Capabilities of Extended State Observer for Estimating Uncertainties

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1 9 Ameican Contol Confeence Hyatt Regency Rivefont, St Louis, MO, USA June -, 9 ThC4 Capabilities of Extended State Obseve fo Estimating Uncetainties Xiaoxia Yang and Yi Huang Abstact The capabilities of extended state obseve ESO fo estimating uncetainties is discussed in this pape The scope of the distubances that can be obseved by LESO with bounded obseving eos is given The obseving eos fo seveal typical distubances constant distubance, sine distubance, amp distubance, and squae wave distubance ae futhe analyzed It is demonstated that ESO can deal with a lage class of distubances Finally, the esults ae tested by simulations equation can be witten in the state fom as ẋ t x t, ẋ n t x n t, ẋ n t x n+ t+bu, ẋ n+ t h, I INTRODUCTION Contol design fo the systems with uncetainties is a longstanding fundamental issue in automatic contol The uncetainties, which ae univesal in pactice, usually stem fom two souces: intenal paamete o stuctue uncetainty and extenal distubance uncetainty Lots of contol methods have been poposed centeing on this issue, such as the widely used PID contol [], adaptive contol [], obust contol [] and distubance-accommodation contol [4] etc What s moe, many distubance estimating techniques appeaed, such as unknown input obseve UIO [5], petubation obseve POB [6], the distubance obseve DOB [7], etc Owing to the less dependence on model infomation, stong capabilities fo distubance eection and simple contol stuctue, the active distubance eection contol ADRC[8][9][][] attacted many eseaches attention The key in ADRC is the use of extended state obseve ESO fo on-line estimating the total uncetainties, which lumps the intenal nonlinea and uncetain dynamics and the extenal distubance Hence the uncetainties of the system can be compensated actively The idea of ESO can be demonstated in the following single-input and single-output system: { x n fx n t,x n t,,xt,ωt,t+but, y xt whee n is the ode of the plant, y is the output, u is the input, b is a constant, ωt is the extenal distubance, f is an unknown function which can be viewed as the total uncetainties o distubances of the system, both intenal and extenal Intoduce h d f/dt If the function f is nonsmooth, h denotes the genealized deivative of f Teat the uncetainty f as an extended state of the system, the Xiaoxia Yang and Yi Huang ae with the Key Laboatoy of Systems and Contol, Academy of Mathematics and Systems Science, Chinese Academy of Sciences 984-yxx@6com whee X [x,,x n+ ] τ R n+ epesents the state of the system The ESO fo estimating both the states and the extended state fo the uncetain system can be given as follows[9][] ż z β f ale,α,δ, ż n z n β n f ale,α n,δ, ż n z n+ β n f ale,α n,δ+bu, ż n+ β n+ f ale,α n+,δ, whee Z [z,,z n+ ] τ R n+,e z x and β i i n+ ae the state of ESO, the obseving eo and the obseve gains, espectively, { e f ale,α,δ α sgne, e > δ e/δ α α,δ >,othewise ESO is designed to have the popety: z i t x i ti n+ It should be noted that takes the fom of the classical Luenbege Obseve, when α i i n+on the othe hand, is consistent with the sliding mode obseve, when α i i n+ The idea of ESO has been used to deal with vaious kinds of engineeing poblems, such as flight contol, web tension contol, chemical pocess contol etc [][4][5] At the same time, many eseaches ae of inteest in the stability analysis of ADRC and ESO [][6][7][8] Refeence [6][7]discussed the capabilities of the linea ESO withα i i n+, which is given as: ż z β e, ż n z n β n e, ż n z n+ β n e,+bu, ż n+ β n+ e, The paametes ae chosen in a special way as s n+ +β s n + +β n+ s+ω n+, whee ω denotes the bandwidth of the LESO 4 It was poved that if f is diffeentiable with espect to t and h ḟ is bounded, then the LESO 4 can estimate ft with bounded eo In [8], the tacking eos /9/$5 9 AACC 7

2 of the LESO 4 wee studied in the case n, unde the assumptions that ft is bounded, piecewise continuous and in each continuous subinteval ḟt is bounded In conclusion, the existing esults ae all based on the assumption that h ḟ is bounded In this pape, the capabilities of LESO 4 will be analyzed fo a wide scope of uncetain function ft The main esult is that it can estimate ft with bounded eo if eithe h d f/dt is bounded o f is bounded This eniches the existing theoetical analysis fo ESO Fo example, accoding to the esult, the squae wave distubance, which usually exists in pactice, can be dealt with by ESO with bounded eos since f is bounded while its deivative h is unbounded Moeove, the obseving eos fo a set of uncetainties, which often appea in pactice, ae analyzed in this pape The pape is oganized as follows In Section II, the scope of the uncetainties that can be obseved by LESO 4 with bounded obseving eos is discussed Then the obseving eos fo seveal typical foms of uncetainties ae futhe analyzed in Section III Simulation esults ae pesented in section IV and the concluding emaks ae given in section V II MAIN RESULTS Intoducing the obseving eos: E [e,,e n+ ] T Z X whee e n+ z n+ x n+ z n+ f is the obseving eo of the uncetainty, the eo dynamics can be got fom and 4 : ė e β e, which can be ewitten as whee A ė n e n β n e, ė n e n+ β n e, ė n+ β n+ e h, 5 Ė AE + Bh 6 β β β n β n+ lving 6, one can get,b Et e At E+e At e Aτ Bhτdτ 7 Next, the conditions fo ft o ht, unde which Et is bounded will be discussed Since the paametes β i i n+ can be chosen such that A is Huwitz, the following discussions ae all based on the fact that A is Huwitz with eal negative eigenvalues Hence, the fist pat of the ight side of 7 will convege to zeo as long as E is bounded, that is eat E 8 Then the analysis will be concentated on the second pat of 7, that is Gt e At τ Bhτdτ 9 Povided that A has diffeent eal eigenvalues λ i < i, whee n i is the multiplicity of λ i, n i n+, then e At τ can be expessed as: e At τ e λ it τ n i S i! whee S i R n+ n+ is the constant matix detemined by A It s obviously that A has n + diffeent eigenvalues if n+, and A has n+ identical eigenvalues if, which is ust the situation as [6][7][8] discussed Then Gt can be unified in the following fom Gt e At τ Bhτdτ n i S i B e λit τ hτdτ! Befoe discussing the popety of Gt, a lemma is given as follows Lemma Fo λ <, the following fomula holds Poof Let t τ s, then k e λt τ dτ k e λt τ dτ t k! λ k+ s k e λs dt s s k e λs ds Next the mathematical induction will be utilized to pove When k, s k e λs ds e λs ds e λt, λ k! λ k+ Since λ <, it follows that e λt Hence, e λs ds λ, holds Assume that holds when k m, that is s m e λs ds m! λ m+, 7

3 Then when k m+, t s m+ e λs ds s m+ de λs λ t m+ e λt m+ λ Since λ <, it follows that tm+ e λt s m+ e λs ds s m e λs ds λ m+ s m e λs ds λ m+ m! λ m+ m+! λ m+ Theefoe holds when k m+ QED The following theoem declaes the scope of distubance f that can be estimated by the LESO 4 with bounded eo Et Theoem Et is bounded if at least one of the following two conditions is satisfied: a h M fo a constant M and all t b f M fo a constant M and all t Poof If popety a is satisfied, fom, Gt has the uppe bound as: Gt n i S i B n i S i B M e λit τ hτdτ! e λit τ dτ! Since A is Huwitz, λ i < i Accoding to lemma we can get Gt n i S i B M λ i + 4 Theefoe, fom 7 8 and 4, Et is bounded If popety b is satisfied, fom, Gt has the uppe bound as: Gt + + n i S i B n i S i B e λit τ hτdτ! e λit τ d fτ! n i S i B e λit τ fτ t! n i λi S i B!! e λit τ fτdτ + + S i B ft n i S i B! n i S i B t! eλ it f λi! e λit τ fτdτ unde the supposition that when < Since f M, similaly, by applying lemma, it can be poved that n i λi S i B! + e λit τ fτdτ! Gt S i B M + n i S i B M λ i n i S i B M λ i 5 Then, fom 7 8 and 5, Et is bounded QED Conditions a and b coves a wide scope of uncetainties in the engineeing pactice, such as constant distubance, squae wave distubance etc The uncetainties descibed in the stability analysis in [][6][7][8] all satisfy condition a Howeve, the squae wave distubance satisfies b but not a Both the uppe bounds in 4 and 5 can povide some guide fo ESO s paamete design in pactice To futhe illustate this, the special case when is studied fo simplicity In this case, S B C C λ λ λ,,,n 6 whee C k k+ k! k, if the condition a is satisfied, it follows that n Gt λ M + C, 7 λ λ 7

4 and if the condition b is satisfied, thee is n Gt M + M λ C λ 8 Accoding to 7, the lage λ is, the smalle the bound will be What s moe, the decay ate of the i th i,,n+ component is O 8 shows that, the λ n+ i fist n components of the bound will decease as λ gows with the decay ate O i,,n The n + th λ n+ i component in 8, which is the bound fo estimating the uncetainty, is a constant Theoem poves that the obseving eos of LESO 4 ae bounded when the uncetainty satisfies the condition a o b Fo some kinds of typical uncetainties, moe accuate bounds of the obseving eos can be obtained Next, the bounds fo a set of typical distubances which often happen in engineeing applications will be discussed III OBSERVING ERRORS OF SEVERAL TYPICAL UNCERTAINTIES A f is a constant distubance Constant uncetain distubance is the most simple and common case in pactice The widely used PID contolle can teat constant distubance vey well because the use of integato Next we will show that the obseving eos of LESO can convege to zeo if f is an uncetain constant Assume f C, whee C is a uncetain constant Then h ḟ Fom 9, thee is Gt, and Et, which means z n+ t C Theefoe the constant uncetainty can be pecisely estimated by ESO B f is the sinuous function The sinuous distubance can be viewed as the continuous and peiodic uncetain dynamics exist in the system Assume f Msinωt, M and ω ae the amplitude and fequency of the sinuous function espectively Obviously it satisfies popety a in Theoem because h Mω Fom 9, we can obtain Gt Mω Integating 9 by pats yields e At τ Bcosωτdτ 9 Gt MBsinωt MA MA Bcosωt + ω ω eat B A ω Gt Since matix I + A ω Gt I + A ω is positive definite, it is invetible Then MBsinωt MA ω MA Bcosωt + ω eat B I Et A I + ω B M + + A M AB ω ω Equation shows that the bounds of the obseving eos ae elated to the fequency ω, the amplitude M and the Huwitz matix A C f is a amp function The amp function can be utilized to simulate the inceasing distubance in the system Assume f ct, whee c is an uncetain constant Then h c It satisfies popety a in Theoem It follows that 7 Gt c e At τ Bdτ ca e At B ca B Et ca B In most existing esults fo contol methods of uncetain systems, the extenal distubance is often assumed to be bounded Howeve, equation eveals that LESO can obseve amp uncetainty with constant eo although the amp function will becoming infinite as the time going to infinite D f is the squae wave function The squae wave function epesents a typical kind of uncetainties, fo example, load change, which exists in many engineeing systems Actually it is a typical example, which only satisfies condition b but not a in theoem Next, a moe accuate bound fo the obseving eos will be pesented fo the squae wave uncetainties, compaed to that in 8 Assume f is a squae wave function as follows { L t [k T,k T ft,k Z + L t [k T,kT whee L and T ae the amplitude and peiod of f espectively Since hkt is infinite, the condition a is not satisfied Howeve, since ft L fo all t, it satisfies condition b Next the bound of Gt will be analyzed by two steps Step : When t [k T,kT, that is ft L, it follows that Gt e At τ Bhτdτ e At τ Bd fτ B ft e At B f + k i+t i i+t k i+t i L de At τ B+ it k T L de At τ B L de At τ B

5 Fist, conside the thid pat of : Since k i k i+t i it k i L de At τ B 4 e At it T e At it BL e At T e k At i e iat BL e iat I e AT I e kat and I e AT is invetible, so k i e iat I e AT I e kat 5 Applying 5 into 4 gives Similaly, k i+t i it L de At τ B 6 e At T e At I e AT I e kat BL e At I e AT e AT I e AT+kT + e kat BL k i+t i i+t L de At τ B e At I e AT 7 e AT e AT e kat + e A kt+t BL Applying 6 7 to, we can obtain Gt BL e At BL e At I e AT e AT I e AT+kT + e kat BL +e At I e AT e AT e AT e kat + e A kt+t BL + I e At kt+t BL e At I + e A kt+t BL+e At I e AT I e AT I + e A kt+t BL Intoducing t k T t, t < T, thee is Gt e AT I + e AT e At + e A t BL 8 Hence, Gt e AT I + e AT e A t BL 9 Step : When t [kt,k + T, that is ft L, intoducing t kt t, it can be similaly deduced that Hence, Gt e AT I + e AT e A t e At BL Gt e AT I + e AT e A t BL Equations 9 and show that, the obseving eo Gt has a close elationship with t, which is the distance between t and the step time the time when f steps fom L to L o fom L to L In each peiod, Gt will decease as t gows when t exceeds a shot peiod of time IV SIMULATION Because second-ode systems occupy an impotant place in the engineeing and pactice, the following second-ode system with single-input u and single- is consideed to test the capability of ESO fo estimating uncetainties ẋ t x t, ẋ t f + bu, x,x, y x whee b The ADRC law is designed to egulate the to a constant when the following set of uncetainties exist in the system espectively: f ; f sint; f t; f sgnsinπt The ADRC law is designed to be u k x k x, 4 whee k,k The ESO is designed as ż z β z x, ż β z x +u, 5 ż β z x, To implement the simulation, we use the Eule discete method with the step size τ In ode to guaantee the convegence of the digital calculation fo ESO, β is usually chosen as β τ The paametes in 5 ae chosen as β, β 48, β 64 Remak In the simulation, the bandwidth of ESO 5 is designed to be aound 6Hz, which is lage than the fequency of distubances in This will guaantee the convegence of ESO in a shot time Since the high fequency noise always exists in the pactice, the bandwidth of ESO should be detemined accoding to the pactical bandwidth 74

6 it One can consult [][4][5] to see how ESO is used in pactical engineeing poblems The simulation esults ae demonstated in Fig to Fig Fig is the when the diffeent distubances ae added It shows that ADRC 45 has stong ability fo contol uncetain systems Its ability can be explained by Fig and Fig Fig is the output of ESO 5, which is designed to estimate the uncetainty f It can be seen that ESO 5 has an excellent capabilities to estimate a wide kind of uncetainties Fig is the static egulating eo yt > and the obseving eo f It can be seen that: When f, the obseving eo f conveges to zeo, and the egulating eo y conveges to zeo When f sint, both the obseving eo f and the egulating eo y ae bounded When f t, as fig shows, although the distubance inceases with the time t, successfully appoaches the inceasing distubance and the obseving eo f is bounded Hence, the egulating eo y is bounded 4 When f sgnsinπt, both the obseving eo f and the egulating eo y ae bounded Futhemoe, although an obseving eo happens at the time when f steps, it deceased with t until the next step The simulation Fig ywhen f ywhen fsint ywhen ft Times ywhen fsgnsinπt The with the diffeent distubances in the system fsint ft Fig fsgnsinπt The output of ESO and the distubance f y when f y when fsint y when ft Times Fig y when fsgnsinπt f when f when fsint when ft when fsgnsinπt Times The egulating eos and obseving eos esults show that ESO has stong capabilities fo estimating uncetainties Because the design of ESO is completely independent of the distubance, a ESO with the same stuctue and same paametes can be used to deal with a lage class of distubances V CONCLUSIONS In this pape, the capabilities of ESO fo estimating uncetainties ae analyzed It demonstates that the eo of ESO is bounded if the uncetainty f is bounded o its deivative o genealized deivative is bounded Futhemoe, the obseving eos ae analyzed fo seveal typical kinds of uncetainties and the analysis esults ae tested by simulations The esults eniched the existing ones fom the theoetical pespective REFERENCES [] K JAstom and T Hagglund, PID contolles: Theoy, design and tuningreseach Tiangle Pak, NC Instument ciety of Ameica, 995 [] KJAstom and B Wittenmak, Adaptive Contol, nd ed Reading, MA: Addison-Wesley, 995 [] K Zhou, J C Doyle, and K Glove, Robust and optimal Contol: Pentice-Hall, 996 [4] C D Johnson, Accommodation of extenal distubances in linea egulato and sevomechanism poblem, IEEE Tansactions on automatic contol, vol, Ac-6, no 6, pp65-644, 97 [5] G Basile and G Mao, On the obsevability of linea, time-invaiant systems with unknown inputs, Jounal of optimization theoy and applications, vol, no 6,pp4-45, 969 [6] S Kwon and W K Chung, A discete-time design and analysis of petubation obseve fo motion contol applications,ieee Tansactions on contol systems technology, vol, no, pp99-47, [7] E Schive and J van Dik, Distubance obseves fo igid mechanical systems: Equivalence, stability and design,asme Jounal of dynamics systems, measuement and contol, vol 4, no 4, pp59-548, [8] Jingqing Han, Active distubance eection contolle and its applications, Contol and Decision, Vol,No,pp9-,998in Chinese [9] ZGao, Y Huang, and J Han, An altenative paadigm fo contol system design, Poc of IEEE confeence on decision and contol, pp , [] Z Gao, Active distubance eection contol: A paadigm shift in feedback contol system design Poc of the 6 Ameican Contol Confeence, pp 99-45, 6 [] Jingqing Han, Fom PID to Active Distubance Reection Contol,IEEE Tansactions on Industial Electonics, Vol56, No, pp9-96,9 [] Jingqing Han, A class of extended state obseves fo uncetain systems,contol and Decision, Vol No, 995 in Chinese [] Yi Huang, Kekang Xu, Jingqing Han, James Lam, Flight contol design using extended state obseve and non-smooth feedback,poceedings of the 4th IEEE Confeence on Decision and Contol, Olando, Floida USA,pp-8, [4] Wankun Zhou and Zhiqiang Gao, An active distubance eection appoach to tension and velocity egulation in web pocessing lines, 6th IEEE intenational confeence on contol and applications, pp , 7 [5] Zhongzhou Chen, Qing Zheng, Zhiqiang Gao, Active distubance eection contol of chemical pocesse,6th IEEE intenational confeence on contol and applications, pp , 7 [6] Qing Zheng, Linda Q Gao, and Zhiqiang Gao, On stability analysis of active distubance eection contol fo nonlinea time-vaing plant with unknown dynamics Poceedings of the 46th IEEE Confeence on Decision and Contol, pp5-56,7 [7] Qing Zheng, Zhiqiang Gao, On estimation of plant dynamics and distubance fom input-output data in eal tim,6th IEEE Intenational Confeence Contol Applications, pp67-7, 7 [8] DYoo, SS-TYAU, Z Gao, Optimal fast tacking obseve bandwidth of the linea extended state obseve,intenational Jounal of Contol, vol8, No, pp-, Januay, 7 75

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