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feedforward control of fuzzy relational systems
FuzzySets and Systems35 (1990) 105-113 North-Holland
ANALYSIS AND FEEDBACK/FEEDFORWARD OF F U Z Z Y R E L A T I O N A L SYSTEMS
105
CONTROL
Chen-Wei XU Department of Automatic Control, Kunming Institute of Technology, Kunming, P.R. China Received March 1988; Revised December 1988 Abstract: Based on the analysisof a given fuzzyrelational model of an open-loop system, the
design problem of a fuzzy feedback/feedforward controller is discussed. A fuzzy feedback/feedforwardcontrollaw is proposed and some properties of the resultingclosed-loop system are investigated. The proposed fuzzy control strategy is easy to implement. Both tracking and regulationproblems can be treated by the method. Keywords: Controltheory;relations; feedback/feedforwardcontrol.
1. Introduction The development of fuzzy control theory seems to follow behind that of fuzzy control applications: we already have several successful applications of fuzzy control techniques [4, 6, 8], but we do not have a satisfactory fuzzy control theory so far. Although some people may say that a major advantage of fuzzy control (or linguistic control) is that the design of the controller requires, instead of the plant model as in regular cases, only the knowledge (experience) of the operators who run the plant, and thus precise analysis of the fuzzy control system makes no sense at all, it is argued here that precise analysis of the fuzzy system does not make sense, but approximate analysis of the system may make sense. Two reasons can be provided for our point of view. Firstly, the operators' knowledge about the control of the plant incorporates the understanding of the dynamic/static behavior of the plant, i.e., an approximate plant model is virtually built up inside his brain, and his control decision will certainly be made on the basis of this approximate model, although probably in an unconscious manner. Secondly, the development of fuzzy system identification technique make it possible to build up fuzzy models (approximate models) for dynamic systems. Furthermore, the operators' control strategy may not be optimal in most cases. It seems that the first trial toward fuzzy control system analysis was done by Kickert and Mamdani [5]. The main disadvantage of their method is that it is too precise - it requires a precise mathematical model of the plant. One may ask why we have to use fuzzy control if a mathematical model is already available. In contrast to [5], Braae and Rutherford made an attempt to analyse fuzzy control systems at the semantic level [1]. Their method may be effective in the cases where very vague results can be accepted. However it may be useless when good approximation is needed. 0165-0114/90/$3.50 ¢~ 1990, ElsevierSciencePublishersB.V. (North-Holland)
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Chen-Wei Xu
Many researchers have used fuzzy relation equations to describe fuzzy systems. The reason, to our opinion, is twofold. Firstly, fuzzy relation equations reflect the fuzziness caused by ill-definition. Secondly, relatively extensive studies have been performed on fuzzy relation equations [10, 12]. Tong investigated some properties of fuzzy relational control systems [14, 15]. Czogala and Pedrycz proposed a design procedure for fuzzy relational controllers [2], which is essentially based on solving fuzzy relation equations and thus suffers from two major difficulties: (1) the existence and the uniqueness of the solution cannot in general be guaranteed, and (2) in order to take a deterministic control action according to the fuzzy control signal, the fuzzy signal is desired to have several features (e.g. the membership curve has a unique peak; the greatest grade of the membership is above some threshold etc.). However, the proposed controller cannot guarantee this. This paper considers feedback/feedforward control problems of fuzzy relation systems. On analysing some properties of the open-loop system, a feedback/feedforward control law is presented. Under some appropriate assumptions, the (fuzzy) output and control variables of the closed-loop system will have several desirable properties. The given control law is both theoretically clear and easy to implement. The text is organized as follows. The feedback/feedforward control problem is presented in Section 2. In Section 3 some discussions are carried out and the feedback/feedforward control law is given. Section 4 is the conclusion.
2. The fuzzy feedback/feedforward control problem Consider the open-loop fuzzy system (1)
Y, = Yt-dl ° Ut-d2 ° Vt-d3 ° Rp
where Yt, ut and vt are fuzzy output, control and disturbance variables at time t respectively. Rp is a fuzzy relation and '0' stands for max-min composition. Let Y, U and V be universes of discours with Yt • F ( Y ) , ut • F(U) and vt • F ( V ) , and Rp • F ( Y x Y x U x V), where "Yt • F ( Y ) " means "Yt is a fuzzy set defined on Y", etc., and Y x Y x U x V represents the Cartesian product of Y, Y, U and V. For convenience, let all the universes appearing in this paper be discrete and have the same cardinality IYI--IUI--IVI--n, i.e., Y = {yl . . . . . yn}, U = { u l , . . . , un}, V = {vl . . . . . vn}. The generic element of Y is denoted by yi (or xi) or y (or x), and that of U, V are denoted by ui, vi (or u, v) respectively. The membership functions of y , u,, v, and Rp are denoted by Yt(Y), u,(u), v,(v) and Rp(y, x, u, v) respectively. The membership function description of the openloop system (1) is then given by Yt(Y) = V V V [y,-d,(X) ^ Ut-d2(U) ^ Vt-a3(V) ^ Rp(y, X, u, v)l, X
U
U
where v = max, ^ = min.
Vy • Y,
(2)
Feedback/feedforward control
107
Ut-d 2 - Yt
l
Vt-d 3
y~ R
P
Fig. 1. Structure of the closed-loopsystem.
Consider now a feedback/feedforward controller of the form
(3)
ut = F(y;+d~,Y,+d~_dl,Vt+d~_d~,Rp)
where F is a mapping which represents the control law to be determined, and y~ • F(Y) is the fuzzy referential input (or set point) at time instant t with membership function y',(y). Obviously a membership function description for (3) may be written as
ut(u) =f[Yt+d~(Y), Yt+d2-d,(X), Vt+d2--a,(V), Rp(y, x, u, v)l
Vu • U.
(4)
Combining the open-loop system (2) and the controller (3) results in the closed-loop system
Y, = Y,-d, o F ( y ; ,
Yt-dl, 7Jt-d3, R p ) o iJt_d3 o Rp.
(5)
The structure of the closed-loop system (5) is shown in Figure 1. For the closed-loop system (5), the feedback/feedforward control problem is stated as follows: Given f~lzzy relation Rp and time delays d~ > 0, d2 > 0 and d3 > 0, determine the feedback/feedforward control law F(-) in Eq. (3) such that for any time t and given disturbance v,, Yt will be equal to y~ in a fuzzy sense which will be given later. For the controller (3) to be realizable, it is necessary to assume that d2 ~ dl
and
de ~
(6)
3. Analysis of the open-loop system and determination of the feedback/feedforward control law Analysis of the open-loop system is necessary before the control law can be determined.
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Definition 1. System (1) is said to be ro-reachable in y = yi at time t if for any Yt-d, and Vt-d3, there exists a ut-d~ such that yt(yi) >1ro,
ro • [0, 1].
(7)
Definition 2. System (1) is said to be ro-reachable at time instant t if it is ro-reachable at time t for all yi • Y. The notion of ro-reachability is now briefly explained. If system (1) is to-reachable, then for any to-normal (see Definition 4 below) Yt-d, and Vt-d,, we can always find a fuzzy control u,_d~ such that the fuzzy output y, is transferred to a given point yi in the sense that the membership curve of y, has a unique peak at yi, a fact that might be linguistically interpreted as the statement "Yt is about yi". Thus the notion of 'ro-reachability' plays a role in fuzzy systems like 'controllability' in linear systems. Theorem 1. System (1) is ro-reachable at time t, if and only if (iff) (i) There exists x • ¥, such that
yt-dl(X) >I to,
(8a)
and there exists v • V, such that
(8b)
v,_d3(v) >I to. (ii) For all x, y • Y and v • V, there exists u • U, such that Rp(y, x, u, v) >I ro.
(9)
Proof. Suppose both (i) and (ii) hold. Then there exist x" • X, v" • V, such that yt-d,(X") >1ro and ~,-d3(V") ~ ro. Also there exists u" such that Rp(y, x", u", v") >I to. If ut-d2 is chosen in such a way that Ut-d2(U") >I ro,
(10)
then according to Eq. (2) we have y,(y)~ro.
(11)
On the other hand, if either (i) or (ii) does not hold, we obviously have for any U t -- d 2
Yt(Y) < r0 for some y • Y.
(12)
Definition 3. System (1) is said to be ro-reachable if it is to-reachable at any time instant t. Deliailion 4. Let X be a universe with generic element xi, and x • F ( X ) be a fuzzy set with membership function x(xi). (a) x is said to be ro-unimodal, iff 3! x O • X ,
x(xO)>~ro and x(xilxO)
(3! = there exists one and only one).
(13a)
Feedback /feedforward control
109
(b) x is further said to be ro-normal iff (i) x is ro-unimodal, and (ii) x is convex, i.e. for all i, j, k, xi <~xj <<-xk,
x(xj) >~min[x(xi), x(xk)].
(13b)
Note that the notion of normality suggested here is slightly different from the one widely used (see e.g. [3]). Corollary 1. If Yt-d, and vt-d3 are ro-unimodal, then system (1) /s ro-reachable iff Eq. (9) holds. So when sometimes we say "Rp is ro-reachable", we really mean that Eq. (9) holds. Proof. Obvious according to Theorem 1.
[]
Deliailion 5. The projection of Rp on Y x Y x V is denoted by R, i.e. R = proj[Rp; Y x Y x V] ~ F ( Y × Y x V).
(14)
The membership function of R is denoted by R(yi, xj, vk) or simply rl/k, i, j, k ~ N = {1 . . . . . n}. Equation (14) can then be written in terms of ri/k,
rijk = R(yi, x], vk) = max Rp(yi, xj, vk, ul) ul
= Rp(yi, x], vk, ulok ),
lok ~ N, Vi, j, k e N .
(15)
Further assume that for each (i, j, k) pair, l~/k is unique, i.e., for any i, j, k ~ N, 3! l~/k, such that
Rp(yi, xj, vk, ulj/k) > Rp(yi, xj, vk, ul)
Vl • l~/k.
(16)
Corollary 2. I f Rp is to-reachable, then Vi, j , k ~ N ,
(17)
ri/k>~ro .
Proof. Obvious according to Eqs. (9) and (16).
[]
Delhnition 6. Define a fuzzy relation Rj ~ F ( Y ) by
Rl = Yt-dz Vt_d3 0 R,
(18a)
o
or equivalently,
Rl(yi) = V V [Yt-d,(xj) A Vt-d3(vk) A R(yi, xj, ok)], xj vk
i ~ N,
(18b)
where Rl(yi) is the membership function of Rl.
Theorem 2. I f Yt-dl and t~t_d3 are ro-unimodal, and hgt(yt-d,) = max Yt-d,(X) = y~-d,(xp), X
hgt(v,-d3) = max Vt_d3(V) = V,_d3(Vq), 0
Chen-Wei Xu
110
then Rt(yi)=ripq,
i=1 .....
n; p e N ,
qeN.
(19)
PraoL Let Yt-d, and vt-d3 be ro-unimodal, and y~-a,(xp) >I to, y,-d,(X) < ro for all x 4:xp, and Vt_d3(Vq) ~ tO, Vt_d3(V) < ro for all v ~ vq. Obviously, Vi,
y,-d,(xp) A ~3t_d3(13q) A ripq ~ ro,
Vi,
Yt_d,(Xj)^v,-d~(vk)^rok
(20a) Vj=p orj4:p ork4=q.
(20b)
Combining Eqs. (18) and (20) gives R l ( y i ) = yt-d,(xp) ^ Vt-d3(vq) ^ ripq,
i e N.
(21)
Since Y,-d, and vt-a3 are fuzzy measurements for output and disturbance respectively, they can always be normalized such that Vi,
y,-d,(xp) >t ripq and V,_d3(vq) >I r~pq.
(22)
Hence, Rx(yi) = rit,q,
i e N.
[]
(23)
Note that the fuzzy set R1 is an intermediate result which gives a relationship between rOk and ulok that is useful in determining ut.
Assumption 1. Vp, q e N, Ulipq ~ uljpq,
Vi, j e N a n d i 4=j.
(24)
Note that this assumption implies that for any given Y,-a, and Ut-d~, there is a unique relation between elements yi and u/~,q corresponding to hgt(yt) and hgt(u~_d2) respectively. In other words, for any given Yt-d,, there is a unique causality between ut-a2 and y, in a sense. Therefore this assumption is, to some extent, reasonable. Theorem 3. Suppose: (i) Assumption 1 holds; (ii) Rp is ro-reachable; (iii) Yt-dl, U,-d2 and Vt-d3 are all ro-unimodal; and
y,-d,(x/), hgt(v,_d3) = v,_d~(vk), and hgt(u,-d2) = u,-d2(ulok). Then the Yt, as computed by Eq. (1), will be ro-unimodal, and hgt(yt)= yt(yi) >I ro. (iv) hgt(y,_d,) =
ProoL If conditions (i)-(iv) are met, we obviously have according to Eq. (2), Yl(Y) ~>ro for x = xj, v = vk, u = ulok, y = yi, Yt(Y) < ro otherwise. []
(25a) (25b)
Definition 7. Define a fuzzy set R2 E F ( Y ) by R 2 -- R 1 N y~
(26a)
Feedback/feedforward control
111
or equivalently, R z ( y i ) = R , ( y i ) ^ y't(yi) = r,pq ^ y;(yi),
i ~ N,
(26b)
where R2(yi) denotes the membership function of R2. T h e o r e m 4. Suppose R 2 is ro-unimodal, Rp is ro-reachable and Y't is ro-unimodal, and suppose hgt(y;) = y't(ym ) >I ro. We immediately conclude that
(27) (28)
R 2 ( y m ) = rmpq ^ y ' ( y m ) >I ro, R z ( y i ) = ro,q ^ y ; ( y i ) < ro,
Vi =/:m.
[]
T h e o r e m 5. I f Rp is to-reachable, Yt-dt, Vt-d3 and y'~ are ro-unimodal, and U,-d2 is determined such that ut-a~(u) = f o
f ° r u = ul°'°' i = l . . . . otherwise.
, n,
(29)
Then ut-a: is ro-unimodal, and the (ro-unimodal) yt given by Eq. (1) satisfies
hgt(yt) = y t ( y m ) when hgt(y;) = y't(ym). Proof. It can be seen, from Eqs. (27) through (29), that U,-d~(U) >I ro for u = ul,,w¢, U,-d2(U) < ro otherwise.
(30a) (30b)
Since U/mp¢is unique for given p and q, Ut-d~ will be ro-unimodal. Further, if Rp is ro-reachable and U,-d~ is given by Eq. (29), we have according to Eq. (2), Yt(Y) >~ro for x = xp, v = vq, u = ul,,,pq, y = ym, Yt(Y) < ro
otherwise.
(31a) (31b)
Here the result of Theorem 3 is utilised. [] With the previous analysis, the feedback/feedforward control law f(-) can now be determined. If the open-loop system is r0-reachable and Assumption 1 holds, then ut can be determined based o n Rp and (r0-unimodal)Yt+ae-d,, V,+ae--d3 and Y;+a~ in the following way: (i) Compute R using Eq. (14). (ii) Compute R1 by R1 = Yt-dz-dl o Vt+d2_d3 o R.
(32)
(iii) Compute R2 by (33)
R2 = R1 N Y~+d,.
(iv) Compute u, by u,(u) __ { R2(yi)
u = ulo,q, i = l ..... n,
otherwise, where ul#,q is determined by Eq. (15).
(34)
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Chen-Wei Xu
The ut produced by steps (i)-(iv) will (a) be ro-unimodal, (b) result in an ro-unimodal Yt+d2, and (c) result in a Yt+d2 such that hgt(yt+d2) = Yt+d2(ym) if hgt(y~+d2) = y;+d2(ym). An r0-unimodal fuzzy set is relatively easy to be defuzzified- to be converted into a nonfuzzy quantity, which is usually required in implementing fuzzy control.
Remark 1. If Rp is time-varying, it can be identified on-line and the control ut can be determined on the basis of the estimated Rp. This is similar to a self-tuning controller. Remark 2. The control law given by steps (i)-(iv) fits both tracking (y~ #: constant) and regulation (y; -- constant) problems. When Re is time-invariant and y~ = constant, the control algorithm may be simplified further. Remark 3. Although the assumption [YI = IV] = [U[ has been made in the previous discussion, the result of the discussion will still be valid in case
Igl ~ IVI ~ IUI. Remark 4. When Assumption 1 does not hold, the control ut is no longer r0-unimodal (li~k in Eq. (15) is no longer unique). In such a case, the nonfuzzy control quantity may be chosen as the mean value of the ulmpq's. Remark 5. If the ut, Yt+d2-d I and ~Jt+d2-d3 a r e all r0-normal the resulting Yt+d2 will also be r0-normal.
4. Conclusions Mainly on the basis of reachability of the open-loop fuzzy system, a fuzzy feedback/feedforward control law is presented. The concepts of 'disturbance decoupling' and 'predictive control', which are borrowed from linear system theory, are incorporated in the determination of the control law. The open-loop system considered in this article is relatively simple, and more complicated problems may be considered in the future. For instance, Definition 1 is a 'one-step teachability' definition and real system is likely to be 'k (>1)-step reachable', etc.
References [1] M. Braae and D.A. Rutherford, Theoretical and linguistic aspects of the fuzzy logic controller, Automatica 15 (1979) 553-557. [2] E. Czogala and W. Pedrycz, On identification in fuzzy systems and its applications in control problems, Fuzzy Sets and Systems 6 (1981) 73-83. [3] D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications (Academic Press, New York, 1980).
Feedback /feedforward control
I13
[4] L.P. Holmblad and J.J. Ostergaard, Control of a cement kiln by fuzzy logic,in: M.M. Gupta and E. Sanchez, Eds., Fuzzy Information and Decision Processes (North-Holland, Amsterdam, 1982) 389-399. [5] W.J.M. Kickert and E.H. Mamdani, Analysis of fuzzy logic controller, Fuzzy Sets and Systems 1 (1978) 29-44. [6] P.J. King and E.H. Mamdani, The application of fuzzy control systems to industrial processes, Automatica 13 (1977) 235-242. [7] E.H. Mamdani and S. Assilian, An experiment in linguistic synthesis wiry a fuzzy logic controller, in: E.H. Mamdani and B.R. Gaines, Eds., Fuzzy Reasoning and Its Applications (Academic, London, 1981) 311-323. [8] N.J. Mandic, E.M. Scharf and E.H. Mamdani, Practical application of a heuristic fuzzy rule-based controller to the dynamic control of a robot arm, lEE Proc. It. D 132 (1985) 190-203. [9] M. Higashi and G.J. Klir, Identification of fuzzy relation systems, IEEE Trans. Systems Man Cybernet. 14 (1984) 349-355. [10] C.P. Pappis and M. Sugeno, Fuzzy relational equations and the inverse problem, Fuzzy Sets and Systems 15 (1985) 79-90. [11] W. Pedrycz, An identification algorithm in fuzzy relational systems, Fuzzy Sets and Systems 13 (1984) 153-167. [12] E. Sanchez, Resolution of composite relation equations, Inform. and Control 30 (1976) 38-48. [13] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modelling and control, IEEE Trans. Systems Man Cybernet. 15 (1985) 116--132. [14] R.M. Tong, Analysis and control of fuzzy systems using finite discrete relations, lnternat. J. Control 27 (1978) 431-40. [15] R.M. Tong, Some properties of fuzzy feedback systems, IEEE Trans. Systems Man Cybernet. 10 (1980) 327-330. [16] L.A. Zadeh, A fuzzy set theoretic interpretation of linguistic hedges, J. Cybernet. 2 (1972) 4-34.
ANALYSIS AND FEEDBACK/FEEDFORWARD OF F U Z Z Y R E L A T I O N A L SYSTEMS
105
CONTROL
Chen-Wei XU Department of Automatic Control, Kunming Institute of Technology, Kunming, P.R. China Received March 1988; Revised December 1988 Abstract: Based on the analysisof a given fuzzyrelational model of an open-loop system, the
design problem of a fuzzy feedback/feedforward controller is discussed. A fuzzy feedback/feedforwardcontrollaw is proposed and some properties of the resultingclosed-loop system are investigated. The proposed fuzzy control strategy is easy to implement. Both tracking and regulationproblems can be treated by the method. Keywords: Controltheory;relations; feedback/feedforwardcontrol.
1. Introduction The development of fuzzy control theory seems to follow behind that of fuzzy control applications: we already have several successful applications of fuzzy control techniques [4, 6, 8], but we do not have a satisfactory fuzzy control theory so far. Although some people may say that a major advantage of fuzzy control (or linguistic control) is that the design of the controller requires, instead of the plant model as in regular cases, only the knowledge (experience) of the operators who run the plant, and thus precise analysis of the fuzzy control system makes no sense at all, it is argued here that precise analysis of the fuzzy system does not make sense, but approximate analysis of the system may make sense. Two reasons can be provided for our point of view. Firstly, the operators' knowledge about the control of the plant incorporates the understanding of the dynamic/static behavior of the plant, i.e., an approximate plant model is virtually built up inside his brain, and his control decision will certainly be made on the basis of this approximate model, although probably in an unconscious manner. Secondly, the development of fuzzy system identification technique make it possible to build up fuzzy models (approximate models) for dynamic systems. Furthermore, the operators' control strategy may not be optimal in most cases. It seems that the first trial toward fuzzy control system analysis was done by Kickert and Mamdani [5]. The main disadvantage of their method is that it is too precise - it requires a precise mathematical model of the plant. One may ask why we have to use fuzzy control if a mathematical model is already available. In contrast to [5], Braae and Rutherford made an attempt to analyse fuzzy control systems at the semantic level [1]. Their method may be effective in the cases where very vague results can be accepted. However it may be useless when good approximation is needed. 0165-0114/90/$3.50 ¢~ 1990, ElsevierSciencePublishersB.V. (North-Holland)
106
Chen-Wei Xu
Many researchers have used fuzzy relation equations to describe fuzzy systems. The reason, to our opinion, is twofold. Firstly, fuzzy relation equations reflect the fuzziness caused by ill-definition. Secondly, relatively extensive studies have been performed on fuzzy relation equations [10, 12]. Tong investigated some properties of fuzzy relational control systems [14, 15]. Czogala and Pedrycz proposed a design procedure for fuzzy relational controllers [2], which is essentially based on solving fuzzy relation equations and thus suffers from two major difficulties: (1) the existence and the uniqueness of the solution cannot in general be guaranteed, and (2) in order to take a deterministic control action according to the fuzzy control signal, the fuzzy signal is desired to have several features (e.g. the membership curve has a unique peak; the greatest grade of the membership is above some threshold etc.). However, the proposed controller cannot guarantee this. This paper considers feedback/feedforward control problems of fuzzy relation systems. On analysing some properties of the open-loop system, a feedback/feedforward control law is presented. Under some appropriate assumptions, the (fuzzy) output and control variables of the closed-loop system will have several desirable properties. The given control law is both theoretically clear and easy to implement. The text is organized as follows. The feedback/feedforward control problem is presented in Section 2. In Section 3 some discussions are carried out and the feedback/feedforward control law is given. Section 4 is the conclusion.
2. The fuzzy feedback/feedforward control problem Consider the open-loop fuzzy system (1)
Y, = Yt-dl ° Ut-d2 ° Vt-d3 ° Rp
where Yt, ut and vt are fuzzy output, control and disturbance variables at time t respectively. Rp is a fuzzy relation and '0' stands for max-min composition. Let Y, U and V be universes of discours with Yt • F ( Y ) , ut • F(U) and vt • F ( V ) , and Rp • F ( Y x Y x U x V), where "Yt • F ( Y ) " means "Yt is a fuzzy set defined on Y", etc., and Y x Y x U x V represents the Cartesian product of Y, Y, U and V. For convenience, let all the universes appearing in this paper be discrete and have the same cardinality IYI--IUI--IVI--n, i.e., Y = {yl . . . . . yn}, U = { u l , . . . , un}, V = {vl . . . . . vn}. The generic element of Y is denoted by yi (or xi) or y (or x), and that of U, V are denoted by ui, vi (or u, v) respectively. The membership functions of y , u,, v, and Rp are denoted by Yt(Y), u,(u), v,(v) and Rp(y, x, u, v) respectively. The membership function description of the openloop system (1) is then given by Yt(Y) = V V V [y,-d,(X) ^ Ut-d2(U) ^ Vt-a3(V) ^ Rp(y, X, u, v)l, X
U
U
where v = max, ^ = min.
Vy • Y,
(2)
Feedback/feedforward control
107
Ut-d 2 - Yt
l
Vt-d 3
y~ R
P
Fig. 1. Structure of the closed-loopsystem.
Consider now a feedback/feedforward controller of the form
(3)
ut = F(y;+d~,Y,+d~_dl,Vt+d~_d~,Rp)
where F is a mapping which represents the control law to be determined, and y~ • F(Y) is the fuzzy referential input (or set point) at time instant t with membership function y',(y). Obviously a membership function description for (3) may be written as
ut(u) =f[Yt+d~(Y), Yt+d2-d,(X), Vt+d2--a,(V), Rp(y, x, u, v)l
Vu • U.
(4)
Combining the open-loop system (2) and the controller (3) results in the closed-loop system
Y, = Y,-d, o F ( y ; ,
Yt-dl, 7Jt-d3, R p ) o iJt_d3 o Rp.
(5)
The structure of the closed-loop system (5) is shown in Figure 1. For the closed-loop system (5), the feedback/feedforward control problem is stated as follows: Given f~lzzy relation Rp and time delays d~ > 0, d2 > 0 and d3 > 0, determine the feedback/feedforward control law F(-) in Eq. (3) such that for any time t and given disturbance v,, Yt will be equal to y~ in a fuzzy sense which will be given later. For the controller (3) to be realizable, it is necessary to assume that d2 ~ dl
and
de ~
(6)
3. Analysis of the open-loop system and determination of the feedback/feedforward control law Analysis of the open-loop system is necessary before the control law can be determined.
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Chen-Wei X u
Definition 1. System (1) is said to be ro-reachable in y = yi at time t if for any Yt-d, and Vt-d3, there exists a ut-d~ such that yt(yi) >1ro,
ro • [0, 1].
(7)
Definition 2. System (1) is said to be ro-reachable at time instant t if it is ro-reachable at time t for all yi • Y. The notion of ro-reachability is now briefly explained. If system (1) is to-reachable, then for any to-normal (see Definition 4 below) Yt-d, and Vt-d,, we can always find a fuzzy control u,_d~ such that the fuzzy output y, is transferred to a given point yi in the sense that the membership curve of y, has a unique peak at yi, a fact that might be linguistically interpreted as the statement "Yt is about yi". Thus the notion of 'ro-reachability' plays a role in fuzzy systems like 'controllability' in linear systems. Theorem 1. System (1) is ro-reachable at time t, if and only if (iff) (i) There exists x • ¥, such that
yt-dl(X) >I to,
(8a)
and there exists v • V, such that
(8b)
v,_d3(v) >I to. (ii) For all x, y • Y and v • V, there exists u • U, such that Rp(y, x, u, v) >I ro.
(9)
Proof. Suppose both (i) and (ii) hold. Then there exist x" • X, v" • V, such that yt-d,(X") >1ro and ~,-d3(V") ~ ro. Also there exists u" such that Rp(y, x", u", v") >I to. If ut-d2 is chosen in such a way that Ut-d2(U") >I ro,
(10)
then according to Eq. (2) we have y,(y)~ro.
(11)
On the other hand, if either (i) or (ii) does not hold, we obviously have for any U t -- d 2
Yt(Y) < r0 for some y • Y.
(12)
Definition 3. System (1) is said to be ro-reachable if it is to-reachable at any time instant t. Deliailion 4. Let X be a universe with generic element xi, and x • F ( X ) be a fuzzy set with membership function x(xi). (a) x is said to be ro-unimodal, iff 3! x O • X ,
x(xO)>~ro and x(xilxO)
(3! = there exists one and only one).
(13a)
Feedback /feedforward control
109
(b) x is further said to be ro-normal iff (i) x is ro-unimodal, and (ii) x is convex, i.e. for all i, j, k, xi <~xj <<-xk,
x(xj) >~min[x(xi), x(xk)].
(13b)
Note that the notion of normality suggested here is slightly different from the one widely used (see e.g. [3]). Corollary 1. If Yt-d, and vt-d3 are ro-unimodal, then system (1) /s ro-reachable iff Eq. (9) holds. So when sometimes we say "Rp is ro-reachable", we really mean that Eq. (9) holds. Proof. Obvious according to Theorem 1.
[]
Deliailion 5. The projection of Rp on Y x Y x V is denoted by R, i.e. R = proj[Rp; Y x Y x V] ~ F ( Y × Y x V).
(14)
The membership function of R is denoted by R(yi, xj, vk) or simply rl/k, i, j, k ~ N = {1 . . . . . n}. Equation (14) can then be written in terms of ri/k,
rijk = R(yi, x], vk) = max Rp(yi, xj, vk, ul) ul
= Rp(yi, x], vk, ulok ),
lok ~ N, Vi, j, k e N .
(15)
Further assume that for each (i, j, k) pair, l~/k is unique, i.e., for any i, j, k ~ N, 3! l~/k, such that
Rp(yi, xj, vk, ulj/k) > Rp(yi, xj, vk, ul)
Vl • l~/k.
(16)
Corollary 2. I f Rp is to-reachable, then Vi, j , k ~ N ,
(17)
ri/k>~ro .
Proof. Obvious according to Eqs. (9) and (16).
[]
Delhnition 6. Define a fuzzy relation Rj ~ F ( Y ) by
Rl = Yt-dz Vt_d3 0 R,
(18a)
o
or equivalently,
Rl(yi) = V V [Yt-d,(xj) A Vt-d3(vk) A R(yi, xj, ok)], xj vk
i ~ N,
(18b)
where Rl(yi) is the membership function of Rl.
Theorem 2. I f Yt-dl and t~t_d3 are ro-unimodal, and hgt(yt-d,) = max Yt-d,(X) = y~-d,(xp), X
hgt(v,-d3) = max Vt_d3(V) = V,_d3(Vq), 0
Chen-Wei Xu
110
then Rt(yi)=ripq,
i=1 .....
n; p e N ,
qeN.
(19)
PraoL Let Yt-d, and vt-d3 be ro-unimodal, and y~-a,(xp) >I to, y,-d,(X) < ro for all x 4:xp, and Vt_d3(Vq) ~ tO, Vt_d3(V) < ro for all v ~ vq. Obviously, Vi,
y,-d,(xp) A ~3t_d3(13q) A ripq ~ ro,
Vi,
Yt_d,(Xj)^v,-d~(vk)^rok
(20a) Vj=p orj4:p ork4=q.
(20b)
Combining Eqs. (18) and (20) gives R l ( y i ) = yt-d,(xp) ^ Vt-d3(vq) ^ ripq,
i e N.
(21)
Since Y,-d, and vt-a3 are fuzzy measurements for output and disturbance respectively, they can always be normalized such that Vi,
y,-d,(xp) >t ripq and V,_d3(vq) >I r~pq.
(22)
Hence, Rx(yi) = rit,q,
i e N.
[]
(23)
Note that the fuzzy set R1 is an intermediate result which gives a relationship between rOk and ulok that is useful in determining ut.
Assumption 1. Vp, q e N, Ulipq ~ uljpq,
Vi, j e N a n d i 4=j.
(24)
Note that this assumption implies that for any given Y,-a, and Ut-d~, there is a unique relation between elements yi and u/~,q corresponding to hgt(yt) and hgt(u~_d2) respectively. In other words, for any given Yt-d,, there is a unique causality between ut-a2 and y, in a sense. Therefore this assumption is, to some extent, reasonable. Theorem 3. Suppose: (i) Assumption 1 holds; (ii) Rp is ro-reachable; (iii) Yt-dl, U,-d2 and Vt-d3 are all ro-unimodal; and
y,-d,(x/), hgt(v,_d3) = v,_d~(vk), and hgt(u,-d2) = u,-d2(ulok). Then the Yt, as computed by Eq. (1), will be ro-unimodal, and hgt(yt)= yt(yi) >I ro. (iv) hgt(y,_d,) =
ProoL If conditions (i)-(iv) are met, we obviously have according to Eq. (2), Yl(Y) ~>ro for x = xj, v = vk, u = ulok, y = yi, Yt(Y) < ro otherwise. []
(25a) (25b)
Definition 7. Define a fuzzy set R2 E F ( Y ) by R 2 -- R 1 N y~
(26a)
Feedback/feedforward control
111
or equivalently, R z ( y i ) = R , ( y i ) ^ y't(yi) = r,pq ^ y;(yi),
i ~ N,
(26b)
where R2(yi) denotes the membership function of R2. T h e o r e m 4. Suppose R 2 is ro-unimodal, Rp is ro-reachable and Y't is ro-unimodal, and suppose hgt(y;) = y't(ym ) >I ro. We immediately conclude that
(27) (28)
R 2 ( y m ) = rmpq ^ y ' ( y m ) >I ro, R z ( y i ) = ro,q ^ y ; ( y i ) < ro,
Vi =/:m.
[]
T h e o r e m 5. I f Rp is to-reachable, Yt-dt, Vt-d3 and y'~ are ro-unimodal, and U,-d2 is determined such that ut-a~(u) = f o
f ° r u = ul°'°' i = l . . . . otherwise.
, n,
(29)
Then ut-a: is ro-unimodal, and the (ro-unimodal) yt given by Eq. (1) satisfies
hgt(yt) = y t ( y m ) when hgt(y;) = y't(ym). Proof. It can be seen, from Eqs. (27) through (29), that U,-d~(U) >I ro for u = ul,,w¢, U,-d2(U) < ro otherwise.
(30a) (30b)
Since U/mp¢is unique for given p and q, Ut-d~ will be ro-unimodal. Further, if Rp is ro-reachable and U,-d~ is given by Eq. (29), we have according to Eq. (2), Yt(Y) >~ro for x = xp, v = vq, u = ul,,,pq, y = ym, Yt(Y) < ro
otherwise.
(31a) (31b)
Here the result of Theorem 3 is utilised. [] With the previous analysis, the feedback/feedforward control law f(-) can now be determined. If the open-loop system is r0-reachable and Assumption 1 holds, then ut can be determined based o n Rp and (r0-unimodal)Yt+ae-d,, V,+ae--d3 and Y;+a~ in the following way: (i) Compute R using Eq. (14). (ii) Compute R1 by R1 = Yt-dz-dl o Vt+d2_d3 o R.
(32)
(iii) Compute R2 by (33)
R2 = R1 N Y~+d,.
(iv) Compute u, by u,(u) __ { R2(yi)
u = ulo,q, i = l ..... n,
otherwise, where ul#,q is determined by Eq. (15).
(34)
112
Chen-Wei Xu
The ut produced by steps (i)-(iv) will (a) be ro-unimodal, (b) result in an ro-unimodal Yt+d2, and (c) result in a Yt+d2 such that hgt(yt+d2) = Yt+d2(ym) if hgt(y~+d2) = y;+d2(ym). An r0-unimodal fuzzy set is relatively easy to be defuzzified- to be converted into a nonfuzzy quantity, which is usually required in implementing fuzzy control.
Remark 1. If Rp is time-varying, it can be identified on-line and the control ut can be determined on the basis of the estimated Rp. This is similar to a self-tuning controller. Remark 2. The control law given by steps (i)-(iv) fits both tracking (y~ #: constant) and regulation (y; -- constant) problems. When Re is time-invariant and y~ = constant, the control algorithm may be simplified further. Remark 3. Although the assumption [YI = IV] = [U[ has been made in the previous discussion, the result of the discussion will still be valid in case
Igl ~ IVI ~ IUI. Remark 4. When Assumption 1 does not hold, the control ut is no longer r0-unimodal (li~k in Eq. (15) is no longer unique). In such a case, the nonfuzzy control quantity may be chosen as the mean value of the ulmpq's. Remark 5. If the ut, Yt+d2-d I and ~Jt+d2-d3 a r e all r0-normal the resulting Yt+d2 will also be r0-normal.
4. Conclusions Mainly on the basis of reachability of the open-loop fuzzy system, a fuzzy feedback/feedforward control law is presented. The concepts of 'disturbance decoupling' and 'predictive control', which are borrowed from linear system theory, are incorporated in the determination of the control law. The open-loop system considered in this article is relatively simple, and more complicated problems may be considered in the future. For instance, Definition 1 is a 'one-step teachability' definition and real system is likely to be 'k (>1)-step reachable', etc.
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[4] L.P. Holmblad and J.J. Ostergaard, Control of a cement kiln by fuzzy logic,in: M.M. Gupta and E. Sanchez, Eds., Fuzzy Information and Decision Processes (North-Holland, Amsterdam, 1982) 389-399. [5] W.J.M. Kickert and E.H. Mamdani, Analysis of fuzzy logic controller, Fuzzy Sets and Systems 1 (1978) 29-44. [6] P.J. King and E.H. Mamdani, The application of fuzzy control systems to industrial processes, Automatica 13 (1977) 235-242. [7] E.H. Mamdani and S. Assilian, An experiment in linguistic synthesis wiry a fuzzy logic controller, in: E.H. Mamdani and B.R. Gaines, Eds., Fuzzy Reasoning and Its Applications (Academic, London, 1981) 311-323. [8] N.J. Mandic, E.M. Scharf and E.H. Mamdani, Practical application of a heuristic fuzzy rule-based controller to the dynamic control of a robot arm, lEE Proc. It. D 132 (1985) 190-203. [9] M. Higashi and G.J. Klir, Identification of fuzzy relation systems, IEEE Trans. Systems Man Cybernet. 14 (1984) 349-355. [10] C.P. Pappis and M. Sugeno, Fuzzy relational equations and the inverse problem, Fuzzy Sets and Systems 15 (1985) 79-90. [11] W. Pedrycz, An identification algorithm in fuzzy relational systems, Fuzzy Sets and Systems 13 (1984) 153-167. [12] E. Sanchez, Resolution of composite relation equations, Inform. and Control 30 (1976) 38-48. [13] T. Takagi and M. Sugeno, Fuzzy identification of systems and its applications to modelling and control, IEEE Trans. Systems Man Cybernet. 15 (1985) 116--132. [14] R.M. Tong, Analysis and control of fuzzy systems using finite discrete relations, lnternat. J. Control 27 (1978) 431-40. [15] R.M. Tong, Some properties of fuzzy feedback systems, IEEE Trans. Systems Man Cybernet. 10 (1980) 327-330. [16] L.A. Zadeh, A fuzzy set theoretic interpretation of linguistic hedges, J. Cybernet. 2 (1972) 4-34.