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PID type fuzzy controller and parameters adaptive method
ZZ'Y
sets and systems ELSEVIER
Fuzzy Sets and Systems 78 (1996) 23-35
PID type fuzzy controller and parameters adaptive method Wu Zhi Qiao*, Masaharu Mizumoto Department of Management Engineering, Osaka Electro-Communication Universi~, Neyagawa, Osaka 572. Japan
Received October 1994; revised January 1995
Abstract The authors of this paper try to analyze the dynamic behavior of the product-sum crisp type fuzzy controller, revealing that this type of fuzzy controller behaves approximately like a PD controller that may yield steady-state error for the control system. By relating to the conventional PID control theory, we propose a new fuzzy controller structure, namely PID type fuzzy controller which retains the characteristics similar to the conventional PID controller. In order to improve further the performance of the fuzzy controller, we work out a method to tune the parameters of the PID type fuzzy controller on line, producing a parameter adaptive fuzzy controller. Simulation experiments are made to demonstrate the fine performance of these novel fuzzy controller structures. Keywords: Fuzzy controller; PID control; Adaptive control
1. Introduction Among various inference methods used in the fuzzy controller found in literatures [5-8, 13, 14], the most widely used ones in practice are the M a m d a n i method proposed by M a m d a n i and his associates [5] who adopted the Min-max compositional rule of inference based on an interpretation of a control rule as a conjunction of the antecedent and consequent, and the product-sum method proposed by Mizumoto [6, 7] who suggested to introduce the product and arithmetic mean aggregation operators to replace the logical A N D (minimum) and O R (maximum) calculations in the Min-max compositional rule of inference. In the algorithm of a fuzzy controller, the defuzzyfication calculation is also a complicated and time consuming task. Tagagi and Sugeno proposed a crisp type model in which the consequent parts of the fuzzy control rules are crisp functional representation or crisp real numbers in the simplified case instead of fuzzy sets [13, 14]. With this model of crisp real number output, the fuzzy set of the inference consequence will be a discrete fuzzy set with a finite number of points, this can greatly simplify the defuzzification algorithm. Both the Min-max method and the product-sum method are often applied with the crisp output model in a mixed manner. Especially the mixed product-sum crisp model has a fine performance and the simplest algorithm that is very
* Corresponding author. Presently at the Department of ECS, University of Southampton. UK. E-mail: [email protected]. 0165-0114/96/$15.00 © 1996 - Elsevier Science B.V. All rights reserved SSDI 0165-01 14(95)001 15-8
VI(Z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23 35
24
easy to be implemented in hardware system and converted into a fuzzy neural network model. In this paper, we will take account of the product-sum crisp type fuzzy controller. There has not been sounded theoretical method available to analyze a fuzzy controller in literature, while the conventional control theory is highly developed. It is natural for the researchers to apply the conventional theory, mostly linear system theory to solve the nonlinear problem of fuzzy controller and many works have been done in this direction [1,2,4, 16]. In this paper, the authors will make effort to analyze the behavior of the product-sum crisp type fuzzy controller, by relating the fuzzy controller to the conventional PID controller. Having the aid of the well-known classical designing method of PID controller, we propose a new fuzzy controller structures, namely P I D type fuzzy controller which retains the characteristics similar to the conventional PID controller. In order to improve further the performance of the fuzzy controller, we work out a method to tune the parameters of the PID type fuzzy controller on line, producing a parameter adaptive fuzzy controller. The proposed methods promise to improve the performance of the fuzzy controller considerably.
2. Analysis of a product-sum crisp type fuzzy controller Before we conduct the analysis, we briefly describe the crisp type fuzzy controller model mixed with product-sum inference method as follows [6, 13, 14]. Suppose that the fuzzy controller in consideration is a two-input and one-output one. The two inputs to the fuzzy controller are error e and change rate of error ~, and the output of the fuzzy controller (that is the input to the controlled process) is u. The universes of discourses of e, 0 and u are E ~ R,/~ c R and U c R, respectively. We denote the linguistic values of e and ~ as A ~(i ~ I = [ - m .... , - 2, - 1, 0, 1, 2 ..... m]) and Bj (j ~ J = [ - n . . . . . - 2, - 1, 0, 1, 2 .... , n]) respectively. The fuzzy control rules are given as in the form of
if e is Ai and ~ is Bj then u is u d. where uij E U(i e I,j ~ J) is a crisp value instead of a fuzzy subset. The uijs are not necessarily different from each other. The fuzzy controller with such kind of control rules is called crisp type fuzzy controller [13, 14]. If the number of control rules are equal to I x J, the fuzzy control rule base is said to be complete [16]. In the following discussion, we assume that the fuzzy control rule base is complete. Suppose that the membership functions of Ai and B~ are Ai(e) and Bj(d). In a certain control time t, we have the observation values e and ~ for error and the change rate of error respectively, then the truth values of Ai and Bj are A~(e) and Bj(k), (i ~ I, j ~ J). Using the product-sum inference method, the truth value of the antecedent part of a fuzzy control rule will be
f j = Ai(e)Bj(k)
(ieI, jeJ).
(1)
The reasoning from the antecedent part to the consequent part will generate a conclusion fuzzy subset which we denote as C. C will be a discrete fuzzy subset with finite number of points, C = { f j / u u l i ~ I , j ~ J }. applying the center of gravity method to defuzzify the fuzzy set C, the real output of the controller u is given by ~i, j fij Uij
u = - Z i, j f/j
,
(2)
In our study we will employ the triangular membership functions for each fuzzy linguistic value of the error e and the change rate of error 0 as shown in Fig. 1. We denote the cores of fuzzy set Ai as el and those of Bj as
W.Z. Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23 35 Ai-1
Ai
25
di-1 di
Ai+l
e/+l
I I !
ei-1
ei
ei+l
Bj-1
Bj
Bj+I
el-1 uij
ei ei+l
ei-1
dj
di+l
1
Fig. 1. The membership functions of A~and Bj.
Fig. 2. T h e
ej.
We design the membership functions as so, the support sets of Bj are equal to [dj_ 1, ej+ 1], as illustrated in Fig. 1. We have Ai(e) = 1
e
-
-
- -
ei + 1 - - - ,
ei
-
ei+ 1 --c i
ei+ 1
-
e
--e i
Ai+l(e)-
e
-
-
ei
A i are
NET
o n t h e e - ~; plane•
equal to [e~_ 1, ei+ 1] and those of
Ak(e) = 0 (k # (i,i + 1)c I),
ei+ 1 - - e i
for e e [ei,ei+t], and Bj(e)
=
d -- ej - dj + I - - e j
l
for ,; ~ [~j,dj+ On the e N O D E of the Combining
ej+l -- d _ _ , dj + 1 - - e j
d -- dj Bj+I(e
) -
Bt(O) = 0 (t # ( j , j + 1) e J),
e j+ 1 -- dj
1]. 0 plane, we call the set {e,d[e = el, d = ej, i ~ I , j e J } N E T . The N E T is illustrated in Fig. 2. Eq. (1), Eq. (2) can be rewritten
a N E T [16] and the points (ei,di) the
Zg, t (Ak(c) B,(e))Ukt
Zk., Ak(e) Bt (0) In the above equation, only those terms in which both Ak(e) and Bt(d) are non-zero will be non-zero. Under the above condition of the membership functions, at any time, at most two neighborhood membership functions have non-zero degrees for e or d. Therefore, at most only 4 terms in the above equation are left. That is to say, at most only 4 rules are fired one time. For instance, if the inputs of the fuzzy controller are located in the N E T lattice area S = [el,e~+t] x [dj, Oj+l] of the e - d plane, by definition, we have Ak(e) = 0 (k # (i, i + l) ~ I), and Bt(O) = O(t # ( j , j + 1) e J). So the above equation become H -~-
~a--'k=(i.i+ 1) (Ak(c)Bt(e))Ukt t=(j.j+l) Z k = ( i , i + l ) (Ak(e)Bt(d)) t-(i.j+
l)
Obviously, by definition of the membership function, A i ( e ) + A i + l ( e ) = l ( e ~ [ e i , e i + l ] ) , and Bj(O) + B j+ 1(0)= 1(0 ~ [dj, dj+ 1]). With these relations it is easy to check that the denominator of the right-hand side of the above equation is
E
k=(i,i+
t=(j,/+
Ak(e)B,(d) = Ai(e)B~(d) + Ai+ l(e)Bj(d) + Ai(e)Bj+ l(d) + Ai+ l (e)Bj+ l(d) I) 1)
= (Ai(e) + Ai+ l(e))(e~(d) + Bj+ l(d)) =1.
W . Z Qiao, M. Mizumoto / F u z z y Sets and Systems 78 (1996) 2 3 - 3 5
26
Therefore, the output of the fuzzy controller can be simplified as
u =
~
(Ak(e) B,(d))ukt
k=(i,i + l) t = ( j , j + l)
= Ai(e)Bj(d)uij + Ai+ l(e)Bj(~)u,+ l~j + Ai(e)Bj+ l(d)uiu+ 1~ + Ai+ l(e)Bj+ l(~)u,+ 1)u+ 1~ -
----I[__---_lUij+ \ei+l -- e i / k e j + l -- e j / (ei+.._._.__l--e'~[/ ~ - - d j
. . . . . . . ~jj \ei+l -- ei,/\ej+l "~
+ ke,+ , - e-J~,)j+-~l----~jJ u " j + '' +
u,+l)j
(es-_e_i - ")(_ d_--~j ) el+,-ei/kdj+7-~-~;
u"+ ' ) ° + "
(3)
for e ¢ [ e i , e i + 1 ], and for d ¢ [dj, ~j+ 1]. At an arbitrary point on the e - ~ plane, as can be seen from Eqs. (3), the output of the fuzzy controller is a non-linear function of the arguments of e and d. There is no known analytical solutions available to deal with such kind of nonlinearity. A complete treatment of this non-linearity is impossible. Nevertheless, we can adopt the linearization method and carry on an analysis of small deviations from nominal just like the conventional or modern control theory usually do. We denote the input-output relation of the fuzzy controller, that is (3), in the following non-linear model: u = f ( e , d, t).
(4)
When e = ei and d = dj, that is at the NODE(el, d~) of the N E T , it follows from Eq. (3) that the output of the product-sum crisp type fuzzy controller is u = u u.
(5)
Therefore, we can take the result of Eq. (5) at the NODE(el, kj) of the e - ~ plane, as the nominal solution of (4), that is
u = f ( e i , ~, t) = uij. So we can conduct a linearization analysis in a neighborhood of the N O D E of the e - k plane. The difference between these nominal values and some slightly perturbed functions e, d and u can be defined by
~e = e
- - el,
8d = d -- dj, 8 U = bl - - l l i j .
For sufficiently small Be, 8~ and 8u perturbations, Eq. (4) can be approximated by the following linear equations: 8u=
~e
~e+ n
~-~
8~
(6)
n
A neighborhood of each N O D E (or nominal point) will be divided into 4 different quadrants by the two N E T lines that cross at the NODE. For simplicity, we only consider the case of the first quadrant where 5e/> 0 and 8~ ~> 0, that is to say, (el + ~e,~j + ~d)e [el, el+l] x [~j, dj+l]. From (3), we have
(e~,dj)
ei+ 1 - - ei
W.Z. Qiao, M. Mizumoto / F u z ~ Sets and Systems 78 (1996) 2 3 - 3 5
IOf l
27
__ Ri(j+ 1) - - Uij (e,.d,)
ej+ 1 -- ej
'
then, 8u=
~e
Be+
~
80-
n
n
Ui(j+
-U~JSe+ ei
+1
--
1)
Uij ~O,
--
e j+ 1 -- ej
ei
that is l,l
--
Uij
--
R ( i + 1 ) j - - Uij
(e
--
el+ 1 -- ei
ei ) + u i ( j + 1) - - :Igij ,~ e. el+ 1 -- ej
--
Oj).
Therefore b / ( i + 1 ) j - - IAij
I
I,I =
Uij
ei+ l -- ei
e~
U i ( j + 1) - - blij
7---.--
e j + 1 -- e j
-1
|
[A(I+ 1 ) j - - Uij
J
el+ 1 - - e i
Oj +
e+
U i ( j + 1) ej+l
--
-u'~o
-- ej
(7)
= A + P e + DO,
where A
=
Rij
U(i+ l)j
- - Uij
el+ 1 - - ei
ei
Ui(j+ 1) - - Uij . : ej = u o ej + 1 - - ej
Pei
-
-
De j,
p _ U ( i + l ) j -- ~ij ei + 1 - - ei D -
bli(J+ 1) - - Uij
Oj+ ~ -- Oj As we can see that the product-sum crisp type fuzzy controller behaves approximately like a P D controller in the neighborhood of the N O D E point of the N E T of e - 0 plane as illustrated by Eq. (7), where the equivalent proportional and derivative control components are P and D respectively. Actually, we can regard such kind of fuzzy controller as a parameter time-varying P D controller. When the error and the change rate of error changes along the e - 0 plane, from one neighborhood of a N O D E to another, the P D parameters switch from one set to another. We will call this type of fuzzy controller as a P D type fuzzy controller (PDfc). The result of (7) enables us to predict the behaviors of the fuzzy controller according to the conventional PID control theory. Recall that the performance of a conventional PID controller is determined by its proportional parameter Kp, integral parameter K~ and derivative parameter KD. The proportional control law can guarantee the fast response of the control system, the integral control law can eliminate the steady-state error of the control system, and the derivative control law can increase the damping of the system thus reduce the overshoot and oscillating times of the system response. Thus a PID controller when designed properly could yield a system with fast rise time and small overshoot and non-steady-state error. A P or P D controller will yield a steady-state error for the system step response if the controlled plant is a type 0 system. The steady-state error is inversely proportional to Kp, if Kp is too large, the stability of the system may be adversely affected. Since the PDfc behaves approximately like a parameters time-varying P D controller, definitely it will yield a steady-state error when used to control a type 0 plant (see Fig. 7). Just like a conventional P D controller,
28
~Z
Qiao, M. Mizumoto / Fuzz. Sets and Systems 78 (1996) 23-35
the control performance cannot be satisfied. However, we can incorporate the integral control law into the fuzzy controller to improve the performance of the fuzzy controller. In this direction, we will present some methods to overcome the shortcomings of PDfc in the following sections.
3. PID type fuzzy controller structure As illustrated in previous sections, the PDfc approximately behaves like a parameter time-varying P D controller. Since the mathematical models of most industrial process systems are of type 0, obviously there would exist an steady-state error if they are controlled by this kind of fuzzy controller. This characteristic has been stated in the brief review of the PID controller in the previous section. If we want to eliminate the steady-state error of the control system, we can imagine to substitute the input ~ (the change rate of error or the derivative of error) of the fuzzy controller with the integration of error. This will result the fuzzy controller behaving like a parameter time-varying PI controller, thus the steady-state error is expelled by the integration action. However, a PI type fuzzy controller will have a slow rise time if the P parameters are chosen small, and have a large overshoot if the P or I parameters are chosen large. So there may be the time when one wants to introduce not only the integration control but the derivative control to the fuzzy control system, because the derivative control can reduce the overshoot of the system's response so as to improve the control performance. Of course this can be realized by designing a fuzzy controller with three inputs, error, the change rate of error and the integration of error. However, these methods will be hard to implement in practice because of the difficulty in constructing fuzzy control rules. Usually fuzzy control rules are constructed by summarizing the manual control experience of an operator who has been controlling the industrial process skillfully and successfully. The operator intuitively regulates the executor to control the process by watching the error and the change rate of the error between the system's output and the set-point value. It is not the practice for the operator to observe the integration of error. Moreover, adding one input variable will greatly increase the number of control rules, the constructing of fuzzy control rules are even more difficult task and it needs more computation efforts. Hence we may want to design a fuzzy controller that possesses the fine characteristics of the PID controller by using only the error and the change rate of error as its inputs. One way is to have an integrator serially connected to the output of the fuzzy controller as shown in Fig. 3. In Fig. 3, K1 and K 2 a r e scaling factors for e and ~ respectively, and fl is the integral constant. In the proceeding text, for convenience, we did not consider the scaling factors. Here in Fig. 4, when we look at the neighborhood of point in the e - ~ plane, it follows from (7) that the control input to the plant can be approximated by
NODE
u~=flfudt:j3 f(A+PKle+DK2~)dt= flAt+ flK2De+flKiP fedt. A-2
-1 B_~
-1 Fig. 3. The PI type fuzzy control system.
A-l
-0.4 B-1
-0.4
A0
0 Bo
0
(8) A1
0.4 B1
0.4
A2
1 B2
1
Fig. 4. The membership functions of Ai and By for simulation.
W.Z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23 35
29
Table 1 The fuzzy control rules e;-2 e-e e 2 eo el ee
I --0.7 -0.5 -0.3 0
e- i --0.7 --0.4 -0.2 0 0.3
00 -0.5 --0.2 0 0.2 0.5
~;l -0.3 0 0.2 0.4 0.7
0_, 0 0.3 0.5 0.7 1
Hence the fuzzy controller becomes a parameter time-varying PI controller, its equivalent proportional control and integral control components are BK2D and ilK1 P respectively. We call this fuzzy controller as the PI type fuzzy controller (PIfc). We can hope that in a PI type fuzzy control system, the steady-state error becomes zero. To verify the property of the PI type fuzzy controller, we carry out some simulation experiments. Before presenting the simulation, we give a description of the simulation model. In the fuzzy control system shown in Fig. 3, the plant model is a second-order and type 0 system with the following transfer function:
Gts~ =
K ( T 1 S + I)(TzS + 1)'
where K = 16, T1 = 1, and T2 = 0.5. In our simulation experiments, we use the discrete simulation method, the results would be slightly different from that of a continuous system, The sampling time of the system is set to be 0.1 s. For the fuzzy controller, the fuzzy subsets of e and d are defined as shown in Fig. 4. Their cores are ~ei} = {e 2 , e - , , e o , e l , e 2 } = { -
1 , - 0.4,0,0.4,1};
{£;j} = I t e _ 2 , e
1, - 0.4,0,0.4, 1};
1,60,el,02}
= { --
The fuzzy control rules are represented as Table 1. Fig. 5 demonstrates the simulation result of step response of the fuzzy control system with a Plfc. We can see that the steady-state error of the control system becomes zero, but when the integration factor fl is small, the system's response is slow, and when it is too large, there is a high overshoot and serious oscillation. Therefore, we may want to introduce the derivative control law into the fuzzy controller to overcome the overshoot and instability. We propose a controller structure that simply connects the P D type and the PI type fuzzy controller together in parallel. We have the equivalent structure of that by connecting a PI device with the basic fuzzy controller serially as shown in Fig. 6. Where ~ is the weight on P D type fuzzy controller and fi is that on PI type fuzzy controller, the larger a/fi means more emphasis on the derivative control and less emphasis on the integration control, and vice versa. It follows from (7) that the output of the fuzzy controller is IIc
= ~u + fi f u d t = .~(A + P K l e + DKzO) + fl f ( A + P K le + D K 2 d ) d t = aA + fiAt + ( g K I P + flKzD) e + f l K 1 P f e d t 2
+ gK2DO.
(9)
W..Z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23-35
30
1.8
I
I
1.6
I
I
I
I
I
Pl type: K1-0.12, K2-1.3,13 =0,13, T=0.1
o
PI type: K1-0.12, K2=1.3,l !=0.31, T.0.1 PI type: K1,,0.12, K2=1.3,13=I, T.0.1
• O
1.4 1.2 1 0.8 0.6 0.4 0.2
0
0.5
1
1.5
2
2.5
3
3.5
Fig. 5. The step response of PI type fuzzycontrol system.
Fig. 6. The PID type fuzzycontrol system.
Thus the fuzzy controller behaves like a time-varying PID controller, its equivalent proportional control, integral control and derivative control components are ~Kt P + flK2D, flK1P and ~K2D respectively. We call this new controller structure a PID type fuzzy controller (PIDfc). Figs. 7 and 8 are the simulation results of the system's step response of such control system. The influence of ~ and fl to the system performance is illustrated. When ~ > 0 and/3 = 0, meaning that the fuzzy controller behaves like PDfc, there exist a steady-state error (see Fig. 7). When ~ = 0 and fl > 0, meaning that the fuzzy controller behaves like a Plfc, the steady-state error of the system is eliminated but there is a large overshoot and serious oscillation (see Fig. 8). When ~ > 0 and 13 > 0 the fuzzy controller becomes a PIDfc, the overshoot is substantially reduced (see both Figs. 7 and 8). It is possible to get a comparatively good performance by carefully choosing the value of ~ and ft.
4. The parameter adaptive method The PIDfc structure proposed in the previous section has substantially improved the performance of the crisp type fuzzy controller. Also we find that the integration component of the PIDfc has an important role
31
W.Z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23-35 I
1.8
I
I
1.6
|
I
PD
type:Kl=l,
I
K2=l,O.=l,~=0,
1
T-0.I
PlO type: K1=1, K2=l,a =1, ~=0.1, T.0.1
o
+
1.4
1.2 1 0.8 0.6 0.4 0.2 0
0
0.5
I
I
1
I
I
I
1
1.5
2
2.5
3
3.5
Fig. 7. Comparison of PD and PID type fuzzy control system.
1.8
,
1.6
/
,
,
i
,
,
PI type: K1=0.1, K2=l,r~=O, [~=1, T=0.1 PID type: K1=0.1, K2=1,o.=I, p,,1, T=0.1
~
,
o 4.
1.4
0.8
0.6 0.4
0.2
0
I
I
I
I
I
I
I
0.5
1
1.5
2
2,5
3
3.5
Fig. 8. Comparison of PI and PID type fuzzy control system. on the performance of the fuzzy control system. If the integration c o m p o n e n t is too weak, the response is slow, and if the integration c o m p o n e n t is too strong, the system will become unstable. So it is still desirable to m a k e further i m p r o v e m e n t on it. We can imagine that let the equivalent integration c o m p o n e n t of the fuzzy controller vary with time. At the early stage of response, we let it take a larger value, and
W.Z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23-35
32
I
I
I
I
I
I
l
I
I
I
f
1.6
1
0,6
0
o
tl
t2
t
Fig. 9. Different phases of the step response of a control system.
reduce it gradually with time so as to increase the damping of the system and make the system more stable. By this way, we can hope to have a fast rise and a short settling time for the system's response. Fig. 9 shows the step response of a control system. The response process can be divided into different phases by the peak value times. F r o m the start time 0 to the time tl when the first peak value occurs, the error of the system cover the whole universe of discourse. F r o m time tl on, the error of the system's response will no longer go beyond the belt area of interval [ - 61,61 ], where 61 is the absolute peak value at time t l. And at t2, another peak value occurs. F r o m t2 on, the error of the system's response will never go beyond the belt area of interval [ - 62,32]. And so on. We can consider to decrease the integral control component at each peak value time according to the absolute value of each peak. Let us examine the equivalent proportional control, integral control and derivative control components of a PIDfc from (9). These equivalent control components are repeated as follows: proportional:
aK1P +/3K2 D,
integral:
/3K 1P,
derivative;
~K2D.
As can be seen that if we decrease the parameter/3 gradually, the integral control component is decreased so that the damping of the system is increased and the system is more stable. Notice that the proportional component includes the term of the production of/3 and Kz. While decreasing the value of/3 will decrease the proportional control component; thus the reaction of the control system against the error will be slowed down. If in the meanwhile of decreasing/3 we increase Kz in the same rate as/3 is decreased, the equivalent proportional control strength will remain unchanged and the system can always keep quick reaction against the error. Also we can see that when K2 is increased, the equivalent derivative control component will be increased at the same time, this would do no harm to the system's performance, because derivative control law can increase the resistance against the overshoot and oscillation of the system.
33
V~Z. Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23-35
Fig. 10. Block diagram of the parameter adaptive fuzzy controller.
1.8
i
i
t
i'
i
i
i
Kt,,2, K2,,0.5,a ,,1, ]3,,0.1, T,,0.1 Non-adaptive
1.6
o
1.4
1.2 1 0.8 0,6 0.4 0.2
0 0
I
I
I
I
I
I
I
0.5
1
1.5
2
2.5
3
3.5
Fig. 11. Comparison of fuzzy control system with and without adaptive mechanism (I).
Motivated by this idea, we design a parameter adaptive PID type fuzzy controller (PAPIDfc). The parameter adaptive fuzzy controller is composed of a PIDfc, a peak observer and a parameter regulator. Fig. 10 is the block diagram of the PAPIDfc. The basic PIDfc is as described in the previous section. The peak observer keeps watching on the system's output and transmits a signal at each peak time and measures the absolute peak value. The parameter regulator tunes the controllers parameters Ka and/~ simultaneously at each peak time signal and according to the peak value at that time. The algorithm of tuning the scaling constants and the integral gain is as follows: Kzs
w h e r e K2s and/~s are the initial values of Kz and/~ respectively. 6k is the absolute peak value at the peak time
tk(k = 1,2,3 .... ). Figs. 11 and 12 present some simulation results of the fuzzy control system with a PAPIDfc under different initial values of the scaling parameters and output gains. In Fig. 11 when the adaptive mechanism does not
W.Z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23-35
34 |,8
i
i
)
I
I
K1-1, K2-1,a-1,
1.6
I
I
~-0.1, T-0.1
Non-adaptive Adaptive
o ÷
1.4 1.2 I 0.8 0.6 0.4 0.2 0 0
0.5
1
I 1.5
2
2.5
I 3
3.5
Fig. 12. Comparison of fuzzy control system with and without adaptive mechanism (II).
come into action, the system oscillates seriously, and when the adaptive mechanism comes into action, the oscillations are strongly resisted. In Fig. 12, the system without adaptive has a slight oscillation, and the parameter adaptive control yield a non-oscillating system. Generally speaking, the simulation results demonstrate that the PAPIDfc substantially improves the performance of the control system. It can greatly reduce the oscillating times and shorten the settling time of the system. So in practice it is possible to choose a large initial value for fl to let a fast rising of response but not result in instability.
5. Conclusions
We have studied the input-output behavior of the product-sum crisp type fuzzy controller, revealing that this type of fuzzy controller behaves approximately like a parameter time-varying PD controller. Therefore, the analysis and designing of a fuzzy control system can take advantage of the conventional PID control theory. According to the coventional PID control theory, we have been able to propose some improvement methods for the crisp type fuzzy controller. It has been illustrated that the PD type fuzzy controller yields a steady-state error for the type 0 system, the PI type fuzzy controller can eliminate the steady-state error. We proposed a controller structure that combine the features of both PD type and PI type fuzzy controller, obtaining a PID type fuzzy controller which allows the control system to have a fast rise and a small overshoot as well as a short settling time. To improve further the performance of the proposed PID type fuzzy controller, the authors designed a parameter adaptive fuzzy controller. The PID type fuzzy controller can be decomposed into the equivalent proportional control, integral control and the derivative control components. The proposed parameter adaptive fuzzy controller decreases the equivalent integral control component of the fuzzy controller gradually with the system response process time, so as to increase the damping of the system when the system
w . z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23-35
35
is a b o u t to settle down, m e a n w h i l e keeps the p r o p o r t i o n a l c o n t r o l c o m p o n e n t u n c h a n g e d so as to g u a r a n t e e quick reaction against the system's error. W i t h the p a r a m e t e r adaptive fuzzy controller, the oscillation of the system is strongly restrained a n d the settling time is shortened considerably. We have presented the s i m u l a t i o n results to d e m o n s t r a t e the fine performance of the proposed P I D type fuzzy c o n t r o l l e r a n d the p a r a m e t e r adaptive fuzzy controller structure.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]
J.J. Buckley, Fuzzy controllers: further limit theorems for linear control rules, Fuzzy Sets and Systems 36 (1990) 225-233. J.J. Buckley, Sugeno type controllers are universal controllers, Fuzzy Sets and Systems 53 (1993) 299-303. B.P. Graham and R.B. Newell, Fuzzy adaptive control of a first order process, Fuzzy Sets and Systems 31 (1989) 47 65. B. Kosko, Fuzzy systems as universal approximators, Proc. IEEE lnternat. Conf. on Fuzzy Systems, San Diego (1992) 1153-1162. E.H. Mamdani and S. Assilian, An experiment in linguistic synthesis with a fuzzy logic controller, Internat. J. Man-Machine Studies 7 (1975) 1 13. M. Mizumoto, Min-max-gravity method versus product- sum-gravity method for fuzzy controls, Proe. IV IFSA Congress, Brussels, Part E (1991) 127-130. M. Mizumoto, Realization of PID controls by fuzzy control methods, Proc. IEEE lnternat. Conf. on Fuzz), Systems, San Diego (1992) 709 715. M. Mizumoto and H.J. Zimmermann, Comparison of fuzzy reasoning methods, Fuzzy Sets and Systems 8(3) (1982) 253 283. A. Ollero and A.J. Garcia-Cerezo, Direct digital control, auto-turning and supervision using fuzzy logic. Fuzzy Sets and Systems 30 (1989) 135 153. T.J. Procyk and E.H. Mamdani, A linguistic self-organizingprocess controller, Automatica 15 (1979) 15-30. Wu Zhi Qiao et al., A rules self regulating fuzzy controller, Fuzzy Sets and Systems 47(1) (1992) 13-21. S. Shao, Fuzzy self-organizingcontroller and its application for dynamic processes, Fuzzy Sets and Systems 26 (1988) 151-164. T. Takagi and M. Sugeno, Derivation of fuzzy control rules from human operator's control actions, Proc. of the IFAC Conll on Fuzzy InJbrmation, Vol. 1 Marseille, France (1983) 55 60. T. Takagi and M. Sugeno, Fuzzy identification of fuzzy systems and its application to modelling and control, IEEE Trans. Systems Man Cybernet. 15 (1985) 116 132. Hu Jia Yao, Wu Zhi Qiao and Soon Shuo Shang, Static analysis of fuzzy controllers, J. of Beijing Light Industry College (in Chinese) (2) (1988) 12-18. Wang Pei Zhuang, Zhang Hong Min and Xu Wei, Pad-analysis of stability of fuzzy control systems, Fuzzy Sets and Systems 38 (1990) 27--42.
sets and systems ELSEVIER
Fuzzy Sets and Systems 78 (1996) 23-35
PID type fuzzy controller and parameters adaptive method Wu Zhi Qiao*, Masaharu Mizumoto Department of Management Engineering, Osaka Electro-Communication Universi~, Neyagawa, Osaka 572. Japan
Received October 1994; revised January 1995
Abstract The authors of this paper try to analyze the dynamic behavior of the product-sum crisp type fuzzy controller, revealing that this type of fuzzy controller behaves approximately like a PD controller that may yield steady-state error for the control system. By relating to the conventional PID control theory, we propose a new fuzzy controller structure, namely PID type fuzzy controller which retains the characteristics similar to the conventional PID controller. In order to improve further the performance of the fuzzy controller, we work out a method to tune the parameters of the PID type fuzzy controller on line, producing a parameter adaptive fuzzy controller. Simulation experiments are made to demonstrate the fine performance of these novel fuzzy controller structures. Keywords: Fuzzy controller; PID control; Adaptive control
1. Introduction Among various inference methods used in the fuzzy controller found in literatures [5-8, 13, 14], the most widely used ones in practice are the M a m d a n i method proposed by M a m d a n i and his associates [5] who adopted the Min-max compositional rule of inference based on an interpretation of a control rule as a conjunction of the antecedent and consequent, and the product-sum method proposed by Mizumoto [6, 7] who suggested to introduce the product and arithmetic mean aggregation operators to replace the logical A N D (minimum) and O R (maximum) calculations in the Min-max compositional rule of inference. In the algorithm of a fuzzy controller, the defuzzyfication calculation is also a complicated and time consuming task. Tagagi and Sugeno proposed a crisp type model in which the consequent parts of the fuzzy control rules are crisp functional representation or crisp real numbers in the simplified case instead of fuzzy sets [13, 14]. With this model of crisp real number output, the fuzzy set of the inference consequence will be a discrete fuzzy set with a finite number of points, this can greatly simplify the defuzzification algorithm. Both the Min-max method and the product-sum method are often applied with the crisp output model in a mixed manner. Especially the mixed product-sum crisp model has a fine performance and the simplest algorithm that is very
* Corresponding author. Presently at the Department of ECS, University of Southampton. UK. E-mail: [email protected]. 0165-0114/96/$15.00 © 1996 - Elsevier Science B.V. All rights reserved SSDI 0165-01 14(95)001 15-8
VI(Z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23 35
24
easy to be implemented in hardware system and converted into a fuzzy neural network model. In this paper, we will take account of the product-sum crisp type fuzzy controller. There has not been sounded theoretical method available to analyze a fuzzy controller in literature, while the conventional control theory is highly developed. It is natural for the researchers to apply the conventional theory, mostly linear system theory to solve the nonlinear problem of fuzzy controller and many works have been done in this direction [1,2,4, 16]. In this paper, the authors will make effort to analyze the behavior of the product-sum crisp type fuzzy controller, by relating the fuzzy controller to the conventional PID controller. Having the aid of the well-known classical designing method of PID controller, we propose a new fuzzy controller structures, namely P I D type fuzzy controller which retains the characteristics similar to the conventional PID controller. In order to improve further the performance of the fuzzy controller, we work out a method to tune the parameters of the PID type fuzzy controller on line, producing a parameter adaptive fuzzy controller. The proposed methods promise to improve the performance of the fuzzy controller considerably.
2. Analysis of a product-sum crisp type fuzzy controller Before we conduct the analysis, we briefly describe the crisp type fuzzy controller model mixed with product-sum inference method as follows [6, 13, 14]. Suppose that the fuzzy controller in consideration is a two-input and one-output one. The two inputs to the fuzzy controller are error e and change rate of error ~, and the output of the fuzzy controller (that is the input to the controlled process) is u. The universes of discourses of e, 0 and u are E ~ R,/~ c R and U c R, respectively. We denote the linguistic values of e and ~ as A ~(i ~ I = [ - m .... , - 2, - 1, 0, 1, 2 ..... m]) and Bj (j ~ J = [ - n . . . . . - 2, - 1, 0, 1, 2 .... , n]) respectively. The fuzzy control rules are given as in the form of
if e is Ai and ~ is Bj then u is u d. where uij E U(i e I,j ~ J) is a crisp value instead of a fuzzy subset. The uijs are not necessarily different from each other. The fuzzy controller with such kind of control rules is called crisp type fuzzy controller [13, 14]. If the number of control rules are equal to I x J, the fuzzy control rule base is said to be complete [16]. In the following discussion, we assume that the fuzzy control rule base is complete. Suppose that the membership functions of Ai and B~ are Ai(e) and Bj(d). In a certain control time t, we have the observation values e and ~ for error and the change rate of error respectively, then the truth values of Ai and Bj are A~(e) and Bj(k), (i ~ I, j ~ J). Using the product-sum inference method, the truth value of the antecedent part of a fuzzy control rule will be
f j = Ai(e)Bj(k)
(ieI, jeJ).
(1)
The reasoning from the antecedent part to the consequent part will generate a conclusion fuzzy subset which we denote as C. C will be a discrete fuzzy subset with finite number of points, C = { f j / u u l i ~ I , j ~ J }. applying the center of gravity method to defuzzify the fuzzy set C, the real output of the controller u is given by ~i, j fij Uij
u = - Z i, j f/j
,
(2)
In our study we will employ the triangular membership functions for each fuzzy linguistic value of the error e and the change rate of error 0 as shown in Fig. 1. We denote the cores of fuzzy set Ai as el and those of Bj as
W.Z. Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23 35 Ai-1
Ai
25
di-1 di
Ai+l
e/+l
I I !
ei-1
ei
ei+l
Bj-1
Bj
Bj+I
el-1 uij
ei ei+l
ei-1
dj
di+l
1
Fig. 1. The membership functions of A~and Bj.
Fig. 2. T h e
ej.
We design the membership functions as so, the support sets of Bj are equal to [dj_ 1, ej+ 1], as illustrated in Fig. 1. We have Ai(e) = 1
e
-
-
- -
ei + 1 - - - ,
ei
-
ei+ 1 --c i
ei+ 1
-
e
--e i
Ai+l(e)-
e
-
-
ei
A i are
NET
o n t h e e - ~; plane•
equal to [e~_ 1, ei+ 1] and those of
Ak(e) = 0 (k # (i,i + 1)c I),
ei+ 1 - - e i
for e e [ei,ei+t], and Bj(e)
=
d -- ej - dj + I - - e j
l
for ,; ~ [~j,dj+ On the e N O D E of the Combining
ej+l -- d _ _ , dj + 1 - - e j
d -- dj Bj+I(e
) -
Bt(O) = 0 (t # ( j , j + 1) e J),
e j+ 1 -- dj
1]. 0 plane, we call the set {e,d[e = el, d = ej, i ~ I , j e J } N E T . The N E T is illustrated in Fig. 2. Eq. (1), Eq. (2) can be rewritten
a N E T [16] and the points (ei,di) the
Zg, t (Ak(c) B,(e))Ukt
Zk., Ak(e) Bt (0) In the above equation, only those terms in which both Ak(e) and Bt(d) are non-zero will be non-zero. Under the above condition of the membership functions, at any time, at most two neighborhood membership functions have non-zero degrees for e or d. Therefore, at most only 4 terms in the above equation are left. That is to say, at most only 4 rules are fired one time. For instance, if the inputs of the fuzzy controller are located in the N E T lattice area S = [el,e~+t] x [dj, Oj+l] of the e - d plane, by definition, we have Ak(e) = 0 (k # (i, i + l) ~ I), and Bt(O) = O(t # ( j , j + 1) e J). So the above equation become H -~-
~a--'k=(i.i+ 1) (Ak(c)Bt(e))Ukt t=(j.j+l) Z k = ( i , i + l ) (Ak(e)Bt(d)) t-(i.j+
l)
Obviously, by definition of the membership function, A i ( e ) + A i + l ( e ) = l ( e ~ [ e i , e i + l ] ) , and Bj(O) + B j+ 1(0)= 1(0 ~ [dj, dj+ 1]). With these relations it is easy to check that the denominator of the right-hand side of the above equation is
E
k=(i,i+
t=(j,/+
Ak(e)B,(d) = Ai(e)B~(d) + Ai+ l(e)Bj(d) + Ai(e)Bj+ l(d) + Ai+ l (e)Bj+ l(d) I) 1)
= (Ai(e) + Ai+ l(e))(e~(d) + Bj+ l(d)) =1.
W . Z Qiao, M. Mizumoto / F u z z y Sets and Systems 78 (1996) 2 3 - 3 5
26
Therefore, the output of the fuzzy controller can be simplified as
u =
~
(Ak(e) B,(d))ukt
k=(i,i + l) t = ( j , j + l)
= Ai(e)Bj(d)uij + Ai+ l(e)Bj(~)u,+ l~j + Ai(e)Bj+ l(d)uiu+ 1~ + Ai+ l(e)Bj+ l(~)u,+ 1)u+ 1~ -
----I[__---_lUij+ \ei+l -- e i / k e j + l -- e j / (ei+.._._.__l--e'~[/ ~ - - d j
. . . . . . . ~jj \ei+l -- ei,/\ej+l "~
+ ke,+ , - e-J~,)j+-~l----~jJ u " j + '' +
u,+l)j
(es-_e_i - ")(_ d_--~j ) el+,-ei/kdj+7-~-~;
u"+ ' ) ° + "
(3)
for e ¢ [ e i , e i + 1 ], and for d ¢ [dj, ~j+ 1]. At an arbitrary point on the e - ~ plane, as can be seen from Eqs. (3), the output of the fuzzy controller is a non-linear function of the arguments of e and d. There is no known analytical solutions available to deal with such kind of nonlinearity. A complete treatment of this non-linearity is impossible. Nevertheless, we can adopt the linearization method and carry on an analysis of small deviations from nominal just like the conventional or modern control theory usually do. We denote the input-output relation of the fuzzy controller, that is (3), in the following non-linear model: u = f ( e , d, t).
(4)
When e = ei and d = dj, that is at the NODE(el, d~) of the N E T , it follows from Eq. (3) that the output of the product-sum crisp type fuzzy controller is u = u u.
(5)
Therefore, we can take the result of Eq. (5) at the NODE(el, kj) of the e - ~ plane, as the nominal solution of (4), that is
u = f ( e i , ~, t) = uij. So we can conduct a linearization analysis in a neighborhood of the N O D E of the e - k plane. The difference between these nominal values and some slightly perturbed functions e, d and u can be defined by
~e = e
- - el,
8d = d -- dj, 8 U = bl - - l l i j .
For sufficiently small Be, 8~ and 8u perturbations, Eq. (4) can be approximated by the following linear equations: 8u=
~e
~e+ n
~-~
8~
(6)
n
A neighborhood of each N O D E (or nominal point) will be divided into 4 different quadrants by the two N E T lines that cross at the NODE. For simplicity, we only consider the case of the first quadrant where 5e/> 0 and 8~ ~> 0, that is to say, (el + ~e,~j + ~d)e [el, el+l] x [~j, dj+l]. From (3), we have
(e~,dj)
ei+ 1 - - ei
W.Z. Qiao, M. Mizumoto / F u z ~ Sets and Systems 78 (1996) 2 3 - 3 5
IOf l
27
__ Ri(j+ 1) - - Uij (e,.d,)
ej+ 1 -- ej
'
then, 8u=
~e
Be+
~
80-
n
n
Ui(j+
-U~JSe+ ei
+1
--
1)
Uij ~O,
--
e j+ 1 -- ej
ei
that is l,l
--
Uij
--
R ( i + 1 ) j - - Uij
(e
--
el+ 1 -- ei
ei ) + u i ( j + 1) - - :Igij ,~ e. el+ 1 -- ej
--
Oj).
Therefore b / ( i + 1 ) j - - IAij
I
I,I =
Uij
ei+ l -- ei
e~
U i ( j + 1) - - blij
7---.--
e j + 1 -- e j
-1
|
[A(I+ 1 ) j - - Uij
J
el+ 1 - - e i
Oj +
e+
U i ( j + 1) ej+l
--
-u'~o
-- ej
(7)
= A + P e + DO,
where A
=
Rij
U(i+ l)j
- - Uij
el+ 1 - - ei
ei
Ui(j+ 1) - - Uij . : ej = u o ej + 1 - - ej
Pei
-
-
De j,
p _ U ( i + l ) j -- ~ij ei + 1 - - ei D -
bli(J+ 1) - - Uij
Oj+ ~ -- Oj As we can see that the product-sum crisp type fuzzy controller behaves approximately like a P D controller in the neighborhood of the N O D E point of the N E T of e - 0 plane as illustrated by Eq. (7), where the equivalent proportional and derivative control components are P and D respectively. Actually, we can regard such kind of fuzzy controller as a parameter time-varying P D controller. When the error and the change rate of error changes along the e - 0 plane, from one neighborhood of a N O D E to another, the P D parameters switch from one set to another. We will call this type of fuzzy controller as a P D type fuzzy controller (PDfc). The result of (7) enables us to predict the behaviors of the fuzzy controller according to the conventional PID control theory. Recall that the performance of a conventional PID controller is determined by its proportional parameter Kp, integral parameter K~ and derivative parameter KD. The proportional control law can guarantee the fast response of the control system, the integral control law can eliminate the steady-state error of the control system, and the derivative control law can increase the damping of the system thus reduce the overshoot and oscillating times of the system response. Thus a PID controller when designed properly could yield a system with fast rise time and small overshoot and non-steady-state error. A P or P D controller will yield a steady-state error for the system step response if the controlled plant is a type 0 system. The steady-state error is inversely proportional to Kp, if Kp is too large, the stability of the system may be adversely affected. Since the PDfc behaves approximately like a parameters time-varying P D controller, definitely it will yield a steady-state error when used to control a type 0 plant (see Fig. 7). Just like a conventional P D controller,
28
~Z
Qiao, M. Mizumoto / Fuzz. Sets and Systems 78 (1996) 23-35
the control performance cannot be satisfied. However, we can incorporate the integral control law into the fuzzy controller to improve the performance of the fuzzy controller. In this direction, we will present some methods to overcome the shortcomings of PDfc in the following sections.
3. PID type fuzzy controller structure As illustrated in previous sections, the PDfc approximately behaves like a parameter time-varying P D controller. Since the mathematical models of most industrial process systems are of type 0, obviously there would exist an steady-state error if they are controlled by this kind of fuzzy controller. This characteristic has been stated in the brief review of the PID controller in the previous section. If we want to eliminate the steady-state error of the control system, we can imagine to substitute the input ~ (the change rate of error or the derivative of error) of the fuzzy controller with the integration of error. This will result the fuzzy controller behaving like a parameter time-varying PI controller, thus the steady-state error is expelled by the integration action. However, a PI type fuzzy controller will have a slow rise time if the P parameters are chosen small, and have a large overshoot if the P or I parameters are chosen large. So there may be the time when one wants to introduce not only the integration control but the derivative control to the fuzzy control system, because the derivative control can reduce the overshoot of the system's response so as to improve the control performance. Of course this can be realized by designing a fuzzy controller with three inputs, error, the change rate of error and the integration of error. However, these methods will be hard to implement in practice because of the difficulty in constructing fuzzy control rules. Usually fuzzy control rules are constructed by summarizing the manual control experience of an operator who has been controlling the industrial process skillfully and successfully. The operator intuitively regulates the executor to control the process by watching the error and the change rate of the error between the system's output and the set-point value. It is not the practice for the operator to observe the integration of error. Moreover, adding one input variable will greatly increase the number of control rules, the constructing of fuzzy control rules are even more difficult task and it needs more computation efforts. Hence we may want to design a fuzzy controller that possesses the fine characteristics of the PID controller by using only the error and the change rate of error as its inputs. One way is to have an integrator serially connected to the output of the fuzzy controller as shown in Fig. 3. In Fig. 3, K1 and K 2 a r e scaling factors for e and ~ respectively, and fl is the integral constant. In the proceeding text, for convenience, we did not consider the scaling factors. Here in Fig. 4, when we look at the neighborhood of point in the e - ~ plane, it follows from (7) that the control input to the plant can be approximated by
NODE
u~=flfudt:j3 f(A+PKle+DK2~)dt= flAt+ flK2De+flKiP fedt. A-2
-1 B_~
-1 Fig. 3. The PI type fuzzy control system.
A-l
-0.4 B-1
-0.4
A0
0 Bo
0
(8) A1
0.4 B1
0.4
A2
1 B2
1
Fig. 4. The membership functions of Ai and By for simulation.
W.Z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23 35
29
Table 1 The fuzzy control rules e;-2 e-e e 2 eo el ee
I --0.7 -0.5 -0.3 0
e- i --0.7 --0.4 -0.2 0 0.3
00 -0.5 --0.2 0 0.2 0.5
~;l -0.3 0 0.2 0.4 0.7
0_, 0 0.3 0.5 0.7 1
Hence the fuzzy controller becomes a parameter time-varying PI controller, its equivalent proportional control and integral control components are BK2D and ilK1 P respectively. We call this fuzzy controller as the PI type fuzzy controller (PIfc). We can hope that in a PI type fuzzy control system, the steady-state error becomes zero. To verify the property of the PI type fuzzy controller, we carry out some simulation experiments. Before presenting the simulation, we give a description of the simulation model. In the fuzzy control system shown in Fig. 3, the plant model is a second-order and type 0 system with the following transfer function:
Gts~ =
K ( T 1 S + I)(TzS + 1)'
where K = 16, T1 = 1, and T2 = 0.5. In our simulation experiments, we use the discrete simulation method, the results would be slightly different from that of a continuous system, The sampling time of the system is set to be 0.1 s. For the fuzzy controller, the fuzzy subsets of e and d are defined as shown in Fig. 4. Their cores are ~ei} = {e 2 , e - , , e o , e l , e 2 } = { -
1 , - 0.4,0,0.4,1};
{£;j} = I t e _ 2 , e
1, - 0.4,0,0.4, 1};
1,60,el,02}
= { --
The fuzzy control rules are represented as Table 1. Fig. 5 demonstrates the simulation result of step response of the fuzzy control system with a Plfc. We can see that the steady-state error of the control system becomes zero, but when the integration factor fl is small, the system's response is slow, and when it is too large, there is a high overshoot and serious oscillation. Therefore, we may want to introduce the derivative control law into the fuzzy controller to overcome the overshoot and instability. We propose a controller structure that simply connects the P D type and the PI type fuzzy controller together in parallel. We have the equivalent structure of that by connecting a PI device with the basic fuzzy controller serially as shown in Fig. 6. Where ~ is the weight on P D type fuzzy controller and fi is that on PI type fuzzy controller, the larger a/fi means more emphasis on the derivative control and less emphasis on the integration control, and vice versa. It follows from (7) that the output of the fuzzy controller is IIc
= ~u + fi f u d t = .~(A + P K l e + DKzO) + fl f ( A + P K le + D K 2 d ) d t = aA + fiAt + ( g K I P + flKzD) e + f l K 1 P f e d t 2
+ gK2DO.
(9)
W..Z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23-35
30
1.8
I
I
1.6
I
I
I
I
I
Pl type: K1-0.12, K2-1.3,13 =0,13, T=0.1
o
PI type: K1-0.12, K2=1.3,l !=0.31, T.0.1 PI type: K1,,0.12, K2=1.3,13=I, T.0.1
• O
1.4 1.2 1 0.8 0.6 0.4 0.2
0
0.5
1
1.5
2
2.5
3
3.5
Fig. 5. The step response of PI type fuzzycontrol system.
Fig. 6. The PID type fuzzycontrol system.
Thus the fuzzy controller behaves like a time-varying PID controller, its equivalent proportional control, integral control and derivative control components are ~Kt P + flK2D, flK1P and ~K2D respectively. We call this new controller structure a PID type fuzzy controller (PIDfc). Figs. 7 and 8 are the simulation results of the system's step response of such control system. The influence of ~ and fl to the system performance is illustrated. When ~ > 0 and/3 = 0, meaning that the fuzzy controller behaves like PDfc, there exist a steady-state error (see Fig. 7). When ~ = 0 and fl > 0, meaning that the fuzzy controller behaves like a Plfc, the steady-state error of the system is eliminated but there is a large overshoot and serious oscillation (see Fig. 8). When ~ > 0 and 13 > 0 the fuzzy controller becomes a PIDfc, the overshoot is substantially reduced (see both Figs. 7 and 8). It is possible to get a comparatively good performance by carefully choosing the value of ~ and ft.
4. The parameter adaptive method The PIDfc structure proposed in the previous section has substantially improved the performance of the crisp type fuzzy controller. Also we find that the integration component of the PIDfc has an important role
31
W.Z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23-35 I
1.8
I
I
1.6
|
I
PD
type:Kl=l,
I
K2=l,O.=l,~=0,
1
T-0.I
PlO type: K1=1, K2=l,a =1, ~=0.1, T.0.1
o
+
1.4
1.2 1 0.8 0.6 0.4 0.2 0
0
0.5
I
I
1
I
I
I
1
1.5
2
2.5
3
3.5
Fig. 7. Comparison of PD and PID type fuzzy control system.
1.8
,
1.6
/
,
,
i
,
,
PI type: K1=0.1, K2=l,r~=O, [~=1, T=0.1 PID type: K1=0.1, K2=1,o.=I, p,,1, T=0.1
~
,
o 4.
1.4
0.8
0.6 0.4
0.2
0
I
I
I
I
I
I
I
0.5
1
1.5
2
2,5
3
3.5
Fig. 8. Comparison of PI and PID type fuzzy control system. on the performance of the fuzzy control system. If the integration c o m p o n e n t is too weak, the response is slow, and if the integration c o m p o n e n t is too strong, the system will become unstable. So it is still desirable to m a k e further i m p r o v e m e n t on it. We can imagine that let the equivalent integration c o m p o n e n t of the fuzzy controller vary with time. At the early stage of response, we let it take a larger value, and
W.Z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23-35
32
I
I
I
I
I
I
l
I
I
I
f
1.6
1
0,6
0
o
tl
t2
t
Fig. 9. Different phases of the step response of a control system.
reduce it gradually with time so as to increase the damping of the system and make the system more stable. By this way, we can hope to have a fast rise and a short settling time for the system's response. Fig. 9 shows the step response of a control system. The response process can be divided into different phases by the peak value times. F r o m the start time 0 to the time tl when the first peak value occurs, the error of the system cover the whole universe of discourse. F r o m time tl on, the error of the system's response will no longer go beyond the belt area of interval [ - 61,61 ], where 61 is the absolute peak value at time t l. And at t2, another peak value occurs. F r o m t2 on, the error of the system's response will never go beyond the belt area of interval [ - 62,32]. And so on. We can consider to decrease the integral control component at each peak value time according to the absolute value of each peak. Let us examine the equivalent proportional control, integral control and derivative control components of a PIDfc from (9). These equivalent control components are repeated as follows: proportional:
aK1P +/3K2 D,
integral:
/3K 1P,
derivative;
~K2D.
As can be seen that if we decrease the parameter/3 gradually, the integral control component is decreased so that the damping of the system is increased and the system is more stable. Notice that the proportional component includes the term of the production of/3 and Kz. While decreasing the value of/3 will decrease the proportional control component; thus the reaction of the control system against the error will be slowed down. If in the meanwhile of decreasing/3 we increase Kz in the same rate as/3 is decreased, the equivalent proportional control strength will remain unchanged and the system can always keep quick reaction against the error. Also we can see that when K2 is increased, the equivalent derivative control component will be increased at the same time, this would do no harm to the system's performance, because derivative control law can increase the resistance against the overshoot and oscillation of the system.
33
V~Z. Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23-35
Fig. 10. Block diagram of the parameter adaptive fuzzy controller.
1.8
i
i
t
i'
i
i
i
Kt,,2, K2,,0.5,a ,,1, ]3,,0.1, T,,0.1 Non-adaptive
1.6
o
1.4
1.2 1 0.8 0,6 0.4 0.2
0 0
I
I
I
I
I
I
I
0.5
1
1.5
2
2.5
3
3.5
Fig. 11. Comparison of fuzzy control system with and without adaptive mechanism (I).
Motivated by this idea, we design a parameter adaptive PID type fuzzy controller (PAPIDfc). The parameter adaptive fuzzy controller is composed of a PIDfc, a peak observer and a parameter regulator. Fig. 10 is the block diagram of the PAPIDfc. The basic PIDfc is as described in the previous section. The peak observer keeps watching on the system's output and transmits a signal at each peak time and measures the absolute peak value. The parameter regulator tunes the controllers parameters Ka and/~ simultaneously at each peak time signal and according to the peak value at that time. The algorithm of tuning the scaling constants and the integral gain is as follows: Kzs
w h e r e K2s and/~s are the initial values of Kz and/~ respectively. 6k is the absolute peak value at the peak time
tk(k = 1,2,3 .... ). Figs. 11 and 12 present some simulation results of the fuzzy control system with a PAPIDfc under different initial values of the scaling parameters and output gains. In Fig. 11 when the adaptive mechanism does not
W.Z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23-35
34 |,8
i
i
)
I
I
K1-1, K2-1,a-1,
1.6
I
I
~-0.1, T-0.1
Non-adaptive Adaptive
o ÷
1.4 1.2 I 0.8 0.6 0.4 0.2 0 0
0.5
1
I 1.5
2
2.5
I 3
3.5
Fig. 12. Comparison of fuzzy control system with and without adaptive mechanism (II).
come into action, the system oscillates seriously, and when the adaptive mechanism comes into action, the oscillations are strongly resisted. In Fig. 12, the system without adaptive has a slight oscillation, and the parameter adaptive control yield a non-oscillating system. Generally speaking, the simulation results demonstrate that the PAPIDfc substantially improves the performance of the control system. It can greatly reduce the oscillating times and shorten the settling time of the system. So in practice it is possible to choose a large initial value for fl to let a fast rising of response but not result in instability.
5. Conclusions
We have studied the input-output behavior of the product-sum crisp type fuzzy controller, revealing that this type of fuzzy controller behaves approximately like a parameter time-varying PD controller. Therefore, the analysis and designing of a fuzzy control system can take advantage of the conventional PID control theory. According to the coventional PID control theory, we have been able to propose some improvement methods for the crisp type fuzzy controller. It has been illustrated that the PD type fuzzy controller yields a steady-state error for the type 0 system, the PI type fuzzy controller can eliminate the steady-state error. We proposed a controller structure that combine the features of both PD type and PI type fuzzy controller, obtaining a PID type fuzzy controller which allows the control system to have a fast rise and a small overshoot as well as a short settling time. To improve further the performance of the proposed PID type fuzzy controller, the authors designed a parameter adaptive fuzzy controller. The PID type fuzzy controller can be decomposed into the equivalent proportional control, integral control and the derivative control components. The proposed parameter adaptive fuzzy controller decreases the equivalent integral control component of the fuzzy controller gradually with the system response process time, so as to increase the damping of the system when the system
w . z Qiao, M. Mizumoto / Fuzzy Sets and Systems 78 (1996) 23-35
35
is a b o u t to settle down, m e a n w h i l e keeps the p r o p o r t i o n a l c o n t r o l c o m p o n e n t u n c h a n g e d so as to g u a r a n t e e quick reaction against the system's error. W i t h the p a r a m e t e r adaptive fuzzy controller, the oscillation of the system is strongly restrained a n d the settling time is shortened considerably. We have presented the s i m u l a t i o n results to d e m o n s t r a t e the fine performance of the proposed P I D type fuzzy c o n t r o l l e r a n d the p a r a m e t e r adaptive fuzzy controller structure.
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