A Decoupled Inversion-Based Iterative Control ... - Semantic Scholar
2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010
WeB14.5
A Decoupled Inversion-based Iterative Control Approach to Multi-axis Precision Positioning: 3-D Nanopositioning Example♯ Yan Yan, Haiming Wang, and Qingze Zou † Abstract— This article proposes the multi-axis inversionbased (MAIIC) approach. System inverse provides a nature avenue to utilize the priori knowledge of system dynamics in iterative learning control, resulting in rapid convergence as well as exact tracking (for nonminimum-phase systems). The benefits of system inverse, however, are not fully exploited in time-domain ILCs due to the lack of uncertainty quantification. This critical limit was removed in the inversionbased iterative control (IIC) techniques through a frequencydomain formulation. The existing IIC techniques, however, is limited to single-input-single-output (SISO) systems, and the time-domain properties of the IIC techniques are not clear. The contribution of this article is: First, the IIC technique is extended from SISO systems to multi-input-multi-output systems, and the convergence condition is analyzed. Secondly, the time-domain properties of the MAIIC law are discussed. It is shown that the set of tractable frequencies is characterized by the bounds of system uncertainty that are quantifiable in practices, and the truncated MAIIC input-output convergences to the neighborhood of the desired input-output truncated in time, where arbitrarily small tracking error can be obtained by having a large enough truncation time. The proposed MAIIC technique is illustrated in a 3-D nanopositioning experiment using piezoelectric actuators.
I. I NTRODUCTION This paper proposes an inversion-based iterative control (IIC) approach for multi-axis precision positioning. In the last decade, the role of system inverse in iterative learning control (ILC) has been unveiled and recognized [1], [2]. Particularly, the system inverse is utilized in in a frequencydomain framework in the recently developed IIC techniques [3]-[6], which provides a straightforward and systematic avenue to exploit the a priori knowledge of system dynamics on one hand, and account for the dynamics uncertainty on the other. The efficacy of the IIC techniques has been demonstrated in various high-speed precision positioning applications [3], [7]. However, currently the IIC techniques are limited to single-input-single-output (SISO) systems. Moreover, the time-domain properties of the IIC techniques— with their frequency-domain representation—have not been clearly understood. Thus, this paper aims to (1) extend the IIC technique proposed in [3] to multi-input-multi-output (MIMO) systems that have a dominant input to each output channel (i.e., the input that is more influential than the ♯:The work was supported by NSF grants CMMI-0626417 and CAREER award CMMI-0846350. The authors are with the Department of Mechanical Engineering, Iowa state University, Ames, IA 50011 (Yan Yan is currently with the Department of Mechanical Engineering, the Johns Hopkins University, Baltimore, MD 21218) USA. † Corresponding author,
E-mail: [email protected].
978-1-4244-7425-7/10/$26.00 ©2010 AACC
combined cross-coupling inputs), and (2) characterize the time-domain properties of the IIC algorithms. It is advantageous to incorporate system inverse in the ILC framework. We note that in many ILC applications, the a priori knowledge of the system dynamics is available, thereby can be utilized for iterative control. System inversion is a nature choice because the use of system inverse in ILC law results in rapid convergence [1], [6], very much desirable in practical implementations [8]. Moreover, system inverse becomes particularly critical to output tracking of nonminimum-phase systems, for which noncausal ILC law is necessary to achieve precision tracking [2], [1], [9]. The advantages of system inversion, however, are not fully exploited in time-domain based ILC approaches [1], [10], where the system dynamics uncertainty is not quantified. Hence, the iteration coefficient has to be chosen small enough, and thereby, rather conservative, resulting in slow convergence. These system uncertainty related constraints are removed through the development of the IIC techniques [3][6]. By formulating the IIC law in frequency-domain, the bound of system dynamics uncertainty can be quantified in a systematic and straightforward manner [3]-[5] resulting in a rapid convergence. Thus, the utilization of system inverse in the frequency-domain IIC techniques provides features desirable in practical implementations. Limits, however, exist in the existing IIC techniques. The IIC techniques [3]-[6], as presented succinctly in frequency domain, are developed for SISO systems only. The direct extension of the IIC techniques to MIMO systems, however, can results in loss of performance, because of the quantification of the dynamics uncertainty of MIMO systems, however, tends to be over conservative, resulting in not only slow convergence, but also a small set of tractable frequencies. Moreover, although the development of the IIC techniques provides unique insights into the ILC framework by determining the tractable frequencies in terms of system dynamics properties, the set of tractable frequencies as well as the timedomain properties of the IIC techniques are not characterized yet. The effect of such frequency-time truncation on the IIC input as well as the tracking performance is yet to be clarified. Therefore, there exists a need to further extend the IIC techniques. The main contribution of this paper is to extend the IIC technique [3] to square MIMO systems in a straightforward and simple manner. Rather than inverting the transfer matrix of the entire MIMO system, a diagonal matrix consisting of the inverse of the dominant dynamics for each output is used in the proposed multi-axis IIC (MAIIC) law. It is
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shown that the proposed MAIIC converges at frequencies where the diagonal subdynamics has (1) a gain larger than the combined gain of other cross-coupling inputs, and (2) a phase uncertainty smaller than a bound given by the combined cross-coupling dynamics effect. The allowable cross-coupling effect and the set of tractable frequencies are characterized and quantified, and the optimal iteration coefficient is obtained that minimizes the number of iterations for given bound of system uncertainties. Secondly, the time-domain properties of the MAIIC law are addressed for practical implementations. We show that the set of tractable frequencies is compact, thereby time-domain truncation is needed in practical implementation of the MAIIC law. The difference between the MAIIC input-output truncated in time and the desired input-output is bounded by the length of the truncation time window, hence, can be rendered arbitrarily small by having a large enough truncation window. Finally, the proposed MAIIC technique is illustrated in experiments by a 3-D nanopositioning application using piezoelectric actuators. The experimental results obtained demonstrate the efficacy of the proposed approach. II. M ULTI - AXIS I NVERSION - BASED I TERATIVE C ONTROL In this section, we extend the IIC technique originally developed for SISO systems in Ref. [3] to MIMO systems. A. Multi-axis Inversion-based Iterative Control Considering a square MIMO system G( jω ) : C → C n×n , ˆ jω ), (1) Yˆ ( jω ) = G( jω )U( where G11 ( jω ) G12 ( jω ) · · · G1n ( jω ) .. G21 ( jω ) G22 ( jω ) · · · . G( jω ) = (2) .. .. ··· . ··· . ··· · · · Gnn ( jω ) Gn1 ( jω ) represents the actual linear dynamics of the entire MIMO system, and Yˆ ( jω ) = [ yˆ1 ( jω ), yˆ2 ( jω ), · · · , yˆn ( jω ) ]T , (3) ˆ jω ) = [ uˆ1 ( jω ), uˆ2 ( jω ), · · · , uˆn ( jω ) ]T , U( The proposed MAIIC law is described in frequency domain as follows, (4) Uˆ 0 ( jω ) = 0 −1 Uˆ k ( jω ) = Uˆ k−1 ( jω ) + ρ (ω )GI,m ( jω )(Yˆd ( jω ) − Yˆk−1 ( jω )) where GI,m is a diagonal matrix with the diagonal elements being the model of the diagonal subsystems of system G( jω ), obtained in the mth modeling process, i.e., for m ∈ N (N = {1, 2, · · · , n}), GI,m ( jω ) = diag [ G11,m ( jω ), · · · , Gnn,m ( jω ) ] , (5)
ρ ( jω ) = diag[ ρ1 ( jω ), ρ2 ( jω ), · · · , ρn ( jω ) ] (6) is the frequency-dependent iteration coefficient matrix where the diagonal element ρ p ( jω ) ∈ R+ for each p ∈ N. In (4), Uˆ k ( jω ), Yˆk ( jω ) denote the input and the output of the system in the kth iteration, respectively, Uˆ k ( jω ) = [uˆ1,k ( jω ), uˆ2,k ( jω ), · · · , uˆn,k ( jω )]T , (7) Yˆk ( jω ) = [yˆ1,k ( jω ), yˆ2,k ( jω ), · · · , yˆn,k ( jω )]T .
Assumption 1: The desired trajectory Yd (·) is in L2 space, i.e., Yd (·) ∈ L2 , and for system (1), exact output tracking of the desired trajectory Yˆd ( jω ) exists at any given frequency ω , i.e., there exists a desired input U d (·) such that (8) Yˆd ( jω ) = G( jω )Uˆ d ( jω ). Assumption 2: System (1) and its model are both proper, stable, and hyperbolic, i.e., for all p, q ∈ N and m ∈ N, both G pq ( jω ) and G pq,m ( jω ) are proper, stable, and hyperbolic (i.e., have no pure imaginary zeros). Assumption 3: During each iteration, the iterative control input Uk (·) is applied to system (1) under the same initial condition as that for the desired input U d (·). B. Convergence Analysis of the MAIIC Law We shall present the convergence of the proposed MAIIC law in steps. We start with clarifying the convergence condition of the MAIIC law at one given frequency ω , and with no truncation of both the iterative input and the output to finite time window (The truncation constraint will be relaxed later). Lemma 1: Let Assumptions 1 to 3 be satisfied. Moreover, assume that no truncation is applied to either the iterative control input nor the corresponding output. Then, at any given frequency ω , the system output Y k ( jω ) converges to the desired output Yd ( jω ), i.e. limk→∞ Yk ( jω ) = Yd ( jω ), if and only if Λ( jω )∞ < 1,
(9)
E − ρ ( jω )G−1 I,m ( j ω )G( j ω ).
where Λ( jω ) The MAIIC law (4) is decoupled, which substantially simplifies the implementation of the MAIIC law. Next, we define the “tractable” set EG as below. Definition 1: The “tractable” set E G is defined as EG
n
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Ep
p=1
where E p denotes the set of frequencies given by ω ∈ R1 |C p ( jω ) < 1, Ep =
∠∆G pp ( jω ) < arccos (C p ( jω )) ,
(11)
with C p ( jω ) the relative combined cross-axis coupling effect with respect to the gain of the diagonal subsystem in the p th axis, G pp ( jω ) , and ∠∆G pq ( jω ) (for p = q) the phase part of the system uncertainty for subsystem G pq ( jω ), n 1 ∑ G pq ( jω ) , C p ( jω ) G pp ( jω ) q=1 q= p (12) ∠∆G ( jω ) j G pq ( jω ) pq ∆G pq ( jω ) = ∆G pq ( jω ) e . G pq,m ( jω ) Furthermore, we assume that Assumption 4: For the square system G( jω ), the measure of the “tractable” set E G is not zero, i.e.,
µ (EG ) = 0,
(13) R1
where µ (E) is the Lebesgue measure of set E in [11]. The above definition and assumption allow us to quantify the iteration coefficient ρ ( jω ) by the system dynamics uncertainties. We quantify, next, the iteration coefficientfor each individual p-axis.
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Lemma 2: Let Assumption 4 and conditions in Lemma 1 be satisfied, then for any frequency in the “tractable” set, ω ∈ EG , the MAIIC law (4) converges if and only if for any given p ∈ N, the iteration coefficient ρ p ( jω ) is chosen as
ρ p ( jω ) < ρ p,ub ( jω ), where (14) cos ∠∆G pp ( jω ) − C p ( jω ) 2 ρ p,ub ( jω ) . 1 − C p( jω )2 ∆G pp ( jω ) C. Characterization of the “Tractable” Set E G First, we shall define the set of admissible models for any given subsystem dynamics G pq ( jω )
(15) S pq G pq,k ( jω ) ; 0
0, there exists a finite truncation time window, T ∗ ∈ (0, ∞), such that when the truncation window is larger than T ∗ , ∗ EU,d (·)2 , EU,t (·)2 , EU,d (·)2 ≤ ε ,
(37)
where EU,d (t) Udd (t) − Utd (t), EU,t (t) Urd (t) − Utd (t), (38) ∗ EU,d (t) Urd (t) − Udd (t) With the above Lemma 8, the convergence of the truncated MAIIC law is addressed in the following Theorem. Theorem 1: Let Assumptions 1 to 4 be satisfied, and the iteration coefficient ρ p ( jω ) be chosen as in Lemma 5. Then for any given positive constant ε > 0, there exists a finite truncation time T , such that the iterative input-output obtained from the truncated MAIIC law (30) converges to the neighborhood of ε of the directly-truncated desired input Udd (·) and that of the directly-truncated desired output U dd (·) in L2 -space, i.e., d lim EU,k (·)2 lim Ukd (·) − Udd (·) < ε , (39) k→∞ k→∞ 2 d d d lim EY,k (·)2 lim Yk (·) − Yd (·) < ε . (40) k→∞
k→∞
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Remark 3: Design of the Window Function WT (·) The above time-domain analysis of the truncated MAIIC law implies that the design of the truncation window function WT (·) can be important for applications where the implementation time is stringent. For example, the Kaiser window filter technique [13] that trades side-lobe amplitude for mainlobe width can be utilized. III. E XAMPLE : NANOPOSITIONING IN 3-D In this section, we illustrate the proposed MAIIC technique by 3-D nanopositioning using piezo actuators. A. Experiment Scheme The two piezotube actuators for positioning in 3-D (one for the lateral x-y axes, and the other for the vertical z-axis direction) on a scanning probe microscope (SPM, Dimension 3100, Veeco Inc.) were used in the experiments. A pyramid pattern was chosen as the desired trajectory. Tracking of such a pattern required equally-well positioning precision in all xy-z axes. Three different pattern-tracking rates (2 Hz, 10 Hz and 20 Hz) were tested in the experiment, where the patterntracking rate was defined as the rate to traverse the four side triangles once by the order denoted in Fig. 1. With the xy- axes displacement range both at 20 µ m and the vertical one at 2 µ m, the average speeds for these three rates were at 0.39 mm/sec, 1.94 mm/sec and 3.88 mm/sec, respectively. B. Dynamics of the Piezo Actuators for 3-D Positioning To implement the proposed MAIIC technique, the diagonal subdynamics of the x- and z-axes piezo actuators (i.e., the frequency response) in 3-D (e.g., x-to-x) were experimentally measured along with the cross-axis coupling dynamics, as shown in Figs. 2 and 3 To quantify the dynamics uncertainty, frequency responses under different input conditions were acquired for, and the supremum and the infimum of the magnitude uncertainty and the phase
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Fig. 4. The upper bounds of the iteration coefficients κp (ω ) and the iteration coefficients ρselected ( j ω ) used in the experiments for (a) x- and (b) z- axes, respectively, where the vertical green-dashed line at frequency 655 Hz marked the upper bound of the practically tractable set ΩG chosen in the experiment.
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uncertainty were estimated according to (16), then used to quantify the upper bound of the iteration coefficient for each axis (κx ( jω ), κy ( jω ), κz ( jω ))(see (23)). The obtained upper bounds of the iteration coefficients for all the three axes and the iteration coefficients are shown in Fig. 4).
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Fig. 1. The desired trajectories of the pyramid pattern: (a) the 3-D pyramid pattern, and the corresponding desired trajectories in (b) the x-axis, (c) the y-axis, and (d) the z-axis, respectively. The numbers in (a) denotes the order followed in the experiments when traversing the side triangles.
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Experimental results show that significant cross-axis coupling effect existed in the piezo actuators used in this experiment. The lateral to vertical x-y-to-z coupling effects were pronounced (see Figs. 2 and 3). Such a relatively large coupling effect might be caused by the slightly misalignment of the z-axis displacement sensor. In this experiment, the lateral trajectories for the desired pyramid pattern were relatively slow compared to the coupling dynamics between xand y- axis. Thereby, the lateral coupling caused disturbances between x- and y- axes were expected to be small and negligible in the experiments. Hence, the experiment results presented next are focused on the compensation for the lateral to vertical coupling effect. C. Tracking results and discussion The MAIIC technique was applied to track the pyramid pattern by implementing the MAIIC law (4) to all three axes simultaneously. For comparison, the desired pyramid pattern was also tracked by using the DC-Gain method and the IIC technique [3]. In the IIC technique, the IIC law was applied to each axis separately (not simultaneously!) to obtain the control input for each axis individually, and then the control
inputs for all three axes were applied. The tracking results obtained by using the MAIIC technique and the DC-Gain method are compared in Fig. 5 for the x-axis (y-axis tracking results are omitted), and the vertical z-axis tracking results are compared in Fig. 6 for the MAIIC, the IIC, and the DC-gain methods. Finally, the 3-D pyramid pattern tracking results are also compared for the above three methods with respect to the desired pyramid pattern in Fig. 7. The experiment results demonstrate that multi-axis (3-D) precision tracking can be achieved by using the proposed method. In the lateral x-y axes tracking at low speed, the dynamics effect was relatively small (as the lateral speed was relatively small), so was the cross-axis coupling effect between x-y axes (as explained above). Hence, the tracking error was mainly caused by the hysteresis effect of piezo actuators (see Fig. 5). We note, however, as the patterntracking rate increased to 20 Hz, the dynamics effect became pronounced and augmented to the hysteresis effect, resulting in larger tracking error at ∼20%, as shown in Fig. 5 (c).We note that although the vibrational dynamics effect of the z-axis dynamics itself was significantly removed by using the IIC technique (see Fig. 6 (a2) to (c2)), the cross-axis coupling effect still existed and became oscillatory as the pattern tracking increased to 20 Hz. On the contrary, by using the proposed MAIIC technique, such large x-y-to-z coupling effect was successfully removed. We also note that rapid convergence was achieved in the experiments: With no more
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input-output. The proposed MAIIC technique was illustrated through 3-D nanopositioning using two piezo actuators in experiments. R EFERENCES
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Fig. 5. Comparison of the vertical x tracking results obtained by using the MAIIC technique with those by using DC-gain method and Single-axis IIC technique at 2 Hz (a1), 10 Hz (b1), 20 Hz (c1), and comparison of the corresponding tracking errors at 2 Hz (a2), 10 Hz (b2), 20 Hz (c2). 2 Hz
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Fig. 6. Comparison of the vertical z tracking results obtained by using the MAIIC technique with those by using DC-gain method and Single-axis IIC technique at 2 Hz (a1), 10 Hz (b1), 20 Hz (c1), and comparison of the corresponding tracking errors at 2 Hz (a2), 10 Hz (b2), 20 Hz (c2).
[1] J. Ghosh and B. Paden, “Iterative learning control for nonlinear nonminimum phase plants,” ASME Journal of Dynamic Systems, Measurement and Control, vol. 123, pp. 21–30, 2001. [2] M. Verwoerd, G. Meinsma, and T. de Vries, “On admissible pairs and equivalent feedback – Youla parameterization in iterative learning control,” Automatica, vol. 42, pp. 2079–2089, 2006. [3] S. Tien, Q. Zou, and S. Devasia, “Control of dynamics-coupling effects in piezo-actuator for high-speed AFM operation,” IEEE Trans. on Control Systems Technology, vol. 13, no. 6, pp. 921–931, 2005. [4] Y. Wu and Q. Zou, “Iterative control approach to compensate for both the hysteresis and the dynamics effects of piezo actuators,” IEEE Trans. on Control Systems Technology, vol. 15, pp. 936–944, 2007. [5] Q. Z. Kyongsoo Kim and C. Su, “A new approach to scan-trajectory design and track: AFM force measurement example,” ASME Journal of Dynamic Systems, Measurement and Control, vol. 130, pp. 051005–1 to 051005–10, 2008. [6] K.-S. Kim and Q. Zou, “A model-less inversion-based iterative control technique for output tracking: Piezo actuator example,” in acc, (Seattle, WA), pp. 2710–2715, June 2008. [7] Z. Xu, K. Kim, Q. Zou, and P. Shrotriya, “Broadband measurement of rate-dependent viscoelasticity at nanoscale using scanning probe microscope: Poly(dimethylsiloxane) example,” Applied Physics Letters, vol. 93, no. 13, p. 133103, 2008. [8] Y. Wu and Q. Zou, “An iterative based feedforward-feedback control approach to high-speed atomic force microscope imaging,” ASME Journal of Dynamic Systems, Measurement and Control, 2009. in press. [9] P. B. Goldsmith, “On the equivalence of causal lti iterative learning control and feedback control,” Automatica, vol. 38, no. 4, pp. 703–708, 2002. [10] Y. Ye and D. Wang, “Clean system inversion learning control law,” Automatica, vol. 41, pp. 1549–1556, 2005. [11] W. Rudin, Real and Complex Analysis. New York: McGraw-Hill, third ed., 1966. [12] W. Ying and Q. Zou, “Robust inversion-based 2-DOF control design for output tracking: Piezoelectric-actuator example,” IEEE Trans. on Control Systems Technology, 2009. in print. [13] J. F. Kaiser and R. W. Schafer, “On the use of the i0 -shih window for spectrum analysis,” IEEE Trans. on Accoustics, Speech, and Signal Processing, vol. ASSP-28, no. 1, pp. 105–107, 1980.
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IV. C ONCLUSION In this article, the multi-axis inversion-based iterative control (MAIIC) technique was proposed by extending the IIC technique from SISO systems to MIMO systems. It was shown that by only using the dominant dynamics for each output in the iterative control algorithm, the control of each output channel was decoupled in the MAIIC. The convergence condition of the MAIIC was characterized in frequency domain, and the set of tractable frequencies was quantified by the bounds of system uncertainties. Moreover, the optimal iteration coefficient in the MAIIC law was obtained, and the analysis of the time domain properties of the MAIIC law showed that the MAIIC law with timedomain truncation convergences to the truncated desired
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than three iterations, the RMS-tracking error was reduced to ∼1% in both lateral x − y axes tracking and vertical z-axes tracking Therefore, the combined tracking precision of the 3-D pyramid pattern illustrated the advantages of the MAIIC technique in 3-D nanopositioning control (see Fig. 7))
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Fig. 7. Comparison of the vertical z tracking results obtained by using the MAIIC technique with those by using DC-gain method and Single-axis IIC technique at 2 Hz (a1), 10 Hz (b1), 20 Hz (c1), and comparison of the corresponding tracking errors at 2 Hz (a2), 10 Hz (b2), 20 Hz (c2).
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A Decoupled Inversion-based Iterative Control Approach to Multi-axis Precision Positioning: 3-D Nanopositioning Example♯ Yan Yan, Haiming Wang, and Qingze Zou † Abstract— This article proposes the multi-axis inversionbased (MAIIC) approach. System inverse provides a nature avenue to utilize the priori knowledge of system dynamics in iterative learning control, resulting in rapid convergence as well as exact tracking (for nonminimum-phase systems). The benefits of system inverse, however, are not fully exploited in time-domain ILCs due to the lack of uncertainty quantification. This critical limit was removed in the inversionbased iterative control (IIC) techniques through a frequencydomain formulation. The existing IIC techniques, however, is limited to single-input-single-output (SISO) systems, and the time-domain properties of the IIC techniques are not clear. The contribution of this article is: First, the IIC technique is extended from SISO systems to multi-input-multi-output systems, and the convergence condition is analyzed. Secondly, the time-domain properties of the MAIIC law are discussed. It is shown that the set of tractable frequencies is characterized by the bounds of system uncertainty that are quantifiable in practices, and the truncated MAIIC input-output convergences to the neighborhood of the desired input-output truncated in time, where arbitrarily small tracking error can be obtained by having a large enough truncation time. The proposed MAIIC technique is illustrated in a 3-D nanopositioning experiment using piezoelectric actuators.
I. I NTRODUCTION This paper proposes an inversion-based iterative control (IIC) approach for multi-axis precision positioning. In the last decade, the role of system inverse in iterative learning control (ILC) has been unveiled and recognized [1], [2]. Particularly, the system inverse is utilized in in a frequencydomain framework in the recently developed IIC techniques [3]-[6], which provides a straightforward and systematic avenue to exploit the a priori knowledge of system dynamics on one hand, and account for the dynamics uncertainty on the other. The efficacy of the IIC techniques has been demonstrated in various high-speed precision positioning applications [3], [7]. However, currently the IIC techniques are limited to single-input-single-output (SISO) systems. Moreover, the time-domain properties of the IIC techniques— with their frequency-domain representation—have not been clearly understood. Thus, this paper aims to (1) extend the IIC technique proposed in [3] to multi-input-multi-output (MIMO) systems that have a dominant input to each output channel (i.e., the input that is more influential than the ♯:The work was supported by NSF grants CMMI-0626417 and CAREER award CMMI-0846350. The authors are with the Department of Mechanical Engineering, Iowa state University, Ames, IA 50011 (Yan Yan is currently with the Department of Mechanical Engineering, the Johns Hopkins University, Baltimore, MD 21218) USA. † Corresponding author,
E-mail: [email protected].
978-1-4244-7425-7/10/$26.00 ©2010 AACC
combined cross-coupling inputs), and (2) characterize the time-domain properties of the IIC algorithms. It is advantageous to incorporate system inverse in the ILC framework. We note that in many ILC applications, the a priori knowledge of the system dynamics is available, thereby can be utilized for iterative control. System inversion is a nature choice because the use of system inverse in ILC law results in rapid convergence [1], [6], very much desirable in practical implementations [8]. Moreover, system inverse becomes particularly critical to output tracking of nonminimum-phase systems, for which noncausal ILC law is necessary to achieve precision tracking [2], [1], [9]. The advantages of system inversion, however, are not fully exploited in time-domain based ILC approaches [1], [10], where the system dynamics uncertainty is not quantified. Hence, the iteration coefficient has to be chosen small enough, and thereby, rather conservative, resulting in slow convergence. These system uncertainty related constraints are removed through the development of the IIC techniques [3][6]. By formulating the IIC law in frequency-domain, the bound of system dynamics uncertainty can be quantified in a systematic and straightforward manner [3]-[5] resulting in a rapid convergence. Thus, the utilization of system inverse in the frequency-domain IIC techniques provides features desirable in practical implementations. Limits, however, exist in the existing IIC techniques. The IIC techniques [3]-[6], as presented succinctly in frequency domain, are developed for SISO systems only. The direct extension of the IIC techniques to MIMO systems, however, can results in loss of performance, because of the quantification of the dynamics uncertainty of MIMO systems, however, tends to be over conservative, resulting in not only slow convergence, but also a small set of tractable frequencies. Moreover, although the development of the IIC techniques provides unique insights into the ILC framework by determining the tractable frequencies in terms of system dynamics properties, the set of tractable frequencies as well as the timedomain properties of the IIC techniques are not characterized yet. The effect of such frequency-time truncation on the IIC input as well as the tracking performance is yet to be clarified. Therefore, there exists a need to further extend the IIC techniques. The main contribution of this paper is to extend the IIC technique [3] to square MIMO systems in a straightforward and simple manner. Rather than inverting the transfer matrix of the entire MIMO system, a diagonal matrix consisting of the inverse of the dominant dynamics for each output is used in the proposed multi-axis IIC (MAIIC) law. It is
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shown that the proposed MAIIC converges at frequencies where the diagonal subdynamics has (1) a gain larger than the combined gain of other cross-coupling inputs, and (2) a phase uncertainty smaller than a bound given by the combined cross-coupling dynamics effect. The allowable cross-coupling effect and the set of tractable frequencies are characterized and quantified, and the optimal iteration coefficient is obtained that minimizes the number of iterations for given bound of system uncertainties. Secondly, the time-domain properties of the MAIIC law are addressed for practical implementations. We show that the set of tractable frequencies is compact, thereby time-domain truncation is needed in practical implementation of the MAIIC law. The difference between the MAIIC input-output truncated in time and the desired input-output is bounded by the length of the truncation time window, hence, can be rendered arbitrarily small by having a large enough truncation window. Finally, the proposed MAIIC technique is illustrated in experiments by a 3-D nanopositioning application using piezoelectric actuators. The experimental results obtained demonstrate the efficacy of the proposed approach. II. M ULTI - AXIS I NVERSION - BASED I TERATIVE C ONTROL In this section, we extend the IIC technique originally developed for SISO systems in Ref. [3] to MIMO systems. A. Multi-axis Inversion-based Iterative Control Considering a square MIMO system G( jω ) : C → C n×n , ˆ jω ), (1) Yˆ ( jω ) = G( jω )U( where G11 ( jω ) G12 ( jω ) · · · G1n ( jω ) .. G21 ( jω ) G22 ( jω ) · · · . G( jω ) = (2) .. .. ··· . ··· . ··· · · · Gnn ( jω ) Gn1 ( jω ) represents the actual linear dynamics of the entire MIMO system, and Yˆ ( jω ) = [ yˆ1 ( jω ), yˆ2 ( jω ), · · · , yˆn ( jω ) ]T , (3) ˆ jω ) = [ uˆ1 ( jω ), uˆ2 ( jω ), · · · , uˆn ( jω ) ]T , U( The proposed MAIIC law is described in frequency domain as follows, (4) Uˆ 0 ( jω ) = 0 −1 Uˆ k ( jω ) = Uˆ k−1 ( jω ) + ρ (ω )GI,m ( jω )(Yˆd ( jω ) − Yˆk−1 ( jω )) where GI,m is a diagonal matrix with the diagonal elements being the model of the diagonal subsystems of system G( jω ), obtained in the mth modeling process, i.e., for m ∈ N (N = {1, 2, · · · , n}), GI,m ( jω ) = diag [ G11,m ( jω ), · · · , Gnn,m ( jω ) ] , (5)
ρ ( jω ) = diag[ ρ1 ( jω ), ρ2 ( jω ), · · · , ρn ( jω ) ] (6) is the frequency-dependent iteration coefficient matrix where the diagonal element ρ p ( jω ) ∈ R+ for each p ∈ N. In (4), Uˆ k ( jω ), Yˆk ( jω ) denote the input and the output of the system in the kth iteration, respectively, Uˆ k ( jω ) = [uˆ1,k ( jω ), uˆ2,k ( jω ), · · · , uˆn,k ( jω )]T , (7) Yˆk ( jω ) = [yˆ1,k ( jω ), yˆ2,k ( jω ), · · · , yˆn,k ( jω )]T .
Assumption 1: The desired trajectory Yd (·) is in L2 space, i.e., Yd (·) ∈ L2 , and for system (1), exact output tracking of the desired trajectory Yˆd ( jω ) exists at any given frequency ω , i.e., there exists a desired input U d (·) such that (8) Yˆd ( jω ) = G( jω )Uˆ d ( jω ). Assumption 2: System (1) and its model are both proper, stable, and hyperbolic, i.e., for all p, q ∈ N and m ∈ N, both G pq ( jω ) and G pq,m ( jω ) are proper, stable, and hyperbolic (i.e., have no pure imaginary zeros). Assumption 3: During each iteration, the iterative control input Uk (·) is applied to system (1) under the same initial condition as that for the desired input U d (·). B. Convergence Analysis of the MAIIC Law We shall present the convergence of the proposed MAIIC law in steps. We start with clarifying the convergence condition of the MAIIC law at one given frequency ω , and with no truncation of both the iterative input and the output to finite time window (The truncation constraint will be relaxed later). Lemma 1: Let Assumptions 1 to 3 be satisfied. Moreover, assume that no truncation is applied to either the iterative control input nor the corresponding output. Then, at any given frequency ω , the system output Y k ( jω ) converges to the desired output Yd ( jω ), i.e. limk→∞ Yk ( jω ) = Yd ( jω ), if and only if Λ( jω )∞ < 1,
(9)
E − ρ ( jω )G−1 I,m ( j ω )G( j ω ).
where Λ( jω ) The MAIIC law (4) is decoupled, which substantially simplifies the implementation of the MAIIC law. Next, we define the “tractable” set EG as below. Definition 1: The “tractable” set E G is defined as EG
n
(10)
Ep
p=1
where E p denotes the set of frequencies given by ω ∈ R1 |C p ( jω ) < 1, Ep =
∠∆G pp ( jω ) < arccos (C p ( jω )) ,
(11)
with C p ( jω ) the relative combined cross-axis coupling effect with respect to the gain of the diagonal subsystem in the p th axis, G pp ( jω ) , and ∠∆G pq ( jω ) (for p = q) the phase part of the system uncertainty for subsystem G pq ( jω ), n 1 ∑ G pq ( jω ) , C p ( jω ) G pp ( jω ) q=1 q= p (12) ∠∆G ( jω ) j G pq ( jω ) pq ∆G pq ( jω ) = ∆G pq ( jω ) e . G pq,m ( jω ) Furthermore, we assume that Assumption 4: For the square system G( jω ), the measure of the “tractable” set E G is not zero, i.e.,
µ (EG ) = 0,
(13) R1
where µ (E) is the Lebesgue measure of set E in [11]. The above definition and assumption allow us to quantify the iteration coefficient ρ ( jω ) by the system dynamics uncertainties. We quantify, next, the iteration coefficientfor each individual p-axis.
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Lemma 2: Let Assumption 4 and conditions in Lemma 1 be satisfied, then for any frequency in the “tractable” set, ω ∈ EG , the MAIIC law (4) converges if and only if for any given p ∈ N, the iteration coefficient ρ p ( jω ) is chosen as
ρ p ( jω ) < ρ p,ub ( jω ), where (14) cos ∠∆G pp ( jω ) − C p ( jω ) 2 ρ p,ub ( jω ) . 1 − C p( jω )2 ∆G pp ( jω ) C. Characterization of the “Tractable” Set E G First, we shall define the set of admissible models for any given subsystem dynamics G pq ( jω )
(15) S pq G pq,k ( jω ) ; 0
0, there exists a finite truncation time window, T ∗ ∈ (0, ∞), such that when the truncation window is larger than T ∗ , ∗ EU,d (·)2 , EU,t (·)2 , EU,d (·)2 ≤ ε ,
(37)
where EU,d (t) Udd (t) − Utd (t), EU,t (t) Urd (t) − Utd (t), (38) ∗ EU,d (t) Urd (t) − Udd (t) With the above Lemma 8, the convergence of the truncated MAIIC law is addressed in the following Theorem. Theorem 1: Let Assumptions 1 to 4 be satisfied, and the iteration coefficient ρ p ( jω ) be chosen as in Lemma 5. Then for any given positive constant ε > 0, there exists a finite truncation time T , such that the iterative input-output obtained from the truncated MAIIC law (30) converges to the neighborhood of ε of the directly-truncated desired input Udd (·) and that of the directly-truncated desired output U dd (·) in L2 -space, i.e., d lim EU,k (·)2 lim Ukd (·) − Udd (·) < ε , (39) k→∞ k→∞ 2 d d d lim EY,k (·)2 lim Yk (·) − Yd (·) < ε . (40) k→∞
k→∞
2
Remark 3: Design of the Window Function WT (·) The above time-domain analysis of the truncated MAIIC law implies that the design of the truncation window function WT (·) can be important for applications where the implementation time is stringent. For example, the Kaiser window filter technique [13] that trades side-lobe amplitude for mainlobe width can be utilized. III. E XAMPLE : NANOPOSITIONING IN 3-D In this section, we illustrate the proposed MAIIC technique by 3-D nanopositioning using piezo actuators. A. Experiment Scheme The two piezotube actuators for positioning in 3-D (one for the lateral x-y axes, and the other for the vertical z-axis direction) on a scanning probe microscope (SPM, Dimension 3100, Veeco Inc.) were used in the experiments. A pyramid pattern was chosen as the desired trajectory. Tracking of such a pattern required equally-well positioning precision in all xy-z axes. Three different pattern-tracking rates (2 Hz, 10 Hz and 20 Hz) were tested in the experiment, where the patterntracking rate was defined as the rate to traverse the four side triangles once by the order denoted in Fig. 1. With the xy- axes displacement range both at 20 µ m and the vertical one at 2 µ m, the average speeds for these three rates were at 0.39 mm/sec, 1.94 mm/sec and 3.88 mm/sec, respectively. B. Dynamics of the Piezo Actuators for 3-D Positioning To implement the proposed MAIIC technique, the diagonal subdynamics of the x- and z-axes piezo actuators (i.e., the frequency response) in 3-D (e.g., x-to-x) were experimentally measured along with the cross-axis coupling dynamics, as shown in Figs. 2 and 3 To quantify the dynamics uncertainty, frequency responses under different input conditions were acquired for, and the supremum and the infimum of the magnitude uncertainty and the phase
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Fig. 4. The upper bounds of the iteration coefficients κp (ω ) and the iteration coefficients ρselected ( j ω ) used in the experiments for (a) x- and (b) z- axes, respectively, where the vertical green-dashed line at frequency 655 Hz marked the upper bound of the practically tractable set ΩG chosen in the experiment.
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Fig. 3. Comparison of the frequency responses of the z-axis piezo actuator dynamics and the coupling dynamics from the lateral x- and y- axes piezo actuators to the z-axis displacement.
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uncertainty were estimated according to (16), then used to quantify the upper bound of the iteration coefficient for each axis (κx ( jω ), κy ( jω ), κz ( jω ))(see (23)). The obtained upper bounds of the iteration coefficients for all the three axes and the iteration coefficients are shown in Fig. 4).
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Fig. 1. The desired trajectories of the pyramid pattern: (a) the 3-D pyramid pattern, and the corresponding desired trajectories in (b) the x-axis, (c) the y-axis, and (d) the z-axis, respectively. The numbers in (a) denotes the order followed in the experiments when traversing the side triangles.
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Experimental results show that significant cross-axis coupling effect existed in the piezo actuators used in this experiment. The lateral to vertical x-y-to-z coupling effects were pronounced (see Figs. 2 and 3). Such a relatively large coupling effect might be caused by the slightly misalignment of the z-axis displacement sensor. In this experiment, the lateral trajectories for the desired pyramid pattern were relatively slow compared to the coupling dynamics between xand y- axis. Thereby, the lateral coupling caused disturbances between x- and y- axes were expected to be small and negligible in the experiments. Hence, the experiment results presented next are focused on the compensation for the lateral to vertical coupling effect. C. Tracking results and discussion The MAIIC technique was applied to track the pyramid pattern by implementing the MAIIC law (4) to all three axes simultaneously. For comparison, the desired pyramid pattern was also tracked by using the DC-Gain method and the IIC technique [3]. In the IIC technique, the IIC law was applied to each axis separately (not simultaneously!) to obtain the control input for each axis individually, and then the control
inputs for all three axes were applied. The tracking results obtained by using the MAIIC technique and the DC-Gain method are compared in Fig. 5 for the x-axis (y-axis tracking results are omitted), and the vertical z-axis tracking results are compared in Fig. 6 for the MAIIC, the IIC, and the DC-gain methods. Finally, the 3-D pyramid pattern tracking results are also compared for the above three methods with respect to the desired pyramid pattern in Fig. 7. The experiment results demonstrate that multi-axis (3-D) precision tracking can be achieved by using the proposed method. In the lateral x-y axes tracking at low speed, the dynamics effect was relatively small (as the lateral speed was relatively small), so was the cross-axis coupling effect between x-y axes (as explained above). Hence, the tracking error was mainly caused by the hysteresis effect of piezo actuators (see Fig. 5). We note, however, as the patterntracking rate increased to 20 Hz, the dynamics effect became pronounced and augmented to the hysteresis effect, resulting in larger tracking error at ∼20%, as shown in Fig. 5 (c).We note that although the vibrational dynamics effect of the z-axis dynamics itself was significantly removed by using the IIC technique (see Fig. 6 (a2) to (c2)), the cross-axis coupling effect still existed and became oscillatory as the pattern tracking increased to 20 Hz. On the contrary, by using the proposed MAIIC technique, such large x-y-to-z coupling effect was successfully removed. We also note that rapid convergence was achieved in the experiments: With no more
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input-output. The proposed MAIIC technique was illustrated through 3-D nanopositioning using two piezo actuators in experiments. R EFERENCES
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Fig. 5. Comparison of the vertical x tracking results obtained by using the MAIIC technique with those by using DC-gain method and Single-axis IIC technique at 2 Hz (a1), 10 Hz (b1), 20 Hz (c1), and comparison of the corresponding tracking errors at 2 Hz (a2), 10 Hz (b2), 20 Hz (c2). 2 Hz
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Fig. 6. Comparison of the vertical z tracking results obtained by using the MAIIC technique with those by using DC-gain method and Single-axis IIC technique at 2 Hz (a1), 10 Hz (b1), 20 Hz (c1), and comparison of the corresponding tracking errors at 2 Hz (a2), 10 Hz (b2), 20 Hz (c2).
[1] J. Ghosh and B. Paden, “Iterative learning control for nonlinear nonminimum phase plants,” ASME Journal of Dynamic Systems, Measurement and Control, vol. 123, pp. 21–30, 2001. [2] M. Verwoerd, G. Meinsma, and T. de Vries, “On admissible pairs and equivalent feedback – Youla parameterization in iterative learning control,” Automatica, vol. 42, pp. 2079–2089, 2006. [3] S. Tien, Q. Zou, and S. Devasia, “Control of dynamics-coupling effects in piezo-actuator for high-speed AFM operation,” IEEE Trans. on Control Systems Technology, vol. 13, no. 6, pp. 921–931, 2005. [4] Y. Wu and Q. Zou, “Iterative control approach to compensate for both the hysteresis and the dynamics effects of piezo actuators,” IEEE Trans. on Control Systems Technology, vol. 15, pp. 936–944, 2007. [5] Q. Z. Kyongsoo Kim and C. Su, “A new approach to scan-trajectory design and track: AFM force measurement example,” ASME Journal of Dynamic Systems, Measurement and Control, vol. 130, pp. 051005–1 to 051005–10, 2008. [6] K.-S. Kim and Q. Zou, “A model-less inversion-based iterative control technique for output tracking: Piezo actuator example,” in acc, (Seattle, WA), pp. 2710–2715, June 2008. [7] Z. Xu, K. Kim, Q. Zou, and P. Shrotriya, “Broadband measurement of rate-dependent viscoelasticity at nanoscale using scanning probe microscope: Poly(dimethylsiloxane) example,” Applied Physics Letters, vol. 93, no. 13, p. 133103, 2008. [8] Y. Wu and Q. Zou, “An iterative based feedforward-feedback control approach to high-speed atomic force microscope imaging,” ASME Journal of Dynamic Systems, Measurement and Control, 2009. in press. [9] P. B. Goldsmith, “On the equivalence of causal lti iterative learning control and feedback control,” Automatica, vol. 38, no. 4, pp. 703–708, 2002. [10] Y. Ye and D. Wang, “Clean system inversion learning control law,” Automatica, vol. 41, pp. 1549–1556, 2005. [11] W. Rudin, Real and Complex Analysis. New York: McGraw-Hill, third ed., 1966. [12] W. Ying and Q. Zou, “Robust inversion-based 2-DOF control design for output tracking: Piezoelectric-actuator example,” IEEE Trans. on Control Systems Technology, 2009. in print. [13] J. F. Kaiser and R. W. Schafer, “On the use of the i0 -shih window for spectrum analysis,” IEEE Trans. on Accoustics, Speech, and Signal Processing, vol. ASSP-28, no. 1, pp. 105–107, 1980.
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IV. C ONCLUSION In this article, the multi-axis inversion-based iterative control (MAIIC) technique was proposed by extending the IIC technique from SISO systems to MIMO systems. It was shown that by only using the dominant dynamics for each output in the iterative control algorithm, the control of each output channel was decoupled in the MAIIC. The convergence condition of the MAIIC was characterized in frequency domain, and the set of tractable frequencies was quantified by the bounds of system uncertainties. Moreover, the optimal iteration coefficient in the MAIIC law was obtained, and the analysis of the time domain properties of the MAIIC law showed that the MAIIC law with timedomain truncation convergences to the truncated desired
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than three iterations, the RMS-tracking error was reduced to ∼1% in both lateral x − y axes tracking and vertical z-axes tracking Therefore, the combined tracking precision of the 3-D pyramid pattern illustrated the advantages of the MAIIC technique in 3-D nanopositioning control (see Fig. 7))
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m)
Desired trajectory
Fig. 7. Comparison of the vertical z tracking results obtained by using the MAIIC technique with those by using DC-gain method and Single-axis IIC technique at 2 Hz (a1), 10 Hz (b1), 20 Hz (c1), and comparison of the corresponding tracking errors at 2 Hz (a2), 10 Hz (b2), 20 Hz (c2).
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