Model Inversion Architectures for Settle Time Applications with Uncertainty Brian P. Rigney, Lucy Y. Pao, and Dale A. Lawrence Abstract— We compare two common model inversion architectures, plant inverse (PI) and closed-loop inverse (CLI), by evaluating their ability to achieve settle time performance improvements. The plant models of interest are discretetime, single-input single-output (SISO), linear time-invariant (LTI), nonminimum phase (NMP), and uncertain. We use a simple algebraic analysis to show that PI and CLI yield the same desired to actual output dynamics if the plant is minimum phase. Using a stable inverse approximation when the plant is certain but NMP, the same algebraic analysis shows that CLI achieves superior settle time performance relative to PI when the settle boundaries are tight. Simulation and experimental data are used to derive conclusions when the plant is NMP and uncertain. We show that CLI has superior performance over PI for our plant dynamics of interest when low frequency parametric uncertainty is present. For higher frequency unstructured uncertainty, the distinction between the two inversion architectures is negligible.
I. I NTRODUCTION We are interested in increasing the performance for small, repeated, point-to-point maneuvers. Disk drive applications such as repetitious single-track movements would benefit from this performance increase, as would other applications including automated manufacturing and space-based imaging. The disk drive terminology for these point-to-point maneuvers is seeks, which we generically apply to all the applications of interest. Seek performance is measured by settle time, ts , defined as the time from the start of the maneuver until the measured position reaches and stays within an acceptable distance from the target. Small improvements in each seek’s settle time compound and cause a substantial time savings over many repetitions. We focus on discrete-time, linear time-invariant (LTI), single-input single-output (SISO) descriptions of the applications’ plant dynamics. These dynamics nominally include a rigid-body mode and higher-order structural resonances, although structured and unstructured uncertainty in the plant population force any fixed model to be in error across the set. The plant sets typically show structured uncertainty at lower frequencies, and much larger, unstructured variation in the high frequency dynamics. It is extremely difficult to fit This work has been supported through research grants from Maxtor Corporation and the National Science Foundation (CMS-0201495). B. P. Rigney is a Ph.D. student in the Electrical and Computer Engineering Dept., University of Colorado, Boulder, CO 80309-0425 and an employee of Cornice Corporation, Longmont, Colorado 80503,
[email protected]
L. Y. Pao is a Professor with the Electrical and Computer Engineering Dept., University of Colorado,
[email protected] D. A. Lawrence is an Associate Professor with the Aerospace Engineering Sciences Dept., University of Colorado,
[email protected]
uff r
+ -
C
+
+
u
P
y
Fig. 1. Block diagram of error feedback compensator C, with reference input r, feedforward input uf f , output trajectory y, and plant dynamics P .
a low-order parametric model in the higher frequency range that is accurate across the population. A further complicating factor in the plant sets is nonminimum phase (NMP) zero dynamics. NMP dynamics can arise when the sensors and actuators are noncollocated [1], a configuration common in our applications of interest. For example, disk drives have the magnetic reader position sensor and voice-coil actuator on opposite ends of the flexible actuator arm. NMP zeros in discrete time dynamics can also result from fast sample rates and high relative degree [2]. As we will see, NMP zeros require special treatment when using model inversion algorithms. The applications’ have large actuator command authority, with actuator commands that do not approach the saturation limits for small motions. Further, saturating actuator commands can excite unmodeled dynamics and lengthen settle time. Thus for short seeks, time-optimal strategies with saturated commands, as in [3] and [4], do not directly apply. Finally, the applications’ feedback compensator C is usually designed for regulation purposes using knowledge of the plant dynamics P , the disturbance and noise spectra, and performance metrics on the regulated state. Any permanent change to the feedback compensator for seek purposes could negatively affect regulation. Further, temporary changes to the feedback compensator during seeks may require complicated switching to remove transients. Hence, improvements to settle time are best accomplished through exogenous inputs r and uf f , pictured in Fig. 1. We desire the exogenous input signals (r, uf f ) which provide settle performance improvements for each plant within the population. Our proposed solution to this problem uses a combination of model inversion and reference command generation, depicted in Fig. 2. The closed-loop system from Fig. 1 has been recast as the two input, one output system G. We use an LTI system V and a desired command profile yd to parameterize the input signals (r, uf f ). If yd is a fixed trajectory and the motion is repetitive, many output tracking techniques exist to design V , including Iterative Learning Control (ILC) [5] [6] and model inversion [7]-[19]. We are
Po-1 yd Fig. 2. Block diagram summarizing the proposed solution: parameterize (r, uf f ) with the desired output trajectory, yd , and a model inverse V . The system F modifies yd to improve settle performance.
interested in improving settle performance, not tracking error with fixed yd , and thus use the system F to modify both the duration and shape of yd in order to decrease ts . In this case, the model inversion techniques are a more desirable choice for V than ILC because, given a parametric model inverse, they will generate signals (r, uf f ) for any yd . Many questions arise in the pursuit of this approach. While the choices for F and NMP inversion algorithm are crucial, they are outside the scope of this paper. Instead, we focus here on two common architecture choices for V , plant inverse (PI) and closed-loop inverse (CLI), and evaluate their performance in the presence of uncertain P . The evaluation includes a simplified system F which computes a series of yd trajectories of decreasing duration and increasing aggressiveness. The following two sections develop the PI and CLI inversion architectures and present the zero phase error tracking controller (ZPETC) method for NMP system inversion. Sections IV through VI present comparison data from algebraic, simulation, and experimental points of view. The final section, VII, discusses conclusions from the comparison data and motivates further work in the pursuit of our settle time reduction goal.
e
+
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uff +
+
u
y
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Fig. 3. Block diagram of the plant inverse (PI) feedforward architecture. Po−1 is the plant inverse based on a nominal model and P is the true plant.
yd
HCL o-1
r+
e -
C
u
P
y
−1 Fig. 4. Block diagram of the closed-loop inverse (CLI) architecture. HCLo is the closed-loop inverse based on the nominal plant model.
causing yd − y to be nonzero and exciting the closed-loop dynamics. This can have drastic effects on ts because the closed-loop dynamics may be much slower than desired for seeking. B. Closed-Loop Inverse (CLI) Architecture
II. I NVERSION A RCHITECTURES
The closed-loop inverse architecture is pictured in Fig. 4. −1 HCLo is the inverse of the closed-loop system using the nominal plant model, while HCL is the true closed-loop system. Examples of CLI in the literature include [14]-[18], all of which use CLI to track a fixed yd reference trajectory. Referring to Fig. 2, the model inverse V which implements the CLI architecture is ¸ ¸ · · 0 uf f . (2) = VCLI yd , where VCLI = −1 r HCLo
We are interested in determining the non-square model inverse V from Fig. 2. This is challenging because many multiple input, multiple output (MIMO) inversion schemes require the number of inputs to equal the number of outputs [7], [8]. Fortunately, constraining the internal architecture of V can convert the original problem into SISO inversion. We now investigate two common architectures for V which result in SISO systems for inversion: the plant inverse (PI) architecture and the closed-loop inverse (CLI) architecture.
−1 −1 When HCLo = HCL , the r computed by the inverse system causes y to exactly match yd . Again, this is unrealistic given uncertainty in P . While both architectures use output feedback to reduce plant modeling error sensitivity, it is unclear if there are sensitivity reduction advantages with PI or CLI for our applications. Sections IV through VI will therefore focus on the settle time comparison between PI and CLI when P is uncertain.
A. Plant Inverse (PI) Architecture The plant inverse feedforward architecture with error feedback, as in Fig. 3, is routinely used in industry and has been investigated by many in the literature (e.g., [8]-[11]). The model inverse V which implements this architecture is · ¸ · −1 ¸ uf f Po = VP I yd , where VP I = , (1) r 1 and Po−1 is the plant inverse based on a nominal model. The subscript o will be used throughout this paper to denote the nominal model. If Po−1 is the exact inverse of the plant dynamics (Po−1 = P −1 ), the feedforward system creates a uf f input which forces the plant output y to track the desired output yd without error. Of course, structured and unstructured uncertainty in P make exact inversion impossible,
III. NMP I NVERSION A LGORITHMS We have yet to discuss how to compute the nominal plant and closed-loop inverses in VP I and VCLI . This is complicated by the fact that Po is NMP, which also causes HCLo to be NMP because C is an internally stabilizing compensator. NMP systems are more difficult to treat with model inversion methods because the inverse system is unstable and can lead to unbounded plant inputs. The available algorithms for NMP inversion generally fall into one of two possible categories: direct treatment of the NMP zeros or stable approximation. Direct inversion methods attempt to use the unstable model-inverse directly but maintain bounded plant inputs by either pre-loading the initial conditions [11], using noncausal plant inputs [8],[10],[12], or adjusting the desired reference trajectory
[13]. Alternatively, approximate inversion techniques replace the unstable inverse system with a stable approximation [14][18]. We are interested in investigating inversion architectures here, not NMP inversion algorithms, and we choose the Zero Phase Error Tracking Controller (ZPETC), which was developed in [14] and is discussed below. A. Zero Phase Error Tracking Controller (ZPETC) The ZPETC model inversion method provides a stable approximation to the unstable NMP inverse [14]-[17]. This well-known, straightforward technique is also computationally feasible for our applications and allows us to focus this study on the algebraic, simulation, and experimental architecture comparisons. The ZPETC method is based on canceling all poles and minimum phase zeros, while canceling the phase shift (but not the gain change) induced by the NMP zeros. Using the PI architecture as an example, we must first partition the nominal plant into acceptable and unacceptable zeros for inversion Po (z −1 ) =
z −d Bco (z −1 )Buo (z −1 ) , Ao (z −1 )
(3)
where d represents the plant delay, Bco is the numerator polynomial with all invertible zeros, Ao contains all poles, and Buo is the uninvertible zeros polynomial Buo (z −1 ) = buo0 + buo1 z −1 + · · · + buon z −n .
(4)
n is thus the order of the polynomial for the uninvertible zeros. The ZPETC method then results in a stable approximation for Po−1 [14] Po−1 (z −1 ) =
∗ z d+n Ao (z −1 )Buo (z −1 ) , Bco (z −1 )[Buo (1)]2
(5)
= buon + buo(n−1) z
−1
+ · · · + buo0 z
−n
In this section, we compare closed-loop system dynamics for the PI and CLI architectures in Figs. 3 and 4. We develop the general algebraic relationships for any stable approximate inversion scheme, including ZPETC as well as other techniques [15],[18],[19]. Motivated by the plant partitioning scheme in (3), we write the nominal and true plant dynamics, respectively, as Po =
Bc Bu Bco Buo , P = . Ao A
(7)
For convenience, we will assume any plant delay d is contained within the Bc and Bco polynomials. We also assume a minimum phase compensator described by CN , CD
C=
(8)
where CN and CD are the numerator and denominator ∗ polynomials, respectively. We let Buo denote some stable approximation to the inverse of Buo ∗ Buo ≈
1 . Buo
(9)
∗ The exact form of Buo is unnecessary for this algebraic ∗ analysis. (For ZPETC, Buo is a polynomial approximation.) Finally, the nominal plant inverse can be written as
Po−1 =
∗ Ao Buo . Bco
(10)
Given these definitions, the PI system dynamics from input yd to output y are ¯ ¶ µ ∗ y ¯¯ CD Ao Buo + CN Bco HCL , (11) = y d ¯P I CN Bco where HCL is the true closed-loop dynamics
∗ where Buo is ∗ Buo (z −1 )
IV. A LGEBRAIC C OMPARISONS
.
(6)
We have used the PI architecture as an example, but the same process can be applied to the CLI architecture to produce a stable approximation of the nominal closed-loop inverse dynamics. ZPETC results in a noncausal filter with n + d samples of advance. This advance amounts to pre-actuation of the plant and requires the yd trajectory to be known in advance, as well as the seek start time. In the applications of interest, the yd trajectory is completely specified before the seek but the seek start time is unknown. Therefore, each seek will need to wait n + d samples after the seek start time before motion begins. This is equivalent to multiplying (5) by the delay z −(n+d) to force Po−1 to be causal. This delay lengthens the settle time and is included in all of our simulation and experimental results.
HCL =
CN Bc Bu . CD A + CN Bc Bu
(12)
Conveniently, (11) separates the PI system dynamics into two factors, where only the second factor, HCL , contains uncertainty. The CLI system dynamics from yd to y can similarly be expressed as ¯ ¶ µ ∗ ∗ y ¯¯ CD Ao Buo + CN Bco Buo Buo HCL .(13) = yd ¯CLI CN Bco Comparing (13) with (11), we again see that all uncertainty enters the CLI system dynamics through the multiplicative factor HCL . The first factors in (11) and (13) are similar but not exactly the same, and do not show any obvious superiority of one architecture over another. By making certain assumptions on P , we can separate the possibilities into the following three categories and provide further insight.
If we further assume that the modeling error is zero (A = Ao and Bc = Bco ), (14) becomes unity and we achieve perfect tracking with either architecture. B. Nonminimum Phase, Certain Plant Dynamics ∗ When P is NMP, Bu , Buo , and Buo are no longer equal to 1. We can still simplify (11) and (13) because A = Ao , Bc = Bco , and Bu = Buo with certain P . The resulting yd to y closed-loop systems are ¯ y ¯¯ CD ABu∗ Bu + CN Bc Bu = (15) y d ¯P I CD A + CN B c B u ¯ y ¯¯ = Bu∗ Bu (16) yd ¯CLI
This case is discussed in detail in [19], where it is shown that the CLI architecture achieves superior settle time performance when compared to the PI architecture. This is a result of (16) having no pole dynamics and a finite impulse response (FIR). We will see a simulation example of this special case in the Section V. C. Nonminimum Phase, Uncertain Plant Dynamics In this case, there are no simplifications on P and we are left with the full forms of (11) and (13). After some manipulation, we can relate these equations through ¯ ¯ y ¯¯ y ¯¯ ∗ Buo ) . (17) = + HCL (1 − Buo y d ¯P I yd ¯CLI The choice of NMP inversion algorithm will affect the ∗ ∗ Buo has zero Buo . Using ZPETC, 1 − Buo term 1 − Buo phase shift at all frequencies, and zero magnitude at low frequencies that approaches unity as frequency increases. The true closed-loop dynamics HCL for our systems of interest act in an opposite manner, with unity magnitude at low frequencies that approaches zero magnitude at extremely high frequencies. The exact makeup of these two factors in the middle frequency range will determine how closely the PI dynamics match the CLI dynamics in (17). We therefore rely on simulation and experimental data in Sections V and VI to compare the PI and CLI architectures with realistic plant and compensator dynamics. V. S IMULATION C OMPARISONS The PI and CLI yd to y system dynamics are complicated for plants with uncertainty, as described in (11) and (13). In this section, we use simulation comparisons to shed light on the differences between the two. The comparison simulations are implementations of Figs. 3 and 4, with a 4th -order NMP nominal plant model Po used in the inverse filter and a
Magnitude [dB]
The yd to y closed-loop systems greatly simplify when P and Po are assumed minimum phase. In this case, ∗ Bu = Buo = Buo = 1, and the two architectures yield the same yd to y dynamics ¯ ¯ y ¯¯ y ¯¯ (CD Ao + CN Bco ) = = HCL . (14) y d ¯P I yd ¯CLI (CN Bco )
Low Freq. Error High Freq. Unmod.
50 0
Low Freq. Error High Freq. Unmod. CP
−50
o
1
10
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A. Minimum Phase Plant Dynamics
2
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100 0 −100 1
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10 10 Frequency [Hz]
Fig. 5. Loop gain frequency response for the nominal plant (solid line), plant with low frequency modeling error (dashed line), and plant with high frequency unmodeled resonance mode (dotted line).
3rd -order compensator C. The nominal plant model is a least squares fit to empirical input-output data from an experimental disk drive system. The model has 2 NMP zeros outside the unit circle at z1,2 = −0.911±j0.556, resulting in n = 2 in (5). The compensator has a sample rate of 15 kHz, includes an integrator, and is designed to provide 45◦ of phase margin, 8 dB of gain margin, and a loop bandwidth of 600 Hz. Fig. 5 shows the nominal loop gain CPo . We compare the PI and CLI architectures for three different models of the true plant P : 1) P = Po : This case was investigated algebraically in (15) and (16). 2) Low frequency modeling error: The true plant’s low frequency mode has a shifted ωn . This parametric structured uncertainty is very typical in the applications of interest. 3) High frequency unmodeled mode: An extra resonance mode, not present in Po , occurs in P at high frequency. Again, we are interested in this type of unstructured high frequency uncertainty because it is common in the application plant sets. Fig. 5 shows the frequency response for the three loop gain models of interest. We wish to evaluate CLI and PI over a sequence of yd signals of increasing aggressiveness. We simply derive yd from the double integral of a constant acceleration pulse followed by a constant deceleration pulse. By decreasing the duration and increasing the magnitude of the pulses, we can alter the duration and thus the aggressiveness of yd . The simulation comparisons investigate the minimum achievable ts as we decrement the yd duration from 8 to 1 unit of normalized time. A. Nominal Plant Simulation While there is no modeling error in this case, the NMP stable approximation results in an imperfect cancellation between Po−1 and P . Fig. 6 shows the normalized actual settle time of y, denoted ta , versus the normalized desired settle time of yd , denoted td , as we decrement the duration of yd . Ideally, ta = td , but ta for both PI and CLI is offset
9
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ta
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Fig. 6. PI and CLI normalized actual settle times, ta , as a function of normalized desired settle time, td , for the P = Po special case.
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Settle Boundary 1.05 Position
1.05
1
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1 yd
y
d
0.95
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Fig. 8. PI and CLI normalized ta as a function of normalized td for the low frequency modeling error case.
1.1
Position
5 td
td
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0.9 0
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Fig. 7. PI and CLI normalized actual and desired output trajectories for the most aggressive desired settle time td = 1 with P = Po .
Fig. 9. PI and CLI normalized output trajectories for td = 1. The true plant model P has low frequency parametric modeling error.
from the ideal line. This offset results from a pre-actuation delay of 4 samples for both CLI and PI. Referring to (5), both Po and HCLo have 2 NMP zeros (n = 2) and 2 delay samples (d = 2). Modulo this delay, both the PI and CLI ta curves follow the ideal as we contract yd , and achieve the same minimum normalized settle time of 2.25 at td = 1. This seems to contradict the differences in (15) and (16). By investigating the plant output trajectories, we discover the cause of the similarity between PI and CLI settle performance. Fig. 7 shows the PI and CLI output trajectories for the most aggressive td trajectory (td = 1). Also included in this plot are the ±5% settle boundaries used in all simulations and experiments. As expected from the FIR closed-loop system dynamics, the CLI output trajectory perfectly tracks the final yd location after a fixed number of samples. The PI output trajectory shows some amount of overshoot consistent with excitation of the dominant closed-loop system mode. Again, this is expected because (15) includes HCL . Even though PI does not track yd as well as CLI, both architectures provide the same settle time performance because the settle boundaries are large enough to contain the PI overshoot. This case is investigated in much more depth using tighter settle boundaries in [19] for plants without uncertainty.
must now rely on the more complex relationships in (11) and (13) to describe the closed-loop system dynamics. Again, we use the normalized actual versus desired settle time plot in Fig. 8 to compare PI and CLI performance. We now see a deterioration in performance relative to the ideal behavior and a difference between the two architectures. CLI achieves a faster normalized settle time (ta = 3.5 at td = 2.75) than PI (ta = 5 at td = 1). The output trajectories in Fig. 9 show an increase in the excitation of the closed-loop dynamics with PI relative to CLI. To explain this difference, we plot the frequency response of the yd to y system dynamics from (11) and (13) in Fig. 10. The phase shift due to the pre-actuation delay has been artificially removed from this response to better see differences in phase error. The perfect tracking frequency response should have unity magnitude and zero phase shift at all frequencies. Both PI and CLI show magnitude roll-off at high frequencies, which is an artifact of the ZPETC stable approximation algorithm. While PI has less magnitude error at the highest frequencies, the system gain is very small and the yd trajectory has little energy in this region. At lower frequencies near 1 kHz, the PI architecture has 8.5◦ of phase error, while CLI has less than 2◦ . The lower frequency deviations between PI and CLI have a larger effect on the settle time results because the (r, uf f ) inputs generally have more energy at lower frequencies. This explains why CLI is able to achieve smaller ta .
B. Low Frequency Modeling Error Simulation In order to investigate realistic low frequency modeling error in the applications, we shift the natural frequency of the low frequency mode from 90 Hz to 40 Hz in P . We
8
−10
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−20
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−30 2 10
6 3
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Magnitude [dB]
9
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5
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4 Ideal
20 0 −20
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1 1 2
10
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Fig. 10. The PI and CLI yd to y closed-loop frequency responses for the plant model with low frequency parametric modeling error.
4
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Fig. 11. PI and CLI normalized ta as a function of normalized td for the high frequency unmodeled mode case.
1.25
C. High Frequency Unmodeled Dynamics Simulation
CLI
Position
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1.1 Settle Boundary
1.05 1 0.95 0.9 0
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We now seek to experimentally verify the simulation conclusions from Section V . The experimental setup is the same disk drive system after which Po was modeled previously. Therefore, we apply the same ZPETC filters used from the PI and CLI simulations. Fig. 14 is the normalized actual settle time plot, which looks very similar to the high frequency unmodeled dynamics simulation case in Fig. 11. CLI and PI show little difference in achievable settle time. Figs. 15 and 16 show the experimentally measured and simulated output trajectories for two yd input trajectories: td = 7.75 and td = 1. Simulation and experiment agree well for the slower seeks, but deviate for the aggressive td = 1 seeks. We do not see the extremely low frequency settle trajectories observed in Fig. 9 with low frequency modeling error. Instead, we notice higher frequency overshoot for the aggressive seeks with some evidence of a high frequency mode oscillation. The overshoot and excitation of this high frequency mode are slightly less with CLI.
6
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10 0 −10
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−30 2 10
3
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100 0 −100 2
VI. E XPERIMENTAL C OMPARISONS
4 Time
Fig. 12. PI and CLI normalized output trajectories for td = 1. The true plant model P has high frequency unmodeled dynamics.
Phase [deg]
High frequency unmodeled dynamics are also common across the plant populations. In order to simulate the effect of unmodeled dynamics, we augment the true plant P with an extra resonance mode (ωn = 3.5 kHz and ζ = 0.04). Fig. 11 shows the resulting actual settle times as we decrement yd . The deterioration in performance relative to the ideal line is less pronounced than in the low frequency modeling error case, and there is less difference between PI and CLI. The minimum normalized ta for PI is 4 which occurs at td = 1.5, and ta = 3.5 at td = 1 for CLI. The output trajectories for the most aggressive seek (td = 1) are plotted in Fig. 12. Both architectures have large overshoot, and also 3.5 kHz content in the settle trajectories. PI shows slightly more excitation of the 3.5 kHz unmodeled mode, which is confirmed by the yd to y frequency response in Fig. 13. The magnitude error shows more peaking in PI at the unmodeled mode frequency. The unmodeled mode also causes significant lower frequency magnitude and phase error for both PI and CLI. This results in excitation of the closedloop dynamics and large overshoot, but at a higher frequency than the low frequency modeling error case.
yd
1.2
10
3
10 Frequency [Hz]
Fig. 13. The PI and CLI yd to y closed-loop frequency responses for the plant model with high frequency unmodeled dynamics.
We do not have a perfect model of the experimental hardware, but we can make some conclusions about the hardware based on the similarities between this experimental data and the simulation data. There is no evidence of the extremely low frequency settle trajectories, as we saw in the low frequency modeling error simulation. This would suggest our model of the hardware matches well at low frequency. We do see settle trajectories for the most aggressive seeks which are very similar to the high frequency error simulations. Thus, we expect this deviation is caused by higher frequency modeling error. At this time, it is unknown if this high frequency model mismatch could be corrected through better identification or adaptation.
9 8 7
ta
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CLI−Exp PI−Exp
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Fig. 14. Experimental PI and CLI normalized actual settle times, ta , as a function of normalized desired settle time, td .
PI−Sim PI−Exp
1.2
1.1 Position
Settle Boundary 1
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Fig. 15. PI experimental and simulated normalized output trajectories for both td = 1 and td = 7.75.
CLI−Sim CLI−Exp
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Settle Boundary 1
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Fig. 16. CLI experimental and simulated normalized output trajectories for both td = 1 and td = 7.75.
VII. C ONCLUSIONS AND F UTURE W ORK The purpose of this paper is to compare the performance of plant-inverse (PI) and closed-loop inverse (CLI) architectures for settle time reduction using stable approximate NMP system inversion. Through algebraic, simulation, and experimental analysis, we can make several conclusions about the PI and CLI architectures for our plant dynamics of interest. First, there are no differences between these architectures if the plant P is minimum phase (MP). It is only when P has NMP zeros that we see a difference. Second, when P is NMP and certain, there is a clear advantage to using CLI over PI. The CLI closed-loop dynamics have a finite impulse response (FIR), which leads to an output trajectory y that perfectly reaches the desired target position after a fixed number of samples. Finally, when P is NMP and uncertain, the two architectures are related through (17). In simulation, we show
that low frequency modeling errors exaggerate the settle time differences between PI and CLI for our plant dynamics of interest. In this case, CLI clearly outperforms PI. When the modeling errors are at higher frequencies, simulation and experimental results show that the advantage of using CLI for settle time improvements becomes negligible. Future work will focus on adding on-line adaptation to our model-inverse using repetitive information. As shown in our simulation results, plant modeling errors, especially at lower frequencies, can cause a significant reduction in settle time performance. We would like to recover the P = Po settle time performance over the population of plant dynamics through a parameter adaptive model-inverse system focused on lower and middle frequency structured uncertainty. Unfortunately, the model inverse will always be a reduced order approximation to the true plant dynamics because of high frequency unmodeled modes. When aggressive yd trajectories begin to excite these unmodeled modes, it may be possible to use the repetitive information to adaptively shape yd and improve settle performance further. R EFERENCES [1] D. K. Miu, Mechatronics, Springer-Verlag, New York, 1993. ˚ om, P. Hagander, and J. Sternby, “Zeros of sampled-data [2] K. J. Astr¨ systems,” Automatica, 20(1), 1984. [3] M. Athans and P. L. Falb, Optimal Control, MacGraw Hill, New York, 1966. [4] H. T. Ho, “Fast servo bang-bang seek control,” IEEE Trans. Mag., 33(6), Nov. 1997. [5] K. L. Moore, Iterative Learning Control for Deterministic Systems, Springer-Verlag, London, 1993. [6] J. Ghosh and B. Paden, “A pseudoinverse-based iterative learning control,” IEEE Trans. Automat. Contr., 47(5), May 2002. [7] L. Silverman, “Inversion of multivariable linear systems,” IEEE Trans. Automat. Contr., 14(3), Jun. 1969. [8] S. Devasia, D. Chen, and B. Paden, “Nonlinear inversion-based output tracking,” IEEE Trans. Automat. Contr., 41(7), Jul. 1996. [9] S. Devasia, “Should model-based inverse inputs be used as feedforward under plant uncertainty?” IEEE Trans. Automat. Contr., 47(11), Nov. 2002. [10] Q. Zou and S. Devasia, “Preview-based stable-inversion for output tracking,” Proc. American Control Conf., San Diego, CA, Jun. 1999. [11] L. Lanari and J. Wen, “Feedforward calculation in tracking control of flexible robots,” Proc. 30th IEEE Conf. Decision Control, Brighton, U.K., 1991. [12] L. R. Hunt, G. Meyer, and R. Su, “Noncausal inverses for linear systems,” IEEE Trans. Automat. Contr., 41(4), Apr. 1996. [13] M. Benosman and G. Le Vey, “Stable inversion of SISO nonminimum phase linear systems through output planning: an experimental application to the one-link flexible manipulator,” IEEE Trans. Ctrl. Sys. Tech., 11(4), Jul. 2003. [14] M. Tomizuka, “Zero phase error tracking algorithm for digital control,” ASME J. Dyn. Sys., Meas., Contr., 109, Mar. 1987. [15] E. Gross and M. Tomizuka, “Experimental beam tip tracking control with a truncated series approximation to uncancelable dynamics,” IEEE Trans. Ctrl. Sys. Tech., 2(4), Dec. 1994. [16] B. Haack and M. Tomizuka, “The effect of adding zeros to feedforward controllers,” ASME J. Dyn. Sys., Meas., Contr., 113, Mar. 1991. [17] D. Torfs, J. De Schutter, and J. Swevers, “Extended bandwidth zero phase error tracking control of nonminimal phase systems,” ASME J. Dyn. Sys., Meas., Contr., 114, Sep. 1992. [18] D. Torfs, R. Vuerinckx, J. Swevers, and J. Schoukens, “Comparison of two feedforward design methods aiming at accurate trajectory tracking of the end point of a flexible robot arm,” IEEE Trans. Ctrl. Sys. Tech., 11(4), Jul. 2003. [19] B. Rigney, L. Pao, and D. Lawrence, “Settle time performance comparisons of stable approximate model inversion techniques,” Proc. American Control Conf., Minneapolis, MN, Jun. 2006.
