2010 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 30-July 02, 2010
FrB08.1
Adaptive-Delay Combined Feedforward/Feedback Control for Raster Tracking with Applications to AFMs Jeffrey A. Butterworth, Lucy Y. Pao, and Daniel Y. Abramovitch Abstract— In previous work, we evaluated the performance of two control architectures applied to atomic force microscopes (AFM) [1]. Experimental results in [1] indicated that the closed-loop-injection (FFCLI) architecture outperformed the plant-injection (FFPI) architecture when using a specific model-inversion feedforward technique for the tracking of a raster pattern. Empirical work suggested that a nontraditional variation upon the experimentally inferior FFPI architecture may allow it to track a raster pattern at a performance level in the neighborhood of the FFCLI architecture. This variation is manifested as additional delay inserted in the feedforward control system. An online adaptive technique is used to determine the required amount of additional delay. Experimental results show that the performance level of the FFCLI architecture and the adaptive-delay FFPI architecture are comparable.
I. INTRODUCTION Several research groups have investigated improving upon the performance of feedback-only AFM control by combining it with a feedforward filter. In particular, two distinct combined feedforward/feedback architectures have been developed and have appeared in the AFM literature: the plant-injection and closed-loop-injection architectures. This paper is an extension of a previous paper [1], in which we demonstrated the experimental superiority of the closedloop-injection architecture over the plant-injection architecture in AFM raster tracking applications. When using the feedforward closed-loop-injection (FFCLI) architecture, FP in Fig. 1 is set to zero and the feedforward filter FCL acts on the reference signal ahead of the closed-loop system. When using the feedforward plantinjection (FFPI) architecture, FP in Fig. 1 is designed to perform as the feedforward filter while FCL is set depending on the type of controller used for FP . Typically, FCL is the identity or a delay function in the FFPI architecture. In this paper, we expand upon the experimental results in [1] and demonstrate how augmenting the plant-injection architecture with an adaptive-delay algorithm can yield a performance equivalent to the closed-loop-injection architecture. When using the adaptive-delay variation upon the FFPI architecture, FP is designed as a model-inverse filter and FCL becomes an adaptive-delay filter (as shown in Fig. 7). This work was supported in part by Agilent Technologies, Inc. and the US National Science Foundation (NSF Grant CMMI-0700877). J. A. Butterworth is a graduate student, and L. Y. Pao is a professor; both are with the Dept. of Electrical, Computer, and Energy Engineering at the University of Colorado at Boulder, Boulder, CO 80309 USA,
[email protected],
[email protected] D. Y. Abramovitch is a senior research engineer in the Nanotechnology Group at Agilent Laboratories, 5301 Stevens Creek Blvd., M/S: 4U-SB, Santa Clara, CA 95051 USA,
[email protected]
978-1-4244-7425-7/10/$26.00 ©2010 AACC
The significance of this result is that it provides: (a) a simple example of how one might jointly design the two feedforward filters FCL and FP and (b) the flexibility of the use of either of the two different feedforward architectures without sacrificing performance. This will be of substantial value as we investigate the use of adaptive methods (similar to those discussed in [2] and [3]) with either architecture to update the appropriate model-inverse filter FCL or FP in future work. II. PLANT-INJECTION ARCHITECTURE Recent work in AFMs has shown improvements in AFM performance when feedback controllers are combined with a feedforward controller in the plant-injection architecture [4]– [8]. In general, this architecture ensures stability through the feedback controller C while the feedforward controller FP further increases tracking performance, disturbance rejection, and robustness to model uncertainties. Similar to work in [9], we examine the transfer function from the desired input xd to the output x of the FFPI system of Fig. 1. Assuming FCL is unity, the transfer function is X(z) P (z)FP (z) + P (z)C(z) = . (1) Xd (z) 1 + P (z)C(z) FFPI
It should be clear that if the feedforward filter FP (z) = 0, then (1) reduces to the common expression for the dynamics of a feedback-only closed-loop system, HCL (z) =
P (z)C(z) . 1 + P (z)C(z)
(2)
When FP (z) is equal to the inverse of P (z), then (1) becomes the identity and we are able to track any desired input perfectly. Of course, this assumes a perfect inverse of P (z) exists and is practically implementable, and we will discuss this further in Subsection II-A. For convenience we drop the argument z, and define the plant dynamics P , the feedback controller C, and the
FP xd
FCL
xff
ex
C
uxff uxfb
ux
P
x
Fig. 1. A single-input single-output (SISO) block diagram of an AFM’s x-direction consisting of a feedback compensator C, a plant-injection feedforward controller FP , and a closed-loop-injection feedforward controller FCL . When using the closed-loop-injection architecture, FP is set to zero. In contrast, when using the plant-injection architecture, FCL is set to the identity (or a delay in the case of model-inverse control when the relative degree of the plant is greater than zero).
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FP xd
xff
z−r
ex
C
uxff uxfb
(containing the unstable (noninvertible) zeros): ux
P
ˆ ˆs (z)B ˆu (z) B B(z) Pˆ (z) = = . ˆ ˆ A(z) A(z)
x
Fig. 2. The plant-injection configuration (when using a ZMETC feedforward controller for FP ) of Fig. 1. The relative degree of the plant P is r and a delay of z −r is included in the ZMETC design of FP to ensure it has a causal implementation. The z −r delay block in the FCL position is then required to ensure the synchronous arrival of the xf f and uxf f signals to the system.
ˆ The polynomial A(z) contains all the poles of the model of ˆu has the form the plant. B ˆu (z) = bun z n + bu(n−1) z n−1 + · · · + bu0 B
B FP N CN , and FP = . , C= A CD FP D
P˜ −1 (z) =
(3)
The subscripts N and D indicate numerator or denominator polynomials. A and B represent polynomials defining the plant poles and zeros, respectively. Using the definitions in (3), we can further reduce (2) and (1) to HCL =
BCN and ACD + BCN
X(z) BFP N CD + BCN FP D . = Xd (z) F F P I FP D (ACD + BCN )
(4) (5)
Equation (5) provides us with an overall transfer function for the FFPI architecture regardless of the design of FP . Below, we design FP using a model-inverse technique.
A. FFPI: Feedforward Model-Inverse Control Devasia and others have studied model-inverse methods applied in the plant-injection architecture for AFMs [1], [4], [5], [8], [10], where C in Fig. 1 is some feedback controller and FP is set approximately equal to Pˆ −1 , the inverse of the plant model. Ideally, FP is equal to Pˆ −1 , or even better, FP is equal to P −1 . However, often the existence of nonminimum-phase zeros in the plant force a stable approximate inverse to be used in place of the exact inverse. In AFMs, the plant and hence model of piezo scanners typically have nonminimum-phase zeros due to the existence of non-collocated sensing and actuation and flexible modes. Many control systems researchers, working in a myriad of application areas, have developed and used stable approximate inversions in order to implement model-inverse based control on nonminium-phase systems; this work has been applied to both the FFPI and FFCLI architectures [3], [5], [9], [11], [12]. Various stable approximate model-inversion techniques exist, but here, we focus on the simple and effective zeromagnitude-error tracking controller (ZMETC): that has appeared in [3], [9], [13]. This is a cousin of Tomizuka’s popular zero-phase-error tracking controller [12]. To design a ZMETC stable approximate model inverse of a nonminimum-phase plant, write the model of the plant ˆs (containdynamics as in (6), partitioning the polynomial B ˆu ing the stable (invertible) zeros) from the polynomial B
(7)
where n is the number of nonminimum-phase zeros. The ZMETC technique then yields a stable approximation of the inverse of the plant
feedforward controller FP as a ratio of polynomials as in P =
(6)
ˆ A(z) , ˆ ˆu∗ (z) Bs (z)B
(8)
where the ∼ indicates an approximate inverse (as compared ˆu∗ (z) is to a ∧ indicating the model of the plant), and B ˆu∗ (z) = bu0 z n + bu1 z n−1 + · · · + bun . B
(9)
Note that the difference between (7) and (9) is the “flipping” of the coefficients. This action is equivalent to reflecting ˆu (z) in (7) into the unit circle to 1/zi the roots zi of B ˆ ∗ (z) in (9) is also and then “advancing” by z n so that B u a polynomial in z. Setting FP equal to P˜ −1 (z) and FCL equal to unity or a delay block (discussed further below) in Fig. 1 would constitute a model-inverse based control using the plant-injection architecture. If the relative degree r of P (z) is greater than zero, the resulting P˜ −1 (z) will be noncausal. Additional delay equal to r will have to be added in order for FP to be implementable in a causal way: FP = z −r P˜ −1 (z) =
ˆ z −r A(z) . ˆs (z)B ˆu∗ (z) B
(10)
If r > 0, then the feedforward block FCL should be defined not as unity, but rather as a delay block equal to z −r (see Fig. 2). Setting FCL = z −r changes the transfer function in (1) to X(z) P (z)FP (z) + z −r P (z)C(z) . (11) = Xd (z) 1 + P (z)C(z) FFPI
If (a) there exist no nonminimum-phase zeros in the plant, (b) the model exactly matches the plant (Pˆ (z) = P (z)), and (c) r = 0, then the output of the system exactly tracks the desired input xd and the feedback loop is not excited. If the first two conditions (a) and (b) hold and r > 0, then the output of the system exactly tracks the desired input xd with a delay of r. If we assume a nonminimum-phase, perfectly known plant (Pˆ (z) = P (z)) with relative degree r and a minimum phase, and exactly proper C, we can apply the above discussion on ZMETC and write (4) and (11) as
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HCL =
(Bs CN )(Bu ) BsCL BuCL = and ACL ACD + Bs Bu CN
X(z) Xd (z) F F P I
=
Bu ACD + Bs Bu CN Bu∗ z r Bu∗ (ACD + Bs Bu CN )
=
ACD + Bs Bu∗ CN HCL . z r Bu∗ Bs CN
(12)
(13)
Ψ(z) =
Bs Bu∗ CN
ACD + , ACD + Bs Bu CN
(14)
Position (µm)
5
F F CLI
If we make the assumption that a stable inverse of HCL exists, we can see that setting FCL equal to that stable inverse would result in perfect tracking.
A. FFCLI: Feedforward Model-Inverse Control The creation of a ZMETC model-inverse controller for the FFCLI architecture follows very closely to the ZMETC procedure described in Subsection II-A for the FFPI architecture. The major difference is the use of ZMETC to create a stable approximate model inverse of the closedloop dynamics (HCL (z)) and setting FCL in Figs. 1 equal to ˜ −1 (z). Following the technique that approximate inverse H CL described in Subsection II-A, and making the assumption that the feedback controller C is stable, minimum phase, and exactly proper, we obtain an approximate stable inverse of HCL (z): −1 ˜ CL FCL = z −r H =
z −r AˆCL ˆ∗ ˆsCL B B
uCL
=
ˆu CN ) ˆ D +B ˆs B z −r (AC ˆs CN )(B ˆu∗ ) (B (17)
where (12) defines ACL and BsCL . For minimum phase C, ∗ BuCL is equal to the same Bu∗ from Subsection II-A. Because C is assumed to be exactly proper, the relative degree r remains the same as before as well. With C stable, minimum phase, and exactly proper, and further assuming the nonminimum-phase plant is known with certainty, we can use (12) and (17) to rewrite (16): z −r (ACD + Bs Bu CN ) (Bs CN )Bu X(z) = , Xd (z) ACD + Bs Bu CN (Bs CN )(Bu∗ ) F F CLI
z −r Bu . Bu∗
(18)
Comparing (18) with (15) for the FFPI architecture, we see one of the advantages of FFCLI over FFPI. In (18), the
0
−5
FFPI
It should be clear that if all plant zeros are minimum phase, then (15) reduces to a delay block indicating perfectly delayed tracking.
III. CLOSED-LOOP-INJECTION ARCHITECTURE When compared to plant-injection, the closed-loopinjection’s superior ability to perform under the presence of uncertainties in a disk drive application [9] provides motivation for application to AFM control. The same feedforward techniques overviewed in Section II are briefly discussed specifically for the closed-loop-injection architecture below. In this architecture, we set FP in Fig. 1 equal to zero and design FCL accordingly. We can formulate an overall transfer function of the FFCLI system from the desired input xd to the output x: X(z) = HCL FCL . (16) Xd (z)
x(t)
10
we can rewrite (13) in the compact form of (15) that will be valuable to us when we compare with the corresponding transfer function in the FFCLI architecture in Section III and IV. X(z) z −r Bu (15) = Ψ(z) Xd (z) Bu∗
=
xd (t − k∗ Ts )
xd (t)
Now, if we define
−10 0
k* = 40 0.005
0.01
0.015 0.02 Time (sec)
0.025
Fig. 3. Example of simulation results using PID feedback only. Three curves appear in the figure: the xd (t) input, the xd (t − k∗ Ts ) used for the performance metrics Je and Jm , and the x(t) output. In this case, k∗ (displayed in the lower left corner of the plot) is 40.
transfer function from xd to x reduces to a transfer function with unity gain at all frequencies and possibly some phase delay. This is in contrast to (15) of the FFPI architecture which also includes the factor Ψ(z) defined in (14). We will look more closely at the effect of Ψ(z) in Section IV. Like the FFPI architecture, we can see that when HCL (z) is minimum-phase and the model Pˆ (z) is known with certainty, (18) becomes unity or a delay block. In this case, we expect x to perfectly track xd (perhaps with some delay). IV. INITIAL EXPERIMENTAL RESULTS This section presents nonadaptive experimental results of a comparison of the standard FFPI and FFCLI architectures while using ZMETC model-inverse feedforward control. First, we discuss a metric used to compare the results, then we provide experimental data showing the performance of each architecture. All laboratory work was performed on the x direction of a nPoint NPXY100A x-y piezoscanner stage, and controllers were implemented on a Texas Instruments digital signal processor. Using frequency response methods, we obtained a 4th-order discrete-time model of the stage (given in [14]). The model includes two nonminimum-phase zeros that challenge the performance of control designs and limit the effectiveness of the model-inverse based feedforward methods. The relative degree r = 2, and the sample rate for this model and all associated controllers is 25 kHz. A. Performance Metrics When discussing the performance of the tracking of a raster scan in AFMs, it is important to recall the overall goal of AFMs: to create a quality image in a timely manner. But this goal requires a definition of a “quality image” when referring to an x-y raster scan. Focusing on the x direction, an ideal controller would cause the system output x(t) to track the desired raster pattern xd (t) flawlessly. This suggests that a mean-square-error metric might be informative in defining the performance of a controller. However, for imaging, meansquare error might not give us the best measurement. This is because phase lag in the raster scan used for AFM imaging is not nearly as critical as consistently tracking the magnitude. Ultimately, this means that perfectly delayed tracking is
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better than imperfect timely tracking if we know the delay well. So, to identify that delay, we examine one period T of the raster scan after a time tss after which all transients (from initial conditions for example) have died out, and define the integer variable k ∗ as Z tss +T 2 ∗ k = arg min k
xd (t − kTs ) − x(t) dt.
(19)
tss
Here, k is also an integer and is defined on [0, TTs ] where Ts = 40µsec is the controller sample period. k ∗ is defined to be an integer as it represents actual implementation in a digital controller. Specifically, k ∗ reflects the discrete delay in the system. One area of ongoing work investigates the benefits of using a non-integer k ∗ to define these performance metrics. In the meantime, we can use the integer k ∗ to define two metrics that disregard the phase lag in the system and emphasize magnitude tracking: Z tss +T 2 ∗ Je =
xd (t − k Ts ) − x(t) dt
(20)
tss
Jm =
max
t∈[tss ,tss +T )
2
xd (t − k∗ Ts ) − x(t)
(21)
For further clarification, we provide Fig. 3 which is an example simulation of a 100Hz raster scan input into a PID feedback-only control loop. We have chosen a simple PID feedback-only simulation as an example for its clear delineation of each line. We see the actual 100Hz raster scan input xd (t) in solid black, and the shifted 100Hz input xd (t−k ∗ Ts ) in dashed black. From Fig. 3, it is clear that the PID feedback-only controller lacks the bandwidth to be able to track the 100Hz raster scan. Images created while using this raster scan would be highly distorted at the edges.
to-noise ratio, signal saturation, vibrations inherent to the piezos and more. V. AN ADAPTIVE-DELAY FFPI ALGORITHM A. Motivation for Adaptive-Delay Approach Some experimental testing showed that the performance of the FFPI architecture could be improved upon if the value of the r delay in Fig. 2 was allowed to increase. Reserving r to be the relative degree of the plant (and the delay required in the causal design of a ZMETC filter), and introducing a new variable τ that represents the delay in the FCL block, we found that performance improvements continued with each increment of τ until a point at which the performance began to degrade again. Note that the optimal value of τ is likely not an integer. Nonetheless, an integer τ was assumed due to its effectiveness and lack of complexity in implementation. We can show that an optimal choice of τ exists by reworking the definition of Ψ from (14). If we start with (11), but replace the r in the FCL block with τ , we arrive at P (z)FP (z) + z −τ P (z)C(z) X(z) = . (22) Xd (z) 1 + P (z)C(z) τ FFPI
Following the same general steps described in Subsection II-A, we can rewrite the overall transfer function of this τ delay variation on the FFPI architecture as # " X(z) z −r Bu ACD + Bs Bu∗ CN z r−τ = . (23) Xd (z) Bu∗ ACD + Bs Bu CN τ FFPI
−r
In (23), we notice the same unity-gain factor z B ∗Bu from u (15) and (18), and we define the portion in brackets as Ψτ (z) =
B. Nonadaptive Experimental Results In Fig. 4, we provide experimental results comparing the FFPI and FFCLI architectures. In this comparison, the exact same 5th-order H∞ feedback controller C (given in [14]) is used for both experiments. Although neither is perfect, the reader should notice the FFCLI architecture’s ability to track the supplied 99.2 Hz raster pattern better than the FFPI architecture. Specifically, the FFPI architecture overshoots the corners of the raster pattern. In contrast, the FFCLI architecture tracks the corners and the rest of the pattern with Je and Jm metrics two orders and one order of magnitude less than those of the FFPI architecture, respectively. The overshoot associated with the plant-injection architecture can be understood by looking at the frequency response of the transfer function Ψ(z) of (14) for this particular case as shown in Fig. 5. In particular, the magnitude of Ψ at 99.2 Hz (the raster frequency) in Fig. 5 is 1.24 (1.86 dB) which is in the neighborhood of the overshoot magnitude shown in Fig. 4(a). From this plot, we can see that under these conditions, the FFCLI architecture will always outperform FFPI except at low frequencies (which are trivial as feedforward filters are not needed to track raster patterns well at these rates) and in simulation at very high frequencies. The high-frequency rasters are not realistic for hardware comparisons because of many problems including low signal-
ACD + Bs Bu∗ CN z r−τ . ACD + Bs Bu CN
(24)
The τ subscript indicates that this transfer function is not the same as (14). Using Ψτ (z), we can study how the magnitude of its frequency response changes as we vary τ . The choice of τ that provides a frequency response of Ψτ (z) closest to unity magnitude will be the optimal. Recall that Fig. 5 shows the magnitude and phase of Ψτ (z) = 2. In Fig. 6, we expand upon Fig. 5 and provide new frequency responses plots for the magnitude of Ψτ (z) as we vary τ . In Fig. 6, the original case (τ = 2) is shown in black. As τ increase, the magnitude of Ψτ (z) becomes closer to unity. At τ = 16 (shown in magenta), the magnitude is closest to providing unity gain for all frequencies. Beyond τ = 16, the magnitudes trend away from unity gain for all frequencies. Under these circumstances, setting τ = 16 should lead to improved FFPI tracking performance. Note that no integer choice of τ leads to Ψτ (z) being unity magnitude for all frequencies. The data used in creating Fig. 6 suggests that the “optimal” choice for τ is actually a value that is less than 16, but greater than 15. Having a non-integer τ will complicate implementation, so it is constrained to be a integer value. Further, the data for Ψτ (z) shown in Fig. 6 is based only upon the model of our plant and our feedback controller, so we cannot assume that τ = 16 will be the ideal choice for implementation on our actual hardware. However, it is
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xd (t − k∗ Ts )
2
2
1
1
0
−1
x(t)
0
−1
−2
−2 k* = 9
0
xd (t − k∗ Ts )
xd (t)
x(t)
Position (µm)
Position (µm)
xd (t)
k* = 16
0.005
0.01
0.015 0.02 Time (sec)
0.025
0.03
0
(a) FFPI: Je = 1.12 × 10−3 and Jm = 5.66 × 10−1
0.005
0.01
0.015 0.02 Time (sec)
0.025
0.03
(b) FFCLI: Je = 5.48 × 10−5 and Jm = 2.67 × 10−2
Fig. 4. Experimental results for (a) ZMETC plant-injection feedforward and (b) ZMETC closed-loop-injection feedforward control when given a 99.2 Hz raster pattern. Similar to Fig. 3, three curves appear in each figure: xd (t), xd (t − k∗ Ts ), and x(t). The metric values for each experiment are provided in the corresponding sub-captions (recall metric values closest to zero are superior), and the value for k∗ is provided in the lower left of each sub-figure.
Magnitude (dB)
not too bold to assume that an ideal choice of τ exists and it is likely in the neighborhood of 16. This claim provides motivation to develop an adaptive algorithm that automatically calibrates the system to use the best choice of τ . Note that the experimental results for the FFCLI algorithm shown in Fig. 4(b) indicate a value for k ∗ that is equal to 16, the same value for τ that was suggested by the analysis in Fig. 6. This is likely more than just coincidence, but we note that this value for k ∗ in the FFCLI algorithm can vary as data is recorded. This also provides motivation for future work in which we implement fully adaptive feedforward filters.
4 2 0 −2 1
Phase (degrees)
10
N −1
Ωj =
X
[x(j, n)2 − xf f (j, n)2 ].
(25)
n=0
Here, N is the total number of discrete-time data points that define the full period of a raster pattern, and n is the specific time index within that raster pattern. We have used the notation x(j, n) to indicate the value of the signal x at the nth time index during the j th raster period. Introducing the positive constant µ as a scalar value used to weight the amount of the update to τj+1 , we define our adaptive update equation as τj+1 = round(τj + µΩj )
(26)
where round(·) converts the argument to the nearest integer. Note that constraining τj+1 to be an integer results in the existence of a “dead zone” within the algorithm. Fig. 7 provides a block-diagram representation of the adaptive-delay FFPI architecture as described above. It is identical to the standard FFPI architecture in design and implementation except for the inclusion of adaptive τ .
3
10
4
10
10 0 −10 −20 −30 1
B. The Adaptive-Delay Algorithm Rather than using Fig. 6 to assume that 16 is the optimal choice of τ , and noting that variations in our plant are common, we choose to create an adaptive algorithm that automatically calibrates the system such that it uses the best choice of τ . This is done by comparing the sum of the squared signals x(k) and xf f (k) at the completion of each raster pattern to define an update direction. Defining the variable j as the raster index (or the adaptive update count), we define an update direction as
2
10
10
2
3
10 10 Frequency (Hz)
4
10
Fig. 5. The frequency response of the transfer function Ψ(z) (for our experimental conditions) in (14) which is the factor that differentiates the two overall transfer functions for each architecture (equations (15) and (18)). This plot was created using the model for the plant P and the feedback controller C.
C. Experimental Results of FFPI with Adaptive-Delay Experimental results when using the adaptive-delay FFPI algorithm on the nPoint NPXY100A x-y piezoscanner stage are provided in Fig. 8. In particular, Fig. 8 shows the steadystate results (after τ has completed adapting). Comparing the performance results of the FFCLI architecture in Fig. 4 with those in Fig. 8, we note that the adaptive-delay FFPI algorithm has successfully performed at a level relatively equal to that of the FFCLI results. Due to the presence of noise in the measurements and natural variations in the system, it is difficult to claim that the adaptive-delay FFPI algorithm is better or worse than the FFCLI algorithm. Despite the analysis of Subsection V-A that suggested the optimal τ is 16, in the experimental data presented, the adaptive algorithm actually converged to τ = 15. Some variation from this analysis can be expected as the data used in Fig. 6 is numerically computed from the feedback controller and a linear plant model that was obtained by experimental system identification methods. Further, the converged value for τ
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xd (t − k∗ Ts )
xd (t)
x(t)
2 4
2
τ = 10
2
1
Position (µm)
Magnitude (dB)
3
12
1
14 16
0
−1
18
−1
0
20 −2
−2 k* = 16
−3 2
10
3
0
4
10 Frequency (Hz)
10
0.005
0.01
0.015 0.02 Time (sec)
0.025
0.03
(a) Adaptive τ FFPI: Je = 4.25 × 10−5 and Jm = 2.15 × 10−2 Fig. 6. The magnitude portion of the frequency response of Ψτ (z) as we vary τ . The τ values for each particular curve are shown on the plot. The original case (τ = 2) is shown in black. At τ = 16 (shown in magenta), the magnitude is closest to providing unity gain for all frequencies, and setting τ = 16 will offer the most improved FFPI performance. This plot was created using the model for the plant P and the feedback controller C.
tends to vary between 15 and 16 as data is taken throughout the day. Additionally, we notice that the value for k ∗ in Fig. 8 is 16 and not the converged τ value of 15. These natural variations further support the need for an adaptive algorithm to determine τ and the motivation for additional adaptive methods for other control filters. In Fig. 9, the output signal x (red) and the input signal xd (black) are plotted immediately after activating the adaptive τ algorithm. Additionally, the value of τ (dark green) as it updates is plotted on an auxiliary axis in Fig. 9. Here, the adaptive scaling factor µ = 2.50 was determined to empirically work well. (Larger values of µ risk the stability of the adaptation while smaller values result in slower convergence.) No adaptation occurs during the first raster pattern. As a result, τ = 2 for this entire first period, and the first period looks much like any period of FFPI data in Fig. 4(a). By the second period, the adaptive algorithm has made a large correction with respect to the data from the previous raster pattern. After 3 raster periods (or 2 updates), the algorithm settles at τ = 15 and the output now matches the data in Fig. 8. The initial choice of τ = 2 was selected because that represents the choice of τ as dictated by the ZMETC FFPI algorithm. A more intelligent choice of τ would be 16 as per
FP xd
z−τ
xff
ex
C
uxff uxfb
ux
P
x
xff Adpt. Alg.
x
Fig. 7. The adaptive-delay variation of the plant-injection configuration from Fig. 1. The difference between this arrangement and the traditional FFPI configuration is the addition of an adaptive algorithm that updates the τ -delay block that usually appears in the FCL block of Fig. 1.
Fig. 8. Experimental results for the adaptive delay algorithm implemented on the FFPI architecture once τ has reached the steady-state value of 15. Similar to Fig. 4, three curves appear in the figure: xd (t), xd (t − k∗ Ts ), and x(t). Note the values for the performance metrics are now on the order of the performance levels of the FFCLI architecture in Fig. 4(b).
the analysis of Subsection V-A and Fig. 6, but that choice fails to demonstrate the adaptive capability of the algorithm. As a result, we selected τ = 2 as a initial value to show the adaptive algorithm in action. To further support the value of this adaptive-delay algorithm and the analysis in Sections IV and V, we have also provided (in Fig. 10) ZMETC experimental results in which the desired raster pattern xd (t) frequency was increased to 115.7 Hz. Corresponding to Fig. 5, a higher magnitude trajectory overshoot occurs in the FFPI results shown in Fig. 10(a). The adaptive-delay algorithm is able to correct the trajectory after arriving at τ = 15 in 5 raster periods (or 4 adaptive updates). Further, a 88.6 Hz raster pattern input was also applied to the system and the results were similar: overshoot occurred in the standard FFPI algorithm (as per Fig. 5) and the adaptive-delay algorithm corrected that overshoot. VI. CONCLUSIONS AND FUTURE WORK The key contribution of this paper is the adaptation of a single parameter in an alternate FFPI architecture that results in near unity gain of the overall transfer function which consequently leads to a performance level matching that of FFCLI. Further, FFPI results in [13] suggest that this adaptive algorithm will likely be successful for other model-inverse based designs including H∞ feedforward filters (which, in the frequency domain, will often mimic the appearance of a model-inverse controller.). This work also provides more flexibility in future work to apply adaptive methods for the model-inverse based feedforward filters to provide robustness to variations in the plant. Prior to this work, the experimental results in [1] suggested that we would be likely limited only to the FFCLI architecture in order to achieve the best tracking performance. Now, we have the flexibility to continue to
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Fig. 9. The output signal x (red) and the 99.2 Hz input signal xd (black) are plotted immediately after activating the adaptive τ algorithm. On an auxiliary (right) axis, we also provide the value of τ (dark green) as it updates. After 3 raster periods (or 2 updates), the algorithm settles at τ = 15 and the output now matches the data shown in Fig. 8.
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Fig. 10. Experimental results for (a) ZMETC plant-injection feedforward and (b) ZMETC plant-injection feedforward control with the adaptive-delay algorithm when given a faster 115.7 Hz raster pattern. The data in (b) is recorded after τ has reached the steady-state value of 15 in 5 raster periods (or 4 adaptive updates). In (a), we notice the same trajectory overshoot that is present in Fig. 4(a), but at a higher magnitude that corresponds to the findings in Fig. 5. Note that the adaptive-delay algorithm in (b) has corrected the overshoot present in (a) despite the faster raster input.
investigate both algorithms for adaptive-feedforward modelinverse based feedforward control. Future work will determine which of these two architectures (FFCLI or adaptive-delay FFPI) will be superior for implementing adaptive-feedforward model-inverse based feedforward control. In short, using the adaptive-delay FFPI architecture may be beneficial as we learn more about specific variations natural to the plant. R EFERENCES [1] J. A. Butterworth, L. Y. Pao, and D. Y. Abramovitch, “A comparison of control architectures for atomic force microscopes,” Asian J. Ctrl., vol. 11(2), pp. 175–181, 2009. [2] A. Fleming, “Time-domain adaptive feed-forward control of nanopositioning systems with periodic inputs,” in Proc. Amer. Ctrl. Conf., June 2009, pp. 1676–1681. [3] B. Potsaid, J. T. Wen, M. Unrath, D. Watt, and M. Alpay, “High performance motion tracking control for electronic manufacturing,” ASME J. Dyn. Sys., Meas., & Ctrl., vol. 129, pp. 767–776, Nov. 2007. [4] D. Croft and S. Devasia, “Vibration compensation for high speed scanning tunneling microscopy,” Rev. Sci. Instr., vol. 70(12), pp. 4600– 4605, 1999. [5] D. Croft, G. Shedd, and S. Devasia, “Creep, hysteresis, and vibration compensation for piezoactuators: Atomic force microscopy application,” ASME J. Dyn. Sys., Meas., & Ctrl., vol. 123, pp. 35–43, 2001.
[6] G. Schitter, R. W. Stark, and A. Stemmer, “Fast contact-mode atomic force microscopy on biological specimen by model-based control,” Ultramicroscopy, vol. 100, pp. 253–257, 2004. [7] G. Schitter and A. Stemmer, “Identification and open-loop tracking control of a piezoelectric tube scanner for high-speed scanning-probe microscopy,” IEEE Trans. Ctrl. Sys. Tech., vol. 12, pp. 449–454, 2004. [8] Q. Zou and S. Devasia, “Preview-based optimal inversion for output tracking: Application to scanning tunneling microscopy,” IEEE Trans. Ctrl. Sys. Tech., vol. 12, no. 3, pp. 375–386, May 2004. [9] B. P. Rigney, L. Y. Pao, and D. A. Lawrence, “Nonminimum phase dynamic inversion for settle time applications,” IEEE Trans. Ctrl. Sys. Tech., vol. 17, no. 5, pp. 989–1005, Sept. 2009. [10] S. Tien, Q. Zou, and S. Devasia, “Iterative control of dynamicscoupling-caused errors in piezoscanners during high-speed AFM operation,” IEEE Trans. Ctrl. Sys. Tech., vol. 13(6), pp. 921–931, 2005. [11] K. K. Leang and S. Devasia, “Feedback-linearized inverse feedforward for creep, hysteresis, and vibration compensation in AFM piezoactuators,” IEEE Trans. Ctrl. Sys. Tech., vol. 15(5), pp. 927–935, 2007. [12] M. Tomizuka, “Zero phase error tracking algorithm for digital control,” ASME J. Dyn. Sys., Meas., & Ctrl., vol. 109, pp. 65–68, March 1987. [13] J. A. Butterworth, L. Y. Pao, and D. Y. Abramovitch, “Architectures for tracking control in atomic force microscopes,” in Proc. IFAC World Cong., July 2008, pp. 8236–8250. [14] ——, “GOALI: Control architectures and adaptive model-inverse based methods for nonminimum-phase uncertain systems, with applications to atomic force microscopes,” in Proc. of 2009 NSF Eng. Research and Innovation Conf., June 2009.
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