Abstract : For a class of nonlinear processes, a nonlinear internal model control method based on the TS fuzzy model is proposed. Fuzzy modeling is performed using a genetic algorithm and fuzzy clustering method, solving the problem of establishing accurate models and their inverse models in nonlinear internal model control methods. The TS model and inverse model of the process are obtained through fuzzy identification, and an internal model controller is designed based on them. Finally, the method is applied to the control of a class of nonlinear processes, and simulation results demonstrate its effectiveness.
Keywords : Genetic algorithm; fuzzy modeling; parameter identification; internal model control
Chinese Library Classification Number: TP391.9 Document Identification Code : A
A Fuzzy Internal Model Control Algorithm Based on Genetic Algorithm
Zhang Xin-fa, Zhao Zhi-cheng
(School of Electronic Information Engineering of Taiyuan University of Science and Technology, Taiyuan, 030024, China)
Abstract: Considering a class nonlinear process, the nonlinear internal model method based on TS Fuzzy model is put forward in this paper. Using the genetic algorithm and fuzzy clustering method for fuzzy modeling, we solve the difficult problem which is model and inverse model in the nonlinear internal model control. By fuzzy identification achieving TS model and inverse model of the process, we design internal model controller based on this. This method is applied to the control for a class of nonlinear process; the simulation result shows the effectiveness of the method.
Key words: Genetic algorithm; Fuzzy modeling; Parameter identification; Internal model control
Internal Model Control (IMC) [1] is a new control strategy based on process mathematical models for control design. It has been valued by the control community for its simple design, good control performance and superiority in system analysis. Economou et al. [2] extended it to nonlinear systems in 1986, providing a very effective way for nonlinear system control.
Obtaining the model and inverse model of the process is the key issue in realizing internal model control. For nonlinear processes, even if the inverse model exists, it is often not easy to obtain directly. At present, the identification method of nonlinear systems has been widely studied. Reference [3] uses RBF neural network to identify nonlinear systems. Since it is a local approximation network, it is easy to get trapped in local minima during network training, so the global optimal solution cannot be obtained. Reference [4] proposes an improved BP neural network identification method, which uses a trained linear model to approximate the nonlinear system. The selection of the model has a great influence on the identification result. If the selected model error is too large, the satisfactory identification effect will not be achieved.
As a general approximator, the TS fuzzy model[5] treats a nonlinear system as the sum of the products of several linear subsystems and their weights, which is easy to express the dynamic characteristics of complex and nonlinear systems. At the same time, it can also apply the linear system control theory to the control of nonlinear systems, thus becoming a research hotspot. The TS model is identified by using the system's input-output data, including structural identification and parameter identification. Structural identification is used to determine the premise fuzzy rules of the TS model, and parameter identification is used to determine the parameters of the conclusion part. Reference [6] uses the clustering method to divide the fuzzy space. Each fuzzy subspace represents a fuzzy rule, but the number of fuzzy spaces is mainly determined by experience and lacks reliable theoretical basis. Reference [7] uses the least squares method, which only considers the identification accuracy. This can easily cause overfitting of the data and poor generalization ability.
To effectively overcome the shortcomings of previous methods, this paper applies a genetic algorithm to the parameter identification problem of the TS model. First, fuzzy clustering and least squares methods are used for coarse identification of the model. Then, the genetic algorithm is applied to simultaneously optimize the premise and conclusion parameters, thereby establishing an accurate TS model of the nonlinear process. Finally, the fuzzy process model and its inverse model obtained from the fuzzy identification are introduced into the internal model control method, and simulation results verify the effectiveness of the proposed method.
1. Identification of TS Fuzzy Model
1.1 TS Fuzzy Model
Takagi and Sugeno proposed the famous TS fuzzy model in 1985[8], which is described in the following form:
1.2 Prerequisite Structure and Parameter Identification of TS Fuzzy Model
The premise structure and parameters of the TS fuzzy model are determined by using the fuzzy C-means algorithm (FCM). The FCM algorithm can be expressed as minimizing the following objective function [9]:
1.3 TS Fuzzy Model Conclusion Parameter Identification
After determining the premise structure and the conclusion structure, the least squares method [10] can be used to roughly identify the conclusion parameters of the TS fuzzy model, fit the input-output data in each range of the premise structure into a first-order polynomial function, and thus obtain the rough conclusion parameters, thereby determining the range of conclusion parameters to be optimized by the genetic algorithm.
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2. Optimization of TS Fuzzy Model Based on Genetic Algorithm
2.1 Genetic manipulation
(1) Common encoding methods mainly include binary encoding and real number encoding. Using binary encoding requires converting binary to decimal, which not only introduces quantization errors but also reduces the optimization speed. However, real number encoding directly performs genetic operations on the original parameters, which not only improves the solution accuracy but also speeds up the optimization process. Since both the premise parameters and the conclusion parameters need to be optimized at the same time, and the number of parameters is large, real number encoding is adopted.
(2) Fitness Function Design In genetic algorithms, the fitness function is the basis for performing selection operations. In order to achieve the goal of optimization, the fitness function can generally be obtained by transforming the objective function. Here, the parameters of the TS fuzzy model are identified, and the mean square error can be used as the objective function:
Clearly, the greater the fitness of an individual, the greater the probability of it being selected.
(4) Crossover operation: In order to avoid destroying the superior individuals in the population, a single-point arithmetic crossover operator is used.
(5) Mutation Operation: In order to prevent the optimization from converging to a suboptimal solution too early, the mutation rate should be appropriately increased as the number of generations increases. The mutation rate can be represented by a function:
2.2 Parameter Optimization Steps Based on Genetic Algorithm
The process of optimizing the precursor and consequent parameters of the TS model using a genetic algorithm is as follows:
The range of values for these parameters can be determined based on the methods described above;
Step 2: Determine the population, number of generations, and mutation probability, given by equation (9), and initialize the population;
Step 3: Use equation (7) to determine the evaluation function;
Step 4: Perform genetic operations such as selection, crossover, and mutation;
If the algorithm termination condition is met in Step 5, then stop; otherwise, proceed to Step 3.
3. Fuzzy Internal Model Control
3.1 Internal Mold Control Structure
Figure 1. Structure of the internal model control system based on the TS model
Fig.1 Internal model control system structure based on TS model
3.2 Design of Internal Mold Controller
4. Simulation Study
The approximate mathematical model of the controlled process is [12]
Figure 2 Step response of FCM-FIMC and GA-FIMC methods
Fig.2 Step response of FCM-FIMC and GA-FIMC
Figure 3 shows the step responses of the perturbation-based FCM-FIMC and GA-FIMC methods.
Fig.3 Step response of FCM-FIMC and GA-FIMC with disturbance
Figure 4. Nonlinear system parameter perturbation and step response with negative step disturbance.
Fig.4 Step response of nonlinear system with perturbation parameters and disturbance
5. Conclusion
This paper applies genetic algorithms to the modeling of TS fuzzy models. Building upon the TS model parameter identification methods using FCM and least squares, this paper utilizes a genetic algorithm to simultaneously optimize both premise and conclusion parameters. This leverages the advantages of genetic algorithms—fast optimization speed and resistance to local optima—to establish an accurate TS fuzzy model. Then, the fuzzy model and its identification are introduced into internal model control, and an internal model controller based on the TS fuzzy model is designed. Simulation results show that the GA-FIMC method significantly outperforms the FCM-FIMC method. Furthermore, this method not only ensures good tracking performance but also maintains good robustness under external disturbances or system parameter perturbations.
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About the author: Zhang Xinfa (1982—), male, master's student, whose main research direction is computer measurement and control systems and devices.
Mailing Address: P.O. Box 673, Taiyuan University of Science and Technology, No. 66 Wuliu Road, Wanbailin District, Taiyuan City
Postal code: 030024
Contact number: 15035139584
Email: [email protected]
