Classifications

• • G—PHYSICS
  • G05—CONTROLLING; REGULATING
  • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
  • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
  • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
  • G05B13/0265—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion
  • G05B13/0275—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric the criterion being a learning criterion using fuzzy logic only
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
  • G06N5/00—Computing arrangements using knowledge-based models
  • G06N5/04—Inference or reasoning models
  • G06N5/048—Fuzzy inferencing
• • G—PHYSICS
  • G06—COMPUTING OR CALCULATING; COUNTING
  • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
  • G06N7/00—Computing arrangements based on specific mathematical models
  • G06N7/02—Computing arrangements based on specific mathematical models using fuzzy logic
  • G06N7/023—Learning or tuning the parameters of a fuzzy system

Definitions

• FRBS Fuzzy Rule-Based System
• FRBS are used in many applications.
• FRBS are used in many applications.
• these systems are easily interpretable by a human being, which is one of the most remarkable of their characteristics.
• a linguistic variable v of a FRBS is defined as a 4-tuple:
• ⁇ ⁇ is the name of the language variable T v the set of linguistic terms
• Dom v the field of digital variable definition associated with v (a variable in the classical sense that underlies the linguistic variable v)
• ⁇ ⁇ is a semantic rule that associates with each term t GT V a fuzzy set or membership function defined on Dom v .
• ⁇ ⁇ ( ⁇ ) is replaced by ⁇ ⁇ because v is implicitly given by t.
• ⁇ ⁇ ⁇ ) is therefore the membership function associated with t, which for a point x of Dom v returns a degree of membership, element of [0,1]: ie Dom v ⁇ [0,1] -
• optimization means the use of so-called optimization methods and more particularly constraint optimization methods, ie aimed at adapting a device to improve its operation while respecting the constraints related to this operation or imposed by the user.
• optimization methods require means to evaluate the operation of said device, for example by being coupled to the actual device on which said device operates, to a simulator of this real device or to a database, and means to verify the satisfaction of said constraints.
• the optimization process is very specific to the type of FRBS used, it is developed ad hoc by the user, it does not have a generic character, it therefore requires a great technical expertise of the user for its design.
• An object of the invention is to optimize the membership functions associated with the linguistic variables of a FRBS, without decreasing interpretability, in the sense defined by the user through the constraints that it imposes on the optimization process. , thanks to the specifications hereafter called initial specifications and secondary specifications.
• Another object of the invention is to provide a formal generic framework underlying the method that is the subject of the invention, which allows the user to impose on the optimization process its constraints expressing the interpretability, without the need to know neither the operation of said optimization process nor the way of representing, applying or checking these constraints.
• a device for optimizing a fuzzy inference system or FRBS comprising:
• topological-geometric processing means delivering a topological signature, comprising first means for calculating a geometric signature, by calculating regions in which the order of the membership functions associated with linguistic variables of said system of fuzzy inference is constant, from said initial specifications, and second means for calculating a topological signature, by calculating topological invariants, from said geometrical signature; and
• Such a device makes it possible to optimize a FRBS, without decreasing interpretability.
• Such a device also provides a formal generic framework that allows the user to impose on the optimization process its constraints expressing the interpretability, without having to know the operation of said optimization process, nor the way of representing, apply or verify these constraints.
• a first function providing an FRBS belonging to the determined family from a set of parameters, the first function being defined later in the initial specification 12;
• a second function providing a real number of cost or vector of real numbers of costs, from a FRBS of said family, the second function being defined later in the initial specification 13, depending on whether the optimization module OPT is respectively single or multi-criteria;
• the device comprises second secondary specification input means, outputting geometric specifications, from said geometrical signature, wherein said optimization means are adapted to determine said associated membership functions. to the variables linguistic constraints imposed by said geometrical specifications.
• the device gives a great flexibility to the user, who can add, if he considers it necessary, additional geometrical constraints imposed by said geometrical specifications resulting from said secondary specifications and from said geometrical signature, on the solutions that the algorithm optimization needs to provide.
• said optimization means comprise means for verifying the realization of said topological constraints imposed by the topological signature resulting from the topological-geometric module.
• the device can verify automatically, without user intervention, whether the solutions provided by the optimization algorithm satisfy said topological constraints.
• said verification means are adapted to further verify the achievement of said geometric constraints imposed by said geometrical specifications.
• the device can verify automatically, without user intervention, whether the solutions provided by the optimization algorithm satisfy said topological constraints and said geometric constraints.
• said constraint realization verification means comprise other topologico-geometric processing means copies of said topological-geometric processing means.
• said initial specifications include:
• a first function providing a fuzzy inference system belonging to said family determined from one of said sets of parameters
• a second function providing a real number of cost or a vector of real numbers of cost determined from a fuzzy inference system of said family
• the user can provide a reference FRBS encoding his expert knowledge, some parameters of which are to be optimized by the invention.
• the optimization means are adapted to perform the single or multi-criteria optimization in order to implement an optimization under constraints respectively of a real number of cost or a vector of real numbers of costs.
• a topologico-geometric processing comprising a first calculation of a geometrical signature, by calculating regions in which the order of the membership functions associated with linguistic variables of said fuzzy inference system is constant, starting from said initial specifications , and a second calculation of a signature topological, by calculating topological invariants, from said geometrical signature;
• the method further comprises a step of entering secondary specifications from said geometric signature, which together determine geometric specifications and wherein said optimizing step determines the inference system. Fuzzy further optimized from geometric constraints imposed by said geometric specifications.
• the method further comprises a step of verifying the achievement of said topological constraints.
• said verification step further verifies the realization of said geometric constraints.
• said verification step uses another topological-geometric processing step copy of said topological-geometric processing step.
• FIG. 1 diagrammatically illustrates a device for optimizing a FRBS according to one aspect of the invention
• FIG. 2 schematically illustrates a device according to Figure 1, the constraint is further determined from geometric specifications, according to one aspect of the invention
• FIG. 3 schematically illustrates a device according to FIG. 1, further comprising a stress verification module, according to one aspect of the invention
• - Figure 4 schematically illustrates a device according to Figure 2, further comprising a stress verification module, according to one aspect of the invention
• FIG. 5 schematically illustrates an exemplary embodiment of the VERIF_C verification module of FIGS. 3 and 4;
• FIG. 6 schematically illustrates an exemplary membership function associated with the linguistic term "lukewarm"
• FIG. 7a illustrates an example of membership functions to be optimized
• FIG. 7b schematically illustrates a possible situation after optimization of the membership functions of FIG. 7a, which is more efficient but can not be interpreted in the user's sense;
• FIG. 8 illustrates a restriction imposed on a triangular membership function, according to the state of the art
• FIGS. 9, 10 and 11 illustrate membership functions associated with a user-defined "inside temperature of the dwelling" linguistic variable according to one aspect of the invention
• FIG. 12 illustrates all the membership functions of the embodiments of FIGS. 9, 10 and 11 associated with the same linguistic variable "interior temperature of the dwelling", according to one aspect of the invention
• FIG. 13 illustrates a obtaining of regions with colors associated with a relative order between the membership functions constant over a region, from the fuzzy partition of FIG. 12, according to one aspect of the invention
• FIG. 14 illustrates a linguistic variable delivered by the device DISP, optimized and interpretable according to the user, obtained thanks to the initial linguistic variable provided by the user represented in FIG. 12;
• FIG. 15 diagrammatically illustrates another example of application of the invention in the case of a spatial linguistic variable.
• the methods of the state of the art only specify ad-hoc constraints for a specific family of FRBS, without giving a method or generic system to obtain them from any FRBS.
• the present approach is based on a precise framework in which it is possible to codify the notion of interpretability (or at least a very important part of it: the readability of fuzzy partitions) in very different cases. From this framework is defined an optimization of a FRBS maintaining its interpretability, in which the constraints necessary for this maintenance are automatically extracted from the initial FRBS as well as, optionally, some secondary constraints set by the user.
• FIG. 1 shows a device DISP for adapting a FRBS comprising:
• a topologico-geometric treatment module TRT_TG comprising a first calculation module CALC1 of a geometric signature by calculating regions in which the order of the membership functions of each of the linguistic variables of the FRBS is constant, starting from the initial specifications; and a second calculation module CALC2 of a topological signature by calculating topological invariants from the geometrical signature; and
• a single or multi-criteria optimization module OPT outputting said optimized FRBS, from the topological signature determined by the module TRT_TG and respectively from one or more functions to be optimized determined by the initial specifications, transmitted by the first input module ENT1.
• the device DISP can be provided with a second input module ENT2 with secondary specifications providing, at the output, geometrical specifications from said secondary specifications and said geometrical signature, and the module d OPT optimization may be adapted to determine said optimized FRBS further from the geometrical specifications provided by the second input module ENT2.
• the device of FIG. 1 may comprise an optimization module OPT provided with a verification module VERIF_C for performing constraints from the topological signature.
• the device of FIG. 2 may comprise an optimization module provided with a verification module VERIF_C of realization of constraints in addition starting from the geometrical specifications.
• FIG. 5 illustrates an exemplary embodiment of verification module VERIF_C of FIGS. 3 and 4, comprising other topological-geometric processing means TRT_TGbis, copy of topological-geometric processing means TRT_TG, which outputs the topological signature T a ) and the geometrical signature ç (a) of a possible solution ⁇ corresponding to the parameters determining the optimized FRBS supplied at the output of the optimization module OPT.
• the verification module VERIF_C comprises a first comparison module COMP1 of the topological signature T p) of the initial admissible point p and of the topological signature T a) of the possible solution to which outputs an information S1 indicating their equivalence (validated condition ) or not (condition not validated).
• the verification module VERIF_C comprises a second comparison module COMP2 of the imposed geometrical specifications delivered by the second input module ENT2 and the geometrical signature ç (a) of the possible solution to which outputs information S2 indicating the realization ( condition validated) or not (condition not validated) conditions imposed by said geometrical specifications for.
• the initial specifications include:
• a first function ⁇ supplying an FRBS belonging to the family ⁇ determined from a set of parameters, the first function ⁇ being defined later in the initial specification 12; a second function providing a real number of cost or vector of real numbers of costs /, from an FRBS of said family ⁇ , the second function being defined later in the initial specification 13, depending on whether the optimization module OPT is respectively single or multi-criteria; and
• the optimization module OPT mono or multi-criteria can, for example, implement an optimization under constraints respectively of a real number of cost or a vector of real numbers of costs. For example, the methods described in the document Dechter, Rina (2003) "Constraint processing" by Morgan Kaufmann.
• the OPT module provides a set of parameters that determines a single optimized FRBS.
• the OPT module can provide several sets of parameters each determining a solution FRBS, for example those located on the Pareto front.
• each component of the vector of real numbers of cost is a criterion whose value is to be optimized.
• a solution is said on the Pareto front when it is not possible to improve it according to one of the criteria without deteriorating it according to at least one of the other criteria.
• Geometric and topological analysis is performed from an initial FRBS.
• the domain of certain linguistic variables of this FRBS, initially specified by the user, is partitioned so that each region is given by the different order relationships between the membership functions associated with said linguistic variables. These regions and Associated order relationships determine the geometric and topological features to be considered.
• ⁇ ⁇ , ..., ⁇ ⁇ represent the parameters to be considered in the optimization made by the optimization module OPT, a £ GA T , where T represents a topological space, which corresponds for example to R with the usual topology ( see the reference "Topology”, J. Munkres, Prentice Hall, 2nd edition, 2000, for standard definitions);
• ⁇ ( ⁇ ) represents the function ⁇ ⁇ A ⁇ ⁇ in which ⁇ represents the set of all possible FRBS.
• ⁇ ( ⁇ ) determines a FRBS from the parameters ⁇ EA, and in particular determines the membership functions that occur in each of the linguistic variables.
• the same symbol ⁇ is used to denote the FRBS and the function that provides a FRBS from a parameter set â.
• T V represents the set of linguistic terms associated with a linguistic variable v G ⁇ ⁇ ;
• (R, e) represents a colored region, associated with a linguistic variable v G ⁇ ⁇ , in which R represents a connected subset or region of Dom v , and e represents a color associated with v.
• a colored region, or associated with a color represents a connected region for which the order of the membership functions is constant or homogeneous
• P (Y Q , ..., Y n ) represents a set proposition, Y 0 , ..., Y n representing subsets of a set Y.
• Said set proposition is defined as a combination of conjunctions ⁇ and / or of disjunctions V of relations of inclusions between certain combinations of meetings U and / or intersections ⁇ of sets Y 0 , ..., Y n .
• £ ⁇ ( ⁇ ) represents the geometric signature of ⁇ relative to the linguistic variable v which consists of a collection of colored regions associated with v, in which v G ⁇ ⁇ and ⁇ G ⁇ . A more precise definition is given later.
• ⁇ ⁇ ⁇ represents the topological signature of ⁇ relative to the linguistic variable v defined from groups of homology resulting from the geometric signature of ⁇ relative to v, in which v G ⁇ ⁇ and ⁇ G ⁇ . A more precise definition is given later.
• £ ( ⁇ ) represents the geometric signature of ⁇ , which function at each v G ⁇ ⁇ associates the object Q v ( ⁇ ).
• ⁇ ⁇ ) represents the topological signature of ⁇ , which function at each v G ⁇ ⁇ associates the object ⁇ ⁇ ⁇ ).
• topological signatures when evoking topological characteristics, reference is made to properties related to topological signatures, and when evoking geometric characteristics, reference is made to properties related to geometric signatures.
• Geometric signatures determine topological signatures.
• the user To determine the relative interpretability of a user, ie the constraints that the output of the optimization module OPT must satisfy, the user provides an initial set of parameters p GA which determines a FRBS which he considers interpretable. From the set of parameters p, the topologico-geometric processing module TRT_TG calculates in the first calculation module CALC1 its geometric signature ç (p), then, from this, in the second calculation module CALC2, its topological signature T p).
• the geometric signature is therefore an object that contains more information than the topological signature.
• the present invention imposes a single minimal constraint for a possible solution to be valid: that its topological signature coincides, or rather be equivalent (notion specified later), to that of the initial admissible point, ie that T a) ⁇ T p) .
• the user can consider other conditions (or constraints) C a) concerning the geometrical characteristics of a. The form that these conditions may take is specified later.
• the initial specifications entered using the first ENT1 input module include:
• - 11 the space ⁇ of the possible FRBS and the space A defined above which the parameters â belong, and in which they can vary.
• - I2 a function ⁇ (): A ⁇ ⁇ which determines a FRBS from a parameter set â.
• - 13 a function /: ⁇ ⁇ R which gives the performance of a FRBS ⁇ G ⁇ ; if it is a problem of multi-criteria optimization, it is possible to consider /: ⁇ ⁇ M N , with n> 1.
• This function / can be given in different ways, for example from a simulator.
• the initial specifications entered using the first input module ENT1 also include specifications concerning the notion of interpretability:
• an element p GA called initial admissible point, as well as a subset of linguistic variables ⁇ ° ⁇ ( ⁇ ) _ ⁇ ⁇ ⁇ ( ⁇ ) , that is to say a subset of variables ⁇ ° ⁇ ( ⁇ ) included in the set of linguistic variables ⁇ ⁇ ( ⁇ ) associated with FRBS ⁇ ();
• the topologico-geometric processing module TRT_TG thus automatically extracts the geometric signature and the topological signature of p, determined by the linguistic variables belonging to ⁇ ° ⁇ ( ⁇ ) .
• the topological-geometric processing module TRT_TG has the task of extracting the geometric and topological characteristics of the initial admissible point provided by the user. Its output is double: ç (p) the geometric signature of p, and T p), its topological signature.
• the first calculation module CALC1 calculates regions from the relative orders of the membership functions ⁇ ⁇ of each linguistic variable belonging to ⁇ ° ⁇ ( ⁇ ) and outputs the geometric signature ç (p). This output is firstly used as input of the second calculation module CALC2 responsible for calculating certain topological invariants and outputting T p), and secondly transmitted to the second input module ENT2, optional shown in Figure 2 , so that the user can, if he wishes, introduce secondary specifications that from the geometric signature will give geometric specifications.
• the permissible point p determines completely, from the function ⁇ provided in the initial specifications, a FRBS (p) G ⁇ .
• the FRBS ⁇ ( ⁇ ) determines the membership function ⁇ ⁇ ⁇ : Dom v ⁇ [0,1] for each t GT v .
• the first calculation module CALC1 of the topological-geometric processing module TRT_TG calculates colored regions, ie for which the order of the membership functions is constant, consisting in partitioning Dom v into related subspaces. R £ such as
• the first calculation module CALC1 partitions the domain Dom v into disjoint related regions, such that the order of the membership functions in each region is homogeneous or constant.
• the first calculation module CALC1 can, for example, discretize the domain Dom v in the form of a grid whose each node represents a discrete point of the domain Dom v , and the nodes are connected to each other while preserving the topology of the Dom domain v , for example using the technique described in the document "A new characterization of simple elements in a tetrahedral mesh" of I.BIoch, J.Pescatore, and L. Garnero -. Graphical Models 67, pp. 260-284, Elsevier 2005, then calculate the value of each membership function in each node, and from this information give a color to each node.
• Q v ⁇ p This collection of colored regions.
• the geometric signature ç (p) is the union of these Q v ⁇ p), or more precisely the function that associates with each linguistic variable v G ⁇ ° ⁇ ( ⁇ ) the collection Q v (p).
• the first calculation module CALC1 of the topologico-geometric processing module TRT_TG transmits to the user, via the second input module ENT2, the geometric signature Q ⁇ p), so that it can, if necessary, introduce secondary geometrical specifications that it wishes to impose on the solution that the optimization module OPT must output.
• the second calculation module CALC2 of the topological-geometric processing module TRT_TG analyzes the colored regions ⁇ R, e) EQ v (p) and calculates certain topological invariants, which impose a minimum constraint to the solutions, possibly through the verification module VERIF_C of realization of the constraints.
• the second calculation module CALC2 outputs the topological signature, supposed to capture the topological information contained in the geometric signature of the initial admissible point p.
• the goal is to capture all the topological information of each Q v ⁇ p).
• this definition means that it is possible to continuously deform the regions of Q v ⁇ ) to obtain those of Q v ⁇ a), and reciprocally continuously deform the regions of Q v ⁇ a) to obtain those from v (p).
• topological signature behaves very badly at the algorithmic level: in general, the verification of ⁇ ⁇ ⁇ ) ⁇ T v p) is not even decidable.
• This type of problem is well known in the field of algorithmic topology, in which homology groups H n of the treated spaces are used as the main tool for studying the topology.
• H n of the treated spaces are used as the main tool for studying the topology.
• These are topological invariants of algebraic character, which capture a very important part of the topological information (for more precision, refer to A. Zomorodian's document, "Topology for Computing Cambridge Monographs on Applied and Computational Mathematics", T 16 Cambridge University Press, 2005).
• topological invariants proposed as topological signature of p in the present invention are therefore defined from homology groups.
• other topological invariants may be considered as a topological signature.
• the topological signature is defined as follows.
• n EN outputs the homology group H n (R K ).
• the topological signature T p) of p is the application which, for ve ⁇ ° ⁇ ( ⁇ ). outputs T v p).
• the second calculation module CALC2 of the topological-geometric processing module TRT_TG uses the information of ç (p).
• the first part F_1 is trivial, since this information is already stored in ç (p).
• it is necessary to integrate in the processing module TRT_TG known calculation methods for the computation of the groups of homology for example starting from the discretization grid of the domain Dom v resulting from the module CALC1 by using the document "An iterative algorithm for homology computation on simplicial shapes" by D. Boltcheva, D. Canino, S. Merino Aceituno, J.-C. Leon, L. De Floriani, and F. Hriady, Computer-Aided Design 43 (11) : 1457-1467 (201 1).
• bijection h mentioned in this definition is not unique. This can happen for example when there are two different regions of the same color.
• the user may want to impose secondary conditions such as for example: - "the point x G Dom v belongs to the region that has the color e", if in Q v p) there is only one region having the color e;
• the user can then specify for each linguistic variable v G ⁇ ° ⁇ ( ⁇ ) by means of the second input module ENT2, a set proposition P ⁇ Xi, - > E m ) such that
• the optional verification module VERIF_C of constraint realization verifies that a solution â satisfies the conditterterpretability_1 and Condlnterpretability_2 conditions previously described:
• Hg QH j QH v The conditterterpretability_1 condition is true if H j ⁇ 0, which one assures in 1 .2) when one finds a h G HJ.
• Cond I nterpretabil ity_2 is true if Hg ⁇ 0, which one ensures in 2.1) when one finds a h G Hg. In other words, it is necessary that 1 .1) be true so that 1 .2) can be true and that 1 .2) be true so that 2.1) can be true.
• active elements that can affect the physical condition of the home, such as changing the temperature of the home, such as the heating system.
• the method used to evaluate inputs and operate the active elements consists of an FRBS.
• the sensors measure the physical state of the house, the measured values are transmitted to one or more calculators that simulate a FRBS, which in response transmits commands to the active elements that act on the physical state of the house.
• the overall goal is to minimize the energy consumption used by the thermal system to heat or cool the home.
• the FRBS consists of fuzzy rules combining membership functions, which, from the inputs (sensor measurements) provide a decision (control actions).
• Xi are input linguistic variables, such as “Outside temperature” or “floor heating temperature” and there is an output linguistic variable such as "Floor circuit circulator action”.
• EFAi and EFB are linguistic terms such as “Cold”, “Hot”, “Strongly Negative”, “Lowly Negative”, “Lowly Positive” or “Highly Positive”.
• a set of such rules is encoded in a processor, which evaluates the inputs provided by the sensors according to these rules and will produce actions accordingly.
• fuzzy set or the equivalent notion of membership function associated with a linguistic term.
• membership functions are functions that go from the domain of a linguistic variable to the interval [0,1] -
• membership function of the language term "lukewarm” could have a triangular shape, with values 1 to 30 0 and 0 to 25 ° and 35 °, as shown in Figure 6.
• the FRBS is implemented on a computer equipped with a processor, a memory, inputs (sensors, simulator, keyboard, mouse) and outputs (actuators, simulator, screen) . It is connected to a system representing the response of a house to the controls (for example, simulator, test bench or instrument house).
• the user generally an expert in the field concerned, in this case an expert in residential thermal regulation, configures the FRBS by introducing the membership functions and the rules by means of the keyboard, the mouse or any other adapted human-machine interface.
• the processor can simulate the FRBS and calculate the actions to be carried out using the fuzzy logic techniques (fuzzy implication, fuzzy conjunction, etc.), based on the values of the sensors provided by the system connected to the FRBS (instrumented house, bench d test or simulator). These actions control the devices of the thermal system or are applied to the actuators of the simulator.
• fuzzy logic techniques fuzzy implication, fuzzy conjunction, etc.
• the present invention allows the user to define what he means by interpretability to integrate it into the optimization.
• TRT_TG which uses a fuzzy partition associated with the fuzzy sets of FIGS. 9, 10 and 11, as illustrated in FIG. 12.
• This fuzzy partition induces 10 Ri regions of dimension 1 and 3 Rj regions of dimension 0, with colors associated with a relative order between the membership functions constant over a region, as illustrated in FIG. 13, which illustrates the link with the figure 12.
• regions Ri are computed by the first calculation module CALC1 of the topologico-geometric processing module TRT_TG, and compose, with its associated colors, the geometric signature Q ⁇ p), presented to the user via a screen .
• the second calculation module CALC2 of the topologico-geometric processing module TRT_TG calculates the topological signature T p) of the linguistic variable "Interior temperature of the dwelling" and stores it internally in terms of groups of homology.
• this topological signature T p determines constraints with the geometrical specifications, so that the solution generated by the optimization module OPT verifies the desired topological and geometrical conditions Condlterpretability_1 and Condlnterpretability_2.
• the computer on which the DISP device is implemented can use a simulator of the thermal system of the home to measure the performance of the solutions generated.
• the solution output by this algorithm consists of a set of modified parameters determining an optimized FRBS, especially in this example of new membership functions correspond to "Cold”, "Hot” and "Extreme”.
• FRBS not restricted to the control domain of a thermal system of a dwelling.
• This can be a FRBS to control a monitoring system, control a factory, a robot, a vehicle, or any process that can be controlled.
• This can also be a FRBS intended for a supervised or semi-supervised classification of objects of the signal, image, text, document, graphs, tree, data type ... this type of classification model is for example described in the document " Pattern Recognition and Machine Learning "from CM. Bishop, Springer 2006.
• the output of the FRBS then makes it possible to determine the class of membership of an object, the function to be optimized then being for example the rate of good classification.
• region R would correspond for example to "Near Paris

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