Automatica. Composite adaptive control for Euler Lagrange systems with additive disturbances

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1 Automatica 46 (21) Contents lists available at ScienceDirect Automatica journal homepage: Brief paper Composite aaptive control for Euler Lagrange systems with aitive isturbances Parag M. Patre, William MacKunis, Marcus Johnson, Warren E. Dixon Department of Mechanical an Aerospace Engineering, University of Floria, Gainesville, FL 32611, USA a r t i c l e i n f o a b s t r a c t Article history: Receive 28 October 28 Receive in revise form 18 September 29 Accepte 29 September 29 Available online 1 November 29 Keywors: Composite aaptation RISE Nonlinear control Lyapunov-be control In a typical aaptive upate law, the rate of aaptation is generally a function of the state feeback error. Ieally, the aaptive upate law woul also inclue some feeback of the parameter estimation error. The esire to inclue some meurable form of the parameter estimation error in the aaptation law resulte in the evelopment of composite aaptive upate laws that are functions of a preiction error an the state feeback. In all previous composite aaptive controllers, the formulation of the preiction error is preicate on the critical sumption that the system uncertainty is linear in the uncertain parameters (LP uncertainty). The presence of aitive isturbances that are not LP woul estroy the preiction error formulation an stability analysis arguments in previous results. In this paper, a new preiction error formulation is constructe through the use of a recently evelope Robust Integral of the Sign of the Error (RISE) technique. The contribution of this esign an sociate stability analysis is that the preiction error can be evelope even with isturbances that o not satisfy the LP sumption (e.g., aitive boune isturbances). A composite aaptive controller is evelope for a general MIMO Euler Lagrange system with mixe structure (i.e., LP) an unstructure uncertainties. A Lyapunov-be stability analysis is use to erive sufficient gain conitions uner which the propose controller yiels semi-global ymptotic tracking. Experimental results are presente to illustrate the approach. 29 Elsevier Lt. All rights reserve. 1. Introuction Aaptive, robust aaptive, an function approximation methos typically use tracking error feeback to upate the aaptive estimates. In general, the use of the tracking error is motivate by the nee for the aaptive upate law to cancel cross-terms in the close-loop tracking error system within a Lyapunov-be analysis. As the tracking error converges, the rate of the upate law also converges, but rawing conclusions about the convergent value (if any) of the parameter upate law is problematic. Ieally, the aaptive upate law woul inclue some estimate of the parameter estimation error a means to prove the parameter estimates This research is supporte in part by the NSF CAREER awar The authors woul like to acknowlege the support of the Department of Energy, grant number DE-FG4-86NE This work w accomplishe part of the DOE University Research Program in Robotics (URPR). A preliminary version of this paper appeare in the American Control Conference, June 1 12, 29, St. Louis, Missouri, USA. This paper w recommene for publication in revise form by Associate Eitor Anrea Serrani uner the irection of Eitor Miroslav Krstic. Corresponing aress: Department of Mechanical an Aerospace Engineering, University of Floria, Rm 312 MAE-A P.O. Box Gainesville, FL , USA. Tel.: ; fax: aresses: parag.patre@gmail.com (P.M. Patre), mackunis@gmail.com (W. MacKunis), marc1518@ufl.eu (M. Johnson), wixon@ufl.eu (W.E. Dixon). converge to the actual values; however, the parameter estimate error is unknown. The esire to inclue some meurable form of the parameter estimation error in the aaptation law resulte in the evelopment of aaptive upate laws that are riven, in part, by a preiction error (Mileton & Goowin, 1988; Morse, 198; Pomet & Praly, 1988; Stry & Isiori, 1989; Slotine & Li, 1991) an also Q-moification techniques (Volyanskyy, Calise, & Yang, 26; Volyanskyy, Haa, & Calise, 28). The preiction error is efine the ifference between the preicte parameter estimate value an the actual system uncertainty. Incluing feeback of the estimation error in the aaptive upate law enables improve parameter estimation. For example, some clsic results (Krstic, Kanellakopoulos, & Kokotovic, 1995; Krstic & Kokotovic, 1995; Slotine & Li, 1991) have proven the parameter estimation error is square integrable an that the parameter estimates may converge to the actual uncertain parameters. Since the preiction error epens on the unmeurable system uncertainty, the swapping lemma (Mileton & Goowin, 1988; Morse, 198; Pomet & Praly, 1988; Stry & Isiori, 1989; Slotine & Li, 1991) is central to the preiction error formulation. The swapping technique (also escribe input or torque filtering in some literature) transforms a ynamic parametric moel into a static form where stanar parameter estimation techniques can be applie. In Krstic an Kokotovic (1995) an Krstic et al. (1995), a nonlinear extension of the swapping lemma w e /$ see front matter 29 Elsevier Lt. All rights reserve. oi:1.116/j.automatica

2 P.M. Patre et al. / Automatica 46 (21) rive, which w use to evelop the moular z-swapping an x- swapping ientifiers via an input-to-state stable (ISS) controller for systems in parametric strict feeback form. The avantages provie by preiction error be aaptive upate laws le to several results that use either the preiction error or a composite of the preiction error an the tracking error (cf. Abiko & Hirzinger, 27; Christoforou, 27; e Queiroz, Dawson, & Agarwal, 1999; Mra & Majalani, 23; Wang & Chen, 21; Zergeroglu, Dixon, Hte, & Dawson, 1999; Christoforou, 21 an the references within). Although preiction error be aaptive upate laws have existe for approximately two ecaes, no stability result h been evelope for systems with aitive boune isturbances with the exception of the result in Bartolini, Ferrara, an Stotsky (1999). However, Bartolini et al. (1999) consiers a linear time invariant system with isturbances an the preiction error is efine only in the sliing moe while the resulting stability is uniformly ultimately boune (UUB). In general, the inclusion of isturbances reuces the steay-state performance of continuous controllers to a UUB result. In aition to a UUB result, the inclusion of isturbances may cause unboune growth of the parameter estimates (Lewis, Aballah, & Dawson, 1993) for tracking error-be aaptive upate laws without the use of projection algorithms or other upate law moifications such σ -moification (Ree & Ioannou, 1989). Problems sociate with the inclusion of isturbances are magnifie for control methos be on preiction error-be upate laws because the formulation of the preiction error requires the swapping (or control filtering) metho. Applying the swapping approach to ynamics with aitive isturbances is problematic because the unknown isturbance terms also get filtere an inclue in the filtere control input. This problem motivates the question of how can a preiction error-be aaptive upate law be evelope for systems with aitive isturbances. To aress this motivating question, a general Euler Lagrangelike MIMO system is consiere with structure an unstructure uncertainties, an a graient-be composite aaptive upate law is evelope that is riven by both the tracking error an the preiction error. The control evelopment is be on the recent continuous Robust Integral of the Sign of the Error (RISE) (Patre, MacKunis, Makkar, & Dixon, 28) technique that w originally evelope in Qu an Xu (22) an Xian, Dawson, e Queiroz, an Chen (24). The RISE architecture is aopte since this metho can accommoate for C 2 isturbances an yiel ymptotic stability. For example, the RISE technique w use in Cai, e Queiroz, an Dawson (26) to evelop a tracking controller for nonlinear systems in the presence of aitive isturbances an parametric uncertainties. Be on the well accepte heuristic notion that the aition of system knowlege in the control structure yiels better performance an reuces control effort, moel-be aaptive an neural network feeforwar elements were ae to the RISE controller in Patre et al. (28) an Patre, MacKunis, Kaiser, an Dixon (28), respectively. In comparison to these approaches that use the RISE metho in the feeback component of the controller, the RISE structure is use in both the feeback an feeforwar elements of the control structure to enable, for the first time, the construction of a preiction error in the presence of aitive isturbances. Specifically, since the swapping metho will result in isturbances in the preiction error (the main obstacle that h previously limite this evelopment), an innovative use of the RISE structure is also employe in the preiction error upate (i.e., the filtere control input estimate). Sufficient gain conitions are evelope uner which this unique ouble RISE controller guarantees semi-global ymptotic tracking. Experimental results are presente to illustrate the performance of the propose approach. The paper is organize follows. Section 2 escribes the ynamic system an the sumptions require for the control evelopment. Section 3 states the control objective an the efines the error states. Section 4 presents the control evelopment an introuces the new RISE-be swapping proceure that is use to efine the preiction error. A Lyapunov-be stability analysis is shown in Section 5 while Section 6 presents experimental results that emonstrate improve performance by the propose metho. Conclusions an future work are escribe in Section Dynamic system Consier a cls of MIMO nonlinear Euler Lagrange systems of the following form: x (m) = f (x, ẋ,..., x (m 1) ) + G(x, ẋ,..., x (m 2) )u + h (t) (1) where ( ) (i) (t) enotes the ith erivative with respect to time, x (i) (t) R n, i =,... (, m 1 are the system states, u (t) R n is the control input, f x, ẋ,..., x (m 1)) R n an G(x, ẋ,..., x (m 2) ) R n n are unknown nonlinear C 2 functions, an h (t) R n enotes a general nonlinear isturbance (e.g., unmoele effects). Throughout the paper, enotes the absolute value of the scalar argument, enotes the stanar Eucliean norm for a vector or the inuce infinity norm for a matrix. The subsequent evelopment is be on the sumption that all the system states are meurable outputs. Moreover, the following sumptions will be exploite in the subsequent evelopment. Assumption 1. G ( ) is symmetric positive efinite, an satisfies the following inequality y(t) R n : g y 2 y T G 1 y ḡ(x, ẋ,..., x (m 2) ) y 2 (2) where g R is a known positive constant, an ḡ(x, ẋ,..., x (m 2) ) R is a known positive function. Assumption 2. The functions G 1 ( ) an f ( ) are secon orer ifferentiable such that G 1, Ġ 1, G 1, f, ḟ, f L if x (i) (t) L, i =, 1,..., m + 1. Assumption 3. The nonlinear isturbance term an its first two time erivatives (i.e., h, ḣ, ḧ) are boune by known constants. Assumption 4. The unknown nonlinearities G 1 ( ) an f ( ) are linear in terms of unknown constant system parameters (i.e., LP). Assumption 5. The esire trajectory x (t) R n is sume to be esigne such that x (i) (t) L, i =, 1,..., m + 2. The esire trajectory x (t) nee not be persistently exciting an can be set to a constant value for the regulation problem. 3. Control objective The objective is to esign a continuous composite aaptive controller which ensures that the system state x (t) tracks a esire time-varying trajectory x (t) espite uncertainties an boune isturbances in the ynamic moel. To quantify this objective, a tracking error, enote by e 1 (t) R n, is efine e 1 x x. (3) To facilitate a compact presentation of the subsequent control evelopment an stability analysis, auxiliary error signals enote

3 142 P.M. Patre et al. / Automatica 46 (21) by e i (t) R n, i = 3,..., m are efine e 2 ė 1 + α 1 e 1, e i ė i 1 + α i 1 e i 1 + e i 2 (4) where α i R, i = 1, 2,..., m 1 enote constant positive control gains. The error signals e i (t), i = 2, 3,..., m can be expresse in terms of e 1 (t) an its time erivatives i 1 e i = b i,j e (j), 1 b i,i 1 = 1 (5) j= where the constant coefficients b i,j R can be evaluate by substituting (5) in (4), an comparing coefficients. A filtere tracking error (Lewis et al., 1993), enote by r(t) R n, is also efine r ė m + α m e m (6) where α m R is a positive, constant control gain. The filtere tracking error r (t) is not meurable since the expression in (6) epens on x (m). 4. Control evelopment To evelop the open-loop tracking error system, the filtere tracking error in (6) is premultiplie by G 1 ( ), an (5) is use to yiel m 1 G 1 r = G 1 b m,j e (j+1) 1 + G 1 α m e m. (7) j= By separating the lt term from the summation, using the fact that b m,m 1 = 1, an making substitutions from (1) an (3), the expression in (7) is rewritten G 1 r = Y θ + S 1 G 1 h u. (8) ( In (8), the auxiliary function S 1 x, ẋ,..., x (m 1), ) t R n is efine ( ) m 2 S 1 G 1 b m,j e (j+1) 1 + α m e m + G 1 x (m) j= G 1 x(m) G 1 f + G 1 Also in (8), Y θ R n is efine f G 1 h + G 1 h. (9) Y θ G 1 x(m) G 1 f (1) where Y (x, ẋ,..., x (m) ) R n p is a esire regression matrix, an θ R p contains the constant unknown system parameters. In (1), the functions G 1 (x, ẋ,..., x (m 2) ) R n n an f (x, ẋ,..., x (m 1) ) R n are efine G 1 G 1 (x, ẋ,..., x (m 2) ) f f (x, ẋ,..., x (m 1) ) RISE-be swapping (11) A meurable form of the preiction error ε (t) R n is efine the ifference between the filtere control input u f (t) R n an the estimate filtere control input û f (t) R n ε u f û f (12) where the filtere control input u f (t) R n is generate by Slotine an Li (1991) u f + ωu f = ωu, u f () = (13) where ω R is a known positive constant, an û f (t) R n is subsequently esigne. The ifferential equation in (13) can be irectly solve to yiel u f = v u, v ωe ωt (14) where v (t) R, an is use to enote the stanar convolution operation. Using (1), the expression in (14) can be rewritten u f = v ( G 1 x (m) G 1 f G 1 h ). (15) Since the system ynamics in (1) inclue non-lp boune isturbances h (t), they also get filtere an inclue in the filtere control input in (15). To compensate for the effects of these isturbances, the typical preiction error formulation is moifie to inclue a RISE-like structure in the esign of the estimate filtere control input. With this motivation, the structure of the open-loop preiction error system is engineere to facilitate the RISE-be esign of the estimate filtere control input. Aing an subtracting the term G 1 x(m) + G 1 f + G 1 h to the expression in (15) an using (1) yiels u f = Y f θ + v S v S + h f (16) where S(x, ẋ,..., x (m) ), S (x, ẋ,..., x (m) ) R n are efine S G 1 x (m) G 1 f G 1 h (17) S G 1 x(m) G 1 f G 1 h, (18) the filtere regressor matrix Y f (x, ẋ,..., x (m) ) R n p is efine Y f v Y, (19) an the isturbance h f (t) R n is efine h f v G 1 h. The term v S(x, ẋ,..., x (m) ) R n in (16) epens on x (m). Using the following property of convolution (Lewis et al., 1993): g 1 ġ 2 = ġ 1 g 2 + g 1 () g 2 g 1 g 2 () (2) an expression inepenent of x (m) can be obtaine. Consier v S = v ( G 1 x (m) G 1 f G 1 h ) which can be rewritten ( ) v S = v t (G 1 x (m 1) ) Ġ 1 x (m 1) G 1 f G 1 h. (21) Applying the property in (2) to the first term of (21) yiels v S = S f + W (22) where the state-epenent terms are inclue in the auxiliary function S f (x, ẋ,..., x (m 1) ) R n, efine S f v ( G 1 x (m 1)) + v () G 1 x (m 1) v Ġ 1 x (m 1) v G 1 f v G 1 h (23) an the terms that epen on the initial states are inclue in W (t) R n, efine W vg 1 ( x (), ẋ (),..., x (m 2) () ) x (m 1) (). (24) Similarly, following the proceure in (21) (24), the expression v S in (16) is evaluate v S = S f + W (25) where S f (x, ẋ,..., x (m 1) ) R n is efine S f v (G 1 x(m 1) ) + v () G 1 x(m 1) v Ġ 1 x(m 1) v G 1 f v G 1 h (26) an W (t) R n is efine W vg 1 (x (), ẋ (),..., x (m 2) ())x (m 1) (). (27) Substituting (22) (27) into (16), an then substituting the resulting expression into (12) yiels ε = Y f θ + S f S f + W W + h f û f. (28)

4 P.M. Patre et al. / Automatica 46 (21) Composite aaptation The composite aaptation for the aaptive estimates ˆθ (t) R p in (39) is given by ˆθ = Γ Ẏ T r + Γ Ẏ T f ε (29) where Γ R p p is a positive efinite, symmetric, constant gain matrix an the filtere regressor matrix Y f (x, ẋ,..., x (m) ) R n p is efine in (19). The upate law in (29) epens on the unmeurable signal r (t), but the parameter estimates are inepenent of r (t) can be shown by irectly solving (29) in Zhang, Dawson, e Queiroz, an Dixon (2) Close-loop preiction error system Be on (3) an the subsequent analysis, the filtere control input estimate is esigne û f = Y f ˆθ + µ 2 (3) where µ 2 (t) R n is a RISE-like term efine µ 2 (t) [k 2 ε(σ ) + β 2 sgn(ε(σ ))]σ (31) where k 2, β 2 R enote constant positive control gains. In a typical preiction error formulation, the estimate filtere control input is esigne to inclue just the first term Y f ˆθ in (3). But previously iscusse, the presence of non-lp isturbances in the system moel results in filtere isturbances in the unmeurable form of the preiction error in (28). Hence, the estimate filtere control input is augmente with an aitional RISE-like term µ 2 (t) to cancel the effects of isturbances in the preiction error an the subsequent esign an stability analysis. Substituting (3) into (28) yiels the following close-loop preiction error system: ε = Y f θ + S f S f + W W + h f µ 2 (32) where θ (t) R p enotes the parameter estimate mismatch efine θ θ ˆθ. (33) To facilitate the subsequent composite aaptive control evelopment an stability analysis, the time erivative of (32) is expresse ε = Ẏ f θ Y f Γ Ẏ T f ε + Ñ 2 + N 2B k 2 ε β 2 sgn(ε) (34) where (29) is utilize. In (34), the unmeurable/unknown auxiliary terms Ñ 2 (e 1, e 2,..., e m, r, t), N 2B (t) R n are efine Ñ 2 Ṡ f Ṡ f Y f Γ Ẏ T r, N 2B Ẇ Ẇ + ḣ f. (35) In a similar manner in Xian et al. (24), the Mean Value Theorem can be use to evelop the following upper boun for the expression in (35): Ñ2(t) ρ 2 ( z ) z (36) where the bouning function ρ 2 ( ) R is a positive, globally invertible, nonecreing function, an z(t) R n(m+1) is efine z(t) [ e T 1 et... ] 2 et m r T T. (37) Using Assumption 3, an the fact that v (t) is a linear, strictly proper, exponentially stable transfer function, the following inequality can be evelope be on the expression in (35) with a similar approach in Lemma 2 of Mileton an Goowin (1988): N 2B (t) ξ (38) where ξ R is a known positive constant Close-loop tracking error system Be on the open-loop error system in (8), the control input is compose of an aaptive feeforwar term plus the RISE feeback term u Y ˆθ + µ 1 (39) where µ 1 (t) R n enotes the RISE feeback term efine µ 1 (t) (k 1 + 1)e m (t) (k 1 + 1)e m () + {(k 1 + 1)α m e m (σ ) + β 1 sgn(e m (σ ))}σ (4) where k 1, β 1 R are positive constant control gains, an α m R w introuce in (6). In (39), ˆθ (t) R p enotes a parameter estimate vector for unknown system parameters θ R p, generate by a subsequently esigne graient-be composite aaptive upate law (Slotine & Li, 1987, 1989; Tang & Arteaga, 1994). The close-loop tracking error system can be evelope by substituting (39) into (8) G 1 r = Y θ + S 1 G 1 h µ 1. (41) To facilitate the subsequent composite aaptive control evelopment an stability analysis, the time erivative of (41) is expresse G 1 ṙ = 1 2 Ġ 1 r + Ẏ θ Y Γ Ẏ T f ε + Ñ 1 + N 1B (k 1 + 1)r β 1 sgn(e m ) e m (42) where (29) w utilize. In (42), the unmeurable/unknown auxiliary terms Ñ 1 (e 1, e 2,..., e m, r, t) an N 1B (t) R n are efine Ñ Ġ 1 r + Ṡ 1 + e m Y Γ Ẏ T r (43) where (29) w use, an N 1B Ġ 1 h G 1 ḣ. (44) The structure of (42) an the introuction of the auxiliary terms in (43) an (44) are motivate by the esire to segregate terms that can be upper boune by state-epenent terms an terms that can be upper boune by constants. In a similar fhion in (36), the following upper boun can be evelope for the expression in (43): Ñ1(t) ρ 1 ( z ) z (45) where the bouning function ρ 1 ( ) R is a positive, globally invertible, nonecreing function, an z(t) R n(m+1) w efine in (37). Using Assumptions 2 an 3, the following inequalities can be evelope be on the expression in (44) an its time erivative: N 1B (t) ζ 1, Ṅ1B (t) ζ 2 (46) where ζ i R, i = 1, 2 are known positive constants.

5 144 P.M. Patre et al. / Automatica 46 (21) Stability analysis Theorem 1. The controller given in (39) an (4) in conjunction with the composite aaptive upate law in (29), where the preiction error is generate from (12), (13), (3) an (31), ensures that all system signals are boune uner a close-loop operation an that the position tracking error an the preiction error are regulate in the sense that e 1 (t) an ε(t) t provie the control gains k 1 an k 2 introuce in (4) an (31) are selecte sufficiently large be on the initial conitions of the system (see the subsequent proof), an the following conitions are satisfie: α m, α m 1 > 1 2, β 1 > ζ α m ζ 2, β 2 > ξ (47) where the gains α m 1 an α m were introuce in (4), β 1 w introuce in (4), β 2 w introuce in (31), ζ 1 an ζ 2 were introuce in (46), an ξ w introuce in (38). Proof. Let D R n(m+2)+p+2 be a omain containing y(t) =, where y(t) R n(m+2)+p+2 is efine y [ z T ε T P 1 P 2 θ T ] T. (48) In (48), the auxiliary function P 1 (t) R is efine n P 1 (t) β 1 e mi () e m () T N 1B () L 1 (τ)τ (49) i=1 where e mi () R enotes the ith element of the vector e m (), an the auxiliary function L 1 (t) R is efine L 1 r T (N 1B β 1 sgn(e m )) (5) where β 1 R is a positive constant chosen accoring to the sufficient conition in (47). Provie the sufficient conition introuce in (47) is satisfie, the following inequality is obtaine (Xian et al., 24): L 1 (τ)τ β 1 n e mi () e m () T N 1B (). (51) i=1 Hence, (51) can be use to conclue that P 1 (t). Also in (48), the auxiliary function P 2 (t) R is efine P 2 (t) L 2 (τ)τ (52) where the auxiliary function L 2 (t) R is efine L 2 ε T (N 2B β 2 sgn(ε)) (53) where β 2 R is a positive constant chosen accoring to the sufficient conition in (47). Provie the sufficient conition introuce in (47) is satisfie, then P 2 (t). Let V L (y, t) : D [, ) R be a continuously ifferentiable, positive efinite function efine V L (y, t) 1 m e T i 2 e i r T G 1 r εt ε i=1 + P 1 + P θ T Γ 1 θ (54) 2 which satisfies the inequalities U 1 (y) V L (y, t) U 2 (y) (55) provie the sufficient conitions introuce in (47) are satisfie. In (55), the continuous positive efinite functions U 1 (y), U 2 (y) R are efine U 1 (y) λ 1 y 2 an U 2 (y) λ 2 (x, ẋ,..., x (m 2) ) y 2, where λ 1, λ 2 (x, ẋ,..., x (m 2) ) R are efine λ 1 1 { { 2 min 1, g, λ }} min Γ 1 (56) { 1 λ 2 max 2 ḡ(x, ẋ,..., x(m 2) ), 1 2 λ { } } max Γ 1, 1 where g, ḡ(x, ẋ,..., x (m 2) ) are introuce in (2), an λ min { } an λ max { } enote the minimum an maximum eigenvalue of the arguments, respectively. Remark 1. From (34), (42), (49), (5), (52) an (53), some of the ifferential equations escribing the close-loop system for which the stability analysis is being performe have iscontinuous righthan sies G 1 ṙ = 1 2 Ġ 1 r + Ẏ θ Y Γ Ẏ T f ε + Ñ 1 + N 1B (k 1 + 1)r β 1 sgn(e m ) e m (57) ε = Ẏ f θ Y f Γ Ẏ T f ε + Ñ 2 + N 2B k 2 ε β 2 sgn(ε) (58) Ṗ 1 = L 1 = r T (N 1B β 1 sgn(e m )) (59) Ṗ 2 = L 2 = ε T (N 2B β 2 sgn(ε)). (6) Let f (y, t) R n(m+2)+p+2 enote the right-han sie of (57) (6). Since the subsequent analysis requires that a solution exists for ẏ = f (y, t), it is important to show the existence an uniqueness of the solution to (57), (59) an (6). As escribe in Polycarpou an Ioannou (1993) an Qu (1998), the existence of Filippov s generalize solution can be establishe for (57) (6). First, note that f (y, t) is continuous except in the set {(y, t) e m = }. Let F (y, t) be a compact, convex, upper semicontinuous set-value map that embes the ifferential equation ẏ = f (y, t) into the ifferential inclusions ẏ F (y, t). From Theorem 2.7 of Qu (1998), an absolute continuous solution exists to ẏ F (y, t) that is a generalize solution to ẏ = f (y, t). A common choice for F (y, t) that satisfies the above conitions is the close convex hull of f (y, t) (Polycarpou & Ioannou, 1993; Qu, 1998). A proof that this choice for F (y, t) is upper semicontinuous is given in Gutman (1979). Moreover, note that the ifferential equation escribing the original close-loop system (i.e., after substituting (39) into (1)) h a continuous right-han sie; thus, satisfying the conition for existence of clsical solutions. After using (4), (6), (29), (34), (42), (49), (5), (52) an (53), the time erivative of (54) can be expresse m V L (y, t) = α i e T i e i + e T m 1 e m r T r k 1 r T r i=1 + r T Ẏ θ + r T Ñ 1 + r T N 1B r T Y Γ Ẏ T f ε β 1 r T sgn(e m ) + ε T Ẏ f θ + ε T Ñ 2 + ε T N 2B k 2 ε T ε ε T Y f Γ Ẏ T f ε β 2ε T sgn(ε) r T (N 1B β 1 sgn(e m )) ε T N 2B + ε T β 2 sgn(ε) θ T Γ 1 (Γ Ẏ T r + Γ Ẏ T f ε). (61) After canceling similar terms, using the fact that a T b 1 2 ( a 2 + b 2 ) for some a, b R n, an using the following upper bouns Y Γ Ẏ T f c 1, Yf Γ Ẏ T f c 2

6 where c 1, c 2 R are positive constants, VL (y, t) is upper boune using the squares of the components of z(t) V L (y, t) λ 3 z 2 k 1 r 2 + r Ñ1 + c 1 ε r + ε (k 2 c 2 ) ε 2, (62) where λ 3 min Ñ2 {α 1, α 2,..., α m 2, α m 1 12, α m 12, 1 }. Letting k 2 = k 2a + k 2b where k 2a, k 2b R are positive constants, using the inequalities in (36) an (45), an completing the squares, the expression in (62) is upper boune P.M. Patre et al. / Automatica 46 (21) V L (y, t) λ 3 z 2 + ρ2 ( z ) z 2 k 2b ε 2 (63) 4k where k R is efine k k 1 (k 2a c 2 ) max {k 1, (k 2a c 2 )}, k 2a > c 2 (64) an ρ( ) R is a positive, globally invertible, nonecreing function efine ρ 2 ( z ) ρ 2 1 ( z ) + (ρ 2( z ) + c 1 ) 2. The expression in (63) can be further upper boune by a continuous, positive semi-efinite function V L (y, t) U(y) = c [ z T ε ] T T 2 y D (65) for some positive constant c, where ( )} D {y (t) R n(m+2)+p+2 y ρ 2 1 λ3 k. Larger values of k will expan the size of the omain D. The inequalities in (55) an (63) can be use to show that V L (y, t) L in D; hence, e i (t) L an ε (t), r(t), θ (t) L in D. The close-loop error systems can now be use to conclue that all the remaining signals are boune in D, an the efinitions for U(y) an z(t) can be use to prove that U(y) is uniformly continuous in D. Let S D enote a set efine ( )) } 2 S {y(t) D U 2 (y(t)) < λ 1 (ρ 2 1 λ3 k. (66) The region of attraction in (66) can be mae arbitrarily large to inclue any initial conitions by increing the control gain k (i.e., a semi-global stability result). Theorem 8.4 of Khalil (22) can now be invoke to state that c [ z T ε ] T T 2 t y() S. (67) Be on the efinition of z(t), (67) can be use to show that e 1 (t), ε(t) t y() S. (68) 6. Experiment A testbe w use to implement the evelope controller. The testbe consists of a circular isc of unknown inertia mounte on a irect-rive switche reluctance motor. A rectangular nylon block w mounte on a pneumatic linear thruster to apply an external friction loa to the rotating isk. A pneumatic regulator maintaine a constant pressure of 2 psi on the circular isk. The ynamics for the testbe are given follows: J q + f ( q) + τ (t) = τ(t) (69) Fig. 1. Tracking error for the propose composite aaptive control law (RISE + CFF). Fig. 2. Control torque for the propose composite aaptive control law (RISE + CFF). where J R enotes the combine inertia of the circular isk an rotor sembly, f ( q) R is the nonlinear friction, an τ (t) R enotes a general nonlinear isturbance (e.g., unmoele effects). The nonlinear friction term f ( q) is sume to be moele a continuously ifferentiable function escribe in Makkar, Hu, Sawyer, an Dixon (27), Patre et al. (28). The esire link trajectory is selecte follows (in egrees): q (t) = 6. sin(1.2t)(1 exp(.1t 3 )). (7) Three ifferent experiments were conucte to emonstrate the efficacy of the propose controller. For each controller, the gains were not retune (i.e., the common control gains remain the same for all controllers). First, no aaptation w use an the controller with only the RISE feeback w implemente. For the secon experiment, the preiction error component of the upate law in (29) w remove, resulting in a stanar graientbe upate law (hereinafter enote RISE + FF). For the thir experiment, the propose composite aaptive controller in (39) (4) (hereinafter enote RISE + CFF) w implemente. The tracking error is shown in Fig. 1. The control torque is shown in Fig. 2 an the aaptive estimates are epicte in Fig. 3. Each experiment w performe five times an the average RMS error an torque values were calculate. The average RMS tracking error (in eg) for the RISE controller is.219, compare to.138 an.12 for the RISE + FF an RISE + CFF (propose), respectively. The average RMS torques (in Nm) for the respective controllers is 31.75, 32.99, an 32.49, which inicate that the propose RISE+CFF controller yiels the lowest RMS error with a similar control effort.

7 146 P.M. Patre et al. / Automatica 46 (21) Fig. 3. Aaptive estimates for the propose composite aaptive control law (RISE + CFF). 7. Conclusion A moel-be feeforwar aaptive component w use in conjunction with the RISE feeback, where the aaptive estimates were generate using a composite upate law riven by both the tracking an preiction error with the motivation of using more information in the aaptive upate law. To account for the effects of non-lp isturbances, the typical preiction error formulation w moifie to inclue a secon RISE-like term in the estimate filtere control input esign. Using a Lyapunov stability analysis, sufficient gain conitions were erive uner which the propose controller yiels semi-global ymptotic stability. Experiments on a rotating isk with externally applie friction inicate that the propose metho yiels better tracking performance with a similar control effort when compare to a typical RISE controller an a RISE controller with a typical graient-be feeforwar term. The ymptotic stability for the propose RISE-be composite aaptive controller comes at the expense of achieving semiglobal stability which requires the initial conition to be within a specifie region of attraction that can be mae larger by increing certain gains subsequently iscusse in Section 5. Development is also provie that proves the preiction error is square integrable; yet, no conclusion can be rawn about the convergence of the parameter estimation error ue to the presence of filtere aitive isturbances in the preiction error. The current evelopment, well all previous RISE controllers, require fullstate feeback. The evelopment of an output feeback result remains an open problem. The propose metho uses a graientbe composite aaptive law with a fixe aaptation gain. Future efforts coul also focus on esigning a composite law with letsquares estimation with time-varying aaptation gain for the consiere cls of systems. References Abiko, S., & Hirzinger, G. (27). An aaptive control for a free-floating space robot by using inverte chain approach. In Proc. IEEE/RSJ int. conf. on intelligent robots an systems (pp ). Bartolini, G., Ferrara, A., & Stotsky, A. A. (1999). Robustness an performance of an inirect aaptive control scheme in presence of boune isturbances. IEEE Transactions on Automatic Control, 44(4), Cai, Z., e Queiroz, M. S., & Dawson, D. M. (26). Robust aaptive ymptotic tracking of nonlinear systems with aitive isturbance. IEEE Transactions on Automatic Control, 51, Christoforou, E. (27). On-line parameter ientification an aaptive control of rigi robots using be reaction forces/torques. In Proc. IEEE int. conf. on robotics an autom. (pp ). e Queiroz, M. S., Dawson, D. M., & Agarwal, M. (1999). Aaptive control of robot manipulators with controller/upate law moularity. Automatica, 35, Gutman, S. (1979). Uncertain ynamical systems a Lyapunov min-max approach. IEEE Transactions on Automatic Control, 24(3), Khalil, H. K. (22). 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IEEE Transactions on Energy Conversion, 16(2), Parag M. Patre w born in Nagpur, Inia. He receive the B. Tech. egree in mechanical engineering from the Inian Institute of Technology Mar, Inia, in 24. Following this he w with Larsen & Toubro Limite, Inia until 25, when he joine the grauate school at the University of Floria. He receive the M.S. an Ph.D. egrees in mechanical engineering in 27 an 29, respectively, an is currently a NASA postoctoral fellow at the NASA Langley Research Center, Hampton, VA. His are of research interest are Lyapunov-be esign an analysis of control methos for uncertain nonlinear systems, robust an aaptive control, control of robots, an neural networks. He h publishe 18 referee journal an conference papers an the monograph RISE- Be Robust an Aaptive Control of Nonlinear Systems.

8 P.M. Patre et al. / Automatica 46 (21) Will MacKunis receive his Ph.D. egree in 29 from the Department of Mechanical an Aerospace Engineering at the University of Floria (UF) a UF Alumni Fellow. After completing his octoral stuies, he w selecte a National Research Council (NRC) Postoctoral Research Associate at the Air Force Research Laboratory Munitions Directorate at Eglin Air Force Be, where he works in the Guiance an Navigation Division. Dr. MacKunis main research interest h been the evelopment an application of Lyapunov-be control techniques for nonlinear mechanical systems with ill-efine ynamic moels. He is the co-author of 26 referee journal an conference papers an the monograph RISE-Be Robust an Aaptive Control of Nonlinear Systems, an his research h been recognize by NASA an the NRC. Marcus Johnson receive his M.S. egree in 28 from the Department of Mechanical an Aerospace Engineering at the University of Floria. He is currently working on his Ph.D. at the University of Floria an his main research interest is the evelopment of Lyapunov-be proofs for optimality of nonlinear aaptive systems. Warren E. Dixon receive his Ph.D. egree in 2 from the Department of Electrical an Computer Engineering from Clemson University. After completing his octoral stuies he w selecte an Eugene P. Wigner Fellow at Oak Rige National Laboratory (ORNL) where he worke in the Robotics an Energetic Systems Group. In 24, Dr. Dixon joine the faculty of the University of Floria in the Mechanical an Aerospace Engineering Department. His main research interest is the evelopment an application of Lyapunov-be control techniques for uncertain nonlinear systems. He is the coauthor of four monographs, an eite collection, six chapters, an over 18 referee journal an conference papers. His work h been recognize by the 29 American Automatic Control Council (AACC) O. Hugo Schuck Awar, 26 IEEE Robotics an Automation Society (RAS) Early Acaemic Career Awar, an NSF CAREER Awar (26 211), 24 DOE Outstaning Mentor Awar, an the 21 ORNL Early Career Awar for Engineering Achievement. Dr. Dixon is a senior member of IEEE. He serves on the IEEE CSS Technical Committee on Intelligent Control, ASME DSC Division Mechatronics an Robotics Technical Committees, is a member of numerous conference program an organizing committees, an serves on the conference eitorial boar for the IEEE CSS an RAS an the ASME DSC. He serve an appointe member to the IEEE CSS Boar of Governors for 28. He is currently an sociate eitor for Automatica, IEEE Transactions on Systems Man an Cybernetics: Part B Cybernetics, International Journal of Robust an Nonlinear Control, an Journal of Robotics.