4 th ITERATIOAL SYMPOSIUM on POWER ELECTROICS - Ee 27 XIV Međunaodni impozijum Enegeta eletonia Ee 27 OVI SAD, REPUBLIC OF SERBIA, ovembe 7 th - 9 th, 27 DESIG OF A ROBUST LIEAR COTROLLER FOR PROPER TRACKIG I FREQUECY DOMAI Jawad Faiz, Mohen Aezoomand * and Ghazanfa Shahgholian * Univeity College of abeei-aam, Tabiz, Ian *Depatment of Electical Engineeing, Ilamic Azad Univeity, ajafabad, Efahan, Ian Abtact: Thi pape peent a novel technique fo deigning a obut contolle in fequency domain. The deigned contolle ha chaacteitic of obut opeation uch a eliminating tubance, educing noie and enabling a pope tacing. Thi technique povide the obut opeation featue fo the poce having tuctued and untuctued uncetaintie. Key Wod: Robut tability/contolle deign/ Tacing capability/stuctued/untuctued uncetaintie. ITRODUCTIO Thee ae ome paamete in dynamic of many eal ytem that ae ometime fficult to detemine peciely and even pactically impoible. Mathematical model of the above mentioned ytem ha uncetainty called tuctued uncetainty. omally, thee paamete affect the ytem behavio ove low fequencie ange. The appoximation ued in modeling o egang the tuctue of the ytem ove high fequencie may eliminate a pat of ytem dynamic within the above mentioned fequencie ange; and thi i called untuctued uncetainty. Thee ae uncetaintie in many pactical ytem that mae fficult to achieve a pope tability and behavio fo cloed loop ytem. Thee ae two following geneal method to confont with uncetainty effect in the contol ytem: I. Adaptive contol in which the ytem paamete ae obtained imultaneouly and paamete of contolle ae then adjuted. II. Robut contol in which a fixed contolle fo woe cae of ytem (inclung uncetainty) i deigned []. In [2], the attactivene of vaiable tate method have been expeed fom mathematical point of view, but it i nown a incomplete in tate feedbac and width of loop band due to neglecting ome point uch a uncetaintie, noie, etc. It ha been alo emphaized that any effot in the contol ytem uing the tate vaiable can lead to the imila fault [3]. Apat fom veity o untuth of thi matte, quantitative feedbac theoy (QFT) i a obut contol technique which wa etablihed by Hoowitz [4]. The QFT i conideed a an efficient technique fo the ytem having tuctued, untuctued and complex uncetainty that ha been ued uccefully fo SISO and MIMO ytem [5]. In compaion with othe obut method (uch a H- infinity), thi method ha ome advantage uch a capability of eliminating tubance, obut tability, limited band width, capability of tudying ytem tanient, pemitting the deigne to etablih a quantitative balance between the contolle behavio and it complexity and having phae data duing the deign poce [, 4]. One of the obtacle in application of the QFT method fo deigning a contol ytem i that it i a gaphical pocedue. Thi dawbac ha been eolved by advancement of compute technology and it ha found vaiou application in aviation indutie, obotic and powe electonic [7, 8]. In pite of the above mentioned advantage of QFT, the capability of the method in tacing cloed loop ytem fo non tep input i not well etablihed. Thi pape peent a obut contolle deign method in the peence of uncetainty. In adtion to the ueful featue of the QFT inclung obut tability, eliminating tubance, noie eduction and limited band width, the method lead to a contolle that tac the non tep input well. Alo the method doe not equie pe filte which can be conideed a an advantageou. 2. STATEMET OF COTROL PROBLEM Conide Fig. whee G() i the tanfe function unde contol with tuctued uncetainty, K() i the contolle, e(t) i the ytem eo and α i the uncetainty vecto due to the ytem paamete defined a follow: Ω [ α,..., α ]; α [ α, α ]}, α Ω R min max q () q i The deign objective of contolle K() i uch that in pite of exiting uncetain paamete in tanfe function G(,, obut tability of cloed loop ytem ha been guaantied. In adtion, it i capable to tac popely, eliminate tubance and educe noie.
3. PROPER BORDER TRACKIG Tanfe function a input to eo within input fequencie domain i a follow: g e (2) + G(j ω, α )K(j ω ) In ode to tac popely, g e mut be mall enough within the input fequencie domain, in othe wod: + K(jω) pp, ω Ω, α Ω (3) whee Ω i the et of fequencie exiting in the input. It i defined fo a mall numbe of ε a follow: L i defined a follow: ω : (jω) Ω (4) L(jω, K(jω) (5) Theefoe, to be able to tac popely, Eqn. 3 can be ewitten a follow: (6) L >> ω Ω, α Ω 4. EFFECT OF OUTPUT DISTURBSCE O THE OUTPUT OF CLOSED LOOP SYSTEM Refeing to Fig., the tanfe function a output to input tubance i a follow: g doy (7) + K(jω) So, in ode to educe the influence of output tubance, g dy at the ytem output mut be a mall a poible, i.e.: << + K( jω) ω Ωdo, α Ω (8) whee Ω d i a et of fequencie content of the output tubance and defined a follow: Ω ω : d (jω) (9) do o conideing Eqn 3 to 6, Eqn. 8 i ewitten a follow: L >> ω Ωdo, α Ω () 5. EFFECT OF OUTPUT DISTURBSCE O THE OUTPUT OF CLOSED LOOP SYSTEM Refeing to Fig., the tanfe function i input d i to output tubance a follow: g y () + K(jω) fo eduction of the input tubance in the ytem output, g y mut be a mall a poible. In othe wod: a ω Ω, α Ω (2) Fig.. Bloc agam of cloed loop ytem whee Ω i a et of fequencie content of the input tubance and defined a follow: Ω ω: d (jω) (3) i The following mut be held in ode to atify Eqn. 2: + K(jω) >> (4) So, Eqn. 8 can be ewitten a follow: L >> ω Ω, α Ω (5) 6. EFFECT OF MEASURED OISE n i () O THE OUTPUT OF CLOSED LOOP SYSTEM Refeing to Fig., the tanfe function of output to meaued noie i a follow: g ny K(jω) (6) + G(j ω, α )K(j ω ) To educe the noie effect in the output, the following mut be atified: g ny << ω Ω, α Ω (7) whee Ω i a et of fequencie content of the meaued noie and defined a follow: Ω ω: n (jω) (8) i In othe wod, the following contion mut be atified fo noie effect eduction: K(jω) << + K(jω) Conequently: ω Ω, α Ω (9) G K(jω) << Ω α Ω (2) ω, Eqn. 2 can be ewitten a follow: L << ω Ω, α Ω (2) 7. SYSTEM SESITIVITY TO PARAMETERS VARIATIOS Refeing to Fig., the tanfe function of cloed loop ytem i a follow: 2
K(jω) T(jω, gy (22) + K(jω) Senitivity of the tanfe function of cloed loop ytem T veu poce tanfe function G i a follow: S T G dt G (23) dg T + a)k(jω) To educe the enitivity to paamete vaiation, the following contion mut be atified: Conequently: << + G ( jω, α )K ( jω ) (24) G K(jω) >> (25) So, in ode to educe the enitivity we have: L >> (26) Regang the contion given in ection 3, fo pope tacing and alo eliminating of the output and input tubance, it i enough to hold contion 6 in the input and output fequency domain. Theefoe, in ode to achieve the capability of tacing and eliminating input and output tubance, the following contion mut be atified: L >> ω Ω, α Ω Ω t t Ω U Ω U Ω (27) in the othe hand, fo eduction of noie Eqn. 22 mut be atified. omally Ω t and Ω contain a et of low fequencie and high fequencie epectively, theefoe Ωt and Ω ae not nomally ovelapped and thee i poibility of holng contion 27 in domain Ω t and contion 22 in domain Ω imultaneouly. Howeve, if Ω t and Ω ae ovelapped, elimination of tubance and capability of tacing by holng contion 27 i pactically moe efficient. In adtion, thi ovelapping can be educed by inclung weighted function W(). In fact, each oppoed contion 2 and 27, i meely atified in a paticula fequency domain. 8. ROBUST STABILITY BORDER (U CTOUR) An impotant objective in the deign of contolle i tabilizing cloed loop ytem; theefoe, K() mut be deigned uch that the tability contion i held. So, it i neceay to ue tool of detemining pope tability fo ichol chat. The aumption i that the poce ha both tuctued and untuctued uncetaintie, whee the tuctued and untuctued uncetainty appea ove low fequencie and high fequencie epectively. To include tuctued cetainty, a poce patten i ued, but to confont the untuctued uncetainty on the cloed loop tanfe function, the following limitation i impoed: L T λ (28) + L do Thi limitation lead to a fobidden egion in ichol chat called high fequency geneal bode o U contou. In fact, thi limitation (Eqn. 28) guaantee a paticula tability magin fo all poce family (with tuctued uncetainty). To obtain the length contou U, the poce tanfe function i conideed a follow: G(, m n + a + b +... + a + a m m (29) n n +... + b + b whee α[, a,, a m-, b,, b n- ] i the poce uncetainty vecto. Suppoe the poce nominal tanfe function i a follow: m + a +... + a + a G(, (3) n + b +... + b + b m m n n In fact, the length of contou U i equal to the length of poce patten in high fequencie, becaue the hape of poce patten fo uncetainty vecto become naowe ove highe fequency. Theefoe: G(, lim (3) jω G(, α ) So, the length of contou U in ichol chat i equal to: max db min db max min (32) db db Fo tabilization of cloed loop ytem having all tuctued and untuctued uncetaintie, the nominal open loop tanfe function mut not ente into U contou. Fig. 2 how a typical U contou. 9. DESIGIG PROPOSED COTROLLER In thi ection, contolle K() of Fig. i deigned uch that in adtion to obut tability of the cloed loop ytem, the equied featue given in ection I to III ae held. t Step: The non compenated ichol chat i plotted fo ffeent value of uncetaintie a uch that the uncetainty behavio i obevable on the ichol chat. ow the nominal poce G i elected a that it ha a minimum value. 2 nd Step: the bode elated to the obut tability, tacing and eliminating tubance ae pecified on ichol chat. 3d Step: A numbe of typical fequencie i elected in the ange of Ω t a uch that it cove the ange of fequency Ω t. Then the point due to the et of elected fequencie on G in ichol chat ae defined. 4 th Step: The nominal open loop ytem tanfe function i loop haping uch that the elevant bode ae atified. 3
Open-Loop Gain (db) 4 3 2 - -2-3 -4-5 db.25 db.5 db -6 - -35-3 -25-2 -5 - -5 db - db - - Fig. 2. A geneal bode of high fequency -2 db -2 db -4 db Finng a uitable L depend on the ill and expeience of the deigne. Geneally deigning begin by open loop tanfe function haping with the poce of nominal tanfe function. Then the gain equied ove low and intemeate fequencie ae added. In the next tep, it i tied to put L in the popoed ange uing contolle Lag, Lead and Lag Lead. Of coue, altenative method uch a Genetic algoithm and othe optimization outine can be ued fo loop haping. Fo a bette deign, the following ae ecommended:. In ode to have poitive gain and poitive magin, L mut be placed in the ight hand ide of contou U. 2. To confine the band width, educe the noie effect and confont with untuctued uncetaintie ove high fequencie, it i neceay that L touche U contou while contou U otate fom ight to the left. 3. To minih the teady tate eo againt tep unit, L mut be deigned uch that / exit within the nominal open loop tanfe function. It i noted that thi i tue only fo tep input and fo othe input a obtained in the imulation, the eo can be educed by atifying the contion L >> (depenng the deigne equiement). Theefoe, the objective i not minihing the eo fo non tep input but the intent i to educe the eo. In ode to confont with the noie effect, it i neceay to chooe the band width a mall a poible, in othe wod to chooe the fequency epone lage than the defined bode fo elimination of tubance and tacing capability ove low fequency (Ω t ), and lowe than the defined bode fo noie eduction ove high fequencie (Ω ). Alo fo obut tability, the epone of the nominal ytem fequency mut not be placed within contou U. A an example, uppoe the poce tanfe function i a follow: G() (33) ( + a) whee a, Є[, ]. The objective i deigning a obut contolle uch that it ha capability of input tacing in the fequency domain Ω Є[,3] and alo eliminating the tubance in domain Ω Ω do Ω. In thi cae, the intention i to have 3dB obut tability bode of cloed loop ytem fo contou U at vaiation of all uncetaintie a and. The ichol chat of non compenated ytem fo vaiation of uncetaintie ha Open-Loop Gain (db) To: Y() 4 2-2 -4-6 -8 ichol Chat Fom: U().25 db.5 db db db - -4-35 -3-25 -2-5 - -5 G - db - - -2 db -2 db -4 db Fig. 3. Diagam of ichol non compenated ytem fo ffeent value of uncetaintie Open-Loop Gain (db) To: Y() 8 6 4 2-2 -4-6 -8 ichol Chat Fom: U().25 db.5 db db db - -4-35 -3-25 -2-5 - -5 Fig. 4. Shaped nominal tanfe function - db - - -2 db -2 db -4 db been hown in Fig. 3. In thi cae, the elected nominal tanfe function G i the mallet agam fom amplitude point of view. So: G () (34) ( + ) In Fig. 4, fequency epone G ha been plotted and given fo a numbe of typical fequencie; it how the tacing capability, tubance elimination and noie eduction bode. In thi example, it i aumed that thee ae the capability of 2 db bode tacing and eliminating tubance and 2 db noie eduction bode. In thi figue, afte detemination of the elevant bode, the ated open loop tanfe function L (jω) i haped though a tial and eo outine a uch that the popoed contion ae atified. The obtained value of L (jω) i a follow: 324( +.342)( +.866) L () (35) ( +.45)( +.286)( + ) 4
whee value of K() i equal to: 324( +.342)( +.866) K() (36) ( +.45)( +.286) Fig. 5 how the cloed loop ytem epone at ffeent uncetainty value fo the following input:.4 Linea Simulation Reult. COCLUSIOS Thi pape peented a novel technique fo deigning a cloed loop ytem that i capable to tac, eliminate tubance and educe the noie in peence of tuctued and untuctued uncetaintie. Thi technique in fequency domain i on the ichol chat that ha the efficient deign featue with QFT method; the output of cloed loop ytem i alo capable to tac the input..2 Step Repone Amplitude To: Y().2.8.6.4 Amplitude.8.6.4.2.2 -.2 5 5 2 25 Time (ec.) Fig. 5. Cloed loop ytem epone fo ffeent uncetainty value -3 x 4.5 4 3.5 3 2.5 2.5.5 Input Ditubance Repone.2.4.6.8.2.4.6.8 2 Fig. 6. Sytem epone at ffeent uncetainty value.4.4 (t) + co( t) + co( 2t) + co(3t) (37) 4 π π 3π Thi figue how that the cloed loop ytem ha tacing capability fo the popoed input. Fig. 6 peent the ytem output epone to input tep tubance fo ffeent uncetainty value. Fig. 7 peent the capability of ytem tubance elimination fo unit tep input at ffeent uncetainty value. -.2..2.3.4.5.6.7.8.9 Time (ec.) Fig. 7. Capability of ytem tubance elimination fo unit tep input at ffeent uncetainty value. REFERECES [] A.C.Zolota, G.D.Haliia, "Optimal deign of PID contolle uing the QFT", IEE Poc. Contol Theoy Appl., Vol. 46, o. 6, pp.585 589, 999. [2] S.F.Wu, W.Wei, M.J.Gimble, "Robut MIMO contol ytem deign uing eigentuctue aignment and QFT", IEE Poc. Contol Theoy Appl., Vol. 5, o. 2, pp.98 29, 24. [3] I.Hoowitz, U.Shaed, "Supeioity of tanfe function ove tate vaiable method in linea time invaiant feedbac ytem deign", IEEE Tan. on Automatic Contol., Vol. AC 2, o., pp.84 97, 975. [4] I.Hoowitz, "Suvey of quantitative feedbac theoy (QFT)", Int. J.Contol, Vol. 53, o. 2, pp.255 29, 99. [5] I.M.Hoowitz, "Quantitative feedbac deign theoy (QFT)", QFT Publication, Vol., 993. [6] C.Bohgeani, Y.Chait, O.Yaniv, "Quantitative feedbac theoy toolbox", The MATH WORKS Inc.994. [7] Z.Y.Wu,.Schofield,C.M.Bingham, D.Howe, D.A.Stone, "Deign of obut cuent tacing contol fo active powe filte", Electic Machine and Dive Confeence, 2. IEMDC, pp.948 953, 2. [8] M.Kapeno,.Sepehi, "Fault toleant contol of a evo hydaulic poitioning ytem with co pot leaage", IEEE Tan. on Cont. Syt. Techn., Vol.3, o., pp.55 6, 25. 5