Patent Publication Number: US-8119044-B1

Title: Device and method for plasticization control of electric injection molding machine

Description:
TECHNICAL FIELD 
     This invention is concerning an apparatus and a method for controlling a plasticizing capability in an electric-motor driven injection molding machine. 
     BACKGROUND ART 
     AC servomotors are becoming used for middle-sized injection molding machines heretofore driven by hydraulic actuators (clamping force&gt;3.5 MN) that have high precision, quick response and higher power which are obtained by performance improvements of permanent magnets and cost reductions. 
     An injection molding machine consists of a plasticizer in which resin pellets are melted by friction heat generated by plasticizing screw revolution and stored at the end of a barrel, an injector in which an amount of melted polymer is injected into a metal mold at a given velocity and a given dwell pressure is applied, and a clamper in which the metal mold is clamped and opened, all using AC servomotors drive system.  FIG. 2  is a view which shows an existing plasticizing mechanism by AC servomotors. 
     On an injection machine base which is fixed on the ground, a movable base is located which moves on a linear slider and both the bases are not shown in  FIG. 2 . All parts except a metal mold  1  shown in  FIG. 2  are mounted on the movable base. By sliding the movable base, the top of a barrel  2  is clamped on the metal mold  1  and vice versa the top of the barrel  2  is separated from the metal mold  1 .  FIG. 2  shows a mode in which the resin pellets are melted by the screw revolution in the plasticizing process. 
     On the movable base, a barrel  2 , a servomotor for injection  3 , a reduction gear  4 , a ball screw  5 , a bearing  6  and a hopper  16  are fixed. A nut  7  of the ball screw  5 , a moving part  8 , a screw  9 , a reduction gear  10 , a servomotor for plasticization  11  and a pressure detector  12  such as a load cell consist of an integral structure. The moving part  8  is mounted on a linear slider  13  so that the integral structure is moved back and forth by the movement of the nut  7 . 
     Rotation of the servomotor for injection  3  is transferred to the ball screw  5  which magnifies a linear force through the reduction gear  4  and the rotation of the ball screw  5  is converted to a linear motion of the nut  7  of the ball screw  5  and through the moving part  8 , a linear motion of the screw  9  and pressure application to the stored melted polymer are realized. Pressure applied to the melted polymer by the screw  9  in the plasticizing process is hereinafter referred to as a screw back pressure. Position of the screw  9  is detected by a rotary encoder  14  mounted on the servomotor for injection  3 . Screw back pressure to the melted polymer stored at the end of the barrel  2  is detected by the pressure detector  12  such as a load cell mounted between the nut  7  and the moving part  8 . The screw  9  is rotated by the servomotor for plasticization  11  through the reduction gear  10  in the plasticizing process in which resin pellets are melted and kneaded and a rotary encoder  15  is mounted on the servomotor for plasticization  11 . 
     Explaining an injection molding process with referent to  FIG. 2 , resin pellets are fed to the screw  9  through the hopper  16  and are melted by the screw  9  rotated by the servomotor for plasticization  11  and the melted polymer is pushed out from the top of the screw  9  and the screw  9  is moved back by the generated screw back pressure. The screw back pressure is a linear force applied to the melted polymer decided by a generated motor torque of the servomotor for injection  3 . The servomotor for plasticization  11  continues to rotate until a given amount of melted polymer necessary for molding a product is stored at the end of the barrel  2  and then the plasticizing process is finished with the stop of the screw revolution. 
     Next the screw  9  is moved forward rapidly by a high-speed revolution of the servomotor for injection  3  and the stored melted polymer at the end of the barrel  2  is injected into a cavity  17  as fast as possible and a given pressure is applied for a given duration at the polymer in the cavity  17  and then the injection process is finished and a molding product with a given figure is taken out from the metal mold  1 . 
     It is necessary to get the melted polymer of homogeneous property in the plasticizing process in order to manufacture good-quality molding products. But as the stored melted polymer at the end of the barrel  2  increases in the plasticizing process, an effective length of the screw  9  for plasticizing the resin pellets decreases as the result of the backward movement of the screw  9  in the barrel  2 . Therefore, the decrease of the effective length of the screw brings about a variation in the property of the melted polymer, that is, the property of the melted polymer generated at the initial stage of plasticization is different from that of the melted polymer generated at the final stage. To make up for this defect, some methods are applied in which a given pattern of screw back pressure corresponding to the backward movement of the screw  9  is realized in the plasticizing process in order to get a homogeneous property of the melted polymer. 
     In patent literatures PTL  1  and PTL  2 , a given screw revolution is realized by a servomotor for plasticization and the speed control of a screw backward movement by a servomotor for injection realizes a given pattern of screw back pressure. 
     In patent literatures PTL  3  and PTL  4 , a constant speed or a given speed pattern of a screw backward movement is realized by a servomotor for injection and the rotation speed control of a screw by a servomotor for plasticization realizes a given pattern of screw back pressure. 
     In patent literatures PTL  5  and PTL  6 , a given pattern of screw back pressure is realized by a motor current (torque) limit control or a motor current (torque) control of a servomotor for injection. 
     In patent literatures PTL  7  and PTL  8 , the position control of a screw by a servomotor for injection realizes a given pattern of screw back pressure. 
     In patent literatures PTL  9  and PTL  10 , a given revolution speed of a screw is realized by a servomotor for plasticization and a speed control of the screw backward movement by a servomotor for injection realizes a given pattern of screw back pressure and in the speed control of the screw backward movement a set value of screw backward speed modified by a control deviation of the screw back pressure is used. 
     In patent literature PTL  11 , the control mode transfer from the first control mode to the second control mode is carried out. In the first control mode a screw revolution control is carried out by a servomotor for plasticization and a screw back pressure control is carried out by a servomotor for injection. In the second control mode a screw back pressure control is carried out by a servomotor for plasticization and a speed control of screw backward movement is carried out by a servomotor for injection. 
     In patent literatures PTL  1 ˜PTL  11 , a screw back pressure control is absolutely necessary in the plasticizing process and a pressure detector is required to realize an accurate control of screw back pressure. 
     In patent literature PTL  12 , a pressure detector with a small dynamic range (0˜15.2 MPa) is used for plasticization and a pressure detector with a large dynamic range (15.2˜304 MPa) is used for injection and pressure application. The control accuracy of screw back pressure in the plasticizing process is improved by using a pressure detector with a smaller dynamic range. 
       FIG. 3  is an explanation drawing which shows a block diagram of an existing plasticizing controller. The plasticizing controller consists of a back pressure controller  20 , a motor controller (servoamplifier) for injection  30 , a screw revolution speed controller  40 , a motor controller (servoamplifier) for plasticization  50  and a pressure detector  12 . 
     The back pressure controller  20  executes a control algorithm at a constant time interval and a discrete-time control is used. The back pressure controller  20  consists of a screw back pressure setting device  21 , a subtracter  22 , an analog/digital (A/D) converter  23 , a pressure controller  24  and a digital/analog (D/A) converter  25 . The pressure detector  12  is connected to the A/D converter  23 . 
     The screw back pressure setting device  21  feeds a time sequence of screw back pressure command P* b  to the subtracter  22 . The pressure detector  12  feeds an actual screw back pressure signal P b  to the subtracter  22  through the A/D converter  23 . The subtracter  22  calculates a back pressure control deviation ΔP b =P* b −P b  and the control deviation ΔP b  is fed to the pressure controller  24 . The pressure controller  24  calculates a motor current demand i* m  for the servomotor for injection  3  from ΔP b  by using PID (Proportional+Integral+Derivative) control algorithm and feeds the demand i* m  to the motor controller for injection  30  through the D/A converter  25 . 
     The motor controller for injection  30  consists of an analog/digital (A/D) converter  31  and a PWM (Pulse Width Modulation) device  32 . The motor controller for injection  30  is connected to the servomotor for injection  3  equipped with a rotary encoder  14 . The A/D converter  31  feeds the motor current demand i* m  from the D/A converter  25  to the PWM device  32 . The PWM device  32  applies three-phase voltage to the servomotor for injection  3  so that the servomotor for injection  3  is driven by the motor current i* m . A linear force by the screw  9  applied to the melted polymer stored at the end of the barrel  2  decided by a generated motor current i* m  (motor torque) realizes a given screw back pressure P* b . 
     The screw revolution speed controller  40  consists of a screw revolution speed setting device  41 . The screw revolution speed setting device  41  feeds a time sequence of screw revolution speed command N* s  to the motor controller for plasticization  50 . 
     The motor controller for plasticization  50  consists of a subtracter  51 , a differentiator  52 , a speed controller  53  and a PWM device  54 . The motor controller for plasticization  50  is connected to the servomotor for plasticization  11  equipped with a rotary encoder  15 . The screw revolution speed command N* s  from the screw revolution speed controller  40  is fed to the subtracter  51 . The rotary encoder  15  mounted on the servomotor for plasticization  11  feeds a pulse train to the differentiator  52 . The differentiator  52  detects an actual screw revolution speed N s  and feeds the speed signal N s  to the subtracter  51 . The subtracter  51  calculates a screw speed control deviation ΔN s =N* s −N s  and feeds the control deviation ΔN s  to the speed controller  53 . The speed controller  53  calculates a motor current demand i* for the servomotor for plasticization  11  from ΔN s  by using PID control algorithm and feeds the demand i* to the PWM device  54 . The PWM device  54  applies three-phase voltage to the servomotor for plasticization  11  so that the servomotor for plasticization  11  is driven by the motor current i* and a given screw revolution speed N* s  is realized. 
     But the usage of the pressure detector in the plasticizing process brings about the following disadvantages.
     (1) A highly reliable pressure detector is very expensive under high pressure circumstances.   (2) Mounting a pressure detector in the cavity or the barrel nozzle part necessitates the troublesome works and the working cost becomes considerable.   (3) Mounting a load cell in an injection shafting alignment from a servomotor for injection to a screw complicates the mechanical structure and degrades the mechanical stiffness of the structure.   (4) A load cell which uses strain gauges as a detection device necessitates an electric protection against noise for weak analog signals. Moreover the works for zero-point and span adjustings of a signal amplifier are necessary (patent literature PTL  13 ).   (5) For the improvement of the control accuracy of screw back pressure, the usage of two kinds of pressure detectors with different dynamic ranges brings about the cost increase (patent literature PTL  12 ).   

     CITATION LIST  
     Patent Literature  
     PTL  1 : Patent 61-37409 
     PTL  2 : Patent 61-217227 
     PTL  3 : Patent 61-72512 
     PTL  4 : Patent 2005-35132 
     PTL  5 : Patent 61-258722 
     PTL  6 : Patent 3-58818 
     PTL  7 : Patent 2-130117 
     PTL  8 : Patent 4-249129 
     PTL  9 : Patent 2-120020 
     PTL  10 : Patent 7-9513 
     PTL  11 : Patent 2002-321264 
     PTL  12 : Patent 2000-351139 
     PTL  13 : Patent 2003-211514 
     Non Patent Literature  
     NPL  1 : H. K. Khalil, Nonlinear Systems, 14.5 High-Gain Observer, Prentice-Hall, (2002), pp. 610-625 
     NPL  2 : B. D. O. Anderson and J. B. Moore, Optimal Control, Linear Quadratic Methods, 7.2 Deterministic Estimator Design, Prentice-Hall, (1990), pp. 168-178 
     NPL  3 : A. M. Dabroom and H. K. Khalil, Discrete-time implementation of high-gain observers for numerical differentiation, Int. J. Control, Vol. 72, No. 17, (1999), pp. 1523-1537 
     NPL  4 : A. M. Dabroom and H. K. Khalil, Output Feedback Sampled-Data Control of Nonlinear Systems Using High-Gain Observers, IEEE trans. Automat. Contr., Vol. 46, No. 11, (2001), pp. 1712-1725 
     SUMMARY OF INVENTION 
     Technical Problem 
     The problem that starts being solved is to realize a plasticizing controller of electric-motor driven injection molding machines which satisfies the requirement that an adequate screw back pressure is applied to the melted polymer stored at the end of the barrel during the plasticizing process without using a pressure detector in order to avoid the five disadvantages described in Background Art resulted by using a pressure detector. 
     Solution to Problem 
     A control of screw back pressure is an effective means to obtain the homogeneous property of melted polymer and to improve the metering accuracy in the plasticizing process. In order to realize an accurate control of screw back pressure without using a pressure detector, a detecting means of screw back pressure which satisfies the following two requirements (A) and (B) is required. (A) The detection means is high-precision. (B) The detection means has very small time-lag. The method of a high-gain observer (non patent literature NPL  1 ) is used as a pressure detecting means which satisfies the above two requirements. The high-gain observer estimates all state variables by using detected variables and satisfies the above two requirements (A) and (B). This is explained by using a simple mathematical model as follows. Equation (1) shows a state equation and an output equation of a simple model. 
                   {     Math   .           ⁢   1     }                                       x   .     1     =     x   2                     x   .     2     =     ϕ   ⁡     (     x   ,   u     )                   y   =     x   1                 x   =     [           x   1               x   2           ]             }           (   1   )               
where x 1 , x 2 : State variables, u: Input variable, y: Output variable, φ(x, u): Nonlinear function of variables x, u. For example x 1  is position variable, x 2  is velocity variable and u is motor current variable. Output variable y and input variable u are supposed to be measurable. The high-gain observer which estimates state x is given by equation (2).
 
                   {     Math   .           ⁢   2     }                                         x   ^     .     1     =         x   ^     2     +       H   1     ⁡     (     y   -       x   ^     1       )                           x   ^     .     2     =         ϕ   0     ⁡     (       x   ^     ,   u     )       +       H   2     ⁡     (     y   -       x   ^     1       )                 }           (   2   )               
where {circumflex over (x)} 1 , {circumflex over (x)} 2 : Estimates of state variables x 1 , x 2 , H 1 , H 2 : Gain constants of the high-gain observer which are larger than 1, φ 0 : Nominal function of φ used in high-gain observer computing. Estimation errors {tilde over (x)} 1 , {tilde over (x)} 2  by using the high-gain observer equation (2) are given by equation (3) from equations (1) and (2).
 
                   {     Math   .           ⁢   3     }                                       x     ~   .       1     =         -     H   1       ⁢       x   ~     1       +       x   ~     2                       x     ~   .       2     =         -     H   2       ⁢       x   ~     1       +     δ   ⁡     (     x   ,     x   ~     ,   u     )                 }           (   3   )                           x   ~     1     =       x   1     -       x   ^     1                       x   ~     2     =       x   2     -       x   ^     2                     δ   ⁡     (     x   ,     x   ~     ,   u     )       =       ϕ   ⁡     (     x   ,   u     )       -       ϕ   0     ⁡     (       x   ^     ,   u     )                 }           (   4   )               
where δ: Model error between the nominal model φ 0  and the true but actually unobtainable function φ. Introducing a positive parameter ε much smaller than 1, H 1 , H 2  are given by equation (5).
 
     
       
         
           
             
               
                 
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     As H 1 , H 2  in equation (5) are large gain constants, equation (2) is called by a high-gain observer. By using equation (5), equation (3) is rewritten as equation (6). 
     
       
         
           
             
               
                 
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     The estimation errors {tilde over (x)} 1 , {tilde over (x)} 2  are replaced by new variables η 1 , η 2  as written in equation (7). 
     
       
         
           
             
               
                 
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     By using equation (7), equation (6) is rewritten as equation (8). 
     
       
         
           
             
               
                 
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     As the parameter ε is much smaller than 1, the effects of model error δ on the estimation errors η 1 , η 2  can be made small enough by equation (8). Thus by using the high-gain observer for a model which has screw back pressure as a state variable, the above requirement (A) “High-precision detection” for a pressure detecting means (paragraph{0028}) is satisfied. 
     When the effects of the model error δ on the estimation errors η 1 , η 2  are neglected, equation (8) is rewritten as equation (9). 
     
       
         
           
             
               
                 
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     When K 1 , K 2  are decided so that conjugate complex eigenvalues λ 1 ,  λ   1  of matrix A have a negative real part, that is, Re(λ 1 )=Re(  λ   1 )&lt;0, the estimate errors η 1 , η 2  are given by equation (11) with initial values η 10 , η 20 . 
                   {     Math   .           ⁢   9     }                                       η   1     ⁡     (   t   )       =       exp   ⁡     (         Re   ⁡     (     λ   1     )       ɛ     ⁢   t     )       ⁢     (           C   1     ⁡     (   t   )       ⁢     η   10       +         C   2     ⁡     (   t   )       ⁢     η   20         )                       η   2     ⁡     (   t   )       =       exp   ⁡     (         Re   ⁡     (     λ   1     )       ɛ     ⁢   t     )       ⁢     (           C   3     ⁡     (   t   )       ⁢     η   10       +         C   4     ⁡     (   t   )       ⁢     η   20         )               }           (   11   )               
where t: Time variable, C 1 (t)˜C 4 (t): Sinusoidal components with constant amplitudes and constant frequency decided by K 1 , K 2 . As Re(λ 1 )&lt;0 and ε is much smaller than 1, equation (11) reveals that the time responses η 1 (t), η 2 (t) of estimation errors tend to zero rapidly. In other words, by using high-gain observer equation (2), the above requirement (B) “Detection with small time-lag” for a pressure detecting means can be satisfied.
 
     Although estimates {circumflex over (x)} 1 , {circumflex over (x)} 2  of all state variables are obtained by equation (2), it is sufficient to get only the estimate {circumflex over (x)} 2  because x 1  is detected as output y. Then the high-gain observer is given by equation (12) (non patent literature NPL  2 ). 
     {Math. 10}
 
 {circumflex over ({dot over (x)}   2   =−H{circumflex over (x)}   2   +H{dot over (y)}+φ   0 ( {circumflex over (x)}   2   , y, u )   (12)
 
where H: Gain constant of the high-gain observer which is larger than 1. As time-derivative term of output y is included in the right-hand side of equation (12), equation (12) cannot be used as computing equation by itself. But it can be shown that the high-gain observer by equation (12) satisfies the above two requirements (A) and (B). Equation (13) is given from the third equation in equation (1).
 
{Math. 11}
 
{dot over (y)}={dot over (x)} 1 =x 2    (13)
 
     Equation (14) is given by using equations (12) and (13). 
     {Math. 12}
 
 {circumflex over ({dot over (x)}   2   =−H{circumflex over (x)}   2   +Hx   2 +φ 0 ( {circumflex over (x)}   2   , y, u )   (14)
 
     By using the second equation of equation (1), equation (15) is given from equation (14). 
     
       
         
           
             
               
                 
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                   16 
                   ) 
                 
               
             
           
         
       
     
     Gain constant H is given by equation (17) by introducing a positive parameter ε much smaller than 1. 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     14 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   H 
                   = 
                   
                     
                       K 
                       ɛ 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       
                         K 
                         &gt; 
                         0 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
     By using equation (17), equation (15) is rewritten as equation (18). 
     {Math. 15}
 
ε {tilde over ({dot over (x)}   2   =−K{tilde over (x)}   2 +εδ( x, {tilde over (x)}   2   , y, u )   (18)
 
     As ε is much smaller than 1, the effect of model error δ on the estimation error {tilde over (x)} 2  can be made small enough from equation (18). Therefore by using the high-gain observer for a model which has screw back pressure as a state variable, the above requirement (A) “High-precision detection” for a pressure detecting means can be satisfied. 
     When the effect of model error δ on the estimation error {tilde over (x)} 2  is neglected, equation (18) is rewritten as equation (19). 
     {Math. 16}
 
ε {tilde over ({dot over (x)}   2   =−K{tilde over (x)}   2    (19)
 
     The estimation error {tilde over (x)} 2  is given by equation (20) from equation (19). 
                   {     Math   .           ⁢   17     }                                 x   ~     2     ⁡     (   t   )       =       exp   ⁡     (       -     K   ɛ       ⁢   t     )       ⁢       x   ~     20               (   20   )               
where {tilde over (x)} 20 : Initial value of {tilde over (x)} 2 . As ε is much smaller than 1, equation (20) reveals that the time response {tilde over (x)} 2 (t) of estimation error tends to zero rapidly. In other words, by using high-gain observer equation (12), the above requirement (B) “Detection with small time-lag” for a pressure detecting means can be satisfied. As in equation (12) the minimum number of state variables to be estimated are included and the measurable state variables are excluded, equation (12) is called by a reduced-order high-gain observer because the order of observer equation (12) is lower than that of observer equation (2).
 
     Then a procedure to modify equation (12) is shown so that the time-derivative term of output y is not appeared. A new variable ŵ is given by equation (21). 
     {Math. 18}
 
 ŵ={circumflex over (x)}   2   −Hy    (21)
 
     By using equation (21), equation (12) is rewritten as equation (22). 
     {Math. 19}
 
 {circumflex over ({dot over (w)}=−H ( ŵ+Hy )+φ 0 ( ŵ, y, u )   (22)
 
     Variable ŵ is calculated by equation (22) and estimate {circumflex over (x)} 2  is obtained by equation (23). 
     {Math. 20}
 
 {circumflex over (x)}   2   =ŵ+Hy    (23)
 
     Procedures of applying a high-gain observer for a model of electric-motor driven injection molding machines which has screw back pressure as a state variable are described in detail in Example to be hereinafter described. 
     Advantageous Effects of Invention 
     By applying a high-gain observer for a model of electric-motor driven injection molding machines which has screw back pressure as a state variable, a high-precision pressure detection with small time-lag becomes possible without using a pressure detector. By using the high-gain observer the requirement for a plasticizing controller of electric-motor driven injection molding machines that an adequate screw back pressure is applied to the melted polymer stored at the end of the barrel during the plasticizing process can be satisfied without using a pressure detector and also the five disadvantages described in Background Art can be avoided. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         FIG. 1  is an explanation drawing of a working example which shows a block diagram of a plasticizing controller for an electric-motor driven injection molding machine according to an embodiment of the present invention. 
         FIG. 2  is a view which shows an existing plasticizing mechanism of an electric-motor driven injection molding machine. 
         FIG. 3  is an explanation drawing which shows a block diagram of an existing plasticizing controller for an electric-motor driven injection molding machine. 
         FIG. 4  is a view which shows a plasticizing mechanism of an electric-motor driven injection molding machine according to an embodiment of the present invention. 
         FIG. 5  is an explanation drawing of a working example which shows computer simulation conditions of the plasticizing process according to an embodiment of the present invention. 
         FIG. 6  is an explanation drawing of a working example which shows computer simulation results of screw back pressure estimation by the high-gain observer according to an embodiment of the present invention. 
     
    
    
     DESCRIPTION OF EMBODIMENT 
     Hereinafter, the embodiment of the present invention on the plasticizing controller of electric-motor driven injection molding machines is described based on the drawings. 
     Example 
       FIG. 4  is a view which shows a plasticizing mechanism of an electric-motor driven injection molding machine without using a pressure detector. As the mechanism in  FIG. 4  consists of the parts with the same reference signs as in  FIG. 2  except a pressure detector  12 , explanations of  FIG. 4  are replaced by those of  FIG. 2  described in Background Art. 
       FIG. 1  is an example of a plasticizing controller of an electric-motor driven injection molding machine using a high-gain observer as a screw back pressure detecting means according to an embodiment of the present invention and shows a block diagram of a system configuration for the plasticizing controller. The plasticizing controller consists of a back pressure controller  60  which contains a high-gain observer  27 , a motor controller (servoamplifier) for injection  70 , a screw revolution speed controller  40  and a motor controller (servoamplifier) for plasticization  50 . 
     The back pressure controller  60  executes a control algorithm at a constant time interval and feeds a discrete-time control demand to the motor controller for injection  70 . The back pressure controller  60  consists of a screw back pressure setting device  21 , a subtracter  22 , a pressure controller  24 , a digital/analog (D/A) converter  25 , an analog/digital (A/D) converter  26  and a high-gain observer  27 . 
     The screw back pressure setting device  21  feeds a time sequence of screw back pressure command P* b  to the subtracter  22 . 
     The actual motor current i m  detected in the motor controller for injection  70  is fed to the high-gain observer  27  through the A/D converter  26 . The backward velocity signal v of the screw  9  is fed to the high-gain observer  27  from the motor controller for injection  70 . The signal v is detected by the pulse train from the rotary encoder  14  mounted on the servomotor for injection  3 . The screw revolution speed N s  from the motor controller for plasticization  50  is fed to the high-gain observer  27 . The high-gain observer  27  executes discrete-time arithmetic expressions which are obtained from a mathematical model of a plasticizing mechanism and outputs an estimate of screw back pressure {circumflex over (P)} b  by using the input signals i m , v and N s . 
     The estimate of screw back pressure {circumflex over (P)} b  is fed to the subtracter  22 . The subtracter  22  calculates the control deviation ΔP b  from screw back pressure command P* b  by equation (24). 
     {Math. 21}
 
Δ P   b   =P*   b   −{circumflex over (P)}   b    (24)
 
     The subtracter  22  feeds the control deviation ΔP b  to the pressure controller  24 . 
     The pressure controller  24  calculates a motor current demand i* m  from ΔP b  by using PID control algorithm and feeds the demand i* m  to the motor controller for injection  70  through the D/A converter  25 . 
     The motor controller for injection  70  consists of an analog/digital (A/D) converter  31 , a PWM device  32 , a current transducer  33  of the servomotor for injection  3  and a differentiator  34 . The motor controller for injection  70  is connected to the servomotor for injection  3  equipped with the rotary encoder  14 . 
     The motor current demand i* m  for the servomotor for injection  3  from the back pressure controller  60  is fed to the A/D converter  31  and the A/D converter  31  feeds the demand i* m  to the PWM device  32 . 
     The PWM device  32  applies three-phase voltage to the servomotor for injection  3  so that the servomotor for injection  3  is driven by the motor current demand i* m . The current transducer  33  of the servomotor  3  detects an actual motor current i m  and the motor current i m  is fed to the A/D converter  26  in the back pressure controller  60 . 
     The differentiator  34  receives the pulse train from the rotary encoder  14  mounted on the servomotor for injection  3 , detects the backward velocity signal v of the screw  9  and feeds the signal v to the high-gain observer  27  in the back pressure controller  60 . 
     The compositions and functions of the screw revolution speed controller  40  and the motor controller for plasticization  50  are already described in detail in Background Art. However, in  FIG. 1  the screw revolution speed N s  is fed to the high-gain observer  27  from the differentiator  52  in the motor controller for plasticization  50 . 
     The high-gain observer  27  outputs an estimate {circumflex over (P)} b  of screw back pressure by using an actual motor current signal i m  of the servomotor for injection  3 , a screw backward velocity signal v and a screw revolution speed N s . The mathematical model of a plasticizing mechanism shown in  FIG. 4  is derived as follows, which is necessary to design the high-gain observer  27 . A motion equation of the servomotor for injection  3  axis is given by equation (25). 
                   {     Math   .           ⁢   22     }                               (       J   M     +     J     G   ⁢           ⁢   1         )     ⁢       ⅆ     ω   m         ⅆ   t         =       T   M     -       r   1     ⁢   F               (   25   )               
where J M : Moment of inertia of motor itself, J G1 : Moment of inertia of motor-side gear, w m : Angular velocity of motor, T M : Motor torque, r 1 : Radius of motor-side gear, F: Transmission force of reduction gear, t: Time variable. A motion equation of the ball screw  5  axis is given by equation (26).
 
                   {     Math   .           ⁢   23     }                               (       J   S     +     J     G   ⁢           ⁢   2         )     ⁢       ⅆ     ω   s         ⅆ   t         =         r   2     ⁢   F     -     T   a               (   26   )               
where J S : Moment of inertia of ball screw axis, J G2 : Moment of inertia of load-side gear, w s : Angular velocity of ball screw axis, r 2 : Radius of load-side gear, T a : Ball screw drive torque. A motion equation of the moving part  8  is given by equations (27) and (28).
 
                   {     Math   .           ⁢   24     }                               W   g     ⁢       ⅆ   v       ⅆ   t         =       F   a     -     F   L     -     μ   ⁢           ⁢   W   ⁢     v        v                      (   27   )                   ⅆ     x   s         ⅆ   t       =   v           (   28   )               
where W: Weight of the moving part  8 , g: Gravity acceleration, v: Backward velocity of the screw (the moving part), x s : Screw position(initial position x s =0), F a : Drive force of the ball screw, F L : Applied force by polymer to the screw, μ: Friction coefficient at the slider. A relation between ball screw drive force F a  and ball screw drive torque T a  is given by equation (29).
 
                   {     Math   .           ⁢   25     }                             T   a     =       l     2   ⁢           ⁢   π       ⁢     1   η     ⁢     F   a               (   29   )               
where l: Ball screw lead, η: Ball screw efficiency. Equations among v, w s  and w m  are given by equation (30).
 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     26 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   v 
                   = 
                   
                     
                       
                         l 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         ω 
                         s 
                       
                     
                     = 
                     
                       
                         l 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           r 
                           1 
                         
                         
                           r 
                           2 
                         
                       
                       ⁢ 
                       
                         ω 
                         m 
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     Applied force to the screw F L  is given by equation (31). 
                   {     Math   .           ⁢   27     }                             F   L     =         A   s     ⁢     P   b       +       C     m   ⁢           ⁢   t       ⁢     v        v          ⁢          v        γ                 (   31   )               
where A s : Screw section area, P b : Screw back pressure, C mt : Friction coefficient between the screw and the barrel surface, γ: Velocity power coefficient. A dynamic equation of screw back pressure P b  is given by equations (32) and (33).
 
                   {     Math   .           ⁢   28     }                                 V   b     β     ⁢       ⅆ     P   b         ⅆ   t         =         A   s     ⁢   v     +     Q   f               (   32   )                 V   b   =V   b0   −A   s   x   s    (33)
 
where V b : Polymer volume at the end of a barrel, V b0 : Initial volume of V b  at the start of plasticizing process, Q f : Plasticizing rate (supply flow rate of melted polymer from the screw to the stored polymer at the end of a barrel), β: Bulk modulus of polymer. The characteristics of the servomotor for injection  3  is given by equation (34).
 
{Math. 29}
 
T M =K T i m    (34)
 
where K T : Motor torque coefficient of the servomotor for injection  3 , i m : Motor current of the servomotor for injection  3 . By using equations (25), (26) and (30) and deleting w s  and F, equation (35) is derived.
 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     30 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       { 
                       
                         
                           J 
                           M 
                         
                         + 
                         
                           J 
                           
                             G 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 J 
                                 S 
                               
                               + 
                               
                                 J 
                                 
                                   G 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             
                               ( 
                               
                                 
                                   r 
                                   1 
                                 
                                 
                                   r 
                                   2 
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                       } 
                     
                     ⁢ 
                     
                       
                         ⅆ 
                         
                           ω 
                           m 
                         
                       
                       
                         ⅆ 
                         t 
                       
                     
                   
                   = 
                   
                     
                       T 
                       M 
                     
                     - 
                     
                       
                         
                           r 
                           1 
                         
                         
                           r 
                           2 
                         
                       
                       ⁢ 
                       
                         T 
                         a 
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     By using equations (27), (29), (30) and (35) and deleting T a  and F a , equation (36) is derived. 
                   {     Math   .           ⁢   31     }                               J   eq     ⁢       ⅆ     ω   m         ⅆ   t         =       T   M     -       l     2   ⁢           ⁢   π       ⁢     1   η     ⁢       r   1       r   2       ⁢     (       F   L     +     μ   ⁢           ⁢   W   ⁢     v        v              )                 (   36   )                 J   eq     =       J   M     +     J     G   ⁢           ⁢   1       +       (       J   S     +     J     G   ⁢           ⁢   1         )     ⁢       (       r   1       r   2       )     2       +       W   g     ⁢       (       r   1       r   2       )     2     ⁢       (     l     2   ⁢           ⁢   π       )     2     ⁢     1   η                 (   37   )               
where J eq : Reduced moment of inertia at motor axis. Equation (36) is the motion equation of a total plasticizing mechanism converted to the motor axis. From equations (28) and (30), equation (38) is derived.
 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     32 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       ⅆ 
                       
                         x 
                         s 
                       
                     
                     
                       ⅆ 
                       t 
                     
                   
                   = 
                   
                     
                       
                         r 
                         1 
                       
                       
                         r 
                         2 
                       
                     
                     ⁢ 
                     
                       l 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         π 
                       
                     
                     ⁢ 
                     
                       ω 
                       m 
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     From equations (31), (34) and (36), the motion equation of a linear motion of the screw is given by equation (39). 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     33 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       J 
                       eq 
                     
                     ⁢ 
                     
                       
                         ⅆ 
                         
                           ω 
                           m 
                         
                       
                       
                         ⅆ 
                         t 
                       
                     
                   
                   = 
                   
                     
                       
                         K 
                         T 
                       
                       ⁢ 
                       
                         i 
                         m 
                       
                     
                     - 
                     
                       
                         l 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         1 
                         η 
                       
                       ⁢ 
                       
                         
                           r 
                           1 
                         
                         
                           r 
                           2 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               A 
                               s 
                             
                             ⁢ 
                             
                               P 
                               b 
                             
                           
                           + 
                           
                             
                               C 
                               
                                 m 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 t 
                               
                             
                             ⁢ 
                             
                               v 
                               
                                  
                                 v 
                                  
                               
                             
                             ⁢ 
                             
                               
                                  
                                 v 
                                  
                               
                               γ 
                             
                           
                           + 
                           
                             μ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             W 
                             ⁢ 
                             
                               v 
                               
                                  
                                 v 
                                  
                               
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     Equation (33) is rewritten as equation (40). 
     {Math. 34}
 
 V   b   =V   b0   −A   s   x   s   =A   s ( x   0   −x   s )   (40)
 
where x 0 : Initial length of the stored melted polymer at the end of the barrel at the start of plasticizing process. By using equations (30) and (40), equation (32) is rewritten as equation (41).
 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     35 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           A 
                           s 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               x 
                               0 
                             
                             - 
                             
                               x 
                               s 
                             
                           
                           ) 
                         
                       
                       β 
                     
                     ⁢ 
                     
                       
                         ⅆ 
                         
                           P 
                           b 
                         
                       
                       
                         ⅆ 
                         t 
                       
                     
                   
                   = 
                   
                     
                       
                         A 
                         s 
                       
                       ⁢ 
                       
                         l 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           r 
                           1 
                         
                         
                           r 
                           2 
                         
                       
                       ⁢ 
                       
                         ω 
                         m 
                       
                     
                     + 
                     
                       Q 
                       f 
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
     The variables in the above equations are made dimensionless. By using dimensionless variables, equation (38) is rewritten as equation (42). 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     36 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       ⅆ 
                       
                         ⅆ 
                         t 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           s 
                         
                         
                           x 
                           max 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         l 
                         
                           2 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         
                           r 
                           1 
                         
                         
                           r 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           
                             ω 
                             max 
                           
                           
                             x 
                             max 
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               ω 
                               m 
                             
                             
                               ω 
                               max 
                             
                           
                           ] 
                         
                       
                     
                     = 
                     
                       
                         
                           v 
                           max 
                         
                         
                           x 
                           max 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             ω 
                             m 
                           
                           
                             ω 
                             max 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   42 
                   ) 
                 
               
             
           
         
       
     
     As a positive rotation of the servomotor for injection  3  corresponds to an injection direction of the screw  9 , screw position x s  and screw backward velocity v become negative in the plasticizing process. v max (&gt;0) is the maximum velocity of screw backward movement in the plasticizing process. w max (&gt;0) is the maximum revolution speed of the servomotor for injection corresponding to v max ·x max (&gt;0) is the maximum stroke of screw backward movement in the plasticizing process. 
     By using dimensionless variables, equation (39) is rewritten as equation (43). 
                           ⁢     {     Math   .           ⁢   37     }                                 J   eq     ⁢     ω   max     ⁢       ⅆ     ⅆ   t       ⁡     [       ω   m       ω   max       ]         =         K   T     ⁢       i   max     ⁡     [       i   m       i   max       ]         -       l     2   ⁢   π       ⁢     1   η     ⁢       r   1       r   2       ⁢     A   s     ⁢       P   max     ⁡     [       P   b       P   max       ]         -       l     2   ⁢   π       ⁢     1   η     ⁢       r   1       r   2       ⁢       [       ω   m       ω   max       ]            [       ω   m       ω   max       ]            ⁢     {         C   mt     ⁢     υ   max   γ     ⁢            [       ω   m       ω   max       ]          γ       +     μ   ⁢           ⁢   W       }                 (   43   )               
where i max : Motor current rating of the servomotor for injection  3 , P max : Maximum screw back pressure. In deriving equation (43), equation (44) is used.
 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     38 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       υ 
                       
                         υ 
                         max 
                       
                     
                     ] 
                   
                   = 
                   
                     [ 
                     
                       
                         ω 
                         m 
                       
                       
                         ω 
                         max 
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
     Equation (43) is rewritten as equation (45). 
                           ⁢     {     Math   .           ⁢   39     }                                 ⅆ     ⅆ   t       ⁡     [       ω   m       ω   max       ]       =           T   Mmax         J   eq     ⁢     ω   max         ⁡     [       i   m       i   max       ]       -       l     2   ⁢   π       ⁢     1   η     ⁢       r   1       r   2       ⁢           A   s     ⁢     P   max           J   eq     ⁢     ω   max         ⁡     [       P   b       P   max       ]         -       l     2   ⁢   π       ⁢     1   η     ⁢       r   1       r   2       ⁢     1       J   eq     ⁢     ω   max         ⁢       [       ω   m       ω   max       ]            [       ω   m       ω   max       ]            ⁢     {         C   mt     ⁢     υ   max   γ     ⁢            [       ω   m       ω   max       ]          γ       +     μ   ⁢           ⁢   W       }                 (   45   )               
where T Mmax =K T i max : Motor rating torque of the servomotor for injection  3 .
 
     By using dimensionless variables, equation (41) is rewritten as equation (46). 
                           ⁢     {     Math   .           ⁢   40     }                                 1   β     ⁢     A   s     ⁢     x   max     ⁢     P   max     ⁢     {       [       x   0       x   max       ]     -     [       x   s       x   max       ]       }     ⁢       ⅆ     ⅆ   t       ⁡     [       P   b       P   max       ]         =       A   s     ⁢     υ   max     ⁢     {       [       ω   m       ω   max       ]     +         υ   fmax       υ   max       ⁡     [       Q   f       Q   max       ]         }               (   46   )               
where Q max =A s v fmax : Maximum plasticizing rate. v fmax (&gt;0) is the screw backward velocity corresponding to the maximum plasticizing rate Q max . Equation (46) is rewritten as equation (47).
 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     { 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       41 
                     
                     } 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       ⅆ 
                       
                         ⅆ 
                         t 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           P 
                           b 
                         
                         
                           P 
                           max 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       β 
                       
                         
                           [ 
                           
                             
                               x 
                               0 
                             
                             
                               x 
                               max 
                             
                           
                           ] 
                         
                         - 
                         
                           [ 
                           
                             
                               x 
                               s 
                             
                             
                               x 
                               max 
                             
                           
                           ] 
                         
                       
                     
                     ⁢ 
                     
                       
                         υ 
                         max 
                       
                       
                         
                           x 
                           max 
                         
                         ⁢ 
                         
                           P 
                           max 
                         
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           [ 
                           
                             
                               ω 
                               m 
                             
                             
                               ω 
                               max 
                             
                           
                           ] 
                         
                         + 
                         
                           
                             
                               υ 
                               fmax 
                             
                             
                               υ 
                               max 
                             
                           
                           ⁡ 
                           
                             [ 
                             
                               
                                 Q 
                                 f 
                               
                               
                                 Q 
                                 max 
                               
                             
                             ] 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
     In general dimensionless plasticizing rate [Q f /Q max ] is a function of dimensionless screw back pressure [P b /P max ] and dimensionless screw revolution speed [N s /N max ] given by equation (48). N max  is the maximum screw revolution speed. 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     42 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         Q 
                         f 
                       
                       
                         Q 
                         max 
                       
                     
                     ] 
                   
                   = 
                   
                     f 
                     ⁡ 
                     
                       ( 
                       
                         
                           [ 
                           
                             
                               P 
                               b 
                             
                             
                               P 
                               max 
                             
                           
                           ] 
                         
                         , 
                         
                           [ 
                           
                             
                               N 
                               s 
                             
                             
                               N 
                               max 
                             
                           
                           ] 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   48 
                   ) 
                 
               
             
           
         
       
     
     The mathematical model of a plasticizing mechanism necessary for designing the high-gain observer  27  is given by equations (49), (50) and (51) by using equations (42), (45), (47) and (48). 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     { 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       43 
                     
                     } 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       ⅆ 
                       
                         ⅆ 
                         t 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           x 
                           s 
                         
                         
                           x 
                           max 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         υ 
                         max 
                       
                       
                         x 
                         max 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           ω 
                           m 
                         
                         
                           ω 
                           max 
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   49 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ⅆ 
                       
                         ⅆ 
                         t 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           ω 
                           m 
                         
                         
                           ω 
                           max 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       
                         
                           T 
                           Mmax 
                         
                         
                           
                             J 
                             eq 
                           
                           ⁢ 
                           
                             ω 
                             max 
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             i 
                             m 
                           
                           
                             i 
                             max 
                           
                         
                         ] 
                       
                     
                     - 
                     
                       
                         l 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         1 
                         η 
                       
                       ⁢ 
                       
                         
                           r 
                           1 
                         
                         
                           r 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               A 
                               s 
                             
                             ⁢ 
                             
                               P 
                               max 
                             
                           
                           
                             
                               J 
                               eq 
                             
                             ⁢ 
                             
                               ω 
                               max 
                             
                           
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               P 
                               b 
                             
                             
                               P 
                               max 
                             
                           
                           ] 
                         
                       
                     
                     - 
                     
                       
                         l 
                         
                           2 
                           ⁢ 
                           π 
                         
                       
                       ⁢ 
                       
                         1 
                         η 
                       
                       ⁢ 
                       
                         
                           r 
                           1 
                         
                         
                           r 
                           2 
                         
                       
                       ⁢ 
                       
                         1 
                         
                           
                             J 
                             eq 
                           
                           ⁢ 
                           
                             ω 
                             max 
                           
                         
                       
                       ⁢ 
                       
                         
                           [ 
                           
                             
                               ω 
                               m 
                             
                             
                               ω 
                               max 
                             
                           
                           ] 
                         
                         
                            
                           
                             [ 
                             
                               
                                 ω 
                                 m 
                               
                               
                                 ω 
                                 max 
                               
                             
                             ] 
                           
                            
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               C 
                               mt 
                             
                             ⁢ 
                             
                               υ 
                               max 
                               γ 
                             
                             ⁢ 
                             
                               
                                  
                                 
                                   [ 
                                   
                                     
                                       ω 
                                       m 
                                     
                                     
                                       ω 
                                       max 
                                     
                                   
                                   ] 
                                 
                                  
                               
                               γ 
                             
                           
                           + 
                           
                             μ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             W 
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   50 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       ⅆ 
                       
                         ⅆ 
                         t 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           P 
                           b 
                         
                         
                           P 
                           max 
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       β 
                       
                         
                           [ 
                           
                             
                               x 
                               0 
                             
                             
                               x 
                               max 
                             
                           
                           ] 
                         
                         - 
                         
                           [ 
                           
                             
                               x 
                               s 
                             
                             
                               x 
                               max 
                             
                           
                           ] 
                         
                       
                     
                     ⁢ 
                     
                       
                         υ 
                         max 
                       
                       
                         
                           x 
                           max 
                         
                         ⁢ 
                         
                           P 
                           max 
                         
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           [ 
                           
                             
                               ω 
                               m 
                             
                             
                               ω 
                               max 
                             
                           
                           ] 
                         
                         + 
                         
                           
                             
                               υ 
                               fmax 
                             
                             
                               υ 
                               max 
                             
                           
                           ⁢ 
                           
                             f 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   [ 
                                   
                                     
                                       P 
                                       b 
                                     
                                     
                                       P 
                                       max 
                                     
                                   
                                   ] 
                                 
                                 , 
                                 
                                   [ 
                                   
                                     
                                       N 
                                       s 
                                     
                                     
                                       N 
                                       max 
                                     
                                   
                                   ] 
                                 
                               
                               ) 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   51 
                   ) 
                 
               
             
           
         
       
     
     When the dimensionless plasticizing rate [Q  f /Q max ] is assumed to be proportional to the screw revolution speed, equation (48) is given by equation (52). 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     44 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     f 
                     ⁡ 
                     
                       ( 
                       
                         
                           [ 
                           
                             
                               P 
                               b 
                             
                             
                               P 
                               max 
                             
                           
                           ] 
                         
                         , 
                         
                           [ 
                           
                             
                               N 
                               s 
                             
                             
                               N 
                               max 
                             
                           
                           ] 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           N 
                           s 
                         
                         
                           N 
                           max 
                         
                       
                       ] 
                     
                     ⁢ 
                     
                       g 
                       ⁡ 
                       
                         ( 
                         
                           [ 
                           
                             
                               P 
                               b 
                             
                             
                               P 
                               max 
                             
                           
                           ] 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   52 
                   ) 
                 
               
             
           
         
       
     
     The following state variables x 1 , x 2  and x 3  defined by equation (53) are introduced. 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     45 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           x 
                           1 
                         
                         = 
                         
                           
                             x 
                             s 
                           
                           
                             x 
                             max 
                           
                         
                       
                     
                     
                       
                         
                           x 
                           2 
                         
                         = 
                         
                           
                             ω 
                             m 
                           
                           
                             ω 
                             max 
                           
                         
                       
                     
                     
                       
                         
                           x 
                           3 
                         
                         = 
                         
                           
                             P 
                             b 
                           
                           
                             P 
                             max 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   53 
                   ) 
                 
               
             
           
         
       
     
     Input variables u 1 , u 2  defined by equation (54) are introduced. u 1 , u 2  are measurable. In the design of high-gain observer  27 , the actual motor current i m  of the servomotor for injection  3  is considered to be equal to motor current demand i* m  because the time lag between i* m  and i m  is very small. 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     46 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     u 
                     1 
                   
                   = 
                   
                     
                       
                         
                           i 
                           m 
                         
                         
                           i 
                           max 
                         
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         u 
                         2 
                       
                     
                     = 
                     
                       
                         N 
                         s 
                       
                       
                         N 
                         max 
                       
                     
                   
                 
               
               
                 
                   ( 
                   54 
                   ) 
                 
               
             
           
         
       
     
     The state variable x 2  is supposed to be measurable and output variable y is defined by equation (55). 
     {Math. 47}
 
y=x 2    (55)
 
     The state equations and the output equation representing equations (49), (50), (51), (52) and (55) are given by equations (56)˜(59). 
     {Math. 48}
 
{dot over (x)} 1 =ax 2   (56)
 
 {dot over (x)}   2   =bx   3 +χ( x   2 )+ cu   1    (57)
 
                       x   .     3     =         d     e   -     x   1         ⁢     {       x   2     +       qu   2     ⁢     g   ⁡     (     x   3     )           }       =     ψ   ⁡     (     x   ,     u   2       )                 (   58   )               y=x 2    (59)
 
                           x   =     [           x   1               x   2               x   3           ]       ⁢                     χ   ⁡     (     x   2     )       =         x   2            x   2            ⁢     (       h   ⁢            x   2          γ       +   p     )                     (   60   )                       a   =       υ   max       x   max               b   =       -     l     2   ⁢   π         ⁢     1   η     ⁢       r   1       r   2       ⁢         A   s     ⁢     P   max           J   eq     ⁢     w   max                       c   =       T   Mmax         J   eq     ⁢     ω   max                 d   =       β   ⁢           ⁢     υ   max           x   max     ⁢     P   max                     e   =       x   0       x   max               h   =       -     l     2   ⁢   π         ⁢     1   η     ⁢       r   1       r   2       ⁢         C   mt     ⁢     υ   max   γ           J   eq     ⁢     ω   max                       p   =       -     l     2   ⁢   π         ⁢     1   η     ⁢       r   1       r   2       ⁢       μ   ⁢           ⁢   W         J   eq     ⁢     ω   max                   q   =       υ   fmax       ω   max               }           (   61   )               
where χ(x 2 ) and ψ(x, u 2 ) are nonlinear functions.
 
     As the output variable y=x 2  represented by equation (59) is measurable, by using the above equation (56) the state variable x 1  is calculated by the following equation (62) and is replaced by a variable y s . The variable y s  represents a dimensionless screw position and an initial value of y s  is assumed to be 0. 
     {Math. 49}
 
 x   1   =∫ax   2   dt=a∫ydt=y   s    (62)
 
     Therefore, the state variable x 1  is removed from the state variables and a new state equation and a new output equation are represented by the following equations (63) and (64) by using equations (57)˜(59). 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     { 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       50 
                     
                     } 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               x 
                               . 
                             
                             2 
                           
                         
                       
                       
                         
                           
                             
                               x 
                               . 
                             
                             3 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                               
                                 b 
                               
                             
                             
                               
                                 0 
                               
                               
                                 0 
                               
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   x 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   x 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   χ 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       x 
                                       2 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 
                                   cu 
                                   1 
                                 
                               
                             
                           
                           
                             
                               
                                 ψ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       2 
                                     
                                     , 
                                     
                                       x 
                                       3 
                                     
                                     , 
                                     
                                       u 
                                       2 
                                     
                                     , 
                                     
                                       y 
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                         
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     A 
                                     11 
                                   
                                 
                                 
                                   
                                     A 
                                     12 
                                   
                                 
                               
                               
                                 
                                   
                                     A 
                                     21 
                                   
                                 
                                 
                                   
                                     A 
                                     22 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                               
                           
                           [ 
                           
                               
                           
                           ⁢ 
                           
                             
                               
                                 
                                   x 
                                   2 
                                 
                               
                             
                             
                               
                                 
                                   x 
                                   3 
                                 
                               
                             
                           
                           ] 
                         
                         + 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     χ 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         x 
                                         2 
                                       
                                       ) 
                                     
                                   
                                   + 
                                   
                                     cu 
                                     1 
                                   
                                 
                               
                             
                             
                               
                                 
                                   ψ 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         x 
                                         2 
                                       
                                       , 
                                       
                                         x 
                                         3 
                                       
                                       , 
                                       
                                         u 
                                         2 
                                       
                                       , 
                                       
                                         y 
                                         s 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   63 
                   ) 
                 
               
             
             
               
                 
                   y 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             1 
                           
                           
                             
                               0 
                               ] 
                             
                           
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 x 
                                 2 
                               
                             
                           
                           
                             
                               
                                 x 
                                 3 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   64 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           χ 
                           ⁡ 
                           
                             ( 
                             
                               x 
                               2 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             
                               x 
                               2 
                             
                             
                                
                               
                                 x 
                                 2 
                               
                                
                             
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 h 
                                 ⁢ 
                                 
                                   
                                      
                                     
                                       x 
                                       2 
                                     
                                      
                                   
                                   γ 
                                 
                               
                               + 
                               p 
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           ψ 
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 2 
                               
                               , 
                               
                                 x 
                                 3 
                               
                               , 
                               
                                 u 
                                 2 
                               
                               , 
                               
                                 y 
                                 s 
                               
                             
                             ) 
                           
                         
                         = 
                         
                           
                             d 
                             
                               e 
                               - 
                               
                                 y 
                                 s 
                               
                             
                           
                           ⁢ 
                           
                             { 
                             
                               
                                 x 
                                 2 
                               
                               + 
                               
                                 
                                   qu 
                                   2 
                                 
                                 ⁢ 
                                 
                                   g 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       x 
                                       3 
                                     
                                     ) 
                                   
                                 
                               
                             
                             } 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   65 
                   ) 
                 
               
             
           
         
       
     
     The variable y s  in the state equation (63) is considered to be a new input variable in addition to the input variables u 1 , u 2  and the input variable y s  is given by the following equation (66) by equation (62). 
     {Math. 51}
 
 y   s   =a∫ydt    (66)
 
     As state variable x 2  is measurable, it is not necessary to estimate state variable x 2 . Therefore, the high-gain observer  27  outputs the estimate of state variable x 3  by using the measurable screw backward velocity signal y=x 2 , the actual motor current u 1  and the screw revolution speed u 2 . The estimate {circumflex over (x)} 3  is given by the following equation (68) (non patent literature NPL  2 ). Input variable y s  in equation (68) is calculated in the high-gain observer  27  by using a time integration method of equation (66) applied to the measurable screw backward velocity y=x 2 . K is a gain constant of the high-gain observer  27 . 
     {Math. 52}
 
 {circumflex over ({dot over (x)}   3 =( A   22   +KA   12 ) {circumflex over (x)}   3   −K{{dot over (y)}−A   11   y−χ   0 ( y )− cu   1   }+A   21   y+ψ   0 ( {circumflex over (x)}   3   , u   2   , y, y   s )   (67)
 
 {circumflex over ({dot over (x)}   3   =Kb{circumflex over (x)}   3   −K{{dot over (y)}−χ   0 ( y )− cu   1 }+ψ 0 ( {circumflex over (x)}   3   , u   2   , y, y   s )   (68)
 
where χ 0 (y), ψ 0 ({circumflex over (x)} 3 , u 2 , y, y s ): Nominal functions of χ(y), ψ({circumflex over (x)} 3 , u 2 , y, y s ), respectively, used in the high-gain observer  27 . Equation (68) is rewritten by equation (69).
 
{Math. 53}
 
 {circumflex over ({dot over (x)}   3   +K{dot over (y)}=Kb{circumflex over (x)}   3   +K{χ   0 ( y )+ cu   1 }+ψ 0 ( {circumflex over (x)}   3   , u   2   , y, y   s )   (69)
 
     A new variable ŵ is introduced by the following equation (70). 
     {Math. 54}
 
 ŵ={circumflex over (x)}   3   +Ky    (70)
 
     The estimate {circumflex over (x)} 3  is given by equations (71) and (72) by using equations (69) and (70). 
     {Math. 55}
 
 {circumflex over ({dot over (w)}=Kb ( ŵ−Ky )+ K{χ   0 ( y )+ cu   1 }+ψ 0 ( ŵ, u   2   , y, y   s )   (71)
 
 {circumflex over (x)}   3   =ŵ−Ky    (72)
 
     A positive parameter ε much smaller than 1 is introduced and the gain constant K is given by equation (73) and a new variable {circumflex over (η)} is introduced by equation (74). 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     56 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   K 
                   = 
                   
                     
                       K 
                       1 
                     
                     ɛ 
                   
                 
               
               
                 
                   ( 
                   73 
                   ) 
                 
               
             
             
               
                 
                   
                     η 
                     ^ 
                   
                   = 
                   
                     ɛ 
                     ⁢ 
                     
                       ω 
                       ^ 
                     
                   
                 
               
               
                 
                   ( 
                   74 
                   ) 
                 
               
             
           
         
       
     
     Equation (71) is rewritten as the following equation (75) by using equations (73) and (74). 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     57 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       η 
                       ^ 
                     
                     . 
                   
                   = 
                   
                     
                       
                         
                           K 
                           1 
                         
                         ɛ 
                       
                       ⁢ 
                       
                         b 
                         ⁡ 
                         
                           ( 
                           
                             
                               η 
                               ^ 
                             
                             - 
                             
                               
                                 K 
                                 1 
                               
                               ⁢ 
                               y 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         K 
                         
                           1 
                           ⁢ 
                           χ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           0 
                         
                       
                       ⁡ 
                       
                         ( 
                         y 
                         ) 
                       
                     
                     + 
                     
                       
                         K 
                         1 
                       
                       ⁢ 
                       
                         cu 
                         1 
                       
                     
                     + 
                     
                       ɛ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           ψ 
                           0 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               η 
                               ^ 
                             
                             , 
                             
                               u 
                               2 
                             
                             , 
                             y 
                             , 
                             
                               y 
                               s 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   75 
                   ) 
                 
               
             
           
         
       
     
     The following equation (76) is given from equation (74). 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     58 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     w 
                     ^ 
                   
                   = 
                   
                     
                       1 
                       ɛ 
                     
                     ⁢ 
                     
                       η 
                       ^ 
                     
                   
                 
               
               
                 
                   ( 
                   76 
                   ) 
                 
               
             
           
         
       
     
     By using equation (76), equation (72) is rewritten as the following equation (77). 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     59 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       x 
                       ^ 
                     
                     3 
                   
                   = 
                   
                     
                       
                         1 
                         ɛ 
                       
                       ⁢ 
                       
                         η 
                         ^ 
                       
                     
                     - 
                     Ky 
                   
                 
               
               
                 
                   ( 
                   77 
                   ) 
                 
               
             
           
         
       
     
     Thus the estimate of state variable {circumflex over (x)} 3  is obtained by the high-gain observer  27 . From equations (66), (75) and (77), the calculation procedures are given by equations (78), (79) and (80). 
     (1) Calculation Procedure 1 
     {Math. 60}
 
 y   s   =a∫ydt    (78)
 
(2) Calculation Procedure 2
 
                   {     Math   .           ⁢   61     }                               η   ^     .     =           K   1     ɛ     ⁢     b   ⁡     (       η   ^     -       K   1     ⁢   y       )         +       K   1     ⁢       χ   0     ⁡     (   y   )         +       K   1     ⁢     cu   1       +       ɛψ   0     ⁡     (       η   ^     ,     u   2     ,   y   ,     y   s       )                 (   79   )               
(3) Calculation Procedure 3
 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     62 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       x 
                       ^ 
                     
                     3 
                   
                   = 
                   
                     
                       1 
                       ɛ 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           η 
                           ^ 
                         
                         - 
                         
                           
                             K 
                             1 
                           
                           ⁢ 
                           y 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   80 
                   ) 
                 
               
             
           
         
       
     
     By the calculation procedure 1, y s  is calculated, by the calculation procedure 2, the estimate {circumflex over (η)} is obtained and by the calculation procedure 3 the estimate {circumflex over (x)} 3  is obtained. 
     Then it is shown that the high-gain observer  27  as the pressure detecting means satisfies the following two requirements (A) and (B) described in Solution to Problem. 
     (A) The detection means is high-precision. 
     (B) The detection means has very small time-lag. 
     If the nominal functions χ 0 (y) and ψ 0 ({circumflex over (η)}, u 2 , y, y s ) in equation (79) are replaced with the true but actually unobtainable functions χ(y) and ψ(η, u 2 , y, y s ), the true value η of the estimate {circumflex over (η)} may be obtained by equation (81). 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     63 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     η 
                     . 
                   
                   = 
                   
                     
                       
                         
                           K 
                           1 
                         
                         ɛ 
                       
                       ⁢ 
                       
                         b 
                         ⁡ 
                         
                           ( 
                           
                             η 
                             - 
                             
                               
                                 K 
                                 1 
                               
                               ⁢ 
                               y 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         K 
                         1 
                       
                       ⁢ 
                       
                         χ 
                         ⁡ 
                         
                           ( 
                           y 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         K 
                         1 
                       
                       ⁢ 
                       
                         cu 
                         1 
                       
                     
                     + 
                     
                       ɛψ 
                       ⁡ 
                       
                         ( 
                         
                           η 
                           , 
                           
                             u 
                             2 
                           
                           , 
                           y 
                           , 
                           
                             y 
                             s 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   81 
                   ) 
                 
               
             
           
         
       
     
     Then the estimate error {tilde over (η)}=η−{circumflex over (η)} is obtained by equation (82) by using equations (79) and (81). 
     {Math. 64}
 
ε{dot over ({tilde over (η)}= K   1   b{tilde over (η)}+εK   1 δ 1 ( y )+ε 2 δ 2 ({tilde over (η)}, u 2 , y, y s )   (82)
 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               
                                 δ 
                                 1 
                               
                               ⁡ 
                               
                                 ( 
                                 y 
                                 ) 
                               
                             
                             = 
                             
                               
                                 χ 
                                 ⁡ 
                                 
                                   ( 
                                   y 
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   χ 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   y 
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                             
                           ⁢ 
                           
                             
                               
                                 δ 
                                 2 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     η 
                                     ~ 
                                   
                                   , 
                                   
                                     u 
                                     2 
                                   
                                   , 
                                   y 
                                   , 
                                   
                                     y 
                                     s 
                                   
                                 
                                 ) 
                               
                             
                             = 
                             
                               
                                 ψ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     η 
                                     , 
                                     
                                       u 
                                       2 
                                     
                                     , 
                                     y 
                                     , 
                                     
                                       y 
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                               - 
                               
                                 
                                   ψ 
                                   0 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       η 
                                       ^ 
                                     
                                     , 
                                     
                                       u 
                                       2 
                                     
                                     , 
                                     y 
                                     , 
                                     
                                       y 
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   83 
                   ) 
                 
               
             
           
         
       
     
     As ε is much smaller than 1, the effects of model errors δ 1  and δ 2  on the estimation error {tilde over (η)} can be made small enough by equation (82). In other words, the high-gain observer  27  satisfies the above requirement (A) “High-precision detection” for screw back pressure estimate {circumflex over (x)} 3  obtained by equations (78), (79) and (80). 
     When the effects of model errors δ 1  and δ 2  on the estimation error {tilde over (η)} are neglected in equation (82), equation (82) is rewritten as equation (84). 
     {Math. 65}
 
ε{dot over ({tilde over (η)}=K 1 b{tilde over (η)}  (84)
 
     The estimate error {tilde over (η)} is given by the following equation (85) from equation (84). 
                   {     Math   .           ⁢   66     }                                     η   ~     ⁡     (   t   )       =       exp   ⁡     (           K   1     ⁢   b     ɛ     ⁢   t     )       ⁢       η   ^     0               (               K   1     &gt;   0     ⁢                   b   &lt;   0           )                 (   85   )               
where t: Time variable, {tilde over (η)} 0 : Initial value of estimate error {tilde over (η)}. As b&lt;0 in the plasticization process of injection molding machines and ε is much smaller than 1, equation (85) reveals that the time response {tilde over (η)}(t) of the estimate error tends to zero rapidly. In other words, the high-gain observer  27  satisfies the above requirement (B) “Detection with small time-lag” for screw back pressure estimate {circumflex over (x)} 3  obtained by equations (78), (79) and (80).
 
     As the back pressure controller  60  executes a control algorithm at a constant time interval Δt, the arithmetic expressions (78), (79) and (80) of the high-gain observer  27  are transformed into the discrete-time arithmetic expressions (non patent literature NPL  3 , NPL  4 ). 
     A new parameter α is introduced and the time interval Δt is expressed by equation (86). 
     {Math. 67}
 
Δt=αε  (86)
 
     A discrete-time expression of a time integration equation (78) can be found by using the standard method of trapezoid rule and is given by the following equation (87). 
     {Math. 68}
 
 y   s ( t   k+1 )= y   s ( t   k )+0.5  aαε{y ( t   k )+ y ( t   k+1 )}  (87)
 
     When function values y s (t k ), y(t k ) at the discrete time t k (k=0, 1, 2, . . . ) are represented by y s (k), y(k), equation (87) is given by the following equation (88). 
     {Math. 69}
 
 y   s ( k+ 1)= y   s ( k )+0.5  aαε{y ( k )+ y ( k+ 1)}  (88)
 
     As the time interval Δt is small, the numerical time integration equation (88) of the output variable y(t) is considered to be high-precision. 
     Next, a discrete-time equivalent of the continuous-time equation (79) can be found by using the standard method of forward rectangular rule which gives the relation between the Laplace-transform operator s representing time-derivative operation and z-transform operator z as follows. 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     70 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   s 
                   = 
                   
                     
                       
                         z 
                         - 
                         1 
                       
                       
                         Δ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                     
                     = 
                     
                       
                         z 
                         - 
                         1 
                       
                       αɛ 
                     
                   
                 
               
               
                 
                   ( 
                   89 
                   ) 
                 
               
             
           
         
       
     
     By using equation (89), equation (79) is rewritten as the following equation (90). 
     
       
         
           
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     { 
                     
                       Math 
                       . 
                       
                           
                       
                       ⁢ 
                       71 
                     
                     } 
                   
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         z 
                         - 
                         1 
                       
                       αɛ 
                     
                     ⁢ 
                     
                       η 
                       ^ 
                     
                   
                   = 
                   
                     
                       
                         
                           K 
                           1 
                         
                         ɛ 
                       
                       ⁢ 
                       
                         b 
                         ⁡ 
                         
                           ( 
                           
                             
                               η 
                               ^ 
                             
                             - 
                             
                               
                                 K 
                                 1 
                               
                               ⁢ 
                               y 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         K 
                         1 
                       
                       ⁢ 
                       
                         
                           χ 
                           0 
                         
                         ⁡ 
                         
                           ( 
                           y 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         K 
                         1 
                       
                       ⁢ 
                       
                         cu 
                         1 
                       
                     
                     + 
                     
                       
                         ɛψ 
                         0 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             η 
                             ^ 
                           
                           , 
                           
                             u 
                             2 
                           
                           , 
                           y 
                           , 
                           
                             y 
                             s 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   90 
                   ) 
                 
               
             
           
         
       
     
     The discrete-time expression of equation (90) similar to equation (88) is given by the following equation (91). 
                                 ⁢     {     Math   .           ⁢   72     }                                                     η   ^     ⁡     (     k   +   1     )       -       η   ^     ⁡     (   k   )         =       α   ⁢           ⁢     K   1     ⁢     b   ⁡     (         η   ^     ⁡     (   k   )       -       K   1     ⁢     y   ⁡     (   k   )           )         +     αɛ   ⁢           ⁢     K   1     ⁢       χ   0     ⁡     (   k   )         +     αɛ   ⁢           ⁢     K   1     ⁢       cu   1     ⁡     (   k   )         +       αɛ   2     ⁢       ψ   0     ⁡     (   k   )                   (   91   )                       ⁢         χ   0     ⁡     (   k   )       =         y   ⁡     (   k   )              y   ⁡     (   k   )              ⁢     (       h   ⁢            y   ⁡     (   k   )            γ       +   p     )                 (   92   )                       ⁢         ψ   0     ⁡     (   k   )       =       d     e   -       y   s     ⁡     (   k   )           ⁢     {       y   ⁡     (   k   )       +         qu   2     ⁡     (   k   )       ⁢   g   ⁢     (         η   ^     ⁡     (   k   )       ,     y   ⁡     (   k   )         )         }                                 
where {circumflex over (η)}(k): Estimate {circumflex over (η)}(t k ) at the discrete-time t k , y(k), u 1 (k), u 2 (k), y s (k): y(t k ), u 1 (t k ), u 2 (t k ), y s (t k ) at the discrete-time t k , χ 0 (k), ψ 0 (k): χ 0 (t k ), ψ 0 (t k ) at the discrete-time t k . χ 0 (k) is given by equation (60) and ψ 0 (k) is given by equation (58). Equation (91) is rewritten as the following equation (93).
 
{Math. 73}
 
{circumflex over (η)}( k+ 1)=(1+α k   1   b ){circumflex over (η)}( k )−α K   1   2   by ( k )+αε K   1 χ 0 ( k )+αε K   1   cu   1 ( k )+αε 2 ψ 0 ( k )   (93)
 
     The discrete-time equivalent of equation (80) is given by equation (94). 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     74 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         x 
                         ^ 
                       
                       3 
                     
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       ɛ 
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             η 
                             ^ 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         - 
                         
                           
                             K 
                             1 
                           
                           ⁢ 
                           
                             y 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   94 
                   ) 
                 
               
             
           
         
       
     
     The high-gain observer  27  obtains screw back pressure estimate {circumflex over (x)} 3 (k) at the discrete-time t k  by executing the arithmetic expressions of equations (88), (93) and (94) at a constant time interval Δt. The high-gain observer  27  by equations (88), (93) and (94) does not estimate the measurable state variable x 2 (k) (screw velocity) and estimates only necessary state variable x 3 (k) (screw back pressure) and so is called by a reduced-order high-gain observer. 
     The results of computer simulation are shown in  FIG. 6  when the high-gain observer  27  is used in the plasticizing process of an electric-motor driven injection molding machine. The constants of the mathematical model are as follows. 
     Maximum stroke of screw backward movement x max =20.0 cm 
     Maximum screw backward velocity v max =2.0 cm/sec 
     Maximum screw back pressure P max 32 19.6 MPa 
     Maximum revolution speed of the servomotor for injection w max =31.67 rad/sec (302.4 rpm) 
     The constants a, b, c and d in equations (56)˜(58) are expressed in equation (95). 
     
       
         
           
             
               
                 
                   { 
                   
                     Math 
                     . 
                     
                         
                     
                     ⁢ 
                     75 
                   
                   } 
                 
               
               
                 
                     
                 
               
             
             
               
                 
                   
                     
                       
                         
                           a 
                           ⁢ 
                             
                           = 
                           
                             0.1000 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sec 
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           b 
                           ⁢ 
                             
                           = 
                           
                             
                               - 
                               4.757 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sec 
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           c 
                           ⁢ 
                             
                           = 
                           
                             24.576 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sec 
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           d 
                           ⁢ 
                             
                           = 
                           
                             9.177 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               sec 
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                     
                   
                   } 
                 
               
               
                 
                   ( 
                   95 
                   ) 
                 
               
             
           
         
       
     
     A monotone decreasing characteristics is used for the characteristics of function g({circumflex over (P)} b /P max ), which decides the plasticizing rate according to the value of {circumflex over (P)} b /P max  at the maximum screw revolution speed N max . The gain constant K of equation (73) used by the high-gain observer  27  is given by equation (96). The data K 1 =0.01, ε=0.01 and Δt=5 msec are used. 
     {Math. 76}
 
K=1.0   (96)
 
       FIG. 5  shows simulation conditions for the screw back pressure control.  FIG. 5(   a ) shows a time sequence of screw revolution speed command N* s =N s  from the screw revolution speed setting device  41 .  FIG. 5(   b ) shows a time sequence of screw back pressure command P* b  from the screw back pressure setting device  21 . 
       FIG. 6  shows time responses of screw back pressure at the control of screw back pressure.  FIG. 6(   a ) shows the time response of screw back pressure when the control of screw back pressure P b  is carried out by the plasticizing controller shown in  FIG. 3  using the pressure detector  12 . The time response of actual screw back pressure P b  shown in  FIG. 6(   a ) agrees well with that of screw back pressure command P* b  shown in  FIG. 5(   b ). 
     When the high-gain observer  27  is supposed to be used under the control system shown in  FIG. 3  and to calculate the estimate of screw back pressure {circumflex over (P)} b  by using the screw velocity signal y(t), the actual motor current signal u 1 (t) of the servomotor for injection and the screw revolution speed signal u 2 (t),  FIG. 6(   b ) shows the time response of the estimated screw back pressure {circumflex over (P)} b /P max . As the time response of screw back pressure in  FIG. 6(   a ) agrees well with that of estimated screw back pressure {circumflex over (P)} b  in  FIG. 6(   b ), it is revealed that the high-gain observer  27  can estimate the screw back pressure exactly with small time-lag. 
       FIG. 6(   c ) shows the time response of screw back pressure P b  when the control of screw back pressure is carried out by the plasticizing controller shown in  FIG. 1  using the high-gain observer  27 . As the time response of screw back pressure P b  shown in  FIG. 6(   a ) using the pressure detector  12  agrees well with that of screw back pressure P b  shown in  FIG. 6(   c ), a good control of screw back pressure can be realized by the high-gain observer  27  without using the pressure detector  12 . 
     INDUSTRIAL APPLICABILITY 
     In the plasticizing control apparatus and the plasticizing control method of electric-motor driven injection molding machines, the following five disadvantages can be avoided by using the estimated screw back pressure obtained by the high-gain observer as a feedback signal of screw back pressure in place of a pressure detector. 
     (1) A highly reliable pressure detector is very expensive under high pressure circumstances. 
     (2) Mounting a pressure detector in the cavity or the barrel nozzle part necessitates the troublesome works and the working cost becomes considerable. 
     (3) Mounting a load cell in an injection shafting alignment from a servomotor for injection to a screw complicates the mechanical structure and degrades the mechanical stiffness of the structure. 
     (4) A load cell which uses strain gauges as a detection device necessitates an electric protection against noise for weak analog signals. Moreover the works for zero-point and span adjustings of a signal amplifier are necessary (patent literature PTL  13 ). 
     (5) For the improvement of the control accuracy of screw back pressure, the usage of two kinds of pressure detectors with different dynamic ranges brings about the cost increase (patent literature PTL  12 ). 
     As the high-gain observer can estimate the screw back pressure exactly with small time-lag, the estimate of screw back pressure obtained by the high-gain observer can be used to monitor the screw back pressure and can be used as a feedback signal in the control system. Thus the high-gain observer of the present invention can be applied to the plasticizing control apparatus and the plasticizing control method of electric-motor driven injection molding machines. 
     
       
         
           
               
             
               
                   
               
               
                 {Reference Signs List } 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
            
               
                 1 
                 Metal mold 
               
               
                 2 
                 Barrel 
               
               
                 3 
                 Servomotor for injection 
               
               
                 4 
                 Reduction gear 
               
               
                 5 
                 Ball screw 
               
               
                 6 
                 Bearing 
               
               
                 7 
                 Nut 
               
               
                 8 
                 Moving part 
               
               
                 9 
                 Screw 
               
               
                 10 
                 Reduction gear 
               
               
                 11 
                 Servomotor for plasticization 
               
               
                 12 
                 Pressure detector 
               
               
                 13 
                 Linear slider 
               
               
                 14 
                 Rotary encoder 
               
               
                 15 
                 Rotary encoder 
               
               
                 16 
                 Hopper 
               
               
                 17 
                 Cavity 
               
               
                 20 
                 Back pressure controller 
               
               
                 21 
                 Screw back pressure setting device 
               
               
                 22 
                 Subtracter 
               
               
                 23 
                 Analog/digital (A/D) converter 
               
               
                 24 
                 Pressure controller 
               
               
                 25 
                 Digital/analog (D/A) converter 
               
               
                 26 
                 Analog/digital (A/D) converter 
               
               
                 27 
                 High-gain observer 
               
               
                 30 
                 Motor controller (servoamplifier) for injection 
               
               
                 31 
                 Analog/digital (A/D) converter 
               
               
                 32 
                 PWM (Pulse Width Modulation) device 
               
               
                 33 
                 Current transducer of the servomotor for injection 
               
               
                 34 
                 Differentiator 
               
               
                 40 
                 Screw revolution speed controller 
               
               
                 41 
                 Screw revolution speed setting device 
               
               
                 50 
                 Motor controller (servoamplifier) for plasticization 
               
               
                 51 
                 Subtracter 
               
               
                 52 
                 Differentiator 
               
               
                 53 
                 Speed controller 
               
               
                 54 
                 PWM (Pulse Width Modulation) device 
               
               
                 60 
                 Back pressure controller 
               
               
                 70 
                 Motor controller (servoamplifier) for injection