A Case Study for Adaptive Robust Precision Motion Control of Systems Preceded by Unknown Dead-zones with Comparative Experiments

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1 Advanced Motion Contro March 2-24, 2, Nagaoka, Japan //$26. 2 IEEE 383 A Case Study for Adaptive Robust Precision Motion Contro of Systems Preceded by Unknown Dead-zones with Comparative Experiments Chuxiong Hu State Key Laboratory of Fuid Power Transmission and Contro Zhejiang University Hangzhou, 327, China Emai: fyfox.hu@gmai.com Bin Yao Schoo of Mechanica Engineering Purdue University West Lafayette, IN 4797, USA Emai: byao@purdue.edu Qingfeng Wang State Key Laboratory of Fuid Power Transmission and Contro Zhejiang University Hangzhou, 327, China Emai: qfwang@zju.edu.cn Abstract The recenty proposed integrated direct/indirect adaptive robust controer (DIARC) for a cass of noninear systems with unknown input dead-zones is combined with desired trajectory compensation to achieve asymptotic stabiity with exceent tracking performance. The agorithm is tested on a inear motor drive system preceded by a simuated non-symmetric dead-zone which is practicay supposed to be unknown. Certain guaranteed robust transient performance and fina tracking accuracy are achieved even when the overa system may be subjected to parametric uncertainties, time-varying disturbances and other uncertain noninearities. Signa noise that affects the adaptation function is aeviated by repacing the noisy state signa with the desired state feedback. Furthermore, asymptotic output tracking is achieved when there is unknown dead-zone noninearity ony. Comparative experimenta resuts obtained vaidate the necessity of dead-zone compensation and the higheffectiveness nature of the proposed approach as we. I. INTRODUCTION A great dea of effort has been devoted to addressing the issue of dead-zone. Specificay, an adaptive dead-zone inverse was proposed in [] but ony bounded output tracking errors are achieved. In [2] and [3], fuzzy-ogic and neura network were used to provide aternative interpretations to the basis functions needed for adaptive dead-zone inversions [] but with no essentia improvement on theoreticay achievabe resuts, i.e., ony bounded output tracking errors are obtained. In [4], [5], [6], the dead-zone was modeed as a combination of a inear input with either an unknown constant gain for symmetric dead-zones or time-varying unknown gain for nonsymmetric ones and a bounded disturbance-ike term. With this formuation, traditiona robust adaptive contro design techniques can be convenienty appied to achieve bounded output tracking errors without expicity exporing the detaied dead-zone characteristics rather than the fact that the deadzone effect can aways be treated as a bounded input disturbance. In [7], smooth basis functions as opposed to the discontinuous ones in [] were expored to provide some approximate inversions of the dead-zone noninearities. In [8], an integrated direct/indirect adaptive robust contro (DIARC) scheme has been deveoped for a cass of SISO The work is supported in part by the US Nationa Science Foundation (No. CMS-656), in part by the Nationa Natura Science Foundation of China (NSFC) under the Joint Research Fund for Overseas Chinese Young Schoars (No ) and the NSFC for Young Schoars (No ). uncertain noninear systems preceded by unknown dead-zone noninearity. Asymptotic output tracking has been achieved even in the presence of unknown dead-zone noninearity, a theoretic resut which cannot be attained in a previous researches on adaptive dead-zone compensation. Furthermore, the proposed DIARC agorithm aso achieves certain guaranteed robust transient performance and fina tracking accuracy even when the overa system may be subjected to other uncertain noninearities and time-varying disturbances. In this paper, the proposed integrated DIARC in [8] is combined with desired trajectory compensation to achieve asymptotic stabiity with exceent tracking performance. This controer has severa impementation merits such as ess onine computation time, reduced effect of measurement noise, and a faster adaptation rate [9]. The agorithm is tested on a inear motor drive system preceded by a simuated nonsymmetric dead-zone which is practicay supposed to be unknown. Certain guaranteed robust transient performance and fina tracking accuracy are achieved even when the overa system may be subjected to parametric uncertainties, time-varying disturbances and other uncertain noninearities. Furthermore, asymptotic output tracking is achieved when there are unknown dead-zone noninearities ony. Comparative experimenta resuts show that the DIARC with the proposed adaptive dead-zone compensation achieves amost the same best tracking performance as that DIARC with compete dead-zone compensation achieves. The comparative resuts aso verify the necessity of dead-zone compensation if highperformance tracking is of major concern. Overa, the highperformance tracking resuts obtained from the experiments vaidate the effectiveness of the proposed DIARC strategy. II. PROBLEM STATEMENT A. System Mode Negecting fast eectrica dynamics, the dynamics of a inear motor preceded by unknown dead-zone can be described as ẍ = Ew(t) Bẋ AS f (ẋ)+ f u () y = x(t), w(t)=d(v(t)) where x represents the position of the inertia oad, v(t) is the input to the unknown dead-zone, w(t) represents the contro input votage to the motor, and y(t) is the output from the

2 384 b m wt () b r m r vt () vt () / m b r b / mr Figure. () A dead-zone mode (2) A perfect dead-zone inverse wt () motor. E is the motor input gain, B is the equivaent viscous friction coefficient, AS f (ẋ) represents the noninear Couomb friction, in which the ampitude A may be unknown but the continuous shape function S f (ẋ) is known, and f u represents the umped uncertain noninearities incuding various mode approximation errors and externa disturbances. The noninearity D(v(t)) is described as a dead-zone characteristic shown in Fig. -(). With input v(t) and output w(t), the dead-zone can be represented as in [] as foows: m r v(t) m r b r for v(t) b r w(t)=d(v(t)) = for b < v(t) < b r (2) m v(t) m b for v(t) b where the parameters m r, m, b r, and b are constants and stand for the right sope, eft sope, right break-point and eft break-point of the dead-zone respectivey. Let E = for that its effect can be considered in the unknown sope m r and m.for notation simpicity, et θ R 6 be the vector of a unknown constant parameters, i.e., θ =[B,A,m r,m r b r,m,m b ] T. X = [x(t),ẋ(t)] is the state vector, X d =[x d (t),ẋ d (t)] is the desired trajectory vector. The contro objective is to design a contro aw v(t) to ensure that a cosed-oop signas are bounded and the pant state vector X tracks the specified desired trajectory X d asymptoticay with certain guaranteed transient responses under the foowing practica assumptions. 2 Assumption : The dead-zone output w(t) is not avaiabe for measurement and the dead-zone parameters are unknown, but their signs are known as: m r >, m >, b r >, b <. Assumption 2: The unknown parameter vector θ is within a known bounded convex set Ω θ. Without oss of generaity, it is assumed θ Ω θ, B min B B max, A min A A max, and < m r m rmax,< m min m m max,<(m r b r ) min m r b r (m r b r ) max, (m b ) min m b (m b ) max <, where B min, B max, A min, A max,, m rmax, m min, m max, (m r b r ) min, (m r b r ) max, (m b ) min, and (m b ) max are a known constants. Assumption 3: The noninearity f u can be bounded by f u δ(x) f d (t) (3) where δ(x) is a known function and f d (t) is an unknown but bounded time-varying function. 2 The foowing nomencature is used throughout this paper: min and max are the minimum vaue and maximum vaue of (t) for a t respectivey, denotes the estimate of, = denotes the estimation error, e.g., θ = θ θ, i is the i-th component of the vector. B. Dead-zone Compensation The essence of compensating dead-zone effect is to empoy a perfect dead-zone inverse function v(t)=d I (w(t)) shown in Fig. -(2) such that D(D I (w(t))) = w(t), w(t). The foowing dead-zone inverse wi be used [8] v(t)=κ + (w d ) w d(t)+ (m r b r ) + κ (w d ) w d(t)+ (m b ) (4) where, (mr b r ),, (m b ) are the estimates of m r, m r b r, m, m b respectivey, w d is the desired contro signa which woud achieve the stated contro objective when there is no dead-zone effect. And κ + (w d ) and κ (w d ) are defined by if (w d (t) > or (w d (t)=& κ + (w d )(t)= v(t ) b v(t ) b r )) (5) ese if (w d (t) < or (w d (t)=& κ (w d )(t)= v(t ) b < v(t ) b r )) (6) ese where b (m r = r b r ) and b = (m b ). The proposed DIARC in section III wi use projection type adaptation aw for a parameter estimates. With those types of parameter adaptation aw and the Assumption 2, the dead-zone parameter estimates, (mr b r ),, and (m b ) are guaranteed to be within their known bounded regions a the time. Thus, in the foowing, it is impicity assumed that b r (t) > and b (t) <, t. With these facts in mind, the foowing emma can be obtained. Lemma : [8] With the dead-zone inverse (4), the error between the actua dead-zone output w by (2) and the desired output w d can be parametrized as w w d = (m r b r )κ + (w d )+ (m b )κ (w d ) w d+ (m r b r ) m r κ + (w d ) w d+ (m b ) m κ (w d )+d(t) where d(t) is a function defined as if κ + (w d )=&v(t) b r m r b r w d+ (m r b r ) m r if κ + (w d )=&< v < b r m b w d+ (m b ) m if κ (w d )=&b < v < if κ (w d )=&v(t) b and bounded above by (m r b r ) max (m r b r ) min m rmax if κ+ (w d )= (m b ) min m min(m b ) max m if κ (w max d )= In addition, d(t) L 2 [, ) when θ L 2 [, ), and d(t) when θ(t) ast. III. INTEGRATED DIRECT/INDIRECT ADAPTIVE ROBUST CONTROLLER (DIARC) SYNTHESIS In this section, the DIARC strategy [] wi be combined with desired trajectory compensation for synthesizing w d (t) and an on-ine parameter adaptation agorithm with conditiona monitoring for the parameter estimates θ. (7) (8) (9)

3 385 A. Projection Type Adaptation Law Structure As in [], [], the first step is to use a projection mapping to keep the parameter estimates within the known bounded set Ω θ as in []. Suppose that the parameter estimate θ is updated by the foowing projection type adaptation aw [2]: θ = Pro j θ (Γτ), θ() Ω θ () where τ is any estimation function and Γ(t) > is any continuousy differentiabe positive symmetric adaptation rate matrix. With this adaptation aw structure, the foowing desirabe properties hod [2]: (P) The parameter estimates are aways within the known bounded set Ω θ, i.e., θ Ω θ, t. Thus, from Assumption 2, t, B min B(t) B max, A min Â(t) A max,< (t) m rmax,< m min (t) m max,<(m r b r ) min (m r b r )(t) (m r b r ) max, and (m b ) min (m b )(t) (m b ) max <. (P2) θ b T (Γ Pro (Γτ) τ), τ () j θ B. Integrated Direct/Indirect ARC Law With the use of the projection type adaptation aw structure (), the parameter estimates are bounded with known bounds, regardess of the estimation function τ to be used. In the foowing, this property wi be used to synthesize an integrated DIARC contro aw for the system () which achieves a guaranteed transient and steady-state output tracking accuracy in genera. Define s(t)= x(t)+λ x(t) with λ > (2) Combining (), (7) and (2), we get ṡ(t)=λ x(t) ẍ d (t) Bẋ AS f (ẋ)+w(t)+ f u = λ x(t) ẍ d (t)+ϕθ p + w d (t)+ (m r b r )κ + (w d ) + (m b )κ (w d ) w d+ (m r b r ) m r κ + (w d ) w d+ (m b ) m κ (w d )+d(t)+ f u (3) where ϕ =[ ẋ, S f (ẋ)], θ p =[B,A] T. The foowing DIARC function is proposed to design w d as foows: w d = w da + w ds,w da = w da + w da2,w ds = w ds + w ds2, w da = (λ x(t) ẍ d (t)) ϕ θ d p,w da2 = d c,w ds = k s s (4) In (4), w da represents the adjustabe mode compensation with the physica parameter estimates θ p updated using an on-ine adaptation agorithm with condition monitoring to be detaied in subsection III-C, ϕ d =[ ẋ d, S f (ẋ d )] is the regressor that depends on the desired trajectory X d (t) ony and thus is free of measurement noise effect. w da2 is a mode compensation term simiar to the fast dynamic compensation type mode compensation used in the DARC designs [], in which d c can be thought as the estimate of the ow frequency component of the umped mode uncertainties. w ds represents the robust contro term in which w ds is a simpe proportiona feedback to stabiize the nomina system and w ds2 is a robust feedback term used to attenuate the effect of various mode uncertainties for a guaranteed robust contro performance in genera. k s is a noninear gain arge enough so that [ k Q = k s k 2 + B + Ag 2 ] λ(b + Ag + λ) 2 λ(b + Ag + λ) 2 λ 3 > (5) where g is defined by S f (ẋ) S f (ẋ d )=g(ẋ,t) x, k 2 is any positive constant gain, k = κ + (w d ) m r + κ (w d ) m which has a vaue of in the absence of dead-zone sopes estimation error. Substituting (4) into (3), ṡ(t)=( B Ag) x ϕ θ d p + d(t)+ f u +[ m r (w ds + w da2 ) m r w da + (m r b r ) m (mr b r ) r ]κ + (w d ) +[ m (w ds + w da2 ) m w da + (m (m b b ) m ) ]κ (w d ) (6) Define a constant d c and time varying (t) such that d c + (t)= ϕ θ d p + d(t)+ f u [ m r κ + (w d ) + m κ (w d )]w da +[ (m r b r ) m (mr b r ) r ]κ + (w d ) (m b +[ (m b ) m ) ]κ (w d )=ϑψ(t)+d(t)+ f u (7) where ϑ = [ B,Ã, m r κ + (w d ) + m κ (w d ),[ (m r b r ) m (mr b r ) (m b r ]κ + (w d ) + [ (m b ) m ) ]κ (w d )], ψ(t) = [ ẋ d, S f (ẋ d ),w da, ] T. Conceptuay, (7) umps the origina system uncertain noninearity f u with the mode uncertainties due to physica parameter estimation errors, and divides it into the static component (or ow frequency component in reaity) d c and the high frequency components (t). The ow frequency component d c wi be compensated through fast adaptation simiar to those in DARC designs [] and DIARC designs [] as foows. Let d cm be any pre-set bound and use this bound to construct the projection type adaptation aw for d c : if d c = Pro j dc (γs)= d c (t) = d cm & d c s > (8) γs ese with γ > and d c ()=. Such an adaptation aw guarantees d c (t) d cm, t. Substituting (7) to (6) and noting (4), ṡ(t) can be written as ṡ(t)=( B Ag) x + k (w ds + w da2 )+d c + (t) =( B Ag) x + k w ds + k w ds2 d c +( k ) d c + (t) (9) Noting Assumptions 2 and 3, and (P), there exists a w ds2 such that the foowing two conditions are satisfied: (i) sw ds2 (ii) s[k w ds2 d c +( k ) d c + (t)] ε + ε d fd 2 (2) where ε and ε d are design parameters which can be arbitrariy sma. Essentiay, (ii) of (2) shows that w ds2 is synthesized to dominate the mode uncertainties coming from both parametric uncertainties and uncertain noninearities, and (i) is to guarantee that w ds2 is dissipative in nature so that it does not interfere with the functionaity of the adaptive contro part w da.

4 386 Remark : One exampe of w ds2 satisfying (2) can be found in the foowing way. Let h be any function satisfying h k max w da2 + ϑ max ψ + d(t) max (2) where kmax = max m rmax, m max m }, ϑ min max = B max B min, ϑ 2max = A max A min, ϑ 3max = max m rmax, m max m min m min }, ϑ 4max = max(m r b r ) max (m r b r ) min + (m rmax ) (m rb r ) max,(m b ) max (m b ) min (m max m min ) (m b ) min m min }, d(t) max = max (m r b r ) max (m r b r ) min (m m rmax, (m b ) min m b ) max min m }. Then, w max ds2 can be chosen as [ ] w ds2 = 4εkmin h2 + 4ε d kmin δ 2 (X) s (22) where kmin = min m rmax, m min m }. It is easy to show that the max above choice of w ds2 does satisfy (2) [], [3]. Theorem : When the DIARC contro aw (4) with the dead-zone inverse (4) and the projection type adaptation aw () is appied, regardess the estimation function τ to be used, in genera, a signas in the resuting cosed oop system are bounded and the output tracking is guaranteed to have a prescribed transient performance and fina tracking accuracy in the sense that the tracking error index s is bounded by V s e λvt Vs 2 + ε + ε d fdmax 2 [ e λvt ] (23) λ v where V s = 2 s2 + 2 λ 2 x 2, λ v = min2k 2,λ} and f dmax represents the L norm of the bounded time function f d (t). C. Parameter adaptation with On-ine Condition Monitoring The reminder of this part is to construct suitabe parameter adaptation agorithms so that an improved fina tracking accuracy - asymptotic output tracking can be obtained in the absence of uncertain noninearities (i.e., assuming f u = in ()) with an emphasis on having a good parameter estimation process as we. In the foowing, we wi make fu use of the fact that the dead-zone can be perfecty ineary parameterized during the working regions of v b r or v b and the on-ine condition monitoring to construct specific estimation functions. Lemma 2: [8] Define a positive constant H as H = max(m r b r ) max (m r b r ) min, (m b ) max (m b ) min }. Then, when w d (t) H, the dead-zone input v(t) by the proposed dead-zone inverse (4) woud satisfy v(t) b r or v(t) b. For notation simpicity, define if > χ + ( )= i f ese, χ if < ( )= (24) i f ese and χ(w d (t)) = χ + (w d (t) H)+χ (w d (t)+h). Then, noting Lemma 2 and mutipying the pant dynamics () by χ(w d (t)), wehave χ (w d (t))ẍ = χ (w d (t)) ( Bx AS f (ẋ) ) +χ + (w d (t) H)(m r v(t) m r b r ) (25) +χ (w d (t)+h)(m v(t) m b ) which is ineary parameterized by the system parameters. The foowing inear regression mode for parameter estimation is then obtained y χ (t)=f T (t)θ (26) where y χ (t)=χ (w d (t))ẍ, F T (k)= [ χ (w d (t))( Bx), χ (w d (t))( AS f (ẋ)), v(t)χ + (w d (t) H), χ + (w d (t) H), v(t)χ (w d (t)+h), χ (w d (t)+h)], θ =[B,A,m r,m r b r,m,m b ] T (27) Note that (26) is in the standard regression form. With this static mode, various estimation agorithms can be used to identify unknown parameters, of which the east square estimation agorithm [4] is given beow to estimate the vaues of B, A, m r, m r b r, m and m b. Let H f (s) be the transfer function of any fiter with a reative degree arger than or equa to (e.g., H f (s) =/(τ f s + )). Then, appying the fiter to both side of (26), we obtains y χ f (t)=f T f (t)θ (28) Appying the east-square estimation agorithm with forgetting and discrete version of the projection (), the foowing formua are obtained to update the parameter estimates as the parameter adaptation agorithm [4]: ς = Ff T θ y χ f = Ff T θ τ = +υtrff T ΓF f } F f ς Γ = κγ +υtrf T f ΓF f } ΓF f F T f Γ, if λ max (Γ(t)) ρ M, otherwise (29) in which ς is caed the prediction error, κ is the forgetting factor, ρ M is the pre-set upper bound for Γ(t), υ with υ = eading to the unnormaized agorithm. With these practica modifications, Γ(t) ρ M I, t. The foowing emma summarizes the properties of these estimators [4]: Lemma 3: [8] With the east squares type adaptation aw with projection (29), in the absence of uncertain noninearities (i.e., f u = in ()), in genera, θ(t) Ω θ, t. In addition, if the foowing persistent excitation (PE) condition is satisfied: t+t t F f F T f dτ κ p I p, f or some κ p > and T > (3) then, the physica parameter estimate θ converge to their true vaues (i.e., θ ast ) and θ L 2 [, ). D. Asymptotic Output Tracking Theorem 2: In the absence of uncertain noninearities (i.e., assuming f u = in ()), when the PE condition (3) is satisfied, an improved steady-state tracking performance asymptotic output tracking is aso achieved, i.e., X(t) and s ast.

5 387 Figure 2. A biaxia inear motor driven gantry system IV. COMPARATIVE EXPERIMENTS The proposed DIARC contro agorithm is impemented on the Y-axis of a precision X-Y Anorad HERC-5-5-AA- B-CC2 gantry driven by LC-5-2 iron core inear motors as shown in Figure2. The position sensors of the gantry are two inear encoders with a resoution of.5μm after quadrature. The veocity signa is obtained by the difference of two consecutive position measurements. Standard off-ine eastsquare identification is performed to obtain the parameters of the Y-axis. To test the earning capabiity of the proposed DIARC agorithms, a 5kg oad is mounted on the motor in experiments and the identified vaues of the parameters are B =.8/s, A =.23m/s, E =.55m/s 2 /V, f N = m/s 2 where f N is the nomina vaue of f u. And w(t) is the output of a dead-zone described by:.9(v.2) forv.2 w = D(v)= for < v <.2 (3) (v ( )) forv Adding the effect of E =.55m/s 2 into the dead-zone, we can get that the approximate accurate vaues are θ = [.8,,23,.395,.674,.55,.55] T. As the simuated deadzone is supposed to be unknown, we set the nomina vaues to be θ N =[.8,,23,.425,.65,.5,.5] T. The bounds of the parametric variations are chosen as θ min =[.7,.,.3,.4,.4,.8] T θ max =[,.4,.6,.9,.8,.4] T (32) Assuming x d =.2 +.8sin(πt π/2)+.7sin(.8πt π/2)+.6sin(.2πt π/2). Initia vaues of pant states are set as X()=[,] T. The contro agorithms are impemented using a dspace DS3 controer board. The controer executes programs at a samping period of T s =.2ms, resuting in a veocity measurement resoution of.25m/sec. The foowing three contro agorithms for the system () are impemented and compared: C: The proposed DIARC adapting the estimates of physica parameters B,A,E, f N with ignoring the existence of the unknown dead-zone (3). C2: The proposed DIARC presented in section II and III. C3: The proposed DIARC in which ony B,A,E, f N are updated based on the proposed on-ine parameter adaptation agorithm and the dead-zone (3) effect is compensated by (4) with actua dead-zone parameter vaues assuming the dead-zone parameters are known. In C2, w ds2 in (4) is given in section III-B. Theoreticay, we shoud use the form of w ds2 = k s2 s with k s2 being a noninear proportiona feedback gain as given in (22) to satisfy the robust performance requirement (2) gobay. In impementation, a arge enough constant feedback gain k s2 is used instead to simpify the resuting contro aw as done in [5]. With this simpification, noting (4), we choose w ds = k s s in the experiments where k s represents the summation of k s and k s2 and is chosen as: k s = 2. The constant λ in (2) is λ = 2. The constant adaptation rate γ in (8) is γ = and the bound of d c is d cm = 2. The initia vaues o θ()=[.8,.23,.425,.65,.5,.5] T, Γ()=I 6. The forgetting factor is α =.2. The constant H in Lemma 2 is H =.7. For a fair comparison, the controer parameters and the initia vaues in case C and C3 are chosen the same as in C2 when they have the same meaning (e.g., w ds in C and C3 are chosen the same as in C2). The initia vaues of the adaptation matrix Γ in (29) for C and C3 is Γ()=I 4 for that the dead-zone parameters are not adapted, consequenty there are B,A,E, f N need to be estimated and their initia vaues are [.8,.23,.55,] T. In C3, the dead-zone (3) are known and can be competey compensated by its inverse (4). The experimenta resuts in terms of a performance indexes after running for one period are given in Tabe I. Overa, C2 and C3 achieve amost the same tracking performances which are better than C. The experimenta resuts for a three cases are shown in Figures 3 6. Specificay, Fig. 3 shows the tracking errors of a three cases. It is aso seen that steadystate output tracking errors of C2 and C3 are amost the same and in the order of 6 m, much ess than that of C which is in the order of 5 m. These resuts agree with the prediction by theory as it is shown in section III that the proposed DIARC in C2 is abe to achieve asymptotic convergence of dead-zone parameter estimates and perfect dead-zone compensation when the estimates converge. This is aso verified by the history of dead-zone parameter estimates shown in Fig. 4. The estimate vaues B =.8, Â =.23, Ê =.58 shown in Fig. 5 are quite accordant with their off-ine identified vaues, and the estimate vaues θ =[.82,.225,.42,.66,.5,.49] T shown in Fig. 4 are accordant with the approximate accurate vaue of θ. Fig. 6 shows the contro inputs v(t) of a three cases, which show the chattering phenomena especiay the sudden jumps due to the dead-zone inverse (4) when the inputs are near zero. Overa, the improved steady-state error of C2 over C vaidates the necessity of dead-zone compensation. And the amost same asymptoticay converging steady-state tracking performance of C2 and C3 vaidates the perfect compensation of unknown dead-zones by the proposed agorithm. Tabe I PERFORMANCES INDEXES Controer C C2 C3 x 2 (μm) x (μm)

6 C: Error (um) C: Input (V) C2: Error (um) C3: Error (um) C2: Input (V) C3: Input (V) Figure 3. Output tracking errors of DIARCs with dead-zone Figure 6. Contro input history of DIARCs with dead-zone B m r m Figure 4. B E Figure 5. A m r b r m b Parameter estimates of DIARC in C2 t (s) f N A t (s) Parameter estimates of DIARC in C3 V. CONCLUSION In this paper, an integrated direct/indirect desired compensation adaptive robust contro scheme has been deveoped for eectrica drive systems having unknown dead-zone effects. Theoreticay, certain guaranteed robust transient performance and steady-state tracking accuracy are achieved even when the overa system may be subjected to parametric uncertainties, time-varying disturbances and other uncertain noninearities. Furthermore, zero steady-state output tracking error is achieved when the system is subjected to unknown parameters and unknown dead-zone noninearity ony. Comparative experimenta resuts are obtained on a inear motor drive system preceded by simuated unknown dead-zone noninearity, which vaidate the effectiveness of the proposed dead-zone compensation scheme. The exceent output tracking performances obtained in the experiments aso verify the high performance nature of the proposed DIARC strategy. REFERENCES [] G. Tao and P. V. Kokotovic, Adaptive contro of pants with unknown dead-zones, IEEE Transactions on Automatic Contro, vo. 39, no., pp , January 994. [2] K. Ma and M. N. Ghasemi-Nejhad, Adaptive contro of fexibe active composite manipuators driven by piezoeectric patches and active struts with dead zones, IEEE Transations on Contro System Technoogy, vo. 6, no. 5, pp , September 28. [3] R. R. Semic and F. L. Lewis, Dead-zone compensation in motion contro systems using neura networks, IEEE Transactions on Automatic Contro, vo. 45, no. 4, pp , Apri, 2. [4] X.-S. Wang, C.-Y. Su and H. Hong, Robust adaptive contro of a cass of inear systems with unknown dead-zone, Automatica, vo. 4, pp , 24. [5] T. P. Zhang and S. S. Ge, Adaptive dynamic surface contro of noninear systems with unknown dead zone in pure feedback form, Automatica, vo. 44, no. 7, pp , Juy, 28. [6] C.-C. Hua, Q.-G. Wang and X.-P. Guan, Adaptive tracking controer design of noninear systems with time deays and unknown dead-zone input, IEEE Transations on Automatic Contro, vo. 53, no. 7, pp , August 28. [7] J. Zhou, C. Wen and Y. Zhang, Adaptive output contro of noninear systems with uncertain dead-zone noninearity, IEEE Transactions on Automatic Contro, vo. 5, no. 3, pp. 54-5, March, 26. [8] C. Hu, B. Yao, and Q. Wang, Integrated direc/indirect adaptive robust contro of a cass of noninear systems with non-symmetric dead-zone input, Automatica. Revision is under review. [9] L. Xu and B. Yao, Adaptive robust precision motion contro of inear motors with negigibe eectrica dynamics: theory and experiments, IEEE/ASME Trans. on Mechatronics, vo. 6, no. 4, pp , 2. [] B. Yao, Integrated direct/indirect adaptive robust contro of SISO noninear systems in semi-strict feedback form, Proceedings of the American Contro conference, Denver, Coorado, pp , June, 23. The O. Hugo Schuck Best Paper (Theory) Award in 24. [] B. Yao, High performance adaptive robust contro of noninear systems: a genera framework and new schemes, Proc. of the IEEE Conference on Decision and Contro, San Diego, Caifornia, pp , 997. [2] M. Krstic, I. Kaneakopouos and P. V. Kokotovic, Noninear and adaptive contro design, Wiey, New York, 995. [3] B. Yao and M. Tomizuka, Adaptive robust contro of SISO noninear systems in a semi-strict feedback form, Automatica, vo. 33, no. 5, pp , 997. [4] I. D. Landau, R. Lozano and M. M saad, Adaptive contro, Springer, New York, 998. [5] B. Yao, F. Bu, J. Reedy, and G. T.-C. Chiu, Adaptive robust contro of singe-rod hydrauic actuators: theory and experiments, IEEE/ASME Transactions on Mechatronics, vo. 5, no., pp. 79 9,