Rui Zhang, Le-peng Song, Jun-ling Yang, and Timothy Hoffman
College of Electronic Information Engineering,
Chongqing University of Science and Technology, Chongqing 401331, China Jerry.zr@163.com, jerryzr.tk@gmail.com
Abstract. The traditional DC double-loop speed control can not effectively over-come non-linear factors; it can not meet the needs of high-precision, high-performance requirements. This paper shows that the Fuzzy Self-Tuning PID Control strategy has been used in DC motor speed control systems to achieve real-time tuning PID parameters and introduces the entire design process of the con-troller. When referencing an actual motor parameter model a simulation model established by Matlab / Simulink, simulation results show that the Fuzzy Self-Tuning PID Controller is superior to the traditional PID control with its accuracy and robustness, thus enhancing its motor dynamics and static performance. From this it is seen that simulation results are concurrent with theoretical research, by verifying the reasonableness of program design, while it achieves an optimum control of control objectives.
Keywords: DC Motor, fuzzy self-tuning PID control, simulink.
1 Introduction
DC motors are widely used having good starting and braking performance in such technologies as electric power drive automatic control systems, such as, rolling mills, mining machines, and mine hoist machines. Most traditional motor control systems use PID control because of its design advantage of simple structure and easy Implementation. However, the motor itself is a non-linear controlled device using many loads that contain nonlinear factors of elasticity and clearance; the traditional PID control cannot overcome the influence of model parameters or substantial changes of loads, including its non-linear factors, etc. It also cannot meet the needs of high-precision, high-performance devices, etc. This paper uses the Fuzzy Self-Tuning PID Control strategy to simulate a DC motor speed control system, and as this paper show, this results in a marked improvement in system reliability, stability, robustness, and dynamic parameters.
The greatest feature of the fuzzy control is that it express the experience and knowledge of the expert as the linguistic control rules, which can control systems, thus, it cannot rely on precise mathematical models of a controlled object in order
B. Cao, T.-F. Li, C.-Y. Zhang (Eds.): Fuzzy Info. and Eng., Volume 2, AISC 62, pp. 967–975. springerlink.com © Springer-Verlag Berlin Heidelberg 2009
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to overcome the influence of non-linear factors; furthermore, it maintains a ro-bustness in the parameters of the controlled object. The fuzzy self-tuning PID control is a kind of online identify object characteristic parameters with modern control, real-time change control strategy, so that indicators of control system quality factor can be maintained within an optimum range to achieve the optimum control of the controlled object, the control effect depends primarily on the accu-racy of identification model [1].
2 Design of the Fuzzy Controller
In the process for DC speed control, a traditional double-loop speed control sys-tem is normally used. A lag PI regulator is used primarily during the engineering design process by having the outer loop (speed loop) as the fundamental factor to determine its control system while the inner loop (current loop) is used to change the operational characteristics of the motor in order to be controlled by the outer loop. Although the structural design of the DC double-loop is simple, reliable, and stable, with the current loop design, the impact of its speed to the current loop and non-linear factors are ignored, such as the actual system and silicon-controlled devices. It is difficult to maintain the quality indicators of control systems in its optimum range.
This paper focuses on the Fuzzy Self-Tuning PID Controller’s parameters. It not only has the same advantages of primary PID control systems, it is simple, easy-to-use, and shows robustness, but it also overcomes the influence of impre-cise mathematical models in controlled objects, including, system non-linear fac-tors. Additionally, it can provide PID parameters real-time tuning, which is an advanced method of intelligence control having better flexibility, adaptability, control accuracy, and robustnes [2].
2.1 Fuzzy Control Model of a Speed Control System
The fuzzy control is a computer controlled system based on Fuzzy-Set Theory, fuzzy linguistic variables, and fuzzy logic inference. It is an intelligent control method that imitates fuzzy inference and decision processes, based on human behavior. Initially, expert experiences are summarized into fuzzy control rules, and then sensor signals are fuzzified determining specified control rules, where fuzzy logic inference is completed. Finally, fuzzy output is de-fuzzified, where the output is sent to actuators to control the object, as shown in Figure 1.
+ - KeFuzzificationEFuzzy controlECFuzzy strategyDefuzzificationKuControlled object de/dtKec
Fig. 1. The structure of the fuzzy controller
DC Motor Speed Control System Simulation Based on Fuzzy Self-tuning PID 969
2.2 Theory and Structure of Self-tuning Fuzzy PID Control System with
Parameter The Self-Tuning Fuzzy PID Control System with parameter can detect and ana-lyze key factors, such as, unpredictable conditions, parameters, delay and interfer-ence in the control process, and conduct online self-tuning of PID parameters Kp, Ki and Kd using the fuzzy inference method, so that optimal adjustment to PID parameters is automatically achieved. [3] The fuzzy control contains two parts: the PID regulator and the fuzzy inference. The fuzzy inference is a fuzzy controller substantially, according to the requirement of the error e and the change of the error ec to the PID parameters self-tuning, modify the PID parameters online with the fuzzy rules, and thus, the controlled object has a good dynamic and static per-formance.
The numerical approximation algorithms is normally used in order to achieve real-time parameters self-tuning with the PID controller in the computer; that is, it replaces the integral with the summation and replaces the difference quotient with the differential quotient, which makes the PID algorithm discrete, as follows:
T
1de(t)
U(k)=Kp{e(t)+e(t)d(t)+Td}
Ti∫dt0
k−1
(1) ≈Kpe(k)+Ki∑e(i)+Kd[e(k)−e(k−1)]
i=0
where U(k) and e(k) are the output and the input (error) of the controller on k-th time sampling, respectively; Kp is the proportional gain, Ti, and Td are the inte-gral, differential time constants, respectively; T is the sampling period; its struc-tural model is shown in Figure 2.
Fuzzy inference KpKiKdr- error dedtPID regulator Controlled objecty Fig. 2. Structure of fuzzy PID controller
3 Design of Self-tuning Fuzzy PID Controller with Parameters
The fuzzy controller is the core of the fuzzy control system. The quality of the fuzzy control system depends on the structure of the fuzzy controller, its fuzzy rules, the synthetic inference algorithm, and the fuzzy strategy.
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3.1 Membership Determination
The Self-Tuning Fuzzy PID Controller with Parameters is based on the error e and the change of the error ec to adjust parameters and achieve PID parameters for online self-tuning with the fuzzy mechanism. In order for the system to have strong robustness when error e and the change of error ec occur, a fuzzy controller having two-inputs and three-outputs is used in the speed loop – Figure 2 shows this basic structure.
The input of the fuzzy controller is a fuzzy signal which is fuzzified from a pre-cision signal, where afterwards, the basic set is converted into the fuzzy-set through quantization factors. The quantization factors of the error and the change of the error are ke=n/emax and kē=n/ēmax , respectively. If the basic set is asymmet-ric, then y=(2n/b-a)(x-(a+b)/2) should be used in the quantization process [4] and we identify the basic set of the error and the change of the error preliminary, ena-bling further identification, while the system adjusts. To simplify for this paper, we define the input and the output variables of the fuzzy controller as a symmetric basic set: according to the experience, we quantify e, ec to the fuzzy-set e, ec = (-3, -2, -1,0,1,2,3), the fuzzy subset of e, ec = (NB, NM, NS, 0, PS, PM, PB) (all subjects are normal distribution). Three output fuzzy sets of linguistic variables respectively are
kp = (-0.3, -0.2, -0.1,0,0.1,0.2,0.3),
ki =- 0.06, -0.04, -0.02,0,0.02, 0.04,0.06), kd = (-3, -2, -1,0,1,2,3).
The fuzzy subset is (NB, NM, NS, 0, PS, PM, PB) (all subjects are normal distribu-tion).Where Z-type membership function and S-type membership function are taken to NB and PM, respectively, and the triangle membership function is taken as the rest. Therefore, the input and the output fuzzy-sets membership of the Fuzzy PID Controller are determined.
3.2 The Principle of PID Parameter Tuning
In general, the structure and algorithm of the PID controller have been identified where the choice of control parameters determine the control quality, based on the experience, considering the system response speed, stability, overshoot, steady-state accuracy, and the effect of kp , ki, kd, to different error e and the change of the error ec in the control process, the tuning rules of the PID control parameters kp, ki, kd are as follows:
1) When│e│ is large, taking a larger kp and a smaller ki (in order to speed up
the system response) so that ki=0 (in order to avoid large overshoot, elimi-nate the effect of integral).
2) When│e│is medium, taking a smaller kp (in order to make the system over-shoot smaller), taking proper ki and kd (especially the value of kd can influ-ence system response particular greatly).
DC Motor Speed Control System Simulation Based on Fuzzy Self-tuning PID 971
3) When│e│is small, taking a larger kp and ki (in order to make the system
steady-state performance better), taking proper value of kd so as to avoid os-cillation near the equilibrium point [5]. 3.3 Design of Fuzzy Control Rule Base
There is a single-variable, two-dimensional fuzzy controller in this system, where, in general, the control rules of the two-dimensional fuzzy controller are in the form of if-then rules and takes the form of [5]:
If e = 'A (i)' and ec = 'B (j)'
Then u = 'C (i × j)'; i = 1,2,3, ..., m; j = 1,2,3 ..., n.
where A (i), B (j), C ( i × j) are fuzzy subsets for the universe of discourse of the error, change of the error and control variable e, ec, u. The control rules of the Self-tuning Fuzzy PID controller with two-input and three-output controller are:
If e = 'NB' and ec = 'NB';
Then kp = 'PB' and ki = 'NB' and kd = 'PS'; R = R1∨R2∨R3 ...∨R (n).
According to the total synthesis rules, the control variable can be obtained by the fuzzy inference:
U=（e′+ec′）T1DR
Based on the operator (expert) experience of long-term practical knowledge, the controller establishes a proper fuzzy rules table, which contains three parameters △Kp, △Ki, △Kd, consisting of 49 control rules, as shown in Table 1 ~ Table 3:
Table 1. Fuzzy rules of △Kp
ec
NB NM NS 0 PS PM PB
NB PB PB PM PM PS 0 0 NM PB PB PM PS PS 0 NS NS PM PM PM PS 0 NS NS
e
0 PM PM PS 0 NS NM NM PS PS PS 0 NS NS NM NM PM PS 0 NS NM NM NM NB PB 0
0 NM NM NM NB NB
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3.4 Defuzzification Strategy
De-fuzzification is converts output linguistic variables into precise numerical values. In this paper, the center of gravity method (COG) is taken to de-fuzzify the subset in the fuzzy controller. The point x∈u is used as a weight coefficient to fuzzy-set membership u(x), the result of de-fuzzification is obtained after the weighted mean. To the discrete universe, let u=u{xi | i=1，2，3…n},
x=
∑
1
n
u(xi)xi
∑
1
n
u(xi) (2)
Table 2. Fuzzy control rules △Ki
ec
NB NM NS 0 PS PM PB
NB NB NB NM NM NS 0 0 NM NB NB NM NS NS 0
0
NS NB NM NS NS 0 PS PS e
0 NM NM NS 0 PS PM PM PS NM NS 0 PS PS PM PB PM 0 0 PS PS PM PB PB PB 0 0 PS PM PM PB PB
Table 3. Fuzzy rules of △Kd
ec
NB NM NS 0 PS PM PB
NB PS NS NB NB NB NM PS NM NS NS NB NM NM NS 0 NS 0 NS NM NM NS NS 0 e
0 0 NS NS NS NS NS 0
PS 0 0 0 0 0 0 0 PM PB NS PS PS PS PS PB PB PB PM PM PM PS PS PB
DC Motor Speed Control System Simulation Based on Fuzzy Self-tuning PID 973
4 Self-tuning Fuzzy PID Controller System with Parameter Simulation
The Matlab fuzzy toolbox provides a dedicated editor for the design of the fuzzy controller. Through the identified input and output linguistic variables assigned table, we can select membership functions to denote the membership of fuzzy linguistic variables (Figure 3).
NBNMNSZPSPMPB1Degree of membership0.80.60.40.20-3-2-10e123
Fig. 3. The curve of membership function
According to the control experience of the DC motor double closed-loop speed control system, the fuzzy control rule table is obtained by editing its control rules under the fuzzy toolbox interface ruleedit, using input control rules in the form of if ... then statements, consisting of a total of 49 rules. With regard to the estab-lished fuzzy inference system, we can open its rules and curved surface observer to view the inference status of its fuzzy rules and output of the curved surface (for example, when e = 0, ec = 0, then kp = 0.00998, ki =- 4.73e-019 , kd =- 0.69).
The de-fuzzy process only needs to establish the de-fuzzification method in the editor; the algorithm will be achieved automatically. It is easy to complete the design process for the fuzzy controller editor of Matlab fuzzy toolbox. After the fuzzy inference system has been completed, it will generate a 'fuzzpi.fis' file, which can be taken to a working space in the system simulation, where it is called as a file to the Simulink fuzzy controller module, and creates a Self-Tuning Fuzzy PID Controller Module with parameters in Simulink, as shown in Figure 4.
5 Simulation Results
In this paper, we selected the thyristor double-loop DC speed control system with a three-phase full-controlled bridge rectifier circuit. The motor parameters are: Pnom = 30kw, Nnom = 1500r/min, Unom = 220V, Inom = 136A. We created a system simulation model using Matlab / Simulink, the results of the starting simulation are shown in Figure 5.
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Fig. 4. The fuzzy self tuning PID simulation
2000Traditional PID15001000Fuzzy self-tuning PID500000.511.522.53
Fig. 5. DC motor speed modulation control system step response curve
It can be seen from Figure 5 that the Fuzzy Self-Tuning PID Control is clearly superior to the traditional PID control without sacrificing the rise time. Further-more, the former has the advantage for fast, high stable precision, and non-overload. It can keep the indicators of the control system quality factors in the optimum range, achieve optimal control of the controlled object, and solve prob-lems of uncertainty, nonlinearity, and time control of the complex control object. Therefore, the Fuzzy Self-Tuning PID Control has better dynamic and static per-formance than the traditional PID control.
6 Conclusion
To solve complex objects control problems of uncertainty, nonlinearity, and time-variances this paper focuses on the design of the Fuzzy Self-Tuning PID Speed System Controller. The simulation experiments proved that it is feasible, and that the motor obtained good dynamic and static performance, maintaining robustness to the parameters of the controlled object. In addition, using the Matlab simulation proved to greatly reduce debugging efforts, showed a powerful ability of simula-tion, increased visualization, and provided an excellent design platform for engi-neering designers.
DC Motor Speed Control System Simulation Based on Fuzzy Self-tuning PID 975
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