WO2016113576A1

WO2016113576A1 - A control system to control precision positioning arrangements

Info

Publication number: WO2016113576A1
Application number: PCT/GB2016/050102
Authority: WO
Inventors: Sumeet Sunil APHALE; Mohammad NAMAVAR
Original assignee: University Court Of The University Of Aberdeen
Priority date: 2015-01-16
Filing date: 2016-01-18
Publication date: 2016-07-21
Also published as: GB201500749D0

Abstract

The present invention relates to a control system to control precision positioning arrangements, such as disk drives, robotic manipulators, micro-grippers, nanopositioners and the like. The control system comprises an input; an output to control the precision positioning arrangement: and a controller having a control loop operating a tracking stage, a control loop operating a pre-filter stage; and a control loop operating a regulator stage.

Description

A CONTROL SYSTEM TO CONTROL

PRECISION POSITIONING ARRANGEMENTS

[001] The present invention relates to a control system to control precision positioning arrangements, such as disk drives, robotic manipulators, micro-grippers , nanopositioners and the like.

[002] Fast, precise positioning at micro and nanometer scales is a key enabling technology in various scientific fields such as nanotechnology, cell-biology, semiconductor manufacturing, chemical engineering and materials science. The positioning bandwidth of known precision positioning arrangements is mainly limited by their first mechanical resonant frequency.

[003] Such precision positioning arrangements are generally operated with a control system that uses closed-loop control techniques having a two-fold stage of damping and tracking control loops to procure the maximum positioning bandwidth. To obtain a larger bandwidth, the typical practice has been to redesign the positioning arrangement to have its first dominant resonant mode at a higher frequency. However, this generally comes at the cost of reduced positioning range.

[004] An object of the present invention is to provide a control system with a new control technique that effectively shifts the frequency associated with the first dominant resonant mode to a higher frequency without requiring any change in the mechanical configuration or design of the positioning arrangement.

[005] According to one aspect of the present invention there is provided a control system to control a precision positioning arrangement, the control system comprising: - an input;

an output to control the precision positioning arrangement: and

a controller having a control loop operating a tracking stage;

a control loop operating a pre-filter stage; a control loop operating a regulator stage.

[006] Experimental results carried out on a nanopositioning platform demonstrate that the new control technique combined with a tracking control loop results in an overall positioning bandwidth that is 40% higher than the resonant frequency of the original arrangement and can control such arrangements to operate in a fast manner.

[007] Further features of the present invention are defined in the subsidiary claims.

[008] The present invention also encompasses a precision positioning arrangement including a control system as herein described .

[009] Examples of the present invention will now be described with reference to the accompanying drawings, in which : -

Figure 1 illustrates the frequency responses of a known positioning arrangement have a controller operating a typical open-loop undamped control and a known positioning arrangement having a controller operating a typical closed- loop damped and tracked control;

Figure 2 illustrates a control system of a proposed positioning arrangement a traditional proportional-integral controller for tracking, resonance-shifting proportional controller, and a known integral resonant controller for damping;

Figure 3 illustrates an abbreviated form of the control loops of figure 2;

Figure 4 illustrates a proposed equivalent regulator form of the transfer function of a combined proportional and damping control loop of a control system of the present invention to control a precision positioning arrangement.

Figure 5 illustrates the control loops for a control system of the present invention including the tracking proportional-integral controller;

Figure 6 illustrates the loop-gain and stability margins of an isolated proportional loop;

Figure 7 illustrates the loop-gain and stability margins of the control system of the present invention;

Figure 8 illustrates the magnitude response of an open loop controller, a controller using the standard integral resonant control loop, and a controller using the control loops of the present invention;

Figure 9 illustrates a comparison of the closed-loop frequency responses of the servo loops of figure 8;

Figure 10 illustrates a recording of the open and closed loop step responses of figure 8;

Figure 11A illustrates time-domain responses for 20 Hz triangle inputs;

Figure 11B illustrates time-domain responses for 50 Hz triangle inputs;

Figure 12 illustrates a filter that implements -C_d(s);

Figure 13 illustrates an analog circuit for the transfer function C_eq(s) ; and

Figure 14 illustrates an analog circuit for the transfer function C_f(s).

[0010] The present invention will be illustrated with reference to piezoelectric actuators in a nanometer positioning arrangement. However, this is by way of example only and the person skilled in the art will readily appreciate that the control technique of the present invention has wider application to other forms of positioning arrangements for precision positioning at micro and nanometer scales such as computer disc drives, medical robotics etc.

[0011] Due to their high stiffness, infinite resolution, compact size and ease of control, piezoelectric actuators are employed in most high-precision micro and nanometer positioning arrangements. They come in two configurations namely: (I) tube type and (ii) platform type. Due to their mechanical design, both types of positioning arrangements exhibit a lightly damped first resonant mode at relatively low frequencies (< few kHz) that dominates their frequency response and acts as a hard limit to the achievable positioning bandwidth.

[0012] In open-loop configuration, the positioning bandwidth is as low as l/100^th of the frequency of this first resonant mode. For example, for the system used to demonstrate the proposed technique in this patent application, the resonant mode is at a frequency of 500 Hz (see figure 8 blowup) . Then, the usable positioning bandwidth would be 5 Hz or less. To increase the positioning bandwidth and to reduce the positioning errors due to piezoelectric actuation induced non-linearities, such as hysteresis and creep, a wide range of closed-loop control techniques are employed.

[0013] Figure 1 shows the frequency responses of a positioning arrangement having a control system which has a typical undamped open-loop control and a positioning arrangement having a control system which has a typical closed-loop control involving a damped and tracked control loop. The figure indicates the resonant mode and the maximum achievable positioning bandwidth when damped and tracked based in a resonant frequency of 507 Hz for consistency. In theory, it is possible to push the damping coefficient to 1. In practice, this is not possible and the typical 3 dB positioning bandwidth (damped and tracked) of the closed-loop system will be significantly lower.

[0014] A common technique for closed-loop control is to damp the resonant mode using a suitable damping controller and to compensate the nonlinear effects using an integral or proportional-integral controller. Damping controllers such as Positive Position Feedback, Positive Velocity and Position Feedback, Resonant Control, Integral Force Feedback, Integral Resonant Control and Robust Control have been successfully implemented to impart substantial damping to the resonant mode whilst US Pat 5,714,831 discloses a controller providing compensation of the nonlinear effects.

[0015] However, such control techniques are unable to extend the positioning bandwidth beyond the first resonant mode of the positioning arrangement. The only means of increasing the positioning bandwidth is to shift the first resonant mode to a higher frequency which may be difficult or undesirable due to reduction in motion range. Moreover, this may not be possible to achieve with pre-existing positioning arrangements.

[0016] The inventors have previously proposed in Proceedings of the 12™ Bi-annual European Control Conference, Zurich, Switzerland, 17 - 19 July 2013 a control technique to improve the performance of a positioning arrangement. The technique involves a resonance-shifting proportional controller operating a resonance-shifting proportional control loop, a standard known integral resonant damping controller operating a damping control loop, and a traditional proportional- integral controller operating a tracking control loop. This is illustrated in figure 2 where k_s is the proportional gain that shifts the system resonance, G(s) is the transfer function of the plant or system or the positioning system, measured from the input (positioning command) to the output (achieved position) , s is the Laplace variable, d is the feed through gain, k_d is the integral resonant damping gain, k_t is the gain of the integral term of the proportional-integral tracking controller, and k_p is the gain of the proportional term of the PI tracking controller.

[0017] Whilst this control technique was found to work well in simulations, in practice, the wide bandwidth of the proportional feedback loop made it impossible to implement.

[0018] The present inventors have realised that although the characteristics of the control loop of the proportional controller are not practical in isolation, the characteristics are significantly more favourable when the characteristics of the control loop of the proportional controller and the control loop of the damping controller are combined .

[0019] In this respect, the complementary sensitivity function of the control loop of the proportional controller can be expressed as:

G_s = _f^ i)

1 *fc_sG(s)

[0020] Similarly, the complementary sensitivity function of the control loop of the damping controller can be expressed as :

c_d G_sm

i -c_dmG_s(s) where

[0021] This more abbreviated form of the control loops of figure 2 is illustrated in figure 3.

[0022] By substituting G_s(s) from equation 1) into equation 2), the transfer function of the combined proportional and damping control loops can be expressed as:

[0023] Removing the factor leads to

G_dm= -₁^_(s)^_(s) which is a transfer function in the form of a regulator and a pre-filter, where the equivalent regulator is

and the pre-filter

[0024] This equivalent regulator form of the combined proportional and damping control loops is illustrated in figure 4. This configuration has the advantage that the pre- filter and regulator is both realizable and stable. In contrast, a standard integral resonant damping controller requires an unstable controller or the use of positive feedback. Furthermore, the stability margins of the equivalent regulator form of the control loop are superior to the isolated proportional control loop.

[0025] The complete or overall control loop of a control system of the present invention to control a precision positioning arrangement, including the proportional-integral controller for tracking, are shown in figure 5 where the shaded portion corresponds to the transfer function G_d(s).

[0026] The performance of a positioning arrangement controlled using a control system having the control loops shown in figure 5 was evaluated on a two-axis serial kinematic nanopositioner designed and constructed at the EasyLab, University of Nevada, Reno. The stage was driven by two 10 mm 200V piezoelectric stack actuators that provided a range of 40 μηι in each axis. The position was measured by a Microsense 6810 capacitive sensor and 6504-01 probe with a sensitivity of 2.5 μπι/V. The stage was driven by two PiezoDrive PDL200 voltage amplifiers with a gain of 20.

[0027] A second-order model of the control system was procured by frequency domain least squares fit. The resulting transfer function was found to be 2.025xl0⁷

G[s)

s² +48.63s+l.0148xl0⁷ [0028] The platform had a dominant resonant mode at 507 Hz and a damping coefficient of 0.0076. A proportional gain of k_s = 1.5 was chosen to increase the resonant frequency from 507 Hz to 1000 Hz. The optimal damping parameters were then determined to be k_d = 6, 000 and d = 1.2. The resulting equivalent regulator was

and the pre-filter was

6000s+4.32xl0⁷

C_f(s)

s²+8400s+8.64xl0⁶ [0029] The loop-gain and stability margins of an isolated proportional loop are shown in figure 6. The maximum stable gain of the proportional feedback was found to be k_s = 0.02 which is only sufficient to achieve an 8 Hz increase in the resonant frequency. The loop gain shows a phase margin of 8 degrees, which is of little practical value. Typically, an acceptable stability margin is a simultaneous gain margin of 3 dB or better and a phase margin of 20 degrees or better. In the case of the isolated proportional loop, though the gain margin is quite high (>11 dB) , the poor phase margin (~8 deg) only allows the proportional feedback gain to be set at K_s = 0.02, which in turn achieves only an 8 Hz increase in the resonant frequency.

[0030] The loop-gain and stability margins of the control system of the present invention are shown in figure 7. The present invention achieves a gain margin of >5 dB and a phase margin of >20 deg even after an increase in proportional gain K_s by two orders of magnitude to 1.5. Even though the proportional gain has been increased by two orders of magnitude, the phase margin is significantly better than the isolated proportional loop shown in figure 6. Moreover, the proportional feedback gain has increased by two orders of magnitude to k_s = 1.5 resulting in an increase in resonant frequency of the system from 507 Hz to 1000 Hz.

[0031] The frequency responses of an open loop system (no damping or tracking incorporated - positioner) , a closed loop controller using the standard integral resonant control (IRC) loop, and a controller using the control loops of the present invention are shown in Figure 8. These do not have the tracking control loop. The standard controller and the controller of the present invention eliminate the resonant peak. However, the controller of the present invention achieves a bandwidth almost twice that of the standard controller and a resonant frequency nearly double that of the open-loop system. Figure 8 shows the magnitude response with the resonance frequency of the open-loop system at 500 Hz, the -3 dB bandwidth of the standard integral resonant implementation at 574.5 Hz and the -3 dB bandwidth of the controller of the present invention at 1139 Hz.

[0032] The tracking controller was tuned to minimize settling time after a step command. The tracking controller for the standard controller C₂ (s) and the controller of the present invention C₂(s) were found to be

650

(s) +0.034

s

840

C₂(s) +0.034

s

[0033] The closed-loop frequency responses of the servo loops are compared in figure 9. These have the tracking control loop included. The figure illustrates the magnitude response with the resonance frequency of the open-loop system at 500 Hz, the -3 dB bandwidth of the standard IRC implementation at 225.2 Hz and the -3 dB bandwidth of the present invention at 711.7 Hz. The controller of the present invention permits an increase in the tracking bandwidth from 225 Hz to 711 Hz, which is 40% higher than the resonant frequency of the arrangement of the open loop positioning system.

[0034] The open and closed loop step responses were recorded to ascertain the improved damping and tracking performance of the present invention. The results are shown in figure 10.

[0035] Time-domain responses for 20 Hz and 50 Hz triangle inputs are shown in figures 11A and 11B respectively. As clearly seen, the present invention delivers a substantially superior positioning performance due to its increased positioning bandwidth.

[0036] Due to the low order of the control technique of the present invention, it is straightforward to implement in both digital (software) and analog forms. There are a number of options for analog implementation of figure 5. One option is to implement the transfer functions C_eq(s) and C_f(s) directly for example, by a state-variable filter. However, this approach may be difficult to tune experimentally. It is preferable to create circuits with component values that are directly related to the control gains.

[0037] In order to implement figure 5 as an analog controller, circuits are required for the transfer functions C_eq(s) and C_f(s) . As can be seen from the equations 6) and 7), they include the parameter C_d(s). A filter that implements -C_d(s) is illustrated in figure 12. The desired transfer function is:

[0038] The circuit transfer function is

I

S + .,_*,_·.

[0039] Since ^_d is positive and d is negative, the equalities are:-

[0040] The transfer function C_eq(s) can be constructed from -C_d(s) as shown in figure 13. The sub-circuit for -C_d(s) is shaded .

[0041] The transfer function C_f(s) can be implemented by noticing that the sub-circuit -C_d(s) is contained in a unity- gain negative feedback loop with an inverting gain. This arrangement can be implemented by the circuit in figure 14.

[0042] With circuit diagrams for C_eq(s) and C_f(s), the block diagram in figure 5 can be implemented with the addition of a subtracter. Once the damping and resonance control loop have been implemented, it is straightforward for a person skilled in the art to implement the outer tracking control loop with well known standard circuits.

[0043] The only known method of increasing positioning bandwidth of precision positioning systems such as nanopositioning platforms has been to change the mechanical design that places the dominant first resonant mode at a higher frequency. This method is time-consuming and in many cases impractical due to material property limitations as well as motion range requirements.

[0044] The present invention provides a new control technique enabling a controller based method of increasing the positioning bandwidth of precision positioning arrangements. The main advantages of the present invention are : -

High gain and phase margins guaranteeing stability. Substantial increase in the positioning bandwidth (as much as 40% of the original resonant freguency of the system) without the necessity of any mechanical modifications.

Parametric design and availability of optimization schemes with respect to user-defined performance indices.

Easy analog implementation with tuning capability.

Claims

1. A control system to control a precision positioning arrangement, the control system comprising: - an input;

an output to control the precision positioning arrangement: and

a controller having

a control loop operating a tracking stage;

2. A control system as claimed in claim 1 wherein the control loop operating the pre-filter stage has a transfer function in the form of

C_fiS) =

1 ^~ CrfCs) where ₌ ^kd

- clk_d and s is the Laplace variable, d is the feed through gain, and k_d is the integral resonant damping gain.

3. A control system as claimed in claim 1 or 2 wherein the control loop operating the regulator stage has a transfer function in the form of

where

and k_s is the resonance shifting gain, s is the Laplace variable, d is the feed through gain, and k_d is the integral resonant damping gain.

4. A control system as claimed in any preceding claim wherein the control loop operating a tracking stage, the control loop operating a pre-filter stage, and the control loop operating a regulator stage are implemented in software.

5. A control system as claimed in any one of claims 1 to 3 wherein the control loop operating a tracking stage, the control loop operating a pre-filter stage, and the control loop operating a regulator stage are implemented in hardware components .

6. A precision positioning arrangement including a control system as claimed in any preceding claim.