基于混沌的自适应模糊PID参数寻优研究_英文_
第20卷第19期 系
统 仿 真 学 报? V ol. 20 No. 19
2008年10月 Journal of System Simulation Oct., 2008
Study on Adaptive Fuzzy PID Parameters Optimization
Based on Chaos
WU Yun-jie, LV Qin, TIAN Da-peng
(Department of Automatic Control , Beihang University, Beijing 100191, China)
Abstract: An optimization algorithm design based on chaotic variable was proposed for adaptive fuzzy PID controller. First, by using the property of chaos ergodicity, a PID controller based on chaos optimization was designed . On the condition of stability, the most excellent parameters could be found with the least performance index by both cursory and precision precise search . These parameters were utilized as the most excellent parameters of the adaptive fuzzy PID controller. Second, according to the adaptive fuzzy method, the control rules and its parameters were determined. By this way, the simulation results show that this control method can be confirmed and reach required production target. Key words: chaos; PID controller; adaptive fuzzy control; parameters optimization
基于混沌的自适应模糊PID 参数寻优研究
吴云洁, 吕 骎, 田大鹏
(北京航空航天大学自动控制系，北京 100191)
摘 要：针对自适应模糊PID 控制器提出了一种基于混沌变量的优化算法。首先，利用混沌运动的遍历性特性，基于混沌优化方法进行PID 控制器设计。在系统稳定的条件下，通过粗调和精确搜索可以找到具有最小参数的最佳控制器。这些参数被用来作为自适应模糊PID 控制器的最佳参数。然后，按照自适应模糊控制方法，控制律及其参数可以被确定。仿真实验结果显示，该方法能够达到所要求的指标，其有效性得到了证明。 关键词：混沌；PID 控制；自适应模糊控制；参数优化
中图分类号：TP273 文献标识码：A 文章编号：1004-731X (2008) 19-5224-04
Introduction
Sometimes in the industrial control, the object of control changes its performance parameters or its system architecture because of load variety and influence factors. Adaptive control system based on modern control theories identifies object’s characteristic parameters on-line and adjusts the control strategy real-time. By this way, adaptive control can keep system index of quality. However, the effect of control depends on the precision of system-model identification. So PID control is used widely in the real industrial control.
Along with the development of computer technology, using artificial intelligence technology, people stone the experience of operating personnel into the computer. Then the automatic PID parameter control can be operated by the computer. Because the experience of operating personnel cannot be denoted accurately and quantitatively, fuzzy theory is a good method to solve the problem. First, using basic theory and method of fuzzy mathematics, the rules and conditions are transformed into fuzzy set. Second, the fuzzy control rules are
Received : 2008-07-18 Revised : 2008-08-08
Foundation items :
National nature science foundation of China, 60304005; Program for new century excellent talents in university, NCET-07-0044 Biographies : Wu Yunjie (1969-), female, Baoding, Hebei province, Ph. D., professor, Ph. D. supervisor, major in servo control, intelligent control; Lv Qin (1981-), male, Jiaxing, Zhejiang province, master, major in servo control; Tian Dapeng (1984-), Tieling Liaoning province, male, Ph. D. candidate, major in servo control.
stoned into the computer as the experience. Third, fuzzy reasoning can be used according to the reality response and regulate the PID parameters automatically. This is adaptive fuzzy PID control. But the selection of the initial values of PID parameters is in the major of the final effect of control. So the question is how to find the best initial values.
Chaos is similar with random motion in deterministic control system. Chaos optimization method has many advantages. Because chaotic motion has some good features such as ergodicity and randomicity, it can search approximate global optimal solution. If chaotic motion is used into the optimization of PID initial parameters, the best parameters must be found more easily than other methods. The standard of which group of parameters is better is index of performance, such as the accumulated error. First, roughness parameters, which have the least index of performance, can be found globally by the chaotic motion. Second, near the roughness parameters, accurate parameters which also have the least performance index can be found by the chaotic motion.
An optimization algorithm design based on chaotic variable is proposed for adaptive fuzzy PID controller. In view of convergence rate and convergence accuracy, a two-phase optimization scheme is proposed in the paper. In phase one, chaos offline optimization scheme is used for global optimization parameter so the best initial values of PID parameters in the index of performance can be found. In phase two, adaptive fuzzy PID parameters based on the chaos
2008年10月 WU Yun-jie, et al ：Study on Adaptive Fuzzy PID Parameters Optimization Based on Chaos Oct., 2008
parameter optimization, are used to control the real object online. By this way, the simulation results show that this control method can be confirmed and reach required production target. And the results show that the chaos optimal control is of high precision, small overshoot, fast response and good robustness.
1 Principle of adaptive fuzzy PID control
Adaptive fuzzy PID controller takes measure- ment error e and rate of error change e Δ as the inputs of the system. By is way, PID parameters can be adapted based on e and e Δ all the time. The PID parameters are changed depending on the fuzzy control rules on-line. Figure 1 shows structure of adaptive fuzzy controller.
Fig. 1 Structure of adaptive fuzzy controller
Adaptive fuzzy PID parameters self-setting depends on the relationship between parameters and e , e Δ. While detecting the error e and rate of error change e Δ, the PID parameters are changed depending on the fuzzy control rules which are online. By this way, system would have satisfied dynamic and static properties.
Integrating system stability, response, overshoot, stable precision and function of p k , i k , d k to system, the relation among the three parameters should be considered in the adaptation of parameters.
The core of fuzzy reasoning design is sum-up of engineer’s technical knowledge and experiences. So the suitable fuzzy control rules list should be set. The list is fuzzy control table of p k , i k , d k . Only the table of p k is below because of the paper’s limit.
Table 1 p k fuzzy control table
e Δ e
NB NM NS AZ PS PM PB
NB PB PB PM PM PS AZ AZ NM PB PB PM PS PS AZ NS NS PM PM PM PS AZ NS NS AZ PM PM PS AZ NS NM NM PS PS PS AZ NS NS NM NM PM PS AZ NS NM NM NM NB PB AZ AZ NM NM NM NB NB
After setting the list of p k , i k , d k , the ranges of error e and
rate of error change e Δ are defined as the domain of fuzzy set: ,{3,2,1,0,1,2,3,}e e Δ=??? (1) Based on the fuzzy control theory, the fuzzy subsets of e and e Δ are ,{,,,,,,}e e NB NM NS AZ PS PM PB Δ= (2) The elements in the subsets mean negative big, negative
middle, negative small, about zero, plus small, plus middle, plus big. The membership functions of p k , i k , d k and e , e Δ can be defined as normal distributions. Depending on the tables of the fuzzy subsets’ membership function and fuzzy control model of the parameters, fuzzy matrixes of PID parameters can be designed. Check corrected parameters in formula (3):
{,}{,}{,}p p p d d d i i i i i i i i i k k e e k k e e k k e e ?????
??
′=+Δ′=+Δ′=+Δ (3) In the online operation, by using fuzzy control rules, the controller completes result check, table lookup and computing.
Although the adaptive fuzzy control method can adjust the parameters online, the initial values of parameters influence the effect of control a lot. So the method of finding the best initial values of parameters is important. Then the PID parameters optimization based on chaos is introduced.
2 Chaotic optimization initial parameters of PID
2.1 Principle block diagram
Figure 2 shows block diagram of chaotic optimization principle. The original values of PID parameters are found by chaos optimization offline, and the globally optimal solutions can be found through two steps of optimization. Then the globally optimal solutions adapt themselves using adaptive fuzzy control.
2.2 The way to chaos, period-doubling bifurcation
In the chaos optimization, the logistic equation (period-doubling bifurcation) is used as the equation of the parameters’ equation of motion.The period of system motion is an ordered state, but in some conditions, the period would be doubled if some of the parameters are changed, which is called period-doubling bifurcation. And the system would loose its period to the chaos.
Fig. 2 Structure of controller based on chaos optimization The function’s magnitude relation between n moment and
n+1 moment is denoted as: 1(1),[0,1]n n n n x x x x μ+=?∈ (4) Set an original value 0x optionally between 0 and 1, the
value n x in any moment later can be calculated by using
Equality 3. The two-periodic motion can be showed in Figure 3.
2008年10月 Journal of System Simulation Oct., 2008
Fig. 3 Two-periodic motion
Iteration from any original value and the trend of final
state is useful. When 3μ<, the final result is on the crossover point of parabola and straight line 1n n x x +=. It is showed by Figure 3. When 3μ=, the system is unstable and appears bifurcation. When 31μ<<, the final results of system jump between two values repeatedly. And this is a period- doubling solution, corresponding to limit cycle of continuous system.
Fig. 4 Chaos motion
When 3.449487743μ=, the system has another period-
doubling bifurcation. The two-period point is unstable and four-period point appears. The eight-period and sixteen- period will appear if μ keep growing. When 3.569945972μ=, chaos motion is showed on Figure 4.
2.3 Performance index In industrial control, the performance index should not only reflect the dynamic characteristic of system but also should contain static characteristic. So the performance index below is adopted. []
|()|max ()t t e t dt J e t =
∫ (5)
After scatter:
()1min ()/max s T i i J t e k T e k =??
=××????????
∑ (6)
In the equalities, T s is the number of sample points, and T
is sample frequency. Depending on the control theory, this
performance index can evaluate the system’s static
characteristic and dynamic characteristic. For instance, it
involves overshoot, response, govern time, and steady state
error. The system’s static characteristic and dynamic
characteristic can be controlled by changing the performance index. For example, if the overshoot is more important than other indexes, the overshoot proportional can be added to the performance index, with a larger weight.
2.4 Chaos optimization method
The formula of PID control is:
1()()()()p d I de t u t k e t e t dt T T dt ?
?=++????∫ (7)
After scatter:
0()()()[()(1)]k
p i d i u k k e k k e i k e k e k ==++??∑ (8)
In the formula, (/)i p s k k T T =×, (/)d p D k k T T =×.
T I is integral time constant. T d is differential time constant. And
T I is sample time constant.
For finding the global optimal solutions of the control parameters, global-rough search and local-fineness search are used to quicken the optimization. Using the logistic equation as the values of chaos motion:
1(1),[0,1]n n n n x x x x μ+=?∈ (9) When 4μ=, the logistic mapping is in chaos. By using chaos characteristic of initial value sensitivity, lots of chaos value orbits can be gotten depending on different initial value.
Steps:
1) Set three variables as the initial variables of the Logistic mapping. Then there are three different paths of the chaos variables. 1,n x , 2,n x and 3,n x corresponding p k , i k and d k .
Because the chaos variables’ ranges are (0,1) , 1,
n x ′ is set as the optimization variable. The PID variables’ ranges are expanded
by getting amplification coefficients. So are 2,
n x ′ and 3,n x ′: 1,111,2,
222,3,333,n n n n n
n x a b x x a b x x a b x
′=+??
′=+??′=+? (10) 2) 1,n x ′,2,n x ′ and 3,
n x ′ are used as p k , i k and d k to control the object. In limited optimization times, if the group of
variables satisfy the stability condition, calculate the
performance index. The variables are set as the 3 suboptimal variables p k , i k and d k . Then the roughness optimization is competed. In the calculation, the condition of system stability is that
all latent roots of the system’s proper equation are on the left side of the virtual axis. Depending on the real part of the closed-loop pole, if real part is nonnegative number, the system is unstable. Else system is stable. By using Matlab software, it
is easy to decide the real part nonnegative or negative. If the system of the variables is unstable, the variables are abnegated.
3) After finding the roughness optimizations, global
optimal solutions should be found based on the roughness
optimizations. Follow the formulas below:
*,1,p n p t n k k Z x =+× (11) x (n +1)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
x/n
Logistic u=3.3 x(0)=0.2
0.10.20.30.40.60.70.80.91
0.5
x (n +1)
0 0.1 0.2 0.3 0.4 0.50.6 0.7 0.8 0.9 1
x/n
Logistic u=4 x(0)=0.2
0.10.20.30.40.60.70.80.910.5
2008年10月 WU Yun-jie, et al ：Study on Adaptive Fuzzy PID Parameters Optimization Based on Chaos Oct., 2008
*,2,i n i t n k k Z x =+× (12) *,3,d n d t n k k Z x =+× (13)
1(1)(01)n t Z Z λλ+=?×<< (14)
In the formulas, λis the time-varying parameter t Z ’s
attenuation factor to reduce the momentum range of chaos
variables gradually. If t Z R <, the loop is end.
In the same way, fineness-search is depended on the
smallest virtual axis. If the group of variables is fit to the
system stability ，and its virtual axis is smallest, register the variables, ,p n k ,,i n k ,,d n k as the global optimal solutions of the PID parameters.
3 Servo system control simulation
Set the approximate transfer function of the positional
servo system: 524() 5.23510()(87.35 1.04710)y s u s s s s ×≈++× (15) In Matlab software, discretize the formula (15). Set the sampling cycle 1ms, depending on the actuality of servo system. The input of the system is step input.
Set 10.5a =, 20a =, 30a =, 10.5b =, 20.1b =, 30.1b = and extend the ranges of the chaos variables to the range of PID parameters. Set sampling point number 100s T =, means chaos optimization is done every 100 steps. The number of roughness optimizations is set 3000. By simulating the optimization, the roughness optimization variables are *0.6P K =, *0.001I K =, *0.005D K =. Set the attenuation factor 0.01λ=, the time-varying parameter t Z ’s initial parameter 10.8Z =, the constant 0.01R =, which means if 10.01t Z +<, the loop is end.
The result of experiment is below.
First, because adaptive fuzzy PID parameters optimization is used in the system, the robustness of system is advanced. In the simulation, in the 300th sampling point, a 0.5 interrupt is added in the input and last 5 sampling cycles. The effect figure
Fig. 5 Efficiency of control and add 0.5 interrupt in 0.3s
In the figure we can see that if there is an interrupt in the input, system can return to normal work condition soon. The control effect is satisfying.
Then compare the adaptive fuzzy PID parameters
optimization with and without chaos optimization. And the initial parameters of PID control and the final control effect are
showed in the Table 2 below.
Table 2 Compare control effects between with and
without chaos optimization
Kp Ki Kd overshoot responsive time
With chaos
optimization
0.426380.000396 1.4171 9.48% 0.114s Without
chaos
0.4 0.0 1.0 14.5% 0.186s The efficiency of the two control methods are showed in
figure 8. The red curve is control efficiency with adaptive fuzzy
chaos optimization and the blue curve is without chaos optimization. With the chaos optimization, the responsive time,
overshot (2% error area), and stable error are all advanced. The
Fig. 6 The errors of the two control methods
Emphasized, the effect of control can be improved by changing the performance index to get the satisfying efficiency of control.4 Conclusion
The method of PID parameters optimization based on chaos uses the behavior of chaos motion to achieve the global optimal solution. Combining with adaptive fuzzy control, the whole control process separates in online part and offline part. By this way, the efficiency of control method has better responsive time, overshot, and stable error.
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