On Stability Analysis of Active Disturbance Rejection Control for Nonlinear Time-Varying Plants with Unknown Dynamics

Transcription

1 Proceedngs of the 46th IEEE Conference on Decson and Control New Orleans, LA, USA, Dec 2-4, 27 ThB76 On Stablty Analyss of Actve Dsturbance Rejecton Control for Nonlnear Tme-Varyng Plants wth Unknown Dynamcs Qng Zheng, Lnda Q Gao 2, and Zhqang Gao,3 Abstract Ths paper concerns wth stablty characterstcs of actve dsturbance rejecton control for nonlnear tmevaryng plants that are largely unknown In partcular, asymptotc stablty s establshed where the plant dynamcs s completely known In the face of large dynamc uncertantes, estmaton and trackng errors are shown to be bounded, wth ther bounds monotonously decreasng wth ther respectve bandwdths Index Terms Lnear actve dsturbance rejecton control, lnear extended state observer, uncertan systems, stablty I INTRODUCTION Most physcal plants n real world are not just nonlnear and tme-varyng but also hghly uncertan Control system desgn for such systems has been the focus of much of the recent developments under the umbrella of robust, adaptve, and nonlnear control Most of the exstng results, however, are obtaned presupposng that a farly detaled and accurate mathematcal model of the plant s avalable The small gan theorem based robustness analyss does allow a small amount of uncertantes n plant dynamcs, but not anywhere near the magntude often encountered n practce As the wellknown control theorst Roger Brockett puts t: If there s no uncertanty n the system, the control, or the envronment, feedback control s largely unnecessary [] The assumpton that a physcal plant, wthout feedback, behaves rather closely as ts mathematcal model descrbes, as the pont of departure n control system desgn, does not reflect ether the ntent of feedback control, or the physcal realty The actve dsturbance rejecton control (ADRC) was proposed as an alternatve paradgm to address ths fundamental ssue [2] The man dfference n the desgn concept pertans to the queston of how much model nformaton s needed Recognzng the vulnerablty of the relance on accurate mathematcal model, there has been a gradual recognton over the years that actve dsturbance estmaton s a vable alternatve to an accurate plant model That s, f the dsturbance, representng the dscrepancy between the plant and ts model, s estmated n real tme, then the plant-model msmatch can be effectvely compensated for, makng the model based desgn tolerant of a large amount of uncertantes The focal pont s how external dsturbance and unknown dynamcs can be estmated Several classes of approach are outlned below, ncludng the unknown nput Center for Advanced Control Technologes, Fenn College of Engneerng, Cleveland State Unversty, Cleveland, OH 445, USA 2 Department of Mathematcs, North Central College, Napervlle, IL 654, USA 3 The Correspondng author E-mal: zgao@csuohoedu Tel: , Fax: observer (UIO) [3]-[], the dsturbance observer (DOB) []-[8], the perturbaton observer (POB) [9]-[22], and the extended state observer (ESO) [2], [23]-[24] UIO s the earlest dsturbance estmator, gong back to 969, where the external dsturbance s formulated as an augmented state and estmated usng a state observer DOB s another man class of dsturbance estmators, based on the nverse of the nomnal transfer functon of the plant, whch s avoded n some of the varatons of DOB [25]- [26] POB s another class of dsturbance estmators, smlar to DOB n concept but formulated n state space n dscrete tme doman It was shown that UIO and DOB are equvalent n terms of dsturbance estmaton [6] but UIO also provdes estmaton of the states Smlar to UIO, ESO s also a state space approach What sets ESO apart from UIO and DOB s that t s conceved to estmate not only the external dsturbance but also plant dynamcs Among the dsturbance estmators, ESO requres the least amount of plant nformaton All estmators above, ncludng UIO, DOB, POB and ESO, prove to be effectve practcal solutons But how fast and n what range the dsturbance and unknown dynamcs can be estmated are not obvous In partcular, both UIO and DOB are orgnally formulated to estmate the external dsturbances but later adopted to estmate the unknown plant dynamcs as well, wth very lttle analytcal support on how ths can be acheved Even when lmted stablty analyss was performed, only boundedness of estmaton or compensaton error was obtaned, whle the actual bound of the error s largely unknown Some robust stablty analyss, based on small gan theorem, was performed for UIO and DOB [6]-[8] but the results tend to be qute conservatve by nature and lmted to lnear and tme-nvarant plants For nonlnear plants, only lmted results on stablty propertes are obtaned for robot manpulators [2], [5] In a rare excepton, the approach proposed by [27] s desgned specfcally to estmate both unknown dynamcs and dsturbance wth asymptotc stablty of the closedloop system frmly establshed But the practcalty method s quckly called nto queston as the observer, hence the stablty proof, requres the use of hgher order dervatves of the output, renderng the system susceptble to nose corrupton In short, for those effectve practcal solutons, there seems to be a lack of rgorous analyss of the estmaton error, especally ts bound But for those methods frmly rooted n mathematcal rgor, the utlty s often questonable Ths paper specfcally addresses ths ssue, pertanng to ESO and the assocated ADRC /7/$25 27 IEEE 35

2 46th IEEE CDC, New Orleans, USA, Dec 2-4, 27 ThB76 ADRC, proposed by Han n 995, s desgned to deal wth those plants wth large amount of uncertantes both n dynamcs and external dsturbances [23] It was further smplfed to lnear ADRC (LADRC), usng lnear ESO (LESO) n [24], whch makes t extremely smple and practcal [28] The purpose of ths paper s to show analytcally how LADRC acheves excellent performance, even when the plant s unknown, nonlnear and tme-varyng The convergence and the bounds of the estmaton and trackng errors are presented In partcular, at one extreme, the asymptotc stablty of LADRC s proved where the accurate mathematcal model of the plant s gven At the other extreme, where the plant dynamcs s largely unknown, the upper bounds of errors are derved The paper s organzed as follows The analyses for the LESO and for LADRC are presented n Secton II and Secton III respectvely, wth and wthout a detaled mathematcal model The paper ends wth a few concludng remarks n Secton IV II ANALYSIS OF LESO ERROR DYNAMICS Consder a generally nonlnear tme-varyng dynamc system wth sngle-nput, u, and sngle-output y, y (n) (t) = f(y (n ) (t),,y (t),w (t)) + bu (t) () where w s the external dsturbance and b s a gven constant Here f ( y (n ) (t),y (n 2) (t),,y (t),w (t) ), or smply denoted as f, represents the nonlnear tme-varyng dynamcs of the plant that s unknown That s, for ths plant, only the order and the parameter b are gven The ADRC s a unque method desgned to tackle ths problem It s centered around estmaton of, and compensaton for, f To ths end, assumng f s dfferentable and let h = f, () can be wrtten n an augmented state space form ẋ = x 2 ẋ n = x n ẋ n = x n+ + bu ẋ n+ = h (x,w) y = x where x = [x,x 2,,x n+ ] T R n+, u R and y R are the state, nput and output of the system, respectvely Any state observer of (2), wll estmate the dervatves of y and f snce the latter s now a state n the extended state model Such observers are known as ESO The convergence of the estmaton error for a partcular ESO, LESO, s shown below (2) A Convergence of the LESO wth the Gven Model of the Plant Wth u and y as nputs and the functon h gven, the LESO of (2) s gven as ˆx = ˆx 2 + l (x ˆx ) ˆx n = ˆx n + l n (x ˆx ) ˆx n = ˆx n+ + l n (x ˆx ) + bu ˆx n+ = l n+ (x ˆx ) + h (ˆx,w) where ˆx = [ˆx, ˆx 2,, ˆx n+ ] T R n+, and l, =, 2,,n+, are the observer gan parameters to be chosen In partcular, let us consder a specal case where the gans are chosen as [l, l 2,,l n+ ] = [ ω o α,α 2 2,, n+ ] α n+ (4) wth ω o > Here α, =, 2,,n +, are selected such that the characterstc polynomal s n+ + α s n + + α n s + α n+ s Hurwtz For smplcty,let s n+ + α s n + +α n s+α n+ = (s + ) n+ where α = (n+)!!(n+ )!, =, 2,, n + Then the characterstc polynomal of (3) s (3) λ o (s) = (s + ω o ) n (5) and ω o, the observer bandwdth, becomes the only tunng parameter of the observer Let x = x ˆx, =,2,,n + From (2) and (3), the observer estmaton error can be shown as x = x 2 ω o α x x n = x n n α n x (6) x n = x n+ n α n x x n+ = h (x,w) h (ˆx,w) n+ α n+ x Now let ε = rewrtten as x ε = ω o Aε + B, =, 2,,n +, then (6) can be h (x,w) h (ˆx,w) α α 2 where A =, B = α n α n+ [ ] T Here A s Hurwtz for the α, =, 2,, n +, chosen above Theorem : Assumng h (x,w) s globally Lpschtz wth respect to x, there exsts a constant ω o >, such that lm x (t) =, =, 2,, n + Proof Snce A s Hurwtz, there exsts a unque postve defnte matrx P such that A T P + PA = I Choose the Lyapunov functon as V (ε) = ε T Pε Hence V (ε) = ω o ε 2 + 2ε T PB (7) h (x,w) h (ˆx,w) n (8) 352

3 46th IEEE CDC, New Orleans, USA, Dec 2-4, 27 ThB76 Snce the functon h(x,w) s globally Lpschtz wth respect to x, that s, there exsts a constant c such that h (x,w) h (ˆx,w) c x ˆx for all x, ˆx, and w,, t follows that 2ε T h (x,w) h (ˆx,w) PB n 2ε T x ˆx PBc n (9) x ˆx x When ω o, one has = = ε 2 +ε2 2 ω2 o +ε2 3 ω4 o + +ε2 n+ ω2n o ε Therefore, we obtan 2ε T h (x,w) h (ˆx,w) PB n c ε 2 () where c = + PBc 2 From (8) and (), one has V (ε) (ω o c) ε 2 () That s, V (ε) < f > c Therefore, lm x (t) =, =, 2,,n +, for ω o > c QED B Convergence of the LESO wth Plant Dynamcs Largely Unknown In many real world scenaros, the plant dynamcs represented by f s mostly unknown In ths case, the LESO n (3) now takes the form of ˆx = ˆx 2 + l (x ˆx ) ˆx n = ˆx n + l n (x ˆx ) ˆx n = ˆx n+ + l n (x ˆx ) + bu ˆx n+ = l n+ (x ˆx ) (2) Consequently, the observer estmaton error n (6) becomes x = x 2 ω o α x x n = x n n α n x x n = x n+ n α n x x n+ = h (x,w) n+ α n+ x and Equaton (7) s now (3) ε = ω o Aε + B h (x,w) n (4) Theorem 2: Assumng h(x,w) s bounded, there exst a constant σ > and a fnte T > such that x (t) σ, =, 2,, n ( +, ) t T > and ω o > ω k o Furthermore, σ = O, for some postve nteger k Proof Solvng (4), t follows that ε (t) = e ω oat ε () + Let p (t) = e A(t τ) B h (x(τ),w) n dτ (5) e A(t τ) B h (x(τ),w) n dτ, (6) snce h (x(τ),w) s bounded, that s, h (x(τ),w) δ, where δ s a postve constant, for =, 2,, n +, we have p (t) δ [( A n+ B ) ( + A e At B ) ] (7) For A and B defned n (7), A = α n+ α α n+ α2 α n+, and αn α n+ ( A B ) ν (8) { where ν = max =2,,n+ α n+, α α n+ } Snce A s Hurwtz, there exsts a fnte tme T > such that [ e At] j (9) n+ for all t T,,j =, 2,,n + Hence [ e At B ] ω n+ o (2) for all t T, =, 2,,n + Note that T depends on s s,n+ ω o A Let A = and e At = s n+, s n+,n+ d d,n+ One has d n+, d n+,n+ ( A e At B ) µ n+ (2) for all t { T, =, 2,,n +, where µ = max =2,,n+ α n+, + α α n+ } From (7), (8), and (2), we obtan δν p (t) n+ + δµ 2n+2 (22) for all t T, =,2,,n+ Let ε sum () = ε () + ε 2 () + + ε n+ () It follows that [ e At ε () ] ε sum () n+ (23) for all t T, =, 2,,n + From (5), one has ε (t) [ e At ε () ] + p (t) (24) Let x sum () = x () + x 2 () + + x n+ () Accordng to ε = x and Equatons (22)-(24), we have x (t) x sum () + δν δµ + ω n+ o ω n +2 o ω 2n +3 o = σ (25) for all t T, =, 2,,n + QED 353

4 46th IEEE CDC, New Orleans, USA, Dec 2-4, 27 ThB76 In summary, t has been proven that ) when the plant model s gven, the dynamc system descrbng the estmaton error of the LESO (3) s asymptotcally stable; and 2) n the absence of such model, the estmaton error of the LESO (2) s bounded and ts upper bound monotonously decreases wth the observer bandwdth, as shown n (25) The stablty of LADRC, where LESO s employed, s analyzed next III STABILITY ANALYSIS OF LADRC Assume that the control desgn objectve s to make the output of the plant n () follow a gven, bounded, reference sgnal r, whose dervatves, ṙ, r,,r (n), are also bounded Let [r,r 2,,r n,r n+ ] T = [r,ṙ,, ṙ n, ṙ n ] T Employng the LESO of (2) n the form of (3) or (2), the ADRC control law s gven as u = [k (r ˆx ) + k 2 (r 2 ˆx 2 ) + +k n (r n ˆx n ) ˆx n+ + r n+ ] /b (26) where k, =, 2,,n, are the controller gan parameters selected to make s n +k n s n + +k Hurwtz The closedloop system becomes y (n) (t) = (f ˆx n+ ) + k (r ˆx ) + k 2 (r 2 ˆx 2 ) + + k n (r n ˆx n ) + r n+ (27) Note that wth a well-desgned ESO, the frst term n the rght hand sde (RHS) of (27) s neglgble and the rest of the terms n the RHS of (27) consttutes a generalzed PD controller wth a feedforward term It generally works very well n applcatons but the ssues to be addressed are: ) the stablty of the closed-loop system (27); and 2) the bound of the trackng error Note that the separaton prncpal does not apply here because of the frst term n the RHS of (27) A Convergence of the LADRC wth the Gven Model of the Plant Consder η (t) = Nη (t) + g (t), (28) where η (t) = [η (t),η 2 (t),,η n (t)] T R n,g (t) = [g (t),g 2 (t),,g n (t)] T R n, and N s an n n matrx Lemma : If N s Hurwtz and lm g (t) =, then lm η (t) = Proof In (28), snce lm g (t) =, then for any φ >, there s a fnte tme T 2 > such that g (t) φ for all t T 2 The response of (28) can be wrtten as η (t) = e Nt η () + When t T 2, we have η (t) e Nt η () + e Nt e N(t τ) g (τ)dτ (29) T 2 e Nτ g (τ) dτ + e N(t τ) φdτ (3) T 2 Now consder the thrd term of rght hand sde of (3) For N, there s nonsngular matrx J and block dagonal matrx Λ = block dag {Λ,,Λ m } such that N = JΛJ (3) and each Λ has a sngle egenvalue λ wth ts algebrac multplctes beng q Suppose λ λ 2 λ m Let q = max {q,q 2,,q m } Let us choose or for the matrx norm It follows that e Λ(t τ) e λ (t τ) q (t τ) k, t τ, (32) k= where are postve constants Note that e N(t τ) e J Λ(t τ) J (33) Hence we have η (t) e Nt η () + e Nt +φ J q J T 2 e λ(t τ) (t τ) k dτ k= T 2 = e Nt η () + e Nt T 2 +e λ (t T 2 ) φ J J q λ k+ k j= k= ( ) j k! (k j)! [λ (t T 2 )] k j +φ J q J ( ) k k! λ k+ (34) k= From (34), t can be seen that lm e Nt η () = lm T e Nt 2 = { e λ (t T 2 ) φ J J q lm k j= λ k+ k= ( ) j k! (k j)! [λ (t T 2 )] k j } = Therefore there exsts T 3 > T 2 such that e Nt η () φ, t T 3, T e Nt 2 φ, t T 3, e λ (t T 2 ) φ J { J q λ k+ k= } k ( ) j k! (k j)! [λ (t T 2 )] k j φ, t T 3 j= (35) (36) 354

5 46th IEEE CDC, New Orleans, USA, Dec 2-4, 27 Let c = J J q ( ) k! k Then we have λ k+ k= η (t) (c + 3)φ, t T 3 (37) Snce φ can be arbtrarly small, t can be concluded that lm η (t) = Theorem 3: Assumng h (x,w) s globally Lpschtz wth respect to x, there exst constants ω o > and ω c >, such that the closed-loop system (27) s asymptotcally stable Proof Defne e = r x, =, 2,, n From (26), one has It follows that u = [k (e + x ) + + k n (e n + x n ) (x n+ x n+ ) + r n+ } /b (38) ė = ṙ ẋ = r 2 x 2 = e 2, ė n = ṙ n ẋ n = r n x n = e n, ė n = ṙ n ẋ n = r n+ (x n+ + bu) = k (e + x ) k n (e n + x n ) x n+ (39) Let e = [e,e 2,,e n ] T R n, x = [ x, x 2,, x n+ ] T R n+, then ė (t) = A e e (t) + A x x(t) (4) where A e = and k k 2 k n k n A x = k k 2 k n Snce k, =,2,,n, are selected such that the characterstc polynomal s n + k n s n + + k s Hurwtz, A e s Hurwtz For tunng smplcty, we just let s n + k n s n + + k = (s + ω c ) n where ω c > and n! k = ( )!(n+ )! ωn+ c, =, 2,,n Ths makes ω c, whch s the controller bandwdth, the only tunng parameter to be adjusted for the controller From Theorem, lm A x x(t) = f h (x,w) s globally Lpschtz wth respect to x Snce A e s Hurwtz, accordng to Theorem and Lemma, t can be concluded that: assumng h(x,w) s globally Lpschtz wth respect to x, there exst constants ω o > and ω c >, such that lm e (t) =, =,2,,n QED B Convergence of the LADRC wth Plant Dynamcs Largely Unknown Now we consder the case where the plant dynamcs s unknown and the LESO n the form of (2) s used nstead Theorem 4: Assumng h (x,w) s bounded, there exst a constant ρ > and a fnte tme T 5 > such that e (t) ρ, =, 2,,n, t ( ) T 5 >, ω o >, and ω c > Furthermore, ρ = O for some postve nteger j j Proof Solvng (4), we have e (t) = e A et e() + Accordng to (4) and Theorem 2, one has where k s e A e(t τ) A x x(τ) dτ (4) [A x x(τ)] =,, n = [A x x(τ)] n k s σ = γ, t T (42) = + n k Smlar to Theorem 3, choose = n! k = ( )!(n+ )! ωn+ c, =, 2,,n, such that A e s Hurwtz Defne Ψ = [ γ] T Let ϕ (t) = t eae(t τ) A x x(τ) dτ It follows that ϕ (t) ( A e Ψ ) + ( A e e Aet Ψ ) (43) and ( A e Ψ ) ( A e Ψ ) = γ k = γ n = =2,, n (44) Snce A e s Hurwtz, there exsts a fnte tme T 4 > such that [ e Aet] j (45) n+ for all t T 4,,j =, 2,,n Note that T 4 depends on A e Let T 5 = max {T,T 4 } It follows that ( e Aet Ψ ) γ n+ (46) for all t T 5, =,2,,n, and + n =2 ( A e e Aet Ψ ) ω c n γ ω n+ c k γ ω n+ c =2,,n = for all t T 5 From (43), (44), and (47), we obtan + n k γ =2 γ n + n n+ ϕ (t) = γ ω n+ c =2,,n o o n for all t T 5 Let e Aet = o n o nn e () + e 2 () + + e n () It follows that [ e Aet e () ] e s () ω n+ c ThB76 (47) (48) and e s () = (49) 355

6 46th IEEE CDC, New Orleans, USA, Dec 2-4, 27 ThB76 for all t T 5, =,2,,n From (4), one has e (t) [ e Aet e () ] + ϕ (t) (5) Accordng to (42), (48)-(5), we have e (t) e s () ω n+ c + k sσ ω n c + ( + n e s () + k s σ ω n+ c ) k k s σ =2 2n+ =2,,n = ρ (5) for all t T 5, =,2,,n, where ρ = e max s () + k (+ n k )k sσ sσ =2 n+ ω +, e s()+k s σ c n 2n+ n+ QED In summary, t has been shown that ) wth the gven model of the plant, the closed-loop system (27) s asymptotcally stable; and 2) wth plant dynamcs largely unknown, the trackng error and ts up to (n ) th order dervatves of LADRC are bounded and ther upper bounds monotonously decrease wth the controller bandwdth, as shown n (5) IV CONCLUDING REMARKS Exstng estmators and ther characterstcs are frst summarzed n ths paper, concernng unknown dsturbance and plant dynamcs The man result n ths paper s the analyss of the stablty and trackng characterstcs of a partcular class of such observers, LESO, and the assocated feedback control system, LADRC Both desgn scenaros, wth and wthout a detaled mathematcal model of the plant, are consdered It s shown that the asymptotc stablty s assured n the former and boundedness of the estmaton error and the closed-loop trackng error n the later Furthermore, t s shown that the trackng error monotonously decreases wth the control loop bandwdth Acknowledgment The authors would lke to thank Prof Sally Shao at Cleveland State Unversty for her nsghtful and valuable suggestons REFERENCES [] R Brockett, New Issues n the Mathematcs of Control, Mathematcs Unlmted - 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