论文全文ICEMS

The Research of parallel structure and composite fuzzy control for linear machine based on Mamdani

method

Mingfei Xu

College of Electrical Engineering, southeast University,

Nanjing, China

E-mail:491135410@https://www.360docs.net/doc/4017628946.html,

Haitao Yu

College of Electrical Engineering, southeast University,

Nanjing, China

E-mail:htyu@https://www.360docs.net/doc/4017628946.html,

Abstract—According to the serious nonlinearity and highly coupling of variables in linear machine, which can contribute to the unsatisfactory control effect, therefore, one kind of parallel structure and composite fuzzy controller is proposed. Firstly, the velocity to voltage mathematical model was established. Secondly, through Ziger-Nichlol method mixed with manual fine-tune method, three parameters of PID controller could be accessed based on the model. Finally, the fuzzy controller was designed: the subordinate function of the input variables adopt gauss form; and fuzzy deduction manner of Mamdani method; the subordinate function of the output variables adopt triangular form. After above work, the MATLAB simulation was performed. The simulation indicates that this parallel structure and composite fuzzy controller has better control effect than the traditional PID controller, and the maximum overshoot decreases from 9.6% to 0%, the regulation time ranges from 1.23 seconds to 0.95 seconds, verifying that this fuzzy controller has positive feasibility.

Keywords—linear machine; fuzzy control; PID control; Ziger-Nichlol method; Mamdani method

I.I NTRODUCTION

Compared with the traditional rotary motor, the linear motor mainly has the following characteristics: Firstly, the structure is simple, thus the weight and volume is greatly decreased; Secondly, it has the advantage of high positioning accuracy, if microcomputer control is used, it can also greatly improve the positioning precision of the whole system; Thirdly, it also has the feature of fast reaction rate, high sensitivity; Lastly, it is safe and reliable with long service life while working. The traditional control method can’t balance the steady and dynamic performance requirements due to the variable parameters so that the desired control effect can’t be achieved. In recent years, burgeoning fuzzy control method takes advantage of the control experience of human experts[1], for the control of nonlinear, complex objectives shows good robustness, high control performance, the control method is especially suitable for the linear motor. MATLAB was used for the simulation of established linear machine’s mathematical model, through comparison between the simulation waveform of traditional PID control and composite fuzzy control, it is discovered that the composite fuzzy control has evident advantage over traditional PID control in dynamic performance. This control method is in the guarantee of steady state accuracy. Above all, a kind of effective control method is proposed for the control of linear machine.

II.T HE MATHEMATICAL MODEL OF THE LINEAR MOTOR In the power transmission system, linear motor is a kind of machine which can directly convert electrical energy into kinetic linear motion energy and does not need to install any intermediate mechanical transmission mechanism. It has the advantages of simple structure, high energy conversion efficiency, low noise, excellent operating reliability and so forth. When the rotary motor is cut open along the radial, and then along the circumferential flattened, the linear motor can

be obtained, as shown in figure 1.

Fig.1. rotary motor transfers to linear motor

The stator is primary while the rotor is secondary, according to the kinematic properties and electromagnetic characteristics of linear motor, the linear motor motion equations, the voltage balance equation and the electromagnetic force equations[2] can be obtained in the time domain format:

()

()()()

dI t

U t E t RI t L

dt

=++（1）

()()

m

F t k I t

=（2）

2

2

()()

()()

d x t dx t

m c kx t F t

dt

dt

++=（3） Among the formula,

()()

(),()

E

dx t dx t

E t k V t

dt dt

==.U is the loop voltage, E as the counter electromotive force, R for the coil conductor resistance, I as the loop current, L is the coil conductor inductance, k E as the back EMF constant, x is the displacement of the linear motor, F as the electromagnetic

force coil, k m as the force constant of the linear motor, m as the coil quality, c is the coefficient of friction viscous state, k is the spring stiffness coefficient, v as the linear motion for the linear motor speed.

Apply Laplace transformation to (1),(2),(3) formula ：

()()()()E U s k sx s RI s LsI s =++ （4）

()()m F s k I s = （5）

2()()()()ms x s csx s kx s F s ++= （6）

Along with ()()V s sx s =, By formula (4), (5), (6), the

transfer function between the speed and voltage is:

2()

()()()

m E m k s V s U s k k s R Ls ms cs k =++++ （7）

III. Z IGER -N ICHLOL METHOD PARAMETER TUNING OF PID

CONTROLLER The output of the PID controller m(t) is proportional to the error signal e(t), and is also proportional to the integral of e(t), and is proportional to the first order derivative of e(t). The controller maintains the advantages of enhancing the system steady state performance based on the PI control rule, and it has more advantages in improving the system dynamic performance. Equations for PID controller in the time domain is:

01()()()()t p d i de t m t K e t e t dt T T dt ??

=++????

? （8） In the formula(8), m is the output of the PID controller, e is the input of PID controller, K p is the proportion coefficient of controller, T i is the integral time constant for the controller, T d is the differential time constant.

After years of development, the Ziger-Nichlol method has become a parameter setting, intermediate methodology between experience and the calculation method. This method can determine the parameters very accurately for the controller, then can also be slightly tuned.

The Ziger-Nichlol method consists of two steps: the construction of a close loop control circuit, determination of the stability limit. Controller parameter is calculated according to the formula. Stability limit is determined by the ratio number of element. When the steady state oscillation ouccrs, the limitation will be reached. The critical factor pcrit K and the critical oscillation cycle crit T can be obtained.

When the critical factor

pcrit

K and the critical

oscillation cycle crit T are confirmed, other parameters can be obtained based on the formula in table I:

of manual fine tuning, we can calculate:

5,40,2p I D K K K === IV.

T HE DESIGN METHOD OF FUZZY CONTROLLER BASED

ON M AMDANI METHOD

The block diagram of fuzzy control system is shown in Figure 3:

Fig.3. composite linear motor fuzzy control system

The fuzzy controller in Figure 3 is the two-dimensional fuzzy controller [3], it has two inputs e and e ?

, they are the system error and error change rate (the exact amount); E and EC are the fuzzy set respectively in fuzzy system which reflects language variables of the deviation and deviation change (fuzzy amount); U as output variables; the exact amount of the output of fuzzy controller which has the control action is u ; y as the output of fuzzy control system (the exact amount).

A. Fuzzy processing of error and error change rate

For fuzzy processing of deviation and rate of change which are the linguistic variables, we define positive big as PB, positive medium as PM, positive small as PS, zero as O, negative small as NS, negative medium as NM, negative big as NB. 7 linguistic variables [4] are used to describe the system. The membership function of the error and error change rate belong to the Gauss function. Parameters of each language variable values are shown in Table 2 and Table 3.

Fig.4. Membership function curve of error and error change rate

B. Fuzzy rules and fuzzy reasoning

In this paper, the fuzzy controller has double input and the single output, control rules of it are fuzzy conditional statements as follows:

If E and EC then U

When fuzzy relation of each fuzzy conditional statement R i (i=1,2,... m, where m is the number of sentences)is acquired, due to the "or" relationship between statements, the whole control system of fuzzy relation for fuzzy control rules can be calculated:

121

m

m i i R R R R R ==∨∨???∨=∨ （9）

If fuzzy subsets of E and EC in the input language variable

for fuzzy controller are given, and the fuzzy relation R i (i=1,2,... m) determined by every fuzzy statement based on the control rule is also given. Then fuzzy subset of U of the output language variable domain can be expressed as

12()()()m

U E EC R E EC R E EC R =?∨?∨???∨? 1()m

i i E EC R ==∨? （10）

Among them:

“∨”： the maximum of two numbers.

“?”：direct product, x, y are any two sets, we call {(,)|}X Y x y x X y Y ?=∈∈or is the direct product of x and y.

“ ”：The synthetic operation of relations, it can be expressed as ==Q R S R S M M M M , if ,,Q R S M M M are expressed as []=,=,=Q ij R ik S kj M q M r M s ????????, then:

=1

=()n

ij ik kj k q r s ∨∧，i=1,2,3,…,m; k=1,2,3,…,n;

j=1,2,3,…,p 。

Fuzzy relations: assume that U and V are discourse domain, if (U V)R F ∈?, we can define R is the fuzzy relation between U and V [5].

Fuzzy reasoning method of this control system is the Mamdani method, this method is essentially a kind of synthesis reasoning rule based on likelihood reasoning [6], whereas taking different representations for fuzzy implication relationships, the highlight is that the fuzzy implication relation A B

→is the direct product representation between A and B.

A B A B →=?

The fuzzy controller in this paper is adopted Mamdani method which sets up 49 fuzzy control statements. Each of these fuzzy statements represent a strategy for a specific case [7]. For the fuzzy linear motor ,the control rule table is:

C. The defuzzification processing of voltage changes

After fuzzy deduction of the fuzzy variable of the voltage variation, after the defuzzification for the membership function of it, the exact solutions of the voltage variation will be accessed. The membership function for the voltage is triangle function. Parameters and images are shown in Table 5 and Figure5.

Standard deviation 0.4 0.4 0.4 0.4 0.4 0.4 0.4

Fig.5. Membership function curve of the voltage variation

V. S IMULATION RESULTS AND ANALYSIS

In this paper, the parameters of coreless linear motor in this simulation are shown in Table 6:

3

2()15.9() 6.5 1.6162.6 3.15

V s s

U s s s s =+++ （11） The simulation main circuit as shown in Figure 6, the below part represents the traditional PID control method; and the above part is the composite fuzzy control method. Firstly, the error and error change rate input delivered to the fuzzy controller, and the error input delivered to PID controller, then two parallel output add to a control

voltage, through the transfer function transformed as the displacement, then it will be differential into speed. The fuzzy control system ’s input and output

characteristics surface is shown in Figure 7[8-10]

.

Fig.6. Simulation main circuit

Fig.7. The input and output characteristic curve

The simulation waveform is shown in Figure 8, as can be

seen from the graph that the control effect of compound fuzzy control system is much better than the traditional PID control system, the

overshoot of the compound

control system is relatively subtle, and it can reach the steady state quickly, and the traditional PID control system has subtle fluctuation near

the steady state value. The steady-state error of two kinds

of control systems tends

to be zero, while

the main dynamic performance index as shown in table 7:

Time(s)

v e l o c i t y o f l i n e a r m o t o r

Fig.8. Simulation waveform

VI. C ONCLUSION

In this paper, owing to the characteristics of linear

motor ’s nonlinearity and multivariable highly correlation, also aimed at the shortage of traditional PID control method, a compound fuzzy control system is proposed, using the advantages of fuzzy control and PID control to drive the linear machine. The simulation results show that: the composite fuzzy control system has better performance than the PID control system. This is due to its error and error change rate which can decide the output quantity intelligently, this fuzzy system combined with the advantages of PID control system, then the dynamic performance and better control effect can be obtained. The proposed control scheme is feasible.

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