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Real interpolation method in Automatic Control Systems self-adjustment problem
Abstract
Practice demonstrates that most of existing methods and techniques of adjustment of the automatic control systems cannot provide the appropriate quality of the systems performing. That is why nowadays main attention is paid to adaptive methods and algorithms. In this paper, the Real Interpolation Method is offered for creation of adaptive Automatic Control Systems. It is based on use of a special case of the Laplace transformation, when the complex variable p = δ + jω comes to real part 5. Due to this approach the number of operations is decreased more than twice in comparison with the frequency domain methods and it considerably simplifies algorithms of self-adjustment of Automatic Control Systems.
... Based on the above analysis, this study proposes a solution to synthesize and calibrate the regulator using the real interpolation method. This method has a simple procedure and allows for a reduction in computational volume, while preserving the specific features and effects of conveyors on the system [33][34][35]. ...
... One of the methods developed by А.С. Коновалов and И.А.Орурк [33], determining the function has the form (12) with: -the steady-state value and the corresponding maximum of the step response determined based on the required overshoot σ R and the desired static mode of the system. When , the expression (12) takes the form: ...
... The essence of the synthesis method applying the real interpolation method is to convert the synthetic equation (15) to the form with real arguments [33][34][35], choose the distribution rule of interpolated nodes δ i , for example: (16) with η -the number of nodes or establishment the nodes {δ i } n to coincide with the zero points of a Chebyshev polynomial of the first order of degree η (T n (x)) to increase the total accuracy [36]: (17) with: a-The real parameter is used to correct the synthesis error, and {x i } nroots of equation: T n (x) =0, which is determined by the relation [36]: (18) Note that the feedback coefficients k d can be determined from the condition: (19) with, ΔH d -error in the settling mode. Solving equation (15) with two interpolated points (δ 1 , δ 2 ) in the interval δ ∈ [0.1,2], and the interpolation step: Δδ = 0.1, we get the results in Table 1. ...
Partial differential equations, integral, differential, or other equations describe multi-motor automatic electric drive systems containing elastic conveyor belts. Because of the elastic and distributive nature of the system parameters, the transfer function describing them is often a complex expression, containing not only the arguments as a linear system but also the inertial and transcendental components. This makes the precise control of tension and speed synchronously much more complicated than the centralized parameter system. A promising numerical solution based on the real interpolation method will simplify the procedure for synthesizing control loops while preserving the characteristic properties of objects with distributed parameters. The objective of the study is to propose a feasible solution for synthesizing the regulators based on the real interpolation method; it allows direct operation with the original transfer function containing the inertial and transcendental components. In this paper, we proposed an approach to synthesize the control system for objects with distributed parameters using the real interpolation method to reduce computational capacity and synthesis error while preserving the properties of this object class. Building an experimental model of the two-motor electric drive system containing an elastic conveyor to verify the effectiveness of the proposed algorithm. The simulation and experimental results indicate that the control system with the received regulators operating stably and meets the required quality criteria. It proves the efficiency of the synthesis algorithm based on the real interpolation method.
... These demands are generally made only with respect to one or two indices of goodness of system's functioning. The regulator design on such initial data is observed in123456. Widespread engineering methods of regulators design are reduced to approximation of the design equation, but not to its exact implementation . ...
... In this connection the approach on the basis of the discrete real Laplace transform is well represented [9]. In this work the Real Interpolation Method allowing to work with system models not only in a continuous form offered in [9] but also in a digital form [3], is used as the base of the embedded system software. Implementation of automatic tuning regulator algorithms of the embedded control system and its experimental researches with a direct-current motor as Control Object are observed. ...
... Real Discrete Transformation is generalization of Real Integrated Transformation [3] on lattice functions f (nT 0 ), where f (t) is the transformable time function, T 0 – sampling period, n = 1, 2, . . .. The real image-function F * (δ) of original-function f (kT 0 ) is defined by the formula of direct transformation ...
The possibility of the embedded automatic-control system construction for a self-adjusted regulator design on the basis of dynamic compensation principle is observed. The description of mathematical and algorithmic apparatus of Control Object identification in a digital form resulted in a design of the regulator. Results of natural experiments are given. An analysis of the regulator robustness properties is carried out.
... The problem body refers to definition of Control Object mathematical model. This part of a problem is solved by two waysanalytical and experimental [1]. In the first case the model is searched on the basis of those laws which present the processes in the Object. ...
... It is the fundamental feature which ensures appreciable reducing of computational burden during organization of numerical information processing procedures -more than 2 times in comparison with frequency method [1,9]. The basic advantages of such approaches are following: • numerical methods developed for the considered class of functions are applicable for action with real functions; • it is possible to find F( ) δ not only with the help of time functions, but also with help of Laplace images, replacing a complex variable s on real δ ; • the mathematical models have graphic representations: that considerably increase obviousness of them; ...
... Second way, which is more interesting and attractive, is based on usage of the information about time dynamic characteristic h(t) of control objects. Following expression defines components of Numerical Characteristic in consideration of (Eq.1) and (Eq.3) [1,9]: ...
In this paper the Real Interpolation Method is offered for the solution of systems Identification problem. The Real Interpolation Method allows to create a sufficiently cost-effective algorithmic basis and to solve all complex of Identification problems. It is based on use of a special case of the Laplace transformation, when the complex variable s =δ + jω is degenerated to real value δ.
... However, they have complicated calculation procedures and large errors and are not a common technical standard. Therefore, this study will propose and investigate a solution to synthesize and calibrate the regulator using the real interpolation method [24,25]. This method has a simple procedure, allowing direct manipulation of the original model (1) in the real domain and thus reducing the amount of calculation while preserving the specific features and effects of the objects with distributed parameters. ...
... The synthesis equations corresponding to the singleloop control system ( Figure 6) will have the form: To establish the synthesis algorithm based on the real interpolation method (RIM) [24,25], the synthesis equations (16) are first converted to equivalent form: ...
Stabilizing the depth of the Towed Underwater Vehicle (TUV) under the impact of ocean waves is the first problem when building a control and monitoring system for TUV. Many strategies and control methods have been proposed in different studies, but they are not a general standard. Therefore, this research will propose and investigate a solution for synthesizing the control system to stabilize the TUV based on the real interpolation method (RIM) with the main contents: modeling the Towed cable (TC) - Underwater vehicle (UV) system, proposing the structure of the control system and the synthesis procedure of the regulators based on the RIM, building a synthesis program with different TC lengths. The simulation results show that the TUV exhibits good performance under the impact of sea wave. They can be applied to build the experimental models of TUV.
... It gives the possibility to develop a controller and get the best performance of control system. This paper presents a distributed parameter model approximation method which is based on Real Interpolation method (RIM) [2,3]. ...
... right part of (5) we have the unknown coefficients and known interpolation nodes i .As a result of system of linear algebraic equations solution we get a reduced mathematical model of a plant in terms of a real (2) is performed by another formal replacement of a real variable by a complex variable s[2,3]. ...
At present, many control theory scientists and control engineers work with controller design problem for distributed parameter system because many real industrial systems have distribution of its parameters. The main difficulty is caused by using the mathematical description in terms of irrational transfer functions which doesn't let to use existing design algorithms. It opens the way for approximation techniques. This paper presents a method for distributed parameter model approximation and enables to solve this problem with low computational costs.
... Особенности двух первых методов в основном достаточно ясны, поэтому остановимся на возможностях ВИМ. Его привлекательность в том, что он позволяет перераспределять погрешность на участках малых и больших значений времени [10]. Условной границей между ними можно считать время, соответствующее величине запаздывания. ...
... Matlab is the software, being widespread in the academic environment and finding its application when solving automatic control tasks [12,13]. It is explained, on the one hand, by a big set of the modules and instruments focused on the solution of various tasks of the analysis and synthesis of control systems, and on the other handby convenience of application of a graphical environment of imitating Simulink modeling for solution of these problems [7]. ...
The article contains presentation of basic provisions of the theory of automatic pulse control systems as well as methods of analysis of such systems using the mathematical software widespread in the academic environment. The pulse systems under research are considered as analogues systems interacting among themselves, including sensors, amplifiers, controlled objects, and discrete parts. To describe such systems, one uses a mathematical apparatus of difference equations as well as discrete transfer functions. To obtain a transfer function of the open-loop system, being important from the point of view of the analysis of control systems, one uses mathematical packages Mathcad and Matlab. Despite identity of the obtained result, the way of its achievement from the point of view of user's action is various for the specified means. In particular, Matlab uses a structural model of the control system while Mathcad allows only execution of a chain of operator transforms. It is worth noting that distinctions taking place allow considering transformation of signals during interaction of the linear and continuous parts of the control system from different sides. The latter can be used in an educational process for the best assimilation of the course of the control system theory by students.
... The identification problem consists estimation of transfer function (1) parameters and identification process should have low computational efforts. Chosen identification method is based on real interpolation method [5,6]: ...
This work presents a solution of the identification problem of chemical reactor plant’s mathematical model. The main goal is to obtain a mathematical description of a chemical reactor plant from experimental data, which based on plant’s time response measurements. This data consists sequence of measurements for water jacket temperature and information about control input signal, which is used to govern plant’s behavior.
... There are many methods of the synthesis of regulators for various objects in the modern theory of automatic control. The main ones are the methods of synthesis of PID-regulators [2], the method of frequency synthesis [3], the desired transfer function method [4], the modal method [5] and some others. ...
The article deals with the problem of the synthesis of control system for unstable object. The novelty of the approach has two main elements. Firstly, there is achieved the generalization of the real interpolation method of the synthesis of a wide class of objects, which includes unstable objects. Secondly, the approach allows eliminating the step of factorization of corresponding polynomials. This step is required for the formation of the reference model of automatic control system for unstable object.
... In literature (Goncharov 1995) this method is known as real interpolation method (RIM) ...
The paper considers identification problem of linear dynamic object based on real interpolation method. It is provided the solution of raised problem that gives formalized algorithm for structure and parameters definition using as input data step-response function. Suggested approach gives possibility to build transfer function of analyzed object with pre-determinate level of relevant error in time domain. Numerical example is provided.
This work considers the parametric identification problem of control objects and signals. The problem feature is the necessity of searching for an approximate solution that provides the improved accuracy of the transition process in the high (or low) values of time. This work suggests a variant that satisfies the requirements of accuracy and robustness to solve the problem. Based on using numerical methods to solve the problem, the variant has its own singularities. Firstly, its instrumental variable allows redistributing the maximum error of the approximate solution by the interval of transient process. Secondly, the variant is applied to the dynamic systems, transfer functions of which have fractional-rational, irrational and transcendental expressions. This work also leads the calculations to illustrate the suggested method that obtains an improved accuracy of the approximation in the initial and final fragments of the transient process.
The present paper describes approach to building the ordered sequence of image pixels at
lossless compression. This approach comprises the methods of supplementary virtual pixels, cascade
fragmentation, and code book. Virtual pixels are added to split image into randomly-sized fragments.
The cascade fragmentation method makes it possible to limit the number of stored start values to one.
The code book method allows to minimize the search of subimage optimal bypass, the search
algorithm complexity attaining O(N2). The numeric parameters such as the number of bypass options
and time required to define optimal bypass are given for 6Ч6 image.
Problem of Control Objects Identification by the Real Interpolation Method is observed. The consideration purpose is the creation of the mobile identification device's software oriented on experimental definition of the Control Objects or other elements of Automatic-Control systems mathematical description directly on the worksite. Appropriate algorithms should possess rather small volume of computing operations, matching to demands of real time. It is suggested that the device is used as a tool at systems diagnostic, adjustment of regulators, and other similar procedures. For the purpose of the problem solution accuracy increase and the decrease in volume of computing operations the Interpolation Points are chosen from Fibonacci numbers sequence.
The problem of sampled-data regulator synthesis of automatic control system is considered. For the decision of the problem, the real interpolation method is offered. Two important procedures, which guarantee decreasing of computation costs in comparison with traditional methods, are in its basis. Firstly, a special discrete transformation of time functions is used and derivable functions-images have real argument. Secondly, these functions are replaced with discrete forms, which can describe control system, its elements and signals. The offered way of problem decision is illustrated with numerical example.
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