GA PID control research in inverter motor speed governing system

Abstract

The variable frequency pump controlled motor speed control system was presented based on genetic algorithm of PID parameter optimization method. The simulation results proved that the genetic arithmetic method after optimization of PID controller has better control characteristic than the conventional PID controller, it showed stronger adaptation and robustness for the model mismatch and the load disturbance, it fit with slow time-varying and load disturbances of inverter control of pump controlled motor speed regulating system. Using genetic algorithm was also pointed out the limitations of PID parameters of optimal variable frequency pump control motor speed regulating system.

Keywords

Variable frequency pump motor control speed control system genetic algorithm optimization PID control

1. Introduction

Motor variable frequency speed regulation technology is changing the frequency of the power implement speed regulation of executing agency, the technology of frequency control of motor speed is used in the hydraulic system, it can overcome some disadvantages of hydraulic system, such as simplifying hydraulic pressure loop, reducing the energy loss of hydraulic system, improving the efficiency of system, reducring noise, etc. The most important one is to reduce the energy loss of hydraulic system (including the overflow loss and throttling loss), improve the efficiency of the whole system.

The traditional PID controller is with the certain advantages of its simple structure, the robustness for the model error and easy to operate, despite at present there are so many advanced control method, but the PID controller is still the most widely used. In the PID control, the control effect depends on completely the setting of PID parameters and optimization. Regular setting way of PID parameters include cut-and-trial method or by some way setting, such as Ziegler Nichols setting method, the electric type self-tuning method and simplex method, etc. Using the conventional setting method are often empirical and not the optimal solution.

Genetic algorithm (GA) is a highly efficient global optimization algorithm, without any initial information, only according to the size of the fitness function value implement a genetic operation, it also has the characteristics of the multi-point parallel operation. Genetic Algorithm is used to optimize the PID parameters, it can obtain the global optimal or suboptimal solution. At present the GA are applied to the combinatorial optimization, the machine learning [1], the optimal control [2], and so on. In the field of hydraulic pressure drive control, the the application of GA is used in the hydraulic servo system [3], but for variable frequency hydraulic system, there is no literature to explore the application of GA. Many scholars at home and abroad have done a lot of research on the PID Contro of Pump-control-motor Speed Governing System [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33].

Figure 1.

The principle diagram of the speed control system of variable frequency motor which is controlled by pump.

2. The mathematical model of speed control system of variable frequency motor

Figure 1 is the principle diagram of the speed control system of variable frequency motor which is controlled by pump. From the picture, the transfer function model can be list for Eq. (1) [34]:

$\displaystyle\frac{W}{U}=\frac{K_{1}K_{2}}{(T_{1}s+1)(s^{2}+A_{1}s+A_{2})}$ (1)

In the equation, $W$ expresses the speed of executive element, the type of $U$ expresses the inverter control electricity pressure, $T_{1}$ is the time constant of the inverter and motor, $K_{1}$ is the link gain of the frequency converter and motor, $A_{1}$ and $A_{2}$ show the characteristic parameters of the hydraulic system, $K_{2}$ is the gain of the hydraulic system.

If considering the possible of the mechanical connection between motor and pump gap, the nonlinear friction and compressibility of oil, the speed control system of variable frequency motor which is controlled by pump has always more or less certain delay, so the transfer function models can be revised to Eq. (2):

$\displaystyle\frac{W}{U}=\frac{K_{1}}{K_{2}}{(T_{1}s+1)(s^{2}+A_{1}s+A_{2})}e^% {-\tau s}$ (2)

The role of the external load will make the movement speed of executive element much lower and the speed to the transfer function of the external load F can be represented as Eq. (3):

$\displaystyle\frac{W}{F}=\frac{B_{1}s^{2}+B_{2}s+B_{3}}{(T_{1}s+1)(s^{2}+A_{1}% s+A_{2})}$ (3)

In the Eq. (3), $B_{1}$ , $B_{2}$ and $B_{3}$ reflect the parameters characteristics of the external load.

3. Genetic algorithm (GA) is used for the optimization PID parameters

The GA was based on the theory of natural selection and genetic, the survival of the fittest laws in the biological evolution process are combined with the mechanism of internal chromosomes random information, the global parallel optimization search is done in the genetic space, and the convergence is obtained in the direction of the global optimal solution.

3.1 Coding and genetic operation

Due to the binary coding error exists, so do the real number encoding code, that is, each genes in the chromosome values is in real field. If real number coding is used, the mapping relationship is 1 between genetic space and the solution space, there is no mapping (that is, the encoding/decoding) error and the resulting reducing of GA precision. While coding in real number, the chromosomes are in the form of a vector. For the PID parameter optimization in genetic space, the j ${}^{\text{th}}$ chromosome vector (individual) of the m ${}^{\text{th}}$ generation populations is expressed as Eq. (4):

$\displaystyle C_{j}^{m}=(K_{Pj}^{m},K_{Ij}^{m},K_{Dj}^{m})^{T}$ (4)

Individual selection probability is determined by the ratio method, individual selection is closed by the roulette method.

Crossover operation takes single point arithmetic crossover operation. For example, $C_{j}^{m}$ and $C_{q}^{m}$ are intersected, then they bring out Eq. (5) for new individuals:

$\displaystyle\left.\begin{matrix}C_{j}^{m+1}=\beta\cdot C_{q}^{m}+(1-\beta)% \cdot C_{j}^{m}\\ C_{q}^{m+1}=\beta\cdot C_{j}^{m}+(1-\beta)\cdot C_{q}^{m}\\ \end{matrix}\right\}$ (5)

In the Eq. (5), $\beta$ is represent the random number, $\beta\in[0,1]$ .

Mutation unconventional operation is took in this part. For the chromosome vector $C_{j}^{m+1}=(c_{j}^{m}(0),c_{j}^{m}(i)^{*},\Lambda,c_{j}^{m}(N-1)^{T})$ , each of these genes has the same mutation chance in the gene, the mutation of a single genetic component results is for Eqs (6) and (7):

$\displaystyle C_{j}^{m+1}=(c_{j}^{m}(0),\Lambda,c_{j}^{m}(i)^{*},\Lambda,c_{j}% ^{m}(N-1)^{T})$ (6) $\displaystyle c_{j}^{m}(i)^{*}=\left\{\begin{matrix}c_{j}^{m}(i)+\Delta\left(m% ,UB-c_{j}^{m}(i)\right);\quad\textit{if \_round}(\textit{rand})=0;\\ c_{j}^{m}(i)+\Delta\left(m,c_{j}^{m}(i)-LB\right);\quad\textit{if \_round}(% \textit{rand})=1;\\ \end{matrix}\right.$ (7)

Among them, $U B$ , $L B$ , are respectively the range of upper and lower bounds of $c_{j}^{m}(i)$ . Function $\Delta(m,y)$ is defined as Eq. (8):

$\displaystyle\Delta(m,y)=y\cdot\gamma\cdot\left[1-\frac{m}{M}\right]^{b}$ (8)

In this Eq. (8), $\gamma$ is the random number in the [0, 1]; $M$ is the biggest genetic algebra; $b$ is the parameters which determine its non-uniform degree.

3.2 Fitness function

GA don’t need external information in the evolutionary search, it is only based on fitness function, the fitness value of each individual is used for the search. So it is very important for the selection of the fitness function, it can directly affect the convergence speed of the GA whether it can find the optimal solution. The determination of the objective function of the specific issues and the fitness function need to meet the requirements of single, continuous, nonnegative and maximizing.

ITAE is a comprehensive index, it reflects the weighted integral from the time $t$ to the error $e(t)$ . If the system response is fast, overshoot is small and static error is small, the parameter value is small. the performance index of ITAE is expressed as Eq. (9):

$\displaystyle J=\int_{0}^{\infty}t\cdot|e(t)|dt=\min$ (9)

Because it is based on minimizing the goal, you need to convert it into the biggest form as a fitness function, it can define the fitness function it is Eq. (10):

$\displaystyle\textit{Fitness}=L-J=L-\int_{0}^{\infty}t\cdot|e(t)|dt=\max$ (10)

$L$ is a constant, $L>\int_{0}^{\infty}t\cdot|e(t)|dt$ .

3.3 Description of the optimization algorithm of process

The process of GA exploring PID parameters optimization algorithm is described below.

(1)
Randomly generate a certain initial scale group in a given domain.
(2)
Calculate the fitness value of each individual in the population.
(3)
Genetic operation: selection, crossover and mutation.

➬
Choice: individual selection method is used by the roulette method choice, the choice probability is determined by proportion method.
➭
Cross: choose the single point arithmetic crossover operation.
➮
Variation: unconventional mutation.

(4)
Ensure that whether the evolution algebra is got. If it is got, turn (1), other, the target was output, the process ends at the same time.

Above optimum algorithm evolution algebra serve directly as a condition of end, if the numerical value is too small, it may not be converge, it will increase the evolution numerical value. If only the population fitness value converge, the output target is the optimal solution of problem.
4. Simulation example

For a speed control system of variable frequency motor which is controlled by pump, the calculation is based on the theory of approximate, system model is obtained in Eq. (11):

$\displaystyle\frac{W}{U}=\frac{14999e^{-0.5s}}{8.8s^{3}+108.1s^{2}+83.8s+50.5}$ (11)

GA was described above, it is used to do the PID parameter optimization for this model. it is used to determine the range of optimal parameters optimization, population size, the number of evolutionary generations and the upper limit of fitness function when we do the PID parameter optimization. The range of the optimal parameters $K_{P}$ , $K_{I}$ , $K_{D}$ , are determined by the $P I D$ simulation debugging, they are as follows: $K_{P}\in[0.00001,0.1]$ , $K_{I}\in[0.00001,0.1]$ , $K_{D}\in[0.001,0.1]$ . The upper limit $L$ of the fitness function is also determined by the simulation debugging: $L=2.5e+5$ .

Population size is 30. Evolution algebra startes in 100, the simulation results is not convergent. The evolution algebra changes to 200, the simulation results is convergent, the target: $K_{P}=$ 0.006183, $K_{I}=$ 0.0000918, $K_{D}=$ 0.1372. Fitness function value: Fitness $=$ 2.4892e $+$ 5. The simulation curve is the curve 1 in Fig. 2.

Figure 2.

The simulation curve.

In order to contrast with the effect of conventional PID control, Fig. 2 also shows the simulation curve of the conventional PID control, which is shown in curve 3. With conventional method after repeating adjustment of PID parameters for: $K_{P}=$ 0.0008, $K_{I}=$ 0.00005, $K_{D}=$ 0.05. From the contrast between curve 1 and curve 3, it can be seen that the rise time of curve 1 is about 1.5 s, overshoot is 11; Rising time of curve 3 is about 4.2 s, overshoot is 48, the control quality of PID parameters of GA optimization of is much better than the conventional method. In order to compare two groups of robustness of PID parameters, adding a step load disturbance on the 25 ${}^{\text{th}}$ s, disturbance duration is 4 s, we contrast curve 1 with 3, the GA optimization results have shown the robustness of PID parameters on the load disturbances, it is better than the robustness of the conventional method.

In order to compare the adaptation of two groups PID parameters mode on the mismatch appearance, we increase the leakage, and adjust the parameters of the hydraulic system, the a model becomes Eq. (12):

$\displaystyle\frac{W}{U}=\frac{14999e^{-0.5s}}{8.8s^{3}+111.6s^{2}+126.6s+50.5}$ (12)

The above two groups of PID parameters is used to simulate, the curve are respectively curves 2 and 4 of graph 2. Contrast curves 1 with 2, the rise time of curve 2 is increased about 0.5 s than Curve 1, overshoot volume is increased about 5. The rise time of curve 4 is increased 1 s than 3, overshoot volume is increased 15, this suggests that using GA optimization of PID parameters have showed stronger adaptability than the conventional method in the model mismatch.

4.1 Several discussions

(1)
The GA optimizing PID parameters is a kind of global search, so the range of PID parameters have given a great influence on the simulation results. Big scope is not easy to converge, and it may cause instability and cause the stagnation of the simulation. While simulation is taken, it is needed to combine with trial, in advance to determine the reasonable target scope.
(2)
When only the evolution algebra is given as a condition of the end, the simulation time mainly is decided by evolutionary algebra, the population size and the given simulation time at the section, one of the biggest impaction was the evolution algebra. Evolution algebra is 200, population size is 30, the simulation time section is 40 s (see chart 2), the consumption time is 21 min.
(3)
When ITAE index is used for a fitness function, every evolution generation is equal to the number experiments of the population size, such as population size is 30, the evolution algebra is 200, which is equivalent to 6000 times PID control experiment, so it is very difficult to do the on-line optimization for the actual object.
(4)
The GA optimization effect of the parameters is used in the actual control object depends on the accuracy of the model. For the actual variable frequency hydraulic regulation speed system, due to some soft parameters, the system and the influence of uncertain factors, it is practically difficult to accurately establish the mathematical model of the object, So it is recommended that firstly do the off-line identification in advance to get more accurate mathematical model of the object, then to do the optimizing PID parameters of the GA.

5. Conclusion

The method of variable frequency motor is presented based on the genetic algorithm in this paper, the motor is controlled by pump speed control system of PID parameters optimization. The genetic operation of real-coded are introduced, the search process and realization of the optimization algorithm are researched. The simulation results prove that the GA optimization of variable frequency motor (which is controlled by pump speed control system PID parameters) is effective. Its effectiveness is that comparing with the conventional methods, it has better control characteristic parameters, it showed stronger adaptability and robustness in the conditions of the model mismatch and load disturbances. This method is very suitable for a slow time-varying system and load disturbance variable frequency motor which is controlled by pump speed control system. This article also pointed out the limitation of PID parameters using GA optimization in the system, it is recommended that we firstly do the off-line identification in advance to get more accurate mathematical model of the object, then to do the optimizing PID parameters of the GA. In addition, the GA optimization of PID parameter simulation related matters are discussed.

Footnotes

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China under grant No. 61074147, the Natural Science Foundation of Guangdong Province under grant No. S2011010005059, the Foundation of Enterprise-University-Research Institute Cooperation from Guangdong Province and Ministry of Education of China under grant No. 2012B091000171 and No. 2011B090400460, the Science and Technology Program of Guangdong Province under grant No. 2012B050600028, No. 2014B010118004, No.2015A030401104 and No. 2016A050502060, the Science and Technology Program of Huadu District, Guangzhou under grant No. HD14ZD001, the Science and Technology Program of Guangzhou under grant No. 201604016055, the Nature Science Program of Jiaying University under grant No.2015KJZ05.

References

Wang

X.P.

and Cao

L.M.

, The theory of genetic algorithm, Application and software implementation, Xi’an Jiaotong University Press, China, 2002.

Krohling

R.A.

and Rey

J.P.

, Design of optimal distubnace rejection PID controllers using genetic algorithms, IEEE Transactions on Evolutionary Computation 5(1) (2001), 78–82.

Johnson

K.L.

, Contact Mechanics, Cambridge University Press, Cambridge, 1985.

Hunter

S.C.

, The rolling contact of a rigid cylinder with a viscoelastic half space, Applied Mechanics 28 (1961), 611–617.

Lodewijks

, Dynamics of Belt Systems, Thesis, Delft University of Technology, Delft, 1995.

Bennett

, A history of control engineering 1930–1955, IEEE Control Engineering Series 47 (1993).

Visioli

, Tuning of PID controllers with fuzzy logic, Proc Inst Elect Eng Contr Theory Applicat 148(1) (2001), 1–8.

Krohling

R.A.

and Rey

J.P.

, Design of optimal disturbance rejection PID controllers using genetic algorithm, IEEE Trans Evol Comput 5 (2001), 78–82.

Wang

, Application of adaptive genetic algorithms in PID controller design, J Xi’An Univ Sci and Tech 25(1) (2005), 93–95,

10.

Wang

X.W.

Yao

X.L.

et al, Research on PID controller parameter tuning of fin stabilizer based on PSO algorithm, Control Engineering of China 23(8) (2016), doi:10.14107/j.cnki.kzgc.140862.

11.

Guo

et al., PID controller parameter optimization based on improved glowworm swarm optimization, Computer Applications and Software 34(7) (2017), doi:10.3969/j.issn.1000-386x.2017.07.042.

12.

Zigeler

J.G.

and Nichlos

N.B.

, Optimization setting for automatic controller, Tran ASME 64(11) (1942), 756–769,

13.

Cohen

G.H.

and Coon

G.A.

, Theoretical considerations of retarded control, Transactions of ASME 75(1953), 827–834.

14.

Z.C.

, Parameter tuning method of robust PID controller based on particle swarm optimization algorithm, Control and Instruments in Chemical Industry 33(5) (2006), 22–25, doi:10.3969/j.issn.1000-3932.2006.05.006.

15.

Kennedy

and Eberhart

, Particle swarm optimization, in: Proc IEEE Int Conf Neural Networks, Vol IV, Perth, Australia, 1995, pp. 1942–1948.

16.

Kim

D.H.

, Tuning of PID controller using gain/phase margin and immune algorithm, in: Proceedings of the 2005 IEEE Midnight-Summer Workshop on Soft Computing in Industrial Applications, Helsinki University of Technology, Espoo, Finland, 2005, pp. 68–74.

17.

Kim

D.H.

and Cho

J.H.

, Intelligent tuning of PID controller with disturbance function using immune algorithm, in: IEEE International Conference on Computational Intelligence for Measurement Systems and Applications, Boston, MA, USA, 2004, pp. 286–291.

18.

Kim

D.H.

Hong

W.P.

et al., Auto-tuning of reference model based PID controller using immune algorithm, in: Proceedings of the 2002 Congress on Evolutionary Computation, Honolulu, HI, USA, 2002, pp. 483–488.

19.

Dorigo

and Dicaro

, The ant colony optimization meta-heuristic, in: New Ideas in Optimization, McGraw Hill, London, UK, 1999, pp. 11–32.

20.

Dorigo

Di Caro

and Gambardella

L.M.

, Ant algorithms for discrete optimization, Artificial Life 5(2) (1999), 137–172.

21.

Chiha

Liouane

and Borne

, Multi-objective ant colony optimization to tuning PID controller, International Journal of Engineering 3(2) (2009), 11–16.

22.

Pham

D.T.

Ghanbarzadeh

et al., The bees algorithm, a novel tool for complex optimisation problems, in: Proc 2nd Virtual International Conference on Intelligent Production Machines and Systems, Elsevier (Oxford), 2006, pp. 454–459.

23.

Jones

K.O.

and Bouffe

, Comparison of bees algorithm, ant colony optimization and particle swarm optimization for PID controller tuning, in: Proceedings of the 9th International Conference on Computer Systems and Technologies and Workshop For PhD Students in Computing, Comp Sys Tech 2008, 374, ACM, New York, IIIA. 9-1–IIIA. 9-6.

24.

Shi

and Eberhart

, A modified particle swarm optimist, in: IEEE World Congress on Computational Intelligence, Piscataway, IEEE Press, 1998, pp. 69–73.

25.

Zeng

J.C.

and Cui

Z.H.

, A guaranteed global convergence particle swarm optimizer, Journal of Computer Research and Development 41(8) (2004), 1333–1338.

26.

Long

Liang

X.M.

et al., Dynamic hierarchical hybrid particle swarm optimization algorithm, Control AND Decision 24(6) (2009), 1406–1411.

27.

Lerc

and Kennedy

, The particle swarm: Explosion, stability, and convergence in multi-dimension complex space, IEEE Transactions on Evolutionary Computation 16(1) (2002), 58–73.

28.

Wang

T.K.

, Fuzzy self-tuning PID control of main steam temperature in thermal power boilers based on UMDA, Journal Of Chinese Society Of Power Engineering 37(7) (2017), 546–551.

29.

Feng

S.W.

Yang

Y.S.

et al., Control strategy for giant magnetostrictive actuator based on genetic algorithm tuning PID parameters, Journal of Naval University of Engineering 29(3) (2017), doi:10.7495/j.issn.1009-3486.2017.03.005.

30.

, Control of nonlinear modular mobile robots using PID method with iterative feedback tuning, Machinery Design and Manufacture 5 (2017), doi:10.3969/j.issn.1001-3997.2017.05.061.

31.

Zhang

G.B.

et al., BA-based PID tuning method, Control Engineering of China 24(3) (2017), 548–553, doi:10.14107/j.cnki.kzgc.150617.

32.

Mei

and Luo

, PID parameters optimization with improved cooperative coevolution algorithm, Computer Technology and Development 27(8) (2017), 37–42, doi:10.3969/j.issn.1673-629X.2017.08.008.

33.

Ren

B.S.

and Jiang

X.F.

, Dynamic voltage restorer control based on RBF-PID, Journal of Electrical Engineering 12(8) (2017), 28–33, doi:10.11985/2017.08.005.

34.

May

W.D.

Morris

E.L.

and Atack

, Rolling friction of a hard cylinder over a viscoelastic material, Applied Physics 30 (1959), 1713–1724.