Fractional Order Disturbance Observer based Robust Control

Transcription

1 201 Inernainal Cnference n Indusrial Insrumenain and Cnrl (ICIC) Cllege f Engineering Pune, India. May 28-30, 201 Fracinal Order Disurbance Observer based Rbus Cnrl Bhagyashri Tamhane 1, Amrua Mujumdar 2 and Dr. Shailaja Kurde 3 1 Deparmen f Elecrical Engineering, Cllege f Engineering Pune, India, 2 PEIM, CME, Pune and PhD Suden a Cllege f Engineering Pune, India, 3 Deparmen f Elecrical Engineering, Cllege f Engg, Pune, India 1 aamujumdar@yah.cm, 2 Cbhagyashri@yah.c.in, 3 srk.elec@cep.ac.in Absrac: Cnrl under heavy uncerain cndiins is ne f he ms impran cnrl prblems in ineger as well as fracinal rder sysems. This paper invesigaes a fracinal rder disurbance bserver based n full sae infrmain. This wrk is he exensin f ineger rder disurbance bserver in 111 fracinal dmain. An exra cmpensar based n esimaed disurbance is added he classical sae feedback cnrl. Effeciveness f he prpsed scheme is illusraed by simulain resuls. I. INTRODUCTION Advanced cnrl echniques are researched and being applied fr varius plans encmpassing indusrial cnrl space echnlgy. These advanced cnrl mechanisms are being sudied by researchers wih an bjecive prvide rbus and precise perfrmance wih pimum cnrl effrs. T achieve his aim, rbus cnrl echniques like H CXJ cnrl, feedback linearizain, sliding mde cnrl ec. are being researched since las few decades. Anher apprach which has evlved fr rbus cnrl is esimain f disurbances in he sysem and heir cmpensain. Tha is a signal represening he disurbance and uncerainy is apprximaed and cmprmised by deducing in he cnrl inpu. Varius appraches like unknwn inpu bserver (UIO), perurbain bserver (PO), disurbance bserver ( O), exended sae bserver (ESO), ime delay cnrl (TDC), uncerainy and disurbance esimar (UDE) ec. have been invesigaed ver he years fr disurbance and uncerainy esimain. Ample research is being carried u by varius researchers in mdifying and implemening hese mehdlgies. The disurbance esimain apprach is used nly fr disurbance rejecin, and he verall perfrmance f he clsed lp sysem depends n he nminal cnrl. Therefre, he implemenain f cnrl requires w lps in which he inner lp is jus esimae f he disurbance and he uer lp f cnrl cnains he nminal cnrl which is required achieve he desired perfrmance f he sysem. Fracinal calculus is a name fr he hery f inegrals and derivaives f nn-ineger rders. I suggess a paradigm shif frm ineger rder varians fracinal nes [2], [3], [4]. Fracinal calculus finds is majr applicain in cnrl dmain []. I has been prved ha fracinal calculus describes mdel mre accuraely han he ineger cunerpar. Wih he develpmen f fracinal calculus, i is bserved ha fracinal differenial equains can describe memry and geneic characerisics mre accuraely. In fac he dynamics f ms f he physical plans are msly fracinal, alhugh fr many f hem he fracinaliy is very lw. Ineger perars f cnveninal cnrl sraegies like PID, fuzzy, neural ec have als been replaced using fracinal varians enhancing he scpe f cnrl hery. Nw sliding mde cnrl (SMC) is als exended using fracinal calculus uilizing advanages f bh he heries. Fracinal mdeling f sysems has necessiaed he implemenain f esimain appraches using fracinal heries. Mahemaical mdels in fracinal dmain have differenial equains f fracinal rder. Hence, ineger rder disurbance esimain appraches cann be used fr fracinal mdels direcly. They require sme mdificains. The basic disurbance bserver which invlves mahemaical inversin f he plan mdel alng wih a filer has been mdified in [6] sui fracinal dmain apprach. The fracinal disurbance bserver design became n lnger cnservaive, i.e., given he cuff frequency and he desired phase margin, he fracinal rder f he lw pass filer culd be deermined. This fracinal rder disurbance bserver is based n he fracinal rder Q filer. This design has been applied fr cerain applicain like vibrain suppressin in w ineria sysem [7], rbus grinding mill cnrl [8] ec. This DO esimaes mached perurbains and uncerainies. A fracinal disurbance bserver has been prpsed in discree dmain in [9]. Sabiliy analysis f fracinal sysems is similar ineger rder sysems. The exensins f cnveninal sabiliy mehds like Lyapunv analysis have been exended in fracinal dmain [] [11] [12] ec. A nnlinear exensin f DO has been prpsed by W.H. Chen e. al which esimaes sysem uncerainies and disurbances [1] [13]. The scheme f nnlinear DO (NLDO) has been sudied fr varius applicains by many researchers fr linear, nnlinear sysems. In his paper, a fracinal disurbance bserver (FDO) n he lines f [1] is prpsed esimae mached disurbances in sysems. This esimaed disurbance is used deliver a rbus sae feedback cnrl. Sabiliy f he prpsed bserver is esablished using fracinal exensin /1/$ IEEE 1412

2 f Lyapunv mehd. A. Srucure f Paper Sme preliminaries and definiins f fracinal calculus are given in he nex secin. Secin 3 describes he fracinal disurbance bserver alng wih is sabiliy prf. Simulain resuls are given in Secin 4 and Secin cncludes he sudy. II. PRELIMINARIES AND DEFINITIONS OF FRACTIONAL CALCULUS Fracinal calculus uses nn-ineger rders f derivaives and inegrains. Many definiins f hese fracinal rder (FO) derivaives are fund in he lieraure, bu he cmmnly used definiins which are used in engineering applicains are given belw. A. Definiins f FO inegrain and derivaive 1) Riemann Liuville fracinal rder inegrain(rlfi): RLFI f a funcin f () can be deermined using Cauchy's clsed frm fr successive inegrain as Ja f() r a) J ( - T) (,,- l ) f(t)dt (1) where J represens inegrain perar, a E lr+ and rc) is he Eulers gamma funcin defined as 00 r(a) J e- a-1d 2) Definiin f Capu fracinal derivaives (CFD): Capu Derivaive Df wih rder a f a funcin f () is deermined as D"f() e Jm _ dm f() a dm (rn - 1) -s: a < rn, where m is an ineger and a is a real number. D represens differeniain perar. In rder find he ah derivaive using CFD, firs funcin is differeniaed by rnh rder and hen i is (rn - a) fld inegraed. Frm (I) and (2), CFD can be expressed as ed f() (2) 1 j' ( ) m-,,- l dm - T f () T dt r.( Tn - a ) -d m 3) Definiin f Riemann Liuville fracinal derivaive (RLFD): RLFD Df wih rder a f a funcin f() is deermined as dm (Jm-a f()) RL Daj'() (3) dm In rder find he ah derivaive using RLFD, iniially he funcin is (rn - a) fld inegraed and hen i is differeniaed by rnh rder. Frm (1) and (3), RLFD can be expressed as B. The Laplace ransfrm The Laplace ransfrm f he abve menined definiins are given as 1) L[Ja f()] s-a F(s) m- l 2) L[RLDf f()] sa F(s) - L: sk [RD -k- l f()] k l m- l 3) L[eDf f()] sa F(s) - L: s,,-k- l f(k )(O) k l The iniial cndiin arising in case f RLFD is n physical (value f RLFI a O) s RLFD definiin has limiains in erms f is applicains in mdeling. In case f Capu FD, he iniial cndiins have physical meanings eg f(o), j(o) ec. Hence his definiin is mre ppular amng physiciss and engineers. C. Fracinal exensin f Lyapunv Mehd The fracinal rder exensin f Lyapunv direc mehd [] is given as; Le x 0 be an equilibrium pin fr he nn-aunmus fracinal rder sysem. Assume here exiss a Lyapunv funcin V(, x) and class-k funcins Ii, i 1,2,3 saisfying Il(llxll) -s: V(, x) -s: 12 (llxll), DaV(, x) -s: 13 (llxll) (4) where a E (0,1). Then he sysem is asympically sable. III. FRACTIONAL ORDER DISTURBANCE OBSERVER Cnsider dynamics fr a secnd rder fracinal sysem; D"x y() f(x) + gl(x)u + g2(x)d() h(x) Here x is he sae vecr, U is cnrl inpu, d( ) is inpu channel disurbance and y is he upu. f(x),gl(x),g2(x) are he sae funcins. Assumpin: The disurbance d() is cninuus and bunded hence saisfies fr a E (0,1). Here fl is a psiive number. The fracinal disurbance bserver (FDO) fr () is prpsed as; d() Lx() - z() Daz() L(f(x) + gl(x)u + g2(x)d()) Here z is he inernal sae f FDO, L is he bserver gain. Le he errr in disurbance esimain be cnsidered as, e() d- } } () (6) 1413

3 d. The errr dynamics can be derived by aking ah derivaive A. Esimain f Disurbances and subsiuing he FDO dynamics, Dae() A. Sabiliy Analysis Dad- Dad Dad - Da(Lx() - z()) Dad - L(f(x) + gl(x)u + g2(x)d()) +L(f(x) + gl(x)u + g2(x)d()) Dae() Dad - Lg2(X)(d - d) The bserver gain is be chsen such ha he errr dynamics (7) are sable. T analyze he sabiliy, fracinal rder exensin f Lyapunv direc mehd (4) is be used. Cnsider a psiive definie Lyapunv funcin V ( e) e 2. Is ah derivaive is given as per Leibniz rule f fracinal differeniain [II] as fllws, (7) (8) This sysem was simulaed fllw a delayed uni sep cmmand a ime O.1 sec. The sysem is cnrlled using an sae feedback cnrller wih desired ple lcains a -4. and -.3. The prpsed fracinal disurbance bserver can handle a large class f disurbances such as cnsan, sinusidal r sae dependen disurbances. The simulain sudy fr hese disurbances is as shwn belw; Cnsan Disurbance A cnsan signal d 0. 6 was given as disurbance he sysem. Figure 1 shws he pls f acual and esimaed disurbance alng wih is esimain errr. where r(l + a) P f DkeDa-ke. e k r(l+k)r(l -k+a) l I is assumed ha he fllwing inequaliy hlds where N is a psiive number []. Therefre, subsiuing (7) in he Lyapunv funcin, DaV Taking inequaliy, e(dad - Lg2(X)(d - d)) + Nlel e(mlldll - Lg2(x)e) + Nlel w 0.6f.,.: i i c Therefre, as , he esimain errr is bunded by, (9) Thus, wih a prper chice f gain L, he asympic cnvergence f he errr he bunds f he ah derivaive f disurbance can be ensured. IV. SIMULATION RESULTS Cnsider a linear secnd rder fracinal sysem as fllws, Dax Ax + bu + ed () Fig. I: Disurbance Esimain using FDO where, A [ ] b [ ] e [ ] y [ [ ] (I I) d is he added mached disurbance in he sysem (). Sinusidal Disurbance A disurbance signal d 0. sin(w), frequency1.hz was given he sysem. Figure 2 shws he w pls. 1414

4 b is he scaling facr; here b Using his cnrl he sysem was simulaed wih a grwing disurbance d 0. sin(w) + /6, frequency1.hz. The perfrmance f he sysem using cmpensain wih disurbance esimain is shwn belw. Figure 4 shws he evluin f he saes wih and wihu cmpensain and Figure shws he crrespnding cnrl inpus. Fig. 2: Disurbance Esimain using FDO Sae dependen Disurbance A disurbance signal d 0. sin(w)x, frequencyi.hz was given he sysem and he esimain f his signal is shwn in Figure 3. Fig. 4: Evluin f saes W Fig. 3: Disurbance Esimain using FDO B. Rbus sae fe edback cnrl I was bserved ha he saes f he sysem were perurbed n addiin f disurbance in he inpu channel. Hence he cnrl was augmened include he knwledge f disurbance in i. The mdified cnrl is Fig. : Cnrl inpu U ue q + Un where ue q is he equivalen cnrl designed fr nminal sysem (i.e. he sae feedback cnrller) and Un - -b1 d. The esimaed and acual disurbance is shwn in Figure

5 [12] H. Ahn, Y. Chen, and 1. Pdlubny, "Rbus sabiliy es f a class f linear ime-invarian inerval fracinal-rder sysem using lyapunv inequaliy," Science Direc, [13] W.-H.Chen, "Disurbance bserver based cnrl fr nnlinear sysems," IEEEIASME Transacins n Mecharnics, vl. 9, December Fig. 6: Disurbance Esimain using FDO I is clearly seen frm he figures ha cmpensain f he mached disurbance is achieved using he mdified cnrl law based n FDO. V. CONCLUSION A fracinal disurbance bserver is prpsed in his paper alng wih is sabiliy analysis. This FDO has been used esimae he uncerainies and perurbains f a fracinal rder sysem which is laer used augmen he cnrl law. This mdified sae feedback cnrl shws rbusness mached disurbances when cmpared wih is simple cunerpar. Simulain resuls are included suppr he sudy. I is fund ha he FDO can handle a large class f mached disurbances. Thus i can be bserved ha a FDO alng wih cnveninal cnrl sraegies can render rbusness a large class f fracinal rder sysems. REFERENCES [1] W.-H.Chen, D. Balance, P.J.Gawhrp, J. Gribble, and J.O'Reilly, "Nnlinear pid predicive cnrller," lee Prc.-Cnrl TheOlY Appl., vl. 146, Nvember [2] D. C. A. Mnje, D. Y. Chen, and D. B. M. Vinagre, Fracinal-rder Sysems and Cnrls. Springer, 20. [3] S. Das, Funcinal Fracinal Calculus fr Sysem Idenificain and Cnrls. Springer, [4] 1. Pdlubny, Fracinal Di ferenial Equains. Academic Press, 1999, vl [] B. Bandypadhyay and S. Kamal, Sabilizain and Cnrl f Fracinal Order Sysems : ASliding Mde Apprach. Springer Inernainal Publishing, 201. [6] Y. Chen, B. M. Vinagre, and 1. Pdlubny, "Fracinal rder disurbance bserver fr rbus vibrain suppressin," Nnlinear Dynamics, vl. 38, p. 3367, [7] w. Li and Y. Hri, "Vibrain suppressin using single neurn-based pi fuzzy cnrller and fracinal-rder disurbance bserver," Indusrial Elecrnics, IEEE 7)'ansacins n, vl. 4, n. I, pp , Feb [8] L. E. Olivier, 1. K. Craig, and Y. Chen, "Fracinal rder disurbance bserver fr a run-f-mine re milling circui," in IEEE A}i"icn, Sepember [9] S. Kamal and B. Bandypadhyay, "Rbus cnrller design fr discree racinal rder sysem: A disurbance bserver based apprach," in ACODS, Sepember [] Y. Li, Y. Chen, and 1. Pdlubny, "Sabiliy f fracinal-rder nnlinear dynamic sysems: Lyapunv direc mehd and generalized miag-ieffier sabiliy," Elsevier: Cmpuers and Mahemaics wih Applicains, 20. [11] M. O. Efe, "Fracinal fuzzy adapive sliding-mde cnrl f a 2- df direc-drive rb ann," IEEE 7)'ansacins n sysems, man, and cyberneics, Decemeber