Stator phase resistance:
Main inductance:
Slip angular speed:
Mechanical angular speed of the motor:
Number of pole pairs:
Parameters of the CL physical model:
DC motors:
Open access peer-reviewed chapter
Submitted: 28 October 2016 Reviewed: 22 February 2017 Published: 30 August 2017
DOI: 10.5772/68050
Standard analytical methods are often ineffective or even useless for design of nonlinear control systems with imprecisely known parameters. The use of fuzzy logic principles presents one possible way to control such systems which can be used both for modeling and design of the control. The advantage of using this method consists in its simplicity and easy way of developing the algorithm, which in the phase of designing the controllers and also for modeling the features of the designed structures, allows the use of computer technology. Simplicity of the proposed structure (usually with the PI controllers) and determination of their parameters without any need for complex mathematical description present another considerable advantage of the used method. This chapter presents two typical examples of designing the control of nonlinear multi‐input multi‐output (MIMO) systems from the field of mechatronic systems based on fuzzy logic principles.
There are many processes in technological practice the analytical description of which is rather complicated. This can be due to their complexity, nonlinearity, transfer lags, complicated measurement of important parameters, etc. However, information on the performance of these processes can often be obtained experimentally (by suitably chosen measurements or by monitoring their responses to the control activities of the operator). In these situations, fuzzy systems can always be considered as an alternative for system modeling and control.
Applying fuzzy logic in fuzzy controller design is very often a suitable possibility of solving problem issues in control in various fields of industry because these controllers are an effective tool for achieving high‐quality properties of the controlled systems [1–9]. The disadvantage in this case is the unsystematic approach to their synthesis and a relatively demanding analysis of their stability. A fuzzy controller design is primarily based on the fuzzification of its range of inputs and the setting up of rules of its behavior within this range. The behavior of classic fuzzy controllers was designed on basis of linguistic rules obtained from experts. However, this knowledge is not always easily obtained, especially in cases of higher‐order nonlinear systems [10, 11]. For this reason, special attention has been focused in recent years on the design of fuzzy control systems that are not based on the search for expert linguistic rules [12–17]. Control methods based on the controlled system fuzzy model have many modifications that depend on the particular application [6, 13–17], while the quality of the fuzzy model of the controlled system is also of significant importance.
It has been proved that fuzzy modeling can be recognized as one of the nonlinear black‐box modeling techniques [11, 12, 18–20]. When designing a black‐box fuzzy system, it is necessary to identify its qualitative properties only on the basis of experimentally measured data, while neither its structure nor its parameters are known. That often results in problems with inconsistency of the database, problems with covering the entire space of possible inputs, etc. [20–22], which makes the fuzzy model unusable in practical applications. In the design of a black‐box fuzzy model of a dynamic system, a suitable method for the selection of qualitative properties from the collected database always needs to be applied. The functional dependencies between inputs and outputs can then be used for developing a suitable nonparametric fuzzy model of the process that can be applied in the design of their control [23–27].
Two typical examples for designing the control of nonlinear MIMO systems using fuzzy approach are presented in this chapter:
The design of fuzzy PI torque controller of the PI type for a drive with an induction motor, whose parameters and rules are obtained by searching control input of such a vector which is optimal in terms of the selected criterion of optimality.
The design of control for middle part of a continuous line for material processing by tension, where the continuous line presents a nonlinear MIMO system. Its control requires to ensure decoupled control of individual subsystems, because the output quality of the processed material depends directly on quality of the control. The controllers of the subsystems ensuring such decoupling usually are of complex structures and, when designing them by analytic methods, they are often unrealizable. When the continuous line is presented by a fuzzy model, it is possible to design simple controllers of the PI type ensuring high‐quality dynamical properties of the controlled system.
An asynchronous motor represents a strongly nonlinear fifth‐order system, whose good quality vector torque control is solved by relatively complex mathematical transformations and leads to a complicated control structure [28, 29]. Therefore, a fuzzy system for design of torque controller for asynchronous motor drive has been used. The fuzzy controller design is based on the concept of identifying the time sequence of the input signal into the controlled system that will provide the control target in terms of the selected optimality criterion. Fuzzy controller design method is characterized by simplicity, and quality of control is appropriate to the considered drive.
API‐type discrete controller is generally described by the equation:
where
A discrete fuzzy PI controller can be described, for example, by the following rules:
where quantities
Fuzzy rules and fuzzification of the fuzzy PI controller workspace can be identified by means of relations expressed by triplets [
When obtaining the sequence of values of vector
Having found the optimal input sequence
From the obtained triplets [
In the concrete application of the said procedure in a drive with asynchronous motor, we will use its analytical model (see Refs. [28, 29]). If we consider a rotating system which rotates with the frequency of the motor’s stator field (usually marked by coordinates
Used symbols:
AC drive parameters are given in the Appendix.
As a standard, asynchronous motors are supplied from static voltage frequency converters in which the stator frequency and voltage rate are
When connected directly to the power supply network, the motor shows a large increase in torque and also in current (Figure 2).
Let the aim of the torque controller design be to adjust the slip
For finding the optimal input sequence
We search for the input signal sequence
The said optimal input signal sequence was used for generating the database of triplets [
The resulting torque control structure of AM with the designed fuzzy controller is shown in Figure 7.
The start‐up of the drive with desired torque Mz = 30 Nm using the designed fuzzy controller is shown in Figure 8.
The comparison of Figures 5 and 8 shows that the application of the fuzzy controller resulted in the achievement of the desired torque responses of the AM at start‐up.
The fuzzy controller designed in this chapter is a PI controller, and it provides optimal dynamics in terms of the selected criterion Eq. (3). The design procedure consists of three steps:
In the first step, we search for such input sequence of input vector
In the second step, using the database obtained by application of the input vector
The relations obtained in the second step of the design procedure, written down in the form of a suitable input and output signals database (mostly in a table), are described in the third step by means of fuzzy logic principles. This way a fuzzy PI controller similar to an optimal continuous PI controller is constructed. Standard computing means for working with fuzzy systems are employed in this step, such as the fuzzy toolbox in Matlab.
The whole procedure of fuzzy PI controller design was verified by simulation of its properties in the concrete control of a drive with asynchronous motor. The results of simulation experiments show that the controller, in spite of its simplicity and the uncomplicated computer oriented design procedure applied, enables considerable improvement in the control circuit dynamic properties also in case of strongly nonlinear higher order controlled systems.
Typical representative of multi‐motor drive is the middle part of the continuous line, where the individual working machines are coupled with each other through the material. It can be lines for processing continuous flows of material (e.g., sheet metal strips, tubes, processing lines in paper mills, and printing works) by material traction in the field of elastic or plastic deformation, which influences the material’s mechanical properties. It means that the multiple motor drives are complex and coupled MIMO nonlinear systems. Therefore, due to the complexity of their mathematical models, which parameters are difficult to identify, the development of effective control systems is quite complicated task. This chapter presents the design of optimal control of continuous production line using a fuzzy model‐based approach.
The structure of the middle part of the continuous line (further referred to as CL) is shown in Figure 9. The structure includes DC motors powered through static transistor converters TC. The working machines of the line are driven by the motors through gearbox
The described system with the mechanical coupling of two machines presents a third‐order nonlinear MIMO system with two inputs and two outputs (Figure 10), the parameters of the system change depending on the mechanical properties of the material and on the speed of its motion. Defining precise parameters of this nonlinear system analytically presents a rather demanding task, and therefore, it is suitable to use for its description a fuzzy system (model) built only on basis of its measured input/output data.
Various fuzzy system structures consisting of static fuzzy subsystems and their dynamic parts can be found in the literature. In setting up the structure of the fuzzy model of a continuous line, we used its state description, where the given state of the system and the given input allow us to define the subsequent state, which can be expressed mathematically by the following equation:
where
Construction of the CL fuzzy model consists in determining the fuzzy approximation of this function on basis of the obtained CL inputs and outputs database. Considering the choice of CL input, state and output quantities presented in Figure 10, the structure of the proposed CL fuzzy model is shown in Figure 11.
The whole design of CL optimal control consists of two steps:
Step 1. The design of the fuzzy model for the middle section of the continuous line.
The first step in the design of the fuzzy model for the middle section of the continuous line is the establishment of a consistent database from measured inputs and their corresponding outputs, which covers its entire assumed work space and describes the behavior of the modeled system. For establishing a consistent database, we can use, for example, the method of dividing the input range into
The responses of the CL to inputs
The database for CL fuzzy model was generated as demonstrated in Figure 16. With sampling time
This measured database can be used to search for two FIS structures of the given nonlinear system which best describe the measured relations between [
Using the measured database, the particular fuzzy model can be designed by standardly known procedures of cluster analysis and adaptive approaches to improve the quality of modeling and reduce development time. The fundamental features of cluster analysis are reduction of the number of fuzzy rules and provision of good initial rule parameters. For our purpose from the large number of methods for adaptive fuzzy networks development [33–36], we chose the adaptive neuro‐fuzzy inference system (ANFIS) with subtractive clustering [14], which is a fast and robust data analysis method, having the following parameters: range of influence = 0.4, squash factor = 1.25, accept ratio = 0.4, reject ratio = 0.01. Subtractive clustering determines the optimal clusters [34] in a multi‐dimensional input/output space that accurately represent the data [34, 37] and CL behavior. The ANFIS approach uses Gaussian functions for fuzzy sets, linear functions for the rule outputs, and Sugeno’s inference mechanism [15]. The results were two static Sugeno‐type fuzzy systems with two rules for each output quantity as is shown in Figure 17.
The thus obtained fuzzy systems were implemented into the final continuous line fuzzy model structure, as illustrated in Figure 11.
To verify the correctness of the CL fuzzy model, randomly generated signals
The comparison of the fuzzy model outputs and CL physical model outputs for these inputs is shown in Figure 19.
The obtained results confirm that the designed fuzzy model very well approximates the performance of the continuous line also for randomly generated inputs and can be further used for the design of CL control.
Step 2. Design of optimal controller for middle section of continuous line.
The principal aim of CL control consists in achieving good dynamic control of tension in the material, with the speed of material movement being in accord with the pre‐set CL speed. As it has been said above, this is in fact a nonlinear MIMO system an important feature of which is mutual influencing of the individual input and state quantities that can result in bad quality or even in the destruction of the material being processed. This fact makes the controller design methods and their subsequent resulting structures often very complex and presents an obstacle to their wider practical application in industry. Therefore, our aim was to design a simple CL controller that would ensure the desired dynamics in terms of the selected criterion for systems that are only described by input/output relationships, that is, on basis of their fuzzy model.
For control of middle section of CL (for which fuzzy model was designed), we chose the simplest control structure consisting of two standard PI controllers (one for tension control
Processing of material in a CL is usually carried out in operation cycles during which a required amount of prepared material is processed (e.g., a roll of paper, a sheet metal coil.) An operation cycle includes three stages—line start‐up, line running at constant processing speed, and line delayed shut‐off.
The objective of the optimization is to find such vector
where
Let us note that what we are looking for is the extreme of the function of various variables, where the value of the criterial function for the individual vector
Several procedures can be applied for the purpose of optimization (e.g., genetic algorithm methods, and network charts). Thanks to today’s availability of high‐performance computing means, we chose the method of even geometrical division of the parameter space into equal intervals and of systematic searching within the whole range of the space. The advantage of this approach is that we can always identify the global minimum of the function Eq. (6), the disadvantage may be the time and computing demands in case vector
At the start of the optimization process, we determined the initial values of controller parameters
The value of the criterial function for initial values of vector
For finding optimal controller parameters, an m‐file was created in Matlab program environment. At the end of the search process for optimal vector of CL controller parameters, the value of the optimization criterion was
The proposed controller has been verified by experimental measurements on a real system which presented the physical model of the continuous line (the parameters of the CL physical model are specified in the Appendix). Figures 22 and 23 illustrate experimental results of the control of the continuous line middle section for selected operational cycle. In industrial practice, the required tension in the strip of material is set the first and then the line starts up to the desired operational speed.
Figure 22 shows the selected CL operation cycle in which first the desired value of tension in the strip of material is set to 0.8 N and at time 4 s the line starts up to reach the operational speed. At time 20 s, failure
For the control of continuous line tension and velocity, a very simple control structure with two PI controllers was designed. We looked for four optimal parameters in the structure, such that would best satisfy the chosen quadratic optimality criterion for the given operation cycle of the line.
The quality of the designed controllers depends on the quality of the constructed fuzzy model which very well approximates the performance of the modeled system and can be further employed in the design of various CL control structures and also in the identification of non‐measurable additive disturbances influencing the system, principally in real time.
The quality of the proposed controller depends on a large extent on a good quality of the nonlinear system fuzzy model which is constructed in the first step of the design procedure The model is constructed only on basis of suitably measured relations between the system’s inputs and outputs, without the necessity of preliminary knowledge of its internal structure and parameters. The fuzzy model design is based on the basic idea of dynamic system description in state space.
The quality of the proposed controller depends on a large extent on a good quality of the nonlinear system fuzzy model which is constructed in the first step of the design procedure (see Section 4). The model is constructed only on basis of suitably measured relations between the system’s inputs and outputs, without the necessity of preliminary knowledge of its internal structure and parameters. The fuzzy model design is based on the basic idea of dynamic system description in state space.
With this method, no principal limitations for the investigated system’s nonlinearities are defined, and therefore, there is good reason to assume that the presented method will find wide use in multi‐motor drives in steel industry, paper‐making, printing and textile industries, in the production of synthetic fibers and foils in the chemical industry and in other industries.
AC drive parameters:
Submitted: 28 October 2016 Reviewed: 22 February 2017 Published: 30 August 2017
© 2017 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.