives, namely: - minimize the consumption of hydrogen along a known path, - minimize the aging of the system and in particular the aging of the fuel cell 22 and the battery 20. The algorithm is implemented in an electronic card controlling the power flows at the level of the junction box 18. The invention can be used in other systems.For example, the invention can be used to design the optimal control of systems such as: - cooling circuit systems based on heat exchangers to be more robust against the effects of fouling of the exchangers; - electrical machines to be more robust against friction forces. For these two examples, it is the dynamic model that will be modified and adapted to the input, output and state quantities of the system that intervene in the system considered. The invention has many advantages described below. The method for determining the control setpoint proposed by the invention is robust and precise because it is adaptive and is based on real measurements carried out within the system.The method is based on the search for a mathematical model linking the inputs and outputs of the system having a linear form so as to obtain a sufficiently precise model and which allows the implementation of an algorithm for adapting and identifying the parameters. The formulation of the control setpoint as a function of: A(θTp+1), B(θTp+1), C(θTp+1) and D(θTp+1), xTp+1, Q, R and K allows an implementation of the determination algorithm in the relatively simple calculation unit. In particular, the measurement step is fairly quick to carry out and the implementation of the resolution algorithms is inexpensive in terms of calculation time. Thus, the implementation of the method does not a priori pose any real-time implementation problem. Consequently, the method of the invention can be implemented by calculation and control units comprising relatively simple and inexpensive processors.The invention proposes a reliable on-board solution for optimizing the energy consumption and aging of a system, in particular a fuel cell electric vehicle.

Claims

CLAIMS 1 . Procédé de détermination d’au moins une consigne de contrôle ut d’un système comportant à un instant t : - un ensemble de grandeurs d’entrée comportant la consigne de contrôle ut et modélisé sous forme d’un vecteur : Ut, - un ensemble de grandeurs de sortie modélisé sous forme d’un vecteur : Yt, et - un ensemble de grandeurs représentant l’état du système et modélisé sous forme d’un vecteur : xt ; le procédé étant mis en œuvre dans une unité de calcul et comportant une première étape E1 de programmation d’un ensemble of linear differential equations between the input quantities, the g randeurs d’état du système et les grandeurs de sortie sous la forme : [Math.13] - où θ ^ ^ ^ t = {^^ , ^ ^, … , ^^} are slowly changing parameters p résentant une variation dans le temps plus lente que Ut , Yt et xt et - où A(θt), B(θt), C(θt) et D(θt) sont des matrices comportant les paramètres à évolution lente ou des 1, -1 ou 0 le procédé étant caractérisé en ce qu’il comporte : - une étape E4 de détermination de la consigne de contrôle optimale using the quadratic programming method on a time window [T0, T0+NΔT] comprising N samples using the e critère quadratique à optimiser suivant : [Math.14] − ^^) - où xideal est le vecteur représentant l’état cible du système - où ^^ ^ , ^ ^ ^ , et (^^^^^^ − ^^) ^ , are the transposed vectors of the vecteurs ^^, ^^, et (^^^^^^ − ^^)., et - où Q, R et K sont des matrices de pondérations du critère quadratic to optimize, - où N est un nombre entier naturel, the control instruction ut then being obtained for N samples at venir à partir de l’instant présent Tp, la consigne s’exprimant pour chaque échantillon de l’instant Tp+i =Tp+iΔT en fonction de :A(θTp+1), B(θTp+1), C(θTp+1) and D(θTp+1), xTp+1, Q, R and K, and in that it is implemented in a fuel cell system (16) comportant : - une batterie (20), - une pile à combustible (22), et - un moteur électrique (26), ^ ^^^ avec [Math.16] ^ = ^ ^ ^^ ^ ^^ ^^ ^ ^ où : ^ - Ibat est l’intensité du courant électrique dans la batterie (20), - IFC est l’intensité du courant électrique dans la pile à combustible (22), - f(IFC) est une fonction de IFC intervenant dans le calcul de VFC, ec [Math.17] ^ = ^ ^ ^^^ av ^ ^^ ^ ^^ ^ où : - Vbat est la tension électrique dans la batterie (20), - VFC est la tension électrique dans la pile à combustible (22), - mH2 est la consommation de dihydrogène dans la pile à combustible (22). ^^^ avec [Math.18] ^ = ^ ^ ^^ ^ ^^^^ ^ où : - SoC est l’état de charge de la batterie (20), - mH2 est la consommation de dihydrogène dans la pile à combustible (22), - Vloss est la tension liée aux pertes électriques dans le système, et en ce que les matrices A, B, C et D ont la forme suivante : 2 . Procédé selon la revendication précédente, caractérisé en ce qu’il comporte en outre : - une étape E2, entre les étapes E1 et E4, de détermination des paramètres à évolution lente θt = {^^ ^ , ^ ^ ^ , ^ ^ ^, … , ^ ^ ^} thanks to the m inimisation de l’écart e entre les valeurs mesurées des grandeurs de sortie (Yt)mes. et les valeurs calculées à l’aide de l’équation math.13 des grandeurs de sortie Yt : [Math.15] précédente caractérisé en ce que dans l’étape E2, les paramètres à évolution lente θt = {^^ ^ , ^ ^ ^ , ^ ^ ^ , … , ^ ^ ^} are obtained numerically after several iterations of résolution des équations 1 et de minimisation de l’écart e dans la time window [T0, T0+NΔT]. 4 . Procédé selon l’une quelconque des revendications 2 ou 3, caractérisé en ce qu’il comporte une nouvelle itération de l’étape E4 of determining the optimal control setpoint using the m éthode de programmation quadratique lorsque l’un des paramètres à évolution lente θt = {^^ ^ , ^ ^ ^ , ^ ^ ^ ^ , … , ^^} varies.

5. Procédé selon la revendication 4, caractérisé en ce qu’il comporte une étape E5 d’envoi à une unité de commande du système de la optimal control setpoint of the next instant Tp+1 after each new iteration of step E4 of determining the optimal control setpoint.

6. Procédé selon l’une quelconque des revendications 1 à 5, caractérisé en ce qu’il comporte en outre : - une étape E3 après l’étape E2 ou simultanément à l’étape E2, d’observation adaptative comprenant le calcul numérique de l’état du système xt à l’instant t par résolution numérique des équations 1 en prenant en compte l’écart e minimum à l’instant t.

7. Procédé selon l’une quelconque des revendications 1 à 6, caractérisé in that it is implemented in a system exhibiting fouling effects, or frictional forces.

8. Carte électronique dans laquelle le procédé selon l’une quelconque of claims 1 to 7 is implemented.

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