pag. 1 S. Marsili-Libelli: Controllo Fuzzy Why Fuzzy Control? 1 - No detailed mathematical model required 2 -Human experience can be easily incorporated 3 -It is robust 4 -Can adapt to process changes 5 -It is parsimonious because with a limited number of rules can control very complex systems 1 -No detailed insight into the problem 2 -No possibility of transferring "classical" control techniques, at least in the Mamdani approach 3 -It is heuristic (no theoretical justification) at least in the Mamdani form 4 -Cannot be guaranteed to be optimal Advantages Disadvantages

pag. 2 S. Marsili-Libelli: Controllo Fuzzy Generic control loop PROCESS REGULATOR - + Control error Control signal +The regulator is intended to generate a control signal so that the process output is as close as possible to the set-point, even in the presence of disturbances +Usually the regulator uses the control error and its derivative +In the fuzzy case, the inference set is composed of rules with the form

pag. 3 S. Marsili-Libelli: Controllo Fuzzy Structure of the fuzzy regulator It is based on the following quantities o A set of membership functions (both for inputs and outputs) o A fuzzification procedure o A set of rules relating the antecedents (inputs) to the consequents (outputs) o A set of connectives implementing the rules o A defuzzification procedure o A set of normalization/denormalization gains

pag. 4 S. Marsili-Libelli: Controllo Fuzzy Structure of a Fuzzy Regulator The regulation is based on the error e(t) and its derivative e(t). Generally an external integral action is introduced after the fuzzy part to ensure good set-point tracking, because the fuzzy regulator cannot provide this So Sr Se z - 1 Fuzzy control rules + - y sp (k) y(k) Normalization gains Fuzzificazione Defuzzificazione Fuzzy Part Denormalization gain deterministic integrator Process The control rules can be specified according to a general template (metarules)

pag. 5 S. Marsili-Libelli: Controllo Fuzzy Control metarules The control rules must obey some general principles, called metarules These guidelines assure that the regulator consistently provides a stabilising action MR1If the error e(t) and its derivative e(t) are about zero, maintain the present action MR2 If the error e(t) tends to zero with a satisfactory rate (self-correcting), maintain the present action MR3 If the error e(t) is not self-correcting, then the contorl action u(t) depends on the sign and magnitude of e(t) and e(t) MR1If the error e(t) and its derivative e(t) are about zero, maintain the present action MR2 If the error e(t) tends to zero with a satisfactory rate (self-correcting), maintain the present action MR3 If the error e(t) is not self-correcting, then the contorl action u(t) depends on the sign and magnitude of e(t) and e(t) Set-point MR3 MR2 MR1

pag. 6 S. Marsili-Libelli: Controllo Fuzzy Control rule table If the error e(t) and its derivative e(t) are used as antecedents the control implication can be represented by a look-up table The cross between each value of e(t) and e(t) indicates which rule is to be activated. The degree of activation of the control rule depends on the combined degree of truth of the corresponding antecedents. e PB PS ZZ NS NB PB PS ZZ PS PB PS ZZ NS e ZZ PS ZZ NS PS ZZ NS NB ZZ NS NB Linguistic labels of the antecedents PB = Positive Big PS = Positive Small ZZ = Zero NS = Negative Small NB = Negative Big

pag. 7 S. Marsili-Libelli: Controllo Fuzzy Use of the control rule table Example: Suppse that e(t) is PS with dot 0.6: µ PS (e) = 0.6 and e is NS with dot 0.4: µ NS ( e) = 0.4, then the degree of activation of the corresponding rule ZZ is 0.4, using the MIN operator e PB PS ZZ NS NB PB PS ZZ PS PB PS ZZ NS e ZZ PS ZZ NS PS ZZ NS NB ZZ NS NB µ NS ( e) = 0.4 µ PS (e) =

pag. 8 S. Marsili-Libelli: Controllo Fuzzy Of course more than one rule is activated 1 1 e PB PS ZZ NS NB PB PS ZZ PS PB PS ZZ NS ZZ PS ZZ NS PS ZZ NS NB ZZ NS NB

pag. 9 S. Marsili-Libelli: Controllo Fuzzy Normalization/Denormalization Normalization/Denormalization gains are required to conform the variables to their operating ranges Fyzzy Inference Rules supporto normalizzato AntecedentConsequent Input RangeOutput Range supporto normalizzato K i K out Operating range

pag. 10 S. Marsili-Libelli: Controllo Fuzzy Effect of Normalization/Denormalization The choice of N/DN coefficients can have dramatic effects of the regulator performance y u y u time y sp y K in = 0.23 K out = K in = 1.0 K out = 0.2

pag. 11 S. Marsili-Libelli: Controllo Fuzzy Lack of integral action in the fuzzy regulator Unlike deterministic regulators integral action cannot guarantee set-point tracking. This is because of the granularity of the increments provided by the fuzzy regulator If the error is in the ZZ band, only that qualifier will be activated and this yields a 0 output. So for that range of errors no contribution will be summed by the integrator and corrective action will result Rimedy: Shrink the dead-band by increasing the overlapping between mf errore N = 5, b = 0.3 ZZPS PB NSNB du Dead band In this zone the only contribution is given by ZZ Input/Output characteristics of the fuzzy regulator

pag. 12 S. Marsili-Libelli: Controllo Fuzzy a) Proportional-Integral (PI) b) Proportional-Derivative (PD) c) Full PID Formal similarities of PID actions

pag. 13 S. Marsili-Libelli: Controllo Fuzzy Fuzzy tuning of a deterministic PID A set of fuzzy rules can be designed to adjust the PID coefficients during operation y sp + y - + de dt K 1 K 2 Deterministic PID FUZZY TUNING PROCESS Fuzzy rules Defuzz.Fuzz.

pag. 14 S. Marsili-Libelli: Controllo Fuzzy Progetto di Regolatori Fuzzy in Matlab/Simulink RATEO di LIVELLO SISTEMA DI REGOLE INFERIENZIALI FUZZY VALVOLA LIVELLO Insieme di regole fuzzy = regolatore Obiettivo: regolazione del livello di un serbatoio mediante apertura della valvola di alimentazione FLC h sp

pag. 15 S. Marsili-Libelli: Controllo Fuzzy Lambiente Fuzzy Toolbox di MATLAB

pag. 16 S. Marsili-Libelli: Controllo Fuzzy Lo schema Simulink per il problema del serbatoio

pag. 17 S. Marsili-Libelli: Controllo Fuzzy Paragone fra le prestazioni di regolatori fuzzy

pag. 18 S. Marsili-Libelli: Controllo Fuzzy Differenze nei conseguenti no_change close_slow close_fast open_slow open_fast valvola 0+1 MAMDANI no_change close_slow close_fast open_slow open_fast valvola 0+1 SUGENO

pag. 19 S. Marsili-Libelli: Controllo Fuzzy Controllo fuzzy del DO Si vuole mantenere costante la concentrazione di ossigeno disciolto basandosi sullerrore (DO sp – DO) ed agendo sul sistema di aerazione Si definiscono i seguenti qualificatori Errore DO: Negative Big (NB); Negative Small (NS), Zero (Z), Positive Small (PS), Positive Big (PB) Portata daria: Zero (Z), Small (S), Medium (M), Large (L), Very Large (LL) Nota: la portata daria è da intendersi come incremento e non valore assoluto Il regolatore fuzzy comprenderà le seguenti semplici regole: 1. If (DO is Z) then (Ua is M) (1) 2. If (DO is NB) then (Ua is Z) (1) 3. If (DO is PB) then (Ua is LL) (1) 4. If (DO is NS) then (Ua is S) (1) 5. If (DO is PS) then (Ua is L) (1)

pag. 20 S. Marsili-Libelli: Controllo Fuzzy Attivazione delle regole quando lerrore è zero, lincremento di uscita è anchesso zero Vista dellattivazione delle regoleVista della funzione di controllo (ingresso/uscita) questa curva rappresenta luscita (Ua) calcolata dal regolatore fuzzy in funzione dellerrore di DO.

pag. 21 S. Marsili-Libelli: Controllo Fuzzy Schema Simulink con controllo DO fuzzy (1+r)*q ricircolo DO (1+r)*q ricircolo errore 1/s X_int DO_err To Workspace1 Mono To Workspace Sum3 Sum2 Sum Saturation 1/s S_int Product Kd 1 s Integrator Mu*u[1]*u[2]/(Ks+u[1]) Fcn 0 Display [t_in,S_in] Dati di ingresso 1/s DO_int DO_Fuzzy DOsp DO set-point Csat DO sat Clock Kg s+a Attuazione areatori Ka Areazione (1-Y)/Y 1/Y2 q 1/Y1 -1/Y 1/Y modello del processo regolatore fuzzy DO_Fuzzy=readfis('DO_Fuzzy') integratore esterno

pag. 22 S. Marsili-Libelli: Controllo Fuzzy Prestazione del regolatore fuzzy BOD (mg/L) X H (mg/L) DO (mg/L) U a (m 3 /h) tempo (ore)

pag. 23 S. Marsili-Libelli: Controllo Fuzzy Attivazione delle regole durante la regolazione Regole Perc. Attivazione If (DO is Z) then (Ua is M) If (DO is NB) then (Ua is Z) If (DO is PB) then (Ua is LL) If (DO is NS) then (Ua is S) If (DO is PS) then (Ua is L) queste regole sono le meno usate perché quasi mai lerrore è grande (positivo o negativo)

pag. 24 S. Marsili-Libelli: Controllo Fuzzy Bibliografia Yager R.R. e Filev D.P. (1994) Essentials of Fuzzy Modelling and Control, Wiley. Klir. G.J. e T.A. Folger (1988) Fuzzy Sets, Uncertainty, and Information, Prentice-Hall. Ross T.J. (1995) Fuzzy Logic with Engineering Applications, McGraw-Hill. Driankov D, Hellendoorn H., Reinfrank M. (1993) An introduction to Fuzzy Control, Springer-Verlag. Patyra M.J. e Mlynek D.M. (editors) (1996) Fuzzy Logic, Implementation and Applications, Wiley Teubner. Nguyen H.T., Sugeno M., Tong R. Yager R.R. (editors) (1995) Theoretical Aspects of Fuzzy Control, Wiley. Yager R.R. e Zadeh L.A. (editors) (1994) Fuzzy Sets, Neural Networks, and Soft Computing, Van Nostrand Reinhold. Wang, L. X. (1994) Adaptive Fuzzy Systems and Control. PTR Prentice Hall.