WO2005012011A1

WO2005012011A1 - Multiple-input-multiple-output control system for an airmount arrangement

Info

Publication number: WO2005012011A1
Application number: PCT/IB2004/051310
Authority: WO
Inventors: Michael J. Vervoordeldonk
Original assignee: Koninklijke Philips Electronics NV
Current assignee: Koninklijke Philips NV
Priority date: 2003-08-04
Filing date: 2004-07-28
Publication date: 2005-02-10
Anticipated expiration: 2006-02-04

Abstract

A method of and control arrangement for controlling a plurality of airmounts supporting a payload, each airmount receiving an input signal (PS1, PS2) and arranged to control a z-component (Z1, Z2) of the playload. The z-components are measured and the input signals are determined in dependence on multiple z-components. In an embodiment, SISO controllers with decoupled variables are used to control a height value (z) and a rotation value (φ) of a center of gravity (15) of the playload. The SISO controller for the rotation value may have a multiple differentiating action.

Description

Multiple-input-multiple-output control system for an airmount arrangement

The invention relates to a control arrangement for controlling a plurality of airmounts supporting a payload, each airmount receiving an input signal and being arranged to control a z-component of the payload, sensors being provided to measure z-components and the control arrangement being arranged to receive z-component signals from the sensors and determine the input signals.

It is known to support a payload with a plurality of, e.g. three or four airmounts. The payload has a center of gravity that may or may not be above these airmounts. In dependence on the design of the airmounts, the critical height of the center of gravity of the payload where the payload gets unbalanced may be lower or higher. Therefore, strict rules apply with respect to allowable upper limit of the height of the center of gravity above the airmounts. As is known to persons skilled in the art, the softer the airmounts or the smaller the base, i.e., distance between the airmounts, the lower the critical height. And, the higher the actual height of the center of gravity of the payload, the higher the airmounts the stiffer the airmounts or the greater the distance between the airmounts must be designed. Another way to cope with this problem, as is also known from the prior art, is to apply some additional horizontal springs at different heights form the horizontal airmount stiffness, engaging side surfaces of the payload and not-suspended part of the frame opposite to the side surfaces. These springs increase rotational stiffness of the payload and keep it from instability with respect to tilt. However, there may be locations where there is little room for this construction of additional horizontal springs and an additional frame therefore. Moreover, this may be an expensive solution, and this addition will also increase the horizontal stiffness. When one wishes to replace existing airmounts with softer airmounts, the base may already be fixed, etc. A typical example is a suspension of an electron microscope or (parts) of a lithographic apparatus. For improved floor vibration isolation, softer airmounts are preferred. The height of the center of gravity of such an apparatus requires airmounts to be located higher or to be located further apart. However, increasing the heights of the airmounts may form obstacles to an operator of the apparatus, and increasing the base may not be allowable, e.g., due to a conflict with an electronics cabinet or for commercial reasons.

Therefore, it is an object of the invention to provide an improved control of the airmounts such that the sensitivity of a payload to gravitational instability is reduced without substantially increasing vertical stiffness. To that end, the invention provides a control arrangement as defined at the outset characterized in that the control arrangement is arranged to determine multiple input signals in dependence on multiple received z-components. By applying a multiple-input-multiple-output controller controlling the airmounts supporting the payload, it is possible to balance the payload much better than is possible with prior art arrangements since the input pressure values turn out to depend on all z-components and not just the z-component related to the airmount concerned. In embodiments, it is even feasible to balance the payload when formulae (based on the assumption that the airmounts can be considered to be springs) for critical height of the center of gravity would indicate gravitational instability. Moreover, it is possible to perform a post and pre-processing such that decoupled variables, i.e., height and rotation of the center of gravity of the payload, can be controlled independently by two single-input-single-output (SISO) controllers. Then, the payload can be balanced for higher values of the center of gravity of the payload, without substantially affecting the vertical stiffness. In an embodiment, the SISO controller for one of these variables performs a multiple differentiating action. In an embodiment, the invention relates to a method of controlling a plurality of airmounts supporting a payload, each airmount receiving an input signal based on multiple z-sensors, and arranged to control the z-component of the payload and the rotation around a horizontal axis. The invention also relates to a computer program product provided with instructions and data to be loaded by a control arrangement as defined above, to allow the control arrangement to perform the method as defined above. Finally, the invention relates to a data carrier provided with such a computer program product. The invention will be explained with reference to some drawings that are only intended to illustrate the invention and not to limit its scope. The scope is defined by the annexed claims and their technical equivalents only. Fig. 1 shows a payload supported by a plurality of airmounts; Fig. 2 shows a general, schematic block diagram of a multiple-input-multiple- output control arrangement according to the invention; Fig. 3 shows a control arrangement according to the prior art; Fig. 4a shows a schematic view of a plant to be controlled, with an added pre- filter and post-filter resulting in a new model (or decoupled) plant; Fig. 4b shows an embodiment of a control arrangement according to the invention using the decoupled plant of Fig. 4a; Figs. 5a-5h show a transfer function matrix for the original plant; Figs. 6a-6h show a transfer function matrix for the decoupled plant of Fig. 4a.

Fig. 1 shows a payload 13 supported by a plurality of airmounts. The isolators provide a low suspension frequency for the payload in both horizontal and vertical directions.

The payload has a center of gravity 15. Fig. 1 shows two airmounts 1, 2, however, in practice, there must be three or more airmounts to support the payload 13. For the sake of simplicity, the invention will be further explained with reference to the application of two airmounts.. Per airmount one Lorenz motor 14, 16 is provided. The Lorenz motors 14, 16 operate in the z-direction. The airmounts 1 , 2 are provided with respective pistons, 5, 9, have respective volumes Vι, V₂ and pressures Pi, P₂. On top of the pistons 5, 9 brackets 7, 11 are provided to support the payload

13. The airmounts 1, 2 are coupled to a supply line 6, 8 for supplying or draining air (or other gas) to the internal volume of the airmounts 1 , 2 in order to lift or lower the pistons 5, 9 to a predetermined height above ground 17. By means of valves 25, 27 (shown in Fig. 2) input air pressure Psi, Ps₂ may be controlled by a controller to control the heights of the pistons 5, 9. Instead of controlling input air pressures Psi, Ps₂, input air flows may be controlled, or other input signals to control the heights of pistons 5, 9. The supply lines 5, 8 have respective resistances R|, R₂. An x, y, z-axis system is defined having a predetermined origin. A rotation is defined as a rotation around the x-axis. The center of gravity is at height h above the pistons 5, 9. Base is 2a; both isolators are at a horizontal distance "a" from the center of gravity 15. Piston 5 is at a distance z_\ from ground 17, whereas piston 9 is at a distance z₂ from ground 17. In an embodiment, the model of Fig. 1 can be described by the following motion equations that are linearized around an operational point (note that in these equations z and φ are defined relative to this operational point). Now (in the linearized model) z = (zl+z2)/2 and φ = (zl-z2)/2a): md l²Xzllddtt²² Jd φldt = mghφ + aA P_l -aA₂P₂ V_wdP_λ IΛ = - A P_wdzldt -aA_λP_wdφldt ~ P_i + ^ P_st (2.1) K₂ -K. RT RT V₂₀dP₂ /dt = - A_lP₂₀dz/dt + aA₂P₂₀dφ/dt P₂ + -P_i2 R-, R-,

See for an explanation of the variables and parameters the annexed list of references. Figs. 5a-5h show an example of a typical transfer function matrix of the plant

19 from [Psi, Ps₂]^T o [zi, z₂]' for a height h < h_c in the form of Bode diagrams, h_c being the critical value of h where the payload becomes unstable. From Figs. 5a-5h, one can derive that there is strong crosstalk and both

and z₂ are dependent on both Psi and Ps₂. Fig. 2 shows a control arrangement for a pneumatic control of the arrangement shown in Fig. 1. The control arrangement is simplified in the sense that it shows only two input signals, i.e. input air pressure Psi, Ps2, and two output signals, i.e. distances zi, z₂. Of course, there are as many input signals as there are airmounts. In practice, there will be three output signals from three sensors since they provide enough information as to location and orientation. When there are, e.g., four airmounts there are four input signals, however, only three of them being independent. The distances Zι_ z₂ are measured with respective height sensors 3, 4. These distances zi, z₂ are fed back to the input of a controller 21 via respective comparators 29, 31. These comparators 29, 31 receive respective desired distances zι_ , z_2ιd and subtract the distances zj, z₂ as measured, respectively. Output signals from these comparators 29, 31 are supplied to the controller 21 that calculates control signals for two valves 25, 27 that are to be controlled to supply airmounts 1, 2 with respective additional amounts of air pressure Psi, Ps2- Fig. 2 shows airmounts 1, 2 and sensors 3, 4 within a box 19 indicated in a dashed line that can be regarded as the "plant" to be controlled. From a theoretical point of view, controller 21 could degrade into a single- input-single-output control system, i.e., a system in which zi only influences Psi, and z₂ only Ps₂, so as two separate control loops. This is the prior art solution and is depicted in Fig. 3. In Fig. 3, the output of comparator 29 is fed to a PI controller 33, and the output of comparator 31 to a PI controller 35. If required PI controllers 33, 35 may be replaced by P or PID controllers and/or extended with an extra SISO filter such as a low-pass filter. Simulations show that ranges can be found for the parameters of PI controllers 33, 35 for any value of h, provided h < h_c where the system is stable. However, the closer h is to h_c, the smaller that range. Above h_c no values for these parameters can be found where the system is stable. To solve this problem, the present invention proposes to use the multiple- input-multiple-output controller 21 to find values of these parameters for a stable system even if h > h_e and without substantially affecting the stiffness of the system in the z-direction. It will be shown that this multiple-input-multiple-output controller can be setup by, firstly, decoupling "plant" 19 by a pre-filter and a post-filter and, secondly, applying two independent single-input-single-output (SISO) controllers, i.e., one related to z and one related to φ. As will be shown, there are still cross terms, however, they are so small, that there is substantially no crosstalk anymore. Fig. 4a shows a block diagram of a model used to design such a decoupled system. The model shown can be seen as a (virtual) model plant 19'. The model plant 19' comprises a pre-filter P_pre 37 before the plant 19, the plant 19 itself and a post-filter P-^_t 39 after the plant 19. The pre-filter 37 receives input signals P_z and P_φ. P₂ is defined as an input height signal to move the center of gravity 15 in the z-direction while keeping φ = 0. Pφ is defined as an input rotation signal to move the center of gravity 15 in the φ-direction where z = 0. In an embodiment, the input air pressure Psi, Ps2 may, e.g., be calculated by the following matrix operation:

Of course, depending on the "plant" configuration, P_pre may have another value. The parameters z and φ may be calculated by:

Of course, depending on the "plant" configuration, Ppost may have another value. Figs. 6a-6h show an example of a transfer function matrix of the model plant 19' of Fig. 4a from [P_z, P_φ]^τ to [z, φ]¹ for a height h < h_c in the form of Bode diagrams. From Figs. 6a-6h one can derive that z is mainly dependent on P_z and φ mainly on P_φ, i.e., the model of Fig. 4a can, for all practical purposes, be viewed as two independent single-input- single-output systems. Fig. 4b shows how the model of Fig. 4a can be used to design a subcontroller 41 with two SISO controllers 41a and 41b for z and φ, respectively. The pre-filter P_pre 37 and the post-filter Pp_ost 39 are re -arranged to form part of the controller 21 that also comprises the sub-controller 41. Now returning to Fig. 4a, it can be shown that the transfer function from P_z to z, i.e., H_z, that includes the plant 19 may have the following form:

8.8e - 7 H. = s.(s² + 0.047s + 9.1)

This equation shows that the model plant 19' is open loop stable for z. From the point of view of this z-loop, the SISO controller 41a for z, may be a generic second-order filter with one pool pair and one zero pair. In an example, a bandwidth of less than 1 Hz for such a filter can be realized. In general, one will choose the bandwidth to be lower than the eigenfrequency of airmounts 1, 2 and payload 13. Figs. 6a-6h show that, for this example, there is a resonance at a frequency of about 0.5 Hz. The transfer function from P_φ to φ (Fig. 4a), i.e., H_φ, may have the following form:

5.0e -6 H_Λ = ^■ t A=02ιr, (s - 0.28) - (s² +0.335 + 3.6) Due to the term (s-0.28) in the denominator, the model plant 19' is open loop unstable for φ. Moreover, H_φ varies with varying height h of the center of gravity 15. It turns out that it is possible to design SISO controller 41b such that it is stable for a value of h > h_c and is also stable for values h < h_c. This is achieved by designing the SISO controller 41b with a double, or other multiple differentiating action. The following transfer function H_c,_φ of SISO controller 41b is an example of such a double differentiating action:

1.8 l (s + 1.2)² H, ^c ( ls.. + , Λ 1

( /₅ ² +32 + 600)

The bandwidth of this controller is 2 Hz. However, other bandwidths may be designed. As may be evident to persons skilled in the art, the controller 21 as depicted in Fig. 4b may be implemented by one computer only that is loaded with suitable software and performs the calculations of all three boxes shown, i.e., the post-filter 39, the SISO controllers 41a, 41b and the pre-filter 37. I.e., there need not be three different processing units to implement this functionality. The controller may also be designed as two or more communicating sub-processors performing different portions of the functions. All these computer(s) or (sub)processors may be remote from the plant 19. The suitable software may be distributed via suitable data carriers or in any other way. Instead of a software implementation, an implementation using analog circuits may be used. As is evident to persons skilled in he art, the invention as described with reference to Fig. 4b may be implemented in other equivalent ways. Fig. 4b shows how, mathematically, relations between Psi, Ps₂ and zi, z can be derived by dividing controller 21 into separate pre-filter P_pre, SISO controller 41, and post-filter P_poa- However, in an actual implementation of controller 21, the successive actions by these components may be combined such that variables z, φ, P_z and P_φare not expressly calculated anymore, but still mathematically equivalent actions are performed. This equivalent implementation falls within the scope of the annexed claims. LIST OF REFERENCES:

Ai = surface area of piston 5

A = surface area of piston 9

Pi = pressure in airmount 1

P₂ = pressure in airmount 2 Vi = air volume in airmount 1

V₂ = air volume in airmount 2 zi = height of piston 1 z₂ = height of piston 2 m = mass of payload 13 z = vertical displacement of center of gravity 15 of payload 13 φ = rotation of center of gravity 15 of payload 13 around x-axis g = free fall acceleration h = height of center of gravity 15 of payload 13 above pistons 5, 9 a = horizontal distance between pistons 5, 9 and center of gravity Vio = nominal air volume in airmount 1

V₂o = nominal air volume in airmount 2

Pio = nominal pressure in airmount 1

P₂o = nominal pressure in airmount 2

R = gas constant = 289 J/kg K Ri = resistance of supply line 6 to airmount 1

R₂ = resistance of supply line 8 to airmount 2

T = absolute temperature

Psi = input air pressure to airmount 1

Ps₂ = input air pressure to airmount 2 = desired zi (set point)

Z ₎ = desired z (set point)

Claims

CLAIMS:

1. Control arrangement for controlling a plurality of airmounts supporting a payload, each airmount receiving an input signal (Psi, Ps₂) and being arranged to control a z- component (zi, z₂) of said payload, sensors being provided to measure said z-components and said control arrangement being arranged to receive z-component signals from said sensors and determine said input signals, characterized in that said control arrangement is arranged to determine multiple input signals in dependence on multiple received z-components.

2. Control arrangement according to claim 1 , wherein said controller is designed to perform the following functions: - post-filtering said z-components to produce a height value (z) and at least one rotation value (φ) of a center of gravity of said payload; calculating an input height signal (P_z) and at least one input rotation signal (P_φ) from said height value (z) and said at least one rotation value (φ); pre-filtering said input height and rotation signals (P_z, P_φ) to produce said input signals (Psi, Ps₂), said post-filtering and pre-filtering being designed such that said input height signal (P_z) depends substantially only on said height value (z) and each said input rotation signal (P_φ) depends substantially only on one said rotation value (P_φ).

3. Control arrangement according to claim 2, wherein said calculating of said input rotation signal (P_φ) includes performing a transfer function with a multiple differentiating action.

4. A method of controlling a plurality of airmounts supporting a payload, each airmount receiving an input signal (Psi, Ps₂) and being arranged to control a z-component (zi, z₂) of said payload, the method comprising measuring said z-components and determining said input signals characterized by determining multiple input signals in dependence on multiple z-components.

5. Method according to claim 4, comprising: post-filtering said z-components to produce a height value (z) and at least one rotation value (φ) of a center of gravity of said payload; calculating an input height signal (P_z) and at least one input rotation signal (P_φ) from said height value (z) and said at least one rotation value (φ); - pre-filtering said input height and rotation signals (P_z, P_φ) to produce said input signals (Psi, Ps2), said post-filtering and pre-filtering being such that said input height signal (P_z) depends substantially only on said height value (z) and each said input rotation signal (P_φ) depends substantially only on one said rotation value (P_φ).

6. Method according to claim 5, wherein said calculating of said input rotation signal (P_φ) includes performing a transfer function with a multiple differentiating action.

7. Computer program product provided with instructions and data to be loaded by a control arrangement according to any of the claims 1-3, to allow said control arrangement to perform the method according to any of the claims 4-6.

8. Data carrier provided with a computer program product according to claim 7.