Fuzzy Identication from a Grey Box Mo deling Point of View P Lindskog Linkoping University S Linkoping Intro duction The design of mathematical mo dels of complex realworld and typically nonlinear systems is essential in many elds of science and engineering The develop ed mo dels can be used eg to explain the b ehavior of the underlying system as well as for prediction and control purp oses A common approach for building mathematical mo dels is socalled black box modeling Ljung Soderstrom and Stoica as opp osed to more traditional physical mo deling or white box mo deling where everything is considered known a priori from physics Strictly sp eaking a blackbox mo del is designed entirely from data using no physical or verbal insight whatso ever The structure of the mo del is chosen from families that are known to b e very exible and successful in past applications This also means that the mo del parameters lackphysical or verbal signicance they are tuned just to t the observed data as well as p ossible The term blackbox mo deling is sometimes used almost as a synonym to system identication although a much more convenient denition and the one often used is that system identication is the theory of designing mathematical mo dels of dynamical systems from observed data Hence by combining the blackbox approachwithphysical or verbal mo deling in suchaway that certain prior knowledge from the system is taken into account we end up with sp ecial identication pro cedures that commonly are referred to as grey box modeling approaches see eg Bohlin Hangos Two imp ortant facts makesuch metho ds intuitively app ealing In a realworld mo deling situation we never have complete pro cess knowledge There are always uncertain factors aecting the system thus indicating that a complete physical mo del hardly ever can b e constructed However uncertain factors can b e revealed through exp eri ments and at least partlytaken care of by employing suciently exible mo del families The mo deling pro cedure on the other hand allows us to restrict the exibility to comply with the prior knowledge This makes it p ossible to follow at least partly another basic identication principle namely to only estimate what is still unknown Traditional grey box approaches assume that the structure of the mo del is given directly as a parameterized mathematical function which at least partly is based on physical principles However for many realworld systems a great deal of information is provided byhuman exp erts who do not reason in terms of mathematics but instead describ e the system verbally through vague or imprecise statements For example in case it is hard to design a suitable mathematical mo del of a heating system an imp ortant part of its b ehavior can still b e characterized eg through If more energy is suppliedtotheheater element then the temperature wil l increase Because so much human knowledge and exp ertise come in terms of verbal rules a sound engi neering approach is to try to integrate such linguistic information into the identication pro cess A convenient and common way of doing this is to use fuzzy logic concepts in order to cast the verbal knowledge into a conventional mathematical representation a mo del structure which sub sequently can b e netuned using inputoutput data It turns out that the structure so obtained very well can be viewed as a layered network having much in common with an ordinary neural and network see eg Brown and Harris Haykin Roger Jang and Sun Lin Lee Chen As a matter of fact the kinship is so evident that many researchers refer to this approachas neurofuzzy modeling With this in mind the palpable question is what is conceptually gained by this approach compared to standard blackbox neural network mo deling Firstly and contrary to neural networks neurofuzzy mo deling or just fuzzy modeling oers a highlevel structured and convenientway of incorp orating linguistic prior into the mo dels Secondly the basic linguistic knowledge entered is of the form speedt is high In fuzzy mo deling such a prop osition is given a precise mathematical meaning through a basis function memb ership function having parameters asso ciated with the prop ertyhigh thus meaning that the parameters can be assigned reasonable initial values This is imp ortant in that the parameter estimation algorithm which often is iterative can b e started from a point where the risk of getting stuck in an undesired lo cal minimum is reduced compared to if the initial parameters are chosen at random which often is the case for neural networks Thirdlyphysically unsound regions can b e avoided By randomly cho osing initial parameter values in a neural network this cannot b e guaranteed and although regularization see b elow is applied in the estimation phase basis functions corresp onding to unsound regions are seldom removed from the nal mo del which then b ecomes more complex than necessary A fourth p otential advantage comes in terms of extrap olation capabilities While data can b e used to explain certain system features the linguistic exp ert knowledge here the rules can b e employed to pick up other phenomena that are not revealed in the available data Finally the human exp ert who supplied the verbal knowledge can always b e consulted for mo del validation This contribution concentrates on how to maintain these advantages when fuzzy mo deling is complemented with system identication techniques More preciselytheaimistoprovide answers to a numb er of central greyb oxtyp e of questions What kind of mathematical rule base in terpretation is suited when system identication asp ects are also taken into account What parameter estimation algorithms should b e used How can the knowledge provided by the domain exp ert ie the meaning of the rule base b e preserved throughout the parameter estimation step How can dierent nonstructural system features b e built into the mo dels By nonstructural knowledge we mean eg that the step resp onse is known to b e monotone or that the steady state gain curve is monotonic in certain input variables or some other qualitative prop erty To b e able to address these issues werstgive a brief intro duction to the eld of parametric system identication fo cusing mainly on basic concepts ideas and algorithms from which the following sections can depart Sect addresses various fuzzy mo deling matters It is argued that a Mamdani typ e of rule base interpretation Mamdani and Assilian Roger Jang and Sun is suited when the rules are of the form and when identication asp ects are also accounted for The remaining main three questions from ab ove are then considered and answered in Sect whereup on Sect illustrates the usefulness of the suggested framework on a realworld lab oratoryscale application example Some practical asp ects of the prop osed mo deling approach are thereafter discussed in Sect and in Sect we nally put forward some concluding remarks and give a few directions for further research within the fuzzy identication area In fact the considered mo del representation turns out to b e structurally equivalent to a zeroorder TakagiSugeno fuzzy structure Takagi and Sugeno Sugeno and Kang which is just a sp ecial case of the general TakagiSugeno fuzzy mo del family System identication Basic ingredients and notation System identication deals with the problem of how to infer relationships between past input output measurements and future outputs Ljung Soderstrom and Stoica In practice this is a pro cedure that is highly iterative in nature and is made up from three main ingredients the data the mo del structure and the selection criterion all of which include choices that are sub ject to p ersonal judgments The data Z By the rowvector N m y t u t u t z t R m we denote one particular data sample at time t collected from a system having one output and m input signals ie we consider a multi input single output MISO system This restriction is mainly for ease of notation and the extension to multi output MIMO systems is fairly straightforward see Ljung Lee Wang Stacking N consecutive samples on top of each other gives the data matrix T N m T T T z z z N R Z N It is of course crucial that the data reect the imp ortant features of the underlying system This will typically b e the case if the input signals are suciently exciting and if large enough data sets are collected However such a situation is unrealistic in many realworld applications since rstly the exp erimental time is limited and secondlymany of the inputs are restricted to certain signal classes Having to live with this reality it is worth stressing that the problem of having incomplete data very well can b e alleviated considerably by building various prior system prop erties into the mo dels or rather into the applied mo del structure The mo del structure g t It is generally agreed up on that the single most dicult step in identication is that of mo del structure selection Roughly sp eaking the problem can b e divided into three subproblems The rst one is to sp ecify the type of mo del set to use This involves the selection b etween linear and nonlinear representations b etween blackbox grey b oxandphysically parameterized approaches and so forth The next issue is to decide the size of the mo del set This includes the choice of p ossible variables inputs and outputs and combinations of variables to use in the mo dels It also involves xing orders and degrees of the chosen mo del typ es often to some intervals The last item to