IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 6, DECEMBER 2013
1767
A Modeling-Free Inversion-Based Iterative Feedforward Control for Precision Output Tracking of Linear Time-Invariant Systems Kyong-Soo Kim and Qingze Zou
Abstract—In this paper, we propose a modeling-free inversionbased iterative feedforward control (MIIFC) approach for highspeed output tracking of single-input single-output linear timeinvariant systems. The recently developed inversion-based iterative learning control (IIC) techniques provide a straightforward manner to quantify and account for the effect of dynamics uncertainty on iterative learning control performance, thereby arriving at rapid convergence of the iterative control input. However, dynamics model and thereby the modeling process are still needed, and the model quality directly limits the performance of the IIC techniques. The main contribution of this paper is the development of the MIIFC algorithm to eliminate the dynamics modeling process, and significantly improve the tracking performance. The disturbance (measurement noise) effect on the tracking precision is addressed in the convergence analysis of the MIIFC algorithm. The allowable disturbance/noise level to guarantee the convergence is quantified in frequency domain, and the noise level can be estimated through the noise spectrum measured before the whole operation. The MIIFC technique is demonstrated by applying it to the output tracking of a piezotube scanner on an atomic force microscope. The experimental results showed that precision output tracking of a frequency-rich desired trajectory with power spectrum similar to a band-limited white noise can be achieved. Index Terms—Iterative learning control (ILC), nanopositioning control, system inversion.
I. INTRODUCTION MODELING-FREE inversion-based iterative feedforward control (MIIFC) technique for high-speed precision output tracking is proposed. It is noted that precise output tracking at high speed is needed in repetitive applications such as the nano-scale imaging/measurement using atomic force microscope (AFM) [1]–[3], the scanning mechanism in MEMSbased micromirrors [4] or semiconductor manufacturing [5], the quick-return mechanisms and cams in manufacturing [6], and the manufacturing process in rapid prototyping [7] and microforming [8]. For example, in AFM imaging, high-speed pre-
A
Manuscript received August 2, 2011; revised February 20, 2012; accepted July 25, 2012. Date of publication August 31, 2012; date of current version December 11, 2013. Recommended by Technical Editor P. X. Liu. This work was supported by the National Science Foundation under Grants CMMI-0626417 and CAREER Award CMMI-1066055. K.-S. Kim is serving in the Korean Army (e-mail:
[email protected]). Q. Zou is with the Department of Mechanical and Aerospace Engineering, Rutgers, The State University of New Jersey, Piscataway, NJ 08854 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2012.2212912
cise scanning is needed to achieve high-speed imaging, which not only improves the efficiency, but more importantly, enables interrogation of nanoscale dynamic phenomena [1], [9]. It has been demonstrated that iterative learning control (ILC) approach is efficient in repetitive output tracking (e.g., [10]–[14], and a recent review [15] and references therein). One of the main advantages of the ILC approach is the utilization of future output tracking information through noncausality [16]–[18], specially for nonminimum-phase systems [16]–[21]. By embedding the notion of noncausality into system inverse in a frequencydomain framework, the recently developed inversion-based iterative learning control (IIC) techniques [22]–[24] provide a straightforward manner to quantify and account for the effect of dynamics uncertainty on ILC performance, thereby arriving at rapid convergence of the IIC input [25]. However, dynamics model and thereby the related modeling process are still needed in existing IIC techniques, and the model quality directly limits the performance of the IIC techniques. In this paper, such modeling-related constraints are removed through the development of the MIIFC approach, which not only substantially facilitates implementations in practical applications, but also further enhances the robustness and tracking performance. Noncausal ILC is particularly effective in repetitive trajectory tracking and/or disturbance rejection for nonminimumphase systems. It has been recognized that noncausality is essential [17], [18] in ILCs for nonminimum-phase systems. A noncausal ILC law can be obtained, for example, by using advanced output tracking error with a constant iterative control gain [16]. Although the input–output mapping is used for the convergence condition, the system dynamics knowledge is not utilized in the control law. The system dynamics knowledge is utilized in the inversion-based [19], [26], [27] or the similar adjoint-system-based approaches [28]. The key is to realize a stable yet noncausal iterative controller based on the inverse of dynamics model. However, in these time-domain inversionbased ILC methods [19], [28], the model uncertainty has not been quantified, and the iteration coefficient has to be small enough, resulting in slow convergence in practices. Rapid convergence is desirable in practical applications such as nanomechanical property measurement during polymer crystallization using AFM [9]. Thus, there exists a need to further improve the inversion-based noncausal ILC approach. These model-uncertainty-related constraints in time-domain inversion-based ILC methods have been removed in the inversion-based iterative control (IIC) techniques [2], [22], [24], [29]. The development of the IIC techniques, instead of aiming
1083-4435 © 2012 IEEE
1768
IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 18, NO. 6, DECEMBER 2013
for general nonlinear systems [19], [20], explicitly considered the model uncertainty in the iterative controller design for linear systems. Particularly, the amplitude and phase uncertainty of the frequency response are quantified through experiments [2], [22], [24] and utilized in the IIC law through a frequency-domain formulation and implementation scheme. Alternatively, recently the frequency-domain model error has also been converted into time-doman and utilized to improve the robustness of the ILC algorithm through the Q-filter design [30]. The quantification of the model error in the frequency domain not only provides insights into the tractability of the system [22], [29], but also allows the iterative coefficient to be tailored to ensure the convergence and arrive at rapid convergence in practical applications such as high-speed nanopositioning [23], [24] and microforming [8]. The IIC technique has also been integrated with feedback for tracking desired trajectory that varies between iterations [2]. The efficacy of the IIC techniques, however, is directly related to the quality of the dynamics model—large model error results in slow convergence rate as well as smaller tractable frequency range. Even though the model error might be reduced via remodeling, the modeling process itself can be time consuming and prone to errors. Moreover, such an improvementvia-remodeling approach is not preferred and even infeasible in practice, particularly in industrial implementations. Therefore, these modeling-related issues—important for practical industrial implementations—need to be addressed. The main contribution of this paper is the development of the MIIFC technique. First, the inverse model is updated in each iteration by using the measured input–output data. Thus, not only the modeling process and its related issues are removed, but also the quality of the inverse model is improved along the iterations. Second, the effects of disturbance/noise are explicitly addressed in the convergence analysis of the proposed MIIFC algorithm. We show that the error in both the iterative input and the corresponding output tracking can be quantified by the disturbance/noise signal ratio (NSR) relative to the desired trajectory at each frequency. Moreover, the allowable size of the NSR, i.e., the size of the NSR for the MIIFC algorithm to be effective, is quantified. The proposed MIIFC technique is illustrated through experiment implementation to the output tracking of a piezotube actuator on an AFM system. Particularly, a frequency-rich desired trajectory with spectrum similar to band-limited white noise was tracked. The experimental results demonstrate that rapid convergence was achieved, and precision output tracking can be achieved even when the spectrum of the desired trajectory spanned much larger than the bandwidth and the dominant resonant peaks of the system. Therefore, the proposed MIIFC technique extends the IIC approach for improved efficiency and tracking performance, and ease of use, all needed in practice.
uk
yk
System
offline MIIC / IIC
uk+1
We start with briefly reviewing the IIC technique [22], which forms the base for the proposed MIIFC technique.
yk,n yd
Fig. 1. MIIFC and IIC scheme, where y k , n denotes the noise/disturbance augmented to the linear output y k in the kth iteration.
A. Inversion-Based Iterative Control IIC Technique [22]: Recently, an IIC technique [22], [23] was developed to achieve high-speed output tracking in repetitive operations. For a stable, single-input single-output (SISO) linear time-invariant (LTI) system, the IIC law can be described in the frequency domain as (see Fig. 1) u0 (jω) = Gm (jω)−1 yd (jω),
k=0
(1)
uk (jω) = uk −1 (jω) + ρ(ω)Gm (jω)−1 [yd (jω) − yk −1 (jω)] ,
k≥1
(2)
where f (jω) denotes the Fourier transform of the signal f (t), yd (·) denotes the desired output trajectory, yk (·) denotes the output obtained by applying the input uk (·) to the system during the kth iteration, ρ(ω) > 0 is the iterative coefficient, and Gm (jω) denotes the frequency response model of the system. It has been shown that the aforementioned IIC algorithm can lead to exact tracking of the desired trajectory at frequency ω, provided that the modeling error is not too large and the iterative coefficient is appropriately chosen [22]. Lemma 1 [22]: At any given frequency ω, let both the actual dynamics of an SISO LTI system G(jω) and its model Gm (jω) be stable and hyperbolic (i.e., both have no zeros on the jω axis), and the dynamics uncertainty ΔG(jω) be described as G (j ω )
ΔG(jω) =
|G(jω)|ej G(jω) = Gm (jω) |Gm (jω)|ej
= |ΔG(ω)|ej Δ
G (j ω )
G m (j ω )
.
(3)
Then the IIC control law converges at frequency ω to the de
sired input ud (jω) = G(jω)−1 yd (jω), i.e., limk →∞ uk (jω) = ud (jω), if and only if, 1) the iterative coefficient ρ(ω) ∈ R is chosen as
0 < ρ(ω) < ρsu p (ω) =
2 cos ( ΔG(jω)) |ΔG(jω)|
(4)
2) the magnitude of the phase variation is less than π/2, i.e., π . (5) 2 Remark 1: Note that exact tracking stated in Lemma 1 is achieved in the absence of nonperiodic adverse effects such as noise, i.e., the linear mapping yk (jω) = G(jω)uk (jω) is assumed. The noise-like disturbance effect on ILC has been recognized [14], [31]. Due to the random nature of the effects, | ΔG(jω)|
