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Abstract

Many servo systems require micro/nano-level positioning accuracy. This requirement sets a number of challenges from the viewpoint of sensing, actuation, and control algorithms. This article considers control algorithms for precision positioning. We examine how prior knowledge about the parameterization of control structure and the disturbance spectrum should be utilized in the design of control algorithms. An outer-loop inverse-based Youla–Kucera parameterization is built in the article. The presented algorithms are evaluated on a tutorial example of a galvo scanner system.

Keywords

Precision control feedback design Youla–Kucera parameterization disturbance rejection loop shaping galvo scanner

Introduction

A precision servo system aims at accurately positioning the controlled object(s) to follow the desired trajectories. The controlled object here, for instance, can be a read/write head for accessing data in a commercial hard disk drive (HDD),¹ a stage for carrying wafers (called wafer stage) in semiconductor lithography,² or a galvo scanner for deflecting a laser beam in selective laser sintering (SLS) additive manufacturing.³ In all examples, the required position accuracies are quite high. In the HDD example, the scenario can be mimicked by an airplane flying at 5,000,000 mile/h above a 100,000-lane highway to follow the center of a lane whose width is only a fraction of an inch!⁴ Such ultra-high precision is achieved by careful consideration of various disciplines in mechanical engineering. From the viewpoint of system integration, the design elements can be classified to the following four categories:

Hardware and sensing components, such as fluid or air bearings for reduced friction, laser interferometers or high-precision encoders for accurate measurements, and piezo-electronic actuators for fine positioning;

Operation environment, including, for example, friction-isolation tables, clean room, and thermostatic chambers;

Task plans and arrangements, such as well-designed trajectories, repetitive tasks in a manufacturing process, and so on;

Servo control algorithms, such as adaptive control, repetitive control (RC), predictive control, and optimal control.

A well-designed precision system needs optimized considerations in all the above categories. Servo control, as the final step, is responsible for compensating as much as possible the imperfections from the previous three design processes. Such imperfections include (1) hardware imperfection, such as system resonances, delays in motor drivers, and delays in signal acquisition; (2) environmental disturbance, such as turbulent airflow, periodic disturbance from cooling fans in HDD, and structural coupling between mirrors in the galvo scanner; and (3) special errors due to the task nature, such as repeated trajectories in the wafer scanner.

Errors from the above sources present great challenges in reaching position accuracies at the micro/nano-scale. Fortunately, part of them—for example, imperfect motor rotation, repeated trajectories, and fan noises—are repeatable once the hardware and the trajectory are fixed. Other errors—such as environmental vibrations—may vary case by case; they are, however, at least structured and can be compensated by carefully designed servo controllers.

In this tutorial, the loop shaping of the feedback control in precision positioning systems is considered. In the presence of various aforementioned error sources, the feedback loop needs to have the flexibility of providing different closed-loop features for error reduction and guarantee the stability under different loop modifications. To satisfy this requirement, enhanced control at selected frequencies has been adopted by many researchers. Based on the location of servo enhancement, enhanced control can be categorized into three groups to deal with (1) independent disturbance frequencies by, for instance, peak filters,^5,6 adaptive feedforward cancellation,^7,8 disturbance observers,^9–11 and Youla–Kucera (YK) parameterization;^12–15 (2) a fundamental frequency and its integer multiples by means of RC and its variants;^16,17and (3) broadband frequencies through adaptive disturbance rejection,¹⁸ adaptive noise cancellation,¹⁹ and so on. This article discusses recent results in enhanced control to reach the desired servo goals.

A preliminary version of this study was presented in the 2013 American Control Conference.²⁰ This article extends the contents by proposing a new outer-loop inverse-based YK parameterization and presenting a detailed case study on the galvo scanner system. The case study is aimed to validate the proposed control algorithm and illustrate how to apply them in practice. The main contributions of the article are:

Building flexible feedback loop-shaping schemes to compensate various error sources in precision positioning;

Formulating a new outer-loop inverse-based YK parameterization;

Presenting a detailed case study on the galvo scanner system to validate the proposed control algorithm.

The remainder of the article will unfold and discuss several feedback control algorithms including add-on designs for precision positioning. Section “Feedback controller parameterization” reviews some fundamental feedback controller parameterizations. From there, section “Proposed controller factorization” discusses the proposed controller factorizations, that is, inverse-based YK parameterization and its variant. A case study is conducted on a galvo scanner control in section “A case study on a galvo scanner system,” and several comments on feedforward designs are provided in section “Remarks.” Section “Conclusion” concludes the article.

Feedback controller parameterization

This section briefly reviews several fundamental concepts in feedback control and introduces YK parameterization for flexible servo designs.

A precision positioning system is designed to have an accurate plant dynamics P which is usually linear and time-invariant. For a single-input single-output plant controlled by linear controllers (at least at the steady state), the position servo design can essentially be cast as a loop-shaping problem. Consider the block diagram of a standard feedback design in Figure 1. The closed-loop transfer function from the disturbance $d_{o}$ to the plant output y is defined as the sensitivity function $S \overset{Δ}{=} 1 / (1 + P C)$ . The complementary sensitivity function is given by $T \overset{Δ}{=} 1 - S = P C / (1 + P C)$ , that is, T is the transfer function from the reference r to y. The sensitivity function measures the closed-loop disturbance-attenuation property, while the complementary sensitivity function defines how the system responds to the reference input as well as the sensor noise. Consider a typical magnitude response of S in Figure 2. Below the bandwidth $Ω_{c} = 2 π ω_{c}$ , the magnitude of $S (j Ω)$ is less than 1, which infers the attenuation of $d_{o}$ in Figure 1. Due to the fundamental relationship $S + T = 1$ , wherever the magnitude of $S (j Ω)$ is small, $T (j Ω)$ is close to unity, which means that at frequencies where good disturbance rejection is achieved, improved reference tracking is also obtained.

Figure 1.

A standard closed-loop system under feedback control.²¹

Figure 2.

Magnitude response of the sensitivity function in section “Experimentation.”

The bandwidth in Figure 2 cannot be pushed to be arbitrarily large. From the practical perspective, a mechanical system cannot respond to arbitrarily fast control inputs due to hardware (such as motors and gears) limitations. It is common practice to keep the gain of the controller small at high frequencies. From the theoretical perspective, under mild conditions (for continuous-time systems, the relative degree of the loop transfer function $L = P C$ is no less than 2; for discrete-time systems, there is no such constraint), it is inevitable to have $| S (j Ω') | > 1$ at certain frequencies if $| S (j Ω) | < 1$ holds over some other frequency intervals. That is, when certain disturbance components are attenuated, some others are amplified. This is the “waterbed” effect which comes from Bode’s integral theorem, a fundamental result of linear control design.

Well formulated tools such as proportional–integral–derivative (PID), lead-lag, and $H_{\infty}$ controllers are available to achieve a loop shape that is similar to the one in Figure 2. We focus next on how to perform safe and flexible add-on modifications to such a standard baseline loop shape.

A foundational tool suitable for achieving the design goal is the YK parameterization,^14,15 also known as all-stabilizing controller parameterization. If a plant $P = N / D$ can be stabilized by a negative-feedback controller $C = X / Y$ with $(N, D)$ and $(X, Y)$ being coprime pairs over the set $S \overset{Δ}{=} {stable, proper, and rational transfer functions}$ , then any stabilizing feedback controller can be represented by means of the YK parameterization as

C_{a l l} = \frac{X + D Q}{Y - N Q} : Q \in S, Y (\infty) - N (\infty) Q (\infty) \neq 0 .

(1)

Here, a pair $(N, D)$ is called coprime over $S$ , if $N, D \in S$ and there exists $U, V \in S$ such that $U N + V D = 1$ . The inequality $Y (\infty) - N (\infty) Q (\infty) \neq 0$ makes the closed loop to be well-posed. This condition is usually easy to satisfy for practical problems. Consider, for example, the case where $P = z^{- 1}$ and $C = 0.8$ . We have $N = z^{- 1}$ , $D = 1$ , $X = 0.8$ , $Y = 1$ , and $Y (z = \infty) - N (z = \infty) Q (z = \infty) = 1$ .

The sensitivity function with controller (1) in the loop is

\tilde{S} = \frac{1}{1 + P C_{a l l}} = \frac{1}{1 + P C} [1 - \frac{N}{Y} Q],

(2)

which is affine in Q. Therefore, changing Q can directly shape $\tilde{S}$ , which is much more convenient compared to redesigning C in the denominator of $\tilde{S}$ .

For stable P and C, one can simply choose $N = P$ , $D = 1$ , $X = C$ , and $Y = 1$ in equation (1). These are valid coprime factorizations since we can pick, for example, $U = C / (1 + P C) \in S$ and $V = 1 / (1 + P C) \in S$ to make $U N + V D = 1$ . In this case, we obtain the following simplified and easier-to-use version of equations (1) and (2)

\tilde{C} = \frac{C + Q}{1 - P Q} : Q \in S .

(3)

\tilde{S} = \frac{1}{1 + P \tilde{C}} = \frac{1 - P Q}{1 + P C} .

(4)

The remaining design task about choosing Q in equation (4) depends on the plant P and the desired closed-loop response. For example, candidate Q-filter designs can be found using a linear combination of some basis transfer functions (e.g., $\sum_{i = 0}^{k_{Q}} θ_{i} z^{- i}$ in discrete-time schemes), and adaptive/ $H_{\infty}$ control can be applied to find the scaling coefficients for the combination.¹¹ Regardless of what tools are used, to achieve good disturbance attenuation, $\tilde{S}$ in (4) is required to be small, and hence PQ should be close to 1 at the desired frequencies. Consider a block diagram realization in Figure 3. In the inner loop marked by $★$ , Q should then approximate $P^{- 1}$ to make $P Q \approx 1$ .

Figure 3.

A forward-model YK parameterization.

Proposed controller factorization

Instead of a direct inversion of P by Q, this section discusses an alternative scheme that brings additional simplifications to equation (4). Although a general analysis can be made, we focus on precision servo systems and assume that the baseline controller C is stable.

Consider designing inverse-based YK parameterization in two steps: first, to perform a stable model inversion $P^{- 1}$ , and second, to use Q for controlling the amount of loop shaping and disturbance rejection. Of course, a perfect inversion of the plant dynamics is practically impossible, and a nominal version ${\hat{P}}^{- 1}$ has to be adopted. In addition, during discrete-time implementation, the sampled-data transfer function ${\hat{P}}^{- 1} (z)$ is possibly not causal/realizable since a practical plant P usually has delays (suppose m steps of delays) that become $z^{m}$ when inverted. For realizability, consider using instead $z^{- m} {\hat{P}}^{- 1} (z)$ . For the moment, ${\hat{P}}^{- 1} (z)$ is assumed to be stable. This is not difficult to achieve for precision positioning systems. A block diagram of the alternative scheme is shown in Figure 4. Compared to Figure 3, this discrete-time scheme is closer to actual implementation, and modifications have been made to reduce the dependence of the plant information in the $(★)$ loop in Figure 3.

Figure 4.

A discrete-time inverse-based YK parameterization.

The transfer function of the overall feedback controller in Figure 4 is

C_{a l l} (z) = \frac{C (z) + z^{- m} {\hat{P}}^{- 1} (z) Q (z)}{1 - z^{- m} Q (z)},

(5)

where high-gain control (increased control effort) is directly provided by $1 / (1 - z^{- m} Q (z))$ instead of $1 / (1 - P Q)$ in equation (3). This reduced dependence on P is also reflected in the frequency sensitivity function $\tilde{S} (e^{j ω}) = {\tilde{S} (z) |}_{z = e^{j ω}}$ . If $\hat{P} (e^{j ω}) = P (e^{j ω})$ , equation (5) gives

\begin{matrix} \tilde{S} (e^{j ω}) = \frac{1}{1 + P (e^{j ω}) C_{a l l} (e^{j ω})} \\ = \frac{1 - e^{- m j ω} Q (e^{j ω})}{1 + P (e^{j ω}) C (e^{j ω})} \\ = {S_{0} (z) (1 - z^{- m} Q (z)) |}_{z = e^{j ω}} . \end{matrix}

(6)

In some practices, when the plant and built-in controller (e.g., designed by manufacturers in the factory) are lumped together, only signals v and y in Figure 5 are accessible and separating the plant model $P_{e}$ is infeasible. In this circumstance, an outer loop can be designed by adding one positive feedback loop and, at the same time, one negative feedback loop outside the original loop shaping, as shown in Figure 5. With $C = 1$ , the servo performance of the original feedback loop keeps unchanged. Take $L (z) \overset{Δ}{=} P_{e} C_{e}$ as the new plant. This augmented plant model $L (z)$ can be computed from $T_{e} (z)$ , the transfer function from v to y, by $L (z) = T_{e} (z) / (1 - T_{e} (z))$ . Then, the controller $C (= 1)$ can be extended to obtain the desired closed-loop response. In Figure 6, the inverse-based YK parameterization is added on C. With $C (e^{j ω}) = 1$ and $P (e^{j ω})$ replaced by $L (e^{j ω})$ in equations (5) and (6), the outer-loop inverse-based YK parameterization gives

C_{a l l} (z) = \frac{1 + z^{- m} {\hat{L}}^{- 1} (z) Q (z)}{1 - z^{- m} Q (z)} .

(7)

\tilde{S} (e^{j ω}) = {\frac{1}{1 + L (e^{j ω})} (1 - z^{- m} Q (z)) |}_{z = e^{j ω}} .

(8)

Figure 5.

An outer-loop block diagram.

Figure 6.

An outer-loop inverse-based YK parameterization.

Nominal stability of the closed-loop system can be guaranteed given proper coprime factorizations. Summarizing the requirements in each of the four introduced YK schemes yields Table 1.

Table 1.

Nominal stability conditions for the four add-on designs.

YK parameterization	Simplified YK parameterization	Inverse-based YK parameterization	Outer-loop YK parameterization
$Q \in S$	P, C, $Q \in S$	${\hat{P}}^{- 1}$ , C, $Q \in S$	${\hat{L}}^{- 1}$ , $Q \in S$

YK: Youla–Kucera.

When there is (stable and bounded) model uncertainty $Δ (z)$ such that $\hat{P} (z) = P (z) (1 + Δ (z))$ , standard robust stability analysis¹¹ gives that the closed-loop system of the add-on design is stable if and only if both of the following hold:

Nominal stability condition in Table 1 is satisfied, that is, the closed loop is stable when $Δ (z) = 0$ .

Robust stability requirement is met: for any $ω$ , $| Δ (e^{j ω}) \tilde{T} (e^{j ω}) | < 1$ , where $\tilde{T} (e^{j ω}) = 1 - \tilde{S} (e^{j ω})$ .

Remark 1

The above robust stability requirement is not difficult to satisfy. Take the inverse-based YK parameterization in equation (6) as an example, where $\tilde{T} (e^{j ω})$ is given by

\tilde{T} (e^{j ω}) = \frac{\hat{P} (e^{j ω}) C (e^{j ω}) + e^{- m j ω} Q (e^{j ω})}{1 + \hat{P} (e^{j ω}) C (e^{j ω})} .

(9)

If $Q (e^{j ω}) = 0$ , $\tilde{T} (e^{j ω})$ reduces to the original complementary sensitivity function $T_{0} (e^{j ω})$ without the add-on design. We thus have $| Δ (e^{j ω}) T_{0} (e^{j ω}) | < 1$ , which is the robust stability condition of the baseline feedback loop. If $e^{- m j ω} Q (e^{j ω}) = 1$ , $\tilde{T} (e^{j ω}) = 1$ . At these frequencies, $| Δ (e^{j ω}) | < 1$ is required, that is, the mismatch between $P (e^{j ω})$ and $\hat{P} (e^{j ω})$ has to be less than 100%.

In equations (6) and (8), the add-on design narrows down to the design of $1 - z^{- m} Q (z)$ , which is much simpler than $1 - P Q$ in equation (4) and $1 - N Q / Y$ in equation (2). Several general design guides for $Q (z)$ can now be made.

Low-frequency servo enhancement

An intuitive choice for $Q (z)$ is a low-pass filter. In this case, if the delay of the plant is not very large, then the frequency transfer function $1 - e^{- j ω m} Q (e^{j ω})$ would be small when $Q (e^{j ω})$ is approximately one at low frequencies and be close to unity when $| Q (e^{j ω}) | << 1$ at high frequencies. A direct result is that any bias disturbance can be rejected. This is because under the low-pass assumption, $1 - e^{- j m \times 0} Q (e^{j \times 0}) = 1 - Q (e^{j \times 0}) = 0$ at $ω = 0$ . Therefore, an integral action is built into the closed-loop controller.

Narrow-band disturbance rejection

Vibrations are frequency-dependent signals by nature. Since the closed-loop bandwidth cannot be arbitrarily increased, vibrations at frequencies above the servo bandwidth are fundamentally more difficult to handle. Actually, such band-limited disturbances, if strong enough, significantly limit the servo performance. To compensate for such vibrations, some special feedback adjustments are needed. Meantime, we commonly want to keep as much as possible the original loop shape because it is often achieved after careful baseline design. A candidate add-on design is to use the proposed inverse-based YK parameterization and design a notch shape for $1 - z^{- m} Q (z)$ such as the one shown in the bottom plot of Figure 7. To demonstrate the flexibility, five notches are introduced in the magnitude of $1 - z^{- m} Q (z)$ . The first three notches are very close to each other, and the other two are separated at higher frequencies. When $1 - e^{- m j ω} Q (e^{j ω})$ is close to unity, $\tilde{S} (e^{j ω}) = {S_{0} (z) (1 - z^{- m} Q (z)) |}_{z = e^{j ω}}$ is close to $S_{0} (e^{j ω})$ and hence the original loop shape is maintained. At frequencies where $1 - z^{- m} Q (z)$ has very small gains in Figure 7, the overall controller (5) then has high gain, $\tilde{S} (e^{j ω})$ is very small and hence vibrations can be attenuated.

Figure 7.

A Q design example for narrow-band disturbance rejection.

An example about disturbance attenuation in a HDD benchmark problem²² is presented in Figure 8. This benchmark has been used in a number of publications on information storage systems. The compensation scheme is implemented for rejecting two strong vibrations at around 1100 and 1500 Hz. Both vibrations occur at frequencies above the baseline servo bandwidth (1060 Hz) and cannot be attenuated in the standard feedback setting. After compensation, the two originally sharp spectral peaks are actually visually not detectable due to the deep notch shape of $1 - z^{- m} Q (z)$ in Figure 7.

Figure 8.

Frequency spectra of the position error signals in a simulated HDD benchmark: 1 TP (track pitch) = 254 nm in this example.

Take the example of $m = 1$ . The key concept of Q design is to have a notch shape of $1 - z^{- 1} Q (z)$ at certain disturbance frequency $ω_{0} = 2 π f_{0} T_{s}$ , where $T_{s}$ is the sampling time and $f_{0}$ is in Hz. Designing $1 - z^{- 1} Q (z) = (1 - 2 \cos ω_{0} z^{- 1} + z^{- 2}) / (1 - 2 α \cos ω_{0} z^{- 1} + α^{2} z^{- 2})$ and solving for $Q (z)$ gives

Q (z) = \frac{a (α - 1) + (α^{2} - 1) z^{- 1}}{1 + a α z^{- 1} + α^{2} z^{- 2}}, a = - 2 \cos (ω_{0}) .

(10)

Extending equation (10), if multiple notches are desired, with $1 - z^{- 1} Q (z) = A (z^{- 1}) / A (α z^{- 1})$ , $A (α z^{- 1}) = Π_{i = 1}^{n} (1 - 2 \cos (Ω_{i} T_{s}) α z^{- 1} + α^{2} z^{- 2})$ , and $A (z^{- 1}) = {A (α z^{- 1}) |}_{α = 1}$ , letting $Q (z) = B_{Q} (z^{- 1}) / A_{Q} (z^{- 1})$ gives the general solution

\begin{matrix} A_{Q} (z^{- 1}) = 1 + \sum_{i = 1}^{n - 1} a_{i} (α^{i} z^{- i} + α^{2 n - i} z^{- 2 n + i}) \\ + a_{n} α^{n} z^{- n} + α^{2 n} z^{- 2 n} . \\ B_{Q} (z^{- 1}) = \sum_{i = 1}^{2 n} (α^{i} - 1) a_{i} z^{- i + 1} \cdot a_{i} = a_{2 n - i} . \end{matrix}

(11)

Here, $α (< 1)$ controls the width of the attenuation regions and can be taken very close to 1 for narrow-band disturbance rejection. The coefficients $a_{i} (i = 1, 2, \dots, n)$ in equations (10) and (11) determine the notch frequencies of $Q (z)$ . In equation (11), $a_{i}$ and the desired notch frequency $Ω_{i}$ in rad/s are connected by the mapping $A_{Q} (z^{- 1}) = A (α z^{- 1})$ .

When $m > 1$ , several additional steps are needed since $1 - z^{- m} Q (z) = A (z^{- 1}) / A (α z^{- 1})$ would not have a realizable solution. An approach that uses Diophantine identity to address this issue is provided in Chen and Tomizuka.⁹ Specifically, when $m = 2$ , the expression of $Q (z)$ is

Q (z) = \frac{[α^{2} - 1 - a^{2} (α - 1)] - a (α - 1) z^{- 1}}{1 + a α z^{- 1} + α^{2} z^{- 2}},

(12)

where $α$ and a have the same meanings as those in equation (10).

Repetitive control

An example application of RC is for addressing the problem involved in wafer-scanning process, one key step for circuit fabrication in the semiconductor industry. To print the circuit, the wafer is exposed to patterned ultraviolet lights that come through a mask carried by a reticle stage. The wafer stage and the reticle stage move the wafer and the mask in a synchronized manner. Due to the limited size of the lens, only a small part of the wafer is exposed at each scan, and the wafer is moved from one field to another between the scans. The scanning process is repeated until all required areas on the wafer have been exposed under the light. An intuition about RC is that if the same type of disturbance occurs after a fixed period of time, that is, $d (k) = d (k - N_{d})$ where $N_{d}$ is the period of the disturbance, then at the next occurrence of $d (k)$ , we can learn and reduce the resulting error, no matter how it behaves within one period of time. From the spectral perspective, the magnitude response of $1 - z^{- m} Q (z)$ in Figure 9 provides a typical loop shape for RC that $| 1 - e^{- j m ω} Q (e^{j ω}) |$ gives low gains to $| \tilde{S} (e^{j ω}) |$ at the fundamental frequency and its integer multiples. Meanwhile, at other frequencies, $| 1 - e^{- j m ω} Q (e^{j ω}) |$ is approximately unity, yielding no change to $| \tilde{S} (e^{j ω}) |$ . The Q-filter to achieve Figure 9 is

Q (z) = \frac{(1 - α^{N_{d}}) z^{- (N_{d} - m - n_{q})}}{1 - α^{N_{d}} z^{- N_{d}}} z^{- n_{q}} q (z, z^{- 1}),

(13)

where $N_{d}$ and m are as defined previously, $q (z, z^{- 1})$ is a zero-phase low-pass filter with order $n_{q}$ , and $α$ controls the “waterbed” effect by the following theorem.

Figure 9.

A Q design example for repetitive control.

Theorem 1

When $P (e^{j ω}) = \hat{P} (e^{j ω})$ , the worst-case amplification of the non-repetitive disturbance is $(2 / (1 + α^{N_{d}}) - 1) \times 100 %$ for RC using equation (13).^16,23 In conventional RC, the amplification is at most 100%, which corresponds to the case with $α = 0$ in equation (13).

Figure 10 shows the experimental result of the algorithm to reduce the errors represented by the dashed line. With the repetitive learning control, the errors are observed to have reduced by two orders of magnitude compared to the case without compensation.

Figure 10.

Tracking errors with the Q-filter configuration in equation (13).

General band-limited vibration compensation

The loop shaping in section “Narrow-band disturbance rejection” is for narrow-band disturbance rejection. When excitation sources are rich in frequency, a wider attenuation bandwidth of $1 - z^{- m} Q (z)$ is needed and can be achieved by changing the design parameter $α$ in equations (10) and (11), which is related to the width of the $- 3 dB$ passband for $1 - z^{- m} Q (z)$ by $B W \approx (1 - α^{2}) / ((α^{2} + 1) (π T_{s}))$ (Hz).²⁴ As the disturbance attenuation is pushed further, the trade-off amplification due to the waterbed effect also increases. Additional design considerations are therefore necessary for a safe implementation. Figure 11 shows the loop-shaping result on a HDD benchmark example. No large sensitivity amplifications are observed due to: (1) a gain scheduling in the Q-filter with $Q_{implement} = k_{Q} Q (z)$ ( $k_{Q} \leq 1$ ) and (2) the use of a damped notch filter $F_{n f} (z) = A (β z^{- 1}) / A (α z^{- 1}) (α < β < 1)$ in the design of $Q (z)$ . Auxiliary zeros and convex optimization can be applied for specific magnitude constraints on $Q (z)$ .²⁵

Figure 11.

The control of attenuation efforts by means of gain scheduling on Q.

A case study on a galvo scanner system

In this section, we provide the design and analysis steps for implementing the (outer-loop) inverse-based YK parameterization on a galvo scanner platform to verify the effectiveness of the proposed servo design.

As an important subcategory of additive manufacturing, SLS can directly fabricate metallic parts from digital models. In SLS, laser beams are controlled to melt the powder materials following pre-designed trajectories. Rather than moving the laser source, SLS efficiently applies a galvo scanner platform to reflect the laser beams. Besides SLS, the galvo scanner platform is also widely used in other laser-related applications such as laser scanning, laser engraving, and laser welding.

A typical galvo scanner platform is composed of two sets of mirrors, galvanometers, and control systems, as shown in Figure 12. The galvanometer consists of motors to rotate the mirrors and encoders to feedback the mirror position information. The mirror assembly is attached to the end of the motors in a coaxial manner and deflects the laser beam over the angular range of the motor shaft, as shown in Figure 13. $O_{1}$ and $O_{2}$ indicate the centers of the two mirrors, and the corresponding rotation angles are denoted as $γ$ and $δ$ , respectively. $L_{1}$ is the distance between the two mirror centers, and $L_{2}$ represents the distance between mirror $O_{2}$ and the image field O.

Figure 12.

Schematic of hardware platform.

Figure 13.

Schematic diagram and electrical model of galvo scanner.

Plant identification, PID tuning, and discretization

In the electrical model (Figure 13) of a generalized galvo scanner platform, the open-loop transfer function^26,27 relating the input drive current $I (s)$ to the output position $Θ (s)$ is

\frac{Θ (s)}{I (s)} = \frac{K}{K_{s} + D s + J s^{2}},

(14)

where K is the torque constant, $K_{s}$ is the spring constant of the galvo, D is the damping constant, and J is the rotor and mirror inertia. In addition, we have

{\begin{matrix} L \frac{d I}{d t} + R I + E_{b} = V \\ E_{b} = K_{b} \frac{d Θ}{d t} \end{matrix},

(15)

where $E_{b}$ is the back electromotive force (EMF), $K_{b}$ is the back EMF constant, L is the lumped inductance, and R is the lumped resistance. Combining equation (14) with equation (15), one can obtain the transfer function from the input voltage V to the output position $Θ$

\frac{Θ}{V} = \frac{K}{(L J) s^{3} + (R J + D L) s^{2} + (R J + K_{s} L + K_{b} K) s + K_{s} R} .

(16)

With $K_{s} = 0.02 Nm \cdot ra d^{- 1}$ , $D = 1.74 \times 10^{- 6} Ns \cdot m$ , $J = 0.028 kg \cdot m^{2}$ , $K_{b} = 0.005 Nm \cdot A$ , $R = 2.3 ohm$ , $L = 0.001 H$ , and $K = 0.005 Nm \cdot A^{- 1}$ , the plant model in equation (16) reduces to a second-order model²⁶

P (s) = \frac{0.005}{0.0644 s^{2} + 0.0644 s + 0.0460},

(17)

where the term $s^{3}$ in equation (16) is ignored as the coefficient LJ is several orders of magnitude smaller than the other coefficients. To build a baseline feedback loop, a PID controller, based on the Ziegler–Nichols tuning, is designed as²⁸

C (s) = 17760 (1 + \frac{1}{0.0650 s} + 0.0163 s) .

(18)

To implement the discrete-time YK scheme, the continuous-time plant and controller are, respectively, discretized by means of the zero-order-hold (ZOH) discretization and the bilinear transformation

C (z) |_{s = \frac{2}{T_{s}} \frac{1 - z^{- 1}}{1 + z^{- 1}}} = 10^{4} \frac{4.243 z^{2} - 7.767 z + 3.56}{z^{2} - 1.667 z + 0.6667} .

(19)

P_{d} (z) = (1 - z^{- 1}) Z {L^{- 1} {[\frac{P (s)}{s}] |}_{t = k T_{s}}} .

(20)

The transfer function of the discrete-time plant with the sampling time $T_{s} = 0.004 s$ is

P_{d} (z) = 10^{- 7} \frac{6.203 z + 6.195}{z^{2} - 1.996 z + 0.996} .

(21)

YK parameterization implementation

The inverse-based YK parameterization (Figure 4 and equation (5)) is applied to reject disturbances in the galvo scanner platform, such as channel crosstalk, atmospheric turbulence, and environmental vibration. Such disturbances are narrow-band by nature. For demonstration, a proof-of-concept single-frequency disturbance $d (k) = A \sin (2 π f_{0} T_{s} k)$ is considered.

$P^{- 1} (z)$ and $C (z)$ are stable, which means that the prerequisites to implement the inverse-based YK scheme are satisfied. The relative degree m of $P (z)$ in equation (21) is 1. Hence, the Q design in equation (10) is applied. Substituting $f_{0} = 12.5 Hz$ , $A = 1 V$ , $T_{s} = 0.004 s$ , and $α = 0.999$ into equation (10) gives

Q (z) = \frac{0.001902 - 0.001999 z^{- 1}}{1 - 1.9 z^{- 1} + 0.998 z^{- 2}} .

(22)

As shown in Figure 14, by applying high-gain control of equation (5), a notch shape is introduced for $| 1 - z^{- 1} Q (z) |$ at $12.5 Hz$ , where $| C_{a l l} (z) | \to \infty$ and $| \tilde{S} (z) |$ in equation (6) is thus very small, yielding the time-domain output $y (k T_{s}) = 0$ at $12.5 Hz$ . At the other frequencies, $| 1 - z^{- 1} Q (z) |$ is close to unity, that is, $| \tilde{S} (z) |$ approximates $| S_{0} (z) |$ and hence the original loop shape is maintained.

Figure 14.

Magnitude responses of $Q (z)$ , $1 - z^{- m} Q (z)$ , and $C_{a l l} (z)$ .

In regulation control, the $Q (z)$ output w acts as a window to check if the Q-filter works well. A good Q design yields w to cancel the disturbance, that is, $w (k) \approx - d (k)$ . Hence, the inverse-based YK structure acts as an input-disturbance observer. Discussions on this point are provided in Chen and Tomizuka.¹⁶

Figure 15 shows the simulated outputs and control efforts with the Q-filter and the baseline PID controller. The results verify that compared with the baseline controller, the inverse-based YK scheme can fully reject the disturbance at $12.5 Hz$ with increased control efforts.

Figure 15.

Outputs and control efforts of the baseline controller and Q-filter.

Experimentation

Experiments are conducted on the galvo scanner platform³ in Figure 12. During implementation, nonlinearity effects such as slew rate limit and input saturation are insignificant compared with the external disturbance and are thus omitted. A dSPACE DS1104 board connects designs in MATLAB with the servo drivers. The control signal is limited to ±10 V. The system is subjected to broadband random disturbances at a magnitude of about 10 mV. The sampling time $T_{s}$ is $0.025 ms$ , that is, the Nyquist frequency is $20 kHz$ .

A pseudorandom binary sequence signal is used as an input to estimate the plant model. Magnitude responses of the measured and identified plants are shown in Figure 16. The identified plant model is expressed as

P_{d} (z) = \frac{- 0.001945 z + 0.03172}{z^{2} - 1.955 z + 0.9703} .

(23)

Figure 16.

Magnitude response of the measured and identified plant.

One can observe that equations (21) and (23) share the same structure and similar poles with some gain normalization, which attests to the validity of the identified plant model.

Next, we take the galvo scanner and the servo drivers, namely, the lumped plant and controller, as the new augmented plant $L (z)$ to apply the outer-loop inverse-based YK parameterization in Figure 6. The bandwidth $ω_{c}$ of the baseline feedback loop is $1526 Hz$ , as shown in Figure 2. The identified $\hat{L} (z)$ is

\hat{L} (z) = \frac{0.02816 z^{2} + 0.1504 z + 0.1146}{z^{4} - 1.319 z^{3} + 0.929 z^{2} - 0.6073 z - 0.0035} .

(24)

Moving the unstable zero of $\hat{L} (z)$ at $- 4.419$ to $- 0.6$ inside the unit circle and keeping the other zeros and poles unchanged generates a new plant ${\hat{L}}_{s} (z)$ with a stable inverse ${\hat{L}}_{s}^{- 1} (z)$

{\hat{L}}_{s} (z) = \frac{z^{2} + 1.521 z + 0.5526}{10.49 z^{4} - 13.83 z^{3} + 9.037 z - 6.312 z + 0.6124} .

(25)

The magnitude responses of the measure system, the identified $\hat{L} (z)$ , and ${\hat{L}}_{s} (z)$ are very close to each other, as illustrated in Figure 17. The Q-filter in Figure 6 is then designed based on the inverse-stable plant ${\hat{L}}_{s} (z)$ . The relative degree m of ${\hat{L}}_{s} (z)$ is 2. A disturbance with $f_{0} = 5 kHz$ and $A = 0.1 V$ is introduced to the system. $f_{0} (5 kHz)$ is beyond the bandwidth $ω_{c} (1526 Hz)$ and cannot be attenuated by the baseline feedback loop. Also, the Nyquist frequency of $20 kHz$ is large enough to obtain the frequency response of the system at 5 kHz. Substituting $α = 0.9$ into equation (12) gives

Q (z) = \frac{0.01 - 0.1414 z^{- 1}}{1 - 1.273 z^{- 1} + 0.81 z^{- 2}} .

(26)

Figure 17.

Magnitude responses of the measured system, and identified $\hat{L} (z)$ , ${\hat{L}}_{s} (z)$ .

As shown in Figure 18, the control effort with the Q-filter is larger than that with the baseline controller. The disturbance is attenuated remarkably with the outer-loop inverse-based YK parameterization turned on. The performance gain is also clear in the frequency domain: in Figure 19, the spectral peak at 5 kHz is completely removed without visible amplification of disturbances at the other frequencies.

Figure 18.

System outputs and control efforts with/without YK parameterization.

Figure 19.

Spectrum of output $y (k)$ .

Remarks

The plant model and its inverse

The (inverse) model information is essential in the discussed control schemes since it not only provides convenience for feedback design but also is beneficial for tasks such as fault detection and disturbance isolation. Such a model is usually available in industries. By the physical construction, the input to actuators in a precision positioning system is usually a voltage/current signal that is approximately linear with respect to motor torque, and the measurement is commonly an angular or linear position. The inverse system dynamics thus usually has a double-differentiator type of frequency response due to Newton’s law. It is thus not difficult to obtain a stable nominal ${\hat{P}}^{- 1} (z)$ that accurately presents the low-frequency dynamics. Direct differentiation at high frequencies is not practical due to high-frequency resonances. High-frequency unstable zeros, if any, can be addressed by, for instance, shifting them inside the unit circle.

Robust stability

Strictly speaking, after $\hat{P}$ is applied in the controller design, we are implementing a robust version of YK parameterization. Notice that in the examples in Figures 7 and 9, the Q-filters are designed such that their magnitudes are small at high frequencies, namely, the loop behavior at ultra-high frequencies does not change. This is important for implementation, since it is practically impossible to have an exact mathematical model for mechanical systems. The robust stability analysis can give us upper bounds of the plant uncertainties for the closed-loop system to maintain stable. Intuitively, flexible loop shaping can be readily achieved at frequency regions where the model ${\hat{P}}^{- 1} (z)$ accurately reflects the actual system dynamics. At high frequencies where the plant behavior itself cannot be well predicted, the magnitude of $Q (z)$ (and hence the add-on control efforts) is recommended to be kept small for robust stability.

Feedforward control

The feedback perspective for precision servo has been discussed. Due to space limit, detailed feedforward discussions are not included in this tutorial article. Necessity of feedforward control comes from the simple fact that feedback designs have bandwidth limitations. In the feedforward class of control algorithms, model-based design is also of essential importance. Inverse complementary sensitivity and inverse plant dynamics are two common approaches for feedforward design. When the process is repetitive, iterative learning control (ILC) is another powerful tool for error correction in the iteration domain.

Adaptive configuration

When the disturbance spectra is not known (but with known structure), adaptive control can be applied to update the parameters of $Q (z)$ online. Investigation for the case of narrow-band disturbance rejection is provided in Chen and Tomizuka.^9,29

Reference of YK parameterization

For additional materials about YK parameterization, readers can refer to the literature.^12,21,30 In the discrete-time case, the simplest example of Q design is using $Q (z) = θ_{0} + θ_{1} z^{- 1} + θ_{2} z^{- 2} + \dots + θ_{k_{Q}} z^{- k_{Q}}$ , namely, a finite impulse response (FIR) filter and then applying parameter adaptation algorithms to find a set of $θ_{i}$ that minimizes the feedback error. This idea is also used in adaptive inverse control.³¹

Conclusion

In this article, the control methodologies for precision positioning systems have been studied. The central concept is that flexible loop shaping is a convenient tool for addressing various servo problems. Multiple simulation and experimental results on actual engineering problems are used to validate the presented designs. Besides the examples used in this article, the control structure has also been tested under other systems, such as electrical power steering in automotive vehicles²⁴ and active suspension systems.^11,12

Footnotes

Appendix 1

Handling Editor: Yong Chen

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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A tutorial on loop-shaping control methodologies for precision positioning systems

Abstract

Keywords

Introduction

Feedback controller parameterization

Proposed controller factorization

Remark 1

Low-frequency servo enhancement

Narrow-band disturbance rejection

Repetitive control

Theorem 1

General band-limited vibration compensation

A case study on a galvo scanner system

Plant identification, PID tuning, and discretization

YK parameterization implementation

Experimentation

Remarks

The plant model and its inverse

Robust stability

Feedforward control

Adaptive configuration

Reference of YK parameterization

Conclusion

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

References