JP5762069B2 - Vibration suppression control method for pneumatic vibration isolation table system - Google Patents

Description

  The present invention relates to a vibration suppression control method for a pneumatic vibration isolation table system, and in particular, vibration suppression of a pneumatic vibration isolation table system suitable for quickly suppressing large displacement vibration, for example, vibration of several millimeters level by active control. It relates to a control method.

  In the field of manufacturing and measuring precision parts, vibration isolation tables are widely used to suppress vibration from the floor. In particular, the pneumatic vibration isolation table is widely used because it can support a large weight with low energy and has high vibration isolation performance. In addition, various proposals and studies have been made so far on active control of a pneumatic vibration isolation table in order to suppress vibration generated by a direct acting disturbance directly applied to the vibration isolation table.

  Patent Document 1 proposes an active control method and apparatus for a pneumatic vibration isolation table. In addition, the differential pressure information is used to actively control the internal pressure of the air spring using a nozzle flapper type servo valve (Non-Patent Document 1), and an external force is applied to the vibration isolation table using a voice coil motor using an absolute displacement sensor. (Non-Patent Document 2) that improves the vibration damping performance by adding, and feeds back position, velocity, and acceleration information, and also combines feedforward to control the internal pressure of the air spring and speed up positioning (non- Patent Documents 3 and 4) have been proposed. Furthermore, a method of expanding the control band by arranging multiple voice coil motors (Non-Patent Document 5), a method of improving performance by reducing acceleration sensor noise (Non-Patent Documents 6 and 7), and using an auxiliary tank In addition, a method of improving the performance by devising the shape of the air spring (Non-Patent Documents 8 and 9), a method of actively controlling with a low consumption flow rate using a spool type servo valve (Non-Patent Document 10), etc. have been proposed. ing. On the other hand, non-patent documents 11 to 18 have been proposed as prior arts related to the present invention.

JP 2006-112535 A

Kengo Kawashima, Tomonori Kato, Kimon Kim, Go Arai, Kotaro Kanno, Toshiharu Kagawa, “Disturbance Compensation Control of Air Spring Vibration Isolator Using Pressure Differential Meter”, Transactions of the Japan Society of Mechanical Engineers, C, Vol. 76, No. 764 (2010), pp. 861-868 Takashi Kai and Shinji Sakurai, “Control of a 2-DOF Pneumatic Vibration Isolator Using an Absolute Displacement Sensor”, Journal of the Japan Society of Mechanical Engineers, Volume C, Vol. 76, No. 766 (2010), pp. 1489-1495 Shinji Sakurai and Shintaro Tada, “Positioning Control of Pneumatic Systems with Pressure Feedback and Series Compensation”, Journal of Precision Engineering, Vol. 74, No. 7 (2008), pp. 769-774 Shinji Sakurai, Masaomi Yamamoto, Masato Takahashi, “Function of one feedforward compensator for pneumatic positioning”, Journal of Japan Society for Precision Engineering, Vol. 74, No. 11 (2008), pp. 1243-1244 Masahito Takahashi, Shinji Sakurai, Susumu Makino, “Broadband Active Vibration Isolator by Multi-point Actuator Arrangement”, Journal of the Japan Society of Mechanical Engineers, Volume C, Vol. 76, No. 763 (2010), pp. 550-556 Ryo Irie, Masato Takahashi, Shinji Sakurai, “Effect of acceleration sensor noise on control of pneumatic vibration isolator”, Journal of Japan Society for Precision Engineering, Vol. 75, No. 6 (2009), pp. 778-779 Ryo Irie and Shinji Sakurai, “Control of Pneumatic Vibration Isolator Using Integral Signal of Relative Acceleration”, Journal of Precision Engineering, Vol. 74, No. 9 (2008), pp.1002-1006 Masato Takahashi and Shinji Sakurai, “High performance of vibration isolation table using auxiliary tank”, Journal of Japan Society for Precision Engineering, Vol. 75, No. 4 (2009), pp. 542-547 Shinji Sakurai, Taishi Tomii, Masato Takahashi, “Proposal of a model of air spring with reverse taper”, Journal of Precision Engineering, Vol. 74, No. 6 (2008), pp.643-644 Tomonori Kato, Kengo Kawashima, Junichi Sawamoto, Tatsuya Kashiwagi, Toshiharu Kagawa “Control of Pneumatic Active Vibration Isolation Table Using Spool Servo Valve and Laminar Flow High-Speed Flowmeter”, Journal of Precision Engineering, Vol. 72, No. 6 (2006), pp. 772-777 Yohei Hoshino, Yukinori Kobayashi, Hiroshi Chida, Mitsuaki Nakanishi, “Active Control of Vibration Isolation Table Using Pneumatic Actuator”, Transactions of the Japan Society of Mechanical Engineers, C, Vol. 73, No. 727 (2007), pp. 677- 684 Hideto Aizaki, Shinji Sakurai, “Adjustment method of model following control for pneumatic vibration isolator equipped with spool type servo valve and its effect”, Transactions of the Japan Society of Mechanical Engineers, Volume C, Vol.76.76 (20010), pp. 835 -841 Daisuke Akasaka, Yasushi Liu, “Design Method of Stabilized Output Feedback Law for On-Off Control”, Transactions of the Society of Instrument and Control Engineers, Vol. 44, No. 5 (2008), pp. 415-424 Kentaro Hamada, Toru Asai, “Stabilization Control Based on PWM Control Input”, Transactions of the Society of Instrument and Control Engineers, Vol. 42, No. 2 (2006), pp.129-138 Kentaro Hamada, Toru Asai, “Stabilization based on three-value PWM control input”, Transactions of the Society of Instrument and Control Engineers, Vol. 44, No. 6 (2008), pp.483-491 Masato Ishikawa, Ichiro Maruta, Toshiharu Sugie, “Design of Discrete Value Input Control System Using Feedback Modulator”, Transactions of the Society of Instrument and Control Engineers, Vol. 43, No. 1 (2007), pp. 31-36 Naoto Abe, Kaoru Kojima, Control of dead time and distributed constant systems, first edition (2007), p. 34, Corona Gao, J., and Huang, B., “Delay-dependent robust guaranteed cost control of an uncertain linear system with state and input delay”, International Journal of Systems Science, Vol. 36, No. 1 (2005), pp. 19-26

  Previous proposals and research on vibration suppression methods for pneumatic vibration isolation tables are for steady-state microvibration control. On the other hand, in a production line of a factory having processing and measurement processes, there is a problem that when a work is placed on a vibration isolation table, vibration of several millimeters level is generated, and it is not possible to proceed to the processing and measurement processes until the vibration is settled.

  For large displacement vibration control, a method of controlling the internal pressure of an air spring by a pneumatic regulator driven by a stepping motor (Non-Patent Document 11) or a method of controlling a flow rate by a spool type servo valve (Non-Patent Document 12) is used. Although some research has been carried out, these methods cannot provide fast response. The reason is that if a large displacement is to be controlled at high speed, the maximum input / minimum input, which is the saturation region of the manipulated variable, must be used instantaneously, but in linear control, reaching the maximum input / minimum input is delayed. is there.

  In view of such a problem, an object of the present invention is to provide a vibration suppression control method for a pneumatic vibration isolation table system that is suitable for quickly suppressing large displacement vibration, for example, vibration of several millimeters level by active control. It is to propose.

In order to solve the above problems, the pneumatic vibration isolation table system of the present invention is
A vibration isolation table,
An air spring supporting the vibration isolation table;
a pneumatic circuit for supplying and discharging compressed air to and from the air spring via at least one intake on-off valve taking on-off binary and at least one exhaust on-off valve taking on-off binary; ,
Compressed air command for the air spring required to suppress vibration of the vibration isolation table by feeding back information on displacement or speed in the vibration isolation direction of the vibration isolation table and pressure change of the air in the air spring The flow rate is calculated, and the command flow rate is converted into a discrete multivalued operation flow rate obtained by on-off operation of the intake on-off valve and the exhaust on-off valve, and the operation flow rate is given to the air spring. And a control system.

  In the pneumatic vibration isolation table system of the present invention, pneumatic control is performed using an intake on-off valve and an exhaust on-off valve. That is, compressed air is supplied to the air spring by opening the on-off valve for intake and closing the on-off valve for exhaust, and compressed air is discharged from the air spring by closing the intake on-off valve and opening the on-off valve for exhaust. .

  When a plurality of intake on-off valves connected in parallel and a plurality of exhaust on-off valves connected in parallel are provided, by opening one or more of the plurality of intake on-off valves simultaneously Compressed air is supplied to the air spring at a multi-stage supply flow rate, and one or more of the plurality of exhaust on-off valves are simultaneously opened to simultaneously open the air spring at a multi-stage exhaust flow rate. Compressed air is discharged from.

  Conventional servo valves used in microvibration control have good responsiveness to linear range input, but maximum input and minimum input (saturated manipulated variable) are required for large displacement vibration suppression control. Therefore, it is difficult to instantaneously switch from the maximum input to the minimum input or from the minimum input to the maximum input. For this reason, large vibration suppression control cannot be performed with good responsiveness. In the present invention, since air pressure control using an open / close type capable of instantaneously switching between the maximum input and the minimum input is performed, even a vibration with a large amplitude can be suppressed with good responsiveness. Moreover, since the on-off valve, for example, the on-off solenoid valve, is inexpensive, there is an economic advantage that the manufacturing cost of the apparatus can be reduced. Further, when a plurality of intake on-off valves and a plurality of exhaust on-off valves are provided, the intake and exhaust flow rates can be controlled in multiple stages, so that vibration suppression control can be performed with high accuracy.

Here, in the vibration suppression control of the pneumatic vibration isolation table system of the present invention, there is an input dead time because piping is used in the pneumatic control. That is, since a delay occurs from the time when the on-off command of the on-off valve is started until the time when the internal pressure fluctuation of the air spring starts, it is necessary to compensate for this dead time in order to obtain a quick response. Further, since the on-off driving of the on-off valve is performed, linear control cannot be used as in the prior art, and active control in consideration of the on-off operation is necessary. Therefore, as a control problem,
It is necessary to consider two points: (1) the input dead time exists in the pneumatic vibration isolation table, and (2) the on-off operation of the on-off valve is required as the operation amount.

  Regarding the above control problem (1), as a method for compensating the input dead time, a method of performing control by predicting an output that is delayed by the dead time (Smith compensation: Non-Patent Document 17) and dead time are considered. A state feedback rule design method (Gao's method: Non-Patent Document 18) is known. The latter is advantageous in that it is more robust against model errors because it does not use feedforward.

  Regarding the above control problem (2), as a method considering on-off operation, a method of designing a state feedback law having an on-off operation (Non-patent Document 13), a conversion method to PWM type input (Non-Patent Document) 14, 15) have been proposed. However, it cannot be applied as it is to the pneumatic vibration isolation table system of the present invention having a dead time. On the other hand, an FBM (feedback modulator) has been proposed as a technique for converting a continuous input into a discrete input (method by Ishikawa et al .: Non-Patent Document 16). This method improves the performance by providing feedback to the quantizer, and is advantageous in that the on-ff operation can be considered independently of the dead time control method because of the wide class of systems that can be adapted. is there.

  Based on these points, the vibration suppression control method of the pneumatic vibration isolation table system of the present invention applies the Gao method (Non-Patent Document 18) as a method for considering dead time, and considers on-off operation. The method is based on the method of Ishikawa et al. (Non-Patent Document 16) and is characterized by adopting a method modified for further performance improvement.

  Since the on-off valve for pneumatic control is used in the pneumatic vibration isolation table system of the present invention, large displacement vibration generated in the pneumatic vibration isolation table system can be subjected to vibration suppression control at high speed. In particular, large vibrations on the order of several millimeters can be quickly suppressed as compared with conventional mechanisms. In addition, the control by on-off driving of the on-off valve also has an effect that it can be realized at a low cost with excellent high-speed response, compared to linear control using a servo valve.

(A) And (b) is the schematic side view and schematic plan view which show the pneumatic type vibration isolator which is the control object in Embodiment 1 of this invention. (A) is a block diagram which shows the pneumatic circuit of a pneumatic vibration isolator, (b) is explanatory drawing which shows an air spring. It is a schematic block diagram which shows the hardware constitutions of the control system which performs vibration suppression control of a pneumatic type vibration isolator. It is explanatory drawing which shows the dynamic model of a control object. It is a graph which shows the quantization rule of a solenoid valve. It is a block diagram which shows a control object (plant). It is a block diagram which shows the control system which gave the pressure minor feedback. It is a graph which shows an example of the pressure follow-up performance test in the system of FIG. It is a block diagram which shows the control system which performed state feedback. It is a block diagram which shows the control system which added FBM. It is a block diagram which shows the control system which added the dead zone function and the saturation function. It is a block diagram which shows the control system which added the dead zone compensator. It is a graph which shows the computer simulation result in Case1. It is a graph which shows the computer simulation result in the case of Case2. It is a graph which shows the computer simulation result in Case3. It is a graph which shows the computer simulation result in Case4. It is a graph which shows the computer simulation result in Case4. It is a graph which shows the computer simulation result in Case5. It is a graph which shows the experimental result in the case of Case1. It is a graph which shows the experimental result in the case of Case2. It is a graph which shows the experimental result in the case of Case3. It is a graph which shows the experimental result in the case of Case4. It is a graph which shows the experimental result in the case of Case5. (A) And (b) is the schematic side view and schematic plan view which show the pneumatic type vibration isolator which is the control object in Embodiment 2 of this invention. (A) is a block diagram which shows the pneumatic circuit of a pneumatic vibration isolator, (b) is explanatory drawing which shows an air spring. It is a schematic block diagram which shows the hardware constitutions of the control system which performs vibration suppression control of a pneumatic type vibration isolator. It is a graph which shows the quantization rule of a solenoid valve. It is a graph which shows the experimental value and calculation value of variable parameter (gamma). It is a block diagram which shows the calculation procedure of the variable parameter (gamma). It is a graph which shows the quantization rule in the case of 3 value input of a solenoid valve. It is a graph which shows the quantization rule in the case of 5 value input of a solenoid valve. It is a block diagram which shows a control object (plant). It is a graph which shows the response state of the displacement with respect to the operation amount, and the pressure change by the computer simulation using the control object of FIG. 32, and a real machine experiment. It is a graph which shows the response state of the displacement with respect to the operation amount, and the pressure change by the computer simulation using the control object of FIG. 32, and a real machine experiment. It is a block diagram which shows the control system which gave the pressure minor feedback. It is a block diagram which shows the control system which performed state feedback. It is a graph which shows an example of the pressure follow-up performance test in the system of FIG. It is a block diagram which shows the control system which added FBM, a dead zone function, a saturation function, and a dead zone compensator. It is a graph which shows the computer simulation result in the case of ternary on-off control. It is a graph which shows the computer simulation result in the case of 5-value on-off control. It is a graph which shows the experimental result in the case of ternary on-off control. It is a graph which shows the experimental result in the case of 5-value on-off control.

  Embodiments of a vibration suppression control method for a pneumatic vibration isolation table system to which the present invention is applied will be described below with reference to the drawings.

[Embodiment 1]
(Pneumatic vibration isolation table)
FIGS. 1A and 1B are a schematic side view and a schematic plan view showing a pneumatic vibration isolation table which is a control target of the present invention. As shown in these drawings, the pneumatic vibration isolation table 1 includes a gantry 2 and a rectangular contour vibration isolation table 3 horizontally disposed thereon. The vibration isolation table 3 is supported by four air springs 4 at the four corner positions. Each air spring 4 is supplied with compressed air from a compressor 6 via a pneumatic circuit 5. An intake solenoid valve 7 and an exhaust solenoid valve 8, which are open / close solenoid valves, are interposed in the pneumatic circuit 5. Each air spring 4 is provided with a pressure sensor 9 for detecting its internal pressure, and the vibration isolation table 3 has a displacement sensor for detecting its displacement (displacement in the vibration isolation direction, vertical displacement in this example). For example, a laser displacement sensor 10 is attached. An acceleration sensor can be used instead of the displacement sensor. In this case, the acceleration signal can be integrated or filtered to obtain a Z differential signal. In the vibration suppression control, the supply and discharge of the compressed air to each air spring 4 is controlled by controlling the pair of solenoid valves 6 and 7 on and off based on the detected pressure and the detected displacement, and the vibration isolator 3 is vibrated. The force in the restraining direction is applied.

  The intake solenoid valve 7 and the exhaust solenoid valve 8 use K2-100HF-04-NC manufactured by Koganei, and the response time is 3 ms when on and 1 ms when off. The laser displacement sensors 10 that detect the displacement in the vertical direction of the vibration isolation table 3 are arranged in parallel as shown in FIG. 1, for example, and the average of the two is taken to obtain the displacement in the vertical direction. The laser displacement sensor 10 uses LB-02 manufactured by Keyence Corporation. The measurement range is ± 10 mm, the response time is 2 ms, and the resolution is 15 mm. The pressure sensor 9 for detecting the pressure of the air in the air spring 4 uses PS420A manufactured by Koganei, and has a measurement pressure range of 0 to 1 MPa and a response time of 1 ms.

  FIG. 2A is a circuit diagram showing the pneumatic circuit 5 of the pneumatic vibration isolation table 1, and FIG. 2B is an explanatory diagram showing the air spring 4. As shown in FIG. 2 (a), the internal pressure of the tank is previously maintained at, for example, 0.3 to 0.4 MPa by the compressor 6, and for example, 0.2 MPa by the pneumatic regulator 11 (SMW AW10-M5BG-N). Adjust the air pressure. By opening the intake solenoid valve 7 and closing the exhaust solenoid valve 8, compressed air is sent to the air spring 4, and the internal pressure of the air spring 4 can be increased. Conversely, by closing the intake solenoid valve 7 and opening the exhaust solenoid valve 8, the compressed air in the air spring 4 is released to the outside air, and the internal pressure of the air spring 4 can be lowered. In this way, by controlling the air pressure in the air spring 4 by opening and closing the electromagnetic valves 7 and 8, it is possible to apply a damping control force to the vibration isolation table 3. For example, the four air springs 4 are simultaneously driven to control the movement in the vertical translation direction.

  Here, in the pneumatic vibration isolator 1, an intake solenoid valve 7 and an exhaust solenoid valve 8 are connected to an air spring 4 via a pipe A. For this reason, even if the solenoid valves 7 and 8 are opened, the internal pressure of the air spring 4 does not change immediately, and a certain dead time occurs due to the effect of pressure loss in the piping. Further, as shown in FIG. 2B, it is connected to the inflow port 4b of the inner chamber 4a of the air spring 4 and to the buffer tank 4c, and has a mechanical configuration that further enhances vibration insulation performance.

(Control system)
FIG. 3 is a schematic block diagram showing a hardware configuration of a control system that performs vibration suppression control of the pneumatic vibration isolation table 1 having the above-described configuration. In the control system 12, for example, based on the displacement and pressure acquired via the 16-bit A / D converter 13, control calculation is performed in the digital signal processor 15 (DSP: HertDSP6x67 manufactured by mtt) connected to the control personal computer 14. The command voltage is applied to both solenoid valves 7 and 8 via the 16-bit D / A converter 16. The digital control is performed with a sampling time of 0.2 ms, for example. The pneumatic vibration isolation table 1 and the control system 12 constitute a pneumatic vibration isolation table system.

(Control target modeling)
Next, the modeling of the pneumatic vibration isolator 1 that is the control target will be discussed. First, it is limited to the movement of the vibration isolation table 3 only in the vertical direction. In general, the spring constant and the viscosity constant of the air spring 4 change with the pressure change of the air spring 4, but it can be assumed that the pressure change is small and constant. In this case, as in the model shown in FIG. 4, the pneumatic vibration isolation table 1 is considered as a simple one-degree-of-freedom secondary spring / mass / damper system, and the thermodynamic equilibrium condition in the air spring 4 is considered. Thus, the pneumatic vibration isolation table 1 can be regarded as a nonlinear model represented by the equation of motion represented by the following equations (1) and (2).

Formula (1)

Formula (2)
here,
m: Mass of vibration isolation table k: Spring constant c: Viscosity constant S: Contact area between vibration isolation table and air spring P ₀ : Initial pressure (equilibrium pressure)
Rs: gas constant kappa: specific heat ratio Ts: the gas temperature L: dead time z (t): displacement ΔP of anti-vibration table (t): pressure changes from the equilibrium point pressure _{P 0} of the air spring Gd: Mass flow (operation rate )
V ₀₁ : Volume of buffer tank V ₀₂ : Volume of air spring at equilibrium point pressure ΔV (t): Volume change from equilibrium point volume (V ₀₁ + V ₀₂ )

The parameter can be obtained from the free response of the pneumatic vibration isolation table 1 of FIG. Table 1 shows examples of parameter values.

The above equation (2) is an equation assuming that the pressures in the volumes V ₀₁ and V ₀₂ always change at the same time, but the pneumatic vibration isolation table 2 of this example is as shown in FIG. immediately after the air inflow due to the inlet 4b directly below the air spring 4, the pressure in the volume V ₀₂ of the air spring 4 is considered to change before the air is filled in the volume V ₀₁ of the buffer tank 4c, during a transient The pressure change is dominated by the volume V ₀₂ . In consideration of this point, the volume (V ₀₁ + V ₀₂ + ΔV (t)) of the second term in the equation (2) is replaced with (V ₀₂ β + ΔV (t)) and can be corrected as the following equation (3). it can.

Formula (3)

Here, β is an influence coefficient and is a parameter representing the degree of control of the volume V ₀₂ with respect to the pressure change at the time of transient. Further, assuming that the air spring 4 expands and contracts only in the z direction, which is the vertical direction, using the contact area S between the vibration isolation table 3 and the air spring 4, equation (3) becomes the following equation (4).

Formula (4)
However, z ₀₁ (equivalent buffer tank height) and z ₀₂ (equivalent air spring height) are as follows.
z ₀₁ = V ₀₁ / S
z ₀₂ = V ₀₂ / S

On the other hand, the mass flow rate Gd (t) is controlled by opening and closing the solenoid valves 7 and 8. Solenoid valves 7 and 8, depending on the value of the command flow rate _{u p,} to control the flow rate Gd in the Rules of the following formula (5). _That, u p _{= G} + / 2 or _{u p> G} + / 2 when closing the electromagnetic valve 8 opens the solenoid valve 7. _Also, opening the electromagnetic valve 8 closes the solenoid valve 7 if u p = _-G + / 2 or _u p _<-G + / 2.

Formula (5)

As shown in FIG. 5, the solenoid valve element including the solenoid valves 7 and 8 has a saturation function and a quantizer characteristic. That is, the opening and closing states of the solenoid valves 7 and 8 is changed by the operation amount u _p, as a result, the flow rate Gd may take a value of 3 values. Note that the value that Gd can take is generally asymmetric. This is because the differential pressure between the air spring internal pressure and the external air pressure is lower than the differential pressure between the air pressure regulator pressure and the air spring internal pressure.

From the above, the control target model (pneumatic vibration isolation table 1) of the control system 12 is expressed by the equations (1), (4), and (5). Operation weight of command flow rate u _p, the components of the controlled object model is a solenoid valve elements (7,8), the dead time L, and the vibration isolation table 3. FIG. 6 shows a block diagram of a controlled object model (hereinafter sometimes referred to as “plant”). Here, the signals that can be observed are the displacement z of the vibration isolation table and the pressure change ΔP of the air spring. The displacement speed of the vibration isolation table, that is, the differential signal (displacement speed) of z can be obtained by calculating the differential speed online.

(Control system design)
(1) Outline of control system design Control system design is performed according to the following procedure.
Step1: Addition of pressure feedback control (linearization of controlled object model)
In order to facilitate the handling of the controlled object, pressure feedback control is added as shown in FIG. As a result, the controlled object is Equation 6 described later.
Step 2: Consideration of dead time L (state prediction control)
The dead time compensation is performed using the equation (6) as a control target.
Step 3: Consideration of on-off operation The possible value of the flow rate by the on-off operation of the solenoid valve is an input quantized into three values. Therefore, compensation is performed in consideration of the on-off operation to prevent deterioration of response performance.

(2) Step 1: Addition of pressure minor feedback control As shown in FIG. 7, in order to facilitate the handling of the controlled object that is a non-linear model, as shown in FIG. 7, pressure feedback control is applied to the control system of the plant. Give as minor feedback. U _c in the figure are the error pressure change signal to the target pressure change ΔPc and the pressure change [Delta] P. p ₁ is a proportional gain of pressure minor feedback.

  The pressure tracking performance of this control system was confirmed through experiments. FIG. 8 is a graph showing an example of the pressure change at this time. For comparison, a response when the transfer function from the target pressure change ΔPc to the pressure change ΔP is approximated to a first-order lag system with a time constant of 0.02 s is obtained by computer simulation, and is represented by a one-dot chain line in FIG. The target pressure change ΔPc is 5 kPa from 1 to 3 s, and 0 kPa at other times.

If the gain p ₁ is excessively large, the flow rate Gd tends to chatter immediately after the ΔPc is increased to 5 kPa (1-2 s in FIG. 8). A large deviation will occur in the change. Therefore, the value of the gain p ₁ may be determined through actual machine experiments so that the flow rate Gd does not chatter in 1 to 2s of FIG. 8 and the steady pressure deviation does not increase. In the case of this example, p ₁ = 2.2870 × 10 ⁻² may be set.

Looking at the response of FIG. 8, the simulation response when approximated to the experimental response and the first-order lag system shows a very close response in the transient state, although a deviation remains in the steady state. Further, since the time constant of the first-order lag system is sufficiently fast as 0.02 s, the transfer function from ΔPc to ΔP of the closed loop system subjected to pressure feedback can be regarded as approximately 1. Therefore, the operation amount to the command pressure change Delta] Pc in place of the command flow rate u _p, the system after pressure control, x = z, x dot = z dot (first derivative of z) of the state variable, control input u = Delta] Pc Can be expressed by the following formula (6).

Formula (6)

  Here, k is a spring constant of the pneumatic vibration isolation table 1, c is a viscosity constant of the pneumatic vibration isolation table 1, and L is a dead time of pressure. Each parameter can be identified from the free response by the actual experiment to the impulse-like disturbance. An example of the value of each identified parameter is shown in Table 2. Moreover, Formula (6) is controllable.

  As described above, the pneumatic vibration isolation table 1 can be regarded as a linearized control target model expressed by the equation (6), and based on this, the control system of the control system 12 can be designed. Is possible. However, in this example, control by a regulator is assumed to suppress vibrations, and the control law of the following equation (7) is used for the linearized control target defined by equation (6) with F as the state feedback control gain. think of. FIG. 9 shows a block diagram of the control system in this case.

Formula (7)

(3) Step 2: Dead time compensation control system design (state prediction control)
Generally, the state feedback control gain F can be obtained by a method such as a pole placement method or solving an optimal regulator problem. On the other hand, when the dead time L exists, it is necessary to consider the influence. That is, the feedback control gain F is determined in consideration of the effect of the dead time L, and the input u = −Fx (t + L) is applied. As a selection method of F, a method based on a state prediction theory or a Gao method (non-patent document) The method according to 18) is known.

Here, in the method of Gao (Non-Patent Document 18), the stability of the system including the input dead time can be guaranteed, and F taking into account the convergence of the state variable can be obtained by LMI (Linear Matrix Inequality). It is suitable for use in the control system of this example. Therefore, in this example, the following theorem proposed by the method of Gao is used.
[Theorem 1]
If there is a matrix Y and a positive definite symmetric matrix X, Q1, Q2, Q￣ satisfying the LMI of the following equation (8) in the control system defined by the equations (6) and (7), the control system is It becomes global asymptotically stable and satisfies J <J ^* .

Formula (8)

Formula (9)

Formula (10)
However,
U = Ax + XA ^T + BY + Y ^T B ^T + L ₂ BQ￣B ^T
X: = P ⁻¹
Y: = − FP ⁻¹ = −FX
It is.

The evaluation function J and the upper bound function J ^* are expressed by the following equations.
Formula (11)

Formula (12)
Here, Q, R, and N are design parameters, and x (0) is an initial value of a state variable.

As a design procedure using Theorem 1, first, design parameters Q, R, and N of the evaluation function J and the upper bound function J ^* are determined. Next, the LMIs of the equations (8) to (10) are solved, and X, Y, Q1, Q2, and Q￣ are obtained. Finally, if F = −YX−1, F is obtained. A system control system using the obtained F is shown in FIG.

(4) Step 3-1: Control system design that compensates for on-off operation In the system control system of FIG. 9, on-off driving of the operation amount is not considered. However, in practice, the on-off operation of the solenoid valve element in the plant results in a three-value quantized value for the flow rate. Therefore, by adding the proposed FBM (the feedback modulator) to the subsequent gain p ₁ by the method of Ishikawa et al. (Non-Patent Document 16), it is desirable to prevent performance deterioration due to on-off operation.

FIG. 10 shows a block diagram of this system control system. The FBM includes a sampler and a zeroth-order hold S / H, a quantizer σ, and a compensator C (s). u _f is an input to the FBM, u _s is an input to the S / H, and the quantizer σ is a quantizer having saturation characteristics, and is defined by the following equation (13). The quantizer σ is determined to have the same characteristics as the saturation function and the quantizer in the plant to be controlled. The difference between the input u _p quantized by S / H and σ and the input u _s before being quantized is fed back through the compensator C (s) to improve the quantization performance.

Formula (13)

It should be noted that the plant of the present embodiment the saturation value G ₊ and G _- the value of the different, sigma is asymmetric. On the other hand, the error due to S / H and quantization is frequency-shaped by C (s). C (s) is expressed by the following equations (14) and (15).

Formula (14)

Formula (15)

Here, τ> 0 is a design parameter, and it is shown that the input / output stability of the FBM is guaranteed by selecting a value larger than the sampling time. In this example, _{u s} is discretized at the sampling time 0.2ms by S / H, C [z] is assumed to be discretized by a zero order hold the C (s).

Next, the characteristics of the FBM will be described. u pulse transfer function K1 from _f to _{u s} [z] If it is assumed that a proportional gain sigma _k characteristics of sigma by the following equation (16).

Formula (16)

Consider the case where the input frequency ω = 0 and σ (u （s) = 0. This assumes a case where the linear input is quantized to 0 by the quantizer σ because the input to the FBM is a constant value and very small. Then, C [z] = 1 and σk = 0, and | K1 [e ^j0 ] | → ∞. Therefore, even if u _f is a constant and minute value, u _s increases, and u _p is eventually converted to a value of G ₊ or G ₋ by the quantizer σ. In other words, the FBM acts to bring the input quantized by the quantizer σ closer to a linear input, and has characteristics like an integrator for a steady input.

(5) Step 3-2: Method 1 for improving control system design that compensates for on-off operation
When the FBM method was directly applied to the pressure minor feedback, the response became a vibration result. This is partly because the pressure signal contains noise and reacts sensitively to this noise. Furthermore, this is caused by trying to strictly follow the target value by ternary input obtained by relatively coarse quantization. Therefore, in order to prevent vibration input, a dead band function Ψ is added before the gain p ₁ to make the response to noise dull and to avoid strict target tracking and tolerate a certain tracking error.

Although the FBM contains saturation function, the input _{u f} to FBM is _{G +} and G _- if was more size is extremely large value, and the output _{u p} of _{u f} and FBM In some cases, the error becomes large, and an extremely large value is fed back in the FBM, so that the response is deteriorated. Therefore, the saturation limit on the input u _f to advance FBM added by adding a saturation function Γ after gain p _1, modifies the control law so as to prevent the error of u _f and u _p becomes larger. As a result, the response of the pressure feedback is expected to be stable.

The block diagram of FIG. 11 shows a feedback control system obtained by the improved method 1 of this example in which a dead band function and a saturation function are added to compensate for an on-off operation. In the figure, u ₁ and u ₂ are input and output of the dead band function Ψ, and u ₃ and u _f are input and output of the saturation function Γ. It is similar to the previous section (Step 3-1) for FBM and _{p 1.} The dead band function Ψ was expressed by the following equation (17). Here, δ represents the width of the dead zone.

Formula (17)

  The quantizer σ in the FBM also includes a dead band element, but its function is fundamentally different. As described in the previous section (Step 3-1), the FBM cancels the dead zone of the manipulated variable and works to approach linear input. As a result, even for a minute signal in the dead zone, an error is accumulated and compensated for by C [z]. On the other hand, the newly added Ψ is introduced in order to make the small signal in the dead zone insensitive. Therefore, it makes sense to add Ψ outside the FBM.

The saturation function Γ is defined by the following equation (18).
Formula (18)
The above equation (18) corresponds to the upper and lower limit values G ₊ and G ₋ of the manipulated variable in the controlled object.

(6) Step 3-3: Method 2 for improving control system design that compensates for on-off operation
When the above improvement method 1 was applied, there was a case where a deviation remained in the pressure. This is because the signal in the dead zone is removed by the addition of the dead zone function Ψ. Therefore, in order to reduce the pressure deviation, a dead band compensator H is added to the dead band function Ψ, and information removed by the dead band function Ψ is fed back. This is expected to reduce the pressure deviation. FIG. 12 is a block diagram showing a system control system of the improved method 2. Here, the dead band function Ψ and the saturation function Γ are the same as those in the improved method 1. At this time, the dead zone compensator H is expressed by the following equation (19).

Formula (19)
Here, α is a design parameter, and signals before and after the dead band function are u ₁ and u ₂ , and the difference is ξ. The sampling time is 0.2 ms as in the FBM.

Next, the meaning of the parameter α and the design conditions will be described. First, when the pulse transfer function from the input _{u c} until ξ and _K 2 [z], _{u 1} remains in the dead zone, | _{u 1} | <if the _{δ k 2 [z] = z} / (z −α). In this case, the stability condition is | α | <1. Therefore, it is necessary that | α | <1. Next, the characteristics of the dead band compensator H will be considered. The pulse transfer function K ₃ [z] from u _c to u ₁ is expressed by the following equation (20) assuming that the characteristic of the dead band function ψ is a proportional gain ψk.

Formula (20)

  Here, a case where the input frequency ω = 0 and Ψk = 0 is considered. This assumes a case where the input to the dead band function Ψ is a constant value and stays within the dead band. At this time, the gain is expressed by the following equation (21).

Formula (21)

In this case, if α → 1, then | K ₃ [e ^j0 ] | → ∞, and u ₁ increases with respect to a constant value input u _c , so that it functions to overcome the dead zone. This is the same as the effect described in Step 3-1, and a function similar to that of an FBM that brings a quantized input closer to a linear input is provided. On the other hand, when α → 0, | K ₃ [e ^j0 ] | → 1 is lost, and the effect of the dead band compensator H is lost, which is equivalent to the improved method 1 described above. Therefore, the effect of the dead band function Ψ can be adjusted by adjusting α. Therefore, α is adjusted within a range not exceeding 1, and the trade-off of reducing chattering associated with the on-off operation is taken while reducing the deviation. Α can be determined based on the experimental results.

(Control system design and simulation results)
(1) Design conditions Control systems were designed for the following Cases 1 to 5. Each Case is summarized in Table 3.

Case 1: The dead time is compensated by the method described in Step 1, but the on-off operation is not considered. A block diagram of the control system is as shown in FIG.
The parameter values were set to p ₁ = 2.2970 × 10 ⁻² and F = [3.8040 × 10 ³ 1.9499 × 10 ⁵ ].

Case 2: As with Case 1, the dead time is compensated by the method described in Step 1, and the on-off operation is compensated by the FBM of Step 3-1. A block diagram of the control system is as shown in FIG. The parameter value of the FBM was set to τ = 0.5. This is a value larger than the sampling time of 0.2 ms, and is selected as a small value that does not cause chattering in the steady state through the pressure servo experiment. Other parameters p ₁ and F are the same as in Case 1.

Case 3: As with Case 1, the dead time is compensated by the method of Step 1, and the on-off operation is compensated by the improved method 1 of Step 3-2. FIG. 11 is a block diagram of the control system. The width of the dead zone was δ = 800. This was determined by experiment so as not to react to noise of the pressure sensor. Other parameters p ₁ , τ, and F are the same as in Case 2.

Case 4: As with Case 1, the dead time is compensated by the method of Step 1, and the on-off operation is compensated by the improved method 2 of Step 3-3. A block diagram of the control system is shown in FIG. Although the experiment was carried out with the parameter α of the dead zone compensator H being α = 0.99, chattering occurred in the manipulated variable in the steady state. Therefore, α is readjusted so that chattering does not occur in a steady state and the pressure deviation becomes small, and α is reset to 0.3. However, in the computer simulation, the case of α = 0.99 was also performed for comparison. Other parameters p ₁ , τ, δ, and F are the same as in Case 3.

Case5: without compensation for the dead time, subjected to state feedback by gain F ₂ determined by optimal regulator. On the other hand, the on-off operation is compensated by the improved method 2 of Step 3-3 as in Case 4. F2 is
F2 = [3.0947 × 10 ² 8.3895 × 10 ⁵ ]. This is the weight matrix
Q˜ = diag (4.0 × 10 ² 1.5 × 10 ⁵ ), R˜ = 2.0 × 10 ⁻⁷ . α was set to α = 0.3. The other parameters p ₁ , τ, and δ are the same as in Case 4.

(2) Computer simulation results The computer simulation results are shown in FIGS. It should be noted that the control target is the equations (1), (4), and (5), and a digital simulation is performed with a sampling period of 0.2 ms. Also, pressure sensor observation noise is not taken into consideration. The control performance is compared by the time response to the impulse-like disturbance when the control of Case 1 to Case 5 is performed. Impulse disturbance is applied after 1 s from the start of simulation.

First, the effect of on-off operation compensation is verified. In on-off does not consider the operation Case1, flow rate Gd is deviation had remained in displacement because it calculates the manipulated variable u _p as applied to the vibration isolation table 3 as a linear input (z in FIG. 13) .

On the other hand, in Case 2, on-off operation is considered by the FBM, but a large value is fed back in the FBM due to the flow rate limitation in the FBM (u _{f in} FIG. 14), so that it is around 1.5 s. A large displacement deviation occurs. Furthermore, work is integral characteristic of the FBM for stationary input to reduce this deviation, fluctuation of the displacement is generated by the operation amount u _p is applied in the vicinity of 2.5 s (z in Figure 14).

In Case 3, the improved method 1 can prevent the input exceeding the saturation value from entering the FBM (u _{f in} FIG. 15), and the displacement fluctuation seen in Case 2 can be prevented. However, a slight deviation remains because it cannot react to the pressure deviation in the dead zone (z in FIG. 15).

  In Case 4, when α = 0.3, there is no significant improvement in response compared to improved method 1 (FIG. 16), but when α = 0.99, the response is greatly improved. The response was ideal with no significant deviation (FIG. 17). From the above, it can be verified that considering the on-off operation in the improved method 2 is very effective in suppressing disturbance.

  Next, the effect of dead time compensation is verified. In Case 5, since the state feedback gain obtained without considering the dead time is used, the response becomes very oscillating (FIG. 18). On the other hand, Case 4 shown in FIGS. 16 and 17 has an ideal response because it uses a state feedback gain considering dead time. From the above, it was verified that compensating for the dead time is effective in suppressing disturbance.

(Experimental result)
Next, FIG. 19 to FIG. 23 show experimental results when Case 1 to Case 5 are controlled. However, the disturbance was applied by allowing the B-class soft baseball (mass 220 g) to fall freely from a position 30 cm above the vibration isolation table 3. Disturbance was added 1 s after the start of the experiment.

First, Case 1 that verifies the effect of on-off operation compensation does not consider the on-off operation, and thus a deviation remains in the displacement as in the simulation (z in FIG. 19). Although not seen in the simulation, chattering had occurred in the operation amount (u _p and Gd in FIG. 19).

In contrast, working by the operation amount _{u p} in a direction to reduce the deviation in 1~2.1s immediately gave disturbance because of taking into account the on-off operation by FBM in Case2. However, 0.1s, will be applied by the operation amount _{u p} around 2.2s and 3.2s, as with simulated displacement is very small vibrations accidentally occurs with fluctuating (Figure 20). This is because it reacts sensitively to the noise of the pressure sensor, strictly tries to follow the target, so a large value is accumulated in the FBM due to accumulation of minute deviations and addition of compensation input, flow restriction in the FBM. This is thought to be due to feedback.

On the other hand, in Case 3, it is possible to prevent the input exceeding the saturation value from entering the FBM by the improved method 1 (u _{f in} FIG. 21), and to prevent the displacement fluctuation seen in Case 2. This is the same as the simulation result. However, a slight deviation remains because it cannot react to the pressure deviation in the dead zone (z in FIG. 21).

  On the other hand, in Case 4, by adding a dead zone compensator based on the improved method 2, the response is greatly improved in the same manner as the simulation result, and there is no great deviation and no ideal vibration is generated. (FIG. 22). From the above, it was verified that considering the on-off operation in the improved method 2 is very effective in suppressing disturbance.

  Next, the effect of dead time compensation is verified. In Case 5, since the state feedback gain obtained without considering the dead time is used, the response diverges (FIG. 23). However, it was possible to obtain a stable response by adjusting the weight matrix so as to relax the convergence condition of the state variable, but in this case, the response speed was significantly deteriorated. Although details are omitted, the experimental results in which the dead time was compensated by Smith compensation did not diverge, but high vibration damping performance was not obtained.

  On the other hand, since the state feedback gain taking account of the dead time is used in Case 4, the response is ideal as described above. From the above, it was verified that compensating for the dead time is effective in suppressing disturbance.

  As described above, in the control system of the pneumatic vibration isolator in consideration of the input dead time and the on-off operation in this example, the dead zone compensator is provided in the FBM for the on-off operation while taking the dead time into consideration. It was confirmed by computer simulation and experiment that compensation using the added improved method 2 is most effective for disturbance suppression.

  Therefore, according to the method of the present invention, large-displacement vibration generated in the pneumatic vibration isolation table can be suppressed and controlled at high speed by an on-off operation of the solenoid valve. In addition, such control by on-off driving of the valve can be realized with excellent high-speed response and low cost as compared with linear control using a servo valve, and is considered to have high practical value.

[Embodiment 2]
In the above-described first embodiment, an effective control method with a high speed and high vibration reduction effect for disturbance suppression is proposed on the premise of a ternary on-off operation amount using an on-off drive opening / closing valve. On the other hand, in the second embodiment, assuming a case in which a minute flow rate adjustment is performed and suppression of residual vibration with a small amplitude is required, a plurality of on-off drive opening / closing valves are used. Residual vibration can be removed by increasing the amount of operation. In this case, the modeling of the controlled object is also considered, a variable parameter model that considers the frequency characteristics of the manipulated variable is proposed, the prediction accuracy of the targeted characteristics is improved, and a control system with a good visibility can be designed. did. Furthermore, the effectiveness of the proposed method was verified by computer simulation and actual machine experiments.

(Pneumatic vibration isolation table)
FIG. 24 shows the pneumatic vibration isolation table 101 according to the second embodiment, and the vibration isolation table 103 (DT-6050M manufactured by Herz) is supported by four air springs 104. As shown in FIG. 25, in the pneumatic circuit 105, the tank internal pressure is previously maintained at 0.3 to 0.4 MPa by the compressor 106, and the air pressure is adjusted to 0.2 MPa by the pneumatic regulator 111 (SMW AW10-M5BG-N). Adjust it.

  Here, there are two intake solenoid valves 107 (1) and 107 (2) and four exhaust solenoid valves 108 (1), 108 (2), 108 (3), and 108 (4) as solenoid valves. It is equipped. By opening one or both of the intake solenoid valves 107 (1) and 107 (2) and closing the exhaust solenoid valves 108 (1) to 108 (4), compressed air is sent to the air spring 104. The internal pressure of can be increased. Conversely, by closing the intake solenoid valves 107 (1) and 107 (2) and opening one or more of the exhaust solenoid valves 108 (1) to 108 (4), the compressed air in the air spring 104 is reduced. It is discharged to the outside air and the internal pressure of the air spring can be lowered.

As described above, the control force can be applied to the vibration isolation table 103 by changing the air pressure in the air spring 104 by opening and closing the six electromagnetic valves. In order to realize a multistage type on-off operation amount, a plurality of solenoid valves and speed controllers (pneumatic devices that adjust the degree of opening and closing of the flow path) are used for both the intake solenoid valve and the exhaust solenoid valve. . The multistage flow rate is generated by changing the combination of solenoid valves that are opened (turned on) as shown in Table 4. The flow rates G _{2+ to} G ₂₋ are as shown in Table 5.

  In the pneumatic circuit 25, the check valve 140 is used to prevent air from flowing backward to the tips of the intake solenoid valves 107 (1) and 107 (2) during exhaust. The four air springs 104 are simultaneously driven to control the movement in the vertical translation direction. The intake solenoid valves 107 (1) and 107 (2) use K2-100HF-04-NC manufactured by Koganei, and the response time is 3 ms when on and 1 ms when off. The exhaust solenoid valves 108 (1) to 108 (4) use VA01HPSC24-1PE-L5 manufactured by Kuroda Pneumatics, and the response time is 2 ms when on and 2 ms when off. The displacement in the vertical direction of the vibration isolation table is detected by the laser displacement sensor 110. As shown in FIG. 24, the laser displacement sensors 110 are arranged in parallel, and the average of the two is taken to obtain the displacement in the vertical direction. The laser displacement sensor 110 uses LB-02 manufactured by Keyence Corporation, and has a measurement range of ± 10 mm, a response time of 2 ms, and a resolution of 15 μm. The pressure of air in the air spring 104 is detected by a pressure sensor 109. The pressure sensor 109 uses GS610A manufactured by Koganei, and has a measurement pressure range of −100 to 100 kPa and a response time of 2.5 ms.

  As shown in the block diagram of FIG. 26, the overall configuration of the pneumatic vibration isolation table system including the pneumatic vibration isolation table 101 and the control system 112 is the same as that of the first embodiment (FIG. 3). Based on the displacement and pressure acquired through the 16-bit A / D converter, control calculation is performed inside the DSP 35 (Merton HeronDSP6x67), and a command voltage is applied to each solenoid valve through the 16-bit D / A. Digital control is performed with a sampling time of 0.2 ms.

  In the pneumatic circuit 105, the solenoid valves 107 (1), 107 (2), 108 (1) to 108 (4) are connected to the air spring 104 via the pipe A (dotted line region in FIG. 25A). doing. For this reason, even if the solenoid valve is opened, the internal pressure of the air spring does not change immediately, and a certain dead time occurs due to the effect of pressure loss in the piping. Further, as shown in FIG. 25B, the inside of the air spring 104 is connected to the buffer tank 104c, and has a mechanical configuration that further improves the vibration insulation performance.

(Control target modeling)
(1) Fundamental equation of motion Also in the second embodiment, only the motion of the vibration isolation table 103 in the vertical direction is considered. In general, the spring constant and viscosity constant of the air spring 104 change with the pressure change of the air spring 104, but since the pressure change is minute, it can be assumed to be practically constant. At this time, when the thermodynamic equilibrium condition in the air spring 104 is taken into consideration, the equations of motion of the pneumatic vibration isolation table 101 are the following equations (101) and (102).

Formula (101)

Formula (102)

Equation (101), in (102), z (t) is the volume of the pressure _{change, V 01} the buffer tank 4c from vertical displacement, DP (t) is the equilibrium point pressure _{P 0} of the vibration isolation table 3, V ₀₂ is the volume of the air spring 4 at the equilibrium point pressure, DV (t) is the volume change from the equilibrium point volume (V ₀₁ + V ₀₂ ), Gd (t) mass flow rate, m is the mass of the vibration isolation table 103, k ₁ Is the spring constant of the vibration isolation table, c ₁ is the viscosity constant of the vibration isolation table, S is the contact area between the vibration isolation table 103 and the air spring 104, Rs is the gas constant, κ is the specific heat ratio, Ts is the gas temperature, and L is the mass This is the dead time of the flow rate. The parameter can be obtained from a free response by an actual machine experiment of the pneumatic vibration isolation table 101 of FIG. 24 to an impulse-like disturbance. Examples of values for each parameter are shown in Table 6.

(2) Variable parameter γ (t) depending on frequency
Expression (102) is an expression assuming that the pressures in the volumes V ₀₁ and V ₀₂ change at the same time. In this system, as shown in FIG. 25B, the inlet 104b is located directly below the air spring 104. Therefore, it is considered that the pressure in the volume V ₀₂ of the air spring 104 changes immediately after the air inflow and before the volume V ₀₁ of the buffer tank 104 c is filled with air, and the volume V ₀₂ is dominant in the pressure change at the time of transient. . In consideration of this point, the volume (V ₀₁ + V ₀₂ + ΔV (t)) of the first term in the formula (102) is replaced with (V ₀₁ γ (t) + V ₀₂ + ΔV (t)), and the volume of the second term ( V ₀₁ + V ₀₂ + ΔV (t)) is replaced with (V ₀₂ β + ΔV (t)) and corrected as shown in the following equation (103).

Formula (103)

Here, β is an influence coefficient and is a parameter representing the degree of control of the volume V ₀₂ with respect to the pressure change at the time of transient. γ (t) is a variable parameter that depends on the frequency of the manipulated variable, and has a characteristic such that the higher the frequency, the closer to 0, and the lower the frequency, the closer to 1. By setting in this way, it is possible to approximate a physical characteristic in which the pressure in the air spring changes before the pressure in the buffer tank changes when an operation amount of a higher frequency is entered. The characteristic of γ (t) was obtained by the following procedure.

First, with respect to an input Gd having three cycles of three values and a constant frequency as shown in FIG. 27, the value of γ is searched for each frequency f of Gd so that the response of the displacement z and the pressure change ΔP most closely matches the experimental value. FIG. 28 shows a result of obtaining the relationship between γ (t) and the frequency f of Gd by a solid line. The experiment was performed by changing the frequency f to 3 to 125 Hz. As the characteristic of γ (t), the characteristic shown by the solid line in FIG. 28 may be provided. However, since it is an off-line calculation, it is not convenient. Therefore, in order to calculate γ (t) online, the approximate expression of FIG. 28 is obtained and calculated according to the block of FIG. First, the flow rate Gd is time-differentiated by a function f ₀ (approximate differentiator) to calculate a flow rate change rate Gd dot (temporary differentiation). The function f ₀ is the following expression (104).

Formula (104)

Here, £ means Laplace transform. f ₀ is extracting the frequency information of Gd. This is because when the Gd frequency is high, the Gd dot (temporary differentiation) also has a large value on average, and therefore has information similar to the frequency. Further, the absolute value of the first derivative of Gd is input to the function f ₁ . The function f ₁ is the following expression (105).

Formula (105)

However, _{g 1} is a design parameter. Since f ₁ approaches γ (t) → 0 if the absolute value of the first derivative of Gd is large, and approaches γ (t) → 1 if the absolute value of the first derivative of Gd is small, it is close to the characteristic of the solid line in FIG. The characteristic can be given to γ (t). The parameter g ₁ is set to g ₁ = 3000 so that the characteristic of γ (t) approaches the characteristic of the solid line in FIG. The relationship between the average value of γ (t) and the frequency f of Gd at this time is indicated by a one-dot chain line in FIG.

  Furthermore, since the relationship of the volume change ΔVz of the air spring can be approximated to ΔV = hSz by experiment, Equation (3) becomes the following Equation (106) using the contact area S of the vibration isolation table and the air spring and the coefficient h.

Formula (106)
However, z ₀₁ and z ₀₂ are respectively z ₀₁ = V ₀₁ / S,
z ₀₂ = V ₀₂ / S
It is.

(3) Relationship between solenoid valve drive pattern and flow rate The mass flow rate Gd (t) is controlled by opening and closing the solenoid valve. In addition, as shown in Table 1, this system allows ternary input and quinary input depending on the driving pattern of the solenoid valve. The flow rate Gd is controlled by the rule of the following equation (107).

Formula (107)

At this time, the solenoid valve has a characteristic of saturation function and quantizer as shown in FIG. 30, the opening and closing states of the solenoid valve is changed by the operation amount u _p, as a result, the flow rate Gd ternary values Can take. This is the same as in the first embodiment. For 5 value input, the electromagnetic valve, depending on the value of the command flow rate u _p, to control the flow rate Gd in the Rules of the formula (108).

Formula (108)

  At this time, the solenoid valve has the characteristics shown in FIG. Note that the value that Gd can take is asymmetric. This is because the difference between the air spring internal pressure and the external air pressure is lower than the differential pressure between the air pressure regulator pressure and the air spring internal pressure. As described above, the control target is represented by the equations (101), (106), and (107) in the case of the ternary on-off control, and the equations (101) and (101) in the case of the quinary on-off control. 106) and (108).

Further, as shown in the block diagram in FIG. 32, the operation amount is the command flow rate u _p, the components of the controlled object solenoid valves, dead time, a pneumatic anti-vibration table. The observation signals are displacement z and pressure change ΔP. Further, the differential signal z _v (= z dot) of z is obtained by performing an on-line differential speed calculation of z.

(4) Verification experiment of response to basic input The response of the displacement z and the pressure change ΔP by the computer simulation using the control target model and the actual machine experiment with respect to the basic operation amount Gd was compared. The Gd was verified by changing the frequency f of the manipulated variable from 3 to 125 hz, as shown in FIG. The results at f = 62.5 hz and f = 12.5 hz are shown in FIGS. 33 and 34, respectively. For comparison, the response when γ (t) is fixed at γ (t) = 1 is shown by a one-dot chain line. Although a rapid change in ΔP cannot be expressed, it can be seen that the steady values of z and ΔP coincide with each other and modeling can be performed with high accuracy. Therefore, using the variable parameter γ (t) depending on the frequency of Gd is effective for modeling of the present system.

(Control system design)
(1) Outline of control system design Control system design is performed according to the following procedure.
Step 1: Addition of pressure feedback control As shown in FIG. 35, pressure feedback control is added to facilitate the handling of the controlled object. As a result, the controlled object is represented by Expression 110.
Step 2: Consideration of dead time Compensation of dead time is performed with the equation (110) as a control target.
Step 3: Consideration of on-off operation The possible value of the flow rate by the on-off operation of the solenoid valve is an input quantized into three or five values. Therefore, compensation is performed in consideration of the on-off operation to prevent deterioration of response performance. However, in this example, control by a regulator is assumed for vibration suppression, and a control law of the following equation (109) is considered with respect to a control target of equation (110) described later, with F as a state feedback gain. FIG. 36 shows a block diagram of the control system in this case.

Formula (109)

(2) Step 1: Addition of Pressure Feedback Control In order to facilitate the handling of the controlled object, as shown in FIG. 35, pressure feedback control is performed as minor feedback. This is the same as in the first embodiment. U _c in the figure are the error pressure change signal to the target pressure change ΔPc and the pressure change [Delta] P. Further, p ₁ is the proportional gain of the pressure minor feedback. The pressure following performance was confirmed through experiments. FIG. 14 shows the pressure change at this time. For comparison, a response when the transfer function from the target pressure change ΔPc to the pressure change ΔP is approximated to a first-order lag system with a time constant of 0.02 s is obtained by computer simulation, and is represented by a one-dot chain line in FIG.

The target pressure change ΔPc is 5 kPa from 1 to 3 s, and 0 kPa at other times. If the gain p1 is set to an excessive value, the flow rate Gd tends to chatter immediately after ΔPc is increased to 5 kPa (1-2 s in FIG. 37). A big deviation will occur. Therefore the value of p ₁ is 1~2s flow rate Gd in FIG. 37 without causing chattering, were determined through actual experimentation so as not steady pressure deviation is large. As a result, p ₁ = 2.4509 × 10 ⁻⁷ was set.

Looking at the response of FIG. 37, since it has a sufficiently fast follow-up performance, the transfer function from ΔPc to ΔP of the closed loop system subjected to pressure feedback is regarded as approximately 1. Therefore, in the following, the command pressure changes ΔOc instead the operation amount to the command flow rate u _p. Therefore, the system after pressure control is expressed by the following equation (110), where x = z and xdot = z are first-order derivatives as state variables and u = ΔPc as a control input.

Formula (110)
However,
It is.

k ₂ is a spring constant of the anti-vibration table, c ₂ is a viscosity constant of the anti-vibration table. Each parameter is identified from the free response by the real machine experiment to the impulsive disturbance. Table 7 shows the values of the identified parameters. Moreover, Formula (110) is controllable.

(3) Step 2: Consideration of dead time Since it is the same as that in the first embodiment, it is omitted.

(4) Step 3: Consideration of on-off operation In the control system of FIG. 36, on-off driving of the operation amount is not considered. However, in practice, the on-off operation of the solenoid valve in the plant results in a three-value or five-value quantized value of the flow rate. Therefore, as with Step3-1,3-2,3-3 in the case of 1 of the embodiment, by adding FBM (the feedback modulator) to the subsequent gain _{p 1,} the performance associated with the on-off operation Prevent deterioration.

In addition, when trying to strictly follow the target value by ternary input or quinary input obtained by relatively coarse quantization, it becomes a vibration input, so the dead band function Ψ is added before p ₁ . However, since the signal in the dead zone is removed by adding the dead zone function Ψ, a steady deviation occurs in z and ΔP. Therefore, a dead band compensator H is added to the dead band function Ψ, and information removed by the dead band function Ψ is fed back. In addition, a saturation function Γ is added after p _{1 in} order to reduce the FBM quantization error. The block diagram is shown in FIG. u _f is the input of the FBM. u _s is the input to H / H (sampler and zero-order hold). u ₁ and u ₂ are the input and output of the dead band function Ψ, and u ₃ and u _f are the input and output of the saturation function Γ.

  The function σ in the FBM is expressed by the following equation (116) in the ternary on-off control. σ is determined to have the same characteristics as the saturation function and the quantizer in the controlled object.

Formula (116)

Since the saturation values G ₂₊ and G ₂₋ are different, σ is asymmetric. In the 5-value on-off control, the following expression (117) is obtained.

Formula (117)

  The function C (s), the dead band function Ψ, the saturation function Γ, and the dead band compensator H are expressed by the following equations (118) to (122). These are the same as in the first embodiment.

Formula (118)

Formula (119)

Formula (120)

Formula (121)

Formula (122)

(Computer simulation results)
The computer simulation results are shown in FIGS. It should be noted that the control targets are the expressions (101), (106), and (107) in the ternary on-off control, and the expressions (101), (106), and (108) in the quinary on-off control. A digital simulation is performed with a sampling period of 0.2 ms. Also, pressure sensor observation noise is not taken into consideration. The control performance is compared by a time response to an impulse-like disturbance when ternary on-off control and quinary on-off control are performed. Impulse disturbance is applied after 1 s from the start of simulation. The design parameters are p ₁ , F is p ₁ = 2.4509 × 10 −7, and F = [38004, 97496]. For τ, δ, and α, τ = 0.5, δ = 1100, α = 0.2 in the case of ternary on-off control, and τ = 0 in the case of quinary on-off control. 0.003, δ = 300, and α = 0.1.

As shown in FIG. 39, in the ternary on-off control, no input is applied after about 1.3 s, and residual vibration remains. On the other hand, as shown in FIG. 40, in the five-value on-off control, not only the maximum / minimum inputs G ₂₊ and G ₂₋ but after the vicinity of 1.2 s, minute inputs G ₁₊ and G ₁₋ are added. Residual vibration is small. When the maximum amplitude of displacement after 1.5 s is compared, it is 382 μm in the ternary on-off control, whereas it is 154 μm in the quinary on-off control. It can be seen that the residual vibration is greatly removed in the control. From the above, it was verified that the multistage input is effective in removing residual vibration.

(Experimental result)
FIG. 41 and FIG. 42 show experimental results when ternary on-off control and quinary on-off control are performed. However, the disturbance was applied by allowing a B-class soft baseball (mass 220 g) to fall freely from a position 30 cm above the vibration isolation table. Disturbance was added 1 s after the start of the experiment. The design parameters are the same as in computer simulation.

As shown in FIG. 41, in ternary on-off control, no input is applied after about 1.2 s, and residual vibration remains. On the other hand, as shown in FIG. 40, in the five-value on-off control, not only the maximum / minimum inputs G ₂₊ and G _2- but after the vicinity of 1.1 s, minute inputs G ₁₊ and G ₁₋ are added, Residual vibration is small. When the maximum amplitude of displacement after 1.5 s is compared, it is 181 μm in the ternary on-off control, whereas it is 55 μm in the quinary on-off control. In the control, it can be seen that the residual vibration is largely removed as in the computer simulation result. From the above, it was verified that the multistage input is effective in removing residual vibration.

DESCRIPTION OF SYMBOLS 1,101 Pneumatic anti-vibration base 2,102 Base 3, 103 Anti-vibration base 4,104 Air spring 4a, 104a Inner chamber 4b, 104b Inlet 4c, 104c Buffer tank 5, 105 Pneumatic circuit 6, 106 Compressor 7, 107 (1), 107 (2) Intake on / off valve 8, 108 (1) to 108 (4) Exhaust on / off valve 9, 109 Pressure sensor 10, 110 Displacement sensor 11, 111 Regulator 12, 112 Control system 13 A / D converter 14 Personal computer 15 for control DSP
16 D / A converter

Claims (6)

A vibration suppression control method for a pneumatic vibration isolation table system,
The pneumatic vibration isolator system is
A vibration isolation table,
An air spring supporting the vibration isolation table;
a pneumatic circuit for supplying and discharging compressed air to and from the air spring via at least one intake on-off valve taking on-off binary and at least one exhaust on-off valve taking on-off binary; ,
The air required to suppress vibration of the vibration isolation table by feeding back information on displacement (z) or speed in the vibration isolation direction of the vibration isolation table and pressure change (ΔP) of air in the air spring. A command flow rate of compressed air to the spring is calculated, and the command flow rate is converted into a discrete multi-valued operation flow rate obtained by an on-off operation of the intake on-off valve and the exhaust on-off valve. And a control system for providing the air spring with
Apply pressure feedback to the control system as minor feedback,
A command pressure change (ΔPc) is adopted as an operation amount input to the control system, and an error pressure change signal (u _c ) between the command pressure change (ΔPc) and the minor feedback pressure change (ΔP) is converted. through the constant gain _{(p 1),} by conversion to the command flow rate _{(u p),}
The system including the control system and the controlled object after the pressure feedback is given by the equation (A), where x = z, x dot = z first derivative, and u = ΔPc as control inputs. Defined by the equation of state,
Formula (A)
However,
m: Mass of the vibration isolation table k: Spring constant of the air spring c: Viscosity constant of the air spring S: Contact area between the vibration isolation table and the air spring L: Dead time (from the time when the on-off command to the on-off valve is generated, the air spring Response delay until the start of pressure fluctuation within
In order to prevent the pressure control performance of the air spring from deteriorating due to the on-off operation of the on-off valve, a feedback modulator (FBM) is added after the conversion constant gain (p ₁ ) in the control system,
In the feedback modulator (FBM), there are provided a saturation function and a quantizer having the same characteristics as the quantization function and the quantizer representing the quantization rule of the operation flow rate (Gd) obtained by the on-off operation of the on-off valve. In addition, the error of the input signal and output signal of the quantizer is fed back to the input signal,
Pneumatic anti-vibration table vibration suppression control method of the system, which comprises using a flow rate command modulated signal outputted from the feedback modulator (FBM) as the command flow rate (u _p). In claim 1,
In order to prevent the pressure controllability of the air spring from deteriorating due to the dead time (L), F is used as a state feedback control gain for the command pressure change (ΔPc) input to the control system. Apply the state feedback represented by (B),
Formula (B)
Vibration of a pneumatic vibration isolation table system characterized in that a value of a state feedback gain (F) for guaranteeing stability of the control system including the dead time (L) is obtained by LMI (Linear Matrix Inequality) Suppression control method. In claim 1 or 2,
In order to prevent the control by the control system from excessively reacting to the noise component included in the error pressure change signal (u _c ), before the conversion constant gain (p ₁ ) in the control system, A dead band function (Ψ) represented by the formula (C) is added (u ₁ , u ₂ : signals before and after the dead band function, u ₁ = u _c ),
Formula (C)
To suppress the error of the input signal of the feedback modulator FBM _{(u f)} and the output signal _{(u p),} during the conversion constant gain _{(p 1)} and the feedback modulator (FBM), the formula (D ) Is added to the input signal (u ₃ ) supplied to the feedback modulator (FBM) through the conversion constant gain (p1), and the input signal ( u _f ) is supplied to the feedback modulator (FBM) as a vibration suppression control method of the pneumatic vibration isolation table system.
Formula (D)
In claim 3,
In order to suppress the pressure deviation generated in the control result by the control system, a dead band compensator (H) represented by the formula (E) is added to the dead band function (Ψ),
Formula (E)
However,
α: Design parameter ξ: Difference between signals u ₁ and u ₂ before and after the dead band function An input signal (u 1) fed back to the error pressure change signal (u _c ) by feeding back the signal component divided by the dead band function (Ψ) ) Is input to the dead band function (Ψ), the vibration suppression control method for the pneumatic vibration isolation table system. In any one of claims 1 to 4,
The pneumatic circuit includes one intake on-off valve and one exhaust on-off valve,
Depending on the value of the command flow rate (u _p), according to the rules with the characteristics of the saturation function and a quantizer defined by the formula (F), the operation flow (Gd) 3 value for the air spring (G ₊ , G ₋ , 0), and a vibration suppression control method for a pneumatic vibration isolation table system.
Formula (F)
In any one of claims 1 to 4,
The pneumatic circuit includes two intake on-off valves (Val. # 1, Val. # 2) connected in parallel and four exhaust on-off valves (Val. # 3 to Val. # 6) connected in parallel. And
Depending on the value of the command flow rate (u _p), the drive pattern of the on-off operation of the valves shown in the following table, with the characteristics of the saturation function and a quantizer defined by the formula (G) A pneumatic vibration isolation table characterized in that an operation flow rate (Gd: Control input) for the air spring is quantized to five values (G ₂₊ , G ₁₊ , G ₁₋ , G ₂₋ , 0) according to a rule. System vibration suppression control method.
Formula (G)