Practical Fractional-Order Variable-Gain Super-Twisting Control with Application to Wafer Stages of Photolithography Systems
JJOURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1
Practical Fractional-Order Variable-GainSuper-Twisting Control with Application to WaferStages of Photolithography Systems
Zhian Kuang,
Student Member,
Liting Sun, Huijun Gao,
Fellow, IEEE,
Masayoshi Tomizuka,
Life Fellow, IEEE
Abstract —In this paper, a practical fractional-order variable-gain super-twisting algorithm (PFVSTA) is proposed to improvethe tracking performance of wafer stages for semiconductormanufacturing. Based on the sliding mode control (SMC), theproposed PFVSTA enhances the tracking performance fromthree aspects: 1) alleviating the chattering phenomenon viasuper-twisting algorithm and a novel fractional-order slidingsurface (FSS) design, 2) improving the dynamics of states onthe sliding surface with fast response and small overshoots viathe designed novel FSS and 3) compensating for disturbancesvia variable-gain control law. Based on practical conditions, thispaper analyzes the stability of the controller and illustratesthe theoretical principle to compensate for the uncertaintiescaused by accelerations. Moreover, numerical simulations provethe effectiveness of the proposed sliding surface and controlscheme, and they are in agreement with the theoretical analysis.Finally, practice-based comparative experiments are conducted.The results show that the proposed PFVSTA can achieve muchbetter tracking performance than the conventional methods fromvarious perspectives.
Index Terms —fractional-order, variable-gain, super-twistingcontrol, sliding mode control, wafer stage.
NTRODUCTION S EMICONDUCTOR manufacturing involves many preci-sion devices, one of which is the wafer scanner [1]. Awafer scanner is an optomechanical device used to finish thephotolithography task, the principle of which is illustrated inFig. 1 [2]. The reticle, which contains the integrated circuit(IC) patterns on it, is placed onto the reticle stage, and thewafer stage carries the wafer. A fixed laser beam is generatedby the illumination system. As the reticle stage and the waferstage move, the reticle’s IC patterns will be scanned andprojected onto the wafer via lenses. To guarantee high yieldingrate and quality, manufacturers require that both the reticlestage and the wafer stage track the designed trajectories in afast and precise manner, i.e., the overlay error should be withina few nanometers [3].Practically, it is not trivial to achieve such high-precisetracking performance, because of the various disturbancesand the inherent movements of wafer stages. For example,
Zhian Kuang and Huijun Gao are with the Research Institute of IntelligentControl and Systems, Harbin Institute of Technology, Harbin,150001, China.Huijun Gao is also with the State Key Laboratory of Robotics and System,Harbin Institute of Technology, Harbin 150001, China.Zhian Kuang, Liting Sun and Masayoshi Tomizuka are with the MechanicalControl System Lab, Mechanical Engineering Department, University ofCalifornia, Berkeley, CA 94720, USA.Corresponding Author: Huijun Gao(e-mail: [email protected]). Fig. 1. Schematic illustration of a photolithography system, which includes(1) an illumination system, (2) a reticle scanning stage, (3) a reticle, (4) pro-jection lens, (5) a wafer stage, (6) a wafer. disturbances like external vibrations, force ripples, cable com-pliance, and structural vibrations can all occur in the waferstage system [4]. Some of these disturbances are even state-dependent, such as those caused by cables, which makesthe application environment more complex [5]. Moreover, thewafer stage’s dynamic behavior is also position-dependent [5],[6]. As the movement of wafer stages contains both rapidacceleration phases and steady precise scanning phases, howto guarantee the performance for both phases, as well as howto balance the disturbance rejection and measurement noisesensitivity, is also a tricky problem to be solved [7].In the practical control of wafer stages, engineers usuallyapply both feedforward control and feedback control methodto obtain excellent performance. Some researchers focus onthe improvement of feedforward control techniques, such asthe widely used iterative learning control (ILC) [8], [9], [10],[11]. The ILC method requires no plant information and iseasy to be implemented, but the performance will be degradedwhen non-repetitive disturbances exist, such as cable forcesand structure vibrations [4]. The other researchers focus onthe feedback control methods, of which the most extensivelyutilized one is PID control [12]. However, the PID controllerand other linear control methods like H ∞ control can not a r X i v : . [ ee ss . S Y ] F e b OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 2 handle the ”water-bed effect” properly [7], [13]. Therefore,researchers proposed the variable-gain PID controller to fixthis problem [14], [15]. Nevertheless, due to the position-dependent dynamics and disturbances in the wafer stage, theactual performance achieved by these PID control methods islimited [4], [5]. Recently, the sliding mode control (SMC) isimplemented to wafer stages by researchers inspired by itseffectiveness in improving the tracking performance in thepresence of uncertainties and disturbances [4], [5], [16], [17].Results have shown that SMC can guarantee the robustnessto both position-dependent disturbances and non-repetitivedisturbances in the wafer stage system [4], [5]. However,traditional SMC implementations suffer from the well-knownchattering problem due to its variable structure [18], [19] .This shortcoming might significantly deteriorate the achievableperformance of the SMC controller [18].To solve the chattering problem, Levant proposed the super-twisting algorithm (STA) in [20], and STA turns out to be oneof the most effective method [21], [22]. In the traditional STA,the gains in the switching control law are constants, denoted asconstant-gain super twisting algorithm (CGSTA) in this paper.This kind of controller can reduce the chattering phenomenonbut can only deal with the time-invariant and state-independentdisturbances and uncertainties [23], [24], which also suffersfrom the water-bed effect. Based on STA, a variable-gainsuper-twisting algorithm (VGSTA) was proposed in [23] tofurther enhance the performance in terms of introducingvariable parameters in STA. This method provides sufficientcompensation for uncertainties and disturbances, which canpromote the robustness of the system. However, the boundaryfunctions of the uncertainties/disturbances in this method arehard to know, which brings obstacles to design the controllerin the practical applications on wafer stages.Moreover, a common drawback of the two kinds of STAabove is the slow responses due to the use of linear slidingsurfaces. To improve the performance further, researchers haveproposed some advanced sliding surfaces to develop STA fur-ther. For instance, Based on the integral sliding surface (ISS),the integral super-twisting algorithm (ISTA) is derived in [25].This control algorithm guarantees the fast response of thestates on the sliding surface but has a rather significant over-shoot. The fractional-order sliding surface (FSS) has emergedas a useful control scheme towards this issue [26], [27],[28], [29]. As a generalization of the integer-order slidingsurface, it can simultaneously guarantee fast response andsmall overshoot of the dynamics [30], [31]. Some researchesalso show that FSS can help to reduce the chattering phe-nomenon further [32]. Due to these merits, it has been used invarious mechatronic systems with excellent performance [33],[34], [30], and a new fractional-order constant-gain super-twisting algorithm (FCGSTA) is developed based on this [35].However, as this controller only has constant gains, it cannotthoroughly compensate for the uncertainties and disturbances.Moreover, the conventional design of the FCGSTA consistsof too many hyper-parameters, which are hard to tune in theapplication to wafer stages.In ultra-precision motion control tasks on wafer stages, thecontroller needs to satisfy: • fast response to track the reference signal; • robust performance in the presence of disturbances anduncertainties; • high performance in different phases; • practically easy to design.Motivated by such goals, a practical fractional-order variable-gain super-twisting algorithm (PFVSTA) is proposed in thispaper. The proposed method’s sliding surface is designed withthe term of fractional-order (FO) calculus, which can guaranteethe states reach the equilibrium points with fast response andsmall overshoot. Moreover, variable gains are introduced inthe controller so that the uncertainties are handled properly.Specifically, the wafer stage’s acceleration is considered, andthe controller can get better performance than traditionalmethods. Finally, the controller is developed based on thepractical model and is easy to implement. With all thesemerits, different disturbances and uncertainties, as well as thewater-bed effect in the wafer stage system, can be handledappropriately by the proposed method.This paper’s contribution is four-fold: 1) A specially de-signed fractional-order sliding surface is given out. By in-troducing a particular term, the convergence of the stated onthe sliding surface is accelerated, making the controller havebetter performance without increasing tunable parameters. 2)A novel variable-gain switching controller is designed andanalyzed theoretically and practically, which guarantees therobustness of the controller under model uncertainties anddisturbances. By designing a novel form of the super-twistingalgorithm, the sliding variable dynamics is further improved.3) In the designed controller, the influence of the feedforwardterm is analyzed for the wafer stage system. Moreover, withacceleration-based variable gains, the acceleration’s influenceis appropriately handled, so that the precision of the waferstage system is further improved. 4) The proposed controlleris applied and verified in simulations and experiments on areal wafer scanning stage testbed.The remainder of this paper is organized as follows. Sec-tion 2 presents the model of the wafer stage and the designedcontrol strategy. Section 3 presents a stability analysis of theproposed PFVSTA. Simulation and experimental results aregiven in Section 4. Finally, Section 5 concludes the paper.2.. C ONTROLLER D ESIGN
A. Model of Wafer Stage
As shown in Fig. 2, the testbed in this paper is a one-dimensional stage. It is supported via air bearings, and directlydriven by a linear motor along with the sliding guide. Basedon Newton’s second law, the dynamics of the wafer stage canbe written as [36] ˙ v = ( F − f ) /m (1)where v is the velocity of the stage, ˙ v is the acceleration ofthe stage, f denotes the sliding friction, m is the mass of thestage, and F is the actuation force exerted by the linear motor.Benefiting from the air bearings, the sliding friction can beapproximated by f = K v v + d f (2) OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 3
Fig. 2. The wafer stage testbed including three parts: (1) the moving stage,(2) the air bearings, and (3) the linear motor. where K v is the viscous friction coefficient, and d f is theequivalent un-modelled part of the friction, i.e., friction un-certainties.The actuation force F is proportional to the input voltage u and it can be written as F = Qu + d r (3)where Q [ N/V ] is the ratio between the actuation force andinput voltage, and d r stands for unknown disturbances causedby, for instance, the force ripple of the motor [37].Substituting (2) and (3) into (1), the system dynamicsbecome ˙ v = − T v v + Ku + d (4)with T v = K v m , K = Qm and d = d f + d r m .Note that in (4), d is unknown and influences the systemdynamics as a lumped disturbance. Hence, we can obtain thefollowing nominal model: ˙ v = − ¯ T v v + ¯ Ku (5)where ¯ T v and ¯ K are the nominal values of T v and K respectively. The uncertainties are represented in an additiveform as K = ¯ K + ∆ K (6) T v = ¯ T v + ∆ T v (7)where ∆ K and ∆ T v stand for the uncertain parts in K and T v respectively. B. PFVSTA
For the succeeding PFVSTA design, the following assump-tion is imposed:
Assumption 1:
The parametric uncertainties of the modeland the disturbance in the system are bounded, i.e., ∆ K , ∆ T v and d are bounded.In this paper, the tracking error is defined as e = p − r ,where e is the tracking error, p is the actual position of thestage, r stands for the reference position, and it is apparentthat ˙ p = v . The sliding mode controller will be designed onthe basis of e . Here, we have another assumption: Assumption 2:
The actual velocity v and the referencevelocity are Lipschitz functions, i.e., ˙ v and ¨ r are bounded. Fig. 3. Block diagram of the proposed controller
A novel fractional-order sliding surface is presented as s = ˙ e + k D ξ − [ sig ( e ) a ] + k sig ( e ) a , ξ, a ∈ (0 , (8)where k and k are positive constants, sig ( • ) ∗ = sgn ( • ) | • | ∗ , D ∗ is fractional-order calculus operator and its definition isstated as Definition 1 in Appendix A.As shown in Fig. 3, we design the sliding mode controlleron the equivalent-control basis [38] and it is written as u = u eq + u sw (9)where u eq is the equivalent controller and u sw is the switchingcontroller with variable-gain super-twisting algorithm.According to the theory of SMC, the switching control lawis implemented to make the sliding variable s converge to 0,i.e., to make the states converge to the sliding surface. In thispaper, a novel switching controller with VGSTA is addressedas u sw = − h ( ˙ v )¯ K Φ ( s ) − (cid:90) t h ( ˙ v )¯ K Φ ( s ) dt (10)with Φ ( s ) = | s | α ( s ) sgn ( s ) (11) Φ ( s ) = Φ (cid:48) ( s )Φ ( s ) (12) α ( s ) = 4 | s | + 12 ( | s | + 1) (13)where h ( ˙ v ) and h ( ˙ v ) are the variable gains.The equivalent control u eq is implemented to maintainthe states on the sliding surface when no uncertainties anddisturbances are considered. Therefore, by setting ˙ s = 0 andconsidering the model of the wafer stage, u eq can be obtainedas u eq = 1¯ K (cid:18) ¨ r − k dD ξ − [ sig ( e ) a ] dt − k a | e | − aa ˙ e + ¯ T v v (cid:19) . (14) Remark 1:
For the application on wafer stages, the designedsliding surface (8) guarantees a fast and smooth response,which is beneficial for precision. The designed switchingcontroller (10) uses a novel structure to make s converge fastand steadily. The variable-gain structure also has an advantageover the traditional methods, especially when the referenceacceleration is large. From Fig. 3 and (14), it can be seenthat our u eq contains both feedback and feedforward terms. Inthe precision control of wafer stages, feedforward techniques OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 4 are extensively used to enhance the performance [8], [9].The feedforward term in our method not only guarantees thecompletion of the sliding mode controller but also improvesthe performance of the wafer stage. All these merits ofour proposed method will be verified by simulations andexperiments in Section 4..3.. D
YNAMIC A NALYSIS OF
PFVSTAThe dynamics of s can be obtained by substituting (9) intothe natural model (4), ˙ s = − K ¯ K h ( ˙ v )Φ ( s ) − K ¯ K (cid:90) t h ( ˙ v )Φ ( s ) dt + (cid:18) K ¯ K − (cid:19) (cid:18) ¨ r − k dD ξ − [ sig ( e ) a ] dt − k ˙ e (cid:19) + (cid:18) K ¯ K ¯ T v − T v (cid:19) v + d. (15)Equation (15) can be further rewritten as ˙ s = − h ( ˙ v )Φ ( s ) − (cid:90) t h ( ˙ v )Φ ( s ) dt + g ( ˙ v )+ g ( ˙ v ) (16)with g ( ˙ v ) = − ∆ K ¯ K h ( ˙ v )Φ ( s ) + ∆ K ¯ K ¨ r − ∆ K ¯ K k D ξ [ sig ( e ) a ] + d (17)and g ( ˙ v ) = (cid:18) K ¯ K ¯ T v − T v (cid:19) v − ∆ K ¯ K (cid:90) t h ( ˙ v )Φ ( s ) . (18)Based on (17), (18), Assumption 1 and Assumption 2, weobtain that | g ( ˙ v ) | and (cid:12)(cid:12)(cid:12) dg ( v, ˙ v ) dt (cid:12)(cid:12)(cid:12) are bounded with | g ( ˙ v ) | ≤ D | Φ ( s ) | + D (19) (cid:12)(cid:12)(cid:12)(cid:12) dg ( v, ˙ v ) dt (cid:12)(cid:12)(cid:12)(cid:12) ≤ D | ˙ v | + D | Φ ( s ) | (20)where D , D , D and D are positive constants that satisfy D ≥ (cid:12)(cid:12) ∆ K ¯ K (cid:12)(cid:12) | h ( ˙ v ) | , D ≥ (cid:12)(cid:12) ∆ K ¯ K (cid:12)(cid:12) | ¨ r | + (cid:12)(cid:12)(cid:12) k dD ξ − [ sig ( e ) a ] dt (cid:12)(cid:12)(cid:12) + | d | , D ≥ (cid:12)(cid:12) K ¯ K ¯ T v − T v (cid:12)(cid:12) and D ≥ (cid:12)(cid:12) ∆ K ¯ K (cid:12)(cid:12) | h ( ˙ v ) | .The following lemma is useful in the further analysis: Lemma 1: (see [39], [40], [41]) For the fractional-orderintegration operator I αt , the following relation holds: | I αt f ( t ) | ≤ κ | f ( t ) | (21)where κ is a finite positive constant. Theorem 1:
For the system (4), if the variable gains h ( ˙ v ) and h ( ˙ v ) of the controller (9)-(10) are selected as h ( ˙ v ) = p p − p p p (cid:18) ( − p δ ( γ ) + p δ ( ˙ v )) p p p + p δ ( ˙ v ) − p p δ ( γ ) (cid:19) (22) h ( ˙ v ) = 1 p ( p − p h ( ˙ v )) (23) with p p − p > , p > , p < (24) δ ( γ ) = D + D Φ ( γ ) (25) δ ( ˙ v ) = D | ˙ v | + D (26)then the sliding variable s will converge to the region Γ = (cid:8) s (cid:12)(cid:12) | s | ≤ γ (cid:9) , γ > . (27) Proof . For the sake of convenience, rewrite (16) as ˙ s = − h ( ˙ v )Φ ( s ) + z + g ( ˙ v ) (28) ˙ z = − h ( ˙ v )Φ ( s ) + dg ( ˙ v ) dt . (29)Construct a Lyapunov function candidate as V = Θ T P Θ = (cid:2) Φ ( s ) z (cid:3) (cid:20) p p p p (cid:21) (cid:20) Φ ( s ) z (cid:21) (30)where Θ is defined as Θ = (cid:2) Φ ( s ) z (cid:3) T and P is definedas P = (cid:20) p p p p (cid:21) . From (24), we find that P ≥ , whichsatisfies the condition of being selected as a Lyapunov functioncandidate.The derivative of the Lyapunov function candidate is ˙ V = (cid:18) d Θ dt (cid:19) T P Θ + Θ T P d Θ dt . (31)In (31), d Θ dt is calculated as d Θ dt = (cid:2) Φ (cid:48) ( s ) ˙ s ˙ z (cid:3) T = Φ (cid:48) ( s ) (cid:20) l ( ˙ v ) 1 l ( ˙ v ) 0 (cid:21) (cid:20) Φ ( s ) z (cid:21) (32)where l ( ˙ v ) and l ( ˙ v ) are defined as l ( ˙ v ) = − h ( ˙ v ) + g ( ˙ v )Φ ( s ) (33) l ( ˙ v ) = − h ( ˙ v ) + dg ( v ) dt ( s ) (34)respectively for the sake of simplicity.By substituting (32) into (31), we have ˙ V = Φ (cid:48) ( s ) Θ T (cid:20) p l ( ˙ v ) + 2 p l ( ˙ v ) ∗ p + p l ( ˙ v ) + p l ( ˙ v ) 2 p (cid:21) Θ (cid:44) Φ (cid:48) ( s ) Θ T (cid:20) A BB C (cid:21) Θ . (35)Considering the definition of l ( ˙ v ) and l ( ˙ v ) , we have A = − p h ( ˙ v ) − p h ( ˙ v ) + 2 p g ( ˙ v )Φ ( s )+ 2 p dg ( v ) dt Φ ( s ) . (36) OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 5
Substitute (22) and (23) into (36) to obtain A = (cid:18) p p − p (cid:19) h ( ˙ v ) − p p p + 2 p g ( ˙ v )Φ ( s )+ 2 p dg ( ˙ v ) dt ( s )= ( − p δ ( γ ) + p δ ( ˙ v )) p + 2 p (cid:18) g ( ˙ v )Φ ( s ) − δ ( γ ) (cid:19) + 2 p (cid:18) δ ( ˙ v ) + dg ( ˙ v ) dt ( s ) (cid:19) . (37)When the sliding variable s is not in the region Γ , i.e., | s | > γ > , we get that Φ ( γ ) < | Φ ( s ) | . Hence, accordingto (25) and (17), δ ( γ ) = D + D Φ ( γ ) > D + D | Φ ( s ) | ≥ | g ( ˙ v ) || Φ ( s ) | . (38)For δ ( ˙ v ) , we can get that δ ( ˙ v ) = D ˙ v | Φ ( s ) | + D ≥ | Φ ( s ) | (cid:12)(cid:12)(cid:12)(cid:12) dg ( ˙ v ) dt (cid:12)(cid:12)(cid:12)(cid:12) . (39)Considering that p > , p < and substituting (38) and(39) into (36), finally we attain A < . On the other hand, (cid:12)(cid:12)(cid:12)(cid:12)
A BB C (cid:12)(cid:12)(cid:12)(cid:12) = 2 p (2 p l ( ˙ v )) + 4 p l ( ˙ v ) − (cid:18) p g ( ˙ v )Φ ( s ) + p dg ( v ) dt Φ ( s ) (cid:19) = − p p h ( ˙ v ) − p h ( ˙ v )+ 4 p p g ( ˙ v )Φ ( s ) + 4 p dg ( v ) dt Φ ( s ) − (cid:18) p g ( ˙ v )Φ ( s ) + p dg ( v ) dt Φ ( s ) (cid:19) . (40)By substituting (22) and (23) to (40), we obtain (cid:12)(cid:12)(cid:12)(cid:12) A BB C (cid:12)(cid:12)(cid:12)(cid:12) = 4 p (cid:18) dg ( ˙ v ) dt Φ ( s ) + δ ( ˙ v ) (cid:19) − (cid:18) p g ( ˙ v )Φ ( s )+ p dg ( v ) dt Φ ( s ) (cid:19) + 4 p p (cid:18) g ( ˙ v )Φ ( s ) − δ ( γ ) (cid:19) + ( − p δ ( γ ) + p δ ( ˙ v )) > . Then based on Φ (cid:48) ( s ) > when | s | > γ , it is obvious that Φ (cid:48) ( s ) Θ T (cid:20) A BB C (cid:21) Θ < . (41)It has been shown that V is indeed a Lyapunov function.This completes the proof of Theorem 1. Remark 2:
Theorem 1 not only states the sufficient con-ditions for the stability, but also gives the relation betweenthe variable gains h ( ˙ v ) , h ( ˙ v ) and the region Γ . Based onthis relation, we can analyze how the acceleration affect theperformance as follows: Consider there are two states that ˙ v and ˙ v , suppose that | ˙ v | > | ˙ v | and let p , p and p remain the same to make thecomparison fair, then we have h ( ˙ v ) = p p − p p p (cid:18) ( p δ ( γ ) + p δ ( ˙ v )) p p p + p δ ( ˙ v ) − p p δ ( γ ) (cid:19) (42) h ( ˙ v ) = p p − p p p (cid:18) ( p δ ( γ ) + p δ ( ˙ v )) p p p + p δ ( ˙ v ) − p p δ ( γ ) (cid:19) (43)and δ ( ˙ v ) > δ ( ˙ v ) from (26). In constant-gain super-twisting algorithms, h ( ˙ v ) and h ( ˙ v ) have the relation that h ( ˙ v ) = h ( ˙ v ) = C . Thus we have δ ( γ ) < δ ( γ ) and γ > γ , which means that a larger absolute value of acceler-ation brings a larger γ under the constant-gain super-twistingcontrol. Based on this, we can infer that large acceleration hasa negative influence on the precision of the system when theconstant-gain super-twisting controller is applied.As for the variable-gain super-twisting controller, h ( ˙ v ) canbe designed as an increasing function, i.e., h ( ˙ v ) < h ( ˙ v ) .Ideally, if h ( ˙ v ) and h ( ˙ v ) are designed to satisfy that h ( ˙ v ) − h ( ˙ v ) = p p − p p p (cid:18) p (cid:0) δ ( ˙ v ) − δ ( ˙ v ) (cid:1) + p ( δ ( ˙ v ) − δ ( ˙ v ))+ p p δ ( γ )2 ( δ ( ˙ v ) − δ ( ˙ v )) (cid:19) (44)then we have γ = γ , which means that the negative influenceof the acceleration is eliminated completely. In practice, dueto the acceleration is usually time-varying and the variablegains are usually designed with a simple function, (44) cannotalways be satisfied. However, as long as h ( ˙ v ) is designedproperly, it is helpful to reduce the influence that the acceler-ation poses on the precision. Theorem 2:
If the sliding variable s reaches the region Γ ,then the tracking error will converge to the region E , and E = (cid:8) e ( t ) (cid:12)(cid:12) | e ( t ) | ≤ ε (cid:9) (45)where ε is the positive solution of the equation f ( x ) = k κx a − k x a + γ = 0 . (46) Proof . Construct another Lyapunov function candidate W ( t ) = | e ( t ) | . (47)The derivative of W ( t ) is ˙ W ( t ) = sgn ( e ( t )) ˙ e ( t ) . (48)From the definition of the sliding surface (8), ˙ e ( t ) = − k D ξ − ( sig a ( e ( t ))) − k sig a ( e ( t )) + s. (49)Then ˙ W ( t ) can be rewritten as ˙ W ( t ) = sgn ( e ( t )) (cid:16) − k D ξ − ( sig a ( e ( t ))) − k e a ( t ) + s (cid:17) ≤ | k t I − ξ | e ( t ) | a sgn ( e ( t )) | − k | e ( t ) | a + sgn ( e ( t )) s. OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 6
From Lemma 1 and the precondition that | s | ≤ γ , we canobtain ˙ W ( t ) ≤ k κ | e ( t ) | a − k | e ( t ) | a + γ . If the positivesolution of f ( x ) = k κx a − k x a + γ = 0 is denoted as ε ,then it is obvious that when x > ε , f ( x ) > .Thus, when | e ( t ) | > ε , we have ˙ W ( t ) < . (50)This completes the proof. Remark 3:
According to Theorem 2, the boundary of thestable state is the solution of (46). Based on the property offunction f ( x ) , a smaller γ leads to a smaller ε and further,higher precision of the system theoretically, which keepsconsistent with our cognition. Remark 4:
When < a < , the positive solution of f ( x ) =0 always exist. We can assume that there are two functions f ( x ) = k κx a + γ and f ( x ) = k x a . It is apparent that f (0) > f (0) . Because f (cid:48)(cid:48) ( x ) = k κa ( a − x a − < and f (cid:48)(cid:48) ( x ) = k a (cid:0) a − (cid:1) x a − > , there always be f ( ε ) = f ( ε ) , i.e., f ( ε ) = 0 .For a = 1 , the sliding surface becomes s = ˙ e + k D ξ − [ sig ( e )] + k e , which is a traditional fractional-ordersliding surface. Through a similar analysis procedure, we canget that f (cid:48)(cid:48) ( x ) = f (cid:48)(cid:48) ( x ) = 0 , which means that if k , k or κ is selected improperly, a finite ε cannot be obtained, i.e., thestability on the sliding surface cannot be guaranteed.4.. S IMULATIONS AND E XPERIMENTS
In this part, simulations and experiments are conducted toverify the effectiveness of the proposed PFVSTA. To make theresults convincing, six different controllers will be mentionedand compared in different aspects. They are listed here tofacilitate understanding: • the traditional CGSTA [22], u = K (cid:0) ¨ r − k ˙ e + ¯ T v v (cid:1) − h ¯ K Φ ( s ) − (cid:82) t h ¯ K Φ ( s ) dt , Φ ( s ) = | s | sgn ( s ) , Φ ( s ) =Φ (cid:48) ( s )Φ ( s ) . • the VGSTA (also denoted as LVGSTA due to the lin-ear sliding surface) [23], u = K (cid:0) ¨ r − k ˙ e + ¯ T v v (cid:1) − h ( ˙ v )¯ K Φ ( s ) − (cid:82) t h ( ˙ v )¯ K Φ ( s ) dt , Φ ( s ) = | s | sgn ( s )+ h s , Φ ( s ) = Φ (cid:48) ( s )Φ ( s ) . • the variable-gain PID (VGPID) controller used in [15]. • the FCGSTA, u = K (cid:16) ¨ r − k dD ξ − [ sig ( e )] dt − k ˙ e + ¯ T v v (cid:17) − h ¯ K Φ ( s ) − (cid:82) t h ¯ K Φ ( s ) dt , Φ ( s ) = | s | sgn ( s ) + h s , Φ ( s ) = Φ (cid:48) ( s )Φ ( s ) . • the proposed PFVSTA scheme (9)-(14). • the proposed PFVSTA scheme with no feedforwardterm, denoted as incomplete PFVSTA (IFVSTA), u = K (cid:16) − k dD ξ − [ sig ( e ) a ] dt − k a | e | − aa ˙ e + ¯ T v v (cid:17) − h ( ˙ v )¯ K Φ ( s ) − (cid:82) t h ( ˙ v )¯ K Φ ( s ) dt , Φ ( s ) and Φ ( s ) are ob-tained from (11) and (12) respectively. A. Simulation Studies1) Sliding Surface Comparison:
Simulations are con-ducted first to investigate the proposed sliding surface (de-noted as Case 4). Simulations are implemented in MAT-LAB/SIMULINK. For comparison, the simulations about the
Time (s) -100-50050100
Case 1Case 2Case 3Case 4
Fig. 4. Dynamics of e around different sliding surfaces.TABLE IP ARAMETERS OF C ONTROLLERS IN S IMULATIONS .Parameters CGSTA VGSTA PFVSTA, IFVSTA k - - 53 k h . v + 50 0 . v + 50 h . v + 10 0 . v + 10 h - 38 - tracking error on the conventional linear sliding surface(LSS), the integral sliding surface (ISS), and the fractional-order sliding surface (FSS) are conducted. The three casesare denoted as Case 1, Case 2, and Case 3, respectively.According to the literature [23], the LSS is formulated as s = ˙ e + k e , the ISS is s = ˙ e + k (cid:82) t e + k e and theFSS is s = ˙ e + k D ξ − [ sig ( e )] + k e . Without loss ofgenerality, parameters of the sliding surfaces are set as: k = 8 , k = 500 , ξ = 0 . , a = 0 . . To implement the fractional-ordercalculus in simulation, we use the discrete-time version toapproximate the continuous fractional-order calculus [27]. Inpractice, s cannot remain zero if uncertainties and disturbancesexist in the system. Thus, s is set as s = 1 to investigatedifferent sliding surfaces’ performance with uncertainties. Thesimulation results are shown in Fig. 4.From Fig. 4, it is apparent that the tracking error on theLSS converges to the equilibrium point at a prolonged rate.The tracking error on the ISS converges fast, but the trajectoryis with a rather significant overshoot, and it takes a long timeto converge. For the FSS, its trajectory converges faster thanISS, and it has a much smaller overshoot. As for our proposedsliding surface, tracking error converges the fastest, and ithas a small overshoot. The enlarged window demonstratesthe trajectories from the time . s to . s to represent howmuch the uncertainties influence the trajectories of each slidingsurface. From the result, we notice that ISS’s trajectory hasnot yet converged, LSS has a rather large distance to zero, andthe trajectory of the proposed sliding surface is closest to zero.This phenomenon represents that the uncertainty on s has theleast influence on the state on the proposed sliding surface.
2) Controller Comparison in Acceleration Phases:
Basedon the model in Section 2.-A, we establish the simulationmodel of the wafer stage as G ( s ) = Ks ( s + T v ) , of which theinput is the control signal and the output is the position of themover. Parameters are chose as T v = 1 . and K = 3 . ,which were identified from the real system in our previouswork [42]. OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 7 (a1) A cc e l e r a t i o n ( m / s ) Reference (b1) -3-2-10 E rr o r ( m ) -6 CGSTAVGSTAIFVSTAPFVSTA (a2) (b2) -3-2-10 10 -6 Fig. 5. Performance of the controllers in the simulations of accelerationphases (the horizontal axis represents time, unit: s)
The reference accelerations are designed as shown inFig. 5 (a1) and (a2). The reference position is obtained byintegration. Note that the nominal parameters of the model areintended to be set as ¯ T v = 1 and ¯ K = 4 to simulate the modeluncertainties. Band-limited white noise is also introduced asmeasurement noises of the system. We compare four controlalgorithms in this group of simulations: CGSTA, VGSTA,IFVSTA, and PFVSTA. Their parameters are shown in Table Iand the results are shown in Fig. 5. (a) P o s i t i o n ( m ) Reference (b) -505 E rr o r ( m ) -6 CGSTAVGSTAPFVSTAVGPIDIFVSTA -8 Time (s)(c) -10-50510 A cc e l e r a t i o n ( m / s ) Reference
A/D PhasesIP IPSP
Fig. 6. Performance of the controllers in simulational scanning phases.
By comparing Fig. 5 (b1) with (a1) and (b2) with (a2),we can see that tracking errors of all the four controllersincrease with the acceleration. This phenomenon accordswell with our theoretical analysis in Remark 2. We can seethat a larger acceleration results in a larger uncertainty to the system from (20), which will deteriorate the system’sprecision. Results shown in Fig. 5 can also reflect the necessityto investigate how the acceleration of reference influences theperformance. In wafer stage systems, the acceleration phasesusually last less than 0.1 s, and the scanning process beginsjust after the acceleration. Fig. 5 demonstrates that the errorsof most controllers cannot converge to zero in 0.1 s. Thereforethe performance in the acceleration phases will influence thescanning process as well.Besides, from the results, PFVSTA’s error trajectory isalways the smallest throughout the simulations, which showsthat our method has a significant advantage over the othermethods. Moreover, the number of tunable parameters in ourproposed method is the same as other methods. It means thatthe improvement in precision will not bring difficulties to tuneparameters in the controller. Moreover, by comparing PFVSTAand IFVSTA in Fig. 5 (b1) and (b2) and considering thedifference between PFVSTA and IFVSTA, we can concludethat the feedforward term in proposed u eq is necessary toget the excellent performance. This conclusion will also beproven more than one time in the following simulations andexperiments.
3) Controller Comparison in Scanning Process:
The sec-ond group of simulations is conducted to simulate the scanningprocess. As shown in Fig. 6 (a), a typical scanning trajectorycontains at least three phases: the idle phase (IP), the scan-ning phase (SP), and the acceleration/deceleration phase (A/Dphase). The scan length is set as 0.05 m, the first idle phase isset as 0.2 s, the acceleration and deceleration phases are set as0.012 s, the scanning phase is set as 0.4 s, and the maximumacceleration is set as 10 m/s . Particularly, when the movingstage moves 0.23 m (around 0.4 s in this case), an extra stepdisturbance with an amplitude of 0.1 is applied intentionallyto investigate the robustness of controllers.Apart from the four control algorithms in the previousgroup, we also implemented the VGPID controller as a com-parison to study the robustness of the proposed PFVSTA.Parameter of PFVSTA are tuned as k = 33 , k = 10000 , h = 0 .
01 ˙ v + 40 , and h = 0 .
01 ˙ v + 20 . Parameters ofVGSTA are k = 1200 , h = 0 . v + 50 , h = 0 . v + 20 and h = 43 . CGSTA has parameters of k = 1200 , h = 1500 , h = 80 . As for VGPID, the gains for the proportionalterm, integral term and derivative term are K p = × , K i = × and K d = × . The variation of the proportionare ∆ K p = × in the A/D phases and ∆ K p = × in other phases.The results are shown in Fig. 6 (b). We can see that in theA/D phases, CGSTA has the largest error, which proves thatcompared with variable-gain methods, constant gain methodscannot get satisfying performance. This proves further thatthe variable-gain structure is helpful to reduce the influenceof reference acceleration and indirectly prove the rationalityof the variable structure in our proposed u sw (10). From theenlarged window, we can see that the proposed PFVSTA issmooth and precise in the scanning phase, while the VGPIDis not as precise as other methods. Moreover, after the extradisturbance is applied, position errors of CGSTA and VGSTAconverge quickly, while VGPID converges rather slow. This OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 8 shows that compared with the SMC method, the PID methodis easy to be influenced by extra disturbances. As for theproposed PFVSTA method, the position error is imperviousto the disturbance, which shows its excellent robustness.
B. Experimental Setup
Fig. 7. The structure of the experimental system.Fig. 8. The laser system for the measurement of the wafer stage position,including (1) a laser generator, (2) two reflectors, (3) an optical pickup, (4) aninterferometer, (5) a reflecting mirror. (a) is part of wafer stage and (b) is partof laser measurement system. The red arrows stand for the optical path.
We also tested the proposed algorithm on the wafer stageshown in Fig. 2. As we mentioned before, the testbed mainlyconsists of a linear motor and a moving stage. The movingstage is driven by the linear motor along with the slidingguide. Between the sliding guide and the moving stage, airbearings are fixed to reduce the friction. As shown in Fig. 7,the experimental system includes a host computer, a controller,an amplifier, the wafer stage, a laser measurement system,and a sampling card. Control algorithms are implemented withthe LabView Real-Time system running on the controller (afield-programmable gate array, i.e., FPGA, PXI 7831R fromNational Instruments). The control signals for the stages andcounter-masses, generated by the FPGA controllers, are am-plified via separate amplifiers (TA330 from Trust Automationcompany) and then sent to the linear motor of the waferstage. In the meanwhile, the position signal from the lasermeasurement system (in Fig. 8) is sampled by a samplingcard (N1231B from Keysight) and fed back into the controllerrunning on the FPGA.In this paper, two groups of experiments are conducted,which are denoted as Case 1 and Case 2. In both cases, thescan length is set as 0.05 m, the scan velocity is set as 0.1 m/s, the waiting time before the scan is set as 0.2 s, and the holdingtime at the top of the scan is set as 0.1 s. The difference isthat the maximum acceleration is set as 1.5 m/s in Case 1and 10 m/s in Case 2. The reference scanning trajectoriesare displayed in Fig. 9 (a) and Fig. 10 (a) respectively.In either case, four controllers are compared: LVGSTA,FCGSTA, IFVSTA and PFVSTA. For the proposed PFVSTAand IFVSTA, ξ and a are directly chosen as / . Otherparameters are tuned to get the best performance, and theyare k = 100 , k = 2 × , h ( ˙ v ) = 550 | ˙ v | + 13 and h ( ˙ v ) = 10 | ˙ v | + 4 . For the other two control schemes, tomake the comparison fair, all the parameters are also tuned toguarantee that the best performance is attained. C. Experimental Results (a) P o s i t i o n ( m ) Reference (b) -2024 P o s i t i o n E rr o r ( m ) -5 FCGSTALVGSTAIFVSTAPFVSTA
Time (s)(c) -2-1012 A cc e l e r a t i o n ( m / s ) Reference
IP IPIP SP 2SP 1 A/D Phases
Fig. 9. Experimental results of the controllers in Case 1.
Fig. 9 depicts the experimental results in Case 1. FromFig. 9 (b), we can see that all the four controllers have rela-tively larger absolute values of position errors during the A/Dphases while having relatively smaller absolute values duringthe idle phases and scanning phases. We can also see that eitherin idle phases, scanning phases, or A/D phases, the proposedPFVSTA has the smallest error overall. For the FCGSTA, wecan find that it usually has the largest error in the A/D phases,while it has a medium error in the idle and scanning phases.This phenomenon proves that the reference’s acceleration ismore likely to influence the performance of FCGSTA thanLVGSTA and PFVSTA. Considering the differences amongthese controllers, it reflects the variable-gain structure, thedesigned sliding surface (8) and the designed switching controllaw (10) help to improve the performance in A/D phases.Fig. 10 shows the results of the experiments in Case 2. Asthe reference trajectory’s maximum acceleration is much larger
OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 9 than the experiments in Case 1, the corners of the referencetrajectory are apparently sharper. For the position errors, thepeak values of FCGSTA and LVGSTA reach at about 100 µm , while the peak values of IFVSTA are the largest. Theerror of the proposed PFVSTA still keeps the smallest valuein all three different phases. This shows that the PFVSTAremains its high precision even when the reference accelerationis raised (compared with Case 1). From the enlarged windowin Fig. 10 (b), we can also observe that in the scanning phase,PFVSTA is the smoothest in the four controllers, which provesthat the chattering phenomenon is significantly reduced. All ofthese phenomena are consistent with the previous simulationresults and theoretical analysis. (a) P o s i t i o n ( m ) Reference (b) -2024 P o s i t i o n E rr o r ( m ) -4 FCGSTALVGSTAIFVSTAPFVSTA -6 Time (s)(c) -10-50510 A cc e l e r a t i o n ( m / s ) Reference
Fig. 10. Experimental results of the controllers in Case 2.
D. Quantitative Analysis
Quantitative analysis of the experiments is conducted to in-vestigate the results thoroughly. Because the root-mean-square(RMS) error and the maximum (MAX) error are two criticalindicators to describe the precision, we figure them out in eachkind of phases and display the bar charts in Fig. 11. The RMSerror is calculated based on
RM S Error = (cid:113) (cid:80) ni =1 e ( i ) n andthe MAX error is based on M AX Error = max | e ( i ) | , i =1 , , ..., n, where e ( i ) is the sampled position error, i denotesthe serial number and n is the total number of the samplederrors.Fig. 11 (a) shows that of all the three phases, the PFVSTAholds the smallest RMS error in both Case 1 and Case 2.Comparing FCGSTA with LVGSTA, we can see that whenthe acceleration is small in Case 1, differences between errorsof FCGSTA and LVGSTA are not as much as those when theacceleration is large in Case 2. We can attribute the reasonfor the effects of the variable gains. Besides, for the scanning Fig. 11. Statistical characteristics of different controllers under the two cases:(a) the RMS Errors, and (b) the MAX Errors. The RMS Error of IFVSTA inCase 2 is . × − m phases, which we care about the most, the proposed method’sRMS error changes much less between Case 1 and Case 2 thanthe other methods. Due to the combination of the variable gainmethod and FSS, the performance of the proposed PFVSTAtends to be less sensitive to the accelerations in the referencesignal. The contrast between IFVSTA and PFVSTA provesthat our proposed method’s feedforward term is essential toreduce the tracking error.All the maximum errors are depicted in Fig. 11 (b). Notsurprisingly, the PFVSTA still keeps the best precision ac-cording to the statistics. An interesting phenomenon is that thecontrollers’ maximum errors in scanning phases are similar tothose in the A/D phases. Checking back the error trajectoriesin Fig. 9 (b) and Fig. 10 (b), we can figure out the reasonis that the maximum errors always happen at the junctions ofdifferent phases. In this way, the bad performance in the A/Dphases negatively influences the performance in the scanningphases. This proves that investigating the performance in A/Dphases is also important to improve the precision of waferstages. Fig. 12. The ratios between the valid scanning time and the whole scanningtime (shown as narrow bars), and the valid scanning errors (shown as widebars) of different controllers.
Considering that the precision of the scanning phases is the
OURNAL OF L A TEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 10 most significant in the industrial application and that there isan obvious decline of the position error when the scanningphases begin, a possible approach is to let the machine beginto scan after the error drops to a satisfying value to guaranteethe precision. We define the time of the whole scanning phaseas t s . After the position error e enters into a certain range,it never escapes from it. Then half of the minimum width ofthe range is defined as the valid scanning error e s , and thetime it keeps in the range is defined as valid scanning time t p .Based on these definitions, it is apparent that a smaller valueof e s and a larger value of t p /t s means better performancewith higher precision and efficiency. The statistics of e s and t p /t s are demonstrated as bar charts in Fig. 12. It is clear thatin the four scanning phases of two cases, both IFVSTA andPFVSTA have the largest values of t p /t s , which is very closeto 1. Furthermore, PFVSTA has the smallest e s under almostall the situations. This result proves the overall advantage ofthe proposed wafer stage control method once more.5.. C ONCLUSION
In this paper, a fractional-order variable-gain super-twistingcontroller was proposed for the wafer stage system, with anovel sliding surface and a novel variable-gain structure ofthe super-twisting algorithm. Specifically, we considered thepractical situation during the analysis of stability and ana-lyzed the controller’s theoretical precision. These theoreticalanalyses explained the advantages of the proposed method.Moreover, simulation and experimental results of the slidingsurfaces and the whole control algorithm not only correlatedwell with the theoretical analyses but also proved the proposedmethod had at least three merits: the high precision, therobustness to uncertainties, and the small chattering amplitude,which guarantee the excellent performance on wafer stages.Finally, adequate experimental results and their quantitativeanalyses proved the superiority of the proposed algorithm withapplication to wafer stages.A
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