IDENTIFICATION OF NONLINEAR SYSTEMS BY USING ELLIPTIC AND LOXODROMIC METHODS

2016. Т. 16, No 3. С. 159–167 159 Introduction A common application of the computing techniques using distributed computing allows for the efficient solutions of the complicated design, production, and scientific-research problems [1]. One of the main advantages of the distributed computing systems is a high speed of the similar calculations for different data sets [2], which makes it possible to use them in solving the tasks of control, design, predicting, diagnostics, and identification [3–5]. To solve the problem of parameter identification for the elements of the linear systems almost instantaneously, the Kalman filter is generally used. However, the Kalman filter often cannot provide with an adequate parameter valuation for the nonlinear systems especially those described by a complex mathematical model [6–8]. In this case, it seems advisable to use the suggested approach [9] of estimating the mean square deviation (MSD) of the received output signal from the known output signal of the system. Then, according to the model of the nonlinear system being studied, the software code with a build-in table of the known signal values is created in general terms with a set of hypothetical parameter values as the input data of computation procedure. The investigated nonlinear system with the set-up parameters will be simulated as a result of the code execution, and the received output signal of the system will be used to estimate the MSD and make decision of the following looking for the new supposed system parameters. The described method is implemented in Acsocad software solution, which includes simulation and identification tools SimACS as well as identification programs for the linear systems in the time ItACS and frequency IfACS domains. An important benefit of the identification in SimACS is a possibility of simulating the systems with the random interelement coupling, and applying both multithreading-based distributed systems and those using modern OpenCL technology for identification. Besides, the software has a feature of creating its own identification methods as well as a possibility of developing its own identification algorithm based on these methods.


Introduction
A common application of the computing techniques using distributed computing allows for the efficient solutions of the complicated design, production, and scientific-research problems [1].One of the main advantages of the distributed computing systems is a high speed of the similar calculations for different data sets [2], which makes it possible to use them in solving the tasks of control, design, predicting, diagnostics, and identification [3][4][5].
To solve the problem of parameter identification for the elements of the linear systems almost instantaneously, the Kalman filter is generally used.However, the Kalman filter often cannot provide with an adequate parameter valuation for the nonlinear systems especially those described by a complex mathematical model [6][7][8].
In this case, it seems advisable to use the suggested approach [9] of estimating the mean square deviation (MSD) of the received output signal from the known output signal of the system.Then, according to the model of the nonlinear system being studied, the software code with a build-in table of the known signal values is created in general terms with a set of hypothetical parameter values as the input data of computation procedure.The investigated nonlinear system with the set-up parameters will be simulated as a result of the code execution, and the received output signal of the system will be used to estimate the MSD and make decision of the following looking for the new supposed system parameters.
The described method is implemented in Acsocad software solution, which includes simulation and identification tools SimACS as well as identification programs for the linear systems in the time ItACS and frequency IfACS domains.An important benefit of the identification in SimACS is a possibility of simulating the systems with the random interelement coupling, and applying both multithreading-based distributed systems and those using modern OpenCL technology for identification.Besides, the software has a feature of creating its own identification methods as well as a possibility of developing its own identification algorithm based on these methods.

Identification methods and algorithms
Today, a great number of methods of a global MSD minimum search has been developed.However, in order to achieve the acceptable results of identification, a combination of methods with different features should be used which is implemented as an identification algorithm in the software.In this case, the identification methods in Acsocad product can be divided into three classes, such as the start points definition methods, grouping methods, and methods of a global minimum search.
The first class of the methods allows defining a list of parameter (point) sets describing a certain region, which is expected to contain the desired global minimum.Such methods have no need of information about the previous points, so they can be used independently with a maximum computational speed.They include random and uniform fill methods, elliptic method, loxodromic method, and the others.
The second class of the methods allows grouping of the previously obtained points into the areas and receiving an additional point list, which helps to define a global minimum quickly.This class includes the geometrical method based on the concepts of the geolocation method with the desired point determination according to the data on the distances and locations of the other points.
The third class of the methods is oriented to a search for a global minimum according to the previously obtained points.This class contains the explorer method using the ideas of the gradient method of descent with the accelerated motion to the MSD minimum while a search is executed for different points simultaneously.

Studying an object with two unknown parameters
To make a comparative study of the identification methods, an experiment with a technical object was carried out.By signaling cos(t), the output signal with a noise in the time domain was obtained, recorded with an apparatus, and put in a file.The investigated technical object has a mathematical model presented in Acsocad product as a block diagram (Fig. 1) with two unknown parameters P0 and P1.

Fig. 1. The block diagram of the investigated object with two parameters
The real values of two unknown parameters are equal to 3.751 and 7.283 respectively.An output signal Y(t) has been measured within two seconds with a step of 0.1 c.The time dependence of output signal value is shown in Fig. 2.

Fig. 2. The time dependence of output signal of the investigated system cos(t)
1 To identify nonlinear system in Acsocad, two unknown parameters are to be set up in the identification tab and a range is defined, in which the target parameters can vary.The range of P0 parameter search is selected from 0 to 10, and of P1 one is from 0 to 20.
After that, the accessible platforms enabling computation are selected in Acsocad product and the specified platform usage factor is pointed out.Consequently, it is possible to make calculations by using video card for computing a packet of 10 points, for example, and to engage additionally the processor cores for computing the packets with a few number of points.Such an approach allows selecting time-optimal method of computation distributing for each specific computer configuration.
An Acsocad dialogue box with the identification tab set and test results of the identification is shown in Fig. 3.

Fig. 3. The Acsocad dialogue box
The identification executed in Acsocad product by means of standard algorithm, which involves the uniform fill method with a following global MSD minimum search by three explorers, resulted in a computation of 443 points and obtaining the model parameters 3.633 and 7.015 as well as the MSD value equal to 0.0209.
The maximum relative error was 3.7 % for estimating P0 parameter.The global minimum search map is shown in Fig. 4.
The random fill method based on using the normal law of distribution for initial points search allowed to obtain 575 points, and the model parameters equal to 3.671 and 7.184 as well as MSD equal to 0.0209 by mean of random sampling of 200 points with the following running the explorers.The maximum relative error was 2.1% for estimating P0 parameter.The global minimum search map is shown in Fig. 5.
However, along with the uniform and random fill methods, the elliptic and loxodromic methods were developed.Their accuracy and speed indices turned out to be quit different.
The elliptic method permitted to estimate the values of parameters as 3.669 and 7.185 by computing 284 points and calculate MSD equal to 0.0209.The maximum relative error was 2.2% for estimating P0 parameter.The global minimum search map is shown in Fig. 6.In two-dimensional space, the loxodromic method can be represented as a spiral distribution [10] beginning in a centre of a parameter search range, and finishing on boundary values of these ranges.
The values of parameters were estimated as 3.733 and 7.139 by computing 398 points and MSD was calculated equal to 0.022.The maximum relative error was 1.9 % for estimating P0 parameter.The global minimum search map is shown in Fig. 7.

Fig. 7. The minimum search map obtained by loxodromic method
Thus, in two-dimensional space, the elliptic and loxodromic methods provide the results of identification with the maximum relative error of the parameters estimating which is less with respect to the maximum error of the uniform fill method by 1.5 %.

Studying an object with three unknown parameters
A block diagram of a technical object with three unknown parameters is given in Fig. 8.

Fig. 8. The block diagram of investigated object with three parameters
The real values of the first two parameters were equal to 3.751 and 7.283 and the value of the third parameter was 9.312.After that, a setting for three unknown system parameters was done in Acsocad software and a search for the third parameter ranged from 0 to 20.
After the identification, 589 points were calculated by standard algorithm and the model parameters were obtained equal to 3.986, 8.278 и 13.907.The MSD value was estimated equal to 0.0223.The maximum relative error was 49 % for estimating P2 parameter.The global minimum search map obtained by uniform fill method is shown in Fig. 9.
An application of the random fill method permitted to obtain 900 points and the model parameters equal to 3.359, 6.426, and 5.411 and the MSD value of 0.0209 by random sampling of 100 points with the following running the explorers.The maximum relative error was 42 % for estimating P2 parameter.The global minimum search map is shown in Fig. 10.After the identification by using elliptic method, the parameters 3.622, 6.986, and 8.877 were estimated by calculating 1694 points and the MSD value was computed equal to 0.0209.The maximum relative error was 4.7 % for estimating P2 parameter.The global minimum search map for the elliptic method is shown in Fig. 11.By mean of loxodromic method plotting a loxodrome in three-dimensional space, the parameters 3.81, 7.564, and 10.883 were estimated by calculating 1196 points and the MSD value was computed equal to 0.021.The maximum relative error was 17 % for estimating P2 parameter.The global minimum search map for the loxodromic method is shown in Fig. 12.Thus, in three-dimensional space, the elliptic and loxodromic methods provide the results of identification with the maximum error of the parameters estimating which is less with respect to the maximum error of the uniform fill method by 32 %.

A comparative performance evaluation of the methods
The characteristics of the methods applied towards the identification of non-linear systems with two and three unknown parameters are given in Table 1.

Conclusion
The elliptic and loxodromic methods can be applied successfully towards the identification of nonlinear systems while using distributed computing.

Fig. 11 .
Fig. 11.The global minimum search map for the elliptic method

Fig. 12 .
Fig. 12.The global minimum search map for the loxodromic method