Fuzzy Control of a Nonlinear Deterministic System for Different Operating Points

Transcription

1 International Jornal of Electrical & Compter Sciences IJECS-IJE Vol: No: 9 Fzz Control of a Nonlinear Deterministic Sstem for Different Operating Points Gonca Ozmen Koca, Cafer Bal, Firat Universit, Technical Edcation Faclt Department of Electronics and Compter Science 39 Elazig, Tre Tel: Ext: 9 Fax: gozmenoca@gmail.com, caferbal@gmail.com Abstract In this paper, a robst control sstem with the fzz logic approach is proposed. Two linear controllers which have a good performance in the pper and lower operating points are designed in order to control a nonlinear deterministic sstem. The performances of linear controllers are investigated in different operating points. A fzz logic controller, generall perred when the linear controllers are not able to control the sstem with a satisfactor performance, is examined for the nonlinear sstem in different operating points. The obtained reslts in the Matlab/SIMULINK environment show the efficienc of the proposed fzz controller for the nonlinear sstem withot the readjstment of the control parameter for each different operating point. Index Term Fzz logic, nonlinear sstem, robst control. I. INTRODUCTION Since 9, fzz set theor introdced b zadeh [] has a great field of implementation in real engineering applications. A Fzz Logic Control () ensres an appropriate control methods for variet control problems becase of providing an effective method for constrcting linear and nonlinear controllers via the se of heristic information [,]. It is ver important to cope with parameter variations and external noises in man indstrial plants [3,]. A can be perred to ensre robstness against parameter variations and external distrbances since it permits the control law to change [-]. A is also sed to provide efficient control when the precise mathematical models of the plants are not obtainable or the closed loop control problems are ver complex. Fzz controllers are especiall sed for nonlinear control problems in the events of having no good performance of linear controllers or need to different linear controllers in the different conditions [9,]. When the closed-loop sstem behavior changes in different operating point, the different control parameters ma give an adeqate otpt response in each point. Since a sstem ses fzz sets which are described b membership fnctions, it can control the sstem with a satisfactor performance over the different operating point []. In this paper, a is proposed to control a nonlinear deterministic sstem which has different behaviors in different operating points. The non-linear sstem can be controlled b sing linear controller bt it doesn t provide good performance in ever operating point and different linear controllers mst be design for each point. Theore, the sstem can be perred to ensre satisfactor performance in different operating points of the nonlinear deterministic sstem. In the paper, mathematical eqation of the nonlinear sstem is given in section-, and the design of two linear controllers in two operating points is presented in section-3. Then, the design of the proposed fzz controller to provide good performance in different operating points is given in section-. Simlation reslts are given in section- to illstrate the performance of the controller. Finall, in section-, a conclsion smmarizes the effectiveness of the proposed controller. (Write the date on which o sbmitted or paper for review.) This wor was spported in part b the U.S. Department of Commerce nder Grant BS3 (sponsor and financial spport acnowledgment goes here). Paper titles shold be written in ppercase and lowercase letters, not all ppercase. Avoid writing long formlas with sbscripts in the title; short formlas that identif the elements are fine (e.g., "Nd Fe B"). Do not write (Invited) in the title. Fll names of athors are perred in the athor field, bt are not reqired. Pt a space between athors initials. F. A. Athor is with the National Institte of Standards and Technolog, Bolder, CO 3 USA (corresponding athor to provide phone: 33-- ; fax: 33--; athor@ bolder.nist.gov). S. B. Athor, Jr., was with Rice Universit, Hoston, TX 77 USA. He is now with the Department of Phsics, Colorado State Universit, Fort Collins, CO 3 USA ( athor@lamar.colostate.ed). T. C. Athor is with the Electrical Engineering Department, Universit of Colorado, Bolder, CO 39 USA, on leave from the National Research Institte for Metals, Tsba, Japan ( athor@nrim.go.jp). II. MATHEMATICAL MODEL OF THE CONTROLLED NOINEAR DETERMINISTIC SYSTEM Let s consider a nonlinear deterministic sstem having different behaviors at different operating points: x x x xx x x where is the control inpt and, are the state variables. The otpt eqation of the nonlinear sstem is given b () -99 IJECS-IJE Agst IJE

2 International Jornal of Electrical & Compter Sciences IJECS-IJE Vol: No: x The sstem described b eqation () and () can be linearized [] abot the operating point in order to show how the sstem needs different linear controller at different operating points. The linearized sstem can be given b ( s )( s ) where is the desired otpt of the sstem which can be assmed as the otpt at operating point. It can be clearl seen from eqation (3) that the different control action is reqired at different vale of the becase of changing sstem pole. () (3) different operating points. IV. FUY LOGIC CONTROL The bloc diagram of a sstem for the nonlinear deterministic sstem is given in Fig.. Inpt variables of the are error, change-of-error and the otpt variable is. The error at sing th sampling time can be expressed e( ) ( ) ( ) () where is the desired otpt and is the otpt of the sstem. The change of error can be given b e( ) e( ) de( ) () T s III. DESIGN OF LINEAR CONTROLLERS FOR TWO OPERATING POINTS The bloc diagram of a linear control sstem for the nonlinear deterministic sstem is given in Fig.. An antiwindp integrator strctre is sed to stop over-integration [3]. The operating region of sstem can be chosen as between and 3. Different linear controllers are designed for these pper and lower limit vales of the operating region in order to obtain an otpt response withot overshoot and a settling time (%) less than.s. Linear controllers which are proposed for two operating points can be given b where is the error vale of the previos sampling time and Ts is the sampling period. The membership fnctions for the inpt and otpt of the sstem are shown in Fig.3. The ranges of the inpt membership fnctions are chosen according to maximm vale of the error and change of error that ma occr. The ranges of the otpt membership fnctions are determined b sing bac propagation algorithm []. These ranges are [-3 3] for e, [- ] for de and [- ] for the otpt variable.. The rle base of the sstem is given in Table and derived from the relationship between inpts and otpt of the z - e /Ts de Fzzifier Rle Base and Inference The Linear Controller Δ Anti-windp The Nonlinear Integrator Sstem Defzzifier Δ Anti-windp Integrator The Nonlinear Sstem 3( z.99)( z.997) G ( z )( z.) (.a) ( z.)( z.99) G 3 ( z )( z.) (.b) The first linear controller is designed to obtain satisfactor performance near to and the second linear controller similarl is designed to provide good performance near to 3. The nmber of controllers can be increased to Fig.. The strctre of a Linear Control Sstem obtain satisfactor performance in each operating points between and 3. It is clear that the Fzz Logic Control can be proposed to compte accrate control action for ever operating point b sing membership fnctions instead of sing different linear controllers at sstem. The th rle has the form of IF e is de is Fig.. The strctre of a Fzz Control Sstem F e AND F de THEN is F e and singleton otpt membership fnctions. w (,..., M ) (7) F de are the interval tpe- fzz sets and w is -99 IJECS-IJE Agst IJE

3 International Jornal of Electrical & Compter Sciences IJECS-IJE Vol: No: NB NK YS PK PB (a) NB NK YS PK PB e, de linear controllers are sed for two operating points ( and 3 ). It is clearl seen from the figres that for the first linear controller and for 3 the second linear controller provide satisfactor performances. On the other hand, the oscillator otpt response to a step erence is obtained b sing the second linear controller. In this wa, as -.. (b) Δ Fig. 3. a) Inpt membership fnctions b) otpt membership fnctions for the sstem. In the defzzifier, a crisp otpt is obtained from the resltant fzz set b sing center of area (COA) method. This method can be expressed as de TABLE I THE RULE BASE OF THE SYSTEM.. Fig.. The otpt () to a step erence e O PL PL PL PL PL PL PL 3 M n M n n n n () where n is the center of the n th implied otpt fzz set and n is the membership degree of. Finall, the crisp otpt of the change of the control signal is obtained with defzzification process. After this nit, an anti-windp integrator strctre is sed to stop over-integration []. V. SIMULATION RESULTS The simlation sampling period is choosen as 3 T s,x sec. Simlations are performed sing a Rnge- Ktta fifth order integration method. Firstl, the closed loop sstem given in Fig. is simlated for the well tned linear control method. The responses of the otpt () are shown in Fig. and Fig.. The designed two Fig.. The otpt () to a step erence 3 seen from the Fig., the performance of the first linear controller for step erence 3 is not satisfactor. This means that different linear controllers shold be design for ever operating point to have a satisfactor performance for the nonlinear deterministic sstem. -99 IJECS-IJE Agst IJE

4 International Jornal of Electrical & Compter Sciences IJECS-IJE Vol: No: G 3 Secondl, the is proposed to ensre satisfactor performance in ever operating point between the pper and lower operating limits. As shown in Fig. 3 trianglar membership fnctions are sed for inpt variables and singleton membership fnctions are sed for the otpt variable respectivel. The range of the membership fnctions are determined b sing bac propagation algorithm. The performances of the sstem are illstrated in Fig. for and 3. G Similarl, the performance of the proposed sstem and 3.. G 3 Fig. 7. The otpt () responses of the closed loop control sstems to a step erence ( ). G (a) 3 Fig.. The otpt () responses of the closed loop control sstems to a step erence ( 3 ) (b) Fig.. The otpt () response of the closed loop sstem to a step erence a), b) 3 two linear controllers are illstrated for different operating points between the limit vales in the Fig.7, Fig., Fig.9 and As seen in Fig. 7,, 9,, the proposed sstem ensres satisfactor performance for the given operating points while the linear controllers reslt in nacceptable control performance. VI. CONCLUSION This paper considers the controller design to obtain satisfactor performance at ever operating point between the pper and lower operating limits for a nonlinear deterministic sstem. The sstem can be controlled b designing different linear controllers for different operating points bt this is not practical becase of time consming. The fzz logic control approach is investigated to ensre good performance all over the operating region b sing the error and the change of the -99 IJECS-IJE Agst IJE

5 International Jornal of Electrical & Compter Sciences IJECS-IJE Vol: No: 3 Fig. 9. The otpt () responses of the closed loop control sstems to a step erence ( ) G 3 G 3 G [] R. Palm, D. D. Drianov, and H. Hellendoorn, Model Based Fzz Control, Springer-Verlag, 997. [] L. X. Wang, A Corse in Fzz Sstems and Control, Prentice Hall, 997. [] J.M. Mendel, Uncertain Rle-Based Fzz Logic Sstems: Introdction and New Directions, Upper Saddle River, NJ: Prentice Hall PTR,. [7] J.J.E. Slotine, and W. Li, Applied Nonlinear Control, Englewood Cliffs, NJ: Prentice-Hall, 99. [] G. F. Franlin, J. D. Powell, and A. E Naeini, Feedbac Control of Dnamic Sstems, Addison-Wesle, 99. [9] B. J. Choi, S. W. Kwa, and B. K. Kim, Design of a Single-Inpt Fzz Logic Controller and Its Properties, Fzz Sets Ss.,, pp. 99-3, 999. [].H. Apolat and G.M. Asher, A practical approach to the design of robst speed controllers for machine drives, IEEE Trans. on Indstrial Electronics, 7(), pp. 3-3,. [].H. Apolat, G.M. Asher and J.C. Clare, Eqivalence of fzz and classical controllers: An approach to fzz control design, th Eropean Conference on Power Electronics and Applications (EPE 99), CD-ROM 9-7--, Lasanne, Switzerland, Sep., 999. []. H. Apolat, Application of Fzz-Sliding Mode Control and Electronic Load Emlation to the Robst Control of Motor Drives, Phd Thesis, Universit of Nottingham, 999. [3] G. O. Koca, and. H. Apolat, Blanı mantı ve erişim ralı alaşımını llanara bir eletrili aracın daanılı hız denetimi, Gazi Üniversitesi Teni Eğitim Faültesi Politeni Dergisi, 9( ), pp. 93-3,. [] M. Brown, and C. Harris, Nerofzz Adaptive Modelling and Control, Prentice Hall, 99. G Fig.. The otpt () responses of the closed loop control sstems to a step erence ( 3 ) error as inpt variables. Ths, the proposed fzz logic control method provides robst control action instead of designing linear controller for each different operating point. Gonca Ozmen Koca, received the Ph.D. degree in Electric and Electronic Engineering Department of Firat Universit. Her interested areas are control of electrical drives, fzz logic, sliding mode control, intelligent control techniqes and mechatronics sstems. Cafer Bal, received the B.Sc and M.Sc degrees in electronic and compter edcation from Firat Universit, Elazig, Tre, in 997 and respectivel. He is crrentl woring towards to the Ph.D degree in electrical and electronic engineering and he is a lectrer at the same niversit. He research interests inclde speed sensorless control of electrical drives, artificial neral networs and fzz control, edcational tools and virtal laboratories. The performances of the controllers are comparativel given in the simlation reslts. The proposed control method is not onl examined at limits of the operating points bt also its performance is investigated at for points between the pper and lower limits in the operating region. The obtained reslts verif the robst control performance at ever operating point for the proposed fzz logic controller. REFERENCES [] L. A. adeh, Fzz Sets, Information and Control, (3), pp , 9. [] K. M. Passino, Fzz Control, Addison Wesle Longman, 99. [3] D. D. Drianov, H. Hellendoorn, and M. Reinfran, An Introdction to Fzz Control, Berlin-Heilderberg, German:Springer-Verlag, IJECS-IJE Agst IJE