OBSERVER-BASED CONTROL OF A DUAL-STAGE PIEZOELECTRIC SCANNER

Abstract

Despite its proven success in a wide variety of applications, the atomic force microscope (AFM) remains limited by its slow imaging rate. One approach to overcome this challenge is to rely on algorithmic approaches that reduce the imaging time not by scanning faster but by scanning less. Such schemes are particularly useful on older instruments as they can provide significant gains despite the existing (slow) hardware. At the same time, algorithms for sub-sampling can yield even greater improvements in imaging rate when combined with advanced scanners that can be retrofitted into the system. In this work, we focus on the use of a dual-stage piezoelectric scanner coupled with a particular scanning algorithm known as Local Circular Scan (LCS). LCS drives the tip of the AFM along a circular trajectory while using feedback to center that circle on a sample edge and to move the circle along the feature, thus reducing imaging time by concentrating the samples to the region of interest. Dual-stage systems are well-suited to LCS as the algorithm is naturally described in terms of a high-frequency, short range path (the scanning circle) and a slower, long range path (the track along the sample). However, control of the scanner is not straightforward as the system is multi-input, single-output. Here we establish controllability and observability of the scanning stage, allowing us to develop individual controllers for the long-range and short-range actuators through the principle of separation. We then use an internal model controller for the short range actuator to track a sinusoidal input (to generate the circular motion) and a state-space set-point tracking controller for the long range actuator. The results are demonstrated through simulation.

1. INTRODUCTION

The Atomic Force Microscope (AFM) is a powerful instrument that allows the user to interrogate with nanometer-scale precision a large variety of surface properties, including topology, material moduli, and surface potential [1–4]. In addition to its continued success in imaging structural properties of samples, it is increasingly being used to study dynamics at the nanometer scale [5–7]. However, despite recent advances in imaging speed, the slow frame rate of AFM remains a limitation. While a handful of current instruments can image on the order of 1–10 frames per second, there is a large installed base of older instruments as well as lower-cost newer instruments with much slower imaging speeds.

Despite a limited frame rate, AFM continues to find application in studying dynamic events, particularly in the biological sciences [6, 8, 9]. As a result, there is a long history of research and development in high-speed AFM (HS-AFM). In general, there are three main approaches for HS-AFM: improving system dynamics, using advanced controller designs, and employing alternative scan paths. Since the AFM is a mechanical microscope, modifications to the physical components can yield substantial increases in operating speed. For instance, the development of smaller cantilevers and faster actuators allows for more rapid imaging [10–12]. Similarly, advanced controllers, such as adaptive proportional-integral-derivative (PID), feedforward compensators, and multi-notch filters, allow for closed-loop bandwidths that approach or even exceed the resonant peaks of the system [13–15]. Finally, because AFMs primarily utilize piezoelectric positioning stages that are highly resonant by nature, consideration of the harmonic content of waveforms that drive scan trajectories can also lead to increased scanning rates [16, 17], though the effectiveness of these methods relies on fast vertical positioning [18].

A complementary class of approaches seeks to improve the imaging rate by reducing the amount of sampling needed. This has been done through sub-sampling methods, combined with reconstruction algorithms [19–23] as well as through schemes that use feedback to focus the measurements to an area of interest [24–26]. In this work, we focus on one such algorithm, termed Local Circular Scan (LCS) [27,28]. LCS is a scheme designed for imaging edges (such as cell boundaries) or string-like samples (such as biopolymers). During imaging, the tip is driven along a circular path and the measurements are used in real time to center the circle on the sample edge and to track that edge as it evolves in space. For this class of samples, the reduction in sampling yields an order-of-magnitude or better improvement in imaging rate without an increase in scanning speed.

To achieve even faster imaging with AFM, algorithms such as LCS can be combined with novel actuators and controllers. In particular, dual-stage actuators are a natural fit to the scanning path of LCS. Dual-stage piezoelectric actuators consist of a low-speed, long-range actuator (LRA) coupled in series with a high-speed, short-range actuator (SRA) [29]. This allows them to achieve both long-range and high-speed, high-resolution motion [30], making them appealing in, for example, disk drive systems [31]. The LCS scan pattern can be broken down to a repetitive motion along a small circular track, suitable for the SRA, and a longer path defined by the sample being tracked, sutiable for the LRA. Ideally, then, the actuator would be controlled by two independent control loops. However, dual-stage systems are multi-input, single-output (MISO) systems as only the final end position of the serial chain of actuators is measured.

Our approach, illustrated in Fig. 1, is to utilize the principle of separation. In this figure, the combined output Y_out goes through an observer to produce the estimated states. The blocks D_LRA and D_SRA denote the (possibly dynamic) compensators for each subsystem and take in the estimated states and the desired reference signals, Y_dLRA and Y_dSRA, respectively, to produce the control signals. For this scheme to work, we must first establish controllability and observability of the MISO system. With this in hand, the observer and the two controllers can be designed independently of each other.

Figure 1:

Block diagram of the basic control scheme. The dual-stage actuator (in the blue box) produces a single scalar output. An observer is used to produce the individual states of the LRA and SRA. These state estimates are then used in independently-designed controllers for each axis. Under this scheme, the LRA and SRA can be used to achieve different goals.

Based on the LCS imaging approach, the controller for the SRA should allow it to track a sinusoidal trajectory so that, when combined with a second dual-stage actuator in an orthogonal axis, circular motion of the tip is achieved. With this in mind, we will utilize an internal model controller for the SRA. The LRA can then be dedicated to tracking the path of the sample being imaged. Of course, this path is not known a priori but is generated in real time by the LCS algorithm based on the measured data. Under the assumption that the rate of change of the path of the sample is slow relative to the closed-loop dynamics (which can be enforced by the user by selecting the speed of motion along the sample), we will use simple state-space set-point regulation for the LRA.

In this paper, we first review the LCS algorithm and describe dual-stage actuators in Sec. 2. In Sec. 3, we consider a generic MIMO system defined by a collection of MIMO systems whose outputs are summed together, establishing results on both observability and controllability for the joint system. With minimality established, we focus in Sec. 4 on the dual stage setting, developing an observer and then applying the principle of separation to develop separate controllers for the LRA and SRA sub-systems. Finally, in Sec. 5, we demonstrate the approach through simulation before providing concluding remarks in Sec. 6.

2. LCS algorithm and dual-stage scanners

2.1. LCS Overview

The fundamental idea of LCS is to drive the tip along a circular trajectory, using the detected edges of the feature of interest of the sample to center the circle on that feature while also moving along the trajectory defined by the sample. While limited to samples that are string-like in nature, such as grating edges, cell boundaries, or biopolymers, the approach can yield an order of magnitude or better reduction in imaging time without any increase in tip speed.

Consider a moving, local reference frame attached to the center of a circle and a thin sample such as a biopolymer. If the sample intersects with the circle, there will be four detection points, two where the tip steps up onto the sample and two where it steps down (see the top image of Fig. 2 for reference). A simple controller then moves the center of the scanning circle so that the circle is centered on the sample and is moved along the path defined by the sample. For samples whose widths are larger than the scanning circle, such as the edges of cells, there will be only two detections but the algorithm is otherwise the same. A typical trajectory is illustrated in the simulation shown in the top image of Fig. 2 with different choices of the circle radius and step size along the sample to illustrate the flexiblity of the scanning scheme. LCS has been demonstrated experimentally on both gratings of a variety heights and biopolymer samples, typically yielding an order of magnitude or better reduction in imaging time [28]. Typical images from LCS scans of two grating samples are shown in the bottom images of Fig. 2. Note that the checkerboard pattern indicates there is no information as the tip of the instrument never scanned those regions.

Figure 2:

LCS algorithm. (top) Illustration of typical scan path and effect of user parameter choice. Blue, sinusoidal curve is the simulated sample and black is the resulting tip trajectory. (bottom) Experimental results. LCS was used to track the edges of a circular and square grating sample with 20 nm high features. Images reproduced from [27] and [32].

2.2. Dual-stage systems

Dual-stage actuators, diagrammed in Fig. 3, consist of a low-speed LRA serially connected with a high-speed SRA [30]. They are often modeled as two tandem spring-mass-damper systems. Existing results typically filter the reference and output signals based on frequency and, possibly, spatial content [29,30]. When applied to LCS, however, it is natural to use the SRA to follow the circular portion of the trajectory and the LRA to track the underlying sample. As dual-stage actuators are MISO systems, providing only a single measurement signal of the overall position of the tandem actuators, how to achieve independent control of the LRA and SRA is not immediately obvious. This is possible under a state-space approach if the system is minimal, that is, if it is controllable and observable. Therefore, in the next section we turn our attention to these properties.

Figure 3:

Illustration of dual-stage actuator.

3. Observer-Based Control of Dual-Stage

While our interest is primarily in a dual-stage system, in this section we begin by considering the more general case of N–stages, with each stage a general multi-input, multi-output system, before focusing on the dual-stage setting. We establish both controllability and observability when the systems are connected by adding their outputs as illustrated in Fig. 4.

Figure 4:

Block diagram of a generic N-stage system.

The results below are developed in large part from the work of CT Chen in [33]. The fundamental idea of his approach is to analyze the joint system using a canonical form. We first introduce the necessary notation.

Consider, then, a collection of independent MIMO systems, given by

$$S_i:{\dot{x}}_i=A_i{x}_i+B_i{u}_i,$$ (1a)

$$y_i=C_i{x}_i,$$ (1b)

where $x_i\in {ℝ}^{n_i}$, $u_i\in {ℝ}^{p_i}$, and $y_i\in {ℝ}^q$. Because the output of each system has the same size, we can connect them as shown in Fig. 4 to produce the composite system

$$S_c:\dot{x}=A_{c}x+B_{c}u,$$ (2a)

$$y_c=C_{c}x,$$ (2b)

where the system matrices for S_c are given by

$$A_c=\left[\begin{array}{llll}{A}_1\hfill & \hfill & \hfill & \hfill \\ \hfill & A_2\hfill & \hfill & \hfill \\ \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & A_N\hfill \end{array}\right],B_c=\left[\begin{array}{llll}{B}_1\hfill & \hfill & \hfill & \hfill \\ \hfill & B_2\hfill & \hfill & \hfill \\ \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & B_N\hfill \end{array}\right],$$ (3a)

$$C_c=\left[\begin{array}{cccc}{C}_1& C_2& \cdots & C_N\end{array}\right].$$ (3b)

In [33], the system properties of (2) were analyzed by first expressing the system in a basis where A_c is in Jordan canonical form. To that end, let λ₁, λ₂, …, λ_m denote the m distinct eigenvalues of A_c. The system matrices of the combined system then have the form

$$A_c=\left[\begin{array}{llll}{A}_{c_1}\hfill & \hfill & \hfill & \hfill \\ \hfill & A_{c_2}\hfill & \hfill & \hfill \\ \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & A_{c_m}\hfill \end{array}\right],B_c=\left[\begin{array}{c}{B}_{c_1}\\ ⋮\\ B_{c_m}\end{array}\right],$$ (4a)

$$C_c=\left[\begin{array}{ccc}{C}_{c_1}& \cdots & C_{c_m}\end{array}\right],$$ (4b)

where A_ci denotes the set of Jordan blocks associated with the eigenvalue λ_i. Letting r(i) denote the number of Jordan blocks associated to eigenvalue λ_i, we have

$$A_{c_i}=\left[\begin{array}{llll}{A}_{i1}\hfill & \hfill & \hfill & \hfill \\ \hfill & A_{i2}\hfill & \hfill & \hfill \\ \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & \hfill & A_{ir\left(i\right)}\hfill \end{array}\right],B_{c_i}=\left[\begin{array}{c}{B}_{i1}\\ ⋮\\ B_{ir(i)}\end{array}\right],$$ (5a)

$$C_{c_i}=\left[\begin{array}{ccc}{C}_{i1}& \cdots & C_{ir\left(i\right)}\end{array}\right].$$ (5b)

where the first sub-index refers to a specific eigenvalue and the second to a specific Jordan block inside that A_ci. In the sequel, c_kij will denote the k^th column in the output matrix of the j^th Jordan block of the i^th distinct eigenvalue (in accordance with the notation set out in [33]). Specifically, c_1ij and c_lij will denote the first and last columns in that block. Similarly, b_1ij and b_lij will denote the first and last rows in the input matrix of the j^th Jordan block of the i^th distinct eigenvalue. Note that the while the value of l depends on the particular Jordan block so that l = l(i, j), we omit explicitly noting this to avoid (further) cluttering the notation. The size of the j^th Jordan block of the i^th distinct eigenvalue is n_ij and the total size is $n={\sum }_{i=1}^N{\sum }_{j=1}^{r\left(i\right)}{n}_{ij}$.

Observability of the general combined system (2) is established as follows. Note that this is independent of any particular interconnection of subsystems and is based on the Jordan form description in (4). It is thus important to keep in mind that subscripts here refer not to subsystems but rather to eigenvalues and Jordan blocks of the combined description. The proof of the following theorem can be found in [33].

Theorem 1.

A system S is observable if and only if, for i = 1, 2, ⋯, m, the set of q-dimensional column vectors

$$c_{1i1},c_{1i2},\cdots ,c_{1ir\left(i\right)}$$

is a linearly independent set.

For ease of reference, we collect the set of column vectors in Thm. 1 into a matrix $\widehat{C}$.

Remark 1.

One important special case (for our purposes) is when q = 1, that is when the subsystems and combined system have scalar outputs. In that case $\widehat{C}$ consists of a single row. The theorem above can then only hold if $\widehat{C}$ is a scalar.

Let us now turn our attention to the composite system (2). Our goal is to establish results based on properties of the original subsystems rather than on the combined Jordan form description. Controllability is straightforward to establish from the Kalman rank condition.

Theorem 2.

The composite system S_cin (2) is controllable if and only if the individual subsystems S_i, i = 1, 2, …, N, are controllable.

Proof.

Given the structure of the composite system matrices in (3), we have

$$\begin{gathered} \left[\begin{array}{llllll}{B}_c\hfill & A_c{B}_c\hfill & \cdots \hfill & A_c^n\hfill & -\hfill & 1B_c\hfill \end{array}\right] \\ =\left[\begin{array}{llllllllll}{B}_1\hfill & \hfill & \hfill & A_1{B}_1\hfill & \hfill & \hfill & \hfill & A_1^{n-1}{B}_1\hfill & \hfill & \hfill \\ \hfill & \ddots \hfill & \hfill & \hfill & \ddots \hfill & \hfill & \ddots \hfill & \hfill & \ddots \hfill & \hfill \\ \hfill & \hfill & B_N\hfill & \hfill & \hfill & A_N{B}_N\hfill & \hfill & \hfill & \hfill & A_N^{n-1}{B}_N\hfill \end{array}\right]. \end{gathered}$$

Assume first that the individual systems are controllable. Clearly, in the matrix above the columns corresponding to each subsystem are independent of the others. Controllability of the individual systems then immediately implies this matrix is full rank and thus the combined systems is controllable.

Next, assume that the combined system is controllable. The matrix above is full rank. Again, due to the independence of the columns corresponding to each subsystem, it immediately follows that the subsystems are controllable.□

We now turn to observability. In the sequel, we use Λ_i to denote the set of eigenvalues for subsystem i. For simplicity, we focus on the single-output setting that is relevant to our particular application.

Theorem 3.

Consider a composite system S_cof the form (2) with q = 1. Suppose the subsystems are observable. Then S_c is observable if and only if

$${\Lambda }_1\cap {\Lambda }_2\cap \cdots \cap \cdots {\Lambda }_N=\varnothing .$$

Proof.

We begin with necessity and proceed by contradiction. Assume, then, that S_c is observable and that at least two of the sub-systems share an eigenvalue. Then, in the Jordan form of the combined system there are at least two Jordan blocks corresponding to that eigenvalue (one coming from each subsystem). Without loss of generality, let this be the first eigenvalue in A_c. Then the collection c₁₁₁,·⋯, c_11r(i) has at least two elements in it. But then, as noted in Remark 1, Thm. 1 cannot hold and thus the combined system cannot be observable, contradicting the original assumption. Therefore the sub-systems cannot share any eigenvalues.

For sufficiency, assume the intersection of the eigenvalue sets is empty. Then the Jordan form of the combined system is simply the combination of the Jordan forms of the sub-systems and checking for observability is equivalent to checking the condition of Thm. 1 for each sub-system independently. Because the sub-systems are each observable, S_c is as well.□

To highlight these results, consider a simple example system with

$$\begin{gathered} A_1=\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right],A_2=\left[\begin{array}{cc}1& 1\\ 0& 2\end{array}\right],B_1=\left[\begin{array}{l}0\hfill \\ 1\hfill \end{array}\right],B_2=\left[\begin{array}{l}1\hfill \\ 1\hfill \end{array}\right], \\ {C}_1=C_2=\left[\begin{array}{cc}1& 0\end{array}\right]. \end{gathered}$$

Clearly these two systems share are individually controllable and observable. By Thm. 2, the combined system is also controllable. Checking the Kalman rank condition, we find that

$$\left[\begin{array}{cccc}{B}_c& A_c{B}_c& A_c^2{B}_c& A_c^3{B}_c\end{array}\right]=\left[\begin{array}{llllllll}0\hfill & 0\hfill & 1\hfill & 0\hfill & 2\hfill & 0\hfill & 3\hfill & 0\hfill \\ 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill \\ 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill & 0\hfill & 1\hfill \\ 0\hfill & 1\hfill & 0\hfill & 2\hfill & 0\hfill & 2\hfill & 0\hfill & 2\hfill \end{array}\right]$$

is indeed full rank, verifying controllability. However, because the systems share the eigenvalue 1, by Thm. 3, the combined system is not observable. Indeed, a quick calculation shows that

$${\left[\begin{array}{cccc}{C}_c^T& A_c^T{C}_c^T& {\left(A_c^T\right)}^2{C}_c^T& {\left(A^T\right)}_c^3{C}_c^T\end{array}\right]}^T=\left[\begin{array}{cccc}1& 0& 1& 0\\ 1& 1& 1& 1\\ 1& 2& 1& 3\\ 1& 3& 1& 7\end{array}\right]$$

clearly has rank three, and thus the combined system is indeed not observable. If, however, we change the second subsystem to

$$A_2=\left[\begin{array}{cc}2& 1\\ 0& 2\end{array}\right]$$

there is no longer a shared eigenvalue. This new system is observable and thus, by Thm. 3, the combined system is also observable. To verify, we calculate

$${\left[\begin{array}{cccc}{C}_c^T& A_c^T{C}_c^T& {\left(A_c^T\right)}^2{C}_c^T& {\left(A^T\right)}_c^3{C}_c^T\end{array}\right]}^T=\left[\begin{array}{cccc}1& 0& 1& 0\\ 1& 1& 2& 1\\ 1& 2& 4& 4\\ 1& 3& 8& 12\end{array}\right]$$

which has full rank.

4. Controller and observer design for dual-stage LCS

With both controllability and observability established in Sec. 3, we turn our attention in this section back to the LCS scheme for high-speed AFM and to the design of the observer and controllers for a dual-stage system suitable for that approach.

4.1. Observer

We use a standard Luenberger observer of the form

$$\dot{\widehat{x}}=A\widehat{x}+Bu-L\left(y-\widehat{y}\right),$$ (6a)

$$\widehat{y}=C\widehat{x}.$$ (6b)

Defining the observer error as $e=x-\widehat{x}$ leads to the usual error dynamics of the observer,

$$\dot{e}=\left(A-LC\right)e.$$ (7)

Thus, selecting the observer gain L such that A−LC is Hurwitz ensures the estimated state converges to the true value. In this work we select L such that the poles of A−LC are approximately five times larger than any closed loop poles of the controlled system so that the estimation error dies away quickly relative to the other system dynamics.

4.2. SRA controller design

Under LCS with dual-stage actuation, the goal is to use the SRA to drive the tip of the AFM along a circular trajectory. As shown in Fig. 5, we achieve tracking of a given sinusoid using the internal model principle (IMP). This form of the IMP essentially acts as integral control and achieves zero tracking error for the sinusoidal inputs.

Figure 5:

Block diagram of the SRA controller branch. The estimated state is used to generate the estimated output. This is compared against the desired output to drive the internal model. The actual control is generated as state feedback from the combined SRA and IMP system.

Recall that the SRA is assumed to be a single-input, single-output (SISO) system. The dynamics in state space form are given by

$${\dot{x}}_{SRA}=A_{SRA}{x}_{SRA}+B_{SRA}{u}_{SRA},$$ (8a)

$$y_{SRA}=C_{SRA}{x}_{SRA}.$$ (8b)

To use the IMP, we introduce a new state, x_d, which evolves according to

$$x_d=A_d{x}_d+B_d{e}_d$$ (9a)

$$y_d=C_d{x}_d$$ (9b)

where, with ω₀ denoting the desired sinusoidal frequency, the system matrices are

$$A_d=\left[\begin{array}{cc}0& 1\\ -{\omega }_0^2& 0\end{array}\right],B_d=\left[\begin{array}{c}0\\ {\omega }_0\end{array}\right].$$ (10)

Here the drive signal e is the output error e_d = y_d − y_SRA. (Of course, since y_SRA is not directly observable, we use instead the estimated output ${\widehat{y}}_{SRA}=C_{SRA}{\widehat{x}}_{SRA}$.)

To design a controller, we first define the combined system with dynamics

$$\left[\begin{array}{c}\dot{x}\\ {\dot{x}}_d\end{array}\right]=\left[\begin{array}{cc}A& 0_{n\times 2}\\ B_{d}C& A_d\end{array}\right]\left[\begin{array}{c}x\\ x_d\end{array}\right]+\left[\begin{array}{c}B\\ 0_{2\times 1}\end{array}\right]u+\left[\begin{array}{c}{0}_{n\times 1}\\ B_d\end{array}\right]r,$$ (11a)

$$y=\left[\begin{array}{cc}{C}_1& 0_{1\times 2}\end{array}\right]\left[\begin{array}{c}x\\ x_d\end{array}\right].$$ (11b)

Control of the combined system is achieved with state feedback

$$u_{SRA}=-K_{SRA}{x}_{SRA}-K_d{x}_d.$$ (12)

The control gains can be selected in a variety of ways; in this work we use simple pole placement. As with y_SRA, in implementation the actual state x_SRA is replaced with the estimated state ${\widehat{x}}_{SRA}$.

4.3. LRA controller design

As described in Sec. 2, the LCS algorithm produces a desired path for the center of the scanning circle. The role of the LRA controller is then to track this path. Under the assumption that motion along the sample trajectory is slow relative to the controller dynamics, we will use simple set point regulation for the LRA. Under this approach, diagrammed in Fig. 6, y_dLRA is the desired setpoint (generated by the LCS algorithm). The control law is

$$u_{LRA}=g{y}_d-K_{LRA}{x}_{LRA}.$$ (13)

The “accumated gain” g yields setpoint tracking when

$$g=\left[\begin{array}{cc}{K}_{LRA}& I\end{array}\right]{\left[\begin{array}{cc}A& B\\ C& 0\end{array}\right]}^{-1}\left[\begin{array}{l}0\hfill \\ I\hfill \end{array}\right].$$ (14)

Note that in general this controller is sensitive to modeling error and in practice one of a variety of more robust controllers should be chosen. This version is selected here primarily for simple demonstration of the overall approach.

Figure 6:

Block diagram of the LRA controller branch. State feedback is applied based on both the (estimated) state of the LRA and the desired setpoint y_LRA.

4.4. On the principle of separation

The control approach taken here relies on the principle of separation, namely that the controllers can be designed as if the true state is known and then control performed using the estimated state without loss of stability. This property is well known to hold for minimal linear, time-invariant systems (and thus for the dual-stage actuator). However, since we desired independent control of the LRA and SRA sub-systems despite the single, combined output measurement, it is worth considering the property for our specific setup.

To place this on a somewhat general setting, let x₁ denote the combined state of the LRA and its associated controller and x₂ the combined state of the SRA and its controller. Under the specific controllers in Sec. 4.2 and 4.3, x₁ is simply x_LRA while x₂ is the combination of x_SRA and x_d. We assume state feedback control with gains K₁ and K₂ respectively. Define the observer errors

$$e_1=x_{LRA}-{\widehat{x}}_{LRA},e_2=x_{SRA}-{\widehat{x}}_{SRA}.$$ (15)

Then, the combined dynamics are given by

$$\begin{gathered} \left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\\ {\dot{e}}_1\\ {\dot{e}}_2\end{array}\right]=\left[\begin{array}{cccc}{A}_1-B_1{K}_1& 0& B_1{K}_1& 0\\ 0& A_2-B_2{K}_2& 0& B_2{K}_2\\ 0& 0& A_1-L_1{C}_1& -L_1{C}_2\\ 0& 0& -L_2{C}_1& A_2-L_2{C}_2\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ e_1\\ e_2\end{array}\right] \\ +\left[\begin{array}{cc}{B}_1& 0\\ 0& B_2\\ 0& 0\\ 0& 0\end{array}\right]\left[\begin{array}{c}{u}_1\\ u_2\end{array}\right] \end{gathered}$$ (16a)

$$y=\left[\begin{array}{cccc}{C}_1& C_2& 0& 0\end{array}\right]\left[\begin{array}{l}{x}_1\hfill \\ x_2\hfill \\ e_1\hfill \\ e_2\hfill \end{array}\right]$$ (16b)

The principle of separation is evident in the block diagonal structure of the combined system, implying that the controllers and observer can be designed independently. In addition, the structure shows that the controllers in the two subsystems x₁ and x₂ can be designed independently of each other. Due to the summed output, however, the observer must be designed for the joint system simultaneously.

5. Simulation

To demonstrate the observer-based control of the dual-stage system and its application to the LCS algorithm, in this section we present simulation results using a dual-stage system (loosely) modeled after the physical system described in [34].

5.1. System model

As shown in Fig. 7, the LRA and SRA subsystems are both described using second-order systems with the LRA having a resonance at 1450 Hz and the SRA at 11 kHz.

Figure 7:

Bode plots for the LRA (black) and SRA (gray).

To approximate how one might derive models in practice, simple second order SISO transfer functions were defined and then converted to state space form in Matlab. The state space model for the LRA was given by

$$\begin{gathered} A_{LRA}=\left[\begin{array}{cc}-18.2& -10132.2\\ 8192& 0\end{array}\right],B_{LRA}=\left[\begin{array}{c}32\\ 0\end{array}\right], \\ {C}_{LRA}=\left[\begin{array}{cc}0& 31.7\end{array}\right], \end{gathered}$$

while the state space model for the SRA was

$$\begin{gathered} A_{SRA}=\left[\begin{array}{cc}-23499.1& -72895.3\\ 65536& 0\end{array}\right],B_{SRA}=\left[\begin{array}{c}8\\ 0\end{array}\right], \\ {C}_{SRA}=\left[\begin{array}{cc}0& 9.1\end{array}\right]. \end{gathered}$$

Note that these systems are both controllable and observable and do not share any eigenvalues. Thus the requirements of Thm. 2 and Thm. 3 are met.

5.2. Controller and observer design

Control of the LRA was done using the set-point regulation controller of (13). The desired closed loop poles, chosen to be slow relative to the natural frequency of the LRA, were selected to be

$$p_1=-200,p_2=-300.$$

Using pole placement, this let to a feedback gain of

$$K_{LRA}=\left[\begin{array}{c}15.0-316.4\end{array}\right].$$ (17)

Using this in (14) yields the accumulator gain

$$g=\mathrm{0.0072.}$$ (18)

The desired sinusoid for the SRA was selected to be

$$y_{d_{SRA}}=0.03\text{sin}\left(2\pi 200t\right).$$ (19)

Control of the SRA was done using the IMP controller in (12). As with the LRA, the desired poles for the closed loop SRA system were selected to be slow relative to the natural dynamics (but fast relative to the desired frequency) and were given by

$$p_1=-4000,p_2=-6000.$$

The other two poles in (11) were selected empirically (based on visual inspection of the controller performance) to be

$$p_1=-10+i,p_2=-10-i$$

Using pole placement, the corresponding gains in (12) were

$$K_{SRA}=[-1687.4-9065.4],K_d=\left[\begin{array}{c}-1256.60.0159\end{array}\right].$$ (20)

With the controller designs in place, the observer was designed to be fast relative to the closed loop dynamics by placing the observer poles approximately fives times farther away from the origin than those of the closed loop system, leading to an observer gain of

$$L=\left[\begin{array}{c}-51.0127.3-6647.0-5570.4\end{array}\right].$$ (21)

5.3. Simulation results

We performed two different simulations. In the first, we drove the single axis, dual stage actuator using

$$y_{d_{SRA}}=0.03\text{sin}\left(2\pi 200t\right),y_{d_{LRA}}=1,t\ge 0.$$ (22)

The result is shown in Fig. 8. The combined output is shown in Fig. 8a) while the tracking error for the two subsystems is shown in Figs. 8b and 8c. The results show good tracking in both the LRA and SRA.

Figure 8:

Results of the first simulation with desired trajectory given by (22). (a) The combined response (blue) clearly shows the desired behavior. (b) The LRA tracking error quickly converges to zero with a first order response. (c) The SRA also quickly converges with second order characteristics driven by the IMP.

For the second simulation, a two-axis dual-stage scanner was simulated by simply repeating the same model for the second axis. To demonstrate using the system as part of a LCS scan, the two SRAs were driven according to

$$y_{d_{SR{A}_1}}=0.03\text{sin}\left(2\pi 200t\right),y_{d_{SR{A}_2}}=0.03\text{cos}\left(2\pi 200t\right),$$

so that the short range actuators traced out a circle of radius 0.03 units at a rate of 200 Hz. The long range actuators were driven along an Archimedean spiral, a trajectory designed to loosely resemble a biopolymer on a surface. The resulting overall trajectory of the scanner is shown in Fig. 9. The system started at (0,0), proceeded clockwise. The combined path follows a circle centered on the desired LRA path (in red). Note that the scanner moves faster along the path as it proceeds along the spiral, illustrating that the final shape of the output path depends on the relative speed of the LRA and SRA elements. While the SRAs continue to trace out a circle, the final output trajectory will begin to lose the character of a circle centered on the desired center-axis as the speed is increased and the time scales of the LRA and SRA subsystems are no longer well-separated.

Figure 9:

Simulated trajectory for a two-axis dual-stage scanner. (red) Desired trajectory for the LRAs. (blue) Final path of the two-stage system under the designed observers and controllers.

6. Conclusion

This paper considered the control of a dual-stage scanning system with a focus on trajectories relevant to local circular scanning for high speed imaging in AFM. Controllability and observability results were established for a general N-stage system whose outputs are summed to produce a single signal. Control for the LCS system was achieved using an observer and separate, independent controllers in the long range and short range actuators. In future work, this approach will be tested on a physical dual-actuation stage designed for high-speed AFM.