Available online at www.sciencedirect.com
ScienceDirect Procedia Computer Science 102 (2016) 302 – 308
12th International Conference on Application of Fuzzy Systems and Soft Computing, ICAFS 2016, 29-30 August 2016, Vienna, Austria
Synthesis of the optimal fuzzy T-S controller for the mobile robot using the chaos theory S.M. Jafarova*, E.R.Zeynalovb, A.M. Mustafayevab b
a Azerbaijan State Oil and Industry University, Azadlig av. 16/21, Baku, AZ1000, Azerbaijan Institute of Control Systems of ANAS, Bakhtiyar Vahabzadeh st., 9, Baku, AZ1141, Azerbaijan
Abstract The automatic synthesis method of the optimal T-S controller has been proposed for nonlinear multiply dynamic object, omnidirectional mobile robot by using the chaos theory. A system of automatic adjustment of parameters of fuzzy controllers has been developed providing high-quality vector control parameters moving multiply nonlinear objects. © by by Elsevier B.V.B.V. This is an open access article under the CC BY-NC-ND license © 2016 2016The TheAuthors. Authors.Published Published Elsevier (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ICAFS 2016. Peer-review under responsibility of the Organizing Committee of ICAFS 2016 Keywords: omnidirectional mobile robot; chaos theory; fuzzy T-S controller; overshoot; integral square error; damping degree
1. Introduction There are a series of non-linear dynamic objects (or systems), which is chaotic-nonregular1-3. As it is well known, phenomena of this type, such as dissipative Lawrence system3, is called "deterministic chaos", "internal stochastic" or just "chaos" in the scientific literature. In chaos theory it is studied the dynamic regularities, generated movement of a nonlinear system, the evolution of the system in time, the determination of the current state of the system at a given prehistory. In such systems dynamic chaotic processes (irregular movement), that is the phenomenon can be characterized as "desirable" (positive) or "undesirable" (negative)1,3.
* Corresponding author. Tel.:.+994706366608 E-mail address:
[email protected]
1877-0509 © 2016 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the Organizing Committee of ICAFS 2016 doi:10.1016/j.procs.2016.09.405
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It is known that, if there are irregular-chaotic transients, then it is tried to deal with such phenomena in control system of nonlinear objects. In other words, in such cases it is achieved to eliminate undesirable irregular movement-chaotic controller by changing the structure and parameters of the control2,3. By using the chaos theory, in a dynamic system it is specifically tried to create irregular- chaotic movements in order to solve some problems, in particular, the definition of some optimum values of the parameters of control systems. Moreover, irregularchaotic motion control system can be accomplished by introducing a special non-linear element. As examples of the application of chaos theory in order to solve fuzzy optimization problems, mathematical programming and the synthesis of the parameters of fuzzy term-set could be shown1-4. In particular, in4 the synthesis problem of the automated system control regulators for the non-linear object solved with application of the elements of chaos theory. However, in the well-known works during synthesis of parameters of controller was carried out by using the "integral square error" criteria of quality. However, there are some non-linear objects with irregular free movement, in particular, manipulators and mobile robots control system1,4,6,11, in which the vector optimality criterion, such as the minimum of overshoot, a maximality of stability degree, speed, minimality of integral square error must be satisfied. In the present work, it has been proposed the method of designing automatic synthesis problem of intellectual control system with using of the elements of chaos theory, i.e. tuning parameters of knowledge base T-S controller, which provides the optimal quality of the vector criterion. 2. Statement of the automatic synthesis problem of fuzzy T-S controller Let the control object is a mobile robot that moves in all directions (omnidirectional-OD), a mathematical model, of which is described by nonlinear differential equation6:
(2.1) where is state vector, i.e., velocities on the coordinate axes OXw, OYw and on the rotation angle of the mobile robot platform, y is output coordinates of the mobile robot on the coordinate axes and - control variables generated by the corresponding wheel drives. By carrying out a series of the rotation angle, simple transformations, a mathematical model of the omnidirectional mobile robot can be described by nonlinear differential equation:
(2.2)
Nonlinear control object of the type (2.2), can be described by fuzzy model with sufficient accuracy: Ri: øf where
is
, Then
and
(2.3a)
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(2.3b)
defined
through
the
-the unit matrix of size 3x3. Coefficients of differential equation in (2.3a), are , , , relevant parameters real OD mobile robot with the
following expressions4:
- 0.0599,
0.2901,
(2.4)
For nonlinear object- OD mobile robot (2.2) - (2.4) the control law is chosen in the intelligent controllers class with two elastic feedback, i.e. for dosing errors output variables and state variables (velocity). Considering the selected class of controllers and requirements on the quality criteria, the control law of mobile robot is implemented by the following fuzzy T-S model Ri: If
is about
, Then
Where is control error on the output coordinates errors on output coordinates of the mobile robot are determined from the following expressions:
(2.5) . Also
(2.6) Here -is the reference signal. For easy comparison of the effectiveness of the proposed synthesis method of optimum tuning parameters of in the linguistic control rules (2.3) and (2.5), controller by the known work5,6, fuzzy term –sets membership functions of initial term of fuzzy sets are selected in the triangular form, which can be defined by the following formulas:
(2.7)
Considering selected structure of fuzzy control system of mobile robot (2.2) - (2.4), according to output errors of fuzzy T-S regulators can be summarized and state variables (2.5), (2.6) the problem of parametric synthesis as follows. of the term fuzzy sets of linguistic According to the phase variables and error, as well as the parameters rules, it is required to determine such optimal matrices , of fuzzy T-S regulators which, while transferring
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of the systems from the initial state to the final state would assure optimality of the vector quality criterion, i.e. the minimum of integral square error (2.8), high damping degree of the oscillations (2.9) and minimum overshoot transient (2.10) in the control system
,
,
(2.8)
(2.9)
,
(2.10)
(2.11) Since the real ODMR must be controlled, and therefore we can assume that there are set of tuning matrices of errors and state variables and parameters of fuzzy term sets M. 3. Decisions of the synthesis problem of the tuning parameters of fuzzy T-S controllers for OD mobile robot It should be noted, that on the basis of analytical methods1-11 the solution of above posed synthesis problem of the parameters of fuzzy regulator is very difficult, or impossible due to vector criteria. Therefore, the solution of abovementioned synthesis problem of the parameters of fuzzy controllers Ki according to the vector criterion K can be implemented on the basis of chaos theory. We will carry out the synthesis of controller parameters Ki and fuzzy term sets by iterative procedure in accordance with the generalized functional scheme. The solution, i.e., the automatic synthesis of parameters- controller matrices according to the error, state variables (2.5) and fuzzy term set (2.7) is carried out by the following proposed procedure. The first stage is construction of the mathematical model of a closed automatic control system of the mobile robot corresponding to (2.2), (2.3) and (2.5) - (2.7). Then, on the basis of expressions (2.2) - (2.7) recurrent discrete model is developed for the computer simulation of the controller. In order to estimate vector, the quality criterias are developed on the basis of (2.8) - (2.11). Sampling time is chosen. At the second stage initial and final states, the time of observation for each experimenter (2-5 times longer than transients), the number of experiments, the initial assessment of the quality criterion are selected for each experiment. At the third stage generators of chaotic number-matrix of 3x size are developed for regulators according to errors phase variables and parameters of fuzzy term sets -. Then, in the chaotic number generators tuning parameters matrices errors control state variables and parameters of fuzzy term sets (2.7) are changed randomly. The initial values , are set from the feasible area (2.11). Number of generation and the last two corresponding values of parameters- settings matrices of controllers are fixed. These values are assigned to the corresponding tuning parameter of fuzzy controllers according to the error and state variables. Setting of the controller are changed via generators of chaotic numbers. For simplicity and the explanation of the analysis of cause-effect relationships of the synthesis process of parameters parameters term-set assumed to be constant and are chosen by experts. In the process of setting up the so-called mode 2, parameters of fuzzy term-sets of controllers are also changed (set) using the generators of chaotic numbers. Modeling of closed CS is performed after assigning of the initial values of the parameters of fuzzy T-S controller and the corresponding generators of chaotic numbers.
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After the j -th experiment, i.e. modeling of CS on the observation interval the vector value of the control quality is estimated(calculated): integral square errorsin each variable in accordance J with (2.8); degree of attenuation and transient overshoot J in each variable. The next step is to determine the optimal vector values of the quality criterion , in which simple comparison of their values of j-1-th experiment is made, i.e.
(3.1)
(3.2)
(3.3) ,
Then function
,
and
and
are assigned
,
,
,
i.ɟ.
(3.4)
Thus it can be noted that the above algorithm of parameter synthesis of fuzzy T-S regulators allows virtually a very large number of simulation experiments (almost infinite), as in the optimization criterion (2.8) - (2.10) it is necessary to remember the results of only last six experiments. It can be concluded that the synthesis problem of the parameters of fuzzy T-S controllers has been solved. The and the parameters of the fuzzy term-sets values of tuning parameters of fuzzy T-S regulators, i.e, matrices are set from the switch block 10, inputs that are connected to the block 9. øn block 9 it is formed 61 (2ɯ(3ɯ3)+7) of generator of the chaotic number. Generation of consecutive chaotic number is carried out by the following logistics functions with different initial conditions:
,
,
(3.5)
,
[א5.72,5.75]
- quality indices of transient processes of control system are evaluated in In the experiments the vector, block 6 in accordance with formulas (2.8) - (2.10). Evaluation of vector indices, for example are transmitted to the optimization block 7. In this block experiment number is defined with the best-optimal estimation vector quality of the transition process and the experiment number and values of tuning parameters (array) of regulators are remembered according to the phase variable and output . It is noted that the generation of chaotic fuzzy numbers with type of triangular is carried error out in accordance with similar functions to (3,5). According to the formula (2.7) the parameters of the fuzzy term-set are defined by the relevant generators of chaotic numbers : , . The optimal value of the estimates of vector quality the number of the experiment , variation of the parameters of fuzzy regulator are stored in memory block 8. The use of a sufficiently large number of simulation experiments allows to escape from local optimal solutions,
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i.e. to determine almost global optimum evaluation vector quality indices and to synthesize the bestoptimal values of tuning parameters of regulators of the phase ( ) and output variables. It is noted that the sampling step of generators are significant to determine the global optimum. Switch block 10 is designed to switch the system from the experimental settings mode in real work mode of of fuzzy regulators (blocks 2 and control system of mobile robot with overlift (synthesized) parameters 3). According to output variables setpoint effects are formed in the block 5 corresponding to individual functions or to the following formulas:
4. Experimental results Technical realization of the above proposed system of automatic synthesis of parameters of fuzzy controllers is implemented using M-language in MATLAB (SøMULøNK packages, FUZZY Logic Toolbox) condition. The following values are accepted as initial conditions for the phase and output variables making the
During computer dimensionless time simulation experiments 20, 6 sampling step interval are selected (Fixed Step) -0.01 As a result of simulation experiments the following optimal values of fuzzy controllers settings are defined:
The following shows the summative evaluation of the quality of synthesized by the proposed approach1,6, with the same initial conditions.
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The experimental results showed that the synthesized CS mobile robot ODMR is relatively efficient than synthesized system by well-known paper1,4,6. Since the total system control integral quadratic error synthesized by a known method1,6 was 0.2605 and above at 0.2113 the proposed method, i.e., quality is improved by about 18.8%. 5. Conclusions The results of experimental studies have shown that the proposed automatic synthesis system of fuzzy regulator parameters with using chaos theory allows to solve control synthesis problem of multivariable and nonlinear dynamic objects relatively efficiently. The proposed method is implemented in MATLAB. References 1.
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