High Precision Feedback Control Design for Dual-Actuator Systems

Transcription

1 Proceeding of the 25 IEEE Conference on Control Alication Toronto, Canada, Augut 28-31, 25 TC1.2 High Preciion Feedback Control Deign for Dual-Actuator Sytem Tielong Shen and Minyue Fu Abtract Dual-actuator control ytem are deigned by iggybacking a mall econdary actuator onto a rimary actuator. Yet the control deign for uch ytem i challenging due to the limited dynamic range of the econdary actuator and it couling with the rimary actuator. In thi aer, we rooe a new control deign aroach for dual-actuator ytem. We firt how that, through a imle coordinate tranformation, a dual-actuator ytem can be earated into two ubytem, one of them controlled only by the rimary actuator, which can be deigned uing conventional method, and the other one involving tracking control uing the econdary actuator. We then devie a witching control trategy for the econdary actuator to achieve high reciion control. Two dual-actuator ytem are ued to demontrate the ue of dual actuator and effectivene of the new deign aroach. I. INTRODUCTION The tructural characteritic of a dual-actuator ytem i that there are two cacaded actuator oerating along a common axi. The rimary actuator, called rimary actuator, i of long range but with oor accuracy and low reone time. The minor actuator, called econdary actuator, i tyically iggybacked onto the rimary actuator and deliver much higher reciion and fater reone but ha a limited dynamic range. A good examle of dual-actuator ytem i the o-called dual-tage hard dik drive ervo ytem which conit of a voice coil motor a the rimary actuator and a iezo-electric tranducer a the econdary actuator [1]-[2]. Other examle include VLSI chi manufacturing machine [3], [4] and high reciion machining tool [5], [6]. Although the mechanical deign of dual-actuator ytem aear to be imle and the functionalitie of each actuator aear to be qualitatively clear, it i a challenging tak to coordinate the two actuator to yield an otimal erformance. The main obtacle i that econdary actuator tyically ha a very limited motion range. A naive deign aroach i to tune the rimary actuator to achieve a rough oitioning or tracking and then to activate the econdary actuator when the rimary actuator reache it teady tate. Thi aroach aear to make ene becaue the econdary actuator doe not have much influence when the tracking error i large; and converely, the econdary actuator hould take over when the rimary actuator ha exhauted it ability to reduce the tracking error. However, uch an aroach i inadequate for at leat two reaon: 1) The overall reone time i tyically exceively long; 2) In many alication the ytem i ubject to diturbance or the tracking reference ignal i Tielong Shen i with the Deartment of Mechanical Engineering, Sohia Univerity, Kioicho 7-1, Chiyoda-ku, Tokyo , Jaan. Minyue Fu i with the School of Electrical Engineering and Comuter Science, The Univerity of Newcatle, Callaghan, NSW 238, Autralia. time-varying, which imlie that the rimary actuator can never be in a teady tate. Many other deign aroache alo exit. For examle, a common deign aroach for dual-tage hard dik drive ytem i to deign two earate control loo, one for each actuator by auming that the interaction between the two control loo i negligible. The two loo can oerate in either erie [7] or arallel [8]. In uch a deign, the rimary and econdary actuator are aigned to control low and frequency reone, reectively. However, it i well known that thi control aroach tyically reult in oor tability erformance due to the couling between the two actuator. Indeed, the mot difficult aect of the dualactuator control deign i to figure out how to coordinate the two actuator in a region where both actuator have ome but limited control authoritie. In thi aer, we rooe a new control deign aroach for dual-actuator ytem. Our reearch tart with an intereting obervation that many dual-actuator ytem can be effectively decouled through a imle coordinate tranformation. The reult of it i that we have two ubytem, a rimary ytem and a econdary ytem behaving a follow. The rimary ytem i aroximately indeendent of the econdary ytem and i urely driven by the rimary actuator. The controller of the rimary ytem can be deigned uing tandard mean, and thu i not of rimary interet. On the other hand, the econdary ytem i controlled by the econdary actuator and the couling from the rimary ytem i only through the reference ignal for oitioning or tracking. Motivated by the obervation above, we then focu on the control deign for the econdary actuator. Modeling the econdary actuator a a aturated controller, thi roblem can be cated a a tracking control roblem with an inut aturation. Exiting deign aroache range from anti-windu comenator [9]-[1] to Riccati equation or linear matrix inequality baed aroache [11]-[13]. In our deign aroach, we chooe to otimize a quadratic erformance function. To get around the difficulty with the inut aturation, we exloit a witching control trategy which alie different control gain when the tate of the ytem lie in different region. Thee region are characterized by neted ellioid with outer ellioid correonding to looer erformance bound and the inner one to tighter bound. A the tracking error converge, the controller gradually witche from low erformance gain to high erformance one. Two deign examle are ued to demontrate our deign aroach. The firt one i a imle two-ma ytem exemlifying a ingleaxi linear-motion dual-actuator ytem. The econd one i a dual-tage ynchronization ytem commonly ued in the micro-manufacturing indutry /5/$2. 25 IEEE 956

2 y r u Dual-Actuator C u Σ Σ z x y Next, we how that the model (1) can be decouled into two ubytem via a imle coordinate tranformation. Thi i motivated by the fact that the rimary ytem tyically ha a much larger ma than the econdary ytem. Let Fig. 2. Fig. 1. BASE Tyical Dual-Actuator Control Sytem u z 1 m y= x c k c k 1 σ[ u ] A two-ma model for Dual-Actuator Sytem (Linear Motion) II. MODELING OF DUAL-ACTUATOR SYSTEMS A dual-actuator ytem can be deicted in Fig. 1, where Σ and Σ denote the rimary and econdary ytem (or ubytem), reectively. The rimary actuator u drive the rimary ytem directly and the econdary actuator u control the econdary ytem but it alo inject a counteraction directly onto the rimary ytem. The tate vector of the two ubytem are denoted by x and z, reectively. The control objective i to deign a uitable controller C uch that the outut y of the econdary ytem track a given reference ignal y r with ome recribed erformance ecification. A dual-actuator ytem with ingle-axi linear motion can be rereented by a two-ma model a hown in Fig. 2. The ma, daming coefficient, ring contant, oition and velocity of the rimary ytem are denoted by m,c, k, z 1 and z 2 reectively. Their counterart for the econdary ytem are denoted by m,c, k, x 1 and x 2, reectively. The outut y equal x 1. In thi model, the rimary ytem i attached to a fixed bae. However, additional dynamic to the bae and the interaction with them can be modeled into the actuating function u. The dynamic equation of the ytem are given a follow: ẋ 1 = x 2 ż 1 = z 2 m ż 2 = u σ[u ] c z 2 k z 1 f(x 1,x 2,z 1,z 2 ) m ẋ 2 = σ[u ] f(x 1,x 2,z 1,z 2 ) (1) where f(x 1,x 2,z 1,z 2 )=c (x 2 z 2 )k (x 1 z 1 l ) rereent the interaction force between two actuator, l > denote the offet ditance between two mae, and σ[ ] i a aturation function defined by σ[u] = a u a (u >a) ( u a) (u < a) m a> (2) z 1 ξ 1 = m x 1 m z 1, z 2 ξ 2 = m m ξ 1 (3) That i, we chooe the ma center ξ 1 and it derivative a a new coordinate for the rimary ytem. It follow that ξ 2 = bu (t) a 2 ξ 2 (t) a 1 ξ 1 (t)ɛ(t) (4) where b =(m m ) 1,a 1 = k m m 1,a 2 = c m m 1 and ɛ(t) =m (k x 1 c x 2 )/(m 2 m m ) decribe the couling force from the motion of the econdary ytem. Due to the fact that m m, we can imly ignore the ɛ(t) term. It follow that the dynamic of the rimary ytem i aroximately decouled from that of the econdary ytem and the deign of u (t) can be done uing any conventional method. On the other hand, if we define e 1 = x 1 y r and e 2 = ė 1 = x 2 ẏ r, then ė 2 = g{σ[u (t)] d(t)} h 2 e 2 (t) h 1 e 1 (t) (5) where g = m 1,h 1 = k m 1,h 2 = c m 1 and d(t) =c (ẏ r z 2 )k [r (z 1 l )] m ÿ r. (6) The model (5) now become the model for the econdary ytem. We oberve that the couling ignal d(t) i dominated by the error between the tate of the rimary actuator (z 1,z 2 ) and the deired tate (y r l, ẏ r ) and the acceleration command m ÿ r. If the error or the acceleration command i large, it i not oible to comletely comenate d(t) by uing u (t). It i only when d(t) i within a mall region (in comarion with a) that the econdary actuator ha ome meaningful control effect. The obervation above mean the following. We can ue the inut of the rimary actuator to drive the ma center uch that the new reference ignal d(t) converge to a mall region a quickly a oible. Prior to uch convergence of d(t), the econdary actuator hould be turned off becaue it doe not have much effect. The control gain of u (t) hould be gradually increaed when d(t) become mall. Thi motivate the idea of a witching control trategy for the econdary actuator, which we dicu in the next two ection. III. CONTROL DESIGN FOR SECONDARY ACTUATOR In thi ection, we focu on the deign roblem for the econdary ytem in a dual-actuator ytem. In view of the dicuion in Section II, we conider the following model: ẋ = Ax b(σ[u] d(t)), x() = x (7) where x IR n i the tate with initial condition x, u IR i the control inut, d(t) IR i a given reference ignal, and A IR n n and b IR n 1 are given. Without lo of generality, we aume that the aturation level i a =1. Roughly eaking, the deign goal i to find a tate feedback control law uch that the tate x i driven from it initial 957

3 tate x() to the origin with a re-ecified erformance cot or le and that the region of uch initial tate i maximal. It i obviou that if d(t) > 1 hold eritently, it i not oible to achieve the deign goal. Therefore, it i neceary for the rimary actuator the bring the teady tate value of d(t) to be within [ 1, 1]. In view of thi, we aume that the control action of the rimary actuator ha been alied rior to t =uch that d(t) d, t for ome d (, 1). Under the aumtion, we can comenate the diturbance by introducing a feed-forward term in u(t) a follow: Then we can rewrite (7) a u(t) =v(t)d(t) (8) ẋ = Ax bσ d [v], x() = x (9) where σ d [ ] i defined a in (2) with a =1 d(t) >. We now conider the following quadratic cot function J(x,v)= {x T Qx rσ 2 d[v]}dt (1) for ome Q = Q T and r > with (A, Q 1 2 ) being detectable, where σ d [v] =σ[u] d(t). We eek to minimize J(x,v) by uing a linear tate feedback v = Kx. If there i no aturation, it i well-known that the otimal olution for v i given by v = r 1 b T P x (11) and the minimum quadratic cot i x T P x, where P = P T > i the olution to the following Riccati equation: A T P P A Q r 1 P bb T P = (12) In the reence of aturation, the otimal v i difficult to comute. To get around thi difficulty, we follow the aroach in [12] by arameterizing the controller by uing a bound on v. More reciely, given, we retrict the v to be uch that v (1 d ) (13) That i, act a an over-aturation bound. It i eay to verify that for any v contrained by (13), σ d [v] lie in the following ector bound a hown in Fig. 3 where δ(v) =σ d [v] 1 v; δ(v) 2 v (14) 1 = 2(1 d ) 2(1 d ), 2 = 2(1 d ) (15) Now, for a given >, conider a Lyaunov function candidate of the form V (x) =x T P x for ome P = P T to be determined and define Ω = A T P P A Q r 1 P bb T P (16) 1 v 2 v σ d [v] (The thick line) 1 v 1 d 1 d(t) 1 v 2 v v 1 d Fig. 3. Sector Bound for σ d [v] It i traightforward to verify that where J(x,v) = lim V (x ) V (x(t )) T V (x ) ( V x T Qx rσ 2 d[v])dt ϕ(x, v, δ(v))dt ϕ(x, v, δ(v)) = x T (A T P P A Q)x rσd 2 [v]2xt P bσ d [v] = x T Ω x r( 1 v δ(v) v ) 2 (17) Thi imlie that if ϕ(x, v, δ(v)) along the trajectory of the ytem (9) with the feedback control, we have J(x,v) V (x ) (18) Uing the analyi above, we formulate the following (relaxed) auxiliary otimal control roblem: For a given, deign P and v to minimize V (x ) ubject to ϕ(x, v, δ(v)) for all x R n and all δ( ) atifying the following ector bound: δ(v) 2 v (19) In addition, determine the larget invariant et of the form X = {x : x T P x µ 2 } (2) uch that for any x X, x(t) X and v(t) 1 d for all t and J(x,v) V (x ). Theorem 1: Conider the ytem in (9), the quadratic cot function in (1) and a given bound on the level of over-aturation. Define = 2 / 1. Suoe the equation A T P P A r 1 (1 2 )P bb T P Q = (21) ha a olution P >. Then the otimal K and the aociated X for the auxiliary otimal control roblem are given by K = 1 1 r 1 b T P (22) µ = 2(1 d ) 2 r (23) b T P b 958

4 Proof: We firt olve min v δ(v) 2 v for a given x IR n. It i traightforward to comute δ(v) 2 v = x T Ω x r{( 1 v v ) ( 1 v v )v 2 2 v2 } where v = r 1 b T P x. We conider two cae: 1) 1 v 2 v v and 2) 1 v 2 v v. For the firt cae, we have δ(v) 2 v = x T Ω x r{( 1 v v ) ( 1 v v )v 2 2v 2 } = x T Ω x r(v v ) 2 The lat te i obtained by uing 1 2 = 1. The minimizing v ubject to 1 v 2 v v i given by v = 1 1 v, then min v: 1v 2 v v δ(v) 2 v = x T Ω x r( 1 1 v v ) 2 = x T Ω x r (v ) 2 = x T Ω x where Ω = A T P P A Q (1 2 )P bb T P. For the econd cae, we have δ(v) 2 v = x T Ω x r{( 1 v v ) ( 1 v v )v 2 2v 2 } = x T Ω x r(( 1 2 )v v ) 2 It follow that min v: 1v 2 v v δ(v) 2 v = x T Ω x r(( 1 2 ) 1 1 v v ) 2 = x T Ω x r (v ) 2 = x T Ω x with the ame otimal v. Hence, we conclude that the otimal v i 1 1 v which i the ame a (22). From the above dicuion, we know that given x and v, δ(v) xt Ω x (x, v) 2 v where (x, v) with (x, 1 1 v )=. Next, we need to minimize x T P x ubject to or equivalently, x T Ω x (x, v) x T Ω x (x, v) We note that the uer bound of Ω increae a (x, v) decreae. Alo, P ha a well-known monotonicity, i.e., it decreae when Ω increae. Hence, the otimal P i obtained by minimizing (x, v) and maximizing Ω, i.e., by etting v a in (22) and Ω =which lead to (21). In order to characterize the invariant et X, all we need to do i to determine the larget µ uch that max Kx = 1 x T P x µ 2 1 r 1 B T P x =1 d (24) Let η = P 1/2 x. Then, max Kx = max r 1 1 x T P x µ 2 1 η µ BT P 1/2 η (25) The otimizing η i given by µ η = P 1/2 B P 1/2 b (26) Subtituting x = P 1/2 η into (24), we obtain (23). Remark 1: If =, the Riccati equation (21) and the control gain (22) recover the reult in (12) and (11) for no aturation. The aociated invariant et i given by X := {x : x T P x µ }, µ = 2(1 d )r 2 (27) b T P b IV. NESTED SWITCHING CONTROL Roughly eaking, a ractical way to avoid or reduce inut aturation i to make the controller gain mall when the tate i large and to increae the gain when the tate i mall. Switching control with a equence of neted control law i motivated by thi baic idea. To do thi, a equence of control law need to be deigned in uch a way that the invariant et are neted. More reciely, we need to find the controller K i,i =, 1,,N uch that the correonding invariant et X i = {x : x T P i x µ 2 i },i =, 1,,N, atify X X 1 X N. Furthermore, we alo demand P <P 1 < <P N o that the quadratic erformance get imroved gradually when the control gain witche from K N to K. The deired neting roertie above can indeed be achieved by uing the deign method in the reviou ection with different value of. Thi i due to the monotonicity of the Riccati equation (21). Indeed, we can rewrite (21) a A T S S A r 1 (1 )S BB T S Q = (28) where S =(1 )P and Q =(1 )Q. Uing S and Q, the invariant et (2) can be exreed a X = {x : x T P x (1 d ) 2 r 2 } (1 )b T S B = {x : x T S x (1 d ) 2 r 2 } (29) b T S B Theorem 2: The olution S to equation (28) i monotonically decreaing in >, i.e., S ε <S ε,if <ε. Conequently, X ha the following neting roerty: Similarly, we have X X ε, < ε (3) P <P ε, < ε (31) Proof: It uffice to etablih the reult above for ufficiently mall ε>. It i eay to verify that = /[2(1 959

5 d )] i monotonically increaing in. Thu, for any ε>, we can write ε 2(1 d ) ε = ε 1 with ome ε 1 >. It follow that the Riccati equation (28) for ε can be rereented a y(t) time[ec] y(t) Fig. 4. Time reone (y(t)[m]) time[ec] A T S ε S ε A r 1 (1 ε 1 )S ε BB T S ε (1 ε 1 )Q = (32) Let E = S S ε. From (28) and (32), we have à T E Eà r 1 (1 ε 1 )EBB T E ε 1 Q ε 1 S BB T S = (33) where à = A r 1 (1 ε 1 )BB T S. From (28), we know that A r 1 (1 )BB T S i Hurwitz. Therefore, A r 1 (1 ε 1 )BB T S i alo Hurwitz when ε 1 (or equivalently, ε) i ufficiently mall. It follow from (33) and the detectability of (A, Q 1/2 ) that E>, i.e. S >S ε. The neting roerty of X then follow from (29). The monotonicity of P i i roved imilarly. Baed-on the neting roerty of the invariant et X, we can aly Theorem 1 to deign a equence of control gain K i (= K i ),i =, 1,,N, for a equence of overaturation bound = < 1 < 2 < < N, and witch the gain to K i whenever x X i (= X i ) but i outide of X i 1 (= X i 1 ). The next reult how the erformance imrovement by uing thi witching control. Theorem 3: Suoe the witching controller above i alied to the ytem in (9) with x X N. Let τ i be the time intance K i i witched on, i =, 1,,N. In articular, τ N =. Then, N 1 J(x,v)=x T P N x x T (τ i ) P i x(τ i ) (34) i= where P k = P i1 P i >, i =, 1,,N 1. Proof: Following the roof of Theorem 1, we have x T Qx rσd 2 [K ix] d { x T P i x } (35) dt along the trajectory of x(t),t [τ i1,τ i ). Integrating the inequality above yield (34). V. DESIGN EXAMPLES A. Single-axi Dual-Stage Poitioning control Recall the dual-tage oitioning ytem in Fig. 2. The control roblem i to drive the outut y to a given command y r. The hyical arameter of the ytem are hown in Table I. The aturation level for the econdary tage i a =.2 and the offet between the two tage i l =.5. Recaling the control inut and the inut matrix a.2u u and.2b B, the ytem can be rewritten a model (1) with the aturation level a =1. Chooe Q and r a [ ] 5 Q =, r =.1 (36) 1 Fig. 5. State trajectorie In order to how the effectivene of the neted witching control trategy, we are mainly intereted in the control deign for the econdary tage. The rimary actuator i imly controlled by a PID controller with the gain k = 5., k I =5, k d =. To deign the econdary tage, we take d =.1 and chooe the over-aturation level to be 4 = 5, 3 = 1, 2 =5, 1 =1and =. Thi lead to the witching feedback control gain K 4 =[ ] K 3 =[ ] K 2 =[ ] (37) K 1 =[ ] K =[ ] The correonding domain of attraction have µ 4 = 1.145,µ 3 =.169,µ 2 =.449,µ 1 =.18 and µ =.51. The correonding Lyaunov matrice for µ 4 and µ, for examle, are given by, reectively, P 4 = [ ],P = [ Simulation reult are hown in Fig. 4-5 (the left lot) which demontrate the te reone to the et-oint command y r =.5. The domain of attraction are alo hown in Fig. 5. To comare the erformance of the rooed witching controller, we alo how the reone (the right lot in Fig. 4-5) when the econdary tage i controlled by a P controller with otimized gain k =.45, k i =, k d =. It i clear from the imulation that the rooed witching controller can drive the outut y to y r much fater than a PID controller. B. Synchronizing control of multi-tage Synchronization of two or more mechanical machine i very imortant when the machine mut cooerate. Steingand-laer canning ytem are tyical machine of thi kind widely ued in the emiconductor manufacturing indutry. ] 96

6 y r Fig. 6. e (Synchr. err) u C PID u PID Primary Stage z Mater-Stage Dual-Stage Secondary Stage Synchronizing control ytem with dual-actuator x y m y TABLE I PHYSICAL PARAMETERS OF THE DUAL-STAGE Ma[kg] Sring[N/m] Damer[N/m] Primary-Stage Secondary-Stage TABLE II PHYSICAL PARAMETERS OF THE STAGES Ma[kg] Sring[N/m] Damer[N/m] Mater-Stage Primary-Stage Secondary-Stage Fig. 6 how a tyical etu of uch a ytem. In thi etu, a wafer i laced on the mater tage (called wafer tage) and a reticle are laced on a lave tage (called reticle tage). The two tage are required to move in ynchrony with high reciion along the ame axi o that the content in the reticle can be exoed to the wafer. In order to achieve the high reciion in ynchronization, a dual-tage i uually ued for the reticle tage. The control of the econdary tage i targeted at tracking the motion of the wafer-tage. The arameter of the ytem ued in the imulation are hown in Table II. For imlicity, the arameter of the dualtage ytem i choen to be the ame a in Examle 1. Conequently, the ame witching controller for Examle 1 i ued here. However, the PID controller for the rimary tage i adjuted to have k = 1,k i =15,k d =to imrove the ynchronizing erformance. Two different command ignal of y r, ram and ine wave, are ued in the imulation. The reone of the mater tage are hown in Fig. 7 (the lot in the to), and the correonding ynchronization error are hown in Fig. 7 (the lot in the middle). To comare the effectivene of the rooed ynchronization aroach where the reticle tage i actuated by the dualtage with the witching controller, we alo how in Fig. 7 (the lat two lot) the ynchronization error when the reticle tage i driven by a ingle actuator, i.e. the econdary tage i fixed on the rimary tage. It i clear from Fig. 7 that the rooed witching controller erform much better. Mater Stage Synchro Error x 1-7 time[ec] x 1-3 time[ec] PI Syn Error time[ec] Mater Stage x 1-5 time[ec] Synchro Error x 1-3 time[ec] Syn Error PI time[ec] Fig. 7. Synchronization with the rooed control and PID control (the mater tage oition y m(t)[m] and the ynchronization error e(t)[m]) VI. CONCLUSIONS For dual-actuator ytem, an effective way to achieve fine control quality i to deign the control trategy with delicate enitivity, due to the rimary and the econdary actuator have different range and different reone roertie. Particulary, the econdary actuator i uually aturated and of mall range. In thi aer, we have hown that a feaible method to thi end i to focu the deign arm on the econdary actuator, ince the control can be decouled by a imle coordinate change. We rooed a neted witching control aroach for the econdary actuator which take the aturation and the interaction from the rimary actuator into account. REFERENCES [1] R. B. Evan et. al., Piezoelectric micro-actuator for dual tage control, IEEE Tran. Magnetic, vol. 35, , [2] S. H. Lee, S. E. Baek and Y. H. Kim, Deign of a dual tage actuator control ytem with dicrite-time liding mode for hard dik drive, IEEE Conf. Deciion and Control, Sydney, , 2. [3] S. Makinouchi, Y. Hayahi and S. Kamiya, New tage ytem for te-and-reead caning teer, Tranaction of The Jaan Society for Preciion Engineering, Vol.61, No.12, , [4] M. Tomizuka, J.-S. Hu, T.-C. Chin and T. Kamano, Synchronization of Two Motion Control Axe Under Adative Feedforward Control, Journal of Dynamic Sytem, Meaurement and Control, Vol.114, No.2, , [5] T. Kamano, T. Suzuki, N. Iuchi and M. Tomizuka, Adative Feedforward Controller for Synchronization of Two Axe Poitioning Sytem, Tranaction of the Society of Intrument and Control Engineer, Vol.29, No.7, , [6] L.-F. Yang and W.-H. Chang, Synchronization of Twin-Gyro Preceion Under Cro-Couled Adative Feedforward Control, Journal of Guidance, Control, and Dynamic, Vol.19, No.3, , [7] S. J. Schhroeck and W. C. Mener, On control deign for time invariant dual-inut ingle-ouut ytem, Proc. American Control Conf., , [8] S. J. Schhroeck, W. C. Mener and R. J. McNab, On comaator deign for linear time invariant dual-inut ingle-ouut ytem, IEEE Tran. Mechatoronic, vol. 5-57, Mar. 21. [9] W. Niu and M. Tomizuka, An anti-windu deign for the aymtotic tracking of linear ytem ubject to actuator aturation, Proc. American Control Conf., Philadehia, , [1] N. Kaoor, A. Teel and P. Daoutidi, An anti-windu deign for linear ytem with inut aturation, Automatica, vol.34, , [11] A. Saberi, Z. Lin and A. R. Teel, Control of linear ytem with aturation actuator, IEEE Tran. Auto. Contr.,, vol.41, , [12] M. Fu, Linear Quadratic Control with Inut Saturation, in Perective in Robut Control, ed.s.o.r.moheimani, Sringer, 21. [13] T. Iwaaki and M. Fu, Regional H 2 erformance ynthei, in Actuator Saturation Control, Ed. V. Kaila and K. M. Grigoriadi, Marcel Dekker, Inc., New York,