Sliding Mode Based Self-Tuning PID Controller for Second Order Systems

Transcription

1 Proceeng of the Internatonal Symoum of Mechanm an Machne Scence, 07 AzC IFToMM Azerbajan Techncal Unverty -4 Setember 07, Bau, Azerbajan Slng Moe Bae Self-Tunng PID Controller for Secon Orer Sytem Aler BAYRAK Deartment of Electrcal & Electronc Engneerng Abant Izzet Bayal Unverty Goloy, Bolu 480 TURKEY E-mal: Abtract In th aer, a lng moe bae elf-tunng PID controller rooe for econ orer ytem. Whle evelong the controller, t aume that the ytem moel ha a art whch contan nonlnear term mlar to PID tructure. The controller an uate rule for PID arameter are obtane from Lyaunov tablty analy. Numercal mulaton are conucte on a Twn-Rotor Mult-Inut Mult-Outut Sytem (TRMS moel to how the erformance of the rooe controller. Keywor: Slng Moe Controller, Self-Tunng PID.. Introucton PID control the mot referre control technque n nutral alcaton ue to t mle tructure an convenence n mlementaton []. However, the effectvene of the PID controller bae on the accurate electon of t arameter. Dete the goo erformance reult n lnear ytem, the electon of the arameter mght be very ffcult an tme watng wth the re of nonlnearte of the ytem. To eal wth th roblem many aroache of elf-tunng PID controller have been reente tll toay. Thee aroache can be ve nto two man categore: moel bae aroache an rule-bae aroache. In moel bae aroache, the tunng mechanm bae on the nowlege of the ytem moel []. In rule bae aroache, the tunng bae on ome otmzaton or etmaton rule wthout moel nowlege, whch bacally mmc an exerence oerator behavor []. A goo urvey can be foun n [] on th toc. In the lterature, many tue can be foun on elftunng PID controller an t alcaton. In [3], An et. al. reente a elf-tunng metho for PID controller bae on the theory of aatve nteracton for the quarotor ytem. In [4], a elf-tunng PID control cheme bae on uort vector machne (SVM an artcle warm otmzaton (PSO were reente. Jang an Jang rooe a fuzzy bae elf-tunng PID controller for temerature control [5]. Zheng et. al. ue fuzzy moule to tune PID controller arameter accorng to the error an change n error [6]. In [7] an [8], genetc algorthm wa utlze to tune the PID arameter. Na reente a tuy on water level control of a nuclear team generator wth PID controller of whch arameter were tune by moel rectve control (MPC [9]. In [], leat quare uort vector machne entfer wa utlze to tune arameter of PID controller. Fan et. al. ue neural networ to tune PID controller for oton tracng of a neumatc artfcal mucle [0]. Gunogu an Komurgaz reente a elf-tunng algorthm for PID controller bae on aatve nteracton aroach []. In [], Howell an Bet ue contnuou acton renforcement learnng automata (CARLA metho to tune the PID controller arameter whle controllng engne le-ee. In [3], Shh an Teng egne a elf-tunng PID controller by ung ntegral of tmeweghte abolute error (ITAE otmal control rncle an the ole-lacement aroach to control oton of a ervo-cylner. Dong an Mo reente moel reference aatve PID controller for motor control ytem wth baclah [4]. In [5], Chama et. al. reente an aatve PID controller combne wth lng moe controller for uncertan nonlnear ytem. Chang an Yan rooe an aatve PID controller bae on lng moe controller for uncertan chaotc ytem [6]. Kuo et. al. reente an aatve lng moe controller wth PID tunng metho for a cla of uncertan ytem [7]. In th aer, a lng moe bae elf-tunng PID controller rooe for uncertan econ orer ytem. Dfferent from the lterature, t aume that the moel contan nonlnear term mlar to PID tructure. The controller an uate rule for PID arameter are obtane from Lyaunov tablty analy. Numercal mulaton are conucte on a Twn-Rotor Mult-Inut Mult-Outut Sytem (TRMS moel to tet the erformance of controller an arameter uate rule. The ret of the aer organze a follow; the ytem moel reente n Secton. Control an arameter uate rule egn are reente n Secton 3. Numercal mulaton reult are gven n Secton 4. 93

2 Proceeng of the Internatonal Symoum of Mechanm an Machne Scence, 07 AzC IFToMM Azerbajan Techncal Unverty -4 Setember 07, Bau, Azerbajan Fnally conclung remar are reente n Secton 5.. Sytem Moel The followng econ orer ytem conere n th aer, ( x ( t f (x u( ( x ( t x t x(t = [x (t, x (t] T tate vector, u(t R control gnal. The functon f( :R R aume n the form of f ( t x g( x x ( t x ( t x (. (3 g( :R R unnown functon,, an are unnown ytem arameter. Aumton : It aume that the functon g( boune a (x g (4 ρ nown. Aumton : It aume that the ytem arameter,, an are n nown boune regon. Aumton 3: It aume that x(t avalable an contnuou. 3. Control an Parameter Uate Rule Degn The objectve of the controller to utlze that trac a ere trajectory whle uatng PID arameter. To acheve th objectve, the error ytem egne a follow, x ( t x ( t x x (5 x ( t x x (6 x an x are ere trajectore. To contruct lng moe controller, the fltere error gnal egne a x x x. (7 The ervatve of (7, whch wll be utlze later, x x x x g x x x u x x. (8 x The control nut egne a u u PID u R (9 u R lng art of the controller an u ˆ x ˆ x ˆ x (0 ˆ, PID ˆ an ˆ are etmate of, ve, reectvely. By ubttutng the (9 an (0 n (8, t obtane a x g x x x ˆ x x x x x x ˆ ˆ ˆ ur (, ˆ, ˆ. ( The Lyaunov functon n (3 utlze to contruct uate rule for PID gan an egn u R V. (3 The ervatve of (3 obtane a V ( x g ˆ x ˆ x ˆ x ur x ( ˆ x x x ˆ ( x ( x ˆ (4 From (4, the uate rule of ˆ, an ˆ are electe a n (5 to elmnate the term wth gan error. ˆ x, ˆ x, ˆ x (5 After ubttuton of (5 n (4, V obtane a ˆ V x ( x x g ˆ x ˆ x ˆ x u ( (6 The nut gnal u R houl be egne to mae V negatve. To acheve th uroe, u R wll be nvetgate by earatng nto three term a u R u u u. (7 3 u, egne a to elmnate frt two term n (6 a u x x x gn(, R R (8 To elmnate the term g n (6, the conton n aumton can be utlze. From (4 the followng nequalty can be obtane g, R (9 By ung (9, u egne a follow 94

3 Proceeng of the Internatonal Symoum of Mechanm an Machne Scence, 07 AzC IFToMM Azerbajan Techncal Unverty -4 Setember 07, Bau, Azerbajan a, u. (0 By ubttutng (8 an (0 n (6, V obtane ( V L u 3 L ˆ ˆ ˆ x x x. ( An uer boun for L can be efne a L x x x. (3, m an are uer boun of, an, reectvely. Hence, the followng nequalty can be wrtten L (4 L m Remar : In (3, L m may go to nfnty for x 0 nce ntegral term. But t houl be et n mn that the man nterete term L m. So f t can be roven that (t converge to 0, fat enough, then, t can be aume that the term L m tay boune. So L m can be accete a boune. In the ret of the aer, t wll be roven that (t converge to 0 wth a tunable rate. From (4,. (5 V L So, u 3 can be obtane a Th lea u L m m u 3 3 (6 V. (7 From (7, t can be a that,, an ˆ are boune. To how that (t goe to zero wth reect to tme, (t houl be nvetgate n ee by tang the tme ervatve of a t ˆ ˆ ( x g x x x u x x x. (8 By ubttutng (9 an (5 n (8, t obtane a, ( x x x t. (9 If electe a x x x, an reectvely, (9 obtane a t whch lea (30 (3 (3, (33 are lower boun of, an, (34. (35 From (35, t een that tartng from any ntal conton, the tate trajectory reache to the urface n a fnte tme maller than an then ( t 0 / converge to x (t exonentally wth a tme contant equal to /λ [8]. 4. Numercal Smulaton The erformance of the control law n (9 an uate rule n (5 were evaluate by conuctng numercal mulaton by ung the ynamc moel of a -DOF helcoter whch nown a TRMS. Durng the mulaton, the arameter value of nut gnal were electe a λ=ag(30,30, =ag(,, ρ=[ ] T, L m = [ ] T. The ntal value of gan etmate were et to =[5 5] T, =[5 5] T an =[5 5] T. The ntal oton of the axe were x(0=[ ] T n raan an the ere oton were electe a x =[ ] T n raan. In the numercal mulaton, t wa oberve that the control law erforme atfactorly. The oton error an the control nut of yaw an tch axe are reente n Fgure,, 3 an 4, reectvely. The PID gan etmate are gven n Fgure 5, 6 an 7. A can be een n the fgure, both the yaw an the tch error are rven to the vcnty of zero. 95

4 Proceeng of the Internatonal Symoum of Mechanm an Machne Scence, 07 AzC IFToMM Azerbajan Techncal Unverty -4 Setember 07, Bau, Azerbajan egnng the controller, t wa aume that the ytem moel contan nonlnear term mlar to PID tructure. The controller an uate rule for PID arameter were obtane from Lyaunov tablty analy. The effectvene of the controller an uate rule were evaluate by conuctng numercal mulaton an acheve atfactory reult. Fg.. Yaw ax oton error Fg. 4. Inut gnal for tch ax Fg.. Ptch ax oton error Fg. 5. K etmate for yaw ax (to an tch ax (bottom Fg. 3. Inut gnal for yaw ax 5. Concluon In th aer, a lng moe bae elf-tunng PID controller wa egne for econ orer ytem. Whle 96

5 Proceeng of the Internatonal Symoum of Mechanm an Machne Scence, 07 AzC IFToMM Azerbajan Techncal Unverty -4 Setember 07, Bau, Azerbajan Fg. 6. K etmate for yaw ax (to an tch ax (bottom Fg. 7. K etmate for yaw ax (to an tch ax (bottom Reference [] S. Wanfeng, Z. Shengun, an S. Yajng, Aatve PID controller bae on onlne LSSVM entfcaton, n Int. Conf. on Avance Intellgent Mechatronc, July 008, age [] K. J. Atrom an T. Hagglun, PID Controller: Theory, Degn an Tunng, NC, USA: Intrument Socety of Amerca, 995. [3] An, S., Yuan, S., L, H., Self-tunng of PID controller egn by aatve nteracton for quarotor UAV, n Proc. Of IEEE Chnee Guance, Navgaton an Control Conference, Nanjng, Chna, Augut 06, age [4] J. Lu, C. Yang, B. Peng, R. Wan, X. Han, an Ma, Self-tunng PID Control Scheme wth Swarm Intellgence Bae on Suort Vector Machne, n Proc. of IEEE Int. Conf. on Mechatronc an Automaton, Tanjn, Chna, Augut 04, age [5] K. Jang, W. Jang, an X. Jang, Degn of an ntellgent temerature control ytem bae on the fuzzy elf-tunng PID, Procea Engneerng, vol. 43, age 307 3, 0. [6] J. mng Zheng, S. un Zhao, an S. guo We, Alcaton of elf-tunng fuzzy PID controller for a SRM rect rve volume control hyraulc re, Control Engneerng Practce, vol. 7, no., age , 009. [7] C. L an J. Lan, The alcaton of mmune genetc algorthm n PID arameter otmzaton for level control ytem, n IEEE Int. Conf. on Automaton an Logtc, Aug 007, age [8] Sreeanth, P., Har, A., Genetc Algorthm Bae Self Tunng Regulator for ball an hoo ytem, n Proc. Of Conference on Emergng Devce an Smart Sytem, Tamlnau, Ina, March 06, age [9] M. G. Na, Auto-tune PID controller ung a moel rectve control metho for the team generator water level, IEEE Tran. on Nuclear Scence, vol. 48, no. 5, age , Oct 00. [0] J. Fan, J. Zhong, J. Zhao an Y. Zhu, BP neural networ tune PID controller for oton tracng of a neumatc artfcal mucle, Technology an Health Care, vol. 3, no., age S3 S38, 05. [] T. Gunog an G. Comoro, Aatve PID controller egn by ung aatve nteracton aroach theory, n Int. Conf. on Electrc Power an Energy Converon Sytem, Oct 03, age 5. [] M. Howell an M. Bet, On-lne PID tunng for engne le-ee control ung contnuou acton renforcement learnng automata, Control Engneerng Practce, vol. 8, no., age 47 54, 000. [3] M.-C. Shh an S.-I. Teng, Pneumatc ervocylner oton control by PID-elf-tunng controller, JSME nter. journal. Ser. C, Dynamc, control, robotc, egn an manufacturng, vol. 37, no. 3, age , 994. [4] J. Dong an B. Mo, The aatve PID controller egn for motor control ytem wth baclah, n Inter. Conf. on Intellgent Control an Informaton Proceng, June 03, age [5] T. Chama, P. Jrawattana, an T. Rauee, Robut aatve PID controller for a cla of uncertan nonlnear ytem: An alcaton for ee tracng control of an SI engne, Mathematcal Problem n Engneerng, 05. [6] W.-D. Chang an J.-J. Yan, Aatve robut PID controller egn bae on a lng moe for uncertan chaotc ytem, Chao, Solton & Fractal, vol. 6, age 67 75, 005. [7] T. C. Kuo, Y. J. Huang, C. Y. Chen, an C. H. Chang, Aatve lng moe control wth PID tunng for uncertan ytem, Engneerng Letter, vol. 6, no. 3, age 3 35, 008. [8] J.-J. E. Slotne an W. L, Ale Nonlnear Control. Englewoo Clff: NJ:Prentce Hall,