Applications of Intelligent Hybrid Systems in Matlab

This paper deals problem of intelligent hybrid systems. Intelligent systems include neural networks (NN), fuzzy systems (FS) and genetic algorithms (GA). Each of these intelligent systems has certain properties (ability of learning, modelling, classifying, obtaining empirical rules, solving optimizing tasks ...) fitting specific kind of applications. Combination of these intelligent systems creates neuro-fuzzy system, fuzzy-GA system, neuro-GA system and these systems together are called hybrid intelligent systems (HIS). For these purposes, there was created program in Matlab, where were made several demo applications for several HIS in the field of system modelling and control. 1 Principles of Hybrid intelligent systems Several modern applications are realised by intelligent technologies as are neural networks (NN), fuzzy systems (FS) and genetic algorithms (GA). Each of these intelligent systems has certain properties (ability of learning, modelling, classifying, obtaining empirical rules, solving optimizing tasks ...) fitting specific kind of applications. Table 1 presents a comparison of different intelligent system [1]. Table 1: COMPARISON OF FUZZY SYSTEMS (FS), NEURAL NETWORKS (NN) AND GENETIC ALGORITHMS (GA) FS NN GA Knowledge representation good bad rather bad Uncertainty tolerance good good good Imprecision tolerance good good good Adaptability rather bad good good Learning ability bad good good Explanation ability good bad rather bad Knowledge discovery and data mining rather bad good rather good Maintain ability rather good good rather good Noticed, that in many real applications we would need not only to acquire knowledge from various sources, but also to combine different intelligent technologies. The need for such a combination has led to the emergence of hybrid intelligent systems (HIS). A hybrid intelligent system is one that combines at least two intelligent systems, i.e. combination of these intelligent systems creates neuro-fuzzy system, fuzzy-GA system, neuro-GA system [1]. The block scheme of HIS is depictured in Fig. 1. For example, HIS can represent following forms. Neuro-fuzzy system is realized as a neural network, in which fuzzy system parameters are encoded in several layers. Using network learning ability the parameters can be adapted, hence the system is called adaptive neural fuzzy inference system (ANFIS). Fuzzy-GA system provides fuzzy system parameters optimization using GA. NeuroGA system provides neural network parameters optimization using GA. For example, in Matlab were made several demo applications for several HIS in the field of system modelling and control. Figure 1: The block scheme of Hybrid Intelligent Systems 2 Genetic algorithms Genetic algorithm (GA) is a powerful stochastic-based search/optimization approach, which mimics the evolution in the nature. It is described in e.g. [2, 3, 4, 5] and others. A general scheme of a GA can be described by following steps (Fig. 2): 1. Initialization of the population of chromosomes (set of randomly generated chromosomes). 2. Evaluation of the cost function (fitness) for all chromosomes. 3. Selection of parent chromosomes. 4. Crossover and mutation of the parents → children. 5. Completion of the new population from the new children and selected members of the old population. Jump to the step 2. Figure 2: Block scheme of genetic algorithm Genetic algorithms fall into the optimization techniques, which are able to find global optimum of the function. They can be used to modeling or control linear or non-linear systems using fuzzy logic or neural networks. When the neural networks are used, the wanted parameters in chromosomes can be connections in the neural network, values of weights and biases or the both. In the case of fuzzy logic, the wanted parameters are parameters of membership functions, base of rules or the both. In modeling of a system, the optimized function is the cost function: ∑ ∑ = = − = = N i i m i N i i y y e J 1 1 , (1) where y is output from the system, ym is output from the model of system, e is model error and N is number of patterns. In control of a system, the optimized function is the cost function: ∑ ∑ = = − = = N i i i N i i y r e J 1 1 , (2) where r is reference variable, y is controlled output, e is control error and N is number of patterns. In the both of cases, the minimum of fitness is searching. Fitness is represented by the cost function or in the case of control, by the modified cost function, which can be penalized for example by derivation of process output y, or by measure or derivation of control action u. 3 Modelling of nonlinear function using ANFIS Neuro-fuzzy system represents connection of numerical data and linguistic representation of knowledge. The system is characterized by transparency as fuzzy systems and learning ability as neural networks. The structure of a neuro-fuzzy system is similar to a multi-layer neural network. In general, neuro-fuzzy system has input and output layers, and three hidden layers that represent membership functions and fuzzy rules. Encoded fuzzy system in several layers of neural network can be in form Mamdani or Sugeno fuzzy interface model. Using network learning ability, the parameters can be adapted, hence the system is called adaptive neural fuzzy inference system (ANFIS) [1, 6]. ANFIS represents Sugeno fuzzy model, which fuzzy rules can be expressed in the following form: IF x1 is A1 AND x2 is A2 .... AND xm is Am THEN y=f(x1,x2,...,xm) where x1, x2, ..., xm are input variables, A1, A2, ..., Am are fuzzy sets and y is either a constant or a linear function of the input variables. ANFIS is represented by a six-layer feedforward neural network, which architecture is displayed in Fig. 3 [1]. Figure 3: The architecture of ANFIS ANFIS is trained using I/O data and backpropagation algorithm. By train algorithm are optimized fuzzification neurons parameters, mask of inference system and defuzzification neurons parameters [1]. For example, ANFIS is used for modelling of nonlinear 3D function with name Peaks in following form: ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − − − + − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − + − − − − = 2 2 ) 1 ( ) 2 2 ( 5 3 2 2 3 1 ) 5 ( 10 ) 1 ( 2 ) 1 ( 3 y x y x e e y x x y x e x z (3) Peaks function neuro-fuzzy model has been created in Matlab. There has been made training and testing data set, both in the range 3 ; 3 , − ∈ y x and interval between each two consecutive input samples on the both axes was for training data 0.5 and for testing data 0.25 (Fig. 4). a) b) Figure 4: Training data a) and Testing data b) ANFIS structure has been created using order genfis1 in Matlab [6]. The order designs initial Sugeno type fuzzy inference system using a grid partition. There has been chosen 7 gbell membership functions for both of the inputs x and y. data = [x y z]; in_fis = genfis1(data,[7 7],char('gbellmf','gbellmf')); There has been used matlab order anfis for neuro-fuzzy system training, which uses hybrid learning algorithm to identify the membership function parameters of single-output, Sugeno type fuzzy inference systems (FIS). A combination of least-squares and backpropagation gradient descent methods are used for training FIS membership function parameters to model a given set of input/output data [6]. There has been employed 200 training epochs for training. epoch_n = 200; out_fis = anfis(data,in_fis,epoch_n); Trained neuro-fuzzy system (ANFIS) next has been tested on training and testing data. There has been used order evalfis to enumerate output from the system [6]. outfissim = evalfis([x y],out_fis); There has been calculated sum of least-squares errors of outputs from modelled system and Peaks function for training and testing data. ∑ = − = N i i i sim z z e 1 2 ) _ ( , (4) Where z is output from Peaks function, z_sim is output from modelled neuro-fuzzy system and N is number of samples. There has been attained these error values: training error e_tren = 0.0015 and testing error e_test = 11.1034. On Fig. 5, there are shown output from ANFIS for training a) and testing b) and errors of training c) and testing d).