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Learning Feedforward Control Using a Dilated B-Spline Network: Frequency Domain Analysis and Design YangQuan Chen, Senior Member, IEEE, Kevin L. Moore, Senior Member, IEEE, and Vikas Bahl
Abstract—This paper presents a frequency-domain analysis and design approach for a learning feedforward controller (LFFC) using a dilated B-spline network. The LFFC acts as an add-on element to the existing feedback controller (FBC). The LFFC signal is updated iteratively based on the FBC signal of the previous iteration as the task repeats. Similar to proportional–integral– derivative controller tuning, there are only two parameters to adjust: The B-spline support width and the learning gain. The effect of dilation in the B-spline network is discussed. Detailed design formulae are given based on a stability analysis. As an illustration, simulation results on the path tracking control of a wheeled mobile robot are presented. Index Terms—B-splines network (BSN), dilated splines, iterative learning control (ILC), learning controller design, learning feedforward control (LFFC), stability analysis.
I. INTRODUCTION
L
EARNING feedforward control (LFFC) [1]–[5] is a variant of iterative learning control (ILC) [6], [7]. The basic ideas are the same in the sense that LFFC and ILC are both considered to be a value-added block for enhancing the feedback control performance of some classes of systems by capitalizing on the repetitiveness of these systems’ operation. The general LFFC scheme is illustrated in Fig. 1, where FBC is the given desired stands for “feedback controller” and output trajectory to be tracked. The scheme works as follows. th repetitive operation, the feedforward control After the and the feedback control signal are to be signal stored in the memory bank for constructing the feedforward . The stored control signal at the next repetition, i.e., feedback control signal is filtered through and multiplied by a learning gain . Note that the general filter is the key LFFC element. The correct selection of this filter and its parameters is the key to a successful LFFC scheme. LFFC shown in Fig. 1, is effectively an iterative learning controller. Detailed literature reviews on ILC research can be found in [7] and [8]. Most of the existing work has focused on the analysis issue of ILC schemes. However, the convergence conditions found in the literature are typically not sufficient for acManuscript received February 4, 2002; revised June 10, 2002. This work was supported in part by the U.S. Army Automotive and Armaments Command (TACOM) Intelligent Mobility Program under Agreement DAAE07-95-3-0023. The authors are with the Center for Self-Organizing and Intelligent Systems (CSOIS), Department of Electrical and Computer Engineering, Utah State University, Logan, UT 84322-4160 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TNN.2004.824268
Fig. 1. Block diagram of LFFC using a general filter H (z; z
).
tual ILC applications. Therefore, in recent years increasing efforts have been made on the design issue of ILC. These can be observed from the latest books [9], [8] and the dedicated ILC web server [10]. A recent survey on ILC design issue [11] has documented various practically tested design schemes, though mainly for robotic manipulators. However, to draw attention from industry, the existing design techniques are still not sufficiently attractive as compared to the successful use of proportional–integral–derivative (PID) controllers in industries. To be popularly accepted, ILC design should be attacked in a similar way as that of PI/PID design, where there are only two or three parameters to tune. This is one of the primary goals of this paper—to develop a form of LFFC, and thus, effectively an ILC scheme, that admits tuning rules using only two parameters. The approach we take in the paper to developing a two-parameter tunable LFFC is to consider a particular form of the shown in Fig. 1. Specifically, we congeneral filter sider using a B-splines network (BSN). In general, a BSN is a function
where is the BSN output, is the input, is a basis function (also called a membership function or function evaluation), are B-spline weights. The region of the input space on and which the basis functions are nonzero is called the support of
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Fig. 2. Second-order B-splines and the filtering process.
the basis. Generally, the support of a membership function is not equal to the whole input space and often the supports the basis functions do not overlap more than 50% each other. When the supports of the basis functions do overlap more than 50% each other, we say that the BSN is dilated. In this paper, following the lead from a series of works in the literature, we consider th-order BSNs whose basis functions are piecewise polyno. The BSN we use, a second-order mial functions of order dilated BSN, is illustrated in Fig. 2, which shows how it can be regarded as a filter. For our purposes, the input space is simply the time . The output of the BSN, which we will later use as , is a weighted sum of all our feedforward control signal B-spline “evaluations” (1) where is the sampling period, is the time index, denotes the th repetitive operation as in (2), is the weight for B-spline in iteration , and is the function evaluation, or membership, of the th B-spline. The is not equal to zero, the part of the input space for which support of our B-Spline, is . For our LFFC or ILC problem, with a time interval of interest , suppose there are (for in Fig. 2) equally-spaced B-splines. Then, one example, . gets the B-spline support (or width) to be Moreover, with the sampling period , we know that within one B-spline, there are samples, where . Finally, the B-splines are placed on the domain of the input of the BSN in such a way that at each input value the sum of the memberships of all basis functions equals unity. It has been shown in the literature that a LFFC scheme as shown in Fig. 1, using a B-spline network for the filter, as shown in Fig. 2, with a weight update discussed in the next section, leads to a simple design procedure with only two tuning knobs: the B-splines support ( , shown in Fig. 2) and the learning gain ( , shown in Fig. 1) [1], [3]–[5]. Based on the known results reported on LFFC and the above discussions about ILC design issues, it can be argued that LFFC is a good design example for ILC. Using experimentally verified empirical design formula for and , satisfactory experimental results were obtained in [3] and [12]. Performance optimization is also considered for LFFC using the clustering and regularization techniques in [13] and
[14]. Despite these results, a number of research topics remain. For instance, state-dependent repeatable disturbances such as the cogging effect in linear motor control [3] can be naturally considered within the LFFC scheme, since in LFFC a general neural network could be used as the filter to learn the desired feedforward control signal. However, although a general neural network can be used in LFFC, so far only BSNs have been considered. Second, commonly considered in the references mentioned is the BSN driven by time instead of state(s). Third, theoretical support for the stability analysis is still missing for general settings of LFFC. The only stability analysis can be found in [4], [5] for a special LFFC scheme using a time-driven BSN and with assumptions about the plant to be controlled that: 1) the initial feedforward is with a triangular waveform; and 2) the learning updating law is in continuous time form. Finally, a systematic design procedure does not yet exist. This paper proposes to address the stability analysis and design of the LFFC scheme using a BSN, taking the perspective of ILC. A new stability analysis is performed in the frequency domain using noncausal filtering thinking with new design formulas close to [4] and [5], but under relaxed assumptions. Another major contribution of this paper is to demonstrate how to perform the stability analysis for BSN-based LFFC schemes in frequency domain. In other words, this paper presents a frequency domain analysis for a class of neural networks which has not been reported in the literature. Within the framework of this paper, the dilation effect in BSN can be discussed explicitly and analytically. This is, to our best knowledge, new in neural network literature. The ideas are also applied to a path following control problem for an omnidirectional vehicle (ODV) developed at Utah State University (USU), Logan [15]. This approach was presented in [16] using dilation 2 BSN-LFFC scheme without stability analysis and detailed derivation. The BSN LFFC scheme was also extended in [17] to a frequency domain adaptive version using time-frequency approach (TFA). In this paper, we concentrate on the detailed derivation and stability analysis. The remainder of the paper is organized as follows. Section II formulates the LFFC problem using the second-order B-splines network. In Section III, a general stability analysis is performed with LFFC as a special case. Details on the LFFC design for the case of BSN with no dilation are given in Section IV. Section V presents the LFFC design methods for the case of a BSN with dilation 2. As an illustration, a simulation example is included for the path tracking control of a wheeled mobile robot. Finally, Section VII concludes this paper. II. LFFC USING SECOND-ORDER BSN LFFC was originally proposed for motion systems that are subject to reproducible disturbances, , that may depend on the state of the process , and that experience random distur[14]. In practice, the reproducible disturbances are bances, compensated by a feed-forward controller. Normally, a feed-forward controller is designed on the basis of an accurate model of the process. To be less dependent on an accurate process model, the feed-forward controller can be implemented as a neural network trained during control operations. The random disturbances are compensated by a feedback controller. When
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random disturbances are small, the feedback controller does not actually determine the final performance of the controlled system. Rather, it is designed mainly for robustness considerations, which does not require an accurate process model. Our LFFC scheme uses a BSN neural network to generate the shown in Fig. 1. The inputs feedforward control signal to the neural network can more generally include the feedback , time or state , and reference or decontrol signal . When repetitive motions are performed, sired trajectory all state dependent disturbances can also be considered time-periodic. Therefore, it is a common practice to choose the time as the input of neural network to get the feed-forward control signal, leading to the scheme shown in Fig. 1. Thus, the overall control signal at the th repetitive operation is given by (2) where is the sampling period and . Here the system is assumed to execute the same task repeatedly. For each repetition, the duration is . The total number of sampling points . We use a second-order BSN, as illustrated in Fig. 2. The output of the BSN, which is the feedforward control signal , is a weighted sum of all B-spline evaluations, as formulated in (1) From one repetitive operation to the next one, the weights are updated by (3) where
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, and the scheme the transfer functions of the plant, , are used. This is not strange because in controller, engineering practice, to design a control system, it is very common and fundamental to have an approximate linear model for frequencies below a frequency of interest, say, . In fact, for a feedback-controlled system, it is almost certain that its frequency response can be well-approximated by a , the closed-loop transfer function. linear system, i.e., in Therefore, at this point, it should be understood that, Fig. 1 is the major linear part of the plant. The plant itself may actually be nonlinear with uncertainty. As will be clear in the following analysis, the LFFC is robust to the model mismatch in terms of the LFFC convergence condition. That is, it is easy for the LFFC scheme to converge even with inexact knowledge about the plant model. The stability analysis of the LFFC scheme is in the sense that approaches to a fixed signal as increases and meanfor all over the fixed time interval . while, This is summarized in the following theorem, where “ ” denotes the standard Fourier transform. Theorem III.1: A linear system shown in Fig. 1 is controlled by an existing feedback controller, which performs a given task repeatedly. A general scheme (6) is applied as a learning feedforward controller (LFFC). There exists a real constant and such that the learning process is convergent a filter and, furthermore (7)
is given by [1], [3], [4] (4)
Clearly, from (1) and (3), it can be seen that the overall system is in a “feedforward-feedback” configuration. Furthermore, one can write the LFFC updating law in a more familiar ILC-type of form
(5)
where vergence rate is given by
and
. The con(8)
where
is the closed-loop transfer function with . Theorem III.1 implies that the LFFC is essentially applied to invert the plant to be controlled in an iterative manner. Since a linear system is considered, in the sequel, frequency domain notion is used. The updating law (6) becomes (9)
Obviously, in the previous updating equation, the linear manipu, already stored in the memory, actually lation of the signal can be in any filter form. Therefore, in general, the LFFC updating law can be written in the following filter form:
. Now, we proceed to present a where proof of Theorem III.1. Proof: From Fig. 1, the feedback signal can be written as
(6)
(10)
is a filter, which can be noncausal since the can be arbitrarily manipulated. Note, here, is not necessarily a zero-phase filter (ZPF). In what follows, we shall present a general discussion on the stability condition for the LFFC in general form (6). Then, we derive expression for the BSN based LFFC a detailed scheme (5) and present a set of design formulae for this specific LFFC scheme. where pre-stored
III. GENERAL STABILITY ANALYSIS We now consider the stability analysis of the scheme in Fig. 1. Before we proceed, we comment that in the LFFC
The learning updating law (9) becomes (11) Iterating (11), one obtains (12) Since essentially has a low pass filter characteristics, it is clearly possible to choose a suitable and such that (8) is true. Therefore, there exists a design of and a choice of such that (8) holds and from
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Fig. 3. Bode plot for first and second-order center-symmetrical B-Spline (FIR) filters.
(12)
and moreover, for all as . Remark III.1: It is implied in (9) that the initial condition of each iteration should be the same. Remark III.2: From the above proof, it can be seen that the which means that learning convergence is independent on the initial feedforward control can be chosen arbitrarily. Howis practically set to 0 because no prior knowledge on ever, is available. It is noted that in [4], the stability determining . Our result relaxes analysis depends on an assumption on this assumption. Remark III.3: Although the stability analysis is in linear system framework, as mentioned before, the plant itself may actually be nonlinear with uncertainty. However, the model mismatch can be tolerated by the LFFC convergence condition (8). In practice, it is easy for the LFFC scheme to converge even with the inexact knowledge about the plant model. Moreover, since the LFFC scheme works in a finite time duration and the system is reinitialized at the beginning of each iteration, the can be relaxed if the minimum phase requirement of system does not have a finite escape time. IV. DESIGNING THE SECOND-ORDER BSN-BASED LFFC WITH NO DILATION Before designing the LFFC using the second-order BSN with no dilation, we need to derive the detailed form of from (5). , First, it is important to observe that at any time instant there are at most two adjacent B-splines involved or activated in
(5). Referring to Fig. 2, the two active B-splines are B-spline and . From and where number is a flooring operator. From Fig. 2, we denote (13) Clearly,
and
. Also, note that (14)
Hence, (5) can be written as
(15) It is important to observe that does not equal 0 only in and similarly for in . We can drop in (15) and write in filter form for , i.e., (16) is a where zero-phase filter for the discrete center-symmetric second-order
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B-spline. Note that the coefficients in malized to 1. The sum of the coefficients of By denoting that
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are not noris .
(17) one can observe that is a filter with a unit gain nor. Similarly, we have malized at
(18) Therefore, (15) can be simplified as Fig. 4.
Magnitude Bode plot for A(!; a
; d) (no dilation).
(19) Comparing LFFC updating law (6), we see from (19) that
(20) . It should be noted that where . phase filter only at We can get an analytical expression for by
is a zero[18] given
(21) Now, by substituting (21) into (20) one obtains the learning updating law (6) in the following form: (22)
Fig. 5. Phase Bode plot for A(!; a
; d) (no dilation).
where (23) (24) as a variable ranging from 0 to , for different , we can draw three-dimensional plots for and . Note that and share the same phase plot due to the fact that is an FIR zero phase filter (center-symmetrical second-order B-spline with support ). Fig. 3 shows the Bode plot comparison between first and second-order center-symmetrical B-Spline (FIR) filters. Clearly, second-order B-Spline exhibits better frequency response compared to the first-order one. Note that the B-Spline filters are low-pass. Figs. 4–5 are magnitude and phase Bode plots for . We can observe from Figs. 4–5 that is almost a zero-phase unit gain filter in the low frequency ). When the frequency increases, the range (e.g., phase distortion may go up to and the gain may drop to 0 as shown in Figs. 4–5. The overall frequency responses are shown in Figs. 6 and 5. Note, of learning filter is not a ZPF for all frequencies but it is close to ZPF in low frequencies. Taking
Fig. 6. Magnitude Bode plot for the overall learning filter dilation).
H (z; z
) (no
A. Design of To design a suitable , we need to study the overall . We suppose that the signal passing learning filter will be attenuated by . Numerically, through for a given starting from (23), we can solve by using a single line of MATLAB script:
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Fig. 7. Critical values for !d=2 (no dilation).
is roughly available. Let and . From the general learning stability condition (8), we can get
to assume that at least ' . We can then obtain versus for given values a plot for the critical values of as shown in Fig. 7 which can be used to design . of The concept for designing a suitable here is that the learning filter is used to attenuate the high frequency signals. is a low-pass filter as It has already been clear that shown in Fig. 6. Suppose we need to attenuate the noise above . For example, if the attenuate is at a frequency least 50% ( dB), then, referring to Fig. 7, we choose such that (25) Note that, from Fig. 7, 2.7831 is the maximal value cor. We can make more aggressive responding to choices and in turn get a conservative minimum . For example, ( dB), referring to Fig. 7, if we choose
which is very close to the formula
(27) and, further (28) (29) When the frequency range for the desired trajectory is pracand . Suppose that the feedback tically low, is small. The learning controller is well designed such that gain can then be simply designed as
(26)
(30)
given in [4].
Note that the BSN weight updating law (4) is normalized as also indicated in (17). The BSN weight updating law used in [4] is simply
B. Design of Having chosen a suitable merely based on the inforand noise attenuation requirement , mation of we now can choose the learning gain . To do so, we need
(31)
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Fig. 8. B-spline network with dilation 2.
which is not normalized compared to (4). In this case, in order to apply the design formula (29) or (30), by observing the sum of weights (14), we need to set (32)
Fig. 9.
Magnitude Bode plot for A (!; a
; d) (dilation 2).
Therefore, using the BSN weight updating law (31) and applying (29) or (30), we have (33) which is close the design formula given in [4]. Remark IV.1: As mentioned in [4], their formula is conservative. This means that a larger can actually be used. With the design formula (33) of this paper, the design of may no longer be conservative. V. DESIGNING THE SECOND-ORDER BSN-BASED LFFC WITH DILATION 2 The second-order B-splines network with dilation 2 is illustrated in Fig. 8 where the dark thick lines are the dilated B-splines with reference to Fig. 2. For example, between B-splines and the dilated B-spline is and between and is . At any time instant , there are B-splines at most four B-splines activated (e.g., in Fig. 8, the activated are , and ). Similar to the case B-splines at time but the interval of with no dilation, here we still use discussion is from point A to point B in Fig. 8. Taking B-spline as an example, should be shifted steps left or back to apply the center-symmetrical B-spline filter with its weight or membership value of . The overall in (23) is that learning filter
Fig. 10.
Phase Bode plot for A (!; a
; d) (dilation 2).
(34) where Fig. 11. Magnitude Bode plot for the overall learning filter (dilation 2).
H (z; z
)
(35) Figs. 9–10 are magnitude and phase Bode plots for . We should note that the axis- is with the same meaning as in Figs. 4–5 which corresponds to the half support . Therefore, we have two parts in Figs. 9 or 10 each width part corresponds to the quarter support width . Compared
to Figs. 4–5, we can observe from Figs. 9–10 that has a better low pass filter property in wider frequency range only. The overall and the worst phase torsion is within are shown in frequency responses of learning filter shown in Figs. 11 and 10 is Figs. 11 and 10. Note, almost a ZPF for all frequencies.
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Fig. 12.
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Critical values for !d=2 (dilation 2).
Based on the overall learning filter for the dilation 2 situation, we can follow the steps in Section IV and derive the design formulas for and . A. Design of Similar to Fig. 7, we can obtain a plot for the critical values of versus for given values of as shown in Fig. 12 which can be used to design . When the attenuate is at least 50% ( dB), referring to Fig. 12, we choose such that
(36)
Note that, from Fig. 12, 2.9481 is the maximal value corre. We can make a more aggressive sponding to choice and in turn get a conservative minimum . For example, ( dB), referring to Fig. 12 if we choose
(37)
given in which is even closer to the formula [4]. We can see here that the dilation of 2 changes little on the design of . However, since it has significant improvement in
Fig. 13.
Mechanical and vetronics layout of ODIS.
phase distortion, the dilation will positively contribute to the design of , as shown in what follows. B. Design of Consider the BSN weight updating law (31) used in [4]. Due to dilation 2, we may have maximum four B-splines activated at the same time as mentioned previous. The sum of weights in (31) at any time instant is 2 for the dilation 2 situation. Therefore, we should set (38)
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Fig. 14.
Behavior control system architecture of ODIS.
Fig. 15.
Spatial path following: desired versus actual.
and then applying (29) or (30) gives
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VI. APPLICATION TO USU ODIS PATH FOLLOWING CONTROL (39)
Note that due to dilation 2 operation, we are more confident to and . set
The USU ODIS robot is a small, man-portable mobile robotic system that can be used for autonomous or semi-autonomous inspection under vehicles in a parking area [15]. Customers for such a system include military police (MP) and other law enforcement and security entities. The robot features: 1) three
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Fig. 16.
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Spatial tracking error time history comparison.
“smart wheels” in which both the speed and direction of the wheel can be independently controlled; 2) a vehicle electronic capability that includes multiple processors; and 3) a sensor array with a laser, sonar and IR sensors, and a video camera. ODIS employs a novel parameterized command language for intelligent behavior generation. A key feature of the ODIS control system is the use of an object recognition system that fits models to sensor data. These models are then used as input parameters to the motion and behavior control commands. Fig. 13 shows the mechanical layout of the ODIS robot. The robot is 9.8-cm tall and weighs approximately 20 kgs. Key ODIS subsystems include its mechanical, vehicle electronics (vetronics) and sensor systems. For a more detailed description, see [19]. Fig. 14 shows the behavior control architecture that has been developed. Starting from the “inside out,” the control architecture contains two inner motion-control loops. The inner most loop is the wheel-level control, which acts to drive each smart wheel to its desired steering and drive speed set-points. The wheel-level controller uses simple PID control algorithms. Around the inner loop is the path-tracking controller. This loop derives the set points need by the wheel-level control in order to force the vehicle to follow a desired path in space, where a path is defined as an arc in inertial space (with a prescribed velocity along the arc) and an associated vehicle yaw motion. The path-tracking controller uses a newly-developed spatial tracking control algorithm that is described in more detail in [20].
The simulation study in this section is to demonstrate that using dilated BSN-based LFFC scheme in parallel to the existing -controller [20] improves the spatial path tracking/ following accuracy. The major objective is to minimize the spatial path tracking error (e.g., with little or no “cut corners”) while keeping the moving speed high. The simulation model implemented in MATLAB/Simulink and StateFlow has been experimentally validated and calibrated. A typical tough path (at maximum speed 0.75 m/s) is shown in Fig. 15 where the optimal PI-type -controller generates larger error than desired during the path transition. When a dilated BSN LFFC (with dilation 2, and ) is used, the tracking performance is improved significantly after ninth iterative learning as shown in Fig. 15. This can also be clearly observed from the spatial tracking error time history comparison shown in Fig. 16. Fig. 17 shows the RMS value versus learning iteration number where the monotonic convergence is obvious although we have included some random noise in tNote that, this simulation shows that, with little knowledge about the complicated nonlinear mobile robot model, the LFFC can still outplay the best spatial-type PI-controller. Moreover, when the model mismatch is not too large, by using a cautiously small learning gain , it is always possible to perform better and better from iteration to iteration at the price of slower convergence rate. In practice, the tradeoff between the size of the model mismatch and the rate of LFFC convergence can be proximately tuned with ease. Quantitatively, the optimal tradeoff is a subject under our investigation.he simulation model.
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Fig. 17.
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RMS of versus learning iteration number.
VII. CONCLUSION The major contribution of this paper is the frequency-domain analysis and design approach for LFFC using dilated BSN. The LFFC acts as an add-on element to the FBC for control performance enhancement. The LFFC signal is updated iteratively based on the FBC signal of previous iteration as the task repeats. Similar to PID controller setting, there are only two parameters to tune: The B-spline support width and the learning gain. The effect of dilation is discussed. Detailed design formulae are given based on frequency domain stability analysis. Simulation results on the path tracking control of a wheeled mobile robot are included for illustration. ACKNOWLEDGMENT The authors would like to thank the Associate Editor and the anonymous reviewers for their constructive comments, as well as M. Davidson for his long-term efforts in creating the MasterSim used in Section VI. REFERENCES [1] J. G. Starrenburg, W. T. C. van Luenen, W. Oelen, and J. van Amerongen, “Learning feedforward controller for a mobile robot vehicle,” Control Eng. Pract., vol. 14, no. 9, pp. 1221–1230, 1996. [2] T. J. A. de Vries, W. J. R. Velthuis, and J. van Amerongen, “Learning feed forward control of a flexible beam,” in Proc. IEEE Int. Symp. Intelligent Control ISIC’96, Dearborn, MI, 1996, pp. 103–108. [3] G. Otten, T. J. A. de Vries, J. van Amerongen, A. M. Rankers, and E. W. Gaal, “Linear motor motion control using a learning feedforward controller,” IEEE/ASME Trans. Mechatron., vol. 2, pp. 179–187, June 1997. [4] W. J. R. Velthuis, T. J. A. de Vries, P. Schaak, and E. W. Gaal, “Stability analysis of learning feed-forward control,” Automatica, vol. 36, no. 12, pp. 1889–1895, 2000.
[5] W. J. R. Velthuis, “Learning feed-forward control—Theory, design and applications,” Ph.D. dissertation, Univ. Twente, Enschede, The Netherlands, 2000. [6] S. Arimoto, S. Kawamura, and F. Miyazaki, “Bettering operation of robots by learning,” J. Robot. Syst., vol. 1, no. 2, pp. 123–140, 1984. [7] K. L. Moore, Iterative Learning Control for Deterministic Systems. New York: Springer-Verlag, 1993. [8] Y. Chen and C. Wen, Iterative Learning Control: Convergence, Robustness and Applications. London, U.K.: Springer-Verlag, 1999. [9] Z. Bien and J.-X. Xu, Iterative Learning Control—Analysis, Design, Integration and Applications. Norwell, MA: Kluwer, 1998. [10] Y. Chen. (1998, Oct.) Dedicated web server for iterative learning control research [Online]. Available: http://www.csois.usu.edu/ilc [11] R. W. Longman, “Designing iterative learning and repetitive controllers,” in Iterative Learning Control—Analysis, Design, Integration and Application, Z. Bien and J.-X. Xu, Eds. Norwell, MA: Kluwer, 1998, ch. 6, pp. 107–145. [12] W. J. R. Velthuis, T. J. A. de Vries, and E. Gaal, “Experimental verification of the stability analysis of learning feed-forward control,” in Proc. 37th IEEE Conf. Decision Control, Tampa, FL, 1998, pp. 1225–1229. [13] W. J. R. Velthuis, T. J. A. de Vries, and J. van Amerongen, “Performance optimization of learning feed forward control,” in Proc. IFAC Symp. AI in Real Time Control (AIRTC’97), Kuala Lumpur, Malaysia, 1997, pp. 391–396. [14] W. J. R. Velthuis, T. J. A. de Vries, and M. Haring, “Regularization in learning feed-forward control,” in Proc. 6th Int. Conf. Control, Automation, Robotics and Vision, Singapore, 2000, pp. 406.1–406.6. [15] K. L. Moore and N. S. Flann, “A six-wheeled omnidirectional autonomous mobile robot,” IEEE Control Syst., vol. 20, pp. 53–66, Dec. 2000. [16] Y. Chen, K. L. Moore, and V. Bahl, “Improved path following of USU ODIS by learning feedforward controller using dilated B-spline network,” in Proc. 2001 IEEE Int. Symp. Computational Intelligence in Robotics Automation (IEEE CIRA 2001), Banff, AB, Canada, Aug. 2001, pp. 59–64. [17] Y. Chen and K. L. Moore, “Frequency domain adaptive learning feedforward control,” in Proc. 2001 IEEE Int. Symp. Computational Intelligence in Robotics Automation (IEEE CIRA 2001), Banff, AB, Canada, Aug. 2001, pp. 396–401. [18] M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Signal Processing Mag., vol. 16, pp. 22–38, June 1999.
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[19] K. Moore, N. Flann, S. Rich, M. Frandsen, Y. Chung, J. Martin, M. Davidson, R. Maxfield, and C. Wood, “Implementation of an omni-directional robotic inspection system (ODIS),” presented at the Proc. SPIE Conf. Robotic Semi-Robotic Ground Vehicle Technology, Orlando, FL, May 2001. [20] M. Davidson and V. Bahl, “The scalar -controller: A spatial path tracking approach for ODV, Ackerman, and differentially-steered autonomous wheeled mobile robots,” in Proc. IEEE Int. Conf. Robotics Automation, Seoul, Korea, 2001, pp. 175–180.
YangQuan Chen (S’95–SM’98) received the B.S. degree in industrial automation from the University of Science and Technology of Beijing (USTB), Beijing, China, in 1985, the M.S. degree in automatic control from the Beijing Institute of Technology (BIT), Beijing, China, in 1989, and the Ph.D. degree in control and instrumentation, Nanyang Technological University (NTU), Singapore, in 1998. He is currently an Assistant Professor of Electrical and Computer Engineering at Utah State University, Logan. His current research interests include iterative learning and repetitive control, robust and optimal control, distributed control systems, fractional-order dynamic systems and control, computational intelligence, intelligent mechatronic systems, and visual servoing/tracking. He holds ten granted and four pending U.S. patents. He has authored/coauthored more than 130 papers in refereed journals and conferences and more than 50 industrial technical reports. He is the coauthor of the research monograph Iterative Learning Control: Convergence, Robustness and Applications (New York: Springer-Verlag, 1999) and the textbook System Simulation: Techniques and Applications Based on MATLAB/Simulink (Beijing, China: Tsinghua Univ. Press, 2002). Dr. Chen is an Associate Editor on the Conference Editorial Board of the IEEE Control Systems Society.
IEEE TRANSACTIONS ON NEURAL NETWORKS, VOL. 15, NO. 2, MARCH 2004
Kevin L. Moore (S’80–M’82) received the B.S. and M.S. degrees in electrical engineering from Louisiana State University, Baton Rouge, and the University of Southern California, Los Angeles, in 1982 and 1983, respectively, and the Ph.D. degree in electrical engineering, with an emphasis in control theory, from Texas A&M University, College Station, in 1989. He is currently a Professor of Electrical and Computer Engineering and Director of the Center for Self-Organizing and Intelligent Systems at Utah State University. His research interests include iterative learning control theory, autonomous robotics, and applications of control to industrial and mechatronic systems. He is the author of the research monograph Iterative Learning Control for Deterministic Systems (New York: Springer-Verlag, 1993), coauthor of the book Modeling, Sensing, and Control of Gas Metal Arc Welding (Oxford, U.K: Elsevier Science Ltd., 2003). Dr. Moore currently serves as an Associate Editor for the IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY.
Vikas Bahl received the B.S. and M.S. degrees in electrical engineering from Mangalore University, Mangalore, India, and Utah State University, Logan, in 1995 and 2002, respectively. He is currently working toward the Ph.D. degree in electrical engineering, with an emphasis in control systems and robotics, from Utah State University. His research interests include path tracking and path planning for autonomous robots, actuator control, and nonlinear system modeling.
