The paper concerns  the time-optimal control of objects described by a  differential equation 
representing  the  second  law  of Newtonian  mechanics  and  taking  into  account  a  discon-
tinuous  model  of  resistances  to  motion.  Such  a  task  has  broad  technical  applications, 
especially  in  robotics.  A  fuzzy  approach  is  used  to  design  a  suboptimal  closed-loop  con-
trol  structure,  convenient  in  practice  thanks  to  its  many  advantages,  especially  in  respect 
to  robustness

Kulczycki, Piotr

Kóczy, László T.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODICA POLYTECHNIC.~ SER. EL. EI"G. VOL. 42, NO. 2, PP. 221-232 (1998) FUZZY SUBOPTIMAL FEEDBACK CONTROLLER Piotr KULCZYCKl x and Laszl6 T. Koczy XX • Faculty of Electrical Engineering Cracow University of Technology ul. Warszawska 24 PL-31-15.j Cracow, Poland Department of Telecommunications and Telematics Technical University of Budapest H-1521 Budapest, Hungary Received: Jan. 20, 1998 Abstract The paper concerns the time-optimal control of objects described by a differential equation representing the second law of :\"ewtonian mechanics and taking into account a discon-tinuous model of resistances to motion. Such a task has broad technical applications, especiaily in robotics. A fuzzy approach is used to design a suboptimal closed-loop con-trol structure, convenient in practice thanks to its many advantages, especially in respect to robustness. Keywords: time-optimal control, fuzzy dynamical system, industrial manipulators and robots, suboptimal feedback controller, fuzzy decision rules. 1. Introduction Consider the following differential equation: j)(t) = h(·) +u(t) . (1 ) Ifu represents a control and y denotes the position of an object, then the above dependence expresses the second law of ;,'\ewtonian mechanics, with the model of resistances to motion given by the function h. Thus, an equa-tion of type (1) provides a useful basis for the analysis of an exceptionally broad class of technical devices often encountered in engineering practice, particularly industrial manipulators and robots. The time-optimal control that yields the minimum operation time for such plants has a direct impact on their efficiency. In this paper a concept based on fuzzy logic (KACPRZYK, 1986; KLIR and FOLGER, 1988) is introduced. The function h will be proposed in a form inspired by the physical perspective: h(-) = - V sgn (fJ(t)) . (2) The sgn mapping describes the discontinuous nature - with respect to ve-locity - of friction phenomena: however, the fuzzy set V regards - as a fuzzy 222 P. Kt:LCZYCKl and L. T. KOCZY uncertainty - the dependence of motion resistances on a number of factors, e.g. position, also velocity, or even temperature and disturbances. which are usually omitted in the traditional approach due to the necessity to simplify the model. By its very nature, the fuzzy approach offers the possibility to describe a complex reality with a precision that exceeds classical modelling techniques. Allowing for a certain discomfort resulting from the uncertainty introduced into the model, one may achieve a characteristic that is essential in modern engineering: robustness of the designed control system. Finally, let: (A) [xo, YoV E R2 and [XT, YTV E R2 represent initial and target states. respectively: (B) Il denote a fuzzy set with a support such that supp(I/) C [0,1): (C) the difference equation Xj+l Yj+l Xj + Yj, Yj - y'sgn (Yj) + Uj \vith the initial condition [ Xo] [ Xo ] Ya Yo for j = 0,1, ... (3) (4) (5) describe the dynamics of a system with the fuzzy state [Xj, YjV, sub-mitted to a control Uj with values limited to the interval [-1, 1]. The goal of this paper is to design a useful suboptimal (in respect to time) feedback controller, whose values due to practical requirements - directly depend only on the valid state of the object, obtained by a real-time mea-surement process. 2. SOIne Auxiliary Considerations In the following section, an auxiliary task will be considered, Let at this point the fuzzy set 'V be reduced to a real number, denoted hereafter by lE, Due to assumption (B), U' E [0, I), Suppose that [x+, Y+V and [;L, Y-V are unique solutions (Kl'LCZYCKL 1996) of the ordinary differential equation related to system (3)-(4): y(t) , u(t) le sgn (y(t)) , (6) (7) with the condition [.1'+(0), y+(O)V = [x_ (0), y-(O)V = [;1'y, yyV, defined on the interval (-x,O], and generated by the controlu == +1 or u == -1, FUZZY SUBOPTI.\IAL FEEDBACK CONTROLLER respectively. Moreover. consider {[x+(t), y+(t)f for t < O} , {[x-(t), y_(t)]T for t < O} ; 223 (8) (9) therefore, these are the sets of all states \vhich can be brought to the target [XT, YTV by the control u == +1 oru == 1, respectively. Let also: R+ = {[ ~] such that there exists [~:] E K , with x < XX and y = yx} , { [ Yx ] such that there exists ') with x > XX and y = yXf' (10) (11) \vhere l( = lC U {[XT, YTV} U K+. The time-optimal control is expressed by the following formula (KGLCZYCJ<L 1992): {-I if [:c(t), y(t)V E (R_ U lC) , u(t) = ll r (.r(t), y(t)) = I 0 ~f [x~:), Y(:)l~ = ~XT' YT]~ , -d If [:t:\L), Y(i)] E \R+ U 1\+) , (12) and the set K constitutes a switching curve (Fig. 1,2). In the time-optimal feedback controiler equations, i.e. formulas (8)-(12), the parameter1.L' inten'enes, because it influences the form of the tra-jectories [x+, Y+V, [x_, Y-V and therefore also the shape of the switching curve K, But in the fuzzy system, its value is of course not uniquely de-fined. The analysis of the s?stem sensitivity to the value of the parameter (KULCZYCJ<L 1992), which is briefly presented in the following, will then be of great importance. Thus, the value of the parameter w occurring in the object is still denoted as w; however. the value assumed in feedback controller equations will be marked by li'. The case where the second coordinate of the target state is equal to zero, i.e. with YT 0, will be considered first. If vV =w, the control is time-optimal (Fig. 1). The state of the system is brought to the s\vitching curve, and being permanently included in this curve hereafter, it reaches the target in a minimal and finite time. The trajectory representative for the case lV > w is shown in Fig. J. As a result of its having oscillations around the target, over-regulations occur in the system. The target is reached in a finite time. 224 P. KULCZYCKI and L. T. KOCZY 1 Fig. 1. Feedback controller and representative trajectories in the case YT = 0 K~ Fig. 2. Feedback controller and representative trajectories in the case YT =f. 0 Fig. 4 shows trajectories representative for the case W < w. After the switching curve is crossed, the sliding trajectories (SLOTI:-;'E and LEE. 1991) appear in the system. Here, too, the target is reached in a finite time. In both of the last two cases, i.e. with VV f. w, the time to reach the target increases from the optimal more or less proportionally to the difference between the values lV and w. FUZZY SUBOPTIMAL FEEDBACK CONTROLLER 225 Pi Fig. 3. Trajectory representative for VV > w in the case YT = 0 t'tJ j ) I sliding ~ [XT] trajectories-~ 1'1 \ ~~sliding I \ trajectories I ;- ~ - / K. Fig. 4· Trajectories representative for W < w in the case YT = 0 The remaInIng case Yr i= 0 will be presented now. Let in particular Yr > 0; investigations for Yr < 0 can be made analogously. If Hi = w (Fig. 2), the considerations are identical as before for YT = O. In the case W > w (Pig. 5), the trajectories occurring in the system create a limit cycle: the target is not reached. 226 P. KfjLCZYCKi and L. T. KOCZY Finally. in the case l·V < u; (Fig. 6) only some of the trajectories (marked on Pig. 6 with arrows) reach the target in a finite time. Other trajectories attain only the end point [XE, YE]T, which is the intersection of the axis x and the switching curve; the state does not reach then the target. Sliding trajectories occur on the s\yitching curve. ( !)] I Fig. 5. Trajectories representative for ~F > 11., in the case YT i=- 0 3. Sub-Optimal Feedback Controller for a Fuzzy System In this section the fuzzy system (3)-(.5). \vhich is the subject of the present paper, will be considered. The parameter w. introduced ill the pre,'ious section, happens to be a fuzzy set in the problem worked out here. A fuzzy set naturally cannot be used directly to define a control in a rea! system. For this reason, some elements of fuzzy decision theory (KACPRZ'y·K. 1986) will be used. Its aim is to make the optimal selection of one element from all possible decisions on the basis of a membership function. Let the following be given: a fuzzy set Z (with the membership func-tion flZ : R --7 [0, x)) representing the state of reality. a non-empty set D of possible decisions and a loss function I : D x R --7 R U {±x} , (13) where the values l(d, z) can be interpreted as losses occurring in the hypo-thetical case when the fuzzy set Z is reduced to the real number z. and the FuZZY SuBOPTIMAL FEEDBACK CONTROLLER 227 PI ~sliding I j \ trai\tories )' !---I / [~~l \ Fig. 6. Trajectories representative for H- < lL in the case YT :f. 0 decision d has been made. Denote by lm : D -+ R U {±x} the minimax loss function I m (d) = sup I ( d, z) . (14) zE supp (Z) If additionally for every dE D the integral I l(d, Z)/lZ(Z) dz exists, suppose R also the Bayes loss function 10 : D -+ R U {±oc} defined as lb(d) = J l(d, z)/lz(z) dz .. R Every element dTn E D such that inf lm (d) dED .. (1.5 ) ( 16) is called a minimax decision. and analogously, every element db E D such that lb(db) = inf lb(d) (17) dED is called a Bayes decision. The above procedures for obtaining these elements are said to be minimax and Bayes rules, respectively. The main difference between the above rules is in their interpretation. This results directly from the forms of the functions lm and 10: the 'pessimistic' minimax rule assumes the occurrence of the most unfavorable state of reality and counteracts it, while the Bayes rule is more flexible. 228 ' P. KULCZYCKI and L. T, KOCZY In the problem of a time-optimal control investigated here, the param-eter \iV E D C [0,1) assumed in the feedback controller equations will be treated as a decision, while the number 1/ occurring in system (3)-(4), as the fuzzy state of reality. The loss function is defined for (W, w) E D x R, and its values are related to the time to reach the target, if in the feedback controller equations the parameter n; was assumed, but hypothetically in the object the value w is occurring. Again the case YT = 0 will be considered first. The following sugges-tions for the determination of the value of the parameter ~1l result from the analysis of the auxiliary problem presented in the previous section. If over-regulations can be allowed, it is worthwhile using the Bayes rule with real values for the loss function. Such a choice is possible because the determination of the parameter \iV value that is either smaller, equal, or greater than wallows the system state to be brought to the target in a finite time. (HO\vever, this time increases more or less proportionally to the difference between the values W' and w.) If over-regulations are not allowed, this determination should be car-ried out on the basis of the minimax rule, assuming infinite values of the loss function for Hr > It·. This enables the over-regulations to be avoided, because they occur only if VV > lE. Let now YT f O. The case YT > 0 will be considered; investigations for YT < 0 are analogous. The case H1 = lE is impossible to obtain in practice. However, the determination of the parameter \iV value that is either greater or smaller than w precludes reaching the target from any initial state, because of the occurrence of the cycle (Fig. 5) or existence of the end point (Fig. 6). In the proposed feedback controller, the switching curve I( will be divided into three parts. The division points \Vill be the target and the point of inter-section with the axis x. For every part there will be differently determined values of the parameter'IV. which for particular parts are defined in the follmying as HT1 , Wz and \iV3. The value of the parameter lVl, i.e. the one which defined the part of the switching curve I( . or I( for y E [YT, x) also Fig. 8), should be determined using the minimax rule with infinite vaiues of the loss function for 1V > w. This choice is made in order to avoid the generation of a limit cycle, which appears when the value of the parameter vV1 is greater than lE. If, however, this value is smaller than U', the state of the system is brought to the target in a finite time. For the determination of the value of the parameter n;2 defining the part K+ for Y E [0, YT], it is necessary to apply the minimax rule \vith infinite values of the loss function for l'V <wo This is because an overly large yalue of the parameter VV2 allows the state to be brought to the part defined by the parameter VV1 , which, as was demonstrated above, can be successfully determined. An overly small one. however, causes the occurrence of the end point, whose existence is not admissible from the point of view of utility. FL:ZZY SfjBOPT/yIAL FEEDBACK CO.'lTROLLER 229 Finally. the value of the parameter }V3 defining the part 1(+ for y E (-x, 0] can be obtained using the Bayes rule with real values of the loss function. Both overly small and large val ues of this parameter are ac-ceptable, because this allows the state to be brought to the parts defined by the parameters H'j and IF2 , which can be successfully determined as shown above. Suppose. as an example. that the fuzzy set F has the support of the form supp(V) = [u.:x. wx] C [0,1), and moreover. let its membership function /.LV be continuous and positive in the interval (lL' x , u,.X). The loss function (13) \\'ill be described by the following formula: I 1'1'. w = . ..-. ' {-pOV - w) if IV - w < 0, ( .) q(W w) If W - w > 0 , (18) \vhere p, q E R u {x}; however, only one of them can be infinite. In this case. let x . 0 = O. According to the above assumptions. it is accepted that D = [wx. u· x ]. \Vith the fixed value of the parameter H', from the definitions of min-imax and Bayes loss functions (14) and (1.5) the follmving results, re-spectively: m ax ( { - p (IV - le x ) , q (IV - U' x)}) . (19 ) W u J q(T .. V -u·)p.v(w) dIe - J p(TV - W)Jiv(w) du·. (20) w. \j' If p = x. then from Eq. (19) it can be obtaineu that the infimum of the function IrE on the set D is realized by (21 ) and if q = x. then this infimum is assumed for (:22) The values TV indicated by formulas (21) and (22) constitute the desired minimax decision with infinite values of the loss function for IV < 1L' and H' > lC, respectively. However. \\'ith real positive values p and q. the function 10 is differ-entiable in the set (u'x, W X ): therefore one obtains (RCD!:";, 1974: Theorem 6.20): \\' IF 1~(W) = p J fi'v(w) dw + q J Ji\;(w) du,·. (23) U·· 230 P. KULCZYCKI and L.. T. K6CZY and analogously Ib'(vV) = (p+ g)f-1V(W). (241 , / Using formula (2:3), the equivalence of the following conditions can be proved by elementary transformations: 0, (25) H' w* J flv(W) du' (26) 'lL':II Formula (24) implies that the function lb' is posltl\'e in the set eu'x, u'X): therefore, the function lo is here strictly convex. Because 0 < ufo < 1, . , . Eg. (26), equivalent to condition (2.5), is fulfilled only in one point: in this point, then, the function ID assumes its minimum. global in the set D = [lUx, wx] due to the continuity of this function in the points u'x and LCX• The value VV that fulfills condition (26) constitutes the desired Bayes decision \vith real values of the loss function. To obtain its value one can us€ t.he kernel estimators technique. according to the algorithm presented in papers (KCLCZYCh:] and SCHI0LER. 1994: SCHl0LER and Knczych:l, 1997) . To summarize. in accordance with the considerations stated before, if the values of the parameters IF or W1 . IV]. W3 should be determined due to the minimax rule with infinite values of the loss function for TV < u; or IV > lC, or the Bayes rule with real values of this function. then they can be obtained from formulas (21), (22) and (26). respectively. If one possesses the obtained values 11' or lr1• IT> H'3. the feedback conlroller equations can be calculated. Thus. the equations of the switching curw [".- take on the form . r for y c .x) . (27) .) .r 1)- -. , +.q' 2(1 - \\2) for i) c [0 .. !JTj . (28) " y-for y ~ ( :x; , 0] , (29) 2(1 + IV}) ill the case UT > O. (Forrnulct (:27) definf's the set K _. while dependencies (28) and (29). the set K+.) For i)T < O. the equations are analogous, If iJT = 0, one should substit ute. in formulcts (27) and (:29). n' = H'I = 11'3 (dependence (28) has no meaniIli!, here). The sets IL and R-!- constitute adequate areas resulting from the section of the plane R2 lly the curve [(. according to formulas (10) (tne! (11). For the sets [\' . [\'+. IL. IL obtained FUZZY SUBOPT1.\iAL FEEDBACK CONTROLLER X J 231 Fig. 1. Fuzzy feedback controller and empirically obtained trajectories III the case YT = 0 Fig. 8. Fuzzy feedback controller and empirically obtained trajectories In the case YT -:j=. 0 232 P. Kt:LCZYCKI and L. T. KOCZY above. the value of the control is simply defined by equation -1 o +1 if the state of the object belongs to the set R_ u le , if the state of the object equals [XT. YTV . if the state of the object belongs to the set R+ U 1(+ . (30) Figs. 7 and 8 provide an illustration of the results obtained above. 4. Conclusions In this paper a fuzzy approach to solve the time-optimal control problem has been investigated. Theoretical considerations led to the design of a closed-loop control system, convenient in engineering practice. A large number of completed empirical examinations confirmed the presented material, and proved the correct operation of the system (see Fig. 7 and 8). The tar-get state \\'as reached in every case \yith the assumed precision of the scale 0.1-0 . .sYc of the initial state norm. If it \\'as assumed that over-regulations were unacceptable. they did not occur during the control process. The constructed control system turned out to be only slightly sensitive to the inaccuracy resulting from identification and the occurrence of perturbations. This should be emphasized as a very valuable property of uncertain. espe-cially fuzzy. control systems. References !\ACPRZYK. J. (1986): Zbiory rozmytc w analizic syslemowej. P\Y=';. \ \·ilI·saw. KUR. (; . .1. FOLGER. T. A. (1988): Fuzzy Sets. Lncertainty. and Information. Prenticc-HalL Englewood Cliffs. [3J K CLCZ YCKI. P. (199'2): Czasowooptymalne sterowanic stochastyczne nieciaglym uk-ladem dynamicznym. WPK. Cracow. r41 Kl·LCZYCKI. P. (1996): Some Remarks on Solutions of Discontinuous Differential Equations Applied in Automatic Control. Industri(J1 .Uathem(Jtics. \'01. 46. :\0. 2. pp.119128. [.s] I<CLCZYCKI. P. - SCIlIOLER. H. (1994): Bayes Decision Lsing :\eural :\etworks. Proc. 10th IFAC Symposium on System Identification, July 4-6,1994. Copenhagen. \'01. Ill. pp. 477 -482. [6] RCDI:\. \V. (1974): Real and Complex .-\nalysis. ),lcGraw-HilI, :\ew York. [7] SCHlOLER. H. KCLCZYCKl. P. (1997): :\eural :\etwork for Estimating Conditional Distributions. IEEE Transactionss on Scural SetlL'orb. \'01. 8, :\0 .. 5. pp. 101.5-1025. [8J SLOTI:\E. 1.-J. E. LI. \V. (1991): .-\pplied :\onlinear Contro!' Prentice-HalL Engle-wood Cliffs. 

FUZZY SUBOPTIMAL FEEDBACK CONTROLLER

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FUZZY SUBOPTIMAL FEEDBACK CONTROLLER

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