Constructing Fuzzy Controllers with B Spline Models

We interpret a type of fuzzy controller as an inter polator of B spline hypersurfaces B spline basis func tions of di erent orders are regarded as a class of mem bership functions MFs with some special properties These properties lead to several interesting conclusions about fuzzy controllers if such membership functions are employed to specify the linguistic terms of the input variables We show that by appropriately designing the rule base C continuity of the output can be achieved n is the order of the B spline basis functions The issues of function approximation and heuristic control using such a fuzzy system are also discussed Introduction Recently fuzzy logic control FLC has been suc cessfully applied to a wide range of control problems and has demonstrated some advantages e g in e ciency of developing control software appropriate pro cessing of imprecise sensor data and real time charac teristics However as pointed out in one obstacle to the wide acceptance for industrial applica tions is that it is still not clear how membership func tions defuzzi cation procedures contribute either in combination or as stand alone factors to the per formance of the FLC Two important related issues are Quality of fuzzy controllers In practical applica tions the smoothness of the controller output is one of the most important design requirements This applies both to the control of very complex systems such as the speed control of automated trains as well as simple actuators like electrical motors whose life expectancy directly depends on the smoothness of the controller output Unfor tunately in general cases smoothness cannot be guaranteed and is frequently hard to determine for a given controller Guidelines for choosing membership functions Up to now there exist no convincing guidelines for the successful design of fuzzy controllers In particular this pertains to the choice of a concrete membership function In various fuzzy control applications membership functions of triangular or trapezoidal shape are utilised because of the simplicity of speci cation and the satisfying results But the question still remains can the control performance be improved by choosing a certain set of membership functions These two issues can be addressed by comparing B spline models with a fuzzy logic controller In our previous work we compared splines and a fuzzy controller with single input single output SISO struc tures In this paper the multi input single output MISO controller is considered Periodical non uniform B spline basis functions are interpreted as membership functions Aspects of function approxi mation and heuristic control are discussed Some Previous Work 2.1. Advances in Fuzzy Control Several authors have shown that fuzzy controllers are universal approximators Wang presents a universal approximator by using Gaussian member ship functions product fuzzy conjunction and centre of average defuzzi cation Buckley has shown that a modi cation of Sugeno type fuzzy controllers are universal approximators Kosko and Dickerson intro duced additive fuzzy systems to generally describe fuzzy controllers which use the addition of THEN parts of red rules to determine the crisp output They then proved that an additive fuzzy system uniformly A MIMO rule base is normally divided into several MISO rule bases Synonyms Takagi Sugeno IDM Inference and Defuzzi ca tion Method Tsukamoto method weighted mean approximates f X Y if X is compact and f is continuous Two successful applications in commercial controller and process control are given in one is the OM RON temperature controller chapt the other is a gas red water heater chapt The member ship functions are selected as only triangles and each pair overlaps Can these be generalised as design rules The work in shows that the triangular membership functions with overlap level produce the zero value of the reconstruction error Further questions are Are there other forms of suitable membership functions Should the overlap of the fuzzy sets for linguistic terms ful ll certain constraints 2.2. The Popularity of B-Splines To solve the problem of numerical approximation for smoothing statistical data Basis Splines B Splines were introduced by I J Schoenberg B splines were used later by R F Riesenfeld and W J Gordon in CAGD for curve and surface representation Due to their versatility based on only low order polynomi als and their straightforward computation B splines have become more and more popular Nowadays B spline techniques represent one of the most important trends in CAD CAM areas they have been extensively applied in modelling free shape curves and surfaces Recently splines have also been proposed for neural network modelling and control Although B splines have been mainly used in o line modelling and fuzzy techniques lend themselves to on line control some interesting common points can still be found Our previous paper pointed out that the B spline basis functions and the membership func tions of a linguistic variable are both normalised over lapping function hulls Splines and fuzzy controllers possess good interpolation features The synthesis of a smooth curve with spline functions can easily be associ ated with the defuzzi cation process These points are the main motivation for our work on utilising B splines to design fuzzy controllers B Spline Basis Functions vs Member ship Functions We consider the membership functions which are used in the context of specifying linguistic terms val ues or labels of input variables of a fuzzy controller In the following basis functions of Non Uniform B Splines NUBS are summarised and compared with the membership functions We also use B functions for the NUBS basis functions 3.1. NUBS B-Functions Given a sequence of ordered parameters x x x xm xm xm n the normalised B functions Ni n of order n are de ned as Ni n x for xi x xi otherwise if n x xi xi n xi Ni n x xi n x xi n xi Ni n x if n with i m Three important properties of B functions are partition of unity Pn i Ni n x positivity Ni n x for x C continuity If the knots fxig are pairwise di erent from each other then Ni n x C i e Ni n x is n times con tinuously di erentiable 3.2. Overview of MFs of B-Function Type The B functions are employed to specify the linguis tic terms knots are chosen to be di erent from each other periodical model Visually the selection of n the order of the B functions determines the following factors of the fuzzy sets for modelling the linguistic terms Table 3.3. Partition of the Input Variable into Support Intervals It is assumed that linguistic terms are to be used to cover x xm the universe of an input variable x of a fuzzy controller m is chosen according to how ne this input variable should be partitioned by con sidering an appropriate granularity to achieve a trade o between the precision of the control approximation and the complexity of the rule base If we want to use B functions Ni n i m as linguistic terms then rst x xm is partitioned into m intervals xi xi xi xi i m Fig In or der to maintain the partition of unity some more B functions should be added at the both ends of x xm They are called marginal B functions and de ne the virtual linguistic terms in the following Marginal B functions are to be de ned on the left end which need additional n intervals adjacent x They are xi xi i n These intervals may be selected as if they have the same order n degree shape width overlap rectangular