Patent Publication Number: US-10325788-B2

Title: Temperature controller of semiconductor wafer and temperature control method of semiconductor wafer

Description:
The entire disclosure of Japanese Patent Application No. 2016-057440 filed Mar. 22, 2016 is expressly incorporated by reference herein. 
     TECHNICAL FIELD 
     The present invention relates to a temperature controller of a semiconductor wafer and a temperature control method of a semiconductor wafer. 
     BACKGROUND ART 
     A process for treating a semiconductor wafer (e.g., a silicon wafer) used for a semiconductor includes controlling an in-plane temperature distribution of the semiconductor wafer as desired while controlling a temperature of the semiconductor wafer to a temperature setpoint. 
     Accordingly, the temperature of the semiconductor wafer is typically controlled to a desired temperature by providing a plurality of temperature adjusters and control loops to the respective temperature adjusters and independently controlling the temperature adjusters with the control loops. As one of such multi-input multi-output control methods, a model following servo control method is used. 
     The model following servo control method includes: selecting a model (e.g., a second-order lag system) having dynamic characteristics generally desired as a reference model; and providing a setpoint response to make a control variable to follow a step response of the reference model as a reference trajectory. 
     In order to improve a throughput in the process, it is desirable to intentionally saturate the manipulated variable to reach a setpoint at the maximum speed in the shortest time. 
     Patent Literature 1 (JP2016-12228A) discloses a technology of generating a reference model based on a response in which, in a plurality of control loops, a manipulated variable of a first control loop having the slowest response speed is defined as 100% and the rest of the control loops is controlled to follow the first control loop; with use of the generated reference model, searching for a manipulated variable pattern, using a predetermined evaluation function, for reaching the temperature setpoint in the shortest time; giving a manipulated variable of the searched optimal manipulated variable pattern to the reference model to obtain an output; and using the obtained output as a reference model output. 
     In the technology of Patent Literature 1, a single reference model is generated based on the dynamic characteristics in heating to provide the reference model output to be followed. 
     However, in a thermo-module such as Peltier element, the dynamic characteristics sometimes differ in heating and cooling. Since the dynamic characteristics in the heating exhibit a larger output, when the reference model output for the heating is used in the cooling, the reference model output precedes an actual output, so that uniformity cannot be kept. 
     When the reference model output for the cooling is used, the response speed becomes slow although the uniformity is ensured. 
     SUMMARY OF THE INVENTION 
     An object of the invention is to provide a temperature controller and a temperature control method of a semiconductor wafer which allow a temperature of the semiconductor wafer to follow a reference model output irrespective of a heating or a cooling and reach a temperature setpoint in and around the shortest time while keeping uniformity. 
     An aspect of the invention is a temperature controller of a semiconductor wafer using a plurality of temperature adjusters usable for heating and cooling the semiconductor wafer, the temperature controller configured to control temperatures of the respective temperature adjusters to adjust temperatures of the semiconductor wafer, the temperature controller including: a plurality of control loops respectively set in the plurality of temperature adjusters; a plurality of temperature detectors provided to the respective control loops and configured to detect the temperatures of the semiconductor wafer adjusted by the respective temperature adjusters; and a manipulated variable calculator configured to calculate manipulated variables to be respectively given to the temperature adjusters of the control loops based on the temperatures detected by the temperature detectors, the manipulated variable calculator including: a reference model generated based on a response in which, in the plurality of control loops, a manipulated variable of a first control loop having the slowest response speed is defined as 100% and the rest of the control loops is controlled to follow the first control loop; an optimal manipulated variable pattern searching unit configured to search online or in advance offline an optimal manipulated variable pattern for reaching a temperature setpoint in the shortest time of period, using the reference model; a reference model output generator configured to generate a reference model output to give to the reference model a manipulated variable of the optimal manipulated variable pattern searched by the optimal manipulated variable pattern searching unit; a model following servo calculator configured to use the reference model output obtained by the reference model output generator; and a reference model selecting unit configured to select one of a predetermined heating reference model and a predetermined cooling reference model depending on whether the temperature adjustment by the plurality of temperature adjusters is for heating or cooling, and apply the selected heating or cooling reference model to the reference model. 
     In the above aspect of the invention, it is preferable that the reference model selecting unit is configured to select the reference model based on: a heating or cooling starting temperature when the manipulated variable is switched; a temperature setpoint; and an equilibrium temperature. 
     In the above aspect of the invention, it is preferable to further include a reference model cluster and an optimal control gain cluster of a number of combinations of heating and cooling in the plurality of control loops configured to heat or cool, in which it is preferable that the reference model selecting unit selects the reference model from the reference model cluster to output the reference model to the reference model output generator and selects an optimal control gain from the optimal control gain cluster to output the optimal control gain to the model following servo calculator. 
     Another aspect of the invention is a temperature control method of a semiconductor wafer using a plurality of temperature adjusters usable for heating and cooling the semiconductor wafer, the temperature control method of controlling temperatures of the plurality of temperature adjusters using control loops respectively set in the temperature adjusters, the temperature control method including: generating a reference model based on a response in which, in the control loops, a manipulated variable of a first control loop having the slowest response speed is defined as 100% and the rest of the control loops is controlled to follow the first control loop; selecting a heating reference model or a cooling reference model depending on whether a to-be-controlled temperature adjuster of the plurality of temperature adjusters is to be heated or to be cooled; searching for, online or in advance offline using the reference model, an optimal manipulated variable pattern for reaching a temperature setpoint in the shortest time; giving to the selected reference model a manipulated variable of the searched optimal manipulated variable pattern to generate a reference model output; and using the generated reference model output to perform a model following servo control. 
     According to the above aspects of the invention, the reference model output can be generated depending on a heating responsiveness or a cooling responsiveness of the temperature adjusters since the reference model selecting unit is provided, so that the reference model output can be followed irrespective of a heating or a cooling and a temperature setpoint can be reached in and around the shortest time while keeping uniformity. 
    
    
     
       BRIEF DESCRIPTION OF DRAWING(S) 
         FIG. 1  is a block diagram showing a temperature adjustment apparatus according to an exemplary embodiment of the invention. 
         FIG. 2A  is a cross-sectional view showing the arrangement of a temperature adjuster and a temperature sensor in the exemplary embodiment. 
         FIG. 2B  is a plan view showing the arrangement of the temperature adjuster and the temperature sensor in the exemplary embodiment. 
         FIG. 3  is a block diagram showing a configuration of a controller configured to control the temperature adjustment apparatus in the exemplary embodiment. 
         FIG. 4  is a graph for explaining a reference model in the exemplary embodiment. 
         FIG. 5A  is a diagram for explaining generation of the reference model in the exemplary embodiment. 
         FIG. 5B  is another diagram for explaining generation of the reference model in the exemplary embodiment. 
         FIG. 6  is a flow chart for explaining an operation in the exemplary embodiment. 
         FIG. 7  is a flow chart for explaining another operation in the exemplary embodiment. 
         FIG. 8  is a diagram showing a control system used for simulation for checking effects in the exemplary embodiment. 
         FIG. 9  is a plan view showing the control system used for simulation for checking the effects in the exemplary embodiment. 
         FIG. 10  is a graph showing the maximum response waveform of the temperature adjuster in heating. 
         FIG. 11  is a graph showing the maximum response waveform of the temperature adjuster in cooling. 
         FIG. 12  is a graph showing a comparison between a heating responsiveness and a cooling responsiveness of the temperature adjuster. 
         FIG. 13  is a graph showing simulation results in the related art. 
         FIG. 14  is a graph showing simulation results in the exemplary embodiment. 
     
    
    
     DESCRIPTION OF EMBODIMENT(S) 
     Exemplary embodiment(s) of the invention will be described below with reference to the attached drawings. 
     [1] Structure of Temperature Adjustment Apparatus  1   
       FIG. 1  shows a temperature adjustment apparatus  1  according to a first exemplary embodiment of the invention. The temperature adjustment device  1  controls a temperature of a silicon wafer W placed on a plate-shaped stage  2  to a temperature setpoint to control an in-plane temperature distribution of the silicon wafer W. The temperature adjustment apparatus  1  is used, for instance, in a dry etching process. 
     The temperature adjustment device  1  includes the plate-shaped stage  2  and a temperature adjuster  3 . The temperature adjuster  3 , which is configured to heat or cool a target zone of the stage  2 , is preferably in a form of a thermoelectric conversion element such as a Peltier element. When the temperature adjuster  3  is used only for heating, a heater is usable as the temperature adjuster  3 . 
     The stage  2  is disposed in a vacuum chamber  4 . The silicon wafer W (semiconductor wafer) is placed on the stage  2 . The silicon wafer W is electrostatically held on the stage  2 . It should be noted that helium gas may be delivered between the stage  2  and the silicon wafer W to enhance efficiency in heat transfer between the stage  2  and the silicon wafer W. 
     In the dry etching process, the vacuum chamber  4  is air-purged to be kept at a predetermined low pressure state. 
     In the stage  2 , a plurality of temperature adjusters  3  are disposed as shown in  FIGS. 2A and 2B  so as to adjust the in-plane temperature distribution of the silicon wafer W placed on the stage  2 . 
       FIG. 2A  is a cross-sectional view of the stage  2 . The temperature adjusters  3  are disposed on a base plate  7 . A plate  5  is placed on the temperature adjusters  3 . A plurality of temperature sensors  6  (temperature detectors) are disposed in the plate  5 . 
       FIG. 2B  is a plan view of the stage  2 , showing that the stage  2  is divided into three concentric zones  2 A (a zone  1  described later),  2 B (a zone  2  described later) and  2 C (a zone  3  described later), in each of which the temperature adjusters  3  are disposed. The temperature sensors  6  (temperature detectors) in the plate  5  are disposed at positions corresponding to the temperature adjusters  3 . 
     The zones  2 A,  2 B and  2 C of the stage  2  can be independently heated or cooled by electrifying the temperature adjusters  3 . Accordingly, by adjusting electrification to the temperature adjusters  3  to control the temperature adjusters  3 , the in-plane temperature distribution of the silicon wafer W on the stage  2  is adjustable. The temperature adjusters  3  in each of the zones  2 A,  2 B and  2 C have a plurality of control loops and are controlled by a controller  24  serving as a temperature controller. 
     2. Structure of Controller  24   
       FIG. 3  shows a block diagram showing the controller  24  in the exemplary embodiment. The controller  24  includes a manipulated variable calculator  30  serving as a manipulated variable calculating unit. The manipulated variable calculator  30  includes: a simulator  31  serving as an optimal manipulated variable pattern searching unit; a manipulated variable switching unit  32 ; a reference model  33 ; an error calculator  34  serving as a model following servo calculator; an integrator  35 ; and an error calculator  36 . The simulator  31  and the manipulated variable switching unit  32  define a reference model output generator of the invention. 
     The manipulated variable calculator  30  in the exemplary embodiment further includes: a reference model selecting unit  37  serving as a reference model selecting unit; a reference model cluster  38 ; an optimal control gain cluster  39 ; and an observer  40 . 
     A state feedback gain K 1  is set through the observer  40  for an output from the temperature sensor  6  of the stage  2 . An integral gain K 2  is set for an output from the error calculator  34 . A state feedforward gain K 3  is set for an output from the reference model  33 . An input to each of the components in each of the block diagrams is shown below. 
     SV: temperature setpoint 
     y 0 : starting temperature at a time of switching 
     Teq: equilibrium temperature 
     Xr: state quantity of a reference model 
     Yr: reference trajectory 
     t 1 : switching time 
     Xr: state quantity of a plant 
     Yp: control variable (temperature) 
     Given that the stage  2  (control target) is controlled by multi-input multi-output in a form of an m-input m-output system, a state variable of the control target is expressed by formulae (1) and (2) below.
 
 x   p ( k+ 1)= A   p   x   p ( k )+ B   p   u ( k )  (1)
 
 y ( k )= C   p   x   p ( k )  (2)
 
     wherein 
     x p :n×1, u:m×1, y:m×1, A p :n×n, B p :n×m, C p :m×n each of which shows a vector or a matrix. 
     Herein, the gain matrixes K 1 , K 2  and K 3  are represented by a formula (3) below. depends on an order of the reference model.
 
 K 1 :m×n, K 2 :m×p,K 3 :m×m   (3)
 
     The reference model  33  selectively employs a linear system (a state equation and a transfer function) having a desired response (dynamic characteristics), in which control variables are made to follow a step response without an error. In the multi-input multi-output system, as long as the same reference model  33  is used for all the loops, a response follows the same reference trajectory, so that uniformity among the loops can be expected. 
     The gains (K 1 , K 2 , K 3 ) of each of the components are determined according to an optimal control method. This method is applicable with merits of the optimal control (i.e., stability guaranty and robustness) and without considering decoupling and is suitable for the multi-input multi-output system with interaction such as the control target described herein. 
     In the exemplary embodiment, an ideal response for the requirement of “reaching a temperature setpoint in the shortest time” is obtained in the following manner. 
     As shown in  FIG. 4 , a manipulated variable pattern for reaching a temperature setpoint SV in the shortest time is achieved by accelerating with the maximum manipulated variable to an intermediate point X, decelerating with the minimum manipulated variable from the point X to the temperature setpoint SV, and switching from the minimum manipulated variable to a steady state manipulated variable MVss required for keeping a current level after reaching the temperature setpoint SV. 
     When this manipulated variable pattern is inputted to a plant model, the plant should reach a setpoint in the shortest time. Accordingly, although a step input is only given to a transfer function showing a desirable step response in a typical model following control as shown in  FIG. 5A , it is considered to input a manipulated variable pattern for controlling in the shortest time to the plant model itself as shown in  FIG. 5B . 
     In the multi-input multi-output system, the number of the manipulated variable patterns as shown in  FIG. 5B  needs to be equal to the number of the control loops. Moreover, unless the control variables of the respective control loops are uniform under interaction, the reference trajectory to be followed is also not uniform. Since it is very difficult to make such manipulated variable patterns only in a combination of step signals, the following method is taken. 
     (1) Using a plant model Gp(s) or an actual plant, the maximum speed response is determined while keeping uniformity of the control variables of the respective control loops. A 100% maximum manipulated variable MVmax is given to a first zone at the slowest speed response among three zones. The remaining two zones are each given a manipulated variable at which the remaining two zones realize the maximum speed while keeping uniformity when following the first zone. 
     (2) The control loop having the slowest speed response is selected from response waveforms obtained in (1). Assuming that the slowest speed response of the control loop is a response by a step input, a reference model Gm(s) of a dead time+a first-order lag system is obtained by a system identification. 
     (3) With respect to the reference model Gm(s) obtained in (2), a manipulated variable for achieving an optimal time control is determined by sequential simulation. Herein, an optimal value of a switching time t 1  is searched for as a value for switching from the maximum manipulated variable to a steady state manipulated variable after the elapse of t 1  seconds. 
     As an evaluation function used for searching for the optimal value, IAE (Integral of Absolute Error) (described in detail later) evaluation model is employed to search for a manipulated variable pattern allowable for the minimum evaluation function. 
     (4) By using the reference model Gm(s) for all the inputs and the outputs, the reference trajectory is used in common for all the control loops, so that uniformity of the control variables among the control loops can be expected. It is also possible to prepare reference models corresponding in number to the control loops and set different setpoints to the respective reference models. 
     The simulator  31  serving as the optimal manipulated variable pattern searching unit sequentially searches for a switching time t 1  when the temperature setpoint SV is inputted, and outputs the obtained switching time t 1  to the manipulated variable switching unit  32 . It should be noted that the search may be made online, or alternatively, offline in advance. 
     The manipulated variable switching unit  32  determines the manipulated variable pattern MVref for switching from the maximum manipulated variable MVmax to the steady state manipulated variable MVss, and outputs the manipulated variable pattern MVref to the reference model  33 . Although the optimal time is sequentially searched for online by the simulator  31  in the exemplary embodiment, the optimal time may be obtained in any manner. A table of the manipulated variable pattern for a control target, in which the temperature setpoint SV is associated with the optimal time for reaching the temperature setpoint SV in the shortest time, may be prepared offline in advance. When the temperature setpoint SV is inputted, the optimal manipulated variable pattern MVref may be selected with reference to the table. 
     Using the plant model Gp(s) or the actual plant as described above, the reference model  33  determines the maximum speed response achievable while keeping the control variables of the respective control loops, and gives the 100% manipulated variable to the slowest response speed zone. For instance, the reference model  33  is expressed by a dead time+a first-order lag system as represented by a formula (4) below. 
     
       
         
           
             
               
                 
                   
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     Patent Literature 1 discloses a technology in which only one reference model generated based on results of a heating test is used. However, the thermo-module exhibits different dynamic characteristics depending on heating and cooling. Particularly, a cooling capability is smaller than a heating capability since a generated Joule heat acts for cancelling the cooling. 
     Accordingly, in the exemplary embodiment, considering both of the heating reference model and the cooling reference model, the reference model cluster  38  and the optimal control gain cluster  39  each in number corresponding to eight (2 3 ) combinations of the heating and the cooling of the zones  1  to  3  are prepared and selected by the reference model selecting unit  37 . 
     The reference model collectively for all the zones (three zones) is represented by state equations (5) and (6).
 
 X   r ( k+ 1)= A   r   X   r ( k )+ B   r   U   r ( k )  (5)
 
 Y   r ( k )= C   r   X   r ( k )  (6)
 
     Herein, matrixes A r , B r , C r , X r  and Y r  are represented by formulae (7), (8) and (9). A combination of the heating h and the cooling c is described in a subscript [ ]. 
     
       
         
           
             
               
                 
                   
                     
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     Since there are the eight combinations of the heating and the cooling of the three zones as described above, the matrixes A r , B r  and C r  each include the eight combinations, thereby defining the reference model cluster  38 . For instance, the reference model cluster  38  is represented by a formula (10). Herein, the subscripts 1, 2 . . . 8 of the matrixes A r , B r  and C r  represent the combinations of the heating and the cooling in the formula (7). 
     
       
         
           
             
               
                 
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     Next, state equations of the plant model are assumed as formula (11) and (12).
 
 X   p ( k+ 1)= A   p   X   p ( k )+ B   p   U   p ( k )  (11)
 
 Y   p ( k )= C   p   X   p ( k )  (12)
 
     Augmented system for the model following servo system is represented by formulae (13) and (14). 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                                         1 
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           = 
                           
                             
                               
                                 [ 
                                 
                                   
                                     
                                       
                                         A 
                                         p 
                                       
                                     
                                     
                                       0 
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       0 
                                     
                                     
                                       
                                         A 
                                         r 
                                       
                                     
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       
                                         
                                           - 
                                           
                                             C 
                                             p 
                                           
                                         
                                         ⁢ 
                                         
                                           A 
                                           p 
                                         
                                       
                                     
                                     
                                       
                                         
                                           C 
                                           r 
                                         
                                         ⁢ 
                                         
                                           A 
                                           r 
                                         
                                       
                                     
                                     
                                       I 
                                     
                                   
                                 
                                 ] 
                               
                               ⁢ 
                                 
                               [ 
                               
                                 
                                   
                                     
                                       Δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           X 
                                           p 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           k 
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       Δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         
                                           X 
                                           r 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           k 
                                           ) 
                                         
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       e 
                                       ⁡ 
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             + 
                           
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     B 
                                     p 
                                   
                                 
                               
                               
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     
                                       - 
                                       
                                         C 
                                         p 
                                       
                                     
                                     ⁢ 
                                     
                                       B 
                                       p 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           Δ 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             
                               U 
                               p 
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               A 
                               a 
                             
                             ⁢ 
                             
                               
                                 X 
                                 a 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               B 
                               a 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 U 
                                 p 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     e 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               0 
                             
                             
                               0 
                             
                             
                               I 
                             
                           
                         
                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     X 
                                     p 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 Δ 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   
                                     X 
                                     r 
                                   
                                   ⁡ 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 e 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         C 
                         a 
                       
                       ⁢ 
                       
                         
                           X 
                           a 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     ΔX p (k+1), ΔX r (k+1), and e(k) are respectively represented by formulae (15), (16) and (17) below.
 
Δ X   p ( k+ 1)= X   p ( k+ 1)− X   p ( k )= A   p   ΔX   p ( k )+ B   p   ΔU   p ( k )  (15)
 
Δ X   r ( k+ 1)= X   r ( k+ 1)− X   r ( k )= A   r   ΔX   r ( k )+ B   r   ΔU   r ( k )  (16)
 
 e ( k )= Y   r ( k )− Y   p ( k )  (17)
 
     When an input showing the minimum evaluation function in the formula (17) is obtained with respect to the formulae (13) and (14), the input is represented by a formula (18). 
     
       
         
           
             
               
                 
                   J 
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       ∞ 
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                              
                             
                               e 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                              
                           
                           Q 
                           2 
                         
                         + 
                         
                           
                              
                             
                               Δ 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   U 
                                   p 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                              
                           
                           R 
                           2 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     Δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       
                         U 
                         p 
                       
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         K 
                         1 
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         X 
                         p 
                       
                     
                     + 
                     
                       
                         K 
                         2 
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         
                           X 
                           r 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         K 
                         3 
                       
                       ⁢ 
                       
                         e 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     When the formula (18) is solved with respect to U p (k), U p (k) is represented by a formula (19) below. 
     
       
         
           
             
               
                 
                   
                     
                       U 
                       p 
                     
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         K 
                         1 
                       
                       ⁢ 
                       
                         
                           X 
                           p 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         K 
                         2 
                       
                       ⁢ 
                       
                         
                           X 
                           r 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                     + 
                     
                       
                         K 
                         3 
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             j 
                             = 
                             1 
                           
                           k 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           e 
                           ⁡ 
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
     Given that G11=[K 1  K 2  K 3 ], the gains are obtained by solving a corresponding Riccati equation. Here, when the gain G11 is applied to the reference model cluster represented by the formula (10) as described above, the gain G11 also has 2 3  combinations=eight combinations, which is called an optimal control gain cluster GC exemplarily represented by a formula (20). 
     
       
         
           
             
               
                 
                   GC 
                   = 
                   
                     [ 
                     
                       
                         
                           
                             G 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               11 
                               1 
                             
                           
                         
                       
                       
                         
                           
                             G 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               11 
                               2 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             G 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               11 
                               8 
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
     The simulator  31  selects the switching time t 1  based on a starting temperature y 0  and the temperature setpoint SV and determines a heating or cooling pattern of each of the zones  1  to  3  based on the equilibrium temperature Teq as well as the selected switching time t 1 . The determined pattern is inputted to the reference model cluster  38  and the optimal control gain cluster  39 . 
     The maximum manipulated variable MVmax and the steady state manipulated variable MVss which are to be inputted to the reference model  33  are calculated based on the starting temperature y 0  and the temperature setpoint SV and inputted to the reference model selected from the reference model cluster  38 . Simultaneously, the model following servo control is executed using the gains K 1 , K 2  and K 3  selected from the optimal control gain cluster  39 . It should be noted that the observer  40  as shown in  FIG. 3  is an estimator configured to estimate an internal state quantity of a control target (plant) unobservable with a sensor. 
     3. Operations in Embodiment(s) 
     Next, operations of a temperature control method of a semiconductor wafer in the exemplary embodiment will be described with reference to flow charts shown in  FIGS. 6 and 7 . 
     The manipulated variable calculator  30  of the controller  24  drives and initializes an apparatus (Step S 1 ). 
     The manipulated variable calculator  30  judges whether the temperature setpoint SV is changed or not (Step S 2 ). When the temperature setpoint SV is not changed, the manipulated variable calculator  30  generates a reference model output yr in Step S 7  described later and continues controlling. 
     When the temperature setpoint SV is changed, the reference model selecting unit  37  acquires the temperature setpoint SV, the starting temperature y 0  and the equilibrium temperature Teq (Step S 3 ). 
     Based on the acquired the temperature setpoint SV, the starting temperature y 0  and the equilibrium temperature Teq, the reference model selecting unit  37  selects a reference model and an optimal gain and applies the reference model and the optimal gain to the reference model  33  (Step S 4 ). 
     The reference model and the optimal gain are selected according to the flow chart shown in  FIG. 7 . 
     Firstly, as shown in  FIG. 7 , the reference model selecting unit  37  judges whether the judgment of the heating or the cooling is made when the manipulated variable is at the maximum manipulated variable MVmax or the steady state manipulated variable MVss in a single time of heating or cooling process (Step S 41 ). 
     When the reference model selecting unit  37  determines that the judgment of the heating or the cooling is made when the manipulated variable is at the maximum manipulated variable MVmax (Step S 41 : No), the reference model selecting unit  37  judges which of the starting temperature y 0  in the heating or the cooling and the temperature setpoint SV is higher (Step S 42 ). 
     When the reference model selecting unit  37  determines that the starting temperature y 0  is higher than the temperature setpoint SV (Step S 42 : No), the reference model selecting unit  37  selects the cooling reference model from the reference model cluster  38  and the optimal control gain for cooling from the optimal control gain cluster  39  (Step S 43 ). 
     When the reference model selecting unit  37  determines that the temperature setpoint SV is higher than the starting temperature y 0  (Step S 42 : Yes), the reference model selecting unit  37  selects the heating reference model from the reference model cluster  38  and the optimal control gain for heating from the optimal control gain cluster  39  (Step S 44 ). 
     When the reference model selecting unit  37  determines that the judgment of the heating or the cooling is made when the manipulated variable is at the steady state manipulated variable MVss (Step S 41 : Yes), the reference model selecting unit  37  judges which of the equilibrium temperature Teq and the temperature setpoint SV is higher (Step S 46 ). 
     When the reference model selecting unit  37  determines that the equilibrium temperature Teq is higher than the temperature setpoint SV (Step S 46 : No), the reference model selecting unit  37  selects the cooling reference model from the reference model cluster  38  and the optimal control gain for cooling from the optimal control gain cluster  39  (Step S 47 ). 
     When the reference model selecting unit  37  determines that the temperature setpoint SV is higher than the equilibrium temperature, the reference model selecting unit  37  selects the heating reference model from the reference model cluster  38  and the optimal control gain for heating from the optimal control gain cluster  39  (Step S 48 ). 
     Referring back to  FIG. 6 , the simulator  31  searches for the switching time t 1  based on the starting temperature y 0  and the temperature setpoint SV (Step S 5 ). 
     After the switching time t 1  is searched for, the manipulated variable switching unit  32  inputs MVref=MVmax to the reference model  33 . 
     The reference model  33  multiplies the generated reference model Gm(s) by the manipulated variable pattern MVref to obtain a reference model output and outputs the reference model output (Step S 7 ). 
     The manipulated variable switching unit  32  monitors a timer and judges whether the switching time t 1  has passed (Step S 8 ). When judging that the switching time t 1  has not passed, the manipulated variable switching unit  32  continues to output MVref=MVmax to the reference model  33  (Step S 9 ). 
     When judging that the switching time t 1  has passed, the manipulated variable switching unit  32  sets MVref=MVss (Step S 10 ). 
     Based on the reference model output generated as described above, the model following servo calculator executes model following servo control. 
     4. Check of Effects of Invention by Simulation 
     4-1. Structure of Control System in Simulation 
     Control of a three-input three-output system will be exemplarily described with a simulation result obtained by modeling a control system shown in  FIG. 8 . This control system is a system for controlling a temperature of an aluminum plate of 400×150×t 4  as shown in  FIG. 9  and uses an actuator in a form of three thermo-modules configured to heat and cool the plate. The temperature of the aluminum plate is measured by three type K thermocouples provided near the respective modules. The thermo-module and the thermocouples are intentionally disposed asymmetrically relative to a longitudinal direction of the plate, which is shown in dimensional detail in  FIG. 9 . Zones  1 ,  2  and  3  are defined from the left in the figure. 
     4-2. Problems in Patent Literature 1 
     The reference model in Patent Literature 1 will be described in detail with respect to the searching method of the switching time when the reference model in Patent Literature 1 is defined as a dead time+a first-order lag system represented by a formula (21). 
     
       
         
           
             
               
                 
                   
                     
                       G 
                       mh 
                     
                     ⁡ 
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       44.2 
                       
                         
                           178 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           s 
                         
                         + 
                         1 
                       
                     
                     ⁢ 
                     
                       e 
                       
                         
                           - 
                           4 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         s 
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     By replacing the dead time with a first-order Padé approximation in the formula (21), the formula (21) is represented by a formula (22). 
     
       
         
           
             
               
                 
                   
                     
                       G 
                       mh 
                     
                     ⁡ 
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           - 
                           44.2 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         s 
                       
                       + 
                       22.1 
                     
                     
                       
                         178 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           s 
                           2 
                         
                       
                       + 
                       
                         90 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         s 
                       
                       + 
                       0.5 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     When the formula (22) is further converted to a state equation of a discrete-time system (a sampling time: 0.1 second), the formula (22) is represented by formulae (23) and (24). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               x 
                               
                                 h 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               x 
                               
                                 h 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               1.951 
                             
                             
                               
                                 - 
                                 0.9507 
                               
                             
                           
                           
                             
                               1 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   x 
                                   
                                     h 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   x 
                                   
                                     h 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     2 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               0.25 
                             
                           
                           
                             
                               0 
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         u 
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
             
               
                 
                   
                     y 
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               - 
                               0.09441 
                             
                           
                           
                             0.09926 
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               
                                 x 
                                 
                                   h 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   1 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 x 
                                 
                                   h 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Considering the response in the formulae (23) and (24), since initial conditions differ depending on the starting temperature y 0  in the heating, it is necessary to input two of the starting temperature y 0  and the temperature setpoint SV in order to search for the switching time t 1 . Here, assuming that the temperature is steady at the starting temperature y 0 , the initial conditions x h1 (0) and x h2 (0) at the starting temperature y 0  are represented by a formula (25). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               x 
                               
                                 h 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               x 
                               
                                 h 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 C 
                                 h 
                               
                             
                           
                           
                             
                               
                                 
                                   C 
                                   h 
                                 
                                 ⁢ 
                                 
                                   A 
                                   h 
                                 
                               
                             
                           
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               y 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                           
                         
                         
                           
                             
                               y 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               0 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     A h  and C h  correspond to coefficient matrixes (see formula (26)) in the formulae (23) and (24). 
     
       
         
           
             
               
                 
                   
                     
                       A 
                       h 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             1.951 
                           
                           
                             
                               - 
                               0.9507 
                             
                           
                         
                         
                           
                             1 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
                       B 
                       h 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             0.25 
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ] 
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       C 
                       h 
                     
                     = 
                     
                       [ 
                       
                         
                           
                             
                               - 
                               0.09441 
                             
                           
                           
                             0.09926 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     The switching time t 1  is searched for in the heating (cooling) range expectable according to the formulae (23) to (25). Consequently, a two-dimensional table in which an input is represented by y 0  and SV and an output is represented by the switching time t 1  is obtained. Table 1 shows an example of the two-dimensional table. 
     
       
         
           
               
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                   
                 SV 
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 y0 
                 −10 
                 −5 
                 0 
                 5 
                 10 
                 15 
                 20 
                 25 
                 30 
                 35 
                 40 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
               
               
               
               
               
            
               
                 −15 
                 15.2 
                 32.2 
                 51 
                 72 
                 95.9 
                 123.5 
                 156.4 
                 196.9 
                 249.7 
                 325.6 
                 462.9 
               
               
                 −10 
                 0 
                 16.7 
                 35.6 
                 56.6 
                 80.4 
                 108.1 
                 141 
                 181.5 
                 234.3 
                 310.2 
                 447.4 
               
               
                 −5 
                 0 
                 0 
                 18.7 
                 39.7 
                 63.5 
                 91.2 
                 124.1 
                 164.6 
                 217.4 
                 293.3 
                 430.5 
               
               
                 0 
                 0 
                 0 
                 0 
                 21 
                 44.8 
                 72.5 
                 105.4 
                 145.9 
                 198.7 
                 274.6 
                 411.8 
               
               
                 5 
                 0 
                 0 
                 0 
                 0 
                 23.9 
                 51.5 
                 84.4 
                 124.9 
                 177.7 
                 253.7 
                 390.9 
               
               
                 10 
                 0 
                 0 
                 0 
                 0 
                 0 
                 27.7 
                 60.6 
                 101.1 
                 153.9 
                 229.9 
                 367.1 
               
               
                 15 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 33 
                 73.5 
                 126.3 
                 202.3 
                 339.5 
               
               
                 20 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 40.8 
                 93.6 
                 169.6 
                 306.8 
               
               
                 25 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 53.3 
                 129.3 
                 266.5 
               
               
                 30 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 76.8 
                 214.1 
               
               
                 35 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
                 138.9 
               
               
                   
               
            
           
         
       
     
     In the table showing that, for instance, the starting temperature y 0  being 10 degrees C. and the temperature setpoint SV being 30 degrees C. provide the switching time being 153.9 seconds. Since the input is given at every interval of 5 degrees C., temperatures in the interval are obtained by interpolation, but the interval may be decreased. 
     In the technology of Patent Literature 1, the formula (22) is used as the reference model based on results of the heating test shown in  FIG. 10 . However, since the thermo-module exhibits different dynamic characteristics depending on heating and cooling, the thermo-module exhibits a behavior as shown in  FIG. 11  in the cooling test, the behavior being not always in conformity with a behavior in the heating test. For instance, the characteristics in the cooling shown in  FIG. 11  are represented by a formula (27). 
     
       
         
           
             
               
                 
                   
                     
                       G 
                       mc 
                     
                     ⁡ 
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       18.4 
                       
                         
                           142 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           s 
                         
                         + 
                         1 
                       
                     
                     ⁢ 
                     
                       e 
                       
                         
                           - 
                           4 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         s 
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     The comparison results between step responses according to the formulae (22) and (27) are shown in  FIG. 12 , from which it is found that the step response in the heating is quicker. 
     Because of this difference in the dynamic characteristics, when the thermo-module is controlled to be cooled only using the heating reference model, as exemplarily shown by simulation results in  FIG. 13 , the reference model output precedes in the cooling and the actual outputs PV 1  to PV 3  cannot follow the reference model output since the manipulated variables MV 1  to MV 3  are restricted to saturation, so that the uniformity cannot be ensured. 
     4-3 Simulation in Embodiment(s) 
     In the exemplary embodiment, considering both of the heating reference model and the cooling reference model, the reference model cluster and the optimal control gain cluster are prepared in number corresponding to combinations of 2^ (the number of the zones) (i.e., exponentiating 2 with the number of the zones) of the heating and the cooling of the zones, and are switched in use depending on conditions of the heating or the cooling of each of the zones. 
     Assuming a state in the heating or the cooling of only a single zone, the state depends on a relationship among three temperatures of the heating (cooling) starting temperature y 0 , the temperature setpoint SV, and the equilibrium temperature Teq. For instance, when y 0 &lt;Teq and SV&lt;Teq, the heating reference model is employed when the maximum manipulated variable MVmax is outputted while the cooling reference model is employed when the steady state manipulated variable MVss is outputted. 
     Moreover, since all the zones are possibly heated and cooled, the combinations of 2 3 =8 of the heating and the cooling are conceivable. For instance, given that the temperature setpoint of all the zone is changed from SV 1 =SV 2 =SV 3 =20 degrees C. to SV 1 =10 degrees C., SV 2 =30 degrees C. and SV 3 =50 degrees C., the zone  1  is to be cooled and the zones  2  and  3  are to be heated. 
     When the dead time is replaced with a first-order Padé approximation in the formula (27) in the same manner as in the formula (22), the formula (27) is represented by a formula (28) 
     
       
         
           
             
               
                 
                   
                     
                       G 
                       mc 
                     
                     ⁡ 
                     
                       ( 
                       s 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           - 
                           18.4 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         s 
                       
                       + 
                       9.2 
                     
                     
                       
                         142 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           s 
                           2 
                         
                       
                       + 
                       
                         72 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         s 
                       
                       + 
                       0.5 
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     When the formula (28) is further converted to a state equation of a discrete-time system (the sampling time: 0.1 second) in the same manner as the formulae (23) and (24), the formula (28) is represented by formulae (29) and (30). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               x 
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               x 
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         
                           A 
                           c 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     x 
                                     
                                       c 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     x 
                                     
                                       c 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         
                           B 
                           c 
                         
                         ⁢ 
                         
                           u 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               
                                 0.9505 
                               
                               
                                 
                                   - 
                                   0.005493 
                                 
                               
                             
                             
                               
                                 0.006094 
                               
                               
                                 1 
                               
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     x 
                                     
                                       c 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     x 
                                     
                                       c 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         
                           [ 
                           
                             
                               
                                 0.09751 
                               
                             
                             
                               
                                 0.0003073 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           u 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
             
               
                 
                   
                       
                   
                   ⁢ 
                   
                     
                       y 
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           C 
                           c 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     x 
                                     
                                       c 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     x 
                                     
                                       c 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         
                           [ 
                           
                             
                               
                                 
                                   - 
                                   0.1296 
                                 
                               
                               
                                 1.037 
                               
                             
                           
                           ] 
                         
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     x 
                                     
                                       c 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       1 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     x 
                                     
                                       c 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       2 
                                     
                                   
                                   ⁡ 
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     Next, a method of smoothly switching the state variables at the switching of the heating and the cooling when the output of the maximum manipulated variable MVmax is switched to the output of the steady state manipulated variable MVss during controlling will be described. 
     Sizes of matrixes are as follows according to the formulae (22), (23), (24), (28), (29) and (30). 
     Ah, Ac:2×2 
     Bh, Bc:2×1 
     Ch, Cc:1×2 
     xh, xc:2×1 
     Initial conditions are obtained according to a formula (31) below and represented by a formula (32) below. 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             y 
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             y 
                             ⁡ 
                             
                               ( 
                               1 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               Cx 
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               Cx 
                               ⁡ 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                     = 
                     
                       
                         [ 
                         
                           
                             
                               
                                 Cx 
                                 ⁡ 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 CAx 
                                 ⁡ 
                                 
                                   ( 
                                   0 
                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       = 
                       
                         
                           [ 
                           
                             
                               
                                 C 
                               
                             
                             
                               
                                 CA 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           x 
                           ⁡ 
                           
                             ( 
                             0 
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   31 
                   ) 
                 
               
             
             
               
                 
                   
                     x 
                     ⁡ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               C 
                             
                           
                           
                             
                               CA 
                             
                           
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               y 
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               y 
                               ⁡ 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
     Accordingly, the initial conditions for heating are represented by a formula (33) and the initial conditions for cooling are represented by a formula (34). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               x 
                               
                                 h 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               x 
                               
                                 h 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 C 
                                 h 
                               
                             
                           
                           
                             
                               
                                 
                                   C 
                                   h 
                                 
                                 ⁢ 
                                 
                                   A 
                                   h 
                                 
                               
                             
                           
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               y 
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               y 
                               ⁡ 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               x 
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               x 
                               
                                 c 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁡ 
                             
                               ( 
                               0 
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 C 
                                 c 
                               
                             
                           
                           
                             
                               
                                 
                                   C 
                                   c 
                                 
                                 ⁢ 
                                 
                                   A 
                                   c 
                                 
                               
                             
                           
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               y 
                               ⁡ 
                               
                                 ( 
                                 0 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               y 
                               ⁡ 
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
     Here, it is assumed to switch the state from the heating reference model to the cooling reference model. At the time of switching, since both of the reference models have the same input u(k) and output y(k), the following formulae (35) and (36) are established.
 
 y ( k )= C   h   x   h ( k )= C   c   x   c ( k )  (35)
 
 y ( k+ 1)= C   h   x   h ( k+ 1)= C   h   A   h   x   h ( k )+ C   h   B   h   u ( k )= C   c   A   c   x   c ( k )+ C   c   B   c   u ( k )  (36)
 
     The formulae (35) and (36) are converted to a matrix to be represented by a formula (37). 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             y 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             y 
                             ⁡ 
                             
                               ( 
                               
                                 k 
                                 + 
                                 1 
                               
                               ) 
                             
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   C 
                                   h 
                                 
                               
                             
                             
                               
                                 
                                   
                                     C 
                                     h 
                                   
                                   ⁢ 
                                   
                                     A 
                                     h 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           
                             x 
                             h 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       + 
                       
                         
                           [ 
                           
                             
                               
                                 0 
                               
                             
                             
                               
                                 
                                   
                                     C 
                                     h 
                                   
                                   ⁢ 
                                   
                                     B 
                                     h 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           u 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                     = 
                     
                         
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     C 
                                     c 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       C 
                                       c 
                                     
                                     ⁢ 
                                     
                                       A 
                                       c 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             
                               x 
                               c 
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         + 
                         
                           
                             [ 
                             
                               
                                 
                                   0 
                                 
                               
                               
                                 
                                   
                                     
                                       C 
                                       c 
                                     
                                     ⁢ 
                                     
                                       B 
                                       c 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             u 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
     From the above, when the heating reference model is switched to the cooling reference model, the following formula (38) is established. 
     
       
         
           
             
               
                 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 C 
                                 c 
                               
                             
                           
                           
                             
                               
                                 
                                   C 
                                   c 
                                 
                                 ⁢ 
                                 
                                   A 
                                   c 
                                 
                               
                             
                           
                         
                         ] 
                       
                       ⁢ 
                       
                         
                           x 
                           c 
                         
                         ⁡ 
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   C 
                                   h 
                                 
                               
                             
                             
                               
                                 
                                   
                                     C 
                                     h 
                                   
                                   ⁢ 
                                   
                                     A 
                                     h 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           
                             x 
                             h 
                           
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       + 
                       
                         
                           ( 
                           
                             
                               [ 
                               
                                 
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     
                                       
                                         C 
                                         h 
                                       
                                       ⁢ 
                                       
                                         B 
                                         h 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             - 
                             
                               [ 
                               
                                 
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     
                                       
                                         C 
                                         c 
                                       
                                       ⁢ 
                                       
                                         B 
                                         c 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           u 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     
                       
                         x 
                         c 
                       
                       ⁡ 
                       
                         ( 
                         k 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   C 
                                   c 
                                 
                               
                             
                             
                               
                                 
                                   
                                     C 
                                     c 
                                   
                                   ⁢ 
                                   
                                     A 
                                     c 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         { 
                         
                           
                             
                               [ 
                               
                                 
                                   
                                     
                                       C 
                                       h 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         C 
                                         h 
                                       
                                       ⁢ 
                                       
                                         A 
                                         h 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               
                                 x 
                                 h 
                               
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           + 
                           
                             
                               ( 
                               
                                 
                                   [ 
                                   
                                     
                                       
                                         0 
                                       
                                     
                                     
                                       
                                         
                                           
                                             C 
                                             h 
                                           
                                           ⁢ 
                                           
                                             B 
                                             h 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                                 - 
                                 
                                   [ 
                                   
                                     
                                       
                                         0 
                                       
                                     
                                     
                                       
                                         
                                           
                                             C 
                                             c 
                                           
                                           ⁢ 
                                           
                                             B 
                                             c 
                                           
                                         
                                       
                                     
                                   
                                   ] 
                                 
                               
                               ) 
                             
                             ⁢ 
                             
                               u 
                               ⁡ 
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
     Also when the cooling reference model is switched to the heating reference model, the following formula (39) is established. 
     
       
         
           
             
               
                 
                   
                     
                       x 
                       h 
                     
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 C 
                                 h 
                               
                             
                           
                           
                             
                               
                                 
                                   C 
                                   h 
                                 
                                 ⁢ 
                                 
                                   A 
                                   h 
                                 
                               
                             
                           
                         
                         ] 
                       
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       { 
                       
                         
                           
                             [ 
                             
                               
                                 
                                   
                                     C 
                                     c 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       C 
                                       c 
                                     
                                     ⁢ 
                                     
                                       A 
                                       c 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             
                               x 
                               c 
                             
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 [ 
                                 
                                   
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       
                                         
                                           C 
                                           c 
                                         
                                         ⁢ 
                                         
                                           B 
                                           c 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                               - 
                               
                                 [ 
                                 
                                   
                                     
                                       0 
                                     
                                   
                                   
                                     
                                       
                                         
                                           C 
                                           h 
                                         
                                         ⁢ 
                                         
                                           B 
                                           h 
                                         
                                       
                                     
                                   
                                 
                                 ] 
                               
                             
                             ) 
                           
                           ⁢ 
                           
                             u 
                             ⁡ 
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     The same idea is also employed in an optimal time search simulation for preparing the table of the switching time t 1 . For instance, when y 0 &gt;SV&gt;Teq, the cooling reference model is switched to the heating reference model at the time t 1 . 
     Results of the above simulation are shown in  FIG. 14 . It is observed that the reference model output is suitably followed also in the cooling.