Classifications

• • G—PHYSICS
  • G06—COMPUTING; CALCULATING OR COUNTING
  • G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
  • G06N5/00—Computing arrangements using knowledge-based models
  • G06N5/02—Knowledge representation; Symbolic representation
  • G06N5/022—Knowledge engineering; Knowledge acquisition
• • G—PHYSICS
  • G06—COMPUTING; CALCULATING OR 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

Definitions

• the present invention relates generally to control systems, and more particularly to the design method of intelligent control system structures based on soft computing optimization.
• Description of the Related Art Feedback control systems are widely used to maintain the output of a dynamic system at a desired value in spite of external disturbances that would displace it from the desired value.
• a household space-heating furnace controlled by a thermostat
• the thermostat continuously measures the air temperature inside the house, and when the temperature falls below a desired minimum temperature the thermostat turns the furnace on. When the interior temperature reaches the desired minimum temperature, the thermostat turns the furnace off.
• the thermostat-furnace system maintains the household temperature at a substantially constant value in spite of external disturbances such as a drop in the outside temperature. Similar types of feedback controls are used in many applications.
• a central component in a feedback control system is a controlled object, a machine or a process that can be defined as a "plant", whose output variable is to be controlled.
• the "plant” is the house
• the output variable is the interior air temperature in the house
• the disturbance is the flow of heat (dispersion) through the walls of the house.
• the plant is controlled by a control system.
• the control system is the thermostat in combination with the furnace.
• the thermostat-furnace system uses simple on-off feedback control system to maintain the temperature of the house.
• a feedback control based on a sum of proportional, plus integral, plus derivative feedbacks is often referred as a P(I)D control.
• a P(I)D control system is a linear control system that is based on a dynamic model of the plant.
• a linear dynamic model is obtained in the form of dynamic equations, usually ordinary differential equations.
• the plant is assumed to be relatively linear, time invariant, and stable.
• many real-world plants are time varying, highly non-linear, and unstable.
• the dynamic model may contain parameters (e.g., masses, inductance, aerodynamics coefficients, etc.), which are either only approximately known or depend on a changing environment. If the parameter variation is small and the dynamic model is stable, then the P(I)D controller may be satisfactory. However, if the parameter variation is large or if the dynamic model is unstable, then it is common to add Adaptive or Intelligent (AI) control functions to the P(I)D control system. AI control systems use an optimizer, typically a non-linear optimizer, to program the operation of the P(I)D controller and thereby improve the overall operation of the control system.
• Classical advanced control theory is based on the assumption that all controlled "plants" can be approximated as linear systems near equilibrium points.
• SC Soft Computing
• FNN Fuzzy Neural Networks
• FC Fuzzy Controllers
• the SC optimizer includes a fuzzy inference engine.
• the fuzzy inference engine includes a Fuzzy Neural Network (FNN).
• the SC Optimizer provides Fuzzy Inference System (FIS) structure selection, FIS structure optimization method selection, and Teaching signal selection.
• the user makes the selection of fuzzy model, including one or more of: the number of input and/or output variables; the type of fuzzy inference model (e.g., Mamdani, Sugeno, Tsukamoto, etc.); and the preliminary type of membership functions.
• a Genetic Algorithm is used to optimize linguistic variable parameters and the input-output training patterns.
• a GA is used to optimize the rule base, using the fuzzy model, optimal linguistic variable parameters, and a teaching signal.
• One embodiment includes fine tuning of the FNN. The GA produces a near-optimal FNN.
• the near-optimal FNN can be improved using classical derivative-based optimization procedures.
• One embodiment includes optimization of the FIS structure by using a GA with a fitness function based on a response of the actual plant model.
• One embodiment includes optimization of the FIS structure by a GA with a fitness function based on a response of the actual plant. The result is a specification of an FIS structure that specifies parameters of the optimal FC according to desired requirements.
• Figure 1 is a block diagram of the general structure of a self-organizing intelligent control system based on SC
• Figure 2 is a block diagram of the general structure of a self-organizing intelligent control system based on SC with a SC optimizer.
• Figure 3 shows information flow in the SC optimizer.
• Figure 4 is a flowchart of the SC optimizer.
• Figure 5 shows information levels of the teaching signal and the linguistic variables.
• Figure 6 shows inputs for linguistic variables 1 and 2.
• Figure 7 shows outputs for linguistic variable 1.
• Figure 8 shows the activation history of the membership functions presented in Figures 6 and 7.
• Figure 9 shows the activation history of the membership functions presented in Figures 6 and 7.
• Figure 10 shows the activation history of the membership functions presented in Figures 6 and 7.
• Figure 11 is a diagram showing rule strength versus rule number for 15 rules
• Figure 12A shows the ordered history of the activations of the rules, where the Y-axis corresponds to the rule index, and the X-axis corresponds to the pattern number (t).
• Figure 12B shows the output membership functions, activated in the same points of the teaching signal, corresponding to the activated rules of Figure 12 A.
• Figure 12C shows the corresponding output teaching signal.
• Figure 12D shows the relation between rule index, and the index of the output membership functions it may activate.
• Figure 13 A shows an example of a first complete teaching signal variable.
• Figure 13B shows an example of a second complete teaching signal variable.
• Figure 13C shows an example of a third complete teaching signal variable.
• Figure 13D shows an example of a first reduced teaching signal variable.
• Figure 13E shows an example of a second reduced teaching signal variable.
• Figure 13E shows an example of a third reduced teaching signal variable.
• Figure 14, is a diagram showing rule strength versus rule number for 15 selected rules after second GA optimization.
• Figure 15 shows approximation results using a reduced teaching signal corresponding to the rules from Figure 14.
• Figure 16 shows the complete teaching signal corresponding to the rules from Figure 14.
• Figure 17 shows embodiment with KB evaluation based on approximation error.
• Figure 18 shows embodiment with KB evaluation based on plant dynamics.
• Figure 19 shows optimal control signal acquisition.
• Figure 20 shows teaching signal acquisition form an optimal control signal.
• Figure 2J shows the stochastic excitation as a left subplot showing time history, and a right subplot showing the normalized histogram.
• Figure 22 ' shows the free oscillations under stochastic excitation.
• Figure 23- shows the free oscillations without excitation.
• Figure 24 shows the P(I)D control under stochastic excitation.
• Figure 2.5 shows the P(I)D gains and control force, obtained with P(I)D control under stochastic excitation.
• Figure 26 shows the P(I)D control without excitations.
• Figure 27 shows the P(I)D gains and control force, obtained with P(I)D control without excitation.
• Figure 28 shows the output of plant controlled by P(I)D controller with gains scheduled with SSCQ with minimum of plant entropy production.
• Figure 29 shows the P(I)D gains adjusted with SSCQ with minimum of plant entropy production, and corresponding control force.
• Figure 3,0 shows the output of plant with P(I)D gains adjusted with FC obtained using AFM, and as a teaching signals the results of SSCQ with minimum of plant entropy production.
• Figure 3 * 1 shows the control gains and control force obtained with AFM.
• Figure 32 shows the output of plant with P(I)D gains adjusted with FC obtained using SC optimizer, and as a teaching signals the results of SSCQ with minimum of plant entropy production.
• Figure 33 shows the control gains and control force obtained with SC optimizer.
• Figure 34 shows a comparison of the control gains obtained with SC optimizer and with AFM.
• Figure 35 shows a comparison of the plant controlled variable obtained with SC optimizer and with AFM controller.
• Figure 36 shows the plant entropy obtained with AFM based FC and with SC optimizer based FC.
• Figure 37 shqws the plant entropy production obtained with AFM based FC and with SC optimizer based FC.
• Figure 38 shows the swing dynamic system.
• Figure 39 shows the stochastic excitation used for teaching signal acquisition.
• Figure 4Q ⁇ shows the teaching signal obtained with GA and Approximated with FNN and with SC optimizer.
• Figure 41 shows the control error obtained with different controllers, simulation conditions are the same as was set for teaching signal acquisition.
• Figure 42 shows the control error derivative obtained with different controllers, simulation conditions are the same as was set for teaching signal acquisition.
• Figure 43 shows the controlled state variable dynamics obtained with different controllers, simulation conditions are the same as was set for teaching signal acquisition.
• Figure 44 shows the intended fitness function of the control obtained with different controllers, simulation conditions are the same as was set for teaching signal acquisition.
• Figure 45 shows the intended fitness function of the control obtained with different controllers, simulation conditions are the same as was set for teaching signal. Comparison only between FNN and SC optimizer based control.
• Figure 46 shows the control gains obtained with different controllers, simulation conditions are the same as was set for teaching signal. P(I)D was set up to the constant gains [5 5 5].
• Figure 47 shows the stochastic excitation used for check of the robustness of the obtained KB.
• Figure 4 ⁇ shows the different realization of the stochastic excitation from the same distribution as for teaching signal.
• Figure 49 shows the controlled variable for a new excitation signal.
• Figure 50 shows the coefficient gains for the new excitation signal.
• Figure 5.1 shows the different reference signal.
• Figure 52 shows the simulation results.
• Figure 53 shows the fitness functions.
• Figure 5.4 shows the coefficient gains.
• Figure 5'5 shows the plant and controller entropy.
• Figure 5 ⁇ shows swing motion under fuzzy control with two P(I)D controllers. Motion along Theta-axis under Gaussian stochastic excitation Comparison of P(I)D,FNN and SCO control.
• Figure 57 shows Swing motion under fuzzy control with two P(I)D controllers. Motion along L-axis under non-Gaussian (Rayleigh) stochastic excitation Comparison of P(I)D, FNN and SCO control.
• Figure 58 shows the controlled variable for a new excitation signal.
• FIG. 59 shows Swing motion under fuzzy control with two P(I)D controllers, Motion along Theta-axis under Gaussian stochastic excitation SCO and FNN Control law comparison, control along Theta-axis.
• Figure 59 shows Swing motion under fuzzy control with two P(I)D controllers, Motion along Length-axis under Gaussian stochastic excitation SCO and FNN Control law comparison, Control along Length-axis.
• Figure 60 shows Swing motion under fuzzy control with two P(I)D controllers. SCO and FNN Control force (Theta-axis and Length-axis) comparison.
• Figure 6L shows Swing motion under fuzzy control with two P(I)D controllers, investigation of robustness, Motion along Theta-axis under Gaussian stochastic excitation, comparison of P(I)D, FNN and SCO control.
• Figure 62 shows swing motion under fuzzy control with two P(I)D controllers, investigation of robustness, motion along Length-axis under non-Gaussian (Rayleigh) stochastic excitation, comparison of P(I)D, FNN and SCO control.
• Figure 1 shows a self-organizing control system 100 for controlling a plant based on Soft Computing (SC).
• the control system 100 includes a plant 120, a Simulation System of
• the SSCQ 130 includes a module 132 for calculating a fitness function, such as, in one embodiment, entropy production from of the plant 120, and a control signal output from the P(I)D controller 150.
• the SSCQ 130 also includes a Genetic Algorithm (GA) 131. In one embodiment, a fitness function of the GA 131 is configured to reduce entropy production.
• the FLCS 140 includes a FNN 142 to program a FC 143. An output of the FC 143 is a coefficient gain schedule for the P(I)D controller 150.
• a fitness function 132 in a GA 131 works in a manner similar to an evolutionary process to arrive at a solution which is, hopefully, optimal.
• the GA 131 generates sets of "chromosomes" (that is, possible solutions) and then sorts the chromosomes by evaluating each solution using the fitness function 132.
• the fitness function 132 determines where each solution ranks on a fitness scale. Chromosomes (solutions) which are more fit, are those which correspond to solutions that rate high on the fitness scale. Chromosomes which are less fit, are those which correspond to solutions that rate low on the fitness scale.
• a P(I)D controller 150 has a substantially linear transfer function and thus is based upon a linearized equation of motion for the controlled "plant" 120.
• Prior art GA used to program P(I)D controllers typically use simple fitness functions and thus do not solve the problem of poor controllability typically seen in linearization models. As is the case with most optimizers, the success or failure of the optimization often ultimately depends on the selection of the performance (fitness) function 132.
• the weight vector K is used by a conventional proportional-integral-differential (P(I)D) controller 150 in the generation of a signal ⁇ (K) which is applied to the plant.
• the entropy S( ⁇ (K)) associated to the behavior of the plant on this signal is assumed as a fitness function to minimize.
• the GA is repeated several times at regular time intervals in order to produce a set of weight vectors.
• the vectors generated by the GA 131 are then provided to a FNN 142 and the output of the FNN 142 to a Fuzzy Controller (FC) 143.
• the output of the FC 143 is a collection of gain schedules for the P(Z)Z -controller 150 that controls the plant.
• Figure 2 shows the self-organizing control system of Figure 1, where the FLCS 140 is replaced by an FLCS 240.
• the FLCS 240 includes a SC optimizer 242 configured to program an optimal FC 243.
• the SSCQ 130 finds teaching patterns (input-output pairs) for optimal control by using the GA 131 based on a mathematical model of controlled plant 120 and physical criteria of minimum of entropy production rate.
• the FLCS 240 produces an approximation of the optimal control produces by the SSCQ 130 by programming the optimal FC 243.
• the SSCQ 130 provides acquisition of a robust teaching signal for optimal control.
• the output of SSCQ 130 is the robust teaching signal, which contains the necessary information about the optimal behavior of the plant 120 and corresponding behavior of the control system 200.
• the SC optimizer 242 produces an approximation of the teaching signal by building a Fuzzy Inference System (FIS).
• the output of the SC optimizer 242 includes a Knowledge Base (KB) for the optimal FC 243.
• the optimal FC operates using an optimal KB from the FC 243 including, but not limited to, the number of input-output membership functions, the shapes and parameters of the membership functions, and a set of optimal fuzzy rules based on the membership functions.
• the optimal FC 243 is obtained using a FNN trained using a training method, such as, for example, the error back propagation algorithm.
• the error back propagation algorithm is based on application of the gradient descent method to the structure of the FNN.
• the error is calculated as a difference between the desired output of the FNN and an actual output of the FNN.
• the error is "back propagated" through the layers of the FNN, and the parameters of each neuron of each layer are modified towards the direction of the minimum of the propagated error.
• the back propagation algorithm has a few disadvantages. First, in order to apply the back propagation approach it is necessary to know the complete structure of the FNN prior to the optimization.
• the back propagation algorithm can not be applied to a network with an unknown number of layers or an unknown number nodes. Second, the back propagation process cannot modify the types of the membership functions.
• the back propagation algorithm very often finds only a local optimum close to the initial state rather then the desired global minimum. This occurs because the initial coefficients for the back propagation algorithm are usually generated randomly.
• the error back propagation algorithm is used, in a commercially available Adaptive Fuzzy Modeler (AFM).
• the AFM permits creation of Sugeno 0 order FIS from digital input-output data using the error back propagation algorithm.
• the algorithm of the AFM has two steps. In the first AFM step, a user specifies the parameters of a future FNN. Parameters include the number of inputs and number of outputs and the number of fuzzy sets for each input/output.
• AFM "optimizes" the rule base, using a so-called “let the best rule win” (LBRW) technique.
• LBRW let the best rule win
• the membership functions are fixed as uniformly distributed among the universe of discourse, and the AFM calculates the firing strength of the each rule, eliminating the rules with zero firing strength, and adjusting centers of the consequents of the rules with nonzero firing strength. It is possible during optimization of the rule base to specify the learning rate parameter.
• the AFM also includes an option to build the rule base manually. In this case, user can specify the centroids of the input fuzzy sets, and then the system builds the rule base according to the specified centroids. In the second AFM step, the AFM builds the membership functions. The user can specify the shape factors of the input membership functions.
• Shape factor supported by the AFM include: Gaussian; Isosceles Triangular; and Scalene Triangular.
• the user must also specify the type of fuzzy AND operation in the Sugeno model, either as a product or a minimum.
• AFM starts optimization of the membership function shapes.
• the user can also specify optional parameters to control optimization rate such as a target error and the number of iterations.
• AFM inherits the limitations and weaknesses of the back propagation algorithm described above.
• the user must specify the types of membership functions, the number of membership functions' for each linguistic variable and so on. AFM uses rule number optimization before membership functions optimization, and as a result, the system becomes very often unstable during the membership function optimization phase.
• the SC optimizer 242 provides GA based FNN learning including rule extraction and KB optimization.
• the SC optimizer 242 can use as a teaching signal either an output from the SSCQ 130 and/or output from the plant 120 (or a model of the plant
• the SC optimizer 242 includes (as shown in Figure 3) a fuzzy inference engine in the form of a FNN.
• the SC optimizer also allows FIS structure selection using models, such as, for example, Sugeno FIS order 0 and 1, Mamdani FIS, Tsukamoto FIS, etc.
• the SC optimizer 242 also allows selection of the FIS structure optimization method including optimization of linguistic variables, and/or optimization of the rule base.
• the SC optimizer 242 also allows selection of the teaching signal source, including: the teaching signal as a look up table of input-output patterns; the teaching signal as a fitness function calculated as a dynamic system response; the teaching signal as a fitness function is calculated as a result of control of a real plant; etc.
• output from the SC optimizer 242 can be exported to other programs or systems for simulation or actual control of a plant 130.
• output from the FC optimizer 242 can be exported to a simulation program for simulation of plant dynamic responses, to an online controller (to use in control of a real plant), etc.
• THE STRUCTURE OF THE SC OPTIMIZER Figure 4 is a high-level flowchart 400 for the SC optimizer 242. By way of explanation, and not by way of limitation, the operation of the flowchart divides operation in to four stages, shown as Stages 1, 2, 3, 4, and 5.
• Stage 1 the user selects a fuzzy model by selecting one or parameters such as, for example, the number of input and output variables, the type of fuzzy inference model (Mamdani, Sugeno, Tsukamoto, etc.), and the source of the teaching signal
• a first GA GA1 optimizes linguistic variable parameters, using the information obtained in Stage 1 about the general system configuration, and the input- output training patterns, obtained from the training signal as an input-output table.
• the teaching signal is obtained using structure presented in Figures 19 and 20.
• Stage 3 precedent part of the rule base is created and rules are ranked according to their firing strength. Rules with high firing strength are kept, whereas weak rules with small firing strength are eliminated.
• a second GA (GA2) optimizes a rule base, using the fuzzy model obtained in Stage 1, optimal linguistic variable parameters obtained in Stage 2, selected set of rules obtained in Stage 3 and the teaching signal.
• the structure of FNN is further optimized. In order to reach the optimal structure, the classical derivative-based optimization procedures can be used, with a combination of initial conditions for back propagation, obtained from previous optimization stages.
• the result of Stage 5 is a specification of fuzzy inference structure that is optimal for the plant 120.
• Stage 5 is optional and can be bypassed. If Stage 5 is bypassed, then the FIS structure obtained with the GAs of Stages 2 and 4 is used.
• Stage 5 can be realized as a GA which further optimizes the structure of the linguistic variables, using set of rules obtained in the Stage 3 and 4.
• Stage 4 and Stage 5 selected components of the KB are optimized.
• the consequent part of the rules may be optimized independently for each output in Stage 4.
• membership functions of selected inputs are optimized in Stage 5.
• the SC optimizer 242 uses a GA approach to solve optimization problems related with choosing the number of membership functions, the types and parameters of the membership functions, optimization of fuzzy rules and refinement of KB.
• GA optimizers are often computationally expensive because each chromosome created during genetic operations is evaluated according to a fitness function. For example a GA with a population size of 100 chromosomes evolved 100 generations, may require up to 10000 calculations of the fitness function. Usually this number is smaller, since it is possible to keep track of chromosomes and avoid re-evaluation. Nevertheless, the total number of calculations is typically much greater than the number of evaluations required by some sophisticated classical optimization algorithm. This computational complexity is a payback for the robustness obtained when a GA is used. The large number of evaluations acts as a practical constraint on applications using a GA.
• each time point of the teaching signal there is a correspondence between the input and output parts, indicated as a horizontal line in Figure 5.
• Each component of the teaching signal (input or output) is assigned to a corresponding linguistic variable, in order to explain the signal characteristics using linguistic terms.
• Each linguistic variable is described by some unknown number of membership functions, like “Large”, “Medium”, “Small”, etc.
• Figure 5 shows various relationships between the membership functions and their parameters.
• "Vertical relations" represent the explicit ess of the linguistic representation of the concrete signal, e.g. how the membership functions is related to the concrete linguistic variable. Increasing the number of vertical relations will increase the number of membership functions, and as a result will increase the correspondence between possible states of the original signal, and its linguistic representation.
• a linguistic variable is usually defined as a quintuple: (x,r(x), U,G,M) , where x is the name of the variable, T(x) is a term set of the , that is the set of the names of the linguistic values ofx , with a fuzzy set defined in U as a value, G is a syntax rule for the generation of the names of the values of the x and M is a semantic rule for the association of each value with its meaning.
• x is associated with the signal name from xory
• term set T(x) is defined using vertical relations
• U is a signal range. In some cases one can use normalized teaching signals, then the range of U is [0,1] .
• the syntax rule G in the linguistic variable optimization can be omitted, and replaced by indexing of the corresponding variables and their fuzzy sets.
• LM AX s specified by the user prior to the optimization, based on considerations such as the computational capacity of the available hardware system. Knowing the number of membership functions, it is possible to introduce a constraint on the possibility of activation of each fuzzy set, denoted as p x .
• the overlap parameter takes zero, when there is no overlap between two attached trapezoids. If it is greater than zero then there is some overlap. The areas with higher probability will have in this case "sharper" membership functions.
• the overlap parameter is another candidate for the GA1 search.
• the fuzzy sets obtained in this case will have uniform possibility of activation. Modal values of the fuzzy sets can be selected as points of the highest possibility, if the membership function has unsymmetrical shape, and as a middle of the corresponding trapezoid base in the case of symmetric shape.
• one can set the type of the membership functions for each signal as a third parameter for the GAL
• the relation between the possibility of the fuzzy set and its membership function shape can also be found.
• the possibility of activation of each membership function is calculated as follows:
• T-norm Fuzzy AND
• T-norm denoted as * is a two-place function from [ ⁇ ,l]x[ ⁇ ,l] to [ ⁇ ,l] . It represents a fuzzy intersection operation and can be interpreted as minimum operation, or algebraic product, or bounded product or drastic product.
• S- conorm, denoted by+ is a two-place function, from [ ⁇ ,l]x[ ⁇ ,l] to[ ⁇ ,l] . It represents a fuzzy union operation and can be interpreted as algebraic sum, or bounded sum and drastic sum.
• Typical T-norm and S-conorm operators are presented in the Table 1.
• equation (1.2) defines "vertical relations"; and ifz ' ⁇ /c , then equation (1.2) defines "horizontal relations".
• the measure of the "vertical” and of the "horizontal” relations is a mutual possibility of the occurrence of the membership functions, connected to the correspondent relation.
• the set of the linguistic variables is considered as optimal, when the total measure of "horizontal relations" is maximized, subject to the minimum of the "vertical relations".
• a fitness function for the GA1 which will optimize the number and shape of membership functions as a maximum of the quantity, defined by equation (1.2), with minimum of the quantity, defined by equation (1.1).
• the chromosomes of the GA1 for optimization of linguistic variables according to Equations (1.1) and (1.2) have the following structure: Where: ) e code the number of membership functions for each linguistic variable X Y ⁇ ) ; cc X ⁇ Y) are genes that code the overlap intervals between the membership functions of the corresponding linguistic variable N ; (- ⁇ ) ; and T X ⁇ Y) are genes that code the types of the membership functions for the corresponding linguistic variables. Another approach to the fitness function calculation is based on the Shannon information entropy.
• GA1 will maximize the quantity of mutual information (1.2a), subject to the minimum of the information about each signal (1.1a).
• the combination of information and probabilistic approach can also be used.
• Figure 6 shows results for input variables.
• Figure 7 shows results for output variables.
• Figures 8, 9, 10 show the activation history of the membership functions presented in Figures 6 and 7.
• the lower graphs of Figures 8, 9 and 10 are original signals, normalized into the interval [0, 1] OPTIMAL RULES SELECTION Rule pre-selection algorithm The pre-selection algorithm selects the number of optimal rules and their premise structure prior optimization of the consequent part.
• the total firing strength E ⁇ of the rule the quantity R ⁇ l (t) can be calculated as follows: for a continuous case, and: V ⁇ V ) for a discrete case.
• R Js L Q - Number of rules in complete rule base Quantity
• R Js is important since it shows in a single value the integral characteristic of the rule base. This value can be used as a fitness function which optimizes the shape parameters of the membership functions of the input linguistic variables, and its maximum guaranties that antecedent part of the KB describes well the mutual behavior of the input signals. Note that this quantity coincides with the "horizontal relations," introduced in the previous section, thus it is optimized automatically by GA1.
• the quantities R f s s can be used for selection of the certain number of fuzzy rules.
• Many hardware implementations of FCs have limits that constrain, in one embodiment, the total possible number of rules.
• the algorithm can select L ⁇ L 0 of rules according to a descending order of the quantities R ⁇ . Rules with zero firing strength can be omitted. It is generally advantageous to calculate the history of membership functions activation prior to the calculation of the rule firing strength, since the same fuzzy sets are participating in different rules. In order to reduce the total computational complexity, the membership function calculation is called in the moment t only if its argument x(t) is within its support. For Gaussian-type membership functions, support can be taken as the square root of the variance value ⁇ 2 .
• FIG. 11 An example of the rule pre-selection algorithm is shown in the Figure 11, where the abscissa axis is an index of the rules , and the ordinate axis is a firing strength of the rule R ⁇ .
• Each point represents one rule.
• the KB has 2 inputs and one output.
• a horizontal line shows the threshold level. The threshold level can be selected based on the maximum number of rules desired, based on user inputs, based on statistical data and/or based on other considerations. Rules with relatively high firing strength will be kept, and the remaining rules are eliminated. As is shown in Figure 11, there are rules with zero firing strength. Such rules give no contributions to the control, but may occupy hardware resources and increase computational complexity. Rules with zero firing strength can be eliminated by default.
• the presence of the rules with zero firing strength may indicate the explicitness of the linguistic variables (linguistic variables contain too many membership functions).
• the total number of the rules with zero firing strength can be reduced during membership functions construction of the input variables. This minimization is equal to the minimization of the "vertical relations.”
• This algorithm produces an optimal configuration of the antecedent part of the rules prior to the optimization of the rules. Optimization of the consequential part of KB can be applied directly to the optimal rules only, without unnecessary calculations of the "un- optimal rules".
• This process can also be used to define a search space for the GA (GA2), which finds the output (consequential) part of the rule.
• the history of the activation of the rules can be associated with the history of the activations of membership functions of output variables or with some intervals of the output signal in the Sugeno fuzzy inference case. Thus, it is possible to define which output membership functions can possibly be activated by the certain rule.
• Figure 12A shows the ordered history of the activations of the rules, where the Y-axis corresponds to the rule index, and the X-axis corresponds to the pattern number (t).
• Figure 12B shows the output membership functions, activated in the same points of the teaching signal, corresponding to the activated rules of Figure 12A. Intervals when the same indexes are activated in Figure 12B are uninteresting for rule optimization and can be removed.
• Figure 12C shows the corresponding output teaching signal.
• Figure 12D shows the relation between rule index, and the index of the output membership functions it may activate.
• Figures 13A-F show plots of the teaching signal reduction using analysis of the possible rule configuration for three signal variables.
• Figures 13A-C show the original signals.
• Figures 13D-F show the results of the teaching signal reduction using the rule activation history.
• the number of points in the original signal is about 600.
• the number of points in reduced teaching signal is about 40. Bifurcation points of the signal, as shown in Figure 12B are kept.
• Figure 14 is a diagram showing rule strength versus rule number for 12 selected rules after GA2 optimization.
• Figure 15 shows approximation results using a reduced teaching signal corresponding to the rules from Figure 14.
• Figure 16 shows the complete teaching signal corresponding to the rules from Figure 14.
• the fitness function used in the GA2 depends, at least in part, on the type of the optimized FIS. Examples of fitness functions for the
• Sugeno Model Generally Typical rules in the Sugeno fuzzy model can be expressed as follows: IF X is (/) ⁇ (x,) AND x 2 is ⁇ U) h (x 2 ) AND ... AND x conveyor is ⁇ (l) j x n )
• Typical rules in the first-order Sugeno fuzzy model can be expressed as follows: IF x, is // C0 ⁇ t ⁇ ) AND x 2 is ⁇ ⁇ ! h (x 2 ) AND ... AND x n is (x impart)
• Tsukamoto Model The typical rule in the Tsukamoto FIS is: IF x x is ⁇ (l) h (x x ) AND x 2 is ⁇ U) h (x 2 ) AND ... AND x discharge is ⁇ (l) jn (x n )
• Tsukamoto FIS is ⁇ j ⁇ » ⁇ y ) , where e I is the set of membership functions describing linguistic values of x ⁇ input variable; j 2 e I is the set of membership functions describing linguistic values of x 2 input variable; and so on , j consumer e I m is the set of membership functions describing linguistic values of x grasp input variable; and k e O is the set of monotonic membership functions describing linguistic values of y output variable.
• the output of the Tsukamoto FIS is calculated as follows:
• the Stage 5 refinement process of the KB structure is realized as another GA (GA3), with the search space from the parameters of the linguistic variables.
• the chromosome of GA3 can have the following structure:
• Different fuzzy membership function can have the same number of parameters, for example Gaussian membership functions have two parameters, as a modal value and variance. Iso-scalene triangular membership functions also have two parameters. In this case, it is advantageous to introduce classification of the membership functions regarding the number of parameters, and to introduce to GA3 the possibility to modify not only parameters of the membership functions, but also the type of the membership functions, form the same class. Classification of the fuzzy membership functions regarding the number of parameters is presented in the Table 2.
• GA3 improves fuzzy inference quality in terms of the approximation error, but may cause over learning, making the KB too sensitive to the input.
• a fitness function for rule base optimization is used.
• an information-based fitness function is used.
• the fitness function used for membership function optimization in GA1 is used.
• the refinement algorithm can be applied only to some selected parameters of the KB.
• refinement algorithm can be applied to selected linguistic variables only.
• the structure realizing evaluation procedure of GA2 or GA3 is shown in Figure 17.
• the SC optimizer 17001 sends the KB structure presented in the current chromosome of GA2 or of GA3 to FC 17101.
• An input part of the teaching signal 17102 is provided to the input of the FC 17101.
• the output part of the teaching signal is provided to the positive input of adder 17103.
• An output of the FC 17101 is provided to the negative input of adder 17103.
• the output of adder 17103 is provided to the evaluation function calculation block 17104.
• Output of evaluation function calculation block 17104 is provided to a fitness function input of the SC optimizer 17001, where an evaluation value is assigned to the current chromosome.
• evaluation function calculation block 17104 calculates approximation error as a weighted sum of the outputs of the adder 17103.
• evaluation function calculation block 17104 calculates the information entropy of the normalized approximation error.
• the function / includes the model of an actual plant controlled by the system with FC.
• the plant model in addition to plant dynamics provides for the evaluation function.
• function / might be an actual plant controlled by an adaptive P(I)D controller with coefficient gains scheduled by FC and measurement system provides as an output some performance index of the KB.
• the output of the plant provides data for calculation of the entropy production rate of the plant and of the control system while the plant is controlled by the FC with the structure from the the KB.
• the evaluation function is not necessarily related to the mechanical characteristics of the motion of the plant (such as, for example, in one embodiment control error) but it may reflect requirements from the other viewpoints such as, for example, entropy produced by the system, or harshness and or bad feelings of the operator expressed in terms of the frequency characteristics of the plant dynamic motion and so on.
• Figure 18 shows one embodiment the structure-realizing KB evaluation system based on plant dynamics.
• SC optimizer 18001 provides the KB structure presented in the current chromosome of the GA2 or of the GA3 to an FC 18101.
• the FC is embedded into the KB evaluation system based on plant dynamics 18100.
• the KB evaluation system based on plant dynamics 18100 includes the FC 18101, an adaptive P(I)D controller 18102 which uses the FC 18101 as a scheduler of the coefficient gains, a plant 18103, a stochastic excitation generation system 18104, a measurement system 18105, an adder 18106, and an evaluation function calculation block 18107.
• An output of the P(I)D controller 18102 is provided as a control force to the plant 18103 and as a first input to the evaluation function calculation block 18107.
• Output of the excitation generation system 18104 is provided to the Plant 18103 to simulate an operational environment.
• An output of the Plant 18103 is provided to the measurement system 18105.
• An output of the measurement system 18105 is provided to the negative input of the adder 18106 and together with the reference input Xref forms in adder 18106 control error which is provided as an input to the P(I)D controller 18102 and to the FC 18101.
• An output of the measurement system 18105 is provided as a second input of the evaluation function calculation block 18107.
• the evaluation function calculation block 18107 forms the evaluation function of the KB and provides it to the fitness function input of SC optimizer 18001.
• Fitness function block of SC optimizer 18001 ranks the evaluation value of the KB presented in the current chromosome into the fitness scale according to the current parameters of the GA2 or of the GA3.
• the evaluation function calculation block 18107 forms evaluation function as a minimum of the entropy production rate of the plant 18103 and of the P(I)D controller 18102.
• the evaluation function calculation block 18107 applies Fast Fourier Transformation on one or more outputs of the measurement system 18105, to extract one or more frequency characteristics of the plant output for the evaluation.
• the KB evaluation system based on plant dynamics 18100 uses a nonlinear model of the plant 18103.
• the KB evaluation system based on plant dynamics 18100 is realized as an actual plant with one or more parameters controlled by the adaptive P(I)D controller 18102 with control gains scheduled by the FC 18101.
• plant 18103 is a stable plant.
• plant 18103 is an unstable plant.
• the output of the SC optimizer 18001 is an optimal KB 18002. TEACHING SIGNAL ACQUISITION
• Figure 19 shows optimal control signal acquisition.
• Figure 19 is an embodiment of the system presented in the Figures 1 and 2, where the FLCS 140 is omitted and plant 120 is controlled by the P(I)D controller 150 with coefficient gains scheduled directly by the SSCQ 130.
• the structure presented in Figure 19 contains an SSCQ 19001, which contains an GA
• the chromosomes in the GA0 contain the samples of coefficient gains as [kp,k D ,kj) N .
• the number of samples N corresponds with the number of lines in the future teaching signal.
• Each chromosome of the GA0 is provided to a Buffer 19101 which schedules the P(I)D controller 19102 embedded into the control signal evaluation system based on plant dynamics 19100.
• the control signal evaluation system based on plant dynamics 19100 includes the buffer 19101, the adaptive P(I)D controller 19102 which uses Buffer 19101 as a scheduler of the coefficient gains, the plant 19103, the stochastic excitation generation system 19104, the measurement system 19105, the adder 19106, and the evaluation function calculation block 19107.
• Output of the P(I)D controller 19102 is provided as a control force to the plant 19103 and as a first input to the evaluation function calculation block 19107.
• Output of the excitation generation system 19104 is provided to the Plant 19103 to simulate an operational environment.
• An output of Plant 19103 is provided to the measurement system 19105.
• An output of the measurement system 19105 is provided to the negative input of the adder 19106 and together with the reference input Xref forms in adder 19106 control error which is provided as an input to P(I)D controller 19102.
• An output of the measurement system 19105 is provided as a second input of the evaluation function calculation block 19107.
• the evaluation function calculation block 19107 forms the evaluation function of the control signal and provides it to the fitness function input of the SSCQ 19001.
• the fitness function block of the SSCQ 19001 ranks the evaluation value of the control signal presented in the current chromosome into the fitness scale according to the current parameters of the GA0.
• An output of the SSCQ 19001 is the optimal control signal 19002.
• the teaching for the SC optimizer 242 is obtained from the optimal control signal 19002 as shown in Figure 20.
• the optimal control signal 20001 is provided to the buffer 20101 embedded into the control signal evaluation system based on plant dynamics 20100 and as a first input of the multiplexer 20001.
• Control signal evaluation system based on plant dynamics 20100 includes a buffer 20101, an adaptive P(I)D controller 20102 which uses the buffer 20101 as a scheduler of the coefficient gains, a plant 20103, a stochastic excitation generation system 20104, a measurement system 20105 and an adder 20106.
• On output of the P(I)D controller 20102 is provided as a control force to the plant 20103.
• An output of the excitation generation system 20104 is provided to the plant 20103 to simulate an operational environment.
• An output of plant 20103 is provided to the measurement system 29105.
• An output of the measurement system 20105 is provided to the negative input of the adder 20106 and together with the reference input Xref forms in adder 20106 control error which is provided as an input to P(I)D controller 20102.
• An output of the measurement system 20105 is the optimal plant response 20003.
• the optimal plant response 20003 is provided to the multiplexer 20002.
• the multiplexer 20002 forms the teaching signal by combining the optimal plant response 20003 with the optimal control signal 20001.
• the output of the multiplexer 20002 is the optimal teaching signal 20004 which is provided as an input to SC optimizer 242.
• optimal plant response 20003 can be transformed in a manner that provides better performance of the final FIS.
• high and/or low and/or band pass filter is applied to the measured optimal plant response 20003 prior to optimal teaching signal 20004 formation.
• detrending and/or differentiation and/or integration operation is applied to the measured optimal plant response 20003 prior to optimal teaching signal 20004 formation.
• FIG. 21-37 shows results of fuzzy control of nonlinear dynamic system under stochastic excitation as an illustration of the example of teaching signal approximation with the optimal FC.
• the entropy production rate of the dynamic system is:
• the total energy is: T + U .
• Model parameters used for simulation are:
• Figure 21 shows the stochastic excitation as a left subplot showing time history, and a right subplot showing the normalized histogram.
• Figure 22 shows the free oscillations under stochastic excitation
• Figure 23 shows the free oscillations without excitation
• Figure 24 shows the P(I)D control under stochastic excitation
• Figure 25 shows the P(I)D gains and control force, obtained with P(I)D control under stochastic excitation
• Figure 26 shows the P(I)D control without excitations
• Figure 27 shows the P(I)D gains and control force, obtained with P(I)D control without excitation
• Figure 28 shows the output of plant with P(I)D gains adjusted with SSCQ with minimum of plant entropy production
• Figure 29 shows the P(I)D gains adjusted with SSCQ with minimum of plant entropy production
• Figure 30 shows the output of plant with P(I)D gains adjusted with FC obtained using AFM, and as a teaching signals the
• the previous example showed simulated control of a stable plant.
• the SC optimizer 242 can also be used to optimize a KB for an unstable object as, for in one embodiment, a nonlinear swing dynamic system.
• the nonlinear equations of motion of the swing dynamic system are:
• the fitness function for the unstable swing is configured to minimize the entropy production rate in the plant and to minimize the entropy production rate in the control system.
• Figure 38 shows the swing system and its equations of motion.
• Figure 39 shows the excitation as a band limited white noise. This excitation was used for the teaching signal acquisition.
• Figure 40 shows the results of the approximation of the teaching signal for different values of the control error and for derivative of the control error.
• the "o" symbols in Figure 40 demonstrate the teaching signal.
• the solid line is a result of the approximation of the signal with back propagation-based FNN.
• the thin line in Figure 40 is the result of the approximation of the teaching signal with the SC optimizer 242.
• Table 4 The results of the approximation can be summarized in the following Table 4:
• Table 4 Figures 41, 42, 43, 44, 45, 46 show the simulation results.
• the simulation results can be summarized as follows. Approximation error of the FNN is smaller than approximation error of the SC optimizer, but both values are sufficient.
• For the FNN it is necessary to manually define number of membership functions for each input variable.
• the number of rules obtained with the FNN is greater than number of rules obtained with the SC optimizer.
• the stochastic excitation acting on the system in this case is the same as was used for the preparation of the teaching signal as well as a reference signal.
• Table 5 summarized in Table 5 below
• Both the FNN controller and the SC optimizer-based controller are better than the P( ⁇ )D controller.
• the FNN approximates the teaching signal with redundant accuracy, and, as a result, better performance with the same conditions as used for teaching signal acquisition, but control signals are unstable near equilibrium points.
• the SC optimizer control has better performance with respect to entropy production, and control gains have simpler physical realization.
• the output of the SC optimizer-based controller is stable near equilibrium points.
• the KB prepared with the SC optimizer uses 24 rules, and has almost the same performance (according to the selected fitness function) as the FNN based FC with 64 rules For analysis of the robustness of the simulated FC, the simulations were repeated with a new excitation signal, having longer duration, and different trajectory using the same distribution as was used for the teaching signal acquisition.
• Table 6 shows that both the FNN controller and the SC optimizer-based controller are better than the P(I)D controller regarding fitness function performance. Due to over learning, the FNN controller becomes unstable with unknown excitation, and asymptotically looses control under the intended fitness function.
• the SC optimizer control works better under unknown conditions, thus the FC prepared with a KB produced by the SC optimizer 242 is more robust regarding variations of the excitation signal from the same distribution.
• a new reference signal in introduced as a harmonic signal obtained by the following equation:
• Table 7 shows that the FC prepared with a KB generated by the SC optimizer 242 is more robust in the presence of reference signal variation.
• the SC optimizer creates a robust KB for FC and reduces the number of rules in comparison with a KB created with other approaches.
• the KB created by the SC optimizer 242 automatically has a relatively more optimal number of rules based.
• the KB created by the SC optimizer 242 tends to be smaller and thus more computationally efficient.
• the KB created by the SC optimizer tends to be more robust for excitation signal variation as well as for reference signal variation. Swing dynamic system simulation results. Motion under fuzzy control with two P(I)D Controllers.
• Sugeno 0 FIS with four inputs and six outputs variables.
• Input variables are: control error, derivative of control error for two P(I)D Controllers (along ⁇ and / -axes).
• Output variables are control gains for P(I)D ⁇ and P(I)D / correspondingly. For fuzzy simulation in this case we have chosen fitness function which minimizes a control error.
• Tables 43, 44 and Figures 56, 57, 58, 59 and 60 show the simulation results.
• Table 8 shows dynamic and thermodynamic characteristics of swing motion along ⁇ -axis.
• Table 9 shows dynamic and thermodynamic characteristics of swing motion along / • axis.

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